Group #27179397 — subgroups

Subgroups — #27179397

Order 12928.

548 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 125 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	#20265113	C₂	1	3	yes	maximal, complemented
2	#20265101	C₂	1	2	yes	maximal, complemented
3	#20265099	C₂	1	6	yes	maximal, complemented
4	#20265032	C₂	1	3	yes	maximal
5	#399×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁴×D₁₀₁	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₀₁	C₂²	1	3	yes	complemented
8	C₂×#13108053	C₂²	1	6	yes	complemented
9	C₂×C₂²⋊Dic₁₀₁	C₂²	1	3	yes	—
10	C₂×#13108051	C₂²	1	6	yes	—
11	C₂×#13108051	C₂²	1	6	yes	—
12	#15692162	C₂²	1	3	yes	complemented
13	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	3	yes	complemented
14	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	3	yes	—
15	#95×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
16	C₂³×D₁₀₁	—	4	3	no	—
17	C₂³×D₁₀₁	—	4	3	no	—
18	C₂³×D₁₀₁	Q₈	1	4	yes	—
19	C₂³×D₁₀₁	C₂³	1	2	yes	—
20	C₂²×Dic₁₀₁	C₂³	1	6	yes	—
21	C₄⋊Dic₁₀₁	—	4	3	no	—
22	#13108053	—	4	6	no	—
23	C₂²⋊Dic₁₀₁	—	4	3	no	—
24	#13108051	—	4	6	no	—
25	#13108051	—	4	6	no	—
26	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₁₀₁	C₂³	1	3	yes	—
28	C₂²×C₄×C₁₀₁	C₂³	1	3	yes	—
29	C₄⋊C₄×C₁₀₁	—	4	3	no	—
30	C₂²⋊C₄×C₁₀₁	—	4	3	no	—
31	C₂³×C₁₀₁	—	2	3	no	—
32	C₂³×C₁₀₁	—	2	3	no	—
33	C₂³×C₁₀₁	C₂×Q₈	1	2	yes	—
34	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
35	C₂²×D₁₀₁	—	2	6	no	—
36	C₂²×D₁₀₁	—	2	6	no	—
37	C₂²×D₁₀₁	—	4	12	no	—
38	C₂²×D₁₀₁	—	4	4	no	—
39	C₂²×D₁₀₁	—	4	12	no	—
40	C₂²×D₁₀₁	C₂×Q₈	1	4	yes	—
41	C₂×C₄×C₁₀₁	—	2	3	no	—
42	C₂×C₄×C₁₀₁	—	2	3	no	—
43	C₂×C₄×C₁₀₁	—	2	6	no	—
44	C₂×C₄×C₁₀₁	—	2	6	no	—
45	C₂×Dic₁₀₁	—	2	6	no	—
46	C₂×Dic₁₀₁	—	2	6	no	—
47	C₂×Dic₁₀₁	—	2	6	no	—
48	C₂²×C₁₀₁	—	2	6	no	—
49	C₂²×C₁₀₁	—	4	1	no	—
50	C₂²×C₁₀₁	—	4	3	no	—
51	C₂²×C₁₀₁	C₂²×Q₈	1	1	yes	characteristic
52	C₂²×C₁₀₁	H(5,𝔽₂)	1	3	yes	—
53	C₂²×C₁₀₁	H(5,𝔽₂)	1	3	yes	—
54	C₂×D₁₀₁	—	2	12	no	—
55	C₂×D₁₀₁	—	4	16	no	—
56	C₂×D₁₀₁	—	4	2	no	—
57	C₂×D₁₀₁	—	4	6	no	—
58	C₄×C₁₀₁	—	4	3	no	—
59	C₄×C₁₀₁	—	4	3	no	—
60	Dic₁₀₁	—	4	6	no	—
61	C₂×C₁₀₁	—	4	2	no	—
62	C₂×C₁₀₁	#543	1	3	yes	—
63	C₂×C₁₀₁	#560	1	1	yes	characteristic
64	C₂×C₁₀₁	#560	1	3	yes	—
65	D₁₀₁	—	4	4	no	—
66	#2842	—	101	1	no	maximal, Sylow
67	C₁₀₁	#2842	1	1	yes	characteristic, complemented, Sylow
68	#399	—	101	3	no	—
69	#399	D₁₀₁	1	1	yes	characteristic, complemented
70	#396	—	101	6	no	—
71	#396	—	101	2	no	—
72	#379	—	101	3	no	—
73	C₂⁵	—	101	1	no	—
74	C₂×C₄⋊C₄	—	101	6	no	—
75	C₂×C₄⋊C₄	—	101	3	no	—
76	C₂×C₄⋊C₄	C₂×D₁₀₁	1	3	yes	complemented
77	C₂×C₂²⋊C₄	—	101	3	no	—
78	C₂×C₂²⋊C₄	—	101	6	no	—
79	C₂×C₂²⋊C₄	—	101	6	no	—
80	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	3	yes	—
81	#95	—	101	3	no	—
82	#95	C₂×D₁₀₁	1	1	yes	characteristic, complemented
83	C₂⁴	—	101	4	no	—
84	C₂⁴	—	101	2	no	—
85	C₂⁴	—	404	3	no	—
86	C₂⁴	—	404	3	no	—
87	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
88	C₂²×C₄	—	101	6	no	—
89	C₂²×C₄	C₂²×D₁₀₁	1	3	yes	—
90	C₂²×C₄	C₂²×D₁₀₁	1	3	yes	—
91	C₄⋊C₄	—	4	3	no	—
92	C₄⋊C₄	—	404	6	no	—
93	C₄⋊C₄	—	404	3	no	—
94	C₂²⋊C₄	—	4	3	no	—
95	C₂²⋊C₄	—	404	6	no	—
96	C₂²⋊C₄	—	404	6	no	—
97	C₂²⋊C₄	—	404	3	no	—
98	C₂³	—	2	3	no	—
99	C₂³	—	2	3	no	—
100	C₂³	—	101	4	no	—
101	C₂³	—	404	12	no	—
102	C₂³	—	404	4	no	—
103	C₂³	—	404	12	no	—
104	C₂³	—	202	6	no	—
105	C₂³	—	202	6	no	—
106	C₂³	Q₈×D₁₀₁	1	2	yes	—
107	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
108	C₂×C₄	—	2	3	no	—
109	C₂×C₄	—	2	3	no	—
110	C₂×C₄	—	2	6	no	—
111	C₂×C₄	—	2	6	no	—
112	C₂×C₄	—	202	6	no	—
113	C₂×C₄	—	202	6	no	—
114	C₂×C₄	—	202	6	no	—
115	C₂²	—	2	6	no	—
116	C₂²	—	4	1	no	—
117	C₂²	—	4	3	no	—
118	C₂²	—	404	16	no	—
119	C₂²	—	404	2	no	—
120	C₂²	—	404	6	no	—
121	C₂²	—	202	12	no	—
122	C₂²	C₂×Q₈×D₁₀₁	1	1	yes	characteristic
123	C₂²	#15692300	1	3	yes	—
124	C₂²	#15692301	1	3	yes	—
125	C₄	—	4	3	no	—
126	C₄	—	4	3	no	—
127	C₄	—	404	6	no	—
128	C₂	—	4	2	no	—
129	C₂	—	404	4	no	—
130	C₂	#20265802	1	3	yes	—
131	C₂	#20265914	1	1	yes	characteristic
132	C₂	#20265917	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20265113

C₂

yes

maximal, complemented

#20265101

C₂

yes

maximal, complemented

#20265099

C₂

yes

maximal, complemented

#20265032

C₂

yes

maximal

#399×C₁₀₁

C₂

yes

maximal, characteristic, complemented

C₂⁴×D₁₀₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₀₁

C₂²

yes

complemented

C₂×#13108053

C₂²

yes

complemented

C₂×C₂²⋊Dic₁₀₁

C₂²

yes

—

C₂×#13108051

C₂²

yes

—

C₂×#13108051

C₂²

yes

—

#15692162

C₂²

yes

complemented

C₂×C₄⋊C₄×C₁₀₁

C₂²

yes

complemented

C₂×C₂²⋊C₄×C₁₀₁

C₂²

yes

—

#95×C₁₀₁

C₂²

yes

characteristic, complemented

C₂³×D₁₀₁

—

C₂³×D₁₀₁

—

C₂³×D₁₀₁

Q₈

yes

—

C₂³×D₁₀₁

C₂³

yes

—

C₂²×Dic₁₀₁

C₂³

yes

—

C₄⋊Dic₁₀₁

—

#13108053

—

C₂²⋊Dic₁₀₁

—

#13108051

—

#13108051

—

C₂⁴×C₁₀₁

C₂³

yes

characteristic

C₂²×C₄×C₁₀₁

C₂³

yes

—

C₂²×C₄×C₁₀₁

C₂³

yes

—

C₄⋊C₄×C₁₀₁

—

C₂²⋊C₄×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

C₂×Q₈

yes

—

C₂³×C₁₀₁

C₂⁴

yes

characteristic

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

—

C₂²×D₁₀₁

C₂×Q₈

yes

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂²×Q₈

yes

characteristic

C₂²×C₁₀₁

H(5,𝔽₂)

yes

—

C₂²×C₁₀₁

H(5,𝔽₂)

yes

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#543

yes

—

C₂×C₁₀₁

#560

yes

characteristic

C₂×C₁₀₁

#560

yes

—

D₁₀₁

—

#2842

—

101

maximal, Sylow

C₁₀₁

#2842

yes

characteristic, complemented, Sylow

#399

—

101

—

#399

D₁₀₁

yes

characteristic, complemented

#396

—

101

—

#396

—

101

—

#379

—

101

—

C₂⁵

—

101

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

—

101

—

C₂×C₄⋊C₄

C₂×D₁₀₁

yes

complemented

C₂×C₂²⋊C₄

—

101

—

C₂×C₂²⋊C₄

—

101

—

C₂×C₂²⋊C₄

—

101

—

C₂×C₂²⋊C₄

C₂×D₁₀₁

yes

—

#95

—

101

—

#95

C₂×D₁₀₁

yes

characteristic, complemented

C₂⁴

—

101

—

C₂⁴

—

101

—

C₂⁴

—

404

—

C₂⁴

—

404

—

C₂⁴

C₂²×D₁₀₁

yes

characteristic

C₂²×C₄

—

101

—

C₂²×C₄

C₂²×D₁₀₁

yes

—

C₂²×C₄

C₂²×D₁₀₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

404

—

C₄⋊C₄

—

404

—

C₂²⋊C₄

—

C₂²⋊C₄

—

404

—

C₂²⋊C₄

—

404

—

C₂²⋊C₄

—

404

—

C₂³

—

C₂³

—

100

C₂³

—

101

—

101

C₂³

—

404

—

102

C₂³

—

404

—

103

C₂³

—

404

—

104

C₂³

—

202

—

105

C₂³

—

202

—

106

C₂³

Q₈×D₁₀₁

yes

—

107

C₂³

C₂³×D₁₀₁

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

202

—

113

C₂×C₄

—

202

—

114

C₂×C₄

—

202

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

404

—

119

C₂²

—

404

—

120

C₂²

—

404

—

121

C₂²

—

202

—

122

C₂²

C₂×Q₈×D₁₀₁

yes

characteristic

123

C₂²

#15692300

yes

—

124

C₂²

#15692301

yes

—

125

C₄

—

126

C₄

—

127

C₄

—

404

—

128

C₂

—

129

C₂

—

404

—

130

C₂

#20265802

yes

—

131

C₂

#20265914

yes

characteristic

132

C₂

#20265917

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator, socle, cosocle	C₂³×C₁₀₁	C₂⁴
Fitting	#399×C₁₀₁	C₂
prime radical	C₂⁴×D₁₀₁	C₂²
hypercenter	#399	D₁₀₁
hypocenter	C₁₀₁	#2842
LCS	G ▷ C₂³×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ #399

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₀₁

commutator, socle, cosocle

C₂³×C₁₀₁

C₂⁴

Fitting

#399×C₁₀₁

C₂

prime radical

C₂⁴×D₁₀₁

C₂²

D₁₀₁

C₁₀₁

G ▷ C₂³×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂³ ◁ #399

N	H ≅ G/N	type
1. #20265113	C₂	semidirect
2. #20265101	C₂	semidirect
3. #20265099	C₂	semidirect
5. #399×C₁₀₁	C₂	semidirect
7. C₂×C₄⋊Dic₁₀₁	C₂²	semidirect
8. C₂×#13108053	C₂²	semidirect
12. #15692162	C₂²	semidirect
13. C₂×C₄⋊C₄×C₁₀₁	C₂²	semidirect
15. #95×C₁₀₁	C₂²	semidirect
67. C₁₀₁	#2842	semidirect
69. #399	D₁₀₁	semidirect
76. C₂×C₄⋊C₄	C₂×D₁₀₁	semidirect
82. #95	C₂×D₁₀₁	semidirect

H ≅ G/N

type

1. #20265113

C₂

semidirect

2. #20265101

C₂

semidirect

3. #20265099

C₂

semidirect

5. #399×C₁₀₁

C₂

semidirect

7. C₂×C₄⋊Dic₁₀₁

C₂²

semidirect

8. C₂×#13108053

C₂²

semidirect

12. #15692162

C₂²

semidirect

13. C₂×C₄⋊C₄×C₁₀₁

C₂²

semidirect

15. #95×C₁₀₁

C₂²

semidirect

67. C₁₀₁

#2842

semidirect

69. #399

D₁₀₁

semidirect

76. C₂×C₄⋊C₄

C₂×D₁₀₁

semidirect

82. #95

C₂×D₁₀₁

semidirect

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

subgroup	index
#20265113	2
#20265101	2
#20265099	2
#20265032	2
#399×C₁₀₁	2
#2842	101

subgroup

index

#399×C₁₀₁

101

Subgroups — #27179397

Conjugacy classes of subgroups

Complements

Maximal subgroups