Subgroups — #16848892

Order 3968.

424 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14520956	C₂	1	4	yes	maximal
2	#14520935	C₂	1	8	yes	maximal
3	#14520917	C₂	1	2	yes	maximal
4	C₂×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₈×C₃₁	C₂²	1	2	yes	—
6	C₃₁×C₄⋊Q₈	C₄	1	8	yes	complemented
7	C₂×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
8	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₃₁	C₂²	1	4	yes	—
10	C₂×C₄×Dic₃₁	C₂²	1	4	yes	—
11	C₃₁⋊#95	C₂²	1	8	yes	—
12	C₃₁⋊#95	C₂²	1	4	yes	—
13	C₃₁⋊#95	C₂²	1	8	yes	—
14	C₂×Q₈×C₃₁	—	4	2	no	—
15	C₂×Q₈×C₃₁	—	4	2	no	—
16	C₂×Q₈×C₃₁	C₂×C₄	1	8	yes	—
17	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
18	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
19	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
20	C₂²×Dic₃₁	C₂³	1	8	yes	—
21	C₄⋊C₄×C₃₁	C₂×C₄	1	16	yes	—
22	C₄²×C₃₁	C₂×C₄	1	4	yes	—
23	C₄⋊Dic₃₁	—	4	4	no	—
24	C₄×Dic₃₁	—	4	4	no	—
25	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
26	Q₈×C₃₁	—	4	8	no	—
27	C₂×C₄×C₃₁	—	2	4	no	—
28	C₂×C₄×C₃₁	—	2	4	no	—
29	C₂×C₄×C₃₁	—	2	2	no	—
30	C₂×C₄×C₃₁	—	2	4	no	—
31	C₂×C₄×C₃₁	C₂²×C₄	1	4	yes	—
32	C₂×C₄×C₃₁	C₂²×C₄	1	2	yes	—
33	C₂×C₄×C₃₁	C₂²×C₄	1	8	yes	—
34	C₂×Dic₃₁	—	2	8	no	—
35	C₂×Dic₃₁	—	2	16	no	—
36	#2381	—	31	1	no	maximal, Sylow
37	C₂²×C₃₁	C₂³×C₄	1	1	yes	characteristic
38	C₂²×C₃₁	H(5,𝔽₂)	1	2	yes	—
39	C₂²×C₃₁	2₋¹⁺⁴	1	4	yes	—
40	C₄×C₃₁	—	2	8	no	—
41	C₄×C₃₁	—	2	2	no	—
42	C₄×C₃₁	—	2	4	no	—
43	Dic₃₁	—	4	8	no	—
44	C₂×C₄⋊Q₈	D₃₁	1	1	yes	characteristic
45	#391	—	31	4	no	—
46	#387	—	31	8	no	—
47	#383	—	31	2	no	—
48	C₂×C₃₁	#518	1	1	yes	characteristic
49	C₂×C₃₁	#519	1	2	yes	—
50	C₂×C₃₁	#563	1	4	yes	—
51	C₂²×Q₈	C₂×D₃₁	1	2	yes	—
52	C₄⋊Q₈	Dic₃₁	1	8	yes	complemented
53	C₂×C₄⋊C₄	—	31	4	no	—
54	C₂×C₄⋊C₄	C₂×D₃₁	1	4	yes	—
55	C₂×C₄²	—	31	4	no	—
56	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
57	#95	—	31	8	no	—
58	#95	—	31	4	no	—
59	#95	—	31	8	no	—
60	C₃₁	#2381	1	1	yes	characteristic, complemented, Sylow
61	C₂×Q₈	—	4	2	no	—
62	C₂×Q₈	—	4	2	no	—
63	C₂×Q₈	C₂×Dic₃₁	1	8	yes	—
64	C₂²×C₄	—	31	8	no	—
65	C₂²×C₄	C₂²×D₃₁	1	2	yes	—
66	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
67	C₂²×C₄	C₂²×D₃₁	1	4	yes	—
68	C₄⋊C₄	—	124	4	no	—
69	C₄⋊C₄	C₂×Dic₃₁	1	16	yes	—
70	C₄²	—	124	4	no	—
71	C₄²	C₂×Dic₃₁	1	4	yes	—
72	C₂³	C₂³×D₃₁	1	1	yes	characteristic
73	Q₈	—	4	8	no	—
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	—	2	4	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	2	4	no	—
78	C₂×C₄	—	62	8	no	—
79	C₂×C₄	—	62	16	no	—
80	C₂×C₄	C₂²×Dic₃₁	1	4	yes	—
81	C₂×C₄	C₂²×Dic₃₁	1	2	yes	—
82	C₂×C₄	C₂²×Dic₃₁	1	8	yes	—
83	C₂²	#11758561	1	2	yes	—
84	C₂²	#11758564	1	4	yes	—
85	C₂²	C₂³×Dic₃₁	1	1	yes	characteristic
86	C₄	—	2	2	no	—
87	C₄	—	2	8	no	—
88	C₄	—	2	4	no	—
89	C₄	—	124	8	no	—
90	C₂	#14521524	1	1	yes	characteristic
91	C₂	#14521527	1	2	yes	—
92	C₂	#14521798	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃₁
commutator	C₂²×C₃₁	C₂³×C₄
Fitting	C₂×C₃₁×C₄⋊Q₈	C₂
socle, prime radical, cosocle	C₂³×C₃₁	C₂⁴
hypercenter	C₂×C₄⋊Q₈	D₃₁
hypocenter	C₃₁	#2381
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
6. C₃₁×C₄⋊Q₈	C₄	semidirect
52. C₄⋊Q₈	Dic₃₁	semidirect
60. C₃₁	#2381	semidirect

Maximal subgroups

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

subgroup	index
#14520956	2
#14520935	2
#14520917	2
C₂×C₃₁×C₄⋊Q₈	2
#2381	31