Group #49174299 — subgroups

Subgroups — #49174299

Order 37504.

240 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 70 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	#34037147	C₂	1	1	yes	maximal, characteristic
2	#34037134	C₂	1	1	yes	maximal, characteristic
3	#34036989	C₂	1	4	yes	maximal, complemented
4	C₄⋊Q₈×D₂₉₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#17989812	—	2	1	no	—
6	#22627691	—	2	1	no	—
7	#22627691	C₂²	1	2	yes	—
8	#22627691	C₂²	1	2	yes	—
9	#22627681	—	2	1	no	—
10	C₄×#14556880	—	2	1	no	—
11	C₂×Q₈×D₂₉₃	C₄	1	2	yes	—
12	#22627624	C₄	1	1	yes	characteristic
13	#22627623	—	2	1	no	—
14	#22627623	—	2	1	no	—
15	#22627622	C₂²	1	2	yes	—
16	C₄⋊C₄×D₂₉₃	—	2	1	no	—
17	C₄⋊C₄×D₂₉₃	—	2	1	no	—
18	C₄²×D₂₉₃	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₂₉₃	C₄	1	1	yes	characteristic
20	#17989812	—	4	2	no	—
21	C₂×#14556880	—	2	2	no	—
22	C₂×#14556880	—	2	2	no	—
23	C₂×Dic₅₈₆	—	2	2	no	—
24	C₂×Dic₅₈₆	C₂×C₄	1	2	yes	—
25	Q₈×D₂₉₃	—	4	4	no	—
26	C₂×C₄×D₂₉₃	—	2	2	no	—
27	C₂×C₄×D₂₉₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×D₂₉₃	D₄	1	1	yes	characteristic
29	C₂×C₄×D₂₉₃	D₄	1	1	yes	characteristic
30	C₄⋊Dic₂₉₃	—	2	1	no	—
31	C₄⋊Dic₂₉₃	—	2	1	no	—
32	#17989781	—	2	2	no	—
33	#17989781	—	2	2	no	—
34	C₄×Dic₂₉₃	D₄	1	1	yes	characteristic
35	C₄×Dic₂₉₃	D₄	1	1	yes	characteristic
36	C₄×Dic₂₉₃	C₂³	1	1	yes	characteristic
37	C₂×Q₈×C₂₉₃	C₂×C₄	1	2	yes	—
38	C₄⋊C₄×C₂₉₃	—	2	1	no	—
39	C₄⋊C₄×C₂₉₃	—	2	1	no	—
40	C₄²×C₂₉₃	C₂×C₄	1	1	yes	characteristic
41	#14556880	—	4	2	no	—
42	#14556880	—	4	2	no	—
43	C₂²×D₂₉₃	C₂²⋊C₄	1	1	yes	characteristic
44	Dic₅₈₆	—	4	4	no	—
45	Dic₅₈₆	—	4	2	no	—
46	C₂×Dic₂₉₃	—	2	2	no	—
47	C₂×Dic₂₉₃	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×Dic₂₉₃	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₂₉₃	C₂×D₄	1	1	yes	characteristic
50	C₄×D₂₉₃	—	2	2	no	—
51	C₄×D₂₉₃	—	2	2	no	—
52	C₄×D₂₉₃	—	4	4	no	—
53	C₄×D₂₉₃	SD₁₆	1	2	yes	—
54	C₄×D₂₉₃	Q₁₆	1	2	yes	—
55	Q₈×C₂₉₃	—	4	2	no	—
56	C₂×C₄×C₂₉₃	—	2	2	no	—
57	C₂×C₄×C₂₉₃	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₂₉₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₂₉₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₂₉₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂×D₂₉₃	—	2	1	no	—
62	C₂×D₂₉₃	—	2	1	no	—
63	C₂×D₂₉₃	#103	1	2	yes	—
64	C₄×C₂₉₃	—	2	1	no	—
65	C₄×C₂₉₃	—	2	1	no	—
66	C₄×C₂₉₃	—	4	2	no	—
67	C₄×C₂₉₃	#103	1	2	yes	—
68	Dic₂₉₃	—	4	2	no	—
69	Dic₂₉₃	#100	1	2	yes	—
70	Dic₂₉₃	C₂×SD₁₆	1	1	yes	characteristic
71	Dic₂₉₃	C₂×Q₁₆	1	1	yes	characteristic
72	Dic₂₉₃	#137	1	2	yes	—
73	C₂×C₂₉₃	C₂×#100	1	1	yes	characteristic
74	C₂×C₂₉₃	C₂×#103	1	1	yes	characteristic
75	C₂×C₂₉₃	#419	1	1	yes	characteristic
76	D₂₉₃	—	2	2	no	—
77	C₂₉₃	#1548	1	1	yes	characteristic, complemented, Sylow
78	#1548	—	293	1	no	maximal, Sylow
79	C₂×C₄⋊Q₈	—	293	1	no	—
80	#431	—	293	1	no	—
81	#423	—	293	1	no	—
82	#333	—	293	4	no	—
83	C₂²×Q₈	—	293	2	no	—
84	C₂×M₁₆	—	586	1	no	—
85	C₄⋊Q₈	—	293	2	no	—
86	C₄⋊Q₈	—	293	1	no	—
87	C₄⋊Q₈	—	586	1	no	—
88	C₄⋊Q₈	—	586	1	no	—
89	C₄⋊Q₈	#11924078	1	1	yes	characteristic
90	C₂×C₄⋊C₄	—	586	1	no	—
91	C₂×C₄⋊C₄	—	586	1	no	—
92	C₂×C₄²	—	293	1	no	—
93	C₄⋊C₈	—	293	2	no	—
94	C₄⋊C₈	—	293	2	no	—
95	C₄⋊C₈	—	586	1	no	—
96	C₂²⋊C₈	—	586	1	no	—
97	C₄×C₈	—	586	1	no	—
98	C₂×Q₈	—	293	2	no	—
99	C₂×Q₈	—	586	2	no	—
100	C₂×Q₈	—	1172	4	no	—
101	C₂×Q₈	C₂×#11924078	1	2	yes	—
102	C₂²×C₄	—	293	1	no	—
103	C₂²×C₄	—	293	1	no	—
104	C₂²×C₄	—	293	1	no	—
105	C₂²×C₄	—	586	2	no	—
106	M₁₆	—	1172	2	no	—
107	C₂×C₈	—	586	2	no	—
108	C₂×C₈	—	586	2	no	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	586	2	no	—
112	C₄⋊C₄	—	586	2	no	—
113	C₄⋊C₄	—	586	1	no	—
114	C₄⋊C₄	—	586	1	no	—
115	C₄²	—	293	1	no	—
116	C₄²	—	293	1	no	—
117	C₄²	—	293	1	no	—
118	C₄²	C₂×#11924078	1	1	yes	characteristic
119	C₂³	—	293	1	no	—
120	Q₈	—	4	2	no	—
121	Q₈	—	1172	2	no	—
122	Q₈	—	1172	4	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	293	1	no	—
125	C₂×C₄	—	293	1	no	—
126	C₂×C₄	—	293	1	no	—
127	C₂×C₄	—	293	2	no	—
128	C₂×C₄	—	293	2	no	—
129	C₂×C₄	—	586	2	no	—
130	C₂×C₄	—	586	2	no	—
131	C₂×C₄	—	586	2	no	—
132	C₂×C₄	—	1172	4	no	—
133	C₂×C₄	C₂²⋊#11924078	1	1	yes	characteristic
134	C₂×C₄	C₂²⋊#11924078	1	1	yes	characteristic
135	C₂×C₄	C₂²×#11924078	1	1	yes	characteristic
136	C₈	—	1172	2	no	—
137	C₈	—	1172	2	no	—
138	C₂²	—	293	2	no	—
139	C₂²	—	586	1	no	—
140	C₂²	—	586	1	no	—
141	C₂²	C₂×C₂²⋊#11924078	1	1	yes	characteristic
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	4	2	no	—
145	C₄	—	293	2	no	—
146	C₄	—	293	1	no	—
147	C₄	—	293	1	no	—
148	C₄	—	293	2	no	—
149	C₄	—	1172	2	no	—
150	C₄	#22627687	1	2	yes	—
151	C₂	—	586	2	no	—
152	C₂	D₂₉₃⋊#100	1	1	yes	characteristic
153	C₂	C₂×#22627687	1	1	yes	characteristic
154	C₂	#34037122	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#34037147

C₂

yes

maximal, characteristic

#34037134

C₂

yes

maximal, characteristic

#34036989

C₂

yes

maximal, complemented

C₄⋊Q₈×D₂₉₃

C₂

yes

maximal, characteristic

C₂×#17989812

—

#22627691

—

#22627691

C₂²

yes

—

#22627691

C₂²

yes

—

#22627681

—

C₄×#14556880

—

C₂×Q₈×D₂₉₃

C₄

yes

—

#22627624

C₄

yes

—

—

C₂²

yes

—

C₄⋊C₄×D₂₉₃

—

C₄⋊C₄×D₂₉₃

—

C₄²×D₂₉₃

C₂²

yes

characteristic

C₄⋊Q₈×C₂₉₃

C₄

yes

—

—

—

C₂×Dic₅₈₆

—

C₂×Dic₅₈₆

C₂×C₄

yes

—

Q₈×D₂₉₃

—

C₂×C₄×D₂₉₃

—

C₂×C₄×D₂₉₃

C₂×C₄

yes

characteristic

C₂×C₄×D₂₉₃

D₄

yes

characteristic

C₂×C₄×D₂₉₃

D₄

yes

characteristic

C₄⋊Dic₂₉₃

—

C₄⋊Dic₂₉₃

—

#17989781

—

#17989781

—

C₄×Dic₂₉₃

D₄

yes

characteristic

C₄×Dic₂₉₃

D₄

yes

characteristic

C₄×Dic₂₉₃

C₂³

yes

characteristic

C₂×Q₈×C₂₉₃

C₂×C₄

yes

—

C₄⋊C₄×C₂₉₃

—

C₄⋊C₄×C₂₉₃

—

C₄²×C₂₉₃

C₂×C₄

yes

characteristic

#14556880

—

#14556880

—

C₂²×D₂₉₃

C₂²⋊C₄

yes

characteristic

Dic₅₈₆

—

Dic₅₈₆

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

C₂²⋊C₄

yes

characteristic

C₂×Dic₂₉₃

C₂×D₄

yes

characteristic

C₂×Dic₂₉₃

C₂×D₄

yes

characteristic

C₄×D₂₉₃

—

C₄×D₂₉₃

—

C₄×D₂₉₃

—

C₄×D₂₉₃

SD₁₆

yes

—

C₄×D₂₉₃

Q₁₆

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

—

C₂×D₂₉₃

—

C₂×D₂₉₃

#103

yes

—

C₄×C₂₉₃

—

C₄×C₂₉₃

—

C₄×C₂₉₃

—

C₄×C₂₉₃

#103

yes

—

Dic₂₉₃

—

Dic₂₉₃

#100

yes

—

Dic₂₉₃

C₂×SD₁₆

yes

characteristic

Dic₂₉₃

C₂×Q₁₆

yes

characteristic

Dic₂₉₃

#137

yes

—

C₂×C₂₉₃

yes

C₂×C₂₉₃

yes

C₂×C₂₉₃

yes

D₂₉₃

—

C₂₉₃

#1548

yes

characteristic, complemented, Sylow

#1548

—

293

maximal, Sylow

C₂×C₄⋊Q₈

—

293

—

#431

—

293

—

#423

—

293

—

#333

—

293

—

C₂²×Q₈

—

293

—

C₂×M₁₆

—

586

—

C₄⋊Q₈

—

293

—

C₄⋊Q₈

—

293

—

C₄⋊Q₈

—

586

—

C₄⋊Q₈

—

586

—

C₄⋊Q₈

#11924078

yes

characteristic

C₂×C₄⋊C₄

—

586

—

C₂×C₄⋊C₄

—

586

—

C₂×C₄²

—

293

—

C₄⋊C₈

—

293

—

C₄⋊C₈

—

293

—

C₄⋊C₈

—

586

—

C₂²⋊C₈

—

586

—

C₄×C₈

—

586

—

C₂×Q₈

—

293

—

C₂×Q₈

—

586

—

100

C₂×Q₈

—

1172

—

101

C₂×Q₈

C₂×#11924078

yes

—

102

C₂²×C₄

—

293

—

103

C₂²×C₄

—

293

—

104

C₂²×C₄

—

293

—

105

C₂²×C₄

—

586

—

106

M₁₆

—

1172

—

107

C₂×C₈

—

586

—

108

C₂×C₈

—

586

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

586

—

112

C₄⋊C₄

—

586

—

113

C₄⋊C₄

—

586

—

114

C₄⋊C₄

—

586

—

115

C₄²

—

293

—

116

C₄²

—

293

—

117

C₄²

—

293

—

118

C₄²

C₂×#11924078

yes

characteristic

119

C₂³

—

293

—

120

Q₈

—

121

Q₈

—

1172

—

122

Q₈

—

1172

—

123

C₂×C₄

—

124

C₂×C₄

—

293

—

125

C₂×C₄

—

293

—

126

C₂×C₄

—

293

—

127

C₂×C₄

—

293

—

128

C₂×C₄

—

293

—

129

C₂×C₄

—

586

—

130

C₂×C₄

—

586

—

131

C₂×C₄

—

586

—

132

C₂×C₄

—

1172

—

133

C₂×C₄

C₂²⋊#11924078

yes

characteristic

134

C₂×C₄

C₂²⋊#11924078

yes

characteristic

135

C₂×C₄

C₂²×#11924078

yes

characteristic

136

C₈

—

1172

—

137

C₈

—

1172

—

138

C₂²

—

293

—

139

C₂²

—

586

—

140

C₂²

—

586

—

141

C₂²

C₂×C₂²⋊#11924078

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

293

—

146

C₄

—

293

—

147

C₄

—

293

—

148

C₄

—

293

—

149

C₄

—

1172

—

150

C₄

#22627687

yes

—

151

C₂

—

586

—

152

C₂

D₂₉₃⋊#100

yes

characteristic

153

C₂

C₂×#22627687

yes

characteristic

154

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	C₂²⋊C₄
cosocle	C₄×Dic₂₉₃	C₂³
hypercenter	C₄⋊Q₈	#11924078
hypocenter	C₂₉₃	#1548
LCS	G ▷ C₂×C₄×C₂₉₃ ▷ C₂²×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

C₂²⋊C₄

cosocle

C₄×Dic₂₉₃

C₂³

C₄⋊Q₈

C₂₉₃

G ▷ C₂×C₄×C₂₉₃ ▷ C₂²×C₂₉₃ ▷ C₂₉₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
3. #34036989	C₂	semidirect
77. C₂₉₃	#1548	semidirect

H ≅ G/N

type

3. #34036989

C₂

semidirect

77. C₂₉₃

#1548

semidirect

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

subgroup	index
#34037147	2
#34037134	2
#34036989	2
C₄⋊Q₈×D₂₉₃	2
#1548	293

subgroup

index

C₄⋊Q₈×D₂₉₃

293

Subgroups — #49174299

Conjugacy classes of subgroups

Complements

Maximal subgroups