Group #27172071 — subgroups

Subgroups — #27172071

Order 12928.

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

normal subgroups only

This group has 280 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×C₄×Dic₂₀₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#15692166	C₂	1	1	yes	maximal, characteristic
3	#20264931	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₀₂	C₄	1	1	yes	characteristic
6	C₂²×#11717579	C₂²	1	2	yes	—
7	C₄×Dic₂₀₂	—	2	2	no	—
8	C₄×Dic₂₀₂	—	2	2	no	—
9	C₂×C₄⋊Dic₁₀₁	C₄	1	1	yes	characteristic
10	C₂×#13108053	C₄	1	2	yes	—
11	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
12	#15692166	—	2	2	no	—
13	C₂²×C₈×C₁₀₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₁₀₁	—	2	2	no	—
16	C₂×Dic₂₀₂	—	2	2	no	—
17	C₂×Dic₂₀₂	—	2	4	no	—
18	C₂²×Dic₁₀₁	C₂×C₄	1	2	yes	—
19	C₂²×Dic₁₀₁	C₂×C₄	1	2	yes	—
20	C₂×#11717579	—	2	2	no	—
21	C₂×#11717579	—	2	2	no	—
22	C₂×#11717579	Q₈	1	4	yes	complemented
23	C₄⋊Dic₁₀₁	—	2	2	no	—
24	#13108053	—	2	4	no	—
25	C₄×Dic₁₀₁	—	2	2	no	—
26	C₄×Dic₁₀₁	—	2	2	no	—
27	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₀₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
30	C₂×C₈×C₁₀₁	—	2	2	no	—
31	C₂×C₈×C₁₀₁	—	2	4	no	—
32	C₄²×C₁₀₁	D₄	1	2	yes	—
33	C₄²×C₁₀₁	D₄	1	2	yes	—
34	C₂³×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₀₁	—	2	2	no	—
36	C₂×C₄×C₁₀₁	—	2	2	no	—
37	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₀₁	C₂×Q₈	1	2	yes	—
42	C₂×C₄×C₁₀₁	#41	1	2	yes	—
43	C₂×Dic₁₀₁	—	2	2	no	—
44	C₂×Dic₁₀₁	—	2	2	no	—
45	C₂×Dic₁₀₁	—	2	2	no	—
46	C₂×Dic₁₀₁	—	2	4	no	—
47	C₂×Dic₁₀₁	M₁₆	1	4	yes	—
48	Dic₂₀₂	—	2	8	no	—
49	C₈×C₁₀₁	—	4	2	no	—
50	#11717579	—	2	4	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	2	no	—
57	C₄×C₁₀₁	C₂²⋊Q₈	1	2	yes	—
58	C₄×C₁₀₁	#124	1	1	yes	characteristic
59	C₄×C₁₀₁	C₄⋊Q₈	1	1	yes	characteristic
60	Dic₁₀₁	—	2	4	no	—
61	Dic₁₀₁	—	4	2	no	—
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	D₁₀₁	1	1	yes	characteristic
70	C₂×#97	—	101	1	no	—
71	#336	—	101	4	no	—
72	C₂²×Q₈	—	101	1	no	—
73	C₂²×C₈	—	101	2	no	—
74	C₂²×C₈	C₂×D₁₀₁	1	2	yes	—
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	2	no	—
80	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
81	C₄⋊C₈	—	2	2	no	—
82	#97	—	202	2	no	—
83	C₂×Q₈	—	202	4	no	—
84	C₂×Q₈	—	202	2	no	—
85	C₂²×C₄	—	101	2	no	—
86	C₂²×C₄	—	101	2	no	—
87	C₂²×C₄	C₄×D₁₀₁	1	1	yes	characteristic
88	C₂²×C₄	C₄×D₁₀₁	1	1	yes	characteristic
89	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	4	no	—
92	C₂×C₈	—	101	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	—	202	2	no	—
98	C₄²	—	202	2	no	—
99	C₄²	D₄₀₄	1	2	yes	—
100	C₄²	C₂²⋊D₁₀₁	1	2	yes	—
101	C₂³	C₂×C₄×D₁₀₁	1	1	yes	characteristic
102	Q₈	—	202	8	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	101	4	no	—
106	C₂×C₄	—	202	2	no	—
107	C₂×C₄	—	202	4	no	—
108	C₂×C₄	—	202	2	no	—
109	C₂×C₄	—	202	2	no	—
110	C₂×C₄	#13108051	1	2	yes	—
111	C₂×C₄	#13108051	1	2	yes	—
112	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
113	C₂×C₄	C₂×D₄₀₄	1	1	yes	characteristic
114	C₂×C₄	Q₈×D₁₀₁	1	2	yes	—
115	C₂×C₄	#13108073	1	2	yes	—
116	C₈	—	4	2	no	—
117	C₈	—	202	4	no	—
118	C₂²	C₂×#13108051	1	1	yes	characteristic
119	C₂²	#15692220	1	2	yes	—
120	C₂²	M₁₆×D₁₀₁	1	2	yes	—
121	C₂²	#15692261	1	2	yes	—
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	404	2	no	—
125	C₄	—	202	4	no	—
126	C₄	#15692235	1	2	yes	—
127	C₄	#15692242	1	1	yes	characteristic
128	C₄	#15692253	1	1	yes	characteristic
129	C₂	#20265082	1	1	yes	characteristic
130	C₂	#20265140	1	1	yes	characteristic
131	C₂	#20265143	1	1	yes	characteristic
132	C₂	#20265306	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₂₀₂

C₂

yes

maximal, characteristic

C₂×#15692166

C₂

yes

maximal, characteristic

#20264931

C₂

yes

maximal

C₂×C₄⋊C₈×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×Dic₂₀₂

C₄

yes

characteristic

C₂²×#11717579

C₂²

yes

—

C₄×Dic₂₀₂

—

C₄×Dic₂₀₂

—

C₂×C₄⋊Dic₁₀₁

C₄

yes

characteristic

C₂×#13108053

C₄

yes

—

C₂×C₄×Dic₁₀₁

C₂²

yes

—

#15692166

—

C₂²×C₈×C₁₀₁

C₂²

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₁

—

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂²×Dic₁₀₁

C₂×C₄

yes

—

C₂²×Dic₁₀₁

C₂×C₄

yes

—

C₂×#11717579

—

C₂×#11717579

—

C₂×#11717579

Q₈

yes

complemented

C₄⋊Dic₁₀₁

—

#13108053

—

C₄×Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₂×C₈×C₁₀₁

—

C₂×C₈×C₁₀₁

—

C₄²×C₁₀₁

D₄

yes

—

C₄²×C₁₀₁

D₄

yes

—

C₂³×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

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

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

M₁₆

yes

—

Dic₂₀₂

—

C₈×C₁₀₁

—

#11717579

—

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

—

C₄×C₁₀₁

C₂²⋊Q₈

yes

—

C₄×C₁₀₁

#124

yes

characteristic

C₄×C₁₀₁

C₄⋊Q₈

yes

characteristic

Dic₁₀₁

—

Dic₁₀₁

—

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

D₁₀₁

yes

characteristic

C₂×#97

—

101

—

#336

—

101

—

C₂²×Q₈

—

101

—

C₂²×C₈

—

101

—

C₂²×C₈

C₂×D₁₀₁

yes

—

C₄×Q₈

—

202

—

C₄×Q₈

—

202

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄²

—

101

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

C₄⋊C₈

—

#97

—

202

—

C₂×Q₈

—

202

—

C₂×Q₈

—

202

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

C₄×D₁₀₁

yes

characteristic

C₂²×C₄

C₄×D₁₀₁

yes

characteristic

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

101

—

C₂×C₈

—

202

—

C₂×C₈

—

202

—

C₄⋊C₄

—

202

—

C₄⋊C₄

—

202

—

C₄²

—

202

—

C₄²

—

202

—

C₄²

D₄₀₄

yes

—

100

C₄²

C₂²⋊D₁₀₁

yes

—

101

C₂³

C₂×C₄×D₁₀₁

yes

characteristic

102

Q₈

—

202

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

101

—

106

C₂×C₄

—

202

—

107

C₂×C₄

—

202

—

108

C₂×C₄

—

202

—

109

C₂×C₄

—

202

—

110

C₂×C₄

#13108051

yes

—

111

C₂×C₄

#13108051

yes

—

112

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

113

C₂×C₄

C₂×D₄₀₄

yes

characteristic

114

C₂×C₄

Q₈×D₁₀₁

yes

—

115

C₂×C₄

#13108073

yes

—

116

C₈

—

117

C₈

—

202

—

118

C₂²

C₂×#13108051

yes

characteristic

119

C₂²

#15692220

yes

—

120

C₂²

M₁₆×D₁₀₁

yes

—

121

C₂²

#15692261

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

404

—

125

C₄

—

202

—

126

C₄

#15692235

yes

—

127

C₄

#15692242

yes

characteristic

128

C₄

#15692253

yes

characteristic

129

C₂

#20265082

yes

characteristic

130

C₂

#20265140

yes

characteristic

131

C₂

#20265143

yes

characteristic

132

C₂

#20265306

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₀₁
commutator, socle, prime radical	C₂³×C₁₀₁	C₂²×C₄
Fitting	C₂×C₄⋊C₈×C₁₀₁	C₂
cosocle	C₂²×C₄×C₁₀₁	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₀₁
hypocenter	C₁₀₁	#1985
LCS	G ▷ C₂³×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₀₁

commutator, socle, prime radical

C₂³×C₁₀₁

C₂²×C₄

Fitting

C₂×C₄⋊C₈×C₁₀₁

C₂

cosocle

C₂²×C₄×C₁₀₁

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₀₁

C₁₀₁

G ▷ C₂³×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
22. C₂×#11717579	Q₈	semidirect
67. C₁₀₁	#1985	semidirect

H ≅ G/N

type

22. C₂×#11717579

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×C₄×Dic₂₀₂	2
C₂×#15692166	2
#20264931	2
C₂×C₄⋊C₈×C₁₀₁	2
#1985	101

subgroup

index

C₂×C₄×Dic₂₀₂

C₂×#15692166

#20264931

C₂×C₄⋊C₈×C₁₀₁

#1985

101

Subgroups — #27172071

Conjugacy classes of subgroups

Complements

Maximal subgroups