Group #46825763 — subgroups

Subgroups — #46825763

Order 34432.

188 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 76 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×#22575520	C₂	1	2	yes	maximal
2	#31770474	C₂	1	1	yes	maximal, characteristic
3	C₂²×#14524137	—	2	2	no	—
4	#22575520	—	2	2	no	—
5	#22575520	C₄	1	4	yes	complemented
6	C₂×C₄⋊Dic₂₆₉	C₄	1	1	yes	characteristic
7	C₂×#16864381	—	2	2	no	—
8	C₂×C₄×Dic₂₆₉	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊C₄×C₂₆₉	C₄	1	1	yes	characteristic
10	C₂×#14524137	—	2	4	no	—
11	C₂×#14524137	—	4	2	no	—
12	C₂×#14524137	—	4	2	no	—
13	C₂²×Dic₂₆₉	—	2	1	no	—
14	C₂²×Dic₂₆₉	D₄	1	2	yes	—
15	C₄⋊Dic₂₆₉	—	4	1	no	—
16	#16864381	—	4	2	no	—
17	#16864381	—	4	2	no	—
18	C₄×Dic₂₆₉	C₂×C₄	1	2	yes	—
19	C₄×Dic₂₆₉	D₄	1	1	yes	characteristic
20	C₄×Dic₂₆₉	Q₈	1	1	yes	characteristic
21	C₂²×C₄×C₂₆₉	—	2	1	no	—
22	C₂²×C₄×C₂₆₉	C₂×C₄	1	1	yes	characteristic
23	C₄⋊C₄×C₂₆₉	—	4	1	no	—
24	#14524137	—	4	4	no	—
25	C₂×Dic₂₆₉	—	4	1	no	—
26	C₂×Dic₂₆₉	—	4	2	no	—
27	C₂×Dic₂₆₉	C₂²⋊C₄	1	2	yes	—
28	C₂×Dic₂₆₉	C₄⋊C₄	1	2	yes	—
29	C₂×Dic₂₆₉	D₈	1	2	yes	—
30	C₂×Dic₂₆₉	SD₁₆	1	2	yes	—
31	C₂×Dic₂₆₉	SD₁₆	1	2	yes	—
32	C₂×Dic₂₆₉	Q₁₆	1	2	yes	—
33	C₂³×C₂₆₉	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₂₆₉	—	2	1	no	—
35	C₂×C₄×C₂₆₉	—	2	1	no	—
36	C₂×C₄×C₂₆₉	—	4	1	no	—
37	C₂×C₄×C₂₆₉	—	4	2	no	—
38	C₂×C₄×C₂₆₉	C₄²	1	1	yes	characteristic
39	C₂×C₄×C₂₆₉	C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₂₆₉	#95	1	1	yes	characteristic
41	C₂²×C₂₆₉	#100	1	1	yes	characteristic
42	C₂²×C₂₆₉	#101	1	1	yes	characteristic
43	C₂²×C₂₆₉	#102	1	2	yes	—
44	C₂²×C₂₆₉	#103	1	2	yes	—
45	C₄×C₂₆₉	—	2	1	no	—
46	C₄×C₂₆₉	—	2	1	no	—
47	C₄×C₂₆₉	—	8	1	no	—
48	Dic₂₆₉	—	8	1	no	—
49	Dic₂₆₉	#102	1	2	yes	—
50	Dic₂₆₉	#103	1	2	yes	—
51	Dic₂₆₉	#106	1	2	yes	—
52	Dic₂₆₉	#107	1	2	yes	—
53	C₂×C₂₆₉	#331	1	1	yes	characteristic
54	C₂×C₂₆₉	#332	1	2	yes	—
55	C₂×C₂₆₉	#333	1	1	yes	characteristic
56	C₂×C₂₆₉	#340	1	2	yes	—
57	C₂×C₂₆₉	#343	1	1	yes	characteristic
58	C₂₆₉	#1296	1	1	yes	characteristic, complemented, Sylow
59	#1296	—	269	1	no	maximal, Sylow
60	C₂×C₄⋊C₈	—	269	2	no	—
61	#384	—	269	1	no	—
62	C₂²×C₈	—	538	2	no	—
63	C₂×C₄⋊C₄	—	269	1	no	—
64	C₂×C₄⋊C₄	—	538	2	no	—
65	C₂×C₄⋊C₄	#11761640	1	1	yes	characteristic
66	C₂×C₄²	—	269	1	no	—
67	C₄⋊C₈	—	269	4	no	—
68	C₄⋊C₈	—	538	2	no	—
69	C₂²×C₄	—	2	1	no	—
70	C₂²×C₄	—	269	2	no	—
71	C₂²×C₄	—	538	1	no	—
72	C₂²×C₄	C₂×#11761640	1	1	yes	characteristic
73	C₂×C₈	—	538	4	no	—
74	C₂×C₈	—	1076	2	no	—
75	C₂×C₈	—	1076	2	no	—
76	C₄⋊C₄	—	4	1	no	—
77	C₄⋊C₄	—	1076	1	no	—
78	C₄⋊C₄	—	1076	2	no	—
79	C₄⋊C₄	—	1076	2	no	—
80	C₄²	—	269	2	no	—
81	C₄²	—	269	1	no	—
82	C₄²	—	269	1	no	—
83	C₂³	C₂²⋊#11761640	1	1	yes	characteristic
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	4	1	no	—
87	C₂×C₄	—	4	2	no	—
88	C₂×C₄	—	269	2	no	—
89	C₂×C₄	—	269	2	no	—
90	C₂×C₄	—	269	2	no	—
91	C₂×C₄	—	269	2	no	—
92	C₂×C₄	—	269	2	no	—
93	C₂×C₄	—	269	2	no	—
94	C₂×C₄	—	1076	1	no	—
95	C₂×C₄	—	1076	2	no	—
96	C₂×C₄	C₄×#11761640	1	1	yes	characteristic
97	C₂×C₄	C₄⋊#11761640	1	1	yes	characteristic
98	C₈	—	1076	4	no	—
99	C₂²	(C₂×#11761640)⋊C₄	1	1	yes	characteristic
100	C₂²	#22575513	1	1	yes	characteristic
101	C₂²	#22575514	1	1	yes	characteristic
102	C₂²	#22575515	1	2	yes	—
103	C₂²	#22575516	1	2	yes	—
104	C₄	—	2	1	no	—
105	C₄	—	2	1	no	—
106	C₄	—	8	1	no	—
107	C₄	—	269	2	no	—
108	C₄	—	269	2	no	—
109	C₄	—	269	2	no	—
110	C₄	—	269	2	no	—
111	C₄	—	2152	1	no	—
112	C₂	#31771463	1	1	yes	characteristic
113	C₂	#31771464	1	2	yes	—
114	C₂	#31771465	1	1	yes	characteristic
115	C₂	#31771478	1	2	yes	—
116	C₂	#31771485	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22575520

C₂

yes

maximal

#31770474

C₂

yes

maximal, characteristic

C₂²×#14524137

—

#22575520

—

#22575520

C₄

yes

complemented

C₂×C₄⋊Dic₂₆₉

C₄

yes

characteristic

C₂×#16864381

—

C₂×C₄×Dic₂₆₉

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₆₉

C₄

yes

—

—

—

C₂²×Dic₂₆₉

—

C₂²×Dic₂₆₉

D₄

yes

—

C₄⋊Dic₂₆₉

—

#16864381

—

#16864381

—

C₄×Dic₂₆₉

C₂×C₄

yes

—

C₄×Dic₂₆₉

D₄

yes

characteristic

C₄×Dic₂₆₉

Q₈

yes

characteristic

C₂²×C₄×C₂₆₉

—

C₂²×C₄×C₂₆₉

C₂×C₄

yes

characteristic

C₄⋊C₄×C₂₆₉

—

#14524137

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

—

C₂×Dic₂₆₉

C₂²⋊C₄

yes

—

C₂×Dic₂₆₉

C₄⋊C₄

yes

—

C₂×Dic₂₆₉

D₈

yes

—

C₂×Dic₂₆₉

SD₁₆

yes

—

C₂×Dic₂₆₉

SD₁₆

yes

—

C₂×Dic₂₆₉

Q₁₆

yes

—

C₂³×C₂₆₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

—

C₂×C₄×C₂₆₉

C₄²

yes

characteristic

C₂×C₄×C₂₆₉

C₄⋊C₄

yes

C₂²×C₂₆₉

yes

C₂²×C₂₆₉

yes

C₂²×C₂₆₉

yes

C₂²×C₂₆₉

yes

—

C₂²×C₂₆₉

#103

yes

—

C₄×C₂₆₉

—

C₄×C₂₆₉

—

C₄×C₂₆₉

—

Dic₂₆₉

—

Dic₂₆₉

#102

yes

—

Dic₂₆₉

#103

yes

—

Dic₂₆₉

#106

yes

—

Dic₂₆₉

#107

yes

—

C₂×C₂₆₉

#331

yes

characteristic

C₂×C₂₆₉

#332

yes

—

C₂×C₂₆₉

#333

yes

characteristic

C₂×C₂₆₉

#340

yes

—

C₂×C₂₆₉

#343

yes

characteristic

C₂₆₉

#1296

yes

characteristic, complemented, Sylow

#1296

—

269

maximal, Sylow

C₂×C₄⋊C₈

—

269

—

#384

—

269

—

C₂²×C₈

—

538

—

C₂×C₄⋊C₄

—

269

—

C₂×C₄⋊C₄

—

538

—

C₂×C₄⋊C₄

#11761640

yes

characteristic

C₂×C₄²

—

269

—

C₄⋊C₈

—

269

—

C₄⋊C₈

—

538

—

C₂²×C₄

—

C₂²×C₄

—

269

—

C₂²×C₄

—

538

—

C₂²×C₄

C₂×#11761640

yes

characteristic

C₂×C₈

—

538

—

C₂×C₈

—

1076

—

C₂×C₈

—

1076

—

C₄⋊C₄

—

C₄⋊C₄

—

1076

—

C₄⋊C₄

—

1076

—

C₄⋊C₄

—

1076

—

C₄²

—

269

—

C₄²

—

269

—

C₄²

—

269

—

C₂³

C₂²⋊#11761640

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

269

—

C₂×C₄

—

269

—

C₂×C₄

—

269

—

C₂×C₄

—

269

—

C₂×C₄

—

269

—

C₂×C₄

—

269

—

C₂×C₄

—

1076

—

C₂×C₄

—

1076

—

C₂×C₄

yes

C₂×C₄

yes

C₈

—

1076

—

C₂²

(C₂×#11761640)⋊C₄

yes

characteristic

100

C₂²

#22575513

yes

characteristic

101

C₂²

#22575514

yes

characteristic

102

C₂²

#22575515

yes

—

103

C₂²

#22575516

yes

—

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

—

269

—

108

C₄

—

269

—

109

C₄

—

269

—

110

C₄

—

269

—

111

C₄

—

2152

—

112

C₂

#31771463

yes

characteristic

113

C₂

#31771464

yes

—

114

C₂

#31771465

yes

characteristic

115

C₂

#31771478

yes

—

116

C₂

#31771485

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²⋊#11761640
commutator	C₂×C₄×C₂₆₉	C₄²
Frattini	C₂²×C₄	C₂×#11761640
Fitting	C₂×C₄⋊C₄×C₂₆₉	C₄
socle, prime radical	C₂³×C₂₆₉	C₂²⋊C₄
cosocle	C₂×C₄×Dic₂₆₉	C₂²
hypercenter	C₂×C₄⋊C₄	#11761640
hypocenter	C₂₆₉	#1296
LCS	G ▷ C₂×C₄×C₂₆₉ ▷ C₂²×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂²⋊#11761640

commutator

C₂×C₄×C₂₆₉

C₄²

Frattini

C₂²×C₄

C₂×#11761640

Fitting

C₂×C₄⋊C₄×C₂₆₉

C₄

socle, prime radical

C₂³×C₂₆₉

C₂²⋊C₄

cosocle

C₂×C₄×Dic₂₆₉

C₂²

C₂×C₄⋊C₄

C₂₆₉

G ▷ C₂×C₄×C₂₆₉ ▷ C₂²×C₂₆₉ ▷ C₂₆₉

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
5. #22575520	C₄	semidirect
58. C₂₆₉	#1296	semidirect

H ≅ G/N

type

5. #22575520

C₄

semidirect

58. C₂₆₉

#1296

semidirect

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

subgroup	index
C₂×#22575520	2
#31770474	2
#1296	269

subgroup

index

C₂×#22575520

#31770474

#1296

269

Subgroups — #46825763

Conjugacy classes of subgroups

Complements

Maximal subgroups