Subgroups — C₁₇×#18020 #16875764

Order 4352.

642 conjugacy classes of subgroups.

The subgroup lattice has 608 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×#2994	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₇×#2974	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₇×#1594	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₇×#1592	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₇×#1583	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₇×#1582	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₇×#1581	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	C₁₇×SD₁₆⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₇×#1573	C₂	1	1	yes	maximal, characteristic, complemented
13	C₁₇×#1494	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₁₇×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#1452	C₂	1	1	yes	maximal, characteristic, complemented
16	C₁₇×#529	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
18	C₁₇×#484	C₄	1	1	yes	characteristic
19	C₁₇×#481	C₄	1	1	yes	characteristic
20	C₁₇×#478	C₄	1	1	yes	characteristic
21	C₁₇×#478	C₄	1	1	yes	characteristic
22	C₁₇×#463	C₄	1	1	yes	characteristic
23	C₁₇×#463	C₄	1	1	yes	characteristic
24	C₁₇×#450	C₄	1	1	yes	characteristic
25	C₁₇×#447	C₄	1	1	yes	characteristic
26	C₁₇×#446	C₂²	1	1	yes	characteristic, complemented
27	C₈×Q₈×C₁₇	C₂²	1	1	yes	characteristic, complemented
28	C₄×SD₁₆×C₁₇	C₂²	1	1	yes	characteristic
29	C₄×SD₁₆×C₁₇	C₂²	1	1	yes	characteristic
30	C₄×D₈×C₁₇	C₂²	1	1	yes	characteristic
31	C₄×D₈×C₁₇	C₂²	1	1	yes	characteristic
32	C₁₇×#436	C₂²	1	1	yes	characteristic
33	C₁₇×#436	C₂²	1	1	yes	characteristic
34	C₁₇×#436	C₂²	1	1	yes	characteristic, complemented
35	C₁₇×#435	—	2	1	no	—
36	C₁₇×#435	—	2	1	no	—
37	C₁₇×#435	—	2	1	no	—
38	C₈×D₄×C₁₇	—	2	1	no	—
39	C₈×D₄×C₁₇	C₂²	1	1	yes	characteristic
40	C₈×D₄×C₁₇	C₂²	1	1	yes	characteristic
41	C₈×D₄×C₁₇	C₂²	1	1	yes	characteristic, complemented
42	C₁₇×#433	—	2	1	no	—
43	C₁₇×#432	C₂²	1	1	yes	characteristic
44	C₁₇×#431	C₂²	1	1	yes	characteristic
45	C₁₇×#431	C₂²	1	1	yes	characteristic
46	C₁₇×#424	—	2	1	no	—
47	C₂×C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
48	C₁₇×#407	—	2	1	no	—
49	C₂×C₁₇×C₂²⋊C₈	—	2	1	no	—
50	C₂×C₄×C₈×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	C₁₇×#322	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	C₂²×C₈×C₁₇	—	2	1	no	—
77	C₂²×C₈×C₁₇	—	2	1	no	—
78	C₂²×C₈×C₁₇	—	2	1	no	—
79	C₂²×C₈×C₁₇	—	2	1	no	—
80	C₂²×C₈×C₁₇	—	2	1	no	—
81	C₁₇×#126	—	2	1	no	—
82	C₁₇×#125	C₂×C₄	1	1	yes	characteristic
83	C₁₇×#124	C₂×C₄	1	1	yes	characteristic
84	C₁₇×#123	—	2	1	no	—
85	C₁₇×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
86	C₁₇×#121	C₂×C₄	1	1	yes	characteristic
87	C₄×Q₈×C₁₇	C₂³	1	1	yes	characteristic
88	C₄×D₄×C₁₇	—	2	1	no	—
89	C₄×D₄×C₁₇	—	2	1	no	—
90	C₄×D₄×C₁₇	—	2	1	no	—
91	C₄×D₄×C₁₇	C₂×C₄	1	1	yes	characteristic
92	C₄×D₄×C₁₇	C₂×C₄	1	1	yes	characteristic
93	C₄×D₄×C₁₇	C₂³	1	1	yes	characteristic
94	C₄×D₄×C₁₇	C₂³	1	1	yes	characteristic
95	C₄×D₄×C₁₇	C₂³	1	1	yes	characteristic
96	C₁₇×#117	—	2	1	no	—
97	C₂×C₄⋊C₄×C₁₇	C₂×C₄	1	1	yes	characteristic
98	C₂×C₂²⋊C₄×C₁₇	—	2	1	no	—
99	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
100	C₁₇×#107	C₂×C₄	1	1	yes	characteristic
101	C₁₇×#107	C₂×C₄	1	1	yes	characteristic
102	C₁₇×#106	C₂×C₄	1	1	yes	characteristic
103	C₁₇×#106	C₂×C₄	1	1	yes	characteristic
104	C₁₇×C₄⋊C₈	—	2	1	no	—
105	C₁₇×C₄⋊C₈	—	2	1	no	—
106	C₁₇×C₄⋊C₈	—	2	1	no	—
107	C₁₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
108	C₁₇×C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
109	C₁₇×C₄⋊C₈	D₄	1	1	yes	characteristic
110	C₁₇×C₄⋊C₈	D₄	1	1	yes	characteristic
111	C₁₇×C₄⋊C₈	C₂³	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×#103	C₂×C₄	1	1	yes	characteristic
116	C₁₇×#103	C₂×C₄	1	1	yes	characteristic
117	C₁₇×#102	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×C₂²⋊C₈	—	2	1	no	—
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	C₂×C₄	1	1	yes	characteristic
131	C₁₇×C₂²⋊C₈	C₂×C₄	1	1	yes	characteristic
132	C₁₇×C₂²⋊C₈	D₄	1	1	yes	characteristic
133	C₁₇×C₂²⋊C₈	D₄	1	1	yes	characteristic
134	C₁₇×#97	—	2	1	no	—
135	C₁₇×#97	—	2	1	no	—
136	C₄×C₈×C₁₇	—	2	1	no	—
137	C₄×C₈×C₁₇	—	2	1	no	—
138	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
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	C₂⁴×C₁₇	—	2	1	no	—
145	C₂×Q₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
146	C₂×D₄×C₁₇	—	2	2	no	—
147	C₂×D₄×C₁₇	—	2	1	no	—
148	C₂×D₄×C₁₇	—	2	1	no	—
149	C₂×D₄×C₁₇	—	4	1	no	—
150	C₂×D₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
151	C₂×D₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
152	C₂×D₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
153	C₂²×C₄×C₁₇	—	2	1	no	—
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	—	8	1	no	—
160	C₂²×C₄×C₁₇	—	8	1	no	—
161	C₂²×C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
162	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₁₇	#41	1	1	yes	characteristic
164	SD₁₆×C₁₇	—	2	2	no	—
165	SD₁₆×C₁₇	—	4	1	no	—
166	D₈×C₁₇	—	2	2	no	—
167	D₈×C₁₇	—	4	1	no	—
168	M₁₆×C₁₇	—	8	1	no	—
169	M₁₆×C₁₇	—	8	1	no	—
170	M₁₆×C₁₇	—	8	1	no	—
171	C₂×C₈×C₁₇	—	2	1	no	—
172	C₂×C₈×C₁₇	—	2	1	no	—
173	C₂×C₈×C₁₇	—	2	1	no	—
174	C₂×C₈×C₁₇	—	2	1	no	—
175	C₂×C₈×C₁₇	—	2	1	no	—
176	C₂×C₈×C₁₇	—	2	1	no	—
177	C₂×C₈×C₁₇	—	4	2	no	—
178	C₂×C₈×C₁₇	—	8	1	no	—
179	C₂×C₈×C₁₇	—	8	1	no	—
180	C₂×C₈×C₁₇	—	8	1	no	—
181	C₂×C₈×C₁₇	—	8	1	no	—
182	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
183	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
184	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
185	C₂×C₈×C₁₇	C₂²×C₄	1	1	yes	characteristic
186	C₂×C₈×C₁₇	C₂×D₄	1	1	yes	characteristic
187	C₂×C₈×C₁₇	C₂×D₄	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	C₄⋊C₄×C₁₇	—	2	1	no	—
191	C₄⋊C₄×C₁₇	—	2	1	no	—
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	C₂²×C₄	1	1	yes	characteristic
196	C₄⋊C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
197	C₄⋊C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
198	C₄⋊C₄×C₁₇	C₂²×C₄	1	1	yes	characteristic
199	C₂²⋊C₄×C₁₇	—	2	1	no	—
200	C₂²⋊C₄×C₁₇	—	2	1	no	—
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	C₄²×C₁₇	—	2	1	no	—
209	C₄²×C₁₇	C₂²×C₄	1	1	yes	characteristic
210	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
211	C₄²×C₁₇	#41	1	1	yes	characteristic
212	C₄²×C₁₇	C₂⁴	1	1	yes	characteristic
213	#18020	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
214	C₂³×C₁₇	—	2	1	no	—
215	C₂³×C₁₇	—	2	1	no	—
216	C₂³×C₁₇	—	2	1	no	—
217	C₂³×C₁₇	—	4	2	no	—
218	C₂³×C₁₇	—	4	2	no	—
219	C₂³×C₁₇	—	8	1	no	—
220	C₂³×C₁₇	C₄×D₄	1	1	yes	characteristic
221	Q₈×C₁₇	—	2	1	no	—
222	Q₈×C₁₇	—	2	1	no	—
223	D₄×C₁₇	—	2	1	no	—
224	D₄×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	—	4	2	no	—
229	D₄×C₁₇	—	4	2	no	—
230	D₄×C₁₇	#131	1	2	yes	—
231	C₂×C₄×C₁₇	—	2	2	no	—
232	C₂×C₄×C₁₇	—	2	1	no	—
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	—	4	1	no	—
242	C₂×C₄×C₁₇	—	4	1	no	—
243	C₂×C₄×C₁₇	—	4	1	no	—
244	C₂×C₄×C₁₇	—	4	1	no	—
245	C₂×C₄×C₁₇	—	4	1	no	—
246	C₂×C₄×C₁₇	—	4	1	no	—
247	C₂×C₄×C₁₇	—	8	1	no	—
248	C₂×C₄×C₁₇	—	8	1	no	—
249	C₂×C₄×C₁₇	—	8	1	no	—
250	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
251	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
252	C₂×C₄×C₁₇	C₄×D₄	1	1	yes	characteristic
253	C₂×C₄×C₁₇	C₂³×C₄	1	1	yes	characteristic
254	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
255	C₂×C₄×C₁₇	C₂×#41	1	1	yes	characteristic
256	C₈×C₁₇	—	2	1	no	—
257	C₈×C₁₇	—	2	1	no	—
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	—	4	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	#3105	C₂×C₁₇	1	1	yes	characteristic
271	#2994	C₂×C₁₇	1	1	yes	characteristic, complemented
272	#2974	C₂×C₁₇	1	1	yes	characteristic, complemented
273	#1594	C₂×C₁₇	1	1	yes	characteristic, complemented
274	#1592	C₂×C₁₇	1	1	yes	characteristic, complemented
275	#1583	C₂×C₁₇	1	1	yes	characteristic, complemented
276	#1582	C₂×C₁₇	1	1	yes	characteristic, complemented
277	#1581	C₂×C₁₇	1	1	yes	characteristic, complemented
278	#1581	C₂×C₁₇	1	1	yes	characteristic, complemented
279	#1580	C₂×C₁₇	1	1	yes	characteristic, complemented
280	SD₁₆⋊C₈	C₂×C₁₇	1	1	yes	characteristic, complemented
281	#1573	C₂×C₁₇	1	1	yes	characteristic, complemented
282	#1494	C₂×C₁₇	1	1	yes	characteristic, complemented
283	C₂×D₄⋊C₈	C₂×C₁₇	1	1	yes	characteristic, complemented
284	#1452	C₂×C₁₇	1	1	yes	characteristic, complemented
285	C₂²×C₁₇	—	2	1	no	—
286	C₂²×C₁₇	—	2	1	no	—
287	C₂²×C₁₇	—	2	2	no	—
288	C₂²×C₁₇	—	4	1	no	—
289	C₂²×C₁₇	—	4	1	no	—
290	C₂²×C₁₇	—	4	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	—	8	4	no	—
297	C₂²×C₁₇	#444	1	2	yes	—
298	C₂²×C₁₇	C₂×C₄×D₄	1	1	yes	characteristic
299	C₄×C₁₇	—	2	1	no	—
300	C₄×C₁₇	—	2	1	no	—
301	C₄×C₁₇	—	2	1	no	—
302	C₄×C₁₇	—	2	1	no	—
303	C₄×C₁₇	—	4	1	no	—
304	C₄×C₁₇	—	4	1	no	—
305	C₄×C₁₇	—	4	1	no	—
306	C₄×C₁₇	—	4	1	no	—
307	C₄×C₁₇	—	4	1	no	—
308	C₄×C₁₇	C₂×#131	1	1	yes	characteristic
309	C₄×C₁₇	#568	1	1	yes	characteristic
310	C₄×C₁₇	#576	1	1	yes	characteristic
311	C₄×C₁₇	#577	1	1	yes	characteristic
312	#529	C₂²×C₁₇	1	1	yes	characteristic
313	C₂×C₄×D₄	C₂²×C₁₇	1	1	yes	characteristic
314	#484	C₄×C₁₇	1	1	yes	characteristic
315	#481	C₄×C₁₇	1	1	yes	characteristic
316	#478	C₄×C₁₇	1	1	yes	characteristic
317	#478	C₄×C₁₇	1	1	yes	characteristic
318	#463	C₄×C₁₇	1	1	yes	characteristic
319	#463	C₄×C₁₇	1	1	yes	characteristic
320	#450	C₄×C₁₇	1	1	yes	characteristic
321	#447	C₄×C₁₇	1	1	yes	characteristic
322	#446	C₂²×C₁₇	1	1	yes	characteristic, complemented
323	C₈×Q₈	C₂²×C₁₇	1	1	yes	characteristic, complemented
324	C₄×SD₁₆	C₂²×C₁₇	1	1	yes	characteristic
325	C₄×SD₁₆	C₂²×C₁₇	1	1	yes	characteristic
326	C₄×D₈	C₂²×C₁₇	1	1	yes	characteristic
327	C₄×D₈	C₂²×C₁₇	1	1	yes	characteristic
328	#436	C₂²×C₁₇	1	1	yes	characteristic
329	#436	C₂²×C₁₇	1	1	yes	characteristic
330	#436	C₂²×C₁₇	1	1	yes	characteristic, complemented
331	#435	—	2	1	no	—
332	#435	—	2	1	no	—
333	#435	—	2	1	no	—
334	C₈×D₄	—	2	1	no	—
335	C₈×D₄	C₂²×C₁₇	1	1	yes	characteristic
336	C₈×D₄	C₂²×C₁₇	1	1	yes	characteristic
337	C₈×D₄	C₂²×C₁₇	1	1	yes	characteristic, complemented
338	#433	—	2	1	no	—
339	#432	C₂²×C₁₇	1	1	yes	characteristic
340	#431	C₂²×C₁₇	1	1	yes	characteristic
341	#431	C₂²×C₁₇	1	1	yes	characteristic
342	#424	—	2	1	no	—
343	C₂×C₄⋊C₈	C₂²×C₁₇	1	1	yes	characteristic
344	#407	—	2	1	no	—
345	C₂×C₂²⋊C₈	—	2	1	no	—
346	C₂×C₄×C₈	C₂²×C₁₇	1	1	yes	characteristic
347	#335	C₂²×C₁₇	1	1	yes	characteristic, complemented
348	#335	C₂²×C₁₇	1	1	yes	characteristic, complemented
349	#334	C₂²×C₁₇	1	1	yes	characteristic, complemented
350	#334	C₂²×C₁₇	1	1	yes	characteristic, complemented
351	Q₈⋊C₈	C₂²×C₁₇	1	1	yes	characteristic, complemented
352	Q₈⋊C₈	C₂²×C₁₇	1	1	yes	characteristic, complemented
353	D₄⋊C₈	C₂²×C₁₇	1	1	yes	characteristic
354	D₄⋊C₈	C₂²×C₁₇	1	1	yes	characteristic
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, complemented
358	D₄⋊C₈	C₂²×C₁₇	1	1	yes	characteristic, complemented
359	#322	C₂²×C₁₇	1	1	yes	characteristic, complemented
360	#322	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	C₂×C₁₇	—	2	2	no	—
364	C₂×C₁₇	—	4	2	no	—
365	C₂×C₁₇	—	8	1	no	—
366	C₂×C₁₇	—	8	1	no	—
367	C₂×C₁₇	#2933	1	1	yes	characteristic
368	C₂×C₁₇	#2946	1	1	yes	characteristic
369	C₂×C₁₇	C₂×#444	1	1	yes	characteristic
370	C₂²×D₄	C₂×C₄×C₁₇	1	1	yes	characteristic
371	C₂³×C₄	—	2	1	no	—
372	C₂×SD₁₆	C₂×C₄×C₁₇	1	1	yes	characteristic
373	C₂×SD₁₆	C₂×C₄×C₁₇	1	1	yes	characteristic
374	C₂×D₈	C₂×C₄×C₁₇	1	1	yes	characteristic
375	C₂×D₈	C₂×C₄×C₁₇	1	1	yes	characteristic
376	C₂×M₁₆	—	2	1	no	—
377	C₂×M₁₆	—	2	1	no	—
378	C₂×M₁₆	—	2	1	no	—
379	C₂²×C₈	—	2	1	no	—
380	C₂²×C₈	—	2	1	no	—
381	C₂²×C₈	—	2	1	no	—
382	C₂²×C₈	—	2	1	no	—
383	C₂²×C₈	—	2	1	no	—
384	#126	—	2	1	no	—
385	#125	C₂×C₄×C₁₇	1	1	yes	characteristic
386	#124	C₂×C₄×C₁₇	1	1	yes	characteristic
387	#123	—	2	1	no	—
388	C₂²⋊Q₈	C₂×C₄×C₁₇	1	1	yes	characteristic
389	#121	C₂×C₄×C₁₇	1	1	yes	characteristic
390	C₄×Q₈	C₂³×C₁₇	1	1	yes	characteristic
391	C₄×D₄	—	2	1	no	—
392	C₄×D₄	—	2	1	no	—
393	C₄×D₄	—	2	1	no	—
394	C₄×D₄	C₂×C₄×C₁₇	1	1	yes	characteristic
395	C₄×D₄	C₂×C₄×C₁₇	1	1	yes	characteristic
396	C₄×D₄	C₂³×C₁₇	1	1	yes	characteristic
397	C₄×D₄	C₂³×C₁₇	1	1	yes	characteristic
398	C₄×D₄	C₂³×C₁₇	1	1	yes	characteristic
399	#117	—	2	1	no	—
400	C₂×C₄⋊C₄	C₂×C₄×C₁₇	1	1	yes	characteristic
401	C₂×C₂²⋊C₄	—	2	1	no	—
402	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
403	#107	C₂×C₄×C₁₇	1	1	yes	characteristic
404	#107	C₂×C₄×C₁₇	1	1	yes	characteristic
405	#106	C₂×C₄×C₁₇	1	1	yes	characteristic
406	#106	C₂×C₄×C₁₇	1	1	yes	characteristic
407	C₄⋊C₈	—	2	1	no	—
408	C₄⋊C₈	—	2	1	no	—
409	C₄⋊C₈	—	2	1	no	—
410	C₄⋊C₈	C₂×C₄×C₁₇	1	1	yes	characteristic
411	C₄⋊C₈	C₂×C₄×C₁₇	1	1	yes	characteristic
412	C₄⋊C₈	D₄×C₁₇	1	1	yes	characteristic
413	C₄⋊C₈	D₄×C₁₇	1	1	yes	characteristic
414	C₄⋊C₈	C₂³×C₁₇	1	1	yes	characteristic
415	C₄⋊C₈	C₂³×C₁₇	1	1	yes	characteristic
416	C₄⋊C₈	C₂³×C₁₇	1	1	yes	characteristic
417	C₄⋊C₈	C₂³×C₁₇	1	1	yes	characteristic
418	#103	C₂×C₄×C₁₇	1	1	yes	characteristic
419	#103	C₂×C₄×C₁₇	1	1	yes	characteristic
420	#102	C₂×C₄×C₁₇	1	1	yes	characteristic
421	#102	C₂×C₄×C₁₇	1	1	yes	characteristic
422	#102	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	C₂²⋊C₈	—	2	1	no	—
427	C₂²⋊C₈	—	2	1	no	—
428	C₂²⋊C₈	—	2	1	no	—
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	C₂×C₄×C₁₇	1	1	yes	characteristic
434	C₂²⋊C₈	C₂×C₄×C₁₇	1	1	yes	characteristic
435	C₂²⋊C₈	D₄×C₁₇	1	1	yes	characteristic
436	C₂²⋊C₈	D₄×C₁₇	1	1	yes	characteristic
437	#97	—	2	1	no	—
438	#97	—	2	1	no	—
439	C₄×C₈	—	2	1	no	—
440	C₄×C₈	—	2	1	no	—
441	C₄×C₈	C₂³×C₁₇	1	1	yes	characteristic
442	C₄×C₈	C₂³×C₁₇	1	1	yes	characteristic
443	C₄×C₈	C₂³×C₁₇	1	1	yes	characteristic
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	C₁₇	#18020	1	1	yes	characteristic, direct product factor, Sylow
448	C₂⁴	—	2	1	no	—
449	C₂×Q₈	C₂²×C₄×C₁₇	1	1	yes	characteristic
450	C₂×D₄	—	2	2	no	—
451	C₂×D₄	—	2	1	no	—
452	C₂×D₄	—	2	1	no	—
453	C₂×D₄	—	4	1	no	—
454	C₂×D₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
455	C₂×D₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
456	C₂×D₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
457	C₂²×C₄	—	2	1	no	—
458	C₂²×C₄	—	2	1	no	—
459	C₂²×C₄	—	2	1	no	—
460	C₂²×C₄	—	2	1	no	—
461	C₂²×C₄	—	2	1	no	—
462	C₂²×C₄	—	2	1	no	—
463	C₂²×C₄	—	8	1	no	—
464	C₂²×C₄	—	8	1	no	—
465	C₂²×C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
466	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
467	C₂²×C₄	#41×C₁₇	1	1	yes	characteristic
468	SD₁₆	—	2	2	no	—
469	SD₁₆	—	4	1	no	—
470	D₈	—	2	2	no	—
471	D₈	—	4	1	no	—
472	M₁₆	—	8	1	no	—
473	M₁₆	—	8	1	no	—
474	M₁₆	—	8	1	no	—
475	C₂×C₈	—	2	1	no	—
476	C₂×C₈	—	2	1	no	—
477	C₂×C₈	—	2	1	no	—
478	C₂×C₈	—	2	1	no	—
479	C₂×C₈	—	2	1	no	—
480	C₂×C₈	—	2	1	no	—
481	C₂×C₈	—	4	2	no	—
482	C₂×C₈	—	8	1	no	—
483	C₂×C₈	—	8	1	no	—
484	C₂×C₈	—	8	1	no	—
485	C₂×C₈	—	8	1	no	—
486	C₂×C₈	C₂²×C₄×C₁₇	1	1	yes	characteristic
487	C₂×C₈	C₂²×C₄×C₁₇	1	1	yes	characteristic
488	C₂×C₈	C₂²×C₄×C₁₇	1	1	yes	characteristic
489	C₂×C₈	C₂²×C₄×C₁₇	1	1	yes	characteristic
490	C₂×C₈	C₂×D₄×C₁₇	1	1	yes	characteristic
491	C₂×C₈	C₂×D₄×C₁₇	1	1	yes	characteristic
492	C₂×C₈	C₂×D₄×C₁₇	1	1	yes	characteristic
493	C₂×C₈	C₂×D₄×C₁₇	1	1	yes	characteristic
494	C₄⋊C₄	—	2	1	no	—
495	C₄⋊C₄	—	2	1	no	—
496	C₄⋊C₄	—	2	1	no	—
497	C₄⋊C₄	—	2	1	no	—
498	C₄⋊C₄	—	2	1	no	—
499	C₄⋊C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
500	C₄⋊C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
501	C₄⋊C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
502	C₄⋊C₄	C₂²×C₄×C₁₇	1	1	yes	characteristic
503	C₂²⋊C₄	—	2	1	no	—
504	C₂²⋊C₄	—	2	1	no	—
505	C₂²⋊C₄	—	2	1	no	—
506	C₂²⋊C₄	—	2	1	no	—
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	C₄²	—	2	1	no	—
513	C₄²	C₂²×C₄×C₁₇	1	1	yes	characteristic
514	C₄²	C₂×D₄×C₁₇	1	1	yes	characteristic
515	C₄²	#41×C₁₇	1	1	yes	characteristic
516	C₄²	C₂⁴×C₁₇	1	1	yes	characteristic
517	C₂³	—	2	1	no	—
518	C₂³	—	2	1	no	—
519	C₂³	—	2	1	no	—
520	C₂³	—	4	2	no	—
521	C₂³	—	4	2	no	—
522	C₂³	—	8	1	no	—
523	C₂³	C₄×D₄×C₁₇	1	1	yes	characteristic
524	Q₈	—	2	1	no	—
525	Q₈	—	2	1	no	—
526	D₄	—	2	1	no	—
527	D₄	—	2	1	no	—
528	D₄	—	2	1	no	—
529	D₄	—	2	1	no	—
530	D₄	—	2	1	no	—
531	D₄	—	4	2	no	—
532	D₄	—	4	2	no	—
533	D₄	C₁₇×#131	1	2	yes	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	1	no	—
536	C₂×C₄	—	2	1	no	—
537	C₂×C₄	—	2	1	no	—
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	—	4	1	no	—
545	C₂×C₄	—	4	1	no	—
546	C₂×C₄	—	4	1	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	4	1	no	—
550	C₂×C₄	—	8	1	no	—
551	C₂×C₄	—	8	1	no	—
552	C₂×C₄	—	8	1	no	—
553	C₂×C₄	C₄×D₄×C₁₇	1	1	yes	characteristic
554	C₂×C₄	C₄×D₄×C₁₇	1	1	yes	characteristic
555	C₂×C₄	C₄×D₄×C₁₇	1	1	yes	characteristic
556	C₂×C₄	C₂³×C₄×C₁₇	1	1	yes	characteristic
557	C₂×C₄	C₂²×D₄×C₁₇	1	1	yes	characteristic
558	C₂×C₄	C₂×#41×C₁₇	1	1	yes	characteristic
559	C₈	—	2	1	no	—
560	C₈	—	2	1	no	—
561	C₈	—	2	1	no	—
562	C₈	—	2	1	no	—
563	C₈	—	2	1	no	—
564	C₈	—	2	1	no	—
565	C₈	—	2	1	no	—
566	C₈	—	2	1	no	—
567	C₈	—	4	1	no	—
568	C₈	—	4	1	no	—
569	C₈	—	4	1	no	—
570	C₈	—	4	1	no	—
571	C₈	—	4	1	no	—
572	C₈	—	4	1	no	—
573	C₂²	—	2	1	no	—
574	C₂²	—	2	1	no	—
575	C₂²	—	2	2	no	—
576	C₂²	—	4	1	no	—
577	C₂²	—	4	1	no	—
578	C₂²	—	4	1	no	—
579	C₂²	—	4	1	no	—
580	C₂²	—	4	1	no	—
581	C₂²	—	4	1	no	—
582	C₂²	—	4	1	no	—
583	C₂²	—	4	1	no	—
584	C₂²	—	8	4	no	—
585	C₂²	C₁₇×#444	1	2	yes	—
586	C₂²	C₂×C₄×D₄×C₁₇	1	1	yes	characteristic
587	C₄	—	2	1	no	—
588	C₄	—	2	1	no	—
589	C₄	—	2	1	no	—
590	C₄	—	2	1	no	—
591	C₄	—	4	1	no	—
592	C₄	—	4	1	no	—
593	C₄	—	4	1	no	—
594	C₄	—	4	1	no	—
595	C₄	—	4	1	no	—
596	C₄	C₂×C₁₇×#131	1	1	yes	characteristic
597	C₄	C₁₇×#568	1	1	yes	characteristic
598	C₄	C₁₇×#576	1	1	yes	characteristic
599	C₄	C₁₇×#577	1	1	yes	characteristic
600	C₂	—	2	2	no	—
601	C₂	—	4	2	no	—
602	C₂	—	8	1	no	—
603	C₂	—	8	1	no	—
604	C₂	C₁₇×#2933	1	1	yes	characteristic
605	C₂	C₁₇×#2946	1	1	yes	characteristic
606	C₂	C₂×C₁₇×#444	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 ◁ C₁₇×#431 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₇×#2994	C₂	semidirect
3. C₁₇×#2974	C₂	semidirect
4. C₁₇×#1594	C₂	semidirect
5. C₁₇×#1592	C₂	semidirect
6. C₁₇×#1583	C₂	semidirect
7. C₁₇×#1582	C₂	semidirect
8. C₁₇×#1581	C₂	semidirect
9. C₁₇×#1581	C₂	semidirect
10. C₁₇×#1580	C₂	semidirect
11. C₁₇×SD₁₆⋊C₈	C₂	semidirect
12. C₁₇×#1573	C₂	semidirect
13. C₁₇×#1494	C₂	semidirect
14. C₂×C₁₇×D₄⋊C₈	C₂	semidirect
15. C₁₇×#1452	C₂	semidirect
26. C₁₇×#446	C₂²	semidirect
27. C₈×Q₈×C₁₇	C₂²	semidirect
34. C₁₇×#436	C₂²	semidirect
41. 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. C₁₇×#322	C₂²	semidirect
213. #18020	C₁₇	direct (N × H)
271. #2994	C₂×C₁₇	semidirect
272. #2974	C₂×C₁₇	semidirect
273. #1594	C₂×C₁₇	semidirect
274. #1592	C₂×C₁₇	semidirect
275. #1583	C₂×C₁₇	semidirect
276. #1582	C₂×C₁₇	semidirect
277. #1581	C₂×C₁₇	semidirect
278. #1581	C₂×C₁₇	semidirect
279. #1580	C₂×C₁₇	semidirect
280. SD₁₆⋊C₈	C₂×C₁₇	semidirect
281. #1573	C₂×C₁₇	semidirect
282. #1494	C₂×C₁₇	semidirect
283. C₂×D₄⋊C₈	C₂×C₁₇	semidirect
284. #1452	C₂×C₁₇	semidirect
322. #446	C₂²×C₁₇	semidirect
323. C₈×Q₈	C₂²×C₁₇	semidirect
330. #436	C₂²×C₁₇	semidirect
337. C₈×D₄	C₂²×C₁₇	semidirect
347. #335	C₂²×C₁₇	semidirect
348. #335	C₂²×C₁₇	semidirect
349. #334	C₂²×C₁₇	semidirect
350. #334	C₂²×C₁₇	semidirect
351. Q₈⋊C₈	C₂²×C₁₇	semidirect
352. Q₈⋊C₈	C₂²×C₁₇	semidirect
357. D₄⋊C₈	C₂²×C₁₇	semidirect
358. D₄⋊C₈	C₂²×C₁₇	semidirect
359. #322	C₂²×C₁₇	semidirect
360. #322	C₂²×C₁₇	semidirect
361. #322	C₂²×C₁₇	semidirect
362. #322	C₂²×C₁₇	semidirect
447. C₁₇	#18020	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×#2994	2
C₁₇×#2974	2
C₁₇×#1594	2
C₁₇×#1592	2
C₁₇×#1583	2
C₁₇×#1582	2
C₁₇×#1581	2
C₁₇×#1580	2
C₁₇×SD₁₆⋊C₈	2
C₁₇×#1573	2
C₁₇×#1494	2
C₂×C₁₇×D₄⋊C₈	2
C₁₇×#1452	2
#18020	17

Subgroups — C17×#18020 #16875764

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#18020 #16875764