Group #49096239 — subgroups

Subgroups — #49096239

Order 35968.

292 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 133 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×C₄×Dic₅₆₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊#14555886	C₂	1	1	yes	maximal, characteristic
3	#31823217	C₂	1	4	yes	maximal
4	C₂×C₄×C₈×C₂₈₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₅₆₂	C₄	1	1	yes	characteristic
6	C₂²×#14555886	C₂²	1	2	yes	—
7	C₄×Dic₅₆₂	—	2	2	no	—
8	C₄×Dic₅₆₂	—	2	2	no	—
9	C₂×C₄⋊Dic₂₈₁	C₄	1	1	yes	characteristic
10	C₂×#17985117	C₄	1	2	yes	—
11	C₂×C₄×Dic₂₈₁	C₂²	1	2	yes	—
12	C₄⋊#14555886	—	2	2	no	—
13	C₂²×C₈×C₂₈₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₈₁	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₂₈₁	—	2	2	no	—
16	C₂×Dic₅₆₂	—	2	2	no	—
17	C₂×Dic₅₆₂	—	2	2	no	—
18	C₂×Dic₅₆₂	C₈	1	4	yes	complemented
19	C₂²×Dic₂₈₁	C₂×C₄	1	2	yes	—
20	C₂²×Dic₂₈₁	C₂×C₄	1	2	yes	—
21	C₂×#14555886	—	2	2	no	—
22	C₂×#14555886	—	2	4	no	—
23	C₄⋊Dic₂₈₁	—	2	2	no	—
24	#17985117	—	2	4	no	—
25	C₄×Dic₂₈₁	—	2	2	no	—
26	C₄×Dic₂₈₁	—	2	2	no	—
27	C₂²×C₄×C₂₈₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₂₈₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₂₈₁	C₂³	1	1	yes	characteristic
30	C₂×C₈×C₂₈₁	—	2	2	no	—
31	C₂×C₈×C₂₈₁	—	2	2	no	—
32	C₂×C₈×C₂₈₁	Q₈	1	4	yes	complemented
33	C₄²×C₂₈₁	D₄	1	2	yes	—
34	C₄²×C₂₈₁	D₄	1	2	yes	—
35	Dic₅₆₂	—	2	8	no	—
36	C₂×Dic₂₈₁	—	2	2	no	—
37	C₂×Dic₂₈₁	—	2	2	no	—
38	C₂×Dic₂₈₁	—	2	2	no	—
39	C₂×Dic₂₈₁	—	2	4	no	—
40	C₂×Dic₂₈₁	C₂×C₈	1	4	yes	—
41	#14555886	—	4	2	no	—
42	C₂³×C₂₈₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₂₈₁	—	2	2	no	—
44	C₂×C₄×C₂₈₁	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₂₈₁	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₂₈₁	C₂×C₈	1	2	yes	—
47	C₂×C₄×C₂₈₁	M₁₆	1	2	yes	—
48	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₂₈₁	C₂×Q₈	1	2	yes	—
51	C₂×C₄×C₂₈₁	#41	1	2	yes	—
52	C₈×C₂₈₁	—	2	4	no	—
53	C₂²×C₂₈₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₂₈₁	C₄×Q₈	1	2	yes	—
55	C₂²×C₂₈₁	C₂²×C₈	1	1	yes	characteristic
56	C₂²×C₂₈₁	C₂×M₁₆	1	1	yes	characteristic
57	C₂²×C₂₈₁	#131	1	2	yes	—
58	C₄×C₂₈₁	—	2	2	no	—
59	C₄×C₂₈₁	C₂²⋊C₈	1	4	yes	—
60	C₄×C₂₈₁	C₂²⋊Q₈	1	2	yes	—
61	C₄×C₂₈₁	#124	1	1	yes	characteristic
62	C₄×C₂₈₁	C₄⋊Q₈	1	1	yes	characteristic
63	Dic₂₈₁	—	2	4	no	—
64	Dic₂₈₁	—	4	2	no	—
65	C₂×C₂₈₁	#391	1	1	yes	characteristic
66	C₂×C₂₈₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
67	C₂×C₂₈₁	#408	1	1	yes	characteristic
68	C₂×C₂₈₁	C₈×Q₈	1	2	yes	—
69	C₂×C₂₈₁	#446	1	2	yes	—
70	C₂₈₁	#1984	1	1	yes	characteristic, complemented, Sylow
71	#1984	—	281	1	no	maximal, Sylow
72	C₂×C₄×Q₈	—	281	1	no	—
73	C₂×C₄⋊C₈	—	281	1	no	—
74	C₂×C₄×C₈	D₂₈₁	1	1	yes	characteristic
75	#336	—	281	4	no	—
76	C₂²×Q₈	—	281	1	no	—
77	C₂²×C₈	—	281	2	no	—
78	C₂²×C₈	C₂×D₂₈₁	1	2	yes	—
79	C₄×Q₈	—	562	2	no	—
80	C₄×Q₈	—	562	2	no	—
81	C₂×C₄⋊C₄	—	281	2	no	—
82	C₂×C₄⋊C₄	—	281	1	no	—
83	C₂×C₄²	—	281	2	no	—
84	C₂×C₄²	C₂×D₂₈₁	1	1	yes	characteristic
85	C₄⋊C₈	—	562	2	no	—
86	C₄×C₈	—	2	2	no	—
87	C₂×Q₈	—	281	4	no	—
88	C₂×Q₈	—	562	2	no	—
89	C₂×Q₈	—	562	2	no	—
90	C₂²×C₄	—	281	2	no	—
91	C₂²×C₄	—	281	2	no	—
92	C₂²×C₄	C₄×D₂₈₁	1	1	yes	characteristic
93	C₂²×C₄	C₄×D₂₈₁	1	1	yes	characteristic
94	C₂²×C₄	C₂²×D₂₈₁	1	1	yes	characteristic
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	562	2	no	—
98	C₂×C₈	—	562	4	no	—
99	C₂×C₈	Dic₅₆₂	1	4	yes	complemented
100	C₄⋊C₄	—	562	4	no	—
101	C₄⋊C₄	—	562	2	no	—
102	C₄²	—	562	2	no	—
103	C₄²	—	562	2	no	—
104	C₄²	C₂²⋊D₂₈₁	1	2	yes	—
105	C₄²	D₁₁₂₄	1	2	yes	—
106	C₂³	C₂×C₄×D₂₈₁	1	1	yes	characteristic
107	Q₈	—	562	8	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	281	4	no	—
110	C₂×C₄	—	562	2	no	—
111	C₂×C₄	—	562	2	no	—
112	C₂×C₄	—	562	2	no	—
113	C₂×C₄	—	562	4	no	—
114	C₂×C₄	#17985115	1	2	yes	—
115	C₂×C₄	#17985115	1	2	yes	—
116	C₂×C₄	C₈×D₂₈₁	1	2	yes	—
117	C₂×C₄	#17985121	1	2	yes	—
118	C₂×C₄	C₂×C₂²⋊D₂₈₁	1	1	yes	characteristic
119	C₂×C₄	C₂×D₁₁₂₄	1	1	yes	characteristic
120	C₂×C₄	C₂×Dic₅₆₂	1	2	yes	—
121	C₂×C₄	#17985138	1	2	yes	—
122	C₈	—	2	4	no	—
123	C₈	—	1124	2	no	—
124	C₂²	C₂×#17985115	1	1	yes	characteristic
125	C₂²	C₄×Dic₅₆₂	1	2	yes	—
126	C₂²	C₂×C₈×D₂₈₁	1	1	yes	characteristic
127	C₂²	C₂×#17985121	1	1	yes	characteristic
128	C₂²	#22582885	1	2	yes	—
129	C₄	—	2	2	no	—
130	C₄	—	562	4	no	—
131	C₄	—	1124	2	no	—
132	C₄	#22582791	1	4	yes	—
133	C₄	#22582858	1	2	yes	—
134	C₄	#22582867	1	1	yes	characteristic
135	C₄	#22582879	1	1	yes	characteristic
136	C₂	#31823369	1	1	yes	characteristic
137	C₂	C₂×#22582791	1	1	yes	characteristic
138	C₂	#31823429	1	1	yes	characteristic
139	C₂	C₈×Dic₅₆₂	1	2	yes	—
140	C₂	#31823593	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₅₆₂

C₂

yes

maximal, characteristic

C₂×C₄⋊#14555886

C₂

yes

maximal, characteristic

#31823217

C₂

yes

maximal

C₂×C₄×C₈×C₂₈₁

C₂

yes

maximal, characteristic

C₂²×Dic₅₆₂

C₄

yes

characteristic

C₂²×#14555886

C₂²

yes

—

C₄×Dic₅₆₂

—

C₄×Dic₅₆₂

—

C₂×C₄⋊Dic₂₈₁

C₄

yes

characteristic

C₂×#17985117

C₄

yes

—

C₂×C₄×Dic₂₈₁

C₂²

yes

—

C₄⋊#14555886

—

C₂²×C₈×C₂₈₁

C₂²

yes

—

C₂×C₄²×C₂₈₁

C₂²

yes

characteristic

C₄×C₈×C₂₈₁

—

C₂×Dic₅₆₂

—

C₂×Dic₅₆₂

—

C₂×Dic₅₆₂

C₈

yes

complemented

C₂²×Dic₂₈₁

C₂×C₄

yes

—

C₂²×Dic₂₈₁

C₂×C₄

yes

—

C₂×#14555886

—

C₂×#14555886

—

C₄⋊Dic₂₈₁

—

#17985117

—

C₄×Dic₂₈₁

—

C₄×Dic₂₈₁

—

C₂²×C₄×C₂₈₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₈₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₈₁

C₂³

yes

characteristic

C₂×C₈×C₂₈₁

—

C₂×C₈×C₂₈₁

—

C₂×C₈×C₂₈₁

Q₈

yes

complemented

C₄²×C₂₈₁

D₄

yes

—

C₄²×C₂₈₁

D₄

yes

—

Dic₅₆₂

—

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

—

C₂×Dic₂₈₁

C₂×C₈

yes

—

#14555886

—

C₂³×C₂₈₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₈₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₈₁

C₂×C₈

yes

—

C₂×C₄×C₂₈₁

M₁₆

yes

—

C₂×C₄×C₂₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₁

C₂×Q₈

yes

—

C₂×C₄×C₂₈₁

#41

yes

—

C₈×C₂₈₁

—

C₂²×C₂₈₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₈₁

C₄×Q₈

yes

—

C₂²×C₂₈₁

C₂²×C₈

yes

characteristic

C₂²×C₂₈₁

C₂×M₁₆

yes

characteristic

C₂²×C₂₈₁

#131

yes

—

C₄×C₂₈₁

—

C₄×C₂₈₁

C₂²⋊C₈

yes

—

C₄×C₂₈₁

C₂²⋊Q₈

yes

—

C₄×C₂₈₁

#124

yes

characteristic

C₄×C₂₈₁

C₄⋊Q₈

yes

characteristic

Dic₂₈₁

—

Dic₂₈₁

—

C₂×C₂₈₁

#391

yes

characteristic

C₂×C₂₈₁

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₂₈₁

#408

yes

characteristic

C₂×C₂₈₁

C₈×Q₈

yes

—

C₂×C₂₈₁

#446

yes

—

C₂₈₁

#1984

yes

characteristic, complemented, Sylow

#1984

—

281

maximal, Sylow

C₂×C₄×Q₈

—

281

—

C₂×C₄⋊C₈

—

281

—

C₂×C₄×C₈

D₂₈₁

yes

characteristic

#336

—

281

—

C₂²×Q₈

—

281

—

C₂²×C₈

—

281

—

C₂²×C₈

C₂×D₂₈₁

yes

—

C₄×Q₈

—

562

—

C₄×Q₈

—

562

—

C₂×C₄⋊C₄

—

281

—

C₂×C₄⋊C₄

—

281

—

C₂×C₄²

—

281

—

C₂×C₄²

C₂×D₂₈₁

yes

characteristic

C₄⋊C₈

—

562

—

C₄×C₈

—

C₂×Q₈

—

281

—

C₂×Q₈

—

562

—

C₂×Q₈

—

562

—

C₂²×C₄

—

281

—

C₂²×C₄

—

281

—

C₂²×C₄

C₄×D₂₈₁

yes

characteristic

C₂²×C₄

C₄×D₂₈₁

yes

characteristic

C₂²×C₄

C₂²×D₂₈₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

562

—

C₂×C₈

—

562

—

C₂×C₈

Dic₅₆₂

yes

complemented

100

C₄⋊C₄

—

562

—

101

C₄⋊C₄

—

562

—

102

C₄²

—

562

—

103

C₄²

—

562

—

104

C₄²

C₂²⋊D₂₈₁

yes

—

105

C₄²

D₁₁₂₄

yes

—

106

C₂³

C₂×C₄×D₂₈₁

yes

characteristic

107

Q₈

—

562

—

108

C₂×C₄

—

109

C₂×C₄

—

281

—

110

C₂×C₄

—

562

—

111

C₂×C₄

—

562

—

112

C₂×C₄

—

562

—

113

C₂×C₄

—

562

—

114

C₂×C₄

#17985115

yes

—

115

C₂×C₄

#17985115

yes

—

116

C₂×C₄

C₈×D₂₈₁

yes

—

117

C₂×C₄

#17985121

yes

—

118

C₂×C₄

C₂×C₂²⋊D₂₈₁

yes

characteristic

119

C₂×C₄

C₂×D₁₁₂₄

yes

characteristic

120

C₂×C₄

C₂×Dic₅₆₂

yes

—

121

C₂×C₄

#17985138

yes

—

122

C₈

—

123

C₈

—

1124

—

124

C₂²

C₂×#17985115

yes

characteristic

125

C₂²

C₄×Dic₅₆₂

yes

—

126

C₂²

C₂×C₈×D₂₈₁

yes

characteristic

127

C₂²

C₂×#17985121

yes

characteristic

128

C₂²

#22582885

yes

—

129

C₄

—

130

C₄

—

562

—

131

C₄

—

1124

—

132

C₄

#22582791

yes

—

133

C₄

#22582858

yes

—

134

C₄

#22582867

yes

characteristic

135

C₄

#22582879

yes

characteristic

136

C₂

#31823369

yes

characteristic

137

C₂

C₂×#22582791

yes

characteristic

138

C₂

#31823429

yes

characteristic

139

C₂

C₈×Dic₅₆₂

yes

—

140

C₂

#31823593

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₈₁
commutator	C₂²×C₂₈₁	C₂²×C₈
Fitting	C₂×C₄×C₈×C₂₈₁	C₂
socle, prime radical	C₂³×C₂₈₁	C₂²×C₄
cosocle	C₂²×C₄×C₂₈₁	C₂³
hypercenter	C₂×C₄×C₈	D₂₈₁
hypocenter	C₂₈₁	#1984
LCS	G ▷ C₂²×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₈₁

commutator

C₂²×C₂₈₁

C₂²×C₈

Fitting

C₂×C₄×C₈×C₂₈₁

C₂

socle, prime radical

C₂³×C₂₈₁

C₂²×C₄

cosocle

C₂²×C₄×C₂₈₁

C₂³

hypercenter

C₂×C₄×C₈

D₂₈₁

C₂₈₁

G ▷ C₂²×C₂₈₁ ▷ C₂₈₁

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
18. C₂×Dic₅₆₂	C₈	semidirect
32. C₂×C₈×C₂₈₁	Q₈	semidirect
70. C₂₈₁	#1984	semidirect
99. C₂×C₈	Dic₅₆₂	semidirect

H ≅ G/N

type

18. C₂×Dic₅₆₂

C₈

semidirect

32. C₂×C₈×C₂₈₁

Q₈

semidirect

70. C₂₈₁

#1984

semidirect

99. C₂×C₈

Dic₅₆₂

semidirect

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

subgroup	index
C₂×C₄×Dic₅₆₂	2
C₂×C₄⋊#14555886	2
#31823217	2
C₂×C₄×C₈×C₂₈₁	2
#1984	281

subgroup

index

C₂×C₄×Dic₅₆₂

C₂×C₄⋊#14555886

#31823217

C₂×C₄×C₈×C₂₈₁

#1984

281

Subgroups — #49096239

Conjugacy classes of subgroups

Complements

Maximal subgroups