Group #49130191 — subgroups

Subgroups — #49130191

Order 36224.

364 conjugacy classes of subgroups; only the 189 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 189 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	#31827022	C₂	1	1	yes	maximal, characteristic
2	#31826592	C₂	1	4	yes	maximal, complemented
3	#31826589	C₂	1	4	yes	maximal, complemented
4	#31826493	C₂	1	2	yes	maximal
5	#31826423	C₂	1	2	yes	maximal
6	C₂×C₄×C₂₈₃⋊C₈	C₂	1	1	yes	maximal, characteristic
7	#423×C₂₈₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₂₈₃⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₂₈₃⋊C₈	C₂²	1	1	yes	characteristic
10	#22583610	C₄	1	2	yes	—
11	#22583602	C₄	1	2	yes	—
12	#22583589	C₄	1	4	yes	—
13	C₄×Dic₅₆₆	C₂²	1	4	yes	—
14	#22583569	C₂²	1	2	yes	—
15	Dic₂₈₃⋊C₈	C₂²	1	8	yes	—
16	#22583520	C₂²	1	4	yes	—
17	C₄×C₂₈₃⋊C₈	C₂²	1	2	yes	—
18	C₄×C₂₈₃⋊C₈	C₂²	1	2	yes	—
19	C₈×Dic₂₈₃	C₂²	1	4	yes	—
20	C₂×M₁₆×C₂₈₃	C₂²	1	2	yes	—
21	C₂×C₄²×C₂₈₃	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₂₈₃	C₂²	1	4	yes	—
23	C₂×Dic₅₆₆	C₂×C₄	1	4	yes	—
24	C₂×C₂₈₃⋊C₈	—	2	2	no	—
25	C₂×C₂₈₃⋊C₈	Q₈	1	4	yes	complemented
26	C₂×C₂₈₃⋊C₈	C₂³	1	2	yes	—
27	C₂×C₂₈₃⋊C₈	C₂³	1	1	yes	characteristic
28	C₂×C₂₈₃⋊C₈	C₂³	1	1	yes	characteristic
29	C₄⋊Dic₂₈₃	C₂×C₄	1	4	yes	—
30	C₄⋊Dic₂₈₃	C₂×C₄	1	4	yes	—
31	Dic₂₈₃⋊C₄	C₂×C₄	1	8	yes	—
32	C₂²⋊Dic₂₈₃	C₂×C₄	1	4	yes	—
33	C₄×Dic₂₈₃	C₂³	1	4	yes	—
34	C₂²×C₄×C₂₈₃	C₂×C₄	1	2	yes	—
35	C₂²×C₄×C₂₈₃	C₂³	1	1	yes	characteristic
36	M₁₆×C₂₈₃	—	2	4	no	—
37	C₂×C₈×C₂₈₃	C₂³	1	4	yes	—
38	C₄²×C₂₈₃	C₂×C₄	1	1	yes	characteristic
39	C₄²×C₂₈₃	C₂×C₄	1	1	yes	characteristic
40	C₄²×C₂₈₃	C₂³	1	2	yes	—
41	Dic₅₆₆	—	2	8	no	—
42	C₂×Dic₂₈₃	C₂²×C₄	1	8	yes	—
43	C₂₈₃⋊C₈	—	2	2	no	—
44	C₂₈₃⋊C₈	C₂×Q₈	1	2	yes	—
45	C₂₈₃⋊C₈	C₂×Q₈	1	2	yes	—
46	C₂³×C₂₈₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₂₈₃	—	2	4	no	—
48	C₂×C₄×C₂₈₃	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₂₈₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₂₈₃	C₂²×C₄	1	2	yes	—
51	C₂×C₄×C₂₈₃	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₂₈₃	#41	1	2	yes	—
53	C₂×C₄×C₂₈₃	C₂⁴	1	1	yes	characteristic
54	C₈×C₂₈₃	—	2	4	no	—
55	C₂²×C₂₈₃	—	2	2	no	—
56	C₂²×C₂₈₃	C₄×Q₈	1	2	yes	—
57	C₂²×C₂₈₃	C₂³×C₄	1	1	yes	characteristic
58	C₄×C₂₈₃	—	2	2	no	—
59	C₄×C₂₈₃	C₄×Q₈	1	2	yes	—
60	C₄×C₂₈₃	#131	1	4	yes	—
61	C₄×C₂₈₃	C₂²×Q₈	1	1	yes	characteristic
62	C₄×C₂₈₃	C₂×#41	1	1	yes	characteristic
63	Dic₂₈₃	—	2	8	no	—
64	C₂×C₂₈₃	—	2	2	no	—
65	C₂×C₂₈₃	C₂×C₄×Q₈	1	1	yes	characteristic
66	C₂×C₂₈₃	C₂×#131	1	2	yes	—
67	C₂₈₃	#2960	1	1	yes	characteristic, complemented, Sylow
68	#2960	—	283	1	no	maximal, Sylow
69	#533	—	283	1	no	—
70	#446	—	283	4	no	—
71	C₈×Q₈	—	283	4	no	—
72	#424	—	283	2	no	—
73	#423	D₂₈₃	1	1	yes	characteristic
74	#405	—	283	2	no	—
75	C₂×C₄×C₈	—	283	1	no	—
76	C₂×M₁₆	C₂×D₂₈₃	1	2	yes	—
77	C₂²×C₈	—	283	1	no	—
78	C₂²×C₈	—	283	1	no	—
79	C₄⋊Q₈	—	283	2	no	—
80	#125	—	283	2	no	—
81	C₂²⋊Q₈	—	283	4	no	—
82	C₄×Q₈	—	283	4	no	—
83	#117	—	283	2	no	—
84	C₂×C₄²	C₂×D₂₈₃	1	1	yes	characteristic
85	C₄⋊C₈	—	283	8	no	—
86	C₄⋊C₈	C₂×D₂₈₃	1	4	yes	—
87	#97	—	283	4	no	—
88	C₄×C₈	—	283	4	no	—
89	C₄×C₈	—	283	2	no	—
90	C₄×C₈	—	283	2	no	—
91	C₂×Q₈	—	283	4	no	—
92	C₂²×C₄	C₄×D₂₈₃	1	2	yes	—
93	C₂²×C₄	C₂²×D₂₈₃	1	1	yes	characteristic
94	M₁₆	—	2	4	no	—
95	C₂×C₈	—	283	4	no	—
96	C₂×C₈	—	283	1	no	—
97	C₂×C₈	—	283	1	no	—
98	C₂×C₈	—	283	2	no	—
99	C₂×C₈	—	566	2	no	—
100	C₂×C₈	C₂²×D₂₈₃	1	4	yes	—
101	C₄⋊C₄	—	283	8	no	—
102	C₄⋊C₄	—	283	4	no	—
103	C₄⋊C₄	—	283	4	no	—
104	C₂²⋊C₄	—	283	4	no	—
105	C₄²	—	283	4	no	—
106	C₄²	C₄×D₂₈₃	1	1	yes	characteristic
107	C₄²	C₄×D₂₈₃	1	1	yes	characteristic
108	C₄²	C₂²×D₂₈₃	1	2	yes	—
109	C₂³	C₂×C₄×D₂₈₃	1	1	yes	characteristic
110	Q₈	—	566	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	283	8	no	—
113	C₂×C₄	C₂×C₄×D₂₈₃	1	2	yes	—
114	C₂×C₄	C₂×C₄×D₂₈₃	1	1	yes	characteristic
115	C₂×C₄	C₂×C₄×D₂₈₃	1	2	yes	—
116	C₂×C₄	Q₈×D₂₈₃	1	2	yes	—
117	C₂×C₄	#17985391	1	2	yes	—
118	C₂×C₄	C₂³×D₂₈₃	1	1	yes	characteristic
119	C₈	—	2	4	no	—
120	C₈	—	283	2	no	—
121	C₈	—	283	2	no	—
122	C₈	—	566	2	no	—
123	C₂²	—	2	2	no	—
124	C₂²	#22583575	1	2	yes	—
125	C₂²	C₂²×C₄×D₂₈₃	1	1	yes	characteristic
126	C₄	—	2	2	no	—
127	C₄	—	566	8	no	—
128	C₄	#22583575	1	2	yes	—
129	C₄	#22583618	1	4	yes	—
130	C₄	C₂×Q₈×D₂₈₃	1	1	yes	characteristic
131	C₄	C₂×#17985391	1	1	yes	characteristic
132	C₂	—	2	2	no	—
133	C₂	C₂×#22583575	1	1	yes	characteristic
134	C₂	C₂×#22583618	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#31827022

C₂

yes

maximal, characteristic

#31826592

C₂

yes

maximal, complemented

#31826589

C₂

yes

maximal, complemented

#31826493

C₂

yes

maximal

#31826423

C₂

yes

maximal

C₂×C₄×C₂₈₃⋊C₈

C₂

yes

maximal, characteristic

#423×C₂₈₃

C₂

yes

maximal, characteristic

C₂²×C₂₈₃⋊C₈

C₂²

yes

characteristic

C₂²×C₂₈₃⋊C₈

C₂²

yes

characteristic

#22583610

C₄

yes

—

#22583602

C₄

yes

—

#22583589

C₄

yes

—

C₄×Dic₅₆₆

C₂²

yes

—

#22583569

C₂²

yes

—

Dic₂₈₃⋊C₈

C₂²

yes

—

#22583520

C₂²

yes

—

C₄×C₂₈₃⋊C₈

C₂²

yes

—

C₄×C₂₈₃⋊C₈

C₂²

yes

—

C₈×Dic₂₈₃

C₂²

yes

—

C₂×M₁₆×C₂₈₃

C₂²

yes

—

C₂×C₄²×C₂₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₂₈₃

C₂²

yes

—

C₂×Dic₅₆₆

C₂×C₄

yes

—

C₂×C₂₈₃⋊C₈

—

C₂×C₂₈₃⋊C₈

Q₈

yes

complemented

C₂×C₂₈₃⋊C₈

C₂³

yes

—

C₂×C₂₈₃⋊C₈

C₂³

yes

characteristic

C₂×C₂₈₃⋊C₈

C₂³

yes

characteristic

C₄⋊Dic₂₈₃

C₂×C₄

yes

—

C₄⋊Dic₂₈₃

C₂×C₄

yes

—

Dic₂₈₃⋊C₄

C₂×C₄

yes

—

C₂²⋊Dic₂₈₃

C₂×C₄

yes

—

C₄×Dic₂₈₃

C₂³

yes

—

C₂²×C₄×C₂₈₃

C₂×C₄

yes

—

C₂²×C₄×C₂₈₃

C₂³

yes

characteristic

M₁₆×C₂₈₃

—

C₂×C₈×C₂₈₃

C₂³

yes

—

C₄²×C₂₈₃

C₂×C₄

yes

characteristic

C₄²×C₂₈₃

C₂×C₄

yes

characteristic

C₄²×C₂₈₃

C₂³

yes

—

Dic₅₆₆

—

C₂×Dic₂₈₃

C₂²×C₄

yes

—

C₂₈₃⋊C₈

—

C₂₈₃⋊C₈

C₂×Q₈

yes

—

C₂₈₃⋊C₈

C₂×Q₈

yes

—

C₂³×C₂₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₈₃

—

C₂×C₄×C₂₈₃

C₂²×C₄

yes

—

C₂×C₄×C₂₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₈₃

C₂²×C₄

yes

—

C₂×C₄×C₂₈₃

C₂×Q₈

yes

—

C₂×C₄×C₂₈₃

#41

yes

—

C₂×C₄×C₂₈₃

C₂⁴

yes

characteristic

C₈×C₂₈₃

—

C₂²×C₂₈₃

—

C₂²×C₂₈₃

C₄×Q₈

yes

—

C₂²×C₂₈₃

C₂³×C₄

yes

characteristic

C₄×C₂₈₃

—

C₄×C₂₈₃

C₄×Q₈

yes

—

C₄×C₂₈₃

#131

yes

—

C₄×C₂₈₃

C₂²×Q₈

yes

characteristic

C₄×C₂₈₃

C₂×#41

yes

characteristic

Dic₂₈₃

—

C₂×C₂₈₃

—

C₂×C₂₈₃

C₂×C₄×Q₈

yes

characteristic

C₂×C₂₈₃

C₂×#131

yes

—

C₂₈₃

#2960

yes

characteristic, complemented, Sylow

#2960

—

283

maximal, Sylow

#533

—

283

—

#446

—

283

—

C₈×Q₈

—

283

—

#424

—

283

—

#423

D₂₈₃

yes

characteristic

#405

—

283

—

C₂×C₄×C₈

—

283

—

C₂×M₁₆

C₂×D₂₈₃

yes

—

C₂²×C₈

—

283

—

C₂²×C₈

—

283

—

C₄⋊Q₈

—

283

—

#125

—

283

—

C₂²⋊Q₈

—

283

—

C₄×Q₈

—

283

—

#117

—

283

—

C₂×C₄²

C₂×D₂₈₃

yes

characteristic

C₄⋊C₈

—

283

—

C₄⋊C₈

C₂×D₂₈₃

yes

—

#97

—

283

—

C₄×C₈

—

283

—

C₄×C₈

—

283

—

C₄×C₈

—

283

—

C₂×Q₈

—

283

—

C₂²×C₄

C₄×D₂₈₃

yes

—

C₂²×C₄

C₂²×D₂₈₃

yes

characteristic

M₁₆

—

C₂×C₈

—

283

—

C₂×C₈

—

283

—

C₂×C₈

—

283

—

C₂×C₈

—

283

—

C₂×C₈

—

566

—

100

C₂×C₈

C₂²×D₂₈₃

yes

—

101

C₄⋊C₄

—

283

—

102

C₄⋊C₄

—

283

—

103

C₄⋊C₄

—

283

—

104

C₂²⋊C₄

—

283

—

105

C₄²

—

283

—

106

C₄²

C₄×D₂₈₃

yes

characteristic

107

C₄²

C₄×D₂₈₃

yes

characteristic

108

C₄²

C₂²×D₂₈₃

yes

—

109

C₂³

C₂×C₄×D₂₈₃

yes

characteristic

110

Q₈

—

566

—

111

C₂×C₄

—

112

C₂×C₄

—

283

—

113

C₂×C₄

C₂×C₄×D₂₈₃

yes

—

114

C₂×C₄

C₂×C₄×D₂₈₃

yes

characteristic

115

C₂×C₄

C₂×C₄×D₂₈₃

yes

—

116

C₂×C₄

Q₈×D₂₈₃

yes

—

117

C₂×C₄

#17985391

yes

—

118

C₂×C₄

C₂³×D₂₈₃

yes

characteristic

119

C₈

—

120

C₈

—

283

—

121

C₈

—

283

—

122

C₈

—

566

—

123

C₂²

—

124

C₂²

#22583575

yes

—

125

C₂²

C₂²×C₄×D₂₈₃

yes

characteristic

126

C₄

—

127

C₄

—

566

—

128

C₄

#22583575

yes

—

129

C₄

#22583618

yes

—

130

C₄

C₂×Q₈×D₂₈₃

yes

characteristic

131

C₄

C₂×#17985391

yes

characteristic

132

C₂

—

133

C₂

C₂×#22583575

yes

characteristic

134

C₂

C₂×#22583618

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₈₃
commutator, socle	C₂²×C₂₈₃	C₂³×C₄
Fitting	#423×C₂₈₃	C₂
prime radical	C₂³×C₂₈₃	C₂²×C₄
cosocle	C₂×C₄×C₂₈₃	C₂⁴
hypercenter	#423	D₂₈₃
hypocenter	C₂₈₃	#2960
LCS	G ▷ C₂²×C₂₈₃ ▷ C₂₈₃
UCS	C₁ ◁ C₂×C₄ ◁ #423

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₂₈₃

commutator, socle

C₂²×C₂₈₃

C₂³×C₄

Fitting

#423×C₂₈₃

C₂

prime radical

C₂³×C₂₈₃

C₂²×C₄

cosocle

C₂×C₄×C₂₈₃

C₂⁴

D₂₈₃

C₂₈₃

G ▷ C₂²×C₂₈₃ ▷ C₂₈₃

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
2. #31826592	C₂	semidirect
3. #31826589	C₂	semidirect
25. C₂×C₂₈₃⋊C₈	Q₈	semidirect
67. C₂₈₃	#2960	semidirect

H ≅ G/N

type

2. #31826592

C₂

semidirect

3. #31826589

C₂

semidirect

25. C₂×C₂₈₃⋊C₈

Q₈

semidirect

67. C₂₈₃

#2960

semidirect

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

subgroup	index
#31827022	2
#31826592	2
#31826589	2
#31826493	2
#31826423	2
C₂×C₄×C₂₈₃⋊C₈	2
#423×C₂₈₃	2
#2960	283

subgroup

index

C₂×C₄×C₂₈₃⋊C₈

#423×C₂₈₃

#2960

283

Subgroups — #49130191

Conjugacy classes of subgroups

Complements

Maximal subgroups