Group #27179502 — subgroups

Subgroups — #27179502

Order 12928.

524 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 157 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×#15692247	C₂	1	1	yes	maximal, characteristic, complemented
2	#20265084	C₂	1	2	yes	maximal, complemented
3	#20265080	C₂	1	2	yes	maximal
4	#20265071	C₂	1	8	yes	maximal, complemented
5	C₄²×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#124×C₁₀₁	C₂	1	1	yes	maximal, characteristic
7	#15692247	—	2	4	no	—
8	C₂×C₄⋊Dic₁₀₁	C₂²	1	2	yes	—
9	C₂×#13108053	C₂²	1	4	yes	—
10	C₂×C₂²⋊Dic₁₀₁	C₂²	1	8	yes	—
11	C₂×C₄×Dic₁₀₁	C₂²	1	4	yes	—
12	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
13	#15692162	C₂²	1	8	yes	—
14	C₂²×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
15	C₂²×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
16	#124×C₁₀₁	—	2	4	no	—
17	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	4	yes	—
18	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
19	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
20	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
21	C₄⋊Dic₁₀₁	—	2	4	no	—
22	#13108053	—	2	8	no	—
23	C₂²⋊Dic₁₀₁	—	2	16	no	—
24	C₄×Dic₁₀₁	—	2	4	no	—
25	C₄×Dic₁₀₁	—	2	8	no	—
26	C₂⁴×C₁₀₁	C₂³	1	2	yes	—
27	C₂×Q₈×C₁₀₁	—	2	2	no	—
28	C₂×Q₈×C₁₀₁	—	2	4	no	—
29	C₂×D₄×C₁₀₁	—	2	2	no	—
30	C₂×D₄×C₁₀₁	—	2	4	no	—
31	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
32	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
33	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
34	C₂²⋊C₄×C₁₀₁	—	2	8	no	—
35	C₄²×C₁₀₁	D₄	1	4	yes	—
36	C₂³×C₁₀₁	—	2	2	no	—
37	C₂³×C₁₀₁	—	2	4	no	—
38	C₂³×C₁₀₁	—	2	8	no	—
39	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
40	Q₈×C₁₀₁	—	4	4	no	—
41	D₄×C₁₀₁	—	4	4	no	—
42	C₂×C₄×C₁₀₁	—	2	4	no	—
43	C₂×C₄×C₁₀₁	—	2	2	no	—
44	C₂×C₄×C₁₀₁	—	2	4	no	—
45	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
46	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₁₀₁	#41	1	4	yes	—
48	C₂×Dic₁₀₁	—	2	4	no	—
49	C₂×Dic₁₀₁	—	2	4	no	—
50	C₂×Dic₁₀₁	—	2	4	no	—
51	C₂×Dic₁₀₁	—	2	8	no	—
52	C₂×Dic₁₀₁	#41	1	8	yes	—
53	C₂²×C₁₀₁	—	4	2	no	—
54	C₂²×C₁₀₁	—	4	8	no	—
55	C₂²×C₁₀₁	—	4	4	no	—
56	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
57	C₂²×C₁₀₁	C₂×#41	1	4	yes	—
58	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
59	C₄×C₁₀₁	—	2	4	no	—
60	C₄×C₁₀₁	—	4	2	no	—
61	C₄×C₁₀₁	#124	1	4	yes	—
62	Dic₁₀₁	—	2	8	no	—
63	Dic₁₀₁	—	4	4	no	—
64	C₂×C₁₀₁	—	8	2	no	—
65	C₂×C₁₀₁	C₂×#124	1	1	yes	characteristic
66	C₂×C₁₀₁	#529	1	4	yes	—
67	C₂×C₁₀₁	#532	1	2	yes	—
68	#2861	—	101	1	no	maximal, Sylow
69	C₁₀₁	#2861	1	1	yes	characteristic, complemented, Sylow
70	C₂×#125	—	101	1	no	—
71	C₂×#124	D₁₀₁	1	1	yes	characteristic
72	#391	—	101	2	no	—
73	#390	—	101	2	no	—
74	#388	—	101	8	no	—
75	C₄³	—	101	1	no	—
76	C₂²×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
77	C₂²×D₄	C₂×D₁₀₁	1	1	yes	characteristic
78	#125	—	202	4	no	—
79	#124	—	2	4	no	—
80	C₂×C₄⋊C₄	—	101	4	no	—
81	C₂×C₄⋊C₄	—	101	2	no	—
82	C₂×C₂²⋊C₄	—	101	8	no	—
83	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	4	yes	—
84	C₂×C₄²	—	101	4	no	—
85	C₂×C₄²	—	101	2	no	—
86	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
87	#95	—	101	8	no	—
88	C₂⁴	C₂²×D₁₀₁	1	2	yes	—
89	C₂×Q₈	—	2	2	no	—
90	C₂×Q₈	—	2	4	no	—
91	C₂×D₄	—	2	2	no	—
92	C₂×D₄	—	2	4	no	—
93	C₂²×C₄	—	101	4	no	—
94	C₂²×C₄	—	101	4	no	—
95	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
96	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
97	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
98	C₄⋊C₄	—	202	8	no	—
99	C₄⋊C₄	—	202	4	no	—
100	C₂²⋊C₄	—	2	8	no	—
101	C₂²⋊C₄	—	202	16	no	—
102	C₄²	—	202	8	no	—
103	C₄²	—	202	4	no	—
104	C₄²	C₂²⋊D₁₀₁	1	4	yes	—
105	C₂³	—	2	2	no	—
106	C₂³	—	2	8	no	—
107	C₂³	—	2	4	no	—
108	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
109	Q₈	—	4	4	no	—
110	D₄	—	4	4	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	101	8	no	—
115	C₂×C₄	—	202	4	no	—
116	C₂×C₄	—	202	8	no	—
117	C₂×C₄	—	202	4	no	—
118	C₂×C₄	—	202	4	no	—
119	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	4	yes	—
120	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
121	C₂×C₄	#13108073	1	4	yes	—
122	C₂²	—	4	2	no	—
123	C₂²	—	4	8	no	—
124	C₂²	—	4	4	no	—
125	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
126	C₂²	#41×D₁₀₁	1	4	yes	—
127	C₂²	C₂×#13108073	1	2	yes	—
128	C₄	—	2	4	no	—
129	C₄	—	4	2	no	—
130	C₄	—	404	4	no	—
131	C₄	—	202	8	no	—
132	C₄	#15692242	1	4	yes	—
133	C₂	—	8	2	no	—
134	C₂	C₂×#15692242	1	1	yes	characteristic
135	C₂	#20265706	1	4	yes	—
136	C₂	#20265723	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692247

C₂

yes

maximal, characteristic, complemented

#20265084

C₂

yes

maximal, complemented

#20265080

C₂

yes

maximal

#20265071

C₂

yes

maximal, complemented

C₄²×Dic₁₀₁

C₂

yes

maximal, characteristic, complemented

C₂×#124×C₁₀₁

C₂

yes

maximal, characteristic

#15692247

—

C₂×C₄⋊Dic₁₀₁

C₂²

yes

—

C₂×#13108053

C₂²

yes

—

C₂×C₂²⋊Dic₁₀₁

C₂²

yes

—

C₂×C₄×Dic₁₀₁

C₂²

yes

—

C₂×C₄×Dic₁₀₁

C₂²

yes

—

#15692162

C₂²

yes

—

C₂²×Q₈×C₁₀₁

C₂²

yes

characteristic

C₂²×D₄×C₁₀₁

C₂²

yes

characteristic

#124×C₁₀₁

—

C₂×C₂²⋊C₄×C₁₀₁

C₂²

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

C₂²×Dic₁₀₁

C₂³

yes

—

C₂²×Dic₁₀₁

C₂³

yes

—

C₄⋊Dic₁₀₁

—

#13108053

—

C₂²⋊Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₂⁴×C₁₀₁

C₂³

yes

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

C₂³

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₂²⋊C₄×C₁₀₁

—

C₄²×C₁₀₁

D₄

yes

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

C₂⁴

yes

characteristic

Q₈×C₁₀₁

—

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

#41

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

#41

yes

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂²×D₄

yes

characteristic

C₂²×C₁₀₁

C₂×#41

yes

—

C₂²×C₁₀₁

C₂×#41

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#124

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

C₂×#124

yes

characteristic

C₂×C₁₀₁

#529

yes

—

C₂×C₁₀₁

#532

yes

—

#2861

—

101

maximal, Sylow

C₁₀₁

#2861

yes

characteristic, complemented, Sylow

C₂×#125

—

101

—

C₂×#124

D₁₀₁

yes

characteristic

#391

—

101

—

#390

—

101

—

#388

—

101

—

C₄³

—

101

—

C₂²×Q₈

C₂×D₁₀₁

yes

characteristic

C₂²×D₄

C₂×D₁₀₁

yes

characteristic

#125

—

202

—

#124

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

—

101

—

C₂×C₂²⋊C₄

—

101

—

C₂×C₂²⋊C₄

C₂×D₁₀₁

yes

—

C₂×C₄²

—

101

—

C₂×C₄²

—

101

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

#95

—

101

—

C₂⁴

C₂²×D₁₀₁

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

C₂²×D₁₀₁

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

C₄⋊C₄

—

202

—

C₄⋊C₄

—

202

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

202

—

102

C₄²

—

202

—

103

C₄²

—

202

—

104

C₄²

C₂²⋊D₁₀₁

yes

—

105

C₂³

—

106

C₂³

—

107

C₂³

—

108

C₂³

C₂³×D₁₀₁

yes

characteristic

109

Q₈

—

110

D₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

101

—

115

C₂×C₄

—

202

—

116

C₂×C₄

—

202

—

117

C₂×C₄

—

202

—

118

C₂×C₄

—

202

—

119

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

120

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

121

C₂×C₄

#13108073

yes

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

C₂²×C₂²⋊D₁₀₁

yes

characteristic

126

C₂²

#41×D₁₀₁

yes

—

127

C₂²

C₂×#13108073

yes

—

128

C₄

—

129

C₄

—

130

C₄

—

404

—

131

C₄

—

202

—

132

C₄

#15692242

yes

—

133

C₂

—

134

C₂

C₂×#15692242

yes

characteristic

135

C₂

#20265706

yes

—

136

C₂

#20265723

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator, socle, cosocle	C₂³×C₁₀₁	C₂⁴
Fitting	C₂×#124×C₁₀₁	C₂
prime radical	C₂²×D₄×C₁₀₁	C₂²
hypercenter	C₂×#124	D₁₀₁
hypocenter	C₁₀₁	#2861
LCS	G ▷ C₂³×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×#124

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₀₁

commutator, socle, cosocle

C₂³×C₁₀₁

C₂⁴

Fitting

C₂×#124×C₁₀₁

C₂

prime radical

C₂²×D₄×C₁₀₁

C₂²

D₁₀₁

C₁₀₁

G ▷ C₂³×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂³ ◁ C₂×#124

N	H ≅ G/N	type
1. C₂×#15692247	C₂	semidirect
2. #20265084	C₂	semidirect
4. #20265071	C₂	semidirect
5. C₄²×Dic₁₀₁	C₂	semidirect
69. C₁₀₁	#2861	semidirect

H ≅ G/N

type

1. C₂×#15692247

C₂

semidirect

2. #20265084

C₂

semidirect

4. #20265071

C₂

semidirect

5. C₄²×Dic₁₀₁

C₂

semidirect

69. C₁₀₁

#2861

semidirect

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

subgroup	index
C₂×#15692247	2
#20265084	2
#20265080	2
#20265071	2
C₄²×Dic₁₀₁	2
C₂×#124×C₁₀₁	2
#2861	101

subgroup

index

C₄²×Dic₁₀₁

C₂×#124×C₁₀₁

#2861

101

Subgroups — #27179502

Conjugacy classes of subgroups

Complements

Maximal subgroups