Group #27181654 — subgroups

Subgroups — #27181654

Order 12928.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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	#20265734	C₂	1	1	yes	maximal, characteristic
2	#20265587	C₂	1	4	yes	maximal, complemented
3	#20265583	C₂	1	4	yes	maximal, complemented
4	#20265213	C₂	1	4	yes	maximal
5	C₂×#15692166	C₂	1	1	yes	maximal, characteristic
6	#533×C₁₀₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11717579	C₂²	1	2	yes	—
8	#15692255	C₂²	1	2	yes	—
9	#15692247	C₂²	1	2	yes	—
10	#15692234	—	2	2	no	—
11	C₄×Dic₂₀₂	—	2	2	no	—
12	#15692214	C₂²	1	2	yes	—
13	#15692188	C₂²	1	8	yes	—
14	#15692186	C₂²	1	8	yes	—
15	#15692166	C₂²	1	2	yes	—
16	#15692166	C₂²	1	2	yes	—
17	C₄⋊Q₈×C₁₀₁	C₂²	1	2	yes	—
18	#125×C₁₀₁	C₂²	1	2	yes	—
19	C₂²⋊Q₈×C₁₀₁	—	2	2	no	—
20	C₄×Q₈×C₁₀₁	—	2	2	no	—
21	#117×C₁₀₁	C₂²	1	2	yes	—
22	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
23	C₂×Dic₂₀₂	—	2	2	no	—
24	C₂×#11717579	Q₈	1	8	yes	complemented
25	C₂×#11717579	C₂³	1	4	yes	—
26	C₄⋊Dic₁₀₁	—	2	2	no	—
27	C₄⋊Dic₁₀₁	C₂³	1	4	yes	—
28	#13108053	—	2	4	no	—
29	C₂²⋊Dic₁₀₁	—	2	2	no	—
30	C₄×Dic₁₀₁	—	2	2	no	—
31	C₂×Q₈×C₁₀₁	—	2	2	no	—
32	C₂²×C₄×C₁₀₁	D₄	1	2	yes	—
33	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₁₀₁	—	2	4	no	—
35	C₄⋊C₄×C₁₀₁	—	2	2	no	—
36	C₄⋊C₄×C₁₀₁	C₂³	1	4	yes	—
37	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
38	C₄²×C₁₀₁	—	2	2	no	—
39	C₄²×C₁₀₁	D₄	1	2	yes	—
40	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
41	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
42	C₂³×C₁₀₁	C₂×D₄	1	1	yes	characteristic
43	Q₈×C₁₀₁	—	4	4	no	—
44	C₂×C₄×C₁₀₁	—	2	4	no	—
45	C₂×C₄×C₁₀₁	—	4	2	no	—
46	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₀₁	C₂×Q₈	1	4	yes	—
50	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
51	C₂×Dic₁₀₁	—	2	4	no	—
52	Dic₂₀₂	—	4	4	no	—
53	#11717579	C₂×Q₈	1	8	yes	—
54	C₂²×C₁₀₁	—	2	2	no	—
55	C₂²×C₁₀₁	C₄⋊Q₈	1	2	yes	—
56	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
57	C₄×C₁₀₁	—	2	2	no	—
58	C₄×C₁₀₁	—	2	2	no	—
59	C₄×C₁₀₁	—	4	4	no	—
60	C₄×C₁₀₁	C₄⋊Q₈	1	2	yes	—
61	C₄×C₁₀₁	C₂²×Q₈	1	2	yes	—
62	Dic₁₀₁	—	4	4	no	—
63	C₂×C₁₀₁	—	2	2	no	—
64	C₂×C₁₀₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₁₀₁	#575	1	2	yes	—
66	#3162	—	101	1	no	maximal, Sylow
67	C₁₀₁	#3162	1	1	yes	characteristic, complemented, Sylow
68	#533	—	101	1	no	—
69	#533	D₁₀₁	1	1	yes	characteristic
70	#501	—	101	4	no	—
71	#501	—	101	4	no	—
72	#427	—	101	4	no	—
73	C₂×#97	—	101	1	no	—
74	C₂²×C₈	—	101	2	no	—
75	C₄⋊Q₈	—	101	2	no	—
76	C₄⋊Q₈	C₂×D₁₀₁	1	2	yes	—
77	#125	—	101	2	no	—
78	#125	C₂×D₁₀₁	1	2	yes	—
79	C₂²⋊Q₈	—	2	2	no	—
80	C₂²⋊Q₈	—	202	2	no	—
81	C₄×Q₈	—	2	2	no	—
82	C₄×Q₈	—	202	2	no	—
83	#117	—	101	2	no	—
84	#117	C₂×D₁₀₁	1	2	yes	—
85	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
86	#107	—	101	8	no	—
87	#106	—	101	8	no	—
88	#97	—	101	2	no	—
89	#97	—	101	2	no	—
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	202	2	no	—
92	C₂²×C₄	C₂²⋊D₁₀₁	1	2	yes	—
93	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
94	C₂×C₈	—	101	8	no	—
95	C₂×C₈	—	101	4	no	—
96	C₄⋊C₄	—	2	4	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	101	4	no	—
99	C₄⋊C₄	—	202	4	no	—
100	C₄⋊C₄	—	202	2	no	—
101	C₄⋊C₄	C₂²×D₁₀₁	1	4	yes	—
102	C₂²⋊C₄	—	2	2	no	—
103	C₂²⋊C₄	—	202	2	no	—
104	C₄²	—	2	2	no	—
105	C₄²	—	202	2	no	—
106	C₄²	C₂²⋊D₁₀₁	1	2	yes	—
107	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
108	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
109	C₂³	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
110	Q₈	—	4	4	no	—
111	Q₈	—	404	4	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	202	4	no	—
115	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
116	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
117	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
118	C₂×C₄	Q₈×D₁₀₁	1	4	yes	—
119	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
120	C₈	—	101	8	no	—
121	C₂²	—	2	2	no	—
122	C₂²	#15692253	1	2	yes	—
123	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
124	C₄	—	2	2	no	—
125	C₄	—	2	2	no	—
126	C₄	—	4	4	no	—
127	C₄	—	404	4	no	—
128	C₄	#15692253	1	2	yes	—
129	C₄	C₂×Q₈×D₁₀₁	1	2	yes	—
130	C₂	—	2	2	no	—
131	C₂	C₂×#15692253	1	1	yes	characteristic
132	C₂	#20265975	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20265734

C₂

yes

maximal, characteristic

#20265587

C₂

yes

maximal, complemented

#20265583

C₂

yes

maximal, complemented

#20265213

C₂

yes

maximal

C₂×#15692166

C₂

yes

maximal, characteristic

#533×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×#11717579

C₂²

yes

—

#15692255

C₂²

yes

—

#15692247

C₂²

yes

—

#15692234

—

C₄×Dic₂₀₂

—

#15692214

C₂²

yes

—

#15692188

C₂²

yes

—

#15692186

C₂²

yes

—

#15692166

C₂²

yes

—

#15692166

C₂²

yes

—

C₄⋊Q₈×C₁₀₁

C₂²

yes

—

#125×C₁₀₁

C₂²

yes

—

C₂²⋊Q₈×C₁₀₁

—

C₄×Q₈×C₁₀₁

—

#117×C₁₀₁

C₂²

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

C₂×Dic₂₀₂

—

C₂×#11717579

Q₈

yes

complemented

C₂×#11717579

C₂³

yes

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

C₂³

yes

—

#13108053

—

C₂²⋊Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂²×C₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

C₂³

yes

—

C₂²⋊C₄×C₁₀₁

—

C₄²×C₁₀₁

—

C₄²×C₁₀₁

D₄

yes

—

C₄²×C₁₀₁

C₂³

yes

characteristic

C₄²×C₁₀₁

C₂³

yes

characteristic

C₂³×C₁₀₁

C₂×D₄

yes

characteristic

Q₈×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×Q₈

yes

—

C₂×C₄×C₁₀₁

C₂⁴

yes

characteristic

C₂×Dic₁₀₁

—

Dic₂₀₂

—

#11717579

C₂×Q₈

yes

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₄⋊Q₈

yes

—

C₂²×C₁₀₁

C₂²×D₄

yes

characteristic

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₄⋊Q₈

yes

—

C₄×C₁₀₁

C₂²×Q₈

yes

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₀₁

#575

yes

—

#3162

—

101

maximal, Sylow

C₁₀₁

#3162

yes

characteristic, complemented, Sylow

#533

—

101

—

#533

D₁₀₁

yes

characteristic

#501

—

101

—

#501

—

101

—

#427

—

101

—

C₂×#97

—

101

—

C₂²×C₈

—

101

—

C₄⋊Q₈

—

101

—

C₄⋊Q₈

C₂×D₁₀₁

yes

—

#125

—

101

—

#125

C₂×D₁₀₁

yes

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

202

—

C₄×Q₈

—

C₄×Q₈

—

202

—

#117

—

101

—

#117

C₂×D₁₀₁

yes

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

#107

—

101

—

#106

—

101

—

#97

—

101

—

#97

—

101

—

C₂×Q₈

—

C₂×Q₈

—

202

—

C₂²×C₄

C₂²⋊D₁₀₁

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

C₂×C₈

—

101

—

C₂×C₈

—

101

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

101

—

C₄⋊C₄

—

202

—

100

C₄⋊C₄

—

202

—

101

C₄⋊C₄

C₂²×D₁₀₁

yes

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

—

202

—

104

C₄²

—

105

C₄²

—

202

—

106

C₄²

C₂²⋊D₁₀₁

yes

—

107

C₄²

C₂²×D₁₀₁

yes

characteristic

108

C₄²

C₂²×D₁₀₁

yes

characteristic

109

C₂³

C₂×C₂²⋊D₁₀₁

yes

characteristic

110

Q₈

—

111

Q₈

—

404

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

202

—

115

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

116

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

117

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

118

C₂×C₄

Q₈×D₁₀₁

yes

—

119

C₂×C₄

C₂³×D₁₀₁

yes

characteristic

120

C₈

—

101

—

121

C₂²

—

122

C₂²

#15692253

yes

—

123

C₂²

C₂²×C₂²⋊D₁₀₁

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

404

—

128

C₄

#15692253

yes

—

129

C₄

C₂×Q₈×D₁₀₁

yes

—

130

C₂

—

131

C₂

C₂×#15692253

yes

characteristic

132

C₂

#20265975

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₀₁
commutator, cosocle	C₂×C₄×C₁₀₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting	#533×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂²×D₄
prime radical	C₂³×C₁₀₁	C₂×D₄
hypercenter	#533	D₁₀₁
hypocenter	C₁₀₁	#3162
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₀₁

commutator, cosocle

C₂×C₄×C₁₀₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₁

Fitting

#533×C₁₀₁

C₂

socle

C₂²×C₁₀₁

C₂²×D₄

prime radical

C₂³×C₁₀₁

C₂×D₄

D₁₀₁

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #20265587	C₂	semidirect
3. #20265583	C₂	semidirect
24. C₂×#11717579	Q₈	semidirect
67. C₁₀₁	#3162	semidirect

H ≅ G/N

type

2. #20265587

C₂

semidirect

3. #20265583

C₂

semidirect

24. C₂×#11717579

Q₈

semidirect

67. C₁₀₁

#3162

semidirect

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

subgroup	index
#20265734	2
#20265587	2
#20265583	2
#20265213	2
C₂×#15692166	2
#533×C₁₀₁	2
#3162	101

subgroup

index

#533×C₁₀₁

101

Subgroups — #27181654

Conjugacy classes of subgroups

Complements

Maximal subgroups