Group #27172069 — subgroups

Subgroups — #27172069

Order 12928.

292 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 145 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×C₄⋊#11717579	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#11717579	C₂	1	1	yes	maximal, characteristic
3	#20264932	C₂	1	4	yes	maximal
4	C₂×C₄×Q₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11717579	C₂²	1	2	yes	—
6	C₂²×#11717579	C₂²	1	2	yes	—
7	C₄⋊#11717579	—	2	2	no	—
8	C₄×#11717579	—	2	2	no	—
9	C₂²×Q₈×C₁₀₁	C₄	1	1	yes	characteristic
10	C₄×Q₈×C₁₀₁	—	2	2	no	—
11	C₄×Q₈×C₁₀₁	—	2	2	no	—
12	C₂×C₄⋊C₄×C₁₀₁	C₄	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₁₀₁	C₄	1	2	yes	—
14	C₂×C₄²×C₁₀₁	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
16	C₂×#11717579	—	2	2	no	—
17	C₂×#11717579	—	2	2	no	—
18	C₂×#11717579	—	2	2	no	—
19	C₂×#11717579	—	2	4	no	—
20	C₂×#11717579	Q₈	1	4	yes	complemented
21	C₂×Q₈×C₁₀₁	—	2	2	no	—
22	C₂×Q₈×C₁₀₁	—	2	2	no	—
23	C₂×Q₈×C₁₀₁	C₈	1	4	yes	complemented
24	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₁₀₁	—	2	4	no	—
30	C₄⋊C₄×C₁₀₁	—	2	2	no	—
31	C₄²×C₁₀₁	—	2	2	no	—
32	C₄²×C₁₀₁	—	2	2	no	—
33	C₄²×C₁₀₁	D₄	1	2	yes	—
34	C₄²×C₁₀₁	D₄	1	2	yes	—
35	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
36	Q₈×C₁₀₁	—	2	8	no	—
37	C₂×C₄×C₁₀₁	—	2	2	no	—
38	C₂×C₄×C₁₀₁	—	2	2	no	—
39	C₂×C₄×C₁₀₁	—	2	2	no	—
40	C₂×C₄×C₁₀₁	—	2	2	no	—
41	C₂×C₄×C₁₀₁	—	2	4	no	—
42	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₀₁	C₂×C₈	1	4	yes	—
45	C₂×C₄×C₁₀₁	C₂×C₈	1	2	yes	—
46	C₂×C₄×C₁₀₁	M₁₆	1	2	yes	—
47	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₀₁	C₂×Q₈	1	2	yes	—
50	C₂×C₄×C₁₀₁	#41	1	2	yes	—
51	#11717579	—	2	4	no	—
52	#11717579	—	4	2	no	—
53	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₁₀₁	C₄×Q₈	1	2	yes	—
55	C₂²×C₁₀₁	C₂²×C₈	1	1	yes	characteristic
56	C₂²×C₁₀₁	C₂×M₁₆	1	1	yes	characteristic
57	C₂²×C₁₀₁	#131	1	2	yes	—
58	C₄×C₁₀₁	—	2	4	no	—
59	C₄×C₁₀₁	—	2	2	no	—
60	C₄×C₁₀₁	—	4	2	no	—
61	C₄×C₁₀₁	C₂²⋊C₈	1	4	yes	—
62	C₄×C₁₀₁	C₂²⋊Q₈	1	2	yes	—
63	C₄×C₁₀₁	#124	1	1	yes	characteristic
64	C₄×C₁₀₁	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₁₀₁	#391	1	1	yes	characteristic
66	C₂×C₁₀₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
67	C₂×C₁₀₁	#408	1	1	yes	characteristic
68	C₂×C₁₀₁	C₈×Q₈	1	2	yes	—
69	C₂×C₁₀₁	#446	1	2	yes	—
70	#1984	—	101	1	no	maximal, Sylow
71	C₁₀₁	#1984	1	1	yes	characteristic, complemented, Sylow
72	C₂×C₄×Q₈	D₁₀₁	1	1	yes	characteristic
73	C₂×C₄⋊C₈	—	101	1	no	—
74	C₂×C₄×C₈	—	101	1	no	—
75	#336	—	101	4	no	—
76	C₂²×Q₈	Dic₁₀₁	1	1	yes	characteristic
77	C₂²×C₈	—	101	2	no	—
78	C₂²×C₈	—	101	2	no	—
79	C₄×Q₈	—	2	2	no	—
80	C₄×Q₈	—	2	2	no	—
81	C₂×C₄⋊C₄	Dic₁₀₁	1	1	yes	characteristic
82	C₂×C₄⋊C₄	Dic₁₀₁	1	2	yes	—
83	C₂×C₄²	C₂×D₁₀₁	1	2	yes	—
84	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
85	C₄⋊C₈	—	202	2	no	—
86	C₄×C₈	—	202	2	no	—
87	C₂×Q₈	—	2	2	no	—
88	C₂×Q₈	—	2	2	no	—
89	C₂×Q₈	#11717579	1	4	yes	complemented
90	C₂²×C₄	C₂×Dic₁₀₁	1	2	yes	—
91	C₂²×C₄	C₂×Dic₁₀₁	1	1	yes	characteristic
92	C₂²×C₄	C₂×Dic₁₀₁	1	2	yes	—
93	C₂²×C₄	C₂×Dic₁₀₁	1	1	yes	characteristic
94	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
95	C₂×C₈	—	101	4	no	—
96	C₂×C₈	—	202	2	no	—
97	C₂×C₈	—	202	4	no	—
98	C₂×C₈	—	202	2	no	—
99	C₂×C₈	—	202	2	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄²	—	2	2	no	—
103	C₄²	—	2	2	no	—
104	C₄²	C₂²⋊D₁₀₁	1	2	yes	—
105	C₄²	C₂²⋊D₁₀₁	1	2	yes	—
106	C₂³	C₂²×Dic₁₀₁	1	1	yes	characteristic
107	Q₈	—	2	8	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	C₂²⋊Dic₁₀₁	1	2	yes	—
114	C₂×C₄	C₂²⋊Dic₁₀₁	1	2	yes	—
115	C₂×C₄	C₂×#11717579	1	4	yes	—
116	C₂×C₄	C₂×#11717579	1	2	yes	—
117	C₂×C₄	#13108058	1	2	yes	—
118	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
119	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
120	C₂×C₄	Q₈×D₁₀₁	1	2	yes	—
121	C₂×C₄	#13108075	1	2	yes	—
122	C₈	—	404	2	no	—
123	C₈	—	202	4	no	—
124	C₂²	C₂×C₂²⋊Dic₁₀₁	1	1	yes	characteristic
125	C₂²	Q₈×Dic₁₀₁	1	2	yes	—
126	C₂²	C₂²×#11717579	1	1	yes	characteristic
127	C₂²	C₂×#13108058	1	1	yes	characteristic
128	C₂²	#15692262	1	2	yes	—
129	C₄	—	2	4	no	—
130	C₄	—	2	2	no	—
131	C₄	—	4	2	no	—
132	C₄	C₂²⋊#11717579	1	4	yes	—
133	C₄	#15692231	1	2	yes	—
134	C₄	#15692244	1	1	yes	characteristic
135	C₄	#15692253	1	1	yes	characteristic
136	C₂	#20265084	1	1	yes	characteristic
137	C₂	C₂×C₂²⋊#11717579	1	1	yes	characteristic
138	C₂	#20265144	1	1	yes	characteristic
139	C₂	Q₈×#11717579	1	2	yes	—
140	C₂	#20265308	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11717579

C₂

yes

maximal, characteristic

C₂×C₄×#11717579

C₂

yes

maximal, characteristic

#20264932

C₂

yes

maximal

C₂×C₄×Q₈×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×#11717579

C₂²

yes

—

C₂²×#11717579

C₂²

yes

—

C₄⋊#11717579

—

C₄×#11717579

—

C₂²×Q₈×C₁₀₁

C₄

yes

characteristic

C₄×Q₈×C₁₀₁

—

C₄×Q₈×C₁₀₁

—

C₂×C₄⋊C₄×C₁₀₁

C₄

yes

characteristic

C₂×C₄⋊C₄×C₁₀₁

C₄

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

—

—

—

—

Q₈

yes

complemented

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

C₈

yes

complemented

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄²×C₁₀₁

—

C₄²×C₁₀₁

—

C₄²×C₁₀₁

D₄

yes

—

C₄²×C₁₀₁

D₄

yes

—

C₂³×C₁₀₁

C₂²×C₄

yes

characteristic

Q₈×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂×C₈

yes

—

C₂×C₄×C₁₀₁

C₂×C₈

yes

—

C₂×C₄×C₁₀₁

M₁₆

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×Q₈

yes

—

C₂×C₄×C₁₀₁

#41

yes

—

#11717579

—

#11717579

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₁

C₄×Q₈

yes

—

C₂²×C₁₀₁

C₂²×C₈

yes

characteristic

C₂²×C₁₀₁

C₂×M₁₆

yes

characteristic

C₂²×C₁₀₁

#131

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₂²⋊C₈

yes

—

C₄×C₁₀₁

C₂²⋊Q₈

yes

—

C₄×C₁₀₁

#124

yes

characteristic

C₄×C₁₀₁

C₄⋊Q₈

yes

characteristic

C₂×C₁₀₁

#391

yes

characteristic

C₂×C₁₀₁

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₁₀₁

#408

yes

characteristic

C₂×C₁₀₁

C₈×Q₈

yes

—

C₂×C₁₀₁

#446

yes

—

#1984

—

101

maximal, Sylow

C₁₀₁

#1984

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

D₁₀₁

yes

characteristic

C₂×C₄⋊C₈

—

101

—

C₂×C₄×C₈

—

101

—

#336

—

101

—

C₂²×Q₈

Dic₁₀₁

yes

characteristic

C₂²×C₈

—

101

—

C₂²×C₈

—

101

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

Dic₁₀₁

yes

characteristic

C₂×C₄⋊C₄

Dic₁₀₁

yes

—

C₂×C₄²

C₂×D₁₀₁

yes

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

C₄⋊C₈

—

202

—

C₄×C₈

—

202

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

#11717579

yes

complemented

C₂²×C₄

C₂×Dic₁₀₁

yes

—

C₂²×C₄

C₂×Dic₁₀₁

yes

characteristic

C₂²×C₄

C₂×Dic₁₀₁

yes

—

C₂²×C₄

C₂×Dic₁₀₁

yes

characteristic

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

C₂×C₈

—

101

—

C₂×C₈

—

202

—

C₂×C₈

—

202

—

C₂×C₈

—

202

—

C₂×C₈

—

202

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄²

—

103

C₄²

—

104

C₄²

C₂²⋊D₁₀₁

yes

—

105

C₄²

C₂²⋊D₁₀₁

yes

—

106

C₂³

C₂²×Dic₁₀₁

yes

characteristic

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂²⋊Dic₁₀₁

yes

—

114

C₂×C₄

C₂²⋊Dic₁₀₁

yes

—

115

C₂×C₄

C₂×#11717579

yes

—

116

C₂×C₄

C₂×#11717579

yes

—

117

C₂×C₄

#13108058

yes

—

118

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

119

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

120

C₂×C₄

Q₈×D₁₀₁

yes

—

121

C₂×C₄

#13108075

yes

—

122

C₈

—

404

—

123

C₈

—

202

—

124

C₂²

C₂×C₂²⋊Dic₁₀₁

yes

characteristic

125

C₂²

Q₈×Dic₁₀₁

yes

—

126

C₂²

C₂²×#11717579

yes

characteristic

127

C₂²

C₂×#13108058

yes

characteristic

128

C₂²

#15692262

yes

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

C₂²⋊#11717579

yes

—

133

C₄

#15692231

yes

—

134

C₄

#15692244

yes

characteristic

135

C₄

#15692253

yes

characteristic

136

C₂

#20265084

yes

characteristic

137

C₂

C₂×C₂²⋊#11717579

yes

characteristic

138

C₂

#20265144

yes

characteristic

139

C₂

Q₈×#11717579

yes

—

140

C₂

#20265308

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₀₁
commutator	C₂²×C₁₀₁	C₂²×C₈
Fitting	C₂×C₄×Q₈×C₁₀₁	C₂
socle, prime radical	C₂³×C₁₀₁	C₂²×C₄
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	C₂×C₄×Q₈	D₁₀₁
hypocenter	C₁₀₁	#1984
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₀₁

commutator

C₂²×C₁₀₁

C₂²×C₈

Fitting

C₂×C₄×Q₈×C₁₀₁

C₂

socle, prime radical

C₂³×C₁₀₁

C₂²×C₄

cosocle

C₂²×C₄×C₁₀₁

C₂³

hypercenter

C₂×C₄×Q₈

D₁₀₁

C₁₀₁

G ▷ C₂²×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
20. C₂×#11717579	Q₈	semidirect
23. C₂×Q₈×C₁₀₁	C₈	semidirect
71. C₁₀₁	#1984	semidirect
89. C₂×Q₈	#11717579	semidirect

H ≅ G/N

type

20. C₂×#11717579

Q₈

semidirect

23. C₂×Q₈×C₁₀₁

C₈

semidirect

71. C₁₀₁

#1984

semidirect

89. C₂×Q₈

#11717579

semidirect

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

subgroup	index
C₂×C₄⋊#11717579	2
C₂×C₄×#11717579	2
#20264932	2
C₂×C₄×Q₈×C₁₀₁	2
#1984	101

subgroup

index

C₂×C₄×Q₈×C₁₀₁

101

Subgroups — #27172069

Conjugacy classes of subgroups

Complements

Maximal subgroups