Subgroups — #16858457

Order 3968.

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	#14521785	C₂	1	1	yes	maximal, characteristic, complemented
2	#14521758	C₂	1	1	yes	maximal, characteristic, complemented
3	#14521413	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521407	C₂	1	1	yes	maximal, characteristic, complemented
5	#14521402	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521357	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521351	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521327	C₂	1	1	yes	maximal, characteristic, complemented
9	#14521262	C₂	1	1	yes	maximal, characteristic, complemented
10	#14521262	C₂	1	1	yes	maximal, characteristic, complemented
11	#14521178	C₂	1	1	yes	maximal, characteristic, complemented
12	#14521161	C₂	1	1	yes	maximal, characteristic, complemented
13	#14521149	C₂	1	1	yes	maximal, characteristic, complemented
14	D₈×Dic₃₁	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#459	C₂	1	1	yes	maximal, characteristic
16	C₂×D₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×D₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#127	C₂²	1	1	yes	characteristic
19	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
20	#11758513	—	2	1	no	—
21	#11758513	C₂²	1	1	yes	characteristic
22	#11758513	C₂²	1	1	yes	characteristic
23	#11758512	—	2	1	no	—
24	#11758512	—	2	1	no	—
25	D₄×Dic₃₁	—	2	1	no	—
26	D₄×Dic₃₁	C₂²	1	1	yes	characteristic
27	D₄×Dic₃₁	C₂²	1	1	yes	characteristic
28	D₄×Dic₃₁	C₂²	1	1	yes	characteristic
29	D₄×Dic₃₁	C₂²	1	1	yes	characteristic
30	C₂×#92480	C₂²	1	1	yes	characteristic
31	#11758501	—	2	1	no	—
32	C₂×Dic₃₁⋊C₄	—	2	1	no	—
33	#11758467	C₂²	1	1	yes	characteristic
34	#11758465	C₂²	1	1	yes	characteristic
35	#11758455	—	2	1	no	—
36	#11758455	—	2	1	no	—
37	#11758447	C₂²	1	1	yes	characteristic
38	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
39	C₄×Dic₆₂	C₂²	1	1	yes	characteristic
40	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
41	C₃₁×#102	C₂²	1	1	yes	characteristic
42	C₃₁×#102	C₂²	1	1	yes	characteristic
43	#11758420	C₂²	1	1	yes	characteristic
44	#11758420	C₂²	1	1	yes	characteristic
45	#11758420	C₂²	1	1	yes	characteristic
46	#11758420	C₂²	1	1	yes	characteristic
47	#11758420	C₂²	1	1	yes	characteristic
48	#11758406	C₂²	1	1	yes	characteristic
49	#11758405	C₂²	1	1	yes	characteristic
50	#11758404	C₂²	1	1	yes	characteristic
51	#11758404	C₂²	1	1	yes	characteristic
52	#11758403	C₂²	1	1	yes	characteristic
53	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
54	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
55	C₈×Dic₃₁	C₂²	1	1	yes	characteristic
56	#11758398	C₂²	1	1	yes	characteristic
57	#11758395	C₂²	1	1	yes	characteristic
58	#11758391	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₃₁	—	2	1	no	—
60	C₂×D₄×C₃₁	—	2	1	no	—
61	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	—	2	1	no	—
65	C₂²×Dic₃₁	—	2	1	no	—
66	C₂²×Dic₃₁	—	2	1	no	—
67	C₂²×Dic₃₁	—	2	1	no	—
68	C₂²×Dic₃₁	—	2	1	no	—
69	C₂×Dic₆₂	—	2	1	no	—
70	C₂×Dic₆₂	D₄	1	1	yes	characteristic
71	C₂×Dic₆₂	C₂³	1	1	yes	characteristic
72	D₈×C₃₁	—	2	2	no	—
73	D₈×C₃₁	—	2	2	no	—
74	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
77	C₂²⋊C₄×C₃₁	—	2	1	no	—
78	C₄²×C₃₁	C₂³	1	1	yes	characteristic
79	C₂²⋊Dic₃₁	—	2	1	no	—
80	C₂²⋊Dic₃₁	—	2	1	no	—
81	C₂²⋊Dic₃₁	—	2	1	no	—
82	C₂²⋊Dic₃₁	—	2	1	no	—
83	C₂²⋊Dic₃₁	—	2	1	no	—
84	C₂²⋊Dic₃₁	—	2	1	no	—
85	C₂²⋊Dic₃₁	—	2	1	no	—
86	C₂²⋊Dic₃₁	—	2	1	no	—
87	#92480	—	2	2	no	—
88	C₄⋊Dic₃₁	—	2	1	no	—
89	C₄⋊Dic₃₁	D₄	1	1	yes	characteristic
90	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
91	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
92	C₄⋊Dic₃₁	C₂³	1	1	yes	characteristic
93	Dic₃₁⋊C₄	—	2	1	no	—
94	Dic₃₁⋊C₄	—	2	1	no	—
95	Dic₃₁⋊C₄	—	2	1	no	—
96	Dic₃₁⋊C₄	—	2	1	no	—
97	Dic₃₁⋊C₄	—	2	1	no	—
98	Dic₃₁⋊C₄	D₄	1	1	yes	characteristic
99	Dic₃₁⋊C₄	D₄	1	1	yes	characteristic
100	C₄×Dic₃₁	—	2	1	no	—
101	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
102	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
103	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
104	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
105	C₂³×C₃₁	—	2	1	no	—
106	C₂³×C₃₁	—	2	1	no	—
107	C₂³×C₃₁	—	2	1	no	—
108	D₄×C₃₁	—	2	1	no	—
109	D₄×C₃₁	—	2	1	no	—
110	D₄×C₃₁	—	2	1	no	—
111	D₄×C₃₁	—	2	1	no	—
112	D₄×C₃₁	—	2	1	no	—
113	D₄×C₃₁	—	4	2	no	—
114	D₄×C₃₁	—	8	1	no	—
115	D₄×C₃₁	#41	1	2	yes	—
116	C₂×C₄×C₃₁	—	2	1	no	—
117	C₂×C₄×C₃₁	—	8	1	no	—
118	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
121	C₂×Dic₃₁	—	2	1	no	—
122	C₂×Dic₃₁	—	2	1	no	—
123	C₂×Dic₃₁	—	2	1	no	—
124	C₂×Dic₃₁	—	2	1	no	—
125	C₂×Dic₃₁	—	4	2	no	—
126	C₂×Dic₃₁	—	8	1	no	—
127	C₂×Dic₃₁	—	8	1	no	—
128	C₂×Dic₃₁	—	8	1	no	—
129	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
133	Dic₆₂	—	2	1	no	—
134	Dic₆₂	—	2	1	no	—
135	Dic₆₂	—	4	1	no	—
136	Dic₆₂	—	8	1	no	—
137	C₈×C₃₁	—	2	1	no	—
138	C₈×C₃₁	—	2	1	no	—
139	C₃₁⋊C₈	—	2	1	no	—
140	C₃₁⋊C₈	—	2	1	no	—
141	#3370	—	31	1	no	maximal, Sylow
142	C₂²×C₃₁	—	2	2	no	—
143	C₂²×C₃₁	—	4	1	no	—
144	C₂²×C₃₁	—	4	1	no	—
145	C₂²×C₃₁	—	4	1	no	—
146	C₂²×C₃₁	—	4	1	no	—
147	C₂²×C₃₁	—	4	1	no	—
148	C₂²×C₃₁	—	4	1	no	—
149	C₂²×C₃₁	—	4	1	no	—
150	C₂²×C₃₁	—	4	1	no	—
151	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
152	C₄×C₃₁	—	2	1	no	—
153	C₄×C₃₁	—	2	1	no	—
154	C₄×C₃₁	—	4	1	no	—
155	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
156	C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
157	Dic₃₁	—	2	1	no	—
158	Dic₃₁	—	2	1	no	—
159	Dic₃₁	—	2	1	no	—
160	Dic₃₁	—	2	1	no	—
161	Dic₃₁	—	4	1	no	—
162	Dic₃₁	—	4	1	no	—
163	Dic₃₁	—	4	1	no	—
164	Dic₃₁	—	4	1	no	—
165	#559	—	31	1	no	—
166	#554	—	31	1	no	—
167	#490	—	31	1	no	—
168	#489	—	31	1	no	—
169	#488	—	31	1	no	—
170	#479	—	31	1	no	—
171	#478	—	31	1	no	—
172	#474	—	31	1	no	—
173	#463	—	31	1	no	—
174	#463	—	31	1	no	—
175	#459	D₃₁	1	1	yes	characteristic
176	#446	—	31	1	no	—
177	#442	—	31	1	no	—
178	#440	—	31	1	no	—
179	C₄×D₈	—	31	1	no	—
180	C₂×C₃₁	—	4	2	no	—
181	C₂×C₃₁	—	8	1	no	—
182	C₂×C₃₁	—	8	1	no	—
183	C₂×C₃₁	#548	1	1	yes	characteristic
184	C₂×C₃₁	#575	1	1	yes	characteristic
185	C₂×C₃₁	#576	1	1	yes	characteristic
186	C₂×SD₁₆	—	31	1	no	—
187	C₂×D₈	C₂×D₃₁	1	1	yes	characteristic
188	C₂×D₈	C₂×D₃₁	1	1	yes	characteristic
189	C₄⋊Q₈	—	31	1	no	—
190	#127	C₂×D₃₁	1	1	yes	characteristic
191	#125	—	31	1	no	—
192	#125	—	31	1	no	—
193	#124	—	31	1	no	—
194	#124	—	31	1	no	—
195	#124	—	62	1	no	—
196	#123	—	62	1	no	—
197	#123	—	62	1	no	—
198	C₂²⋊Q₈	—	62	1	no	—
199	C₂²⋊Q₈	—	62	1	no	—
200	C₂²⋊Q₈	—	62	1	no	—
201	C₄×Q₈	—	31	1	no	—
202	C₄×Q₈	—	31	1	no	—
203	C₄×D₄	—	31	1	no	—
204	C₄×D₄	—	31	1	no	—
205	C₄×D₄	—	31	1	no	—
206	C₄×D₄	—	31	1	no	—
207	C₄×D₄	—	62	1	no	—
208	C₄×D₄	C₂×D₃₁	1	1	yes	characteristic
209	C₂×C₄⋊C₄	—	62	1	no	—
210	#107	—	31	1	no	—
211	#107	—	31	1	no	—
212	#106	—	31	1	no	—
213	C₄⋊C₈	—	31	1	no	—
214	C₄⋊C₈	—	31	1	no	—
215	C₄⋊C₈	C₂×D₃₁	1	1	yes	characteristic
216	#103	—	31	1	no	—
217	#103	—	31	1	no	—
218	#103	—	31	1	no	—
219	#102	—	31	1	no	—
220	#102	—	31	1	no	—
221	#102	—	31	1	no	—
222	#102	—	31	1	no	—
223	#102	—	31	1	no	—
224	#102	C₂×D₃₁	1	1	yes	characteristic
225	#102	C₂×D₃₁	1	1	yes	characteristic
226	#97	—	31	1	no	—
227	#97	—	31	1	no	—
228	C₄×C₈	—	31	1	no	—
229	C₃₁	#3370	1	1	yes	characteristic, complemented, Sylow
230	C₂×Q₈	—	31	1	no	—
231	C₂×Q₈	—	31	1	no	—
232	C₂×Q₈	—	62	1	no	—
233	C₂×D₄	—	2	1	no	—
234	C₂×D₄	—	2	1	no	—
235	C₂×D₄	C₂²×D₃₁	1	1	yes	characteristic
236	C₂×D₄	C₂²×D₃₁	1	1	yes	characteristic
237	C₂×D₄	C₂²×D₃₁	1	1	yes	characteristic
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	62	1	no	—
240	C₂²×C₄	—	62	1	no	—
241	C₂²×C₄	—	62	1	no	—
242	C₂²×C₄	—	62	1	no	—
243	SD₁₆	—	62	2	no	—
244	D₈	—	2	2	no	—
245	D₈	—	2	2	no	—
246	C₂×C₈	—	31	1	no	—
247	C₂×C₈	—	31	1	no	—
248	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
249	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
250	C₄⋊C₄	—	31	1	no	—
251	C₄⋊C₄	—	31	1	no	—
252	C₄⋊C₄	—	31	1	no	—
253	C₄⋊C₄	—	31	1	no	—
254	C₄⋊C₄	—	31	1	no	—
255	C₄⋊C₄	—	31	1	no	—
256	C₄⋊C₄	—	62	1	no	—
257	C₄⋊C₄	—	62	1	no	—
258	C₄⋊C₄	—	62	1	no	—
259	C₄⋊C₄	—	62	1	no	—
260	C₄⋊C₄	—	62	1	no	—
261	C₄⋊C₄	—	62	1	no	—
262	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
263	C₂²⋊C₄	—	2	1	no	—
264	C₂²⋊C₄	—	62	1	no	—
265	C₂²⋊C₄	—	62	1	no	—
266	C₂²⋊C₄	—	62	1	no	—
267	C₂²⋊C₄	—	62	1	no	—
268	C₂²⋊C₄	—	62	1	no	—
269	C₂²⋊C₄	—	62	1	no	—
270	C₂²⋊C₄	—	62	1	no	—
271	C₂²⋊C₄	—	62	1	no	—
272	C₄²	—	31	1	no	—
273	C₄²	—	31	1	no	—
274	C₄²	—	62	1	no	—
275	C₄²	C₂²×D₃₁	1	1	yes	characteristic
276	C₂³	—	2	1	no	—
277	C₂³	—	2	1	no	—
278	C₂³	—	2	1	no	—
279	Q₈	—	62	1	no	—
280	Q₈	—	62	1	no	—
281	Q₈	—	124	1	no	—
282	Q₈	—	248	1	no	—
283	D₄	—	2	1	no	—
284	D₄	—	2	1	no	—
285	D₄	—	2	1	no	—
286	D₄	—	2	1	no	—
287	D₄	—	2	1	no	—
288	D₄	—	4	2	no	—
289	D₄	—	8	1	no	—
290	D₄	#92497	1	2	yes	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	8	1	no	—
293	C₂×C₄	—	31	1	no	—
294	C₂×C₄	—	31	1	no	—
295	C₂×C₄	—	31	1	no	—
296	C₂×C₄	—	31	1	no	—
297	C₂×C₄	—	62	1	no	—
298	C₂×C₄	—	62	1	no	—
299	C₂×C₄	—	62	1	no	—
300	C₂×C₄	—	62	1	no	—
301	C₂×C₄	—	124	2	no	—
302	C₂×C₄	—	248	1	no	—
303	C₂×C₄	—	248	1	no	—
304	C₂×C₄	—	248	1	no	—
305	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
306	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	62	1	no	—
311	C₈	—	62	1	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	62	1	no	—
326	C₄	—	62	1	no	—
327	C₄	—	62	1	no	—
328	C₄	—	62	1	no	—
329	C₄	—	124	1	no	—
330	C₄	—	124	1	no	—
331	C₄	—	124	1	no	—
332	C₄	—	124	1	no	—
333	C₄	C₂×#92497	1	1	yes	characteristic
334	C₄	#11758567	1	1	yes	characteristic
335	C₂	—	4	2	no	—
336	C₂	—	8	1	no	—
337	C₂	—	8	1	no	—
338	C₂	#14521705	1	1	yes	characteristic
339	C₂	#14521845	1	1	yes	characteristic
340	C₂	#14521850	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₁
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting, prime radical	C₃₁×#459	C₂
socle	C₂²×C₃₁	C₂²×D₄
hypercenter	#459	D₃₁
hypocenter	C₃₁	#3370
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #459

Complements

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

N	H ≅ G/N	type
1. #14521785	C₂	semidirect
2. #14521758	C₂	semidirect
3. #14521413	C₂	semidirect
4. #14521407	C₂	semidirect
5. #14521402	C₂	semidirect
6. #14521357	C₂	semidirect
7. #14521351	C₂	semidirect
8. #14521327	C₂	semidirect
9. #14521262	C₂	semidirect
10. #14521262	C₂	semidirect
11. #14521178	C₂	semidirect
12. #14521161	C₂	semidirect
13. #14521149	C₂	semidirect
14. D₈×Dic₃₁	C₂	semidirect
229. C₃₁	#3370	semidirect

Maximal subgroups

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

subgroup	index
#14521785	2
#14521758	2
#14521413	2
#14521407	2
#14521402	2
#14521357	2
#14521351	2
#14521327	2
#14521262	2
#14521178	2
#14521161	2
#14521149	2
D₈×Dic₃₁	2
C₃₁×#459	2
#3370	31