Group #27186621 — subgroups

Subgroups — #27186621

Order 12928.

238 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 56 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	#20266200	C₂	1	2	yes	maximal, complemented
2	#20266032	C₂	1	4	yes	maximal, complemented
3	#20265725	C₂	1	1	yes	maximal, characteristic
4	#15692322	—	2	2	no	—
5	#15692312	—	2	2	no	—
6	#15692310	C₂²	1	4	yes	—
7	C₂×#13108073	C₄	1	2	yes	—
8	#15692255	C₄	1	1	yes	characteristic
9	#15692242	C₂²	1	2	yes	—
10	#15692226	—	2	2	no	—
11	D₄×Dic₁₀₁	—	2	2	no	—
12	C₄²×D₁₀₁	C₂²	1	1	yes	characteristic
13	#127×C₁₀₁	C₄	1	1	yes	characteristic
14	C₂×#11717581	—	2	4	no	—
15	#13108073	—	4	4	no	—
16	C₂×Dic₂₀₂	C₂×C₄	1	2	yes	—
17	C₂×C₂²⋊D₁₀₁	—	2	2	no	—
18	C₂²×Dic₁₀₁	—	2	2	no	—
19	C₂×C₄×D₁₀₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×D₁₀₁	D₄	1	2	yes	—
21	C₄⋊Dic₁₀₁	—	2	2	no	—
22	C₂²⋊Dic₁₀₁	—	2	4	no	—
23	C₄×Dic₁₀₁	D₄	1	2	yes	—
24	C₄×Dic₁₀₁	C₂³	1	1	yes	characteristic
25	C₂×D₄×C₁₀₁	—	2	2	no	—
26	C₂×D₄×C₁₀₁	C₂×C₄	1	2	yes	—
27	C₄²×C₁₀₁	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₁₀₁	—	2	2	no	—
29	C₂²×D₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
30	D₄×C₁₀₁	—	4	2	no	—
31	D₄×C₁₀₁	—	4	4	no	—
32	C₂×C₄×C₁₀₁	C₂²⋊C₄	1	2	yes	—
33	C₂×C₄×C₁₀₁	C₂²×C₄	1	1	yes	characteristic
34	C₂²⋊D₁₀₁	—	4	4	no	—
35	C₂×Dic₁₀₁	—	2	2	no	—
36	C₂×Dic₁₀₁	—	4	4	no	—
37	C₂×Dic₁₀₁	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
39	C₄×D₁₀₁	—	2	2	no	—
40	C₄×D₁₀₁	—	4	2	no	—
41	Dic₂₀₂	—	4	2	no	—
42	#11717581	—	4	4	no	—
43	C₂²×C₁₀₁	—	4	2	no	—
44	C₂²×C₁₀₁	—	4	4	no	—
45	C₂²×C₁₀₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂×D₁₀₁	—	2	1	no	—
47	C₂×D₁₀₁	—	2	2	no	—
48	C₄×C₁₀₁	—	2	1	no	—
49	C₄×C₁₀₁	—	2	2	no	—
50	Dic₁₀₁	—	2	2	no	—
51	Dic₁₀₁	—	4	2	no	—
52	Dic₁₀₁	#100	1	2	yes	—
53	C₂×C₁₀₁	—	8	2	no	—
54	C₂×C₁₀₁	C₂×#100	1	1	yes	characteristic
55	C₂×C₁₀₁	#421	1	2	yes	—
56	D₁₀₁	—	4	1	no	—
57	#1536	—	101	1	no	maximal, Sylow
58	C₁₀₁	#1536	1	1	yes	characteristic, complemented, Sylow
59	#532	—	101	1	no	—
60	#432	—	101	2	no	—
61	#329	—	101	4	no	—
62	C₂×#41	—	101	2	no	—
63	C₄⋊Q₈	—	101	1	no	—
64	#127	#87938	1	1	yes	characteristic
65	#124	—	101	2	no	—
66	#121	—	202	2	no	—
67	C₄×D₄	—	202	2	no	—
68	C₂×C₄²	—	101	1	no	—
69	C₄⋊C₈	—	202	2	no	—
70	C₂²⋊C₈	—	202	2	no	—
71	#97	—	101	4	no	—
72	#41	—	404	4	no	—
73	C₂×Q₈	—	101	2	no	—
74	C₂×D₄	—	2	2	no	—
75	C₂×D₄	—	202	2	no	—
76	C₂×D₄	C₂×#87938	1	2	yes	—
77	C₂²×C₄	—	101	2	no	—
78	C₂²×C₄	—	101	1	no	—
79	C₂²×C₄	—	202	2	no	—
80	C₂×C₈	—	202	4	no	—
81	C₄⋊C₄	—	202	2	no	—
82	C₂²⋊C₄	—	202	4	no	—
83	C₄²	—	101	2	no	—
84	C₄²	—	101	1	no	—
85	C₄²	C₂×#87938	1	1	yes	characteristic
86	C₂³	—	2	2	no	—
87	C₂³	—	101	1	no	—
88	Q₈	—	404	2	no	—
89	D₄	—	4	2	no	—
90	D₄	—	4	4	no	—
91	D₄	—	404	4	no	—
92	C₂×C₄	—	101	1	no	—
93	C₂×C₄	—	101	2	no	—
94	C₂×C₄	—	404	2	no	—
95	C₂×C₄	—	404	4	no	—
96	C₂×C₄	—	202	2	no	—
97	C₂×C₄	—	202	2	no	—
98	C₂×C₄	C₂²⋊#87938	1	2	yes	—
99	C₂×C₄	C₂²×#87938	1	1	yes	characteristic
100	C₈	—	404	4	no	—
101	C₂²	—	4	2	no	—
102	C₂²	—	4	4	no	—
103	C₂²	—	202	1	no	—
104	C₂²	—	202	2	no	—
105	C₂²	C₂×C₂²⋊#87938	1	1	yes	characteristic
106	C₄	—	2	1	no	—
107	C₄	—	2	2	no	—
108	C₄	—	101	2	no	—
109	C₄	—	404	2	no	—
110	C₄	—	202	2	no	—
111	C₂	—	8	2	no	—
112	C₂	—	404	1	no	—
113	C₂	D₁₀₁⋊#100	1	1	yes	characteristic
114	C₂	#20266175	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20266200

C₂

yes

maximal, complemented

#20266032

C₂

yes

maximal, complemented

#20265725

C₂

yes

maximal, characteristic

#15692322

—

#15692312

—

#15692310

C₂²

yes

—

C₂×#13108073

C₄

yes

—

#15692255

C₄

yes

characteristic

#15692242

C₂²

yes

—

#15692226

—

D₄×Dic₁₀₁

—

C₄²×D₁₀₁

C₂²

yes

characteristic

#127×C₁₀₁

C₄

yes

characteristic

C₂×#11717581

—

#13108073

—

C₂×Dic₂₀₂

C₂×C₄

yes

—

C₂×C₂²⋊D₁₀₁

—

C₂²×Dic₁₀₁

—

C₂×C₄×D₁₀₁

C₂×C₄

yes

characteristic

C₂×C₄×D₁₀₁

D₄

yes

—

C₄⋊Dic₁₀₁

—

C₂²⋊Dic₁₀₁

—

C₄×Dic₁₀₁

D₄

yes

—

C₄×Dic₁₀₁

C₂³

yes

characteristic

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

C₂×C₄

yes

—

C₄²×C₁₀₁

C₂×C₄

yes

characteristic

C₂³×C₁₀₁

—

C₂²×D₁₀₁

C₂²⋊C₄

yes

characteristic

D₄×C₁₀₁

—

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₁

C₂²×C₄

yes

characteristic

C₂²⋊D₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₁₀₁

C₂×D₄

yes

—

C₄×D₁₀₁

—

C₄×D₁₀₁

—

Dic₂₀₂

—

#11717581

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

#100

yes

—

C₂×C₁₀₁

—

C₂×C₁₀₁

C₂×#100

yes

characteristic

C₂×C₁₀₁

#421

yes

—

D₁₀₁

—

#1536

—

101

maximal, Sylow

C₁₀₁

#1536

yes

characteristic, complemented, Sylow

#532

—

101

—

#432

—

101

—

#329

—

101

—

C₂×#41

—

101

—

C₄⋊Q₈

—

101

—

yes

—

101

—

#121

—

202

—

C₄×D₄

—

202

—

C₂×C₄²

—

101

—

C₄⋊C₈

—

202

—

C₂²⋊C₈

—

202

—

#97

—

101

—

#41

—

404

—

C₂×Q₈

—

101

—

C₂×D₄

—

C₂×D₄

—

202

—

C₂×D₄

C₂×#87938

yes

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

202

—

C₂×C₈

—

202

—

C₄⋊C₄

—

202

—

C₂²⋊C₄

—

202

—

C₄²

—

101

—

C₄²

—

101

—

C₄²

C₂×#87938

yes

characteristic

C₂³

—

C₂³

—

101

—

Q₈

—

404

—

D₄

—

D₄

—

D₄

—

404

—

C₂×C₄

—

101

—

C₂×C₄

—

101

—

C₂×C₄

—

404

—

C₂×C₄

—

404

—

C₂×C₄

—

202

—

C₂×C₄

—

202

—

C₂×C₄

C₂²⋊#87938

yes

—

C₂×C₄

C₂²×#87938

yes

characteristic

100

C₈

—

404

—

101

C₂²

—

102

C₂²

—

103

C₂²

—

202

—

104

C₂²

—

202

—

105

C₂²

C₂×C₂²⋊#87938

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

101

—

109

C₄

—

404

—

110

C₄

—

202

—

111

C₂

—

112

C₂

—

404

—

113

C₂

D₁₀₁⋊#100

yes

characteristic

114

C₂

#20266175

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#87938
commutator	C₂×C₄×C₁₀₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×#87938
Fitting	#127×C₁₀₁	C₄
socle	C₂²×C₁₀₁	C₂×C₂²⋊C₄
prime radical	#20265725	C₂
cosocle	C₄×Dic₁₀₁	C₂³
hypercenter	#127	#87938
hypocenter	C₁₀₁	#1536
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #127

subgroup

G/H

center

C₂²

C₂×C₂²⋊#87938

commutator

C₂×C₄×C₁₀₁

C₂²×C₄

Frattini

C₂×C₄

C₂²×#87938

Fitting

#127×C₁₀₁

C₄

socle

C₂²×C₁₀₁

C₂×C₂²⋊C₄

prime radical

#20265725

C₂

cosocle

C₄×Dic₁₀₁

C₂³

C₁₀₁

G ▷ C₂×C₄×C₁₀₁ ▷ C₂²×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #127

N	H ≅ G/N	type
1. #20266200	C₂	semidirect
2. #20266032	C₂	semidirect
58. C₁₀₁	#1536	semidirect

H ≅ G/N

type

1. #20266200

C₂

semidirect

2. #20266032

C₂

semidirect

58. C₁₀₁

#1536

semidirect

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

subgroup	index
#20266200	2
#20266032	2
#20265725	2
#1536	101

subgroup

index

101

Subgroups — #27186621

Conjugacy classes of subgroups

Complements

Maximal subgroups