Group #27164772 — subgroups

Subgroups — #27164772

Order 12736.

196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 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	#20263954	C₂	1	2	yes	maximal, complemented
2	#20263950	C₂	1	2	yes	maximal, complemented
3	#20263950	C₂	1	2	yes	maximal, complemented
4	#20263947	C₂	1	1	yes	maximal, characteristic, complemented
5	#20263943	C₂	1	1	yes	maximal, characteristic
6	#20263940	C₂	1	2	yes	maximal
7	#20263935	C₂	1	2	yes	maximal, complemented
8	#20263933	C₂	1	2	yes	maximal, complemented
9	#125×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₃₉₈	C₂²	1	1	yes	characteristic
11	C₂×Dic₃₉₈	C₂²	1	2	yes	—
12	C₂×D₇₉₆	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₁₉₉	C₂²	1	2	yes	—
14	C₄⋊Dic₁₉₉	C₂²	1	1	yes	characteristic
15	C₄⋊Dic₁₉₉	C₂²	1	1	yes	characteristic
16	C₄⋊Dic₁₉₉	C₂²	1	2	yes	—
17	Dic₁₉₉⋊C₄	C₂²	1	2	yes	—
18	Dic₁₉₉⋊C₄	C₂²	1	2	yes	—
19	Dic₁₉₉⋊C₄	C₂²	1	2	yes	—
20	(C₂×D₁₉₉)⋊C₄	C₂²	1	2	yes	—
21	(C₂×D₁₉₉)⋊C₄	C₂²	1	2	yes	—
22	(C₂×D₁₉₉)⋊C₄	C₂²	1	2	yes	—
23	(C₂×D₁₉₉)⋊C₄	C₂²	1	2	yes	—
24	(C₂×D₁₉₉)⋊C₄	C₂²	1	2	yes	—
25	C₄×Dic₁₉₉	C₂²	1	2	yes	—
26	C₄⋊C₄×C₁₉₉	C₂²	1	2	yes	—
27	C₄⋊C₄×C₁₉₉	C₂²	1	1	yes	characteristic
28	C₄⋊C₄×C₁₉₉	C₂²	1	2	yes	—
29	C₄⋊C₄×C₁₉₉	C₂²	1	1	yes	characteristic
30	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
31	C₂²×D₁₉₉	C₂³	1	2	yes	—
32	Dic₃₉₈	—	4	2	no	—
33	Dic₃₉₈	—	4	1	no	—
34	D₇₉₆	—	4	1	no	—
35	C₂×Dic₁₉₉	C₂³	1	2	yes	—
36	C₂×Dic₁₉₉	C₂³	1	2	yes	—
37	C₂×Dic₁₉₉	C₂³	1	2	yes	—
38	C₄×D₁₉₉	—	4	2	no	—
39	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₁₉₉	C₂³	1	2	yes	—
43	C₂×C₄×C₁₉₉	C₂³	1	2	yes	—
44	C₂²×C₁₉₉	C₂⁴	1	1	yes	characteristic
45	C₂×D₁₉₉	—	2	2	no	—
46	C₂×D₁₉₉	—	2	2	no	—
47	C₂×D₁₉₉	—	2	2	no	—
48	C₄×C₁₉₉	—	2	2	no	—
49	C₄×C₁₉₉	—	2	1	no	—
50	C₄×C₁₉₉	—	2	1	no	—
51	C₄×C₁₉₉	—	2	2	no	—
52	C₄×C₁₉₉	—	2	1	no	—
53	Dic₁₉₉	—	2	2	no	—
54	Dic₁₉₉	—	2	2	no	—
55	Dic₁₉₉	—	2	2	no	—
56	C₂×C₁₉₉	H(5,𝔽₂)	1	1	yes	characteristic
57	C₂×C₁₉₉	2₋¹⁺⁴	1	1	yes	characteristic
58	C₂×C₁₉₉	2₋¹⁺⁴	1	1	yes	characteristic
59	D₁₉₉	—	4	2	no	—
60	C₁₉₉	#563	1	1	yes	characteristic, complemented, Sylow
61	#563	—	199	1	no	maximal, Sylow
62	C₄⋊Q₈	—	199	2	no	—
63	#126	—	199	2	no	—
64	#126	—	199	2	no	—
65	#125	—	199	1	no	—
66	#125	D₁₉₉	1	1	yes	characteristic, complemented
67	#124	—	199	1	no	—
68	#123	—	199	2	no	—
69	C₂²⋊Q₈	—	199	2	no	—
70	C₂²⋊Q₈	—	199	2	no	—
71	C₂×Q₈	—	199	2	no	—
72	C₂×Q₈	—	199	1	no	—
73	C₂×D₄	—	199	1	no	—
74	C₂²×C₄	—	199	2	no	—
75	C₄⋊C₄	—	199	2	no	—
76	C₄⋊C₄	—	199	2	no	—
77	C₄⋊C₄	—	199	2	no	—
78	C₄⋊C₄	—	199	1	no	—
79	C₄⋊C₄	—	199	2	no	—
80	C₄⋊C₄	—	199	1	no	—
81	C₄⋊C₄	C₂×D₁₉₉	1	1	yes	characteristic
82	C₄⋊C₄	C₂×D₁₉₉	1	2	yes	—
83	C₄⋊C₄	C₂×D₁₉₉	1	2	yes	—
84	C₄⋊C₄	C₂×D₁₉₉	1	1	yes	characteristic
85	C₂²⋊C₄	—	199	2	no	—
86	C₂²⋊C₄	—	199	2	no	—
87	C₂²⋊C₄	—	199	2	no	—
88	C₂²⋊C₄	—	199	2	no	—
89	C₂²⋊C₄	—	199	2	no	—
90	C₄²	—	199	2	no	—
91	C₄²	C₂×D₁₉₉	1	1	yes	characteristic
92	C₂³	—	199	2	no	—
93	Q₈	—	796	2	no	—
94	Q₈	—	796	1	no	—
95	D₄	—	796	1	no	—
96	C₂×C₄	—	796	2	no	—
97	C₂×C₄	—	199	2	no	—
98	C₂×C₄	—	199	2	no	—
99	C₂×C₄	—	199	2	no	—
100	C₂×C₄	C₂²×D₁₉₉	1	2	yes	—
101	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
102	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
103	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
104	C₂×C₄	C₂²×D₁₉₉	1	2	yes	—
105	C₂²	—	398	2	no	—
106	C₂²	—	398	2	no	—
107	C₂²	—	398	2	no	—
108	C₂²	C₂³×D₁₉₉	1	1	yes	characteristic
109	C₄	—	2	2	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	2	no	—
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	—	398	2	no	—
115	C₄	—	398	2	no	—
116	C₄	—	398	2	no	—
117	C₂	—	796	2	no	—
118	C₂	#20264001	1	1	yes	characteristic
119	C₂	#20264002	1	1	yes	characteristic
120	C₂	#20264003	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20263954

C₂

yes

maximal, complemented

#20263950

C₂

yes

maximal, complemented

#20263950

C₂

yes

maximal, complemented

#20263947

C₂

yes

maximal, characteristic, complemented

#20263943

C₂

yes

maximal, characteristic

#20263940

C₂

yes

maximal

#20263935

C₂

yes

maximal, complemented

#20263933

C₂

yes

maximal, complemented

#125×C₁₉₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₃₉₈

C₂²

yes

characteristic

C₂×Dic₃₉₈

C₂²

yes

—

C₂×D₇₉₆

C₂²

yes

characteristic

C₂×C₄×D₁₉₉

C₂²

yes

—

C₄⋊Dic₁₉₉

C₂²

yes

characteristic

C₄⋊Dic₁₉₉

C₂²

yes

characteristic

C₄⋊Dic₁₉₉

C₂²

yes

—

Dic₁₉₉⋊C₄

C₂²

yes

—

Dic₁₉₉⋊C₄

C₂²

yes

—

Dic₁₉₉⋊C₄

C₂²

yes

—

(C₂×D₁₉₉)⋊C₄

C₂²

yes

—

(C₂×D₁₉₉)⋊C₄

C₂²

yes

—

(C₂×D₁₉₉)⋊C₄

C₂²

yes

—

(C₂×D₁₉₉)⋊C₄

C₂²

yes

—

(C₂×D₁₉₉)⋊C₄

C₂²

yes

—

C₄×Dic₁₉₉

C₂²

yes

—

C₄⋊C₄×C₁₉₉

C₂²

yes

—

C₄⋊C₄×C₁₉₉

C₂²

yes

characteristic

C₄⋊C₄×C₁₉₉

C₂²

yes

—

C₄⋊C₄×C₁₉₉

C₂²

yes

characteristic

C₄²×C₁₉₉

C₂²

yes

characteristic

C₂²×D₁₉₉

C₂³

yes

—

Dic₃₉₈

—

Dic₃₉₈

—

D₇₉₆

—

C₂×Dic₁₉₉

C₂³

yes

—

C₂×Dic₁₉₉

C₂³

yes

—

C₂×Dic₁₉₉

C₂³

yes

—

C₄×D₁₉₉

—

C₂×C₄×C₁₉₉

C₂³

yes

characteristic

C₂×C₄×C₁₉₉

C₂³

yes

characteristic

C₂×C₄×C₁₉₉

C₂³

yes

characteristic

C₂×C₄×C₁₉₉

C₂³

yes

—

C₂×C₄×C₁₉₉

C₂³

yes

—

C₂²×C₁₉₉

C₂⁴

yes

characteristic

C₂×D₁₉₉

—

C₂×D₁₉₉

—

C₂×D₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

Dic₁₉₉

—

Dic₁₉₉

—

Dic₁₉₉

—

C₂×C₁₉₉

H(5,𝔽₂)

yes

characteristic

C₂×C₁₉₉

2₋¹⁺⁴

yes

characteristic

C₂×C₁₉₉

2₋¹⁺⁴

yes

characteristic

D₁₉₉

—

C₁₉₉

#563

yes

characteristic, complemented, Sylow

#563

—

199

maximal, Sylow

C₄⋊Q₈

—

199

—

#126

—

199

—

#126

—

199

—

#125

—

199

—

#125

D₁₉₉

yes

characteristic, complemented

#124

—

199

—

#123

—

199

—

C₂²⋊Q₈

—

199

—

C₂²⋊Q₈

—

199

—

C₂×Q₈

—

199

—

C₂×Q₈

—

199

—

C₂×D₄

—

199

—

C₂²×C₄

—

199

—

C₄⋊C₄

—

199

—

C₄⋊C₄

—

199

—

C₄⋊C₄

—

199

—

C₄⋊C₄

—

199

—

C₄⋊C₄

—

199

—

C₄⋊C₄

—

199

—

C₄⋊C₄

C₂×D₁₉₉

yes

characteristic

C₄⋊C₄

C₂×D₁₉₉

yes

—

C₄⋊C₄

C₂×D₁₉₉

yes

—

C₄⋊C₄

C₂×D₁₉₉

yes

characteristic

C₂²⋊C₄

—

199

—

C₂²⋊C₄

—

199

—

C₂²⋊C₄

—

199

—

C₂²⋊C₄

—

199

—

C₂²⋊C₄

—

199

—

C₄²

—

199

—

C₄²

C₂×D₁₉₉

yes

characteristic

C₂³

—

199

—

Q₈

—

796

—

Q₈

—

796

—

D₄

—

796

—

C₂×C₄

—

796

—

C₂×C₄

—

199

—

C₂×C₄

—

199

—

C₂×C₄

—

199

—

100

C₂×C₄

C₂²×D₁₉₉

yes

—

101

C₂×C₄

C₂²×D₁₉₉

yes

characteristic

102

C₂×C₄

C₂²×D₁₉₉

yes

characteristic

103

C₂×C₄

C₂²×D₁₉₉

yes

characteristic

104

C₂×C₄

C₂²×D₁₉₉

yes

—

105

C₂²

—

398

—

106

C₂²

—

398

—

107

C₂²

—

398

—

108

C₂²

C₂³×D₁₉₉

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

398

—

115

C₄

—

398

—

116

C₄

—

398

—

117

C₂

—

796

—

118

C₂

#20264001

yes

characteristic

119

C₂

#20264002

yes

characteristic

120

C₂

#20264003

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₉₉
commutator, socle, cosocle	C₂²×C₁₉₉	C₂⁴
Fitting	#125×C₁₉₉	C₂
prime radical	C₂×D₇₉₆	C₂²
hypercenter	#125	D₁₉₉
hypocenter	C₁₉₉	#563
LCS	G ▷ C₂²×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₉₉

commutator, socle, cosocle

C₂²×C₁₉₉

C₂⁴

Fitting

#125×C₁₉₉

C₂

prime radical

C₂×D₇₉₆

C₂²

D₁₉₉

C₁₉₉

G ▷ C₂²×C₁₉₉ ▷ C₁₉₉

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
1. #20263954	C₂	semidirect
2. #20263950	C₂	semidirect
3. #20263950	C₂	semidirect
4. #20263947	C₂	semidirect
7. #20263935	C₂	semidirect
8. #20263933	C₂	semidirect
9. #125×C₁₉₉	C₂	semidirect
60. C₁₉₉	#563	semidirect
66. #125	D₁₉₉	semidirect

H ≅ G/N

type

1. #20263954

C₂

semidirect

2. #20263950

C₂

semidirect

3. #20263950

C₂

semidirect

4. #20263947

C₂

semidirect

7. #20263935

C₂

semidirect

8. #20263933

C₂

semidirect

9. #125×C₁₉₉

C₂

semidirect

60. C₁₉₉

#563

semidirect

66. #125

D₁₉₉

semidirect

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

subgroup	index
#20263954	2
#20263950	2
#20263947	2
#20263943	2
#20263940	2
#20263935	2
#20263933	2
#125×C₁₉₉	2
#563	199

subgroup

index

#125×C₁₉₉

199

Subgroups — #27164772

Conjugacy classes of subgroups

Complements

Maximal subgroups