Group #16854088 — subgroups

Subgroups — #16854088

Order 3968.

504 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 197 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	C₂×#11758446	C₂	1	1	yes	maximal, characteristic
2	#14520956	C₂	1	4	yes	maximal
3	#14520954	C₂	1	4	yes	maximal
4	#14520914	C₂	1	4	yes	maximal
5	C₄²×Dic₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₃₁	C₂²	1	2	yes	—
8	C₂²×Dic₆₂	C₂²	1	2	yes	—
9	C₃₁×C₄⋊Q₈	—	2	4	no	—
10	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
11	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
12	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₃₁	C₂²	1	2	yes	—
15	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
16	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
17	C₂×C₄×Dic₃₁	C₂²	1	2	yes	—
18	#11758446	—	2	4	no	—
19	C₃₁⋊#95	C₂²	1	16	yes	—
20	C₂×Q₈×C₃₁	—	2	4	no	—
21	C₂×Q₈×C₃₁	—	2	8	no	—
22	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
23	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
25	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
26	C₂²×Dic₃₁	C₂³	1	4	yes	—
27	C₂²×Dic₃₁	C₂³	1	4	yes	—
28	C₂×Dic₆₂	—	2	4	no	—
29	C₂×Dic₆₂	—	2	8	no	—
30	C₄⋊C₄×C₃₁	—	2	4	no	—
31	C₄⋊C₄×C₃₁	—	2	4	no	—
32	C₄²×C₃₁	D₄	1	4	yes	—
33	C₄⋊Dic₃₁	—	2	4	no	—
34	C₄⋊Dic₃₁	—	2	4	no	—
35	C₄×Dic₃₁	—	2	4	no	—
36	C₄×Dic₃₁	—	2	4	no	—
37	C₄×Dic₃₁	Q₈	1	8	yes	complemented
38	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
39	Q₈×C₃₁	—	4	8	no	—
40	C₂×C₄×C₃₁	—	2	4	no	—
41	C₂×C₄×C₃₁	—	2	8	no	—
42	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₃₁	C₂×Q₈	1	4	yes	—
46	C₂×C₄×C₃₁	#41	1	4	yes	—
47	C₂×C₄×C₃₁	#41	1	4	yes	—
48	C₂×Dic₃₁	—	2	4	no	—
49	C₂×Dic₃₁	—	2	4	no	—
50	C₂×Dic₃₁	—	2	4	no	—
51	C₂×Dic₃₁	—	2	8	no	—
52	C₂×Dic₃₁	C₂×Q₈	1	8	yes	—
53	Dic₆₂	—	4	8	no	—
54	#2862	—	31	1	no	maximal, Sylow
55	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
56	C₂²×C₃₁	C₂²×Q₈	1	2	yes	—
57	C₂²×C₃₁	C₂×#41	1	2	yes	—
58	C₂²×C₃₁	C₂×#41	1	2	yes	—
59	C₄×C₃₁	—	4	4	no	—
60	C₄×C₃₁	#124	1	4	yes	—
61	C₄×C₃₁	#124	1	4	yes	—
62	C₄×C₃₁	C₄⋊Q₈	1	4	yes	—
63	Dic₃₁	—	2	8	no	—
64	Dic₃₁	—	4	4	no	—
65	C₂×C₄⋊Q₈	—	31	1	no	—
66	C₂×C₄⋊Q₈	D₃₁	1	1	yes	characteristic
67	#391	—	31	4	no	—
68	#391	—	31	4	no	—
69	#382	—	31	4	no	—
70	C₄³	—	31	1	no	—
71	C₂×C₃₁	C₂×#124	1	1	yes	characteristic
72	C₂×C₃₁	C₂×#124	1	1	yes	characteristic
73	C₂×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
74	C₂×C₃₁	#533	1	4	yes	—
75	C₂²×Q₈	—	31	2	no	—
76	C₂²×Q₈	C₂×D₃₁	1	2	yes	—
77	C₄⋊Q₈	—	2	4	no	—
78	C₄⋊Q₈	—	62	4	no	—
79	C₂×C₄⋊C₄	—	31	2	no	—
80	C₂×C₄⋊C₄	—	31	2	no	—
81	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
82	C₂×C₄⋊C₄	C₂×D₃₁	1	2	yes	—
83	C₂×C₄²	—	31	2	no	—
84	C₂×C₄²	—	31	2	no	—
85	C₂×C₄²	—	31	2	no	—
86	C₂×C₄²	C₂×D₃₁	1	1	yes	characteristic
87	#95	—	31	16	no	—
88	C₃₁	#2862	1	1	yes	characteristic, complemented, Sylow
89	C₂×Q₈	—	2	8	no	—
90	C₂×Q₈	—	2	4	no	—
91	C₂×Q₈	—	62	8	no	—
92	C₂×Q₈	—	62	4	no	—
93	C₂²×C₄	—	31	4	no	—
94	C₂²×C₄	—	31	4	no	—
95	C₂²×C₄	C₂²×D₃₁	1	4	yes	—
96	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
97	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₃₁	1	1	yes	characteristic
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄⋊C₄	—	62	4	no	—
102	C₄⋊C₄	—	62	4	no	—
103	C₄²	—	31	8	no	—
104	C₄²	—	62	4	no	—
105	C₄²	—	62	4	no	—
106	C₄²	C₂²⋊D₃₁	1	4	yes	—
107	C₂³	C₂³×D₃₁	1	1	yes	characteristic
108	Q₈	—	4	8	no	—
109	Q₈	—	124	8	no	—
110	C₂×C₄	—	2	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	31	8	no	—
113	C₂×C₄	—	62	4	no	—
114	C₂×C₄	—	62	4	no	—
115	C₂×C₄	—	62	8	no	—
116	C₂×C₄	—	62	4	no	—
117	C₂×C₄	#92497	1	4	yes	—
118	C₂×C₄	Q₈×D₃₁	1	4	yes	—
119	C₂×C₄	#92499	1	4	yes	—
120	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
121	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
122	C₂×C₄	C₂×C₂²⋊D₃₁	1	2	yes	—
123	C₂²	C₂×#92497	1	2	yes	—
124	C₂²	C₂×Q₈×D₃₁	1	2	yes	—
125	C₂²	C₂×#92499	1	2	yes	—
126	C₂²	C₂²×C₂²⋊D₃₁	1	1	yes	characteristic
127	C₄	—	4	4	no	—
128	C₄	—	62	8	no	—
129	C₄	—	124	4	no	—
130	C₄	#11758513	1	4	yes	—
131	C₄	#11758521	1	4	yes	—
132	C₄	#11758524	1	4	yes	—
133	C₂	C₂×#11758513	1	1	yes	characteristic
134	C₂	C₂×#11758524	1	1	yes	characteristic
135	C₂	C₂×#11758521	1	1	yes	characteristic
136	C₂	#14521604	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758446

C₂

yes

maximal, characteristic

#14520956

C₂

yes

maximal

#14520954

C₂

yes

maximal

#14520914

C₂

yes

maximal

C₄²×Dic₃₁

C₂

yes

maximal, characteristic

C₂×C₃₁×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂²×Q₈×C₃₁

C₂²

yes

—

C₂²×Dic₆₂

C₂²

yes

—

C₃₁×C₄⋊Q₈

—

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂×C₄²×C₃₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×C₄⋊Dic₃₁

C₂²

yes

—

C₂×C₄×Dic₃₁

C₂²

yes

—

C₂×C₄×Dic₃₁

C₂²

yes

—

C₂×C₄×Dic₃₁

C₂²

yes

—

#11758446

—

C₃₁⋊#95

C₂²

yes

—

C₂×Q₈×C₃₁

—

C₂×Q₈×C₃₁

—

C₂²×C₄×C₃₁

C₂³

yes

—

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×C₄×C₃₁

C₂³

yes

characteristic

C₂²×Dic₃₁

C₂³

yes

—

C₂²×Dic₃₁

C₂³

yes

—

C₂×Dic₆₂

—

C₂×Dic₆₂

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄²×C₃₁

D₄

yes

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

—

C₄×Dic₃₁

—

C₄×Dic₃₁

—

C₄×Dic₃₁

Q₈

yes

complemented

C₂³×C₃₁

C₂⁴

yes

characteristic

Q₈×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×Q₈

yes

—

C₂×C₄×C₃₁

#41

yes

—

C₂×C₄×C₃₁

#41

yes

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

—

C₂×Dic₃₁

C₂×Q₈

yes

—

Dic₆₂

—

#2862

—

maximal, Sylow

C₂²×C₃₁

C₂²×D₄

yes

characteristic

C₂²×C₃₁

C₂²×Q₈

yes

—

C₂²×C₃₁

C₂×#41

yes

—

C₂²×C₃₁

C₂×#41

yes

—

C₄×C₃₁

—

C₄×C₃₁

#124

yes

—

C₄×C₃₁

#124

yes

—

C₄×C₃₁

C₄⋊Q₈

yes

—

Dic₃₁

—

Dic₃₁

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

D₃₁

yes

—

—

—

C₄³

—

C₂×C₃₁

yes

C₂×C₃₁

yes

C₂×C₃₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₃₁

#533

yes

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₃₁

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₁

yes

—

C₂×C₄⋊C₄

C₂×D₃₁

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₁

yes

characteristic

#95

—

C₃₁

#2862

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₁

yes

—

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₂²×C₄

C₂²×D₃₁

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₄²

C₂²⋊D₃₁

yes

—

107

C₂³

C₂³×D₃₁

yes

characteristic

108

Q₈

—

109

Q₈

—

124

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

#92497

yes

—

118

C₂×C₄

Q₈×D₃₁

yes

—

119

C₂×C₄

#92499

yes

—

120

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

121

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

122

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

123

C₂²

C₂×#92497

yes

—

124

C₂²

C₂×Q₈×D₃₁

yes

—

125

C₂²

C₂×#92499

yes

—

126

C₂²

C₂²×C₂²⋊D₃₁

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

124

—

130

C₄

#11758513

yes

—

131

C₄

#11758521

yes

—

132

C₄

#11758524

yes

—

133

C₂

C₂×#11758513

yes

characteristic

134

C₂

C₂×#11758524

yes

characteristic

135

C₂

C₂×#11758521

yes

characteristic

136

C₂

#14521604

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃₁
commutator, socle, prime radical, cosocle	C₂³×C₃₁	C₂⁴
Fitting	C₂×C₃₁×C₄⋊Q₈	C₂
hypercenter	C₂×C₄⋊Q₈	D₃₁
hypocenter	C₃₁	#2862
LCS	G ▷ C₂³×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₃₁

commutator, socle, prime radical, cosocle

C₂³×C₃₁

C₂⁴

Fitting

C₂×C₃₁×C₄⋊Q₈

C₂

hypercenter

C₂×C₄⋊Q₈

D₃₁

C₃₁

G ▷ C₂³×C₃₁ ▷ C₃₁

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
37. C₄×Dic₃₁	Q₈	semidirect
88. C₃₁	#2862	semidirect

H ≅ G/N

type

37. C₄×Dic₃₁

Q₈

semidirect

88. C₃₁

#2862

semidirect

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

subgroup	index
C₂×#11758446	2
#14520956	2
#14520954	2
#14520914	2
C₄²×Dic₃₁	2
C₂×C₃₁×C₄⋊Q₈	2
#2862	31

subgroup

index

C₄²×Dic₃₁

C₂×C₃₁×C₄⋊Q₈

#2862

Subgroups — #16854088

Conjugacy classes of subgroups

Complements

Maximal subgroups