Group #14521798 — subgroups

Subgroups — #14521798

Order 1984.

196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#11758524	C₂	1	1	yes	maximal, characteristic
3	#11758523	C₂	1	2	yes	maximal, complemented
4	#11758521	C₂	1	1	yes	maximal, characteristic, complemented
5	#11758475	C₂	1	2	yes	maximal, complemented
6	#11758473	C₂	1	2	yes	maximal, complemented
7	#11758471	C₂	1	2	yes	maximal
8	#11758466	C₂	1	2	yes	maximal, complemented
9	#11758453	C₂	1	2	yes	maximal, complemented
10	C₂×Q₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×Q₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×D₁₂₄	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₃₁	C₂²	1	2	yes	—
14	C₂×Dic₆₂	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
16	C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
17	C₄²×C₃₁	C₂²	1	1	yes	characteristic
18	(C₂×D₃₁)⋊C₄	C₂²	1	2	yes	—
19	(C₂×D₃₁)⋊C₄	C₂²	1	2	yes	—
20	(C₂×D₃₁)⋊C₄	C₂²	1	2	yes	—
21	(C₂×D₃₁)⋊C₄	C₂²	1	2	yes	—
22	(C₂×D₃₁)⋊C₄	C₂²	1	2	yes	—
23	C₄⋊Dic₃₁	C₂²	1	2	yes	—
24	Dic₃₁⋊C₄	C₂²	1	2	yes	—
25	Dic₃₁⋊C₄	C₂²	1	2	yes	—
26	Dic₃₁⋊C₄	C₂²	1	2	yes	—
27	Dic₃₁⋊C₄	C₂²	1	2	yes	—
28	Dic₃₁⋊C₄	C₂²	1	2	yes	—
29	C₄×Dic₃₁	C₂²	1	1	yes	characteristic
30	C₄×Dic₃₁	C₂²	1	1	yes	characteristic
31	C₂²×D₃₁	C₂³	1	2	yes	—
32	Q₈×C₃₁	—	4	1	no	—
33	Q₈×C₃₁	—	4	1	no	—
34	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
35	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₃₁	C₂³	1	2	yes	—
37	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₃₁	C₂³	1	2	yes	—
39	C₂×Dic₃₁	C₂³	1	2	yes	—
40	C₂×Dic₃₁	C₂³	1	2	yes	—
41	C₂×Dic₃₁	C₂³	1	2	yes	—
42	D₁₂₄	—	4	1	no	—
43	C₄×D₃₁	—	4	2	no	—
44	Dic₆₂	—	4	1	no	—
45	C₂²×C₃₁	C₂⁴	1	1	yes	characteristic
46	C₂×D₃₁	—	2	2	no	—
47	C₂×D₃₁	—	2	4	no	—
48	C₄×C₃₁	—	2	1	no	—
49	C₄×C₃₁	—	2	2	no	—
50	C₄×C₃₁	—	2	1	no	—
51	C₄×C₃₁	—	2	1	no	—
52	C₄×C₃₁	—	2	2	no	—
53	Dic₃₁	—	2	2	no	—
54	Dic₃₁	—	2	2	no	—
55	Dic₃₁	—	2	2	no	—
56	#563	—	31	1	no	maximal, Sylow
57	C₂×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
58	C₂×C₃₁	2₋¹⁺⁴	1	2	yes	—
59	D₃₁	—	4	2	no	—
60	C₄⋊Q₈	—	31	1	no	—
61	C₄⋊Q₈	D₃₁	1	1	yes	characteristic, complemented
62	#126	—	31	2	no	—
63	#126	—	31	2	no	—
64	#125	—	31	2	no	—
65	#124	—	31	1	no	—
66	#123	—	31	2	no	—
67	C₂²⋊Q₈	—	31	2	no	—
68	C₂²⋊Q₈	—	31	2	no	—
69	C₃₁	#563	1	1	yes	characteristic, complemented, Sylow
70	C₂×Q₈	—	31	1	no	—
71	C₂×Q₈	C₂×D₃₁	1	1	yes	characteristic
72	C₂×Q₈	C₂×D₃₁	1	1	yes	characteristic
73	C₂×D₄	—	31	1	no	—
74	C₂²×C₄	—	31	2	no	—
75	C₄⋊C₄	—	31	2	no	—
76	C₄⋊C₄	—	31	2	no	—
77	C₄⋊C₄	—	31	2	no	—
78	C₄⋊C₄	—	31	2	no	—
79	C₄⋊C₄	—	31	2	no	—
80	C₄⋊C₄	—	31	2	no	—
81	C₄⋊C₄	C₂×D₃₁	1	2	yes	—
82	C₄⋊C₄	C₂×D₃₁	1	2	yes	—
83	C₂²⋊C₄	—	31	2	no	—
84	C₂²⋊C₄	—	31	2	no	—
85	C₂²⋊C₄	—	31	2	no	—
86	C₂²⋊C₄	—	31	2	no	—
87	C₂²⋊C₄	—	31	2	no	—
88	C₄²	—	31	1	no	—
89	C₄²	—	31	1	no	—
90	C₄²	C₂×D₃₁	1	1	yes	characteristic
91	C₂³	—	31	2	no	—
92	Q₈	—	4	1	no	—
93	Q₈	—	4	1	no	—
94	Q₈	—	124	1	no	—
95	D₄	—	124	1	no	—
96	C₂×C₄	—	31	2	no	—
97	C₂×C₄	—	31	2	no	—
98	C₂×C₄	—	31	2	no	—
99	C₂×C₄	—	124	2	no	—
100	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
101	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
102	C₂×C₄	C₂²×D₃₁	1	2	yes	—
103	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
104	C₂×C₄	C₂²×D₃₁	1	2	yes	—
105	C₂²	—	62	2	no	—
106	C₂²	—	62	4	no	—
107	C₂²	C₂³×D₃₁	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	1	no	—
112	C₄	—	2	2	no	—
113	C₄	—	62	2	no	—
114	C₄	—	62	2	no	—
115	C₄	—	62	2	no	—
116	C₂	—	124	2	no	—
117	C₂	#11758561	1	1	yes	characteristic
118	C₂	#11758564	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃₁×C₄⋊Q₈

C₂

yes

maximal, characteristic, complemented

#11758524

C₂

yes

maximal, characteristic

#11758523

C₂

yes

maximal, complemented

#11758521

C₂

yes

maximal, characteristic, complemented

#11758475

C₂

yes

maximal, complemented

#11758473

C₂

yes

maximal, complemented

#11758471

C₂

yes

maximal

#11758466

C₂

yes

maximal, complemented

#11758453

C₂

yes

maximal, complemented

C₂×Q₈×C₃₁

C₂²

yes

characteristic

C₂×Q₈×C₃₁

C₂²

yes

characteristic

C₂×D₁₂₄

C₂²

yes

characteristic

C₂×C₄×D₃₁

C₂²

yes

—

C₂×Dic₆₂

C₂²

yes

characteristic

C₄⋊C₄×C₃₁

C₂²

yes

—

C₄⋊C₄×C₃₁

C₂²

yes

—

C₄²×C₃₁

C₂²

yes

characteristic

(C₂×D₃₁)⋊C₄

C₂²

yes

—

(C₂×D₃₁)⋊C₄

C₂²

yes

—

(C₂×D₃₁)⋊C₄

C₂²

yes

—

(C₂×D₃₁)⋊C₄

C₂²

yes

—

(C₂×D₃₁)⋊C₄

C₂²

yes

—

C₄⋊Dic₃₁

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

C₄×Dic₃₁

C₂²

yes

characteristic

C₄×Dic₃₁

C₂²

yes

characteristic

C₂²×D₃₁

C₂³

yes

—

Q₈×C₃₁

—

Q₈×C₃₁

—

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂×C₄×C₃₁

C₂³

yes

—

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂×C₄×C₃₁

C₂³

yes

—

C₂×Dic₃₁

C₂³

yes

—

C₂×Dic₃₁

C₂³

yes

—

C₂×Dic₃₁

C₂³

yes

—

D₁₂₄

—

C₄×D₃₁

—

Dic₆₂

—

C₂²×C₃₁

C₂⁴

yes

characteristic

C₂×D₃₁

—

C₂×D₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

—

C₂×C₃₁

yes

C₂×C₃₁

2₋¹⁺⁴

yes

—

D₃₁

—

C₄⋊Q₈

—

C₄⋊Q₈

D₃₁

yes

characteristic, complemented

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₃₁

#563

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

C₂×D₃₁

yes

characteristic

C₂×Q₈

C₂×D₃₁

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₃₁

yes

—

C₄⋊C₄

C₂×D₃₁

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂×D₃₁

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

Q₈

—

124

—

D₄

—

124

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

124

—

100

C₂×C₄

C₂²×D₃₁

yes

characteristic

101

C₂×C₄

C₂²×D₃₁

yes

characteristic

102

C₂×C₄

C₂²×D₃₁

yes

—

103

C₂×C₄

C₂²×D₃₁

yes

characteristic

104

C₂×C₄

C₂²×D₃₁

yes

—

105

C₂²

—

106

C₂²

—

107

C₂²

C₂³×D₃₁

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

116

C₂

—

124

—

117

C₂

#11758561

yes

characteristic

118

C₂

#11758564

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₁
commutator, socle, cosocle	C₂²×C₃₁	C₂⁴
Fitting	C₃₁×C₄⋊Q₈	C₂
prime radical	C₂×D₁₂₄	C₂²
hypercenter	C₄⋊Q₈	D₃₁
hypocenter	C₃₁	#563
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₁

commutator, socle, cosocle

C₂²×C₃₁

C₂⁴

Fitting

C₃₁×C₄⋊Q₈

C₂

prime radical

C₂×D₁₂₄

C₂²

hypercenter

C₄⋊Q₈

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. C₃₁×C₄⋊Q₈	C₂	semidirect
3. #11758523	C₂	semidirect
4. #11758521	C₂	semidirect
5. #11758475	C₂	semidirect
6. #11758473	C₂	semidirect
8. #11758466	C₂	semidirect
9. #11758453	C₂	semidirect
61. C₄⋊Q₈	D₃₁	semidirect
69. C₃₁	#563	semidirect

H ≅ G/N

type

1. C₃₁×C₄⋊Q₈

C₂

semidirect

3. #11758523

C₂

semidirect

4. #11758521

C₂

semidirect

5. #11758475

C₂

semidirect

6. #11758473

C₂

semidirect

8. #11758466

C₂

semidirect

9. #11758453

C₂

semidirect

61. C₄⋊Q₈

D₃₁

semidirect

69. C₃₁

#563

semidirect

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

subgroup	index
C₃₁×C₄⋊Q₈	2
#11758524	2
#11758523	2
#11758521	2
#11758475	2
#11758473	2
#11758471	2
#11758466	2
#11758453	2
#563	31

subgroup

index

C₃₁×C₄⋊Q₈

Subgroups — #14521798

Conjugacy classes of subgroups

Complements

Maximal subgroups