Group #30599520 — subgroups

Subgroups — #30599520

Order 15616.

422 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 148 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	#21461755	C₂	1	2	yes	maximal
2	#21461740	C₂	1	4	yes	maximal
3	C₈²×D₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519846	—	2	4	no	—
5	#16840708	C₂²	1	4	yes	—
6	#16840636	—	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	complemented
10	C₈×#92418	C₄	1	1	yes	characteristic, complemented
11	C₈²×C₆₁	C₄	1	1	yes	characteristic, complemented
12	C₂×C₄⋊#6923	—	2	4	no	—
13	C₂×C₄⋊#6923	—	2	8	no	—
14	#14519846	—	4	8	no	—
15	C₂×C₈×D₆₁	D₄	1	4	yes	—
16	C₂×C₈×D₆₁	D₄	1	2	yes	—
17	C₄²×D₆₁	C₂³	1	1	yes	characteristic
18	C₄×#92418	C₂×C₄	1	2	yes	—
19	C₄×#92418	C₂×C₄	1	1	yes	characteristic
20	C₈×Dic₆₁	Q₈	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	C₂²×#6923	—	4	4	no	—
25	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
26	C₂×C₄×D₆₁	C₂×D₄	1	2	yes	—
27	C₄⋊#6923	—	4	8	no	—
28	C₄⋊#6923	—	4	16	no	—
29	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
30	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
31	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
32	C₄×Dic₆₁	C₂×Q₈	1	1	yes	characteristic
33	C₄×Dic₆₁	C₂×Q₈	1	2	yes	—
34	C₂×#92418	C₄⋊C₄	1	4	yes	—
35	C₂×#92418	C₄⋊C₄	1	2	yes	—
36	C₈×D₆₁	—	2	8	no	—
37	C₈×D₆₁	D₈	1	4	yes	—
38	C₈×D₆₁	Q₁₆	1	4	yes	—
39	C₂²×D₆₁	#127	1	1	yes	characteristic
40	C₂×#6923	—	8	4	no	—
41	C₂×#6923	—	8	8	no	—
42	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
44	C₂×Dic₆₁	C₄⋊Q₈	1	2	yes	—
45	C₂×Dic₆₁	C₄⋊Q₈	1	1	yes	characteristic
46	C₄×D₆₁	C₂×D₈	1	2	yes	—
47	C₄×D₆₁	C₂×Q₁₆	1	2	yes	—
48	C₄×D₆₁	#135	1	8	yes	—
49	C₈×C₆₁	—	2	4	no	—
50	C₈×C₆₁	#107	1	4	yes	—
51	#92418	—	2	4	no	—
52	#92418	#107	1	4	yes	—
53	#8816	—	61	1	no	maximal, Sylow
54	C₂²×C₆₁	#384	1	1	yes	characteristic
55	C₂×D₆₁	#493	1	1	yes	characteristic
56	C₂×D₆₁	#494	1	1	yes	characteristic
57	C₂×D₆₁	#495	1	4	yes	—
58	#6923	—	16	4	no	—
59	C₄×C₆₁	C₂×#107	1	2	yes	—
60	C₄×C₆₁	#427	1	4	yes	—
61	Dic₆₁	#499	1	4	yes	—
62	Dic₆₁	#500	1	2	yes	—
63	#1845	—	61	2	no	—
64	#1834	—	61	4	no	—
65	C₂×C₈²	—	61	1	no	—
66	C₂×C₆₁	#1845	1	1	yes	characteristic
67	C₂×C₆₁	#1846	1	2	yes	—
68	D₆₁	#1709	1	2	yes	—
69	D₆₁	#1710	1	2	yes	—
70	C₂×#107	—	122	4	no	—
71	C₂×C₄×C₈	—	61	2	no	—
72	C₂×C₄×C₈	—	61	1	no	—
73	#384	—	61	4	no	—
74	#340	—	122	8	no	—
75	C₈²	—	61	2	no	—
76	C₈²	—	61	1	no	—
77	C₈²	#6923	1	1	yes	characteristic, complemented
78	C₆₁	#8816	1	1	yes	characteristic, complemented, Sylow
79	C₂²×C₈	—	61	2	no	—
80	C₂²×C₈	—	61	4	no	—
81	C₂×C₄⋊C₄	—	122	8	no	—
82	C₂×C₄⋊C₄	—	122	4	no	—
83	C₂×C₄²	—	61	1	no	—
84	#107	—	244	8	no	—
85	C₄×C₈	—	61	2	no	—
86	C₄×C₈	—	61	4	no	—
87	C₄×C₈	—	61	1	no	—
88	C₄×C₈	—	61	2	no	—
89	C₄×C₈	C₂×#6923	1	2	yes	—
90	C₄×C₈	C₂×#6923	1	1	yes	characteristic
91	C₂²×C₄	—	61	1	no	—
92	C₂²×C₄	—	61	2	no	—
93	C₂²×C₄	—	244	4	no	—
94	C₂×C₈	—	61	2	no	—
95	C₂×C₈	—	61	4	no	—
96	C₂×C₈	—	61	4	no	—
97	C₂×C₈	—	61	4	no	—
98	C₂×C₈	—	122	8	no	—
99	C₂×C₈	C₄⋊#6923	1	4	yes	—
100	C₂×C₈	C₄⋊#6923	1	2	yes	—
101	C₄⋊C₄	—	244	8	no	—
102	C₄⋊C₄	—	244	16	no	—
103	C₄²	—	61	1	no	—
104	C₄²	—	61	2	no	—
105	C₄²	C₂²×#6923	1	1	yes	characteristic
106	C₂³	—	61	1	no	—
107	C₂×C₄	—	61	1	no	—
108	C₂×C₄	—	61	2	no	—
109	C₂×C₄	—	61	2	no	—
110	C₂×C₄	—	61	2	no	—
111	C₂×C₄	—	61	8	no	—
112	C₂×C₄	—	488	4	no	—
113	C₂×C₄	—	488	8	no	—
114	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
115	C₂×C₄	C₂×C₄⋊#6923	1	2	yes	—
116	C₈	—	2	4	no	—
117	C₈	—	61	4	no	—
118	C₈	—	122	4	no	—
119	C₈	#14519846	1	4	yes	—
120	C₂²	—	61	1	no	—
121	C₂²	—	61	1	no	—
122	C₂²	—	61	4	no	—
123	C₂²	#16840708	1	1	yes	characteristic
124	C₄	—	61	2	no	—
125	C₄	—	61	4	no	—
126	C₄	—	976	4	no	—
127	C₄	C₂×#14519846	1	2	yes	—
128	C₄	#16840772	1	4	yes	—
129	C₂	—	61	2	no	—
130	C₂	—	61	2	no	—
131	C₂	#21461755	1	1	yes	characteristic
132	C₂	#21461756	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21461755

C₂

yes

maximal

#21461740

C₂

yes

maximal

C₈²×D₆₁

C₂

yes

maximal, characteristic

C₂×#14519846

—

#16840708

C₂²

yes

—

#16840636

—

C₄×C₈×D₆₁

C₂²

yes

—

C₄×C₈×D₆₁

C₂²

yes

characteristic

C₈×#92418

C₄

yes

complemented

C₈×#92418

C₄

yes

characteristic, complemented

C₈²×C₆₁

C₄

yes

characteristic, complemented

C₂×C₄⋊#6923

—

C₂×C₄⋊#6923

—

#14519846

—

C₂×C₈×D₆₁

D₄

yes

—

C₂×C₈×D₆₁

D₄

yes

—

C₄²×D₆₁

C₂³

yes

characteristic

C₄×#92418

C₂×C₄

yes

—

C₄×#92418

C₂×C₄

yes

characteristic

C₈×Dic₆₁

Q₈

yes

—

C₈×Dic₆₁

Q₈

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂²×#6923

—

C₂×C₄×D₆₁

C₂×D₄

yes

characteristic

C₂×C₄×D₆₁

C₂×D₄

yes

—

C₄⋊#6923

—

C₄⋊#6923

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₄²×C₆₁

C₂²×C₄

yes

characteristic

C₄×Dic₆₁

C₂×Q₈

yes

characteristic

C₄×Dic₆₁

C₂×Q₈

yes

—

C₂×#92418

C₄⋊C₄

yes

—

C₂×#92418

C₄⋊C₄

yes

—

C₈×D₆₁

—

C₈×D₆₁

D₈

yes

—

C₈×D₆₁

Q₁₆

yes

—

C₂²×D₆₁

yes

—

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

—

C₂×Dic₆₁

C₄⋊Q₈

yes

—

C₂×Dic₆₁

C₄⋊Q₈

yes

characteristic

C₄×D₆₁

C₂×D₈

yes

—

C₄×D₆₁

C₂×Q₁₆

yes

—

C₄×D₆₁

#135

yes

—

C₈×C₆₁

—

C₈×C₆₁

#107

yes

—

#92418

—

#92418

#107

yes

—

—

C₂²×C₆₁

yes

C₂×D₆₁

yes

C₂×D₆₁

yes

C₂×D₆₁

yes

—

#6923

—

C₄×C₆₁

C₂×#107

yes

—

C₄×C₆₁

#427

yes

—

Dic₆₁

#499

yes

—

Dic₆₁

#500

yes

—

#1845

—

#1834

—

C₂×C₈²

—

C₂×C₆₁

#1845

yes

characteristic

C₂×C₆₁

#1846

yes

—

D₆₁

#1709

yes

—

D₆₁

#1710

yes

—

C₂×#107

—

122

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#384

—

#340

—

122

—

C₈²

—

C₈²

—

C₈²

#6923

yes

characteristic, complemented

C₆₁

#8816

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄²

—

#107

—

244

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#6923

yes

—

C₄×C₈

C₂×#6923

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

244

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

C₄⋊#6923

yes

—

100

C₂×C₈

C₄⋊#6923

yes

—

101

C₄⋊C₄

—

244

—

102

C₄⋊C₄

—

244

—

103

C₄²

—

104

C₄²

—

105

C₄²

C₂²×#6923

yes

characteristic

106

C₂³

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

488

—

113

C₂×C₄

—

488

—

114

C₂×C₄

C₂×C₄⋊#6923

yes

characteristic

115

C₂×C₄

C₂×C₄⋊#6923

yes

—

116

C₈

—

117

C₈

—

118

C₈

—

122

—

119

C₈

#14519846

yes

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

#16840708

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

976

—

127

C₄

C₂×#14519846

yes

—

128

C₄

#16840772

yes

—

129

C₂

—

130

C₂

—

131

C₂

#21461755

yes

characteristic

132

C₂

#21461756

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	#127
cosocle	C₄²×D₆₁	C₂³
hypercenter	C₈²	#6923
hypocenter	C₆₁	#8816
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

C₂²×C₆₁

C₂²×D₆₁

C₄²×D₆₁

C₂³

C₈²

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈²

N	H ≅ G/N	type
9. C₈×#92418	C₄	semidirect
10. C₈×#92418	C₄	semidirect
11. C₈²×C₆₁	C₄	semidirect
77. C₈²	#6923	semidirect
78. C₆₁	#8816	semidirect

H ≅ G/N

type

9. C₈×#92418

C₄

semidirect

10. C₈×#92418

C₄

semidirect

11. C₈²×C₆₁

C₄

semidirect

77. C₈²

#6923

semidirect

78. C₆₁

#8816

semidirect

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

subgroup	index
#21461755	2
#21461740	2
C₈²×D₆₁	2
#8816	61

subgroup

index

#21461755

#21461740

C₈²×D₆₁

#8816

Subgroups — #30599520

Conjugacy classes of subgroups

Complements

Maximal subgroups