Group #49159928 — subgroups

Subgroups — #49159928

Order 37504.

294 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 87 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×#22627544	C₂	1	2	yes	maximal
2	#34036031	C₂	1	1	yes	maximal, characteristic
3	#34035891	C₂	1	2	yes	maximal, complemented
4	#34035884	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#125×C₂₉₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₁₇₂	C₂²	1	1	yes	characteristic
7	C₂²×#14556873	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₂₉₃	C₂²	1	1	yes	characteristic
9	C₂×#17989779	—	2	2	no	—
10	#22627544	—	2	4	no	—
11	#125×C₂₉₃	—	2	4	no	—
12	C₂×C₄⋊C₄×C₂₉₃	—	2	2	no	—
13	C₂×C₄⋊C₄×C₂₉₃	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₉₃	C₂²	1	1	yes	characteristic
15	C₂³×D₂₉₃	—	2	1	no	—
16	C₂×D₁₁₇₂	—	2	2	no	—
17	C₂×D₁₁₇₂	—	2	2	no	—
18	C₂×D₁₁₇₂	—	2	1	no	—
19	C₂×D₁₁₇₂	Q₈	1	2	yes	—
20	C₂²×Dic₂₉₃	—	2	1	no	—
21	C₂×#14556873	—	2	2	no	—
22	C₂×#14556873	—	2	2	no	—
23	C₂×#14556873	—	2	2	no	—
24	C₄⋊Dic₂₉₃	—	2	2	no	—
25	#17989779	—	4	4	no	—
26	C₂²×C₄×C₂₉₃	—	2	2	no	—
27	C₂²×C₄×C₂₉₃	D₄	1	2	yes	—
28	C₂²×C₄×C₂₉₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₂₉₃	—	2	8	no	—
30	C₄⋊C₄×C₂₉₃	—	2	2	no	—
31	C₄⋊C₄×C₂₉₃	D₄	1	4	yes	—
32	C₄²×C₂₉₃	—	2	2	no	—
33	C₂²×D₂₉₃	—	2	2	no	—
34	C₂²×D₂₉₃	—	4	1	no	—
35	C₂²×D₂₉₃	—	4	1	no	—
36	C₂²×D₂₉₃	—	4	2	no	—
37	C₂²×D₂₉₃	—	4	2	no	—
38	D₁₁₇₂	—	2	4	no	—
39	D₁₁₇₂	—	4	2	no	—
40	C₂×Dic₂₉₃	—	4	1	no	—
41	C₂×Dic₂₉₃	—	4	2	no	—
42	#14556873	—	4	2	no	—
43	C₂³×C₂₉₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₂₉₃	—	2	2	no	—
45	C₂×C₄×C₂₉₃	—	2	4	no	—
46	C₂×C₄×C₂₉₃	—	2	2	no	—
47	C₂×C₄×C₂₉₃	—	2	2	no	—
48	C₂×C₄×C₂₉₃	—	4	2	no	—
49	C₂×C₄×C₂₉₃	—	4	2	no	—
50	C₂×C₄×C₂₉₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₂₉₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×C₂₉₃	#41	1	2	yes	—
53	C₂×C₄×C₂₉₃	#41	1	1	yes	characteristic
54	C₂²×C₂₉₃	#120	1	1	yes	characteristic
55	C₂²×C₂₉₃	C₂²⋊Q₈	1	1	yes	characteristic
56	C₂²×C₂₉₃	#124	1	1	yes	characteristic
57	C₂²×C₂₉₃	#135	1	2	yes	—
58	C₂²×C₂₉₃	AGL(1,ℤ/8)	1	2	yes	—
59	C₂×D₂₉₃	—	4	2	no	—
60	C₂×D₂₉₃	—	4	4	no	—
61	C₂×D₂₉₃	—	8	2	no	—
62	C₂×D₂₉₃	—	8	1	no	—
63	C₂×D₂₉₃	—	8	1	no	—
64	C₄×C₂₉₃	—	4	2	no	—
65	C₄×C₂₉₃	—	4	4	no	—
66	C₄×C₂₉₃	C₂²⋊Q₈	1	2	yes	—
67	C₄×C₂₉₃	#124	1	2	yes	—
68	Dic₂₉₃	—	8	1	no	—
69	C₂×C₂₉₃	#393	1	1	yes	characteristic
70	C₂×C₂₉₃	#449	1	2	yes	—
71	C₂×C₂₉₃	#478	1	2	yes	—
72	C₂×C₂₉₃	#488	1	1	yes	characteristic
73	C₂×C₂₉₃	#490	1	1	yes	characteristic
74	D₂₉₃	—	8	2	no	—
75	C₂₉₃	#2025	1	1	yes	characteristic, complemented, Sylow
76	#2025	—	293	1	no	maximal, Sylow
77	C₂×#125	D₂₉₃	1	1	yes	characteristic, complemented
78	C₂×#102	—	293	2	no	—
79	#390	—	293	1	no	—
80	#340	—	293	2	no	—
81	#336	—	293	1	no	—
82	C₂²×D₄	—	293	1	no	—
83	C₂²×C₈	—	293	2	no	—
84	#125	—	2	4	no	—
85	C₂×C₄⋊C₄	—	2	2	no	—
86	C₂×C₄⋊C₄	—	293	1	no	—
87	C₂×C₄⋊C₄	C₂×D₂₉₃	1	2	yes	—
88	C₂×C₂²⋊C₄	—	586	2	no	—
89	C₂×C₄²	C₂×D₂₉₃	1	1	yes	characteristic
90	#102	—	586	4	no	—
91	C₂⁴	—	586	1	no	—
92	C₂×D₄	—	293	2	no	—
93	C₂×D₄	—	586	2	no	—
94	C₂×D₄	—	586	2	no	—
95	C₂×D₄	—	586	1	no	—
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	—	586	1	no	—
98	C₂²×C₄	C₂²⋊D₂₉₃	1	2	yes	—
99	C₂²×C₄	C₂²×D₂₉₃	1	1	yes	characteristic
100	C₂×C₈	—	586	2	no	—
101	C₂×C₈	—	586	2	no	—
102	C₂×C₈	—	586	2	no	—
103	C₄⋊C₄	—	2	8	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	586	2	no	—
106	C₄⋊C₄	C₂²⋊D₂₉₃	1	4	yes	—
107	C₂²⋊C₄	—	1172	4	no	—
108	C₄²	—	2	2	no	—
109	C₂³	—	586	2	no	—
110	C₂³	—	1172	1	no	—
111	C₂³	—	1172	1	no	—
112	C₂³	—	1172	2	no	—
113	C₂³	—	1172	2	no	—
114	C₂³	C₂×C₂²⋊D₂₉₃	1	1	yes	characteristic
115	D₄	—	586	4	no	—
116	D₄	—	1172	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	4	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	1172	1	no	—
124	C₂×C₄	—	1172	2	no	—
125	C₂×C₄	C₂×C₂²⋊D₂₉₃	1	2	yes	—
126	C₂×C₄	Q₈×D₂₉₃	1	1	yes	characteristic
127	C₂×C₄	#17989803	1	2	yes	—
128	C₂×C₄	#17989803	1	1	yes	characteristic
129	C₈	—	1172	2	no	—
130	C₂²	—	2344	1	no	—
131	C₂²	—	2344	1	no	—
132	C₂²	—	2344	2	no	—
133	C₂²	—	1172	2	no	—
134	C₂²	—	1172	4	no	—
135	C₂²	#22627593	1	1	yes	characteristic
136	C₂²	#22627600	1	1	yes	characteristic
137	C₂²	#22627613	1	1	yes	characteristic
138	C₂²	#22627644	1	2	yes	—
139	C₂²	#22627649	1	2	yes	—
140	C₄	—	4	2	no	—
141	C₄	—	4	4	no	—
142	C₄	—	2344	1	no	—
143	C₄	#22627600	1	2	yes	—
144	C₄	#22627613	1	2	yes	—
145	C₂	—	2344	2	no	—
146	C₂	#34036040	1	1	yes	characteristic
147	C₂	#34036274	1	2	yes	—
148	C₂	#34036426	1	2	yes	—
149	C₂	#34036486	1	1	yes	characteristic
150	C₂	#34036495	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22627544

C₂

yes

maximal

#34036031

C₂

yes

maximal, characteristic

#34035891

C₂

yes

maximal, complemented

#34035884

C₂

yes

maximal, characteristic, complemented

C₂×#125×C₂₉₃

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₁₇₂

C₂²

yes

characteristic

C₂²×#14556873

C₂²

yes

—

C₂×C₄⋊Dic₂₉₃

C₂²

yes

characteristic

C₂×#17989779

—

#22627544

—

#125×C₂₉₃

—

C₂×C₄⋊C₄×C₂₉₃

—

C₂×C₄⋊C₄×C₂₉₃

C₂²

yes

—

C₂×C₄²×C₂₉₃

C₂²

yes

characteristic

C₂³×D₂₉₃

—

C₂×D₁₁₇₂

—

C₂×D₁₁₇₂

—

C₂×D₁₁₇₂

—

C₂×D₁₁₇₂

Q₈

yes

—

C₂²×Dic₂₉₃

—

C₂×#14556873

—

C₂×#14556873

—

C₂×#14556873

—

C₄⋊Dic₂₉₃

—

#17989779

—

C₂²×C₄×C₂₉₃

—

C₂²×C₄×C₂₉₃

D₄

yes

—

C₂²×C₄×C₂₉₃

C₂³

yes

characteristic

C₄⋊C₄×C₂₉₃

—

C₄⋊C₄×C₂₉₃

—

C₄⋊C₄×C₂₉₃

D₄

yes

—

C₄²×C₂₉₃

—

C₂²×D₂₉₃

—

C₂²×D₂₉₃

—

C₂²×D₂₉₃

—

C₂²×D₂₉₃

—

C₂²×D₂₉₃

—

D₁₁₇₂

—

D₁₁₇₂

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

#14556873

—

C₂³×C₂₉₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

C₂×D₄

yes

—

C₂×C₄×C₂₉₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₉₃

#41

yes

—

C₂×C₄×C₂₉₃

yes

C₂²×C₂₉₃

yes

C₂²×C₂₉₃

C₂²⋊Q₈

yes

C₂²×C₂₉₃

yes

C₂²×C₂₉₃

yes

—

C₂²×C₂₉₃

AGL(1,ℤ/8)

yes

—

C₂×D₂₉₃

—

C₂×D₂₉₃

—

C₂×D₂₉₃

—

C₂×D₂₉₃

—

C₂×D₂₉₃

—

C₄×C₂₉₃

—

C₄×C₂₉₃

—

C₄×C₂₉₃

C₂²⋊Q₈

yes

—

C₄×C₂₉₃

#124

yes

—

Dic₂₉₃

—

C₂×C₂₉₃

#393

yes

characteristic

C₂×C₂₉₃

#449

yes

—

C₂×C₂₉₃

#478

yes

—

C₂×C₂₉₃

yes

C₂×C₂₉₃

yes

D₂₉₃

—

C₂₉₃

#2025

yes

characteristic, complemented, Sylow

#2025

—

293

maximal, Sylow

C₂×#125

D₂₉₃

yes

characteristic, complemented

C₂×#102

—

293

—

#390

—

293

—

#340

—

293

—

#336

—

293

—

C₂²×D₄

—

293

—

C₂²×C₈

—

293

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

293

—

C₂×C₄⋊C₄

C₂×D₂₉₃

yes

—

C₂×C₂²⋊C₄

—

586

—

C₂×C₄²

C₂×D₂₉₃

yes

characteristic

#102

—

586

—

C₂⁴

—

586

—

C₂×D₄

—

293

—

C₂×D₄

—

586

—

C₂×D₄

—

586

—

C₂×D₄

—

586

—

C₂²×C₄

—

C₂²×C₄

—

586

—

C₂²×C₄

C₂²⋊D₂₉₃

yes

—

C₂²×C₄

C₂²×D₂₉₃

yes

characteristic

100

C₂×C₈

—

586

—

101

C₂×C₈

—

586

—

102

C₂×C₈

—

586

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

586

—

106

C₄⋊C₄

C₂²⋊D₂₉₃

yes

—

107

C₂²⋊C₄

—

1172

—

108

C₄²

—

109

C₂³

—

586

—

110

C₂³

—

1172

—

111

C₂³

—

1172

—

112

C₂³

—

1172

—

113

C₂³

—

1172

—

114

C₂³

C₂×C₂²⋊D₂₉₃

yes

characteristic

115

D₄

—

586

—

116

D₄

—

1172

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

1172

—

124

C₂×C₄

—

1172

—

125

C₂×C₄

C₂×C₂²⋊D₂₉₃

yes

—

126

C₂×C₄

Q₈×D₂₉₃

yes

characteristic

127

C₂×C₄

#17989803

yes

—

128

C₂×C₄

#17989803

yes

characteristic

129

C₈

—

1172

—

130

C₂²

—

2344

—

131

C₂²

—

2344

—

132

C₂²

—

2344

—

133

C₂²

—

1172

—

134

C₂²

—

1172

—

135

C₂²

#22627593

yes

characteristic

136

C₂²

#22627600

yes

characteristic

137

C₂²

#22627613

yes

characteristic

138

C₂²

#22627644

yes

—

139

C₂²

#22627649

yes

—

140

C₄

—

141

C₄

—

142

C₄

—

2344

—

143

C₄

#22627600

yes

—

144

C₄

#22627613

yes

—

145

C₂

—

2344

—

146

C₂

#34036040

yes

characteristic

147

C₂

#34036274

yes

—

148

C₂

#34036426

yes

—

149

C₂

#34036486

yes

characteristic

150

C₂

#34036495

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₉₃
commutator, cosocle	C₂²×C₄×C₂₉₃	C₂³
Frattini	C₂²×C₄	C₂²×D₂₉₃
Fitting	C₂×#125×C₂₉₃	C₂
socle	C₂³×C₂₉₃	C₂×D₄
prime radical	C₂²×D₁₁₇₂	C₂²
hypercenter	C₂×#125	D₂₉₃
hypocenter	C₂₉₃	#2025
LCS	G ▷ C₂²×C₄×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₉₃

commutator, cosocle

C₂²×C₄×C₂₉₃

C₂³

Frattini

C₂²×C₄

C₂²×D₂₉₃

Fitting

C₂×#125×C₂₉₃

C₂

socle

C₂³×C₂₉₃

C₂×D₄

prime radical

C₂²×D₁₁₇₂

C₂²

D₂₉₃

C₂₉₃

G ▷ C₂²×C₄×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
3. #34035891	C₂	semidirect
4. #34035884	C₂	semidirect
5. C₂×#125×C₂₉₃	C₂	semidirect
75. C₂₉₃	#2025	semidirect
77. C₂×#125	D₂₉₃	semidirect

H ≅ G/N

type

3. #34035891

C₂

semidirect

4. #34035884

C₂

semidirect

5. C₂×#125×C₂₉₃

C₂

semidirect

75. C₂₉₃

#2025

semidirect

77. C₂×#125

D₂₉₃

semidirect

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

subgroup	index
C₂×#22627544	2
#34036031	2
#34035891	2
#34035884	2
C₂×#125×C₂₉₃	2
#2025	293

subgroup

index

C₂×#125×C₂₉₃

293

Subgroups — #49159928

Conjugacy classes of subgroups

Complements

Maximal subgroups