Group #35298089 — subgroups

Subgroups — #35298089

Order 21248.

262 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 115 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	#24907601	C₂	1	1	yes	maximal, characteristic
2	#24907594	C₂	1	1	yes	maximal, characteristic
3	#24905322	C₂	1	4	yes	maximal
4	C₈₃×#1450	C₂	1	1	yes	maximal, characteristic
5	#19127725	C₂²	1	2	yes	—
6	#19127567	C₂²	1	2	yes	—
7	#19127466	—	2	2	no	—
8	#19127464	—	2	2	no	—
9	#431×C₈₃	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
11	#322×C₈₃	C₄	1	2	yes	complemented
12	C₈²×C₈₃	C₄	1	2	yes	complemented
13	C₂×#13050574	—	2	2	no	—
14	C₂×C₄⋊Dic₈₃	—	2	2	no	—
15	C₄⋊(C₈₃⋊C₈)	—	2	4	no	—
16	C₈₃⋊#95	—	2	4	no	—
17	C₂²×C₈×C₈₃	D₄	1	2	yes	—
18	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
19	C₄⋊C₈×C₈₃	—	2	2	no	—
20	C₂²⋊C₈×C₈₃	—	2	2	no	—
21	C₄×C₈×C₈₃	C₂×C₄	1	4	yes	—
22	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₈₃	Q₈	1	2	yes	—
25	C₂²×Dic₈₃	—	4	2	no	—
26	#13050574	—	4	4	no	—
27	C₂×C₈₃⋊C₈	—	4	2	no	—
28	C₄⋊Dic₈₃	—	4	4	no	—
29	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
30	C₂²×C₄×C₈₃	#41	1	2	yes	—
31	C₂×C₈×C₈₃	—	2	4	no	—
32	C₂×C₈×C₈₃	—	2	4	no	—
33	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
34	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
35	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
36	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
37	C₄²×C₈₃	#41	1	2	yes	—
38	C₂³×C₈₃	#124	1	1	yes	characteristic
39	C₂×C₄×C₈₃	—	4	2	no	—
40	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₈₃	#117	1	2	yes	—
42	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
43	C₂×C₄×C₈₃	#125	1	2	yes	—
44	C₂×C₄×C₈₃	#135	1	4	yes	—
45	C₂×Dic₈₃	—	8	4	no	—
46	C₂×Dic₈₃	—	8	2	no	—
47	C₈×C₈₃	—	2	2	no	—
48	C₈×C₈₃	—	2	2	no	—
49	C₈×C₈₃	—	4	4	no	—
50	C₈₃⋊C₈	—	8	2	no	—
51	C₂²×C₈₃	—	2	2	no	—
52	C₂²×C₈₃	#382	1	1	yes	characteristic
53	C₂²×C₈₃	#488	1	2	yes	—
54	C₄×C₈₃	—	2	2	no	—
55	C₄×C₈₃	—	2	2	no	—
56	C₄×C₈₃	#427	1	2	yes	—
57	C₄×C₈₃	#488	1	2	yes	—
58	Dic₈₃	—	16	2	no	—
59	#8791	—	83	1	no	maximal, Sylow
60	C₂×C₈₃	—	2	2	no	—
61	C₂×C₈₃	#1833	1	2	yes	—
62	C₂×C₈₃	#1835	1	1	yes	characteristic
63	#1837	—	83	1	no	—
64	#1835	—	83	1	no	—
65	#1450	D₈₃	1	1	yes	characteristic
66	#1286	—	83	4	no	—
67	C₈₃	#8791	1	1	yes	characteristic, complemented, Sylow
68	#431	C₂×D₈₃	1	2	yes	—
69	#423	—	83	2	no	—
70	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
71	#382	—	83	2	no	—
72	#341	—	166	2	no	—
73	#340	—	166	2	no	—
74	#322	Dic₈₃	1	2	yes	complemented
75	C₈²	Dic₈₃	1	2	yes	complemented
76	C₂×M₁₆	—	166	2	no	—
77	C₂²×C₈	D₃₃₂	1	2	yes	—
78	C₂×C₄⋊C₄	—	166	2	no	—
79	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
80	C₄⋊C₈	—	2	2	no	—
81	C₄⋊C₈	—	166	4	no	—
82	C₂²⋊C₈	—	2	2	no	—
83	C₄×C₈	Dic₁₆₆	1	2	yes	—
84	C₄×C₈	C₂×Dic₈₃	1	4	yes	—
85	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
86	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
87	#95	—	166	4	no	—
88	C₂²×C₄	—	332	2	no	—
89	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
90	C₂²×C₄	#13050590	1	2	yes	—
91	M₁₆	—	332	4	no	—
92	C₂×C₈	—	2	4	no	—
93	C₂×C₈	—	2	4	no	—
94	C₂×C₈	—	332	2	no	—
95	C₂×C₈	C₄⋊Dic₈₃	1	2	yes	—
96	C₂×C₈	C₄⋊Dic₈₃	1	2	yes	—
97	C₄⋊C₄	—	332	4	no	—
98	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
99	C₄²	C₂×Dic₁₆₆	1	1	yes	characteristic
100	C₄²	#13050590	1	2	yes	—
101	C₂³	#14581780	1	1	yes	characteristic
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	664	4	no	—
104	C₂×C₄	—	664	2	no	—
105	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
106	C₂×C₄	#14581751	1	2	yes	—
107	C₂×C₄	#14581780	1	1	yes	characteristic
108	C₂×C₄	#14581784	1	2	yes	—
109	C₂×C₄	#14581814	1	4	yes	—
110	C₈	—	2	2	no	—
111	C₈	—	2	2	no	—
112	C₈	—	4	4	no	—
113	C₈	—	664	2	no	—
114	C₂²	—	2	2	no	—
115	C₂²	#19127567	1	1	yes	characteristic
116	C₂²	#19128056	1	2	yes	—
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	1328	2	no	—
120	C₄	#19127739	1	2	yes	—
121	C₄	#19128056	1	2	yes	—
122	C₂	—	2	2	no	—
123	C₂	#24907587	1	2	yes	—
124	C₂	#24907594	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#24907601

C₂

yes

maximal, characteristic

#24907594

C₂

yes

maximal, characteristic

#24905322

C₂

yes

maximal

C₈₃×#1450

C₂

yes

maximal, characteristic

#19127725

C₂²

yes

—

#19127567

C₂²

yes

—

#19127466

—

#19127464

—

#431×C₈₃

C₂²

yes

—

C₂×C₄×C₈×C₈₃

C₂²

yes

characteristic

#322×C₈₃

C₄

yes

complemented

C₈²×C₈₃

C₄

yes

complemented

C₂×#13050574

—

C₂×C₄⋊Dic₈₃

—

C₄⋊(C₈₃⋊C₈)

—

C₈₃⋊#95

—

C₂²×C₈×C₈₃

D₄

yes

—

C₂×C₄²×C₈₃

C₂³

yes

characteristic

C₄⋊C₈×C₈₃

—

C₂²⋊C₈×C₈₃

—

C₄×C₈×C₈₃

C₂×C₄

yes

—

C₄×C₈×C₈₃

C₂×C₄

yes

characteristic

C₄×C₈×C₈₃

C₂×C₄

yes

characteristic

C₄×C₈×C₈₃

Q₈

yes

—

C₂²×Dic₈₃

—

#13050574

—

C₂×C₈₃⋊C₈

—

C₄⋊Dic₈₃

—

C₂²×C₄×C₈₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₈₃

#41

yes

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

—

C₂×C₈×C₈₃

C₄⋊C₄

yes

—

C₂×C₈×C₈₃

C₄⋊C₄

yes

—

C₄²×C₈₃

C₂²×C₄

yes

characteristic

C₄²×C₈₃

C₂×Q₈

yes

characteristic

C₄²×C₈₃

#41

yes

—

C₂³×C₈₃

#124

yes

characteristic

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₈₃

#117

yes

—

C₂×C₄×C₈₃

#124

yes

characteristic

C₂×C₄×C₈₃

#125

yes

—

C₂×C₄×C₈₃

#135

yes

—

C₂×Dic₈₃

—

C₂×Dic₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈×C₈₃

—

C₈₃⋊C₈

—

C₂²×C₈₃

—

C₂²×C₈₃

#382

yes

characteristic

C₂²×C₈₃

#488

yes

—

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

#427

yes

—

C₄×C₈₃

#488

yes

—

Dic₈₃

—

#8791

—

maximal, Sylow

C₂×C₈₃

—

C₂×C₈₃

#1833

yes

—

C₂×C₈₃

yes

—

—

D₈₃

yes

characteristic

#1286

—

C₈₃

#8791

yes

characteristic, complemented, Sylow

#431

C₂×D₈₃

yes

—

#423

—

C₂×C₄×C₈

C₂×D₈₃

yes

characteristic

#382

—

#341

—

166

—

#340

—

166

—

#322

Dic₈₃

yes

complemented

C₈²

Dic₈₃

yes

complemented

C₂×M₁₆

—

166

—

C₂²×C₈

D₃₃₂

yes

—

C₂×C₄⋊C₄

—

166

—

C₂×C₄²

C₂²×D₈₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

166

—

C₂²⋊C₈

—

C₄×C₈

Dic₁₆₆

yes

—

C₄×C₈

C₂×Dic₈₃

yes

—

C₄×C₈

C₂×Dic₈₃

yes

characteristic

C₄×C₈

C₂×Dic₈₃

yes

characteristic

#95

—

166

—

C₂²×C₄

—

332

—

C₂²×C₄

C₂×D₃₃₂

yes

characteristic

C₂²×C₄

#13050590

yes

—

M₁₆

—

332

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

332

—

C₂×C₈

C₄⋊Dic₈₃

yes

—

C₂×C₈

C₄⋊Dic₈₃

yes

—

C₄⋊C₄

—

332

—

C₄²

C₂²×Dic₈₃

yes

characteristic

C₄²

C₂×Dic₁₆₆

yes

characteristic

100

C₄²

#13050590

yes

—

101

C₂³

#14581780

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

664

—

104

C₂×C₄

—

664

—

105

C₂×C₄

C₂×C₄⋊Dic₈₃

yes

characteristic

106

C₂×C₄

#14581751

yes

—

107

C₂×C₄

#14581780

yes

characteristic

108

C₂×C₄

#14581784

yes

—

109

C₂×C₄

#14581814

yes

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

664

—

114

C₂²

—

115

C₂²

#19127567

yes

characteristic

116

C₂²

#19128056

yes

—

117

C₄

—

118

C₄

—

119

C₄

—

1328

—

120

C₄

#19127739

yes

—

121

C₄

#19128056

yes

—

122

C₂

—

123

C₂

#24907587

yes

—

124

C₂

#24907594

yes

characteristic

	subgroup	G/H
center	C₂²	#19127567
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1450	C₂
socle	C₂²×C₈₃	#382
prime radical	C₂³×C₈₃	#124
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1450	D₈₃
hypocenter	C₈₃	#8791
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

subgroup

G/H

center

C₂²

#19127567

commutator

C₄²×C₈₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₈₃

Fitting

C₈₃×#1450

C₂

C₂²×C₈₃

C₂³×C₈₃

C₂×C₄²×C₈₃

C₂³

D₈₃

C₈₃

G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

N	H ≅ G/N	type
11. #322×C₈₃	C₄	semidirect
12. C₈²×C₈₃	C₄	semidirect
67. C₈₃	#8791	semidirect
74. #322	Dic₈₃	semidirect
75. C₈²	Dic₈₃	semidirect

H ≅ G/N

type

11. #322×C₈₃

C₄

semidirect

12. C₈²×C₈₃

C₄

semidirect

67. C₈₃

#8791

semidirect

74. #322

Dic₈₃

semidirect

75. C₈²

Dic₈₃

semidirect

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

subgroup	index
#24907601	2
#24907594	2
#24905322	2
C₈₃×#1450	2
#8791	83

subgroup

index

C₈₃×#1450

Subgroups — #35298089

Conjugacy classes of subgroups

Complements

Maximal subgroups