Group #38765129 — subgroups

Subgroups — #2057×C₁₉₉ #38765129

Order 25472.

282 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 116 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⋊Q₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#107×C₁₉₉	C₂	1	2	yes	maximal
3	#389×C₁₉₉	C₂	1	1	yes	maximal, characteristic
4	#340×C₁₉₉	C₂	1	2	yes	maximal
5	#336×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₁₉₉	—	2	1	no	—
7	C₂²×C₈×C₁₉₉	C₂²	1	2	yes	—
8	C₄⋊Q₈×C₁₉₉	—	2	4	no	—
9	C₂×C₄⋊C₄×C₁₉₉	—	2	1	no	—
10	C₂×C₄⋊C₄×C₁₉₉	—	2	1	no	—
11	C₂×C₄⋊C₄×C₁₉₉	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊C₄×C₁₉₉	C₂²	1	2	yes	—
13	C₂×C₄⋊C₄×C₁₉₉	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
15	#107×C₁₉₉	—	2	4	no	—
16	#95×C₁₉₉	—	2	1	no	—
17	C₂×Q₈×C₁₉₉	—	2	4	no	—
18	C₂×Q₈×C₁₉₉	—	4	2	no	—
19	C₂×Q₈×C₁₉₉	—	4	2	no	—
20	C₂²×C₄×C₁₉₉	—	2	1	no	—
21	C₂²×C₄×C₁₉₉	—	2	2	no	—
22	C₂²×C₄×C₁₉₉	—	2	1	no	—
23	C₂²×C₄×C₁₉₉	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₉₉	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₁₉₉	—	2	2	no	—
27	C₂×C₈×C₁₉₉	—	2	2	no	—
28	C₂×C₈×C₁₉₉	Q₈	1	4	yes	complemented
29	C₄⋊C₄×C₁₉₉	—	2	4	no	—
30	C₄⋊C₄×C₁₉₉	—	2	2	no	—
31	C₄⋊C₄×C₁₉₉	—	2	1	no	—
32	C₄⋊C₄×C₁₉₉	—	2	2	no	—
33	C₄⋊C₄×C₁₉₉	—	4	2	no	—
34	C₄⋊C₄×C₁₉₉	D₄	1	4	yes	—
35	C₄⋊C₄×C₁₉₉	Q₈	1	2	yes	complemented
36	C₄²×C₁₉₉	—	2	2	no	—
37	C₂³×C₁₉₉	C₂×D₄	1	1	yes	characteristic
38	Q₈×C₁₉₉	—	4	8	no	—
39	C₂×C₄×C₁₉₉	—	2	1	no	—
40	C₂×C₄×C₁₉₉	—	2	1	no	—
41	C₂×C₄×C₁₉₉	—	2	1	no	—
42	C₂×C₄×C₁₉₉	—	2	1	no	—
43	C₂×C₄×C₁₉₉	—	2	4	no	—
44	C₂×C₄×C₁₉₉	—	2	2	no	—
45	C₂×C₄×C₁₉₉	—	4	1	no	—
46	C₂×C₄×C₁₉₉	—	4	2	no	—
47	C₂×C₄×C₁₉₉	—	4	2	no	—
48	C₂×C₄×C₁₉₉	—	4	1	no	—
49	C₂×C₄×C₁₉₉	—	4	1	no	—
50	C₂×C₄×C₁₉₉	—	4	1	no	—
51	C₂×C₄×C₁₉₉	—	4	1	no	—
52	C₂×C₄×C₁₉₉	D₈	1	2	yes	—
53	C₂×C₄×C₁₉₉	Q₁₆	1	2	yes	—
54	C₂×C₄×C₁₉₉	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₁₉₉	C₂×Q₈	1	2	yes	—
56	C₂×C₄×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₄×C₁₉₉	#41	1	1	yes	characteristic
58	C₈×C₁₉₉	—	2	4	no	—
59	C₂²×C₁₉₉	#120	1	1	yes	characteristic
60	C₂²×C₁₉₉	C₂²⋊Q₈	1	1	yes	characteristic
61	C₂²×C₁₉₉	C₄⋊Q₈	1	1	yes	characteristic
62	C₂²×C₁₉₉	C₂×D₈	1	1	yes	characteristic
63	C₂²×C₁₉₉	C₂×Q₁₆	1	1	yes	characteristic
64	C₂²×C₁₉₉	AGL(1,ℤ/8)	1	1	yes	characteristic
65	C₂²×C₁₉₉	#137	1	1	yes	characteristic
66	C₄×C₁₉₉	—	2	2	no	—
67	C₄×C₁₉₉	—	2	2	no	—
68	C₄×C₁₉₉	—	4	4	no	—
69	C₄×C₁₉₉	—	4	2	no	—
70	C₄×C₁₉₉	—	8	1	no	—
71	C₄×C₁₉₉	C₂²⋊Q₈	1	2	yes	—
72	C₄×C₁₉₉	C₄⋊Q₈	1	2	yes	—
73	C₂×C₁₉₉	#395	1	1	yes	characteristic
74	C₂×C₁₉₉	#447	1	1	yes	characteristic
75	C₂×C₁₉₉	#451	1	1	yes	characteristic
76	C₂×C₁₉₉	#474	1	1	yes	characteristic
77	C₂×C₁₉₉	#477	1	1	yes	characteristic
78	C₂×C₁₉₉	#500	1	1	yes	characteristic
79	C₂×C₁₉₉	#501	1	1	yes	characteristic
80	C₁₉₉	#2057	1	1	yes	characteristic, direct product factor, Sylow
81	#2057	C₁₉₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂×C₄⋊Q₈	C₂×C₁₉₉	1	1	yes	characteristic
83	C₂×#107	C₂×C₁₉₉	1	2	yes	—
84	#389	C₂×C₁₉₉	1	1	yes	characteristic
85	#340	C₂×C₁₉₉	1	2	yes	—
86	#336	C₂×C₁₉₉	1	1	yes	characteristic
87	C₂²×Q₈	—	2	1	no	—
88	C₂²×C₈	C₂²×C₁₉₉	1	2	yes	—
89	C₄⋊Q₈	—	2	4	no	—
90	C₂×C₄⋊C₄	—	2	1	no	—
91	C₂×C₄⋊C₄	—	2	1	no	—
92	C₂×C₄⋊C₄	C₂²×C₁₉₉	1	1	yes	characteristic
93	C₂×C₄⋊C₄	C₂²×C₁₉₉	1	2	yes	—
94	C₂×C₄⋊C₄	C₂²×C₁₉₉	1	1	yes	characteristic
95	C₂×C₄²	C₂²×C₁₉₉	1	1	yes	characteristic
96	#107	—	2	4	no	—
97	#95	—	2	1	no	—
98	C₂×Q₈	—	2	4	no	—
99	C₂×Q₈	—	4	2	no	—
100	C₂×Q₈	—	4	2	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	2	2	no	—
103	C₂²×C₄	—	2	1	no	—
104	C₂²×C₄	D₄×C₁₉₉	1	1	yes	characteristic
105	C₂²×C₄	D₄×C₁₉₉	1	1	yes	characteristic
106	C₂²×C₄	C₂³×C₁₉₉	1	1	yes	characteristic
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	Q₈×C₁₉₉	1	4	yes	complemented
110	C₄⋊C₄	—	2	4	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	—	4	2	no	—
115	C₄⋊C₄	D₄×C₁₉₉	1	4	yes	—
116	C₄⋊C₄	Q₈×C₁₉₉	1	2	yes	complemented
117	C₄²	—	2	2	no	—
118	C₂³	C₂×D₄×C₁₉₉	1	1	yes	characteristic
119	Q₈	—	4	8	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	D₈×C₁₉₉	1	2	yes	—
134	C₂×C₄	Q₁₆×C₁₉₉	1	2	yes	—
135	C₂×C₄	C₂×D₄×C₁₉₉	1	2	yes	—
136	C₂×C₄	C₂×Q₈×C₁₉₉	1	2	yes	—
137	C₂×C₄	C₂×Q₈×C₁₉₉	1	1	yes	characteristic
138	C₂×C₄	#41×C₁₉₉	1	1	yes	characteristic
139	C₈	—	2	4	no	—
140	C₂²	#120×C₁₉₉	1	1	yes	characteristic
141	C₂²	C₂²⋊Q₈×C₁₉₉	1	1	yes	characteristic
142	C₂²	C₄⋊Q₈×C₁₉₉	1	1	yes	characteristic
143	C₂²	C₂×D₈×C₁₉₉	1	1	yes	characteristic
144	C₂²	C₂×Q₁₆×C₁₉₉	1	1	yes	characteristic
145	C₂²	AGL(1,ℤ/8)×C₁₉₉	1	1	yes	characteristic
146	C₂²	#137×C₁₉₉	1	1	yes	characteristic
147	C₄	—	2	2	no	—
148	C₄	—	2	2	no	—
149	C₄	—	4	4	no	—
150	C₄	—	4	2	no	—
151	C₄	—	8	1	no	—
152	C₄	C₂²⋊Q₈×C₁₉₉	1	2	yes	—
153	C₄	C₄⋊Q₈×C₁₉₉	1	2	yes	—
154	C₂	#395×C₁₉₉	1	1	yes	characteristic
155	C₂	#447×C₁₉₉	1	1	yes	characteristic
156	C₂	#451×C₁₉₉	1	1	yes	characteristic
157	C₂	#474×C₁₉₉	1	1	yes	characteristic
158	C₂	#477×C₁₉₉	1	1	yes	characteristic
159	C₂	#500×C₁₉₉	1	1	yes	characteristic
160	C₂	#501×C₁₉₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊Q₈×C₁₉₉

C₂

yes

maximal, characteristic

C₂×#107×C₁₉₉

C₂

yes

maximal

#389×C₁₉₉

C₂

yes

maximal, characteristic

#340×C₁₉₉

C₂

yes

maximal

#336×C₁₉₉

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₉₉

—

C₂²×C₈×C₁₉₉

C₂²

yes

—

C₄⋊Q₈×C₁₉₉

—

C₂×C₄⋊C₄×C₁₉₉

—

C₂×C₄⋊C₄×C₁₉₉

—

C₂×C₄⋊C₄×C₁₉₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₉₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₉₉

C₂²

yes

characteristic

C₂×C₄²×C₁₉₉

C₂²

yes

characteristic

#107×C₁₉₉

—

#95×C₁₉₉

—

C₂×Q₈×C₁₉₉

—

C₂×Q₈×C₁₉₉

—

C₂×Q₈×C₁₉₉

—

C₂²×C₄×C₁₉₉

—

C₂²×C₄×C₁₉₉

—

C₂²×C₄×C₁₉₉

—

C₂²×C₄×C₁₉₉

D₄

yes

characteristic

C₂²×C₄×C₁₉₉

D₄

yes

characteristic

C₂²×C₄×C₁₉₉

C₂³

yes

characteristic

C₂×C₈×C₁₉₉

—

C₂×C₈×C₁₉₉

—

C₂×C₈×C₁₉₉

Q₈

yes

complemented

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

D₄

yes

—

C₄⋊C₄×C₁₉₉

Q₈

yes

complemented

C₄²×C₁₉₉

—

C₂³×C₁₉₉

C₂×D₄

yes

characteristic

Q₈×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

D₈

yes

—

C₂×C₄×C₁₉₉

Q₁₆

yes

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

—

C₂×C₄×C₁₉₉

C₂×Q₈

yes

—

C₂×C₄×C₁₉₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₉

#41

yes

characteristic

C₈×C₁₉₉

—

C₂²×C₁₉₉

#120

yes

characteristic

C₂²×C₁₉₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₉₉

C₄⋊Q₈

yes

characteristic

C₂²×C₁₉₉

C₂×D₈

yes

characteristic

C₂²×C₁₉₉

C₂×Q₁₆

yes

C₂²×C₁₉₉

yes

C₂²×C₁₉₉

yes

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

C₂²⋊Q₈

yes

—

C₄×C₁₉₉

C₄⋊Q₈

yes

—

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₁₉₉

yes

characteristic, direct product factor, Sylow

#2057

C₁₉₉

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₄⋊Q₈

C₂×C₁₉₉

yes

characteristic

C₂×#107

C₂×C₁₉₉

yes

—

#389

C₂×C₁₉₉

yes

characteristic

#340

C₂×C₁₉₉

yes

—

#336

C₂×C₁₉₉

yes

characteristic

C₂²×Q₈

—

C₂²×C₈

C₂²×C₁₉₉

yes

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₁₉₉

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₁₉₉

yes

—

C₂×C₄⋊C₄

C₂²×C₁₉₉

yes

characteristic

C₂×C₄²

C₂²×C₁₉₉

yes

characteristic

#107

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

D₄×C₁₉₉

yes

characteristic

105

C₂²×C₄

D₄×C₁₉₉

yes

characteristic

106

C₂²×C₄

C₂³×C₁₉₉

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

Q₈×C₁₉₉

yes

complemented

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

D₄×C₁₉₉

yes

—

116

C₄⋊C₄

Q₈×C₁₉₉

yes

complemented

117

C₄²

—

118

C₂³

C₂×D₄×C₁₉₉

yes

characteristic

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

D₈×C₁₉₉

yes

—

134

C₂×C₄

Q₁₆×C₁₉₉

yes

—

135

C₂×C₄

C₂×D₄×C₁₉₉

yes

—

136

C₂×C₄

C₂×Q₈×C₁₉₉

yes

—

137

C₂×C₄

C₂×Q₈×C₁₉₉

yes

characteristic

138

C₂×C₄

#41×C₁₉₉

yes

characteristic

139

C₈

—

140

C₂²

#120×C₁₉₉

yes

characteristic

141

C₂²

C₂²⋊Q₈×C₁₉₉

yes

characteristic

142

C₂²

C₄⋊Q₈×C₁₉₉

yes

characteristic

143

C₂²

C₂×D₈×C₁₉₉

yes

characteristic

144

C₂²

C₂×Q₁₆×C₁₉₉

yes

characteristic

145

C₂²

AGL(1,ℤ/8)×C₁₉₉

yes

characteristic

146

C₂²

#137×C₁₉₉

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

C₂²⋊Q₈×C₁₉₉

yes

—

153

C₄

C₄⋊Q₈×C₁₉₉

yes

—

154

C₂

#395×C₁₉₉

yes

characteristic

155

C₂

#447×C₁₉₉

yes

characteristic

156

C₂

#451×C₁₉₉

yes

characteristic

157

C₂

#474×C₁₉₉

yes

characteristic

158

C₂

#477×C₁₉₉

yes

characteristic

159

C₂

#500×C₁₉₉

yes

characteristic

160

C₂

#501×C₁₉₉

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₉₉	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₁₉₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉₉ ◁ C₂×C₄²×C₁₉₉ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₁₉₉

C₂×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₁₉₉

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₉₉ ◁ C₂×C₄²×C₁₉₉ ◁ G

N	H ≅ G/N	type
28. C₂×C₈×C₁₉₉	Q₈	semidirect
35. C₄⋊C₄×C₁₉₉	Q₈	semidirect
80. C₁₉₉	#2057	direct (N × H)
81. #2057	C₁₉₉	direct (N × H)
109. C₂×C₈	Q₈×C₁₉₉	semidirect
116. C₄⋊C₄	Q₈×C₁₉₉	semidirect

H ≅ G/N

type

28. C₂×C₈×C₁₉₉

Q₈

semidirect

35. C₄⋊C₄×C₁₉₉

Q₈

semidirect

80. C₁₉₉

#2057

direct (N × H)

81. #2057

C₁₉₉

direct (N × H)

109. C₂×C₈

Q₈×C₁₉₉

semidirect

116. C₄⋊C₄

Q₈×C₁₉₉

semidirect

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

subgroup	index
C₂×C₄⋊Q₈×C₁₉₉	2
C₂×#107×C₁₉₉	2
#389×C₁₉₉	2
#340×C₁₉₉	2
#336×C₁₉₉	2
#2057	199

subgroup

index

C₂×C₄⋊Q₈×C₁₉₉

C₂×#107×C₁₉₉

#389×C₁₉₉

#340×C₁₉₉

#336×C₁₉₉

#2057

199

Subgroups — #2057×C₁₉₉ #38765129

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2057×C199 #38765129

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2057×C₁₉₉ #38765129