Group #21454071 — subgroups

Subgroups — #21454071

Order 7808.

580 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 125 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	#16839675	C₂	1	6	yes	maximal, complemented
2	#16839674	C₂	1	3	yes	maximal, complemented
3	#16839673	C₂	1	2	yes	maximal, complemented
4	#16839617	C₂	1	3	yes	maximal
5	C₆₁×#393	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁴×D₆₁	C₂²	1	1	yes	characteristic
7	C₂²×Dic₁₂₂	C₂²	1	3	yes	complemented
8	C₂×C₂²⋊Dic₆₁	C₂²	1	3	yes	—
9	C₂×#11758198	C₂²	1	6	yes	—
10	C₂×#11758198	C₂²	1	6	yes	—
11	#14519684	C₂²	1	6	yes	complemented
12	#14519684	C₂²	1	3	yes	complemented
13	C₂²×Q₈×C₆₁	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₂²⋊C₄×C₆₁	C₂²	1	3	yes	—
15	#95×C₆₁	C₂²	1	3	yes	complemented
16	C₂⁴×C₆₁	C₂³	1	1	yes	characteristic
17	C₂³×D₆₁	—	4	3	no	—
18	C₂³×D₆₁	—	4	3	no	—
19	C₂³×D₆₁	Q₈	1	4	yes	complemented
20	C₂³×D₆₁	C₂³	1	2	yes	—
21	C₂×Q₈×C₆₁	—	4	3	no	—
22	C₂²×C₄×C₆₁	C₂³	1	3	yes	—
23	C₂²×C₄×C₆₁	C₂³	1	3	yes	—
24	C₂²×Dic₆₁	C₂³	1	6	yes	—
25	C₂×Dic₁₂₂	—	4	3	no	—
26	C₂×Dic₁₂₂	—	4	6	no	—
27	C₂²⋊C₄×C₆₁	—	4	3	no	—
28	C₂²⋊Dic₆₁	—	4	3	no	—
29	#11758198	—	4	6	no	—
30	#11758198	—	4	6	no	—
31	C₂³×C₆₁	—	2	3	no	—
32	C₂³×C₆₁	—	2	3	no	—
33	C₂³×C₆₁	C₂×Q₈	1	2	yes	—
34	C₂³×C₆₁	C₂⁴	1	1	yes	characteristic
35	C₂²×D₆₁	—	2	6	no	—
36	C₂²×D₆₁	—	2	6	no	—
37	C₂²×D₆₁	—	4	12	no	—
38	C₂²×D₆₁	—	4	12	no	—
39	C₂²×D₆₁	—	4	4	no	—
40	C₂²×D₆₁	C₂×Q₈	1	4	yes	—
41	Q₈×C₆₁	—	4	4	no	—
42	C₂×C₄×C₆₁	—	2	6	no	—
43	C₂×C₄×C₆₁	—	2	6	no	—
44	C₂×C₄×C₆₁	—	2	3	no	—
45	C₂×C₄×C₆₁	—	2	3	no	—
46	C₂×Dic₆₁	—	2	6	no	—
47	C₂×Dic₆₁	—	2	6	no	—
48	C₂×Dic₆₁	—	2	6	no	—
49	Dic₁₂₂	—	4	12	no	—
50	C₂²×C₆₁	—	2	6	no	—
51	C₂²×C₆₁	—	4	3	no	—
52	C₂²×C₆₁	—	4	1	no	—
53	C₂²×C₆₁	C₂²×Q₈	1	1	yes	characteristic
54	C₂²×C₆₁	H(5,𝔽₂)	1	3	yes	—
55	C₂²×C₆₁	H(5,𝔽₂)	1	3	yes	—
56	C₂×D₆₁	—	2	12	no	—
57	C₂×D₆₁	—	4	16	no	—
58	C₂×D₆₁	—	4	6	no	—
59	C₂×D₆₁	—	4	2	no	—
60	C₄×C₆₁	—	4	3	no	—
61	C₄×C₆₁	—	4	3	no	—
62	Dic₆₁	—	4	6	no	—
63	#2840	—	61	1	no	maximal, Sylow
64	C₂×C₆₁	—	4	2	no	—
65	C₂×C₆₁	#543	1	3	yes	—
66	C₂×C₆₁	H(3,𝔽₄)	1	3	yes	—
67	C₂×C₆₁	H(3,𝔽₄)	1	1	yes	characteristic
68	D₆₁	—	4	4	no	—
69	#393	—	61	2	no	—
70	#393	—	61	3	no	—
71	#393	—	61	6	no	—
72	#393	D₆₁	1	1	yes	characteristic, complemented
73	#379	—	61	3	no	—
74	C₆₁	#2840	1	1	yes	characteristic, complemented, Sylow
75	C₂⁵	—	61	1	no	—
76	C₂²×Q₈	—	61	3	no	—
77	C₂²×Q₈	C₂×D₆₁	1	1	yes	characteristic, complemented
78	C₂×C₂²⋊C₄	—	61	3	no	—
79	C₂×C₂²⋊C₄	—	61	6	no	—
80	C₂×C₂²⋊C₄	—	61	6	no	—
81	C₂×C₂²⋊C₄	C₂×D₆₁	1	3	yes	—
82	#95	—	61	6	no	—
83	#95	—	61	3	no	—
84	#95	C₂×D₆₁	1	3	yes	complemented
85	C₂⁴	—	61	2	no	—
86	C₂⁴	—	61	4	no	—
87	C₂⁴	—	244	3	no	—
88	C₂⁴	—	244	3	no	—
89	C₂⁴	C₂²×D₆₁	1	1	yes	characteristic
90	C₂×Q₈	—	4	3	no	—
91	C₂×Q₈	—	244	3	no	—
92	C₂×Q₈	—	244	6	no	—
93	C₂²×C₄	—	61	6	no	—
94	C₂²×C₄	C₂²×D₆₁	1	3	yes	—
95	C₂²×C₄	C₂²×D₆₁	1	3	yes	—
96	C₂²⋊C₄	—	4	3	no	—
97	C₂²⋊C₄	—	244	6	no	—
98	C₂²⋊C₄	—	244	6	no	—
99	C₂²⋊C₄	—	244	3	no	—
100	C₂³	—	2	3	no	—
101	C₂³	—	2	3	no	—
102	C₂³	—	61	4	no	—
103	C₂³	—	122	6	no	—
104	C₂³	—	122	6	no	—
105	C₂³	—	244	12	no	—
106	C₂³	—	244	12	no	—
107	C₂³	—	244	4	no	—
108	C₂³	Q₈×D₆₁	1	2	yes	—
109	C₂³	C₂³×D₆₁	1	1	yes	characteristic
110	Q₈	—	4	4	no	—
111	Q₈	—	244	12	no	—
112	C₂×C₄	—	2	3	no	—
113	C₂×C₄	—	2	3	no	—
114	C₂×C₄	—	2	6	no	—
115	C₂×C₄	—	2	6	no	—
116	C₂×C₄	—	122	6	no	—
117	C₂×C₄	—	122	6	no	—
118	C₂×C₄	—	122	6	no	—
119	C₂²	—	2	6	no	—
120	C₂²	—	4	3	no	—
121	C₂²	—	4	1	no	—
122	C₂²	—	122	12	no	—
123	C₂²	—	244	16	no	—
124	C₂²	—	244	6	no	—
125	C₂²	—	244	2	no	—
126	C₂²	C₂×Q₈×D₆₁	1	1	yes	characteristic
127	C₂²	#14519822	1	3	yes	—
128	C₂²	#14519823	1	3	yes	—
129	C₄	—	4	3	no	—
130	C₄	—	4	3	no	—
131	C₄	—	244	6	no	—
132	C₂	—	4	2	no	—
133	C₂	—	244	4	no	—
134	C₂	#16840387	1	3	yes	—
135	C₂	#16840504	1	3	yes	—
136	C₂	#16840505	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16839675

C₂

yes

maximal, complemented

#16839674

C₂

yes

maximal, complemented

#16839673

C₂

yes

maximal, complemented

#16839617

C₂

yes

maximal

C₆₁×#393

C₂

yes

maximal, characteristic, complemented

C₂⁴×D₆₁

C₂²

yes

characteristic

C₂²×Dic₁₂₂

C₂²

yes

complemented

C₂×C₂²⋊Dic₆₁

C₂²

yes

—

C₂×#11758198

C₂²

yes

—

C₂×#11758198

C₂²

yes

—

#14519684

C₂²

yes

complemented

#14519684

C₂²

yes

complemented

C₂²×Q₈×C₆₁

C₂²

yes

characteristic, complemented

C₂×C₂²⋊C₄×C₆₁

C₂²

yes

—

#95×C₆₁

C₂²

yes

complemented

C₂⁴×C₆₁

C₂³

yes

characteristic

C₂³×D₆₁

—

C₂³×D₆₁

—

C₂³×D₆₁

Q₈

yes

complemented

C₂³×D₆₁

C₂³

yes

—

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

C₂³

yes

—

C₂²×C₄×C₆₁

C₂³

yes

—

C₂²×Dic₆₁

C₂³

yes

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂²⋊C₄×C₆₁

—

C₂²⋊Dic₆₁

—

#11758198

—

#11758198

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

C₂×Q₈

yes

—

C₂³×C₆₁

C₂⁴

yes

characteristic

C₂²×D₆₁

—

C₂²×D₆₁

—

C₂²×D₆₁

—

C₂²×D₆₁

—

C₂²×D₆₁

—

C₂²×D₆₁

C₂×Q₈

yes

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂²×Q₈

yes

characteristic

C₂²×C₆₁

H(5,𝔽₂)

yes

—

C₂²×C₆₁

H(5,𝔽₂)

yes

—

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

Dic₆₁

—

#2840

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

#543

yes

—

C₂×C₆₁

H(3,𝔽₄)

yes

—

C₂×C₆₁

H(3,𝔽₄)

yes

characteristic

D₆₁

—

—

—

—

D₆₁

yes

characteristic, complemented

#379

—

C₆₁

#2840

yes

characteristic, complemented, Sylow

C₂⁵

—

C₂²×Q₈

—

C₂²×Q₈

C₂×D₆₁

yes

characteristic, complemented

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₆₁

yes

—

#95

—

#95

—

#95

C₂×D₆₁

yes

complemented

C₂⁴

—

C₂⁴

—

C₂⁴

—

244

—

C₂⁴

—

244

—

C₂⁴

C₂²×D₆₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

244

—

C₂×Q₈

—

244

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

244

—

C₂²⋊C₄

—

244

—

C₂²⋊C₄

—

244

—

100

C₂³

—

101

C₂³

—

102

C₂³

—

103

C₂³

—

122

—

104

C₂³

—

122

—

105

C₂³

—

244

—

106

C₂³

—

244

—

107

C₂³

—

244

—

108

C₂³

Q₈×D₆₁

yes

—

109

C₂³

C₂³×D₆₁

yes

characteristic

110

Q₈

—

111

Q₈

—

244

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

122

—

117

C₂×C₄

—

122

—

118

C₂×C₄

—

122

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

122

—

123

C₂²

—

244

—

124

C₂²

—

244

—

125

C₂²

—

244

—

126

C₂²

C₂×Q₈×D₆₁

yes

characteristic

127

C₂²

#14519822

yes

—

128

C₂²

#14519823

yes

—

129

C₄

—

130

C₄

—

131

C₄

—

244

—

132

C₂

—

133

C₂

—

244

—

134

C₂

#16840387

yes

—

135

C₂

#16840504

yes

—

136

C₂

#16840505

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₁
commutator, socle	C₂³×C₆₁	C₂⁴
Fitting	C₆₁×#393	C₂
hypercenter	#393	D₆₁
hypocenter	C₆₁	#2840
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ #393

subgroup

G/H

center, Frattini

C₂³

C₂³×D₆₁

commutator, socle

C₂³×C₆₁

C₂⁴

Fitting

C₆₁×#393

C₂

D₆₁

C₆₁

G ▷ C₂³×C₆₁ ▷ C₆₁

C₁ ◁ C₂³ ◁ #393

N	H ≅ G/N	type
1. #16839675	C₂	semidirect
2. #16839674	C₂	semidirect
3. #16839673	C₂	semidirect
5. C₆₁×#393	C₂	semidirect
7. C₂²×Dic₁₂₂	C₂²	semidirect
11. #14519684	C₂²	semidirect
12. #14519684	C₂²	semidirect
13. C₂²×Q₈×C₆₁	C₂²	semidirect
15. #95×C₆₁	C₂²	semidirect
19. C₂³×D₆₁	Q₈	semidirect
72. #393	D₆₁	semidirect
74. C₆₁	#2840	semidirect
77. C₂²×Q₈	C₂×D₆₁	semidirect
84. #95	C₂×D₆₁	semidirect

H ≅ G/N

type

1. #16839675

C₂

semidirect

2. #16839674

C₂

semidirect

3. #16839673

C₂

semidirect

5. C₆₁×#393

C₂

semidirect

7. C₂²×Dic₁₂₂

C₂²

semidirect

11. #14519684

C₂²

semidirect

12. #14519684

C₂²

semidirect

13. C₂²×Q₈×C₆₁

C₂²

semidirect

15. #95×C₆₁

C₂²

semidirect

19. C₂³×D₆₁

Q₈

semidirect

72. #393

D₆₁

semidirect

74. C₆₁

#2840

semidirect

77. C₂²×Q₈

C₂×D₆₁

semidirect

84. #95

C₂×D₆₁

semidirect

Subgroups — #21454071

Conjugacy classes of subgroups

Complements