Subgroups — #14540197

Order 2176.

358 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763430	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763414	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763117	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763110	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763100	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763090	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763062	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763055	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763045	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763038	C₂	1	1	yes	maximal, characteristic, complemented
11	#11763025	C₂	1	1	yes	maximal, characteristic, complemented
12	#11763025	C₂	1	1	yes	maximal, characteristic, complemented
13	#11763010	C₂	1	1	yes	maximal, characteristic, complemented
14	#11763005	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#424	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×M₁₆×C₁₇	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
18	C₁₇×#117	C₂²	1	1	yes	characteristic
19	#10587616	C₂²	1	1	yes	characteristic
20	#10587616	C₂²	1	1	yes	characteristic
21	#10587616	C₂²	1	1	yes	characteristic
22	#10587610	C₂²	1	1	yes	characteristic
23	C₂×D₁₃₆	C₂²	1	1	yes	characteristic
24	C₂×D₁₃₆	C₂²	1	1	yes	characteristic
25	C₂×#63219	C₂²	1	1	yes	characteristic
26	C₂×#63219	C₂²	1	1	yes	characteristic
27	#10587584	—	2	1	no	—
28	#10587582	—	2	1	no	—
29	#10587580	—	2	1	no	—
30	#10587580	—	2	1	no	—
31	#10587571	—	2	1	no	—
32	#10587570	—	2	1	no	—
33	#10587570	—	2	1	no	—
34	#10587569	—	2	1	no	—
35	#10587561	C₂²	1	1	yes	characteristic
36	#10587560	C₂²	1	1	yes	characteristic
37	C₄×D₆₈	C₂²	1	1	yes	characteristic
38	C₄×D₆₈	C₂²	1	1	yes	characteristic
39	C₄×D₆₈	C₂²	1	1	yes	characteristic
40	#10587556	C₂²	1	1	yes	characteristic
41	#10587555	C₂²	1	1	yes	characteristic
42	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
43	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
44	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
45	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
46	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
47	#10587492	C₂²	1	1	yes	characteristic
48	#10587492	C₂²	1	1	yes	characteristic
49	#10587492	C₂²	1	1	yes	characteristic
50	#10587492	C₂²	1	1	yes	characteristic
51	#10587492	C₂²	1	1	yes	characteristic
52	#10587492	C₂²	1	1	yes	characteristic
53	#10587489	C₂²	1	1	yes	characteristic
54	#10587489	C₂²	1	1	yes	characteristic
55	#10587488	C₂²	1	1	yes	characteristic
56	#10587488	C₂²	1	1	yes	characteristic
57	#10587487	C₂²	1	1	yes	characteristic
58	#10587487	C₂²	1	1	yes	characteristic
59	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
60	C₂×C₂²⋊D₁₇	—	2	1	no	—
61	C₂×C₂²⋊D₁₇	—	2	1	no	—
62	C₂×C₂²⋊D₁₇	—	2	1	no	—
63	C₂×D₆₈	—	2	1	no	—
64	C₂×D₆₈	—	2	1	no	—
65	C₂×D₆₈	C₂³	1	1	yes	characteristic
66	C₂×D₆₈	C₂³	1	1	yes	characteristic
67	C₂×D₆₈	C₂³	1	1	yes	characteristic
68	C₂×C₄×D₁₇	—	2	1	no	—
69	C₂×C₄×D₁₇	—	2	1	no	—
70	C₂×C₄×D₁₇	—	2	1	no	—
71	C₂×Dic₃₄	—	2	1	no	—
72	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
73	M₁₆×C₁₇	—	4	1	no	—
74	C₂×C₈×C₁₇	—	4	1	no	—
75	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
76	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
80	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
81	C₂²⋊C₄×C₁₇	D₄	1	1	yes	characteristic
82	C₂²⋊C₄×C₁₇	D₄	1	1	yes	characteristic
83	C₄²×C₁₇	C₂³	1	1	yes	characteristic
84	C₄²×C₁₇	C₂³	1	1	yes	characteristic
85	C₂²⋊Dic₁₇	—	2	1	no	—
86	#63227	—	2	1	no	—
87	#63227	—	2	1	no	—
88	#63227	—	2	1	no	—
89	#63227	—	2	1	no	—
90	#63227	—	2	1	no	—
91	#63227	—	2	1	no	—
92	#63227	—	2	1	no	—
93	#63227	—	2	1	no	—
94	C₄⋊Dic₁₇	—	2	1	no	—
95	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
96	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
97	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
98	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
99	#63225	—	2	1	no	—
100	#63225	—	2	1	no	—
101	#63225	—	2	1	no	—
102	#63225	—	2	1	no	—
103	C₄×Dic₁₇	—	2	1	no	—
104	D₁₃₆	—	4	1	no	—
105	D₁₃₆	—	4	1	no	—
106	#63219	—	4	1	no	—
107	#63219	—	4	1	no	—
108	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₁₇	—	2	1	no	—
110	C₂²×D₁₇	—	2	1	no	—
111	C₂²×D₁₇	—	2	1	no	—
112	C₂×C₄×C₁₇	—	2	1	no	—
113	C₂×C₄×C₁₇	—	2	1	no	—
114	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
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	—	8	1	no	—
121	C₂²⋊D₁₇	—	8	1	no	—
122	C₂²⋊D₁₇	—	8	1	no	—
123	C₂×Dic₁₇	—	2	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	D₆₈	—	2	1	no	—
129	D₆₈	—	2	1	no	—
130	D₆₈	—	2	1	no	—
131	D₆₈	—	2	1	no	—
132	D₆₈	—	2	1	no	—
133	D₆₈	—	2	1	no	—
134	D₆₈	—	8	1	no	—
135	D₆₈	—	8	1	no	—
136	C₄×D₁₇	—	8	1	no	—
137	C₄×D₁₇	—	8	1	no	—
138	C₄×D₁₇	—	8	1	no	—
139	Dic₃₄	—	2	1	no	—
140	Dic₃₄	—	2	1	no	—
141	Dic₃₄	—	8	1	no	—
142	C₈×C₁₇	—	2	1	no	—
143	C₈×C₁₇	—	2	1	no	—
144	C₈×C₁₇	—	2	1	no	—
145	C₈×C₁₇	—	2	1	no	—
146	#3257	—	17	1	no	maximal, Sylow
147	C₂²×C₁₇	—	2	1	no	—
148	C₂²×C₁₇	—	2	1	no	—
149	C₂²×C₁₇	—	2	1	no	—
150	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
151	C₂×D₁₇	—	4	1	no	—
152	C₂×D₁₇	—	4	1	no	—
153	C₂×D₁₇	—	4	1	no	—
154	C₂×D₁₇	—	4	1	no	—
155	C₂×D₁₇	—	4	1	no	—
156	C₂×D₁₇	—	4	1	no	—
157	C₂×D₁₇	—	4	1	no	—
158	C₂×D₁₇	—	4	1	no	—
159	C₂×D₁₇	—	4	1	no	—
160	C₄×C₁₇	—	2	1	no	—
161	C₄×C₁₇	—	2	1	no	—
162	C₄×C₁₇	—	2	1	no	—
163	C₄×C₁₇	—	2	1	no	—
164	C₄×C₁₇	—	2	1	no	—
165	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
166	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
167	Dic₁₇	—	4	1	no	—
168	Dic₁₇	—	4	1	no	—
169	Dic₁₇	—	4	1	no	—
170	Dic₁₇	—	4	1	no	—
171	Dic₁₇	—	4	1	no	—
172	#542	—	17	1	no	—
173	#540	—	17	1	no	—
174	#479	—	17	1	no	—
175	#478	—	17	1	no	—
176	#476	—	17	1	no	—
177	#474	—	17	1	no	—
178	#469	—	17	1	no	—
179	#468	—	17	1	no	—
180	#466	—	17	1	no	—
181	#465	—	17	1	no	—
182	#463	—	17	1	no	—
183	#463	—	17	1	no	—
184	#460	—	17	1	no	—
185	#459	—	17	1	no	—
186	#424	D₁₇	1	1	yes	characteristic, complemented
187	C₂×C₁₇	—	4	1	no	—
188	C₂×C₁₇	#537	1	1	yes	characteristic
189	C₂×C₁₇	#576	1	1	yes	characteristic
190	C₂×C₁₇	#577	1	1	yes	characteristic
191	D₁₇	—	8	1	no	—
192	D₁₇	—	8	1	no	—
193	D₁₇	—	8	1	no	—
194	C₂×SD₁₆	—	17	1	no	—
195	C₂×SD₁₆	—	17	1	no	—
196	C₂×D₈	—	17	1	no	—
197	C₂×D₈	—	17	1	no	—
198	C₂×M₁₆	C₂×D₁₇	1	1	yes	characteristic
199	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
200	C₄⋊Q₈	—	17	1	no	—
201	#127	—	17	1	no	—
202	#126	—	34	1	no	—
203	#125	—	17	1	no	—
204	#124	—	17	1	no	—
205	#124	—	34	1	no	—
206	#123	—	34	1	no	—
207	#123	—	34	1	no	—
208	#123	—	34	1	no	—
209	C₂²⋊Q₈	—	17	1	no	—
210	C₂²⋊Q₈	—	34	1	no	—
211	#121	—	17	1	no	—
212	#121	—	17	1	no	—
213	#121	—	17	1	no	—
214	#121	—	34	1	no	—
215	#121	—	34	1	no	—
216	C₄×Q₈	—	17	1	no	—
217	C₄×D₄	—	17	1	no	—
218	C₄×D₄	—	17	1	no	—
219	C₄×D₄	—	17	1	no	—
220	#117	C₂×D₁₇	1	1	yes	characteristic
221	#107	—	17	1	no	—
222	#107	—	17	1	no	—
223	#106	—	17	1	no	—
224	#106	—	17	1	no	—
225	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
226	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
227	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
228	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
229	#103	—	17	1	no	—
230	#103	—	17	1	no	—
231	#102	—	17	1	no	—
232	#102	—	17	1	no	—
233	#102	—	17	1	no	—
234	#102	—	17	1	no	—
235	#102	—	17	1	no	—
236	#102	—	17	1	no	—
237	C₁₇	#3257	1	1	yes	characteristic, complemented, Sylow
238	C₂×Q₈	—	17	1	no	—
239	C₂×Q₈	—	34	1	no	—
240	C₂×D₄	—	17	1	no	—
241	C₂×D₄	—	17	1	no	—
242	C₂×D₄	—	17	1	no	—
243	C₂×D₄	—	34	1	no	—
244	C₂×D₄	—	34	1	no	—
245	C₂×D₄	—	34	1	no	—
246	C₂×D₄	—	34	1	no	—
247	C₂×D₄	—	34	1	no	—
248	C₂²×C₄	—	34	1	no	—
249	C₂²×C₄	—	34	1	no	—
250	C₂²×C₄	—	34	1	no	—
251	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
252	SD₁₆	—	68	1	no	—
253	SD₁₆	—	68	1	no	—
254	D₈	—	68	1	no	—
255	D₈	—	68	1	no	—
256	M₁₆	—	4	1	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
259	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
260	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
261	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
262	C₄⋊C₄	—	17	1	no	—
263	C₄⋊C₄	—	17	1	no	—
264	C₄⋊C₄	—	17	1	no	—
265	C₄⋊C₄	—	17	1	no	—
266	C₄⋊C₄	—	34	1	no	—
267	C₄⋊C₄	—	34	1	no	—
268	C₄⋊C₄	—	34	1	no	—
269	C₄⋊C₄	—	34	1	no	—
270	C₄⋊C₄	—	34	1	no	—
271	C₄⋊C₄	D₆₈	1	1	yes	characteristic
272	C₄⋊C₄	D₆₈	1	1	yes	characteristic
273	C₂²⋊C₄	—	34	1	no	—
274	C₂²⋊C₄	—	34	1	no	—
275	C₂²⋊C₄	—	34	1	no	—
276	C₂²⋊C₄	—	34	1	no	—
277	C₂²⋊C₄	—	34	1	no	—
278	C₂²⋊C₄	—	34	1	no	—
279	C₂²⋊C₄	—	34	1	no	—
280	C₂²⋊C₄	—	34	1	no	—
281	C₂²⋊C₄	—	34	1	no	—
282	C₂²⋊C₄	D₆₈	1	1	yes	characteristic
283	C₂²⋊C₄	D₆₈	1	1	yes	characteristic
284	C₄²	—	34	1	no	—
285	C₄²	C₂²×D₁₇	1	1	yes	characteristic
286	C₄²	C₂²×D₁₇	1	1	yes	characteristic
287	C₂³	—	34	1	no	—
288	C₂³	—	34	1	no	—
289	C₂³	—	34	1	no	—
290	C₂³	C₂×D₆₈	1	1	yes	characteristic
291	Q₈	—	34	1	no	—
292	Q₈	—	34	1	no	—
293	Q₈	—	136	1	no	—
294	D₄	—	34	1	no	—
295	D₄	—	34	1	no	—
296	D₄	—	34	1	no	—
297	D₄	—	34	1	no	—
298	D₄	—	34	1	no	—
299	D₄	—	34	1	no	—
300	D₄	—	136	1	no	—
301	D₄	—	136	1	no	—
302	D₄	—	136	1	no	—
303	D₄	—	136	1	no	—
304	D₄	—	136	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	34	1	no	—
308	C₂×C₄	—	34	1	no	—
309	C₂×C₄	—	34	1	no	—
310	C₂×C₄	—	34	1	no	—
311	C₂×C₄	—	34	1	no	—
312	C₂×C₄	—	136	1	no	—
313	C₂×C₄	—	136	1	no	—
314	C₂×C₄	—	136	1	no	—
315	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
316	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
317	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
318	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
319	C₂×C₄	C₂×D₆₈	1	1	yes	characteristic
320	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	2	1	no	—
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	68	1	no	—
329	C₂²	—	68	1	no	—
330	C₂²	—	68	1	no	—
331	C₂²	—	68	1	no	—
332	C₂²	—	68	1	no	—
333	C₂²	—	68	1	no	—
334	C₂²	—	68	1	no	—
335	C₂²	—	68	1	no	—
336	C₂²	—	68	1	no	—
337	C₂²	C₂²×D₆₈	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	68	1	no	—
344	C₄	—	68	1	no	—
345	C₄	—	68	1	no	—
346	C₄	—	68	1	no	—
347	C₄	—	68	1	no	—
348	C₄	#10587691	1	1	yes	characteristic
349	C₄	#10587694	1	1	yes	characteristic
350	C₂	—	4	1	no	—
351	C₂	—	136	1	no	—
352	C₂	—	136	1	no	—
353	C₂	—	136	1	no	—
354	C₂	#11763390	1	1	yes	characteristic
355	C₂	#11763614	1	1	yes	characteristic
356	C₂	#11763621	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₆₈
commutator, cosocle	C₂×C₄×C₁₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₇
Fitting	C₁₇×#424	C₂
socle	C₂²×C₁₇	C₂²×D₄
hypercenter	#424	D₁₇
hypocenter	C₁₇	#3257
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #424

Complements

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

N	H ≅ G/N	type
1. #11763430	C₂	semidirect
2. #11763414	C₂	semidirect
3. #11763117	C₂	semidirect
4. #11763110	C₂	semidirect
5. #11763100	C₂	semidirect
6. #11763090	C₂	semidirect
7. #11763062	C₂	semidirect
8. #11763055	C₂	semidirect
9. #11763045	C₂	semidirect
10. #11763038	C₂	semidirect
11. #11763025	C₂	semidirect
12. #11763025	C₂	semidirect
13. #11763010	C₂	semidirect
14. #11763005	C₂	semidirect
15. C₁₇×#424	C₂	semidirect
186. #424	D₁₇	semidirect
237. C₁₇	#3257	semidirect

Maximal subgroups

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

subgroup	index
#11763430	2
#11763414	2
#11763117	2
#11763110	2
#11763100	2
#11763090	2
#11763062	2
#11763055	2
#11763045	2
#11763038	2
#11763025	2
#11763010	2
#11763005	2
C₁₇×#424	2
#3257	17