Group #49095299 — subgroups

Subgroups — #49095299

Order 35968.

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

normal subgroups only

This group has 272 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	#31823487	C₂	1	2	yes	maximal
2	C₂×#22582798	C₂	1	2	yes	maximal
3	#31823232	C₂	1	2	yes	maximal
4	C₂×C₄×Q₈×C₂₈₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#17985122	C₂²	1	4	yes	—
6	C₄⋊#14555886	—	2	2	no	—
7	C₄⋊#14555886	—	2	2	no	—
8	#22582798	—	2	4	no	—
9	C₂²×Q₈×C₂₈₁	C₂²	1	1	yes	characteristic
10	C₄×Q₈×C₂₈₁	—	2	2	no	—
11	C₄×Q₈×C₂₈₁	—	2	2	no	—
12	C₂×C₄⋊C₄×C₂₈₁	C₄	1	2	yes	—
13	C₂×C₄⋊C₄×C₂₈₁	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₂₈₁	C₄	1	2	yes	—
15	C₂×C₄²×C₂₈₁	C₂²	1	1	yes	characteristic
16	#17985122	—	2	4	no	—
17	#17985122	—	2	4	no	—
18	C₂×#14555886	—	2	4	no	—
19	C₂×Q₈×C₂₈₁	—	2	2	no	—
20	C₂×Q₈×C₂₈₁	—	2	2	no	—
21	C₂×Q₈×C₂₈₁	D₄	1	2	yes	—
22	C₂×Q₈×C₂₈₁	Q₈	1	2	yes	—
23	C₂²×C₄×C₂₈₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₂₈₁	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₂₈₁	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₂₈₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₂₈₁	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₂₈₁	—	2	4	no	—
29	C₄⋊C₄×C₂₈₁	—	2	2	no	—
30	C₄²×C₂₈₁	—	2	2	no	—
31	C₄²×C₂₈₁	—	2	2	no	—
32	C₄²×C₂₈₁	D₄	1	2	yes	—
33	C₄²×C₂₈₁	D₄	1	2	yes	—
34	#14555886	—	4	4	no	—
35	C₂³×C₂₈₁	C₂²×C₄	1	1	yes	characteristic
36	Q₈×C₂₈₁	—	2	4	no	—
37	Q₈×C₂₈₁	—	2	4	no	—
38	C₂×C₄×C₂₈₁	—	2	2	no	—
39	C₂×C₄×C₂₈₁	—	2	2	no	—
40	C₂×C₄×C₂₈₁	—	2	2	no	—
41	C₂×C₄×C₂₈₁	—	2	2	no	—
42	C₂×C₄×C₂₈₁	—	4	1	no	—
43	C₂×C₄×C₂₈₁	—	4	2	no	—
44	C₂×C₄×C₂₈₁	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₈₁	C₂²⋊C₄	1	2	yes	—
46	C₂×C₄×C₂₈₁	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₂₈₁	C₄⋊C₄	1	4	yes	—
48	C₂×C₄×C₂₈₁	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₂₈₁	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₂₈₁	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₂₈₁	#41	1	2	yes	—
52	C₂²×C₂₈₁	—	2	1	no	—
53	C₂²×C₂₈₁	—	2	1	no	—
54	C₂²×C₂₈₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₂₈₁	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₂₈₁	#117	1	1	yes	characteristic
57	C₄×C₂₈₁	—	2	4	no	—
58	C₄×C₂₈₁	—	2	1	no	—
59	C₄×C₂₈₁	—	2	2	no	—
60	C₄×C₂₈₁	—	4	2	no	—
61	C₄×C₂₈₁	#101	1	2	yes	complemented
62	C₄×C₂₈₁	#121	1	2	yes	—
63	C₄×C₂₈₁	C₂²⋊Q₈	1	2	yes	—
64	C₂×C₂₈₁	—	2	1	no	—
65	C₂×C₂₈₁	—	2	1	no	—
66	C₂×C₂₈₁	#380	1	1	yes	characteristic
67	C₂×C₂₈₁	C₂×#101	1	1	yes	characteristic
68	C₂×C₂₈₁	#413	1	1	yes	characteristic
69	C₂₈₁	#1801	1	1	yes	characteristic, complemented, Sylow
70	#1801	—	281	1	no	maximal, Sylow
71	C₂×C₄×Q₈	D₂₈₁	1	1	yes	characteristic
72	#423	—	281	2	no	—
73	C₂×#101	—	281	2	no	—
74	#343	—	281	2	no	—
75	C₂²×Q₈	C₂×D₂₈₁	1	1	yes	characteristic
76	C₂×M₁₆	—	281	4	no	—
77	C₄×Q₈	—	2	2	no	—
78	C₄×Q₈	—	2	2	no	—
79	C₂×C₄⋊C₄	Dic₂₈₁	1	2	yes	—
80	C₂×C₄⋊C₄	C₂×D₂₈₁	1	1	yes	characteristic
81	C₂×C₄²	Dic₂₈₁	1	2	yes	—
82	C₂×C₄²	C₂×D₂₈₁	1	1	yes	characteristic
83	C₄⋊C₈	—	562	2	no	—
84	C₄⋊C₈	—	562	2	no	—
85	#101	—	562	4	no	—
86	C₂×Q₈	—	2	2	no	—
87	C₂×Q₈	—	2	2	no	—
88	C₂×Q₈	D₁₁₂₄	1	2	yes	—
89	C₂×Q₈	Dic₅₆₂	1	2	yes	—
90	C₂²×C₄	C₂×Dic₂₈₁	1	1	yes	characteristic
91	C₂²×C₄	C₂×Dic₂₈₁	1	2	yes	—
92	C₂²×C₄	C₂×Dic₂₈₁	1	2	yes	—
93	C₂²×C₄	C₂×Dic₂₈₁	1	1	yes	characteristic
94	C₂²×C₄	C₂²×D₂₈₁	1	1	yes	characteristic
95	M₁₆	—	562	4	no	—
96	M₁₆	—	562	4	no	—
97	C₂×C₈	—	562	4	no	—
98	C₄⋊C₄	—	2	4	no	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄²	—	2	2	no	—
101	C₄²	—	2	2	no	—
102	C₄²	C₂²⋊D₂₈₁	1	2	yes	—
103	C₄²	C₂²⋊D₂₈₁	1	2	yes	—
104	C₂³	C₂²×Dic₂₈₁	1	1	yes	characteristic
105	Q₈	—	2	4	no	—
106	Q₈	—	2	4	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	C₂²⋊Dic₂₈₁	1	1	yes	characteristic
114	C₂×C₄	C₂²⋊Dic₂₈₁	1	2	yes	—
115	C₂×C₄	C₂²⋊Dic₂₈₁	1	1	yes	characteristic
116	C₂×C₄	C₄⋊Dic₂₈₁	1	4	yes	—
117	C₂×C₄	C₂×C₂²⋊D₂₈₁	1	2	yes	—
118	C₂×C₄	C₂×D₁₁₂₄	1	1	yes	characteristic
119	C₂×C₄	C₂×Dic₅₆₂	1	1	yes	characteristic
120	C₂×C₄	#17985138	1	2	yes	—
121	C₈	—	1124	4	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	C₂×C₂²⋊Dic₂₈₁	1	1	yes	characteristic
125	C₂²	C₂×C₄⋊Dic₂₈₁	1	1	yes	characteristic
126	C₂²	#22582838	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	2	2	no	—
129	C₄	—	2	4	no	—
130	C₄	—	4	2	no	—
131	C₄	#22582798	1	2	yes	complemented
132	C₄	#22582853	1	2	yes	—
133	C₄	#22582858	1	2	yes	—
134	C₂	—	2	1	no	—
135	C₂	—	2	1	no	—
136	C₂	#31823323	1	1	yes	characteristic
137	C₂	C₂×#22582798	1	1	yes	characteristic
138	C₂	#31823447	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#31823487

C₂

yes

maximal

C₂×#22582798

C₂

yes

maximal

#31823232

C₂

yes

maximal

C₂×C₄×Q₈×C₂₈₁

C₂

yes

maximal, characteristic

C₂×#17985122

C₂²

yes

—

C₄⋊#14555886

—

C₄⋊#14555886

—

#22582798

—

C₂²×Q₈×C₂₈₁

C₂²

yes

characteristic

C₄×Q₈×C₂₈₁

—

C₄×Q₈×C₂₈₁

—

C₂×C₄⋊C₄×C₂₈₁

C₄

yes

—

C₂×C₄⋊C₄×C₂₈₁

C₂²

yes

characteristic

C₂×C₄²×C₂₈₁

C₄

yes

—

C₂×C₄²×C₂₈₁

C₂²

yes

—

—

—

C₂×Q₈×C₂₈₁

—

C₂×Q₈×C₂₈₁

—

C₂×Q₈×C₂₈₁

D₄

yes

—

C₂×Q₈×C₂₈₁

Q₈

yes

—

C₂²×C₄×C₂₈₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₈₁

C₂×C₄

yes

—

C₂²×C₄×C₂₈₁

C₂×C₄

yes

—

C₂²×C₄×C₂₈₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₈₁

C₂³

yes

characteristic

C₄⋊C₄×C₂₈₁

—

C₄⋊C₄×C₂₈₁

—

C₄²×C₂₈₁

—

C₄²×C₂₈₁

—

C₄²×C₂₈₁

D₄

yes

—

C₄²×C₂₈₁

D₄

yes

—

#14555886

—

C₂³×C₂₈₁

C₂²×C₄

yes

characteristic

Q₈×C₂₈₁

—

Q₈×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

—

C₂×C₄×C₂₈₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₈₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₈₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₈₁

C₄⋊C₄

yes

—

C₂×C₄×C₂₈₁

C₂×D₄

yes

—

C₂×C₄×C₂₈₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₈₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₈₁

#41

yes

—

C₂²×C₂₈₁

—

C₂²×C₂₈₁

—

C₂²×C₂₈₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₈₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₈₁

#117

yes

characteristic

C₄×C₂₈₁

—

C₄×C₂₈₁

—

C₄×C₂₈₁

—

C₄×C₂₈₁

—

C₄×C₂₈₁

#101

yes

complemented

C₄×C₂₈₁

#121

yes

—

C₄×C₂₈₁

C₂²⋊Q₈

yes

—

C₂×C₂₈₁

—

C₂×C₂₈₁

—

C₂×C₂₈₁

yes

C₂×C₂₈₁

yes

C₂×C₂₈₁

yes

C₂₈₁

yes

characteristic, complemented, Sylow

#1801

—

281

maximal, Sylow

C₂×C₄×Q₈

D₂₈₁

yes

characteristic

#423

—

281

—

C₂×#101

—

281

—

#343

—

281

—

C₂²×Q₈

C₂×D₂₈₁

yes

characteristic

C₂×M₁₆

—

281

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

Dic₂₈₁

yes

—

C₂×C₄⋊C₄

C₂×D₂₈₁

yes

characteristic

C₂×C₄²

Dic₂₈₁

yes

—

C₂×C₄²

C₂×D₂₈₁

yes

characteristic

C₄⋊C₈

—

562

—

C₄⋊C₈

—

562

—

#101

—

562

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₁₁₂₄

yes

—

C₂×Q₈

Dic₅₆₂

yes

—

C₂²×C₄

C₂×Dic₂₈₁

yes

characteristic

C₂²×C₄

C₂×Dic₂₈₁

yes

—

C₂²×C₄

C₂×Dic₂₈₁

yes

—

C₂²×C₄

C₂×Dic₂₈₁

yes

characteristic

C₂²×C₄

C₂²×D₂₈₁

yes

characteristic

M₁₆

—

562

—

M₁₆

—

562

—

C₂×C₈

—

562

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄²

—

101

C₄²

—

102

C₄²

C₂²⋊D₂₈₁

yes

—

103

C₄²

C₂²⋊D₂₈₁

yes

—

104

C₂³

C₂²×Dic₂₈₁

yes

characteristic

105

Q₈

—

106

Q₈

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂²⋊Dic₂₈₁

yes

characteristic

114

C₂×C₄

C₂²⋊Dic₂₈₁

yes

—

115

C₂×C₄

C₂²⋊Dic₂₈₁

yes

characteristic

116

C₂×C₄

C₄⋊Dic₂₈₁

yes

—

117

C₂×C₄

C₂×C₂²⋊D₂₈₁

yes

—

118

C₂×C₄

C₂×D₁₁₂₄

yes

characteristic

119

C₂×C₄

C₂×Dic₅₆₂

yes

characteristic

120

C₂×C₄

#17985138

yes

—

121

C₈

—

1124

—

122

C₂²

—

123

C₂²

—

124

C₂²

C₂×C₂²⋊Dic₂₈₁

yes

characteristic

125

C₂²

C₂×C₄⋊Dic₂₈₁

yes

characteristic

126

C₂²

#22582838

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

#22582798

yes

complemented

132

C₄

#22582853

yes

—

133

C₄

#22582858

yes

—

134

C₂

—

135

C₂

—

136

C₂

#31823323

yes

characteristic

137

C₂

C₂×#22582798

yes

characteristic

138

C₂

#31823447

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₂₈₁
commutator, prime radical	C₂³×C₂₈₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₈₁
Fitting	C₂×C₄×Q₈×C₂₈₁	C₂
socle	C₂²×C₂₈₁	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₂₈₁	C₂³
hypercenter	C₂×C₄×Q₈	D₂₈₁
hypocenter	C₂₈₁	#1801
LCS	G ▷ C₂³×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₂₈₁

commutator, prime radical

C₂³×C₂₈₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₈₁

Fitting

C₂×C₄×Q₈×C₂₈₁

C₂

socle

C₂²×C₂₈₁

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₂₈₁

C₂³

hypercenter

C₂×C₄×Q₈

D₂₈₁

hypocenter

C₂₈₁

#1801

LCS

G ▷ C₂³×C₂₈₁ ▷ C₂×C₂₈₁ ▷ C₂₈₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
61. C₄×C₂₈₁	#101	semidirect
69. C₂₈₁	#1801	semidirect
131. C₄	#22582798	semidirect

H ≅ G/N

type

61. C₄×C₂₈₁

#101

semidirect

69. C₂₈₁

#1801

semidirect

131. C₄

#22582798

semidirect

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

subgroup	index
#31823487	2
C₂×#22582798	2
#31823232	2
C₂×C₄×Q₈×C₂₈₁	2
#1801	281

subgroup

index

C₂×C₄×Q₈×C₂₈₁

281

Subgroups — #49095299

Conjugacy classes of subgroups

Complements

Maximal subgroups