Group #27170191 — subgroups

Subgroups — #27170191

Order 12928.

240 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 57 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	#20265553	C₂	1	1	yes	maximal, characteristic
2	#20265377	C₂	1	2	yes	maximal, complemented
3	#20264921	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₈⋊#11717579	C₂	1	2	yes	maximal, complemented
5	#559×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13108061	—	2	2	no	—
7	C₂×#13108059	—	2	2	no	—
8	#15692252	C₂²	1	1	yes	characteristic
9	C₄⋊#11717579	C₂²	1	2	yes	—
10	#15692175	—	2	2	no	—
11	C₄×#11717579	C₂²	1	1	yes	characteristic
12	#127×C₁₀₁	C₂²	1	1	yes	characteristic
13	#124×C₁₀₁	—	2	2	no	—
14	#123×C₁₀₁	—	2	2	no	—
15	C₄×Q₈×C₁₀₁	C₂²	1	2	yes	—
16	C₄×D₄×C₁₀₁	—	2	2	no	—
17	C₂×D₄₀₄	—	2	2	no	—
18	C₂×D₄₀₄	—	2	1	no	—
19	#13108061	—	4	4	no	—
20	#13108059	—	4	4	no	—
21	C₂×#11717579	—	2	1	no	—
22	C₂×#11717579	—	2	2	no	—
23	C₂×Q₈×C₁₀₁	D₄	1	2	yes	—
24	C₂×D₄×C₁₀₁	—	2	2	no	—
25	C₂×D₄×C₁₀₁	D₄	1	2	yes	—
26	C₂²×C₄×C₁₀₁	—	2	2	no	—
27	C₄⋊C₄×C₁₀₁	—	2	2	no	—
28	C₄⋊C₄×C₁₀₁	D₄	1	2	yes	—
29	C₂²⋊C₄×C₁₀₁	—	2	4	no	—
30	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
31	C₄²×C₁₀₁	—	2	2	no	—
32	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
33	C₂³×C₁₀₁	—	2	2	no	—
34	C₂²×D₁₀₁	—	4	1	no	—
35	Q₈×C₁₀₁	—	2	2	no	—
36	Q₈×C₁₀₁	—	2	2	no	—
37	D₄×C₁₀₁	—	4	2	no	—
38	D₄×C₁₀₁	—	4	2	no	—
39	D₄×C₁₀₁	—	4	2	no	—
40	C₂×C₄×C₁₀₁	—	2	2	no	—
41	C₂×C₄×C₁₀₁	—	2	2	no	—
42	C₂×C₄×C₁₀₁	—	8	2	no	—
43	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
45	D₄₀₄	—	4	2	no	—
46	D₄₀₄	—	4	2	no	—
47	D₄₀₄	—	4	2	no	—
48	#11717579	—	2	2	no	—
49	#11717579	—	4	2	no	—
50	C₂²×C₁₀₁	—	4	4	no	—
51	C₂²×C₁₀₁	—	4	2	no	—
52	C₂²×C₁₀₁	#120	1	1	yes	characteristic
53	C₂×D₁₀₁	—	8	2	no	—
54	C₂×D₁₀₁	—	8	1	no	—
55	C₄×C₁₀₁	—	2	1	no	—
56	C₄×C₁₀₁	—	4	2	no	—
57	C₄×C₁₀₁	—	4	2	no	—
58	C₄×C₁₀₁	#135	1	2	yes	—
59	C₄×C₁₀₁	AGL(1,ℤ/8)	1	2	yes	—
60	C₂×C₁₀₁	—	8	2	no	—
61	C₂×C₁₀₁	#449	1	2	yes	—
62	C₂×C₁₀₁	#453	1	1	yes	characteristic
63	D₁₀₁	—	16	1	no	—
64	#1620	—	101	1	no	maximal, Sylow
65	C₁₀₁	#1620	1	1	yes	characteristic, complemented, Sylow
66	#559	D₁₀₁	1	1	yes	characteristic, complemented
67	#493	—	101	1	no	—
68	#460	—	101	2	no	—
69	#331	—	101	1	no	—
70	Q₈⋊C₈	—	101	2	no	—
71	C₂×SD₁₆	—	202	2	no	—
72	C₂×D₈	—	202	2	no	—
73	#127	—	101	1	no	—
74	#127	C₂×D₁₀₁	1	1	yes	characteristic
75	#124	—	2	2	no	—
76	#123	—	2	2	no	—
77	C₄×Q₈	C₂×D₁₀₁	1	2	yes	—
78	C₄×D₄	—	2	2	no	—
79	C₄⋊C₈	—	101	2	no	—
80	#102	—	202	2	no	—
81	C₄×C₈	—	101	1	no	—
82	C₂×Q₈	C₂²⋊D₁₀₁	1	2	yes	—
83	C₂×D₄	—	2	2	no	—
84	C₂×D₄	—	202	2	no	—
85	C₂×D₄	—	202	1	no	—
86	C₂×D₄	C₂²⋊D₁₀₁	1	2	yes	—
87	C₂²×C₄	—	2	2	no	—
88	SD₁₆	—	404	4	no	—
89	D₈	—	404	4	no	—
90	C₂×C₈	—	202	2	no	—
91	C₂×C₈	—	202	1	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	C₂²⋊D₁₀₁	1	2	yes	—
94	C₂²⋊C₄	—	2	4	no	—
95	C₂²⋊C₄	—	2	2	no	—
96	C₄²	—	2	2	no	—
97	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
98	C₂³	—	2	2	no	—
99	C₂³	—	404	1	no	—
100	Q₈	—	2	2	no	—
101	Q₈	—	2	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	404	2	no	—
106	D₄	—	404	2	no	—
107	D₄	—	404	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
113	C₈	—	404	2	no	—
114	C₈	—	202	2	no	—
115	C₂²	—	4	4	no	—
116	C₂²	—	4	2	no	—
117	C₂²	—	808	2	no	—
118	C₂²	—	808	1	no	—
119	C₂²	#15692224	1	1	yes	characteristic
120	C₄	—	2	1	no	—
121	C₄	—	4	2	no	—
122	C₄	—	4	2	no	—
123	C₄	#15692275	1	2	yes	—
124	C₄	#15692280	1	2	yes	—
125	C₂	—	8	2	no	—
126	C₂	—	1616	1	no	—
127	C₂	#20265322	1	2	yes	—
128	C₂	#20265345	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20265553

C₂

yes

maximal, characteristic

#20265377

C₂

yes

maximal, complemented

#20264921

C₂

yes

maximal, characteristic, complemented

Q₈⋊#11717579

C₂

yes

maximal, complemented

#559×C₁₀₁

C₂

yes

maximal, characteristic, complemented

C₂×#13108061

—

C₂×#13108059

—

#15692252

C₂²

yes

characteristic

C₄⋊#11717579

C₂²

yes

—

#15692175

—

C₄×#11717579

C₂²

yes

characteristic

#127×C₁₀₁

C₂²

yes

characteristic

#124×C₁₀₁

—

#123×C₁₀₁

—

C₄×Q₈×C₁₀₁

C₂²

yes

—

C₄×D₄×C₁₀₁

—

C₂×D₄₀₄

—

C₂×D₄₀₄

—

—

—

—

—

C₂×Q₈×C₁₀₁

D₄

yes

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

D₄

yes

—

C₂²×C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄⋊C₄×C₁₀₁

D₄

yes

—

C₂²⋊C₄×C₁₀₁

—

C₂²⋊C₄×C₁₀₁

—

C₄²×C₁₀₁

—

C₄²×C₁₀₁

C₂³

yes

characteristic

C₂³×C₁₀₁

—

C₂²×D₁₀₁

—

Q₈×C₁₀₁

—

Q₈×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

D₄₀₄

—

D₄₀₄

—

D₄₀₄

—

#11717579

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

#120

yes

characteristic

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#135

yes

—

C₄×C₁₀₁

AGL(1,ℤ/8)

yes

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#449

yes

—

C₂×C₁₀₁

#453

yes

characteristic

D₁₀₁

—

#1620

—

101

maximal, Sylow

C₁₀₁

#1620

yes

characteristic, complemented, Sylow

#559

D₁₀₁

yes

characteristic, complemented

#493

—

101

—

#460

—

101

—

#331

—

101

—

Q₈⋊C₈

—

101

—

C₂×SD₁₆

—

202

—

C₂×D₈

—

202

—

#127

—

101

—

#127

C₂×D₁₀₁

yes

characteristic

#124

—

#123

—

C₄×Q₈

C₂×D₁₀₁

yes

—

C₄×D₄

—

C₄⋊C₈

—

101

—

#102

—

202

—

C₄×C₈

—

101

—

C₂×Q₈

C₂²⋊D₁₀₁

yes

—

C₂×D₄

—

C₂×D₄

—

202

—

C₂×D₄

—

202

—

C₂×D₄

C₂²⋊D₁₀₁

yes

—

C₂²×C₄

—

SD₁₆

—

404

—

D₈

—

404

—

C₂×C₈

—

202

—

C₂×C₈

—

202

—

C₄⋊C₄

—

C₄⋊C₄

C₂²⋊D₁₀₁

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×D₁₀₁

yes

characteristic

C₂³

—

C₂³

—

404

—

100

Q₈

—

101

Q₈

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

404

—

106

D₄

—

404

—

107

D₄

—

404

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

113

C₈

—

404

—

114

C₈

—

202

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

808

—

118

C₂²

—

808

—

119

C₂²

#15692224

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

#15692275

yes

—

124

C₄

#15692280

yes

—

125

C₂

—

126

C₂

—

1616

—

127

C₂

#20265322

yes

—

128

C₂

#20265345

yes

characteristic

	subgroup	G/H
center	C₂²	#15692224
commutator, cosocle	C₄²×C₁₀₁	C₂³
Frattini	C₄²	C₂²×D₁₀₁
Fitting	#559×C₁₀₁	C₂
socle	C₂²×C₁₀₁	#120
prime radical	#20265553	C₂
hypercenter	#559	D₁₀₁
hypocenter	C₁₀₁	#1620
LCS	G ▷ C₄²×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #559

subgroup

G/H

center

C₂²

#15692224

commutator, cosocle

C₄²×C₁₀₁

C₂³

Frattini

C₄²

C₂²×D₁₀₁

Fitting

#559×C₁₀₁

C₂

C₂²×C₁₀₁

C₂

D₁₀₁

C₁₀₁

G ▷ C₄²×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #559

N	H ≅ G/N	type
2. #20265377	C₂	semidirect
3. #20264921	C₂	semidirect
4. Q₈⋊#11717579	C₂	semidirect
5. #559×C₁₀₁	C₂	semidirect
65. C₁₀₁	#1620	semidirect
66. #559	D₁₀₁	semidirect

H ≅ G/N

type

2. #20265377

C₂

semidirect

3. #20264921

C₂

semidirect

4. Q₈⋊#11717579

C₂

semidirect

5. #559×C₁₀₁

C₂

semidirect

65. C₁₀₁

#1620

semidirect

66. #559

D₁₀₁

semidirect

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

subgroup	index
#20265553	2
#20265377	2
#20264921	2
Q₈⋊#11717579	2
#559×C₁₀₁	2
#1620	101

subgroup

index

#559×C₁₀₁

101

Subgroups — #27170191

Conjugacy classes of subgroups

Complements

Maximal subgroups