Group #21456342 — subgroups

Subgroups — #21456342

Order 7808.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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	#16840315	C₂	1	2	yes	maximal
2	#16840170	C₂	1	4	yes	maximal, complemented
3	#16840169	C₂	1	4	yes	maximal, complemented
4	#16839800	C₂	1	1	yes	maximal, characteristic
5	#16839798	C₂	1	2	yes	maximal
6	C₂×#14519687	C₂	1	1	yes	maximal, characteristic
7	C₆₁×#427	C₂	1	1	yes	maximal, characteristic
8	C₂²×#92418	C₂²	1	1	yes	characteristic
9	#14519776	C₂²	1	4	yes	—
10	#14519768	C₂²	1	4	yes	—
11	#14519754	—	2	4	no	—
12	C₄×Dic₁₂₂	—	2	4	no	—
13	#14519736	C₂²	1	2	yes	—
14	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
15	#14519711	C₂²	1	2	yes	—
16	#14519710	C₂²	1	4	yes	—
17	#14519709	C₂²	1	2	yes	—
18	#14519708	C₂²	1	4	yes	—
19	#14519687	C₂²	1	2	yes	—
20	#14519687	C₂²	1	2	yes	—
21	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
22	#117×C₆₁	C₂²	1	2	yes	—
23	#107×C₆₁	C₂²	1	2	yes	—
24	#106×C₆₁	C₂²	1	2	yes	—
25	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
26	C₂²×Dic₆₁	D₄	1	2	yes	—
27	C₂×Dic₁₂₂	—	2	4	no	—
28	C₂×C₈×C₆₁	Q₈	1	4	yes	complemented
29	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
30	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
31	C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
32	C₂²⋊C₄×C₆₁	—	2	2	no	—
33	C₄²×C₆₁	—	2	2	no	—
34	C₂²⋊Dic₆₁	—	2	2	no	—
35	C₄⋊Dic₆₁	—	2	4	no	—
36	C₄⋊Dic₆₁	C₂³	1	4	yes	—
37	#11758196	—	2	4	no	—
38	#11758196	—	2	4	no	—
39	C₄×Dic₆₁	—	2	2	no	—
40	C₄×Dic₆₁	D₄	1	2	yes	—
41	C₄×Dic₆₁	C₂³	1	2	yes	—
42	C₂×#92418	Q₈	1	4	yes	complemented
43	C₂×#92418	C₂³	1	2	yes	—
44	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₆₁	—	2	4	no	—
46	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
48	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
49	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₆₁	—	2	4	no	—
51	C₂×Dic₆₁	—	4	2	no	—
52	C₂×Dic₆₁	C₂×D₄	1	4	yes	—
53	Dic₁₂₂	—	4	8	no	—
54	C₈×C₆₁	C₂×Q₈	1	2	yes	—
55	C₈×C₆₁	C₂×Q₈	1	2	yes	—
56	#92418	C₂×Q₈	1	4	yes	—
57	C₂²×C₆₁	—	2	2	no	—
58	C₂²×C₆₁	C₄⋊Q₈	1	2	yes	—
59	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
60	C₄×C₆₁	—	4	4	no	—
61	C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
62	C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
63	C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
64	Dic₆₁	—	2	4	no	—
65	Dic₆₁	—	4	4	no	—
66	#3162	—	61	1	no	maximal, Sylow
67	C₂×C₆₁	—	2	2	no	—
68	C₂×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
69	C₂×C₆₁	#575	1	2	yes	—
70	#533	—	61	2	no	—
71	#501	—	61	4	no	—
72	#501	—	61	4	no	—
73	#427	—	61	1	no	—
74	#427	—	61	2	no	—
75	#427	D₆₁	1	1	yes	characteristic
76	C₂×#97	—	61	1	no	—
77	C₆₁	#3162	1	1	yes	characteristic, complemented, Sylow
78	C₂²×C₈	—	61	1	no	—
79	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
80	C₄⋊Q₈	—	61	4	no	—
81	#125	—	61	4	no	—
82	C₂²⋊Q₈	—	122	4	no	—
83	C₄×Q₈	—	122	4	no	—
84	#117	—	61	2	no	—
85	#117	C₂×D₆₁	1	2	yes	—
86	C₂×C₄²	—	61	1	no	—
87	#107	—	61	2	no	—
88	#107	—	61	4	no	—
89	#107	C₂×D₆₁	1	2	yes	—
90	#106	—	61	2	no	—
91	#106	—	61	4	no	—
92	#106	C₂×D₆₁	1	2	yes	—
93	#97	—	61	2	no	—
94	#97	—	61	2	no	—
95	C₂×Q₈	—	122	4	no	—
96	C₂²×C₄	—	61	2	no	—
97	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
98	C₂×C₈	—	61	4	no	—
99	C₂×C₈	—	61	2	no	—
100	C₂×C₈	Dic₁₂₂	1	4	yes	complemented
101	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
102	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
103	C₄⋊C₄	—	61	4	no	—
104	C₄⋊C₄	—	122	4	no	—
105	C₄⋊C₄	—	122	4	no	—
106	C₄⋊C₄	—	122	4	no	—
107	C₄⋊C₄	C₂²×D₆₁	1	4	yes	—
108	C₂²⋊C₄	—	2	2	no	—
109	C₂²⋊C₄	—	122	2	no	—
110	C₄²	—	2	2	no	—
111	C₄²	—	61	2	no	—
112	C₄²	—	61	2	no	—
113	C₄²	—	122	2	no	—
114	C₂³	D₄×D₆₁	1	1	yes	characteristic
115	Q₈	—	244	8	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	61	4	no	—
118	C₂×C₄	—	122	4	no	—
119	C₂×C₄	—	244	2	no	—
120	C₂×C₄	C₂×Dic₁₂₂	1	2	yes	—
121	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
122	C₂×C₄	Q₈×D₆₁	1	2	yes	—
123	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
124	C₈	—	61	4	no	—
125	C₈	C₂×Dic₁₂₂	1	2	yes	—
126	C₈	C₂×Dic₁₂₂	1	2	yes	—
127	C₂²	—	2	2	no	—
128	C₂²	#14519776	1	2	yes	—
129	C₂²	C₂×D₄×D₆₁	1	1	yes	characteristic
130	C₄	—	4	4	no	—
131	C₄	—	122	4	no	—
132	C₄	—	244	4	no	—
133	C₄	#14519776	1	2	yes	—
134	C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
135	C₄	C₂²×Dic₁₂₂	1	1	yes	characteristic
136	C₂	—	2	2	no	—
137	C₂	C₂×#14519776	1	1	yes	characteristic
138	C₂	#16840559	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16840315

C₂

yes

maximal

#16840170

C₂

yes

maximal, complemented

#16840169

C₂

yes

maximal, complemented

#16839800

C₂

yes

maximal, characteristic

#16839798

C₂

yes

maximal

C₂×#14519687

C₂

yes

maximal, characteristic

C₆₁×#427

C₂

yes

maximal, characteristic

C₂²×#92418

C₂²

yes

characteristic

#14519776

C₂²

yes

—

#14519768

C₂²

yes

—

#14519754

—

C₄×Dic₁₂₂

—

#14519736

C₂²

yes

—

C₂×C₄×Dic₆₁

C₂²

yes

characteristic

#14519711

C₂²

yes

—

#14519710

C₂²

yes

—

#14519709

C₂²

yes

—

#14519708

C₂²

yes

—

#14519687

C₂²

yes

—

#14519687

C₂²

yes

—

C₂²×C₈×C₆₁

C₂²

yes

characteristic

#117×C₆₁

C₂²

yes

—

#107×C₆₁

C₂²

yes

—

#106×C₆₁

C₂²

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

D₄

yes

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

Q₈

yes

complemented

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₄⋊C₄×C₆₁

C₂³

yes

—

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂³

yes

—

#11758196

—

#11758196

—

C₄×Dic₆₁

—

C₄×Dic₆₁

D₄

yes

—

C₄×Dic₆₁

C₂³

yes

—

C₂×#92418

Q₈

yes

complemented

C₂×#92418

C₂³

yes

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂×D₄

yes

—

Dic₁₂₂

—

C₈×C₆₁

C₂×Q₈

yes

—

C₈×C₆₁

C₂×Q₈

yes

—

#92418

C₂×Q₈

yes

—

C₂²×C₆₁

—

C₂²×C₆₁

C₄⋊Q₈

yes

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

C₄⋊Q₈

yes

—

C₄×C₆₁

C₂²×Q₈

yes

characteristic

C₄×C₆₁

C₂²×Q₈

yes

characteristic

Dic₆₁

—

Dic₆₁

—

#3162

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₆₁

#575

yes

—

—

—

—

—

—

D₆₁

yes

characteristic

C₂×#97

—

C₆₁

#3162

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₆₁

yes

characteristic

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

122

—

C₄×Q₈

—

122

—

#117

—

#117

C₂×D₆₁

yes

—

C₂×C₄²

—

#107

—

#107

—

#107

C₂×D₆₁

yes

—

#106

—

#106

—

#106

C₂×D₆₁

yes

—

#97

—

#97

—

C₂×Q₈

—

122

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

Dic₁₂₂

yes

complemented

101

C₂×C₈

C₂²×D₆₁

yes

characteristic

102

C₂×C₈

C₂²×D₆₁

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

122

—

105

C₄⋊C₄

—

122

—

106

C₄⋊C₄

—

122

—

107

C₄⋊C₄

C₂²×D₆₁

yes

—

108

C₂²⋊C₄

—

109

C₂²⋊C₄

—

122

—

110

C₄²

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

122

—

114

C₂³

D₄×D₆₁

yes

characteristic

115

Q₈

—

244

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

122

—

119

C₂×C₄

—

244

—

120

C₂×C₄

C₂×Dic₁₂₂

yes

—

121

C₂×C₄

D₄×D₆₁

yes

characteristic

122

C₂×C₄

Q₈×D₆₁

yes

—

123

C₂×C₄

C₂³×D₆₁

yes

characteristic

124

C₈

—

125

C₈

C₂×Dic₁₂₂

yes

—

126

C₈

C₂×Dic₁₂₂

yes

—

127

C₂²

—

128

C₂²

#14519776

yes

—

129

C₂²

C₂×D₄×D₆₁

yes

characteristic

130

C₄

—

131

C₄

—

122

—

132

C₄

—

244

—

133

C₄

#14519776

yes

—

134

C₄

C₂×Q₈×D₆₁

yes

characteristic

135

C₄

C₂²×Dic₁₂₂

yes

characteristic

136

C₂

—

137

C₂

C₂×#14519776

yes

characteristic

138

C₂

#16840559

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₆₁×#427	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	#427	D₆₁
hypocenter	C₆₁	#3162
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₆₁

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₆₁×#427

C₂

socle

C₂²×C₆₁

C₂²×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
2. #16840170	C₂	semidirect
3. #16840169	C₂	semidirect
28. C₂×C₈×C₆₁	Q₈	semidirect
42. C₂×#92418	Q₈	semidirect
77. C₆₁	#3162	semidirect
100. C₂×C₈	Dic₁₂₂	semidirect

H ≅ G/N

type

2. #16840170

C₂

semidirect

3. #16840169

C₂

semidirect

28. C₂×C₈×C₆₁

Q₈

semidirect

42. C₂×#92418

Q₈

semidirect

77. C₆₁

#3162

semidirect

100. C₂×C₈

Dic₁₂₂

semidirect

Subgroups — #21456342

Conjugacy classes of subgroups

Complements