Subgroups — #14541535

Order 2176.

358 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763542	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763515	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763172	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763167	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763166	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763119	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763088	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763025	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763023	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763003	C₂	1	1	yes	maximal, characteristic
11	#11762936	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762920	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762908	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#63228	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#478	C₂	1	1	yes	maximal, characteristic, complemented
16	C₁₇×#125	C₂²	1	1	yes	characteristic
17	C₄×D₄×C₁₇	C₂²	1	1	yes	characteristic
18	#10587626	C₂²	1	1	yes	characteristic
19	#10587622	C₂²	1	1	yes	characteristic
20	C₂×#63228	C₂²	1	1	yes	characteristic
21	#10587615	—	2	1	no	—
22	C₄×C₂²⋊D₁₇	—	2	1	no	—
23	C₂×D₁₃₆	C₂²	1	1	yes	characteristic
24	C₂×#63219	C₂²	1	1	yes	characteristic
25	#10587583	—	2	1	no	—
26	#10587582	—	2	1	no	—
27	#10587582	—	2	1	no	—
28	#10587579	C₂²	1	1	yes	characteristic
29	#10587579	C₂²	1	1	yes	characteristic
30	#10587579	C₂²	1	1	yes	characteristic
31	C₄⋊C₄×D₁₇	—	2	1	no	—
32	#10587576	C₂²	1	1	yes	characteristic
33	#10587574	C₂²	1	1	yes	characteristic
34	#10587573	C₂²	1	1	yes	characteristic
35	#10587571	—	2	1	no	—
36	#10587569	—	2	1	no	—
37	#10587561	C₂²	1	1	yes	characteristic
38	C₄×D₆₈	C₂²	1	1	yes	characteristic
39	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
40	C₁₇×#107	C₂²	1	1	yes	characteristic
41	C₁₇×#106	C₂²	1	1	yes	characteristic
42	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₁₇×#102	C₂²	1	1	yes	characteristic
44	C₁₇×#102	C₂²	1	1	yes	characteristic
45	#10587503	C₂²	1	1	yes	characteristic
46	#10587492	C₂²	1	1	yes	characteristic
47	#10587487	C₂²	1	1	yes	characteristic
48	#10587486	C₂²	1	1	yes	characteristic
49	#10587486	C₂²	1	1	yes	characteristic
50	#10587485	C₂²	1	1	yes	characteristic
51	#10587485	C₂²	1	1	yes	characteristic
52	#10587481	C₂²	1	1	yes	characteristic
53	#10587481	C₂²	1	1	yes	characteristic
54	#10587480	C₂²	1	1	yes	characteristic
55	#10587480	C₂²	1	1	yes	characteristic
56	#10587480	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	C₂²×C₄×C₁₇	—	2	1	no	—
61	C₂×C₂²⋊D₁₇	—	2	1	no	—
62	C₂×C₂²⋊D₁₇	—	2	1	no	—
63	C₂×D₆₈	C₂³	1	1	yes	characteristic
64	C₂×D₆₈	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₁₇	—	2	1	no	—
66	C₂×C₄×D₁₇	—	2	1	no	—
67	C₂×C₄×D₁₇	—	2	1	no	—
68	C₂×C₄×D₁₇	—	2	1	no	—
69	C₂×Dic₃₄	—	2	1	no	—
70	C₂×Dic₃₄	D₄	1	1	yes	characteristic
71	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₁₇	—	2	1	no	—
75	C₄⋊C₄×C₁₇	—	2	1	no	—
76	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₁₇	C₂³	1	1	yes	characteristic
79	C₂²⋊C₄×C₁₇	—	2	1	no	—
80	C₄²×C₁₇	C₂³	1	1	yes	characteristic
81	C₂²⋊Dic₁₇	—	2	1	no	—
82	C₂²⋊Dic₁₇	—	2	1	no	—
83	#63228	—	2	2	no	—
84	#63227	—	2	1	no	—
85	#63227	—	2	1	no	—
86	#63227	—	2	1	no	—
87	#63227	—	2	1	no	—
88	#63227	—	2	1	no	—
89	#63227	—	2	1	no	—
90	C₄⋊Dic₁₇	—	2	1	no	—
91	C₄⋊Dic₁₇	D₄	1	1	yes	characteristic
92	C₄⋊Dic₁₇	C₂³	1	1	yes	characteristic
93	#63225	—	2	1	no	—
94	#63225	—	2	1	no	—
95	#63225	—	2	1	no	—
96	#63225	D₄	1	1	yes	characteristic
97	#63225	D₄	1	1	yes	characteristic
98	C₄×Dic₁₇	—	2	1	no	—
99	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
100	C₄×Dic₁₇	C₂³	1	1	yes	characteristic
101	C₂×#3622	C₂³	1	1	yes	characteristic
102	C₂×#3622	C₂³	1	1	yes	characteristic
103	D₁₃₆	—	2	2	no	—
104	#63219	—	2	2	no	—
105	C₂³×C₁₇	—	2	1	no	—
106	C₂²×D₁₇	—	2	1	no	—
107	C₂²×D₁₇	—	2	1	no	—
108	D₄×C₁₇	—	2	1	no	—
109	D₄×C₁₇	—	2	1	no	—
110	C₂×C₄×C₁₇	—	2	1	no	—
111	C₂×C₄×C₁₇	—	2	1	no	—
112	C₂×C₄×C₁₇	—	2	1	no	—
113	C₂×C₄×C₁₇	—	8	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	C₂⁴	1	1	yes	characteristic
117	C₂²⋊D₁₇	—	4	2	no	—
118	C₂²⋊D₁₇	—	8	1	no	—
119	C₂×Dic₁₇	—	2	1	no	—
120	C₂×Dic₁₇	—	2	1	no	—
121	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
125	D₆₈	—	2	1	no	—
126	D₆₈	—	2	1	no	—
127	D₆₈	—	2	1	no	—
128	D₆₈	#41	1	2	yes	—
129	C₄×D₁₇	—	4	2	no	—
130	C₄×D₁₇	—	8	1	no	—
131	C₄×D₁₇	—	8	1	no	—
132	C₄×D₁₇	—	8	1	no	—
133	Dic₃₄	—	2	1	no	—
134	Dic₃₄	—	2	1	no	—
135	Dic₃₄	—	4	1	no	—
136	Dic₃₄	—	8	1	no	—
137	C₈×C₁₇	—	2	1	no	—
138	C₈×C₁₇	—	2	1	no	—
139	#3622	—	2	1	no	—
140	#3622	—	2	1	no	—
141	#3370	—	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	C₂²×D₄	1	1	yes	characteristic
146	C₂×D₁₇	—	2	2	no	—
147	C₂×D₁₇	—	4	1	no	—
148	C₂×D₁₇	—	4	1	no	—
149	C₂×D₁₇	—	4	1	no	—
150	C₂×D₁₇	—	4	1	no	—
151	C₂×D₁₇	—	4	1	no	—
152	C₄×C₁₇	—	2	1	no	—
153	C₄×C₁₇	—	2	1	no	—
154	C₄×C₁₇	—	4	1	no	—
155	C₄×C₁₇	—	4	1	no	—
156	C₄×C₁₇	—	4	1	no	—
157	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
158	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
159	Dic₁₇	—	2	1	no	—
160	Dic₁₇	—	2	1	no	—
161	Dic₁₇	—	2	1	no	—
162	Dic₁₇	—	2	1	no	—
163	Dic₁₇	—	4	1	no	—
164	Dic₁₇	—	4	1	no	—
165	#559	—	17	1	no	—
166	#554	—	17	1	no	—
167	#490	—	17	1	no	—
168	#489	—	17	1	no	—
169	#488	—	17	1	no	—
170	#479	—	17	1	no	—
171	#478	D₁₇	1	1	yes	characteristic, complemented
172	#474	—	17	1	no	—
173	#463	—	17	1	no	—
174	#463	—	17	1	no	—
175	#459	—	17	1	no	—
176	#446	—	17	1	no	—
177	#442	—	17	1	no	—
178	#440	—	17	1	no	—
179	C₄×D₈	—	17	1	no	—
180	C₂×C₁₇	—	8	1	no	—
181	C₂×C₁₇	#548	1	1	yes	characteristic
182	C₂×C₁₇	#575	1	1	yes	characteristic
183	C₂×C₁₇	#576	1	1	yes	characteristic
184	D₁₇	—	4	2	no	—
185	D₁₇	—	8	1	no	—
186	C₂×SD₁₆	—	17	1	no	—
187	C₂×D₈	—	17	1	no	—
188	C₂×D₈	—	17	1	no	—
189	C₄⋊Q₈	—	17	1	no	—
190	#127	—	17	1	no	—
191	#125	—	17	1	no	—
192	#125	C₂×D₁₇	1	1	yes	characteristic
193	#124	—	17	1	no	—
194	#124	—	17	1	no	—
195	#124	—	34	1	no	—
196	#123	—	34	1	no	—
197	#123	—	34	1	no	—
198	C₂²⋊Q₈	—	34	1	no	—
199	C₂²⋊Q₈	—	34	1	no	—
200	C₂²⋊Q₈	—	34	1	no	—
201	C₄×Q₈	—	17	1	no	—
202	C₄×Q₈	—	17	1	no	—
203	C₄×D₄	—	17	1	no	—
204	C₄×D₄	—	17	1	no	—
205	C₄×D₄	—	17	1	no	—
206	C₄×D₄	—	17	1	no	—
207	C₄×D₄	—	34	1	no	—
208	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
209	C₂×C₄⋊C₄	—	34	1	no	—
210	#107	—	17	1	no	—
211	#107	C₂×D₁₇	1	1	yes	characteristic
212	#106	C₂×D₁₇	1	1	yes	characteristic
213	C₄⋊C₈	—	17	1	no	—
214	C₄⋊C₈	—	17	1	no	—
215	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
216	#103	—	17	1	no	—
217	#103	—	17	1	no	—
218	#103	—	17	1	no	—
219	#102	—	17	1	no	—
220	#102	—	17	1	no	—
221	#102	—	17	1	no	—
222	#102	—	17	1	no	—
223	#102	—	17	1	no	—
224	#102	C₂×D₁₇	1	1	yes	characteristic
225	#102	C₂×D₁₇	1	1	yes	characteristic
226	#97	—	17	1	no	—
227	#97	—	17	1	no	—
228	C₄×C₈	—	17	1	no	—
229	C₁₇	#3370	1	1	yes	characteristic, complemented, Sylow
230	C₂×Q₈	—	17	1	no	—
231	C₂×Q₈	—	17	1	no	—
232	C₂×Q₈	—	34	1	no	—
233	C₂×D₄	—	17	1	no	—
234	C₂×D₄	—	17	1	no	—
235	C₂×D₄	—	34	1	no	—
236	C₂×D₄	—	34	1	no	—
237	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	34	1	no	—
240	C₂²×C₄	—	34	1	no	—
241	C₂²×C₄	—	34	1	no	—
242	C₂²×C₄	—	34	1	no	—
243	SD₁₆	—	34	2	no	—
244	D₈	—	34	2	no	—
245	D₈	—	34	2	no	—
246	C₂×C₈	—	17	1	no	—
247	C₂×C₈	—	17	1	no	—
248	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
249	C₂×C₈	C₂²×D₁₇	1	1	yes	characteristic
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	2	1	no	—
252	C₄⋊C₄	—	17	1	no	—
253	C₄⋊C₄	—	17	1	no	—
254	C₄⋊C₄	—	17	1	no	—
255	C₄⋊C₄	—	17	1	no	—
256	C₄⋊C₄	—	34	1	no	—
257	C₄⋊C₄	—	34	1	no	—
258	C₄⋊C₄	—	34	1	no	—
259	C₄⋊C₄	—	34	1	no	—
260	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
261	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
262	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
263	C₂²⋊C₄	—	2	1	no	—
264	C₂²⋊C₄	—	34	1	no	—
265	C₂²⋊C₄	—	34	1	no	—
266	C₂²⋊C₄	—	34	1	no	—
267	C₂²⋊C₄	—	34	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	C₄²	—	17	1	no	—
273	C₄²	—	17	1	no	—
274	C₄²	—	34	1	no	—
275	C₄²	C₂²×D₁₇	1	1	yes	characteristic
276	C₂³	—	2	1	no	—
277	C₂³	—	34	1	no	—
278	C₂³	—	34	1	no	—
279	Q₈	—	34	1	no	—
280	Q₈	—	34	1	no	—
281	Q₈	—	68	1	no	—
282	Q₈	—	136	1	no	—
283	D₄	—	2	1	no	—
284	D₄	—	2	1	no	—
285	D₄	—	17	2	no	—
286	D₄	—	34	1	no	—
287	D₄	—	34	1	no	—
288	D₄	—	34	1	no	—
289	D₄	—	68	2	no	—
290	D₄	—	136	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	8	1	no	—
295	C₂×C₄	—	17	1	no	—
296	C₂×C₄	—	17	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	—	68	2	no	—
302	C₂×C₄	—	136	1	no	—
303	C₂×C₄	—	136	1	no	—
304	C₂×C₄	—	136	1	no	—
305	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
306	C₂×C₄	D₄×D₁₇	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	34	1	no	—
311	C₈	—	34	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	—	34	2	no	—
316	C₂²	—	68	1	no	—
317	C₂²	—	68	1	no	—
318	C₂²	—	68	1	no	—
319	C₂²	—	68	1	no	—
320	C₂²	—	68	1	no	—
321	C₂²	C₂×D₄×D₁₇	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	34	1	no	—
328	C₄	—	34	1	no	—
329	C₄	—	34	1	no	—
330	C₄	—	34	1	no	—
331	C₄	—	68	1	no	—
332	C₄	—	68	1	no	—
333	C₄	#10587694	1	1	yes	characteristic
334	C₄	#41×D₁₇	1	1	yes	characteristic
335	C₂	—	8	1	no	—
336	C₂	—	68	2	no	—
337	C₂	—	136	1	no	—
338	C₂	#11763469	1	1	yes	characteristic
339	C₂	#11763606	1	1	yes	characteristic
340	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×#478	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#11763003	C₂
hypercenter	#478	D₁₇
hypocenter	C₁₇	#3370
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
1. #11763542	C₂	semidirect
2. #11763515	C₂	semidirect
3. #11763172	C₂	semidirect
4. #11763167	C₂	semidirect
5. #11763166	C₂	semidirect
6. #11763119	C₂	semidirect
7. #11763088	C₂	semidirect
8. #11763025	C₂	semidirect
9. #11763023	C₂	semidirect
11. #11762936	C₂	semidirect
12. #11762920	C₂	semidirect
13. #11762908	C₂	semidirect
14. C₄×#63228	C₂	semidirect
15. C₁₇×#478	C₂	semidirect
171. #478	D₁₇	semidirect
229. C₁₇	#3370	semidirect

Maximal subgroups

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

subgroup	index
#11763542	2
#11763515	2
#11763172	2
#11763167	2
#11763166	2
#11763119	2
#11763088	2
#11763025	2
#11763023	2
#11763003	2
#11762936	2
#11762920	2
#11762908	2
C₄×#63228	2
C₁₇×#478	2
#3370	17