Group #14521740 — subgroups

Subgroups — #14521740

Order 1984.

214 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#11758523	C₂	1	1	yes	maximal, characteristic, complemented
3	#11758523	C₂	1	1	yes	maximal, characteristic, complemented
4	#11758471	C₂	1	2	yes	maximal
5	#11758469	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄⋊C₄×D₃₁	C₂	1	1	yes	maximal, characteristic, complemented
7	#11758466	C₂	1	2	yes	maximal, complemented
8	#11758453	C₂	1	2	yes	maximal, complemented
9	C₄×D₁₂₄	C₂	1	1	yes	maximal, characteristic
10	#11758449	C₂	1	2	yes	maximal, complemented
11	#11758447	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×D₁₂₄	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₃₁	C₂²	1	2	yes	—
15	C₂×C₄×D₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₃₁	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	C₄²×C₃₁	C₂²	1	2	yes	—
21	(C₂×D₃₁)⋊C₄	C₂²	1	1	yes	characteristic
22	(C₂×D₃₁)⋊C₄	C₂²	1	2	yes	—
23	(C₂×D₃₁)⋊C₄	C₂²	1	1	yes	characteristic
24	(C₂×D₃₁)⋊C₄	C₂²	1	2	yes	—
25	(C₂×D₃₁)⋊C₄	C₂²	1	2	yes	—
26	C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
27	C₄⋊Dic₃₁	C₂²	1	1	yes	characteristic
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	2	yes	—
33	Dic₃₁⋊C₄	C₂²	1	2	yes	—
34	C₄×Dic₃₁	C₂²	1	2	yes	—
35	C₂²×D₃₁	C₂³	1	1	yes	characteristic
36	C₂²×D₃₁	C₂³	1	1	yes	characteristic
37	Q₈×C₃₁	—	2	2	no	—
38	C₂×C₄×C₃₁	C₂³	1	2	yes	—
39	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₃₁	C₂³	1	2	yes	—
42	C₂×C₄×C₃₁	C₂³	1	1	yes	characteristic
43	C₂×Dic₃₁	C₂³	1	1	yes	characteristic
44	C₂×Dic₃₁	C₂³	1	2	yes	—
45	C₂×Dic₃₁	C₂³	1	2	yes	—
46	C₂×Dic₃₁	C₂³	1	1	yes	characteristic
47	D₁₂₄	—	2	2	no	—
48	C₄×D₃₁	—	2	4	no	—
49	C₄×D₃₁	—	2	2	no	—
50	C₄×D₃₁	—	2	2	no	—
51	C₂²×C₃₁	C₂⁴	1	1	yes	characteristic
52	C₂×D₃₁	—	2	1	no	—
53	C₂×D₃₁	—	2	1	no	—
54	C₂×D₃₁	—	2	1	no	—
55	C₂×D₃₁	—	2	1	no	—
56	C₂×D₃₁	—	2	1	no	—
57	C₂×D₃₁	#41	1	2	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	C₄×C₃₁	#41	1	2	yes	—
63	Dic₃₁	—	2	1	no	—
64	Dic₃₁	—	2	1	no	—
65	Dic₃₁	—	2	2	no	—
66	Dic₃₁	—	2	2	no	—
67	#552	—	31	1	no	maximal, Sylow
68	C₂×C₃₁	C₂×#41	1	1	yes	characteristic
69	C₂×C₃₁	C₂×#41	1	1	yes	characteristic
70	C₂×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
71	D₃₁	—	2	2	no	—
72	D₃₁	—	4	1	no	—
73	#126	—	31	2	no	—
74	#125	—	31	2	no	—
75	#125	—	31	1	no	—
76	#123	—	31	2	no	—
77	C₂²⋊Q₈	—	31	1	no	—
78	C₂²⋊Q₈	—	31	1	no	—
79	C₄×Q₈	D₃₁	1	1	yes	characteristic, complemented
80	C₄×D₄	—	31	1	no	—
81	#117	—	31	1	no	—
82	#117	—	31	2	no	—
83	C₂×C₄⋊C₄	—	31	1	no	—
84	C₃₁	#552	1	1	yes	characteristic, complemented, Sylow
85	C₂×Q₈	C₂×D₃₁	1	1	yes	characteristic
86	C₂×D₄	—	31	1	no	—
87	C₂²×C₄	—	31	2	no	—
88	C₂²×C₄	—	31	1	no	—
89	C₂²×C₄	—	31	1	no	—
90	C₄⋊C₄	—	31	2	no	—
91	C₄⋊C₄	—	31	2	no	—
92	C₄⋊C₄	—	31	2	no	—
93	C₄⋊C₄	—	31	2	no	—
94	C₄⋊C₄	—	31	2	no	—
95	C₄⋊C₄	—	31	1	no	—
96	C₄⋊C₄	—	31	1	no	—
97	C₄⋊C₄	—	31	1	no	—
98	C₄⋊C₄	C₂×D₃₁	1	1	yes	characteristic
99	C₄⋊C₄	C₂×D₃₁	1	2	yes	—
100	C₂²⋊C₄	—	31	1	no	—
101	C₂²⋊C₄	—	31	2	no	—
102	C₂²⋊C₄	—	31	2	no	—
103	C₂²⋊C₄	—	31	1	no	—
104	C₂²⋊C₄	—	31	2	no	—
105	C₄²	—	31	2	no	—
106	C₄²	C₂×D₃₁	1	2	yes	—
107	C₄²	C₂×D₃₁	1	1	yes	characteristic
108	C₂³	—	31	1	no	—
109	C₂³	—	31	1	no	—
110	Q₈	—	2	2	no	—
111	D₄	—	62	2	no	—
112	C₂×C₄	—	31	1	no	—
113	C₂×C₄	—	31	2	no	—
114	C₂×C₄	—	31	1	no	—
115	C₂×C₄	—	31	2	no	—
116	C₂×C₄	—	62	4	no	—
117	C₂×C₄	—	62	2	no	—
118	C₂×C₄	—	62	2	no	—
119	C₂×C₄	C₂²×D₃₁	1	2	yes	—
120	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
121	C₂×C₄	C₂²×D₃₁	1	2	yes	—
122	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
123	C₂×C₄	C₂²×D₃₁	1	1	yes	characteristic
124	C₂²	—	31	2	no	—
125	C₂²	—	62	1	no	—
126	C₂²	—	62	1	no	—
127	C₂²	—	62	1	no	—
128	C₂²	—	62	1	no	—
129	C₂²	—	62	1	no	—
130	C₂²	C₂³×D₃₁	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	1	no	—
133	C₄	—	2	2	no	—
134	C₄	—	2	1	no	—
135	C₄	—	62	1	no	—
136	C₄	—	62	1	no	—
137	C₄	—	62	2	no	—
138	C₄	—	62	2	no	—
139	C₄	#92495	1	2	yes	—
140	C₂	—	62	2	no	—
141	C₂	—	124	1	no	—
142	C₂	C₂×#92495	1	1	yes	characteristic
143	C₂	#11758564	1	1	yes	characteristic
144	C₂	#41×D₃₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Q₈×C₃₁

C₂

yes

maximal, characteristic, complemented

#11758523

C₂

yes

maximal, characteristic, complemented

#11758523

C₂

yes

maximal, characteristic, complemented

#11758471

C₂

yes

maximal

#11758469

C₂

yes

maximal, characteristic, complemented

C₄⋊C₄×D₃₁

C₂

yes

maximal, characteristic, complemented

#11758466

C₂

yes

maximal, complemented

#11758453

C₂

yes

maximal, complemented

C₄×D₁₂₄

C₂

yes

maximal, characteristic

#11758449

C₂

yes

maximal, complemented

#11758447

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₃₁

C₂²

yes

characteristic

C₂×D₁₂₄

C₂²

yes

characteristic

C₂×C₄×D₃₁

C₂²

yes

—

C₂×C₄×D₃₁

C₂²

yes

characteristic

C₂×C₄×D₃₁

C₂²

yes

characteristic

C₄⋊C₄×C₃₁

C₂²

yes

—

C₄⋊C₄×C₃₁

C₂²

yes

characteristic

C₄²×C₃₁

C₂²

yes

characteristic

C₄²×C₃₁

C₂²

yes

—

(C₂×D₃₁)⋊C₄

C₂²

yes

characteristic

(C₂×D₃₁)⋊C₄

C₂²

yes

—

(C₂×D₃₁)⋊C₄

C₂²

yes

characteristic

(C₂×D₃₁)⋊C₄

C₂²

yes

—

(C₂×D₃₁)⋊C₄

C₂²

yes

—

C₄⋊Dic₃₁

C₂²

yes

characteristic

C₄⋊Dic₃₁

C₂²

yes

characteristic

C₄⋊Dic₃₁

C₂²

yes

characteristic

Dic₃₁⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

Dic₃₁⋊C₄

C₂²

yes

—

C₄×Dic₃₁

C₂²

yes

—

C₂²×D₃₁

C₂³

yes

characteristic

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

—

C₂×Dic₃₁

C₂³

yes

characteristic

D₁₂₄

—

C₄×D₃₁

—

C₄×D₃₁

—

C₄×D₃₁

—

C₂²×C₃₁

C₂⁴

yes

characteristic

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

#41

yes

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#41

yes

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

Dic₃₁

—

#552

—

maximal, Sylow

C₂×C₃₁

C₂×#41

yes

characteristic

C₂×C₃₁

C₂×#41

yes

characteristic

C₂×C₃₁

2₋¹⁺⁴

yes

characteristic

D₃₁

—

D₃₁

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

D₃₁

yes

characteristic, complemented

C₄×D₄

—

#117

—

#117

—

C₂×C₄⋊C₄

—

C₃₁

#552

yes

characteristic, complemented, Sylow

C₂×Q₈

C₂×D₃₁

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×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

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

—

104

C₂²⋊C₄

—

105

C₄²

—

106

C₄²

C₂×D₃₁

yes

—

107

C₄²

C₂×D₃₁

yes

characteristic

108

C₂³

—

109

C₂³

—

110

Q₈

—

111

D₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₂²×D₃₁

yes

—

120

C₂×C₄

C₂²×D₃₁

yes

characteristic

121

C₂×C₄

C₂²×D₃₁

yes

—

122

C₂×C₄

C₂²×D₃₁

yes

characteristic

123

C₂×C₄

C₂²×D₃₁

yes

characteristic

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

C₂³×D₃₁

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

#92495

yes

—

140

C₂

—

141

C₂

—

124

—

142

C₂

C₂×#92495

yes

characteristic

143

C₂

#11758564

yes

characteristic

144

C₂

#41×D₃₁

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₁
commutator, socle, cosocle	C₂²×C₃₁	C₂⁴
Fitting	C₄×Q₈×C₃₁	C₂
prime radical	C₂×D₁₂₄	C₂²
hypercenter	C₄×Q₈	D₃₁
hypocenter	C₃₁	#552
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

C₄×Q₈×C₃₁

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. C₄×Q₈×C₃₁	C₂	semidirect
2. #11758523	C₂	semidirect
3. #11758523	C₂	semidirect
5. #11758469	C₂	semidirect
6. C₄⋊C₄×D₃₁	C₂	semidirect
7. #11758466	C₂	semidirect
8. #11758453	C₂	semidirect
10. #11758449	C₂	semidirect
11. #11758447	C₂	semidirect
79. C₄×Q₈	D₃₁	semidirect
84. C₃₁	#552	semidirect

H ≅ G/N

type

1. C₄×Q₈×C₃₁

C₂

semidirect

2. #11758523

C₂

semidirect

3. #11758523

C₂

semidirect

5. #11758469

C₂

semidirect

6. C₄⋊C₄×D₃₁

C₂

semidirect

7. #11758466

C₂

semidirect

8. #11758453

C₂

semidirect

10. #11758449

C₂

semidirect

11. #11758447

C₂

semidirect

79. C₄×Q₈

D₃₁

semidirect

84. C₃₁

#552

semidirect

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

subgroup	index
C₄×Q₈×C₃₁	2
#11758523	2
#11758471	2
#11758469	2
C₄⋊C₄×D₃₁	2
#11758466	2
#11758453	2
C₄×D₁₂₄	2
#11758449	2
#11758447	2
#552	31

subgroup

index

C₄×Q₈×C₃₁

C₄⋊C₄×D₃₁

C₄×D₁₂₄

Subgroups — #14521740

Conjugacy classes of subgroups

Complements

Maximal subgroups