Subgroups — #14521053

Order 1984.

190 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₂₄	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#92474	C₂	1	1	yes	maximal, characteristic, complemented
4	#11758397	C₂	1	4	yes	maximal, complemented
5	C₂³×D₃₁	—	2	1	no	—
6	C₂²×C₄×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×D₁₂₄	—	2	1	no	—
8	C₂×D₁₂₄	—	2	2	no	—
9	C₂×D₁₂₄	C₄	1	4	yes	complemented
10	C₂×D₁₂₄	C₂²	1	2	yes	—
11	C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
12	C₂²⋊C₄×C₃₁	—	2	1	no	—
13	C₄²×C₃₁	—	2	1	no	—
14	#92474	—	2	2	no	—
15	C₂×C₃₁⋊C₈	C₂²	1	2	yes	complemented
16	C₂³×C₃₁	D₄	1	1	yes	characteristic
17	C₂²×D₃₁	—	2	4	no	—
18	C₂²×D₃₁	—	2	2	no	—
19	C₂²×D₃₁	—	4	4	no	—
20	C₂×C₄×C₃₁	—	2	2	no	—
21	C₂×C₄×C₃₁	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₃₁	D₄	1	2	yes	—
23	C₂×C₄×C₃₁	D₄	1	1	yes	characteristic
24	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
25	D₁₂₄	—	2	4	no	—
26	D₁₂₄	—	2	2	no	—
27	D₁₂₄	C₂×C₄	1	4	yes	—
28	C₃₁⋊C₈	—	2	2	no	—
29	C₂²×C₃₁	—	2	2	no	—
30	C₂²×C₃₁	C₂²⋊C₄	1	2	yes	—
31	C₂²×C₃₁	C₂×D₄	1	1	yes	characteristic
32	C₂×D₃₁	—	2	4	no	—
33	C₂×D₃₁	—	4	8	no	—
34	C₂×D₃₁	—	4	4	no	—
35	C₄×C₃₁	—	4	2	no	—
36	C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
37	C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
38	C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
39	#418	—	31	1	no	maximal, Sylow
40	C₂×C₃₁	—	2	2	no	—
41	C₂×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
43	D₃₁	—	4	4	no	—
44	C₂²×D₄	—	31	1	no	—
45	C₂×M₁₆	—	31	1	no	—
46	#117	D₃₁	1	1	yes	characteristic, complemented
47	#102	—	31	4	no	—
48	C₃₁	#418	1	1	yes	characteristic, complemented, Sylow
49	C₂⁴	—	62	1	no	—
50	C₂×D₄	—	31	4	no	—
51	C₂×D₄	—	31	2	no	—
52	C₂×D₄	—	62	2	no	—
53	C₂×D₄	—	62	1	no	—
54	C₂²×C₄	C₂×D₃₁	1	1	yes	characteristic
55	M₁₆	—	62	2	no	—
56	C₂×C₈	—	31	2	no	—
57	C₄⋊C₄	C₂×D₃₁	1	2	yes	complemented
58	C₂²⋊C₄	—	2	1	no	—
59	C₄²	—	2	1	no	—
60	C₂³	—	62	4	no	—
61	C₂³	—	62	2	no	—
62	C₂³	—	124	4	no	—
63	C₂³	C₂²⋊D₃₁	1	1	yes	characteristic
64	D₄	—	31	4	no	—
65	D₄	—	62	4	no	—
66	D₄	—	62	2	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	C₄×D₃₁	1	2	yes	—
69	C₂×C₄	D₁₂₄	1	2	yes	—
70	C₂×C₄	C₂²⋊D₃₁	1	1	yes	characteristic
71	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
72	C₈	—	62	2	no	—
73	C₂²	—	2	2	no	—
74	C₂²	—	62	4	no	—
75	C₂²	—	124	8	no	—
76	C₂²	—	124	4	no	—
77	C₂²	(C₂×D₃₁)⋊C₄	1	2	yes	—
78	C₂²	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
79	C₄	—	4	2	no	—
80	C₄	(C₂×D₃₁)⋊C₄	1	2	yes	—
81	C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
82	C₄	C₂×D₁₂₄	1	1	yes	characteristic
83	C₂	—	2	2	no	—
84	C₂	—	124	4	no	—
85	C₂	C₂×(C₂×D₃₁)⋊C₄	1	1	yes	characteristic
86	C₂	#11758526	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₃₁
commutator	C₄×C₃₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃₁
Fitting	C₃₁×#117	C₂
socle	C₂²×C₃₁	C₂×D₄
prime radical	C₂²×D₁₂₄	C₂
cosocle	C₂×C₄×C₃₁	C₂³
hypercenter	#117	D₃₁
hypocenter	C₃₁	#418
LCS	G ▷ C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
2. C₃₁×#117	C₂	semidirect
3. C₂×#92474	C₂	semidirect
4. #11758397	C₂	semidirect
9. C₂×D₁₂₄	C₄	semidirect
11. C₄⋊C₄×C₃₁	C₂²	semidirect
15. C₂×C₃₁⋊C₈	C₂²	semidirect
46. #117	D₃₁	semidirect
48. C₃₁	#418	semidirect
57. C₄⋊C₄	C₂×D₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₁₂₄	2
C₃₁×#117	2
C₂×#92474	2
#11758397	2
#418	31