Group #27172746 — subgroups

Subgroups — #27172746

Order 12928.

228 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 109 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×#15692195	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#13108049	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊#13108049	C₂	1	4	yes	maximal, complemented
4	C₂×#131×C₁₀₁	C₂	1	1	yes	maximal, characteristic
5	#15692195	—	2	2	no	—
6	C₂×#13108049	—	2	2	no	—
7	C₂×#13108049	C₂²	1	2	yes	—
8	C₂×#13108049	C₂²	1	2	yes	—
9	C₂×#41×C₁₀₁	C₄	1	1	yes	characteristic
10	#131×C₁₀₁	—	2	2	no	—
11	#131×C₁₀₁	—	2	2	no	—
12	C₂×M₁₆×C₁₀₁	C₄	1	1	yes	characteristic
13	C₂×M₁₆×C₁₀₁	C₄	1	2	yes	—
14	C₂²×C₈×C₁₀₁	C₂²	1	2	yes	—
15	C₂²×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
16	#13108049	—	2	2	no	—
17	#13108049	—	2	2	no	—
18	#41×C₁₀₁	—	2	2	no	—
19	#41×C₁₀₁	—	2	2	no	—
20	C₂×Q₈×C₁₀₁	C₈	1	1	yes	characteristic
21	C₂×D₄×C₁₀₁	C₈	1	2	yes	—
22	C₂×D₄×C₁₀₁	C₈	1	1	yes	characteristic
23	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
25	M₁₆×C₁₀₁	—	2	2	no	—
26	M₁₆×C₁₀₁	—	2	4	no	—
27	C₂×C₈×C₁₀₁	—	2	2	no	—
28	C₂×C₈×C₁₀₁	—	2	2	no	—
29	C₂×C₈×C₁₀₁	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₈×C₁₀₁	D₄	1	2	yes	—
32	C₂×C₈×C₁₀₁	D₄	1	2	yes	—
33	C₂×C₈×C₁₀₁	C₂³	1	1	yes	characteristic
34	C₂³×C₁₀₁	C₂×C₈	1	2	yes	—
35	C₂³×C₁₀₁	C₂×C₈	1	1	yes	characteristic
36	Q₈×C₁₀₁	—	2	2	no	—
37	D₄×C₁₀₁	—	2	4	no	—
38	D₄×C₁₀₁	—	2	2	no	—
39	C₂×C₄×C₁₀₁	—	2	2	no	—
40	C₂×C₄×C₁₀₁	—	2	2	no	—
41	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₁₀₁	C₂×C₈	1	2	yes	—
43	C₂×C₄×C₁₀₁	C₂×C₈	1	1	yes	characteristic
44	C₂×C₄×C₁₀₁	M₁₆	1	2	yes	—
45	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
46	C₈×C₁₀₁	—	2	2	no	—
47	C₈×C₁₀₁	C₂²⋊C₄	1	2	yes	—
48	C₈×C₁₀₁	C₂×D₄	1	1	yes	characteristic
49	C₈×C₁₀₁	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₁₀₁	—	2	2	no	—
51	C₂²×C₁₀₁	—	2	2	no	—
52	C₂²×C₁₀₁	—	2	4	no	—
53	C₂²×C₁₀₁	C₂²⋊C₈	1	2	yes	—
54	C₂²×C₁₀₁	C₂²×C₈	1	1	yes	characteristic
55	C₄×C₁₀₁	—	2	2	no	—
56	C₄×C₁₀₁	C₂²⋊C₈	1	2	yes	—
57	C₄×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₄×C₁₀₁	C₂×M₁₆	1	1	yes	characteristic
59	C₂×C₁₀₁	—	2	2	no	—
60	C₂×C₁₀₁	—	4	2	no	—
61	C₂×C₁₀₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
62	C₂×C₁₀₁	#504	1	2	yes	—
63	#2113	—	101	1	no	maximal, Sylow
64	C₁₀₁	#2113	1	1	yes	characteristic, complemented, Sylow
65	C₂×#131	D₁₀₁	1	1	yes	characteristic
66	C₂×M₃₂	—	101	1	no	—
67	C₂²×C₁₆	—	101	1	no	—
68	C₂²⋊C₁₆	—	101	4	no	—
69	C₂×#41	Dic₁₀₁	1	1	yes	characteristic
70	#131	—	2	2	no	—
71	#131	—	2	2	no	—
72	C₂×M₁₆	Dic₁₀₁	1	1	yes	characteristic
73	C₂×M₁₆	Dic₁₀₁	1	2	yes	—
74	C₂²×C₈	C₂×D₁₀₁	1	2	yes	—
75	C₂²×C₈	C₂×D₁₀₁	1	1	yes	characteristic
76	M₃₂	—	202	2	no	—
77	C₂×C₁₆	—	101	2	no	—
78	C₂×C₁₆	—	101	2	no	—
79	C₂×C₁₆	—	202	2	no	—
80	#41	—	2	2	no	—
81	#41	—	2	2	no	—
82	C₂×Q₈	#11717579	1	1	yes	characteristic
83	C₂×D₄	#11717579	1	2	yes	—
84	C₂×D₄	#11717579	1	1	yes	characteristic
85	C₂²×C₄	C₂×Dic₁₀₁	1	2	yes	—
86	C₂²×C₄	C₂×Dic₁₀₁	1	1	yes	characteristic
87	M₁₆	—	2	2	no	—
88	M₁₆	—	2	4	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	C₂×Dic₁₀₁	1	2	yes	—
92	C₂×C₈	C₂×Dic₁₀₁	1	1	yes	characteristic
93	C₂×C₈	C₂²⋊D₁₀₁	1	2	yes	—
94	C₂×C₈	C₂²⋊D₁₀₁	1	2	yes	—
95	C₂×C₈	C₂²×D₁₀₁	1	1	yes	characteristic
96	C₁₆	—	202	2	no	—
97	C₁₆	—	202	2	no	—
98	C₂³	C₂×#11717579	1	2	yes	—
99	C₂³	C₂×#11717579	1	1	yes	characteristic
100	Q₈	—	2	2	no	—
101	D₄	—	2	4	no	—
102	D₄	—	2	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	C₂²⋊Dic₁₀₁	1	2	yes	—
106	C₂×C₄	C₂×#11717579	1	2	yes	—
107	C₂×C₄	C₂×#11717579	1	1	yes	characteristic
108	C₂×C₄	#13108058	1	2	yes	—
109	C₂×C₄	C₂²×Dic₁₀₁	1	1	yes	characteristic
110	C₈	—	2	2	no	—
111	C₈	C₂²⋊Dic₁₀₁	1	2	yes	—
112	C₈	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
113	C₈	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
114	C₂²	—	2	2	no	—
115	C₂²	—	2	4	no	—
116	C₂²	—	2	2	no	—
117	C₂²	C₂²⋊#11717579	1	2	yes	—
118	C₂²	C₂²×#11717579	1	1	yes	characteristic
119	C₄	—	2	2	no	—
120	C₄	C₂²⋊#11717579	1	2	yes	—
121	C₄	C₂×C₂²⋊Dic₁₀₁	1	1	yes	characteristic
122	C₄	C₂×#13108058	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	4	2	no	—
125	C₂	C₂×C₂²⋊#11717579	1	1	yes	characteristic
126	C₂	#20265594	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692195

C₂

yes

maximal, characteristic, complemented

C₂²×#13108049

C₂

yes

maximal, characteristic, complemented

C₂²⋊#13108049

C₂

yes

maximal, complemented

C₂×#131×C₁₀₁

C₂

yes

maximal, characteristic

#15692195

—

C₂×#13108049

—

C₂×#13108049

C₂²

yes

—

C₂×#13108049

C₂²

yes

—

C₂×#41×C₁₀₁

C₄

yes

characteristic

#131×C₁₀₁

—

#131×C₁₀₁

—

C₂×M₁₆×C₁₀₁

C₄

yes

characteristic

C₂×M₁₆×C₁₀₁

C₄

yes

—

C₂²×C₈×C₁₀₁

C₂²

yes

—

C₂²×C₈×C₁₀₁

C₂²

yes

characteristic

#13108049

—

#13108049

—

#41×C₁₀₁

—

#41×C₁₀₁

—

C₂×Q₈×C₁₀₁

C₈

yes

characteristic

C₂×D₄×C₁₀₁

C₈

yes

—

C₂×D₄×C₁₀₁

C₈

yes

characteristic

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

characteristic

M₁₆×C₁₀₁

—

M₁₆×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

C₂×C₄

yes

—

C₂×C₈×C₁₀₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₀₁

D₄

yes

—

C₂×C₈×C₁₀₁

D₄

yes

—

C₂×C₈×C₁₀₁

C₂³

yes

characteristic

C₂³×C₁₀₁

C₂×C₈

yes

—

C₂³×C₁₀₁

C₂×C₈

yes

characteristic

Q₈×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂×C₈

yes

—

C₂×C₄×C₁₀₁

C₂×C₈

yes

characteristic

C₂×C₄×C₁₀₁

M₁₆

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₈×C₁₀₁

—

C₈×C₁₀₁

C₂²⋊C₄

yes

—

C₈×C₁₀₁

C₂×D₄

yes

characteristic

C₈×C₁₀₁

C₂×D₄

yes

characteristic

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂²⋊C₈

yes

—

C₂²×C₁₀₁

C₂²×C₈

yes

characteristic

C₄×C₁₀₁

—

C₄×C₁₀₁

C₂²⋊C₈

yes

—

C₄×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₀₁

C₂×M₁₆

yes

characteristic

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₁₀₁

#504

yes

—

#2113

—

101

maximal, Sylow

C₁₀₁

#2113

yes

characteristic, complemented, Sylow

C₂×#131

D₁₀₁

yes

characteristic

C₂×M₃₂

—

101

—

C₂²×C₁₆

—

101

—

C₂²⋊C₁₆

—

101

—

C₂×#41

Dic₁₀₁

yes

characteristic

#131

—

#131

—

C₂×M₁₆

Dic₁₀₁

yes

characteristic

C₂×M₁₆

Dic₁₀₁

yes

—

C₂²×C₈

C₂×D₁₀₁

yes

—

C₂²×C₈

C₂×D₁₀₁

yes

characteristic

M₃₂

—

202

—

C₂×C₁₆

—

101

—

C₂×C₁₆

—

101

—

C₂×C₁₆

—

202

—

#41

—

#41

—

C₂×Q₈

#11717579

yes

characteristic

C₂×D₄

#11717579

yes

—

C₂×D₄

#11717579

yes

characteristic

C₂²×C₄

C₂×Dic₁₀₁

yes

—

C₂²×C₄

C₂×Dic₁₀₁

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₁₀₁

yes

—

C₂×C₈

C₂×Dic₁₀₁

yes

characteristic

C₂×C₈

C₂²⋊D₁₀₁

yes

—

C₂×C₈

C₂²⋊D₁₀₁

yes

—

C₂×C₈

C₂²×D₁₀₁

yes

characteristic

C₁₆

—

202

—

C₁₆

—

202

—

C₂³

C₂×#11717579

yes

—

C₂³

C₂×#11717579

yes

characteristic

100

Q₈

—

101

D₄

—

102

D₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂²⋊Dic₁₀₁

yes

—

106

C₂×C₄

C₂×#11717579

yes

—

107

C₂×C₄

C₂×#11717579

yes

characteristic

108

C₂×C₄

#13108058

yes

—

109

C₂×C₄

C₂²×Dic₁₀₁

yes

characteristic

110

C₈

—

111

C₈

C₂²⋊Dic₁₀₁

yes

—

112

C₈

C₂×C₂²⋊D₁₀₁

yes

characteristic

113

C₈

C₂×C₂²⋊D₁₀₁

yes

characteristic

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

C₂²⋊#11717579

yes

—

118

C₂²

C₂²×#11717579

yes

characteristic

119

C₄

—

120

C₄

C₂²⋊#11717579

yes

—

121

C₄

C₂×C₂²⋊Dic₁₀₁

yes

characteristic

122

C₄

C₂×#13108058

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

C₂×C₂²⋊#11717579

yes

characteristic

126

C₂

#20265594

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₀₁
commutator, socle	C₂²×C₁₀₁	C₂²×C₈
Fitting	C₂×#131×C₁₀₁	C₂
prime radical	C₂×#41×C₁₀₁	C₄
cosocle	C₂×C₈×C₁₀₁	C₂³
hypercenter	C₂×#131	D₁₀₁
hypocenter	C₁₀₁	#2113
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#131

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₁₀₁

commutator, socle

C₂²×C₁₀₁

C₂²×C₈

Fitting

C₂×#131×C₁₀₁

C₂

prime radical

C₂×#41×C₁₀₁

C₄

cosocle

C₂×C₈×C₁₀₁

C₂³

D₁₀₁

C₁₀₁

G ▷ C₂²×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂×C₈ ◁ C₂×#131

N	H ≅ G/N	type
1. C₂×#15692195	C₂	semidirect
2. C₂²×#13108049	C₂	semidirect
3. C₂²⋊#13108049	C₂	semidirect
64. C₁₀₁	#2113	semidirect

H ≅ G/N

type

1. C₂×#15692195

C₂

semidirect

2. C₂²×#13108049

C₂

semidirect

3. C₂²⋊#13108049

C₂

semidirect

64. C₁₀₁

#2113

semidirect

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

subgroup	index
C₂×#15692195	2
C₂²×#13108049	2
C₂²⋊#13108049	2
C₂×#131×C₁₀₁	2
#2113	101

subgroup

index

C₂×#131×C₁₀₁

101

Subgroups — #27172746

Conjugacy classes of subgroups

Complements

Maximal subgroups