Group #21447170 — subgroups

Subgroups — #21447170

Order 7808.

230 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 75 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	#16840393	C₂	1	1	yes	maximal, characteristic
2	#16839804	C₂	1	2	yes	maximal
3	#16839627	C₂	1	1	yes	maximal, characteristic
4	#16839523	C₂	1	2	yes	maximal
5	C₆₁×#343	C₂	1	1	yes	maximal, characteristic
6	#14519776	—	2	2	no	—
7	#14519768	—	2	2	no	—
8	#14519756	—	2	2	no	—
9	#14519736	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
12	#14519711	—	2	2	no	—
13	#14519709	—	2	2	no	—
14	#14519684	—	2	2	no	—
15	C₂×M₁₆×C₆₁	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₆₁	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₆₁	D₄	1	2	yes	—
18	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
19	C₂²×Dic₆₁	—	2	1	no	—
20	C₂²×Dic₆₁	—	2	1	no	—
21	C₂×Dic₁₂₂	—	2	2	no	—
22	M₁₆×C₆₁	—	2	2	no	—
23	M₁₆×C₆₁	Q₈	1	4	yes	complemented
24	C₂×C₈×C₆₁	—	2	2	no	—
25	C₄⋊C₄×C₆₁	—	2	2	no	—
26	C₂²⋊Dic₆₁	—	2	2	no	—
27	C₄⋊Dic₆₁	—	2	2	no	—
28	C₄⋊Dic₆₁	—	2	2	no	—
29	C₄⋊Dic₆₁	D₄	1	2	yes	—
30	C₄⋊Dic₆₁	D₄	1	2	yes	—
31	#11758196	—	2	4	no	—
32	C₄×Dic₆₁	—	2	2	no	—
33	C₄×Dic₆₁	—	2	1	no	—
34	C₄×Dic₆₁	Q₈	1	2	yes	complemented
35	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₆₁	—	2	2	no	—
37	C₂×C₄×C₆₁	—	4	2	no	—
38	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
42	C₂×Dic₆₁	—	2	2	no	—
43	C₂×Dic₆₁	—	2	2	no	—
44	C₂×Dic₆₁	—	2	2	no	—
45	C₂×Dic₆₁	—	4	1	no	—
46	C₂×Dic₆₁	—	4	1	no	—
47	C₂×Dic₆₁	—	4	2	no	—
48	C₂×Dic₆₁	—	4	1	no	—
49	Dic₁₂₂	—	8	2	no	—
50	C₈×C₆₁	—	2	4	no	—
51	C₂²×C₆₁	—	2	1	no	—
52	C₂²×C₆₁	—	2	1	no	—
53	C₂²×C₆₁	#120	1	1	yes	characteristic
54	C₂²×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
55	C₂²×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
56	C₄×C₆₁	—	4	2	no	—
57	C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
58	C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
59	Dic₆₁	—	4	2	no	—
60	Dic₆₁	—	4	2	no	—
61	Dic₆₁	—	4	2	no	—
62	Dic₆₁	—	8	1	no	—
63	#2061	—	61	1	no	maximal, Sylow
64	C₂×C₆₁	—	2	1	no	—
65	C₂×C₆₁	—	2	1	no	—
66	C₂×C₆₁	#395	1	1	yes	characteristic
67	C₂×C₆₁	#454	1	1	yes	characteristic
68	C₂×C₆₁	#455	1	1	yes	characteristic
69	#544	—	61	1	no	—
70	#428	—	61	2	no	—
71	#382	—	61	1	no	—
72	#343	D₆₁	1	1	yes	characteristic
73	#339	—	61	2	no	—
74	C₆₁	#2061	1	1	yes	characteristic, complemented, Sylow
75	C₂×M₁₆	C₂×D₆₁	1	2	yes	—
76	C₄⋊Q₈	—	122	2	no	—
77	#125	—	122	2	no	—
78	C₂²⋊Q₈	—	122	2	no	—
79	#117	—	61	2	no	—
80	C₂×C₄⋊C₄	—	61	1	no	—
81	C₂×C₄⋊C₄	C₂×D₆₁	1	1	yes	characteristic
82	C₂×C₄²	—	61	1	no	—
83	#107	—	122	2	no	—
84	#106	—	122	2	no	—
85	#95	—	122	2	no	—
86	C₂×Q₈	—	122	2	no	—
87	C₂²×C₄	—	122	1	no	—
88	C₂²×C₄	—	122	1	no	—
89	C₂²×C₄	D₂₄₄	1	2	yes	—
90	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
91	M₁₆	—	2	2	no	—
92	M₁₆	Dic₁₂₂	1	4	yes	complemented
93	C₂×C₈	—	2	2	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	61	2	no	—
96	C₄⋊C₄	—	61	2	no	—
97	C₄⋊C₄	—	122	4	no	—
98	C₄⋊C₄	—	122	2	no	—
99	C₄⋊C₄	—	122	2	no	—
100	C₂²⋊C₄	—	122	2	no	—
101	C₄²	—	61	2	no	—
102	C₄²	—	122	2	no	—
103	C₄²	—	122	1	no	—
104	C₂³	C₂×D₂₄₄	1	1	yes	characteristic
105	Q₈	—	488	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	122	2	no	—
109	C₂×C₄	—	122	2	no	—
110	C₂×C₄	—	122	2	no	—
111	C₂×C₄	—	244	2	no	—
112	C₂×C₄	—	244	1	no	—
113	C₂×C₄	—	244	1	no	—
114	C₂×C₄	—	244	1	no	—
115	C₂×C₄	C₂×Dic₁₂₂	1	2	yes	—
116	C₂×C₄	D₄×D₆₁	1	2	yes	—
117	C₂×C₄	#11758224	1	1	yes	characteristic
118	C₂×C₄	Q₈×D₆₁	1	1	yes	characteristic
119	C₈	—	2	4	no	—
120	C₂²	—	2	1	no	—
121	C₂²	—	2	1	no	—
122	C₂²	#14519747	1	1	yes	characteristic
123	C₂²	#14519757	1	1	yes	characteristic
124	C₂²	#14519777	1	1	yes	characteristic
125	C₄	—	4	2	no	—
126	C₄	—	488	1	no	—
127	C₄	—	244	2	no	—
128	C₄	—	244	2	no	—
129	C₄	—	244	2	no	—
130	C₄	#14519754	1	2	yes	—
131	C₄	#14519776	1	2	yes	—
132	C₂	—	2	1	no	—
133	C₂	—	2	1	no	—
134	C₂	#16839683	1	1	yes	characteristic
135	C₂	#16839937	1	1	yes	characteristic
136	C₂	#16839942	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840393

C₂

yes

maximal, characteristic

#16839804

C₂

yes

maximal

#16839627

C₂

yes

maximal, characteristic

#16839523

C₂

yes

maximal

C₆₁×#343

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

C₂×C₄⋊Dic₆₁

C₂²

yes

characteristic

C₂×C₄×Dic₆₁

C₂²

yes

—

—

—

C₂×M₁₆×C₆₁

C₂²

yes

—

C₂×C₄⋊C₄×C₆₁

C₂²

yes

characteristic

C₂²×C₄×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₂×Dic₁₂₂

—

M₁₆×C₆₁

—

M₁₆×C₆₁

Q₈

yes

complemented

C₂×C₈×C₆₁

—

C₄⋊C₄×C₆₁

—

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

D₄

yes

—

C₄⋊Dic₆₁

D₄

yes

—

#11758196

—

C₄×Dic₆₁

—

C₄×Dic₆₁

—

C₄×Dic₆₁

Q₈

yes

complemented

C₂³×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₁

#41

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

C₈×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#120

yes

characteristic

C₂²×C₆₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₆₁

C₄⋊Q₈

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

C₂²⋊Q₈

yes

—

C₄×C₆₁

C₄⋊Q₈

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

#2061

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

—

—

D₆₁

yes

characteristic

#339

—

C₆₁

#2061

yes

characteristic, complemented, Sylow

C₂×M₁₆

C₂×D₆₁

yes

—

C₄⋊Q₈

—

122

—

#125

—

122

—

C₂²⋊Q₈

—

122

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

characteristic

C₂×C₄²

—

#107

—

122

—

#106

—

122

—

#95

—

122

—

C₂×Q₈

—

122

—

C₂²×C₄

—

122

—

C₂²×C₄

—

122

—

C₂²×C₄

D₂₄₄

yes

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

M₁₆

—

M₁₆

Dic₁₂₂

yes

complemented

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

100

C₂²⋊C₄

—

122

—

101

C₄²

—

102

C₄²

—

122

—

103

C₄²

—

122

—

104

C₂³

C₂×D₂₄₄

yes

characteristic

105

Q₈

—

488

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

122

—

109

C₂×C₄

—

122

—

110

C₂×C₄

—

122

—

111

C₂×C₄

—

244

—

112

C₂×C₄

—

244

—

113

C₂×C₄

—

244

—

114

C₂×C₄

—

244

—

115

C₂×C₄

C₂×Dic₁₂₂

yes

—

116

C₂×C₄

D₄×D₆₁

yes

—

117

C₂×C₄

#11758224

yes

characteristic

118

C₂×C₄

Q₈×D₆₁

yes

characteristic

119

C₈

—

120

C₂²

—

121

C₂²

—

122

C₂²

#14519747

yes

characteristic

123

C₂²

#14519757

yes

characteristic

124

C₂²

#14519777

yes

characteristic

125

C₄

—

126

C₄

—

488

—

127

C₄

—

244

—

128

C₄

—

244

—

129

C₄

—

244

—

130

C₄

#14519754

yes

—

131

C₄

#14519776

yes

—

132

C₂

—

133

C₂

—

134

C₂

#16839683

yes

characteristic

135

C₂

#16839937

yes

characteristic

136

C₂

#16839942

yes

characteristic

	subgroup	G/H
center	C₂²	#14519747
commutator	C₂²×C₄×C₆₁	C₂³
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₆₁×#343	C₂
socle	C₂²×C₆₁	#120
hypercenter	#343	D₆₁
hypocenter	C₆₁	#2061
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

subgroup

G/H

center

C₂²

#14519747

commutator

C₂²×C₄×C₆₁

C₂³

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₆₁×#343

C₂

C₂²×C₆₁

D₆₁

C₆₁

G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

N	H ≅ G/N	type
23. M₁₆×C₆₁	Q₈	semidirect
34. C₄×Dic₆₁	Q₈	semidirect
74. C₆₁	#2061	semidirect
92. M₁₆	Dic₁₂₂	semidirect

H ≅ G/N

type

23. M₁₆×C₆₁

Q₈

semidirect

34. C₄×Dic₆₁

Q₈

semidirect

74. C₆₁

#2061

semidirect

92. M₁₆

Dic₁₂₂

semidirect

Subgroups — #21447170

Conjugacy classes of subgroups

Complements