Group #38770008 — subgroups

Subgroups — #38770008

Order 25472.

246 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 71 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	#27164026	C₂	1	2	yes	maximal
2	#27163923	C₂	1	1	yes	maximal, characteristic
3	#27163787	C₂	1	1	yes	maximal, characteristic
4	#27163777	C₂	1	2	yes	maximal
5	#544×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₉₈	C₂²	1	1	yes	characteristic
7	C₂×#15691870	C₂²	1	2	yes	—
8	C₂×C₄×Dic₁₉₉	C₂²	1	1	yes	characteristic
9	#20263878	—	2	2	no	—
10	#20263878	—	2	2	no	—
11	C₁₉₉⋊#95	—	2	2	no	—
12	C₄⋊Q₈×C₁₉₉	—	2	2	no	—
13	#125×C₁₉₉	—	2	2	no	—
14	C₂²⋊Q₈×C₁₉₉	—	2	2	no	—
15	#117×C₁₉₉	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₁₉₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₃₉₈	—	2	2	no	—
18	C₂×Dic₃₉₈	—	2	1	no	—
19	C₂×Dic₃₉₈	—	2	2	no	—
20	C₂×Dic₃₉₈	Q₈	1	2	yes	complemented
21	C₂²×Dic₁₉₉	—	2	1	no	—
22	C₂²×Dic₁₉₉	—	2	1	no	—
23	#15691870	—	2	2	no	—
24	#15691870	—	2	2	no	—
25	C₂×C₁₉₉⋊C₈	—	2	2	no	—
26	C₄×Dic₁₉₉	—	2	2	no	—
27	C₂×Q₈×C₁₉₉	—	2	2	no	—
28	C₂²×C₄×C₁₉₉	D₄	1	2	yes	—
29	C₂²×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₁₉₉	—	2	2	no	—
31	C₄⋊C₄×C₁₉₉	—	2	4	no	—
32	C₄⋊C₄×C₁₉₉	—	2	2	no	—
33	C₄⋊C₄×C₁₉₉	D₄	1	2	yes	—
34	C₄⋊C₄×C₁₉₉	D₄	1	2	yes	—
35	C₂²⋊C₄×C₁₉₉	—	2	2	no	—
36	C₄²×C₁₉₉	—	2	2	no	—
37	Dic₃₉₈	—	2	4	no	—
38	Dic₃₉₈	—	4	2	no	—
39	C₂×Dic₁₉₉	—	2	2	no	—
40	C₂×Dic₁₉₉	—	2	2	no	—
41	C₂×Dic₁₉₉	—	4	1	no	—
42	C₂×Dic₁₉₉	—	4	1	no	—
43	C₂×Dic₁₉₉	—	4	1	no	—
44	C₂×Dic₁₉₉	—	4	1	no	—
45	C₁₉₉⋊C₈	—	4	2	no	—
46	C₂³×C₁₉₉	C₂×D₄	1	1	yes	characteristic
47	Q₈×C₁₉₉	—	8	2	no	—
48	C₂×C₄×C₁₉₉	—	2	2	no	—
49	C₂×C₄×C₁₉₉	—	2	2	no	—
50	C₂×C₄×C₁₉₉	—	2	2	no	—
51	C₂×C₄×C₁₉₉	—	4	2	no	—
52	C₂×C₄×C₁₉₉	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄×C₁₉₉	#41	1	2	yes	—
55	C₂×C₄×C₁₉₉	#41	1	1	yes	characteristic
56	C₂²×C₁₉₉	—	2	1	no	—
57	C₂²×C₁₉₉	—	2	1	no	—
58	C₂²×C₁₉₉	#120	1	1	yes	characteristic
59	C₂²×C₁₉₉	C₂²⋊Q₈	1	1	yes	characteristic
60	C₂²×C₁₉₉	#124	1	1	yes	characteristic
61	C₄×C₁₉₉	—	4	2	no	—
62	C₄×C₁₉₉	—	4	2	no	—
63	C₄×C₁₉₉	—	4	2	no	—
64	C₄×C₁₉₉	C₂²⋊Q₈	1	2	yes	—
65	C₄×C₁₉₉	#124	1	2	yes	—
66	Dic₁₉₉	—	4	2	no	—
67	Dic₁₉₉	—	4	2	no	—
68	C₂×C₁₉₉	—	2	1	no	—
69	C₂×C₁₉₉	—	2	1	no	—
70	C₂×C₁₉₉	#393	1	1	yes	characteristic
71	C₂×C₁₉₉	#455	1	1	yes	characteristic
72	C₂×C₁₉₉	#456	1	1	yes	characteristic
73	C₁₉₉	#2028	1	1	yes	characteristic, complemented, Sylow
74	#2028	—	199	1	no	maximal, Sylow
75	#544	D₁₉₉	1	1	yes	characteristic
76	#419	—	199	2	no	—
77	#391	—	199	1	no	—
78	#343	—	199	1	no	—
79	#339	—	199	2	no	—
80	C₂²×Q₈	—	199	1	no	—
81	C₂×M₁₆	—	199	2	no	—
82	C₄⋊Q₈	—	2	2	no	—
83	#125	—	2	2	no	—
84	C₂²⋊Q₈	—	2	2	no	—
85	#117	C₂×D₁₉₉	1	2	yes	—
86	C₂×C₄⋊C₄	C₂×D₁₉₉	1	1	yes	characteristic
87	C₂×C₄²	—	199	1	no	—
88	#103	—	398	2	no	—
89	#103	—	398	2	no	—
90	#95	—	398	2	no	—
91	C₂×Q₈	—	2	2	no	—
92	C₂×Q₈	—	398	2	no	—
93	C₂×Q₈	—	398	2	no	—
94	C₂×Q₈	—	398	1	no	—
95	C₂×Q₈	—	199	2	no	—
96	C₂²×C₄	—	398	1	no	—
97	C₂²×C₄	—	398	1	no	—
98	C₂²×C₄	C₂²⋊D₁₉₉	1	2	yes	—
99	C₂²×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
100	M₁₆	—	398	2	no	—
101	M₁₆	—	398	2	no	—
102	C₂×C₈	—	398	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	2	4	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	C₂²⋊D₁₉₉	1	2	yes	—
107	C₄⋊C₄	C₂²⋊D₁₉₉	1	2	yes	—
108	C₂²⋊C₄	—	2	2	no	—
109	C₄²	—	2	2	no	—
110	C₄²	—	398	2	no	—
111	C₂³	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
112	Q₈	—	8	2	no	—
113	Q₈	—	796	2	no	—
114	Q₈	—	398	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	796	1	no	—
120	C₂×C₄	—	796	1	no	—
121	C₂×C₄	—	796	1	no	—
122	C₂×C₄	—	796	1	no	—
123	C₂×C₄	—	398	2	no	—
124	C₂×C₄	—	398	2	no	—
125	C₂×C₄	C₂×C₂²⋊D₁₉₉	1	2	yes	—
126	C₂×C₄	Q₈×D₁₉₉	1	1	yes	characteristic
127	C₂×C₄	#15691887	1	1	yes	characteristic
128	C₂×C₄	#15691887	1	2	yes	—
129	C₈	—	796	2	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	#20263924	1	1	yes	characteristic
133	C₂²	#20263931	1	1	yes	characteristic
134	C₂²	#20263944	1	1	yes	characteristic
135	C₄	—	4	2	no	—
136	C₄	—	4	2	no	—
137	C₄	—	4	2	no	—
138	C₄	—	796	2	no	—
139	C₄	—	796	2	no	—
140	C₄	#20263931	1	2	yes	—
141	C₄	#20263944	1	2	yes	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#27163929	1	1	yes	characteristic
145	C₂	#27164196	1	1	yes	characteristic
146	C₂	#27164201	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27164026

C₂

yes

maximal

#27163923

C₂

yes

maximal, characteristic

#27163787

C₂

yes

maximal, characteristic

#27163777

C₂

yes

maximal

#544×C₁₉₉

C₂

yes

maximal, characteristic

C₂²×Dic₃₉₈

C₂²

yes

characteristic

C₂×#15691870

C₂²

yes

—

C₂×C₄×Dic₁₉₉

C₂²

yes

characteristic

#20263878

—

#20263878

—

C₁₉₉⋊#95

—

C₄⋊Q₈×C₁₉₉

—

#125×C₁₉₉

—

C₂²⋊Q₈×C₁₉₉

—

#117×C₁₉₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₉₉

C₂²

yes

characteristic

C₂×Dic₃₉₈

—

C₂×Dic₃₉₈

—

C₂×Dic₃₉₈

—

C₂×Dic₃₉₈

Q₈

yes

complemented

C₂²×Dic₁₉₉

—

C₂²×Dic₁₉₉

—

#15691870

—

#15691870

—

C₂×C₁₉₉⋊C₈

—

C₄×Dic₁₉₉

—

C₂×Q₈×C₁₉₉

—

C₂²×C₄×C₁₉₉

D₄

yes

—

C₂²×C₄×C₁₉₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

D₄

yes

—

C₄⋊C₄×C₁₉₉

D₄

yes

—

C₂²⋊C₄×C₁₉₉

—

C₄²×C₁₉₉

—

Dic₃₉₈

—

Dic₃₉₈

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₁₉₉⋊C₈

—

C₂³×C₁₉₉

C₂×D₄

yes

characteristic

Q₈×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

—

C₂×C₄×C₁₉₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₉

#41

yes

—

C₂×C₄×C₁₉₉

#41

yes

characteristic

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

#120

yes

characteristic

C₂²×C₁₉₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₉₉

#124

yes

characteristic

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

C₂²⋊Q₈

yes

—

C₄×C₁₉₉

#124

yes

—

Dic₁₉₉

—

Dic₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₁₉₉

yes

characteristic, complemented, Sylow

#2028

—

199

maximal, Sylow

#544

D₁₉₉

yes

characteristic

#419

—

199

—

#391

—

199

—

#343

—

199

—

#339

—

199

—

C₂²×Q₈

—

199

—

C₂×M₁₆

—

199

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

#117

C₂×D₁₉₉

yes

—

C₂×C₄⋊C₄

C₂×D₁₉₉

yes

characteristic

C₂×C₄²

—

199

—

#103

—

398

—

#103

—

398

—

#95

—

398

—

C₂×Q₈

—

C₂×Q₈

—

398

—

C₂×Q₈

—

398

—

C₂×Q₈

—

398

—

C₂×Q₈

—

199

—

C₂²×C₄

—

398

—

C₂²×C₄

—

398

—

C₂²×C₄

C₂²⋊D₁₉₉

yes

—

C₂²×C₄

C₂²×D₁₉₉

yes

characteristic

100

M₁₆

—

398

—

101

M₁₆

—

398

—

102

C₂×C₈

—

398

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

C₂²⋊D₁₉₉

yes

—

107

C₄⋊C₄

C₂²⋊D₁₉₉

yes

—

108

C₂²⋊C₄

—

109

C₄²

—

110

C₄²

—

398

—

111

C₂³

C₂×C₂²⋊D₁₉₉

yes

characteristic

112

Q₈

—

113

Q₈

—

796

—

114

Q₈

—

398

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

796

—

120

C₂×C₄

—

796

—

121

C₂×C₄

—

796

—

122

C₂×C₄

—

796

—

123

C₂×C₄

—

398

—

124

C₂×C₄

—

398

—

125

C₂×C₄

C₂×C₂²⋊D₁₉₉

yes

—

126

C₂×C₄

Q₈×D₁₉₉

yes

characteristic

127

C₂×C₄

#15691887

yes

characteristic

128

C₂×C₄

#15691887

yes

—

129

C₈

—

796

—

130

C₂²

—

131

C₂²

—

132

C₂²

#20263924

yes

characteristic

133

C₂²

#20263931

yes

characteristic

134

C₂²

#20263944

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

796

—

139

C₄

—

796

—

140

C₄

#20263931

yes

—

141

C₄

#20263944

yes

—

142

C₂

—

143

C₂

—

144

C₂

#27163929

yes

characteristic

145

C₂

#27164196

yes

characteristic

146

C₂

#27164201

yes

characteristic

	subgroup	G/H
center	C₂²	#20263924
commutator, cosocle	C₂²×C₄×C₁₉₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₉₉
Fitting	#544×C₁₉₉	C₂
socle	C₂²×C₁₉₉	#120
prime radical	C₂³×C₁₉₉	C₂×D₄
hypercenter	#544	D₁₉₉
hypocenter	C₁₉₉	#2028
LCS	G ▷ C₂²×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #544

subgroup

G/H

center

C₂²

#20263924

commutator, cosocle

C₂²×C₄×C₁₉₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₉₉

Fitting

#544×C₁₉₉

C₂

socle

C₂²×C₁₉₉

#120

prime radical

C₂³×C₁₉₉

C₂×D₄

D₁₉₉

C₁₉₉

G ▷ C₂²×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #544

N	H ≅ G/N	type
20. C₂×Dic₃₉₈	Q₈	semidirect
73. C₁₉₉	#2028	semidirect

H ≅ G/N

type

20. C₂×Dic₃₉₈

Q₈

semidirect

73. C₁₉₉

#2028

semidirect

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

subgroup	index
#27164026	2
#27163923	2
#27163787	2
#27163777	2
#544×C₁₉₉	2
#2028	199

subgroup

index

#544×C₁₉₉

199

Subgroups — #38770008

Conjugacy classes of subgroups

Complements

Maximal subgroups