Group #14521674 — subgroups

Subgroups — #14521674

Order 1984.

242 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 103 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×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#11758523	C₂	1	2	yes	maximal, complemented
3	C₂×(C₂×D₃₁)⋊C₄	C₂	1	1	yes	maximal, characteristic
4	#11758501	C₂	1	1	yes	maximal, characteristic, complemented
5	#11758474	C₂	1	2	yes	maximal, complemented
6	#11758473	C₂	1	2	yes	maximal, complemented
7	#11758473	C₂	1	2	yes	maximal, complemented
8	C₂²⋊C₄×D₃₁	C₂	1	2	yes	maximal
9	#11758455	C₂	1	2	yes	maximal, complemented
10	C₂³×D₃₁	C₂²	1	1	yes	characteristic
11	C₂×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
12	C₂²×C₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×Dic₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₃₁	C₂²	1	2	yes	—
15	C₂×C₄×D₃₁	C₂²	1	2	yes	—
16	C₂×Dic₆₂	C₂²	1	2	yes	complemented
17	C₂×Dic₆₂	C₂²	1	1	yes	characteristic, complemented
18	C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
19	C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
20	C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
21	C₂²⋊Dic₃₁	C₂²	1	2	yes	—
22	(C₂×D₃₁)⋊C₄	C₂²	1	2	yes	—
23	(C₂×D₃₁)⋊C₄	C₂²	1	2	yes	—
24	(C₂×D₃₁)⋊C₄	C₂²	1	2	yes	—
25	(C₂×D₃₁)⋊C₄	C₂²	1	2	yes	—
26	C₄⋊Dic₃₁	C₂²	1	2	yes	complemented
27	C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic, complemented
28	Dic₃₁⋊C₄	C₂²	1	2	yes	complemented
29	Dic₃₁⋊C₄	C₂²	1	2	yes	complemented
30	Dic₃₁⋊C₄	C₂²	1	2	yes	complemented
31	C₂³×C₃₁	C₂³	1	1	yes	characteristic
32	C₂²×D₃₁	—	4	1	no	—
33	C₂²×D₃₁	—	4	1	no	—
34	C₂²×D₃₁	Q₈	1	4	yes	complemented
35	C₂²×D₃₁	C₂³	1	2	yes	—
36	Q₈×C₃₁	—	4	1	no	—
37	C₂×C₄×C₃₁	—	4	1	no	—
38	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₃₁	C₂³	1	2	yes	—
40	C₂×C₄×C₃₁	C₂³	1	2	yes	—
41	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
42	C₂×Dic₃₁	—	4	1	no	—
43	C₂×Dic₃₁	C₂³	1	2	yes	—
44	C₂×Dic₃₁	C₂³	1	2	yes	—
45	C₂×Dic₃₁	C₂³	1	2	yes	—
46	C₄×D₃₁	—	4	2	no	—
47	C₄×D₃₁	—	4	2	no	—
48	Dic₆₂	—	4	1	no	—
49	Dic₆₂	—	4	2	no	—
50	C₂²×C₃₁	—	2	1	no	—
51	C₂²×C₃₁	—	2	1	no	—
52	C₂²×C₃₁	C₂×Q₈	1	2	yes	—
53	C₂²×C₃₁	C₂⁴	1	1	yes	characteristic
54	C₂×D₃₁	—	2	2	no	—
55	C₂×D₃₁	—	2	2	no	—
56	C₂×D₃₁	—	4	4	no	—
57	C₂×D₃₁	C₂×Q₈	1	4	yes	—
58	C₄×C₃₁	—	2	2	no	—
59	C₄×C₃₁	—	2	1	no	—
60	C₄×C₃₁	—	2	2	no	—
61	C₄×C₃₁	—	2	1	no	—
62	Dic₃₁	—	2	2	no	—
63	Dic₃₁	—	2	2	no	—
64	Dic₃₁	—	2	2	no	—
65	#543	—	31	1	no	maximal, Sylow
66	C₂×C₃₁	—	2	2	no	—
67	C₂×C₃₁	C₂²×Q₈	1	1	yes	characteristic
68	C₂×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
69	C₂×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
70	D₃₁	—	2	4	no	—
71	C₂²⋊Q₈	—	31	2	no	—
72	C₂²⋊Q₈	—	31	1	no	—
73	C₂²⋊Q₈	—	31	2	no	—
74	C₂²⋊Q₈	—	31	2	no	—
75	C₂²⋊Q₈	—	31	2	no	—
76	C₂²⋊Q₈	—	31	2	no	—
77	C₂²⋊Q₈	D₃₁	1	1	yes	characteristic, complemented
78	C₂×C₂²⋊C₄	—	31	1	no	—
79	C₂×C₂²⋊C₄	—	31	2	no	—
80	C₃₁	#543	1	1	yes	characteristic, complemented, Sylow
81	C₂⁴	—	31	1	no	—
82	C₂×Q₈	—	31	1	no	—
83	C₂×Q₈	—	31	2	no	—
84	C₂×Q₈	C₂×D₃₁	1	1	yes	characteristic, complemented
85	C₂²×C₄	—	31	1	no	—
86	C₂²×C₄	—	31	2	no	—
87	C₂²×C₄	—	31	2	no	—
88	C₂²×C₄	C₂×D₃₁	1	1	yes	characteristic
89	C₄⋊C₄	—	31	2	no	—
90	C₄⋊C₄	—	31	2	no	—
91	C₄⋊C₄	—	31	2	no	—
92	C₄⋊C₄	—	31	1	no	—
93	C₄⋊C₄	—	31	2	no	—
94	C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic, complemented
95	C₄⋊C₄	C₂×D₃₁	1	2	yes	complemented
96	C₂²⋊C₄	—	31	2	no	—
97	C₂²⋊C₄	—	31	2	no	—
98	C₂²⋊C₄	—	31	2	no	—
99	C₂²⋊C₄	—	31	2	no	—
100	C₂²⋊C₄	—	31	2	no	—
101	C₂²⋊C₄	C₂×D₃₁	1	2	yes	—
102	C₂³	—	31	2	no	—
103	C₂³	—	31	4	no	—
104	C₂³	—	124	1	no	—
105	C₂³	—	124	1	no	—
106	C₂³	C₂²×D₃₁	1	1	yes	characteristic
107	Q₈	—	4	1	no	—
108	Q₈	—	124	1	no	—
109	Q₈	—	124	2	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	31	2	no	—
112	C₂×C₄	—	31	2	no	—
113	C₂×C₄	—	31	2	no	—
114	C₂×C₄	—	124	2	no	—
115	C₂×C₄	—	124	2	no	—
116	C₂×C₄	—	124	1	no	—
117	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
118	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
119	C₂×C₄	C₂²×D₃₁	1	2	yes	—
120	C₂×C₄	C₂²×D₃₁	1	2	yes	—
121	C₂²	—	2	1	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	31	4	no	—
124	C₂²	—	62	2	no	—
125	C₂²	—	62	2	no	—
126	C₂²	—	124	4	no	—
127	C₂²	Q₈×D₃₁	1	2	yes	—
128	C₂²	C₂³×D₃₁	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	2	1	no	—
131	C₄	—	2	2	no	—
132	C₄	—	2	2	no	—
133	C₄	—	62	2	no	—
134	C₄	—	62	2	no	—
135	C₄	—	62	2	no	—
136	C₂	—	2	2	no	—
137	C₂	—	62	4	no	—
138	C₂	#11758561	1	1	yes	characteristic
139	C₂	C₂×Q₈×D₃₁	1	1	yes	characteristic
140	C₂	#11758566	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃₁×C₂²⋊Q₈

C₂

yes

maximal, characteristic, complemented

#11758523

C₂

yes

maximal, complemented

C₂×(C₂×D₃₁)⋊C₄

C₂

yes

maximal, characteristic

#11758501

C₂

yes

maximal, characteristic, complemented

#11758474

C₂

yes

maximal, complemented

#11758473

C₂

yes

maximal, complemented

#11758473

C₂

yes

maximal, complemented

C₂²⋊C₄×D₃₁

C₂

yes

maximal

#11758455

C₂

yes

maximal, complemented

C₂³×D₃₁

C₂²

yes

characteristic

C₂×Q₈×C₃₁

C₂²

yes

characteristic, complemented

C₂²×C₄×C₃₁

C₂²

yes

characteristic

C₂²×Dic₃₁

C₂²

yes

characteristic

C₂×C₄×D₃₁

C₂²

yes

—

C₂×C₄×D₃₁

C₂²

yes

—

C₂×Dic₆₂

C₂²

yes

complemented

C₂×Dic₆₂

C₂²

yes

characteristic, complemented

C₄⋊C₄×C₃₁

C₂²

yes

characteristic, complemented

C₄⋊C₄×C₃₁

C₂²

yes

complemented

C₂²⋊C₄×C₃₁

C₂²

yes

—

C₂²⋊Dic₃₁

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

complemented

C₄⋊Dic₃₁

C₂²

yes

characteristic, complemented

Dic₃₁⋊C₄

C₂²

yes

complemented

Dic₃₁⋊C₄

C₂²

yes

complemented

Dic₃₁⋊C₄

C₂²

yes

complemented

C₂³×C₃₁

C₂³

yes

characteristic

C₂²×D₃₁

—

C₂²×D₃₁

—

C₂²×D₃₁

Q₈

yes

complemented

C₂²×D₃₁

C₂³

yes

—

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂×C₄×C₃₁

C₂³

yes

—

C₂×C₄×C₃₁

C₂³

yes

—

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

C₂³

yes

—

C₂×Dic₃₁

C₂³

yes

—

C₂×Dic₃₁

C₂³

yes

—

C₄×D₃₁

—

C₄×D₃₁

—

Dic₆₂

—

Dic₆₂

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂×Q₈

yes

—

C₂²×C₃₁

C₂⁴

yes

characteristic

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

C₂×Q₈

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

#543

—

maximal, Sylow

C₂×C₃₁

—

C₂×C₃₁

C₂²×Q₈

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

D₃₁

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

D₃₁

yes

characteristic, complemented

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₃₁

#543

yes

characteristic, complemented, Sylow

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₃₁

yes

characteristic, complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₃₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₃₁

yes

characteristic, complemented

C₄⋊C₄

C₂×D₃₁

yes

complemented

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

C₂×D₃₁

yes

—

102

C₂³

—

103

C₂³

—

104

C₂³

—

124

—

105

C₂³

—

124

—

106

C₂³

C₂²×D₃₁

yes

characteristic

107

Q₈

—

108

Q₈

—

124

—

109

Q₈

—

124

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

124

—

115

C₂×C₄

—

124

—

116

C₂×C₄

—

124

—

117

C₂×C₄

C₂²×D₃₁

yes

characteristic

118

C₂×C₄

C₂²×D₃₁

yes

characteristic

119

C₂×C₄

C₂²×D₃₁

yes

—

120

C₂×C₄

C₂²×D₃₁

yes

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

124

—

127

C₂²

Q₈×D₃₁

yes

—

128

C₂²

C₂³×D₃₁

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₂

—

137

C₂

—

138

C₂

#11758561

yes

characteristic

139

C₂

C₂×Q₈×D₃₁

yes

characteristic

140

C₂

#11758566

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₁
commutator, socle, cosocle	C₂²×C₃₁	C₂⁴
Fitting	C₃₁×C₂²⋊Q₈	C₂
prime radical	C₂³×D₃₁	C₂²
hypercenter	C₂²⋊Q₈	D₃₁
hypocenter	C₃₁	#543
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₃₁

commutator, socle, cosocle

C₂²×C₃₁

C₂⁴

Fitting

C₃₁×C₂²⋊Q₈

C₂

prime radical

C₂³×D₃₁

C₂²

hypercenter

C₂²⋊Q₈

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. C₃₁×C₂²⋊Q₈	C₂	semidirect
2. #11758523	C₂	semidirect
4. #11758501	C₂	semidirect
5. #11758474	C₂	semidirect
6. #11758473	C₂	semidirect
7. #11758473	C₂	semidirect
9. #11758455	C₂	semidirect
11. C₂×Q₈×C₃₁	C₂²	semidirect
16. C₂×Dic₆₂	C₂²	semidirect
17. C₂×Dic₆₂	C₂²	semidirect
18. C₄⋊C₄×C₃₁	C₂²	semidirect
19. C₄⋊C₄×C₃₁	C₂²	semidirect
26. C₄⋊Dic₃₁	C₂²	semidirect
27. C₄⋊Dic₃₁	C₂²	semidirect
28. Dic₃₁⋊C₄	C₂²	semidirect
29. Dic₃₁⋊C₄	C₂²	semidirect
30. Dic₃₁⋊C₄	C₂²	semidirect
34. C₂²×D₃₁	Q₈	semidirect
77. C₂²⋊Q₈	D₃₁	semidirect
80. C₃₁	#543	semidirect
84. C₂×Q₈	C₂×D₃₁	semidirect
94. C₄⋊C₄	C₂×D₃₁	semidirect
95. C₄⋊C₄	C₂×D₃₁	semidirect

H ≅ G/N

type

1. C₃₁×C₂²⋊Q₈

C₂

semidirect

2. #11758523

C₂

semidirect

4. #11758501

C₂

semidirect

5. #11758474

C₂

semidirect

6. #11758473

C₂

semidirect

7. #11758473

C₂

semidirect

9. #11758455

C₂

semidirect

11. C₂×Q₈×C₃₁

C₂²

semidirect

16. C₂×Dic₆₂

C₂²

semidirect

17. C₂×Dic₆₂

C₂²

semidirect

18. C₄⋊C₄×C₃₁

C₂²

semidirect

19. C₄⋊C₄×C₃₁

C₂²

semidirect

26. C₄⋊Dic₃₁

C₂²

semidirect

27. C₄⋊Dic₃₁

C₂²

semidirect

28. Dic₃₁⋊C₄

C₂²

semidirect

29. Dic₃₁⋊C₄

C₂²

semidirect

30. Dic₃₁⋊C₄

C₂²

semidirect

34. C₂²×D₃₁

Q₈

semidirect

77. C₂²⋊Q₈

D₃₁

semidirect

80. C₃₁

#543

semidirect

84. C₂×Q₈

C₂×D₃₁

semidirect

94. C₄⋊C₄

C₂×D₃₁

semidirect

95. C₄⋊C₄

C₂×D₃₁

semidirect

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

subgroup	index
C₃₁×C₂²⋊Q₈	2
#11758523	2
C₂×(C₂×D₃₁)⋊C₄	2
#11758501	2
#11758474	2
#11758473	2
C₂²⋊C₄×D₃₁	2
#11758455	2
#543	31

subgroup

index

C₃₁×C₂²⋊Q₈

C₂×(C₂×D₃₁)⋊C₄

C₂²⋊C₄×D₃₁

Subgroups — #14521674

Conjugacy classes of subgroups

Complements

Maximal subgroups