Group #24908451 — subgroups

Subgroups — #24908451

Order 10624.

260 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 97 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×#14581792	C₂	1	1	yes	maximal, characteristic
2	C₄×#13050574	C₂	1	1	yes	maximal, characteristic
3	#19127469	C₂	1	4	yes	maximal
4	#423×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₆₆	C₄	1	2	yes	—
6	C₂×#13050574	C₂²	1	2	yes	—
7	#14581792	—	2	1	no	—
8	#14581792	—	2	1	no	—
9	#14581792	—	2	1	no	—
10	#14581792	—	2	1	no	—
11	C₂×C₄⋊Dic₈₃	C₄	1	2	yes	—
12	C₂×C₄⋊Dic₈₃	C₂²	1	2	yes	—
13	#14581703	—	2	1	no	—
14	C₄×C₈₃⋊C₈	—	2	1	no	—
15	C₂×M₁₆×C₈₃	C₂²	1	2	yes	—
16	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
17	C₄⋊C₈×C₈₃	—	2	1	no	—
18	C₄⋊C₈×C₈₃	—	2	1	no	—
19	C₂×Dic₁₆₆	—	2	2	no	—
20	C₂×Dic₁₆₆	—	2	2	no	—
21	C₂×Dic₁₆₆	—	4	4	no	—
22	C₂²×Dic₈₃	C₂×C₄	1	4	yes	—
23	#13050574	—	2	2	no	—
24	#13050574	Q₈	1	4	yes	complemented
25	C₂×C₈₃⋊C₈	—	2	2	no	—
26	C₄⋊Dic₈₃	—	2	2	no	—
27	C₄⋊Dic₈₃	—	2	2	no	—
28	C₄⋊Dic₈₃	—	2	2	no	—
29	C₄⋊Dic₈₃	—	2	2	no	—
30	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
33	M₁₆×C₈₃	—	2	4	no	—
34	C₂×C₈×C₈₃	—	2	2	no	—
35	C₄²×C₈₃	D₄	1	1	yes	characteristic
36	C₄²×C₈₃	D₄	1	1	yes	characteristic
37	C₄²×C₈₃	D₄	1	1	yes	characteristic
38	C₄²×C₈₃	D₄	1	1	yes	characteristic
39	C₂³×C₈₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₈₃	—	2	2	no	—
41	C₂×C₄×C₈₃	—	2	2	no	—
42	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₈₃	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₈₃	#41	1	2	yes	—
50	C₂×Dic₈₃	—	2	4	no	—
51	C₂×Dic₈₃	—	4	4	no	—
52	Dic₁₆₆	—	4	4	no	—
53	Dic₁₆₆	—	4	4	no	—
54	C₈×C₈₃	—	4	2	no	—
55	C₈₃⋊C₈	—	2	4	no	—
56	C₂²×C₈₃	—	2	1	no	—
57	C₂²×C₈₃	—	2	1	no	—
58	C₂²×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₈₃	C₄×Q₈	1	2	yes	—
60	C₄×C₈₃	—	2	1	no	—
61	C₄×C₈₃	—	2	1	no	—
62	C₄×C₈₃	#100	1	2	yes	—
63	C₄×C₈₃	#101	1	2	yes	—
64	C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
65	C₄×C₈₃	#124	1	1	yes	characteristic
66	C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
67	Dic₈₃	—	4	4	no	—
68	C₂×C₈₃	—	2	2	no	—
69	C₂×C₈₃	#391	1	1	yes	characteristic
70	C₂×C₈₃	C₂×#100	1	1	yes	characteristic
71	C₂×C₈₃	C₂×#101	1	1	yes	characteristic
72	#1997	—	83	1	no	maximal, Sylow
73	C₈₃	#1997	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₄⋊Q₈	—	83	1	no	—
75	#423	D₈₃	1	1	yes	characteristic
76	C₄×M₁₆	—	83	1	no	—
77	#343	—	83	4	no	—
78	C₂²×Q₈	—	83	2	no	—
79	C₂×M₁₆	—	83	2	no	—
80	C₂×M₁₆	C₂×D₈₃	1	2	yes	—
81	C₄⋊Q₈	—	166	1	no	—
82	C₄⋊Q₈	—	166	1	no	—
83	C₄⋊Q₈	—	166	1	no	—
84	C₄⋊Q₈	—	166	1	no	—
85	C₂×C₄⋊C₄	—	83	2	no	—
86	C₂×C₄⋊C₄	—	83	2	no	—
87	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
88	C₄⋊C₈	—	2	1	no	—
89	C₄⋊C₈	—	2	1	no	—
90	#97	—	166	1	no	—
91	C₄×C₈	—	166	1	no	—
92	C₂×Q₈	—	332	4	no	—
93	C₂×Q₈	—	166	2	no	—
94	C₂×Q₈	—	166	2	no	—
95	C₂²×C₄	—	83	4	no	—
96	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
97	C₂²×C₄	C₄×D₈₃	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
99	M₁₆	—	2	4	no	—
100	M₁₆	—	83	4	no	—
101	M₁₆	—	166	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₂×C₈	—	166	2	no	—
104	C₄⋊C₄	—	166	2	no	—
105	C₄⋊C₄	—	166	2	no	—
106	C₄⋊C₄	—	166	2	no	—
107	C₄⋊C₄	—	166	2	no	—
108	C₄²	D₃₃₂	1	1	yes	characteristic
109	C₄²	D₃₃₂	1	1	yes	characteristic
110	C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
111	C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
112	C₂³	C₂×C₄×D₈₃	1	1	yes	characteristic
113	Q₈	—	332	4	no	—
114	Q₈	—	332	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	332	4	no	—
118	C₂×C₄	—	166	4	no	—
119	C₂×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
120	C₂×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
121	C₂×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
122	C₂×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
123	C₂×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
124	C₂×C₄	C₂×D₃₃₂	1	1	yes	characteristic
125	C₂×C₄	Q₈×D₈₃	1	2	yes	—
126	C₂×C₄	#13050589	1	2	yes	—
127	C₈	—	4	2	no	—
128	C₈	—	166	4	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
132	C₂²	#14581757	1	2	yes	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	332	4	no	—
136	C₄	#14581708	1	2	yes	—
137	C₄	#14581710	1	2	yes	—
138	C₄	#14581772	1	2	yes	—
139	C₄	#14581779	1	1	yes	characteristic
140	C₄	#14581790	1	1	yes	characteristic
141	C₂	—	2	2	no	—
142	C₂	#19127606	1	1	yes	characteristic
143	C₂	C₂×#14581708	1	1	yes	characteristic
144	C₂	C₂×#14581710	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14581792

C₂

yes

maximal, characteristic

C₄×#13050574

C₂

yes

maximal, characteristic

#19127469

C₂

yes

maximal

#423×C₈₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₆₆

C₄

yes

—

C₂×#13050574

C₂²

yes

—

—

—

—

—

C₂×C₄⋊Dic₈₃

C₄

yes

—

C₂×C₄⋊Dic₈₃

C₂²

yes

—

#14581703

—

C₄×C₈₃⋊C₈

—

C₂×M₁₆×C₈₃

C₂²

yes

—

C₂×C₄²×C₈₃

C₂²

yes

characteristic

C₄⋊C₈×C₈₃

—

C₄⋊C₈×C₈₃

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

—

C₂²×Dic₈₃

C₂×C₄

yes

—

#13050574

—

#13050574

Q₈

yes

complemented

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₈₃

C₂³

yes

characteristic

M₁₆×C₈₃

—

C₂×C₈×C₈₃

—

C₄²×C₈₃

D₄

yes

characteristic

C₄²×C₈₃

D₄

yes

characteristic

C₄²×C₈₃

D₄

yes

characteristic

C₄²×C₈₃

D₄

yes

characteristic

C₂³×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×Q₈

yes

—

C₂×C₄×C₈₃

#41

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₈₃

C₄×Q₈

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#100

yes

—

C₄×C₈₃

#101

yes

—

C₄×C₈₃

C₂²⋊Q₈

yes

—

C₄×C₈₃

#124

yes

characteristic

C₄×C₈₃

C₄⋊Q₈

yes

characteristic

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₈₃

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

#423

D₈₃

yes

characteristic

C₄×M₁₆

—

#343

—

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₈₃

yes

—

C₄⋊Q₈

—

166

—

C₄⋊Q₈

—

166

—

C₄⋊Q₈

—

166

—

C₄⋊Q₈

—

166

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₈₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

166

—

C₄×C₈

—

166

—

C₂×Q₈

—

332

—

C₂×Q₈

—

166

—

C₂×Q₈

—

166

—

C₂²×C₄

—

C₂²×C₄

C₄×D₈₃

yes

characteristic

C₂²×C₄

C₄×D₈₃

yes

characteristic

C₂²×C₄

C₂²×D₈₃

yes

characteristic

M₁₆

—

100

M₁₆

—

101

M₁₆

—

166

—

102

C₂×C₈

—

103

C₂×C₈

—

166

—

104

C₄⋊C₄

—

166

—

105

C₄⋊C₄

—

166

—

106

C₄⋊C₄

—

166

—

107

C₄⋊C₄

—

166

—

108

C₄²

D₃₃₂

yes

characteristic

109

C₄²

D₃₃₂

yes

characteristic

110

C₄²

C₂²⋊D₈₃

yes

characteristic

111

C₄²

C₂²⋊D₈₃

yes

characteristic

112

C₂³

C₂×C₄×D₈₃

yes

characteristic

113

Q₈

—

332

—

114

Q₈

—

332

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

332

—

118

C₂×C₄

—

166

—

119

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

characteristic

120

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

characteristic

121

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

characteristic

122

C₂×C₄

(C₂×D₈₃)⋊C₄

yes

characteristic

123

C₂×C₄

C₂×C₂²⋊D₈₃

yes

characteristic

124

C₂×C₄

C₂×D₃₃₂

yes

characteristic

125

C₂×C₄

Q₈×D₈₃

yes

—

126

C₂×C₄

#13050589

yes

—

127

C₈

—

128

C₈

—

166

—

129

C₂²

—

130

C₂²

—

131

C₂²

C₂×(C₂×D₈₃)⋊C₄

yes

characteristic

132

C₂²

#14581757

yes

—

133

C₄

—

134

C₄

—

135

C₄

—

332

—

136

C₄

#14581708

yes

—

137

C₄

#14581710

yes

—

138

C₄

#14581772

yes

—

139

C₄

#14581779

yes

characteristic

140

C₄

#14581790

yes

characteristic

141

C₂

—

142

C₂

#19127606

yes

characteristic

143

C₂

C₂×#14581708

yes

characteristic

144

C₂

C₂×#14581710

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×(C₂×D₈₃)⋊C₄
commutator, prime radical	C₂³×C₈₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₈₃
Fitting	#423×C₈₃	C₂
socle	C₂²×C₈₃	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₈₃	C₂³
hypercenter	#423	D₈₃
hypocenter	C₈₃	#1997
LCS	G ▷ C₂³×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂×(C₂×D₈₃)⋊C₄

commutator, prime radical

C₂³×C₈₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₈₃

Fitting

#423×C₈₃

C₂

socle

C₂²×C₈₃

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₂³×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
24. #13050574	Q₈	semidirect
73. C₈₃	#1997	semidirect

H ≅ G/N

type

24. #13050574

Q₈

semidirect

73. C₈₃

#1997

semidirect

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

subgroup	index
C₂×#14581792	2
C₄×#13050574	2
#19127469	2
#423×C₈₃	2
#1997	83

subgroup

index

#423×C₈₃

Subgroups — #24908451

Conjugacy classes of subgroups

Complements

Maximal subgroups