Group #27179488 — subgroups

Subgroups — #27179488

Order 12928.

752 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 197 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	C₂×#15692255	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#15692242	C₂	1	4	yes	maximal, complemented
3	#20265080	C₂	1	8	yes	maximal, complemented
4	C₄²×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#127×C₁₀₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₀₂	C₂²	1	2	yes	—
7	#15692255	—	2	4	no	—
8	#15692242	—	2	16	no	—
9	C₂×C₄⋊Dic₁₀₁	C₂²	1	4	yes	—
10	C₂×C₂²⋊Dic₁₀₁	C₂²	1	16	yes	—
11	C₂×C₄×Dic₁₀₁	C₂²	1	4	yes	—
12	C₂×C₄×Dic₁₀₁	C₂²	1	2	yes	—
13	C₂²×D₄×C₁₀₁	C₂²	1	4	yes	—
14	C₂²×D₄×C₁₀₁	C₂²	1	2	yes	—
15	#127×C₁₀₁	—	2	4	no	—
16	C₂×C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₀₂	—	2	8	no	—
18	C₂×Dic₂₀₂	—	2	4	no	—
19	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
20	C₂²×Dic₁₀₁	C₂³	1	4	yes	—
21	C₄⋊Dic₁₀₁	—	2	8	no	—
22	C₂²⋊Dic₁₀₁	—	2	32	no	—
23	C₄×Dic₁₀₁	—	2	8	no	—
24	C₄×Dic₁₀₁	D₄	1	8	yes	complemented
25	C₂⁴×C₁₀₁	C₂³	1	4	yes	—
26	C₂×D₄×C₁₀₁	—	2	8	no	—
27	C₂×D₄×C₁₀₁	—	2	4	no	—
28	C₂×D₄×C₁₀₁	—	2	8	no	—
29	C₂×D₄×C₁₀₁	—	2	16	no	—
30	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
31	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
32	C₄²×C₁₀₁	D₄	1	4	yes	—
33	C₂³×C₁₀₁	—	2	4	no	—
34	C₂³×C₁₀₁	—	2	8	no	—
35	C₂³×C₁₀₁	—	2	16	no	—
36	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
37	D₄×C₁₀₁	—	4	16	no	—
38	D₄×C₁₀₁	—	4	8	no	—
39	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
40	C₂×C₄×C₁₀₁	C₂×D₄	1	4	yes	—
41	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
42	C₂×C₄×C₁₀₁	#41	1	8	yes	—
43	C₂×Dic₁₀₁	—	2	8	no	—
44	C₂×Dic₁₀₁	—	2	8	no	—
45	C₂×Dic₁₀₁	—	2	4	no	—
46	C₂×Dic₁₀₁	C₂×D₄	1	8	yes	—
47	Dic₂₀₂	—	4	8	no	—
48	C₂²×C₁₀₁	—	4	8	no	—
49	C₂²×C₁₀₁	—	4	4	no	—
50	C₂²×C₁₀₁	—	4	16	no	—
51	C₂²×C₁₀₁	C₂²×D₄	1	2	yes	—
52	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
53	C₂²×C₁₀₁	C₂×#41	1	4	yes	—
54	C₄×C₁₀₁	#124	1	8	yes	complemented
55	C₄×C₁₀₁	#127	1	4	yes	—
56	Dic₁₀₁	—	2	8	no	—
57	Dic₁₀₁	—	4	4	no	—
58	C₂×C₁₀₁	—	8	4	no	—
59	C₂×C₁₀₁	C₂×#124	1	2	yes	—
60	C₂×C₁₀₁	C₂×#127	1	1	yes	characteristic
61	C₂×C₁₀₁	#532	1	4	yes	—
62	#2858	—	101	1	no	maximal, Sylow
63	C₁₀₁	#2858	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₄⋊Q₈	—	101	1	no	—
65	C₂×#127	D₁₀₁	1	1	yes	characteristic
66	C₂×#124	—	101	4	no	—
67	#390	—	101	8	no	—
68	C₄³	—	101	1	no	—
69	C₂²×Q₈	—	101	2	no	—
70	C₂²×D₄	C₂×D₁₀₁	1	2	yes	—
71	C₂²×D₄	C₂×D₁₀₁	1	4	yes	—
72	C₄⋊Q₈	—	202	4	no	—
73	#127	—	2	4	no	—
74	#124	—	202	16	no	—
75	C₂×C₄⋊C₄	—	101	4	no	—
76	C₂×C₂²⋊C₄	—	101	16	no	—
77	C₂×C₄²	—	101	2	no	—
78	C₂×C₄²	—	101	4	no	—
79	C₂×C₄²	C₂×D₁₀₁	1	1	yes	characteristic
80	C₂⁴	C₂²×D₁₀₁	1	4	yes	—
81	C₂×Q₈	—	202	8	no	—
82	C₂×Q₈	—	202	4	no	—
83	C₂×D₄	—	2	8	no	—
84	C₂×D₄	—	2	8	no	—
85	C₂×D₄	—	2	4	no	—
86	C₂×D₄	—	2	16	no	—
87	C₂²×C₄	—	101	4	no	—
88	C₂²×C₄	—	101	4	no	—
89	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
90	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
91	C₄⋊C₄	—	202	8	no	—
92	C₂²⋊C₄	—	202	32	no	—
93	C₄²	—	101	8	no	—
94	C₄²	—	202	8	no	—
95	C₄²	C₂²⋊D₁₀₁	1	4	yes	—
96	C₂³	—	2	8	no	—
97	C₂³	—	2	4	no	—
98	C₂³	—	2	16	no	—
99	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
100	Q₈	—	404	8	no	—
101	D₄	—	4	16	no	—
102	D₄	—	4	8	no	—
103	C₂×C₄	—	101	8	no	—
104	C₂×C₄	—	202	8	no	—
105	C₂×C₄	—	202	4	no	—
106	C₂×C₄	—	202	8	no	—
107	C₂×C₄	D₄×D₁₀₁	1	4	yes	—
108	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	4	yes	—
109	C₂×C₄	C₂×C₂²⋊D₁₀₁	1	2	yes	—
110	C₂×C₄	#13108073	1	8	yes	—
111	C₂²	—	4	8	no	—
112	C₂²	—	4	4	no	—
113	C₂²	—	4	16	no	—
114	C₂²	C₂×D₄×D₁₀₁	1	2	yes	—
115	C₂²	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
116	C₂²	C₂×#13108073	1	4	yes	—
117	C₄	—	404	4	no	—
118	C₄	—	202	8	no	—
119	C₄	#15692242	1	8	yes	complemented
120	C₄	#15692251	1	4	yes	—
121	C₂	—	8	4	no	—
122	C₂	C₂×#15692242	1	2	yes	—
123	C₂	C₂×#15692251	1	1	yes	characteristic
124	C₂	#20265725	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15692255

C₂

yes

maximal, characteristic, complemented

C₂×#15692242

C₂

yes

maximal, complemented

#20265080

C₂

yes

maximal, complemented

C₄²×Dic₁₀₁

C₂

yes

maximal, characteristic, complemented

C₂×#127×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×Dic₂₀₂

C₂²

yes

—

#15692255

—

#15692242

—

C₂×C₄⋊Dic₁₀₁

C₂²

yes

—

C₂×C₂²⋊Dic₁₀₁

C₂²

yes

—

C₂×C₄×Dic₁₀₁

C₂²

yes

—

C₂×C₄×Dic₁₀₁

C₂²

yes

—

C₂²×D₄×C₁₀₁

C₂²

yes

—

C₂²×D₄×C₁₀₁

C₂²

yes

—

#127×C₁₀₁

—

C₂×C₄²×C₁₀₁

C₂²

yes

characteristic

C₂×Dic₂₀₂

—

C₂×Dic₂₀₂

—

C₂²×Dic₁₀₁

C₂³

yes

—

C₂²×Dic₁₀₁

C₂³

yes

—

C₄⋊Dic₁₀₁

—

C₂²⋊Dic₁₀₁

—

C₄×Dic₁₀₁

—

C₄×Dic₁₀₁

D₄

yes

complemented

C₂⁴×C₁₀₁

C₂³

yes

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

—

C₂²×C₄×C₁₀₁

C₂³

yes

characteristic

C₂²×C₄×C₁₀₁

C₂³

yes

—

C₄²×C₁₀₁

D₄

yes

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

C₂⁴

yes

characteristic

D₄×C₁₀₁

—

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

C₂×D₄

yes

—

C₂×C₄×C₁₀₁

#41

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

C₂×D₄

yes

—

Dic₂₀₂

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂²×D₄

yes

—

C₂²×C₁₀₁

C₂²×D₄

yes

characteristic

C₂²×C₁₀₁

C₂×#41

yes

—

C₄×C₁₀₁

#124

yes

complemented

C₄×C₁₀₁

#127

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

C₂×#124

yes

—

C₂×C₁₀₁

C₂×#127

yes

characteristic

C₂×C₁₀₁

#532

yes

—

#2858

—

101

maximal, Sylow

C₁₀₁

#2858

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

101

—

C₂×#127

D₁₀₁

yes

characteristic

C₂×#124

—

101

—

#390

—

101

—

C₄³

—

101

—

C₂²×Q₈

—

101

—

C₂²×D₄

C₂×D₁₀₁

yes

—

C₂²×D₄

C₂×D₁₀₁

yes

—

C₄⋊Q₈

—

202

—

#127

—

#124

—

202

—

C₂×C₄⋊C₄

—

101

—

C₂×C₂²⋊C₄

—

101

—

C₂×C₄²

—

101

—

C₂×C₄²

—

101

—

C₂×C₄²

C₂×D₁₀₁

yes

characteristic

C₂⁴

C₂²×D₁₀₁

yes

—

C₂×Q₈

—

202

—

C₂×Q₈

—

202

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

C₂²×D₁₀₁

yes

characteristic

C₂²×C₄

C₂²×D₁₀₁

yes

—

C₄⋊C₄

—

202

—

C₂²⋊C₄

—

202

—

C₄²

—

101

—

C₄²

—

202

—

C₄²

C₂²⋊D₁₀₁

yes

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂³×D₁₀₁

yes

characteristic

100

Q₈

—

404

—

101

D₄

—

102

D₄

—

103

C₂×C₄

—

101

—

104

C₂×C₄

—

202

—

105

C₂×C₄

—

202

—

106

C₂×C₄

—

202

—

107

C₂×C₄

D₄×D₁₀₁

yes

—

108

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

109

C₂×C₄

C₂×C₂²⋊D₁₀₁

yes

—

110

C₂×C₄

#13108073

yes

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂×D₄×D₁₀₁

yes

—

115

C₂²

C₂²×C₂²⋊D₁₀₁

yes

characteristic

116

C₂²

C₂×#13108073

yes

—

117

C₄

—

404

—

118

C₄

—

202

—

119

C₄

#15692242

yes

complemented

120

C₄

#15692251

yes

—

121

C₂

—

122

C₂

C₂×#15692242

yes

—

123

C₂

C₂×#15692251

yes

characteristic

124

C₂

#20265725

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator, socle, cosocle	C₂³×C₁₀₁	C₂⁴
Fitting, prime radical	C₂×#127×C₁₀₁	C₂
hypercenter	C₂×#127	D₁₀₁
hypocenter	C₁₀₁	#2858
LCS	G ▷ C₂³×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂×#127

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₀₁

commutator, socle, cosocle

C₂³×C₁₀₁

C₂⁴

Fitting, prime radical

C₂×#127×C₁₀₁

C₂

D₁₀₁

C₁₀₁

G ▷ C₂³×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂³ ◁ C₂×#127

N	H ≅ G/N	type
1. C₂×#15692255	C₂	semidirect
2. C₂×#15692242	C₂	semidirect
3. #20265080	C₂	semidirect
4. C₄²×Dic₁₀₁	C₂	semidirect
24. C₄×Dic₁₀₁	D₄	semidirect
54. C₄×C₁₀₁	#124	semidirect
63. C₁₀₁	#2858	semidirect
119. C₄	#15692242	semidirect

H ≅ G/N

type

1. C₂×#15692255

C₂

semidirect

2. C₂×#15692242

C₂

semidirect

3. #20265080

C₂

semidirect

4. C₄²×Dic₁₀₁

C₂

semidirect

24. C₄×Dic₁₀₁

D₄

semidirect

54. C₄×C₁₀₁

#124

semidirect

63. C₁₀₁

#2858

semidirect

119. C₄

#15692242

semidirect

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

subgroup	index
C₂×#15692255	2
C₂×#15692242	2
#20265080	2
C₄²×Dic₁₀₁	2
C₂×#127×C₁₀₁	2
#2858	101

subgroup

index

C₂×#15692255

C₂×#15692242

#20265080

C₄²×Dic₁₀₁

C₂×#127×C₁₀₁

#2858

101

Subgroups — #27179488

Conjugacy classes of subgroups

Complements

Maximal subgroups