Group #14521757 — subgroups

Subgroups — #14521757

Order 1984.

228 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 107 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	1	yes	maximal, characteristic, complemented
3	Q₈×Dic₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₂²⋊D₃₁	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Dic₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#11758474	C₂	1	2	yes	maximal, complemented
7	C₄⋊C₄×D₃₁	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	(C₂²⋊D₃₁)⋊C₄	C₂	1	2	yes	maximal
11	#11758455	C₂	1	2	yes	maximal, complemented
12	C₂×Q₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊D₃₁	C₂²	1	1	yes	characteristic
15	C₂²×Dic₃₁	C₂²	1	2	yes	—
16	C₂×C₄×D₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×D₃₁	C₂²	1	2	yes	—
18	C₂×Dic₆₂	C₂²	1	2	yes	—
19	C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
21	C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
22	C₂²⋊Dic₃₁	C₂²	1	1	yes	characteristic
23	(C₂×D₃₁)⋊C₄	C₂²	1	2	yes	—
24	(C₂×D₃₁)⋊C₄	C₂²	1	1	yes	characteristic
25	C₄⋊Dic₃₁	C₂²	1	2	yes	—
26	C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
27	C₄⋊Dic₃₁	C₂²	1	2	yes	—
28	C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
29	Dic₃₁⋊C₄	C₂²	1	2	yes	—
30	Dic₃₁⋊C₄	C₂²	1	2	yes	—
31	Dic₃₁⋊C₄	C₂²	1	2	yes	—
32	Dic₃₁⋊C₄	C₂²	1	1	yes	characteristic
33	C₄×Dic₃₁	C₂²	1	2	yes	—
34	C₄×Dic₃₁	C₂²	1	1	yes	characteristic
35	C₂³×C₃₁	C₂³	1	1	yes	characteristic
36	C₂²×D₃₁	C₂³	1	1	yes	characteristic
37	Q₈×C₃₁	—	2	2	no	—
38	C₂×C₄×C₃₁	—	2	2	no	—
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×C₄×C₃₁	C₂³	1	1	yes	characteristic
43	C₂²⋊D₃₁	Q₈	1	4	yes	complemented
44	C₂×Dic₃₁	—	4	2	no	—
45	C₂×Dic₃₁	C₂³	1	2	yes	—
46	C₂×Dic₃₁	C₂³	1	1	yes	characteristic
47	C₂×Dic₃₁	C₂³	1	1	yes	characteristic
48	C₂×Dic₃₁	C₂³	1	2	yes	—
49	C₂×Dic₃₁	C₂³	1	1	yes	characteristic
50	C₄×D₃₁	—	2	2	no	—
51	C₄×D₃₁	—	4	2	no	—
52	Dic₆₂	—	4	2	no	—
53	C₂²×C₃₁	—	2	1	no	—
54	C₂²×C₃₁	—	2	1	no	—
55	C₂²×C₃₁	C₂×Q₈	1	2	yes	—
56	C₂²×C₃₁	C₂⁴	1	1	yes	characteristic
57	C₂×D₃₁	—	2	1	no	—
58	C₂×D₃₁	—	2	1	no	—
59	C₂×D₃₁	C₂×Q₈	1	2	yes	—
60	C₄×C₃₁	—	2	1	no	—
61	C₄×C₃₁	—	2	2	no	—
62	C₄×C₃₁	—	2	2	no	—
63	C₄×C₃₁	#41	1	2	yes	—
64	Dic₃₁	—	2	1	no	—
65	Dic₃₁	—	2	1	no	—
66	Dic₃₁	—	2	2	no	—
67	Dic₃₁	—	2	2	no	—
68	Dic₃₁	C₂×Q₈	1	2	yes	—
69	#554	—	31	1	no	maximal, Sylow
70	C₂×C₃₁	—	2	2	no	—
71	C₂×C₃₁	C₂²×Q₈	1	1	yes	characteristic
72	C₂×C₃₁	C₂×#41	1	1	yes	characteristic
73	C₂×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
74	D₃₁	—	2	2	no	—
75	C₄⋊Q₈	—	31	1	no	—
76	#125	—	31	2	no	—
77	C₂²⋊Q₈	—	31	1	no	—
78	C₂²⋊Q₈	—	31	2	no	—
79	C₂²⋊Q₈	—	31	2	no	—
80	C₂²⋊Q₈	D₃₁	1	1	yes	characteristic, complemented
81	C₄×Q₈	—	31	1	no	—
82	C₄×D₄	—	31	1	no	—
83	C₄×D₄	—	31	2	no	—
84	C₂×C₄⋊C₄	—	31	1	no	—
85	C₂×C₄⋊C₄	—	31	1	no	—
86	C₃₁	#554	1	1	yes	characteristic, complemented, Sylow
87	C₂×Q₈	—	31	2	no	—
88	C₂×Q₈	C₂×D₃₁	1	1	yes	characteristic
89	C₂×D₄	—	31	1	no	—
90	C₂²×C₄	—	31	2	no	—
91	C₂²×C₄	—	31	1	no	—
92	C₂²×C₄	—	31	2	no	—
93	C₂²×C₄	C₂×D₃₁	1	1	yes	characteristic
94	C₄⋊C₄	—	31	2	no	—
95	C₄⋊C₄	—	31	1	no	—
96	C₄⋊C₄	—	31	2	no	—
97	C₄⋊C₄	—	31	2	no	—
98	C₄⋊C₄	—	31	1	no	—
99	C₄⋊C₄	—	31	2	no	—
100	C₄⋊C₄	—	31	1	no	—
101	C₄⋊C₄	—	31	2	no	—
102	C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic
103	C₄⋊C₄	C₂×D₃₁	1	2	yes	—
104	C₂²⋊C₄	—	31	1	no	—
105	C₂²⋊C₄	—	31	1	no	—
106	C₂²⋊C₄	—	31	2	no	—
107	C₂²⋊C₄	C₂×D₃₁	1	2	yes	—
108	C₄²	—	31	2	no	—
109	C₄²	—	31	1	no	—
110	C₂³	—	31	1	no	—
111	C₂³	C₂²×D₃₁	1	1	yes	characteristic
112	Q₈	—	2	2	no	—
113	Q₈	—	124	2	no	—
114	D₄	—	31	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	31	1	no	—
117	C₂×C₄	—	31	1	no	—
118	C₂×C₄	—	31	2	no	—
119	C₂×C₄	—	31	1	no	—
120	C₂×C₄	—	31	2	no	—
121	C₂×C₄	—	62	2	no	—
122	C₂×C₄	—	124	2	no	—
123	C₂×C₄	—	124	2	no	—
124	C₂×C₄	C₂²×D₃₁	1	2	yes	—
125	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
126	C₂×C₄	C₂²×D₃₁	1	2	yes	—
127	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
128	C₂²	—	2	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	31	2	no	—
131	C₂²	—	62	1	no	—
132	C₂²	—	62	1	no	—
133	C₂²	Q₈×D₃₁	1	2	yes	—
134	C₂²	C₂³×D₃₁	1	1	yes	characteristic
135	C₄	—	2	1	no	—
136	C₄	—	2	2	no	—
137	C₄	—	2	2	no	—
138	C₄	—	31	2	no	—
139	C₄	—	62	1	no	—
140	C₄	—	62	2	no	—
141	C₄	—	62	2	no	—
142	C₄	—	62	1	no	—
143	C₄	#92497	1	2	yes	—
144	C₂	—	2	2	no	—
145	C₂	—	62	2	no	—
146	C₂	C₂×#92497	1	1	yes	characteristic
147	C₂	C₂×Q₈×D₃₁	1	1	yes	characteristic
148	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, characteristic, complemented

Q₈×Dic₃₁

C₂

yes

maximal, characteristic, complemented

C₄×C₂²⋊D₃₁

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₃₁

C₂

yes

maximal, characteristic, complemented

#11758474

C₂

yes

maximal, complemented

C₄⋊C₄×D₃₁

C₂

yes

maximal, characteristic, complemented

#11758466

C₂

yes

maximal, complemented

#11758465

C₂

yes

maximal, characteristic, complemented

(C₂²⋊D₃₁)⋊C₄

C₂

yes

maximal

#11758455

C₂

yes

maximal, complemented

C₂×Q₈×C₃₁

C₂²

yes

characteristic

C₂²×C₄×C₃₁

C₂²

yes

characteristic

C₂×C₂²⋊D₃₁

C₂²

yes

characteristic

C₂²×Dic₃₁

C₂²

yes

—

C₂×C₄×D₃₁

C₂²

yes

characteristic

C₂×C₄×D₃₁

C₂²

yes

—

C₂×Dic₆₂

C₂²

yes

—

C₄⋊C₄×C₃₁

C₂²

yes

characteristic

C₄⋊C₄×C₃₁

C₂²

yes

—

C₂²⋊C₄×C₃₁

C₂²

yes

—

C₂²⋊Dic₃₁

C₂²

yes

characteristic

(C₂×D₃₁)⋊C₄

C₂²

yes

—

(C₂×D₃₁)⋊C₄

C₂²

yes

characteristic

C₄⋊Dic₃₁

C₂²

yes

—

C₄⋊Dic₃₁

C₂²

yes

characteristic

C₄⋊Dic₃₁

C₂²

yes

—

C₄⋊Dic₃₁

C₂²

yes

characteristic

Dic₃₁⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

characteristic

C₄×Dic₃₁

C₂²

yes

—

C₄×Dic₃₁

C₂²

yes

characteristic

C₂³×C₃₁

C₂³

yes

characteristic

C₂²×D₃₁

C₂³

yes

characteristic

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂³

yes

—

C₂×C₄×C₃₁

C₂³

yes

—

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂×C₄×C₃₁

C₂³

yes

characteristic

C₂²⋊D₃₁

Q₈

yes

complemented

C₂×Dic₃₁

—

C₂×Dic₃₁

C₂³

yes

—

C₂×Dic₃₁

C₂³

yes

characteristic

C₂×Dic₃₁

C₂³

yes

characteristic

C₂×Dic₃₁

C₂³

yes

—

C₂×Dic₃₁

C₂³

yes

characteristic

C₄×D₃₁

—

C₄×D₃₁

—

Dic₆₂

—

C₂²×C₃₁

—

C₂²×C₃₁

—

C₂²×C₃₁

C₂×Q₈

yes

—

C₂²×C₃₁

C₂⁴

yes

characteristic

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

C₂×Q₈

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#41

yes

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

C₂×Q₈

yes

—

#554

—

maximal, Sylow

C₂×C₃₁

—

C₂×C₃₁

C₂²×Q₈

yes

characteristic

C₂×C₃₁

C₂×#41

yes

characteristic

C₂×C₃₁

H(5,𝔽₂)

yes

characteristic

D₃₁

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

D₃₁

yes

characteristic, complemented

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₃₁

#554

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

C₂×D₃₁

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₃₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

C₂×D₃₁

yes

characteristic

103

C₄⋊C₄

C₂×D₃₁

yes

—

104

C₂²⋊C₄

—

105

C₂²⋊C₄

—

106

C₂²⋊C₄

—

107

C₂²⋊C₄

C₂×D₃₁

yes

—

108

C₄²

—

109

C₄²

—

110

C₂³

—

111

C₂³

C₂²×D₃₁

yes

characteristic

112

Q₈

—

113

Q₈

—

124

—

114

D₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

124

—

123

C₂×C₄

—

124

—

124

C₂×C₄

C₂²×D₃₁

yes

—

125

C₂×C₄

C₂²×D₃₁

yes

characteristic

126

C₂×C₄

C₂²×D₃₁

yes

—

127

C₂×C₄

C₂²×D₃₁

yes

characteristic

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

Q₈×D₃₁

yes

—

134

C₂²

C₂³×D₃₁

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

#92497

yes

—

144

C₂

—

145

C₂

—

146

C₂

C₂×#92497

yes

characteristic

147

C₂

C₂×Q₈×D₃₁

yes

characteristic

148

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	C₂×C₂²⋊D₃₁	C₂²
hypercenter	C₂²⋊Q₈	D₃₁
hypocenter	C₃₁	#554
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

C₂×C₂²⋊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
3. Q₈×Dic₃₁	C₂	semidirect
5. C₂×C₄⋊Dic₃₁	C₂	semidirect
6. #11758474	C₂	semidirect
7. C₄⋊C₄×D₃₁	C₂	semidirect
8. #11758466	C₂	semidirect
9. #11758465	C₂	semidirect
11. #11758455	C₂	semidirect
43. C₂²⋊D₃₁	Q₈	semidirect
80. C₂²⋊Q₈	D₃₁	semidirect
86. C₃₁	#554	semidirect

H ≅ G/N

type

1. C₃₁×C₂²⋊Q₈

C₂

semidirect

2. #11758523

C₂

semidirect

3. Q₈×Dic₃₁

C₂

semidirect

5. C₂×C₄⋊Dic₃₁

C₂

semidirect

6. #11758474

C₂

semidirect

7. C₄⋊C₄×D₃₁

C₂

semidirect

8. #11758466

C₂

semidirect

9. #11758465

C₂

semidirect

11. #11758455

C₂

semidirect

43. C₂²⋊D₃₁

Q₈

semidirect

80. C₂²⋊Q₈

D₃₁

semidirect

86. C₃₁

#554

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
Q₈×Dic₃₁	2
C₄×C₂²⋊D₃₁	2
C₂×C₄⋊Dic₃₁	2
#11758474	2
C₄⋊C₄×D₃₁	2
#11758466	2
#11758465	2
(C₂²⋊D₃₁)⋊C₄	2
#11758455	2
#554	31

subgroup

index

C₃₁×C₂²⋊Q₈

#11758523

Q₈×Dic₃₁

C₄×C₂²⋊D₃₁

C₂×C₄⋊Dic₃₁

C₄⋊C₄×D₃₁

(C₂²⋊D₃₁)⋊C₄

Subgroups — #14521757

Conjugacy classes of subgroups

Complements

Maximal subgroups