Group #49158708 — subgroups

Subgroups — #49158708

Order 37504.

564 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂³×Dic₅₈₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#22627542	C₂	1	4	yes	maximal, complemented
3	#34036093	C₂	1	1	yes	maximal, characteristic
4	#407×C₂₉₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₅₈₆	—	2	8	no	—
6	C₂²×Dic₅₈₆	—	2	2	no	—
7	C₂²×Dic₅₈₆	—	2	2	no	—
8	C₂²×Dic₅₈₆	C₄	1	2	yes	—
9	C₂²×Dic₅₈₆	C₂²	1	2	yes	—
10	C₂³×Dic₂₉₃	C₄	1	2	yes	—
11	C₂×#17989786	C₂²	1	2	yes	—
12	#22627542	—	2	8	no	—
13	C₂²⋊#14556873	—	2	2	no	—
14	C₂³×C₄×C₂₉₃	C₂²	1	1	yes	characteristic
15	C₂×M₁₆×C₂₉₃	C₂²	1	2	yes	—
16	C₂²⋊C₈×C₂₉₃	—	2	2	no	—
17	C₂×Dic₅₈₆	—	2	16	no	—
18	C₂×Dic₅₈₆	—	2	16	no	—
19	C₂×Dic₅₈₆	—	2	8	no	—
20	C₂×Dic₅₈₆	—	2	4	no	—
21	C₂×Dic₅₈₆	—	2	4	no	—
22	C₂×Dic₅₈₆	—	2	2	no	—
23	C₂×Dic₅₈₆	—	2	2	no	—
24	C₂×Dic₅₈₆	D₄	1	8	yes	—
25	C₂²×Dic₂₉₃	—	2	8	no	—
26	C₂²×Dic₂₉₃	—	2	4	no	—
27	C₂²×Dic₂₉₃	C₂×C₄	1	4	yes	—
28	#17989786	—	2	4	no	—
29	C₂×#14556873	—	2	2	no	—
30	C₂⁴×C₂₉₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₂₉₃	—	2	4	no	—
32	C₂²×C₄×C₂₉₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₂₉₃	D₄	1	2	yes	—
34	C₂²×C₄×C₂₉₃	D₄	1	2	yes	—
35	C₂²×C₄×C₂₉₃	C₂³	1	1	yes	characteristic
36	M₁₆×C₂₉₃	—	2	4	no	—
37	C₂×C₈×C₂₉₃	—	2	2	no	—
38	Dic₅₈₆	—	2	16	no	—
39	Dic₅₈₆	—	2	8	no	—
40	Dic₅₈₆	—	2	8	no	—
41	C₂×Dic₂₉₃	—	2	16	no	—
42	C₂×Dic₂₉₃	—	2	4	no	—
43	C₂×Dic₂₉₃	—	2	4	no	—
44	C₂×Dic₂₉₃	C₂²⋊C₄	1	8	yes	—
45	#14556873	—	4	2	no	—
46	C₂³×C₂₉₃	—	2	4	no	—
47	C₂³×C₂₉₃	—	2	2	no	—
48	C₂³×C₂₉₃	C₂²⋊C₄	1	2	yes	—
49	C₂³×C₂₉₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₂₉₃	—	2	2	no	—
51	C₂×C₄×C₂₉₃	—	2	4	no	—
52	C₂×C₄×C₂₉₃	—	2	4	no	—
53	C₂×C₄×C₂₉₃	C₂²⋊C₄	1	2	yes	—
54	C₂×C₄×C₂₉₃	C₂×D₄	1	4	yes	—
55	C₂×C₄×C₂₉₃	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₂₉₃	C₂×D₄	1	1	yes	characteristic
57	C₈×C₂₉₃	—	4	2	no	—
58	C₂²×C₂₉₃	—	2	8	no	—
59	C₂²×C₂₉₃	—	2	2	no	—
60	C₂²×C₂₉₃	—	2	2	no	—
61	C₂²×C₂₉₃	—	2	2	no	—
62	C₂²×C₂₉₃	#101	1	4	yes	complemented
63	C₂²×C₂₉₃	C₂×C₂²⋊C₄	1	2	yes	—
64	C₂²×C₂₉₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₄×C₂₉₃	—	2	2	no	—
66	C₄×C₂₉₃	#120	1	2	yes	—
67	C₄×C₂₉₃	#120	1	2	yes	—
68	Dic₂₉₃	—	2	8	no	—
69	C₂×C₂₉₃	—	2	4	no	—
70	C₂×C₂₉₃	—	2	2	no	—
71	C₂×C₂₉₃	#379	1	1	yes	characteristic
72	C₂×C₂₉₃	C₂×#101	1	2	yes	—
73	C₂₉₃	#1784	1	1	yes	characteristic, complemented, Sylow
74	#1784	—	293	1	no	maximal, Sylow
75	C₂³×Q₈	—	293	1	no	—
76	C₂×#101	—	293	4	no	—
77	#407	—	293	1	no	—
78	#407	D₂₉₃	1	1	yes	characteristic
79	C₂²×Q₈	—	293	2	no	—
80	C₂²×Q₈	—	293	2	no	—
81	C₂²×Q₈	—	586	8	no	—
82	C₂²×Q₈	—	586	2	no	—
83	C₂²×Q₈	—	586	2	no	—
84	C₂³×C₄	—	293	2	no	—
85	C₂³×C₄	C₂×D₂₉₃	1	1	yes	characteristic
86	C₂×M₁₆	—	293	2	no	—
87	C₂×M₁₆	C₂×D₂₉₃	1	2	yes	—
88	#101	—	586	8	no	—
89	C₂²⋊C₈	—	2	2	no	—
90	C₂²⋊C₈	—	586	2	no	—
91	C₂⁴	C₄×D₂₉₃	1	1	yes	characteristic
92	C₂×Q₈	—	293	8	no	—
93	C₂×Q₈	—	586	8	no	—
94	C₂×Q₈	—	586	16	no	—
95	C₂×Q₈	—	586	16	no	—
96	C₂×Q₈	—	586	4	no	—
97	C₂×Q₈	—	586	4	no	—
98	C₂×Q₈	—	586	2	no	—
99	C₂×Q₈	—	586	2	no	—
100	C₂²×C₄	—	2	4	no	—
101	C₂²×C₄	—	293	4	no	—
102	C₂²×C₄	—	586	8	no	—
103	C₂²×C₄	—	586	4	no	—
104	C₂²×C₄	C₄×D₂₉₃	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊D₂₉₃	1	2	yes	—
106	C₂²×C₄	D₁₁₇₂	1	2	yes	—
107	C₂²×C₄	C₂²×D₂₉₃	1	1	yes	characteristic
108	M₁₆	—	2	4	no	—
109	M₁₆	—	586	4	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	586	2	no	—
112	C₂³	—	2	4	no	—
113	C₂³	—	2	2	no	—
114	C₂³	#17989779	1	2	yes	—
115	C₂³	C₂×C₄×D₂₉₃	1	1	yes	characteristic
116	Q₈	—	586	16	no	—
117	Q₈	—	586	8	no	—
118	Q₈	—	586	8	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	2	4	no	—
122	C₂×C₄	—	293	8	no	—
123	C₂×C₄	—	586	16	no	—
124	C₂×C₄	—	586	4	no	—
125	C₂×C₄	—	586	4	no	—
126	C₂×C₄	#17989779	1	2	yes	—
127	C₂×C₄	D₄×D₂₉₃	1	4	yes	—
128	C₂×C₄	C₂×C₂²⋊D₂₉₃	1	1	yes	characteristic
129	C₂×C₄	C₂×D₁₁₇₂	1	1	yes	characteristic
130	C₈	—	4	2	no	—
131	C₈	—	1172	2	no	—
132	C₂²	—	2	8	no	—
133	C₂²	—	2	2	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	2	2	no	—
136	C₂²	#22627542	1	4	yes	complemented
137	C₂²	C₂²⋊C₄×D₂₉₃	1	2	yes	—
138	C₂²	C₂×#17989779	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	586	8	no	—
141	C₄	#22627592	1	2	yes	—
142	C₄	#22627594	1	2	yes	—
143	C₂	—	2	4	no	—
144	C₂	—	2	2	no	—
145	C₂	#34035984	1	1	yes	characteristic
146	C₂	C₂×#22627542	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×Dic₅₈₆

C₂

yes

maximal, characteristic

C₂×#22627542

C₂

yes

maximal, complemented

#34036093

C₂

yes

maximal, characteristic

#407×C₂₉₃

C₂

yes

maximal, characteristic

C₂²×Dic₅₈₆

—

C₂²×Dic₅₈₆

—

C₂²×Dic₅₈₆

—

C₂²×Dic₅₈₆

C₄

yes

—

C₂²×Dic₅₈₆

C₂²

yes

—

C₂³×Dic₂₉₃

C₄

yes

—

C₂×#17989786

C₂²

yes

—

#22627542

—

C₂²⋊#14556873

—

C₂³×C₄×C₂₉₃

C₂²

yes

characteristic

C₂×M₁₆×C₂₉₃

C₂²

yes

—

C₂²⋊C₈×C₂₉₃

—

C₂×Dic₅₈₆

—

C₂×Dic₅₈₆

—

C₂×Dic₅₈₆

—

C₂×Dic₅₈₆

—

C₂×Dic₅₈₆

—

C₂×Dic₅₈₆

—

C₂×Dic₅₈₆

—

C₂×Dic₅₈₆

D₄

yes

—

C₂²×Dic₂₉₃

—

C₂²×Dic₂₉₃

—

C₂²×Dic₂₉₃

C₂×C₄

yes

—

#17989786

—

C₂×#14556873

—

C₂⁴×C₂₉₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₉₃

—

C₂²×C₄×C₂₉₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₉₃

D₄

yes

—

C₂²×C₄×C₂₉₃

D₄

yes

—

C₂²×C₄×C₂₉₃

C₂³

yes

characteristic

M₁₆×C₂₉₃

—

C₂×C₈×C₂₉₃

—

Dic₅₈₆

—

Dic₅₈₆

—

Dic₅₈₆

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

C₂²⋊C₄

yes

—

#14556873

—

C₂³×C₂₉₃

—

C₂³×C₂₉₃

—

C₂³×C₂₉₃

C₂²⋊C₄

yes

—

C₂³×C₂₉₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₉₃

C₂×D₄

yes

—

C₂×C₄×C₂₉₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉₃

C₂×D₄

yes

characteristic

C₈×C₂₉₃

—

C₂²×C₂₉₃

—

C₂²×C₂₉₃

—

C₂²×C₂₉₃

—

C₂²×C₂₉₃

—

C₂²×C₂₉₃

#101

yes

complemented

C₂²×C₂₉₃

C₂×C₂²⋊C₄

yes

—

C₂²×C₂₉₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₉₃

—

C₄×C₂₉₃

#120

yes

—

C₄×C₂₉₃

#120

yes

—

Dic₂₉₃

—

C₂×C₂₉₃

—

C₂×C₂₉₃

—

C₂×C₂₉₃

#379

yes

characteristic

C₂×C₂₉₃

C₂×#101

yes

—

C₂₉₃

#1784

yes

characteristic, complemented, Sylow

#1784

—

293

maximal, Sylow

C₂³×Q₈

—

293

—

C₂×#101

—

293

—

#407

—

293

—

#407

D₂₉₃

yes

characteristic

C₂²×Q₈

—

293

—

C₂²×Q₈

—

293

—

C₂²×Q₈

—

586

—

C₂²×Q₈

—

586

—

C₂²×Q₈

—

586

—

C₂³×C₄

—

293

—

C₂³×C₄

C₂×D₂₉₃

yes

characteristic

C₂×M₁₆

—

293

—

C₂×M₁₆

C₂×D₂₉₃

yes

—

#101

—

586

—

C₂²⋊C₈

—

C₂²⋊C₈

—

586

—

C₂⁴

C₄×D₂₉₃

yes

characteristic

C₂×Q₈

—

293

—

C₂×Q₈

—

586

—

C₂×Q₈

—

586

—

C₂×Q₈

—

586

—

C₂×Q₈

—

586

—

C₂×Q₈

—

586

—

C₂×Q₈

—

586

—

C₂×Q₈

—

586

—

100

C₂²×C₄

—

101

C₂²×C₄

—

293

—

102

C₂²×C₄

—

586

—

103

C₂²×C₄

—

586

—

104

C₂²×C₄

C₄×D₂₉₃

yes

characteristic

105

C₂²×C₄

C₂²⋊D₂₉₃

yes

—

106

C₂²×C₄

D₁₁₇₂

yes

—

107

C₂²×C₄

C₂²×D₂₉₃

yes

characteristic

108

M₁₆

—

109

M₁₆

—

586

—

110

C₂×C₈

—

111

C₂×C₈

—

586

—

112

C₂³

—

113

C₂³

—

114

C₂³

#17989779

yes

—

115

C₂³

C₂×C₄×D₂₉₃

yes

characteristic

116

Q₈

—

586

—

117

Q₈

—

586

—

118

Q₈

—

586

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

293

—

123

C₂×C₄

—

586

—

124

C₂×C₄

—

586

—

125

C₂×C₄

—

586

—

126

C₂×C₄

#17989779

yes

—

127

C₂×C₄

D₄×D₂₉₃

yes

—

128

C₂×C₄

C₂×C₂²⋊D₂₉₃

yes

characteristic

129

C₂×C₄

C₂×D₁₁₇₂

yes

characteristic

130

C₈

—

131

C₈

—

1172

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

#22627542

yes

complemented

137

C₂²

C₂²⋊C₄×D₂₉₃

yes

—

138

C₂²

C₂×#17989779

yes

characteristic

139

C₄

—

140

C₄

—

586

—

141

C₄

#22627592

yes

—

142

C₄

#22627594

yes

—

143

C₂

—

144

C₂

—

145

C₂

#34035984

yes

characteristic

146

C₂

C₂×#22627542

yes

—

	subgroup	G/H
center	C₂²	C₂×#17989779
commutator	C₂³×C₂₉₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₉₃
Fitting	#407×C₂₉₃	C₂
socle	C₂²×C₂₉₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₂₉₃	C₂×C₄
cosocle	C₂²×C₄×C₂₉₃	C₂³
hypercenter	#407	D₂₉₃
hypocenter	C₂₉₃	#1784
LCS	G ▷ C₂³×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂²

C₂×#17989779

commutator

C₂³×C₂₉₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₉₃

Fitting

#407×C₂₉₃

C₂

socle

C₂²×C₂₉₃

C₂×C₂²⋊C₄

prime radical

C₂⁴×C₂₉₃

C₂×C₄

cosocle

C₂²×C₄×C₂₉₃

C₂³

D₂₉₃

C₂₉₃

G ▷ C₂³×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
2. C₂×#22627542	C₂	semidirect
62. C₂²×C₂₉₃	#101	semidirect
73. C₂₉₃	#1784	semidirect
136. C₂²	#22627542	semidirect

H ≅ G/N

type

2. C₂×#22627542

C₂

semidirect

62. C₂²×C₂₉₃

#101

semidirect

73. C₂₉₃

#1784

semidirect

136. C₂²

#22627542

semidirect

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

subgroup	index
C₂³×Dic₅₈₆	2
C₂×#22627542	2
#34036093	2
#407×C₂₉₃	2
#1784	293

subgroup

index

C₂³×Dic₅₈₆

C₂×#22627542

#34036093

#407×C₂₉₃

#1784

293

Subgroups — #49158708

Conjugacy classes of subgroups

Complements

Maximal subgroups