Group #27164762 — subgroups

Subgroups — #27164762

Order 12736.

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

normal subgroups only

This group has 220 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	#20263947	C₂	1	1	yes	maximal, characteristic, complemented
2	#20263943	C₂	1	1	yes	maximal, characteristic, complemented
3	#20263939	C₂	1	2	yes	maximal, complemented
4	#20263937	C₂	1	2	yes	maximal, complemented
5	#20263932	C₂	1	2	yes	maximal, complemented
6	#20263931	C₂	1	2	yes	maximal, complemented
7	#20263928	C₂	1	2	yes	maximal, complemented
8	#20263927	C₂	1	2	yes	maximal, complemented
9	#124×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₃₉₈	C₂²	1	1	yes	characteristic
11	C₂×D₇₉₆	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₁₉₉	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₁₉₉	C₂²	1	2	yes	—
14	C₂²×Dic₁₉₉	C₂²	1	2	yes	—
15	C₂×C₄×D₁₉₉	C₂²	1	2	yes	—
16	C₄⋊Dic₁₉₉	C₂²	1	1	yes	characteristic
17	C₄⋊Dic₁₉₉	C₂²	1	1	yes	characteristic
18	C₄⋊Dic₁₉₉	C₂²	1	2	yes	—
19	Dic₁₉₉⋊C₄	C₂²	1	2	yes	—
20	Dic₁₉₉⋊C₄	C₂²	1	2	yes	—
21	Dic₁₉₉⋊C₄	C₂²	1	2	yes	—
22	C₂²⋊Dic₁₉₉	C₂²	1	2	yes	—
23	(C₂×D₁₉₉)⋊C₄	C₂²	1	2	yes	—
24	(C₂×D₁₉₉)⋊C₄	C₂²	1	2	yes	—
25	(C₂×D₁₉₉)⋊C₄	C₂²	1	2	yes	—
26	C₂×Q₈×C₁₉₉	C₂²	1	1	yes	characteristic
27	C₂×D₄×C₁₉₉	C₂²	1	1	yes	characteristic
28	C₂²⋊C₄×C₁₉₉	C₂²	1	2	yes	—
29	C₂²⋊C₄×C₁₉₉	C₂²	1	2	yes	—
30	C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
31	C₂²×D₁₉₉	C₂³	1	2	yes	—
32	Dic₃₉₈	—	4	1	no	—
33	D₇₉₆	—	4	1	no	—
34	C₂²⋊D₁₉₉	—	4	2	no	—
35	C₂²⋊D₁₉₉	—	4	2	no	—
36	C₂×Dic₁₉₉	—	4	2	no	—
37	C₂×Dic₁₉₉	C₂³	1	2	yes	—
38	C₂×Dic₁₉₉	C₂³	1	2	yes	—
39	C₂×Dic₁₉₉	C₂³	1	2	yes	—
40	C₄×D₁₉₉	—	4	2	no	—
41	C₂³×C₁₉₉	C₂³	1	2	yes	—
42	Q₈×C₁₉₉	—	4	1	no	—
43	D₄×C₁₉₉	—	4	1	no	—
44	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
45	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
46	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₁₉₉	C₂³	1	2	yes	—
48	C₂²×C₁₉₉	—	2	2	no	—
49	C₂²×C₁₉₉	—	2	2	no	—
50	C₂²×C₁₉₉	—	2	2	no	—
51	C₂²×C₁₉₉	C₂⁴	1	1	yes	characteristic
52	C₂×D₁₉₉	—	2	2	no	—
53	C₂×D₁₉₉	—	2	2	no	—
54	C₂×D₁₉₉	—	2	2	no	—
55	C₄×C₁₉₉	—	2	1	no	—
56	C₄×C₁₉₉	—	2	2	no	—
57	C₄×C₁₉₉	—	2	1	no	—
58	C₄×C₁₉₉	—	2	1	no	—
59	Dic₁₉₉	—	2	2	no	—
60	Dic₁₉₉	—	2	2	no	—
61	Dic₁₉₉	—	2	2	no	—
62	C₂×C₁₉₉	—	4	2	no	—
63	C₂×C₁₉₉	H(5,𝔽₂)	1	1	yes	characteristic
64	C₂×C₁₉₉	H(5,𝔽₂)	1	1	yes	characteristic
65	C₂×C₁₉₉	2₋¹⁺⁴	1	1	yes	characteristic
66	D₁₉₉	—	4	2	no	—
67	C₁₉₉	#562	1	1	yes	characteristic, complemented, Sylow
68	#562	—	199	1	no	maximal, Sylow
69	#125	—	199	1	no	—
70	#124	—	199	1	no	—
71	#124	D₁₉₉	1	1	yes	characteristic, complemented
72	#123	—	199	2	no	—
73	#123	—	199	2	no	—
74	C₂²⋊Q₈	—	199	2	no	—
75	C₂²⋊Q₈	—	199	2	no	—
76	#121	—	199	2	no	—
77	#121	—	199	2	no	—
78	C₂×Q₈	—	199	1	no	—
79	C₂×Q₈	C₂×D₁₉₉	1	1	yes	characteristic
80	C₂×D₄	—	199	2	no	—
81	C₂×D₄	—	199	2	no	—
82	C₂×D₄	—	199	1	no	—
83	C₂×D₄	C₂×D₁₉₉	1	1	yes	characteristic
84	C₂²×C₄	—	199	2	no	—
85	C₂²×C₄	—	199	2	no	—
86	C₄⋊C₄	—	199	2	no	—
87	C₄⋊C₄	—	199	2	no	—
88	C₄⋊C₄	—	199	2	no	—
89	C₄⋊C₄	—	199	2	no	—
90	C₄⋊C₄	—	199	1	no	—
91	C₄⋊C₄	—	199	1	no	—
92	C₂²⋊C₄	—	199	2	no	—
93	C₂²⋊C₄	—	199	2	no	—
94	C₂²⋊C₄	—	199	2	no	—
95	C₂²⋊C₄	—	199	2	no	—
96	C₂²⋊C₄	C₂×D₁₉₉	1	2	yes	—
97	C₂²⋊C₄	C₂×D₁₉₉	1	2	yes	—
98	C₄²	C₂×D₁₉₉	1	1	yes	characteristic
99	C₂³	—	199	2	no	—
100	C₂³	C₂²×D₁₉₉	1	2	yes	—
101	Q₈	—	4	1	no	—
102	Q₈	—	796	1	no	—
103	D₄	—	4	1	no	—
104	D₄	—	796	1	no	—
105	D₄	—	796	2	no	—
106	D₄	—	796	2	no	—
107	C₂×C₄	—	796	2	no	—
108	C₂×C₄	—	796	2	no	—
109	C₂×C₄	—	199	2	no	—
110	C₂×C₄	—	199	2	no	—
111	C₂×C₄	—	199	2	no	—
112	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
114	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
115	C₂×C₄	C₂²×D₁₉₉	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	398	2	no	—
120	C₂²	—	398	2	no	—
121	C₂²	—	398	2	no	—
122	C₂²	C₂³×D₁₉₉	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	398	2	no	—
128	C₄	—	398	2	no	—
129	C₄	—	398	2	no	—
130	C₂	—	4	2	no	—
131	C₂	—	796	2	no	—
132	C₂	#20264000	1	1	yes	characteristic
133	C₂	#20264001	1	1	yes	characteristic
134	C₂	#20264002	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20263947

C₂

yes

maximal, characteristic, complemented

#20263943

C₂

yes

maximal, characteristic, complemented

#20263939

C₂

yes

maximal, complemented

#20263937

C₂

yes

maximal, complemented

#20263932

C₂

yes

maximal, complemented

#20263931

C₂

yes

maximal, complemented

#20263928

C₂

yes

maximal, complemented

#20263927

C₂

yes

maximal, complemented

#124×C₁₉₉

C₂

yes

maximal, characteristic, complemented

C₂×Dic₃₉₈

C₂²

yes

characteristic

C₂×D₇₉₆

C₂²

yes

characteristic

C₂×C₂²⋊D₁₉₉

C₂²

yes

—

C₂×C₂²⋊D₁₉₉

C₂²

yes

—

C₂²×Dic₁₉₉

C₂²

yes

—

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

—

C₂²⋊Dic₁₉₉

C₂²

yes

—

(C₂×D₁₉₉)⋊C₄

C₂²

yes

—

(C₂×D₁₉₉)⋊C₄

C₂²

yes

—

(C₂×D₁₉₉)⋊C₄

C₂²

yes

—

C₂×Q₈×C₁₉₉

C₂²

yes

characteristic

C₂×D₄×C₁₉₉

C₂²

yes

characteristic

C₂²⋊C₄×C₁₉₉

C₂²

yes

—

C₂²⋊C₄×C₁₉₉

C₂²

yes

—

C₄²×C₁₉₉

C₂²

yes

characteristic

C₂²×D₁₉₉

C₂³

yes

—

Dic₃₉₈

—

D₇₉₆

—

C₂²⋊D₁₉₉

—

C₂²⋊D₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

C₂³

yes

—

C₂×Dic₁₉₉

C₂³

yes

—

C₂×Dic₁₉₉

C₂³

yes

—

C₄×D₁₉₉

—

C₂³×C₁₉₉

C₂³

yes

—

Q₈×C₁₉₉

—

D₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₂³

yes

characteristic

C₂×C₄×C₁₉₉

C₂³

yes

characteristic

C₂×C₄×C₁₉₉

C₂³

yes

characteristic

C₂×C₄×C₁₉₉

C₂³

yes

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

C₂⁴

yes

characteristic

C₂×D₁₉₉

—

C₂×D₁₉₉

—

C₂×D₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

Dic₁₉₉

—

Dic₁₉₉

—

Dic₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₂×C₁₉₉

2₋¹⁺⁴

yes

characteristic

D₁₉₉

—

C₁₉₉

#562

yes

characteristic, complemented, Sylow

#562

—

199

maximal, Sylow

#125

—

199

—

#124

—

199

—

#124

D₁₉₉

yes

characteristic, complemented

#123

—

199

—

#123

—

199

—

C₂²⋊Q₈

—

199

—

C₂²⋊Q₈

—

199

—

#121

—

199

—

#121

—

199

—

C₂×Q₈

—

199

—

C₂×Q₈

C₂×D₁₉₉

yes

characteristic

C₂×D₄

—

199

—

C₂×D₄

—

199

—

C₂×D₄

—

199

—

C₂×D₄

C₂×D₁₉₉

yes

characteristic

C₂²×C₄

—

199

—

C₂²×C₄

—

199

—

C₄⋊C₄

—

199

—

C₄⋊C₄

—

199

—

C₄⋊C₄

—

199

—

C₄⋊C₄

—

199

—

C₄⋊C₄

—

199

—

C₄⋊C₄

—

199

—

C₂²⋊C₄

—

199

—

C₂²⋊C₄

—

199

—

C₂²⋊C₄

—

199

—

C₂²⋊C₄

—

199

—

C₂²⋊C₄

C₂×D₁₉₉

yes

—

C₂²⋊C₄

C₂×D₁₉₉

yes

—

C₄²

C₂×D₁₉₉

yes

characteristic

C₂³

—

199

—

100

C₂³

C₂²×D₁₉₉

yes

—

101

Q₈

—

102

Q₈

—

796

—

103

D₄

—

104

D₄

—

796

—

105

D₄

—

796

—

106

D₄

—

796

—

107

C₂×C₄

—

796

—

108

C₂×C₄

—

796

—

109

C₂×C₄

—

199

—

110

C₂×C₄

—

199

—

111

C₂×C₄

—

199

—

112

C₂×C₄

C₂²×D₁₉₉

yes

characteristic

113

C₂×C₄

C₂²×D₁₉₉

yes

characteristic

114

C₂×C₄

C₂²×D₁₉₉

yes

characteristic

115

C₂×C₄

C₂²×D₁₉₉

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

398

—

120

C₂²

—

398

—

121

C₂²

—

398

—

122

C₂²

C₂³×D₁₉₉

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

398

—

128

C₄

—

398

—

129

C₄

—

398

—

130

C₂

—

131

C₂

—

796

—

132

C₂

#20264000

yes

characteristic

133

C₂

#20264001

yes

characteristic

134

C₂

#20264002

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₉₉
commutator, socle, cosocle	C₂²×C₁₉₉	C₂⁴
Fitting	#124×C₁₉₉	C₂
hypercenter	#124	D₁₉₉
hypocenter	C₁₉₉	#562
LCS	G ▷ C₂²×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₉₉

commutator, socle, cosocle

C₂²×C₁₉₉

C₂⁴

Fitting

#124×C₁₉₉

C₂

D₁₉₉

C₁₉₉

G ▷ C₂²×C₁₉₉ ▷ C₁₉₉

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. #20263947	C₂	semidirect
2. #20263943	C₂	semidirect
3. #20263939	C₂	semidirect
4. #20263937	C₂	semidirect
5. #20263932	C₂	semidirect
6. #20263931	C₂	semidirect
7. #20263928	C₂	semidirect
8. #20263927	C₂	semidirect
9. #124×C₁₉₉	C₂	semidirect
67. C₁₉₉	#562	semidirect
71. #124	D₁₉₉	semidirect

H ≅ G/N

type

1. #20263947

C₂

semidirect

2. #20263943

C₂

semidirect

3. #20263939

C₂

semidirect

4. #20263937

C₂

semidirect

5. #20263932

C₂

semidirect

6. #20263931

C₂

semidirect

7. #20263928

C₂

semidirect

8. #20263927

C₂

semidirect

9. #124×C₁₉₉

C₂

semidirect

67. C₁₉₉

#562

semidirect

71. #124

D₁₉₉

semidirect

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

subgroup	index
#20263947	2
#20263943	2
#20263939	2
#20263937	2
#20263932	2
#20263931	2
#20263928	2
#20263927	2
#124×C₁₉₉	2
#562	199

subgroup

index

#124×C₁₉₉

199

Subgroups — #27164762

Conjugacy classes of subgroups

Complements

Maximal subgroups