Group #16847324 — subgroups

Subgroups — #16847324

Order 3968.

228 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 93 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×#11758478	C₂	1	1	yes	maximal, characteristic
2	C₂×#11758400	C₂	1	1	yes	maximal, characteristic, complemented
3	(C₂×D₃₁)⋊C₁₆	C₂	1	4	yes	maximal, complemented
4	C₂²×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#92495	C₄	1	1	yes	characteristic
6	C₂²×C₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×#92474	C₄	1	1	yes	characteristic
8	#11758478	—	2	2	no	—
9	#11758478	—	2	2	no	—
10	C₂×#92469	C₄	1	2	yes	—
11	C₂×C₈×D₃₁	C₂²	1	2	yes	—
12	C₂×C₁₆×C₃₁	—	2	2	no	—
13	C₂×C₁₆×C₃₁	C₂²	1	2	yes	—
14	#11758400	—	2	2	no	—
15	C₂×C₃₁⋊C₁₆	C₂²	1	2	yes	—
16	C₂²×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
17	C₂×C₂²⋊D₃₁	C₈	1	2	yes	—
18	#92495	—	2	2	no	—
19	#92495	—	2	2	no	—
20	C₂×D₁₂₄	C₈	1	1	yes	characteristic
21	C₂×C₄×D₃₁	C₂×C₄	1	2	yes	—
22	C₂×Dic₆₂	C₈	1	1	yes	characteristic
23	C₂×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₃₁	D₄	1	2	yes	—
25	C₂×C₈×C₃₁	D₄	1	2	yes	—
26	C₂×C₈×C₃₁	C₂³	1	1	yes	characteristic
27	#92474	—	2	2	no	—
28	C₂×C₃₁⋊C₈	C₂×C₄	1	2	yes	—
29	#92469	—	2	4	no	—
30	C₈×D₃₁	—	2	2	no	—
31	C₈×D₃₁	—	2	2	no	—
32	C₁₆×C₃₁	—	2	2	no	—
33	C₃₁⋊C₁₆	—	2	2	no	—
34	C₂³×C₃₁	C₂×C₈	1	1	yes	characteristic
35	C₂²×D₃₁	C₂×C₈	1	2	yes	—
36	C₂×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
38	C₂×C₄×C₃₁	M₁₆	1	2	yes	—
39	C₂×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
40	C₂²⋊D₃₁	—	2	4	no	—
41	C₂×Dic₃₁	C₂×C₈	1	2	yes	—
42	D₁₂₄	—	2	2	no	—
43	C₄×D₃₁	—	2	2	no	—
44	C₄×D₃₁	—	2	2	no	—
45	Dic₆₂	—	2	2	no	—
46	C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
47	C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
48	C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
49	C₃₁⋊C₈	—	2	2	no	—
50	#2113	—	31	1	no	maximal, Sylow
51	C₂²×C₃₁	—	2	2	no	—
52	C₂²×C₃₁	C₂²⋊C₈	1	2	yes	—
53	C₂²×C₃₁	C₂²×C₈	1	1	yes	characteristic
54	C₂×D₃₁	—	2	2	no	—
55	C₂×D₃₁	—	2	4	no	—
56	C₄×C₃₁	C₂²⋊C₈	1	2	yes	—
57	C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
59	Dic₃₁	—	2	2	no	—
60	C₂×#131	—	31	1	no	—
61	C₂×M₃₂	—	31	1	no	—
62	C₂²×C₁₆	D₃₁	1	1	yes	characteristic, complemented
63	C₂²⋊C₁₆	—	31	4	no	—
64	C₂×C₃₁	—	2	2	no	—
65	C₂×C₃₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
66	C₂×C₃₁	#504	1	2	yes	—
67	D₃₁	—	4	2	no	—
68	C₂×#41	—	31	1	no	—
69	#131	—	62	2	no	—
70	#131	—	62	2	no	—
71	C₂×M₁₆	—	31	1	no	—
72	C₂×M₁₆	—	31	2	no	—
73	C₂²×C₈	—	31	2	no	—
74	C₂²×C₈	C₂×D₃₁	1	1	yes	characteristic
75	M₃₂	—	62	2	no	—
76	C₂×C₁₆	—	2	2	no	—
77	C₂×C₁₆	—	31	2	no	—
78	C₂×C₁₆	C₂×D₃₁	1	2	yes	—
79	C₃₁	#2113	1	1	yes	characteristic, complemented, Sylow
80	#41	—	62	2	no	—
81	#41	—	62	2	no	—
82	C₂×Q₈	—	31	1	no	—
83	C₂×D₄	—	31	2	no	—
84	C₂×D₄	—	31	1	no	—
85	C₂²×C₄	—	31	2	no	—
86	C₂²×C₄	C₄×D₃₁	1	1	yes	characteristic
87	M₁₆	—	62	2	no	—
88	M₁₆	—	62	4	no	—
89	C₂×C₈	—	31	2	no	—
90	C₂×C₈	—	62	2	no	—
91	C₂×C₈	—	62	2	no	—
92	C₂×C₈	C₄×D₃₁	1	1	yes	characteristic
93	C₂×C₈	D₁₂₄	1	2	yes	—
94	C₂×C₈	C₂²⋊D₃₁	1	2	yes	—
95	C₂×C₈	C₂²×D₃₁	1	1	yes	characteristic
96	C₁₆	—	2	2	no	—
97	C₁₆	—	62	2	no	—
98	C₂³	—	31	2	no	—
99	C₂³	C₈×D₃₁	1	1	yes	characteristic
100	Q₈	—	62	2	no	—
101	D₄	—	62	4	no	—
102	D₄	—	62	2	no	—
103	C₂×C₄	—	31	2	no	—
104	C₂×C₄	—	62	2	no	—
105	C₂×C₄	—	62	2	no	—
106	C₂×C₄	C₈×D₃₁	1	1	yes	characteristic
107	C₂×C₄	#92469	1	2	yes	—
108	C₂×C₄	(C₂×D₃₁)⋊C₄	1	2	yes	—
109	C₂×C₄	C₂×C₄×D₃₁	1	1	yes	characteristic
110	C₈	—	62	2	no	—
111	C₈	(C₂×D₃₁)⋊C₄	1	2	yes	—
112	C₈	C₂×D₁₂₄	1	1	yes	characteristic
113	C₈	C₂×C₂²⋊D₃₁	1	1	yes	characteristic
114	C₂²	—	2	2	no	—
115	C₂²	—	62	4	no	—
116	C₂²	—	62	2	no	—
117	C₂²	(C₂×D₃₁)⋊C₈	1	2	yes	—
118	C₂²	C₂×C₈×D₃₁	1	1	yes	characteristic
119	C₄	—	62	2	no	—
120	C₄	(C₂×D₃₁)⋊C₈	1	2	yes	—
121	C₄	C₂×#92469	1	1	yes	characteristic
122	C₄	C₂×(C₂×D₃₁)⋊C₄	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	124	2	no	—
125	C₂	C₂×(C₂×D₃₁)⋊C₈	1	1	yes	characteristic
126	C₂	#14521465	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11758478

C₂

yes

maximal, characteristic

C₂×#11758400

C₂

yes

maximal, characteristic, complemented

(C₂×D₃₁)⋊C₁₆

C₂

yes

maximal, complemented

C₂²×C₁₆×C₃₁

C₂

yes

maximal, characteristic, complemented

C₂×#92495

C₄

yes

characteristic

C₂²×C₈×C₃₁

C₂²

yes

characteristic

C₂×#92474

C₄

yes

—

—

C₄

yes

—

C₂×C₈×D₃₁

C₂²

yes

—

C₂×C₁₆×C₃₁

—

C₂×C₁₆×C₃₁

C₂²

yes

—

#11758400

—

C₂×C₃₁⋊C₁₆

C₂²

yes

—

C₂²×C₄×C₃₁

C₂×C₄

yes

characteristic

C₂×C₂²⋊D₃₁

C₈

yes

—

#92495

—

#92495

—

C₂×D₁₂₄

C₈

yes

characteristic

C₂×C₄×D₃₁

C₂×C₄

yes

—

C₂×Dic₆₂

C₈

yes

characteristic

C₂×C₈×C₃₁

C₂×C₄

yes

characteristic

C₂×C₈×C₃₁

D₄

yes

—

C₂×C₈×C₃₁

D₄

yes

—

C₂×C₈×C₃₁

C₂³

yes

characteristic

#92474

—

C₂×C₃₁⋊C₈

C₂×C₄

yes

—

#92469

—

C₈×D₃₁

—

C₈×D₃₁

—

C₁₆×C₃₁

—

C₃₁⋊C₁₆

—

C₂³×C₃₁

C₂×C₈

yes

characteristic

C₂²×D₃₁

C₂×C₈

yes

—

C₂×C₄×C₃₁

C₂²⋊C₄

yes

—

C₂×C₄×C₃₁

C₂×C₈

yes

characteristic

C₂×C₄×C₃₁

M₁₆

yes

—

C₂×C₄×C₃₁

C₂²×C₄

yes

characteristic

C₂²⋊D₃₁

—

C₂×Dic₃₁

C₂×C₈

yes

—

D₁₂₄

—

C₄×D₃₁

—

C₄×D₃₁

—

Dic₆₂

—

C₈×C₃₁

C₂²⋊C₄

yes

—

C₈×C₃₁

C₂×D₄

yes

characteristic

C₈×C₃₁

C₂×D₄

yes

characteristic

C₃₁⋊C₈

—

#2113

—

maximal, Sylow

C₂²×C₃₁

—

C₂²×C₃₁

C₂²⋊C₈

yes

—

C₂²×C₃₁

C₂²×C₈

yes

characteristic

C₂×D₃₁

—

C₂×D₃₁

—

C₄×C₃₁

C₂²⋊C₈

yes

—

C₄×C₃₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₃₁

C₂×M₁₆

yes

characteristic

Dic₃₁

—

C₂×#131

—

C₂×M₃₂

—

C₂²×C₁₆

D₃₁

yes

characteristic, complemented

C₂²⋊C₁₆

—

C₂×C₃₁

—

C₂×C₃₁

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₃₁

#504

yes

—

D₃₁

—

C₂×#41

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₃₁

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₃₁

yes

—

C₃₁

#2113

yes

characteristic, complemented, Sylow

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₃₁

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₃₁

yes

characteristic

C₂×C₈

D₁₂₄

yes

—

C₂×C₈

C₂²⋊D₃₁

yes

—

C₂×C₈

C₂²×D₃₁

yes

characteristic

C₁₆

—

C₁₆

—

C₂³

—

C₂³

C₈×D₃₁

yes

characteristic

100

Q₈

—

101

D₄

—

102

D₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₈×D₃₁

yes

characteristic

107

C₂×C₄

#92469

yes

—

108

C₂×C₄

(C₂×D₃₁)⋊C₄

yes

—

109

C₂×C₄

C₂×C₄×D₃₁

yes

characteristic

110

C₈

—

111

C₈

(C₂×D₃₁)⋊C₄

yes

—

112

C₈

C₂×D₁₂₄

yes

characteristic

113

C₈

C₂×C₂²⋊D₃₁

yes

characteristic

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

(C₂×D₃₁)⋊C₈

yes

—

118

C₂²

C₂×C₈×D₃₁

yes

characteristic

119

C₄

—

120

C₄

(C₂×D₃₁)⋊C₈

yes

—

121

C₄

C₂×#92469

yes

characteristic

122

C₄

C₂×(C₂×D₃₁)⋊C₄

yes

characteristic

123

C₂

—

124

C₂

—

124

—

125

C₂

C₂×(C₂×D₃₁)⋊C₈

yes

characteristic

126

C₂

#14521465

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₃₁
commutator, socle	C₂²×C₃₁	C₂²×C₈
Fitting	C₂²×C₁₆×C₃₁	C₂
prime radical	C₂×#92495	C₄
cosocle	C₂×C₈×C₃₁	C₂³
hypercenter	C₂²×C₁₆	D₃₁
hypocenter	C₃₁	#2113
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₃₁

commutator, socle

C₂²×C₃₁

C₂²×C₈

Fitting

C₂²×C₁₆×C₃₁

C₂

prime radical

C₂×#92495

C₄

cosocle

C₂×C₈×C₃₁

C₂³

hypercenter

C₂²×C₁₆

D₃₁

C₃₁

G ▷ C₂²×C₃₁ ▷ C₃₁

C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
2. C₂×#11758400	C₂	semidirect
3. (C₂×D₃₁)⋊C₁₆	C₂	semidirect
4. C₂²×C₁₆×C₃₁	C₂	semidirect
62. C₂²×C₁₆	D₃₁	semidirect
79. C₃₁	#2113	semidirect

H ≅ G/N

type

2. C₂×#11758400

C₂

semidirect

3. (C₂×D₃₁)⋊C₁₆

C₂

semidirect

4. C₂²×C₁₆×C₃₁

C₂

semidirect

62. C₂²×C₁₆

D₃₁

semidirect

79. C₃₁

#2113

semidirect

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

subgroup	index
C₂×#11758478	2
C₂×#11758400	2
(C₂×D₃₁)⋊C₁₆	2
C₂²×C₁₆×C₃₁	2
#2113	31

subgroup

index

C₂×#11758478

C₂×#11758400

(C₂×D₃₁)⋊C₁₆

C₂²×C₁₆×C₃₁

#2113

Subgroups — #16847324

Conjugacy classes of subgroups

Complements

Maximal subgroups