Group #21454106 — subgroups

Subgroups — #21454106

Order 7808.

1068 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 237 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	C₂×#14519756	C₂	1	3	yes	maximal, complemented
2	#16839635	C₂	1	6	yes	maximal, complemented
3	#16839622	C₂	1	3	yes	maximal, complemented
4	#16839618	C₂	1	2	yes	maximal
5	C₂⁴×C₄×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₂₂	C₂²	1	1	yes	characteristic, complemented
7	#14519756	—	2	12	no	—
8	C₂×C₄⋊Dic₆₁	C₂²	1	3	yes	complemented
9	C₂×#11758196	C₂²	1	6	yes	complemented
10	C₂×C₂²⋊Dic₆₁	C₂²	1	6	yes	—
11	C₂×C₂²⋊Dic₆₁	C₂²	1	6	yes	—
12	#14519684	C₂²	1	6	yes	complemented
13	C₂⁵×C₆₁	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₆₁	—	2	12	no	—
15	C₂³×C₄×C₆₁	C₂²	1	3	yes	—
16	C₂³×C₄×C₆₁	C₂²	1	3	yes	—
17	C₂⁴×C₆₁	—	2	6	no	—
18	C₂⁴×C₆₁	—	2	6	no	—
19	C₂⁴×C₆₁	Q₈	1	4	yes	complemented
20	C₂⁴×C₆₁	C₂³	1	3	yes	—
21	C₂²×C₄×C₆₁	—	2	32	no	—
22	C₂²×C₄×C₆₁	—	2	6	no	—
23	C₂²×C₄×C₆₁	—	2	12	no	—
24	C₂²×C₄×C₆₁	—	2	12	no	—
25	C₂²×C₄×C₆₁	D₄	1	12	yes	—
26	C₂²×C₄×C₆₁	C₂³	1	3	yes	—
27	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
28	C₂²×Dic₆₁	C₂³	1	2	yes	—
29	C₂²×Dic₆₁	C₂³	1	6	yes	—
30	C₂×Dic₁₂₂	—	2	6	no	—
31	C₂²⋊Dic₆₁	—	2	12	no	—
32	C₂²⋊Dic₆₁	—	2	12	no	—
33	C₄⋊Dic₆₁	—	2	6	no	—
34	#11758196	—	2	12	no	—
35	C₂³×C₆₁	—	2	8	no	—
36	C₂³×C₆₁	—	2	24	no	—
37	C₂³×C₆₁	—	2	24	no	—
38	C₂³×C₆₁	—	2	6	no	—
39	C₂³×C₆₁	—	2	6	no	—
40	C₂³×C₆₁	C₂×D₄	1	6	yes	—
41	C₂³×C₆₁	C₂×Q₈	1	6	yes	—
42	C₂³×C₆₁	#41	1	6	yes	—
43	C₂³×C₆₁	C₂⁴	1	1	yes	characteristic
44	C₂×C₄×C₆₁	—	2	24	no	—
45	C₂×C₄×C₆₁	—	2	6	no	—
46	C₂×C₄×C₆₁	—	2	24	no	—
47	C₂×C₄×C₆₁	C₂×D₄	1	6	yes	—
48	C₂×C₄×C₆₁	C₂×D₄	1	6	yes	—
49	C₂×Dic₆₁	—	2	6	no	—
50	C₂×Dic₆₁	—	2	6	no	—
51	C₂×Dic₆₁	—	2	12	no	—
52	Dic₁₂₂	—	4	4	no	—
53	C₂²×C₆₁	—	2	32	no	—
54	C₂²×C₆₁	—	2	6	no	—
55	C₂²×C₆₁	—	2	12	no	—
56	C₂²×C₆₁	—	2	6	no	—
57	C₂²×C₆₁	—	2	12	no	—
58	C₂²×C₆₁	C₂²⋊Q₈	1	12	yes	complemented
59	C₂²×C₆₁	C₂²×D₄	1	3	yes	—
60	C₂²×C₆₁	C₂²×Q₈	1	1	yes	characteristic
61	C₂²×C₆₁	C₂×#41	1	3	yes	—
62	C₄×C₆₁	—	2	6	no	—
63	C₄×C₆₁	#120	1	4	yes	—
64	Dic₆₁	—	4	2	no	—
65	Dic₆₁	—	4	6	no	—
66	#2848	—	61	1	no	maximal, Sylow
67	C₂×C₆₁	—	2	12	no	—
68	C₂×C₆₁	C₂×#120	1	1	yes	characteristic
69	C₂×C₆₁	C₂×C₂²⋊Q₈	1	3	yes	—
70	C₂×C₆₁	#525	1	3	yes	—
71	C₂⁴×C₄	D₆₁	1	1	yes	characteristic
72	C₂×C₂²⋊Q₈	—	61	3	no	—
73	#385	—	61	6	no	—
74	#380	—	61	3	no	—
75	#379	—	61	2	no	—
76	C₆₁	#2848	1	1	yes	characteristic, complemented, Sylow
77	C₂⁵	C₂×D₆₁	1	1	yes	characteristic
78	C₂²×Q₈	—	61	1	no	—
79	C₂³×C₄	—	2	12	no	—
80	C₂³×C₄	C₂×D₆₁	1	3	yes	—
81	C₂³×C₄	C₂×D₆₁	1	3	yes	—
82	C₂²⋊Q₈	—	122	12	no	—
83	C₂×C₄⋊C₄	—	61	3	no	—
84	C₂×C₄⋊C₄	—	61	6	no	—
85	C₂×C₂²⋊C₄	—	61	6	no	—
86	C₂×C₂²⋊C₄	—	61	6	no	—
87	#95	—	61	6	no	—
88	C₂⁴	—	2	6	no	—
89	C₂⁴	—	2	6	no	—
90	C₂⁴	Dic₁₂₂	1	4	yes	complemented
91	C₂⁴	C₂²×D₆₁	1	3	yes	—
92	C₂×Q₈	—	122	6	no	—
93	C₂²×C₄	—	2	32	no	—
94	C₂²×C₄	—	2	6	no	—
95	C₂²×C₄	—	2	12	no	—
96	C₂²×C₄	—	2	12	no	—
97	C₂²×C₄	—	61	6	no	—
98	C₂²×C₄	—	61	2	no	—
99	C₂²×C₄	C₂²⋊D₆₁	1	12	yes	—
100	C₂²×C₄	C₂²×D₆₁	1	3	yes	—
101	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
102	C₄⋊C₄	—	122	12	no	—
103	C₄⋊C₄	—	122	6	no	—
104	C₂²⋊C₄	—	122	12	no	—
105	C₂²⋊C₄	—	122	12	no	—
106	C₂³	—	2	8	no	—
107	C₂³	—	2	24	no	—
108	C₂³	—	2	6	no	—
109	C₂³	—	2	6	no	—
110	C₂³	—	2	24	no	—
111	C₂³	C₂×Dic₁₂₂	1	6	yes	—
112	C₂³	#11758222	1	6	yes	—
113	C₂³	C₂×C₂²⋊D₆₁	1	6	yes	—
114	C₂³	C₂³×D₆₁	1	1	yes	characteristic
115	Q₈	—	244	4	no	—
116	C₂×C₄	—	2	24	no	—
117	C₂×C₄	—	2	6	no	—
118	C₂×C₄	—	2	24	no	—
119	C₂×C₄	—	122	6	no	—
120	C₂×C₄	—	122	6	no	—
121	C₂×C₄	—	122	12	no	—
122	C₂×C₄	C₂×C₂²⋊D₆₁	1	6	yes	—
123	C₂×C₄	C₂×C₂²⋊D₆₁	1	6	yes	—
124	C₂²	—	2	32	no	—
125	C₂²	—	2	6	no	—
126	C₂²	—	2	12	no	—
127	C₂²	—	2	12	no	—
128	C₂²	—	2	6	no	—
129	C₂²	#14519756	1	12	yes	complemented
130	C₂²	C₂²×C₂²⋊D₆₁	1	3	yes	—
131	C₂²	C₂²×Dic₁₂₂	1	1	yes	characteristic
132	C₂²	C₂×#11758222	1	3	yes	—
133	C₄	—	2	6	no	—
134	C₄	—	244	2	no	—
135	C₄	—	244	6	no	—
136	C₄	#14519746	1	4	yes	—
137	C₂	—	2	12	no	—
138	C₂	C₂×#14519746	1	1	yes	characteristic
139	C₂	C₂×#14519756	1	3	yes	—
140	C₂	#16840273	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519756

C₂

yes

maximal, complemented

#16839635

C₂

yes

maximal, complemented

#16839622

C₂

yes

maximal, complemented

#16839618

C₂

yes

maximal

C₂⁴×C₄×C₆₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₂₂

C₂²

yes

characteristic, complemented

#14519756

—

C₂×C₄⋊Dic₆₁

C₂²

yes

complemented

C₂×#11758196

C₂²

yes

complemented

C₂×C₂²⋊Dic₆₁

C₂²

yes

—

C₂×C₂²⋊Dic₆₁

C₂²

yes

—

#14519684

C₂²

yes

complemented

C₂⁵×C₆₁

C₂²

yes

characteristic

C₂³×C₄×C₆₁

—

C₂³×C₄×C₆₁

C₂²

yes

—

C₂³×C₄×C₆₁

C₂²

yes

—

C₂⁴×C₆₁

—

C₂⁴×C₆₁

—

C₂⁴×C₆₁

Q₈

yes

complemented

C₂⁴×C₆₁

C₂³

yes

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

C₂³

yes

—

C₂²×Dic₆₁

C₂³

yes

—

C₂×Dic₁₂₂

—

C₂²⋊Dic₆₁

—

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

#11758196

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

—

C₂³×C₆₁

C₂×D₄

yes

—

C₂³×C₆₁

C₂×Q₈

yes

—

C₂³×C₆₁

#41

yes

—

C₂³×C₆₁

C₂⁴

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂²⋊Q₈

yes

complemented

C₂²×C₆₁

C₂²×D₄

yes

—

C₂²×C₆₁

C₂²×Q₈

yes

characteristic

C₂²×C₆₁

C₂×#41

yes

—

C₄×C₆₁

—

C₄×C₆₁

#120

yes

—

Dic₆₁

—

Dic₆₁

—

#2848

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

C₂×#120

yes

characteristic

C₂×C₆₁

C₂×C₂²⋊Q₈

yes

—

C₂×C₆₁

#525

yes

—

C₂⁴×C₄

D₆₁

yes

characteristic

C₂×C₂²⋊Q₈

—

#385

—

#380

—

#379

—

C₆₁

#2848

yes

characteristic, complemented, Sylow

C₂⁵

C₂×D₆₁

yes

characteristic

C₂²×Q₈

—

C₂³×C₄

—

C₂³×C₄

C₂×D₆₁

yes

—

C₂³×C₄

C₂×D₆₁

yes

—

C₂²⋊Q₈

—

122

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

#95

—

C₂⁴

—

C₂⁴

—

C₂⁴

Dic₁₂₂

yes

complemented

C₂⁴

C₂²×D₆₁

yes

—

C₂×Q₈

—

122

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₆₁

yes

—

100

C₂²×C₄

C₂²×D₆₁

yes

—

101

C₂²×C₄

C₂²×D₆₁

yes

characteristic

102

C₄⋊C₄

—

122

—

103

C₄⋊C₄

—

122

—

104

C₂²⋊C₄

—

122

—

105

C₂²⋊C₄

—

122

—

106

C₂³

—

107

C₂³

—

108

C₂³

—

109

C₂³

—

110

C₂³

—

111

C₂³

C₂×Dic₁₂₂

yes

—

112

C₂³

#11758222

yes

—

113

C₂³

C₂×C₂²⋊D₆₁

yes

—

114

C₂³

C₂³×D₆₁

yes

characteristic

115

Q₈

—

244

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

122

—

120

C₂×C₄

—

122

—

121

C₂×C₄

—

122

—

122

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

123

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

#14519756

yes

complemented

130

C₂²

C₂²×C₂²⋊D₆₁

yes

—

131

C₂²

C₂²×Dic₁₂₂

yes

characteristic

132

C₂²

C₂×#11758222

yes

—

133

C₄

—

134

C₄

—

244

—

135

C₄

—

244

—

136

C₄

#14519746

yes

—

137

C₂

—

138

C₂

C₂×#14519746

yes

characteristic

139

C₂

C₂×#14519756

yes

—

140

C₂

#16840273

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₁
commutator, socle	C₂³×C₆₁	C₂⁴
Fitting	C₂⁴×C₄×C₆₁	C₂
hypercenter	C₂⁴×C₄	D₆₁
hypocenter	C₆₁	#2848
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₆₁

commutator, socle

C₂³×C₆₁

C₂⁴

Fitting

C₂⁴×C₄×C₆₁

C₂

hypercenter

C₂⁴×C₄

D₆₁

C₆₁

G ▷ C₂³×C₆₁ ▷ C₆₁

C₁ ◁ C₂³ ◁ C₂⁴×C₄

N	H ≅ G/N	type
1. C₂×#14519756	C₂	semidirect
2. #16839635	C₂	semidirect
3. #16839622	C₂	semidirect
6. C₂²×Dic₁₂₂	C₂²	semidirect
8. C₂×C₄⋊Dic₆₁	C₂²	semidirect
9. C₂×#11758196	C₂²	semidirect
12. #14519684	C₂²	semidirect
19. C₂⁴×C₆₁	Q₈	semidirect
58. C₂²×C₆₁	C₂²⋊Q₈	semidirect
76. C₆₁	#2848	semidirect
90. C₂⁴	Dic₁₂₂	semidirect
129. C₂²	#14519756	semidirect

H ≅ G/N

type

1. C₂×#14519756

C₂

semidirect

2. #16839635

C₂

semidirect

3. #16839622

C₂

semidirect

6. C₂²×Dic₁₂₂

C₂²

semidirect

8. C₂×C₄⋊Dic₆₁

C₂²

semidirect

9. C₂×#11758196

C₂²

semidirect

12. #14519684

C₂²

semidirect

19. C₂⁴×C₆₁

Q₈

semidirect

58. C₂²×C₆₁

C₂²⋊Q₈

semidirect

76. C₆₁

#2848

semidirect

90. C₂⁴

Dic₁₂₂

semidirect

129. C₂²

#14519756

semidirect

Subgroups — #21454106

Conjugacy classes of subgroups

Complements