Group #14521676 — subgroups

Subgroups — #14521676

Order 1984.

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

normal subgroups only

This group has 206 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×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#11758523	C₂	1	2	yes	maximal
3	#11758521	C₂	1	2	yes	maximal, complemented
4	#11758473	C₂	1	2	yes	maximal
5	#11758469	C₂	1	1	yes	maximal, characteristic
6	C₄⋊C₄×D₃₁	C₂	1	1	yes	maximal, characteristic
7	#11758467	C₂	1	1	yes	maximal, characteristic, complemented
8	#11758466	C₂	1	2	yes	maximal, complemented
9	#11758465	C₂	1	1	yes	maximal, characteristic, complemented
10	#11758449	C₂	1	1	yes	maximal, characteristic
11	#11758447	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×C₃₁	C₂²	1	2	yes	—
13	C₂×C₄×D₃₁	C₂²	1	2	yes	—
14	C₂×C₄×D₃₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₆₂	C₂²	1	2	yes	—
16	C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
18	C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
19	C₄²×C₃₁	C₂²	1	1	yes	characteristic
20	(C₂×D₃₁)⋊C₄	C₂²	1	1	yes	characteristic
21	(C₂×D₃₁)⋊C₄	C₂²	1	1	yes	characteristic
22	(C₂×D₃₁)⋊C₄	C₂²	1	2	yes	—
23	C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
24	C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
25	C₄⋊Dic₃₁	C₂²	1	2	yes	—
26	Dic₃₁⋊C₄	C₂²	1	2	yes	—
27	Dic₃₁⋊C₄	C₂²	1	2	yes	—
28	Dic₃₁⋊C₄	C₂²	1	1	yes	characteristic
29	Dic₃₁⋊C₄	C₂²	1	1	yes	characteristic
30	Dic₃₁⋊C₄	C₂²	1	2	yes	—
31	Dic₃₁⋊C₄	C₂²	1	2	yes	—
32	Dic₃₁⋊C₄	C₂²	1	2	yes	—
33	C₄×Dic₃₁	C₂²	1	1	yes	characteristic
34	C₄×Dic₃₁	C₂²	1	2	yes	—
35	C₂²×D₃₁	C₂³	1	1	yes	characteristic
36	Q₈×C₃₁	—	4	2	no	—
37	C₂×C₄×C₃₁	C₂³	1	2	yes	—
38	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₃₁	C₂³	1	2	yes	—
41	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
42	C₂×Dic₃₁	C₂³	1	1	yes	characteristic
43	C₂×Dic₃₁	C₂³	1	2	yes	—
44	C₂×Dic₃₁	C₂³	1	1	yes	characteristic
45	C₂×Dic₃₁	C₂³	1	2	yes	—
46	C₂×Dic₃₁	C₂³	1	1	yes	characteristic
47	C₄×D₃₁	—	4	2	no	—
48	C₄×D₃₁	Q₈	1	4	yes	complemented
49	Dic₆₂	—	4	2	no	—
50	C₂²×C₃₁	C₂⁴	1	1	yes	characteristic
51	C₂×D₃₁	—	2	1	no	—
52	C₂×D₃₁	—	2	1	no	—
53	C₂×D₃₁	C₂×Q₈	1	2	yes	—
54	C₄×C₃₁	—	2	2	no	—
55	C₄×C₃₁	—	2	2	no	—
56	C₄×C₃₁	—	2	1	no	—
57	C₄×C₃₁	—	2	1	no	—
58	C₄×C₃₁	C₂×Q₈	1	2	yes	—
59	Dic₃₁	—	2	1	no	—
60	Dic₃₁	—	2	2	no	—
61	Dic₃₁	—	2	1	no	—
62	Dic₃₁	—	2	2	no	—
63	Dic₃₁	C₂×Q₈	1	2	yes	—
64	#544	—	31	1	no	maximal, Sylow
65	C₂×C₃₁	C₂²×Q₈	1	1	yes	characteristic
66	C₂×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
67	C₂×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
68	D₃₁	—	2	2	no	—
69	C₄⋊Q₈	—	31	2	no	—
70	C₄⋊Q₈	—	31	1	no	—
71	C₄⋊Q₈	D₃₁	1	1	yes	characteristic, complemented
72	#125	—	31	2	no	—
73	#125	—	31	1	no	—
74	#125	—	31	1	no	—
75	C₂²⋊Q₈	—	31	2	no	—
76	C₂²⋊Q₈	—	31	2	no	—
77	#117	—	31	1	no	—
78	#117	—	31	1	no	—
79	C₂×C₄⋊C₄	—	31	1	no	—
80	C₃₁	#544	1	1	yes	characteristic, complemented, Sylow
81	C₂×Q₈	—	31	2	no	—
82	C₂×Q₈	C₂×D₃₁	1	2	yes	—
83	C₂²×C₄	—	31	1	no	—
84	C₂²×C₄	—	31	2	no	—
85	C₄⋊C₄	—	31	2	no	—
86	C₄⋊C₄	—	31	1	no	—
87	C₄⋊C₄	—	31	1	no	—
88	C₄⋊C₄	—	31	2	no	—
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	1	no	—
94	C₄⋊C₄	—	31	2	no	—
95	C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic
96	C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic
97	C₄⋊C₄	C₂×D₃₁	1	2	yes	—
98	C₂²⋊C₄	—	31	2	no	—
99	C₂²⋊C₄	—	31	1	no	—
100	C₂²⋊C₄	—	31	1	no	—
101	C₄²	—	31	2	no	—
102	C₄²	—	31	1	no	—
103	C₄²	C₂×D₃₁	1	1	yes	characteristic
104	C₂³	—	31	1	no	—
105	Q₈	—	4	2	no	—
106	Q₈	—	124	2	no	—
107	C₂×C₄	—	31	1	no	—
108	C₂×C₄	—	31	4	no	—
109	C₂×C₄	—	31	1	no	—
110	C₂×C₄	—	31	2	no	—
111	C₂×C₄	—	31	1	no	—
112	C₂×C₄	—	31	2	no	—
113	C₂×C₄	—	124	2	no	—
114	C₂×C₄	C₂²×D₃₁	1	2	yes	—
115	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
116	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
117	C₂×C₄	C₂²×D₃₁	1	2	yes	—
118	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
119	C₂²	—	31	2	no	—
120	C₂²	—	62	1	no	—
121	C₂²	—	62	1	no	—
122	C₂²	C₂³×D₃₁	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	2	no	—
126	C₄	—	2	1	no	—
127	C₄	—	31	2	no	—
128	C₄	—	62	2	no	—
129	C₄	—	62	1	no	—
130	C₄	—	62	1	no	—
131	C₄	—	62	2	no	—
132	C₄	Q₈×D₃₁	1	2	yes	—
133	C₂	—	62	2	no	—
134	C₂	#11758561	1	1	yes	characteristic
135	C₂	C₂×Q₈×D₃₁	1	1	yes	characteristic
136	C₂	#11758564	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃₁×C₄⋊Q₈

C₂

yes

maximal, characteristic, complemented

#11758523

C₂

yes

maximal

#11758521

C₂

yes

maximal, complemented

#11758473

C₂

yes

maximal

#11758469

C₂

yes

maximal, characteristic

C₄⋊C₄×D₃₁

C₂

yes

maximal, characteristic

#11758467

C₂

yes

maximal, characteristic, complemented

#11758466

C₂

yes

maximal, complemented

#11758465

C₂

yes

maximal, characteristic, complemented

#11758449

C₂

yes

maximal, characteristic

#11758447

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₃₁

C₂²

yes

—

C₂×C₄×D₃₁

C₂²

yes

—

C₂×C₄×D₃₁

C₂²

yes

characteristic

C₂×Dic₆₂

C₂²

yes

—

C₄⋊C₄×C₃₁

C₂²

yes

characteristic

C₄⋊C₄×C₃₁

C₂²

yes

—

C₄⋊C₄×C₃₁

C₂²

yes

characteristic

C₄²×C₃₁

C₂²

yes

characteristic

(C₂×D₃₁)⋊C₄

C₂²

yes

characteristic

(C₂×D₃₁)⋊C₄

C₂²

yes

characteristic

(C₂×D₃₁)⋊C₄

C₂²

yes

—

C₄⋊Dic₃₁

C₂²

yes

characteristic

C₄⋊Dic₃₁

C₂²

yes

characteristic

C₄⋊Dic₃₁

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

characteristic

Dic₃₁⋊C₄

C₂²

yes

characteristic

Dic₃₁⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

C₄×Dic₃₁

C₂²

yes

characteristic

C₄×Dic₃₁

C₂²

yes

—

C₂²×D₃₁

C₂³

yes

characteristic

Q₈×C₃₁

—

C₂×C₄×C₃₁

C₂³

yes

—

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂×C₄×C₃₁

C₂³

yes

—

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂×Dic₃₁

C₂³

yes

characteristic

C₂×Dic₃₁

C₂³

yes

—

C₂×Dic₃₁

C₂³

yes

characteristic

C₂×Dic₃₁

C₂³

yes

—

C₂×Dic₃₁

C₂³

yes

characteristic

C₄×D₃₁

—

C₄×D₃₁

Q₈

yes

complemented

Dic₆₂

—

C₂²×C₃₁

C₂⁴

yes

characteristic

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

C₂×Q₈

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂×Q₈

yes

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

C₂×Q₈

yes

—

#544

—

maximal, Sylow

C₂×C₃₁

C₂²×Q₈

yes

characteristic

C₂×C₃₁

H(5,𝔽₂)

yes

characteristic

C₂×C₃₁

2₋¹⁺⁴

yes

characteristic

D₃₁

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

D₃₁

yes

characteristic, complemented

#125

—

#125

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

#117

—

#117

—

C₂×C₄⋊C₄

—

C₃₁

#544

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

C₂×D₃₁

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₃₁

yes

characteristic

C₄⋊C₄

C₂×D₃₁

yes

characteristic

C₄⋊C₄

C₂×D₃₁

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

100

C₂²⋊C₄

—

101

C₄²

—

102

C₄²

—

103

C₄²

C₂×D₃₁

yes

characteristic

104

C₂³

—

105

Q₈

—

106

Q₈

—

124

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

124

—

114

C₂×C₄

C₂²×D₃₁

yes

—

115

C₂×C₄

C₂²×D₃₁

yes

characteristic

116

C₂×C₄

C₂²×D₃₁

yes

characteristic

117

C₂×C₄

C₂²×D₃₁

yes

—

118

C₂×C₄

C₂²×D₃₁

yes

characteristic

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

C₂³×D₃₁

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

Q₈×D₃₁

yes

—

133

C₂

—

134

C₂

#11758561

yes

characteristic

135

C₂

C₂×Q₈×D₃₁

yes

characteristic

136

C₂

#11758564

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₁
commutator, socle, cosocle	C₂²×C₃₁	C₂⁴
Fitting	C₃₁×C₄⋊Q₈	C₂
prime radical	C₂²×D₃₁	C₂³
hypercenter	C₄⋊Q₈	D₃₁
hypocenter	C₃₁	#544
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

C₂²×D₃₁

C₂³

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. #11758521	C₂	semidirect
7. #11758467	C₂	semidirect
8. #11758466	C₂	semidirect
9. #11758465	C₂	semidirect
11. #11758447	C₂	semidirect
48. C₄×D₃₁	Q₈	semidirect
71. C₄⋊Q₈	D₃₁	semidirect
80. C₃₁	#544	semidirect

H ≅ G/N

type

1. C₃₁×C₄⋊Q₈

C₂

semidirect

3. #11758521

C₂

semidirect

7. #11758467

C₂

semidirect

8. #11758466

C₂

semidirect

9. #11758465

C₂

semidirect

11. #11758447

C₂

semidirect

48. C₄×D₃₁

Q₈

semidirect

71. C₄⋊Q₈

D₃₁

semidirect

80. C₃₁

#544

semidirect

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

subgroup	index
C₃₁×C₄⋊Q₈	2
#11758523	2
#11758521	2
#11758473	2
#11758469	2
C₄⋊C₄×D₃₁	2
#11758467	2
#11758466	2
#11758465	2
#11758449	2
#11758447	2
#544	31

subgroup

index

C₃₁×C₄⋊Q₈

C₄⋊C₄×D₃₁

Subgroups — #14521676

Conjugacy classes of subgroups

Complements

Maximal subgroups