Group #30607324 — subgroups

Subgroups — #30607324

Order 15616.

310 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 108 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	#21461756	C₂	1	2	yes	maximal
2	#21460877	C₂	1	4	yes	maximal
3	C₄×C₁₆×D₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519846	C₂²	1	4	yes	—
5	C₂×#14519845	—	2	2	no	—
6	#16840706	—	2	2	no	—
7	C₂×C₁₆×D₆₁	C₂²	1	2	yes	—
8	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
9	C₄×#11758185	C₄	1	1	yes	characteristic, complemented
10	C₁₆×Dic₆₁	C₄	1	2	yes	complemented
11	C₄×C₁₆×C₆₁	C₄	1	1	yes	characteristic, complemented
12	C₂×C₄⋊#6923	—	2	4	no	—
13	#14519846	—	2	8	no	—
14	#14519845	—	4	4	no	—
15	(C₂×#6923)⋊C₄	—	4	4	no	—
16	C₂×C₈×D₆₁	D₄	1	1	yes	characteristic
17	C₂×C₈×D₆₁	C₂³	1	1	yes	characteristic
18	C₄²×D₆₁	D₄	1	1	yes	characteristic
19	C₂×#11758185	C₂×C₄	1	2	yes	—
20	C₁₆×D₆₁	—	2	4	no	—
21	C₄×#92418	Q₈	1	1	yes	characteristic
22	C₈×Dic₆₁	C₂×C₄	1	1	yes	characteristic
23	C₈×Dic₆₁	Q₈	1	1	yes	characteristic
24	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₂²×#6923	—	4	4	no	—
27	C₂×C₄×D₆₁	D₈	1	2	yes	—
28	C₂×C₄×D₆₁	C₂×D₄	1	1	yes	characteristic
29	C₄⋊#6923	—	4	4	no	—
30	C₄⋊#6923	—	4	4	no	—
31	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
33	C₄²×C₆₁	C₄⋊C₄	1	1	yes	characteristic
34	C₄×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
35	C₄×Dic₆₁	Q₁₆	1	2	yes	—
36	C₂×#92418	C₄⋊C₄	1	1	yes	characteristic
37	C₂×#92418	C₂×Q₈	1	1	yes	characteristic
38	C₈×D₆₁	SD₁₆	1	4	yes	—
39	C₈×D₆₁	#41	1	4	yes	—
40	C₁₆×C₆₁	—	2	2	no	—
41	#11758185	—	2	2	no	—
42	C₂²×D₆₁	C₂×D₈	1	1	yes	characteristic
43	C₂×#6923	—	8	4	no	—
44	C₂×#6923	—	8	8	no	—
45	C₂×C₄×C₆₁	#107	1	2	yes	—
46	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×Dic₆₁	#107	1	2	yes	—
48	C₂×Dic₆₁	C₂×Q₁₆	1	1	yes	characteristic
49	C₄×D₆₁	—	2	4	no	—
50	C₄×D₆₁	#124	1	2	yes	—
51	C₄×D₆₁	C₂×SD₁₆	1	2	yes	—
52	C₈×C₆₁	#106	1	2	yes	—
53	C₈×C₆₁	#117	1	2	yes	—
54	#92418	#106	1	2	yes	—
55	#92418	#125	1	2	yes	—
56	#12889	—	61	1	no	maximal, Sylow
57	C₂²×C₆₁	C₂×#107	1	1	yes	characteristic
58	C₂×D₆₁	#486	1	2	yes	—
59	C₂×D₆₁	#508	1	4	yes	—
60	#6923	—	16	4	no	—
61	C₄×C₆₁	—	2	2	no	—
62	C₄×C₆₁	#382	1	1	yes	characteristic
63	C₄×C₆₁	C₂×#106	1	1	yes	characteristic
64	Dic₆₁	—	2	2	no	—
65	Dic₆₁	#487	1	2	yes	—
66	C₂×C₄×C₁₆	—	61	1	no	—
67	#1846	—	61	2	no	—
68	#1380	—	61	4	no	—
69	C₂×C₆₁	#1834	1	1	yes	characteristic
70	C₂×C₆₁	#2158	1	2	yes	—
71	D₆₁	#2242	1	4	yes	—
72	C₂²×C₁₆	—	61	2	no	—
73	C₂×#107	—	61	4	no	—
74	C₂×#106	—	122	2	no	—
75	C₂×C₄×C₈	—	61	1	no	—
76	#382	—	122	2	no	—
77	C₄×C₁₆	—	61	2	no	—
78	C₄×C₁₆	—	61	1	no	—
79	C₄×C₁₆	#6923	1	1	yes	characteristic, complemented
80	C₆₁	#12889	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₈	—	61	1	no	—
82	C₂²×C₈	—	61	1	no	—
83	C₂×C₄⋊C₄	—	122	4	no	—
84	C₂×C₄²	—	61	1	no	—
85	C₂×C₁₆	—	61	2	no	—
86	C₂×C₁₆	—	122	4	no	—
87	C₂×C₁₆	C₂×#6923	1	2	yes	—
88	#107	—	122	8	no	—
89	#106	—	244	4	no	—
90	C₄×C₈	—	61	1	no	—
91	C₄×C₈	—	61	1	no	—
92	C₄×C₈	—	61	1	no	—
93	C₄×C₈	C₂×#6923	1	1	yes	characteristic
94	#95	—	244	4	no	—
95	C₂²×C₄	—	61	1	no	—
96	C₂²×C₄	—	61	2	no	—
97	C₂²×C₄	—	244	4	no	—
98	C₂×C₈	—	61	1	no	—
99	C₂×C₈	—	61	1	no	—
100	C₂×C₈	—	61	4	no	—
101	C₂×C₈	—	61	4	no	—
102	C₂×C₈	C₄⋊#6923	1	1	yes	characteristic
103	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
104	C₄⋊C₄	—	244	4	no	—
105	C₄⋊C₄	—	244	4	no	—
106	C₄²	—	61	2	no	—
107	C₄²	—	61	1	no	—
108	C₄²	C₄⋊#6923	1	1	yes	characteristic
109	C₁₆	—	2	2	no	—
110	C₁₆	—	122	2	no	—
111	C₂³	—	61	1	no	—
112	C₂×C₄	—	61	1	no	—
113	C₂×C₄	—	61	2	no	—
114	C₂×C₄	—	61	2	no	—
115	C₂×C₄	—	61	2	no	—
116	C₂×C₄	—	122	4	no	—
117	C₂×C₄	—	488	4	no	—
118	C₂×C₄	—	488	8	no	—
119	C₂×C₄	#14519846	1	2	yes	—
120	C₂×C₄	C₂×C₄⋊#6923	1	1	yes	characteristic
121	C₈	—	61	2	no	—
122	C₈	—	61	2	no	—
123	C₈	#14519845	1	2	yes	—
124	C₈	#14519856	1	2	yes	—
125	C₂²	—	61	4	no	—
126	C₂²	—	61	2	no	—
127	C₂²	C₂×#14519846	1	1	yes	characteristic
128	C₄	—	2	2	no	—
129	C₄	—	61	2	no	—
130	C₄	—	122	2	no	—
131	C₄	—	976	4	no	—
132	C₄	#16840706	1	1	yes	characteristic
133	C₄	C₂×#14519845	1	1	yes	characteristic
134	C₂	—	61	4	no	—
135	C₂	#21461740	1	1	yes	characteristic
136	C₂	#21462170	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#21461756

C₂

yes

maximal

#21460877

C₂

yes

maximal

C₄×C₁₆×D₆₁

C₂

yes

maximal, characteristic

C₂×#14519846

C₂²

yes

—

C₂×#14519845

—

#16840706

—

C₂×C₁₆×D₆₁

C₂²

yes

—

C₄×C₈×D₆₁

C₂²

yes

characteristic

C₄×#11758185

C₄

yes

characteristic, complemented

C₁₆×Dic₆₁

C₄

yes

complemented

C₄×C₁₆×C₆₁

C₄

yes

characteristic, complemented

C₂×C₄⋊#6923

—

#14519846

—

#14519845

—

(C₂×#6923)⋊C₄

—

C₂×C₈×D₆₁

D₄

yes

characteristic

C₂×C₈×D₆₁

C₂³

yes

characteristic

C₄²×D₆₁

D₄

yes

characteristic

C₂×#11758185

C₂×C₄

yes

—

C₁₆×D₆₁

—

C₄×#92418

Q₈

yes

characteristic

C₈×Dic₆₁

C₂×C₄

yes

characteristic

C₈×Dic₆₁

Q₈

yes

characteristic

C₂×C₁₆×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂²×#6923

—

C₂×C₄×D₆₁

D₈

yes

—

C₂×C₄×D₆₁

C₂×D₄

yes

characteristic

C₄⋊#6923

—

C₄⋊#6923

—

C₂×C₈×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₈×C₆₁

C₂²×C₄

yes

characteristic

C₄²×C₆₁

C₄⋊C₄

yes

characteristic

C₄×Dic₆₁

C₄⋊C₄

yes

characteristic

C₄×Dic₆₁

Q₁₆

yes

—

C₂×#92418

C₄⋊C₄

yes

characteristic

C₂×#92418

C₂×Q₈

yes

characteristic

C₈×D₆₁

SD₁₆

yes

—

C₈×D₆₁

#41

yes

—

C₁₆×C₆₁

—

#11758185

—

C₂²×D₆₁

C₂×D₈

yes

characteristic

C₂×#6923

—

C₂×#6923

—

C₂×C₄×C₆₁

#107

yes

—

C₂×C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₆₁

#107

yes

—

C₂×Dic₆₁

C₂×Q₁₆

yes

characteristic

C₄×D₆₁

—

C₄×D₆₁

#124

yes

—

C₄×D₆₁

C₂×SD₁₆

yes

—

C₈×C₆₁

#106

yes

—

C₈×C₆₁

#117

yes

—

#92418

#106

yes

—

#92418

#125

yes

—

—

C₂²×C₆₁

yes

C₂×D₆₁

yes

—

C₂×D₆₁

#508

yes

—

#6923

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

#487

yes

—

C₂×C₄×C₁₆

—

#1846

—

#1380

—

C₂×C₆₁

#1834

yes

characteristic

C₂×C₆₁

#2158

yes

—

D₆₁

#2242

yes

—

C₂²×C₁₆

—

C₂×#107

—

C₂×#106

—

122

—

C₂×C₄×C₈

—

#382

—

122

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

#6923

yes

characteristic, complemented

C₆₁

#12889

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

122

—

C₂×C₁₆

C₂×#6923

yes

—

#107

—

122

—

#106

—

244

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#6923

yes

characteristic

#95

—

244

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

244

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

C₄⋊#6923

yes

characteristic

103

C₂×C₈

C₂²×#6923

yes

characteristic

104

C₄⋊C₄

—

244

—

105

C₄⋊C₄

—

244

—

106

C₄²

—

107

C₄²

—

108

C₄²

C₄⋊#6923

yes

characteristic

109

C₁₆

—

110

C₁₆

—

122

—

111

C₂³

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

122

—

117

C₂×C₄

—

488

—

118

C₂×C₄

—

488

—

119

C₂×C₄

#14519846

yes

—

120

C₂×C₄

C₂×C₄⋊#6923

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

#14519845

yes

—

124

C₈

#14519856

yes

—

125

C₂²

—

126

C₂²

—

127

C₂²

C₂×#14519846

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

122

—

131

C₄

—

976

—

132

C₄

#16840706

yes

characteristic

133

C₄

C₂×#14519845

yes

characteristic

134

C₂

—

135

C₂

#21461740

yes

characteristic

136

C₂

#21462170

yes

—

	subgroup	G/H
center	C₂²	C₂×#14519846
commutator	C₂×C₈×C₆₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×#6923
Fitting	C₄×C₁₆×C₆₁	C₄
socle	C₂²×C₆₁	C₂×#107
prime radical	C₂²×D₆₁	C₂×D₈
cosocle	C₂×C₈×D₆₁	C₂³
hypercenter	C₄×C₁₆	#6923
hypocenter	C₆₁	#12889
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

subgroup

G/H

center

C₂²

C₂×#14519846

commutator

C₂×C₈×C₆₁

C₂²×C₄

Frattini

C₂×C₈

C₂²×#6923

Fitting

C₄×C₁₆×C₆₁

C₄

socle

C₂²×C₆₁

C₂×#107

prime radical

C₂²×D₆₁

C₂×D₈

cosocle

C₂×C₈×D₆₁

C₂³

C₄×C₁₆

C₆₁

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

N	H ≅ G/N	type
9. C₄×#11758185	C₄	semidirect
10. C₁₆×Dic₆₁	C₄	semidirect
11. C₄×C₁₆×C₆₁	C₄	semidirect
79. C₄×C₁₆	#6923	semidirect
80. C₆₁	#12889	semidirect

H ≅ G/N

type

9. C₄×#11758185

C₄

semidirect

10. C₁₆×Dic₆₁

C₄

semidirect

11. C₄×C₁₆×C₆₁

C₄

semidirect

79. C₄×C₁₆

#6923

semidirect

80. C₆₁

#12889

semidirect

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

subgroup	index
#21461756	2
#21460877	2
C₄×C₁₆×D₆₁	2
#12889	61

subgroup

index

#21461756

#21460877

C₄×C₁₆×D₆₁

#12889

Subgroups — #30607324

Conjugacy classes of subgroups

Complements

Maximal subgroups