Group #49096175 — subgroups

Subgroups — #49096175

Order 35968.

304 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 113 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×#22582811	C₂	1	1	yes	maximal, characteristic
2	C₂×#22582803	C₂	1	2	yes	maximal
3	C₂×#22582789	C₂	1	1	yes	maximal, characteristic
4	#31823370	C₂	1	2	yes	maximal
5	C₂²×Q₁₆×C₂₈₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×#14555886	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₂₈₁	C₂²	1	2	yes	—
8	C₂×C₄×Dic₂₈₁	C₂²	1	1	yes	characteristic
9	#22582811	—	2	2	no	—
10	#22582803	—	2	4	no	—
11	#22582789	—	2	2	no	—
12	#22582786	—	2	4	no	—
13	C₂²×Q₈×C₂₈₁	C₂²	1	2	yes	—
14	C₂×Q₁₆×C₂₈₁	—	2	2	no	—
15	C₂×Q₁₆×C₂₈₁	—	2	2	no	—
16	C₂×Q₁₆×C₂₈₁	C₄	1	4	yes	complemented
17	C₂²×C₈×C₂₈₁	C₂²	1	1	yes	characteristic
18	C₂²×Dic₂₈₁	—	2	2	no	—
19	C₂²×Dic₂₈₁	D₄	1	2	yes	—
20	C₂×#14555886	—	2	1	no	—
21	C₂×#14555886	—	2	1	no	—
22	C₂×#14555886	D₄	1	2	yes	—
23	C₄⋊Dic₂₈₁	—	2	2	no	—
24	C₄⋊Dic₂₈₁	—	2	2	no	—
25	C₄×Dic₂₈₁	—	2	1	no	—
26	C₄×Dic₂₈₁	—	2	1	no	—
27	C₂×Q₈×C₂₈₁	—	2	2	no	—
28	C₂×Q₈×C₂₈₁	—	2	2	no	—
29	C₂×Q₈×C₂₈₁	—	2	2	no	—
30	C₂×Q₈×C₂₈₁	—	2	2	no	—
31	C₂×Q₈×C₂₈₁	—	2	2	no	—
32	C₂×Q₈×C₂₈₁	C₂×C₄	1	4	yes	—
33	C₂²×C₄×C₂₈₁	—	2	2	no	—
34	C₂²×C₄×C₂₈₁	C₂³	1	1	yes	characteristic
35	Q₁₆×C₂₈₁	—	2	8	no	—
36	C₂×C₈×C₂₈₁	C₂×C₄	1	2	yes	—
37	C₂×C₈×C₂₈₁	D₄	1	2	yes	—
38	C₂×C₈×C₂₈₁	D₄	1	2	yes	—
39	C₂×Dic₂₈₁	—	2	2	no	—
40	C₂×Dic₂₈₁	—	2	4	no	—
41	C₂×Dic₂₈₁	—	4	2	no	—
42	C₂×Dic₂₈₁	—	4	4	no	—
43	#14555886	—	2	2	no	—
44	C₂³×C₂₈₁	C₂×D₄	1	1	yes	characteristic
45	Q₈×C₂₈₁	—	2	4	no	—
46	Q₈×C₂₈₁	—	2	4	no	—
47	Q₈×C₂₈₁	—	4	2	no	—
48	Q₈×C₂₈₁	—	4	2	no	—
49	C₂×C₄×C₂₈₁	—	2	4	no	—
50	C₂×C₄×C₂₈₁	—	4	2	no	—
51	C₂×C₄×C₂₈₁	—	4	2	no	—
52	C₂×C₄×C₂₈₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₂₈₁	#41	1	1	yes	characteristic
57	C₂×C₄×C₂₈₁	#41	1	1	yes	characteristic
58	C₈×C₂₈₁	C₂²⋊C₄	1	4	yes	—
59	C₂²×C₂₈₁	C₄×D₄	1	1	yes	characteristic
60	C₂²×C₂₈₁	#121	1	1	yes	characteristic
61	C₂²×C₂₈₁	#127	1	1	yes	characteristic
62	C₂²×C₂₈₁	#137	1	4	yes	—
63	C₄×C₂₈₁	—	4	4	no	—
64	C₄×C₂₈₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₄×C₂₈₁	C₄×D₄	1	1	yes	characteristic
66	C₄×C₂₈₁	#121	1	1	yes	characteristic
67	C₄×C₂₈₁	#124	1	1	yes	characteristic
68	Dic₂₈₁	—	4	2	no	—
69	Dic₂₈₁	—	8	2	no	—
70	C₂×C₂₈₁	#390	1	1	yes	characteristic
71	C₂×C₂₈₁	#441	1	2	yes	—
72	C₂×C₂₈₁	#470	1	2	yes	—
73	C₂×C₂₈₁	#497	1	2	yes	—
74	C₂₈₁	#1971	1	1	yes	characteristic, complemented, Sylow
75	#1971	—	281	1	no	maximal, Sylow
76	C₂²×Q₁₆	D₂₈₁	1	1	yes	characteristic
77	C₂×#106	—	281	1	no	—
78	C₂×#103	—	281	2	no	—
79	C₂×#97	—	281	1	no	—
80	#391	—	281	2	no	—
81	C₂²×Q₈	C₂×D₂₈₁	1	2	yes	—
82	C₂×Q₁₆	—	2	2	no	—
83	C₂×Q₁₆	—	2	2	no	—
84	C₂×Q₁₆	Dic₂₈₁	1	4	yes	complemented
85	C₂²×C₈	—	281	1	no	—
86	C₂²×C₈	C₂×D₂₈₁	1	1	yes	characteristic
87	C₂×C₄⋊C₄	—	281	2	no	—
88	C₂×C₄²	—	281	1	no	—
89	#106	—	562	2	no	—
90	#103	—	562	4	no	—
91	#97	—	562	2	no	—
92	#95	—	562	4	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	—	2	2	no	—
95	C₂×Q₈	—	2	2	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	C₂×Dic₂₈₁	1	4	yes	—
99	C₂²×C₄	—	2	2	no	—
100	C₂²×C₄	—	281	2	no	—
101	C₂²×C₄	—	562	2	no	—
102	C₂²×C₄	C₂²×D₂₈₁	1	1	yes	characteristic
103	Q₁₆	—	2	8	no	—
104	C₂×C₈	—	281	2	no	—
105	C₂×C₈	—	562	1	no	—
106	C₂×C₈	—	562	1	no	—
107	C₂×C₈	C₂×Dic₂₈₁	1	2	yes	—
108	C₂×C₈	C₂²⋊D₂₈₁	1	2	yes	—
109	C₂×C₈	C₂²⋊D₂₈₁	1	2	yes	—
110	C₄⋊C₄	—	562	2	no	—
111	C₄⋊C₄	—	562	2	no	—
112	C₄²	—	562	1	no	—
113	C₄²	—	562	1	no	—
114	C₂³	D₄×D₂₈₁	1	1	yes	characteristic
115	Q₈	—	2	4	no	—
116	Q₈	—	2	4	no	—
117	Q₈	—	4	2	no	—
118	Q₈	—	4	2	no	—
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	562	2	no	—
123	C₂×C₄	—	562	4	no	—
124	C₂×C₄	—	1124	2	no	—
125	C₂×C₄	—	1124	4	no	—
126	C₂×C₄	C₂²×Dic₂₈₁	1	1	yes	characteristic
127	C₂×C₄	D₄×D₂₈₁	1	1	yes	characteristic
128	C₂×C₄	C₂×C₂²⋊D₂₈₁	1	1	yes	characteristic
129	C₂×C₄	C₂×C₂²⋊D₂₈₁	1	1	yes	characteristic
130	C₂×C₄	#17985137	1	1	yes	characteristic
131	C₂×C₄	#17985137	1	1	yes	characteristic
132	C₈	—	562	2	no	—
133	C₈	C₂²⋊Dic₂₈₁	1	4	yes	—
134	C₂²	D₄×Dic₂₈₁	1	1	yes	characteristic
135	C₂²	#22582850	1	1	yes	characteristic
136	C₂²	#22582875	1	1	yes	characteristic
137	C₂²	#22582911	1	4	yes	—
138	C₄	—	4	4	no	—
139	C₄	—	1124	2	no	—
140	C₄	—	2248	2	no	—
141	C₄	C₂×C₂²⋊Dic₂₈₁	1	1	yes	characteristic
142	C₄	D₄×Dic₂₈₁	1	1	yes	characteristic
143	C₄	#22582850	1	1	yes	characteristic
144	C₄	#22582866	1	1	yes	characteristic
145	C₂	#31823366	1	1	yes	characteristic
146	C₂	#31823570	1	2	yes	—
147	C₂	#31823718	1	2	yes	—
148	C₂	#31823856	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22582811

C₂

yes

maximal, characteristic

C₂×#22582803

C₂

yes

maximal

C₂×#22582789

C₂

yes

maximal, characteristic

#31823370

C₂

yes

maximal

C₂²×Q₁₆×C₂₈₁

C₂

yes

maximal, characteristic

C₂²×#14555886

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₈₁

C₂²

yes

—

C₂×C₄×Dic₂₈₁

C₂²

yes

—

—

—

—

C₂²×Q₈×C₂₈₁

C₂²

yes

—

C₂×Q₁₆×C₂₈₁

—

C₂×Q₁₆×C₂₈₁

—

C₂×Q₁₆×C₂₈₁

C₄

yes

complemented

C₂²×C₈×C₂₈₁

C₂²

yes

characteristic

C₂²×Dic₂₈₁

—

C₂²×Dic₂₈₁

D₄

yes

—

C₂×#14555886

—

C₂×#14555886

—

C₂×#14555886

D₄

yes

—

C₄⋊Dic₂₈₁

—

C₄⋊Dic₂₈₁

—

C₄×Dic₂₈₁

—

C₄×Dic₂₈₁

—

C₂×Q₈×C₂₈₁

—

C₂×Q₈×C₂₈₁

—

C₂×Q₈×C₂₈₁

—

C₂×Q₈×C₂₈₁

—

C₂×Q₈×C₂₈₁

—

C₂×Q₈×C₂₈₁

C₂×C₄

yes

—

C₂²×C₄×C₂₈₁

—

C₂²×C₄×C₂₈₁

C₂³

yes

characteristic

Q₁₆×C₂₈₁

—

C₂×C₈×C₂₈₁

C₂×C₄

yes

—

C₂×C₈×C₂₈₁

D₄

yes

—

C₂×C₈×C₂₈₁

D₄

yes

—

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

—

#14555886

—

C₂³×C₂₈₁

C₂×D₄

yes

characteristic

Q₈×C₂₈₁

—

Q₈×C₂₈₁

—

Q₈×C₂₈₁

—

Q₈×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₁

C₂×D₄

yes

C₂×C₄×C₂₈₁

yes

C₂×C₄×C₂₈₁

yes

C₈×C₂₈₁

C₂²⋊C₄

yes

—

C₂²×C₂₈₁

C₄×D₄

yes

C₂²×C₂₈₁

yes

C₂²×C₂₈₁

yes

C₂²×C₂₈₁

yes

—

C₄×C₂₈₁

—

C₄×C₂₈₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₈₁

C₄×D₄

yes

C₄×C₂₈₁

yes

C₄×C₂₈₁

yes

Dic₂₈₁

—

Dic₂₈₁

—

C₂×C₂₈₁

#390

yes

characteristic

C₂×C₂₈₁

#441

yes

—

C₂×C₂₈₁

#470

yes

—

C₂×C₂₈₁

#497

yes

—

C₂₈₁

#1971

yes

characteristic, complemented, Sylow

#1971

—

281

maximal, Sylow

C₂²×Q₁₆

D₂₈₁

yes

characteristic

C₂×#106

—

281

—

C₂×#103

—

281

—

C₂×#97

—

281

—

#391

—

281

—

C₂²×Q₈

C₂×D₂₈₁

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

Dic₂₈₁

yes

complemented

C₂²×C₈

—

281

—

C₂²×C₈

C₂×D₂₈₁

yes

characteristic

C₂×C₄⋊C₄

—

281

—

C₂×C₄²

—

281

—

#106

—

562

—

#103

—

562

—

#97

—

562

—

#95

—

562

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₂₈₁

yes

—

C₂²×C₄

—

100

C₂²×C₄

—

281

—

101

C₂²×C₄

—

562

—

102

C₂²×C₄

C₂²×D₂₈₁

yes

characteristic

103

Q₁₆

—

104

C₂×C₈

—

281

—

105

C₂×C₈

—

562

—

106

C₂×C₈

—

562

—

107

C₂×C₈

C₂×Dic₂₈₁

yes

—

108

C₂×C₈

C₂²⋊D₂₈₁

yes

—

109

C₂×C₈

C₂²⋊D₂₈₁

yes

—

110

C₄⋊C₄

—

562

—

111

C₄⋊C₄

—

562

—

112

C₄²

—

562

—

113

C₄²

—

562

—

114

C₂³

D₄×D₂₈₁

yes

characteristic

115

Q₈

—

116

Q₈

—

117

Q₈

—

118

Q₈

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

562

—

123

C₂×C₄

—

562

—

124

C₂×C₄

—

1124

—

125

C₂×C₄

—

1124

—

126

C₂×C₄

C₂²×Dic₂₈₁

yes

characteristic

127

C₂×C₄

D₄×D₂₈₁

yes

characteristic

128

C₂×C₄

C₂×C₂²⋊D₂₈₁

yes

characteristic

129

C₂×C₄

C₂×C₂²⋊D₂₈₁

yes

characteristic

130

C₂×C₄

#17985137

yes

characteristic

131

C₂×C₄

#17985137

yes

characteristic

132

C₈

—

562

—

133

C₈

C₂²⋊Dic₂₈₁

yes

—

134

C₂²

D₄×Dic₂₈₁

yes

characteristic

135

C₂²

#22582850

yes

characteristic

136

C₂²

#22582875

yes

characteristic

137

C₂²

#22582911

yes

—

138

C₄

—

139

C₄

—

1124

—

140

C₄

—

2248

—

141

C₄

C₂×C₂²⋊Dic₂₈₁

yes

characteristic

142

C₄

D₄×Dic₂₈₁

yes

characteristic

143

C₄

#22582850

yes

characteristic

144

C₄

#22582866

yes

characteristic

145

C₂

#31823366

yes

characteristic

146

C₂

#31823570

yes

—

147

C₂

#31823718

yes

—

148

C₂

#31823856

yes

—

	subgroup	G/H
center	C₂³	D₄×D₂₈₁
commutator	C₂×C₄×C₂₈₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₈₁
Fitting	C₂²×Q₁₆×C₂₈₁	C₂
socle, prime radical	C₂³×C₂₈₁	C₂×D₄
cosocle	C₂²×C₄×C₂₈₁	C₂³
hypercenter	C₂²×Q₁₆	D₂₈₁
hypocenter	C₂₈₁	#1971
LCS	G ▷ C₂×C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂³

D₄×D₂₈₁

commutator

C₂×C₄×C₂₈₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₈₁

Fitting

C₂²×Q₁₆×C₂₈₁

C₂

socle, prime radical

C₂³×C₂₈₁

C₂×D₄

cosocle

C₂²×C₄×C₂₈₁

C₂³

hypercenter

C₂²×Q₁₆

D₂₈₁

hypocenter

C₂₈₁

#1971

LCS

G ▷ C₂×C₄×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
16. C₂×Q₁₆×C₂₈₁	C₄	semidirect
74. C₂₈₁	#1971	semidirect
84. C₂×Q₁₆	Dic₂₈₁	semidirect

H ≅ G/N

type

16. C₂×Q₁₆×C₂₈₁

C₄

semidirect

74. C₂₈₁

#1971

semidirect

84. C₂×Q₁₆

Dic₂₈₁

semidirect

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

subgroup	index
C₂×#22582811	2
C₂×#22582803	2
C₂×#22582789	2
#31823370	2
C₂²×Q₁₆×C₂₈₁	2
#1971	281

subgroup

index

C₂²×Q₁₆×C₂₈₁

281

Subgroups — #49096175

Conjugacy classes of subgroups

Complements

Maximal subgroups