Subgroups — #14521330

Order 1984.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₁₂₄	C₂	1	1	yes	maximal, characteristic
2	#11758446	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#11758409	C₂	1	2	yes	maximal
5	#11758406	C₂	1	2	yes	maximal, complemented
6	C₂×D₁₂₄	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₃₁	—	2	1	no	—
8	C₂×Dic₆₂	—	2	1	no	—
9	C₂×C₈×C₃₁	C₂²	1	2	yes	—
10	C₄²×C₃₁	C₂²	1	1	yes	characteristic
11	(C₂×D₃₁)⋊C₄	—	2	1	no	—
12	C₄⋊Dic₃₁	—	2	1	no	—
13	C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₃₁	C₂²	1	2	yes	—
15	C₂²×D₃₁	—	2	1	no	—
16	C₂×C₄×C₃₁	D₄	1	1	yes	characteristic
17	C₂×C₄×C₃₁	D₄	1	1	yes	characteristic
18	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
19	C₂×Dic₃₁	—	2	2	no	—
20	C₂×Dic₃₁	—	2	1	no	—
21	D₁₂₄	—	2	1	no	—
22	D₁₂₄	Q₈	1	2	yes	complemented
23	C₄×D₃₁	—	4	2	no	—
24	Dic₆₂	—	4	2	no	—
25	C₈×C₃₁	—	2	2	no	—
26	C₂²×C₃₁	C₂×D₄	1	1	yes	characteristic
27	C₂×D₃₁	—	2	2	no	—
28	C₂×D₃₁	—	4	1	no	—
29	C₂×D₃₁	—	4	1	no	—
30	C₄×C₃₁	—	2	1	no	—
31	C₄×C₃₁	D₈	1	2	yes	—
32	C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
33	C₄×C₃₁	#41	1	1	yes	characteristic
34	Dic₃₁	—	4	2	no	—
35	Dic₃₁	—	4	1	no	—
36	#474	—	31	1	no	maximal, Sylow
37	C₂×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
38	C₂×C₃₁	C₂×D₈	1	1	yes	characteristic
39	C₂×C₃₁	#137	1	1	yes	characteristic
40	D₃₁	—	4	2	no	—
41	C₄⋊Q₈	—	31	1	no	—
42	C₄×D₄	—	31	1	no	—
43	#107	—	31	2	no	—
44	C₄⋊C₈	D₃₁	1	1	yes	characteristic, complemented
45	#102	—	31	2	no	—
46	C₃₁	#474	1	1	yes	characteristic, complemented, Sylow
47	C₂×Q₈	—	62	1	no	—
48	C₂×D₄	—	31	1	no	—
49	C₂²×C₄	—	62	1	no	—
50	C₂×C₈	C₂×D₃₁	1	2	yes	—
51	C₄⋊C₄	—	31	2	no	—
52	C₄⋊C₄	—	31	1	no	—
53	C₄⋊C₄	—	62	1	no	—
54	C₂²⋊C₄	—	62	1	no	—
55	C₄²	C₂×D₃₁	1	1	yes	characteristic
56	C₂³	—	62	1	no	—
57	Q₈	—	124	2	no	—
58	D₄	—	31	2	no	—
59	D₄	—	62	1	no	—
60	C₂×C₄	—	62	1	no	—
61	C₂×C₄	—	62	2	no	—
62	C₂×C₄	—	124	2	no	—
63	C₂×C₄	D₁₂₄	1	1	yes	characteristic
64	C₂×C₄	D₁₂₄	1	1	yes	characteristic
65	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
66	C₈	—	2	2	no	—
67	C₂²	—	62	2	no	—
68	C₂²	—	124	1	no	—
69	C₂²	—	124	1	no	—
70	C₂²	C₂×D₁₂₄	1	1	yes	characteristic
71	C₄	—	2	1	no	—
72	C₄	—	124	1	no	—
73	C₄	—	124	2	no	—
74	C₄	D₂₄₈	1	2	yes	—
75	C₄	#92497	1	1	yes	characteristic
76	C₄	Q₈×D₃₁	1	1	yes	characteristic
77	C₂	—	124	2	no	—
78	C₂	#11758474	1	1	yes	characteristic
79	C₂	C₂×D₂₄₈	1	1	yes	characteristic
80	C₂	#11758486	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₂₄
commutator, cosocle	C₂×C₄×C₃₁	C₂³
Frattini	C₂×C₄	C₂²×D₃₁
Fitting	C₃₁×C₄⋊C₈	C₂
socle	C₂²×C₃₁	C₂×D₄
prime radical	C₂×D₁₂₄	C₂²
hypercenter	C₄⋊C₈	D₃₁
hypocenter	C₃₁	#474
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #11758446	C₂	semidirect
3. C₃₁×C₄⋊C₈	C₂	semidirect
5. #11758406	C₂	semidirect
22. D₁₂₄	Q₈	semidirect
44. C₄⋊C₈	D₃₁	semidirect
46. C₃₁	#474	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₁₂₄	2
#11758446	2
C₃₁×C₄⋊C₈	2
#11758409	2
#11758406	2
#474	31