Group #24907266 — subgroups

Subgroups — #24907266

Order 10624.

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

normal subgroups only

This group has 284 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×#14581799	C₂	1	1	yes	maximal, characteristic
2	#19127675	C₂	1	1	yes	maximal, characteristic
3	#19127659	C₂	1	2	yes	maximal, complemented
4	#19127473	C₂	1	2	yes	maximal, complemented
5	#413×C₈₃	C₂	1	1	yes	maximal, characteristic
6	#14581799	—	2	2	no	—
7	#14581799	—	2	2	no	—
8	C₂×#13050574	C₄	1	2	yes	complemented
9	C₂×#13050574	C₂²	1	1	yes	characteristic
10	C₂²×C₈₃⋊C₈	C₄	1	2	yes	complemented
11	C₂²×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
12	#14581751	C₂²	1	2	yes	—
13	#14581707	C₄	1	4	yes	—
14	#14581702	—	2	2	no	—
15	C₈×Dic₈₃	—	2	2	no	—
16	C₂×#41×C₈₃	C₂²	1	1	yes	characteristic
17	C₂×M₁₆×C₈₃	C₂²	1	2	yes	—
18	#101×C₈₃	C₄	1	2	yes	complemented
19	#100×C₈₃	C₄	1	2	yes	complemented
20	#13050574	—	2	2	no	—
21	#13050574	—	2	4	no	—
22	C₂×C₈₃⋊C₈	—	2	2	no	—
23	C₂×C₈₃⋊C₈	—	2	2	no	—
24	C₂×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
25	C₂×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈₃⋊C₈	D₄	1	2	yes	—
28	C₂×C₈₃⋊C₈	D₄	1	2	yes	—
29	C₄⋊Dic₈₃	—	2	2	no	—
30	C₂²⋊Dic₈₃	C₂×C₄	1	4	yes	—
31	C₄×Dic₈₃	—	2	2	no	—
32	#41×C₈₃	—	2	2	no	—
33	#41×C₈₃	—	2	2	no	—
34	C₂×Q₈×C₈₃	C₂×C₄	1	1	yes	characteristic
35	C₂×D₄×C₈₃	C₂×C₄	1	2	yes	—
36	C₂×D₄×C₈₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₈₃	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
39	M₁₆×C₈₃	—	2	2	no	—
40	M₁₆×C₈₃	C₂×C₄	1	4	yes	—
41	C₂×C₈×C₈₃	—	2	2	no	—
42	C₂³×C₈₃	C₄²	1	2	yes	—
43	C₂³×C₈₃	C₂²×C₄	1	1	yes	characteristic
44	Q₈×C₈₃	—	2	2	no	—
45	D₄×C₈₃	—	2	4	no	—
46	D₄×C₈₃	—	2	2	no	—
47	C₂×C₄×C₈₃	—	2	2	no	—
48	C₂×C₄×C₈₃	—	2	2	no	—
49	C₂×C₄×C₈₃	C₄²	1	2	yes	—
50	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₈₃	#41	1	2	yes	—
54	C₂×Dic₈₃	—	2	4	no	—
55	C₈×C₈₃	—	2	4	no	—
56	C₈₃⋊C₈	—	2	2	no	—
57	C₈₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
58	C₈₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
59	C₈₃⋊C₈	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₈₃	—	2	2	no	—
61	C₂²×C₈₃	—	4	1	no	—
62	C₂²×C₈₃	—	4	2	no	—
63	C₂²×C₈₃	C₂×C₄²	1	1	yes	characteristic
64	C₂²×C₈₃	C₄×D₄	1	2	yes	—
65	C₄×C₈₃	—	2	2	no	—
66	C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₄×C₈₃	#117	1	1	yes	characteristic
68	C₄×C₈₃	C₄×D₄	1	2	yes	—
69	Dic₈₃	—	4	4	no	—
70	C₂×C₈₃	—	2	2	no	—
71	C₂×C₈₃	—	2	1	no	—
72	C₂×C₈₃	—	4	2	no	—
73	C₂×C₈₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
74	#1757	—	83	1	no	maximal, Sylow
75	C₈₃	#1757	1	1	yes	characteristic, complemented, Sylow
76	C₂×#131	—	83	1	no	—
77	#413	D₈₃	1	1	yes	characteristic
78	#410	—	83	1	no	—
79	#405	—	83	2	no	—
80	#344	—	83	2	no	—
81	C₂×#41	C₂×D₈₃	1	1	yes	characteristic
82	#131	—	166	2	no	—
83	#131	—	166	2	no	—
84	C₂×M₁₆	—	83	2	no	—
85	C₂×M₁₆	—	83	1	no	—
86	C₂×M₁₆	C₂×D₈₃	1	2	yes	—
87	C₂²×C₈	—	83	1	no	—
88	C₂²×C₈	—	83	2	no	—
89	#117	—	83	2	no	—
90	#101	Dic₈₃	1	2	yes	complemented
91	#100	Dic₈₃	1	2	yes	complemented
92	#99	—	83	4	no	—
93	#97	—	166	2	no	—
94	C₄×C₈	—	166	2	no	—
95	#41	—	2	2	no	—
96	#41	—	2	2	no	—
97	C₂×Q₈	C₂×Dic₈₃	1	1	yes	characteristic
98	C₂×D₄	C₄×D₈₃	1	2	yes	—
99	C₂×D₄	C₂×Dic₈₃	1	1	yes	characteristic
100	C₂²×C₄	C₄×D₈₃	1	2	yes	—
101	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
102	M₁₆	—	2	2	no	—
103	M₁₆	—	166	4	no	—
104	M₁₆	—	166	2	no	—
105	M₁₆	C₂×Dic₈₃	1	4	yes	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	83	1	no	—
108	C₂×C₈	—	83	1	no	—
109	C₂×C₈	—	83	2	no	—
110	C₂×C₈	—	83	2	no	—
111	C₂×C₈	—	83	2	no	—
112	C₂×C₈	—	166	2	no	—
113	C₂×C₈	—	166	2	no	—
114	C₄⋊C₄	—	166	2	no	—
115	C₂²⋊C₄	—	83	4	no	—
116	C₄²	—	166	2	no	—
117	C₂³	C₄×Dic₈₃	1	2	yes	—
118	C₂³	C₂×C₄×D₈₃	1	1	yes	characteristic
119	Q₈	—	2	2	no	—
120	D₄	—	2	4	no	—
121	D₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	166	4	no	—
125	C₂×C₄	C₄×Dic₈₃	1	2	yes	—
126	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
127	C₂×C₄	C₂²×Dic₈₃	1	1	yes	characteristic
128	C₂×C₄	D₄×D₈₃	1	2	yes	—
129	C₂×C₄	#13050589	1	2	yes	—
130	C₈	—	2	4	no	—
131	C₈	—	83	1	no	—
132	C₈	—	83	1	no	—
133	C₈	—	83	2	no	—
134	C₈	—	166	2	no	—
135	C₂²	—	2	2	no	—
136	C₂²	—	4	1	no	—
137	C₂²	—	4	2	no	—
138	C₂²	C₂×C₄×Dic₈₃	1	1	yes	characteristic
139	C₂²	(C₂²⋊D₈₃)⋊C₄	1	2	yes	—
140	C₄	—	2	2	no	—
141	C₄	—	332	4	no	—
142	C₄	C₂²⋊C₄×D₈₃	1	1	yes	characteristic
143	C₄	#14581748	1	1	yes	characteristic
144	C₄	D₄×Dic₈₃	1	2	yes	—
145	C₂	—	2	2	no	—
146	C₂	—	2	1	no	—
147	C₂	—	4	2	no	—
148	C₂	C₂²⋊C₄×Dic₈₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581799

C₂

yes

maximal, characteristic

#19127675

C₂

yes

maximal, characteristic

#19127659

C₂

yes

maximal, complemented

#19127473

C₂

yes

maximal, complemented

#413×C₈₃

C₂

yes

maximal, characteristic

#14581799

—

#14581799

—

C₂×#13050574

C₄

yes

complemented

C₂×#13050574

C₂²

yes

characteristic

C₂²×C₈₃⋊C₈

C₄

yes

complemented

C₂²×C₈₃⋊C₈

C₂²

yes

characteristic

#14581751

C₂²

yes

—

#14581707

C₄

yes

—

#14581702

—

C₈×Dic₈₃

—

C₂×#41×C₈₃

C₂²

yes

characteristic

C₂×M₁₆×C₈₃

C₂²

yes

—

#101×C₈₃

C₄

yes

complemented

#100×C₈₃

C₄

yes

complemented

#13050574

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

C₂×C₄

yes

—

C₂×C₈₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₈₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₈₃⋊C₈

D₄

yes

—

C₂×C₈₃⋊C₈

D₄

yes

—

C₄⋊Dic₈₃

—

C₂²⋊Dic₈₃

C₂×C₄

yes

—

C₄×Dic₈₃

—

#41×C₈₃

—

#41×C₈₃

—

C₂×Q₈×C₈₃

C₂×C₄

yes

characteristic

C₂×D₄×C₈₃

C₂×C₄

yes

—

C₂×D₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×C₄

yes

—

C₂²×C₄×C₈₃

C₂³

yes

characteristic

M₁₆×C₈₃

—

M₁₆×C₈₃

C₂×C₄

yes

—

C₂×C₈×C₈₃

—

C₂³×C₈₃

C₄²

yes

—

C₂³×C₈₃

C₂²×C₄

yes

characteristic

Q₈×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₄²

yes

—

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

#41

yes

—

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₈₃⋊C₈

C₂²⋊C₄

yes

characteristic

C₈₃⋊C₈

C₂²⋊C₄

yes

characteristic

C₈₃⋊C₈

C₂²⋊C₄

yes

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂×C₄²

yes

characteristic

C₂²×C₈₃

C₄×D₄

yes

—

C₄×C₈₃

—

C₄×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₈₃

#117

yes

characteristic

C₄×C₈₃

C₄×D₄

yes

—

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

C₄×C₂²⋊C₄

yes

—

C₈₃

yes

characteristic, complemented, Sylow

C₂×#131

—

#413

D₈₃

yes

—

—

—

C₂×#41

C₂×D₈₃

yes

characteristic

#131

—

166

—

#131

—

166

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₃

yes

—

C₂²×C₈

—

C₂²×C₈

—

#117

—

#101

Dic₈₃

yes

complemented

#100

Dic₈₃

yes

complemented

#99

—

#97

—

166

—

C₄×C₈

—

166

—

#41

—

#41

—

C₂×Q₈

C₂×Dic₈₃

yes

characteristic

C₂×D₄

C₄×D₈₃

yes

—

C₂×D₄

C₂×Dic₈₃

yes

characteristic

100

C₂²×C₄

C₄×D₈₃

yes

—

101

C₂²×C₄

C₂²×D₈₃

yes

characteristic

102

M₁₆

—

103

M₁₆

—

166

—

104

M₁₆

—

166

—

105

M₁₆

C₂×Dic₈₃

yes

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

166

—

113

C₂×C₈

—

166

—

114

C₄⋊C₄

—

166

—

115

C₂²⋊C₄

—

116

C₄²

—

166

—

117

C₂³

C₄×Dic₈₃

yes

—

118

C₂³

C₂×C₄×D₈₃

yes

characteristic

119

Q₈

—

120

D₄

—

121

D₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

166

—

125

C₂×C₄

C₄×Dic₈₃

yes

—

126

C₂×C₄

C₂×C₄×D₈₃

yes

characteristic

127

C₂×C₄

C₂²×Dic₈₃

yes

characteristic

128

C₂×C₄

D₄×D₈₃

yes

—

129

C₂×C₄

#13050589

yes

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₈

—

134

C₈

—

166

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

C₂×C₄×Dic₈₃

yes

characteristic

139

C₂²

(C₂²⋊D₈₃)⋊C₄

yes

—

140

C₄

—

141

C₄

—

332

—

142

C₄

C₂²⋊C₄×D₈₃

yes

characteristic

143

C₄

#14581748

yes

characteristic

144

C₄

D₄×Dic₈₃

yes

—

145

C₂

—

146

C₂

—

147

C₂

—

148

C₂

C₂²⋊C₄×Dic₈₃

yes

characteristic

	subgroup	G/H
center	C₄	C₂²⋊C₄×D₈₃
commutator	C₂²×C₈₃	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	#413×C₈₃	C₂
socle	C₂×C₈₃	C₄×C₂²⋊C₄
prime radical	C₂×#41×C₈₃	C₂²
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	#413	D₈₃
hypocenter	C₈₃	#1757
LCS	G ▷ C₂²×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

subgroup

G/H

center

C₄

C₂²⋊C₄×D₈₃

commutator

C₂²×C₈₃

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

#413×C₈₃

C₂

socle

C₂×C₈₃

C₄×C₂²⋊C₄

prime radical

C₂×#41×C₈₃

C₂²

cosocle

C₂²×C₄×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂²×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

N	H ≅ G/N	type
3. #19127659	C₂	semidirect
4. #19127473	C₂	semidirect
8. C₂×#13050574	C₄	semidirect
10. C₂²×C₈₃⋊C₈	C₄	semidirect
18. #101×C₈₃	C₄	semidirect
19. #100×C₈₃	C₄	semidirect
75. C₈₃	#1757	semidirect
90. #101	Dic₈₃	semidirect
91. #100	Dic₈₃	semidirect

H ≅ G/N

type

3. #19127659

C₂

semidirect

4. #19127473

C₂

semidirect

8. C₂×#13050574

C₄

semidirect

10. C₂²×C₈₃⋊C₈

C₄

semidirect

18. #101×C₈₃

C₄

semidirect

19. #100×C₈₃

C₄

semidirect

75. C₈₃

#1757

semidirect

90. #101

Dic₈₃

semidirect

91. #100

Dic₈₃

semidirect

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

subgroup	index
C₂×#14581799	2
#19127675	2
#19127659	2
#19127473	2
#413×C₈₃	2
#1757	83

subgroup

index

#413×C₈₃

Subgroups — #24907266

Conjugacy classes of subgroups

Complements

Maximal subgroups