Group #38768919 — subgroups

Subgroups — #38768919

Order 25472.

292 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 101 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×#20263914	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊(C₁₉₉⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#27163772	C₂	1	4	yes	maximal, complemented
4	C₂×C₂²⋊C₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₉₉⋊C₈	C₂²	1	2	yes	—
6	#20263914	—	2	2	no	—
7	#20263914	—	2	2	no	—
8	C₂×C₄⋊Dic₁₉₉	C₄	1	1	yes	characteristic
9	C₂×C₄⋊Dic₁₉₉	C₄	1	1	yes	characteristic
10	C₂×C₂²⋊Dic₁₉₉	C₄	1	2	yes	—
11	C₂×C₄×Dic₁₉₉	C₂²	1	2	yes	—
12	C₂²⋊(C₁₉₉⋊C₈)	—	2	2	no	—
13	C₂³×C₄×C₁₉₉	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₉₉	C₂²	1	2	yes	—
15	C₂²⋊C₈×C₁₉₉	—	2	2	no	—
16	C₂²×Dic₁₉₉	C₂×C₄	1	2	yes	—
17	C₂²×Dic₁₉₉	C₂×C₄	1	2	yes	—
18	C₂×C₁₉₉⋊C₈	—	2	2	no	—
19	C₂×C₁₉₉⋊C₈	—	2	4	no	—
20	C₄⋊Dic₁₉₉	—	2	2	no	—
21	C₄⋊Dic₁₉₉	—	2	2	no	—
22	C₂²⋊Dic₁₉₉	—	2	4	no	—
23	C₄×Dic₁₉₉	—	2	2	no	—
24	C₄×Dic₁₉₉	—	2	2	no	—
25	C₂⁴×C₁₉₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₉₉	—	2	4	no	—
27	C₂²×C₄×C₁₉₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₁₉₉	D₄	1	2	yes	—
29	C₂²×C₄×C₁₉₉	D₄	1	2	yes	—
30	C₂²×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₁₉₉	—	2	2	no	—
32	C₂×C₈×C₁₉₉	—	2	4	no	—
33	C₂×Dic₁₉₉	—	2	2	no	—
34	C₂×Dic₁₉₉	—	2	2	no	—
35	C₂×Dic₁₉₉	—	2	4	no	—
36	C₂×Dic₁₉₉	—	2	4	no	—
37	C₁₉₉⋊C₈	—	4	2	no	—
38	C₂³×C₁₉₉	—	2	2	no	—
39	C₂³×C₁₉₉	—	2	2	no	—
40	C₂³×C₁₉₉	—	2	2	no	—
41	C₂³×C₁₉₉	C₂²⋊C₄	1	2	yes	—
42	C₂³×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₉₉	—	2	2	no	—
44	C₂×C₄×C₁₉₉	—	2	4	no	—
45	C₂×C₄×C₁₉₉	—	2	4	no	—
46	C₂×C₄×C₁₉₉	C₂²⋊C₄	1	2	yes	—
47	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₉₉	#41	1	4	yes	—
50	C₈×C₁₉₉	—	4	2	no	—
51	C₂²×C₁₉₉	—	2	2	no	—
52	C₂²×C₁₉₉	—	2	2	no	—
53	C₂²×C₁₉₉	—	2	2	no	—
54	C₂²×C₁₉₉	—	4	4	no	—
55	C₂²×C₁₉₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₁₉₉	#117	1	2	yes	—
57	C₂²×C₁₉₉	#131	1	2	yes	—
58	C₂²×C₁₉₉	#131	1	2	yes	—
59	C₄×C₁₉₉	—	2	2	no	—
60	C₄×C₁₉₉	#123	1	2	yes	—
61	C₄×C₁₉₉	#123	1	2	yes	—
62	Dic₁₉₉	—	4	2	no	—
63	Dic₁₉₉	—	4	2	no	—
64	C₂×C₁₉₉	—	4	2	no	—
65	C₂×C₁₉₉	#381	1	1	yes	characteristic
66	C₂×C₁₉₉	#408	1	1	yes	characteristic
67	C₂×C₁₉₉	#408	1	1	yes	characteristic
68	C₂×C₁₉₉	#433	1	4	yes	—
69	C₁₉₉	#1818	1	1	yes	characteristic, complemented, Sylow
70	#1818	—	199	1	no	maximal, Sylow
71	C₂×#117	—	199	1	no	—
72	C₂×C₂²⋊C₈	—	199	1	no	—
73	C₂×C₂²⋊C₈	D₁₉₉	1	1	yes	characteristic
74	#336	—	199	4	no	—
75	C₂³×C₄	C₂×D₁₉₉	1	1	yes	characteristic
76	C₂²×C₈	—	199	2	no	—
77	C₂²×C₈	C₂×D₁₉₉	1	2	yes	—
78	#117	—	398	2	no	—
79	#117	—	398	2	no	—
80	C₂×C₄⋊C₄	—	199	1	no	—
81	C₂×C₄⋊C₄	—	199	1	no	—
82	C₂×C₂²⋊C₄	—	199	2	no	—
83	C₂×C₄²	—	199	2	no	—
84	C₂²⋊C₈	—	2	2	no	—
85	C₂²⋊C₈	—	398	2	no	—
86	C₂⁴	C₄×D₁₉₉	1	1	yes	characteristic
87	C₂²×C₄	—	2	4	no	—
88	C₂²×C₄	—	199	2	no	—
89	C₂²×C₄	—	199	2	no	—
90	C₂²×C₄	C₄×D₁₉₉	1	1	yes	characteristic
91	C₂²×C₄	C₂²⋊D₁₉₉	1	2	yes	—
92	C₂²×C₄	D₇₉₆	1	2	yes	—
93	C₂²×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	4	no	—
96	C₂×C₈	—	398	2	no	—
97	C₂×C₈	—	398	4	no	—
98	C₄⋊C₄	—	398	2	no	—
99	C₄⋊C₄	—	398	2	no	—
100	C₂²⋊C₄	—	398	4	no	—
101	C₄²	—	398	2	no	—
102	C₄²	—	398	2	no	—
103	C₂³	—	2	2	no	—
104	C₂³	—	2	2	no	—
105	C₂³	—	2	2	no	—
106	C₂³	(C₂×D₁₉₉)⋊C₄	1	2	yes	—
107	C₂³	C₂×C₄×D₁₉₉	1	1	yes	characteristic
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	2	4	no	—
111	C₂×C₄	—	398	4	no	—
112	C₂×C₄	—	398	4	no	—
113	C₂×C₄	—	398	2	no	—
114	C₂×C₄	—	398	2	no	—
115	C₂×C₄	(C₂×D₁₉₉)⋊C₄	1	2	yes	—
116	C₂×C₄	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
117	C₂×C₄	C₂×D₇₉₆	1	1	yes	characteristic
118	C₂×C₄	#15691885	1	4	yes	—
119	C₈	—	4	2	no	—
120	C₈	—	796	2	no	—
121	C₂²	—	2	2	no	—
122	C₂²	—	2	2	no	—
123	C₂²	—	2	2	no	—
124	C₂²	—	4	4	no	—
125	C₂²	C₂×(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
126	C₂²	#20263911	1	2	yes	—
127	C₂²	#20263961	1	2	yes	—
128	C₂²	#20263963	1	2	yes	—
129	C₄	—	2	2	no	—
130	C₄	—	796	2	no	—
131	C₄	—	796	2	no	—
132	C₄	#20263936	1	2	yes	—
133	C₄	#20263939	1	2	yes	—
134	C₂	—	4	2	no	—
135	C₂	#27163879	1	1	yes	characteristic
136	C₂	#27163984	1	1	yes	characteristic
137	C₂	#27163986	1	1	yes	characteristic
138	C₂	#27164076	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20263914

C₂

yes

maximal, characteristic

C₂×C₂²⋊(C₁₉₉⋊C₈)

C₂

yes

maximal, characteristic

#27163772

C₂

yes

maximal, complemented

C₂×C₂²⋊C₈×C₁₉₉

C₂

yes

maximal, characteristic

C₂²×C₁₉₉⋊C₈

C₂²

yes

—

#20263914

—

#20263914

—

C₂×C₄⋊Dic₁₉₉

C₄

yes

characteristic

C₂×C₄⋊Dic₁₉₉

C₄

yes

characteristic

C₂×C₂²⋊Dic₁₉₉

C₄

yes

—

C₂×C₄×Dic₁₉₉

C₂²

yes

—

C₂²⋊(C₁₉₉⋊C₈)

—

C₂³×C₄×C₁₉₉

C₂²

yes

characteristic

C₂²×C₈×C₁₉₉

C₂²

yes

—

C₂²⋊C₈×C₁₉₉

—

C₂²×Dic₁₉₉

C₂×C₄

yes

—

C₂²×Dic₁₉₉

C₂×C₄

yes

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

—

C₄⋊Dic₁₉₉

—

C₄⋊Dic₁₉₉

—

C₂²⋊Dic₁₉₉

—

C₄×Dic₁₉₉

—

C₄×Dic₁₉₉

—

C₂⁴×C₁₉₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₉

—

C₂²×C₄×C₁₉₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₉

D₄

yes

—

C₂²×C₄×C₁₉₉

D₄

yes

—

C₂²×C₄×C₁₉₉

C₂³

yes

characteristic

C₂×C₈×C₁₉₉

—

C₂×C₈×C₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₁₉₉⋊C₈

—

C₂³×C₁₉₉

—

C₂³×C₁₉₉

—

C₂³×C₁₉₉

—

C₂³×C₁₉₉

C₂²⋊C₄

yes

—

C₂³×C₁₉₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

#41

yes

—

C₈×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉₉

#117

yes

—

C₂²×C₁₉₉

#131

yes

—

C₂²×C₁₉₉

#131

yes

—

C₄×C₁₉₉

—

C₄×C₁₉₉

#123

yes

—

C₄×C₁₉₉

#123

yes

—

Dic₁₉₉

—

Dic₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

—

C₁₉₉

#1818

yes

characteristic, complemented, Sylow

#1818

—

199

maximal, Sylow

C₂×#117

—

199

—

C₂×C₂²⋊C₈

—

199

—

C₂×C₂²⋊C₈

D₁₉₉

yes

characteristic

#336

—

199

—

C₂³×C₄

C₂×D₁₉₉

yes

characteristic

C₂²×C₈

—

199

—

C₂²×C₈

C₂×D₁₉₉

yes

—

#117

—

398

—

#117

—

398

—

C₂×C₄⋊C₄

—

199

—

C₂×C₄⋊C₄

—

199

—

C₂×C₂²⋊C₄

—

199

—

C₂×C₄²

—

199

—

C₂²⋊C₈

—

C₂²⋊C₈

—

398

—

C₂⁴

C₄×D₁₉₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

199

—

C₂²×C₄

—

199

—

C₂²×C₄

C₄×D₁₉₉

yes

characteristic

C₂²×C₄

C₂²⋊D₁₉₉

yes

—

C₂²×C₄

D₇₉₆

yes

—

C₂²×C₄

C₂²×D₁₉₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

398

—

C₂×C₈

—

398

—

C₄⋊C₄

—

398

—

C₄⋊C₄

—

398

—

100

C₂²⋊C₄

—

398

—

101

C₄²

—

398

—

102

C₄²

—

398

—

103

C₂³

—

104

C₂³

—

105

C₂³

—

106

C₂³

(C₂×D₁₉₉)⋊C₄

yes

—

107

C₂³

C₂×C₄×D₁₉₉

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

398

—

112

C₂×C₄

—

398

—

113

C₂×C₄

—

398

—

114

C₂×C₄

—

398

—

115

C₂×C₄

(C₂×D₁₉₉)⋊C₄

yes

—

116

C₂×C₄

C₂×C₂²⋊D₁₉₉

yes

characteristic

117

C₂×C₄

C₂×D₇₉₆

yes

characteristic

118

C₂×C₄

#15691885

yes

—

119

C₈

—

120

C₈

—

796

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

C₂×(C₂×D₁₉₉)⋊C₄

yes

characteristic

126

C₂²

#20263911

yes

—

127

C₂²

#20263961

yes

—

128

C₂²

#20263963

yes

—

129

C₄

—

130

C₄

—

796

—

131

C₄

—

796

—

132

C₄

#20263936

yes

—

133

C₄

#20263939

yes

—

134

C₂

—

135

C₂

#27163879

yes

characteristic

136

C₂

#27163984

yes

characteristic

137

C₂

#27163986

yes

characteristic

138

C₂

#27164076

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₉₉
commutator, socle	C₂³×C₁₉₉	C₂²×C₄
Fitting	C₂×C₂²⋊C₈×C₁₉₉	C₂
prime radical	C₂⁴×C₁₉₉	C₂×C₄
cosocle	C₂²×C₄×C₁₉₉	C₂³
hypercenter	C₂×C₂²⋊C₈	D₁₉₉
hypocenter	C₁₉₉	#1818
LCS	G ▷ C₂³×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₉₉

commutator, socle

C₂³×C₁₉₉

C₂²×C₄

Fitting

C₂×C₂²⋊C₈×C₁₉₉

C₂

prime radical

C₂⁴×C₁₉₉

C₂×C₄

cosocle

C₂²×C₄×C₁₉₉

C₂³

hypercenter

C₂×C₂²⋊C₈

D₁₉₉

C₁₉₉

G ▷ C₂³×C₁₉₉ ▷ C₁₉₉

C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
3. #27163772	C₂	semidirect
69. C₁₉₉	#1818	semidirect

H ≅ G/N

type

3. #27163772

C₂

semidirect

69. C₁₉₉

#1818

semidirect

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

subgroup	index
C₂×#20263914	2
C₂×C₂²⋊(C₁₉₉⋊C₈)	2
#27163772	2
C₂×C₂²⋊C₈×C₁₉₉	2
#1818	199

subgroup

index

C₂×#20263914

C₂×C₂²⋊(C₁₉₉⋊C₈)

#27163772

C₂×C₂²⋊C₈×C₁₉₉

#1818

199

Subgroups — #38768919

Conjugacy classes of subgroups

Complements

Maximal subgroups