Subgroups — #14541852

Order 2176.

368 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11763544	C₂	1	1	yes	maximal, characteristic, complemented
2	#11763532	C₂	1	1	yes	maximal, characteristic, complemented
3	#11763203	C₂	1	1	yes	maximal, characteristic, complemented
4	#11763198	C₂	1	1	yes	maximal, characteristic
5	#11763193	C₂	1	1	yes	maximal, characteristic, complemented
6	#11763032	C₂	1	1	yes	maximal, characteristic, complemented
7	#11763032	C₂	1	1	yes	maximal, characteristic, complemented
8	#11763025	C₂	1	1	yes	maximal, characteristic, complemented
9	#11763025	C₂	1	1	yes	maximal, characteristic, complemented
10	#11763015	C₂	1	1	yes	maximal, characteristic, complemented
11	#11763010	C₂	1	1	yes	maximal, characteristic, complemented
12	#11762922	C₂	1	1	yes	maximal, characteristic, complemented
13	#11762917	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#63219	C₂	1	1	yes	maximal, characteristic, complemented
15	C₁₇×#446	C₂	1	1	yes	maximal, characteristic, complemented
16	C₄×Q₈×C₁₇	C₂²	1	1	yes	characteristic
17	#10587633	—	2	1	no	—
18	#10587632	—	2	1	no	—
19	C₂×Dic₆₈	C₂²	1	1	yes	characteristic
20	C₂×Dic₆₈	C₂²	1	1	yes	characteristic
21	C₂×D₁₃₆	C₂²	1	1	yes	characteristic
22	C₂×D₁₃₆	C₂²	1	1	yes	characteristic
23	C₂×#63219	C₂²	1	1	yes	characteristic
24	C₂×#63219	C₂²	1	1	yes	characteristic
25	C₂×#63219	C₂²	1	1	yes	characteristic
26	C₂×#63219	C₂²	1	1	yes	characteristic
27	C₂×#63219	C₂²	1	1	yes	characteristic
28	#10587584	—	2	1	no	—
29	#10587584	—	2	1	no	—
30	#10587580	—	2	1	no	—
31	#10587580	—	2	1	no	—
32	#10587579	—	2	1	no	—
33	#10587578	—	2	1	no	—
34	#10587561	C₂²	1	1	yes	characteristic
35	#10587561	C₂²	1	1	yes	characteristic
36	#10587561	C₂²	1	1	yes	characteristic
37	#10587561	C₂²	1	1	yes	characteristic
38	#10587560	C₂²	1	1	yes	characteristic
39	C₄×D₆₈	C₂²	1	1	yes	characteristic
40	C₄×D₆₈	C₂²	1	1	yes	characteristic
41	C₄×D₆₈	C₂²	1	1	yes	characteristic
42	#10587555	C₂²	1	1	yes	characteristic
43	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
44	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
45	C₄×Dic₃₄	C₂²	1	1	yes	characteristic
46	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
47	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
48	C₁₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
49	C₁₇×#97	C₂²	1	1	yes	characteristic
50	C₁₇×#97	C₂²	1	1	yes	characteristic
51	C₄×C₈×C₁₇	C₂²	1	1	yes	characteristic
52	#10587492	C₂²	1	1	yes	characteristic
53	#10587492	C₂²	1	1	yes	characteristic
54	#10587492	C₂²	1	1	yes	characteristic
55	#10587488	C₂²	1	1	yes	characteristic
56	#10587487	C₂²	1	1	yes	characteristic
57	#10587487	C₂²	1	1	yes	characteristic
58	#10587487	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₁₇	D₄	1	1	yes	characteristic
60	C₂×D₆₈	—	2	1	no	—
61	C₂×D₆₈	—	2	1	no	—
62	C₂×D₆₈	C₂³	1	1	yes	characteristic
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×Dic₃₄	C₂³	1	1	yes	characteristic
69	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
70	C₂×Dic₃₄	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₁₇	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
76	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
77	C₄⋊C₄×C₁₇	D₄	1	1	yes	characteristic
78	C₄²×C₁₇	C₂³	1	1	yes	characteristic
79	C₄²×C₁₇	C₂³	1	1	yes	characteristic
80	C₄²×C₁₇	C₂³	1	1	yes	characteristic
81	#63227	—	2	1	no	—
82	#63227	—	2	1	no	—
83	#63227	—	2	1	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	#63227	—	2	1	no	—
91	#63227	—	2	1	no	—
92	C₄⋊Dic₁₇	—	2	1	no	—
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	C₄×Dic₁₇	—	2	1	no	—
100	C₄×Dic₁₇	—	2	1	no	—
101	C₄×Dic₁₇	—	2	1	no	—
102	Dic₆₈	—	2	2	no	—
103	Dic₆₈	—	4	1	no	—
104	D₁₃₆	—	2	2	no	—
105	D₁₃₆	—	4	1	no	—
106	#63219	—	2	2	no	—
107	#63219	—	4	1	no	—
108	#63219	—	4	1	no	—
109	#63219	—	4	1	no	—
110	#63219	—	4	1	no	—
111	C₂²×D₁₇	—	2	1	no	—
112	C₂²×D₁₇	—	2	1	no	—
113	C₂²×D₁₇	—	2	1	no	—
114	Q₈×C₁₇	—	4	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	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₇	C₂⁴	1	1	yes	characteristic
122	C₂×Dic₁₇	—	2	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	D₆₈	—	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	—	4	2	no	—
134	D₆₈	—	8	1	no	—
135	C₄×D₁₇	—	4	2	no	—
136	C₄×D₁₇	—	8	1	no	—
137	C₄×D₁₇	—	8	1	no	—
138	Dic₃₄	—	2	1	no	—
139	Dic₃₄	—	2	1	no	—
140	Dic₃₄	—	2	1	no	—
141	Dic₃₄	—	2	1	no	—
142	Dic₃₄	—	2	1	no	—
143	Dic₃₄	—	2	1	no	—
144	C₈×C₁₇	—	2	1	no	—
145	C₈×C₁₇	—	2	1	no	—
146	C₈×C₁₇	—	2	1	no	—
147	C₈×C₁₇	#41	1	2	yes	—
148	#3401	—	17	1	no	maximal, Sylow
149	C₂²×C₁₇	C₂²×D₄	1	1	yes	characteristic
150	C₂×D₁₇	—	4	1	no	—
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	C₄×C₁₇	—	2	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	C₂×#41	1	1	yes	characteristic
166	C₄×C₁₇	H(5,𝔽₂)	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	#559	—	17	1	no	—
173	#556	—	17	1	no	—
174	#497	—	17	1	no	—
175	#496	—	17	1	no	—
176	#495	—	17	1	no	—
177	#464	—	17	1	no	—
178	#464	—	17	1	no	—
179	#463	—	17	1	no	—
180	#463	—	17	1	no	—
181	#461	—	17	1	no	—
182	#460	—	17	1	no	—
183	#446	D₁₇	1	1	yes	characteristic, complemented
184	#442	—	17	1	no	—
185	#441	—	17	1	no	—
186	C₄×SD₁₆	—	17	1	no	—
187	C₂×C₁₇	#550	1	1	yes	characteristic
188	C₂×C₁₇	#575	1	1	yes	characteristic
189	C₂×C₁₇	#577	1	1	yes	characteristic
190	D₁₇	—	8	1	no	—
191	D₁₇	—	8	1	no	—
192	D₁₇	—	8	1	no	—
193	C₂×Q₁₆	—	17	1	no	—
194	C₂×Q₁₆	—	17	1	no	—
195	C₂×SD₁₆	—	17	1	no	—
196	C₂×SD₁₆	—	17	1	no	—
197	C₂×SD₁₆	—	17	1	no	—
198	C₂×SD₁₆	—	17	1	no	—
199	C₂×SD₁₆	—	17	1	no	—
200	C₂×D₈	—	17	1	no	—
201	C₂×D₈	—	17	1	no	—
202	C₄⋊Q₈	—	17	1	no	—
203	#127	—	17	1	no	—
204	#126	—	34	1	no	—
205	#126	—	34	1	no	—
206	#124	—	17	1	no	—
207	#124	—	17	1	no	—
208	#124	—	17	1	no	—
209	#124	—	17	1	no	—
210	#124	—	34	1	no	—
211	#123	—	34	1	no	—
212	#123	—	34	1	no	—
213	C₂²⋊Q₈	—	34	1	no	—
214	C₄×Q₈	—	17	1	no	—
215	C₄×Q₈	—	17	1	no	—
216	C₄×Q₈	—	17	1	no	—
217	C₄×Q₈	C₂×D₁₇	1	1	yes	characteristic
218	C₄×D₄	—	17	1	no	—
219	C₄×D₄	—	17	1	no	—
220	C₄×D₄	—	17	1	no	—
221	C₄×D₄	—	34	1	no	—
222	#117	—	34	1	no	—
223	#106	—	17	1	no	—
224	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
225	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
226	C₄⋊C₈	C₂×D₁₇	1	1	yes	characteristic
227	#103	—	17	1	no	—
228	#103	—	17	1	no	—
229	#103	—	17	1	no	—
230	#102	—	17	1	no	—
231	#102	—	17	1	no	—
232	#102	—	17	1	no	—
233	#97	C₂×D₁₇	1	1	yes	characteristic
234	#97	C₂×D₁₇	1	1	yes	characteristic
235	C₄×C₈	C₂×D₁₇	1	1	yes	characteristic
236	C₁₇	#3401	1	1	yes	characteristic, complemented, Sylow
237	C₂×Q₈	—	17	1	no	—
238	C₂×Q₈	—	17	1	no	—
239	C₂×Q₈	—	17	1	no	—
240	C₂×Q₈	D₆₈	1	1	yes	characteristic
241	C₂×D₄	—	17	1	no	—
242	C₂×D₄	—	17	1	no	—
243	C₂×D₄	—	17	1	no	—
244	C₂×D₄	—	34	1	no	—
245	C₂×D₄	—	34	1	no	—
246	C₂²×C₄	—	34	1	no	—
247	C₂²×C₄	—	34	1	no	—
248	C₂²×C₄	—	34	1	no	—
249	Q₁₆	—	34	2	no	—
250	Q₁₆	—	68	1	no	—
251	SD₁₆	—	34	2	no	—
252	SD₁₆	—	68	1	no	—
253	SD₁₆	—	68	1	no	—
254	SD₁₆	—	68	1	no	—
255	SD₁₆	—	68	1	no	—
256	D₈	—	34	2	no	—
257	D₈	—	68	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	—	34	1	no	—
265	C₄⋊C₄	—	34	1	no	—
266	C₄⋊C₄	—	34	1	no	—
267	C₄⋊C₄	—	34	1	no	—
268	C₄⋊C₄	—	34	1	no	—
269	C₄⋊C₄	D₆₈	1	1	yes	characteristic
270	C₄⋊C₄	D₆₈	1	1	yes	characteristic
271	C₄⋊C₄	D₆₈	1	1	yes	characteristic
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	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	—	34	1	no	—
283	C₄²	—	34	1	no	—
284	C₄²	—	34	1	no	—
285	C₄²	—	34	1	no	—
286	C₄²	C₂²×D₁₇	1	1	yes	characteristic
287	C₄²	C₂²×D₁₇	1	1	yes	characteristic
288	C₄²	C₂²×D₁₇	1	1	yes	characteristic
289	C₂³	—	34	1	no	—
290	C₂³	—	34	1	no	—
291	C₂³	—	34	1	no	—
292	Q₈	—	4	1	no	—
293	Q₈	—	34	1	no	—
294	Q₈	—	34	1	no	—
295	Q₈	—	34	1	no	—
296	Q₈	—	34	1	no	—
297	Q₈	—	34	1	no	—
298	Q₈	—	34	1	no	—
299	D₄	—	34	1	no	—
300	D₄	—	34	1	no	—
301	D₄	—	34	1	no	—
302	D₄	—	34	1	no	—
303	D₄	—	34	1	no	—
304	D₄	—	34	1	no	—
305	D₄	—	68	2	no	—
306	D₄	—	136	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	—	68	2	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	C₂×D₆₈	1	1	yes	characteristic
321	C₂×C₄	C₂³×D₁₇	1	1	yes	characteristic
322	C₈	—	2	1	no	—
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	#63256	1	2	yes	—
326	C₂²	—	68	1	no	—
327	C₂²	—	68	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	C₂²×D₆₈	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	68	1	no	—
343	C₄	—	68	1	no	—
344	C₄	—	68	1	no	—
345	C₄	—	68	1	no	—
346	C₄	—	68	1	no	—
347	C₄	C₂×#63256	1	1	yes	characteristic
348	C₄	#10587696	1	1	yes	characteristic
349	C₂	—	136	1	no	—
350	C₂	—	136	1	no	—
351	C₂	—	136	1	no	—
352	C₂	#11763481	1	1	yes	characteristic
353	C₂	#11763609	1	1	yes	characteristic
354	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×#446	C₂
socle	C₂²×C₁₇	C₂²×D₄
prime radical	#11763198	C₂
hypercenter	#446	D₁₇
hypocenter	C₁₇	#3401
LCS	G ▷ C₂×C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

Complements

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

N	H ≅ G/N	type
1. #11763544	C₂	semidirect
2. #11763532	C₂	semidirect
3. #11763203	C₂	semidirect
5. #11763193	C₂	semidirect
6. #11763032	C₂	semidirect
7. #11763032	C₂	semidirect
8. #11763025	C₂	semidirect
9. #11763025	C₂	semidirect
10. #11763015	C₂	semidirect
11. #11763010	C₂	semidirect
12. #11762922	C₂	semidirect
13. #11762917	C₂	semidirect
14. C₄×#63219	C₂	semidirect
15. C₁₇×#446	C₂	semidirect
183. #446	D₁₇	semidirect
236. C₁₇	#3401	semidirect

Maximal subgroups

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

subgroup	index
#11763544	2
#11763532	2
#11763203	2
#11763198	2
#11763193	2
#11763032	2
#11763025	2
#11763015	2
#11763010	2
#11762922	2
#11762917	2
C₄×#63219	2
C₁₇×#446	2
#3401	17