Group #49158876 — subgroups

Subgroups — #49158876

Order 37504.

344 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 117 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	C₂²×C₄⋊Dic₂₉₃	C₂	1	1	yes	maximal, characteristic
2	#34035989	C₂	1	1	yes	maximal, characteristic
3	#34035892	C₂	1	4	yes	maximal, complemented
4	C₂×C₂²⋊C₈×C₂₉₃	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₂₉₃	—	2	1	no	—
6	C₂×C₄⋊Dic₂₉₃	—	2	2	no	—
7	C₂×C₄⋊Dic₂₉₃	—	2	1	no	—
8	C₂×C₄⋊Dic₂₉₃	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₂₉₃	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₂₉₃	C₂²	1	2	yes	—
11	C₂×C₂²⋊Dic₂₉₃	—	2	1	no	—
12	#22627531	—	2	1	no	—
13	C₂³×C₄×C₂₉₃	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₂₉₃	C₂²	1	2	yes	—
15	C₂²⋊C₈×C₂₉₃	—	2	2	no	—
16	C₂²×Dic₂₉₃	—	2	4	no	—
17	C₂²×Dic₂₉₃	—	2	2	no	—
18	C₂²×Dic₂₉₃	—	2	2	no	—
19	C₂²×Dic₂₉₃	—	4	4	no	—
20	C₄⋊Dic₂₉₃	—	2	4	no	—
21	C₄⋊Dic₂₉₃	—	2	2	no	—
22	C₄⋊Dic₂₉₃	—	2	4	no	—
23	C₄⋊Dic₂₉₃	C₂×C₄	1	4	yes	—
24	C₂²⋊Dic₂₉₃	—	4	2	no	—
25	C₂⁴×C₂₉₃	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₂₉₃	—	2	4	no	—
27	C₂²×C₄×C₂₉₃	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₂₉₃	D₄	1	2	yes	—
29	C₂²×C₄×C₂₉₃	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₂₉₃	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₂₉₃	—	2	2	no	—
32	C₂×C₈×C₂₉₃	—	2	4	no	—
33	C₂×Dic₂₉₃	—	2	4	no	—
34	C₂×Dic₂₉₃	—	4	8	no	—
35	C₂×Dic₂₉₃	—	4	4	no	—
36	C₂×Dic₂₉₃	—	4	2	no	—
37	C₂×Dic₂₉₃	—	4	2	no	—
38	C₂×Dic₂₉₃	—	4	2	no	—
39	C₂³×C₂₉₃	—	2	2	no	—
40	C₂³×C₂₉₃	—	2	2	no	—
41	C₂³×C₂₉₃	C₂²⋊C₄	1	2	yes	—
42	C₂³×C₂₉₃	SD₁₆	1	2	yes	—
43	C₂³×C₂₉₃	Q₁₆	1	2	yes	—
44	C₂³×C₂₉₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₉₃	—	2	2	no	—
46	C₂×C₄×C₂₉₃	—	2	4	no	—
47	C₂×C₄×C₂₉₃	—	2	4	no	—
48	C₂×C₄×C₂₉₃	C₂²⋊C₄	1	2	yes	—
49	C₂×C₄×C₂₉₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₂₉₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₂₉₃	#41	1	4	yes	—
52	C₈×C₂₉₃	—	4	2	no	—
53	C₂²×C₂₉₃	—	2	2	no	—
54	C₂²×C₂₉₃	—	2	2	no	—
55	C₂²×C₂₉₃	—	2	4	no	—
56	C₂²×C₂₉₃	—	2	4	no	—
57	C₂²×C₂₉₃	#103	1	4	yes	—
58	C₂²×C₂₉₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₂₉₃	#123	1	2	yes	—
60	C₂²×C₂₉₃	C₂×SD₁₆	1	1	yes	characteristic
61	C₂²×C₂₉₃	C₂×Q₁₆	1	1	yes	characteristic
62	C₂²×C₂₉₃	AGL(1,ℤ/8)	1	2	yes	—
63	C₄×C₂₉₃	—	2	2	no	—
64	C₄×C₂₉₃	#117	1	2	yes	—
65	C₄×C₂₉₃	#123	1	2	yes	—
66	Dic₂₉₃	—	4	4	no	—
67	Dic₂₉₃	—	8	2	no	—
68	C₂×C₂₉₃	—	2	4	no	—
69	C₂×C₂₉₃	#381	1	1	yes	characteristic
70	C₂×C₂₉₃	C₂×#103	1	1	yes	characteristic
71	C₂×C₂₉₃	#418	1	1	yes	characteristic
72	C₂×C₂₉₃	#481	1	2	yes	—
73	C₂×C₂₉₃	#484	1	2	yes	—
74	C₂₉₃	#1824	1	1	yes	characteristic, complemented, Sylow
75	#1824	—	293	1	no	maximal, Sylow
76	C₂²×C₄⋊C₄	—	293	1	no	—
77	C₂×C₂²⋊C₈	D₂₉₃	1	1	yes	characteristic
78	#380	—	293	1	no	—
79	#340	—	293	4	no	—
80	C₂³×C₄	—	586	1	no	—
81	C₂³×C₄	C₂×D₂₉₃	1	1	yes	characteristic
82	C₂²×C₈	C₂×D₂₉₃	1	2	yes	—
83	C₂×C₄⋊C₄	—	293	4	no	—
84	C₂×C₄⋊C₄	—	293	2	no	—
85	C₂×C₄⋊C₄	—	293	2	no	—
86	C₂×C₄⋊C₄	—	586	2	no	—
87	C₂×C₄⋊C₄	—	586	1	no	—
88	C₂×C₂²⋊C₄	—	586	1	no	—
89	C₂²⋊C₈	—	2	2	no	—
90	#95	—	586	1	no	—
91	C₂⁴	D₁₁₇₂	1	1	yes	characteristic
92	C₂²×C₄	—	2	4	no	—
93	C₂²×C₄	—	586	4	no	—
94	C₂²×C₄	—	586	2	no	—
95	C₂²×C₄	—	586	2	no	—
96	C₂²×C₄	—	1172	4	no	—
97	C₂²×C₄	C₄×D₂₉₃	1	2	yes	—
98	C₂²×C₄	C₂²⋊D₂₉₃	1	2	yes	—
99	C₂²×C₄	D₁₁₇₂	1	1	yes	characteristic
100	C₂²×C₄	C₂²×D₂₉₃	1	1	yes	characteristic
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	2	4	no	—
103	C₄⋊C₄	—	293	4	no	—
104	C₄⋊C₄	—	586	4	no	—
105	C₄⋊C₄	—	586	2	no	—
106	C₄⋊C₄	—	586	4	no	—
107	C₂²⋊C₄	—	1172	2	no	—
108	C₂³	—	2	2	no	—
109	C₂³	—	2	2	no	—
110	C₂³	#17989779	1	2	yes	—
111	C₂³	#17989791	1	2	yes	—
112	C₂³	Dic₁₁₇₂	1	2	yes	—
113	C₂³	C₂×D₁₁₇₂	1	1	yes	characteristic
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	586	4	no	—
118	C₂×C₄	—	1172	2	no	—
119	C₂×C₄	—	1172	2	no	—
120	C₂×C₄	—	1172	2	no	—
121	C₂×C₄	—	1172	8	no	—
122	C₂×C₄	—	1172	4	no	—
123	C₂×C₄	#17989779	1	2	yes	—
124	C₂×C₄	C₂×C₄×D₂₉₃	1	1	yes	characteristic
125	C₂×C₄	C₂×C₂²⋊D₂₉₃	1	1	yes	characteristic
126	C₂×C₄	#17989801	1	4	yes	—
127	C₈	—	4	2	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	2	no	—
130	C₂²	—	2	4	no	—
131	C₂²	—	2	4	no	—
132	C₂²	#22627549	1	4	yes	—
133	C₂²	C₂×#17989779	1	1	yes	characteristic
134	C₂²	#22627608	1	2	yes	—
135	C₂²	C₂×#17989791	1	1	yes	characteristic
136	C₂²	C₂×Dic₁₁₇₂	1	1	yes	characteristic
137	C₂²	#22627652	1	2	yes	—
138	C₄	—	2	2	no	—
139	C₄	—	2344	2	no	—
140	C₄	—	1172	4	no	—
141	C₄	#22627580	1	2	yes	—
142	C₄	#22627605	1	2	yes	—
143	C₂	—	2	4	no	—
144	C₂	#34035990	1	1	yes	characteristic
145	C₂	C₂×#22627549	1	1	yes	characteristic
146	C₂	#34036135	1	1	yes	characteristic
147	C₂	#34036448	1	2	yes	—
148	C₂	#34036465	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊Dic₂₉₃

C₂

yes

maximal, characteristic

#34035989

C₂

yes

maximal, characteristic

#34035892

C₂

yes

maximal, complemented

C₂×C₂²⋊C₈×C₂₉₃

C₂

yes

maximal, characteristic

C₂³×Dic₂₉₃

—

C₂×C₄⋊Dic₂₉₃

—

C₂×C₄⋊Dic₂₉₃

—

C₂×C₄⋊Dic₂₉₃

C₄

yes

complemented

C₂×C₄⋊Dic₂₉₃

C₂²

yes

—

C₂×C₄⋊Dic₂₉₃

C₂²

yes

—

C₂×C₂²⋊Dic₂₉₃

—

#22627531

—

C₂³×C₄×C₂₉₃

C₂²

yes

characteristic

C₂²×C₈×C₂₉₃

C₂²

yes

—

C₂²⋊C₈×C₂₉₃

—

C₂²×Dic₂₉₃

—

C₂²×Dic₂₉₃

—

C₂²×Dic₂₉₃

—

C₂²×Dic₂₉₃

—

C₄⋊Dic₂₉₃

—

C₄⋊Dic₂₉₃

—

C₄⋊Dic₂₉₃

—

C₄⋊Dic₂₉₃

C₂×C₄

yes

—

C₂²⋊Dic₂₉₃

—

C₂⁴×C₂₉₃

D₄

yes

characteristic

C₂²×C₄×C₂₉₃

—

C₂²×C₄×C₂₉₃

C₂×C₄

yes

—

C₂²×C₄×C₂₉₃

D₄

yes

—

C₂²×C₄×C₂₉₃

D₄

yes

characteristic

C₂²×C₄×C₂₉₃

C₂³

yes

characteristic

C₂×C₈×C₂₉₃

—

C₂×C₈×C₂₉₃

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂³×C₂₉₃

—

C₂³×C₂₉₃

—

C₂³×C₂₉₃

C₂²⋊C₄

yes

—

C₂³×C₂₉₃

SD₁₆

yes

—

C₂³×C₂₉₃

Q₁₆

yes

—

C₂³×C₂₉₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₉₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉₃

#41

yes

—

C₈×C₂₉₃

—

C₂²×C₂₉₃

—

C₂²×C₂₉₃

—

C₂²×C₂₉₃

—

C₂²×C₂₉₃

—

C₂²×C₂₉₃

#103

yes

—

C₂²×C₂₉₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₉₃

#123

yes

—

C₂²×C₂₉₃

C₂×SD₁₆

yes

characteristic

C₂²×C₂₉₃

C₂×Q₁₆

yes

characteristic

C₂²×C₂₉₃

AGL(1,ℤ/8)

yes

—

C₄×C₂₉₃

—

C₄×C₂₉₃

#117

yes

—

C₄×C₂₉₃

#123

yes

—

Dic₂₉₃

—

Dic₂₉₃

—

C₂×C₂₉₃

—

C₂×C₂₉₃

yes

C₂×C₂₉₃

yes

C₂×C₂₉₃

yes

C₂×C₂₉₃

yes

—

C₂×C₂₉₃

#484

yes

—

C₂₉₃

#1824

yes

characteristic, complemented, Sylow

#1824

—

293

maximal, Sylow

C₂²×C₄⋊C₄

—

293

—

C₂×C₂²⋊C₈

D₂₉₃

yes

characteristic

#380

—

293

—

#340

—

293

—

C₂³×C₄

—

586

—

C₂³×C₄

C₂×D₂₉₃

yes

characteristic

C₂²×C₈

C₂×D₂₉₃

yes

—

C₂×C₄⋊C₄

—

293

—

C₂×C₄⋊C₄

—

293

—

C₂×C₄⋊C₄

—

293

—

C₂×C₄⋊C₄

—

586

—

C₂×C₄⋊C₄

—

586

—

C₂×C₂²⋊C₄

—

586

—

C₂²⋊C₈

—

#95

—

586

—

C₂⁴

D₁₁₇₂

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

586

—

C₂²×C₄

—

586

—

C₂²×C₄

—

586

—

C₂²×C₄

—

1172

—

C₂²×C₄

C₄×D₂₉₃

yes

—

C₂²×C₄

C₂²⋊D₂₉₃

yes

—

C₂²×C₄

D₁₁₇₂

yes

characteristic

100

C₂²×C₄

C₂²×D₂₉₃

yes

characteristic

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

293

—

104

C₄⋊C₄

—

586

—

105

C₄⋊C₄

—

586

—

106

C₄⋊C₄

—

586

—

107

C₂²⋊C₄

—

1172

—

108

C₂³

—

109

C₂³

—

110

C₂³

#17989779

yes

—

111

C₂³

#17989791

yes

—

112

C₂³

Dic₁₁₇₂

yes

—

113

C₂³

C₂×D₁₁₇₂

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

586

—

118

C₂×C₄

—

1172

—

119

C₂×C₄

—

1172

—

120

C₂×C₄

—

1172

—

121

C₂×C₄

—

1172

—

122

C₂×C₄

—

1172

—

123

C₂×C₄

#17989779

yes

—

124

C₂×C₄

C₂×C₄×D₂₉₃

yes

characteristic

125

C₂×C₄

C₂×C₂²⋊D₂₉₃

yes

characteristic

126

C₂×C₄

#17989801

yes

—

127

C₈

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

#22627549

yes

—

133

C₂²

C₂×#17989779

yes

characteristic

134

C₂²

#22627608

yes

—

135

C₂²

C₂×#17989791

yes

characteristic

136

C₂²

C₂×Dic₁₁₇₂

yes

characteristic

137

C₂²

#22627652

yes

—

138

C₄

—

139

C₄

—

2344

—

140

C₄

—

1172

—

141

C₄

#22627580

yes

—

142

C₄

#22627605

yes

—

143

C₂

—

144

C₂

#34035990

yes

characteristic

145

C₂

C₂×#22627549

yes

characteristic

146

C₂

#34036135

yes

characteristic

147

C₂

#34036448

yes

—

148

C₂

#34036465

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₁₇₂
commutator	C₂×C₄×C₂₉₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₉₃
Fitting	C₂×C₂²⋊C₈×C₂₉₃	C₂
socle	C₂³×C₂₉₃	C₂×D₄
prime radical	C₂⁴×C₂₉₃	D₄
cosocle	C₂²×C₄×C₂₉₃	C₂³
hypercenter	C₂×C₂²⋊C₈	D₂₉₃
hypocenter	C₂₉₃	#1824
LCS	G ▷ C₂×C₄×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center

C₂³

C₂×D₁₁₇₂

commutator

C₂×C₄×C₂₉₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₉₃

Fitting

C₂×C₂²⋊C₈×C₂₉₃

C₂

socle

C₂³×C₂₉₃

C₂×D₄

prime radical

C₂⁴×C₂₉₃

D₄

cosocle

C₂²×C₄×C₂₉₃

C₂³

hypercenter

C₂×C₂²⋊C₈

D₂₉₃

hypocenter

C₂₉₃

#1824

LCS

G ▷ C₂×C₄×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
3. #34035892	C₂	semidirect
8. C₂×C₄⋊Dic₂₉₃	C₄	semidirect
74. C₂₉₃	#1824	semidirect

H ≅ G/N

type

3. #34035892

C₂

semidirect

8. C₂×C₄⋊Dic₂₉₃

C₄

semidirect

74. C₂₉₃

#1824

semidirect

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

subgroup	index
C₂²×C₄⋊Dic₂₉₃	2
#34035989	2
#34035892	2
C₂×C₂²⋊C₈×C₂₉₃	2
#1824	293

subgroup

index

C₂²×C₄⋊Dic₂₉₃

#34035989

#34035892

C₂×C₂²⋊C₈×C₂₉₃

#1824

293

Subgroups — #49158876

Conjugacy classes of subgroups

Complements

Maximal subgroups