Group #38769241 — subgroups

Subgroups — #38769241

Order 25472.

324 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 89 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×#20263953	C₂	1	1	yes	maximal, characteristic
2	#27164496	C₂	1	1	yes	maximal, characteristic
3	#27164028	C₂	1	2	yes	maximal
4	#27163779	C₂	1	2	yes	maximal
5	C₂×#101×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₉₈	C₂²	1	1	yes	characteristic
7	#20263953	—	2	2	no	—
8	#20263953	C₄	1	4	yes	complemented
9	Q₈×Dic₁₉₉	—	2	4	no	—
10	#20263914	—	2	2	no	—
11	#20263914	C₂²	1	2	yes	—
12	C₂×Dic₁₉₉⋊C₄	—	2	2	no	—
13	C₂×C₄×Dic₁₉₉	C₂²	1	1	yes	characteristic
14	#20263880	—	2	4	no	—
15	C₂²×Q₈×C₁₉₉	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₁₉₉	C₂²	1	2	yes	—
17	#101×C₁₉₉	—	2	2	no	—
18	C₂×Dic₃₉₈	—	2	1	no	—
19	C₂×Dic₃₉₈	—	2	2	no	—
20	C₂×Dic₃₉₈	—	2	2	no	—
21	C₂×Dic₃₉₈	C₂×C₄	1	2	yes	—
22	C₂²×Dic₁₉₉	—	2	1	no	—
23	C₂²×Dic₁₉₉	—	2	1	no	—
24	C₄⋊Dic₁₉₉	—	2	4	no	—
25	C₄⋊Dic₁₉₉	D₄	1	4	yes	—
26	Dic₁₉₉⋊C₄	—	2	8	no	—
27	C₂²⋊Dic₁₉₉	—	2	2	no	—
28	C₄×Dic₁₉₉	—	2	4	no	—
29	C₄×Dic₁₉₉	—	2	1	no	—
30	C₄×Dic₁₉₉	—	2	2	no	—
31	C₄×Dic₁₉₉	C₂×C₄	1	2	yes	—
32	C₂×Q₈×C₁₉₉	—	2	2	no	—
33	C₂×Q₈×C₁₉₉	—	2	2	no	—
34	C₂×Q₈×C₁₉₉	C₂×C₄	1	2	yes	—
35	C₂×Q₈×C₁₉₉	D₄	1	2	yes	—
36	C₂²×C₄×C₁₉₉	D₄	1	2	yes	—
37	C₂²×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
38	M₁₆×C₁₉₉	—	2	2	no	—
39	M₁₆×C₁₉₉	—	2	2	no	—
40	C₂×C₈×C₁₉₉	—	2	2	no	—
41	Dic₃₉₈	—	2	4	no	—
42	Dic₃₉₈	—	4	2	no	—
43	C₂×Dic₁₉₉	—	2	4	no	—
44	C₂×Dic₁₉₉	—	2	2	no	—
45	C₂×Dic₁₉₉	—	2	2	no	—
46	C₂×Dic₁₉₉	—	2	4	no	—
47	C₂×Dic₁₉₉	—	4	1	no	—
48	C₂×Dic₁₉₉	—	4	1	no	—
49	C₂³×C₁₉₉	C₂×D₄	1	1	yes	characteristic
50	Q₈×C₁₉₉	—	2	4	no	—
51	Q₈×C₁₉₉	—	2	4	no	—
52	C₂×C₄×C₁₉₉	—	2	2	no	—
53	C₂×C₄×C₁₉₉	—	2	2	no	—
54	C₂×C₄×C₁₉₉	C₂²⋊C₄	1	4	yes	—
55	C₂×C₄×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₉₉	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₁₉₉	#41	1	2	yes	—
59	C₈×C₁₉₉	—	4	2	no	—
60	C₂²×C₁₉₉	—	2	2	no	—
61	C₂²×C₁₉₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₁₉₉	#120	1	1	yes	characteristic
63	C₂²×C₁₉₉	#123	1	1	yes	characteristic
64	C₄×C₁₉₉	—	2	4	no	—
65	C₄×C₁₉₉	C₄×D₄	1	2	yes	—
66	C₄×C₁₉₉	#121	1	2	yes	—
67	Dic₁₉₉	—	2	4	no	—
68	Dic₁₉₉	—	4	2	no	—
69	Dic₁₉₉	—	4	4	no	—
70	C₂×C₁₉₉	—	2	1	no	—
71	C₂×C₁₉₉	—	2	1	no	—
72	C₂×C₁₉₉	#386	1	1	yes	characteristic
73	C₂×C₁₉₉	#456	1	2	yes	—
74	C₁₉₉	#1886	1	1	yes	characteristic, complemented, Sylow
75	#1886	—	199	1	no	maximal, Sylow
76	C₂×C₄⋊Q₈	—	199	1	no	—
77	#519	—	199	1	no	—
78	#419	—	199	2	no	—
79	C₂×#101	D₁₉₉	1	1	yes	characteristic
80	#339	—	199	2	no	—
81	C₂²×Q₈	—	199	1	no	—
82	C₂²×Q₈	C₂×D₁₉₉	1	1	yes	characteristic
83	C₂×M₁₆	C₂×D₁₉₉	1	2	yes	—
84	C₄⋊Q₈	—	398	2	no	—
85	C₄⋊Q₈	—	199	4	no	—
86	C₄×Q₈	—	398	4	no	—
87	#117	—	398	2	no	—
88	#117	—	199	2	no	—
89	C₂×C₄⋊C₄	—	398	2	no	—
90	C₂×C₄²	—	199	1	no	—
91	#103	—	398	4	no	—
92	#101	—	2	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	—	2	2	no	—
95	C₂×Q₈	—	398	2	no	—
96	C₂×Q₈	—	398	2	no	—
97	C₂×Q₈	—	398	1	no	—
98	C₂×Q₈	—	199	2	no	—
99	C₂×Q₈	C₄×D₁₉₉	1	2	yes	—
100	C₂×Q₈	C₂²⋊D₁₉₉	1	2	yes	—
101	C₂²×C₄	—	398	1	no	—
102	C₂²×C₄	—	398	1	no	—
103	C₂²×C₄	D₇₉₆	1	2	yes	—
104	C₂²×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
105	M₁₆	—	2	2	no	—
106	M₁₆	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₄⋊C₄	—	398	4	no	—
109	C₄⋊C₄	—	398	8	no	—
110	C₄⋊C₄	—	199	4	no	—
111	C₂²⋊C₄	—	398	2	no	—
112	C₄²	—	398	4	no	—
113	C₄²	—	398	2	no	—
114	C₄²	—	398	1	no	—
115	C₄²	—	199	2	no	—
116	C₂³	C₂×D₇₉₆	1	1	yes	characteristic
117	Q₈	—	2	4	no	—
118	Q₈	—	2	4	no	—
119	Q₈	—	796	2	no	—
120	Q₈	—	398	4	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	796	1	no	—
124	C₂×C₄	—	796	1	no	—
125	C₂×C₄	—	398	2	no	—
126	C₂×C₄	—	398	4	no	—
127	C₂×C₄	—	398	4	no	—
128	C₂×C₄	—	398	2	no	—
129	C₂×C₄	(C₂×D₁₉₉)⋊C₄	1	4	yes	—
130	C₂×C₄	C₂×C₄×D₁₉₉	1	1	yes	characteristic
131	C₂×C₄	D₄×D₁₉₉	1	2	yes	—
132	C₂×C₄	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
133	C₂×C₄	#15691885	1	2	yes	—
134	C₈	—	4	2	no	—
135	C₂²	—	2	2	no	—
136	C₂²	C₂×(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
137	C₂²	#20263925	1	1	yes	characteristic
138	C₂²	#20263939	1	1	yes	characteristic
139	C₄	—	2	4	no	—
140	C₄	—	796	4	no	—
141	C₄	—	796	2	no	—
142	C₄	—	398	4	no	—
143	C₄	(C₂²⋊D₁₉₉)⋊C₄	1	2	yes	—
144	C₄	#20263927	1	2	yes	—
145	C₂	—	2	1	no	—
146	C₂	—	2	1	no	—
147	C₂	#27163895	1	1	yes	characteristic
148	C₂	#27164203	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20263953

C₂

yes

maximal, characteristic

#27164496

C₂

yes

maximal, characteristic

#27164028

C₂

yes

maximal

#27163779

C₂

yes

maximal

C₂×#101×C₁₉₉

C₂

yes

maximal, characteristic

C₂²×Dic₃₉₈

C₂²

yes

characteristic

#20263953

—

#20263953

C₄

yes

complemented

Q₈×Dic₁₉₉

—

#20263914

—

#20263914

C₂²

yes

—

C₂×Dic₁₉₉⋊C₄

—

C₂×C₄×Dic₁₉₉

C₂²

yes

characteristic

#20263880

—

C₂²×Q₈×C₁₉₉

C₂²

yes

characteristic

C₂×M₁₆×C₁₉₉

C₂²

yes

—

#101×C₁₉₉

—

C₂×Dic₃₉₈

—

C₂×Dic₃₉₈

—

C₂×Dic₃₉₈

—

C₂×Dic₃₉₈

C₂×C₄

yes

—

C₂²×Dic₁₉₉

—

C₂²×Dic₁₉₉

—

C₄⋊Dic₁₉₉

—

C₄⋊Dic₁₉₉

D₄

yes

—

Dic₁₉₉⋊C₄

—

C₂²⋊Dic₁₉₉

—

C₄×Dic₁₉₉

—

C₄×Dic₁₉₉

—

C₄×Dic₁₉₉

—

C₄×Dic₁₉₉

C₂×C₄

yes

—

C₂×Q₈×C₁₉₉

—

C₂×Q₈×C₁₉₉

—

C₂×Q₈×C₁₉₉

C₂×C₄

yes

—

C₂×Q₈×C₁₉₉

D₄

yes

—

C₂²×C₄×C₁₉₉

D₄

yes

—

C₂²×C₄×C₁₉₉

C₂³

yes

characteristic

M₁₆×C₁₉₉

—

M₁₆×C₁₉₉

—

C₂×C₈×C₁₉₉

—

Dic₃₉₈

—

Dic₃₉₈

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂³×C₁₉₉

C₂×D₄

yes

characteristic

Q₈×C₁₉₉

—

Q₈×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×D₄

yes

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

#41

yes

—

C₈×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

C₂×C₂²⋊C₄

yes

C₂²×C₁₉₉

yes

C₂²×C₁₉₉

yes

C₄×C₁₉₉

—

C₄×C₁₉₉

C₄×D₄

yes

—

C₄×C₁₉₉

#121

yes

—

Dic₁₉₉

—

Dic₁₉₉

—

Dic₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

#386

yes

characteristic

C₂×C₁₉₉

#456

yes

—

C₁₉₉

#1886

yes

characteristic, complemented, Sylow

#1886

—

199

maximal, Sylow

C₂×C₄⋊Q₈

—

199

—

#519

—

199

—

#419

—

199

—

C₂×#101

D₁₉₉

yes

characteristic

#339

—

199

—

C₂²×Q₈

—

199

—

C₂²×Q₈

C₂×D₁₉₉

yes

characteristic

C₂×M₁₆

C₂×D₁₉₉

yes

—

C₄⋊Q₈

—

398

—

C₄⋊Q₈

—

199

—

C₄×Q₈

—

398

—

#117

—

398

—

#117

—

199

—

C₂×C₄⋊C₄

—

398

—

C₂×C₄²

—

199

—

#103

—

398

—

#101

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

398

—

C₂×Q₈

—

398

—

C₂×Q₈

—

398

—

C₂×Q₈

—

199

—

C₂×Q₈

C₄×D₁₉₉

yes

—

100

C₂×Q₈

C₂²⋊D₁₉₉

yes

—

101

C₂²×C₄

—

398

—

102

C₂²×C₄

—

398

—

103

C₂²×C₄

D₇₉₆

yes

—

104

C₂²×C₄

C₂²×D₁₉₉

yes

characteristic

105

M₁₆

—

106

M₁₆

—

107

C₂×C₈

—

108

C₄⋊C₄

—

398

—

109

C₄⋊C₄

—

398

—

110

C₄⋊C₄

—

199

—

111

C₂²⋊C₄

—

398

—

112

C₄²

—

398

—

113

C₄²

—

398

—

114

C₄²

—

398

—

115

C₄²

—

199

—

116

C₂³

C₂×D₇₉₆

yes

characteristic

117

Q₈

—

118

Q₈

—

119

Q₈

—

796

—

120

Q₈

—

398

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

796

—

124

C₂×C₄

—

796

—

125

C₂×C₄

—

398

—

126

C₂×C₄

—

398

—

127

C₂×C₄

—

398

—

128

C₂×C₄

—

398

—

129

C₂×C₄

(C₂×D₁₉₉)⋊C₄

yes

—

130

C₂×C₄

C₂×C₄×D₁₉₉

yes

characteristic

131

C₂×C₄

D₄×D₁₉₉

yes

—

132

C₂×C₄

C₂×C₂²⋊D₁₉₉

yes

characteristic

133

C₂×C₄

#15691885

yes

—

134

C₈

—

135

C₂²

—

136

C₂²

C₂×(C₂×D₁₉₉)⋊C₄

yes

characteristic

137

C₂²

#20263925

yes

characteristic

138

C₂²

#20263939

yes

characteristic

139

C₄

—

140

C₄

—

796

—

141

C₄

—

796

—

142

C₄

—

398

—

143

C₄

(C₂²⋊D₁₉₉)⋊C₄

yes

—

144

C₄

#20263927

yes

—

145

C₂

—

146

C₂

—

147

C₂

#27163895

yes

characteristic

148

C₂

#27164203

yes

—

	subgroup	G/H
center	C₂²	#20263925
commutator	C₂×C₄×C₁₉₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₉
Fitting	C₂×#101×C₁₉₉	C₂
socle	C₂²×C₁₉₉	#120
prime radical	C₂³×C₁₉₉	C₂×D₄
cosocle	C₂²×C₄×C₁₉₉	C₂³
hypercenter	C₂×#101	D₁₉₉
hypocenter	C₁₉₉	#1886
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

subgroup

G/H

center

C₂²

#20263925

commutator

C₂×C₄×C₁₉₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₉

Fitting

C₂×#101×C₁₉₉

C₂

socle

C₂²×C₁₉₉

#120

prime radical

C₂³×C₁₉₉

C₂×D₄

cosocle

C₂²×C₄×C₁₉₉

C₂³

D₁₉₉

C₁₉₉

G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

N	H ≅ G/N	type
8. #20263953	C₄	semidirect
74. C₁₉₉	#1886	semidirect

H ≅ G/N

type

8. #20263953

C₄

semidirect

74. C₁₉₉

#1886

semidirect

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

subgroup	index
C₂×#20263953	2
#27164496	2
#27164028	2
#27163779	2
C₂×#101×C₁₉₉	2
#1886	199

subgroup

index

C₂×#101×C₁₉₉

199

Subgroups — #38769241

Conjugacy classes of subgroups

Complements

Maximal subgroups