Group #29567112 — subgroups

Subgroups — #29567112

Order 15616.

206 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 103 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	#21444578	C₂	1	2	yes	maximal
2	C₂×C₆₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#16839786	—	2	2	no	—
4	#16839515	—	2	2	no	—
5	#16839515	C₄	1	4	yes	—
6	C₂²×C₁₆×C₆₁	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
8	C₆₁×C₄⋊C₁₆	—	2	2	no	—
9	C₂×#11758194	—	4	2	no	—
10	C₄⋊#92418	—	2	4	no	—
11	C₄⋊#92418	—	4	2	no	—
12	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₆₁	Q₈	1	1	yes	characteristic
14	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
15	C₂×C₁₆×C₆₁	—	2	2	no	—
16	C₂×C₁₆×C₆₁	—	4	2	no	—
17	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
18	C₄×C₈×C₆₁	D₄	1	2	yes	—
19	C₂²×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
20	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
21	C₂²×C₄×C₆₁	Q₁₆	1	1	yes	characteristic
22	C₂×C₈×C₆₁	—	2	1	no	—
23	C₂×C₈×C₆₁	—	2	1	no	—
24	C₂×C₈×C₆₁	—	4	1	no	—
25	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
26	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₈×C₆₁	C₄⋊C₄	1	2	yes	—
28	C₄²×C₆₁	C₂²⋊C₄	1	2	yes	—
29	C₄²×C₆₁	SD₁₆	1	2	yes	—
30	#11758194	—	8	2	no	—
31	#11758194	—	8	2	no	—
32	C₂×#92418	—	4	4	no	—
33	C₁₆×C₆₁	—	4	2	no	—
34	C₂³×C₆₁	#107	1	1	yes	characteristic
35	C₂×C₄×C₆₁	—	2	1	no	—
36	C₂×C₄×C₆₁	—	2	1	no	—
37	C₂×C₄×C₆₁	—	4	1	no	—
38	C₂×C₄×C₆₁	#95	1	1	yes	characteristic
39	C₂×C₄×C₆₁	#102	1	2	yes	—
40	C₂×C₄×C₆₁	#103	1	2	yes	—
41	C₂×C₄×C₆₁	#106	1	1	yes	characteristic
42	C₂×C₄×C₆₁	D₁₆	1	1	yes	characteristic
43	C₂×C₄×C₆₁	SD₃₂	1	2	yes	—
44	C₂×C₄×C₆₁	Q₃₂	1	1	yes	characteristic
45	C₈×C₆₁	—	2	2	no	—
46	C₈×C₆₁	—	2	1	no	—
47	C₈×C₆₁	#100	1	1	yes	characteristic
48	C₈×C₆₁	#101	1	1	yes	characteristic
49	#92418	—	8	4	no	—
50	#7241	—	61	1	no	maximal, Sylow
51	C₂²×C₆₁	—	2	1	no	—
52	C₂²×C₆₁	—	2	1	no	—
53	C₂²×C₆₁	#340	1	1	yes	characteristic
54	C₂²×C₆₁	#366	1	1	yes	characteristic
55	C₂²×C₆₁	#367	1	1	yes	characteristic
56	C₄×C₆₁	—	2	1	no	—
57	C₄×C₆₁	—	2	2	no	—
58	C₄×C₆₁	#338	1	1	yes	characteristic
59	C₄×C₆₁	#343	1	1	yes	characteristic
60	C₄×C₆₁	#357	1	2	yes	—
61	C₄×C₆₁	#358	1	2	yes	—
62	C₂×C₄⋊C₁₆	D₆₁	1	1	yes	characteristic
63	#1567	—	61	2	no	—
64	C₂×C₆₁	—	2	1	no	—
65	C₂×C₆₁	—	2	1	no	—
66	C₂×C₆₁	#1292	1	1	yes	characteristic
67	C₂×C₆₁	#1380	1	1	yes	characteristic
68	C₂×C₆₁	#1385	1	1	yes	characteristic
69	C₂²×C₁₆	Dic₆₁	1	2	yes	—
70	#423	—	122	2	no	—
71	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
72	C₄⋊C₁₆	—	2	2	no	—
73	#335	—	61	4	no	—
74	#335	—	122	2	no	—
75	C₆₁	#7241	1	1	yes	characteristic, complemented, Sylow
76	C₂×M₁₆	—	244	2	no	—
77	C₂²×C₈	Dic₁₂₂	1	1	yes	characteristic
78	C₂²×C₈	C₂×Dic₆₁	1	1	yes	characteristic
79	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
80	C₂×C₁₆	—	2	2	no	—
81	C₂×C₁₆	—	4	2	no	—
82	C₄⋊C₈	—	122	4	no	—
83	C₄⋊C₈	—	244	2	no	—
84	C₄×C₈	C₄×D₆₁	1	2	yes	—
85	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
86	C₂²×C₄	D₄₈₈	1	1	yes	characteristic
87	C₂²×C₄	Dic₂₄₄	1	1	yes	characteristic
88	C₂²×C₄	C₄⋊Dic₆₁	1	1	yes	characteristic
89	M₁₆	—	488	2	no	—
90	M₁₆	—	488	2	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	4	1	no	—
94	C₂×C₈	—	244	4	no	—
95	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
96	C₂×C₈	#11758196	1	2	yes	—
97	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
98	C₄²	#11758190	1	2	yes	—
99	C₄²	#11758198	1	2	yes	—
100	C₁₆	—	4	2	no	—
101	C₂³	#14519711	1	1	yes	characteristic
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	4	1	no	—
105	C₂×C₄	#14519684	1	1	yes	characteristic
106	C₂×C₄	#14519699	1	2	yes	—
107	C₂×C₄	#14519702	1	2	yes	—
108	C₂×C₄	#14519709	1	1	yes	characteristic
109	C₂×C₄	D₉₇₆	1	1	yes	characteristic
110	C₂×C₄	#14519722	1	2	yes	—
111	C₂×C₄	Dic₄₈₈	1	1	yes	characteristic
112	C₈	—	2	2	no	—
113	C₈	—	2	1	no	—
114	C₈	—	488	4	no	—
115	C₈	#14519694	1	1	yes	characteristic
116	C₈	#14519696	1	1	yes	characteristic
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	#16839525	1	1	yes	characteristic
120	C₂²	#16839588	1	1	yes	characteristic
121	C₂²	#16839590	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	1	no	—
124	C₄	#16839520	1	1	yes	characteristic
125	C₄	#16839531	1	1	yes	characteristic
126	C₄	#16839564	1	2	yes	—
127	C₄	#16839567	1	2	yes	—
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#21443694	1	1	yes	characteristic
131	C₂	#21443911	1	1	yes	characteristic
132	C₂	#21443923	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#21444578

C₂

yes

maximal

C₂×C₆₁×C₄⋊C₁₆

C₂

yes

maximal, characteristic

#16839786

—

#16839515

—

#16839515

C₄

yes

—

C₂²×C₁₆×C₆₁

C₄

yes

—

C₂×C₄×C₈×C₆₁

C₂²

yes

characteristic

C₆₁×C₄⋊C₁₆

—

C₂×#11758194

—

C₄⋊#92418

—

C₄⋊#92418

—

C₂²×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₁

Q₈

yes

characteristic

C₂×C₄²×C₆₁

D₄

yes

characteristic

C₂×C₁₆×C₆₁

—

C₂×C₁₆×C₆₁

—

C₄×C₈×C₆₁

C₂×C₄

yes

—

C₄×C₈×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₆₁

D₈

yes

characteristic

C₂²×C₄×C₆₁

Q₁₆

yes

characteristic

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₄²

yes

characteristic

C₂×C₈×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₁

C₄⋊C₄

yes

—

C₄²×C₆₁

C₂²⋊C₄

yes

—

C₄²×C₆₁

SD₁₆

yes

—

#11758194

—

#11758194

—

C₂×#92418

—

C₁₆×C₆₁

—

C₂³×C₆₁

#107

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

#95

yes

characteristic

C₂×C₄×C₆₁

#102

yes

—

C₂×C₄×C₆₁

#103

yes

—

C₂×C₄×C₆₁

#106

yes

characteristic

C₂×C₄×C₆₁

D₁₆

yes

characteristic

C₂×C₄×C₆₁

SD₃₂

yes

—

C₂×C₄×C₆₁

Q₃₂

yes

characteristic

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

yes

C₈×C₆₁

yes

—

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

C₄×C₆₁

#358

yes

—

C₂×C₄⋊C₁₆

D₆₁

yes

characteristic

#1567

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂²×C₁₆

Dic₆₁

yes

—

#423

—

122

—

C₂×C₄×C₈

C₂×D₆₁

yes

characteristic

C₄⋊C₁₆

—

#335

—

#335

—

122

—

C₆₁

#7241

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

244

—

C₂²×C₈

Dic₁₂₂

yes

characteristic

C₂²×C₈

C₂×Dic₆₁

yes

characteristic

C₂×C₄²

D₂₄₄

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

122

—

C₄⋊C₈

—

244

—

C₄×C₈

C₄×D₆₁

yes

—

C₄×C₈

C₂²⋊D₆₁

yes

—

C₂²×C₄

D₄₈₈

yes

characteristic

C₂²×C₄

Dic₂₄₄

yes

characteristic

C₂²×C₄

C₄⋊Dic₆₁

yes

characteristic

M₁₆

—

488

—

M₁₆

—

488

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

244

—

C₂×C₈

C₄×Dic₆₁

yes

characteristic

C₂×C₈

#11758196

yes

—

C₂×C₈

C₂²⋊Dic₆₁

yes

characteristic

C₄²

#11758190

yes

—

C₄²

#11758198

yes

—

100

C₁₆

—

101

C₂³

#14519711

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

#14519684

yes

characteristic

106

C₂×C₄

#14519699

yes

—

107

C₂×C₄

#14519702

yes

—

108

C₂×C₄

#14519709

yes

characteristic

109

C₂×C₄

D₉₇₆

yes

characteristic

110

C₂×C₄

#14519722

yes

—

111

C₂×C₄

Dic₄₈₈

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

—

488

—

115

C₈

#14519694

yes

characteristic

116

C₈

#14519696

yes

characteristic

117

C₂²

—

118

C₂²

—

119

C₂²

#16839525

yes

characteristic

120

C₂²

#16839588

yes

characteristic

121

C₂²

#16839590

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

#16839520

yes

characteristic

125

C₄

#16839531

yes

characteristic

126

C₄

#16839564

yes

—

127

C₄

#16839567

yes

—

128

C₂

—

129

C₂

—

130

C₂

#21443694

yes

characteristic

131

C₂

#21443911

yes

characteristic

132

C₂

#21443923

yes

characteristic

	subgroup	G/H
center	C₂²	#16839525
commutator	C₂×C₈×C₆₁	C₄²
Frattini	C₂×C₄×C₈	C₂×D₆₁
Fitting	C₂×C₆₁×C₄⋊C₁₆	C₂
socle	C₂²×C₆₁	#340
prime radical	C₂³×C₆₁	#107
cosocle	C₂×C₄×C₈×C₆₁	C₂²
hypercenter	C₂×C₄⋊C₁₆	D₆₁
hypocenter	C₆₁	#7241
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

subgroup

G/H

center

C₂²

#16839525

commutator

C₂×C₈×C₆₁

C₄²

Frattini

C₂×C₄×C₈

C₂×D₆₁

Fitting

C₂×C₆₁×C₄⋊C₁₆

C₂

C₂²×C₆₁

C₂³×C₆₁

C₂×C₄×C₈×C₆₁

C₂²

hypercenter

C₂×C₄⋊C₁₆

D₆₁

hypocenter

C₆₁

#7241

LCS

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

N	H ≅ G/N	type
75. C₆₁	#7241	semidirect

H ≅ G/N

type

75. C₆₁

#7241

semidirect

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

subgroup	index
#21444578	2
C₂×C₆₁×C₄⋊C₁₆	2
#7241	61

subgroup

index

#21444578

C₂×C₆₁×C₄⋊C₁₆

#7241

Subgroups — #29567112

Conjugacy classes of subgroups

Complements

Maximal subgroups