Subgroups — C₁₇×#18008 #16875752

Order 4352.

670 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×#3105	C₂	1	1	yes	maximal, characteristic
2	C₁₇×#2969	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×M₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₇×#1592	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#1589	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₇×#1586	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#1585	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₇×#1583	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₇×#1581	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₇×#1580	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₇×#1574	C₂	1	1	yes	maximal, characteristic, complemented
13	C₁₇×#1495	C₂	1	1	yes	maximal, characteristic, complemented
14	C₁₇×#1478	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#1453	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×M₁₆×C₁₇	—	2	1	no	—
17	C₁₇×#529	C₂²	1	1	yes	characteristic
18	C₂×C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
19	C₁₇×#484	C₄	1	1	yes	characteristic
20	C₁₇×#481	C₄	1	1	yes	characteristic
21	C₁₇×#478	C₄	1	1	yes	characteristic
22	C₁₇×#478	C₄	1	1	yes	characteristic
23	C₁₇×#463	C₄	1	1	yes	characteristic
24	C₁₇×#463	C₄	1	1	yes	characteristic
25	C₁₇×#450	C₄	1	1	yes	characteristic
26	C₁₇×#447	C₄	1	1	yes	characteristic
27	C₁₇×#446	C₂²	1	1	yes	characteristic, complemented
28	C₈×Q₈×C₁₇	C₂²	1	1	yes	characteristic, complemented
29	C₄×SD₁₆×C₁₇	C₂²	1	1	yes	characteristic
30	C₄×SD₁₆×C₁₇	C₂²	1	1	yes	characteristic
31	C₄×D₈×C₁₇	C₂²	1	1	yes	characteristic
32	C₄×D₈×C₁₇	C₂²	1	1	yes	characteristic
33	C₁₇×#436	—	2	1	no	—
34	C₁₇×#436	C₂²	1	1	yes	characteristic
35	C₁₇×#436	C₂²	1	1	yes	characteristic
36	C₁₇×#436	C₂²	1	1	yes	characteristic, complemented
37	C₁₇×#435	—	2	1	no	—
38	C₁₇×#435	—	2	1	no	—
39	C₁₇×#435	—	2	1	no	—
40	C₈×D₄×C₁₇	C₂²	1	1	yes	characteristic
41	C₈×D₄×C₁₇	C₂²	1	1	yes	characteristic
42	C₈×D₄×C₁₇	C₂²	1	1	yes	characteristic, complemented
43	C₁₇×#433	—	2	1	no	—
44	C₁₇×#431	C₂²	1	1	yes	characteristic
45	C₁₇×#431	C₂²	1	1	yes	characteristic
46	C₁₇×#431	C₂²	1	1	yes	characteristic
47	C₁₇×#423	C₂²	1	1	yes	characteristic
48	C₁₇×#407	—	2	1	no	—
49	C₁₇×#405	—	2	1	no	—
50	C₄×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
51	C₁₇×#335	C₂²	1	1	yes	characteristic, complemented
52	C₁₇×#335	C₂²	1	1	yes	characteristic, complemented
53	C₁₇×#334	C₂²	1	1	yes	characteristic, complemented
54	C₁₇×#334	C₂²	1	1	yes	characteristic, complemented
55	C₁₇×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
56	C₁₇×Q₈⋊C₈	C₂²	1	1	yes	characteristic, complemented
57	C₁₇×D₄⋊C₈	C₂²	1	1	yes	characteristic
58	C₁₇×D₄⋊C₈	C₂²	1	1	yes	characteristic
59	C₁₇×D₄⋊C₈	C₂²	1	1	yes	characteristic
60	C₁₇×D₄⋊C₈	C₂²	1	1	yes	characteristic
61	C₁₇×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
62	C₁₇×D₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
63	C₁₇×#322	C₂²	1	1	yes	characteristic, complemented
64	C₁₇×#322	C₂²	1	1	yes	characteristic, complemented
65	C₁₇×#322	C₂²	1	1	yes	characteristic, complemented
66	C₈²×C₁₇	C₂²	1	1	yes	characteristic, complemented
67	C₂²×D₄×C₁₇	C₂×C₄	1	1	yes	characteristic
68	C₂³×C₄×C₁₇	—	2	1	no	—
69	C₂×SD₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
70	C₂×SD₁₆×C₁₇	C₂×C₄	1	1	yes	characteristic
71	C₂×D₈×C₁₇	C₂×C₄	1	1	yes	characteristic
72	C₂×D₈×C₁₇	C₂×C₄	1	1	yes	characteristic
73	C₂×M₁₆×C₁₇	—	2	1	no	—
74	C₂×M₁₆×C₁₇	—	2	1	no	—
75	C₂×M₁₆×C₁₇	—	2	1	no	—
76	C₂×M₁₆×C₁₇	—	2	1	no	—
77	C₂×M₁₆×C₁₇	—	4	4	no	—
78	C₂×M₁₆×C₁₇	D₄	1	1	yes	characteristic
79	C₂×M₁₆×C₁₇	D₄	1	1	yes	characteristic
80	C₂²×C₈×C₁₇	—	2	1	no	—
81	C₂²×C₈×C₁₇	—	2	1	no	—
82	C₂²×C₈×C₁₇	—	2	1	no	—
83	C₁₇×#126	—	2	1	no	—
84	C₁₇×#125	C₂×C₄	1	1	yes	characteristic
85	C₁₇×#124	C₂×C₄	1	1	yes	characteristic
86	C₁₇×#123	—	2	1	no	—
87	C₁₇×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
88	C₁₇×#121	C₂×C₄	1	1	yes	characteristic
89	C₄×Q₈×C₁₇	C₂³	1	1	yes	characteristic
90	C₄×D₄×C₁₇	—	2	1	no	—
91	C₄×D₄×C₁₇	—	2	1	no	—
92	C₄×D₄×C₁₇	—	2	1	no	—
93	C₄×D₄×C₁₇	C₂×C₄	1	1	yes	characteristic
94	C₄×D₄×C₁₇	C₂×C₄	1	1	yes	characteristic
95	C₄×D₄×C₁₇	C₂³	1	1	yes	characteristic
96	C₄×D₄×C₁₇	C₂³	1	1	yes	characteristic
97	C₄×D₄×C₁₇	C₂³	1	1	yes	characteristic
98	C₁₇×#117	—	2	1	no	—
99	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
100	C₂×C₂²⋊C₄×C₁₇	—	2	1	no	—
101	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
102	C₁₇×#107	C₂×C₄	1	1	yes	characteristic
103	C₁₇×#107	C₂×C₄	1	1	yes	characteristic
104	C₁₇×#106	C₂×C₄	1	1	yes	characteristic
105	C₁₇×#106	C₂×C₄	1	1	yes	characteristic
106	C₁₇×C₄⋊C₈	—	2	1	no	—
107	C₁₇×C₄⋊C₈	—	2	1	no	—
108	C₁₇×C₄⋊C₈	—	2	1	no	—
109	C₁₇×C₄⋊C₈	—	2	1	no	—
110	C₁₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
111	C₁₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
112	C₁₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
113	C₁₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
114	C₁₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
115	C₁₇×C₄⋊C₈	C₂³	1	1	yes	characteristic
116	C₁₇×#103	C₂×C₄	1	1	yes	characteristic
117	C₁₇×#103	C₂×C₄	1	1	yes	characteristic
118	C₁₇×#102	C₂×C₄	1	1	yes	characteristic
119	C₁₇×#102	C₂×C₄	1	1	yes	characteristic
120	C₁₇×#102	C₂×C₄	1	1	yes	characteristic
121	C₁₇×#102	C₂×C₄	1	1	yes	characteristic
122	C₁₇×#102	C₂×C₄	1	1	yes	characteristic
123	C₁₇×#102	C₂×C₄	1	1	yes	characteristic
124	C₁₇×C₂²⋊C₈	—	2	1	no	—
125	C₁₇×C₂²⋊C₈	—	2	1	no	—
126	C₁₇×C₂²⋊C₈	—	2	1	no	—
127	C₁₇×C₂²⋊C₈	—	2	1	no	—
128	C₁₇×C₂²⋊C₈	—	2	1	no	—
129	C₁₇×C₂²⋊C₈	—	2	1	no	—
130	C₁₇×C₂²⋊C₈	—	2	1	no	—
131	C₁₇×C₂²⋊C₈	—	2	1	no	—
132	C₁₇×C₂²⋊C₈	C₂×C₄	1	1	yes	characteristic
133	C₁₇×C₂²⋊C₈	C₂×C₄	1	1	yes	characteristic
134	C₁₇×#97	—	2	1	no	—
135	C₁₇×#97	D₄	1	1	yes	characteristic
136	C₁₇×#97	D₄	1	1	yes	characteristic
137	C₄×C₈×C₁₇	—	2	1	no	—
138	C₄×C₈×C₁₇	—	2	1	no	—
139	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
140	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
141	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
142	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
143	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
144	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
145	C₂⁴×C₁₇	—	2	1	no	—
146	C₂×Q₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
147	C₂×D₄×C₁₇	—	2	2	no	—
148	C₂×D₄×C₁₇	—	2	1	no	—
149	C₂×D₄×C₁₇	—	2	1	no	—
150	C₂×D₄×C₁₇	—	4	1	no	—
151	C₂×D₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
152	C₂×D₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
153	C₂×D₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
154	C₂²×C₄×C₁₇	—	2	1	no	—
155	C₂²×C₄×C₁₇	—	2	1	no	—
156	C₂²×C₄×C₁₇	—	2	1	no	—
157	C₂²×C₄×C₁₇	—	2	1	no	—
158	C₂²×C₄×C₁₇	—	2	1	no	—
159	C₂²×C₄×C₁₇	—	2	1	no	—
160	C₂²×C₄×C₁₇	—	4	2	no	—
161	C₂²×C₄×C₁₇	—	4	2	no	—
162	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
163	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
165	SD₁₆×C₁₇	—	2	2	no	—
166	SD₁₆×C₁₇	—	4	1	no	—
167	D₈×C₁₇	—	2	2	no	—
168	D₈×C₁₇	—	4	1	no	—
169	M₁₆×C₁₇	—	2	2	no	—
170	M₁₆×C₁₇	—	4	1	no	—
171	M₁₆×C₁₇	—	4	2	no	—
172	M₁₆×C₁₇	—	4	2	no	—
173	M₁₆×C₁₇	—	4	2	no	—
174	M₁₆×C₁₇	—	4	2	no	—
175	C₂×C₈×C₁₇	—	2	1	no	—
176	C₂×C₈×C₁₇	—	2	1	no	—
177	C₂×C₈×C₁₇	—	2	1	no	—
178	C₂×C₈×C₁₇	—	2	1	no	—
179	C₂×C₈×C₁₇	—	2	1	no	—
180	C₂×C₈×C₁₇	—	2	1	no	—
181	C₂×C₈×C₁₇	—	4	2	no	—
182	C₂×C₈×C₁₇	—	4	2	no	—
183	C₂×C₈×C₁₇	—	4	2	no	—
184	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
185	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
186	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
187	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
188	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
189	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
190	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
191	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
192	C₄⋊C₄×C₁₇	—	2	1	no	—
193	C₄⋊C₄×C₁₇	—	2	1	no	—
194	C₄⋊C₄×C₁₇	—	2	1	no	—
195	C₄⋊C₄×C₁₇	—	2	1	no	—
196	C₄⋊C₄×C₁₇	—	2	1	no	—
197	C₄⋊C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
198	C₄⋊C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
199	C₄⋊C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
200	C₄⋊C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
201	C₂²⋊C₄×C₁₇	—	2	1	no	—
202	C₂²⋊C₄×C₁₇	—	2	1	no	—
203	C₂²⋊C₄×C₁₇	—	2	1	no	—
204	C₂²⋊C₄×C₁₇	—	2	1	no	—
205	C₂²⋊C₄×C₁₇	—	2	1	no	—
206	C₂²⋊C₄×C₁₇	—	2	1	no	—
207	C₂²⋊C₄×C₁₇	—	2	1	no	—
208	C₂²⋊C₄×C₁₇	—	2	1	no	—
209	C₂²⋊C₄×C₁₇	—	2	1	no	—
210	C₄²×C₁₇	—	2	1	no	—
211	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
212	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
213	C₄²×C₁₇	#41	1	1	yes	characteristic
214	C₄²×C₁₇	C₂⁴	1	1	yes	characteristic
215	#18008	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
216	C₂³×C₁₇	—	2	1	no	—
217	C₂³×C₁₇	—	2	1	no	—
218	C₂³×C₁₇	—	2	1	no	—
219	C₂³×C₁₇	—	4	2	no	—
220	C₂³×C₁₇	—	4	2	no	—
221	C₂³×C₁₇	—	8	1	no	—
222	C₂³×C₁₇	C₄×D₄	1	1	yes	characteristic
223	Q₈×C₁₇	—	2	1	no	—
224	Q₈×C₁₇	—	2	1	no	—
225	D₄×C₁₇	—	2	1	no	—
226	D₄×C₁₇	—	2	1	no	—
227	D₄×C₁₇	—	2	1	no	—
228	D₄×C₁₇	—	2	1	no	—
229	D₄×C₁₇	—	2	1	no	—
230	D₄×C₁₇	—	4	2	no	—
231	D₄×C₁₇	—	8	1	no	—
232	D₄×C₁₇	#131	1	2	yes	—
233	C₂×C₄×C₁₇	—	2	1	no	—
234	C₂×C₄×C₁₇	—	2	1	no	—
235	C₂×C₄×C₁₇	—	2	1	no	—
236	C₂×C₄×C₁₇	—	2	1	no	—
237	C₂×C₄×C₁₇	—	2	1	no	—
238	C₂×C₄×C₁₇	—	2	1	no	—
239	C₂×C₄×C₁₇	—	2	1	no	—
240	C₂×C₄×C₁₇	—	2	1	no	—
241	C₂×C₄×C₁₇	—	2	1	no	—
242	C₂×C₄×C₁₇	—	4	1	no	—
243	C₂×C₄×C₁₇	—	4	2	no	—
244	C₂×C₄×C₁₇	—	4	2	no	—
245	C₂×C₄×C₁₇	—	4	2	no	—
246	C₂×C₄×C₁₇	—	4	1	no	—
247	C₂×C₄×C₁₇	—	4	1	no	—
248	C₂×C₄×C₁₇	—	4	1	no	—
249	C₂×C₄×C₁₇	—	4	1	no	—
250	C₂×C₄×C₁₇	—	4	1	no	—
251	C₂×C₄×C₁₇	—	4	1	no	—
252	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
253	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
254	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
255	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
256	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
257	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
258	C₈×C₁₇	—	2	1	no	—
259	C₈×C₁₇	—	2	1	no	—
260	C₈×C₁₇	—	2	1	no	—
261	C₈×C₁₇	—	2	1	no	—
262	C₈×C₁₇	—	2	1	no	—
263	C₈×C₁₇	—	2	1	no	—
264	C₈×C₁₇	—	2	1	no	—
265	C₈×C₁₇	—	4	1	no	—
266	C₈×C₁₇	—	4	1	no	—
267	C₈×C₁₇	—	4	1	no	—
268	C₈×C₁₇	—	4	1	no	—
269	C₈×C₁₇	—	4	1	no	—
270	C₈×C₁₇	—	4	1	no	—
271	C₈×C₁₇	AGL(1,ℤ/8)	1	2	yes	—
272	#3105	C₂×C₁₇	1	1	yes	characteristic
273	#2969	C₂×C₁₇	1	1	yes	characteristic, complemented
274	D₄×M₁₆	C₂×C₁₇	1	1	yes	characteristic, complemented
275	#1592	C₂×C₁₇	1	1	yes	characteristic, complemented
276	#1589	C₂×C₁₇	1	1	yes	characteristic, complemented
277	#1586	C₂×C₁₇	1	1	yes	characteristic, complemented
278	#1585	C₂×C₁₇	1	1	yes	characteristic, complemented
279	#1583	C₂×C₁₇	1	1	yes	characteristic, complemented
280	#1581	C₂×C₁₇	1	1	yes	characteristic, complemented
281	#1580	C₂×C₁₇	1	1	yes	characteristic, complemented
282	C₈×SD₁₆	C₂×C₁₇	1	1	yes	characteristic, complemented
283	#1574	C₂×C₁₇	1	1	yes	characteristic, complemented
284	#1495	C₂×C₁₇	1	1	yes	characteristic, complemented
285	#1478	C₂×C₁₇	1	1	yes	characteristic, complemented
286	#1453	C₂×C₁₇	1	1	yes	characteristic, complemented
287	C₂²×C₁₇	—	2	1	no	—
288	C₂²×C₁₇	—	2	2	no	—
289	C₂²×C₁₇	—	2	1	no	—
290	C₂²×C₁₇	—	2	1	no	—
291	C₂²×C₁₇	—	4	1	no	—
292	C₂²×C₁₇	—	4	1	no	—
293	C₂²×C₁₇	—	4	1	no	—
294	C₂²×C₁₇	—	4	1	no	—
295	C₂²×C₁₇	—	4	1	no	—
296	C₂²×C₁₇	—	4	1	no	—
297	C₂²×C₁₇	—	4	1	no	—
298	C₂²×C₁₇	—	4	1	no	—
299	C₂²×C₁₇	—	8	4	no	—
300	C₂²×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
301	C₄×C₁₇	—	2	1	no	—
302	C₄×C₁₇	—	2	1	no	—
303	C₄×C₁₇	—	2	1	no	—
304	C₄×C₁₇	—	2	1	no	—
305	C₄×C₁₇	—	4	1	no	—
306	C₄×C₁₇	—	4	1	no	—
307	C₄×C₁₇	—	4	1	no	—
308	C₄×C₁₇	—	4	1	no	—
309	C₄×C₁₇	—	4	1	no	—
310	C₄×C₁₇	C₂×#131	1	1	yes	characteristic
311	C₄×C₁₇	#568	1	1	yes	characteristic
312	C₄×C₁₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
313	C₄×C₁₇	#575	1	1	yes	characteristic
314	C₂²×M₁₆	—	2	1	no	—
315	#529	C₂²×C₁₇	1	1	yes	characteristic
316	C₂×C₄×D₄	C₂²×C₁₇	1	1	yes	characteristic
317	#484	C₄×C₁₇	1	1	yes	characteristic
318	#481	C₄×C₁₇	1	1	yes	characteristic
319	#478	C₄×C₁₇	1	1	yes	characteristic
320	#478	C₄×C₁₇	1	1	yes	characteristic
321	#463	C₄×C₁₇	1	1	yes	characteristic
322	#463	C₄×C₁₇	1	1	yes	characteristic
323	#450	C₄×C₁₇	1	1	yes	characteristic
324	#447	C₄×C₁₇	1	1	yes	characteristic
325	#446	C₂²×C₁₇	1	1	yes	characteristic, complemented
326	C₈×Q₈	C₂²×C₁₇	1	1	yes	characteristic, complemented
327	C₄×SD₁₆	C₂²×C₁₇	1	1	yes	characteristic
328	C₄×SD₁₆	C₂²×C₁₇	1	1	yes	characteristic
329	C₄×D₈	C₂²×C₁₇	1	1	yes	characteristic
330	C₄×D₈	C₂²×C₁₇	1	1	yes	characteristic
331	#436	—	2	1	no	—
332	#436	C₂²×C₁₇	1	1	yes	characteristic
333	#436	C₂²×C₁₇	1	1	yes	characteristic
334	#436	C₂²×C₁₇	1	1	yes	characteristic, complemented
335	#435	—	2	1	no	—
336	#435	—	2	1	no	—
337	#435	—	2	1	no	—
338	C₈×D₄	C₂²×C₁₇	1	1	yes	characteristic
339	C₈×D₄	C₂²×C₁₇	1	1	yes	characteristic
340	C₈×D₄	C₂²×C₁₇	1	1	yes	characteristic, complemented
341	#433	—	2	1	no	—
342	#431	C₂²×C₁₇	1	1	yes	characteristic
343	#431	C₂²×C₁₇	1	1	yes	characteristic
344	#431	C₂²×C₁₇	1	1	yes	characteristic
345	#423	C₂²×C₁₇	1	1	yes	characteristic
346	#407	—	2	1	no	—
347	#405	—	2	1	no	—
348	C₄×M₁₆	C₂²×C₁₇	1	1	yes	characteristic
349	#335	C₂²×C₁₇	1	1	yes	characteristic, complemented
350	#335	C₂²×C₁₇	1	1	yes	characteristic, complemented
351	#334	C₂²×C₁₇	1	1	yes	characteristic, complemented
352	#334	C₂²×C₁₇	1	1	yes	characteristic, complemented
353	Q₈⋊C₈	C₂²×C₁₇	1	1	yes	characteristic, complemented
354	Q₈⋊C₈	C₂²×C₁₇	1	1	yes	characteristic, complemented
355	D₄⋊C₈	C₂²×C₁₇	1	1	yes	characteristic
356	D₄⋊C₈	C₂²×C₁₇	1	1	yes	characteristic
357	D₄⋊C₈	C₂²×C₁₇	1	1	yes	characteristic
358	D₄⋊C₈	C₂²×C₁₇	1	1	yes	characteristic
359	D₄⋊C₈	C₂²×C₁₇	1	1	yes	characteristic, complemented
360	D₄⋊C₈	C₂²×C₁₇	1	1	yes	characteristic, complemented
361	#322	C₂²×C₁₇	1	1	yes	characteristic, complemented
362	#322	C₂²×C₁₇	1	1	yes	characteristic, complemented
363	#322	C₂²×C₁₇	1	1	yes	characteristic, complemented
364	C₈²	C₂²×C₁₇	1	1	yes	characteristic, complemented
365	C₂×C₁₇	—	4	1	no	—
366	C₂×C₁₇	—	4	2	no	—
367	C₂×C₁₇	—	8	1	no	—
368	C₂×C₁₇	—	8	1	no	—
369	C₂×C₁₇	#2933	1	1	yes	characteristic
370	C₂×C₁₇	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
371	C₂×C₁₇	#2957	1	1	yes	characteristic
372	C₂²×D₄	C₂×C₄×C₁₇	1	1	yes	characteristic
373	C₂³×C₄	—	2	1	no	—
374	C₂×SD₁₆	C₂×C₄×C₁₇	1	1	yes	characteristic
375	C₂×SD₁₆	C₂×C₄×C₁₇	1	1	yes	characteristic
376	C₂×D₈	C₂×C₄×C₁₇	1	1	yes	characteristic
377	C₂×D₈	C₂×C₄×C₁₇	1	1	yes	characteristic
378	C₂×M₁₆	—	2	1	no	—
379	C₂×M₁₆	—	2	1	no	—
380	C₂×M₁₆	—	2	1	no	—
381	C₂×M₁₆	—	2	1	no	—
382	C₂×M₁₆	—	4	4	no	—
383	C₂×M₁₆	D₄×C₁₇	1	1	yes	characteristic
384	C₂×M₁₆	D₄×C₁₇	1	1	yes	characteristic
385	C₂²×C₈	—	2	1	no	—
386	C₂²×C₈	—	2	1	no	—
387	C₂²×C₈	—	2	1	no	—
388	#126	—	2	1	no	—
389	#125	C₂×C₄×C₁₇	1	1	yes	characteristic
390	#124	C₂×C₄×C₁₇	1	1	yes	characteristic
391	#123	—	2	1	no	—
392	C₂²⋊Q₈	C₂×C₄×C₁₇	1	1	yes	characteristic
393	#121	C₂×C₄×C₁₇	1	1	yes	characteristic
394	C₄×Q₈	C₂³×C₁₇	1	1	yes	characteristic
395	C₄×D₄	—	2	1	no	—
396	C₄×D₄	—	2	1	no	—
397	C₄×D₄	—	2	1	no	—
398	C₄×D₄	C₂×C₄×C₁₇	1	1	yes	characteristic
399	C₄×D₄	C₂×C₄×C₁₇	1	1	yes	characteristic
400	C₄×D₄	C₂³×C₁₇	1	1	yes	characteristic
401	C₄×D₄	C₂³×C₁₇	1	1	yes	characteristic
402	C₄×D₄	C₂³×C₁₇	1	1	yes	characteristic
403	#117	—	2	1	no	—
404	C₂×C₄⋊C₄	C₂×C₄×C₁₇	1	1	yes	characteristic
405	C₂×C₂²⋊C₄	—	2	1	no	—
406	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
407	#107	C₂×C₄×C₁₇	1	1	yes	characteristic
408	#107	C₂×C₄×C₁₇	1	1	yes	characteristic
409	#106	C₂×C₄×C₁₇	1	1	yes	characteristic
410	#106	C₂×C₄×C₁₇	1	1	yes	characteristic
411	C₄⋊C₈	—	2	1	no	—
412	C₄⋊C₈	—	2	1	no	—
413	C₄⋊C₈	—	2	1	no	—
414	C₄⋊C₈	—	2	1	no	—
415	C₄⋊C₈	C₂×C₄×C₁₇	1	1	yes	characteristic
416	C₄⋊C₈	C₂×C₄×C₁₇	1	1	yes	characteristic
417	C₄⋊C₈	C₂³×C₁₇	1	1	yes	characteristic
418	C₄⋊C₈	C₂³×C₁₇	1	1	yes	characteristic
419	C₄⋊C₈	C₂³×C₁₇	1	1	yes	characteristic
420	C₄⋊C₈	C₂³×C₁₇	1	1	yes	characteristic
421	#103	C₂×C₄×C₁₇	1	1	yes	characteristic
422	#103	C₂×C₄×C₁₇	1	1	yes	characteristic
423	#102	C₂×C₄×C₁₇	1	1	yes	characteristic
424	#102	C₂×C₄×C₁₇	1	1	yes	characteristic
425	#102	C₂×C₄×C₁₇	1	1	yes	characteristic
426	#102	C₂×C₄×C₁₇	1	1	yes	characteristic
427	#102	C₂×C₄×C₁₇	1	1	yes	characteristic
428	#102	C₂×C₄×C₁₇	1	1	yes	characteristic
429	C₂²⋊C₈	—	2	1	no	—
430	C₂²⋊C₈	—	2	1	no	—
431	C₂²⋊C₈	—	2	1	no	—
432	C₂²⋊C₈	—	2	1	no	—
433	C₂²⋊C₈	—	2	1	no	—
434	C₂²⋊C₈	—	2	1	no	—
435	C₂²⋊C₈	—	2	1	no	—
436	C₂²⋊C₈	—	2	1	no	—
437	C₂²⋊C₈	C₂×C₄×C₁₇	1	1	yes	characteristic
438	C₂²⋊C₈	C₂×C₄×C₁₇	1	1	yes	characteristic
439	#97	—	2	1	no	—
440	#97	D₄×C₁₇	1	1	yes	characteristic
441	#97	D₄×C₁₇	1	1	yes	characteristic
442	C₄×C₈	—	2	1	no	—
443	C₄×C₈	—	2	1	no	—
444	C₄×C₈	C₂³×C₁₇	1	1	yes	characteristic
445	C₄×C₈	C₂³×C₁₇	1	1	yes	characteristic
446	C₄×C₈	C₂³×C₁₇	1	1	yes	characteristic
447	C₄×C₈	C₂³×C₁₇	1	1	yes	characteristic
448	C₄×C₈	C₂³×C₁₇	1	1	yes	characteristic
449	C₄×C₈	C₂³×C₁₇	1	1	yes	characteristic
450	C₁₇	#18008	1	1	yes	characteristic, direct product factor, Sylow
451	C₂⁴	—	2	1	no	—
452	C₂×Q₈	C₂²×C₄×C₁₇	1	1	yes	characteristic
453	C₂×D₄	—	2	2	no	—
454	C₂×D₄	—	2	1	no	—
455	C₂×D₄	—	2	1	no	—
456	C₂×D₄	—	4	1	no	—
457	C₂×D₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
458	C₂×D₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
459	C₂×D₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
460	C₂²×C₄	—	2	1	no	—
461	C₂²×C₄	—	2	1	no	—
462	C₂²×C₄	—	2	1	no	—
463	C₂²×C₄	—	2	1	no	—
464	C₂²×C₄	—	2	1	no	—
465	C₂²×C₄	—	2	1	no	—
466	C₂²×C₄	—	4	2	no	—
467	C₂²×C₄	—	4	2	no	—
468	C₂²×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
469	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
470	C₂²×C₄	#41×C₁₇	1	1	yes	characteristic
471	SD₁₆	—	2	2	no	—
472	SD₁₆	—	4	1	no	—
473	D₈	—	2	2	no	—
474	D₈	—	4	1	no	—
475	M₁₆	—	2	2	no	—
476	M₁₆	—	4	1	no	—
477	M₁₆	—	4	2	no	—
478	M₁₆	—	4	2	no	—
479	M₁₆	—	4	2	no	—
480	M₁₆	—	4	2	no	—
481	C₂×C₈	—	2	1	no	—
482	C₂×C₈	—	2	1	no	—
483	C₂×C₈	—	2	1	no	—
484	C₂×C₈	—	2	1	no	—
485	C₂×C₈	—	2	1	no	—
486	C₂×C₈	—	2	1	no	—
487	C₂×C₈	—	4	2	no	—
488	C₂×C₈	—	4	2	no	—
489	C₂×C₈	—	4	2	no	—
490	C₂×C₈	C₂²×C₄×C₁₇	1	1	yes	characteristic
491	C₂×C₈	C₂²×C₄×C₁₇	1	1	yes	characteristic
492	C₂×C₈	C₂²×C₄×C₁₇	1	1	yes	characteristic
493	C₂×C₈	C₂²×C₄×C₁₇	1	1	yes	characteristic
494	C₂×C₈	C₂×D₄×C₁₇	1	1	yes	characteristic
495	C₂×C₈	C₂×D₄×C₁₇	1	1	yes	characteristic
496	C₂×C₈	C₂×D₄×C₁₇	1	1	yes	characteristic
497	C₂×C₈	C₂×D₄×C₁₇	1	1	yes	characteristic
498	C₄⋊C₄	—	2	1	no	—
499	C₄⋊C₄	—	2	1	no	—
500	C₄⋊C₄	—	2	1	no	—
501	C₄⋊C₄	—	2	1	no	—
502	C₄⋊C₄	—	2	1	no	—
503	C₄⋊C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
504	C₄⋊C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
505	C₄⋊C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
506	C₄⋊C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
507	C₂²⋊C₄	—	2	1	no	—
508	C₂²⋊C₄	—	2	1	no	—
509	C₂²⋊C₄	—	2	1	no	—
510	C₂²⋊C₄	—	2	1	no	—
511	C₂²⋊C₄	—	2	1	no	—
512	C₂²⋊C₄	—	2	1	no	—
513	C₂²⋊C₄	—	2	1	no	—
514	C₂²⋊C₄	—	2	1	no	—
515	C₂²⋊C₄	—	2	1	no	—
516	C₄²	—	2	1	no	—
517	C₄²	C₂²×C₄×C₁₇	1	1	yes	characteristic
518	C₄²	C₂×D₄×C₁₇	1	1	yes	characteristic
519	C₄²	#41×C₁₇	1	1	yes	characteristic
520	C₄²	C₂⁴×C₁₇	1	1	yes	characteristic
521	C₂³	—	2	1	no	—
522	C₂³	—	2	1	no	—
523	C₂³	—	2	1	no	—
524	C₂³	—	4	2	no	—
525	C₂³	—	4	2	no	—
526	C₂³	—	8	1	no	—
527	C₂³	C₄×D₄×C₁₇	1	1	yes	characteristic
528	Q₈	—	2	1	no	—
529	Q₈	—	2	1	no	—
530	D₄	—	2	1	no	—
531	D₄	—	2	1	no	—
532	D₄	—	2	1	no	—
533	D₄	—	2	1	no	—
534	D₄	—	2	1	no	—
535	D₄	—	4	2	no	—
536	D₄	—	8	1	no	—
537	D₄	C₁₇×#131	1	2	yes	—
538	C₂×C₄	—	2	1	no	—
539	C₂×C₄	—	2	1	no	—
540	C₂×C₄	—	2	1	no	—
541	C₂×C₄	—	2	1	no	—
542	C₂×C₄	—	2	1	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	2	1	no	—
546	C₂×C₄	—	2	1	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	4	2	no	—
549	C₂×C₄	—	4	2	no	—
550	C₂×C₄	—	4	2	no	—
551	C₂×C₄	—	4	1	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	1	no	—
556	C₂×C₄	—	4	1	no	—
557	C₂×C₄	C₄×D₄×C₁₇	1	1	yes	characteristic
558	C₂×C₄	C₄×D₄×C₁₇	1	1	yes	characteristic
559	C₂×C₄	C₄×D₄×C₁₇	1	1	yes	characteristic
560	C₂×C₄	C₂³×C₄×C₁₇	1	1	yes	characteristic
561	C₂×C₄	C₂²×D₄×C₁₇	1	1	yes	characteristic
562	C₂×C₄	C₂×#41×C₁₇	1	1	yes	characteristic
563	C₈	—	2	1	no	—
564	C₈	—	2	1	no	—
565	C₈	—	2	1	no	—
566	C₈	—	2	1	no	—
567	C₈	—	2	1	no	—
568	C₈	—	2	1	no	—
569	C₈	—	2	1	no	—
570	C₈	—	4	1	no	—
571	C₈	—	4	1	no	—
572	C₈	—	4	1	no	—
573	C₈	—	4	1	no	—
574	C₈	—	4	1	no	—
575	C₈	—	4	1	no	—
576	C₈	C₁₇×AGL(1,ℤ/8)	1	2	yes	—
577	C₂²	—	2	1	no	—
578	C₂²	—	2	1	no	—
579	C₂²	—	2	2	no	—
580	C₂²	—	2	1	no	—
581	C₂²	—	4	1	no	—
582	C₂²	—	4	1	no	—
583	C₂²	—	4	1	no	—
584	C₂²	—	4	1	no	—
585	C₂²	—	4	1	no	—
586	C₂²	—	4	1	no	—
587	C₂²	—	4	1	no	—
588	C₂²	—	4	1	no	—
589	C₂²	—	8	4	no	—
590	C₂²	C₂×C₄×D₄×C₁₇	1	1	yes	characteristic
591	C₄	—	2	1	no	—
592	C₄	—	2	1	no	—
593	C₄	—	2	1	no	—
594	C₄	—	2	1	no	—
595	C₄	—	4	1	no	—
596	C₄	—	4	1	no	—
597	C₄	—	4	1	no	—
598	C₄	—	4	1	no	—
599	C₄	—	4	1	no	—
600	C₄	C₂×C₁₇×#131	1	1	yes	characteristic
601	C₄	C₁₇×#568	1	1	yes	characteristic
602	C₄	C₂×C₁₇×AGL(1,ℤ/8)	1	1	yes	characteristic
603	C₄	C₁₇×#575	1	1	yes	characteristic
604	C₂	—	4	1	no	—
605	C₂	—	4	2	no	—
606	C₂	—	8	1	no	—
607	C₂	—	8	1	no	—
608	C₂	C₁₇×#2933	1	1	yes	characteristic
609	C₂	C₄×C₁₇×AGL(1,ℤ/8)	1	1	yes	characteristic
610	C₂	C₁₇×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₇
Frattini, cosocle	C₄²	C₂⁴×C₁₇
socle	C₂²×C₁₇	C₂×C₄×D₄
prime radical	C₁₇×#447	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₇ ◁ C₄×M₁₆×C₁₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₁₇×#2969	C₂	semidirect
3. D₄×M₁₆×C₁₇	C₂	semidirect
4. C₁₇×#1592	C₂	semidirect
5. C₁₇×#1589	C₂	semidirect
6. C₁₇×#1586	C₂	semidirect
7. C₁₇×#1585	C₂	semidirect
8. C₁₇×#1583	C₂	semidirect
9. C₁₇×#1581	C₂	semidirect
10. C₁₇×#1580	C₂	semidirect
11. C₈×SD₁₆×C₁₇	C₂	semidirect
12. C₁₇×#1574	C₂	semidirect
13. C₁₇×#1495	C₂	semidirect
14. C₁₇×#1478	C₂	semidirect
15. C₁₇×#1453	C₂	semidirect
27. C₁₇×#446	C₂²	semidirect
28. C₈×Q₈×C₁₇	C₂²	semidirect
36. C₁₇×#436	C₂²	semidirect
42. C₈×D₄×C₁₇	C₂²	semidirect
51. C₁₇×#335	C₂²	semidirect
52. C₁₇×#335	C₂²	semidirect
53. C₁₇×#334	C₂²	semidirect
54. C₁₇×#334	C₂²	semidirect
55. C₁₇×Q₈⋊C₈	C₂²	semidirect
56. C₁₇×Q₈⋊C₈	C₂²	semidirect
61. C₁₇×D₄⋊C₈	C₂²	semidirect
62. C₁₇×D₄⋊C₈	C₂²	semidirect
63. C₁₇×#322	C₂²	semidirect
64. C₁₇×#322	C₂²	semidirect
65. C₁₇×#322	C₂²	semidirect
66. C₈²×C₁₇	C₂²	semidirect
215. #18008	C₁₇	direct (N × H)
273. #2969	C₂×C₁₇	semidirect
274. D₄×M₁₆	C₂×C₁₇	semidirect
275. #1592	C₂×C₁₇	semidirect
276. #1589	C₂×C₁₇	semidirect
277. #1586	C₂×C₁₇	semidirect
278. #1585	C₂×C₁₇	semidirect
279. #1583	C₂×C₁₇	semidirect
280. #1581	C₂×C₁₇	semidirect
281. #1580	C₂×C₁₇	semidirect
282. C₈×SD₁₆	C₂×C₁₇	semidirect
283. #1574	C₂×C₁₇	semidirect
284. #1495	C₂×C₁₇	semidirect
285. #1478	C₂×C₁₇	semidirect
286. #1453	C₂×C₁₇	semidirect
325. #446	C₂²×C₁₇	semidirect
326. C₈×Q₈	C₂²×C₁₇	semidirect
334. #436	C₂²×C₁₇	semidirect
340. C₈×D₄	C₂²×C₁₇	semidirect
349. #335	C₂²×C₁₇	semidirect
350. #335	C₂²×C₁₇	semidirect
351. #334	C₂²×C₁₇	semidirect
352. #334	C₂²×C₁₇	semidirect
353. Q₈⋊C₈	C₂²×C₁₇	semidirect
354. Q₈⋊C₈	C₂²×C₁₇	semidirect
359. D₄⋊C₈	C₂²×C₁₇	semidirect
360. D₄⋊C₈	C₂²×C₁₇	semidirect
361. #322	C₂²×C₁₇	semidirect
362. #322	C₂²×C₁₇	semidirect
363. #322	C₂²×C₁₇	semidirect
364. C₈²	C₂²×C₁₇	semidirect
450. C₁₇	#18008	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₁₇×#3105	2
C₁₇×#2969	2
D₄×M₁₆×C₁₇	2
C₁₇×#1592	2
C₁₇×#1589	2
C₁₇×#1586	2
C₁₇×#1585	2
C₁₇×#1583	2
C₁₇×#1581	2
C₁₇×#1580	2
C₈×SD₁₆×C₁₇	2
C₁₇×#1574	2
C₁₇×#1495	2
C₁₇×#1478	2
C₁₇×#1453	2
#18008	17

Subgroups — C17×#18008 #16875752

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#18008 #16875752