Subgroups — C₁₇×#61406 #16919150

Order 4352.

1986 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×#3506	C₂	1	2	yes	maximal, complemented
2	C₁₇×#3403	C₂	1	4	yes	maximal, complemented
3	C₁₇×#3397	C₂	1	4	yes	maximal, complemented
4	C₁₇×#3396	C₂	1	4	yes	maximal, complemented
5	C₁₇×#3393	C₂	1	2	yes	maximal, complemented
6	C₁₇×#3391	C₂	1	2	yes	maximal, complemented
7	C₁₇×#3155	C₂	1	2	yes	maximal, complemented
8	C₁₇×#3147	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₇×#3045	C₂	1	4	yes	maximal, complemented
10	C₁₇×#3036	C₂	1	2	yes	maximal, complemented
11	C₁₇×#2960	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₇×#2951	C₂	1	2	yes	maximal, complemented
13	C₂×C₄×D₈×C₁₇	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₁₇×AGL(1,ℤ/8)	C₂²	1	4	yes	—
15	C₂×C₁₇×#135	C₂²	1	2	yes	—
16	C₂×C₁₇×#135	C₂²	1	2	yes	—
17	C₂²×D₈×C₁₇	C₂²	1	1	yes	characteristic
18	C₁₇×#559	C₂²	1	8	yes	complemented
19	C₁₇×#556	C₂²	1	4	yes	complemented
20	C₁₇×#555	—	2	2	no	—
21	C₁₇×#553	—	2	4	no	—
22	C₁₇×#550	C₂²	1	4	yes	complemented
23	C₁₇×#547	—	2	2	no	—
24	C₁₇×#538	—	2	4	no	—
25	C₁₇×#535	C₂²	1	4	yes	—
26	C₁₇×#533	C₂²	1	1	yes	characteristic
27	C₁₇×#532	C₂²	1	2	yes	—
28	C₁₇×#525	—	2	4	no	—
29	C₁₇×#520	C₂²	1	4	yes	—
30	C₂×C₄×D₄×C₁₇	C₂²	1	2	yes	—
31	C₁₇×#496	C₂²	1	8	yes	—
32	C₁₇×#495	C₂²	1	4	yes	—
33	C₁₇×#495	C₂²	1	2	yes	complemented
34	C₁₇×#495	C₂²	1	2	yes	complemented
35	C₁₇×#494	C₂²	1	1	yes	characteristic, complemented
36	C₁₇×#493	C₂²	1	4	yes	—
37	C₁₇×#493	C₂²	1	1	yes	characteristic, complemented
38	C₁₇×#492	C₂²	1	2	yes	complemented
39	C₁₇×#464	C₂²	1	8	yes	complemented
40	C₁₇×#463	C₂²	1	8	yes	complemented
41	C₁₇×#463	C₂²	1	8	yes	complemented
42	C₁₇×#461	C₂²	1	4	yes	complemented
43	C₁₇×#460	C₂²	1	8	yes	complemented
44	C₁₇×#459	C₂²	1	4	yes	complemented
45	C₁₇×#459	C₂²	1	8	yes	complemented
46	C₁₇×#446	C₂²	1	2	yes	complemented
47	C₁₇×#446	C₂²	1	2	yes	complemented
48	C₈×Q₈×C₁₇	C₂²	1	2	yes	complemented
49	C₈×Q₈×C₁₇	C₂²	1	2	yes	complemented
50	C₁₇×#440	C₂²	1	8	yes	complemented
51	C₄×Q₁₆×C₁₇	C₂²	1	4	yes	complemented
52	C₄×D₈×C₁₇	C₂²	1	2	yes	—
53	C₄×D₈×C₁₇	C₂²	1	2	yes	—
54	C₄×D₈×C₁₇	C₂²	1	2	yes	—
55	C₄×D₈×C₁₇	C₂²	1	2	yes	—
56	C₄×D₈×C₁₇	C₂²	1	4	yes	complemented
57	C₂×C₁₇×#107	C₂²	1	1	yes	characteristic
58	C₁₇×#424	C₂²	1	2	yes	—
59	C₁₇×#423	C₂²	1	1	yes	characteristic
60	C₁₇×#417	C₂²	1	4	yes	—
61	C₂×C₁₇×#102	C₂²	1	2	yes	—
62	C₁₇×#405	C₂²	1	2	yes	—
63	C₂×C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
64	C₂×#41×C₁₇	—	2	4	no	—
65	C₂×#41×C₁₇	C₂³	1	4	yes	—
66	C₂²×D₄×C₁₇	C₂³	1	2	yes	—
67	C₂³×C₄×C₁₇	—	2	2	no	—
68	C₁₇×AGL(1,ℤ/8)	—	4	8	no	—
69	C₁₇×#135	—	2	8	no	—
70	C₁₇×#135	—	4	4	no	—
71	C₂×Q₁₆×C₁₇	C₂³	1	2	yes	—
72	C₂×Q₁₆×C₁₇	C₂³	1	2	yes	—
73	C₂×SD₁₆×C₁₇	C₂³	1	4	yes	—
74	C₂×SD₁₆×C₁₇	C₂³	1	4	yes	—
75	C₂×SD₁₆×C₁₇	C₂³	1	8	yes	—
76	C₂×D₈×C₁₇	—	2	4	no	—
77	C₂×D₈×C₁₇	C₂³	1	2	yes	—
78	C₂×D₈×C₁₇	C₂³	1	2	yes	—
79	C₂×D₈×C₁₇	C₂³	1	2	yes	—
80	C₂×D₈×C₁₇	C₂³	1	8	yes	—
81	C₂×D₈×C₁₇	C₂³	1	2	yes	—
82	C₂×M₁₆×C₁₇	C₂³	1	2	yes	—
83	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
84	C₂²×C₈×C₁₇	C₂³	1	1	yes	characteristic
85	C₁₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
86	C₁₇×C₄⋊Q₈	D₄	1	1	yes	characteristic
87	C₁₇×C₄⋊Q₈	C₂³	1	2	yes	—
88	C₁₇×#127	C₂³	1	2	yes	—
89	C₁₇×#127	C₂³	1	8	yes	—
90	C₁₇×#126	—	2	8	no	—
91	C₁₇×#125	D₄	1	2	yes	—
92	C₁₇×#124	—	2	8	no	—
93	C₁₇×#124	C₂³	1	8	yes	—
94	C₁₇×#124	C₂³	1	4	yes	—
95	C₁₇×#123	—	2	8	no	—
96	C₁₇×#123	—	2	8	no	—
97	C₁₇×C₂²⋊Q₈	—	2	4	no	—
98	C₁₇×C₂²⋊Q₈	D₄	1	2	yes	—
99	C₁₇×C₂²⋊Q₈	D₄	1	2	yes	—
100	C₁₇×#121	—	2	2	no	—
101	C₁₇×#121	—	2	2	no	—
102	C₁₇×#121	—	2	4	no	—
103	C₁₇×#121	—	2	4	no	—
104	C₁₇×#121	—	2	4	no	—
105	C₁₇×#121	—	2	8	no	—
106	C₁₇×#120	—	2	8	no	—
107	C₄×Q₈×C₁₇	C₂³	1	8	yes	—
108	C₄×Q₈×C₁₇	C₂³	1	2	yes	—
109	C₄×Q₈×C₁₇	C₂³	1	2	yes	—
110	C₄×D₄×C₁₇	—	2	8	no	—
111	C₄×D₄×C₁₇	—	2	8	no	—
112	C₄×D₄×C₁₇	—	2	2	no	—
113	C₄×D₄×C₁₇	—	2	2	no	—
114	C₄×D₄×C₁₇	—	2	2	no	—
115	C₄×D₄×C₁₇	—	2	2	no	—
116	C₄×D₄×C₁₇	C₂³	1	4	yes	—
117	C₄×D₄×C₁₇	C₂³	1	4	yes	—
118	C₄×D₄×C₁₇	C₂³	1	8	yes	—
119	C₁₇×#117	—	2	4	no	—
120	C₁₇×#117	—	2	4	no	—
121	C₁₇×#117	C₂³	1	2	yes	—
122	C₂×C₄⋊C₄×C₁₇	—	2	4	no	—
123	C₂×C₄⋊C₄×C₁₇	C₂³	1	2	yes	—
124	C₂×C₂²⋊C₄×C₁₇	—	2	2	no	—
125	C₂×C₄²×C₁₇	C₂³	1	1	yes	characteristic
126	C₁₇×#107	C₂³	1	2	yes	—
127	C₁₇×#107	C₂³	1	2	yes	—
128	C₁₇×C₄⋊C₈	C₂³	1	2	yes	—
129	C₁₇×C₄⋊C₈	C₂³	1	2	yes	—
130	C₁₇×C₄⋊C₈	C₂³	1	4	yes	—
131	C₁₇×C₄⋊C₈	C₂³	1	4	yes	—
132	C₁₇×#103	C₂³	1	8	yes	—
133	C₁₇×#102	C₂³	1	4	yes	—
134	C₁₇×#102	C₂³	1	4	yes	—
135	C₁₇×#102	C₂³	1	8	yes	—
136	C₁₇×#97	C₂³	1	4	yes	—
137	C₄×C₈×C₁₇	C₂³	1	1	yes	characteristic
138	C₄×C₈×C₁₇	C₂³	1	4	yes	—
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	C₂⁴×C₁₇	—	2	2	no	—
143	#41×C₁₇	—	2	8	no	—
144	#41×C₁₇	—	2	8	no	—
145	#41×C₁₇	—	8	8	no	—
146	C₂×Q₈×C₁₇	C₂×D₄	1	2	yes	—
147	C₂×Q₈×C₁₇	C₂×D₄	1	2	yes	—
148	C₂×Q₈×C₁₇	C₂⁴	1	4	yes	—
149	C₂×D₄×C₁₇	—	2	4	no	—
150	C₂×D₄×C₁₇	—	2	2	no	—
151	C₂×D₄×C₁₇	—	2	2	no	—
152	C₂×D₄×C₁₇	—	2	8	no	—
153	C₂×D₄×C₁₇	—	2	8	no	—
154	C₂×D₄×C₁₇	—	2	2	no	—
155	C₂×D₄×C₁₇	—	2	4	no	—
156	C₂×D₄×C₁₇	—	2	4	no	—
157	C₂×D₄×C₁₇	C₂⁴	1	4	yes	—
158	C₂×D₄×C₁₇	C₂⁴	1	4	yes	—
159	C₂²×C₄×C₁₇	—	2	2	no	—
160	C₂²×C₄×C₁₇	—	2	4	no	—
161	C₂²×C₄×C₁₇	—	2	4	no	—
162	C₂²×C₄×C₁₇	—	2	4	no	—
163	C₂²×C₄×C₁₇	—	2	8	no	—
164	C₂²×C₄×C₁₇	—	8	4	no	—
165	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
168	Q₁₆×C₁₇	—	2	4	no	—
169	Q₁₆×C₁₇	—	4	2	no	—
170	SD₁₆×C₁₇	—	2	8	no	—
171	SD₁₆×C₁₇	—	4	4	no	—
172	SD₁₆×C₁₇	—	4	8	no	—
173	D₈×C₁₇	—	2	4	no	—
174	D₈×C₁₇	—	2	4	no	—
175	D₈×C₁₇	—	2	4	no	—
176	D₈×C₁₇	—	4	2	no	—
177	D₈×C₁₇	—	4	8	no	—
178	M₁₆×C₁₇	—	2	4	no	—
179	C₂×C₈×C₁₇	—	2	2	no	—
180	C₂×C₈×C₁₇	#41	1	4	yes	complemented
181	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
182	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
183	C₂×C₈×C₁₇	C₂⁴	1	4	yes	—
184	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
185	C₂×C₈×C₁₇	C₂⁴	1	1	yes	characteristic
186	C₄⋊C₄×C₁₇	—	2	2	no	—
187	C₄⋊C₄×C₁₇	—	2	8	no	—
188	C₄⋊C₄×C₁₇	—	2	2	no	—
189	C₄⋊C₄×C₁₇	—	2	2	no	—
190	C₄⋊C₄×C₁₇	C₂×D₄	1	2	yes	—
191	C₄⋊C₄×C₁₇	C₂×D₄	1	4	yes	—
192	C₄⋊C₄×C₁₇	C₂×D₄	1	4	yes	—
193	C₄⋊C₄×C₁₇	C₂×D₄	1	2	yes	—
194	C₄⋊C₄×C₁₇	C₂×D₄	1	2	yes	—
195	C₄⋊C₄×C₁₇	C₂×D₄	1	2	yes	—
196	C₄⋊C₄×C₁₇	C₂⁴	1	4	yes	—
197	C₂²⋊C₄×C₁₇	—	2	8	no	—
198	C₂²⋊C₄×C₁₇	—	2	4	no	—
199	C₂²⋊C₄×C₁₇	—	2	4	no	—
200	C₂²⋊C₄×C₁₇	—	2	4	no	—
201	C₂²⋊C₄×C₁₇	—	2	8	no	—
202	C₂²⋊C₄×C₁₇	—	2	8	no	—
203	C₂²⋊C₄×C₁₇	—	2	4	no	—
204	C₂²⋊C₄×C₁₇	C₂×D₄	1	4	yes	—
205	C₄²×C₁₇	—	2	8	no	—
206	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
207	C₄²×C₁₇	C₂×D₄	1	1	yes	characteristic
208	C₄²×C₁₇	C₂⁴	1	1	yes	characteristic
209	C₄²×C₁₇	C₂⁴	1	4	yes	—
210	C₄²×C₁₇	C₂⁴	1	1	yes	characteristic
211	#61406	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
212	C₂³×C₁₇	—	2	4	no	—
213	C₂³×C₁₇	—	2	4	no	—
214	C₂³×C₁₇	—	4	4	no	—
215	C₂³×C₁₇	—	8	2	no	—
216	C₂³×C₁₇	—	8	2	no	—
217	C₂³×C₁₇	C₂²×D₄	1	1	yes	characteristic
218	Q₈×C₁₇	—	2	4	no	—
219	Q₈×C₁₇	—	2	4	no	—
220	Q₈×C₁₇	—	2	4	no	—
221	Q₈×C₁₇	—	4	2	no	—
222	D₄×C₁₇	—	2	4	no	—
223	D₄×C₁₇	—	2	4	no	—
224	D₄×C₁₇	—	2	4	no	—
225	D₄×C₁₇	—	2	4	no	—
226	D₄×C₁₇	—	4	8	no	—
227	D₄×C₁₇	—	4	4	no	—
228	D₄×C₁₇	—	4	4	no	—
229	D₄×C₁₇	—	8	2	no	—
230	D₄×C₁₇	—	8	8	no	—
231	D₄×C₁₇	—	8	8	no	—
232	C₂×C₄×C₁₇	—	2	2	no	—
233	C₂×C₄×C₁₇	—	2	4	no	—
234	C₂×C₄×C₁₇	—	2	4	no	—
235	C₂×C₄×C₁₇	—	4	1	no	—
236	C₂×C₄×C₁₇	—	4	8	no	—
237	C₂×C₄×C₁₇	—	4	8	no	—
238	C₂×C₄×C₁₇	—	8	4	no	—
239	C₂×C₄×C₁₇	—	8	8	no	—
240	C₂×C₄×C₁₇	—	8	2	no	—
241	C₂×C₄×C₁₇	C₂²×D₄	1	4	yes	—
242	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
243	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
244	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
245	C₂×C₄×C₁₇	C₂²×D₄	1	4	yes	—
246	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
247	C₂×C₄×C₁₇	C₂²×D₄	1	1	yes	characteristic
248	C₂×C₄×C₁₇	C₂×#41	1	2	yes	—
249	C₂×C₄×C₁₇	H(5,𝔽₂)	1	2	yes	—
250	C₂×C₄×C₁₇	C₂⁵	1	1	yes	characteristic
251	C₈×C₁₇	—	2	1	no	—
252	C₈×C₁₇	—	2	1	no	—
253	C₈×C₁₇	—	2	4	no	—
254	C₈×C₁₇	C₂×#41	1	2	yes	—
255	C₈×C₁₇	C₂×#41	1	2	yes	—
256	#3506	C₂×C₁₇	1	2	yes	complemented
257	#3403	C₂×C₁₇	1	4	yes	complemented
258	#3397	C₂×C₁₇	1	4	yes	complemented
259	#3396	C₂×C₁₇	1	4	yes	complemented
260	#3393	C₂×C₁₇	1	2	yes	complemented
261	#3391	C₂×C₁₇	1	2	yes	complemented
262	#3155	C₂×C₁₇	1	2	yes	complemented
263	#3147	C₂×C₁₇	1	1	yes	characteristic, complemented
264	#3045	C₂×C₁₇	1	4	yes	complemented
265	#3036	C₂×C₁₇	1	2	yes	complemented
266	#2960	C₂×C₁₇	1	1	yes	characteristic, complemented
267	#2951	C₂×C₁₇	1	2	yes	complemented
268	C₂×C₄×D₈	C₂×C₁₇	1	1	yes	characteristic, complemented
269	C₂²×C₁₇	—	2	1	no	—
270	C₂²×C₁₇	—	2	1	no	—
271	C₂²×C₁₇	—	4	4	no	—
272	C₂²×C₁₇	—	4	4	no	—
273	C₂²×C₁₇	—	4	4	no	—
274	C₂²×C₁₇	—	4	4	no	—
275	C₂²×C₁₇	—	4	4	no	—
276	C₂²×C₁₇	—	4	4	no	—
277	C₂²×C₁₇	—	8	8	no	—
278	C₂²×C₁₇	#550	1	2	yes	—
279	C₂²×C₁₇	C₂³×D₄	1	1	yes	characteristic
280	C₄×C₁₇	—	2	1	no	—
281	C₄×C₁₇	—	2	1	no	—
282	C₄×C₁₇	—	2	4	no	—
283	C₄×C₁₇	—	2	1	no	—
284	C₄×C₁₇	—	2	4	no	—
285	C₄×C₁₇	—	4	4	no	—
286	C₄×C₁₇	—	4	4	no	—
287	C₄×C₁₇	#550	1	2	yes	—
288	C₄×C₁₇	#576	1	2	yes	—
289	C₄×C₁₇	C₂²×#41	1	1	yes	characteristic
290	C₄×C₁₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
291	C₂×AGL(1,ℤ/8)	C₂²×C₁₇	1	4	yes	—
292	C₂×#135	C₂²×C₁₇	1	2	yes	—
293	C₂×#135	C₂²×C₁₇	1	2	yes	—
294	C₂²×D₈	C₂²×C₁₇	1	1	yes	characteristic
295	#559	C₂²×C₁₇	1	8	yes	complemented
296	#556	C₂²×C₁₇	1	4	yes	complemented
297	#555	—	2	2	no	—
298	#553	—	2	4	no	—
299	#550	C₂²×C₁₇	1	4	yes	complemented
300	#547	—	2	2	no	—
301	#538	—	2	4	no	—
302	#535	C₂²×C₁₇	1	4	yes	—
303	#533	C₂²×C₁₇	1	1	yes	characteristic
304	#532	C₂²×C₁₇	1	2	yes	—
305	#525	—	2	4	no	—
306	#520	C₂²×C₁₇	1	4	yes	—
307	C₂×C₄×D₄	C₂²×C₁₇	1	2	yes	—
308	#496	C₂²×C₁₇	1	8	yes	—
309	#495	C₂²×C₁₇	1	4	yes	—
310	#495	C₂²×C₁₇	1	2	yes	complemented
311	#495	C₂²×C₁₇	1	2	yes	complemented
312	#494	C₂²×C₁₇	1	1	yes	characteristic, complemented
313	#493	C₂²×C₁₇	1	4	yes	—
314	#493	C₂²×C₁₇	1	1	yes	characteristic, complemented
315	#492	C₂²×C₁₇	1	2	yes	complemented
316	#464	C₂²×C₁₇	1	8	yes	complemented
317	#463	C₂²×C₁₇	1	8	yes	complemented
318	#463	C₂²×C₁₇	1	8	yes	complemented
319	#461	C₂²×C₁₇	1	4	yes	complemented
320	#460	C₂²×C₁₇	1	8	yes	complemented
321	#459	C₂²×C₁₇	1	4	yes	complemented
322	#459	C₂²×C₁₇	1	8	yes	complemented
323	#446	C₂²×C₁₇	1	2	yes	complemented
324	#446	C₂²×C₁₇	1	2	yes	complemented
325	C₈×Q₈	C₂²×C₁₇	1	2	yes	complemented
326	C₈×Q₈	C₂²×C₁₇	1	2	yes	complemented
327	#440	C₂²×C₁₇	1	8	yes	complemented
328	C₄×Q₁₆	C₂²×C₁₇	1	4	yes	complemented
329	C₄×D₈	C₂²×C₁₇	1	2	yes	—
330	C₄×D₈	C₂²×C₁₇	1	2	yes	—
331	C₄×D₈	C₂²×C₁₇	1	2	yes	—
332	C₄×D₈	C₂²×C₁₇	1	2	yes	—
333	C₄×D₈	C₂²×C₁₇	1	4	yes	complemented
334	C₂×#107	C₂²×C₁₇	1	1	yes	characteristic
335	#424	C₂²×C₁₇	1	2	yes	—
336	#423	C₂²×C₁₇	1	1	yes	characteristic
337	#417	C₂²×C₁₇	1	4	yes	—
338	C₂×#102	C₂²×C₁₇	1	2	yes	—
339	#405	C₂²×C₁₇	1	2	yes	—
340	C₂×C₄×C₈	C₂²×C₁₇	1	1	yes	characteristic
341	C₂×C₁₇	—	2	2	no	—
342	C₂×C₁₇	—	8	4	no	—
343	C₂×C₁₇	—	8	4	no	—
344	C₂×C₁₇	C₂×#550	1	1	yes	characteristic
345	C₂×C₁₇	C₂×#576	1	1	yes	characteristic
346	C₂×C₁₇	#3584	1	1	yes	characteristic
347	C₂×#41	—	2	4	no	—
348	C₂×#41	C₂³×C₁₇	1	4	yes	—
349	C₂²×D₄	C₂³×C₁₇	1	2	yes	—
350	C₂³×C₄	—	2	2	no	—
351	AGL(1,ℤ/8)	—	4	8	no	—
352	#135	—	2	8	no	—
353	#135	—	4	4	no	—
354	C₂×Q₁₆	C₂³×C₁₇	1	2	yes	—
355	C₂×Q₁₆	C₂³×C₁₇	1	2	yes	—
356	C₂×SD₁₆	C₂³×C₁₇	1	4	yes	—
357	C₂×SD₁₆	C₂³×C₁₇	1	4	yes	—
358	C₂×SD₁₆	C₂³×C₁₇	1	8	yes	—
359	C₂×D₈	—	2	4	no	—
360	C₂×D₈	C₂³×C₁₇	1	2	yes	—
361	C₂×D₈	C₂³×C₁₇	1	2	yes	—
362	C₂×D₈	C₂³×C₁₇	1	2	yes	—
363	C₂×D₈	C₂³×C₁₇	1	2	yes	—
364	C₂×D₈	C₂³×C₁₇	1	8	yes	—
365	C₂×M₁₆	C₂³×C₁₇	1	2	yes	—
366	C₂²×C₈	C₂³×C₁₇	1	1	yes	characteristic
367	C₂²×C₈	C₂³×C₁₇	1	1	yes	characteristic
368	C₄⋊Q₈	D₄×C₁₇	1	1	yes	characteristic
369	C₄⋊Q₈	D₄×C₁₇	1	1	yes	characteristic
370	C₄⋊Q₈	C₂³×C₁₇	1	2	yes	—
371	#127	C₂³×C₁₇	1	2	yes	—
372	#127	C₂³×C₁₇	1	8	yes	—
373	#126	—	2	8	no	—
374	#125	D₄×C₁₇	1	2	yes	—
375	#124	—	2	8	no	—
376	#124	C₂³×C₁₇	1	4	yes	—
377	#124	C₂³×C₁₇	1	8	yes	—
378	#123	—	2	8	no	—
379	#123	—	2	8	no	—
380	C₂²⋊Q₈	—	2	4	no	—
381	C₂²⋊Q₈	D₄×C₁₇	1	2	yes	—
382	C₂²⋊Q₈	D₄×C₁₇	1	2	yes	—
383	#121	—	2	2	no	—
384	#121	—	2	2	no	—
385	#121	—	2	4	no	—
386	#121	—	2	4	no	—
387	#121	—	2	4	no	—
388	#121	—	2	8	no	—
389	#120	—	2	8	no	—
390	C₄×Q₈	C₂³×C₁₇	1	8	yes	—
391	C₄×Q₈	C₂³×C₁₇	1	2	yes	—
392	C₄×Q₈	C₂³×C₁₇	1	2	yes	—
393	C₄×D₄	—	2	8	no	—
394	C₄×D₄	—	2	8	no	—
395	C₄×D₄	—	2	2	no	—
396	C₄×D₄	—	2	2	no	—
397	C₄×D₄	—	2	2	no	—
398	C₄×D₄	—	2	2	no	—
399	C₄×D₄	C₂³×C₁₇	1	4	yes	—
400	C₄×D₄	C₂³×C₁₇	1	4	yes	—
401	C₄×D₄	C₂³×C₁₇	1	8	yes	—
402	#117	—	2	4	no	—
403	#117	—	2	4	no	—
404	#117	C₂³×C₁₇	1	2	yes	—
405	C₂×C₄⋊C₄	—	2	4	no	—
406	C₂×C₄⋊C₄	C₂³×C₁₇	1	2	yes	—
407	C₂×C₂²⋊C₄	—	2	2	no	—
408	C₂×C₄²	C₂³×C₁₇	1	1	yes	characteristic
409	#107	C₂³×C₁₇	1	2	yes	—
410	#107	C₂³×C₁₇	1	2	yes	—
411	C₄⋊C₈	C₂³×C₁₇	1	2	yes	—
412	C₄⋊C₈	C₂³×C₁₇	1	2	yes	—
413	C₄⋊C₈	C₂³×C₁₇	1	4	yes	—
414	C₄⋊C₈	C₂³×C₁₇	1	4	yes	—
415	#103	C₂³×C₁₇	1	8	yes	—
416	#102	C₂³×C₁₇	1	4	yes	—
417	#102	C₂³×C₁₇	1	4	yes	—
418	#102	C₂³×C₁₇	1	8	yes	—
419	#97	C₂³×C₁₇	1	4	yes	—
420	C₄×C₈	C₂³×C₁₇	1	1	yes	characteristic
421	C₄×C₈	C₂³×C₁₇	1	1	yes	characteristic
422	C₄×C₈	C₂³×C₁₇	1	1	yes	characteristic
423	C₄×C₈	C₂³×C₁₇	1	1	yes	characteristic
424	C₄×C₈	C₂³×C₁₇	1	4	yes	—
425	C₁₇	#61406	1	1	yes	characteristic, direct product factor, Sylow
426	C₂⁴	—	2	2	no	—
427	#41	—	2	8	no	—
428	#41	—	2	8	no	—
429	#41	—	8	8	no	—
430	C₂×Q₈	C₂×D₄×C₁₇	1	2	yes	—
431	C₂×Q₈	C₂×D₄×C₁₇	1	2	yes	—
432	C₂×Q₈	C₂⁴×C₁₇	1	4	yes	—
433	C₂×D₄	—	2	4	no	—
434	C₂×D₄	—	2	2	no	—
435	C₂×D₄	—	2	2	no	—
436	C₂×D₄	—	2	8	no	—
437	C₂×D₄	—	2	2	no	—
438	C₂×D₄	—	2	8	no	—
439	C₂×D₄	—	2	4	no	—
440	C₂×D₄	—	2	4	no	—
441	C₂×D₄	C₂⁴×C₁₇	1	4	yes	—
442	C₂×D₄	C₂⁴×C₁₇	1	4	yes	—
443	C₂²×C₄	—	2	2	no	—
444	C₂²×C₄	—	2	4	no	—
445	C₂²×C₄	—	2	4	no	—
446	C₂²×C₄	—	2	8	no	—
447	C₂²×C₄	—	2	4	no	—
448	C₂²×C₄	—	8	4	no	—
449	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
450	C₂²×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
451	C₂²×C₄	C₂⁴×C₁₇	1	1	yes	characteristic
452	Q₁₆	—	2	4	no	—
453	Q₁₆	—	4	2	no	—
454	SD₁₆	—	2	8	no	—
455	SD₁₆	—	4	4	no	—
456	SD₁₆	—	4	8	no	—
457	D₈	—	2	4	no	—
458	D₈	—	2	4	no	—
459	D₈	—	2	4	no	—
460	D₈	—	4	2	no	—
461	D₈	—	4	8	no	—
462	M₁₆	—	2	4	no	—
463	C₂×C₈	—	2	2	no	—
464	C₂×C₈	#41×C₁₇	1	4	yes	complemented
465	C₂×C₈	C₂⁴×C₁₇	1	1	yes	characteristic
466	C₂×C₈	C₂⁴×C₁₇	1	1	yes	characteristic
467	C₂×C₈	C₂⁴×C₁₇	1	4	yes	—
468	C₂×C₈	C₂⁴×C₁₇	1	1	yes	characteristic
469	C₂×C₈	C₂⁴×C₁₇	1	1	yes	characteristic
470	C₄⋊C₄	—	2	2	no	—
471	C₄⋊C₄	—	2	8	no	—
472	C₄⋊C₄	—	2	2	no	—
473	C₄⋊C₄	—	2	2	no	—
474	C₄⋊C₄	C₂×D₄×C₁₇	1	2	yes	—
475	C₄⋊C₄	C₂×D₄×C₁₇	1	2	yes	—
476	C₄⋊C₄	C₂×D₄×C₁₇	1	2	yes	—
477	C₄⋊C₄	C₂×D₄×C₁₇	1	4	yes	—
478	C₄⋊C₄	C₂×D₄×C₁₇	1	4	yes	—
479	C₄⋊C₄	C₂×D₄×C₁₇	1	2	yes	—
480	C₄⋊C₄	C₂⁴×C₁₇	1	4	yes	—
481	C₂²⋊C₄	—	2	4	no	—
482	C₂²⋊C₄	—	2	4	no	—
483	C₂²⋊C₄	—	2	8	no	—
484	C₂²⋊C₄	—	2	8	no	—
485	C₂²⋊C₄	—	2	4	no	—
486	C₂²⋊C₄	—	2	8	no	—
487	C₂²⋊C₄	—	2	4	no	—
488	C₂²⋊C₄	C₂×D₄×C₁₇	1	4	yes	—
489	C₄²	—	2	8	no	—
490	C₄²	C₂×D₄×C₁₇	1	1	yes	characteristic
491	C₄²	C₂×D₄×C₁₇	1	1	yes	characteristic
492	C₄²	C₂⁴×C₁₇	1	1	yes	characteristic
493	C₄²	C₂⁴×C₁₇	1	1	yes	characteristic
494	C₄²	C₂⁴×C₁₇	1	4	yes	—
495	C₂³	—	2	4	no	—
496	C₂³	—	2	4	no	—
497	C₂³	—	4	4	no	—
498	C₂³	—	8	2	no	—
499	C₂³	—	8	2	no	—
500	C₂³	C₂²×D₄×C₁₇	1	1	yes	characteristic
501	Q₈	—	2	4	no	—
502	Q₈	—	2	4	no	—
503	Q₈	—	2	4	no	—
504	Q₈	—	4	2	no	—
505	D₄	—	2	4	no	—
506	D₄	—	2	4	no	—
507	D₄	—	2	4	no	—
508	D₄	—	2	4	no	—
509	D₄	—	4	8	no	—
510	D₄	—	4	4	no	—
511	D₄	—	4	4	no	—
512	D₄	—	8	8	no	—
513	D₄	—	8	2	no	—
514	D₄	—	8	8	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	4	no	—
517	C₂×C₄	—	2	4	no	—
518	C₂×C₄	—	4	1	no	—
519	C₂×C₄	—	4	8	no	—
520	C₂×C₄	—	4	8	no	—
521	C₂×C₄	—	8	4	no	—
522	C₂×C₄	—	8	8	no	—
523	C₂×C₄	—	8	2	no	—
524	C₂×C₄	C₂²×D₄×C₁₇	1	4	yes	—
525	C₂×C₄	C₂²×D₄×C₁₇	1	1	yes	characteristic
526	C₂×C₄	C₂²×D₄×C₁₇	1	1	yes	characteristic
527	C₂×C₄	C₂²×D₄×C₁₇	1	1	yes	characteristic
528	C₂×C₄	C₂²×D₄×C₁₇	1	4	yes	—
529	C₂×C₄	C₂²×D₄×C₁₇	1	1	yes	characteristic
530	C₂×C₄	C₂²×D₄×C₁₇	1	1	yes	characteristic
531	C₂×C₄	C₂×#41×C₁₇	1	2	yes	—
532	C₂×C₄	C₁₇×H(5,𝔽₂)	1	2	yes	—
533	C₂×C₄	C₂⁵×C₁₇	1	1	yes	characteristic
534	C₈	—	2	1	no	—
535	C₈	—	2	1	no	—
536	C₈	—	2	4	no	—
537	C₈	C₂×#41×C₁₇	1	2	yes	—
538	C₈	C₂×#41×C₁₇	1	2	yes	—
539	C₂²	—	2	1	no	—
540	C₂²	—	2	1	no	—
541	C₂²	—	4	4	no	—
542	C₂²	—	4	4	no	—
543	C₂²	—	4	4	no	—
544	C₂²	—	4	4	no	—
545	C₂²	—	4	4	no	—
546	C₂²	—	4	4	no	—
547	C₂²	—	8	8	no	—
548	C₂²	C₁₇×#550	1	2	yes	—
549	C₂²	C₂³×D₄×C₁₇	1	1	yes	characteristic
550	C₄	—	2	1	no	—
551	C₄	—	2	4	no	—
552	C₄	—	2	1	no	—
553	C₄	—	2	1	no	—
554	C₄	—	2	4	no	—
555	C₄	—	4	4	no	—
556	C₄	—	4	4	no	—
557	C₄	C₁₇×#550	1	2	yes	—
558	C₄	C₁₇×#576	1	2	yes	—
559	C₄	C₂²×#41×C₁₇	1	1	yes	characteristic
560	C₄	C₂×C₁₇×H(5,𝔽₂)	1	1	yes	characteristic
561	C₂	—	2	2	no	—
562	C₂	—	8	4	no	—
563	C₂	—	8	4	no	—
564	C₂	C₂×C₁₇×#550	1	1	yes	characteristic
565	C₂	C₂×C₁₇×#576	1	1	yes	characteristic
566	C₂	C₁₇×#3584	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₇	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₁₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₇ ◁ C₁₇×#533 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₇×#3506	C₂	semidirect
2. C₁₇×#3403	C₂	semidirect
3. C₁₇×#3397	C₂	semidirect
4. C₁₇×#3396	C₂	semidirect
5. C₁₇×#3393	C₂	semidirect
6. C₁₇×#3391	C₂	semidirect
7. C₁₇×#3155	C₂	semidirect
8. C₁₇×#3147	C₂	semidirect
9. C₁₇×#3045	C₂	semidirect
10. C₁₇×#3036	C₂	semidirect
11. C₁₇×#2960	C₂	semidirect
12. C₁₇×#2951	C₂	semidirect
13. C₂×C₄×D₈×C₁₇	C₂	semidirect
18. C₁₇×#559	C₂²	semidirect
19. C₁₇×#556	C₂²	semidirect
22. C₁₇×#550	C₂²	semidirect
33. C₁₇×#495	C₂²	semidirect
34. C₁₇×#495	C₂²	semidirect
35. C₁₇×#494	C₂²	semidirect
37. C₁₇×#493	C₂²	semidirect
38. C₁₇×#492	C₂²	semidirect
39. C₁₇×#464	C₂²	semidirect
40. C₁₇×#463	C₂²	semidirect
41. C₁₇×#463	C₂²	semidirect
42. C₁₇×#461	C₂²	semidirect
43. C₁₇×#460	C₂²	semidirect
44. C₁₇×#459	C₂²	semidirect
45. C₁₇×#459	C₂²	semidirect
46. C₁₇×#446	C₂²	semidirect
47. C₁₇×#446	C₂²	semidirect
48. C₈×Q₈×C₁₇	C₂²	semidirect
49. C₈×Q₈×C₁₇	C₂²	semidirect
50. C₁₇×#440	C₂²	semidirect
51. C₄×Q₁₆×C₁₇	C₂²	semidirect
56. C₄×D₈×C₁₇	C₂²	semidirect
180. C₂×C₈×C₁₇	#41	semidirect
211. #61406	C₁₇	direct (N × H)
256. #3506	C₂×C₁₇	semidirect
257. #3403	C₂×C₁₇	semidirect
258. #3397	C₂×C₁₇	semidirect
259. #3396	C₂×C₁₇	semidirect
260. #3393	C₂×C₁₇	semidirect
261. #3391	C₂×C₁₇	semidirect
262. #3155	C₂×C₁₇	semidirect
263. #3147	C₂×C₁₇	semidirect
264. #3045	C₂×C₁₇	semidirect
265. #3036	C₂×C₁₇	semidirect
266. #2960	C₂×C₁₇	semidirect
267. #2951	C₂×C₁₇	semidirect
268. C₂×C₄×D₈	C₂×C₁₇	semidirect
295. #559	C₂²×C₁₇	semidirect
296. #556	C₂²×C₁₇	semidirect
299. #550	C₂²×C₁₇	semidirect
310. #495	C₂²×C₁₇	semidirect
311. #495	C₂²×C₁₇	semidirect
312. #494	C₂²×C₁₇	semidirect
314. #493	C₂²×C₁₇	semidirect
315. #492	C₂²×C₁₇	semidirect
316. #464	C₂²×C₁₇	semidirect
317. #463	C₂²×C₁₇	semidirect
318. #463	C₂²×C₁₇	semidirect
319. #461	C₂²×C₁₇	semidirect
320. #460	C₂²×C₁₇	semidirect
321. #459	C₂²×C₁₇	semidirect
322. #459	C₂²×C₁₇	semidirect
323. #446	C₂²×C₁₇	semidirect
324. #446	C₂²×C₁₇	semidirect
325. C₈×Q₈	C₂²×C₁₇	semidirect
326. C₈×Q₈	C₂²×C₁₇	semidirect
327. #440	C₂²×C₁₇	semidirect
328. C₄×Q₁₆	C₂²×C₁₇	semidirect
333. C₄×D₈	C₂²×C₁₇	semidirect
425. C₁₇	#61406	direct (N × H)
464. C₂×C₈	#41×C₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₁₇×#3506	2
C₁₇×#3403	2
C₁₇×#3397	2
C₁₇×#3396	2
C₁₇×#3393	2
C₁₇×#3391	2
C₁₇×#3155	2
C₁₇×#3147	2
C₁₇×#3045	2
C₁₇×#3036	2
C₁₇×#2960	2
C₁₇×#2951	2
C₂×C₄×D₈×C₁₇	2
#61406	17

Subgroups — C17×#61406 #16919150

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#61406 #16919150