Group #24918206 — subgroups

Subgroups — #24918206

Order 10624.

358 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 125 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	#19128256	C₂	1	2	yes	maximal
2	#19128099	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128051	C₂	1	2	yes	maximal, complemented
4	#19127956	C₂	1	4	yes	maximal
5	#19127808	C₂	1	4	yes	maximal, complemented
6	C₄×#13050573	C₂	1	1	yes	maximal, characteristic
7	#494×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13050573	C₂²	1	2	yes	—
9	#14581792	C₂²	1	2	yes	—
10	#14581783	—	2	2	no	—
11	#14581772	—	2	2	no	—
12	#14581768	C₂²	1	4	yes	—
13	Q₈×Dic₈₃	C₂²	1	4	yes	—
14	#14581757	—	2	2	no	—
15	#14581750	—	2	2	no	—
16	C₄²×D₈₃	C₂²	1	1	yes	characteristic
17	#14581724	C₂²	1	4	yes	—
18	#14581716	C₂²	1	8	yes	—
19	#14581702	C₂²	1	2	yes	—
20	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×Q₁₆×C₈₃	C₂²	1	4	yes	—
22	C₄⋊Q₈×C₈₃	C₂²	1	2	yes	—
23	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
24	C₂×Dic₁₆₆	—	2	2	no	—
25	C₂×C₄×D₈₃	D₄	1	2	yes	—
26	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
27	#13050573	—	2	4	no	—
28	C₂×C₈₃⋊C₈	C₂³	1	2	yes	—
29	C₄⋊Dic₈₃	—	2	2	no	—
30	C₄⋊Dic₈₃	—	2	4	no	—
31	C₄⋊Dic₈₃	C₂³	1	4	yes	—
32	Dic₈₃⋊C₄	—	2	4	no	—
33	(C₂×D₈₃)⋊C₄	—	2	4	no	—
34	C₄×Dic₈₃	—	2	4	no	—
35	C₄×Dic₈₃	D₄	1	2	yes	—
36	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
37	C₂×Q₈×C₈₃	C₂³	1	4	yes	—
38	Q₁₆×C₈₃	—	2	8	no	—
39	C₂×C₈×C₈₃	C₂³	1	2	yes	—
40	C₄⋊C₄×C₈₃	—	2	2	no	—
41	C₄⋊C₄×C₈₃	—	2	2	no	—
42	C₄²×C₈₃	C₂³	1	1	yes	characteristic
43	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
44	Q₈×C₈₃	—	2	8	no	—
45	C₂×C₄×C₈₃	—	2	4	no	—
46	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
48	C₂×Dic₈₃	—	2	4	no	—
49	C₂×Dic₈₃	C₂×D₄	1	2	yes	—
50	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
51	C₄×D₈₃	—	2	4	no	—
52	C₄×D₈₃	—	2	2	no	—
53	Dic₁₆₆	—	4	4	no	—
54	C₈×C₈₃	#41	1	4	yes	—
55	C₈₃⋊C₈	—	2	2	no	—
56	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
57	C₂×D₈₃	—	2	1	no	—
58	C₂×D₈₃	—	2	2	no	—
59	C₄×C₈₃	—	4	4	no	—
60	C₄×C₈₃	#137	1	4	yes	—
61	C₄×C₈₃	C₂×#41	1	2	yes	—
62	Dic₈₃	—	2	1	no	—
63	Dic₈₃	—	2	2	no	—
64	Dic₈₃	—	4	4	no	—
65	C₂×C₈₃	#532	1	1	yes	characteristic
66	C₂×C₈₃	C₂×#137	1	2	yes	—
67	D₈₃	—	4	1	no	—
68	#3185	—	83	1	no	maximal, Sylow
69	C₈₃	#3185	1	1	yes	characteristic, complemented, Sylow
70	#533	—	83	2	no	—
71	#498	—	83	1	no	—
72	#494	D₈₃	1	1	yes	characteristic, complemented
73	#487	—	83	2	no	—
74	#470	—	83	4	no	—
75	#441	—	83	4	no	—
76	C₄×M₁₆	—	83	1	no	—
77	C₂×Q₁₆	C₂×D₈₃	1	4	yes	—
78	C₂×M₁₆	—	83	2	no	—
79	C₄⋊Q₈	—	83	2	no	—
80	C₄⋊Q₈	C₂×D₈₃	1	2	yes	—
81	#125	—	166	2	no	—
82	C₂²⋊Q₈	—	83	4	no	—
83	C₂²⋊Q₈	—	166	2	no	—
84	C₄×Q₈	—	83	4	no	—
85	C₄×Q₈	—	166	2	no	—
86	#117	—	166	2	no	—
87	C₂×C₄²	—	83	1	no	—
88	#106	—	83	4	no	—
89	#103	—	83	8	no	—
90	#97	—	83	2	no	—
91	C₄×C₈	—	83	1	no	—
92	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
93	C₂×Q₈	—	166	2	no	—
94	C₂×Q₈	C₂²×D₈₃	1	4	yes	—
95	C₂²×C₄	—	83	2	no	—
96	C₂²×C₄	—	83	1	no	—
97	Q₁₆	—	2	8	no	—
98	M₁₆	—	166	4	no	—
99	C₂×C₈	—	83	2	no	—
100	C₂×C₈	C₂²×D₈₃	1	2	yes	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	83	4	no	—
104	C₄⋊C₄	—	166	4	no	—
105	C₄⋊C₄	—	166	4	no	—
106	C₄⋊C₄	—	166	2	no	—
107	C₂²⋊C₄	—	166	4	no	—
108	C₄²	—	83	2	no	—
109	C₄²	—	83	1	no	—
110	C₄²	—	166	4	no	—
111	C₄²	C₂²×D₈₃	1	1	yes	characteristic
112	C₂³	—	83	1	no	—
113	Q₈	—	2	8	no	—
114	Q₈	—	332	4	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	83	2	no	—
117	C₂×C₄	—	83	1	no	—
118	C₂×C₄	—	166	4	no	—
119	C₂×C₄	—	166	4	no	—
120	C₂×C₄	—	166	2	no	—
121	C₂×C₄	D₄×D₈₃	1	2	yes	—
122	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
123	C₈	—	166	2	no	—
124	C₈	#13050589	1	4	yes	—
125	C₂²	—	166	2	no	—
126	C₂²	—	166	1	no	—
127	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
128	C₄	—	4	4	no	—
129	C₄	—	332	4	no	—
130	C₄	—	166	2	no	—
131	C₄	—	166	1	no	—
132	C₄	#14581824	1	4	yes	—
133	C₄	C₂×#13050589	1	2	yes	—
134	C₂	—	332	1	no	—
135	C₂	#19128249	1	1	yes	characteristic
136	C₂	C₂×#14581824	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#19128256

C₂

yes

maximal

#19128099

C₂

yes

maximal, characteristic, complemented

#19128051

C₂

yes

maximal, complemented

#19127956

C₂

yes

maximal

#19127808

C₂

yes

maximal, complemented

C₄×#13050573

C₂

yes

maximal, characteristic

#494×C₈₃

C₂

yes

maximal, characteristic, complemented

C₂×#13050573

C₂²

yes

—

#14581792

C₂²

yes

—

#14581783

—

#14581772

—

#14581768

C₂²

yes

—

Q₈×Dic₈₃

C₂²

yes

—

#14581757

—

#14581750

—

C₄²×D₈₃

C₂²

yes

characteristic

#14581724

C₂²

yes

—

#14581716

C₂²

yes

—

#14581702

C₂²

yes

—

C₄×C₈₃⋊C₈

C₂²

yes

characteristic

C₂×Q₁₆×C₈₃

C₂²

yes

—

C₄⋊Q₈×C₈₃

C₂²

yes

—

C₄×C₈×C₈₃

C₂²

yes

characteristic

C₂×Dic₁₆₆

—

C₂×C₄×D₈₃

D₄

yes

—

C₂×C₄×D₈₃

C₂³

yes

characteristic

#13050573

—

C₂×C₈₃⋊C₈

C₂³

yes

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

C₂³

yes

—

Dic₈₃⋊C₄

—

(C₂×D₈₃)⋊C₄

—

C₄×Dic₈₃

—

C₄×Dic₈₃

D₄

yes

—

C₄×Dic₈₃

C₂³

yes

characteristic

C₂×Q₈×C₈₃

C₂³

yes

—

Q₁₆×C₈₃

—

C₂×C₈×C₈₃

C₂³

yes

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

C₂³

yes

characteristic

C₂²×D₈₃

C₂×D₄

yes

characteristic

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂⁴

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

C₂×D₄

yes

—

C₂×Dic₈₃

C₂×D₄

yes

characteristic

C₄×D₈₃

—

C₄×D₈₃

—

Dic₁₆₆

—

C₈×C₈₃

#41

yes

—

C₈₃⋊C₈

—

C₂²×C₈₃

C₂²×D₄

yes

characteristic

C₂×D₈₃

—

C₂×D₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#137

yes

—

C₄×C₈₃

C₂×#41

yes

—

Dic₈₃

—

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

#532

yes

characteristic

C₂×C₈₃

C₂×#137

yes

—

D₈₃

—

#3185

—

maximal, Sylow

C₈₃

#3185

yes

characteristic, complemented, Sylow

#533

—

#498

—

#494

D₈₃

yes

characteristic, complemented

#487

—

#470

—

#441

—

C₄×M₁₆

—

C₂×Q₁₆

C₂×D₈₃

yes

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₈₃

yes

—

#125

—

166

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

166

—

C₄×Q₈

—

C₄×Q₈

—

166

—

#117

—

166

—

C₂×C₄²

—

#106

—

#103

—

#97

—

C₄×C₈

—

C₄×C₈

C₂×D₈₃

yes

characteristic

C₂×Q₈

—

166

—

C₂×Q₈

C₂²×D₈₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

Q₁₆

—

M₁₆

—

166

—

C₂×C₈

—

100

C₂×C₈

C₂²×D₈₃

yes

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

166

—

105

C₄⋊C₄

—

166

—

106

C₄⋊C₄

—

166

—

107

C₂²⋊C₄

—

166

—

108

C₄²

—

109

C₄²

—

110

C₄²

—

166

—

111

C₄²

C₂²×D₈₃

yes

characteristic

112

C₂³

—

113

Q₈

—

114

Q₈

—

332

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

166

—

119

C₂×C₄

—

166

—

120

C₂×C₄

—

166

—

121

C₂×C₄

D₄×D₈₃

yes

—

122

C₂×C₄

C₂³×D₈₃

yes

characteristic

123

C₈

—

166

—

124

C₈

#13050589

yes

—

125

C₂²

—

166

—

126

C₂²

—

166

—

127

C₂²

C₂×D₄×D₈₃

yes

characteristic

128

C₄

—

129

C₄

—

332

—

130

C₄

—

166

—

131

C₄

—

166

—

132

C₄

#14581824

yes

—

133

C₄

C₂×#13050589

yes

—

134

C₂

—

332

—

135

C₂

#19128249

yes

characteristic

136

C₂

C₂×#14581824

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#494×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂²×D₈₃	C₂×D₄
hypercenter	#494	D₈₃
hypocenter	C₈₃	#3185
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

C₂×D₄×D₈₃

commutator, cosocle

C₂×C₄×C₈₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₃

Fitting

#494×C₈₃

C₂

socle

C₂²×C₈₃

C₂²×D₄

prime radical

C₂²×D₈₃

C₂×D₄

D₈₃

C₈₃

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
2. #19128099	C₂	semidirect
3. #19128051	C₂	semidirect
5. #19127808	C₂	semidirect
7. #494×C₈₃	C₂	semidirect
69. C₈₃	#3185	semidirect
72. #494	D₈₃	semidirect

H ≅ G/N

type

2. #19128099

C₂

semidirect

3. #19128051

C₂

semidirect

5. #19127808

C₂

semidirect

7. #494×C₈₃

C₂

semidirect

69. C₈₃

#3185

semidirect

72. #494

D₈₃

semidirect

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

subgroup	index
#19128256	2
#19128099	2
#19128051	2
#19127956	2
#19127808	2
C₄×#13050573	2
#494×C₈₃	2
#3185	83

subgroup

index

#494×C₈₃

Subgroups — #24918206

Conjugacy classes of subgroups

Complements

Maximal subgroups