Group #21454890 — subgroups

Subgroups — #21454890

Order 7808.

394 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 197 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	#16839834	C₂	1	8	yes	maximal, complemented
2	#16839819	C₂	1	2	yes	maximal, complemented
3	#16839729	C₂	1	4	yes	maximal, complemented
4	C₆₁×#532	C₂	1	1	yes	maximal, characteristic
5	C₂×#11758194	C₂²	1	4	yes	—
6	C₂²×#92418	C₂²	1	4	yes	—
7	C₄⋊#92418	C₂²	1	4	yes	—
8	C₂²⋊#92418	C₂²	1	4	yes	—
9	C₂²⋊#92418	C₂²	1	8	yes	—
10	#14519688	C₂²	1	4	yes	—
11	C₂×#41×C₆₁	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₆₁	C₄	1	1	yes	characteristic
13	#127×C₆₁	C₄	1	1	yes	characteristic
14	#124×C₆₁	C₄	1	2	yes	—
15	#121×C₆₁	C₄	1	4	yes	—
16	C₄×D₄×C₆₁	C₂²	1	4	yes	—
17	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
18	#41×C₆₁	—	4	2	no	—
19	#41×C₆₁	—	4	2	no	—
20	C₂×Q₈×C₆₁	C₂×C₄	1	2	yes	—
21	C₂×D₄×C₆₁	C₂×C₄	1	4	yes	—
22	C₂×D₄×C₆₁	C₂×C₄	1	4	yes	—
23	C₂×D₄×C₆₁	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₆₁	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
26	C₂²×C₄×C₆₁	C₂³	1	4	yes	—
27	C₄⋊C₄×C₆₁	C₂×C₄	1	4	yes	—
28	C₂²⋊C₄×C₆₁	C₂×C₄	1	8	yes	—
29	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₆₁	C₂³	1	2	yes	—
32	#11758194	—	4	4	no	—
33	C₂×#92418	—	4	4	no	—
34	C₂×#92418	C₂³	1	8	yes	—
35	C₂³×C₆₁	C₂²×C₄	1	4	yes	—
36	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
37	Q₈×C₆₁	—	4	2	no	—
38	D₄×C₆₁	—	4	4	no	—
39	D₄×C₆₁	—	4	2	no	—
40	D₄×C₆₁	—	4	4	no	—
41	C₂×C₄×C₆₁	—	2	4	no	—
42	C₂×C₄×C₆₁	—	2	4	no	—
43	C₂×C₄×C₆₁	—	2	2	no	—
44	C₂×C₄×C₆₁	—	4	2	no	—
45	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₆₁	C₂²×C₄	1	4	yes	—
49	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
50	#92418	—	2	8	no	—
51	C₂²×C₆₁	—	2	1	no	—
52	C₂²×C₆₁	—	2	4	no	—
53	C₂²×C₆₁	—	2	2	no	—
54	C₂²×C₆₁	—	2	8	no	—
55	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
56	C₄×C₆₁	—	2	1	no	—
57	C₄×C₆₁	—	2	4	no	—
58	C₄×C₆₁	—	2	2	no	—
59	C₄×C₆₁	—	2	2	no	—
60	C₄×C₆₁	H(5,𝔽₂)	1	2	yes	—
61	#2980	—	61	1	no	maximal, Sylow
62	C₂×C₆₁	—	4	1	no	—
63	C₂×C₆₁	—	4	4	no	—
64	C₂×C₆₁	#518	1	1	yes	characteristic
65	C₂×C₆₁	#568	1	2	yes	—
66	#532	D₆₁	1	1	yes	characteristic
67	#435	—	61	8	no	—
68	#432	—	61	2	no	—
69	#408	—	61	4	no	—
70	C₆₁	#2980	1	1	yes	characteristic, complemented, Sylow
71	C₂×#41	C₂×D₆₁	1	2	yes	—
72	C₂×M₁₆	—	61	4	no	—
73	C₂²×C₈	—	61	4	no	—
74	C₄⋊Q₈	Dic₆₁	1	1	yes	characteristic
75	#127	Dic₆₁	1	1	yes	characteristic
76	#124	Dic₆₁	1	2	yes	—
77	#121	Dic₆₁	1	4	yes	—
78	C₄×D₄	C₂×D₆₁	1	4	yes	—
79	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
80	C₄⋊C₈	—	61	4	no	—
81	C₂²⋊C₈	—	61	4	no	—
82	C₂²⋊C₈	—	61	8	no	—
83	#97	—	61	4	no	—
84	#41	—	4	2	no	—
85	#41	—	4	2	no	—
86	C₂×Q₈	C₂×Dic₆₁	1	2	yes	—
87	C₂×D₄	C₂×Dic₆₁	1	4	yes	—
88	C₂×D₄	C₂×Dic₆₁	1	4	yes	—
89	C₂×D₄	C₂×Dic₆₁	1	2	yes	—
90	C₂²×C₄	C₂×Dic₆₁	1	2	yes	—
91	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
92	C₂²×C₄	C₂²×D₆₁	1	4	yes	—
93	M₁₆	—	244	4	no	—
94	C₂×C₈	—	61	8	no	—
95	C₂×C₈	—	244	4	no	—
96	C₄⋊C₄	C₂×Dic₆₁	1	4	yes	—
97	C₂²⋊C₄	C₂×Dic₆₁	1	8	yes	—
98	C₄²	C₂×Dic₆₁	1	1	yes	characteristic
99	C₄²	C₂×Dic₆₁	1	1	yes	characteristic
100	C₄²	C₂²×D₆₁	1	2	yes	—
101	C₂³	C₂²×Dic₆₁	1	4	yes	—
102	C₂³	C₂²×Dic₆₁	1	1	yes	characteristic
103	Q₈	—	4	2	no	—
104	D₄	—	4	4	no	—
105	D₄	—	4	2	no	—
106	D₄	—	4	4	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
112	C₂×C₄	C₂²×Dic₆₁	1	2	yes	—
113	C₂×C₄	C₂²×Dic₆₁	1	2	yes	—
114	C₂×C₄	C₂²×Dic₆₁	1	4	yes	—
115	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
116	C₈	—	122	8	no	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	4	no	—
119	C₂²	—	2	2	no	—
120	C₂²	—	2	8	no	—
121	C₂²	C₂³×Dic₆₁	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	2	4	no	—
124	C₄	—	2	2	no	—
125	C₄	—	2	2	no	—
126	C₄	#14519822	1	2	yes	—
127	C₂	—	4	1	no	—
128	C₂	—	4	4	no	—
129	C₂	#16840237	1	1	yes	characteristic
130	C₂	#16840529	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16839834

C₂

yes

maximal, complemented

#16839819

C₂

yes

maximal, complemented

#16839729

C₂

yes

maximal, complemented

C₆₁×#532

C₂

yes

maximal, characteristic

C₂×#11758194

C₂²

yes

—

C₂²×#92418

C₂²

yes

—

C₄⋊#92418

C₂²

yes

—

C₂²⋊#92418

C₂²

yes

—

C₂²⋊#92418

C₂²

yes

—

#14519688

C₂²

yes

—

C₂×#41×C₆₁

C₂²

yes

—

C₄⋊Q₈×C₆₁

C₄

yes

characteristic

#127×C₆₁

C₄

yes

characteristic

#124×C₆₁

C₄

yes

—

#121×C₆₁

C₄

yes

—

C₄×D₄×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

#41×C₆₁

—

#41×C₆₁

—

C₂×Q₈×C₆₁

C₂×C₄

yes

—

C₂×D₄×C₆₁

C₂×C₄

yes

—

C₂×D₄×C₆₁

C₂×C₄

yes

—

C₂×D₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×C₄×C₆₁

C₂³

yes

—

C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₂²⋊C₄×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

C₂³

yes

—

#11758194

—

C₂×#92418

—

C₂×#92418

C₂³

yes

—

C₂³×C₆₁

C₂²×C₄

yes

—

C₂³×C₆₁

C₂²×C₄

yes

characteristic

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂³×C₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

H(5,𝔽₂)

yes

—

#2980

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

#518

yes

characteristic

C₂×C₆₁

#568

yes

—

#532

D₆₁

yes

—

—

—

C₆₁

#2980

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₆₁

yes

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

Dic₆₁

yes

characteristic

#127

Dic₆₁

yes

characteristic

#124

Dic₆₁

yes

—

#121

Dic₆₁

yes

—

C₄×D₄

C₂×D₆₁

yes

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

#41

—

#41

—

C₂×Q₈

C₂×Dic₆₁

yes

—

C₂×D₄

C₂×Dic₆₁

yes

—

C₂×D₄

C₂×Dic₆₁

yes

—

C₂×D₄

C₂×Dic₆₁

yes

—

C₂²×C₄

C₂×Dic₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

C₂²×C₄

C₂²×D₆₁

yes

—

M₁₆

—

244

—

C₂×C₈

—

C₂×C₈

—

244

—

C₄⋊C₄

C₂×Dic₆₁

yes

—

C₂²⋊C₄

C₂×Dic₆₁

yes

—

C₄²

C₂×Dic₆₁

yes

characteristic

C₄²

C₂×Dic₆₁

yes

characteristic

100

C₄²

C₂²×D₆₁

yes

—

101

C₂³

C₂²×Dic₆₁

yes

—

102

C₂³

C₂²×Dic₆₁

yes

characteristic

103

Q₈

—

104

D₄

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂²×Dic₆₁

yes

characteristic

112

C₂×C₄

C₂²×Dic₆₁

yes

—

113

C₂×C₄

C₂²×Dic₆₁

yes

—

114

C₂×C₄

C₂²×Dic₆₁

yes

—

115

C₂×C₄

C₂³×D₆₁

yes

characteristic

116

C₈

—

122

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂³×Dic₆₁

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#14519822

yes

—

127

C₂

—

128

C₂

—

129

C₂

#16840237

yes

characteristic

130

C₂

#16840529

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₆₁
commutator, socle	C₂²×C₆₁	C₂³×C₄
Fitting	C₆₁×#532	C₂
hypercenter	#532	D₆₁
hypocenter	C₆₁	#2980
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ #532

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₆₁

commutator, socle

C₂²×C₆₁

C₂³×C₄

Fitting

C₆₁×#532

C₂

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂×C₄ ◁ #532

N	H ≅ G/N	type
1. #16839834	C₂	semidirect
2. #16839819	C₂	semidirect
3. #16839729	C₂	semidirect
70. C₆₁	#2980	semidirect

H ≅ G/N

type

1. #16839834

C₂

semidirect

2. #16839819

C₂

semidirect

3. #16839729

C₂

semidirect

70. C₆₁

#2980

semidirect

Subgroups — #21454890

Conjugacy classes of subgroups

Complements