Group #30599527 — subgroups

Subgroups — #30599527

Order 15616.

354 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 136 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	#21461758	C₂	1	2	yes	maximal
2	#21461375	C₂	1	4	yes	maximal
3	C₈²×D₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519847	—	2	4	no	—
5	#16840766	C₂²	1	4	yes	—
6	#16840624	—	2	8	no	—
7	C₄×C₈×D₆₁	C₂²	1	2	yes	—
8	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
9	C₈×#92418	C₄	1	2	yes	—
10	C₈×#92418	C₄	1	1	yes	characteristic
11	C₈²×C₆₁	C₄	1	1	yes	characteristic
12	C₂×#11758213	—	2	4	no	—
13	#14519847	—	4	8	no	—
14	C₄⋊#92420	—	2	8	no	—
15	C₂×C₈×D₆₁	D₄	1	2	yes	—
16	C₂×C₈×D₆₁	Q₈	1	4	yes	—
17	C₄²×D₆₁	C₂³	1	1	yes	characteristic
18	C₄×#92418	C₂×C₄	1	1	yes	characteristic
19	C₄×#92418	C₂×C₄	1	2	yes	—
20	C₈×Dic₆₁	D₄	1	4	yes	—
21	C₈×Dic₆₁	Q₈	1	2	yes	—
22	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
25	C₂×C₄×D₆₁	C₂×Q₈	1	2	yes	—
26	C₂×#92420	—	4	4	no	—
27	#11758213	—	4	8	no	—
28	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
29	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
30	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
31	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
32	C₄×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
33	C₂×#92418	C₄⋊C₄	1	2	yes	—
34	C₂×#92418	C₄⋊C₄	1	4	yes	—
35	C₈×D₆₁	—	2	8	no	—
36	C₈×D₆₁	SD₁₆	1	8	yes	—
37	C₂²×D₆₁	C₄⋊Q₈	1	1	yes	characteristic
38	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
40	C₂×Dic₆₁	#127	1	1	yes	characteristic
41	C₂×Dic₆₁	C₄⋊Q₈	1	2	yes	—
42	C₄×D₆₁	—	2	4	no	—
43	C₄×D₆₁	C₂×SD₁₆	1	4	yes	—
44	#92420	—	8	4	no	—
45	C₈×C₆₁	—	2	4	no	—
46	C₈×C₆₁	#106	1	4	yes	—
47	#92418	#106	1	4	yes	—
48	#92418	#108	1	8	yes	complemented
49	#8822	—	61	1	no	maximal, Sylow
50	C₂²×C₆₁	#384	1	1	yes	characteristic
51	C₂×D₆₁	—	2	2	no	—
52	C₂×D₆₁	#498	1	2	yes	—
53	C₄×C₆₁	C₂×#106	1	2	yes	—
54	C₄×C₆₁	C₂×#108	1	4	yes	—
55	Dic₆₁	—	2	2	no	—
56	Dic₆₁	#492	1	2	yes	—
57	#1848	—	61	2	no	—
58	#1566	—	61	4	no	—
59	C₂×C₈²	—	61	1	no	—
60	C₂×C₆₁	#1844	1	1	yes	characteristic
61	C₂×C₆₁	#1848	1	2	yes	—
62	D₆₁	—	2	2	no	—
63	C₂×#108	—	122	4	no	—
64	#423	—	61	4	no	—
65	C₂×C₄×C₈	—	61	2	no	—
66	C₂×C₄×C₈	—	61	1	no	—
67	#334	—	122	8	no	—
68	C₈²	—	61	1	no	—
69	C₈²	—	61	2	no	—
70	C₈²	#6923	1	1	yes	characteristic
71	C₆₁	#8822	1	1	yes	characteristic, complemented, Sylow
72	C₂×M₁₆	—	122	4	no	—
73	C₂²×C₈	—	61	2	no	—
74	C₂²×C₈	—	61	4	no	—
75	C₂×C₄²	—	61	1	no	—
76	#108	—	244	8	no	—
77	C₄⋊C₈	—	122	8	no	—
78	C₄×C₈	—	61	4	no	—
79	C₄×C₈	—	61	2	no	—
80	C₄×C₈	—	61	2	no	—
81	C₄×C₈	—	61	1	no	—
82	C₄×C₈	C₂×#6923	1	2	yes	—
83	C₄×C₈	C₂×#6923	1	1	yes	characteristic
84	C₂²×C₄	—	61	1	no	—
85	C₂²×C₄	—	61	2	no	—
86	M₁₆	—	244	8	no	—
87	C₂×C₈	—	61	2	no	—
88	C₂×C₈	—	61	4	no	—
89	C₂×C₈	—	61	8	no	—
90	C₂×C₈	—	122	8	no	—
91	C₂×C₈	—	244	4	no	—
92	C₂×C₈	C₄⋊#6923	1	4	yes	—
93	C₂×C₈	C₄⋊#6923	1	2	yes	—
94	C₄²	—	61	2	no	—
95	C₄²	—	61	1	no	—
96	C₄²	C₂²×#6923	1	1	yes	characteristic
97	C₂³	—	61	1	no	—
98	C₂×C₄	—	61	1	no	—
99	C₂×C₄	—	61	4	no	—
100	C₂×C₄	—	61	2	no	—
101	C₂×C₄	—	122	4	no	—
102	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
103	C₂×C₄	C₂×C₄⋊#6923	1	2	yes	—
104	C₈	—	2	4	no	—
105	C₈	—	61	4	no	—
106	C₈	—	61	8	no	—
107	C₈	—	488	4	no	—
108	C₈	#14519845	1	4	yes	—
109	C₂²	—	61	2	no	—
110	C₂²	—	122	2	no	—
111	C₂²	#16840708	1	1	yes	characteristic
112	C₄	—	61	2	no	—
113	C₄	—	122	2	no	—
114	C₄	C₂×#14519845	1	2	yes	—
115	C₄	D₆₁⋊#108	1	4	yes	—
116	C₂	—	122	2	no	—
117	C₂	#21461754	1	1	yes	characteristic
118	C₂	#21461759	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21461758

C₂

yes

maximal

#21461375

C₂

yes

maximal

C₈²×D₆₁

C₂

yes

maximal, characteristic

C₂×#14519847

—

#16840766

C₂²

yes

—

#16840624

—

C₄×C₈×D₆₁

C₂²

yes

—

C₄×C₈×D₆₁

C₂²

yes

characteristic

C₈×#92418

C₄

yes

—

C₈×#92418

C₄

yes

characteristic

C₈²×C₆₁

C₄

yes

—

—

—

C₂×C₈×D₆₁

D₄

yes

—

C₂×C₈×D₆₁

Q₈

yes

—

C₄²×D₆₁

C₂³

yes

characteristic

C₄×#92418

C₂×C₄

yes

characteristic

C₄×#92418

C₂×C₄

yes

—

C₈×Dic₆₁

D₄

yes

—

C₈×Dic₆₁

Q₈

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₄×D₆₁

C₂×D₄

yes

characteristic

C₂×C₄×D₆₁

C₂×Q₈

yes

—

C₂×#92420

—

#11758213

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄×Dic₆₁

C₂×D₄

yes

—

C₄×Dic₆₁

C₂×Q₈

yes

characteristic

C₂×#92418

C₄⋊C₄

yes

—

C₂×#92418

C₄⋊C₄

yes

—

C₈×D₆₁

—

C₈×D₆₁

SD₁₆

yes

—

C₂²×D₆₁

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

—

C₂×Dic₆₁

#127

yes

characteristic

C₂×Dic₆₁

C₄⋊Q₈

yes

—

C₄×D₆₁

—

C₄×D₆₁

C₂×SD₁₆

yes

—

#92420

—

C₈×C₆₁

—

C₈×C₆₁

#106

yes

—

#92418

#106

yes

—

yes

—

C₂²×C₆₁

yes

C₂×D₆₁

—

C₂×D₆₁

#498

yes

—

C₄×C₆₁

C₂×#106

yes

—

C₄×C₆₁

C₂×#108

yes

—

Dic₆₁

—

Dic₆₁

#492

yes

—

#1848

—

#1566

—

C₂×C₈²

—

C₂×C₆₁

#1844

yes

characteristic

C₂×C₆₁

#1848

yes

—

D₆₁

—

C₂×#108

—

122

—

#423

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#334

—

122

—

C₈²

—

C₈²

—

C₈²

#6923

yes

characteristic

C₆₁

#8822

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

122

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

#108

—

244

—

C₄⋊C₈

—

122

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#6923

yes

—

C₄×C₈

C₂×#6923

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

244

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

—

244

—

C₂×C₈

C₄⋊#6923

yes

—

C₂×C₈

C₄⋊#6923

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×#6923

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

122

—

102

C₂×C₄

C₂×C₄⋊#6923

yes

characteristic

103

C₂×C₄

C₂×C₄⋊#6923

yes

—

104

C₈

—

105

C₈

—

106

C₈

—

107

C₈

—

488

—

108

C₈

#14519845

yes

—

109

C₂²

—

110

C₂²

—

122

—

111

C₂²

#16840708

yes

characteristic

112

C₄

—

113

C₄

—

122

—

114

C₄

C₂×#14519845

yes

—

115

C₄

D₆₁⋊#108

yes

—

116

C₂

—

122

—

117

C₂

#21461754

yes

characteristic

118

C₂

#21461759

yes

—

	subgroup	G/H
center	C₂²	#16840708
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₄²	C₂²×#6923
Fitting	C₈²×C₆₁	C₄
socle	C₂²×C₆₁	#384
prime radical	C₂²×D₆₁	C₄⋊Q₈
cosocle	C₄²×D₆₁	C₂³
hypercenter	C₈²	#6923
hypocenter	C₆₁	#8822
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

subgroup

G/H

center

C₂²

#16840708

commutator

C₄²×C₆₁

C₂²×C₄

Frattini

C₄²

C₂²×#6923

Fitting

C₈²×C₆₁

C₄

socle

C₂²×C₆₁

#384

prime radical

C₂²×D₆₁

C₄⋊Q₈

cosocle

C₄²×D₆₁

C₂³

C₈²

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈²

N	H ≅ G/N	type
48. #92418	#108	semidirect
71. C₆₁	#8822	semidirect

H ≅ G/N

type

48. #92418

#108

semidirect

71. C₆₁

#8822

semidirect

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

subgroup	index
#21461758	2
#21461375	2
C₈²×D₆₁	2
#8822	61

subgroup

index

#21461758

#21461375

C₈²×D₆₁

#8822

Subgroups — #30599527

Conjugacy classes of subgroups

Complements

Maximal subgroups