Group #27187306 — subgroups

Subgroups — #27187306

Order 12928.

280 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 94 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×#15692322	C₂	1	1	yes	maximal, characteristic
2	C₂×#15692310	C₂	1	1	yes	maximal, characteristic
3	#20266043	C₂	1	4	yes	maximal
4	C₂×Q₈×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11717581	C₂²	1	2	yes	—
6	C₂²×#11717581	C₂²	1	2	yes	—
7	#15692322	—	2	2	no	—
8	#15692310	—	2	2	no	—
9	Q₈×Dic₁₀₁	—	2	2	no	—
10	Q₈×Dic₁₀₁	—	2	2	no	—
11	C₂×C₄⋊Dic₁₀₁	C₄	1	1	yes	characteristic
12	C₂×C₄⋊Dic₁₀₁	C₄	1	2	yes	—
13	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
14	C₂×C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
15	C₂²×Q₈×C₁₀₁	C₄	1	1	yes	characteristic
16	C₂×#11717581	—	2	2	no	—
17	C₂×#11717581	—	2	2	no	—
18	C₂×#11717581	—	2	2	no	—
19	C₂×#11717581	—	2	4	no	—
20	C₂×#11717581	Q₈	1	4	yes	complemented
21	C₂²×Dic₁₀₁	C₂×C₄	1	2	yes	—
22	C₂²×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
24	C₄⋊Dic₁₀₁	—	2	2	no	—
25	C₄⋊Dic₁₀₁	—	2	4	no	—
26	C₄×Dic₁₀₁	—	2	2	no	—
27	C₄×Dic₁₀₁	—	2	2	no	—
28	C₄×Dic₁₀₁	D₄	1	2	yes	—
29	C₄×Dic₁₀₁	D₄	1	2	yes	—
30	C₂×Q₈×C₁₀₁	—	2	2	no	—
31	C₂×Q₈×C₁₀₁	—	2	4	no	—
32	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₀₁	C₂×C₄	1	2	yes	—
34	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
35	Q₈×C₁₀₁	—	2	8	no	—
36	C₂×C₄×C₁₀₁	—	2	2	no	—
37	C₂×C₄×C₁₀₁	—	2	2	no	—
38	C₂×C₄×C₁₀₁	—	2	2	no	—
39	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₀₁	M₁₆	1	4	yes	—
41	C₂×Dic₁₀₁	—	2	2	no	—
42	C₂×Dic₁₀₁	—	2	2	no	—
43	C₂×Dic₁₀₁	—	2	4	no	—
44	C₂×Dic₁₀₁	C₂²⋊C₄	1	2	yes	—
45	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
46	C₂×Dic₁₀₁	C₂×D₄	1	1	yes	characteristic
47	C₂×Dic₁₀₁	C₂×Q₈	1	2	yes	—
48	C₂×Dic₁₀₁	#41	1	2	yes	—
49	#11717581	—	2	4	no	—
50	#11717581	—	4	2	no	—
51	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₁₀₁	C₄×Q₈	1	2	yes	—
53	C₂²×C₁₀₁	C₂×M₁₆	1	2	yes	—
54	C₂²×C₁₀₁	#131	1	2	yes	—
55	C₄×C₁₀₁	—	2	2	no	—
56	C₄×C₁₀₁	—	2	4	no	—
57	Dic₁₀₁	—	2	2	no	—
58	Dic₁₀₁	—	4	2	no	—
59	Dic₁₀₁	C₂²⋊Q₈	1	2	yes	—
60	Dic₁₀₁	#124	1	1	yes	characteristic
61	Dic₁₀₁	C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₁₀₁	#391	1	1	yes	characteristic
63	C₂×C₁₀₁	#407	1	1	yes	characteristic
64	C₂×C₁₀₁	#408	1	1	yes	characteristic
65	C₂×C₁₀₁	#446	1	4	yes	—
66	#1985	—	101	1	no	maximal, Sylow
67	C₁₀₁	#1985	1	1	yes	characteristic, complemented, Sylow
68	C₂×C₄×Q₈	—	101	1	no	—
69	C₂×C₄⋊C₈	—	101	1	no	—
70	C₂×#97	—	101	1	no	—
71	#336	—	101	4	no	—
72	C₂²×Q₈	#87938	1	1	yes	characteristic
73	C₂²×C₈	—	101	2	no	—
74	C₂²×C₈	—	101	2	no	—
75	C₄×Q₈	—	202	2	no	—
76	C₄×Q₈	—	202	2	no	—
77	C₂×C₄⋊C₄	—	101	2	no	—
78	C₂×C₄⋊C₄	—	101	1	no	—
79	C₂×C₄²	—	101	1	no	—
80	C₂×C₄²	—	101	2	no	—
81	C₄⋊C₈	—	202	2	no	—
82	#97	—	202	2	no	—
83	C₂×Q₈	—	2	2	no	—
84	C₂×Q₈	—	2	4	no	—
85	C₂²×C₄	—	101	1	no	—
86	C₂²×C₄	—	101	1	no	—
87	C₂²×C₄	—	101	2	no	—
88	C₂²×C₄	C₂×#87938	1	1	yes	characteristic
89	C₂²×C₄	C₂×#87938	1	2	yes	—
90	C₂×C₈	—	101	4	no	—
91	C₂×C₈	—	202	2	no	—
92	C₂×C₈	—	202	4	no	—
93	C₂×C₈	—	202	2	no	—
94	C₂×C₈	—	202	2	no	—
95	C₄⋊C₄	—	202	4	no	—
96	C₄⋊C₄	—	202	2	no	—
97	C₄²	—	101	2	no	—
98	C₄²	—	101	2	no	—
99	C₄²	—	202	2	no	—
100	C₄²	—	202	2	no	—
101	C₂³	C₂²×#87938	1	1	yes	characteristic
102	Q₈	—	2	8	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	101	1	no	—
107	C₂×C₄	—	101	1	no	—
108	C₂×C₄	—	101	2	no	—
109	C₂×C₄	—	101	2	no	—
110	C₂×C₄	—	101	2	no	—
111	C₂×C₄	—	202	2	no	—
112	C₂×C₄	—	202	4	no	—
113	C₂×C₄	—	202	2	no	—
114	C₂×C₄	C₂²⋊#87938	1	2	yes	—
115	C₂×C₄	#13108083	1	4	yes	—
116	C₈	—	404	2	no	—
117	C₈	—	202	4	no	—
118	C₂²	C₂×C₂²⋊#87938	1	1	yes	characteristic
119	C₂²	Q₈×#87938	1	2	yes	—
120	C₂²	C₂×#13108083	1	2	yes	—
121	C₂²	#15692343	1	2	yes	—
122	C₄	—	2	2	no	—
123	C₄	—	2	4	no	—
124	C₄	—	101	1	no	—
125	C₄	—	101	2	no	—
126	C₄	—	101	1	no	—
127	C₄	—	404	2	no	—
128	C₄	—	202	2	no	—
129	C₂	(C₂×#87938)⋊Q₈	1	1	yes	characteristic
130	C₂	#20266147	1	1	yes	characteristic
131	C₂	#20266152	1	1	yes	characteristic
132	C₂	#20266228	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692322

C₂

yes

maximal, characteristic

C₂×#15692310

C₂

yes

maximal, characteristic

#20266043

C₂

yes

maximal

C₂×Q₈×Dic₁₀₁

C₂

yes

maximal, characteristic

C₂²×#11717581

C₂²

yes

—

C₂²×#11717581

C₂²

yes

—

#15692322

—

#15692310

—

Q₈×Dic₁₀₁

—

Q₈×Dic₁₀₁

—

C₂×C₄⋊Dic₁₀₁

C₄

yes

characteristic

C₂×C₄⋊Dic₁₀₁

C₄

yes

—

C₂×C₄×Dic₁₀₁

C₂²

yes

—

C₂×C₄×Dic₁₀₁

C₂²

yes

characteristic

C₂²×Q₈×C₁₀₁

C₄

yes

—

—

—

—

Q₈

yes

complemented

C₂²×Dic₁₀₁

C₂×C₄

yes

—

C₂²×Dic₁₀₁

C₂×C₄

yes

characteristic

C₂²×Dic₁₀₁

C₂³

yes

characteristic

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₄×Dic₁₀₁

D₄

yes

—

C₄×Dic₁₀₁

D₄

yes

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂³×C₁₀₁

C₂²×C₄

yes

characteristic

Q₈×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

M₁₆

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

C₂²⋊C₄

yes

—

C₂×Dic₁₀₁

C₂×D₄

yes

characteristic

C₂×Dic₁₀₁

C₂×D₄

yes

characteristic

C₂×Dic₁₀₁

C₂×Q₈

yes

—

C₂×Dic₁₀₁

#41

yes

—

#11717581

—

#11717581

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₁

C₄×Q₈

yes

—

C₂²×C₁₀₁

C₂×M₁₆

yes

—

C₂²×C₁₀₁

#131

yes

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

C₂²⋊Q₈

yes

—

Dic₁₀₁

#124

yes

characteristic

Dic₁₀₁

C₄⋊Q₈

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

C₂×C₁₀₁

yes

—

#1985

—

101

maximal, Sylow

C₁₀₁

#1985

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

101

—

C₂×C₄⋊C₈

—

101

—

C₂×#97

—

101

—

#336

—

101

—

C₂²×Q₈

#87938

yes

characteristic

C₂²×C₈

—

101

—

C₂²×C₈

—

101

—

C₄×Q₈

—

202

—

C₄×Q₈

—

202

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄²

—

101

—

C₂×C₄²

—

101

—

C₄⋊C₈

—

202

—

#97

—

202

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

C₂×#87938

yes

characteristic

C₂²×C₄

C₂×#87938

yes

—

C₂×C₈

—

101

—

C₂×C₈

—

202

—

C₂×C₈

—

202

—

C₂×C₈

—

202

—

C₂×C₈

—

202

—

C₄⋊C₄

—

202

—

C₄⋊C₄

—

202

—

C₄²

—

101

—

C₄²

—

101

—

C₄²

—

202

—

100

C₄²

—

202

—

101

C₂³

C₂²×#87938

yes

characteristic

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

101

—

107

C₂×C₄

—

101

—

108

C₂×C₄

—

101

—

109

C₂×C₄

—

101

—

110

C₂×C₄

—

101

—

111

C₂×C₄

—

202

—

112

C₂×C₄

—

202

—

113

C₂×C₄

—

202

—

114

C₂×C₄

C₂²⋊#87938

yes

—

115

C₂×C₄

#13108083

yes

—

116

C₈

—

404

—

117

C₈

—

202

—

118

C₂²

C₂×C₂²⋊#87938

yes

characteristic

119

C₂²

Q₈×#87938

yes

—

120

C₂²

C₂×#13108083

yes

—

121

C₂²

#15692343

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

101

—

125

C₄

—

101

—

126

C₄

—

101

—

127

C₄

—

404

—

128

C₄

—

202

—

129

C₂

(C₂×#87938)⋊Q₈

yes

characteristic

130

C₂

#20266147

yes

characteristic

131

C₂

#20266152

yes

characteristic

132

C₂

#20266228

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂²×#87938
commutator, socle, prime radical	C₂³×C₁₀₁	C₂²×C₄
Fitting	C₂²×Q₈×C₁₀₁	C₄
cosocle	C₂²×Dic₁₀₁	C₂³
hypercenter	C₂²×Q₈	#87938
hypocenter	C₁₀₁	#1985
LCS	G ▷ C₂³×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂²×#87938

commutator, socle, prime radical

C₂³×C₁₀₁

C₂²×C₄

Fitting

C₂²×Q₈×C₁₀₁

C₄

cosocle

C₂²×Dic₁₀₁

C₂³

C₂²×Q₈

C₁₀₁

G ▷ C₂³×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂³ ◁ C₂²×Q₈

N	H ≅ G/N	type
20. C₂×#11717581	Q₈	semidirect
67. C₁₀₁	#1985	semidirect

H ≅ G/N

type

20. C₂×#11717581

Q₈

semidirect

67. C₁₀₁

#1985

semidirect

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

subgroup	index
C₂×#15692322	2
C₂×#15692310	2
#20266043	2
C₂×Q₈×Dic₁₀₁	2
#1985	101

subgroup

index

C₂×Q₈×Dic₁₀₁

101

Subgroups — #27187306

Conjugacy classes of subgroups

Complements

Maximal subgroups