Group #38768683 — subgroups

Subgroups — #38768683

Order 25472.

324 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 129 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×C₄×Dic₃₉₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#20263878	C₂	1	2	yes	maximal
3	C₂×#20263866	C₂	1	1	yes	maximal, characteristic
4	#27163780	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₄×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₉₈	C₂²	1	1	yes	characteristic
7	C₂²×C₁₉₉⋊C₈	C₂²	1	2	yes	—
8	C₄×Dic₃₉₈	—	2	2	no	—
9	C₄×Dic₃₉₈	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₁₉₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₉₉⋊C₄	—	2	1	no	—
12	C₂×C₄×Dic₁₉₉	—	2	1	no	—
13	#20263878	C₄	1	8	yes	complemented
14	#20263866	—	2	2	no	—
15	C₂×C₄⋊C₄×C₁₉₉	C₂²	1	2	yes	—
16	C₂×C₄²×C₁₉₉	—	2	1	no	—
17	C₂×C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
18	#95×C₁₉₉	—	2	1	no	—
19	C₂×Dic₃₉₈	—	2	1	no	—
20	C₂×Dic₃₉₈	—	2	2	no	—
21	C₂×Dic₃₉₈	C₂×C₄	1	4	yes	—
22	C₂×Dic₃₉₈	C₂×C₄	1	2	yes	—
23	C₂²×Dic₁₉₉	—	2	1	no	—
24	C₂²×Dic₁₉₉	—	2	1	no	—
25	C₂×C₁₉₉⋊C₈	—	2	2	no	—
26	C₂×C₁₉₉⋊C₈	—	2	2	no	—
27	C₂×C₁₉₉⋊C₈	C₂×C₄	1	4	yes	—
28	C₄⋊Dic₁₉₉	—	2	1	no	—
29	C₄⋊Dic₁₉₉	C₂×C₄	1	2	yes	—
30	Dic₁₉₉⋊C₄	—	2	4	no	—
31	C₄×Dic₁₉₉	—	2	4	no	—
32	C₂²×C₄×C₁₉₉	—	2	2	no	—
33	C₂²×C₄×C₁₉₉	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₁₉₉	D₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₁₉₉	—	2	2	no	—
37	C₄⋊C₄×C₁₉₉	C₂×C₄	1	4	yes	—
38	C₄²×C₁₉₉	—	4	2	no	—
39	C₄²×C₁₉₉	C₂×C₄	1	2	yes	—
40	C₄²×C₁₉₉	D₄	1	2	yes	—
41	Dic₃₉₈	—	2	2	no	—
42	Dic₃₉₈	—	2	4	no	—
43	Dic₃₉₈	C₄²	1	4	yes	complemented
44	C₂×Dic₁₉₉	—	2	2	no	—
45	C₂×Dic₁₉₉	—	2	4	no	—
46	C₂×Dic₁₉₉	—	2	2	no	—
47	C₂×Dic₁₉₉	—	4	1	no	—
48	C₂×Dic₁₉₉	—	4	1	no	—
49	C₁₉₉⋊C₈	—	2	4	no	—
50	C₂³×C₁₉₉	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₁₉₉	—	2	1	no	—
52	C₂×C₄×C₁₉₉	—	2	2	no	—
53	C₂×C₄×C₁₉₉	—	2	1	no	—
54	C₂×C₄×C₁₉₉	—	2	4	no	—
55	C₂×C₄×C₁₉₉	—	4	2	no	—
56	C₂×C₄×C₁₉₉	—	4	2	no	—
57	C₂×C₄×C₁₉₉	C₂²⋊C₄	1	2	yes	—
58	C₂×C₄×C₁₉₉	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₉₉	C₂²×C₄	1	2	yes	—
61	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₉₉	#41	1	2	yes	—
63	C₂²×C₁₉₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂²×C₁₉₉	C₄×D₄	1	2	yes	—
65	C₂²×C₁₉₉	AGL(1,ℤ/8)	1	1	yes	characteristic
66	C₂²×C₁₉₉	#137	1	2	yes	—
67	C₂²×C₁₉₉	#137	1	1	yes	characteristic
68	C₄×C₁₉₉	—	2	2	no	—
69	C₄×C₁₉₉	—	2	2	no	—
70	C₄×C₁₉₉	—	4	4	no	—
71	C₄×C₁₉₉	C₂×C₄²	1	1	yes	characteristic
72	C₄×C₁₉₉	#117	1	1	yes	characteristic
73	C₄×C₁₉₉	C₄×D₄	1	2	yes	—
74	Dic₁₉₉	—	2	4	no	—
75	Dic₁₉₉	—	4	2	no	—
76	C₂×C₁₉₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₁₉₉	#417	1	1	yes	characteristic
78	C₂×C₁₉₉	#419	1	1	yes	characteristic
79	C₂×C₁₉₉	#440	1	2	yes	—
80	C₂×C₁₉₉	#441	1	2	yes	—
81	C₁₉₉	#1763	1	1	yes	characteristic, complemented, Sylow
82	#1763	—	199	1	no	maximal, Sylow
83	C₂×C₄×Q₈	—	199	1	no	—
84	C₂×#103	—	199	2	no	—
85	C₂×#97	—	199	1	no	—
86	C₄×C₄⋊C₄	D₁₉₉	1	1	yes	characteristic
87	#340	—	199	2	no	—
88	C₂²×Q₈	—	199	1	no	—
89	C₂²×C₈	—	199	2	no	—
90	C₄×Q₈	—	398	2	no	—
91	C₄×Q₈	—	199	4	no	—
92	C₂×C₄⋊C₄	—	398	1	no	—
93	C₂×C₄⋊C₄	—	199	1	no	—
94	C₂×C₄⋊C₄	C₂×D₁₉₉	1	2	yes	—
95	C₂×C₄²	—	2	1	no	—
96	C₂×C₄²	—	398	1	no	—
97	C₂×C₄²	C₂×D₁₉₉	1	1	yes	characteristic
98	#103	—	199	8	no	—
99	#97	—	398	2	no	—
100	#95	—	2	1	no	—
101	C₂×Q₈	—	398	1	no	—
102	C₂×Q₈	—	398	2	no	—
103	C₂×Q₈	—	199	4	no	—
104	C₂×Q₈	—	199	2	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	—	398	1	no	—
107	C₂²×C₄	—	398	1	no	—
108	C₂²×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
109	C₂²×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
110	C₂²×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
111	C₂×C₈	—	398	2	no	—
112	C₂×C₈	—	398	2	no	—
113	C₂×C₈	—	199	4	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	398	4	no	—
116	C₄⋊C₄	—	398	1	no	—
117	C₄⋊C₄	—	199	2	no	—
118	C₄⋊C₄	C₄×D₁₉₉	1	4	yes	—
119	C₄²	—	4	2	no	—
120	C₄²	—	398	4	no	—
121	C₄²	C₄×D₁₉₉	1	2	yes	—
122	C₄²	D₇₉₆	1	2	yes	—
123	C₂³	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
124	Q₈	—	398	2	no	—
125	Q₈	—	398	4	no	—
126	Q₈	—	199	4	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	4	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	796	1	no	—
134	C₂×C₄	—	796	1	no	—
135	C₂×C₄	—	398	2	no	—
136	C₂×C₄	—	398	2	no	—
137	C₂×C₄	—	398	4	no	—
138	C₂×C₄	(C₂×D₁₉₉)⋊C₄	1	2	yes	—
139	C₂×C₄	(C₂×D₁₉₉)⋊C₄	1	2	yes	—
140	C₂×C₄	C₂×C₄×D₁₉₉	1	1	yes	characteristic
141	C₂×C₄	C₂×C₄×D₁₉₉	1	2	yes	—
142	C₂×C₄	C₂×D₇₉₆	1	1	yes	characteristic
143	C₂×C₄	#15691886	1	2	yes	—
144	C₈	—	398	4	no	—
145	C₂²	C₂×(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
146	C₂²	C₄×C₂²⋊D₁₉₉	1	2	yes	—
147	C₂²	#20263981	1	1	yes	characteristic
148	C₂²	#20263985	1	2	yes	—
149	C₂²	#20263986	1	1	yes	characteristic
150	C₄	—	2	2	no	—
151	C₄	—	2	2	no	—
152	C₄	—	4	4	no	—
153	C₄	—	796	2	no	—
154	C₄	—	398	4	no	—
155	C₄	C₄²×D₁₉₉	1	1	yes	characteristic
156	C₄	#20263912	1	1	yes	characteristic
157	C₄	C₄×D₇₉₆	1	2	yes	—
158	C₂	C₄×(C₂×D₁₉₉)⋊C₄	1	1	yes	characteristic
159	C₂	#27164017	1	1	yes	characteristic
160	C₂	#27164026	1	1	yes	characteristic
161	C₂	#27164118	1	2	yes	—
162	C₂	#27164123	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₃₉₈

C₂

yes

maximal, characteristic

C₂×#20263878

C₂

yes

maximal

C₂×#20263866

C₂

yes

maximal, characteristic

#27163780

C₂

yes

maximal

C₄×C₄⋊C₄×C₁₉₉

C₂

yes

maximal, characteristic

C₂²×Dic₃₉₈

C₂²

yes

characteristic

C₂²×C₁₉₉⋊C₈

C₂²

yes

—

C₄×Dic₃₉₈

—

C₄×Dic₃₉₈

C₄

yes

complemented

C₂×C₄⋊Dic₁₉₉

C₂²

yes

characteristic

C₂×Dic₁₉₉⋊C₄

—

C₂×C₄×Dic₁₉₉

—

#20263878

C₄

yes

complemented

#20263866

—

C₂×C₄⋊C₄×C₁₉₉

C₂²

yes

—

C₂×C₄²×C₁₉₉

—

C₂×C₄²×C₁₉₉

C₂²

yes

characteristic

#95×C₁₉₉

—

C₂×Dic₃₉₈

—

C₂×Dic₃₉₈

—

C₂×Dic₃₉₈

C₂×C₄

yes

—

C₂×Dic₃₉₈

C₂×C₄

yes

—

C₂²×Dic₁₉₉

—

C₂²×Dic₁₉₉

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₁₉₉

—

C₄⋊Dic₁₉₉

C₂×C₄

yes

—

Dic₁₉₉⋊C₄

—

C₄×Dic₁₉₉

—

C₂²×C₄×C₁₉₉

—

C₂²×C₄×C₁₉₉

D₄

yes

characteristic

C₂²×C₄×C₁₉₉

D₄

yes

characteristic

C₂²×C₄×C₁₉₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

C₂×C₄

yes

—

C₄²×C₁₉₉

—

C₄²×C₁₉₉

C₂×C₄

yes

—

C₄²×C₁₉₉

D₄

yes

—

Dic₃₉₈

—

Dic₃₉₈

—

Dic₃₉₈

C₄²

yes

complemented

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₁₉₉⋊C₈

—

C₂³×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂²×C₄

yes

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

#41

yes

—

C₂²×C₁₉₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉₉

C₄×D₄

yes

—

C₂²×C₁₉₉

AGL(1,ℤ/8)

yes

characteristic

C₂²×C₁₉₉

#137

yes

—

C₂²×C₁₉₉

#137

yes

characteristic

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

C₂×C₄²

yes

characteristic

C₄×C₁₉₉

#117

yes

characteristic

C₄×C₁₉₉

C₄×D₄

yes

—

Dic₁₉₉

—

Dic₁₉₉

—

C₂×C₁₉₉

C₄×C₂²⋊C₄

yes

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

—

C₂×C₁₉₉

#441

yes

—

C₁₉₉

#1763

yes

characteristic, complemented, Sylow

#1763

—

199

maximal, Sylow

C₂×C₄×Q₈

—

199

—

C₂×#103

—

199

—

C₂×#97

—

199

—

C₄×C₄⋊C₄

D₁₉₉

yes

characteristic

#340

—

199

—

C₂²×Q₈

—

199

—

C₂²×C₈

—

199

—

C₄×Q₈

—

398

—

C₄×Q₈

—

199

—

C₂×C₄⋊C₄

—

398

—

C₂×C₄⋊C₄

—

199

—

C₂×C₄⋊C₄

C₂×D₁₉₉

yes

—

C₂×C₄²

—

C₂×C₄²

—

398

—

C₂×C₄²

C₂×D₁₉₉

yes

characteristic

#103

—

199

—

#97

—

398

—

100

#95

—

101

C₂×Q₈

—

398

—

102

C₂×Q₈

—

398

—

103

C₂×Q₈

—

199

—

104

C₂×Q₈

—

199

—

105

C₂²×C₄

—

106

C₂²×C₄

—

398

—

107

C₂²×C₄

—

398

—

108

C₂²×C₄

C₂²⋊D₁₉₉

yes

characteristic

109

C₂²×C₄

C₂²⋊D₁₉₉

yes

characteristic

110

C₂²×C₄

C₂²×D₁₉₉

yes

characteristic

111

C₂×C₈

—

398

—

112

C₂×C₈

—

398

—

113

C₂×C₈

—

199

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

398

—

116

C₄⋊C₄

—

398

—

117

C₄⋊C₄

—

199

—

118

C₄⋊C₄

C₄×D₁₉₉

yes

—

119

C₄²

—

120

C₄²

—

398

—

121

C₄²

C₄×D₁₉₉

yes

—

122

C₄²

D₇₉₆

yes

—

123

C₂³

C₂×C₂²⋊D₁₉₉

yes

characteristic

124

Q₈

—

398

—

125

Q₈

—

398

—

126

Q₈

—

199

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

796

—

134

C₂×C₄

—

796

—

135

C₂×C₄

—

398

—

136

C₂×C₄

—

398

—

137

C₂×C₄

—

398

—

138

C₂×C₄

(C₂×D₁₉₉)⋊C₄

yes

—

139

C₂×C₄

(C₂×D₁₉₉)⋊C₄

yes

—

140

C₂×C₄

C₂×C₄×D₁₉₉

yes

characteristic

141

C₂×C₄

C₂×C₄×D₁₉₉

yes

—

142

C₂×C₄

C₂×D₇₉₆

yes

characteristic

143

C₂×C₄

#15691886

yes

—

144

C₈

—

398

—

145

C₂²

C₂×(C₂×D₁₉₉)⋊C₄

yes

characteristic

146

C₂²

C₄×C₂²⋊D₁₉₉

yes

—

147

C₂²

#20263981

yes

characteristic

148

C₂²

#20263985

yes

—

149

C₂²

#20263986

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

796

—

154

C₄

—

398

—

155

C₄

C₄²×D₁₉₉

yes

characteristic

156

C₄

#20263912

yes

characteristic

157

C₄

C₄×D₇₉₆

yes

—

158

C₂

C₄×(C₂×D₁₉₉)⋊C₄

yes

characteristic

159

C₂

#27164017

yes

characteristic

160

C₂

#27164026

yes

characteristic

161

C₂

#27164118

yes

—

162

C₂

#27164123

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₉₉
commutator	C₄×C₁₉₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₁₉₉
Fitting	C₄×C₄⋊C₄×C₁₉₉	C₂
socle, prime radical	C₂³×C₁₉₉	C₂×D₄
cosocle	C₂²×C₄×C₁₉₉	C₂³
hypercenter	C₄×C₄⋊C₄	D₁₉₉
hypocenter	C₁₉₉	#1763
LCS	G ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₉₉

commutator

C₄×C₁₉₉

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₁₉₉

Fitting

C₄×C₄⋊C₄×C₁₉₉

C₂

socle, prime radical

C₂³×C₁₉₉

C₂×D₄

cosocle

C₂²×C₄×C₁₉₉

C₂³

hypercenter

C₄×C₄⋊C₄

D₁₉₉

hypocenter

C₁₉₉

#1763

LCS

G ▷ C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
9. C₄×Dic₃₉₈	C₄	semidirect
13. #20263878	C₄	semidirect
43. Dic₃₉₈	C₄²	semidirect
81. C₁₉₉	#1763	semidirect

H ≅ G/N

type

9. C₄×Dic₃₉₈

C₄

semidirect

13. #20263878

C₄

semidirect

43. Dic₃₉₈

C₄²

semidirect

81. C₁₉₉

#1763

semidirect

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

subgroup	index
C₂×C₄×Dic₃₉₈	2
C₂×#20263878	2
C₂×#20263866	2
#27163780	2
C₄×C₄⋊C₄×C₁₉₉	2
#1763	199

subgroup

index

C₂×C₄×Dic₃₉₈

C₄×C₄⋊C₄×C₁₉₉

199

Subgroups — #38768683

Conjugacy classes of subgroups

Complements

Maximal subgroups