Subgroups — #14541289

Order 2176.

360 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763524	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763509	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763218	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763132	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763132	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763131	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763121	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763109	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763089	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763059	C₂	1	1	yes	maximal, characteristic, complemented
11	#11763042	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762882	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762848	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762844	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#442	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₈×C₁₇	C₂²	1	1	yes	characteristic
17	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
18	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
19	#10587624	C₂²	1	1	yes	characteristic
20	#10587624	C₂²	1	1	yes	characteristic
21	#10587624	C₂²	1	1	yes	characteristic
22	#10587624	C₂²	1	1	yes	characteristic
23	#10587616	—	2	1	no	—
24	#10587615	—	2	1	no	—
25	C₄×C₂²⋊D₁₇	—	2	1	no	—
26	#10587612	—	2	1	no	—
27	C₂×#63218	C₂²	1	1	yes	characteristic
28	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
29	#10587578	C₂²	1	1	yes	characteristic
30	#10587578	C₂²	1	1	yes	characteristic
31	#10587578	C₂²	1	1	yes	characteristic
32	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
33	#10587571	—	2	1	no	—
34	#10587571	—	2	1	no	—
35	#10587570	—	2	1	no	—
36	#10587565	—	2	1	no	—
37	C₄×D₆₈	C₂²	1	1	yes	characteristic
38	#10587556	C₂²	1	1	yes	characteristic
39	#10587555	C₂²	1	1	yes	characteristic
40	C₁₇×#106	C₂²	1	1	yes	characteristic
41	C₁₇×#102	C₂²	1	1	yes	characteristic
42	C₁₇×#102	C₂²	1	1	yes	characteristic
43	C₁₇×#97	C₂²	1	1	yes	characteristic
44	#10587503	C₂²	1	1	yes	characteristic
45	#10587503	C₂²	1	1	yes	characteristic
46	#10587503	C₂²	1	1	yes	characteristic
47	#10587503	C₂²	1	1	yes	characteristic
48	#10587491	C₂²	1	1	yes	characteristic
49	#10587491	C₂²	1	1	yes	characteristic
50	#10587489	C₂²	1	1	yes	characteristic
51	#10587489	C₂²	1	1	yes	characteristic
52	#10587488	C₂²	1	1	yes	characteristic
53	#10587488	C₂²	1	1	yes	characteristic
54	#10587479	C₂²	1	1	yes	characteristic
55	#10587478	C₂²	1	1	yes	characteristic
56	#10587478	C₂²	1	1	yes	characteristic
57	#10587478	C₂²	1	1	yes	characteristic
58	C₄⋊#3622	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₇	—	2	1	no	—
62	C₂²×C₄×C₁₇	—	2	1	no	—
63	C₂×C₂²⋊D₁₇	—	2	1	no	—
64	C₂×C₂²⋊D₁₇	—	2	1	no	—
65	C₂×C₂²⋊D₁₇	—	2	1	no	—
66	C₂×C₂²⋊D₁₇	—	2	1	no	—
67	C₂×D₆₈	D₄	1	1	yes	characteristic
68	C₂×C₄×D₁₇	—	2	1	no	—
69	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
70	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
71	C₂×Dic₃₄	—	2	1	no	—
72	D₈×C₁₇	—	2	2	no	—
73	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
77	C₂²⋊C₄×C₁₇	—	2	1	no	—
78	C₂²⋊C₄×C₁₇	—	2	1	no	—
79	C₄²×C₁₇	C₂³	1	1	yes	characteristic
80	C₂²⋊Dic₁₇	—	2	1	no	—
81	C₂²⋊Dic₁₇	—	2	1	no	—
82	C₂²⋊Dic₁₇	—	2	1	no	—
83	C₂²⋊Dic₁₇	—	2	1	no	—
84	#63227	—	2	1	no	—
85	#63227	—	2	1	no	—
86	#63227	—	2	1	no	—
87	#63227	D₄	1	1	yes	characteristic
88	#63227	D₄	1	1	yes	characteristic
89	C₄⋊Dic₁₇	—	2	1	no	—
90	C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
91	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
92	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
93	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
94	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
95	#63225	—	2	1	no	—
96	#63225	—	2	1	no	—
97	#63225	—	2	1	no	—
98	C₄×Dic₁₇	—	2	1	no	—
99	C₄×Dic₁₇	—	2	1	no	—
100	C₄×Dic₁₇	—	2	1	no	—
101	C₂×#3622	C₂³	1	1	yes	characteristic
102	C₂×#3622	C₂³	1	1	yes	characteristic
103	#63218	—	2	2	no	—
104	C₈×D₁₇	—	2	2	no	—
105	C₂³×C₁₇	—	2	1	no	—
106	C₂³×C₁₇	—	2	1	no	—
107	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
108	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
109	D₄×C₁₇	—	2	1	no	—
110	D₄×C₁₇	—	2	1	no	—
111	D₄×C₁₇	—	2	1	no	—
112	D₄×C₁₇	—	2	1	no	—
113	C₂×C₄×C₁₇	—	2	1	no	—
114	C₂×C₄×C₁₇	—	2	1	no	—
115	C₂×C₄×C₁₇	—	4	2	no	—
116	C₂×C₄×C₁₇	—	8	1	no	—
117	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
120	C₂²⋊D₁₇	—	4	2	no	—
121	C₂²⋊D₁₇	—	8	1	no	—
122	C₂²⋊D₁₇	—	8	1	no	—
123	C₂²⋊D₁₇	—	8	1	no	—
124	C₂×Dic₁₇	—	2	1	no	—
125	C₂×Dic₁₇	—	2	1	no	—
126	C₂×Dic₁₇	—	2	1	no	—
127	C₂×Dic₁₇	—	2	1	no	—
128	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
130	D₆₈	—	4	1	no	—
131	C₄×D₁₇	—	2	1	no	—
132	C₄×D₁₇	—	2	1	no	—
133	C₄×D₁₇	—	2	1	no	—
134	C₄×D₁₇	—	2	1	no	—
135	C₄×D₁₇	—	8	1	no	—
136	Dic₃₄	—	8	1	no	—
137	C₈×C₁₇	—	2	1	no	—
138	C₈×C₁₇	#41	1	2	yes	—
139	#3622	—	2	1	no	—
140	#3622	—	2	1	no	—
141	#3347	—	17	1	no	maximal, Sylow
142	C₂²×C₁₇	—	4	1	no	—
143	C₂²×C₁₇	—	4	1	no	—
144	C₂²×C₁₇	—	4	1	no	—
145	C₂²×C₁₇	—	4	1	no	—
146	C₂²×C₁₇	—	4	1	no	—
147	C₂²×C₁₇	—	4	1	no	—
148	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
149	C₂×D₁₇	—	2	1	no	—
150	C₂×D₁₇	—	2	1	no	—
151	C₂×D₁₇	—	2	1	no	—
152	C₂×D₁₇	—	2	1	no	—
153	C₂×D₁₇	—	2	1	no	—
154	C₂×D₁₇	—	2	1	no	—
155	C₄×C₁₇	—	2	1	no	—
156	C₄×C₁₇	—	2	1	no	—
157	C₄×C₁₇	—	4	1	no	—
158	C₄×C₁₇	—	4	1	no	—
159	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
160	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
161	Dic₁₇	—	2	1	no	—
162	Dic₁₇	—	2	1	no	—
163	Dic₁₇	—	4	1	no	—
164	Dic₁₇	—	4	1	no	—
165	Dic₁₇	—	4	1	no	—
166	Dic₁₇	—	4	1	no	—
167	#555	—	17	1	no	—
168	#553	—	17	1	no	—
169	#501	—	17	1	no	—
170	#482	—	17	1	no	—
171	#482	—	17	1	no	—
172	#482	—	17	1	no	—
173	#480	—	17	1	no	—
174	#478	—	17	1	no	—
175	#474	—	17	1	no	—
176	#469	—	17	1	no	—
177	#466	—	17	1	no	—
178	#442	D₁₇	1	1	yes	characteristic, complemented
179	#435	—	17	1	no	—
180	#428	—	17	1	no	—
181	#427	—	17	1	no	—
182	C₂×C₁₇	—	8	1	no	—
183	C₂×C₁₇	—	8	1	no	—
184	C₂×C₁₇	#547	1	1	yes	characteristic
185	C₂×C₁₇	#575	1	1	yes	characteristic
186	C₂×C₁₇	#576	1	1	yes	characteristic
187	D₁₇	—	4	1	no	—
188	D₁₇	—	4	1	no	—
189	C₂×D₈	C₂×D₁₇	1	1	yes	characteristic
190	C₂×M₁₆	—	17	1	no	—
191	C₂²×C₈	—	17	1	no	—
192	C₄⋊Q₈	—	17	1	no	—
193	#126	—	34	1	no	—
194	#125	—	17	1	no	—
195	#124	—	34	1	no	—
196	#124	—	34	1	no	—
197	#123	—	34	1	no	—
198	#121	—	17	1	no	—
199	#121	—	17	1	no	—
200	#121	—	17	1	no	—
201	#121	—	17	1	no	—
202	#121	—	34	1	no	—
203	#121	—	34	1	no	—
204	C₄×D₄	—	17	1	no	—
205	C₄×D₄	—	34	1	no	—
206	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
207	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
208	#117	—	17	1	no	—
209	#117	—	17	1	no	—
210	#117	—	17	1	no	—
211	#117	—	34	1	no	—
212	C₂×C₄⋊C₄	—	17	1	no	—
213	#107	—	17	1	no	—
214	#107	—	17	1	no	—
215	#107	—	17	1	no	—
216	#107	—	17	1	no	—
217	#107	—	17	1	no	—
218	#106	—	17	1	no	—
219	#106	—	17	1	no	—
220	#106	—	17	1	no	—
221	#106	C₂×D₁₇	1	1	yes	characteristic
222	C₄⋊C₈	—	17	1	no	—
223	#102	—	17	1	no	—
224	#102	—	17	1	no	—
225	#102	—	17	1	no	—
226	#102	—	17	1	no	—
227	#102	C₂×D₁₇	1	1	yes	characteristic
228	#102	C₂×D₁₇	1	1	yes	characteristic
229	C₂²⋊C₈	—	17	1	no	—
230	C₂²⋊C₈	—	17	1	no	—
231	#97	C₂×D₁₇	1	1	yes	characteristic
232	C₁₇	#3347	1	1	yes	characteristic, complemented, Sylow
233	C₂×Q₈	—	34	1	no	—
234	C₂×D₄	—	17	1	no	—
235	C₂×D₄	—	34	1	no	—
236	C₂×D₄	—	34	1	no	—
237	C₂×D₄	—	34	1	no	—
238	C₂×D₄	—	34	1	no	—
239	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
240	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	17	1	no	—
244	C₂²×C₄	—	17	1	no	—
245	C₂²×C₄	—	34	1	no	—
246	D₈	—	2	2	no	—
247	M₁₆	—	34	2	no	—
248	C₂×C₈	—	17	1	no	—
249	C₂×C₈	—	17	1	no	—
250	C₂×C₈	—	34	2	no	—
251	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
252	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
253	C₄⋊C₄	—	17	1	no	—
254	C₄⋊C₄	—	17	1	no	—
255	C₄⋊C₄	—	17	1	no	—
256	C₄⋊C₄	—	17	1	no	—
257	C₄⋊C₄	—	17	1	no	—
258	C₄⋊C₄	—	34	1	no	—
259	C₄⋊C₄	—	34	1	no	—
260	C₄⋊C₄	—	34	1	no	—
261	C₄⋊C₄	—	34	1	no	—
262	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
263	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	17	1	no	—
267	C₂²⋊C₄	—	17	1	no	—
268	C₂²⋊C₄	—	34	1	no	—
269	C₂²⋊C₄	—	34	1	no	—
270	C₂²⋊C₄	—	34	1	no	—
271	C₂²⋊C₄	—	34	1	no	—
272	C₂²⋊C₄	—	34	1	no	—
273	C₂²⋊C₄	—	34	1	no	—
274	C₂²⋊C₄	—	34	1	no	—
275	C₄²	—	34	1	no	—
276	C₄²	—	34	1	no	—
277	C₄²	—	34	1	no	—
278	C₄²	C₂²×D₁₇	1	1	yes	characteristic
279	C₂³	—	2	1	no	—
280	C₂³	—	2	1	no	—
281	C₂³	—	17	1	no	—
282	C₂³	—	17	1	no	—
283	Q₈	—	136	1	no	—
284	D₄	—	2	1	no	—
285	D₄	—	2	1	no	—
286	D₄	—	2	1	no	—
287	D₄	—	2	1	no	—
288	D₄	—	68	1	no	—
289	D₄	—	68	2	no	—
290	D₄	—	136	1	no	—
291	D₄	—	136	1	no	—
292	D₄	—	136	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	8	1	no	—
297	C₂×C₄	—	17	1	no	—
298	C₂×C₄	—	17	1	no	—
299	C₂×C₄	—	34	1	no	—
300	C₂×C₄	—	34	1	no	—
301	C₂×C₄	—	34	1	no	—
302	C₂×C₄	—	34	1	no	—
303	C₂×C₄	—	34	1	no	—
304	C₂×C₄	—	34	1	no	—
305	C₂×C₄	—	34	1	no	—
306	C₂×C₄	—	34	1	no	—
307	C₂×C₄	—	136	1	no	—
308	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
309	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	34	1	no	—
313	C₈	—	34	1	no	—
314	C₈	#63252	1	2	yes	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	34	1	no	—
322	C₂²	—	34	1	no	—
323	C₂²	—	34	1	no	—
324	C₂²	—	34	1	no	—
325	C₂²	—	34	1	no	—
326	C₂²	—	34	1	no	—
327	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	34	1	no	—
333	C₄	—	34	1	no	—
334	C₄	—	68	1	no	—
335	C₄	—	68	1	no	—
336	C₄	—	68	1	no	—
337	C₄	—	68	1	no	—
338	C₄	C₂×#63252	1	1	yes	characteristic
339	C₄	#10587697	1	1	yes	characteristic
340	C₂	—	8	1	no	—
341	C₂	—	8	1	no	—
342	C₂	—	68	1	no	—
343	C₂	—	68	1	no	—
344	C₂	#11763464	1	1	yes	characteristic
345	C₂	#11763605	1	1	yes	characteristic
346	C₂	#11763613	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₇
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#442	C₂
socle	C₂²×C₁₇	C₂²×D₄
hypercenter	#442	D₁₇
hypocenter	C₁₇	#3347
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #442

Complements

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

N	H ≅ G/N	type
1. #11763524	C₂	semidirect
2. #11763509	C₂	semidirect
3. #11763218	C₂	semidirect
4. #11763132	C₂	semidirect
5. #11763132	C₂	semidirect
6. #11763131	C₂	semidirect
7. #11763121	C₂	semidirect
8. #11763109	C₂	semidirect
9. #11763089	C₂	semidirect
10. #11763059	C₂	semidirect
11. #11763042	C₂	semidirect
12. #11762882	C₂	semidirect
13. #11762848	C₂	semidirect
14. #11762844	C₂	semidirect
15. C₁₇×#442	C₂	semidirect
178. #442	D₁₇	semidirect
232. C₁₇	#3347	semidirect

Maximal subgroups

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

subgroup	index
#11763524	2
#11763509	2
#11763218	2
#11763132	2
#11763131	2
#11763121	2
#11763109	2
#11763089	2
#11763059	2
#11763042	2
#11762882	2
#11762848	2
#11762844	2
C₁₇×#442	2
#3347	17