Group #27180264 — subgroups

Subgroups — #27180264

Order 12928.

394 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 221 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	Q₈×#11717579	C₂	1	2	yes	maximal, complemented
2	#20265255	C₂	1	6	yes	maximal, complemented
3	#20265230	C₂	1	3	yes	maximal, complemented
4	#20265201	C₂	1	3	yes	maximal, complemented
5	C₄×#41×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#13108058	C₂²	1	6	yes	—
7	C₄⋊#11717579	C₂²	1	1	yes	characteristic
8	C₄⋊#11717579	C₂²	1	3	yes	—
9	C₄⋊#11717579	C₂²	1	6	yes	—
10	C₂²⋊#11717579	C₂²	1	6	yes	—
11	C₄×#11717579	C₂²	1	6	yes	—
12	C₂×#41×C₁₀₁	C₄	1	1	yes	characteristic
13	C₄×Q₈×C₁₀₁	C₄	1	1	yes	characteristic
14	C₄×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
15	C₄×D₄×C₁₀₁	C₄	1	3	yes	—
16	C₄×D₄×C₁₀₁	C₂²	1	3	yes	—
17	#117×C₁₀₁	C₄	1	3	yes	—
18	C₂×C₄²×C₁₀₁	C₂²	1	3	yes	—
19	#13108058	—	4	6	no	—
20	C₂×#11717579	C₂³	1	2	yes	—
21	C₂×#11717579	C₂³	1	6	yes	—
22	#41×C₁₀₁	—	2	4	no	—
23	C₂×Q₈×C₁₀₁	C₂×C₄	1	1	yes	characteristic
24	C₂×D₄×C₁₀₁	C₂×C₄	1	3	yes	—
25	C₂²×C₄×C₁₀₁	C₂×C₄	1	3	yes	—
26	C₂²×C₄×C₁₀₁	C₂×C₄	1	3	yes	—
27	C₂²×C₄×C₁₀₁	C₂³	1	3	yes	—
28	C₄⋊C₄×C₁₀₁	C₂×C₄	1	3	yes	—
29	C₄⋊C₄×C₁₀₁	C₂×C₄	1	3	yes	—
30	C₂²⋊C₄×C₁₀₁	C₂×C₄	1	6	yes	—
31	C₄²×C₁₀₁	C₂×C₄	1	3	yes	—
32	C₄²×C₁₀₁	C₂×C₄	1	3	yes	—
33	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
34	C₄²×C₁₀₁	C₂³	1	3	yes	—
35	C₂³×C₁₀₁	C₂²×C₄	1	3	yes	—
36	Q₈×C₁₀₁	M₁₆	1	4	yes	complemented
37	D₄×C₁₀₁	—	2	6	no	—
38	C₂×C₄×C₁₀₁	—	2	6	no	—
39	C₂×C₄×C₁₀₁	—	2	6	no	—
40	C₂×C₄×C₁₀₁	—	2	3	no	—
41	C₂×C₄×C₁₀₁	—	2	3	no	—
42	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₀₁	C₂²×C₄	1	3	yes	—
44	C₂×C₄×C₁₀₁	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₁₀₁	C₂²×C₄	1	3	yes	—
46	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₀₁	C₂⁴	1	1	yes	characteristic
48	#11717579	—	2	6	no	—
49	#11717579	—	2	2	no	—
50	C₂²×C₁₀₁	—	2	3	no	—
51	C₂²×C₁₀₁	—	2	3	no	—
52	C₂²×C₁₀₁	—	2	3	no	—
53	C₂²×C₁₀₁	C₂³×C₄	1	1	yes	characteristic
54	C₄×C₁₀₁	—	2	3	no	—
55	C₄×C₁₀₁	—	2	1	no	—
56	C₄×C₁₀₁	—	2	3	no	—
57	C₄×C₁₀₁	C₂×M₁₆	1	6	yes	—
58	C₄×C₁₀₁	#131	1	2	yes	—
59	C₄×C₁₀₁	H(5,𝔽₂)	1	1	yes	characteristic
60	C₄×C₁₀₁	2₋¹⁺⁴	1	1	yes	characteristic
61	C₂×C₁₀₁	—	4	3	no	—
62	C₂×C₁₀₁	#520	1	1	yes	characteristic
63	C₂×C₁₀₁	C₂²×M₁₆	1	1	yes	characteristic
64	C₂×C₁₀₁	C₂×#131	1	1	yes	characteristic
65	#2991	—	101	1	no	maximal, Sylow
66	C₁₀₁	#2991	1	1	yes	characteristic, complemented, Sylow
67	C₄×#41	D₁₀₁	1	1	yes	characteristic
68	C₈×Q₈	—	101	2	no	—
69	#436	—	101	6	no	—
70	#431	—	101	3	no	—
71	#423	—	101	3	no	—
72	C₂×#41	Dic₁₀₁	1	1	yes	characteristic
73	C₂×M₁₆	—	101	6	no	—
74	C₄×Q₈	Dic₁₀₁	1	1	yes	characteristic
75	C₄×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
76	C₄×D₄	Dic₁₀₁	1	3	yes	—
77	C₄×D₄	C₂×D₁₀₁	1	3	yes	—
78	#117	Dic₁₀₁	1	3	yes	—
79	C₂×C₄²	C₂×D₁₀₁	1	3	yes	—
80	C₄⋊C₈	—	101	3	no	—
81	C₄⋊C₈	—	101	6	no	—
82	C₄⋊C₈	—	101	1	no	—
83	C₂²⋊C₈	—	101	6	no	—
84	C₄×C₈	—	101	6	no	—
85	#41	—	2	4	no	—
86	C₂×Q₈	C₂×Dic₁₀₁	1	1	yes	characteristic
87	C₂×D₄	C₂×Dic₁₀₁	1	3	yes	—
88	C₂²×C₄	C₂×Dic₁₀₁	1	3	yes	—
89	C₂²×C₄	C₂×Dic₁₀₁	1	3	yes	—
90	C₂²×C₄	C₂²×D₁₀₁	1	3	yes	—
91	M₁₆	—	404	6	no	—
92	C₂×C₈	—	101	2	no	—
93	C₂×C₈	—	101	6	no	—
94	C₄⋊C₄	C₂×Dic₁₀₁	1	3	yes	—
95	C₄⋊C₄	C₂×Dic₁₀₁	1	3	yes	—
96	C₂²⋊C₄	C₂×Dic₁₀₁	1	6	yes	—
97	C₄²	C₂×Dic₁₀₁	1	3	yes	—
98	C₄²	C₂×Dic₁₀₁	1	3	yes	—
99	C₄²	C₂²×D₁₀₁	1	3	yes	—
100	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
101	C₂³	C₂²×Dic₁₀₁	1	3	yes	—
102	Q₈	#13108058	1	4	yes	complemented
103	D₄	—	2	6	no	—
104	C₂×C₄	—	2	3	no	—
105	C₂×C₄	—	2	3	no	—
106	C₂×C₄	—	2	6	no	—
107	C₂×C₄	—	2	6	no	—
108	C₂×C₄	C₂²×Dic₁₀₁	1	1	yes	characteristic
109	C₂×C₄	C₂²×Dic₁₀₁	1	3	yes	—
110	C₂×C₄	C₂²×Dic₁₀₁	1	3	yes	—
111	C₂×C₄	C₂²×Dic₁₀₁	1	3	yes	—
112	C₂×C₄	C₂²×Dic₁₀₁	1	1	yes	characteristic
113	C₂×C₄	C₂³×D₁₀₁	1	1	yes	characteristic
114	C₈	—	202	6	no	—
115	C₈	—	202	2	no	—
116	C₂²	—	2	3	no	—
117	C₂²	—	2	3	no	—
118	C₂²	—	2	3	no	—
119	C₂²	C₂³×Dic₁₀₁	1	1	yes	characteristic
120	C₄	—	2	3	no	—
121	C₄	—	2	1	no	—
122	C₄	—	2	3	no	—
123	C₄	C₂×#13108058	1	6	yes	—
124	C₄	#15692262	1	2	yes	—
125	C₄	#15692301	1	1	yes	characteristic
126	C₄	#15692302	1	1	yes	characteristic
127	C₂	—	4	3	no	—
128	C₂	#20265660	1	1	yes	characteristic
129	C₂	C₂²×#13108058	1	1	yes	characteristic
130	C₂	C₂×#15692262	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#11717579

C₂

yes

maximal, complemented

#20265255

C₂

yes

maximal, complemented

#20265230

C₂

yes

maximal, complemented

#20265201

C₂

yes

maximal, complemented

C₄×#41×C₁₀₁

C₂

yes

maximal, characteristic

C₂×#13108058

C₂²

yes

—

C₄⋊#11717579

C₂²

yes

characteristic

C₄⋊#11717579

C₂²

yes

—

C₄⋊#11717579

C₂²

yes

—

C₂²⋊#11717579

C₂²

yes

—

C₄×#11717579

C₂²

yes

—

C₂×#41×C₁₀₁

C₄

yes

characteristic

C₄×Q₈×C₁₀₁

C₄

yes

characteristic

C₄×Q₈×C₁₀₁

C₂²

yes

characteristic

C₄×D₄×C₁₀₁

C₄

yes

—

C₄×D₄×C₁₀₁

C₂²

yes

—

#117×C₁₀₁

C₄

yes

—

C₂×C₄²×C₁₀₁

C₂²

yes

—

#13108058

—

C₂×#11717579

C₂³

yes

—

C₂×#11717579

C₂³

yes

—

#41×C₁₀₁

—

C₂×Q₈×C₁₀₁

C₂×C₄

yes

characteristic

C₂×D₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

C₂×C₄

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

—

C₄⋊C₄×C₁₀₁

C₂×C₄

yes

—

C₄⋊C₄×C₁₀₁

C₂×C₄

yes

—

C₂²⋊C₄×C₁₀₁

C₂×C₄

yes

—

C₄²×C₁₀₁

C₂×C₄

yes

—

C₄²×C₁₀₁

C₂×C₄

yes

—

C₄²×C₁₀₁

C₂³

yes

characteristic

C₄²×C₁₀₁

C₂³

yes

—

C₂³×C₁₀₁

C₂²×C₄

yes

—

Q₈×C₁₀₁

M₁₆

yes

complemented

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂²×C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂⁴

yes

characteristic

#11717579

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂³×C₄

yes

characteristic

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₂×M₁₆

yes

—

C₄×C₁₀₁

#131

yes

—

C₄×C₁₀₁

H(5,𝔽₂)

yes

characteristic

C₄×C₁₀₁

2₋¹⁺⁴

yes

characteristic

C₂×C₁₀₁

—

C₂×C₁₀₁

#520

yes

characteristic

C₂×C₁₀₁

C₂²×M₁₆

yes

C₂×C₁₀₁

yes

—

101

maximal, Sylow

C₁₀₁

#2991

yes

characteristic, complemented, Sylow

C₄×#41

D₁₀₁

yes

characteristic

C₈×Q₈

—

101

—

#436

—

101

—

#431

—

101

—

#423

—

101

—

C₂×#41

Dic₁₀₁

yes

characteristic

C₂×M₁₆

—

101

—

C₄×Q₈

Dic₁₀₁

yes

characteristic

C₄×Q₈

C₂×D₁₀₁

yes

characteristic

C₄×D₄

Dic₁₀₁

yes

—

C₄×D₄

C₂×D₁₀₁

yes

—

#117

Dic₁₀₁

yes

—

C₂×C₄²

C₂×D₁₀₁

yes

—

C₄⋊C₈

—

101

—

C₄⋊C₈

—

101

—

C₄⋊C₈

—

101

—

C₂²⋊C₈

—

101

—

C₄×C₈

—

101

—

#41

—

C₂×Q₈

C₂×Dic₁₀₁

yes

characteristic

C₂×D₄

C₂×Dic₁₀₁

yes

—

C₂²×C₄

C₂×Dic₁₀₁

yes

—

C₂²×C₄

C₂×Dic₁₀₁

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

—

M₁₆

—

404

—

C₂×C₈

—

101

—

C₂×C₈

—

101

—

C₄⋊C₄

C₂×Dic₁₀₁

yes

—

C₄⋊C₄

C₂×Dic₁₀₁

yes

—

C₂²⋊C₄

C₂×Dic₁₀₁

yes

—

C₄²

C₂×Dic₁₀₁

yes

—

C₄²

C₂×Dic₁₀₁

yes

—

C₄²

C₂²×D₁₀₁

yes

—

100

C₄²

C₂²×D₁₀₁

yes

characteristic

101

C₂³

C₂²×Dic₁₀₁

yes

—

102

Q₈

#13108058

yes

complemented

103

D₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂²×Dic₁₀₁

yes

characteristic

109

C₂×C₄

C₂²×Dic₁₀₁

yes

—

110

C₂×C₄

C₂²×Dic₁₀₁

yes

—

111

C₂×C₄

C₂²×Dic₁₀₁

yes

—

112

C₂×C₄

C₂²×Dic₁₀₁

yes

characteristic

113

C₂×C₄

C₂³×D₁₀₁

yes

characteristic

114

C₈

—

202

—

115

C₈

—

202

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

C₂³×Dic₁₀₁

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

C₂×#13108058

yes

—

124

C₄

#15692262

yes

—

125

C₄

#15692301

yes

characteristic

126

C₄

#15692302

yes

characteristic

127

C₂

—

128

C₂

#20265660

yes

characteristic

129

C₂

C₂²×#13108058

yes

characteristic

130

C₂

C₂×#15692262

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₁
commutator, socle	C₂²×C₁₀₁	C₂³×C₄
Fitting	C₄×#41×C₁₀₁	C₂
prime radical	C₂×#41×C₁₀₁	C₄
cosocle	C₂×C₄×C₁₀₁	C₂⁴
hypercenter	C₄×#41	D₁₀₁
hypocenter	C₁₀₁	#2991
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₀₁

commutator, socle

C₂²×C₁₀₁

C₂³×C₄

Fitting

C₄×#41×C₁₀₁

C₂

prime radical

C₂×#41×C₁₀₁

C₄

cosocle

C₂×C₄×C₁₀₁

C₂⁴

hypercenter

C₄×#41

D₁₀₁

C₁₀₁

G ▷ C₂²×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂×C₄ ◁ C₄×#41

N	H ≅ G/N	type
1. Q₈×#11717579	C₂	semidirect
2. #20265255	C₂	semidirect
3. #20265230	C₂	semidirect
4. #20265201	C₂	semidirect
36. Q₈×C₁₀₁	M₁₆	semidirect
66. C₁₀₁	#2991	semidirect
102. Q₈	#13108058	semidirect

H ≅ G/N

type

1. Q₈×#11717579

C₂

semidirect

2. #20265255

C₂

semidirect

3. #20265230

C₂

semidirect

4. #20265201

C₂

semidirect

36. Q₈×C₁₀₁

M₁₆

semidirect

66. C₁₀₁

#2991

semidirect

102. Q₈

#13108058

semidirect

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

subgroup	index
Q₈×#11717579	2
#20265255	2
#20265230	2
#20265201	2
C₄×#41×C₁₀₁	2
#2991	101

subgroup

index

C₄×#41×C₁₀₁

101

Subgroups — #27180264

Conjugacy classes of subgroups

Complements

Maximal subgroups