Group #30594531 — subgroups

Subgroups — #30594531

Order 15616.

200 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 66 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	#21461376	C₂	1	2	yes	maximal
2	#21447574	C₂	1	1	yes	maximal, characteristic
3	#16840765	—	2	2	no	—
4	#16840626	C₄	1	4	yes	—
5	#16840625	—	2	2	no	—
6	C₂×#14519717	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
8	#16839580	—	2	2	no	—
9	C₂×M₃₂×C₆₁	C₄	1	1	yes	characteristic
10	C₂×#11758217	—	4	2	no	—
11	#14519844	—	4	2	no	—
12	C₄⋊#92420	—	2	4	no	—
13	C₂²×#92418	Q₈	1	1	yes	characteristic
14	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
15	#14519717	—	4	1	no	—
16	C₂×#11758185	—	2	1	no	—
17	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
18	C₈×Dic₆₁	D₄	1	2	yes	—
19	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
20	M₃₂×C₆₁	—	4	1	no	—
21	C₂×C₁₆×C₆₁	—	2	1	no	—
22	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
23	C₂²×Dic₆₁	SD₁₆	1	2	yes	—
24	#11758217	—	8	4	no	—
25	C₂×#92420	—	4	4	no	—
26	C₂×C₈×C₆₁	—	2	1	no	—
27	C₂×C₈×C₆₁	—	2	1	no	—
28	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
29	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
30	C₄×Dic₆₁	C₂²⋊C₄	1	2	yes	—
31	C₄×Dic₆₁	D₈	1	1	yes	characteristic
32	C₄×Dic₆₁	Q₁₆	1	1	yes	characteristic
33	C₂×#92418	—	4	1	no	—
34	C₂×#92418	C₄⋊C₄	1	2	yes	—
35	C₁₆×C₆₁	—	4	1	no	—
36	#11758185	—	4	1	no	—
37	C₂³×C₆₁	#106	1	1	yes	characteristic
38	C₂×C₄×C₆₁	—	2	1	no	—
39	C₂×C₄×C₆₁	—	2	1	no	—
40	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
41	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
42	C₂×Dic₆₁	—	2	2	no	—
43	C₂×Dic₆₁	—	2	2	no	—
44	C₂×Dic₆₁	#102	1	2	yes	—
45	C₂×Dic₆₁	#103	1	2	yes	—
46	#92420	—	8	4	no	—
47	C₈×C₆₁	—	2	1	no	—
48	C₈×C₆₁	#100	1	1	yes	characteristic
49	C₈×C₆₁	#101	1	1	yes	characteristic
50	#92418	—	2	2	no	—
51	#7245	—	61	1	no	maximal, Sylow
52	C₂²×C₆₁	—	2	1	no	—
53	C₂²×C₆₁	—	2	2	no	—
54	C₂²×C₆₁	#340	1	1	yes	characteristic
55	C₄×C₆₁	—	2	1	no	—
56	C₄×C₆₁	#338	1	1	yes	characteristic
57	C₄×C₆₁	#343	1	1	yes	characteristic
58	Dic₆₁	—	2	2	no	—
59	Dic₆₁	#361	1	2	yes	—
60	Dic₆₁	#362	1	2	yes	—
61	#2150	—	61	1	no	—
62	#1567	—	61	2	no	—
63	C₂×C₆₁	—	4	1	no	—
64	C₂×C₆₁	#1292	1	1	yes	characteristic
65	C₂×C₆₁	#1387	1	2	yes	—
66	C₂×M₃₂	—	61	1	no	—
67	C₂×M₃₂	#6923	1	1	yes	characteristic
68	#423	—	122	2	no	—
69	C₂×C₄×C₈	—	61	1	no	—
70	C₄⋊C₁₆	—	122	2	no	—
71	#335	—	61	4	no	—
72	#335	—	122	2	no	—
73	C₆₁	#7245	1	1	yes	characteristic, complemented, Sylow
74	C₂×M₁₆	—	244	2	no	—
75	C₂²×C₈	—	61	1	no	—
76	C₂²×C₈	C₂×#6923	1	1	yes	characteristic
77	C₂×C₄²	—	61	1	no	—
78	M₃₂	—	4	1	no	—
79	M₃₂	—	244	1	no	—
80	C₂×C₁₆	—	2	1	no	—
81	C₂×C₁₆	—	122	1	no	—
82	C₄⋊C₈	—	122	4	no	—
83	C₄⋊C₈	—	244	2	no	—
84	C₄×C₈	—	61	2	no	—
85	C₄×C₈	—	61	2	no	—
86	C₂²×C₄	—	61	2	no	—
87	C₂²×C₄	C₄⋊#6923	1	1	yes	characteristic
88	M₁₆	—	488	4	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	61	2	no	—
92	C₂×C₈	—	244	1	no	—
93	C₂×C₈	—	244	4	no	—
94	C₂×C₈	C₄×#6923	1	1	yes	characteristic
95	C₂×C₈	C₂²⋊#6923	1	1	yes	characteristic
96	C₄²	—	61	2	no	—
97	C₄²	—	61	1	no	—
98	C₄²	—	61	1	no	—
99	C₁₆	—	4	1	no	—
100	C₁₆	—	244	1	no	—
101	C₂³	#14519845	1	1	yes	characteristic
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	61	2	no	—
105	C₂×C₄	—	61	2	no	—
106	C₂×C₄	—	122	2	no	—
107	C₂×C₄	—	122	2	no	—
108	C₂×C₄	(C₂×#6923)⋊C₄	1	1	yes	characteristic
109	C₂×C₄	#14519846	1	1	yes	characteristic
110	C₈	—	2	1	no	—
111	C₈	—	122	2	no	—
112	C₈	—	488	4	no	—
113	C₈	#14519837	1	1	yes	characteristic
114	C₈	#14519838	1	1	yes	characteristic
115	C₂²	—	2	2	no	—
116	C₂²	—	2	1	no	—
117	C₂²	#16840636	1	1	yes	characteristic
118	C₄	—	2	1	no	—
119	C₄	—	61	2	no	—
120	C₄	—	61	2	no	—
121	C₄	—	122	2	no	—
122	C₄	#16840631	1	1	yes	characteristic
123	C₄	#16840642	1	1	yes	characteristic
124	C₂	—	4	1	no	—
125	C₂	#21460689	1	1	yes	characteristic
126	C₂	#21460894	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21461376

C₂

yes

maximal

#21447574

C₂

yes

maximal, characteristic

#16840765

—

#16840626

C₄

yes

—

#16840625

—

C₂×#14519717

C₄

yes

characteristic

C₂×C₈×Dic₆₁

C₂²

yes

characteristic

#16839580

—

C₂×M₃₂×C₆₁

C₄

yes

—

—

—

C₂²×#92418

Q₈

yes

characteristic

C₂×C₄×Dic₆₁

D₄

yes

characteristic

#14519717

—

C₂×#11758185

—

C₈×Dic₆₁

C₂×C₄

yes

—

C₈×Dic₆₁

D₄

yes

—

C₂²×C₈×C₆₁

C₂×C₄

yes

characteristic

M₃₂×C₆₁

—

C₂×C₁₆×C₆₁

—

C₂²×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂²×Dic₆₁

SD₁₆

yes

—

#11758217

—

C₂×#92420

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄²

yes

characteristic

C₂×C₈×C₆₁

C₂²⋊C₄

yes

characteristic

C₄×Dic₆₁

C₂²⋊C₄

yes

—

C₄×Dic₆₁

D₈

yes

characteristic

C₄×Dic₆₁

Q₁₆

yes

characteristic

C₂×#92418

—

C₂×#92418

C₄⋊C₄

yes

—

C₁₆×C₆₁

—

#11758185

—

C₂³×C₆₁

#106

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

yes

C₂×C₄×C₆₁

yes

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

#102

yes

—

C₂×Dic₆₁

#103

yes

—

#92420

—

C₈×C₆₁

—

C₈×C₆₁

yes

C₈×C₆₁

yes

—

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#340

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

#361

yes

—

Dic₆₁

#362

yes

—

#2150

—

#1567

—

C₂×C₆₁

—

C₂×C₆₁

#1292

yes

characteristic

C₂×C₆₁

#1387

yes

—

C₂×M₃₂

—

C₂×M₃₂

#6923

yes

characteristic

#423

—

122

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

122

—

#335

—

#335

—

122

—

C₆₁

#7245

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

244

—

C₂²×C₈

—

C₂²×C₈

C₂×#6923

yes

characteristic

C₂×C₄²

—

M₃₂

—

M₃₂

—

244

—

C₂×C₁₆

—

C₂×C₁₆

—

122

—

C₄⋊C₈

—

122

—

C₄⋊C₈

—

244

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

C₄⋊#6923

yes

characteristic

M₁₆

—

488

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

244

—

C₂×C₈

—

244

—

C₂×C₈

C₄×#6923

yes

characteristic

C₂×C₈

C₂²⋊#6923

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₁₆

—

100

C₁₆

—

244

—

101

C₂³

#14519845

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

122

—

107

C₂×C₄

—

122

—

108

C₂×C₄

(C₂×#6923)⋊C₄

yes

characteristic

109

C₂×C₄

#14519846

yes

characteristic

110

C₈

—

111

C₈

—

122

—

112

C₈

—

488

—

113

C₈

#14519837

yes

characteristic

114

C₈

#14519838

yes

characteristic

115

C₂²

—

116

C₂²

—

117

C₂²

#16840636

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

—

122

C₄

#16840631

yes

characteristic

123

C₄

#16840642

yes

characteristic

124

C₂

—

125

C₂

#21460689

yes

characteristic

126

C₂

#21460894

yes

—

	subgroup	G/H
center	C₂²	#16840636
commutator	C₂×C₈×C₆₁	C₄²
Frattini	C₂²×C₈	C₂×#6923
Fitting	C₂×M₃₂×C₆₁	C₄
socle	C₂²×C₆₁	#340
prime radical	C₂³×C₆₁	#106
cosocle	C₂×C₈×Dic₆₁	C₂²
hypercenter	C₂×M₃₂	#6923
hypocenter	C₆₁	#7245
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₂²

#16840636

commutator

C₂×C₈×C₆₁

C₄²

Frattini

C₂²×C₈

C₂×#6923

Fitting

C₂×M₃₂×C₆₁

C₄

C₂²×C₆₁

C₂³×C₆₁

C₂×C₈×Dic₆₁

C₂²

C₂×M₃₂

C₆₁

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
73. C₆₁	#7245	semidirect

H ≅ G/N

type

73. C₆₁

#7245

semidirect

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

subgroup	index
#21461376	2
#21447574	2
#7245	61

subgroup

index

#21461376

#21447574

#7245

Subgroups — #30594531

Conjugacy classes of subgroups

Complements

Maximal subgroups