Group #16858385 — subgroups

Subgroups — #16858385

Order 3968.

336 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 139 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	Q₈×Dic₆₂	C₂	1	1	yes	maximal, characteristic
2	#14521399	C₂	1	3	yes	maximal
3	#14521344	C₂	1	3	yes	maximal
4	#14521256	C₂	1	3	yes	maximal
5	Q₈×C₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₄×#92482	C₂	1	3	yes	maximal
7	C₃₁×#557	C₂	1	1	yes	maximal, characteristic
8	C₃₁×C₄⋊Q₈	C₂²	1	3	yes	—
9	C₃₁×#125	—	2	3	no	—
10	C₄×Q₈×C₃₁	—	2	1	no	—
11	C₄×Q₈×C₃₁	C₂²	1	3	yes	—
12	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
13	Q₈×Dic₃₁	—	2	1	no	—
14	C₂×#92482	C₂²	1	3	yes	—
15	#11758465	—	2	3	no	—
16	#11758446	C₂²	1	3	yes	—
17	C₄×Dic₆₂	C₂²	1	3	yes	—
18	#11758423	C₂²	1	6	yes	—
19	#11758423	C₂²	1	3	yes	—
20	#11758398	C₂²	1	1	yes	characteristic
21	#11758395	C₂²	1	3	yes	—
22	C₄⋊(C₃₁⋊C₈)	C₂²	1	3	yes	—
23	C₄×C₃₁⋊C₈	C₂²	1	3	yes	—
24	C₂×Q₈×C₃₁	D₄	1	1	yes	characteristic
25	C₂×Q₈×C₃₁	C₂³	1	3	yes	—
26	C₂×Dic₆₂	—	2	3	no	—
27	C₂×Dic₆₂	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₃₁	—	2	3	no	—
29	C₄⋊C₄×C₃₁	—	2	3	no	—
30	C₄⋊C₄×C₃₁	—	2	3	no	—
31	C₄⋊C₄×C₃₁	D₄	1	3	yes	—
32	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
33	C₄²×C₃₁	—	2	3	no	—
34	C₄²×C₃₁	C₂³	1	3	yes	—
35	#92482	—	2	6	no	—
36	C₄⋊Dic₃₁	—	2	3	no	—
37	C₄⋊Dic₃₁	C₂³	1	3	yes	—
38	Dic₃₁⋊C₄	—	2	3	no	—
39	C₄×Dic₃₁	—	2	3	no	—
40	C₂×C₃₁⋊C₈	C₂³	1	1	yes	characteristic
41	C₂×C₃₁⋊C₈	C₂³	1	3	yes	—
42	Q₈×C₃₁	—	2	3	no	—
43	Q₈×C₃₁	—	2	3	no	—
44	Q₈×C₃₁	Q₁₆	1	4	yes	complemented
45	C₂×C₄×C₃₁	—	2	1	no	—
46	C₂×C₄×C₃₁	—	2	3	no	—
47	C₂×C₄×C₃₁	C₂×D₄	1	3	yes	—
48	C₂×C₄×C₃₁	C₂×D₄	1	3	yes	—
49	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₃₁	—	2	1	no	—
51	C₂×Dic₃₁	—	2	3	no	—
52	Dic₆₂	—	2	1	no	—
53	Dic₆₂	—	8	3	no	—
54	Dic₆₂	#41	1	2	yes	—
55	C₃₁⋊C₈	—	2	3	no	—
56	C₃₁⋊C₈	—	2	1	no	—
57	#3363	—	31	1	no	maximal, Sylow
58	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
59	C₄×C₃₁	—	2	3	no	—
60	C₄×C₃₁	—	4	3	no	—
61	C₄×C₃₁	—	4	1	no	—
62	C₄×C₃₁	C₂×Q₁₆	1	6	yes	—
63	C₄×C₃₁	C₂×#41	1	1	yes	characteristic
64	C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
65	Dic₃₁	—	2	2	no	—
66	Dic₃₁	—	4	3	no	—
67	Q₈²	—	31	1	no	—
68	#557	D₃₁	1	1	yes	characteristic
69	#487	—	31	3	no	—
70	#477	—	31	3	no	—
71	#462	—	31	3	no	—
72	C₈×Q₈	—	31	1	no	—
73	C₄×Q₁₆	—	31	3	no	—
74	C₂×C₃₁	#548	1	1	yes	characteristic
75	C₂×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
76	C₂×C₃₁	#577	1	1	yes	characteristic
77	C₂×Q₁₆	—	31	3	no	—
78	C₄⋊Q₈	—	31	3	no	—
79	C₄⋊Q₈	—	62	3	no	—
80	C₄⋊Q₈	C₂×D₃₁	1	3	yes	—
81	#125	—	2	3	no	—
82	C₄×Q₈	—	2	1	no	—
83	C₄×Q₈	—	31	3	no	—
84	C₄×Q₈	—	62	1	no	—
85	C₄×Q₈	C₂×D₃₁	1	3	yes	—
86	C₄×Q₈	C₂×D₃₁	1	1	yes	characteristic
87	#107	—	31	3	no	—
88	C₄⋊C₈	—	31	3	no	—
89	#103	—	31	3	no	—
90	#103	—	31	6	no	—
91	#103	—	31	1	no	—
92	C₄×C₈	—	31	3	no	—
93	C₃₁	#3363	1	1	yes	characteristic, complemented, Sylow
94	C₂×Q₈	—	31	1	no	—
95	C₂×Q₈	—	62	3	no	—
96	C₂×Q₈	C₂²⋊D₃₁	1	1	yes	characteristic
97	C₂×Q₈	C₂²×D₃₁	1	3	yes	—
98	Q₁₆	—	62	6	no	—
99	C₂×C₈	—	31	1	no	—
100	C₂×C₈	—	31	3	no	—
101	C₄⋊C₄	—	2	3	no	—
102	C₄⋊C₄	—	2	3	no	—
103	C₄⋊C₄	—	2	3	no	—
104	C₄⋊C₄	—	31	3	no	—
105	C₄⋊C₄	—	62	3	no	—
106	C₄⋊C₄	—	62	3	no	—
107	C₄⋊C₄	C₂²⋊D₃₁	1	3	yes	—
108	C₄⋊C₄	C₂²×D₃₁	1	1	yes	characteristic
109	C₄²	—	2	3	no	—
110	C₄²	—	62	3	no	—
111	C₄²	C₂²×D₃₁	1	3	yes	—
112	Q₈	—	2	3	no	—
113	Q₈	—	2	3	no	—
114	Q₈	—	31	2	no	—
115	Q₈	—	62	1	no	—
116	Q₈	—	248	3	no	—
117	Q₈	#92482	1	4	yes	complemented
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	3	no	—
120	C₂×C₄	—	62	1	no	—
121	C₂×C₄	—	62	3	no	—
122	C₂×C₄	C₂×C₂²⋊D₃₁	1	3	yes	—
123	C₂×C₄	C₂×C₂²⋊D₃₁	1	3	yes	—
124	C₂×C₄	C₂³×D₃₁	1	1	yes	characteristic
125	C₈	—	62	3	no	—
126	C₈	—	62	1	no	—
127	C₂²	C₂²×C₂²⋊D₃₁	1	1	yes	characteristic
128	C₄	—	2	3	no	—
129	C₄	—	4	1	no	—
130	C₄	—	4	3	no	—
131	C₄	—	62	2	no	—
132	C₄	—	124	3	no	—
133	C₄	C₂×#92482	1	6	yes	—
134	C₄	#41×D₃₁	1	1	yes	characteristic
135	C₄	#11758567	1	1	yes	characteristic
136	C₂	#14521706	1	1	yes	characteristic
137	C₂	C₂²×#92482	1	1	yes	characteristic
138	C₂	#14521857	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₆₂

C₂

yes

maximal, characteristic

#14521399

C₂

yes

maximal

#14521344

C₂

yes

maximal

#14521256

C₂

yes

maximal

Q₈×C₃₁⋊C₈

C₂

yes

maximal, characteristic

C₄×#92482

C₂

yes

maximal

C₃₁×#557

C₂

yes

maximal, characteristic

C₃₁×C₄⋊Q₈

C₂²

yes

—

C₃₁×#125

—

C₄×Q₈×C₃₁

—

C₄×Q₈×C₃₁

C₂²

yes

—

C₄×Q₈×C₃₁

C₂²

yes

characteristic

Q₈×Dic₃₁

—

C₂×#92482

C₂²

yes

—

#11758465

—

#11758446

C₂²

yes

—

C₄×Dic₆₂

C₂²

yes

—

#11758423

C₂²

yes

—

#11758423

C₂²

yes

—

#11758398

C₂²

yes

characteristic

#11758395

C₂²

yes

—

C₄⋊(C₃₁⋊C₈)

C₂²

yes

—

C₄×C₃₁⋊C₈

C₂²

yes

—

C₂×Q₈×C₃₁

D₄

yes

characteristic

C₂×Q₈×C₃₁

C₂³

yes

—

C₂×Dic₆₂

—

C₂×Dic₆₂

C₂³

yes

characteristic

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

—

C₄⋊C₄×C₃₁

D₄

yes

—

C₄⋊C₄×C₃₁

C₂³

yes

characteristic

C₄²×C₃₁

—

C₄²×C₃₁

C₂³

yes

—

#92482

—

C₄⋊Dic₃₁

—

C₄⋊Dic₃₁

C₂³

yes

—

Dic₃₁⋊C₄

—

C₄×Dic₃₁

—

C₂×C₃₁⋊C₈

C₂³

yes

characteristic

C₂×C₃₁⋊C₈

C₂³

yes

—

Q₈×C₃₁

—

Q₈×C₃₁

—

Q₈×C₃₁

Q₁₆

yes

complemented

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂×D₄

yes

—

C₂×C₄×C₃₁

C₂⁴

yes

characteristic

C₂×Dic₃₁

—

C₂×Dic₃₁

—

Dic₆₂

—

Dic₆₂

—

Dic₆₂

#41

yes

—

C₃₁⋊C₈

—

C₃₁⋊C₈

—

#3363

—

maximal, Sylow

C₂²×C₃₁

C₂²×D₄

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

—

C₄×C₃₁

C₂×Q₁₆

yes

—

C₄×C₃₁

C₂×#41

yes

characteristic

C₄×C₃₁

2₋¹⁺⁴

yes

characteristic

Dic₃₁

—

Dic₃₁

—

Q₈²

—

#557

D₃₁

yes

—

—

—

C₈×Q₈

—

C₄×Q₁₆

—

C₂×C₃₁

#548

yes

characteristic

C₂×C₃₁

C₂²×Q₁₆

yes

characteristic

C₂×C₃₁

#577

yes

characteristic

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃₁

yes

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₃₁

yes

—

C₄×Q₈

C₂×D₃₁

yes

characteristic

#107

—

C₄⋊C₈

—

#103

—

#103

—

#103

—

C₄×C₈

—

C₃₁

#3363

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₃₁

yes

characteristic

C₂×Q₈

C₂²×D₃₁

yes

—

Q₁₆

—

C₂×C₈

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

C₂²⋊D₃₁

yes

—

108

C₄⋊C₄

C₂²×D₃₁

yes

characteristic

109

C₄²

—

110

C₄²

—

111

C₄²

C₂²×D₃₁

yes

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

Q₈

—

116

Q₈

—

248

—

117

Q₈

#92482

yes

complemented

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

123

C₂×C₄

C₂×C₂²⋊D₃₁

yes

—

124

C₂×C₄

C₂³×D₃₁

yes

characteristic

125

C₈

—

126

C₈

—

127

C₂²

C₂²×C₂²⋊D₃₁

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

124

—

133

C₄

C₂×#92482

yes

—

134

C₄

#41×D₃₁

yes

characteristic

135

C₄

#11758567

yes

characteristic

136

C₂

#14521706

yes

characteristic

137

C₂

C₂²×#92482

yes

characteristic

138

C₂

#14521857

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₃₁
commutator, cosocle	C₂×C₄×C₃₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₁
Fitting	C₃₁×#557	C₂
socle, prime radical	C₂²×C₃₁	C₂²×D₄
hypercenter	#557	D₃₁
hypocenter	C₃₁	#3363
LCS	G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #557

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₃₁

commutator, cosocle

C₂×C₄×C₃₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₁

Fitting

C₃₁×#557

C₂

socle, prime radical

C₂²×C₃₁

C₂²×D₄

D₃₁

C₃₁

G ▷ C₂×C₄×C₃₁ ▷ C₂×C₃₁ ▷ C₃₁

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #557

N	H ≅ G/N	type
44. Q₈×C₃₁	Q₁₆	semidirect
93. C₃₁	#3363	semidirect
117. Q₈	#92482	semidirect

H ≅ G/N

type

44. Q₈×C₃₁

Q₁₆

semidirect

93. C₃₁

#3363

semidirect

117. Q₈

#92482

semidirect

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

subgroup	index
Q₈×Dic₆₂	2
#14521399	2
#14521344	2
#14521256	2
Q₈×C₃₁⋊C₈	2
C₄×#92482	2
C₃₁×#557	2
#3363	31

subgroup

index

Q₈×Dic₆₂

Q₈×C₃₁⋊C₈

C₃₁×#557

Subgroups — #16858385

Conjugacy classes of subgroups

Complements

Maximal subgroups