Group #35316151 — subgroups

Subgroups — #35316151

Order 21248.

222 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 87 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	C₄×Dic₆₆₄	C₂	1	1	yes	maximal, characteristic
2	C₈×Dic₃₃₂	C₂	1	2	yes	maximal
3	#24905537	C₂	1	1	yes	maximal, characteristic
4	#24905428	C₂	1	2	yes	maximal
5	C₈₃×#1311	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₆₆₄	C₄	1	1	yes	characteristic
7	C₈×Dic₁₆₆	—	2	2	no	—
8	C₄×Dic₃₃₂	C₂²	1	2	yes	—
9	#19127527	C₄	1	1	yes	characteristic
10	#19127506	C₄	1	2	yes	—
11	#19127454	C₂²	1	2	yes	—
12	#19127433	—	2	2	no	—
13	C₄×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
15	C₈²×C₈₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₃₃₂	C₂×C₄	1	2	yes	—
17	C₄×Dic₁₆₆	—	2	2	no	—
18	Dic₆₆₄	C₈	1	4	yes	complemented
19	#14581726	C₂×C₄	1	2	yes	—
20	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
21	Dic₈₃⋊C₈	—	4	2	no	—
22	#14581717	—	2	2	no	—
23	C₈×Dic₈₃	—	4	2	no	—
24	C₂×C₁₆×C₈₃	—	2	1	no	—
25	C₂×C₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
27	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
28	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₆₆	—	2	2	no	—
31	Dic₃₃₂	—	2	2	no	—
32	Dic₃₃₂	C₂×C₈	1	4	yes	—
33	C₂×C₈₃⋊C₈	—	4	2	no	—
34	C₄⋊Dic₈₃	—	2	2	no	—
35	Dic₈₃⋊C₄	—	4	2	no	—
36	C₄×Dic₈₃	—	4	2	no	—
37	C₂×C₈×C₈₃	—	2	1	no	—
38	C₂×C₈×C₈₃	D₈	1	2	yes	—
39	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
41	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₈₃	—	2	2	no	—
43	C₁₆×C₈₃	—	2	1	no	—
44	C₁₆×C₈₃	C₂×C₈	1	2	yes	—
45	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
46	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
47	C₂×C₄×C₈₃	#135	1	1	yes	characteristic
48	C₂×Dic₈₃	—	4	2	no	—
49	C₂×Dic₈₃	—	4	2	no	—
50	Dic₁₆₆	—	2	4	no	—
51	Dic₁₆₆	—	4	2	no	—
52	C₈×C₈₃	—	2	1	no	—
53	C₈×C₈₃	—	2	2	no	—
54	C₈×C₈₃	—	4	1	no	—
55	C₈×C₈₃	C₂²×C₈	1	1	yes	characteristic
56	C₈×C₈₃	#131	1	1	yes	characteristic
57	C₈₃⋊C₈	—	8	2	no	—
58	C₂²×C₈₃	C₄×D₈	1	1	yes	characteristic
59	C₄×C₈₃	—	2	1	no	—
60	C₄×C₈₃	C₈×D₄	1	1	yes	characteristic
61	C₄×C₈₃	#443	1	1	yes	characteristic
62	C₄×C₈₃	#510	1	2	yes	—
63	Dic₈₃	—	4	4	no	—
64	Dic₈₃	—	8	2	no	—
65	#12044	—	83	1	no	maximal, Sylow
66	C₂×C₈₃	C₈×D₈	1	1	yes	characteristic
67	C₂×C₈₃	#2176	1	1	yes	characteristic
68	C₂×C₈₃	#2179	1	1	yes	characteristic
69	C₄×Q₃₂	—	83	1	no	—
70	C₈×Q₁₆	—	83	2	no	—
71	#1372	—	83	1	no	—
72	#1332	—	83	2	no	—
73	#1311	D₈₃	1	1	yes	characteristic
74	C₈₃	#12044	1	1	yes	characteristic, complemented, Sylow
75	C₂×Q₃₂	—	83	1	no	—
76	C₈×Q₈	—	166	2	no	—
77	C₄×Q₁₆	—	83	2	no	—
78	#366	—	83	1	no	—
79	#358	—	83	2	no	—
80	C₄×C₁₆	C₂×D₈₃	1	1	yes	characteristic
81	C₄×C₁₆	C₂×D₈₃	1	1	yes	characteristic
82	#335	—	83	2	no	—
83	Q₈⋊C₈	—	166	2	no	—
84	C₈²	C₂×D₈₃	1	1	yes	characteristic
85	C₂×Q₁₆	—	83	2	no	—
86	C₄×Q₈	—	166	2	no	—
87	Q₃₂	—	83	4	no	—
88	C₂×C₁₆	—	2	1	no	—
89	C₂×C₁₆	C₄×D₈₃	1	1	yes	characteristic
90	C₂×C₁₆	C₄×D₈₃	1	1	yes	characteristic
91	#107	—	83	2	no	—
92	C₄⋊C₈	—	332	2	no	—
93	C₄⋊C₈	—	166	2	no	—
94	#103	—	166	2	no	—
95	C₄×C₈	—	332	2	no	—
96	C₄×C₈	D₃₃₂	1	1	yes	characteristic
97	C₄×C₈	D₃₃₂	1	1	yes	characteristic
98	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
99	C₂×Q₈	—	166	2	no	—
100	Q₁₆	—	83	4	no	—
101	Q₁₆	—	166	2	no	—
102	C₂×C₈	—	2	1	no	—
103	C₂×C₈	—	332	2	no	—
104	C₂×C₈	D₆₆₄	1	2	yes	—
105	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
106	C₂×C₈	#13050590	1	1	yes	characteristic
107	C₄⋊C₄	—	332	2	no	—
108	C₄⋊C₄	—	166	2	no	—
109	C₄²	—	332	2	no	—
110	C₄²	C₂×D₃₃₂	1	1	yes	characteristic
111	C₁₆	—	2	2	no	—
112	C₁₆	—	2	1	no	—
113	C₁₆	C₈×D₈₃	1	2	yes	—
114	Q₈	—	332	2	no	—
115	Q₈	—	166	4	no	—
116	C₂×C₄	—	332	2	no	—
117	C₂×C₄	—	332	2	no	—
118	C₂×C₄	C₄×D₃₃₂	1	1	yes	characteristic
119	C₂×C₄	C₂×D₆₆₄	1	1	yes	characteristic
120	C₂×C₄	#14581814	1	1	yes	characteristic
121	C₈	—	2	1	no	—
122	C₈	—	2	2	no	—
123	C₈	—	4	1	no	—
124	C₈	—	664	2	no	—
125	C₈	C₂×C₈×D₈₃	1	1	yes	characteristic
126	C₈	#14581798	1	1	yes	characteristic
127	C₂²	C₄×D₆₆₄	1	1	yes	characteristic
128	C₄	—	2	1	no	—
129	C₄	—	332	4	no	—
130	C₄	—	664	2	no	—
131	C₄	C₈×D₃₃₂	1	1	yes	characteristic
132	C₄	#19127819	1	1	yes	characteristic
133	C₄	#19128144	1	2	yes	—
134	C₂	C₈×D₆₆₄	1	1	yes	characteristic
135	C₂	#24909335	1	1	yes	characteristic
136	C₂	#24909350	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Dic₆₆₄

C₂

yes

maximal, characteristic

C₈×Dic₃₃₂

C₂

yes

maximal

#24905537

C₂

yes

maximal, characteristic

#24905428

C₂

yes

maximal

C₈₃×#1311

C₂

yes

maximal, characteristic

C₂×Dic₆₆₄

C₄

yes

characteristic

C₈×Dic₁₆₆

—

C₄×Dic₃₃₂

C₂²

yes

—

#19127527

C₄

yes

characteristic

#19127506

C₄

yes

—

#19127454

C₂²

yes

—

#19127433

—

C₄×C₁₆×C₈₃

C₂²

yes

characteristic

C₄×C₁₆×C₈₃

C₂²

yes

characteristic

C₈²×C₈₃

C₂²

yes

characteristic

C₂×Dic₃₃₂

C₂×C₄

yes

—

C₄×Dic₁₆₆

—

Dic₆₆₄

C₈

yes

complemented

#14581726

C₂×C₄

yes

—

C₄⋊(C₈₃⋊C₈)

—

Dic₈₃⋊C₈

—

#14581717

—

C₈×Dic₈₃

—

C₂×C₁₆×C₈₃

—

C₂×C₁₆×C₈₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₈₃

C₂×C₄

yes

characteristic

C₄×C₈×C₈₃

D₄

yes

characteristic

C₄×C₈×C₈₃

D₄

yes

characteristic

C₄×C₈×C₈₃

C₂³

yes

characteristic

C₂×Dic₁₆₆

—

Dic₃₃₂

—

Dic₃₃₂

C₂×C₈

yes

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

Dic₈₃⋊C₄

—

C₄×Dic₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

D₈

yes

—

C₂×C₈×C₈₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₈₃

#41

yes

characteristic

C₄²×C₈₃

C₂×D₄

yes

characteristic

C₁₆×C₈₃

—

C₁₆×C₈₃

—

C₁₆×C₈₃

C₂×C₈

yes

—

C₂×C₄×C₈₃

C₄×D₄

yes

characteristic

C₂×C₄×C₈₃

C₂×D₈

yes

characteristic

C₂×C₄×C₈₃

#135

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

C₂²×C₈

yes

characteristic

C₈×C₈₃

#131

yes

characteristic

C₈₃⋊C₈

—

C₂²×C₈₃

C₄×D₈

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

C₈×D₄

yes

C₄×C₈₃

yes

C₄×C₈₃

yes

—

Dic₈₃

—

Dic₈₃

—

#12044

—

maximal, Sylow

C₂×C₈₃

C₈×D₈

yes

C₂×C₈₃

yes

C₂×C₈₃

yes

C₄×Q₃₂

—

C₈×Q₁₆

—

#1372

—

#1332

—

#1311

D₈₃

yes

characteristic

C₈₃

#12044

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₈×Q₈

—

166

—

C₄×Q₁₆

—

#366

—

#358

—

C₄×C₁₆

C₂×D₈₃

yes

characteristic

C₄×C₁₆

C₂×D₈₃

yes

characteristic

#335

—

Q₈⋊C₈

—

166

—

C₈²

C₂×D₈₃

yes

characteristic

C₂×Q₁₆

—

C₄×Q₈

—

166

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₈₃

yes

characteristic

C₂×C₁₆

C₄×D₈₃

yes

characteristic

#107

—

C₄⋊C₈

—

332

—

C₄⋊C₈

—

166

—

#103

—

166

—

C₄×C₈

—

332

—

C₄×C₈

D₃₃₂

yes

characteristic

C₄×C₈

D₃₃₂

yes

characteristic

C₄×C₈

C₂²×D₈₃

yes

characteristic

C₂×Q₈

—

166

—

100

Q₁₆

—

101

Q₁₆

—

166

—

102

C₂×C₈

—

103

C₂×C₈

—

332

—

104

C₂×C₈

D₆₆₄

yes

—

105

C₂×C₈

C₂×C₄×D₈₃

yes

characteristic

106

C₂×C₈

#13050590

yes

characteristic

107

C₄⋊C₄

—

332

—

108

C₄⋊C₄

—

166

—

109

C₄²

—

332

—

110

C₄²

C₂×D₃₃₂

yes

characteristic

111

C₁₆

—

112

C₁₆

—

113

C₁₆

C₈×D₈₃

yes

—

114

Q₈

—

332

—

115

Q₈

—

166

—

116

C₂×C₄

—

332

—

117

C₂×C₄

—

332

—

118

C₂×C₄

C₄×D₃₃₂

yes

characteristic

119

C₂×C₄

C₂×D₆₆₄

yes

characteristic

120

C₂×C₄

#14581814

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

664

—

125

C₈

C₂×C₈×D₈₃

yes

characteristic

126

C₈

#14581798

yes

characteristic

127

C₂²

C₄×D₆₆₄

yes

characteristic

128

C₄

—

129

C₄

—

332

—

130

C₄

—

664

—

131

C₄

C₈×D₃₃₂

yes

characteristic

132

C₄

#19127819

yes

characteristic

133

C₄

#19128144

yes

—

134

C₂

C₈×D₆₆₄

yes

characteristic

135

C₂

#24909335

yes

characteristic

136

C₂

#24909350

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₆₆₄
commutator	C₈×C₈₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	C₈₃×#1311	C₂
socle, prime radical	C₂²×C₈₃	C₄×D₈
cosocle	C₄×C₈×C₈₃	C₂³
hypercenter	#1311	D₈₃
hypocenter	C₈₃	#12044
LCS	G ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂×C₄

C₂×D₆₆₄

commutator

C₈×C₈₃

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₈₃

Fitting

C₈₃×#1311

C₂

socle, prime radical

C₂²×C₈₃

C₄×D₈

cosocle

C₄×C₈×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1311

N	H ≅ G/N	type
18. Dic₆₆₄	C₈	semidirect
74. C₈₃	#12044	semidirect

H ≅ G/N

type

18. Dic₆₆₄

C₈

semidirect

74. C₈₃

#12044

semidirect

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

subgroup	index
C₄×Dic₆₆₄	2
C₈×Dic₃₃₂	2
#24905537	2
#24905428	2
C₈₃×#1311	2
#12044	83

subgroup

index

C₄×Dic₆₆₄

C₈×Dic₃₃₂

#24905537

#24905428

C₈₃×#1311

#12044

Subgroups — #35316151

Conjugacy classes of subgroups

Complements

Maximal subgroups