Group #27179505 — subgroups

Subgroups — #27179505

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×#15692243	C₂	1	1	yes	maximal, characteristic
2	#20265082	C₂	1	2	yes	maximal, complemented
3	#20265078	C₂	1	2	yes	maximal
4	#20265067	C₂	1	8	yes	maximal, complemented
5	C₄²×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#125×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Dic₂₀₂	C₂²	1	1	yes	characteristic
8	C₂²×D₄₀₄	C₂²	1	1	yes	characteristic
9	#15692243	—	2	4	no	—
10	C₂×#13108051	C₂²	1	8	yes	—
11	C₂×#13108051	C₂²	1	4	yes	—
12	C₂×C₄×Dic₁₀₁	C₂²	1	4	yes	—
13	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
14	#15692162	C₂²	1	8	yes	—
15	#125×C₁₀₁	—	2	4	no	—
16	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
18	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
19	C₂³×D₁₀₁	C₂³	1	2	yes	—
20	C₂×Dic₂₀₂	—	2	2	no	—
21	C₂×Dic₂₀₂	—	2	4	no	—
22	C₂×D₄₀₄	—	2	2	no	—
23	C₂×D₄₀₄	—	2	4	no	—
24	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
25	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
26	#13108051	—	2	8	no	—
27	#13108051	—	2	16	no	—
28	C₄×Dic₁₀₁	—	2	4	no	—
29	C₄×Dic₁₀₁	—	2	8	no	—
30	C₂²×C₄×C₁₀₁	C₂³	1	4	yes	—
31	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
32	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₁₀₁	—	2	8	no	—
34	C₄⋊C₄×C₁₀₁	—	2	4	no	—
35	C₄²×C₁₀₁	D₄	1	4	yes	—
36	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
37	C₂²×D₁₀₁	—	2	2	no	—
38	C₂²×D₁₀₁	—	2	8	no	—
39	C₂²×D₁₀₁	—	2	4	no	—
40	C₂×C₄×C₁₀₁	—	2	4	no	—
41	C₂×C₄×C₁₀₁	—	2	4	no	—
42	C₂×C₄×C₁₀₁	—	2	8	no	—
43	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
44	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₀₁	#41	1	4	yes	—
46	C₂×Dic₁₀₁	—	2	4	no	—
47	C₂×Dic₁₀₁	—	2	4	no	—
48	C₂×Dic₁₀₁	—	2	2	no	—
49	C₂×Dic₁₀₁	—	2	4	no	—
50	C₂×Dic₁₀₁	#41	1	8	yes	—
51	D₄₀₄	—	4	4	no	—
52	Dic₂₀₂	—	4	4	no	—
53	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
54	C₂²×C₁₀₁	C₂×#41	1	4	yes	—
55	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
56	C₂×D₁₀₁	—	4	2	no	—
57	C₂×D₁₀₁	—	4	8	no	—
58	C₂×D₁₀₁	—	4	4	no	—
59	C₄×C₁₀₁	—	2	4	no	—
60	C₄×C₁₀₁	—	4	4	no	—
61	C₄×C₁₀₁	#124	1	4	yes	—
62	Dic₁₀₁	—	2	8	no	—
63	Dic₁₀₁	—	4	2	no	—
64	C₂×C₁₀₁	C₂×#124	1	1	yes	characteristic
65	C₂×C₁₀₁	#529	1	4	yes	—
66	C₂×C₁₀₁	#532	1	2	yes	—
67	D₁₀₁	—	8	2	no	—
68	#2861	—	101	1	no	maximal, Sylow
69	C₁₀₁	#2861	1	1	yes	characteristic, complemented, Sylow
70	C₂×#125	D₁₀₁	1	1	yes	characteristic, complemented
71	C₂×#124	—	101	1	no	—
72	#391	—	101	2	no	—
73	#390	—	101	2	no	—
74	#388	—	101	8	no	—
75	C₄³	—	101	1	no	—
76	C₂²×Q₈	—	101	1	no	—
77	C₂²×D₄	—	101	1	no	—
78	#125	—	2	4	no	—
79	#124	—	202	4	no	—
80	C₂×C₄⋊C₄	C₂×D₁₀₁	1	4	yes	—
81	C₂×C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
82	C₂×C₂²⋊C₄	—	101	4	no	—
83	C₂×C₂²⋊C₄	—	101	8	no	—
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	—	101	2	no	—
89	C₂×Q₈	—	202	2	no	—
90	C₂×Q₈	—	202	4	no	—
91	C₂×D₄	—	202	2	no	—
92	C₂×D₄	—	202	4	no	—
93	C₂²×C₄	—	101	4	no	—
94	C₂²×C₄	—	101	2	no	—
95	C₂²×C₄	C₂²×D₁₀₁	1	4	yes	—
96	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
97	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
98	C₄⋊C₄	—	2	8	no	—
99	C₄⋊C₄	—	2	4	no	—
100	C₂²⋊C₄	—	202	16	no	—
101	C₂²⋊C₄	—	202	8	no	—
102	C₄²	—	202	8	no	—
103	C₄²	—	202	4	no	—
104	C₄²	C₂²⋊D₁₀₁	1	4	yes	—
105	C₂³	—	202	2	no	—
106	C₂³	—	202	8	no	—
107	C₂³	—	202	4	no	—
108	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
109	Q₈	—	404	4	no	—
110	D₄	—	404	4	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	2	8	no	—
114	C₂×C₄	—	101	8	no	—
115	C₂×C₄	—	202	2	no	—
116	C₂×C₄	—	202	4	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	#13108075	1	4	yes	—
122	C₂²	—	404	2	no	—
123	C₂²	—	404	8	no	—
124	C₂²	—	404	4	no	—
125	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
126	C₂²	#41×D₁₀₁	1	4	yes	—
127	C₂²	C₂×#13108075	1	2	yes	—
128	C₄	—	2	4	no	—
129	C₄	—	4	4	no	—
130	C₄	—	404	2	no	—
131	C₄	—	202	8	no	—
132	C₄	#15692244	1	4	yes	—
133	C₂	—	808	2	no	—
134	C₂	C₂×#15692244	1	1	yes	characteristic
135	C₂	#20265712	1	4	yes	—
136	C₂	#20265723	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692243

C₂

yes

maximal, characteristic

#20265082

C₂

yes

maximal, complemented

#20265078

C₂

yes

maximal

#20265067

C₂

yes

maximal, complemented

C₄²×Dic₁₀₁

C₂

yes

maximal, characteristic, complemented

C₂×#125×C₁₀₁

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₂₀₂

C₂²

yes

characteristic

C₂²×D₄₀₄

C₂²

yes

characteristic

#15692243

—

C₂×#13108051

C₂²

yes

—

C₂×#13108051

C₂²

yes

—

C₂×C₄×Dic₁₀₁

C₂²

yes

—

C₂×C₄×Dic₁₀₁

C₂²

yes

—

#15692162

C₂²

yes

—

#125×C₁₀₁

—

C₂×C₄⋊C₄×C₁₀₁

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₁

C₂²

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

C₂³×D₁₀₁

C₂³

yes

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂×D₄₀₄

—

C₂×D₄₀₄

—

C₂²×Dic₁₀₁

C₂³

yes

—

C₂²×Dic₁₀₁

C₂³

yes

—

#13108051

—

#13108051

—

C₄×Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₂²×C₄×C₁₀₁

C₂³

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄²×C₁₀₁

D₄

yes

—

C₂³×C₁₀₁

C₂⁴

yes

characteristic

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

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

—

D₄₀₄

—

Dic₂₀₂

—

C₂²×C₁₀₁

C₂²×D₄

yes

characteristic

C₂²×C₁₀₁

C₂×#41

yes

—

C₂²×C₁₀₁

C₂×#41

yes

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#124

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

C₂×#124

yes

characteristic

C₂×C₁₀₁

#529

yes

—

C₂×C₁₀₁

#532

yes

—

D₁₀₁

—

#2861

—

101

maximal, Sylow

C₁₀₁

#2861

yes

characteristic, complemented, Sylow

C₂×#125

D₁₀₁

yes

characteristic, complemented

C₂×#124

—

101

—

#391

—

101

—

#390

—

101

—

#388

—

101

—

C₄³

—

101

—

C₂²×Q₈

—

101

—

C₂²×D₄

—

101

—

#125

—

#124

—

202

—

C₂×C₄⋊C₄

C₂×D₁₀₁

yes

—

C₂×C₄⋊C₄

C₂×D₁₀₁

yes

—

C₂×C₂²⋊C₄

—

101

—

C₂×C₂²⋊C₄

—

101

—

C₂×C₄²

—

101

—

C₂×C₄²

—

101

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

#95

—

101

—

C₂⁴

—

101

—

C₂×Q₈

—

202

—

C₂×Q₈

—

202

—

C₂×D₄

—

202

—

C₂×D₄

—

202

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

C₂²×D₁₀₁

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

100

C₂²⋊C₄

—

202

—

101

C₂²⋊C₄

—

202

—

102

C₄²

—

202

—

103

C₄²

—

202

—

104

C₄²

C₂²⋊D₁₀₁

yes

—

105

C₂³

—

202

—

106

C₂³

—

202

—

107

C₂³

—

202

—

108

C₂³

C₂³×D₁₀₁

yes

characteristic

109

Q₈

—

404

—

110

D₄

—

404

—

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

#13108075

yes

—

122

C₂²

—

404

—

123

C₂²

—

404

—

124

C₂²

—

404

—

125

C₂²

C₂²×C₂²⋊D₁₀₁

yes

characteristic

126

C₂²

#41×D₁₀₁

yes

—

127

C₂²

C₂×#13108075

yes

—

128

C₄

—

129

C₄

—

130

C₄

—

404

—

131

C₄

—

202

—

132

C₄

#15692244

yes

—

133

C₂

—

808

—

134

C₂

C₂×#15692244

yes

characteristic

135

C₂

#20265712

yes

—

136

C₂

#20265723

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator, socle, cosocle	C₂³×C₁₀₁	C₂⁴
Fitting	C₂×#125×C₁₀₁	C₂
prime radical	C₂²×D₄₀₄	C₂²
hypercenter	C₂×#125	D₁₀₁
hypocenter	C₁₀₁	#2861
LCS	G ▷ C₂³×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₀₁

commutator, socle, cosocle

C₂³×C₁₀₁

C₂⁴

Fitting

C₂×#125×C₁₀₁

C₂

prime radical

C₂²×D₄₀₄

C₂²

D₁₀₁

C₁₀₁

G ▷ C₂³×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
2. #20265082	C₂	semidirect
4. #20265067	C₂	semidirect
5. C₄²×Dic₁₀₁	C₂	semidirect
6. C₂×#125×C₁₀₁	C₂	semidirect
69. C₁₀₁	#2861	semidirect
70. C₂×#125	D₁₀₁	semidirect

H ≅ G/N

type

2. #20265082

C₂

semidirect

4. #20265067

C₂

semidirect

5. C₄²×Dic₁₀₁

C₂

semidirect

6. C₂×#125×C₁₀₁

C₂

semidirect

69. C₁₀₁

#2861

semidirect

70. C₂×#125

D₁₀₁

semidirect

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

subgroup	index
C₂×#15692243	2
#20265082	2
#20265078	2
#20265067	2
C₄²×Dic₁₀₁	2
C₂×#125×C₁₀₁	2
#2861	101

subgroup

index

C₄²×Dic₁₀₁

C₂×#125×C₁₀₁

#2861

101

Subgroups — #27179505

Conjugacy classes of subgroups

Complements

Maximal subgroups