Group #35308900 — subgroups

Subgroups — #35308900

Order 21248.

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

normal subgroups only

This group has 406 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	#24907632	C₂	1	3	yes	maximal
2	#24905344	C₂	1	3	yes	maximal
3	C₈₃×#2844	C₂	1	1	yes	maximal, characteristic
4	C₂×#14581725	—	2	3	no	—
5	C₂×#14581723	—	2	3	no	—
6	C₂×#14581703	C₂²	1	3	yes	—
7	#19127571	C₂²	1	1	yes	characteristic
8	#400×C₈₃	—	2	3	no	—
9	#395×C₈₃	—	2	3	no	—
10	#382×C₈₃	C₂²	1	3	yes	—
11	C₂²×C₈₃⋊C₈	—	2	3	no	—
12	C₂×C₄⋊Dic₈₃	—	2	3	no	—
13	#14581725	—	4	6	no	—
14	#14581723	—	4	6	no	—
15	#14581703	—	2	3	no	—
16	#14581703	Q₈	1	6	yes	complemented
17	C₂²×Q₈×C₈₃	—	2	1	no	—
18	C₂×C₄⋊C₄×C₈₃	—	2	3	no	—
19	C₂×C₄⋊C₄×C₈₃	D₄	1	3	yes	—
20	C₂×C₄⋊C₄×C₈₃	D₄	1	3	yes	—
21	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
22	#95×C₈₃	—	2	3	no	—
23	#95×C₈₃	—	2	1	no	—
24	#95×C₈₃	—	2	6	no	—
25	C₂²×Dic₈₃	—	4	1	no	—
26	C₂×C₈₃⋊C₈	—	2	6	no	—
27	C₂×C₈₃⋊C₈	—	2	6	no	—
28	C₂×C₈₃⋊C₈	—	2	6	no	—
29	C₄⋊Dic₈₃	—	4	3	no	—
30	C₄⋊Dic₈₃	—	4	3	no	—
31	C₂×Q₈×C₈₃	—	8	3	no	—
32	C₂²×C₄×C₈₃	—	2	3	no	—
33	C₂²×C₄×C₈₃	—	2	3	no	—
34	C₂²×C₄×C₈₃	C₂×D₄	1	3	yes	—
35	C₄⋊C₄×C₈₃	—	4	3	no	—
36	C₄⋊C₄×C₈₃	—	4	3	no	—
37	C₄⋊C₄×C₈₃	—	8	3	no	—
38	C₄²×C₈₃	C₂×Q₈	1	3	yes	—
39	C₄²×C₈₃	#41	1	1	yes	characteristic
40	C₂³×C₈₃	#120	1	1	yes	characteristic
41	Q₈×C₈₃	—	8	4	no	—
42	C₂×C₄×C₈₃	—	2	3	no	—
43	C₂×C₄×C₈₃	—	2	3	no	—
44	C₂×C₄×C₈₃	—	4	3	no	—
45	C₂×C₄×C₈₃	—	4	3	no	—
46	C₂×C₄×C₈₃	—	4	6	no	—
47	C₂×C₄×C₈₃	—	4	6	no	—
48	C₂×C₄×C₈₃	C₂²⋊Q₈	1	3	yes	—
49	C₂×C₄×C₈₃	C₄⋊Q₈	1	3	yes	—
50	C₂×Dic₈₃	—	8	3	no	—
51	C₈₃⋊C₈	—	2	12	no	—
52	C₂²×C₈₃	#395	1	1	yes	characteristic
53	C₂²×C₈₃	#453	1	3	yes	—
54	C₂²×C₈₃	#456	1	3	yes	—
55	C₄×C₈₃	—	2	3	no	—
56	C₄×C₈₃	—	2	3	no	—
57	C₄×C₈₃	—	8	3	no	—
58	C₄×C₈₃	—	8	3	no	—
59	Dic₈₃	—	16	1	no	—
60	#10583	—	83	1	no	maximal, Sylow
61	C₂×C₈₃	#1655	1	1	yes	characteristic
62	C₂×C₈₃	#1660	1	3	yes	—
63	C₂×C₈₃	#2060	1	3	yes	—
64	#2844	D₈₃	1	1	yes	characteristic
65	#1847	—	83	3	no	—
66	#1295	—	83	3	no	—
67	C₈₃	#10583	1	1	yes	characteristic, complemented, Sylow
68	C₂×#107	—	166	3	no	—
69	C₂×#106	—	166	3	no	—
70	C₂×#97	—	83	3	no	—
71	#400	—	2	3	no	—
72	#395	—	2	3	no	—
73	#384	—	83	1	no	—
74	#382	C₂×D₈₃	1	3	yes	—
75	C₂²×Q₈	—	2	1	no	—
76	C₂²×C₈	—	166	3	no	—
77	C₂×C₄⋊C₄	—	2	3	no	—
78	C₂×C₄⋊C₄	—	166	3	no	—
79	C₂×C₄⋊C₄	C₂²⋊D₈₃	1	3	yes	—
80	C₂×C₄⋊C₄	C₂²⋊D₈₃	1	3	yes	—
81	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
82	#107	—	332	6	no	—
83	#106	—	332	6	no	—
84	#97	—	83	6	no	—
85	#97	—	166	3	no	—
86	#95	—	2	3	no	—
87	#95	—	2	1	no	—
88	#95	—	2	6	no	—
89	C₂×Q₈	—	8	3	no	—
90	C₂²×C₄	—	2	3	no	—
91	C₂²×C₄	—	2	3	no	—
92	C₂²×C₄	—	332	1	no	—
93	C₂²×C₄	C₂×C₂²⋊D₈₃	1	3	yes	—
94	C₂×C₈	—	166	6	no	—
95	C₂×C₈	—	166	6	no	—
96	C₂×C₈	—	166	6	no	—
97	C₄⋊C₄	—	4	3	no	—
98	C₄⋊C₄	—	4	3	no	—
99	C₄⋊C₄	—	8	3	no	—
100	C₄⋊C₄	—	332	3	no	—
101	C₄⋊C₄	—	332	3	no	—
102	C₄²	Q₈×D₈₃	1	3	yes	—
103	C₄²	#13050591	1	1	yes	characteristic
104	C₂³	#14581761	1	1	yes	characteristic
105	Q₈	—	8	4	no	—
106	C₂×C₄	—	2	3	no	—
107	C₂×C₄	—	2	3	no	—
108	C₂×C₄	—	4	3	no	—
109	C₂×C₄	—	4	3	no	—
110	C₂×C₄	—	4	6	no	—
111	C₂×C₄	—	4	6	no	—
112	C₂×C₄	—	664	3	no	—
113	C₂×C₄	#14581768	1	3	yes	—
114	C₂×C₄	#14581790	1	3	yes	—
115	C₈	—	166	12	no	—
116	C₂²	#19127620	1	1	yes	characteristic
117	C₂²	#19127869	1	3	yes	—
118	C₂²	#19127884	1	3	yes	—
119	C₄	—	2	3	no	—
120	C₄	—	2	3	no	—
121	C₄	—	8	3	no	—
122	C₄	—	8	3	no	—
123	C₄	—	1328	1	no	—
124	C₂	#24906752	1	1	yes	characteristic
125	C₂	#24906774	1	3	yes	—
126	C₂	#24908805	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#24907632

C₂

yes

maximal

#24905344

C₂

yes

maximal

C₈₃×#2844

C₂

yes

maximal, characteristic

C₂×#14581725

—

C₂×#14581723

—

C₂×#14581703

C₂²

yes

—

#19127571

C₂²

yes

characteristic

#400×C₈₃

—

#395×C₈₃

—

#382×C₈₃

C₂²

yes

—

C₂²×C₈₃⋊C₈

—

C₂×C₄⋊Dic₈₃

—

—

—

—

Q₈

yes

complemented

C₂²×Q₈×C₈₃

—

C₂×C₄⋊C₄×C₈₃

—

C₂×C₄⋊C₄×C₈₃

D₄

yes

—

C₂×C₄⋊C₄×C₈₃

D₄

yes

—

C₂×C₄²×C₈₃

C₂³

yes

characteristic

#95×C₈₃

—

#95×C₈₃

—

#95×C₈₃

—

C₂²×Dic₈₃

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

—

C₂×Q₈×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

—

C₂²×C₄×C₈₃

C₂×D₄

yes

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄²×C₈₃

C₂×Q₈

yes

—

C₄²×C₈₃

yes

C₂³×C₈₃

yes

Q₈×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₈₃

C₄⋊Q₈

yes

—

C₂×Dic₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

#395

yes

characteristic

C₂²×C₈₃

#453

yes

—

C₂²×C₈₃

#456

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

—

Dic₈₃

—

—

C₂×C₈₃

yes

C₂×C₈₃

yes

—

C₂×C₈₃

#2060

yes

—

#2844

D₈₃

yes

characteristic

#1847

—

#1295

—

C₈₃

#10583

yes

characteristic, complemented, Sylow

C₂×#107

—

166

—

C₂×#106

—

166

—

C₂×#97

—

—

—

—

C₂×D₈₃

yes

—

C₂²×Q₈

—

C₂²×C₈

—

166

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

166

—

C₂×C₄⋊C₄

C₂²⋊D₈₃

yes

—

C₂×C₄⋊C₄

C₂²⋊D₈₃

yes

—

C₂×C₄²

C₂²×D₈₃

yes

characteristic

#107

—

332

—

#106

—

332

—

#97

—

#97

—

166

—

#95

—

#95

—

#95

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

332

—

C₂²×C₄

C₂×C₂²⋊D₈₃

yes

—

C₂×C₈

—

166

—

C₂×C₈

—

166

—

C₂×C₈

—

166

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

332

—

101

C₄⋊C₄

—

332

—

102

C₄²

Q₈×D₈₃

yes

—

103

C₄²

#13050591

yes

characteristic

104

C₂³

#14581761

yes

characteristic

105

Q₈

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

664

—

113

C₂×C₄

#14581768

yes

—

114

C₂×C₄

#14581790

yes

—

115

C₈

—

166

—

116

C₂²

#19127620

yes

characteristic

117

C₂²

#19127869

yes

—

118

C₂²

#19127884

yes

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

1328

—

124

C₂

#24906752

yes

characteristic

125

C₂

#24906774

yes

—

126

C₂

#24908805

yes

—

	subgroup	G/H
center	C₂³	#14581761
commutator, cosocle	C₂×C₄²×C₈₃	C₂³
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#2844	C₂
socle, prime radical	C₂³×C₈₃	#120
hypercenter	#2844	D₈₃
hypocenter	C₈₃	#10583
LCS	G ▷ C₂×C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2844

subgroup

G/H

center

C₂³

#14581761

commutator, cosocle

C₂×C₄²×C₈₃

C₂³

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₈₃×#2844

C₂

C₂³×C₈₃

D₈₃

C₈₃

G ▷ C₂×C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2844

N	H ≅ G/N	type
16. #14581703	Q₈	semidirect
67. C₈₃	#10583	semidirect

H ≅ G/N

type

16. #14581703

Q₈

semidirect

67. C₈₃

#10583

semidirect

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

subgroup	index
#24907632	2
#24905344	2
C₈₃×#2844	2
#10583	83

subgroup

index

#24907632

#24905344

C₈₃×#2844

#10583

Subgroups — #35308900

Conjugacy classes of subgroups

Complements

Maximal subgroups