Subgroups — #16858293

Order 3968.

374 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	#14521767	C₂	1	1	yes	maximal, characteristic, complemented
2	#14521699	C₂	1	1	yes	maximal, characteristic
3	#14521453	C₂	1	1	yes	maximal, characteristic, complemented
4	#14521386	C₂	1	1	yes	maximal, characteristic, complemented
5	#14521385	C₂	1	1	yes	maximal, characteristic, complemented
6	#14521371	C₂	1	1	yes	maximal, characteristic, complemented
7	#14521333	C₂	1	1	yes	maximal, characteristic, complemented
8	#14521327	C₂	1	1	yes	maximal, characteristic, complemented
9	#14521294	C₂	1	1	yes	maximal, characteristic, complemented
10	#14521293	C₂	1	1	yes	maximal, characteristic, complemented
11	#14521140	C₂	1	1	yes	maximal, characteristic, complemented
12	#14521124	C₂	1	1	yes	maximal, characteristic, complemented
13	#14521092	C₂	1	1	yes	maximal, characteristic, complemented
14	#14521088	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#482	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#92497	—	2	1	no	—
17	C₃₁×#121	C₂²	1	1	yes	characteristic
18	C₃₁×#117	C₂²	1	1	yes	characteristic
19	#11758515	C₂²	1	1	yes	characteristic
20	#11758512	—	2	1	no	—
21	D₄×Dic₃₁	C₂²	1	1	yes	characteristic
22	C₂×#92480	C₂²	1	1	yes	characteristic
23	C₄×C₂²⋊D₃₁	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
25	#11758501	C₂²	1	1	yes	characteristic
26	C₂×#92474	C₂²	1	1	yes	characteristic
27	C₂×#92469	C₂²	1	1	yes	characteristic
28	#11758474	C₂²	1	1	yes	characteristic
29	#11758469	C₂²	1	1	yes	characteristic
30	C₄⋊C₄×D₃₁	C₂²	1	1	yes	characteristic
31	#11758465	C₂²	1	1	yes	characteristic
32	#11758465	C₂²	1	1	yes	characteristic
33	#11758464	C₂²	1	1	yes	characteristic
34	#11758462	—	2	1	no	—
35	#11758460	—	2	1	no	—
36	#11758456	—	2	1	no	—
37	#11758455	—	2	1	no	—
38	#11758453	—	2	1	no	—
39	C₄×Dic₆₂	—	2	1	no	—
40	C₃₁×#107	C₂²	1	1	yes	characteristic
41	C₃₁×#106	C₂²	1	1	yes	characteristic
42	C₃₁×#102	C₂²	1	1	yes	characteristic
43	C₃₁×#102	C₂²	1	1	yes	characteristic
44	C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
45	#11758420	C₂²	1	1	yes	characteristic
46	(C₂×D₃₁)⋊C₈	C₂²	1	1	yes	characteristic
47	#11758406	C₂²	1	1	yes	characteristic
48	#11758406	C₂²	1	1	yes	characteristic
49	#11758404	C₂²	1	1	yes	characteristic
50	#11758404	C₂²	1	1	yes	characteristic
51	Dic₃₁⋊C₈	C₂²	1	1	yes	characteristic
52	C₈×Dic₃₁	C₂²	1	1	yes	characteristic
53	#11758398	C₂²	1	1	yes	characteristic
54	#11758396	C₂²	1	1	yes	characteristic
55	#11758396	C₂²	1	1	yes	characteristic
56	#11758395	C₂²	1	1	yes	characteristic
57	#11758395	C₂²	1	1	yes	characteristic
58	#11758395	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₃₁	—	2	1	no	—
60	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₃₁	—	2	1	no	—
63	C₂×C₂²⋊D₃₁	D₄	1	1	yes	characteristic
64	C₂²×Dic₃₁	—	2	1	no	—
65	C₂²×Dic₃₁	—	2	1	no	—
66	#92497	—	4	4	no	—
67	C₂×C₄×D₃₁	—	2	1	no	—
68	C₂×C₄×D₃₁	C₂³	1	1	yes	characteristic
69	C₂×Dic₆₂	—	2	1	no	—
70	C₂×Dic₆₂	—	2	1	no	—
71	C₂×Dic₆₂	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
77	C₂²⋊C₄×C₃₁	—	2	1	no	—
78	C₂²⋊C₄×C₃₁	—	2	1	no	—
79	C₄²×C₃₁	—	2	1	no	—
80	C₂²⋊Dic₃₁	—	2	1	no	—
81	C₂²⋊Dic₃₁	—	2	1	no	—
82	C₂²⋊Dic₃₁	—	2	1	no	—
83	C₂²⋊Dic₃₁	D₄	1	1	yes	characteristic
84	#92480	—	2	2	no	—
85	(C₂×D₃₁)⋊C₄	—	2	1	no	—
86	(C₂×D₃₁)⋊C₄	—	2	1	no	—
87	(C₂×D₃₁)⋊C₄	D₄	1	1	yes	characteristic
88	C₄⋊Dic₃₁	—	2	1	no	—
89	C₄⋊Dic₃₁	—	2	1	no	—
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	C₄×Dic₃₁	—	2	1	no	—
100	C₄×Dic₃₁	—	2	1	no	—
101	C₄×Dic₃₁	C₂³	1	1	yes	characteristic
102	#92474	—	2	2	no	—
103	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
104	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
105	#92469	—	2	2	no	—
106	C₂³×C₃₁	—	2	1	no	—
107	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
108	C₂²×D₃₁	C₂×D₄	1	1	yes	characteristic
109	D₄×C₃₁	—	2	1	no	—
110	D₄×C₃₁	—	2	1	no	—
111	D₄×C₃₁	—	4	2	no	—
112	C₂×C₄×C₃₁	—	2	1	no	—
113	C₂×C₄×C₃₁	—	2	1	no	—
114	C₂×C₄×C₃₁	—	2	1	no	—
115	C₂×C₄×C₃₁	—	2	1	no	—
116	C₂×C₄×C₃₁	—	2	1	no	—
117	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
119	C₂²⋊D₃₁	—	2	2	no	—
120	C₂²⋊D₃₁	—	4	2	no	—
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	—	4	2	no	—
127	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₃₁	C₂×D₄	1	1	yes	characteristic
130	C₄×D₃₁	—	2	1	no	—
131	C₄×D₃₁	—	2	1	no	—
132	C₄×D₃₁	—	4	2	no	—
133	Dic₆₂	—	2	1	no	—
134	Dic₆₂	—	2	1	no	—
135	Dic₆₂	—	4	2	no	—
136	Dic₆₂	—	4	2	no	—
137	C₈×C₃₁	—	2	1	no	—
138	C₈×C₃₁	—	2	1	no	—
139	C₃₁⋊C₈	—	2	1	no	—
140	C₃₁⋊C₈	#41	1	2	yes	complemented
141	#3353	—	31	1	no	maximal, Sylow
142	C₂²×C₃₁	—	2	1	no	—
143	C₂²×C₃₁	—	2	1	no	—
144	C₂²×C₃₁	—	2	1	no	—
145	C₂²×C₃₁	—	4	1	no	—
146	C₂²×C₃₁	—	4	1	no	—
147	C₂²×C₃₁	—	4	1	no	—
148	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
149	C₂×D₃₁	—	2	1	no	—
150	C₂×D₃₁	—	2	1	no	—
151	C₂×D₃₁	—	2	1	no	—
152	C₄×C₃₁	—	2	1	no	—
153	C₄×C₃₁	—	4	1	no	—
154	C₄×C₃₁	—	4	1	no	—
155	C₄×C₃₁	—	4	1	no	—
156	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
157	C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
158	Dic₃₁	—	2	1	no	—
159	Dic₃₁	—	2	1	no	—
160	Dic₃₁	—	4	1	no	—
161	Dic₃₁	—	4	1	no	—
162	Dic₃₁	—	4	1	no	—
163	Dic₃₁	—	4	1	no	—
164	Dic₃₁	AGL(1,ℤ/8)	1	2	yes	—
165	#556	—	31	1	no	—
166	#547	—	31	1	no	—
167	#500	—	31	1	no	—
168	#484	—	31	1	no	—
169	#484	—	31	1	no	—
170	#482	—	31	1	no	—
171	#482	D₃₁	1	1	yes	characteristic, complemented
172	#475	—	31	1	no	—
173	#474	—	31	1	no	—
174	#468	—	31	1	no	—
175	#468	—	31	1	no	—
176	C₄×SD₁₆	—	31	1	no	—
177	#436	—	31	1	no	—
178	#428	—	31	1	no	—
179	#428	—	31	1	no	—
180	C₂×C₃₁	—	4	1	no	—
181	C₂×C₃₁	—	8	1	no	—
182	C₂×C₃₁	#547	1	1	yes	characteristic
183	C₂×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
184	C₂×C₃₁	#578	1	1	yes	characteristic
185	D₃₁	—	4	1	no	—
186	C₂×#41	—	62	1	no	—
187	C₂×SD₁₆	—	31	1	no	—
188	C₂×M₁₆	—	31	1	no	—
189	C₂×M₁₆	—	31	1	no	—
190	C₄⋊Q₈	—	31	1	no	—
191	C₄⋊Q₈	—	31	1	no	—
192	#126	—	62	1	no	—
193	#126	—	62	1	no	—
194	#124	—	62	1	no	—
195	#123	—	62	1	no	—
196	#123	—	62	1	no	—
197	C₂²⋊Q₈	—	31	1	no	—
198	C₂²⋊Q₈	—	31	1	no	—
199	C₂²⋊Q₈	—	62	1	no	—
200	#121	—	31	1	no	—
201	#121	C₂×D₃₁	1	1	yes	characteristic
202	C₄×Q₈	—	31	1	no	—
203	C₄×Q₈	—	62	1	no	—
204	C₄×D₄	—	31	1	no	—
205	C₄×D₄	—	31	1	no	—
206	#117	—	31	1	no	—
207	#117	C₂×D₃₁	1	1	yes	characteristic
208	C₂×C₄⋊C₄	—	31	1	no	—
209	C₂×C₄⋊C₄	—	31	1	no	—
210	#107	—	31	1	no	—
211	#107	—	31	1	no	—
212	#107	—	31	1	no	—
213	#107	—	31	1	no	—
214	#107	—	31	1	no	—
215	#107	C₂×D₃₁	1	1	yes	characteristic
216	#106	—	31	1	no	—
217	#106	—	31	1	no	—
218	#106	C₂×D₃₁	1	1	yes	characteristic
219	C₄⋊C₈	—	31	1	no	—
220	#103	—	31	1	no	—
221	#103	—	31	1	no	—
222	#103	—	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	C₂²⋊C₈	—	31	1	no	—
227	C₂²⋊C₈	C₂×D₃₁	1	1	yes	characteristic
228	C₄×C₈	—	31	1	no	—
229	C₃₁	#3353	1	1	yes	characteristic, complemented, Sylow
230	#41	—	124	4	no	—
231	C₂×Q₈	—	31	1	no	—
232	C₂×Q₈	—	62	1	no	—
233	C₂×Q₈	—	62	1	no	—
234	C₂×D₄	—	2	1	no	—
235	C₂×D₄	—	31	1	no	—
236	C₂×D₄	—	62	1	no	—
237	C₂×D₄	C₂²×D₃₁	1	1	yes	characteristic
238	C₂²×C₄	—	31	1	no	—
239	C₂²×C₄	—	62	1	no	—
240	C₂²×C₄	—	62	1	no	—
241	C₂²×C₄	—	62	1	no	—
242	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
243	SD₁₆	—	62	2	no	—
244	M₁₆	—	62	2	no	—
245	M₁₆	—	62	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	—	62	1	no	—
255	C₄⋊C₄	—	62	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	C₂²×D₃₁	1	1	yes	characteristic
262	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
263	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	31	1	no	—
267	C₂²⋊C₄	—	31	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	C₂²⋊C₄	—	62	1	no	—
273	C₄²	—	2	1	no	—
274	C₄²	—	31	1	no	—
275	C₄²	—	62	1	no	—
276	C₄²	—	62	1	no	—
277	C₂³	—	2	1	no	—
278	C₂³	—	31	1	no	—
279	C₂³	D₄×D₃₁	1	1	yes	characteristic
280	Q₈	—	62	1	no	—
281	Q₈	—	62	1	no	—
282	Q₈	—	124	2	no	—
283	Q₈	—	124	2	no	—
284	D₄	—	2	1	no	—
285	D₄	—	2	1	no	—
286	D₄	—	4	2	no	—
287	D₄	—	62	2	no	—
288	D₄	—	124	2	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	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	—	62	1	no	—
302	C₂×C₄	—	62	1	no	—
303	C₂×C₄	—	124	2	no	—
304	C₂×C₄	—	124	2	no	—
305	C₂×C₄	—	124	2	no	—
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	—	31	2	no	—
311	C₈	—	62	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	62	1	no	—
319	C₂²	—	62	1	no	—
320	C₂²	—	62	1	no	—
321	C₂²	C₂×D₄×D₃₁	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	31	2	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	#41×D₃₁	1	1	yes	characteristic
334	C₄	#11758567	1	1	yes	characteristic
335	C₂	—	4	1	no	—
336	C₂	—	8	1	no	—
337	C₂	—	124	1	no	—
338	C₂	#14521697	1	1	yes	characteristic
339	C₂	AGL(1,ℤ/8)×D₃₁	1	1	yes	characteristic
340	C₂	#14521863	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₁
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₃₁×#482	C₂
socle	C₂²×C₃₁	C₂²×D₄
prime radical	#14521699	C₂
hypercenter	#482	D₃₁
hypocenter	C₃₁	#3353
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #482

Complements

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

N	H ≅ G/N	type
1. #14521767	C₂	semidirect
3. #14521453	C₂	semidirect
4. #14521386	C₂	semidirect
5. #14521385	C₂	semidirect
6. #14521371	C₂	semidirect
7. #14521333	C₂	semidirect
8. #14521327	C₂	semidirect
9. #14521294	C₂	semidirect
10. #14521293	C₂	semidirect
11. #14521140	C₂	semidirect
12. #14521124	C₂	semidirect
13. #14521092	C₂	semidirect
14. #14521088	C₂	semidirect
15. C₃₁×#482	C₂	semidirect
140. C₃₁⋊C₈	#41	semidirect
171. #482	D₃₁	semidirect
229. C₃₁	#3353	semidirect

Maximal subgroups

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

subgroup	index
#14521767	2
#14521699	2
#14521453	2
#14521386	2
#14521385	2
#14521371	2
#14521333	2
#14521327	2
#14521294	2
#14521293	2
#14521140	2
#14521124	2
#14521092	2
#14521088	2
C₃₁×#482	2
#3353	31