Group #49174969 — subgroups

Subgroups — #49174969

Order 37504.

276 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 78 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	#34037088	C₂	1	1	yes	maximal, characteristic
2	#34037073	C₂	1	1	yes	maximal, characteristic
3	#34037002	C₂	1	4	yes	maximal
4	C₄⋊Q₈×D₂₉₃	C₂	1	1	yes	maximal, characteristic
5	C₂×D₂₉₃⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊#11924078	C₂²	1	2	yes	—
7	#22627679	—	2	1	no	—
8	#22627679	—	2	1	no	—
9	(C₂×#11924078)⋊C₄	—	2	1	no	—
10	(C₂×#11924078)⋊C₄	—	2	1	no	—
11	C₂×Q₈×D₂₉₃	—	2	1	no	—
12	#22627624	C₄	1	1	yes	characteristic, complemented
13	#22627623	—	2	1	no	—
14	#22627623	C₄	1	2	yes	complemented
15	#22627622	—	2	1	no	—
16	C₄⋊C₄×D₂₉₃	—	2	1	no	—
17	C₄⋊C₄×D₂₉₃	C₂²	1	2	yes	—
18	C₄²×D₂₉₃	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₂₉₃	C₄	1	1	yes	characteristic, complemented
20	C₂²×#11924078	—	2	2	no	—
21	D₂₉₃⋊C₈	—	2	2	no	—
22	D₂₉₃⋊C₈	Q₈	1	4	yes	complemented
23	C₂×#14556880	—	2	2	no	—
24	C₄⋊#11924078	—	2	4	no	—
25	C₂×Dic₅₈₆	—	2	2	no	—
26	C₂×Dic₅₈₆	—	2	1	no	—
27	Q₈×D₂₉₃	—	4	4	no	—
28	C₂×C₄×D₂₉₃	—	2	2	no	—
29	C₂×C₄×D₂₉₃	D₄	1	1	yes	characteristic
30	C₂×C₄×D₂₉₃	D₄	1	1	yes	characteristic
31	C₂×C₄×D₂₉₃	C₂³	1	1	yes	characteristic
32	C₄⋊Dic₂₉₃	—	2	1	no	—
33	C₄⋊Dic₂₉₃	C₂×C₄	1	2	yes	—
34	#17989781	—	2	2	no	—
35	#17989781	—	2	2	no	—
36	C₄×Dic₂₉₃	C₂×C₄	1	1	yes	characteristic
37	C₄×Dic₂₉₃	D₄	1	1	yes	characteristic
38	C₄×Dic₂₉₃	D₄	1	1	yes	characteristic
39	C₂×Q₈×C₂₉₃	—	2	1	no	—
40	C₄⋊C₄×C₂₉₃	—	2	1	no	—
41	C₄⋊C₄×C₂₉₃	C₂×C₄	1	2	yes	—
42	C₄²×C₂₉₃	C₂×C₄	1	1	yes	characteristic
43	C₂×#11924078	—	4	2	no	—
44	C₂×#11924078	—	4	2	no	—
45	C₂×#11924078	—	4	2	no	—
46	#14556880	—	2	4	no	—
47	C₂²×D₂₉₃	C₂×D₄	1	1	yes	characteristic
48	Dic₅₈₆	—	4	4	no	—
49	Dic₅₈₆	—	4	2	no	—
50	C₂×Dic₂₉₃	—	2	2	no	—
51	C₂×Dic₂₉₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×Dic₂₉₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×Dic₂₉₃	C₂×D₄	1	1	yes	characteristic
54	C₄×D₂₉₃	—	2	2	no	—
55	C₄×D₂₉₃	—	2	2	no	—
56	C₄×D₂₉₃	—	4	4	no	—
57	C₄×D₂₉₃	C₂×Q₈	1	2	yes	—
58	C₄×D₂₉₃	#41	1	2	yes	—
59	Q₈×C₂₉₃	—	4	2	no	—
60	C₂×C₄×C₂₉₃	—	2	2	no	—
61	C₂×C₄×C₂₉₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₂₉₃	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₂₉₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₂₉₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂×D₂₉₃	#124	1	1	yes	characteristic
66	C₂×D₂₉₃	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×D₂₉₃	AGL(1,ℤ/8)	1	2	yes	—
68	C₂×D₂₉₃	#137	1	2	yes	—
69	#11924078	—	8	2	no	—
70	C₄×C₂₉₃	—	2	1	no	—
71	C₄×C₂₉₃	—	2	1	no	—
72	C₄×C₂₉₃	—	4	2	no	—
73	C₄×C₂₉₃	C₄×Q₈	1	2	yes	—
74	Dic₂₉₃	—	2	1	no	—
75	Dic₂₉₃	—	2	1	no	—
76	Dic₂₉₃	—	4	2	no	—
77	Dic₂₉₃	C₂²⋊Q₈	1	2	yes	—
78	C₂×C₂₉₃	#391	1	1	yes	characteristic
79	C₂×C₂₉₃	#418	1	1	yes	characteristic
80	C₂×C₂₉₃	#419	1	1	yes	characteristic
81	D₂₉₃	#489	1	2	yes	—
82	D₂₉₃	#501	1	2	yes	—
83	C₂₉₃	#1993	1	1	yes	characteristic, complemented, Sylow
84	#1993	—	293	1	no	maximal, Sylow
85	C₂×C₄⋊Q₈	—	293	1	no	—
86	C₂×#97	—	293	1	no	—
87	#382	—	293	1	no	—
88	#340	—	293	4	no	—
89	C₂²×Q₈	—	586	1	no	—
90	C₂²×C₈	—	293	2	no	—
91	C₄⋊Q₈	—	293	2	no	—
92	C₄⋊Q₈	—	293	1	no	—
93	C₄⋊Q₈	—	586	1	no	—
94	C₄⋊Q₈	—	586	1	no	—
95	C₄⋊Q₈	#11924078	1	1	yes	characteristic, complemented
96	C₂×C₄⋊C₄	—	293	2	no	—
97	C₂×C₄⋊C₄	—	293	2	no	—
98	C₂×C₄⋊C₄	—	586	1	no	—
99	C₂×C₄²	—	293	1	no	—
100	#97	—	586	1	no	—
101	#97	—	586	1	no	—
102	#95	—	586	1	no	—
103	#95	—	586	1	no	—
104	C₂×Q₈	—	2	1	no	—
105	C₂×Q₈	—	586	2	no	—
106	C₂×Q₈	—	586	1	no	—
107	C₂×Q₈	—	1172	4	no	—
108	C₂²×C₄	—	293	1	no	—
109	C₂²×C₄	—	293	1	no	—
110	C₂²×C₄	—	293	1	no	—
111	C₂²×C₄	—	586	2	no	—
112	C₂²×C₄	—	586	2	no	—
113	C₂×C₈	—	293	4	no	—
114	C₂×C₈	—	586	2	no	—
115	C₂×C₈	—	586	2	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	—	293	2	no	—
118	C₄⋊C₄	—	586	2	no	—
119	C₄⋊C₄	—	586	2	no	—
120	C₄⋊C₄	—	586	4	no	—
121	C₄⋊C₄	—	586	1	no	—
122	C₄⋊C₄	C₂×#11924078	1	2	yes	—
123	C₄²	—	293	1	no	—
124	C₄²	—	293	1	no	—
125	C₄²	—	293	1	no	—
126	C₄²	C₂×#11924078	1	1	yes	characteristic
127	C₂³	—	293	1	no	—
128	Q₈	—	4	2	no	—
129	Q₈	—	1172	2	no	—
130	Q₈	—	1172	4	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	293	1	no	—
133	C₂×C₄	—	293	2	no	—
134	C₂×C₄	—	293	2	no	—
135	C₂×C₄	—	293	1	no	—
136	C₂×C₄	—	293	1	no	—
137	C₂×C₄	—	586	2	no	—
138	C₂×C₄	—	586	2	no	—
139	C₂×C₄	—	586	2	no	—
140	C₂×C₄	—	1172	4	no	—
141	C₂×C₄	—	1172	2	no	—
142	C₂×C₄	—	1172	2	no	—
143	C₂×C₄	—	1172	2	no	—
144	C₂×C₄	C₂²⋊#11924078	1	1	yes	characteristic
145	C₂×C₄	C₂²⋊#11924078	1	1	yes	characteristic
146	C₂×C₄	C₂²×#11924078	1	1	yes	characteristic
147	C₈	—	586	4	no	—
148	C₂²	—	293	1	no	—
149	C₂²	—	293	1	no	—
150	C₂²	—	293	2	no	—
151	C₂²	—	293	2	no	—
152	C₂²	C₂×C₂²⋊#11924078	1	1	yes	characteristic
153	C₄	—	2	1	no	—
154	C₄	—	2	1	no	—
155	C₄	—	4	2	no	—
156	C₄	—	293	2	no	—
157	C₄	—	2344	2	no	—
158	C₄	—	586	1	no	—
159	C₄	—	586	1	no	—
160	C₄	—	1172	2	no	—
161	C₄	Q₈×#11924078	1	2	yes	—
162	C₂	—	293	2	no	—
163	C₂	—	293	2	no	—
164	C₂	(C₂×#11924078)⋊Q₈	1	1	yes	characteristic
165	C₂	#34037121	1	1	yes	characteristic
166	C₂	#34037122	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#34037088

C₂

yes

maximal, characteristic

#34037073

C₂

yes

maximal, characteristic

#34037002

C₂

yes

maximal

C₄⋊Q₈×D₂₉₃

C₂

yes

maximal, characteristic

C₂×D₂₉₃⋊C₈

C₂²

yes

—

C₂×C₄⋊#11924078

C₂²

yes

—

—

—

—

—

C₂×Q₈×D₂₉₃

—

#22627624

C₄

yes

characteristic, complemented

#22627623

—

#22627623

C₄

yes

complemented

#22627622

—

C₄⋊C₄×D₂₉₃

—

C₄⋊C₄×D₂₉₃

C₂²

yes

—

C₄²×D₂₉₃

C₂²

yes

characteristic

C₄⋊Q₈×C₂₉₃

C₄

yes

characteristic, complemented

C₂²×#11924078

—

D₂₉₃⋊C₈

—

D₂₉₃⋊C₈

Q₈

yes

complemented

C₂×#14556880

—

C₄⋊#11924078

—

C₂×Dic₅₈₆

—

C₂×Dic₅₈₆

—

Q₈×D₂₉₃

—

C₂×C₄×D₂₉₃

—

C₂×C₄×D₂₉₃

D₄

yes

characteristic

C₂×C₄×D₂₉₃

D₄

yes

characteristic

C₂×C₄×D₂₉₃

C₂³

yes

characteristic

C₄⋊Dic₂₉₃

—

C₄⋊Dic₂₉₃

C₂×C₄

yes

—

#17989781

—

#17989781

—

C₄×Dic₂₉₃

C₂×C₄

yes

characteristic

C₄×Dic₂₉₃

D₄

yes

characteristic

C₄×Dic₂₉₃

D₄

yes

characteristic

C₂×Q₈×C₂₉₃

—

C₄⋊C₄×C₂₉₃

—

C₄⋊C₄×C₂₉₃

C₂×C₄

yes

—

C₄²×C₂₉₃

C₂×C₄

yes

—

—

—

—

C₂²×D₂₉₃

C₂×D₄

yes

characteristic

Dic₅₈₆

—

Dic₅₈₆

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₉₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₉₃

C₂×D₄

yes

characteristic

C₄×D₂₉₃

—

C₄×D₂₉₃

—

C₄×D₂₉₃

—

C₄×D₂₉₃

C₂×Q₈

yes

—

C₄×D₂₉₃

#41

yes

—

Q₈×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₉₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₉₃

C₂²×C₄

yes

characteristic

C₂²×C₂₉₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₂₉₃

#124

yes

characteristic

C₂×D₂₉₃

C₄⋊Q₈

yes

characteristic

C₂×D₂₉₃

AGL(1,ℤ/8)

yes

—

C₂×D₂₉₃

#137

yes

—

#11924078

—

C₄×C₂₉₃

—

C₄×C₂₉₃

—

C₄×C₂₉₃

—

C₄×C₂₉₃

C₄×Q₈

yes

—

Dic₂₉₃

—

Dic₂₉₃

—

Dic₂₉₃

—

Dic₂₉₃

C₂²⋊Q₈

yes

—

C₂×C₂₉₃

yes

C₂×C₂₉₃

yes

C₂×C₂₉₃

yes

D₂₉₃

yes

—

D₂₉₃

#501

yes

—

C₂₉₃

#1993

yes

characteristic, complemented, Sylow

#1993

—

293

maximal, Sylow

C₂×C₄⋊Q₈

—

293

—

C₂×#97

—

293

—

#382

—

293

—

#340

—

293

—

C₂²×Q₈

—

586

—

C₂²×C₈

—

293

—

C₄⋊Q₈

—

293

—

C₄⋊Q₈

—

293

—

C₄⋊Q₈

—

586

—

C₄⋊Q₈

—

586

—

C₄⋊Q₈

#11924078

yes

characteristic, complemented

C₂×C₄⋊C₄

—

293

—

C₂×C₄⋊C₄

—

293

—

C₂×C₄⋊C₄

—

586

—

C₂×C₄²

—

293

—

100

#97

—

586

—

101

#97

—

586

—

102

#95

—

586

—

103

#95

—

586

—

104

C₂×Q₈

—

105

C₂×Q₈

—

586

—

106

C₂×Q₈

—

586

—

107

C₂×Q₈

—

1172

—

108

C₂²×C₄

—

293

—

109

C₂²×C₄

—

293

—

110

C₂²×C₄

—

293

—

111

C₂²×C₄

—

586

—

112

C₂²×C₄

—

586

—

113

C₂×C₈

—

293

—

114

C₂×C₈

—

586

—

115

C₂×C₈

—

586

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

293

—

118

C₄⋊C₄

—

586

—

119

C₄⋊C₄

—

586

—

120

C₄⋊C₄

—

586

—

121

C₄⋊C₄

—

586

—

122

C₄⋊C₄

C₂×#11924078

yes

—

123

C₄²

—

293

—

124

C₄²

—

293

—

125

C₄²

—

293

—

126

C₄²

C₂×#11924078

yes

characteristic

127

C₂³

—

293

—

128

Q₈

—

129

Q₈

—

1172

—

130

Q₈

—

1172

—

131

C₂×C₄

—

132

C₂×C₄

—

293

—

133

C₂×C₄

—

293

—

134

C₂×C₄

—

293

—

135

C₂×C₄

—

293

—

136

C₂×C₄

—

293

—

137

C₂×C₄

—

586

—

138

C₂×C₄

—

586

—

139

C₂×C₄

—

586

—

140

C₂×C₄

—

1172

—

141

C₂×C₄

—

1172

—

142

C₂×C₄

—

1172

—

143

C₂×C₄

—

1172

—

144

C₂×C₄

C₂²⋊#11924078

yes

characteristic

145

C₂×C₄

C₂²⋊#11924078

yes

characteristic

146

C₂×C₄

C₂²×#11924078

yes

characteristic

147

C₈

—

586

—

148

C₂²

—

293

—

149

C₂²

—

293

—

150

C₂²

—

293

—

151

C₂²

—

293

—

152

C₂²

C₂×C₂²⋊#11924078

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

293

—

157

C₄

—

2344

—

158

C₄

—

586

—

159

C₄

—

586

—

160

C₄

—

1172

—

161

C₄

Q₈×#11924078

yes

—

162

C₂

—

293

—

163

C₂

—

293

—

164

C₂

(C₂×#11924078)⋊Q₈

yes

characteristic

165

C₂

#34037121

yes

characteristic

166

C₂

#34037122

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#11924078
commutator	C₂×C₄×C₂₉₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#11924078
Fitting	C₄⋊Q₈×C₂₉₃	C₄
socle	C₂²×C₂₉₃	C₂×C₂²⋊C₄
prime radical	C₂²×D₂₉₃	C₂×D₄
cosocle	C₂×C₄×D₂₉₃	C₂³
hypercenter	C₄⋊Q₈	#11924078
hypocenter	C₂₉₃	#1993
LCS	G ▷ C₂×C₄×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#11924078

commutator

C₂×C₄×C₂₉₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#11924078

Fitting

C₄⋊Q₈×C₂₉₃

C₄

socle

C₂²×C₂₉₃

C₂×C₂²⋊C₄

prime radical

C₂²×D₂₉₃

C₂×D₄

cosocle

C₂×C₄×D₂₉₃

C₂³

C₄⋊Q₈

C₂₉₃

G ▷ C₂×C₄×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
12. #22627624	C₄	semidirect
14. #22627623	C₄	semidirect
19. C₄⋊Q₈×C₂₉₃	C₄	semidirect
22. D₂₉₃⋊C₈	Q₈	semidirect
83. C₂₉₃	#1993	semidirect
95. C₄⋊Q₈	#11924078	semidirect

H ≅ G/N

type

12. #22627624

C₄

semidirect

14. #22627623

C₄

semidirect

19. C₄⋊Q₈×C₂₉₃

C₄

semidirect

22. D₂₉₃⋊C₈

Q₈

semidirect

83. C₂₉₃

#1993

semidirect

95. C₄⋊Q₈

#11924078

semidirect

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

subgroup	index
#34037088	2
#34037073	2
#34037002	2
C₄⋊Q₈×D₂₉₃	2
#1993	293

subgroup

index

C₄⋊Q₈×D₂₉₃

293

Subgroups — #49174969

Conjugacy classes of subgroups

Complements

Maximal subgroups