Group #27183810 — subgroups

Subgroups — #27183810

Order 12928.

336 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 141 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	#20265906	C₂	1	1	yes	maximal, characteristic
2	#20265529	C₂	1	3	yes	maximal
3	#20265469	C₂	1	3	yes	maximal
4	#20265387	C₂	1	3	yes	maximal
5	Q₈×#11717579	C₂	1	1	yes	maximal, characteristic
6	C₄×#13108064	C₂	1	3	yes	maximal
7	Q₈²×C₁₀₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#13108064	C₂²	1	3	yes	—
9	#15692255	C₂²	1	3	yes	—
10	#15692246	—	2	3	no	—
11	Q₈×Dic₁₀₁	—	2	1	no	—
12	C₄×Dic₂₀₂	C₂²	1	3	yes	—
13	#15692188	C₂²	1	3	yes	—
14	C₄⋊#11717579	C₂²	1	3	yes	—
15	#15692179	C₂²	1	1	yes	characteristic
16	#15692178	C₂²	1	3	yes	—
17	#15692178	C₂²	1	6	yes	—
18	C₄×#11717579	C₂²	1	3	yes	—
19	C₄⋊Q₈×C₁₀₁	—	2	3	no	—
20	C₄⋊Q₈×C₁₀₁	C₂²	1	3	yes	—
21	C₄×Q₈×C₁₀₁	—	2	1	no	—
22	C₄×Q₈×C₁₀₁	C₂²	1	3	yes	—
23	C₄×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
24	C₂×Dic₂₀₂	C₂³	1	3	yes	—
25	#13108064	—	2	6	no	—
26	C₂×#11717579	C₂³	1	1	yes	characteristic
27	C₂×#11717579	C₂³	1	3	yes	—
28	C₄⋊Dic₁₀₁	—	2	3	no	—
29	C₄⋊Dic₁₀₁	—	2	3	no	—
30	C₄⋊Dic₁₀₁	C₂³	1	1	yes	characteristic
31	#13108053	—	2	3	no	—
32	C₄×Dic₁₀₁	—	2	3	no	—
33	C₂×Q₈×C₁₀₁	—	2	3	no	—
34	C₂×Q₈×C₁₀₁	D₄	1	1	yes	characteristic
35	C₂×Q₈×C₁₀₁	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₁₀₁	—	2	3	no	—
37	C₄⋊C₄×C₁₀₁	—	2	3	no	—
38	C₄⋊C₄×C₁₀₁	D₄	1	3	yes	—
39	C₄⋊C₄×C₁₀₁	C₂³	1	3	yes	—
40	C₄²×C₁₀₁	—	2	3	no	—
41	C₄²×C₁₀₁	C₂³	1	3	yes	—
42	Q₈×C₁₀₁	—	2	1	no	—
43	Q₈×C₁₀₁	—	8	3	no	—
44	Q₈×C₁₀₁	Q₁₆	1	4	yes	complemented
45	Q₈×C₁₀₁	#41	1	2	yes	—
46	C₂×C₄×C₁₀₁	—	2	1	no	—
47	C₂×C₄×C₁₀₁	—	2	3	no	—
48	C₂×C₄×C₁₀₁	C₂×D₄	1	3	yes	—
49	C₂×C₄×C₁₀₁	C₂×D₄	1	3	yes	—
50	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
51	C₂×Dic₁₀₁	—	2	1	no	—
52	C₂×Dic₁₀₁	—	2	3	no	—
53	Dic₂₀₂	—	2	3	no	—
54	Dic₂₀₂	—	2	3	no	—
55	#11717579	—	2	3	no	—
56	#11717579	—	2	1	no	—
57	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₁₀₁	—	2	2	no	—
59	C₄×C₁₀₁	—	2	3	no	—
60	C₄×C₁₀₁	—	4	3	no	—
61	C₄×C₁₀₁	C₂×Q₁₆	1	6	yes	—
62	C₄×C₁₀₁	C₂×#41	1	1	yes	characteristic
63	C₄×C₁₀₁	2₋¹⁺⁴	1	1	yes	characteristic
64	Dic₁₀₁	—	4	3	no	—
65	Dic₁₀₁	—	4	1	no	—
66	C₂×C₁₀₁	#548	1	1	yes	characteristic
67	C₂×C₁₀₁	C₂²×Q₁₆	1	1	yes	characteristic
68	C₂×C₁₀₁	#577	1	1	yes	characteristic
69	#3363	—	101	1	no	maximal, Sylow
70	C₁₀₁	#3363	1	1	yes	characteristic, complemented, Sylow
71	Q₈²	D₁₀₁	1	1	yes	characteristic
72	#557	—	101	1	no	—
73	#487	—	101	3	no	—
74	#477	—	101	3	no	—
75	#462	—	101	3	no	—
76	C₈×Q₈	—	101	1	no	—
77	C₄×Q₁₆	—	101	3	no	—
78	C₂×Q₁₆	—	101	3	no	—
79	C₄⋊Q₈	—	2	3	no	—
80	C₄⋊Q₈	—	101	3	no	—
81	C₄⋊Q₈	C₂×D₁₀₁	1	3	yes	—
82	#125	—	202	3	no	—
83	C₄×Q₈	—	2	1	no	—
84	C₄×Q₈	—	101	3	no	—
85	C₄×Q₈	—	202	1	no	—
86	C₄×Q₈	C₂×D₁₀₁	1	3	yes	—
87	C₄×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
88	#107	—	101	3	no	—
89	C₄⋊C₈	—	101	3	no	—
90	#103	—	101	1	no	—
91	#103	—	101	3	no	—
92	#103	—	101	6	no	—
93	C₄×C₈	—	101	3	no	—
94	C₂×Q₈	—	2	3	no	—
95	C₂×Q₈	—	101	3	no	—
96	C₂×Q₈	C₂²⋊D₁₀₁	1	1	yes	characteristic
97	C₂×Q₈	C₂²×D₁₀₁	1	1	yes	characteristic
98	Q₁₆	—	202	6	no	—
99	C₂×C₈	—	101	1	no	—
100	C₂×C₈	—	101	3	no	—
101	C₄⋊C₄	—	2	3	no	—
102	C₄⋊C₄	—	2	3	no	—
103	C₄⋊C₄	—	101	1	no	—
104	C₄⋊C₄	—	202	3	no	—
105	C₄⋊C₄	—	202	3	no	—
106	C₄⋊C₄	—	202	3	no	—
107	C₄⋊C₄	C₂²⋊D₁₀₁	1	3	yes	—
108	C₄⋊C₄	C₂²×D₁₀₁	1	3	yes	—
109	C₄²	—	2	3	no	—
110	C₄²	—	202	3	no	—
111	C₄²	C₂²×D₁₀₁	1	3	yes	—
112	Q₈	—	2	1	no	—
113	Q₈	—	8	3	no	—
114	Q₈	—	202	3	no	—
115	Q₈	—	202	3	no	—
116	Q₈	#13108064	1	4	yes	complemented
117	Q₈	#13108075	1	2	yes	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	3	no	—
120	C₂×C₄	—	202	1	no	—
121	C₂×C₄	—	202	3	no	—
122	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	3	yes	—
123	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	3	yes	—
124	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
125	C₈	—	202	3	no	—
126	C₈	—	202	1	no	—
127	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
128	C₄	—	2	3	no	—
129	C₄	—	2	2	no	—
130	C₄	—	4	3	no	—
131	C₄	—	404	1	no	—
132	C₄	—	404	3	no	—
133	C₄	C₂×#13108064	1	6	yes	—
134	C₄	C₂×#13108075	1	1	yes	characteristic
135	C₄	#15692303	1	1	yes	characteristic
136	C₂	#20265838	1	1	yes	characteristic
137	C₂	C₂²×#13108064	1	1	yes	characteristic
138	C₂	#20265988	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20265906

C₂

yes

maximal, characteristic

#20265529

C₂

yes

maximal

#20265469

C₂

yes

maximal

#20265387

C₂

yes

maximal

Q₈×#11717579

C₂

yes

maximal, characteristic

C₄×#13108064

C₂

yes

maximal

Q₈²×C₁₀₁

C₂

yes

maximal, characteristic

C₂×#13108064

C₂²

yes

—

#15692255

C₂²

yes

—

#15692246

—

Q₈×Dic₁₀₁

—

C₄×Dic₂₀₂

C₂²

yes

—

#15692188

C₂²

yes

—

C₄⋊#11717579

C₂²

yes

—

#15692179

C₂²

yes

characteristic

#15692178

C₂²

yes

—

#15692178

C₂²

yes

—

C₄×#11717579

C₂²

yes

—

C₄⋊Q₈×C₁₀₁

—

C₄⋊Q₈×C₁₀₁

C₂²

yes

—

C₄×Q₈×C₁₀₁

—

C₄×Q₈×C₁₀₁

C₂²

yes

—

C₄×Q₈×C₁₀₁

C₂²

yes

characteristic

C₂×Dic₂₀₂

C₂³

yes

—

#13108064

—

C₂×#11717579

C₂³

yes

characteristic

C₂×#11717579

C₂³

yes

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

C₂³

yes

characteristic

#13108053

—

C₄×Dic₁₀₁

—

C₂×Q₈×C₁₀₁

—

C₂×Q₈×C₁₀₁

D₄

yes

characteristic

C₂×Q₈×C₁₀₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

D₄

yes

—

C₄⋊C₄×C₁₀₁

C₂³

yes

—

C₄²×C₁₀₁

—

C₄²×C₁₀₁

C₂³

yes

—

Q₈×C₁₀₁

—

Q₈×C₁₀₁

—

Q₈×C₁₀₁

Q₁₆

yes

complemented

Q₈×C₁₀₁

#41

yes

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂⁴

yes

characteristic

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

Dic₂₀₂

—

Dic₂₀₂

—

#11717579

—

#11717579

—

C₂²×C₁₀₁

C₂²×D₄

yes

characteristic

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₂×Q₁₆

yes

—

C₄×C₁₀₁

C₂×#41

yes

characteristic

C₄×C₁₀₁

2₋¹⁺⁴

yes

characteristic

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

#548

yes

characteristic

C₂×C₁₀₁

C₂²×Q₁₆

yes

C₂×C₁₀₁

yes

—

101

maximal, Sylow

C₁₀₁

#3363

yes

characteristic, complemented, Sylow

Q₈²

D₁₀₁

yes

characteristic

#557

—

101

—

#487

—

101

—

#477

—

101

—

#462

—

101

—

C₈×Q₈

—

101

—

C₄×Q₁₆

—

101

—

C₂×Q₁₆

—

101

—

C₄⋊Q₈

—

C₄⋊Q₈

—

101

—

C₄⋊Q₈

C₂×D₁₀₁

yes

—

#125

—

202

—

C₄×Q₈

—

C₄×Q₈

—

101

—

C₄×Q₈

—

202

—

C₄×Q₈

C₂×D₁₀₁

yes

—

C₄×Q₈

C₂×D₁₀₁

yes

characteristic

#107

—

101

—

C₄⋊C₈

—

101

—

#103

—

101

—

#103

—

101

—

#103

—

101

—

C₄×C₈

—

101

—

C₂×Q₈

—

C₂×Q₈

—

101

—

C₂×Q₈

C₂²⋊D₁₀₁

yes

characteristic

C₂×Q₈

C₂²×D₁₀₁

yes

characteristic

Q₁₆

—

202

—

C₂×C₈

—

101

—

100

C₂×C₈

—

101

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

101

—

104

C₄⋊C₄

—

202

—

105

C₄⋊C₄

—

202

—

106

C₄⋊C₄

—

202

—

107

C₄⋊C₄

C₂²⋊D₁₀₁

yes

—

108

C₄⋊C₄

C₂²×D₁₀₁

yes

—

109

C₄²

—

110

C₄²

—

202

—

111

C₄²

C₂²×D₁₀₁

yes

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

202

—

115

Q₈

—

202

—

116

Q₈

#13108064

yes

complemented

117

Q₈

#13108075

yes

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

202

—

121

C₂×C₄

—

202

—

122

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

123

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

124

C₂×C₄

C₂³×D₁₀₁

yes

characteristic

125

C₈

—

202

—

126

C₈

—

202

—

127

C₂²

C₂²×C₂²⋊D₁₀₁

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

404

—

132

C₄

—

404

—

133

C₄

C₂×#13108064

yes

—

134

C₄

C₂×#13108075

yes

characteristic

135

C₄

#15692303

yes

characteristic

136

C₂

#20265838

yes

characteristic

137

C₂

C₂²×#13108064

yes

characteristic

138

C₂

#20265988

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₀₁
commutator, cosocle	C₂×C₄×C₁₀₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₁
Fitting	Q₈²×C₁₀₁	C₂
socle, prime radical	C₂²×C₁₀₁	C₂²×D₄
hypercenter	Q₈²	D₁₀₁
hypocenter	C₁₀₁	#3363
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈²

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₀₁

commutator, cosocle

C₂×C₄×C₁₀₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₁

Fitting

Q₈²×C₁₀₁

C₂

socle, prime radical

C₂²×C₁₀₁

C₂²×D₄

hypercenter

Q₈²

D₁₀₁

hypocenter

C₁₀₁

#3363

LCS

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈²

N	H ≅ G/N	type
44. Q₈×C₁₀₁	Q₁₆	semidirect
70. C₁₀₁	#3363	semidirect
116. Q₈	#13108064	semidirect

H ≅ G/N

type

44. Q₈×C₁₀₁

Q₁₆

semidirect

70. C₁₀₁

#3363

semidirect

116. Q₈

#13108064

semidirect

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

subgroup	index
#20265906	2
#20265529	2
#20265469	2
#20265387	2
Q₈×#11717579	2
C₄×#13108064	2
Q₈²×C₁₀₁	2
#3363	101

subgroup

index

Q₈²×C₁₀₁

101

Subgroups — #27183810

Conjugacy classes of subgroups

Complements

Maximal subgroups