Group #21456328 — subgroups

Subgroups — #21456328

Order 7808.

356 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 145 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	#16840317	C₂	1	2	yes	maximal
2	#16840165	C₂	1	2	yes	maximal, complemented
3	#16840162	C₂	1	4	yes	maximal, complemented
4	#16840160	C₂	1	1	yes	maximal, characteristic, complemented
5	#16839797	C₂	1	4	yes	maximal
6	C₄×C₈×D₆₁	C₂	1	1	yes	maximal, characteristic
7	C₆₁×#498	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₆₁	C₂²	1	2	yes	—
9	#14519777	C₂²	1	2	yes	—
10	#14519768	C₂²	1	4	yes	—
11	#14519757	—	2	2	no	—
12	#14519753	—	2	2	no	—
13	Q₈×Dic₆₁	—	2	2	no	—
14	#14519742	—	2	2	no	—
15	#14519735	C₂²	1	4	yes	—
16	C₄²×D₆₁	C₂²	1	1	yes	characteristic
17	#14519710	C₂²	1	8	yes	—
18	#14519709	C₂²	1	4	yes	—
19	C₄×#92418	C₂²	1	1	yes	characteristic
20	C₈×Dic₆₁	C₂²	1	2	yes	—
21	C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
22	#106×C₆₁	C₂²	1	4	yes	—
23	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
24	C₂×Q₈×C₆₁	—	2	2	no	—
25	C₂×C₄×D₆₁	D₄	1	2	yes	—
26	C₂×C₄×D₆₁	C₂³	1	1	yes	characteristic
27	C₂×Dic₁₂₂	—	2	2	no	—
28	C₂×C₈×C₆₁	C₂³	1	2	yes	—
29	C₄⋊C₄×C₆₁	—	2	2	no	—
30	C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
31	C₄²×C₆₁	C₂³	1	1	yes	characteristic
32	#11758198	—	2	4	no	—
33	C₄⋊Dic₆₁	—	2	2	no	—
34	C₄⋊Dic₆₁	—	2	4	no	—
35	C₄⋊Dic₆₁	C₂³	1	4	yes	—
36	#11758196	—	2	4	no	—
37	C₄×Dic₆₁	—	2	4	no	—
38	C₄×Dic₆₁	D₄	1	2	yes	—
39	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
40	C₂×#92418	C₂³	1	2	yes	—
41	C₈×D₆₁	Q₈	1	8	yes	complemented
42	C₂²×D₆₁	C₂×D₄	1	1	yes	characteristic
43	Q₈×C₆₁	—	4	4	no	—
44	C₂×C₄×C₆₁	—	2	4	no	—
45	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
47	C₂×Dic₆₁	—	2	4	no	—
48	C₂×Dic₆₁	C₂×D₄	1	2	yes	—
49	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
50	C₄×D₆₁	—	2	4	no	—
51	C₄×D₆₁	C₂×Q₈	1	4	yes	—
52	Dic₁₂₂	—	4	4	no	—
53	C₈×C₆₁	C₂×Q₈	1	4	yes	—
54	#92418	C₂×Q₈	1	4	yes	—
55	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
56	C₂×D₆₁	—	2	2	no	—
57	C₂×D₆₁	C₄⋊Q₈	1	2	yes	—
58	C₄×C₆₁	—	4	4	no	—
59	C₄×C₆₁	#135	1	4	yes	—
60	C₄×C₆₁	C₂²×Q₈	1	2	yes	—
61	Dic₆₁	—	2	2	no	—
62	Dic₆₁	—	4	4	no	—
63	Dic₆₁	C₄⋊Q₈	1	2	yes	—
64	#3160	—	61	1	no	maximal, Sylow
65	C₂×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₆₁	C₂×#135	1	2	yes	—
67	D₆₁	—	2	2	no	—
68	#533	—	61	2	no	—
69	#500	—	61	2	no	—
70	#499	—	61	4	no	—
71	#498	—	61	1	no	—
72	#498	D₆₁	1	1	yes	characteristic, complemented
73	#427	—	61	4	no	—
74	C₂×C₄×C₈	—	61	1	no	—
75	C₆₁	#3160	1	1	yes	characteristic, complemented, Sylow
76	C₂²×C₈	—	61	2	no	—
77	C₄⋊Q₈	—	61	2	no	—
78	C₄⋊Q₈	C₂×D₆₁	1	2	yes	—
79	#125	—	61	4	no	—
80	C₂²⋊Q₈	—	122	2	no	—
81	C₂²⋊Q₈	—	122	2	no	—
82	C₄×Q₈	—	122	2	no	—
83	C₄×Q₈	—	122	2	no	—
84	#117	—	61	4	no	—
85	C₂×C₄²	—	61	1	no	—
86	#107	—	61	8	no	—
87	#106	—	61	4	no	—
88	#106	C₂×D₆₁	1	4	yes	—
89	C₄×C₈	—	61	2	no	—
90	C₄×C₈	—	61	1	no	—
91	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	122	2	no	—
94	C₂²×C₄	—	61	1	no	—
95	C₂²×C₄	—	61	2	no	—
96	C₂×C₈	—	61	2	no	—
97	C₂×C₈	—	61	8	no	—
98	C₂×C₈	C₂²×D₆₁	1	2	yes	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	61	4	no	—
101	C₄⋊C₄	—	122	4	no	—
102	C₄⋊C₄	—	122	4	no	—
103	C₄⋊C₄	—	122	2	no	—
104	C₄⋊C₄	C₂²×D₆₁	1	4	yes	—
105	C₂²⋊C₄	—	122	4	no	—
106	C₄²	—	61	2	no	—
107	C₄²	—	61	1	no	—
108	C₄²	—	122	4	no	—
109	C₄²	C₂²×D₆₁	1	1	yes	characteristic
110	C₂³	—	61	1	no	—
111	Q₈	—	4	4	no	—
112	Q₈	—	244	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	61	1	no	—
115	C₂×C₄	—	61	4	no	—
116	C₂×C₄	—	61	2	no	—
117	C₂×C₄	—	122	4	no	—
118	C₂×C₄	—	122	4	no	—
119	C₂×C₄	D₄×D₆₁	1	2	yes	—
120	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
121	C₈	—	61	4	no	—
122	C₈	Q₈×D₆₁	1	4	yes	—
123	C₂²	—	61	2	no	—
124	C₂²	—	122	2	no	—
125	C₂²	C₂×D₄×D₆₁	1	1	yes	characteristic
126	C₄	—	4	4	no	—
127	C₄	—	61	2	no	—
128	C₄	—	122	2	no	—
129	C₄	—	244	4	no	—
130	C₄	#14519798	1	4	yes	—
131	C₄	C₂×Q₈×D₆₁	1	2	yes	—
132	C₂	—	122	2	no	—
133	C₂	C₄⋊Q₈×D₆₁	1	1	yes	characteristic
134	C₂	C₂×#14519798	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16840317

C₂

yes

maximal

#16840165

C₂

yes

maximal, complemented

#16840162

C₂

yes

maximal, complemented

#16840160

C₂

yes

maximal, characteristic, complemented

#16839797

C₂

yes

maximal

C₄×C₈×D₆₁

C₂

yes

maximal, characteristic

C₆₁×#498

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₆₁

C₂²

yes

—

#14519777

C₂²

yes

—

#14519768

C₂²

yes

—

#14519757

—

#14519753

—

Q₈×Dic₆₁

—

#14519742

—

#14519735

C₂²

yes

—

C₄²×D₆₁

C₂²

yes

characteristic

#14519710

C₂²

yes

—

#14519709

C₂²

yes

—

C₄×#92418

C₂²

yes

characteristic

C₈×Dic₆₁

C₂²

yes

—

C₄⋊Q₈×C₆₁

C₂²

yes

—

#106×C₆₁

C₂²

yes

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×C₄×D₆₁

D₄

yes

—

C₂×C₄×D₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

C₂³

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂³

yes

—

C₄²×C₆₁

C₂³

yes

characteristic

#11758198

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂³

yes

—

#11758196

—

C₄×Dic₆₁

—

C₄×Dic₆₁

D₄

yes

—

C₄×Dic₆₁

C₂³

yes

characteristic

C₂×#92418

C₂³

yes

—

C₈×D₆₁

Q₈

yes

complemented

C₂²×D₆₁

C₂×D₄

yes

characteristic

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂×D₄

yes

—

C₂×Dic₆₁

C₂×D₄

yes

characteristic

C₄×D₆₁

—

C₄×D₆₁

C₂×Q₈

yes

—

Dic₁₂₂

—

C₈×C₆₁

C₂×Q₈

yes

—

#92418

C₂×Q₈

yes

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

C₄⋊Q₈

yes

—

C₄×C₆₁

—

C₄×C₆₁

#135

yes

—

C₄×C₆₁

C₂²×Q₈

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

C₄⋊Q₈

yes

—

#3160

—

maximal, Sylow

C₂×C₆₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₆₁

C₂×#135

yes

—

D₆₁

—

—

—

—

—

D₆₁

yes

characteristic, complemented

#427

—

C₂×C₄×C₈

—

C₆₁

#3160

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₆₁

yes

—

#125

—

C₂²⋊Q₈

—

122

—

C₂²⋊Q₈

—

122

—

C₄×Q₈

—

122

—

C₄×Q₈

—

122

—

#117

—

C₂×C₄²

—

#107

—

#106

—

#106

C₂×D₆₁

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₆₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

122

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×D₆₁

yes

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

122

—

102

C₄⋊C₄

—

122

—

103

C₄⋊C₄

—

122

—

104

C₄⋊C₄

C₂²×D₆₁

yes

—

105

C₂²⋊C₄

—

122

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

122

—

109

C₄²

C₂²×D₆₁

yes

characteristic

110

C₂³

—

111

Q₈

—

112

Q₈

—

244

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

122

—

118

C₂×C₄

—

122

—

119

C₂×C₄

D₄×D₆₁

yes

—

120

C₂×C₄

C₂³×D₆₁

yes

characteristic

121

C₈

—

122

C₈

Q₈×D₆₁

yes

—

123

C₂²

—

124

C₂²

—

122

—

125

C₂²

C₂×D₄×D₆₁

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

122

—

129

C₄

—

244

—

130

C₄

#14519798

yes

—

131

C₄

C₂×Q₈×D₆₁

yes

—

132

C₂

—

122

—

133

C₂

C₄⋊Q₈×D₆₁

yes

characteristic

134

C₂

C₂×#14519798

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₆₁×#498	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	#498	D₆₁
hypocenter	C₆₁	#3160
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

C₂×D₄×D₆₁

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₆₁×#498

C₂

socle

C₂²×C₆₁

C₂²×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
2. #16840165	C₂	semidirect
3. #16840162	C₂	semidirect
4. #16840160	C₂	semidirect
7. C₆₁×#498	C₂	semidirect
41. C₈×D₆₁	Q₈	semidirect
72. #498	D₆₁	semidirect
75. C₆₁	#3160	semidirect

H ≅ G/N

type

2. #16840165

C₂

semidirect

3. #16840162

C₂

semidirect

4. #16840160

C₂

semidirect

7. C₆₁×#498

C₂

semidirect

41. C₈×D₆₁

Q₈

semidirect

72. #498

D₆₁

semidirect

75. C₆₁

#3160

semidirect

Subgroups — #21456328

Conjugacy classes of subgroups

Complements