Subgroups — #14521605

Order 1984.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	#11758473	C₂	1	4	yes	maximal
3	#11758469	C₂	1	2	yes	maximal
4	#11758465	C₂	1	2	yes	maximal, complemented
5	#11758464	C₂	1	4	yes	maximal, complemented
6	C₄²×D₃₁	C₂	1	1	yes	maximal, characteristic
7	#11758447	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₃₁	C₂²	1	2	yes	—
9	C₂×C₄×D₃₁	C₂²	1	1	yes	characteristic
10	C₂×Dic₆₂	C₂²	1	4	yes	—
11	C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
12	C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
13	C₄²×C₃₁	C₂²	1	1	yes	characteristic
14	(C₂×D₃₁)⋊C₄	C₂²	1	4	yes	—
15	C₄⋊Dic₃₁	C₂²	1	2	yes	—
16	Dic₃₁⋊C₄	C₂²	1	4	yes	—
17	Dic₃₁⋊C₄	C₂²	1	4	yes	—
18	C₄×Dic₃₁	C₂²	1	2	yes	—
19	C₄×Dic₃₁	C₂²	1	1	yes	characteristic
20	C₄×Dic₃₁	C₂²	1	4	yes	—
21	C₂²×D₃₁	C₂³	1	1	yes	characteristic
22	C₂×C₄×C₃₁	C₂³	1	2	yes	—
23	C₂×C₄×C₃₁	C₂³	1	4	yes	—
24	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
25	C₂×Dic₃₁	C₂³	1	4	yes	—
26	C₂×Dic₃₁	C₂³	1	1	yes	characteristic
27	C₂×Dic₃₁	C₂³	1	2	yes	—
28	C₄×D₃₁	—	2	4	no	—
29	C₄×D₃₁	Q₈	1	4	yes	complemented
30	Dic₆₂	—	2	8	no	—
31	C₂²×C₃₁	C₂⁴	1	1	yes	characteristic
32	C₂×D₃₁	—	2	2	no	—
33	C₂×D₃₁	C₂×Q₈	1	2	yes	—
34	C₄×C₃₁	—	2	4	no	—
35	C₄×C₃₁	—	2	2	no	—
36	C₄×C₃₁	C₂×Q₈	1	2	yes	—
37	Dic₃₁	—	2	4	no	—
38	Dic₃₁	C₂×Q₈	1	2	yes	—
39	Dic₃₁	#41	1	4	yes	—
40	#533	—	31	1	no	maximal, Sylow
41	C₂×C₃₁	C₂²×Q₈	1	1	yes	characteristic
42	C₂×C₃₁	C₂×#41	1	2	yes	—
43	D₃₁	—	2	2	no	—
44	C₄⋊Q₈	—	31	2	no	—
45	#125	—	31	1	no	—
46	#125	D₃₁	1	1	yes	characteristic, complemented
47	C₂²⋊Q₈	—	31	4	no	—
48	C₄×Q₈	—	31	4	no	—
49	#117	—	31	2	no	—
50	C₂×C₄²	—	31	1	no	—
51	C₃₁	#533	1	1	yes	characteristic, complemented, Sylow
52	C₂×Q₈	—	31	4	no	—
53	C₂²×C₄	—	31	1	no	—
54	C₂²×C₄	—	31	2	no	—
55	C₄⋊C₄	—	31	4	no	—
56	C₄⋊C₄	—	31	4	no	—
57	C₄⋊C₄	—	31	2	no	—
58	C₄⋊C₄	C₂×D₃₁	1	2	yes	—
59	C₄⋊C₄	C₂×D₃₁	1	4	yes	—
60	C₂²⋊C₄	—	31	4	no	—
61	C₄²	—	31	2	no	—
62	C₄²	—	31	1	no	—
63	C₄²	—	31	4	no	—
64	C₄²	C₂×D₃₁	1	1	yes	characteristic
65	C₂³	—	31	1	no	—
66	Q₈	—	62	8	no	—
67	C₂×C₄	—	31	1	no	—
68	C₂×C₄	—	31	2	no	—
69	C₂×C₄	—	31	4	no	—
70	C₂×C₄	—	31	4	no	—
71	C₂×C₄	—	62	4	no	—
72	C₂×C₄	C₂²×D₃₁	1	2	yes	—
73	C₂×C₄	C₂²×D₃₁	1	4	yes	—
74	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
75	C₂²	—	31	2	no	—
76	C₂²	—	62	2	no	—
77	C₂²	C₂³×D₃₁	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	2	4	no	—
80	C₄	—	31	2	no	—
81	C₄	—	31	4	no	—
82	C₄	—	62	4	no	—
83	C₄	Q₈×D₃₁	1	2	yes	—
84	C₂	—	62	2	no	—
85	C₂	C₂×Q₈×D₃₁	1	1	yes	characteristic
86	C₂	#41×D₃₁	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₁
commutator, socle, cosocle	C₂²×C₃₁	C₂⁴
Fitting	C₃₁×#125	C₂
prime radical	C₂²×D₃₁	C₂³
hypercenter	#125	D₃₁
hypocenter	C₃₁	#533
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ #125

Complements

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

N	H ≅ G/N	type
1. C₃₁×#125	C₂	semidirect
4. #11758465	C₂	semidirect
5. #11758464	C₂	semidirect
7. #11758447	C₂	semidirect
29. C₄×D₃₁	Q₈	semidirect
46. #125	D₃₁	semidirect
51. C₃₁	#533	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#125	2
#11758473	2
#11758469	2
#11758465	2
#11758464	2
C₄²×D₃₁	2
#11758447	2
#533	31