Group #27169864 — subgroups

Subgroups — #27169864

Order 12928.

254 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 67 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	#20265725	C₂	1	1	yes	maximal, characteristic
2	#20265199	C₂	1	2	yes	maximal, complemented
3	#20264925	C₂	1	2	yes	maximal, complemented
4	#20264921	C₂	1	1	yes	maximal, characteristic, complemented
5	#331×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#13108073	C₄	1	2	yes	—
7	C₂×#13108057	—	2	2	no	—
8	#15692255	C₂²	1	1	yes	characteristic
9	#15692242	C₄	1	2	yes	—
10	#15692226	—	2	2	no	—
11	D₄×Dic₁₀₁	—	2	2	no	—
12	C₄²×D₁₀₁	C₂²	1	1	yes	characteristic
13	C₄⋊#11717579	C₂²	1	2	yes	—
14	#15692184	—	2	2	no	—
15	#127×C₁₀₁	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₀₁	C₂²	1	2	yes	—
17	#13108073	—	4	4	no	—
18	C₂×Dic₂₀₂	C₂×C₄	1	2	yes	—
19	C₂×C₂²⋊D₁₀₁	—	2	2	no	—
20	C₂²×Dic₁₀₁	—	2	2	no	—
21	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₁₀₁	D₄	1	2	yes	—
23	#13108057	—	4	4	no	—
24	C₂×#11717579	—	2	2	no	—
25	C₄⋊Dic₁₀₁	—	2	2	no	—
26	C₂²⋊Dic₁₀₁	—	2	4	no	—
27	C₄×Dic₁₀₁	C₂×C₄	1	1	yes	characteristic
28	C₄×Dic₁₀₁	D₄	1	2	yes	—
29	C₂×D₄×C₁₀₁	—	2	2	no	—
30	C₂×D₄×C₁₀₁	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₁₀₁	—	2	2	no	—
32	C₄²×C₁₀₁	C₂³	1	1	yes	characteristic
33	C₂³×C₁₀₁	—	2	2	no	—
34	C₂²×D₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
35	D₄×C₁₀₁	—	4	2	no	—
36	D₄×C₁₀₁	—	4	2	no	—
37	D₄×C₁₀₁	—	4	2	no	—
38	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
40	C₂²⋊D₁₀₁	—	4	4	no	—
41	C₂×Dic₁₀₁	—	2	2	no	—
42	C₂×Dic₁₀₁	—	4	4	no	—
43	C₂×Dic₁₀₁	C₂²⋊C₄	1	2	yes	—
44	C₂×Dic₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
45	C₄×D₁₀₁	—	2	2	no	—
46	C₄×D₁₀₁	—	2	2	no	—
47	C₄×D₁₀₁	—	2	2	no	—
48	Dic₂₀₂	—	4	2	no	—
49	C₈×C₁₀₁	—	4	2	no	—
50	#11717579	—	4	2	no	—
51	C₂²×C₁₀₁	—	4	2	no	—
52	C₂²×C₁₀₁	—	4	4	no	—
53	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂×D₁₀₁	—	2	1	no	—
55	C₂×D₁₀₁	—	2	2	no	—
56	C₄×C₁₀₁	—	2	1	no	—
57	C₄×C₁₀₁	AGL(1,ℤ/8)	1	2	yes	—
58	C₄×C₁₀₁	#137	1	2	yes	—
59	Dic₁₀₁	—	2	2	no	—
60	Dic₁₀₁	—	4	2	no	—
61	Dic₁₀₁	#100	1	2	yes	—
62	C₂×C₁₀₁	—	8	2	no	—
63	C₂×C₁₀₁	C₂×#100	1	1	yes	characteristic
64	C₂×C₁₀₁	#417	1	2	yes	—
65	D₁₀₁	—	4	1	no	—
66	#1560	—	101	1	no	maximal, Sylow
67	C₁₀₁	#1560	1	1	yes	characteristic, complemented, Sylow
68	#532	—	101	1	no	—
69	#423	—	101	2	no	—
70	#333	—	101	2	no	—
71	#331	—	101	1	no	—
72	#331	D₁₀₁	1	1	yes	characteristic, complemented
73	C₂×#41	—	101	2	no	—
74	C₂×M₁₆	—	202	2	no	—
75	C₄⋊Q₈	—	101	1	no	—
76	#127	C₂×D₁₀₁	1	1	yes	characteristic
77	#124	—	101	2	no	—
78	#121	—	202	2	no	—
79	C₄×D₄	—	202	2	no	—
80	C₂×C₄²	—	101	1	no	—
81	C₄⋊C₈	—	101	2	no	—
82	C₄⋊C₈	—	202	2	no	—
83	C₄⋊C₈	C₂×D₁₀₁	1	2	yes	—
84	#41	—	404	4	no	—
85	C₂×Q₈	—	101	2	no	—
86	C₂×D₄	—	2	2	no	—
87	C₂×D₄	—	202	2	no	—
88	C₂×D₄	C₄×D₁₀₁	1	2	yes	—
89	C₂²×C₄	—	101	1	no	—
90	C₂²×C₄	—	101	2	no	—
91	C₂²×C₄	—	202	2	no	—
92	M₁₆	—	404	4	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	202	2	no	—
95	C₄⋊C₄	—	202	2	no	—
96	C₂²⋊C₄	—	202	4	no	—
97	C₄²	—	101	1	no	—
98	C₄²	—	101	2	no	—
99	C₄²	C₂²×D₁₀₁	1	1	yes	characteristic
100	C₂³	—	2	2	no	—
101	C₂³	—	101	1	no	—
102	Q₈	—	404	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	4	2	no	—
106	D₄	—	404	4	no	—
107	C₂×C₄	—	101	1	no	—
108	C₂×C₄	—	101	2	no	—
109	C₂×C₄	—	404	4	no	—
110	C₂×C₄	—	202	2	no	—
111	C₂×C₄	—	202	2	no	—
112	C₂×C₄	—	202	2	no	—
113	C₂×C₄	—	202	2	no	—
114	C₂×C₄	C₂×C₄×D₁₀₁	1	1	yes	characteristic
115	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
116	C₈	—	4	2	no	—
117	C₈	—	404	2	no	—
118	C₂²	—	4	2	no	—
119	C₂²	—	4	4	no	—
120	C₂²	—	202	1	no	—
121	C₂²	—	202	2	no	—
122	C₂²	C₂²⋊C₄×D₁₀₁	1	1	yes	characteristic
123	C₄	—	2	1	no	—
124	C₄	—	101	2	no	—
125	C₄	—	404	2	no	—
126	C₄	—	202	2	no	—
127	C₄	#15692279	1	2	yes	—
128	C₄	#15692284	1	2	yes	—
129	C₂	—	8	2	no	—
130	C₂	—	404	1	no	—
131	C₂	#100×D₁₀₁	1	1	yes	characteristic
132	C₂	#20265175	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20265725

C₂

yes

maximal, characteristic

#20265199

C₂

yes

maximal, complemented

#20264925

C₂

yes

maximal, complemented

#20264921

C₂

yes

maximal, characteristic, complemented

#331×C₁₀₁

C₂

yes

maximal, characteristic, complemented

C₂×#13108073

C₄

yes

—

C₂×#13108057

—

#15692255

C₂²

yes

characteristic

#15692242

C₄

yes

—

#15692226

—

D₄×Dic₁₀₁

—

C₄²×D₁₀₁

C₂²

yes

characteristic

C₄⋊#11717579

C₂²

yes

—

#15692184

—

#127×C₁₀₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₀₁

C₂²

yes

—

#13108073

—

C₂×Dic₂₀₂

C₂×C₄

yes

—

C₂×C₂²⋊D₁₀₁

—

C₂²×Dic₁₀₁

—

C₂×C₄×D₁₀₁

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₁

D₄

yes

—

#13108057

—

C₂×#11717579

—

C₄⋊Dic₁₀₁

—

C₂²⋊Dic₁₀₁

—

C₄×Dic₁₀₁

C₂×C₄

yes

characteristic

C₄×Dic₁₀₁

D₄

yes

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

C₂×C₄

yes

—

C₂×C₈×C₁₀₁

—

C₄²×C₁₀₁

C₂³

yes

characteristic

C₂³×C₁₀₁

—

C₂²×D₁₀₁

C₂²⋊C₄

yes

characteristic

D₄×C₁₀₁

—

D₄×C₁₀₁

—

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂²⋊D₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

C₂²⋊C₄

yes

—

C₂×Dic₁₀₁

C₂²⋊C₄

yes

characteristic

C₄×D₁₀₁

—

C₄×D₁₀₁

—

C₄×D₁₀₁

—

Dic₂₀₂

—

C₈×C₁₀₁

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

AGL(1,ℤ/8)

yes

—

C₄×C₁₀₁

#137

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

#100

yes

—

C₂×C₁₀₁

—

C₂×C₁₀₁

C₂×#100

yes

characteristic

C₂×C₁₀₁

#417

yes

—

D₁₀₁

—

#1560

—

101

maximal, Sylow

C₁₀₁

#1560

yes

characteristic, complemented, Sylow

#532

—

101

—

#423

—

101

—

#333

—

101

—

#331

—

101

—

#331

D₁₀₁

yes

characteristic, complemented

C₂×#41

—

101

—

C₂×M₁₆

—

202

—

C₄⋊Q₈

—

101

—

#127

C₂×D₁₀₁

yes

characteristic

#124

—

101

—

#121

—

202

—

C₄×D₄

—

202

—

C₂×C₄²

—

101

—

C₄⋊C₈

—

101

—

C₄⋊C₈

—

202

—

C₄⋊C₈

C₂×D₁₀₁

yes

—

#41

—

404

—

C₂×Q₈

—

101

—

C₂×D₄

—

C₂×D₄

—

202

—

C₂×D₄

C₄×D₁₀₁

yes

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

202

—

M₁₆

—

404

—

C₂×C₈

—

C₂×C₈

—

202

—

C₄⋊C₄

—

202

—

C₂²⋊C₄

—

202

—

C₄²

—

101

—

C₄²

—

101

—

C₄²

C₂²×D₁₀₁

yes

characteristic

100

C₂³

—

101

C₂³

—

101

—

102

Q₈

—

404

—

103

D₄

—

104

D₄

—

105

D₄

—

106

D₄

—

404

—

107

C₂×C₄

—

101

—

108

C₂×C₄

—

101

—

109

C₂×C₄

—

404

—

110

C₂×C₄

—

202

—

111

C₂×C₄

—

202

—

112

C₂×C₄

—

202

—

113

C₂×C₄

—

202

—

114

C₂×C₄

C₂×C₄×D₁₀₁

yes

characteristic

115

C₂×C₄

D₄×D₁₀₁

yes

—

116

C₈

—

117

C₈

—

404

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

202

—

121

C₂²

—

202

—

122

C₂²

C₂²⋊C₄×D₁₀₁

yes

characteristic

123

C₄

—

124

C₄

—

101

—

125

C₄

—

404

—

126

C₄

—

202

—

127

C₄

#15692279

yes

—

128

C₄

#15692284

yes

—

129

C₂

—

130

C₂

—

404

—

131

C₂

#100×D₁₀₁

yes

characteristic

132

C₂

#20265175

yes

—

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₁₀₁
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₄²	C₂²×D₁₀₁
Fitting	#331×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂×C₂²⋊C₄
prime radical	#20265725	C₂
cosocle	C₄²×C₁₀₁	C₂³
hypercenter	#331	D₁₀₁
hypocenter	C₁₀₁	#1560
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #331

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₁₀₁

commutator

C₂×C₄×C₁₀₁

C₂²×C₄

Frattini

C₄²

C₂²×D₁₀₁

Fitting

#331×C₁₀₁

C₂

socle

C₂²×C₁₀₁

C₂×C₂²⋊C₄

prime radical

#20265725

C₂

cosocle

C₄²×C₁₀₁

C₂³

D₁₀₁

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #331

N	H ≅ G/N	type
2. #20265199	C₂	semidirect
3. #20264925	C₂	semidirect
4. #20264921	C₂	semidirect
5. #331×C₁₀₁	C₂	semidirect
67. C₁₀₁	#1560	semidirect
72. #331	D₁₀₁	semidirect

H ≅ G/N

type

2. #20265199

C₂

semidirect

3. #20264925

C₂

semidirect

4. #20264921

C₂

semidirect

5. #331×C₁₀₁

C₂

semidirect

67. C₁₀₁

#1560

semidirect

72. #331

D₁₀₁

semidirect

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

subgroup	index
#20265725	2
#20265199	2
#20264925	2
#20264921	2
#331×C₁₀₁	2
#1560	101

subgroup

index

#331×C₁₀₁

101

Subgroups — #27169864

Conjugacy classes of subgroups

Complements

Maximal subgroups