Subgroups — #14541183

Order 2176.

366 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	#11763512	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763509	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763208	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763136	C₂	1	1	yes	maximal, characteristic, complemented
5	#11763131	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763129	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763126	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763110	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763044	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763044	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#63228	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762874	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762844	C₂	1	1	yes	maximal, characteristic, complemented
14	#107×D₁₇	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	#10587624	C₂²	1	1	yes	characteristic
19	#10587624	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×C₈×D₁₇	C₂²	1	1	yes	characteristic
24	C₂×C₈×D₁₇	C₂²	1	1	yes	characteristic
25	#10587584	—	2	1	no	—
26	#10587581	—	2	1	no	—
27	#10587581	C₂²	1	1	yes	characteristic
28	#10587581	C₂²	1	1	yes	characteristic
29	#10587580	—	2	1	no	—
30	#10587579	—	2	1	no	—
31	#10587578	C₂²	1	1	yes	characteristic
32	#10587578	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
34	C₄⋊C₄×D₁₇	C₂²	1	1	yes	characteristic
35	#10587570	—	2	1	no	—
36	#10587565	—	2	1	no	—
37	C₄×D₆₈	C₂²	1	1	yes	characteristic
38	C₄×D₆₈	C₂²	1	1	yes	characteristic
39	#10587556	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	#10587492	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	#10587488	C₂²	1	1	yes	characteristic
52	#10587480	C₂²	1	1	yes	characteristic
53	#10587479	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	C₂²×C₄×C₁₇	—	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	D₄	1	1	yes	characteristic
66	C₂×D₆₈	C₂³	1	1	yes	characteristic
67	C₂×C₄×D₁₇	—	2	1	no	—
68	C₂×C₄×D₁₇	—	2	1	no	—
69	C₂×C₄×D₁₇	—	2	1	no	—
70	C₂×C₄×D₁₇	C₂³	1	1	yes	characteristic
71	C₂×C₄×D₁₇	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	D₄	1	1	yes	characteristic
90	#63227	D₄	1	1	yes	characteristic
91	C₄⋊Dic₁₇	D₄	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	#63225	—	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	C₈×D₁₇	—	2	2	no	—
104	C₈×D₁₇	—	2	2	no	—
105	C₂³×C₁₇	—	2	1	no	—
106	C₂²×D₁₇	—	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	C₂×C₄×C₁₇	—	2	1	no	—
112	C₂×C₄×C₁₇	—	2	1	no	—
113	C₂×C₄×C₁₇	—	2	1	no	—
114	C₂×C₄×C₁₇	—	8	1	no	—
115	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
118	C₂²⋊D₁₇	—	4	2	no	—
119	C₂²⋊D₁₇	—	8	1	no	—
120	C₂×Dic₁₇	—	2	1	no	—
121	C₂×Dic₁₇	—	2	1	no	—
122	C₂×Dic₁₇	—	2	1	no	—
123	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₁₇	C₂×D₄	1	1	yes	characteristic
125	D₆₈	—	2	2	no	—
126	D₆₈	—	2	1	no	—
127	D₆₈	—	2	1	no	—
128	D₆₈	—	4	2	no	—
129	D₆₈	—	8	1	no	—
130	C₄×D₁₇	—	2	1	no	—
131	C₄×D₁₇	—	2	1	no	—
132	C₄×D₁₇	—	2	1	no	—
133	C₄×D₁₇	—	2	1	no	—
134	C₄×D₁₇	—	8	1	no	—
135	C₄×D₁₇	—	8	1	no	—
136	C₄×D₁₇	—	8	1	no	—
137	C₈×C₁₇	—	2	1	no	—
138	C₈×C₁₇	—	2	1	no	—
139	#3622	—	2	1	no	—
140	#3622	#41	1	2	yes	—
141	#3339	—	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	1	no	—
147	C₂×D₁₇	—	2	1	no	—
148	C₂×D₁₇	—	2	1	no	—
149	C₂×D₁₇	—	2	1	no	—
150	C₂×D₁₇	—	2	1	no	—
151	C₂×D₁₇	—	4	1	no	—
152	C₂×D₁₇	—	4	1	no	—
153	C₂×D₁₇	—	4	1	no	—
154	C₂×D₁₇	#135	1	2	yes	—
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	—	4	1	no	—
160	C₄×C₁₇	C₂×#41	1	1	yes	characteristic
161	C₄×C₁₇	2₋¹⁺⁴	1	1	yes	characteristic
162	Dic₁₇	—	2	1	no	—
163	Dic₁₇	—	2	1	no	—
164	Dic₁₇	—	4	1	no	—
165	Dic₁₇	—	4	1	no	—
166	Dic₁₇	—	4	1	no	—
167	#553	—	17	1	no	—
168	#553	—	17	1	no	—
169	#499	—	17	1	no	—
170	#482	—	17	1	no	—
171	#482	—	17	1	no	—
172	#481	—	17	1	no	—
173	#481	—	17	1	no	—
174	#478	—	17	1	no	—
175	#478	D₁₇	1	1	yes	characteristic, complemented
176	#466	—	17	1	no	—
177	#466	—	17	1	no	—
178	C₄×D₈	—	17	1	no	—
179	C₈×D₄	—	17	1	no	—
180	#427	—	17	1	no	—
181	C₂×#107	—	17	1	no	—
182	C₂×C₁₇	—	8	1	no	—
183	C₂×C₁₇	#547	1	1	yes	characteristic
184	C₂×C₁₇	C₂×#135	1	1	yes	characteristic
185	C₂×C₁₇	#576	1	1	yes	characteristic
186	D₁₇	—	2	2	no	—
187	D₁₇	—	4	1	no	—
188	D₁₇	—	8	1	no	—
189	C₂×D₈	—	17	1	no	—
190	C₂²×C₈	—	17	1	no	—
191	C₂²×C₈	—	17	1	no	—
192	#126	—	34	1	no	—
193	#126	—	34	1	no	—
194	#125	—	17	1	no	—
195	#125	C₂×D₁₇	1	1	yes	characteristic
196	#123	—	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	—	17	1	no	—
206	C₄×D₄	—	34	1	no	—
207	C₄×D₄	—	34	1	no	—
208	C₄×D₄	C₂×D₁₇	1	1	yes	characteristic
209	#117	—	17	1	no	—
210	#117	—	17	1	no	—
211	C₂×C₄⋊C₄	—	17	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	C₂×D₁₇	1	1	yes	characteristic
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	C₂×D₁₇	1	1	yes	characteristic
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	C₄×C₈	—	17	1	no	—
232	C₁₇	#3339	1	1	yes	characteristic, complemented, Sylow
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	—	34	1	no	—
238	C₂×D₄	—	34	1	no	—
239	C₂×D₄	C₂²×D₁₇	1	1	yes	characteristic
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	17	1	no	—
242	C₂²×C₄	—	17	1	no	—
243	C₂²×C₄	—	34	1	no	—
244	C₂²×C₄	—	34	1	no	—
245	C₂²×C₄	—	34	1	no	—
246	D₈	—	34	2	no	—
247	C₂×C₈	—	17	1	no	—
248	C₂×C₈	—	17	1	no	—
249	C₂×C₈	—	34	2	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	—	2	1	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	17	1	no	—
256	C₄⋊C₄	—	17	1	no	—
257	C₄⋊C₄	—	17	1	no	—
258	C₄⋊C₄	—	17	1	no	—
259	C₄⋊C₄	—	34	1	no	—
260	C₄⋊C₄	—	34	1	no	—
261	C₄⋊C₄	—	34	1	no	—
262	C₄⋊C₄	—	34	1	no	—
263	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
264	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
265	C₄⋊C₄	C₂²×D₁₇	1	1	yes	characteristic
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	17	1	no	—
268	C₂²⋊C₄	—	17	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	C₂²⋊C₄	—	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	—	17	1	no	—
281	C₂³	—	17	1	no	—
282	C₂³	—	34	1	no	—
283	D₄	—	2	1	no	—
284	D₄	—	2	1	no	—
285	D₄	—	34	2	no	—
286	D₄	—	34	1	no	—
287	D₄	—	34	1	no	—
288	D₄	—	68	2	no	—
289	D₄	—	68	2	no	—
290	D₄	—	136	1	no	—
291	D₄	—	136	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	8	1	no	—
296	C₂×C₄	—	17	1	no	—
297	C₂×C₄	—	17	1	no	—
298	C₂×C₄	—	34	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	—	136	1	no	—
306	C₂×C₄	—	136	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	—	2	1	no	—
313	C₈	—	17	2	no	—
314	C₈	—	34	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	17	2	no	—
319	C₂²	—	34	1	no	—
320	C₂²	—	34	1	no	—
321	C₂²	—	34	1	no	—
322	C₂²	—	34	1	no	—
323	C₂²	—	34	1	no	—
324	C₂²	—	68	1	no	—
325	C₂²	—	68	1	no	—
326	C₂²	—	68	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	—	4	1	no	—
333	C₄	—	34	1	no	—
334	C₄	—	34	1	no	—
335	C₄	—	68	1	no	—
336	C₄	—	68	1	no	—
337	C₄	—	68	1	no	—
338	C₄	#10587694	1	1	yes	characteristic
339	C₄	#41×D₁₇	1	1	yes	characteristic
340	C₂	—	8	1	no	—
341	C₂	—	34	2	no	—
342	C₂	—	68	1	no	—
343	C₂	—	136	1	no	—
344	C₂	#11763461	1	1	yes	characteristic
345	C₂	#135×D₁₇	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×#478	C₂
socle	C₂²×C₁₇	C₂²×D₄
hypercenter	#478	D₁₇
hypocenter	C₁₇	#3339
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. #11763512	C₂	semidirect
2. #11763509	C₂	semidirect
3. #11763208	C₂	semidirect
4. #11763136	C₂	semidirect
5. #11763131	C₂	semidirect
6. #11763129	C₂	semidirect
7. #11763126	C₂	semidirect
8. #11763110	C₂	semidirect
9. #11763044	C₂	semidirect
10. #11763044	C₂	semidirect
11. C₄×#63228	C₂	semidirect
12. #11762874	C₂	semidirect
13. #11762844	C₂	semidirect
14. #107×D₁₇	C₂	semidirect
15. C₁₇×#478	C₂	semidirect
175. #478	D₁₇	semidirect
232. C₁₇	#3339	semidirect

Maximal subgroups

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

subgroup	index
#11763512	2
#11763509	2
#11763208	2
#11763136	2
#11763131	2
#11763129	2
#11763126	2
#11763110	2
#11763044	2
C₄×#63228	2
#11762874	2
#11762844	2
#107×D₁₇	2
C₁₇×#478	2
#3339	17