Group #24905591 — subgroups

Subgroups — #24905591

Order 10624.

162 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 53 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	#19127685	C₂	1	1	yes	maximal, characteristic
2	#19127668	C₂	1	1	yes	maximal, characteristic
3	#410×C₈₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13050574	—	2	1	no	—
5	C₂×#13050574	C₄	1	1	yes	characteristic, complemented
6	C₂²×C₈₃⋊C₈	C₄	1	1	yes	characteristic, complemented
7	#14581711	C₄	1	2	yes	complemented
8	#14581709	—	2	1	no	—
9	C₂²⋊(C₈₃⋊C₈)	—	2	1	no	—
10	C₂²⋊(C₈₃⋊C₈)	—	2	1	no	—
11	C₂×#41×C₈₃	C₂²	1	1	yes	characteristic
12	#117×C₈₃	—	2	1	no	—
13	#99×C₈₃	—	2	1	no	—
14	#99×C₈₃	C₄	1	2	yes	—
15	#13050574	—	2	2	no	—
16	#13050574	—	4	1	no	—
17	#13050574	—	4	1	no	—
18	C₂×C₈₃⋊C₈	—	2	1	no	—
19	C₂×C₈₃⋊C₈	—	2	1	no	—
20	C₂×C₈₃⋊C₈	—	4	1	no	—
21	C₂×C₈₃⋊C₈	—	4	1	no	—
22	#41×C₈₃	—	4	1	no	—
23	#41×C₈₃	—	4	1	no	—
24	C₂×Q₈×C₈₃	C₂×C₄	1	1	yes	characteristic
25	C₂×D₄×C₈₃	C₂×C₄	1	1	yes	characteristic
26	C₂×D₄×C₈₃	D₄	1	1	yes	characteristic
27	C₂×D₄×C₈₃	Q₈	1	1	yes	characteristic
28	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
31	C₄⋊C₄×C₈₃	—	4	1	no	—
32	C₂²⋊C₄×C₈₃	—	2	2	no	—
33	C₄²×C₈₃	—	4	1	no	—
34	C₂³×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
35	C₂³×C₈₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂³×C₈₃	C₄⋊C₄	1	1	yes	characteristic
37	Q₈×C₈₃	—	4	1	no	—
38	D₄×C₈₃	—	4	1	no	—
39	D₄×C₈₃	—	4	1	no	—
40	D₄×C₈₃	—	4	1	no	—
41	C₂×C₄×C₈₃	—	2	1	no	—
42	C₂×C₄×C₈₃	—	2	1	no	—
43	C₂×C₄×C₈₃	—	2	1	no	—
44	C₂×C₄×C₈₃	—	2	1	no	—
45	C₂×C₄×C₈₃	—	2	1	no	—
46	C₂×C₄×C₈₃	—	2	1	no	—
47	C₂×C₄×C₈₃	—	4	2	no	—
48	C₂×C₄×C₈₃	C₄²	1	1	yes	characteristic
49	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₈₃	C₄⋊C₄	1	1	yes	characteristic
52	C₈₃⋊C₈	—	4	2	no	—
53	C₈₃⋊C₈	—	4	1	no	—
54	C₈₃⋊C₈	—	4	1	no	—
55	C₂²×C₈₃	—	2	1	no	—
56	C₂²×C₈₃	—	2	1	no	—
57	C₂²×C₈₃	—	2	1	no	—
58	C₂²×C₈₃	—	4	1	no	—
59	C₂²×C₈₃	—	4	1	no	—
60	C₂²×C₈₃	—	4	1	no	—
61	C₂²×C₈₃	#95	1	1	yes	characteristic
62	C₄×C₈₃	—	2	1	no	—
63	C₄×C₈₃	—	2	1	no	—
64	C₄×C₈₃	—	2	1	no	—
65	C₄×C₈₃	—	8	2	no	—
66	C₄×C₈₃	#99	1	1	yes	characteristic
67	C₄×C₈₃	#99	1	1	yes	characteristic
68	C₂×C₈₃	—	2	1	no	—
69	C₂×C₈₃	—	4	1	no	—
70	C₂×C₈₃	—	4	1	no	—
71	C₂×C₈₃	—	4	1	no	—
72	C₂×C₈₃	#342	1	1	yes	characteristic
73	#1395	—	83	1	no	maximal, Sylow
74	C₈₃	#1395	1	1	yes	characteristic, complemented, Sylow
75	#413	—	83	1	no	—
76	#410	D₈₃	1	1	yes	characteristic
77	#408	—	83	1	no	—
78	C₂×#41	C₂×D₈₃	1	1	yes	characteristic
79	C₂×M₁₆	—	83	1	no	—
80	C₂×M₁₆	—	166	1	no	—
81	C₂²×C₈	—	83	1	no	—
82	#117	—	2	1	no	—
83	#101	—	83	2	no	—
84	#100	—	166	1	no	—
85	#99	—	2	1	no	—
86	#99	Dic₈₃	1	2	yes	—
87	C₂²⋊C₈	—	166	1	no	—
88	C₂²⋊C₈	—	166	1	no	—
89	#41	—	4	1	no	—
90	#41	—	4	1	no	—
91	C₂×Q₈	C₄×D₈₃	1	1	yes	characteristic
92	C₂×D₄	Dic₁₆₆	1	1	yes	characteristic
93	C₂×D₄	C₂×Dic₈₃	1	1	yes	characteristic
94	C₂×D₄	C₂²⋊D₈₃	1	1	yes	characteristic
95	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
96	C₂²×C₄	D₃₃₂	1	1	yes	characteristic
97	C₂²×C₄	C₂²⋊D₈₃	1	1	yes	characteristic
98	M₁₆	—	332	1	no	—
99	M₁₆	—	332	1	no	—
100	M₁₆	—	166	2	no	—
101	C₂×C₈	—	332	1	no	—
102	C₂×C₈	—	332	1	no	—
103	C₂×C₈	—	166	1	no	—
104	C₂×C₈	—	166	1	no	—
105	C₄⋊C₄	—	4	1	no	—
106	C₂²⋊C₄	—	2	2	no	—
107	C₄²	—	4	1	no	—
108	C₂³	C₂²⋊Dic₈₃	1	1	yes	characteristic
109	C₂³	Dic₈₃⋊C₄	1	1	yes	characteristic
110	C₂³	C₄⋊Dic₈₃	1	1	yes	characteristic
111	Q₈	—	4	1	no	—
112	D₄	—	4	1	no	—
113	D₄	—	4	1	no	—
114	D₄	—	4	1	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	C₄×Dic₈₃	1	1	yes	characteristic
123	C₂×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
124	C₂×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
125	C₂×C₄	Dic₈₃⋊C₄	1	1	yes	characteristic
126	C₈	—	332	2	no	—
127	C₈	—	332	1	no	—
128	C₈	—	332	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	4	1	no	—
133	C₂²	—	4	1	no	—
134	C₂²	—	4	1	no	—
135	C₂²	C₈₃⋊#95	1	1	yes	characteristic
136	C₄	—	2	1	no	—
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	8	2	no	—
140	C₄	#14581706	1	1	yes	characteristic
141	C₄	#14581706	1	1	yes	characteristic
142	C₂	—	2	1	no	—
143	C₂	—	4	1	no	—
144	C₂	—	4	1	no	—
145	C₂	—	4	1	no	—
146	C₂	#19127467	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#19127685

C₂

yes

maximal, characteristic

#19127668

C₂

yes

maximal, characteristic

#410×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050574

—

C₂×#13050574

C₄

yes

characteristic, complemented

C₂²×C₈₃⋊C₈

C₄

yes

characteristic, complemented

#14581711

C₄

yes

complemented

#14581709

—

C₂²⋊(C₈₃⋊C₈)

—

C₂²⋊(C₈₃⋊C₈)

—

C₂×#41×C₈₃

C₂²

yes

characteristic

#117×C₈₃

—

#99×C₈₃

—

#99×C₈₃

C₄

yes

—

#13050574

—

#13050574

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

#41×C₈₃

—

#41×C₈₃

—

C₂×Q₈×C₈₃

C₂×C₄

yes

characteristic

C₂×D₄×C₈₃

C₂×C₄

yes

characteristic

C₂×D₄×C₈₃

D₄

yes

characteristic

C₂×D₄×C₈₃

Q₈

yes

characteristic

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₂²×C₄×C₈₃

D₄

yes

characteristic

C₄⋊C₄×C₈₃

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

—

C₂³×C₈₃

C₂²⋊C₄

yes

characteristic

C₂³×C₈₃

C₄⋊C₄

yes

characteristic

C₂³×C₈₃

C₄⋊C₄

yes

characteristic

Q₈×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

D₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₄²

yes

characteristic

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₄⋊C₄

yes

characteristic

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

#95

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

yes

C₄×C₈₃

yes

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

—

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

#413

—

#410

D₈₃

yes

characteristic

#408

—

C₂×#41

C₂×D₈₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

166

—

C₂²×C₈

—

#117

—

#101

—

#100

—

166

—

#99

—

#99

Dic₈₃

yes

—

C₂²⋊C₈

—

166

—

C₂²⋊C₈

—

166

—

#41

—

#41

—

C₂×Q₈

C₄×D₈₃

yes

characteristic

C₂×D₄

Dic₁₆₆

yes

characteristic

C₂×D₄

C₂×Dic₈₃

yes

characteristic

C₂×D₄

C₂²⋊D₈₃

yes

characteristic

C₂²×C₄

C₄×D₈₃

yes

characteristic

C₂²×C₄

D₃₃₂

yes

characteristic

C₂²×C₄

C₂²⋊D₈₃

yes

characteristic

M₁₆

—

332

—

M₁₆

—

332

—

100

M₁₆

—

166

—

101

C₂×C₈

—

332

—

102

C₂×C₈

—

332

—

103

C₂×C₈

—

166

—

104

C₂×C₈

—

166

—

105

C₄⋊C₄

—

106

C₂²⋊C₄

—

107

C₄²

—

108

C₂³

C₂²⋊Dic₈₃

yes

characteristic

109

C₂³

Dic₈₃⋊C₄

yes

characteristic

110

C₂³

C₄⋊Dic₈₃

yes

characteristic

111

Q₈

—

112

D₄

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

C₄×Dic₈₃

yes

characteristic

123

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

characteristic

124

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

characteristic

125

C₂×C₄

Dic₈₃⋊C₄

yes

characteristic

126

C₈

—

332

—

127

C₈

—

332

—

128

C₈

—

332

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

C₈₃⋊#95

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

#14581706

yes

characteristic

141

C₄

#14581706

yes

characteristic

142

C₂

—

143

C₂

—

144

C₂

—

145

C₂

—

146

C₂

#19127467

yes

characteristic

	subgroup	G/H
center	C₄	#14581706
commutator	C₂×C₄×C₈₃	C₄²
Frattini	C₂×#41	C₂×D₈₃
Fitting	#410×C₈₃	C₂
socle	C₂×C₈₃	#342
prime radical, cosocle	C₂×#41×C₈₃	C₂²
hypercenter	#410	D₈₃
hypocenter	C₈₃	#1395
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×#41 ◁ #410

subgroup

G/H

center

C₄

#14581706

commutator

C₂×C₄×C₈₃

C₄²

Frattini

C₂×#41

C₂×D₈₃

Fitting

#410×C₈₃

C₂

socle

C₂×C₈₃

#342

prime radical, cosocle

C₂×#41×C₈₃

C₂²

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂²×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×#41 ◁ #410

N	H ≅ G/N	type
5. C₂×#13050574	C₄	semidirect
6. C₂²×C₈₃⋊C₈	C₄	semidirect
7. #14581711	C₄	semidirect
74. C₈₃	#1395	semidirect

H ≅ G/N

type

5. C₂×#13050574

C₄

semidirect

6. C₂²×C₈₃⋊C₈

C₄

semidirect

7. #14581711

C₄

semidirect

74. C₈₃

#1395

semidirect

Subgroups — #24905591

Conjugacy classes of subgroups

Complements