Group #49159176 — subgroups

Subgroups — #49159176

Order 37504.

324 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 89 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×#22627622	C₂	1	1	yes	maximal, characteristic
2	#34036607	C₂	1	1	yes	maximal, characteristic
3	#34036139	C₂	1	2	yes	maximal
4	#34035890	C₂	1	2	yes	maximal
5	C₂×#101×C₂₉₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₅₈₆	C₂²	1	1	yes	characteristic
7	#22627622	—	2	2	no	—
8	#22627622	C₄	1	4	yes	complemented
9	Q₈×Dic₂₉₃	—	2	4	no	—
10	#22627583	—	2	2	no	—
11	#22627583	C₂²	1	2	yes	—
12	C₂×#17989781	—	2	2	no	—
13	C₂×C₄×Dic₂₉₃	C₂²	1	1	yes	characteristic
14	#22627549	—	2	4	no	—
15	C₂²×Q₈×C₂₉₃	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₂₉₃	C₂²	1	2	yes	—
17	#101×C₂₉₃	—	2	2	no	—
18	C₂×Dic₅₈₆	—	2	1	no	—
19	C₂×Dic₅₈₆	—	2	2	no	—
20	C₂×Dic₅₈₆	—	2	2	no	—
21	C₂×Dic₅₈₆	C₂×C₄	1	2	yes	—
22	C₂²×Dic₂₉₃	—	2	1	no	—
23	C₂²×Dic₂₉₃	—	2	1	no	—
24	C₄⋊Dic₂₉₃	—	2	4	no	—
25	C₄⋊Dic₂₉₃	D₄	1	4	yes	—
26	#17989781	—	2	8	no	—
27	C₂²⋊Dic₂₉₃	—	2	2	no	—
28	C₄×Dic₂₉₃	—	2	4	no	—
29	C₄×Dic₂₉₃	—	2	2	no	—
30	C₄×Dic₂₉₃	—	2	1	no	—
31	C₄×Dic₂₉₃	C₂×C₄	1	2	yes	—
32	C₂×Q₈×C₂₉₃	—	2	2	no	—
33	C₂×Q₈×C₂₉₃	—	2	2	no	—
34	C₂×Q₈×C₂₉₃	C₂×C₄	1	2	yes	—
35	C₂×Q₈×C₂₉₃	D₄	1	2	yes	—
36	C₂²×C₄×C₂₉₃	D₄	1	2	yes	—
37	C₂²×C₄×C₂₉₃	C₂³	1	1	yes	characteristic
38	M₁₆×C₂₉₃	—	2	2	no	—
39	M₁₆×C₂₉₃	—	2	2	no	—
40	C₂×C₈×C₂₉₃	—	2	2	no	—
41	Dic₅₈₆	—	2	4	no	—
42	Dic₅₈₆	—	4	2	no	—
43	C₂×Dic₂₉₃	—	2	4	no	—
44	C₂×Dic₂₉₃	—	2	2	no	—
45	C₂×Dic₂₉₃	—	2	2	no	—
46	C₂×Dic₂₉₃	—	2	4	no	—
47	C₂×Dic₂₉₃	—	4	1	no	—
48	C₂×Dic₂₉₃	—	4	1	no	—
49	C₂³×C₂₉₃	C₂×D₄	1	1	yes	characteristic
50	Q₈×C₂₉₃	—	2	4	no	—
51	Q₈×C₂₉₃	—	2	4	no	—
52	C₂×C₄×C₂₉₃	—	2	2	no	—
53	C₂×C₄×C₂₉₃	—	2	2	no	—
54	C₂×C₄×C₂₉₃	C₂²⋊C₄	1	4	yes	—
55	C₂×C₄×C₂₉₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₂₉₃	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₂₉₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₂₉₃	#41	1	2	yes	—
59	C₈×C₂₉₃	—	4	2	no	—
60	C₂²×C₂₉₃	—	2	2	no	—
61	C₂²×C₂₉₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₂₉₃	#120	1	1	yes	characteristic
63	C₂²×C₂₉₃	#123	1	1	yes	characteristic
64	C₄×C₂₉₃	—	2	4	no	—
65	C₄×C₂₉₃	C₄×D₄	1	2	yes	—
66	C₄×C₂₉₃	#121	1	2	yes	—
67	Dic₂₉₃	—	2	4	no	—
68	Dic₂₉₃	—	4	2	no	—
69	Dic₂₉₃	—	4	4	no	—
70	C₂×C₂₉₃	—	2	1	no	—
71	C₂×C₂₉₃	—	2	1	no	—
72	C₂×C₂₉₃	#386	1	1	yes	characteristic
73	C₂×C₂₉₃	#456	1	2	yes	—
74	C₂₉₃	#1886	1	1	yes	characteristic, complemented, Sylow
75	#1886	—	293	1	no	maximal, Sylow
76	C₂×C₄⋊Q₈	—	293	1	no	—
77	#519	—	293	1	no	—
78	#419	—	293	2	no	—
79	C₂×#101	D₂₉₃	1	1	yes	characteristic
80	#339	—	293	2	no	—
81	C₂²×Q₈	—	293	1	no	—
82	C₂²×Q₈	C₂×D₂₉₃	1	1	yes	characteristic
83	C₂×M₁₆	C₂×D₂₉₃	1	2	yes	—
84	C₄⋊Q₈	—	293	4	no	—
85	C₄⋊Q₈	—	586	2	no	—
86	C₄×Q₈	—	586	4	no	—
87	#117	—	293	2	no	—
88	#117	—	586	2	no	—
89	C₂×C₄⋊C₄	—	586	2	no	—
90	C₂×C₄²	—	293	1	no	—
91	#103	—	586	4	no	—
92	#101	—	2	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	—	2	2	no	—
95	C₂×Q₈	—	293	2	no	—
96	C₂×Q₈	—	586	1	no	—
97	C₂×Q₈	—	586	2	no	—
98	C₂×Q₈	—	586	2	no	—
99	C₂×Q₈	C₄×D₂₉₃	1	2	yes	—
100	C₂×Q₈	C₂²⋊D₂₉₃	1	2	yes	—
101	C₂²×C₄	—	586	1	no	—
102	C₂²×C₄	—	586	1	no	—
103	C₂²×C₄	D₁₁₇₂	1	2	yes	—
104	C₂²×C₄	C₂²×D₂₉₃	1	1	yes	characteristic
105	M₁₆	—	2	2	no	—
106	M₁₆	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₄⋊C₄	—	293	4	no	—
109	C₄⋊C₄	—	586	4	no	—
110	C₄⋊C₄	—	586	8	no	—
111	C₂²⋊C₄	—	586	2	no	—
112	C₄²	—	293	2	no	—
113	C₄²	—	586	4	no	—
114	C₄²	—	586	2	no	—
115	C₄²	—	586	1	no	—
116	C₂³	C₂×D₁₁₇₂	1	1	yes	characteristic
117	Q₈	—	2	4	no	—
118	Q₈	—	2	4	no	—
119	Q₈	—	586	4	no	—
120	Q₈	—	1172	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	586	4	no	—
124	C₂×C₄	—	586	2	no	—
125	C₂×C₄	—	586	4	no	—
126	C₂×C₄	—	586	2	no	—
127	C₂×C₄	—	1172	1	no	—
128	C₂×C₄	—	1172	1	no	—
129	C₂×C₄	#17989779	1	4	yes	—
130	C₂×C₄	C₂×C₄×D₂₉₃	1	1	yes	characteristic
131	C₂×C₄	D₄×D₂₉₃	1	2	yes	—
132	C₂×C₄	C₂×C₂²⋊D₂₉₃	1	1	yes	characteristic
133	C₂×C₄	#17989801	1	2	yes	—
134	C₈	—	4	2	no	—
135	C₂²	—	2	2	no	—
136	C₂²	C₂×#17989779	1	1	yes	characteristic
137	C₂²	#22627594	1	1	yes	characteristic
138	C₂²	#22627608	1	1	yes	characteristic
139	C₄	—	2	4	no	—
140	C₄	—	586	4	no	—
141	C₄	—	1172	2	no	—
142	C₄	—	1172	4	no	—
143	C₄	#22627584	1	2	yes	—
144	C₄	#22627596	1	2	yes	—
145	C₂	—	2	1	no	—
146	C₂	—	2	1	no	—
147	C₂	#34036006	1	1	yes	characteristic
148	C₂	#34036314	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22627622

C₂

yes

maximal, characteristic

#34036607

C₂

yes

maximal, characteristic

#34036139

C₂

yes

maximal

#34035890

C₂

yes

maximal

C₂×#101×C₂₉₃

C₂

yes

maximal, characteristic

C₂²×Dic₅₈₆

C₂²

yes

characteristic

#22627622

—

#22627622

C₄

yes

complemented

Q₈×Dic₂₉₃

—

#22627583

—

#22627583

C₂²

yes

—

C₂×#17989781

—

C₂×C₄×Dic₂₉₃

C₂²

yes

characteristic

#22627549

—

C₂²×Q₈×C₂₉₃

C₂²

yes

characteristic

C₂×M₁₆×C₂₉₃

C₂²

yes

—

#101×C₂₉₃

—

C₂×Dic₅₈₆

—

C₂×Dic₅₈₆

—

C₂×Dic₅₈₆

—

C₂×Dic₅₈₆

C₂×C₄

yes

—

C₂²×Dic₂₉₃

—

C₂²×Dic₂₉₃

—

C₄⋊Dic₂₉₃

—

C₄⋊Dic₂₉₃

D₄

yes

—

#17989781

—

C₂²⋊Dic₂₉₃

—

C₄×Dic₂₉₃

—

C₄×Dic₂₉₃

—

C₄×Dic₂₉₃

—

C₄×Dic₂₉₃

C₂×C₄

yes

—

C₂×Q₈×C₂₉₃

—

C₂×Q₈×C₂₉₃

—

C₂×Q₈×C₂₉₃

C₂×C₄

yes

—

C₂×Q₈×C₂₉₃

D₄

yes

—

C₂²×C₄×C₂₉₃

D₄

yes

—

C₂²×C₄×C₂₉₃

C₂³

yes

characteristic

M₁₆×C₂₉₃

—

M₁₆×C₂₉₃

—

C₂×C₈×C₂₉₃

—

Dic₅₈₆

—

Dic₅₈₆

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂³×C₂₉₃

C₂×D₄

yes

characteristic

Q₈×C₂₉₃

—

Q₈×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

—

C₂×C₄×C₂₉₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉₃

#41

yes

—

C₈×C₂₉₃

—

C₂²×C₂₉₃

—

C₂²×C₂₉₃

C₂×C₂²⋊C₄

yes

C₂²×C₂₉₃

yes

C₂²×C₂₉₃

yes

C₄×C₂₉₃

—

C₄×C₂₉₃

C₄×D₄

yes

—

C₄×C₂₉₃

#121

yes

—

Dic₂₉₃

—

Dic₂₉₃

—

Dic₂₉₃

—

C₂×C₂₉₃

—

C₂×C₂₉₃

—

C₂×C₂₉₃

#386

yes

characteristic

C₂×C₂₉₃

#456

yes

—

C₂₉₃

#1886

yes

characteristic, complemented, Sylow

#1886

—

293

maximal, Sylow

C₂×C₄⋊Q₈

—

293

—

#519

—

293

—

#419

—

293

—

C₂×#101

D₂₉₃

yes

characteristic

#339

—

293

—

C₂²×Q₈

—

293

—

C₂²×Q₈

C₂×D₂₉₃

yes

characteristic

C₂×M₁₆

C₂×D₂₉₃

yes

—

C₄⋊Q₈

—

293

—

C₄⋊Q₈

—

586

—

C₄×Q₈

—

586

—

#117

—

293

—

#117

—

586

—

C₂×C₄⋊C₄

—

586

—

C₂×C₄²

—

293

—

#103

—

586

—

#101

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

293

—

C₂×Q₈

—

586

—

C₂×Q₈

—

586

—

C₂×Q₈

—

586

—

C₂×Q₈

C₄×D₂₉₃

yes

—

100

C₂×Q₈

C₂²⋊D₂₉₃

yes

—

101

C₂²×C₄

—

586

—

102

C₂²×C₄

—

586

—

103

C₂²×C₄

D₁₁₇₂

yes

—

104

C₂²×C₄

C₂²×D₂₉₃

yes

characteristic

105

M₁₆

—

106

M₁₆

—

107

C₂×C₈

—

108

C₄⋊C₄

—

293

—

109

C₄⋊C₄

—

586

—

110

C₄⋊C₄

—

586

—

111

C₂²⋊C₄

—

586

—

112

C₄²

—

293

—

113

C₄²

—

586

—

114

C₄²

—

586

—

115

C₄²

—

586

—

116

C₂³

C₂×D₁₁₇₂

yes

characteristic

117

Q₈

—

118

Q₈

—

119

Q₈

—

586

—

120

Q₈

—

1172

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

586

—

124

C₂×C₄

—

586

—

125

C₂×C₄

—

586

—

126

C₂×C₄

—

586

—

127

C₂×C₄

—

1172

—

128

C₂×C₄

—

1172

—

129

C₂×C₄

#17989779

yes

—

130

C₂×C₄

C₂×C₄×D₂₉₃

yes

characteristic

131

C₂×C₄

D₄×D₂₉₃

yes

—

132

C₂×C₄

C₂×C₂²⋊D₂₉₃

yes

characteristic

133

C₂×C₄

#17989801

yes

—

134

C₈

—

135

C₂²

—

136

C₂²

C₂×#17989779

yes

characteristic

137

C₂²

#22627594

yes

characteristic

138

C₂²

#22627608

yes

characteristic

139

C₄

—

140

C₄

—

586

—

141

C₄

—

1172

—

142

C₄

—

1172

—

143

C₄

#22627584

yes

—

144

C₄

#22627596

yes

—

145

C₂

—

146

C₂

—

147

C₂

#34036006

yes

characteristic

148

C₂

#34036314

yes

—

	subgroup	G/H
center	C₂²	#22627594
commutator	C₂×C₄×C₂₉₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₉₃
Fitting	C₂×#101×C₂₉₃	C₂
socle	C₂²×C₂₉₃	#120
prime radical	C₂³×C₂₉₃	C₂×D₄
cosocle	C₂²×C₄×C₂₉₃	C₂³
hypercenter	C₂×#101	D₂₉₃
hypocenter	C₂₉₃	#1886
LCS	G ▷ C₂×C₄×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

subgroup

G/H

center

C₂²

#22627594

commutator

C₂×C₄×C₂₉₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₉₃

Fitting

C₂×#101×C₂₉₃

C₂

socle

C₂²×C₂₉₃

#120

prime radical

C₂³×C₂₉₃

C₂×D₄

cosocle

C₂²×C₄×C₂₉₃

C₂³

D₂₉₃

C₂₉₃

G ▷ C₂×C₄×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#101

N	H ≅ G/N	type
8. #22627622	C₄	semidirect
74. C₂₉₃	#1886	semidirect

H ≅ G/N

type

8. #22627622

C₄

semidirect

74. C₂₉₃

#1886

semidirect

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

subgroup	index
C₂×#22627622	2
#34036607	2
#34036139	2
#34035890	2
C₂×#101×C₂₉₃	2
#1886	293

subgroup

index

C₂×#101×C₂₉₃

293

Subgroups — #49159176

Conjugacy classes of subgroups

Complements

Maximal subgroups