Subgroups — #14521799

Order 1984.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#124	C₂	1	1	yes	maximal, characteristic
2	#11758521	C₂	1	2	yes	maximal, complemented
3	#11758512	C₂	1	2	yes	maximal
4	#11758456	C₂	1	4	yes	maximal, complemented
5	#11758455	C₂	1	4	yes	maximal, complemented
6	#11758447	C₂	1	2	yes	maximal, complemented
7	C₂×Q₈×C₃₁	C₂²	1	1	yes	characteristic
8	C₂×D₄×C₃₁	C₂²	1	1	yes	characteristic
9	C₂²×Dic₃₁	C₂²	1	2	yes	—
10	C₂×Dic₆₂	C₂²	1	2	yes	—
11	C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
12	C₄²×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²⋊Dic₃₁	C₂²	1	4	yes	—
14	C₂²⋊Dic₃₁	C₂²	1	2	yes	—
15	C₄⋊Dic₃₁	C₂²	1	2	yes	—
16	C₄⋊Dic₃₁	C₂²	1	2	yes	—
17	Dic₃₁⋊C₄	C₂²	1	4	yes	—
18	Dic₃₁⋊C₄	C₂²	1	4	yes	—
19	Dic₃₁⋊C₄	C₂²	1	4	yes	—
20	C₄×Dic₃₁	C₂²	1	2	yes	—
21	C₂³×C₃₁	C₂³	1	2	yes	—
22	Q₈×C₃₁	—	4	1	no	—
23	D₄×C₃₁	—	4	1	no	—
24	C₂×C₄×C₃₁	C₂³	1	2	yes	—
25	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
26	C₂×C₄×C₃₁	C₂³	1	2	yes	—
27	C₂×Dic₃₁	—	4	2	no	—
28	C₂×Dic₃₁	C₂³	1	4	yes	—
29	C₂×Dic₃₁	C₂³	1	4	yes	—
30	Dic₆₂	—	4	2	no	—
31	C₂²×C₃₁	—	2	2	no	—
32	C₂²×C₃₁	—	2	4	no	—
33	C₂²×C₃₁	C₂⁴	1	1	yes	characteristic
34	C₄×C₃₁	—	2	2	no	—
35	C₄×C₃₁	—	2	1	no	—
36	C₄×C₃₁	—	2	2	no	—
37	Dic₃₁	—	2	4	no	—
38	Dic₃₁	—	2	4	no	—
39	#563	—	31	1	no	maximal, Sylow
40	C₂×C₃₁	—	4	2	no	—
41	C₂×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
42	C₂×C₃₁	2₋¹⁺⁴	1	2	yes	—
43	C₄⋊Q₈	—	31	2	no	—
44	#126	—	31	4	no	—
45	#125	—	31	2	no	—
46	#124	D₃₁	1	1	yes	characteristic
47	#123	—	31	2	no	—
48	C₂²⋊Q₈	—	31	4	no	—
49	C₃₁	#563	1	1	yes	characteristic, complemented, Sylow
50	C₂×Q₈	—	31	2	no	—
51	C₂×Q₈	C₂×D₃₁	1	1	yes	characteristic
52	C₂×D₄	C₂×D₃₁	1	1	yes	characteristic
53	C₂²×C₄	—	31	2	no	—
54	C₄⋊C₄	—	31	4	no	—
55	C₄⋊C₄	—	31	4	no	—
56	C₄⋊C₄	—	31	4	no	—
57	C₄⋊C₄	—	31	2	no	—
58	C₄⋊C₄	—	31	2	no	—
59	C₂²⋊C₄	—	31	4	no	—
60	C₂²⋊C₄	—	31	2	no	—
61	C₂²⋊C₄	C₂×D₃₁	1	4	yes	—
62	C₄²	—	31	2	no	—
63	C₄²	C₂×D₃₁	1	1	yes	characteristic
64	C₂³	C₂²×D₃₁	1	2	yes	—
65	Q₈	—	4	1	no	—
66	Q₈	—	124	2	no	—
67	D₄	—	4	1	no	—
68	C₂×C₄	—	31	4	no	—
69	C₂×C₄	—	31	4	no	—
70	C₂×C₄	—	124	2	no	—
71	C₂×C₄	C₂²×D₃₁	1	2	yes	—
72	C₂×C₄	C₂²×D₃₁	1	2	yes	—
73	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
74	C₂²	—	2	2	no	—
75	C₂²	—	2	4	no	—
76	C₂²	C₂³×D₃₁	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	2	2	no	—
79	C₄	—	2	2	no	—
80	C₄	—	62	4	no	—
81	C₄	—	62	4	no	—
82	C₂	—	4	2	no	—
83	C₂	#11758561	1	1	yes	characteristic
84	C₂	#11758567	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₁
commutator, socle, cosocle	C₂²×C₃₁	C₂⁴
Fitting	C₃₁×#124	C₂
prime radical	C₂×D₄×C₃₁	C₂²
hypercenter	#124	D₃₁
hypocenter	C₃₁	#563
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ #124

Complements

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

N	H ≅ G/N	type
2. #11758521	C₂	semidirect
4. #11758456	C₂	semidirect
5. #11758455	C₂	semidirect
6. #11758447	C₂	semidirect
49. C₃₁	#563	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#124	2
#11758521	2
#11758512	2
#11758456	2
#11758455	2
#11758447	2
#563	31