Subgroups — #16847228

Order 3968.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14521098	C₂	1	3	yes	maximal, complemented
2	#14521012	C₂	1	3	yes	maximal
3	C₃₁×#338	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×C₄×D₃₁	C₂²	1	1	yes	characteristic
5	C₂×M₁₆×C₃₁	C₂²	1	3	yes	—
6	C₂×#92474	C₂²	1	3	yes	—
7	(C₂×D₃₁)⋊C₈	—	2	6	no	—
8	#11758407	—	2	6	no	—
9	C₂³×D₃₁	Q₈	1	1	yes	characteristic
10	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
11	C₂²×Dic₃₁	Q₈	1	1	yes	characteristic
12	C₂×C₄×D₃₁	—	2	3	no	—
13	C₂×C₄×D₃₁	—	2	3	no	—
14	M₁₆×C₃₁	—	2	6	no	—
15	C₂×C₈×C₃₁	D₄	1	6	yes	—
16	#92474	—	2	6	no	—
17	C₂×C₃₁⋊C₈	—	2	3	no	—
18	C₂³×C₃₁	C₂×Q₈	1	1	yes	characteristic
19	C₂²×D₃₁	—	2	3	no	—
20	C₂²×D₃₁	—	4	2	no	—
21	C₂×C₄×C₃₁	C₂×D₄	1	3	yes	—
22	C₂×C₄×C₃₁	#41	1	3	yes	—
23	C₂×Dic₃₁	—	2	3	no	—
24	C₄×D₃₁	—	4	3	no	—
25	C₄×D₃₁	—	4	1	no	—
26	C₈×C₃₁	—	2	6	no	—
27	C₃₁⋊C₈	—	4	3	no	—
28	#2091	—	31	1	no	maximal, Sylow
29	C₂²×C₃₁	—	4	1	no	—
30	C₂²×C₃₁	C₂²⋊Q₈	1	3	yes	—
31	C₂×D₃₁	—	2	6	no	—
32	C₂×D₃₁	—	4	3	no	—
33	C₂×D₃₁	—	4	1	no	—
34	C₄×C₃₁	#123	1	3	yes	—
35	C₄×C₃₁	#127	1	1	yes	characteristic
36	Dic₃₁	—	4	1	no	—
37	#430	—	31	3	no	—
38	#407	—	31	3	no	—
39	#338	D₃₁	1	1	yes	characteristic, complemented
40	C₂×C₃₁	—	2	3	no	—
41	C₂×C₃₁	#399	1	1	yes	characteristic
42	D₃₁	—	4	2	no	—
43	C₂³×C₄	—	31	1	no	—
44	C₂×M₁₆	—	31	3	no	—
45	C₂×M₁₆	C₂×D₃₁	1	3	yes	—
46	#108	—	62	6	no	—
47	C₂²⋊C₈	—	62	6	no	—
48	C₃₁	#2091	1	1	yes	characteristic, complemented, Sylow
49	C₂⁴	—	31	1	no	—
50	C₂²×C₄	—	31	1	no	—
51	C₂²×C₄	—	62	3	no	—
52	C₂²×C₄	—	62	3	no	—
53	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
54	M₁₆	—	2	6	no	—
55	M₁₆	—	62	6	no	—
56	C₂×C₈	—	62	3	no	—
57	C₂×C₈	D₁₂₄	1	6	yes	—
58	C₂³	—	62	3	no	—
59	C₂³	—	124	2	no	—
60	C₂³	Q₈×D₃₁	1	1	yes	characteristic
61	C₂×C₄	—	62	3	no	—
62	C₂×C₄	—	124	1	no	—
63	C₂×C₄	—	124	3	no	—
64	C₂×C₄	C₂×D₁₂₄	1	3	yes	—
65	C₂×C₄	#92497	1	3	yes	—
66	C₈	—	2	6	no	—
67	C₈	—	124	3	no	—
68	C₂²	—	4	1	no	—
69	C₂²	—	62	6	no	—
70	C₂²	—	124	3	no	—
71	C₂²	—	124	1	no	—
72	C₂²	#11758474	1	3	yes	—
73	C₄	—	124	1	no	—
74	C₄	#11758451	1	1	yes	characteristic
75	C₄	#11758463	1	3	yes	—
76	C₂	—	2	3	no	—
77	C₂	—	124	2	no	—
78	C₂	#14520986	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#11758451
commutator, cosocle	C₂²×C₄×C₃₁	C₂³
Frattini	C₂²×C₄	C₂²×D₃₁
Fitting	C₃₁×#338	C₂
socle	C₂×C₃₁	#399
prime radical	C₂³×D₃₁	Q₈
hypercenter	#338	D₃₁
hypocenter	C₃₁	#2091
LCS	G ▷ C₂²×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #338

Complements

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

N	H ≅ G/N	type
1. #14521098	C₂	semidirect
3. C₃₁×#338	C₂	semidirect
39. #338	D₃₁	semidirect
48. C₃₁	#2091	semidirect

Maximal subgroups

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

subgroup	index
#14521098	2
#14521012	2
C₃₁×#338	2
#2091	31