Subgroups — #14540697

Order 2176.

386 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	#11763527	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763458	C₂	1	1	yes	maximal, characteristic
3	#11763158	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763150	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763149	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763087	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763050	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763049	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763018	C₂	1	1	yes	maximal, characteristic, complemented
10	#11762972	C₂	1	1	yes	maximal, characteristic, complemented
11	#11762971	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762900	C₂	1	1	yes	maximal, characteristic, complemented
13	C₈×C₂²⋊D₁₇	C₂	1	1	yes	maximal, characteristic, complemented
14	#11762802	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#416	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#41×C₁₇	C₂²	1	1	yes	characteristic
17	C₂×#63254	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₁₇	C₂²	1	1	yes	characteristic
19	C₁₇×#117	C₂²	1	1	yes	characteristic
20	#10587632	C₂²	1	1	yes	characteristic
21	#10587630	C₂²	1	1	yes	characteristic
22	C₂×#63231	C₂²	1	1	yes	characteristic
23	#10587625	—	2	1	no	—
24	#10587621	—	2	1	no	—
25	D₄×Dic₁₇	C₂²	1	1	yes	characteristic
26	C₂×#63229	C₂²	1	1	yes	characteristic
27	#10587615	—	2	1	no	—
28	C₄×C₂²⋊D₁₇	C₂²	1	1	yes	characteristic
29	#10587610	C₂²	1	1	yes	characteristic
30	#10587610	C₂²	1	1	yes	characteristic
31	C₂×#63225	—	2	1	no	—
32	C₂×Dic₆₈	C₂²	1	1	yes	characteristic
33	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
34	#10587583	C₂²	1	1	yes	characteristic
35	#10587578	C₂²	1	1	yes	characteristic
36	#10587574	C₂²	1	1	yes	characteristic
37	#10587573	C₂²	1	1	yes	characteristic
38	#10587571	—	2	1	no	—
39	#10587565	—	2	1	no	—
40	#10587562	—	2	1	no	—
41	C₄×Dic₃₄	—	2	1	no	—
42	C₁₇×#103	C₂²	1	1	yes	characteristic
43	C₁₇×#103	C₂²	1	1	yes	characteristic
44	C₁₇×#102	C₂²	1	1	yes	characteristic
45	C₁₇×#102	C₂²	1	1	yes	characteristic
46	#10587506	C₂²	1	1	yes	characteristic
47	#10587503	C₂²	1	1	yes	characteristic
48	C₂²⋊#3622	C₂²	1	1	yes	characteristic
49	#10587491	C₂²	1	1	yes	characteristic
50	#10587489	C₂²	1	1	yes	characteristic
51	#10587487	C₂²	1	1	yes	characteristic
52	#10587487	C₂²	1	1	yes	characteristic
53	#10587485	C₂²	1	1	yes	characteristic
54	C₈×Dic₁₇	C₂²	1	1	yes	characteristic
55	#10587481	C₂²	1	1	yes	characteristic
56	#10587481	C₂²	1	1	yes	characteristic
57	#10587479	C₂²	1	1	yes	characteristic
58	#10587478	C₂²	1	1	yes	characteristic
59	#41×C₁₇	—	2	2	no	—
60	#41×C₁₇	—	4	1	no	—
61	C₂×Q₈×C₁₇	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₁₇	—	2	1	no	—
63	C₂×D₄×C₁₇	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₁₇	—	2	1	no	—
65	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
66	C₂×C₂²⋊D₁₇	—	2	1	no	—
67	C₂×C₂²⋊D₁₇	D₄	1	1	yes	characteristic
68	C₂²×Dic₁₇	—	2	1	no	—
69	C₂²×Dic₁₇	—	2	1	no	—
70	#63254	—	2	2	no	—
71	#63254	—	4	1	no	—
72	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
73	C₂×Dic₃₄	—	2	1	no	—
74	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
75	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
76	C₂×C₈×C₁₇	—	4	1	no	—
77	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
81	C₂²⋊C₄×C₁₇	—	2	1	no	—
82	C₄²×C₁₇	—	2	1	no	—
83	C₂²⋊Dic₁₇	—	2	1	no	—
84	C₂²⋊Dic₁₇	—	2	1	no	—
85	C₂²⋊Dic₁₇	—	2	1	no	—
86	C₂²⋊Dic₁₇	D₄	1	1	yes	characteristic
87	#63231	—	4	1	no	—
88	#63229	—	2	2	no	—
89	#63227	—	2	1	no	—
90	#63227	—	2	1	no	—
91	#63227	—	2	1	no	—
92	#63227	D₄	1	1	yes	characteristic
93	C₄⋊Dic₁₇	—	2	1	no	—
94	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
95	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
96	#63225	—	2	1	no	—
97	#63225	—	2	1	no	—
98	#63225	—	2	1	no	—
99	#63225	—	2	1	no	—
100	#63225	—	2	1	no	—
101	#63225	—	2	1	no	—
102	#63225	—	2	1	no	—
103	#63225	D₄	1	1	yes	characteristic
104	C₄×Dic₁₇	—	2	1	no	—
105	C₄×Dic₁₇	—	2	1	no	—
106	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
107	C₂×#3622	C₂³	1	1	yes	characteristic
108	C₂×#3622	C₂³	1	1	yes	characteristic
109	Dic₆₈	—	4	1	no	—
110	C₈×D₁₇	—	4	1	no	—
111	C₂³×C₁₇	—	2	1	no	—
112	C₂³×C₁₇	C₂×D₄	1	1	yes	characteristic
113	C₂²×D₁₇	C₂×D₄	1	1	yes	characteristic
114	Q₈×C₁₇	—	2	1	no	—
115	Q₈×C₁₇	—	2	1	no	—
116	D₄×C₁₇	—	2	1	no	—
117	D₄×C₁₇	—	4	2	no	—
118	D₄×C₁₇	#41	1	2	yes	—
119	C₂×C₄×C₁₇	—	2	1	no	—
120	C₂×C₄×C₁₇	—	2	1	no	—
121	C₂×C₄×C₁₇	—	2	1	no	—
122	C₂×C₄×C₁₇	—	2	1	no	—
123	C₂×C₄×C₁₇	—	2	1	no	—
124	C₂×C₄×C₁₇	—	4	2	no	—
125	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
127	C₂²⋊D₁₇	—	2	2	no	—
128	C₂²⋊D₁₇	—	4	2	no	—
129	C₂×Dic₁₇	—	2	1	no	—
130	C₂×Dic₁₇	—	2	1	no	—
131	C₂×Dic₁₇	—	2	1	no	—
132	C₂×Dic₁₇	—	2	1	no	—
133	C₂×Dic₁₇	—	4	2	no	—
134	C₂×Dic₁₇	—	4	2	no	—
135	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
137	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
138	C₄×D₁₇	—	2	1	no	—
139	C₄×D₁₇	—	2	1	no	—
140	Dic₃₄	—	2	1	no	—
141	Dic₃₄	—	2	1	no	—
142	Dic₃₄	—	2	1	no	—
143	Dic₃₄	—	2	1	no	—
144	Dic₃₄	—	4	2	no	—
145	C₈×C₁₇	—	2	1	no	—
146	C₈×C₁₇	—	2	1	no	—
147	#3622	—	2	1	no	—
148	#3622	—	2	1	no	—
149	#3302	—	17	1	no	maximal, Sylow
150	C₂²×C₁₇	—	2	2	no	—
151	C₂²×C₁₇	—	2	1	no	—
152	C₂²×C₁₇	—	2	1	no	—
153	C₂²×C₁₇	—	2	1	no	—
154	C₂²×C₁₇	—	4	1	no	—
155	C₂²×C₁₇	—	4	1	no	—
156	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
157	C₂×D₁₇	—	2	1	no	—
158	C₂×D₁₇	—	2	1	no	—
159	C₂×D₁₇	—	2	1	no	—
160	C₄×C₁₇	—	2	1	no	—
161	C₄×C₁₇	—	4	1	no	—
162	C₄×C₁₇	—	4	1	no	—
163	C₄×C₁₇	—	4	1	no	—
164	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
165	C₄×C₁₇	H(5,𝔽₂)	1	1	yes	characteristic
166	Dic₁₇	—	2	1	no	—
167	Dic₁₇	—	2	1	no	—
168	Dic₁₇	—	4	1	no	—
169	Dic₁₇	—	4	1	no	—
170	Dic₁₇	—	4	1	no	—
171	Dic₁₇	—	4	1	no	—
172	Dic₁₇	#135	1	2	yes	—
173	#556	—	17	1	no	—
174	#547	—	17	1	no	—
175	#487	—	17	1	no	—
176	#485	—	17	1	no	—
177	#485	—	17	1	no	—
178	#474	—	17	1	no	—
179	#467	—	17	1	no	—
180	#467	—	17	1	no	—
181	#462	—	17	1	no	—
182	#452	—	17	1	no	—
183	#452	—	17	1	no	—
184	C₄×SD₁₆	—	17	1	no	—
185	C₈×D₄	—	17	1	no	—
186	#416	—	17	1	no	—
187	#416	D₁₇	1	1	yes	characteristic, complemented
188	C₂×C₁₇	—	4	1	no	—
189	C₂×C₁₇	—	4	2	no	—
190	C₂×C₁₇	#546	1	1	yes	characteristic
191	C₂×C₁₇	C₂×#135	1	1	yes	characteristic
192	C₂×C₁₇	#578	1	1	yes	characteristic
193	D₁₇	—	4	1	no	—
194	C₂×#41	—	17	1	no	—
195	C₂×#41	C₂×D₁₇	1	1	yes	characteristic
196	C₂×Q₁₆	—	17	1	no	—
197	C₂×Q₁₆	—	17	1	no	—
198	C₂×SD₁₆	—	17	1	no	—
199	C₂²×C₈	—	17	1	no	—
200	C₂²×C₈	C₂×D₁₇	1	1	yes	characteristic
201	C₄⋊Q₈	—	17	1	no	—
202	C₄⋊Q₈	—	17	1	no	—
203	#126	—	34	1	no	—
204	#126	—	34	1	no	—
205	#124	—	34	1	no	—
206	#123	—	34	1	no	—
207	#123	—	34	1	no	—
208	C₂²⋊Q₈	—	17	1	no	—
209	C₂²⋊Q₈	—	17	1	no	—
210	C₂²⋊Q₈	—	17	1	no	—
211	C₂²⋊Q₈	—	17	1	no	—
212	#121	—	34	1	no	—
213	C₄×Q₈	—	17	1	no	—
214	C₄×Q₈	—	34	1	no	—
215	C₄×D₄	—	17	1	no	—
216	C₄×D₄	—	17	1	no	—
217	#117	—	17	1	no	—
218	#117	C₂×D₁₇	1	1	yes	characteristic
219	C₂×C₄⋊C₄	—	34	1	no	—
220	#107	—	17	1	no	—
221	#107	—	17	1	no	—
222	#106	—	17	1	no	—
223	C₄⋊C₈	—	17	1	no	—
224	#103	—	17	1	no	—
225	#103	—	17	1	no	—
226	#103	—	17	1	no	—
227	#103	—	17	1	no	—
228	#103	—	17	1	no	—
229	#103	C₂×D₁₇	1	1	yes	characteristic
230	#103	C₂×D₁₇	1	1	yes	characteristic
231	#102	—	17	1	no	—
232	#102	C₂×D₁₇	1	1	yes	characteristic
233	#102	C₂×D₁₇	1	1	yes	characteristic
234	C₂²⋊C₈	—	17	1	no	—
235	C₂²⋊C₈	—	17	1	no	—
236	C₄×C₈	—	17	1	no	—
237	C₁₇	#3302	1	1	yes	characteristic, complemented, Sylow
238	#41	—	2	2	no	—
239	#41	—	4	1	no	—
240	#41	—	34	2	no	—
241	#41	—	68	1	no	—
242	C₂×Q₈	—	17	1	no	—
243	C₂×Q₈	—	17	1	no	—
244	C₂×Q₈	—	34	1	no	—
245	C₂×Q₈	C₂²×D₁₇	1	1	yes	characteristic
246	C₂×D₄	—	2	1	no	—
247	C₂×D₄	—	17	1	no	—
248	C₂×D₄	—	34	1	no	—
249	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
250	C₂²×C₄	—	2	1	no	—
251	C₂²×C₄	—	17	1	no	—
252	C₂²×C₄	—	34	1	no	—
253	C₂²×C₄	—	34	1	no	—
254	C₂²×C₄	C₂²×D₁₇	1	1	yes	characteristic
255	Q₁₆	—	68	1	no	—
256	Q₁₆	—	68	1	no	—
257	SD₁₆	—	34	2	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	—	17	1	no	—
260	C₂×C₈	—	17	1	no	—
261	C₂×C₈	—	68	1	no	—
262	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
263	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
264	C₄⋊C₄	—	17	1	no	—
265	C₄⋊C₄	—	17	1	no	—
266	C₄⋊C₄	—	17	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	—	34	1	no	—
272	C₄⋊C₄	—	34	1	no	—
273	C₄⋊C₄	—	34	1	no	—
274	C₄⋊C₄	—	34	1	no	—
275	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	17	1	no	—
279	C₂²⋊C₄	—	17	1	no	—
280	C₂²⋊C₄	—	34	1	no	—
281	C₂²⋊C₄	—	34	1	no	—
282	C₂²⋊C₄	—	34	1	no	—
283	C₂²⋊C₄	—	34	1	no	—
284	C₂²⋊C₄	—	34	1	no	—
285	C₂²⋊C₄	—	34	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	—	17	1	no	—
288	C₄²	—	34	1	no	—
289	C₄²	—	34	1	no	—
290	C₂³	—	2	1	no	—
291	C₂³	—	17	1	no	—
292	C₂³	D₄×D₁₇	1	1	yes	characteristic
293	Q₈	—	2	1	no	—
294	Q₈	—	2	1	no	—
295	Q₈	—	34	1	no	—
296	Q₈	—	34	1	no	—
297	Q₈	—	34	1	no	—
298	Q₈	—	34	1	no	—
299	Q₈	—	68	2	no	—
300	D₄	—	2	1	no	—
301	D₄	—	4	2	no	—
302	D₄	—	34	2	no	—
303	D₄	—	68	2	no	—
304	D₄	#63252	1	2	yes	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	17	1	no	—
312	C₂×C₄	—	17	1	no	—
313	C₂×C₄	—	17	1	no	—
314	C₂×C₄	—	34	1	no	—
315	C₂×C₄	—	34	1	no	—
316	C₂×C₄	—	34	1	no	—
317	C₂×C₄	—	34	1	no	—
318	C₂×C₄	—	34	1	no	—
319	C₂×C₄	—	34	1	no	—
320	C₂×C₄	—	68	2	no	—
321	C₂×C₄	—	68	2	no	—
322	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
323	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
324	C₈	—	2	1	no	—
325	C₈	—	2	1	no	—
326	C₈	—	34	1	no	—
327	C₈	—	34	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	2	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	4	1	no	—
334	C₂²	—	34	1	no	—
335	C₂²	—	34	1	no	—
336	C₂²	—	34	1	no	—
337	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	4	1	no	—
342	C₄	—	17	2	no	—
343	C₄	—	34	1	no	—
344	C₄	—	34	1	no	—
345	C₄	—	68	1	no	—
346	C₄	—	68	1	no	—
347	C₄	—	68	1	no	—
348	C₄	—	68	1	no	—
349	C₄	C₂×#63252	1	1	yes	characteristic
350	C₄	#10587691	1	1	yes	characteristic
351	C₂	—	4	1	no	—
352	C₂	—	4	2	no	—
353	C₂	—	68	1	no	—
354	C₂	#11763449	1	1	yes	characteristic
355	C₂	#135×D₁₇	1	1	yes	characteristic
356	C₂	#11763623	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×#416	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#11763458	C₂
hypercenter	#416	D₁₇
hypocenter	C₁₇	#3302
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #416

Complements

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

N	H ≅ G/N	type
1. #11763527	C₂	semidirect
3. #11763158	C₂	semidirect
4. #11763150	C₂	semidirect
5. #11763149	C₂	semidirect
6. #11763087	C₂	semidirect
7. #11763050	C₂	semidirect
8. #11763049	C₂	semidirect
9. #11763018	C₂	semidirect
10. #11762972	C₂	semidirect
11. #11762971	C₂	semidirect
12. #11762900	C₂	semidirect
13. C₈×C₂²⋊D₁₇	C₂	semidirect
14. #11762802	C₂	semidirect
15. C₁₇×#416	C₂	semidirect
187. #416	D₁₇	semidirect
237. C₁₇	#3302	semidirect

Maximal subgroups

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

subgroup	index
#11763527	2
#11763458	2
#11763158	2
#11763150	2
#11763149	2
#11763087	2
#11763050	2
#11763049	2
#11763018	2
#11762972	2
#11762971	2
#11762900	2
C₈×C₂²⋊D₁₇	2
#11762802	2
C₁₇×#416	2
#3302	17