Group #27170216 — subgroups

Subgroups — #27170216

Order 12928.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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	#20265893	C₂	1	1	yes	maximal, characteristic, complemented
2	#20265551	C₂	1	1	yes	maximal, characteristic, complemented
3	#20265373	C₂	1	2	yes	maximal, complemented
4	#20264904	C₂	1	2	yes	maximal, complemented
5	#333×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13108061	—	2	2	no	—
7	C₂×D₈₀₈	—	2	2	no	—
8	#15692252	C₂²	1	1	yes	characteristic
9	#15692244	—	2	2	no	—
10	#15692230	—	2	2	no	—
11	C₄×D₄₀₄	C₂²	1	2	yes	—
12	#15692213	—	2	2	no	—
13	#15692177	—	2	2	no	—
14	C₄×#11717579	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₀₁	C₂²	1	2	yes	—
17	C₂×D₄₀₄	—	2	2	no	—
18	C₂×D₄₀₄	—	2	1	no	—
19	C₂×D₄₀₄	—	2	2	no	—
20	C₂×D₄₀₄	D₄	1	2	yes	—
21	C₂×C₄×D₁₀₁	—	2	2	no	—
22	#13108061	—	4	4	no	—
23	D₈₀₈	—	4	4	no	—
24	C₂×#11717579	—	2	1	no	—
25	C₄⋊Dic₁₀₁	D₄	1	2	yes	—
26	#13108051	—	2	4	no	—
27	#13108051	—	2	2	no	—
28	C₄×Dic₁₀₁	—	2	2	no	—
29	C₂×Q₈×C₁₀₁	D₄	1	2	yes	—
30	C₂×C₈×C₁₀₁	—	2	2	no	—
31	C₄⋊C₄×C₁₀₁	—	2	2	no	—
32	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
33	C₂²×D₁₀₁	—	2	2	no	—
34	C₂²×D₁₀₁	—	4	1	no	—
35	Q₈×C₁₀₁	—	4	2	no	—
36	C₂×C₄×C₁₀₁	—	2	2	no	—
37	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
38	C₂×C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
39	C₂×Dic₁₀₁	—	2	2	no	—
40	D₄₀₄	—	2	2	no	—
41	D₄₀₄	—	2	2	no	—
42	D₄₀₄	—	4	2	no	—
43	D₄₀₄	—	4	2	no	—
44	D₄₀₄	—	4	2	no	—
45	D₄₀₄	—	8	2	no	—
46	C₄×D₁₀₁	—	4	2	no	—
47	C₄×D₁₀₁	—	4	2	no	—
48	C₈×C₁₀₁	—	4	2	no	—
49	#11717579	—	2	2	no	—
50	C₂²×C₁₀₁	#120	1	1	yes	characteristic
51	C₂×D₁₀₁	—	4	2	no	—
52	C₂×D₁₀₁	—	4	2	no	—
53	C₂×D₁₀₁	—	4	2	no	—
54	C₂×D₁₀₁	—	8	2	no	—
55	C₂×D₁₀₁	—	8	1	no	—
56	C₄×C₁₀₁	—	2	1	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	Dic₁₀₁	—	4	2	no	—
61	C₂×C₁₀₁	#449	1	2	yes	—
62	C₂×C₁₀₁	#453	1	1	yes	characteristic
63	D₁₀₁	—	8	2	no	—
64	D₁₀₁	—	16	1	no	—
65	#1624	—	101	1	no	maximal, Sylow
66	C₁₀₁	#1624	1	1	yes	characteristic, complemented, Sylow
67	#555	—	101	1	no	—
68	#492	—	101	1	no	—
69	#459	—	101	2	no	—
70	#333	D₁₀₁	1	1	yes	characteristic, complemented
71	D₄⋊C₈	—	101	2	no	—
72	C₂×SD₁₆	—	202	2	no	—
73	C₂×D₈	—	202	2	no	—
74	C₄⋊Q₈	C₂×D₁₀₁	1	1	yes	characteristic
75	#127	—	101	1	no	—
76	#124	—	202	2	no	—
77	#121	—	202	2	no	—
78	C₄×D₄	—	101	2	no	—
79	#117	—	202	2	no	—
80	C₄⋊C₈	C₂×D₁₀₁	1	2	yes	—
81	#102	—	202	2	no	—
82	C₄×C₈	—	101	1	no	—
83	C₂×Q₈	D₄₀₄	1	2	yes	—
84	C₂×D₄	—	101	2	no	—
85	C₂×D₄	—	202	2	no	—
86	C₂×D₄	—	202	2	no	—
87	C₂×D₄	—	202	1	no	—
88	C₂²×C₄	—	202	2	no	—
89	SD₁₆	—	404	4	no	—
90	D₈	—	404	4	no	—
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	202	1	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	101	2	no	—
95	C₂²⋊C₄	—	202	4	no	—
96	C₂²⋊C₄	—	202	2	no	—
97	C₄²	—	202	2	no	—
98	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
99	C₂³	—	404	1	no	—
100	C₂³	—	202	2	no	—
101	Q₈	—	4	2	no	—
102	D₄	—	808	2	no	—
103	D₄	—	404	2	no	—
104	D₄	—	404	2	no	—
105	D₄	—	404	2	no	—
106	D₄	—	202	2	no	—
107	D₄	—	202	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	404	2	no	—
110	C₂×C₄	—	404	2	no	—
111	C₂×C₄	—	202	2	no	—
112	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
113	C₂×C₄	C₂×D₄₀₄	1	1	yes	characteristic
114	C₈	—	4	2	no	—
115	C₈	—	202	2	no	—
116	C₂²	—	808	2	no	—
117	C₂²	—	808	1	no	—
118	C₂²	—	404	2	no	—
119	C₂²	—	404	2	no	—
120	C₂²	—	404	2	no	—
121	C₂²	#15692225	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	4	2	no	—
124	C₄	—	404	2	no	—
125	C₄	#15692278	1	2	yes	—
126	C₄	#15692282	1	2	yes	—
127	C₂	—	808	2	no	—
128	C₂	—	1616	1	no	—
129	C₂	#20265325	1	2	yes	—
130	C₂	#20265347	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20265893

C₂

yes

maximal, characteristic, complemented

#20265551

C₂

yes

maximal, characteristic, complemented

#20265373

C₂

yes

maximal, complemented

#20264904

C₂

yes

maximal, complemented

#333×C₁₀₁

C₂

yes

maximal, characteristic, complemented

C₂×#13108061

—

C₂×D₈₀₈

—

#15692252

C₂²

yes

characteristic

#15692244

—

#15692230

—

C₄×D₄₀₄

C₂²

yes

—

#15692213

—

#15692177

—

C₄×#11717579

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₁

C₂²

yes

—

C₂×D₄₀₄

—

C₂×D₄₀₄

—

C₂×D₄₀₄

—

C₂×D₄₀₄

D₄

yes

—

C₂×C₄×D₁₀₁

—

#13108061

—

D₈₀₈

—

C₂×#11717579

—

C₄⋊Dic₁₀₁

D₄

yes

—

#13108051

—

#13108051

—

C₄×Dic₁₀₁

—

C₂×Q₈×C₁₀₁

D₄

yes

—

C₂×C₈×C₁₀₁

—

C₄⋊C₄×C₁₀₁

—

C₄²×C₁₀₁

C₂³

yes

characteristic

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

Q₈×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₂×Dic₁₀₁

—

D₄₀₄

—

D₄₀₄

—

D₄₀₄

—

D₄₀₄

—

D₄₀₄

—

D₄₀₄

—

C₄×D₁₀₁

—

C₄×D₁₀₁

—

C₈×C₁₀₁

—

#11717579

—

C₂²×C₁₀₁

#120

yes

characteristic

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#135

yes

—

C₄×C₁₀₁

AGL(1,ℤ/8)

yes

—

Dic₁₀₁

—

C₂×C₁₀₁

#449

yes

—

C₂×C₁₀₁

#453

yes

characteristic

D₁₀₁

—

D₁₀₁

—

#1624

—

101

maximal, Sylow

C₁₀₁

#1624

yes

characteristic, complemented, Sylow

#555

—

101

—

#492

—

101

—

#459

—

101

—

#333

D₁₀₁

yes

characteristic, complemented

D₄⋊C₈

—

101

—

C₂×SD₁₆

—

202

—

C₂×D₈

—

202

—

C₄⋊Q₈

C₂×D₁₀₁

yes

characteristic

#127

—

101

—

#124

—

202

—

#121

—

202

—

C₄×D₄

—

101

—

#117

—

202

—

C₄⋊C₈

C₂×D₁₀₁

yes

—

#102

—

202

—

C₄×C₈

—

101

—

C₂×Q₈

D₄₀₄

yes

—

C₂×D₄

—

101

—

C₂×D₄

—

202

—

C₂×D₄

—

202

—

C₂×D₄

—

202

—

C₂²×C₄

—

202

—

SD₁₆

—

404

—

D₈

—

404

—

C₂×C₈

—

C₂×C₈

—

202

—

C₄⋊C₄

—

C₄⋊C₄

—

101

—

C₂²⋊C₄

—

202

—

C₂²⋊C₄

—

202

—

C₄²

—

202

—

C₄²

C₂²×D₁₀₁

yes

characteristic

C₂³

—

404

—

100

C₂³

—

202

—

101

Q₈

—

102

D₄

—

808

—

103

D₄

—

404

—

104

D₄

—

404

—

105

D₄

—

404

—

106

D₄

—

202

—

107

D₄

—

202

—

108

C₂×C₄

—

109

C₂×C₄

—

404

—

110

C₂×C₄

—

404

—

111

C₂×C₄

—

202

—

112

C₂×C₄

D₄×D₁₀₁

yes

—

113

C₂×C₄

C₂×D₄₀₄

yes

characteristic

114

C₈

—

115

C₈

—

202

—

116

C₂²

—

808

—

117

C₂²

—

808

—

118

C₂²

—

404

—

119

C₂²

—

404

—

120

C₂²

—

404

—

121

C₂²

#15692225

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

404

—

125

C₄

#15692278

yes

—

126

C₄

#15692282

yes

—

127

C₂

—

808

—

128

C₂

—

1616

—

129

C₂

#20265325

yes

—

130

C₂

#20265347

yes

characteristic

	subgroup	G/H
center	C₂²	#15692225
commutator, cosocle	C₄²×C₁₀₁	C₂³
Frattini	C₄²	C₂²×D₁₀₁
Fitting	#333×C₁₀₁	C₂
socle	C₂²×C₁₀₁	#120
hypercenter	#333	D₁₀₁
hypocenter	C₁₀₁	#1624
LCS	G ▷ C₄²×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

#15692225

commutator, cosocle

C₄²×C₁₀₁

C₂³

Frattini

C₄²

C₂²×D₁₀₁

Fitting

#333×C₁₀₁

C₂

C₂²×C₁₀₁

D₁₀₁

C₁₀₁

G ▷ C₄²×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
1. #20265893	C₂	semidirect
2. #20265551	C₂	semidirect
3. #20265373	C₂	semidirect
4. #20264904	C₂	semidirect
5. #333×C₁₀₁	C₂	semidirect
66. C₁₀₁	#1624	semidirect
70. #333	D₁₀₁	semidirect

H ≅ G/N

type

1. #20265893

C₂

semidirect

2. #20265551

C₂

semidirect

3. #20265373

C₂

semidirect

4. #20264904

C₂

semidirect

5. #333×C₁₀₁

C₂

semidirect

66. C₁₀₁

#1624

semidirect

70. #333

D₁₀₁

semidirect

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

subgroup	index
#20265893	2
#20265551	2
#20265373	2
#20264904	2
#333×C₁₀₁	2
#1624	101

subgroup

index

#333×C₁₀₁

101

Subgroups — #27170216

Conjugacy classes of subgroups

Complements

Maximal subgroups