Group #16845324 — subgroups

Subgroups — #16845324

Order 3968.

178 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 57 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	#14521401	C₂	1	1	yes	maximal, characteristic, complemented
2	#14521399	C₂	1	1	yes	maximal, characteristic, complemented
3	#14521397	C₂	1	1	yes	maximal, characteristic
4	#14521397	C₂	1	1	yes	maximal, characteristic
5	#14521396	C₂	1	1	yes	maximal, characteristic
6	C₈×C₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#487	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	#11758451	C₂²	1	1	yes	characteristic
11	#11758446	C₂²	1	1	yes	characteristic
12	C₃₁×#103	—	2	2	no	—
13	C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	#11758423	—	2	2	no	—
15	#11758409	—	2	2	no	—
16	#11758398	—	2	2	no	—
17	#11758397	—	2	2	no	—
18	#11758397	—	2	2	no	—
19	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
20	C₄×C₃₁⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₃₁	—	2	1	no	—
22	C₂×Q₈×C₃₁	—	2	1	no	—
23	C₂×D₁₂₄	—	2	1	no	—
24	C₂×D₁₂₄	—	2	1	no	—
25	C₂×D₁₂₄	—	2	1	no	—
26	C₂×Dic₆₂	—	2	1	no	—
27	C₂×C₈×C₃₁	D₄	1	2	yes	—
28	C₄⋊C₄×C₃₁	—	2	1	no	—
29	C₄⋊C₄×C₃₁	—	2	1	no	—
30	C₄⋊C₄×C₃₁	—	2	1	no	—
31	C₄⋊C₄×C₃₁	—	2	1	no	—
32	C₄²×C₃₁	C₂³	1	1	yes	characteristic
33	C₄⋊Dic₃₁	—	2	1	no	—
34	C₄⋊Dic₃₁	—	2	1	no	—
35	C₂×C₃₁⋊C₈	D₄	1	2	yes	—
36	C₂×C₃₁⋊C₈	D₄	1	2	yes	—
37	C₂²×D₃₁	—	4	1	no	—
38	Q₈×C₃₁	—	4	2	no	—
39	Q₈×C₃₁	—	4	2	no	—
40	C₂×C₄×C₃₁	—	4	1	no	—
41	C₂×C₄×C₃₁	—	4	1	no	—
42	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
45	C₂×Dic₃₁	—	4	1	no	—
46	D₁₂₄	—	4	2	no	—
47	D₁₂₄	—	4	2	no	—
48	D₁₂₄	—	4	2	no	—
49	Dic₆₂	—	4	2	no	—
50	C₈×C₃₁	—	2	2	no	—
51	C₃₁⋊C₈	—	2	2	no	—
52	C₃₁⋊C₈	—	2	2	no	—
53	#1711	—	31	1	no	maximal, Sylow
54	C₂²×C₃₁	#127	1	1	yes	characteristic
55	C₂×D₃₁	—	8	1	no	—
56	C₂×D₃₁	—	8	1	no	—
57	C₂×D₃₁	—	8	1	no	—
58	C₄×C₃₁	—	8	1	no	—
59	C₄×C₃₁	—	8	1	no	—
60	C₄×C₃₁	#135	1	2	yes	—
61	C₄×C₃₁	#135	1	2	yes	—
62	C₄×C₃₁	#135	1	2	yes	—
63	Dic₃₁	—	8	1	no	—
64	#487	—	31	1	no	—
65	#487	—	31	1	no	—
66	#487	D₃₁	1	1	yes	characteristic, complemented
67	#486	—	31	1	no	—
68	#486	—	31	1	no	—
69	#486	—	31	1	no	—
70	C₈²	—	31	1	no	—
71	C₂×C₃₁	#495	1	1	yes	characteristic
72	C₂×C₃₁	#495	1	1	yes	characteristic
73	C₂×C₃₁	#495	1	1	yes	characteristic
74	D₃₁	—	16	1	no	—
75	C₄⋊Q₈	—	31	1	no	—
76	C₄⋊Q₈	C₂×D₃₁	1	1	yes	characteristic
77	C₄⋊Q₈	C₂×D₃₁	1	1	yes	characteristic
78	#127	—	31	1	no	—
79	#103	—	2	2	no	—
80	#103	—	62	2	no	—
81	#103	—	62	2	no	—
82	#102	—	62	2	no	—
83	#102	—	62	2	no	—
84	#102	—	62	2	no	—
85	C₄×C₈	—	31	1	no	—
86	C₄×C₈	—	31	1	no	—
87	C₄×C₈	C₂×D₃₁	1	1	yes	characteristic
88	C₃₁	#1711	1	1	yes	characteristic, complemented, Sylow
89	C₂×Q₈	—	2	1	no	—
90	C₂×Q₈	—	2	1	no	—
91	C₂×Q₈	—	62	1	no	—
92	C₂×D₄	—	62	1	no	—
93	C₂×D₄	—	62	1	no	—
94	C₂×D₄	—	62	1	no	—
95	C₂×C₈	—	31	2	no	—
96	C₂×C₈	—	31	2	no	—
97	C₂×C₈	C₂²⋊D₃₁	1	2	yes	—
98	C₄⋊C₄	—	2	1	no	—
99	C₄⋊C₄	—	2	1	no	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	62	1	no	—
103	C₄⋊C₄	—	62	1	no	—
104	C₄²	C₂²×D₃₁	1	1	yes	characteristic
105	C₂³	—	124	1	no	—
106	Q₈	—	4	2	no	—
107	Q₈	—	4	2	no	—
108	Q₈	—	124	2	no	—
109	D₄	—	124	2	no	—
110	D₄	—	124	2	no	—
111	D₄	—	124	2	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	—	4	1	no	—
114	C₂×C₄	—	124	1	no	—
115	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
116	C₂×C₄	D₄×D₃₁	1	1	yes	characteristic
117	C₂×C₄	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
118	C₈	—	2	2	no	—
119	C₈	—	62	2	no	—
120	C₈	—	62	2	no	—
121	C₂²	—	248	1	no	—
122	C₂²	—	248	1	no	—
123	C₂²	—	248	1	no	—
124	C₂²	#11758517	1	1	yes	characteristic
125	C₄	—	8	1	no	—
126	C₄	—	8	1	no	—
127	C₄	—	248	1	no	—
128	C₄	#11758493	1	2	yes	—
129	C₄	#11758496	1	2	yes	—
130	C₄	#11758527	1	2	yes	—
131	C₂	—	496	1	no	—
132	C₂	#14521430	1	1	yes	characteristic
133	C₂	#14521431	1	1	yes	characteristic
134	C₂	#14521434	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14521401

C₂

yes

maximal, characteristic, complemented

#14521399

C₂

yes

maximal, characteristic, complemented

#14521397

C₂

yes

maximal, characteristic

#14521397

C₂

yes

maximal, characteristic

#14521396

C₂

yes

maximal, characteristic

C₈×C₃₁⋊C₈

C₂

yes

maximal, characteristic, complemented

C₃₁×#487

C₂

yes

maximal, characteristic, complemented

C₃₁×C₄⋊Q₈

C₂²

yes

characteristic

C₃₁×C₄⋊Q₈

C₂²

yes

characteristic

#11758451

C₂²

yes

characteristic

#11758446

C₂²

yes

characteristic

C₃₁×#103

—

C₄×C₈×C₃₁

C₂²

yes

—

—

—

—

—

C₄×C₃₁⋊C₈

C₂²

yes

characteristic

C₄×C₃₁⋊C₈

C₂²

yes

characteristic

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂×D₁₂₄

—

C₂×D₁₂₄

—

C₂×D₁₂₄

—

C₂×Dic₆₂

—

C₂×C₈×C₃₁

D₄

yes

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

C₂³

yes

characteristic

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₂×C₃₁⋊C₈

D₄

yes

—

C₂×C₃₁⋊C₈

D₄

yes

—

C₂²×D₃₁

—

Q₈×C₃₁

—

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₃₁

C₂×D₄

yes

characteristic

C₂×Dic₃₁

—

D₁₂₄

—

D₁₂₄

—

D₁₂₄

—

Dic₆₂

—

C₈×C₃₁

—

C₃₁⋊C₈

—

C₃₁⋊C₈

—

—

C₂²×C₃₁

yes

C₂×D₃₁

—

C₂×D₃₁

—

C₂×D₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

#135

yes

—

C₄×C₃₁

#135

yes

—

C₄×C₃₁

#135

yes

—

Dic₃₁

—

#487

—

#487

—

#487

D₃₁

yes

characteristic, complemented

#486

—

#486

—

#486

—

C₈²

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

yes

D₃₁

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃₁

yes

characteristic

C₄⋊Q₈

C₂×D₃₁

yes

—

—

—

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₃₁

yes

characteristic

C₃₁

#1711

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₃₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄²

C₂²×D₃₁

yes

characteristic

105

C₂³

—

124

—

106

Q₈

—

107

Q₈

—

108

Q₈

—

124

—

109

D₄

—

124

—

110

D₄

—

124

—

111

D₄

—

124

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

124

—

115

C₂×C₄

D₄×D₃₁

yes

characteristic

116

C₂×C₄

D₄×D₃₁

yes

characteristic

117

C₂×C₄

C₂×C₂²⋊D₃₁

yes

characteristic

118

C₈

—

119

C₈

—

120

C₈

—

121

C₂²

—

248

—

122

C₂²

—

248

—

123

C₂²

—

248

—

124

C₂²

#11758517

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

248

—

128

C₄

#11758493

yes

—

129

C₄

#11758496

yes

—

130

C₄

#11758527

yes

—

131

C₂

—

496

—

132

C₂

#14521430

yes

characteristic

133

C₂

#14521431

yes

characteristic

134

C₂

#14521434

yes

characteristic

	subgroup	G/H
center	C₂²	#11758517
commutator, cosocle	C₄²×C₃₁	C₂³
Frattini	C₄²	C₂²×D₃₁
Fitting	C₃₁×#487	C₂
socle	C₂²×C₃₁	#127
prime radical	#11758451	C₂²
hypercenter	#487	D₃₁
hypocenter	C₃₁	#1711
LCS	G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #487

subgroup

G/H

center

C₂²

#11758517

commutator, cosocle

C₄²×C₃₁

C₂³

Frattini

C₄²

C₂²×D₃₁

Fitting

C₃₁×#487

C₂

C₂²×C₃₁

C₂²

D₃₁

C₃₁

G ▷ C₄²×C₃₁ ▷ C₂²×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #487

N	H ≅ G/N	type
1. #14521401	C₂	semidirect
2. #14521399	C₂	semidirect
6. C₈×C₃₁⋊C₈	C₂	semidirect
7. C₃₁×#487	C₂	semidirect
66. #487	D₃₁	semidirect
88. C₃₁	#1711	semidirect

H ≅ G/N

type

1. #14521401

C₂

semidirect

2. #14521399

C₂

semidirect

6. C₈×C₃₁⋊C₈

C₂

semidirect

7. C₃₁×#487

C₂

semidirect

66. #487

D₃₁

semidirect

88. C₃₁

#1711

semidirect

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

subgroup	index
#14521401	2
#14521399	2
#14521397	2
#14521396	2
C₈×C₃₁⋊C₈	2
C₃₁×#487	2
#1711	31

subgroup

index

C₈×C₃₁⋊C₈

C₃₁×#487

#1711

Subgroups — #16845324

Conjugacy classes of subgroups

Complements

Maximal subgroups