Group #21445198 — subgroups

Subgroups — #21445198

Order 7808.

150 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16840054	C₂	1	1	yes	maximal, characteristic
2	#16839969	C₂	1	1	yes	maximal, characteristic
3	C₈×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic
4	#16839513	C₂	1	1	yes	maximal, characteristic
5	#16839511	C₂	1	1	yes	maximal, characteristic
6	#16839508	C₂	1	1	yes	maximal, characteristic
7	C₆₁×#498	C₂	1	1	yes	maximal, characteristic
8	C₂×#11758202	—	2	1	no	—
9	C₄×Dic₁₂₂	C₂²	1	1	yes	characteristic
10	#14519710	—	2	1	no	—
11	C₄⋊#92418	C₂²	1	1	yes	characteristic
12	C₄⋊#92418	C₂²	1	1	yes	characteristic
13	C₄⋊#92418	C₂²	1	1	yes	characteristic
14	#14519706	—	2	1	no	—
15	#14519701	—	2	1	no	—
16	#14519700	—	2	1	no	—
17	C₈×Dic₆₁	—	2	1	no	—
18	C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
20	#106×C₆₁	—	2	1	no	—
21	#106×C₆₁	—	2	1	no	—
22	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₆₁	—	2	1	no	—
24	C₂×Q₈×C₆₁	—	2	1	no	—
25	C₂×Dic₁₂₂	D₄	1	1	yes	characteristic
26	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
27	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
28	C₄⋊C₄×C₆₁	—	2	1	no	—
29	C₄⋊C₄×C₆₁	—	2	1	no	—
30	C₄⋊C₄×C₆₁	—	2	1	no	—
31	C₄⋊C₄×C₆₁	—	2	1	no	—
32	C₄²×C₆₁	C₂³	1	1	yes	characteristic
33	#11758202	—	4	2	no	—
34	C₄⋊Dic₆₁	D₄	1	1	yes	characteristic
35	#11758196	—	2	1	no	—
36	C₄×Dic₆₁	—	2	1	no	—
37	C₂×#92418	—	2	1	no	—
38	C₂×#92418	—	2	1	no	—
39	C₂×#92418	—	2	1	no	—
40	Q₈×C₆₁	—	4	1	no	—
41	Q₈×C₆₁	—	4	1	no	—
42	Q₈×C₆₁	—	8	1	no	—
43	C₂×C₄×C₆₁	—	4	1	no	—
44	C₂×C₄×C₆₁	—	4	1	no	—
45	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
48	C₂×Dic₆₁	—	2	1	no	—
49	C₂×Dic₆₁	—	2	1	no	—
50	Dic₁₂₂	—	2	1	no	—
51	Dic₁₂₂	SD₁₆	1	2	yes	—
52	C₈×C₆₁	—	2	1	no	—
53	C₈×C₆₁	Q₁₆	1	2	yes	complemented
54	#92418	—	4	1	no	—
55	#92418	—	4	1	no	—
56	#92418	—	4	1	no	—
57	C₂²×C₆₁	#121	1	1	yes	characteristic
58	C₄×C₆₁	—	2	1	no	—
59	C₄×C₆₁	—	8	1	no	—
60	C₄×C₆₁	—	8	1	no	—
61	C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
62	C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
63	C₄×C₆₁	#135	1	1	yes	characteristic
64	C₄×C₆₁	#137	1	1	yes	characteristic
65	Dic₆₁	—	2	2	no	—
66	Dic₆₁	—	4	1	no	—
67	#1672	—	61	1	no	maximal, Sylow
68	C₂×C₆₁	#462	1	1	yes	characteristic
69	C₂×C₆₁	#465	1	1	yes	characteristic
70	C₂×C₆₁	#471	1	1	yes	characteristic
71	#498	D₆₁	1	1	yes	characteristic
72	#477	—	61	1	no	—
73	#462	—	61	1	no	—
74	C₈×Q₈	—	61	1	no	—
75	#334	—	61	1	no	—
76	#333	—	61	1	no	—
77	#332	—	61	1	no	—
78	C₆₁	#1672	1	1	yes	characteristic, complemented, Sylow
79	C₂×Q₁₆	—	122	1	no	—
80	C₄⋊Q₈	C₂×D₆₁	1	1	yes	characteristic
81	C₄⋊Q₈	C₂×D₆₁	1	1	yes	characteristic
82	C₄×Q₈	—	61	1	no	—
83	#107	—	122	1	no	—
84	#106	—	2	1	no	—
85	#106	—	2	1	no	—
86	C₄⋊C₈	—	61	1	no	—
87	C₄⋊C₈	—	61	1	no	—
88	C₄⋊C₈	—	61	1	no	—
89	C₄⋊C₈	—	122	1	no	—
90	#103	—	122	1	no	—
91	#103	—	122	1	no	—
92	C₄×C₈	—	122	1	no	—
93	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
94	C₂×Q₈	—	2	1	no	—
95	C₂×Q₈	—	2	1	no	—
96	C₂×Q₈	—	61	1	no	—
97	Q₁₆	—	244	2	no	—
98	C₂×C₈	—	122	1	no	—
99	C₂×C₈	—	122	1	no	—
100	C₂×C₈	—	122	1	no	—
101	C₂×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
102	C₂×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	2	1	no	—
107	C₄⋊C₄	—	61	1	no	—
108	C₄⋊C₄	—	122	1	no	—
109	C₄²	—	122	1	no	—
110	C₄²	C₂²×D₆₁	1	1	yes	characteristic
111	Q₈	—	4	1	no	—
112	Q₈	—	4	1	no	—
113	Q₈	—	8	1	no	—
114	Q₈	—	61	2	no	—
115	Q₈	—	122	1	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	122	1	no	—
119	C₂×C₄	—	122	1	no	—
120	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
121	C₂×C₄	#11758224	1	1	yes	characteristic
122	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	244	1	no	—
125	C₈	—	244	1	no	—
126	C₈	—	244	1	no	—
127	C₈	#11758202	1	2	yes	complemented
128	C₂²	#14519748	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	8	1	no	—
131	C₄	—	8	1	no	—
132	C₄	—	122	2	no	—
133	C₄	—	244	1	no	—
134	C₄	SD₁₆×D₆₁	1	1	yes	characteristic
135	C₄	C₂×#11758202	1	1	yes	characteristic
136	C₄	#14519798	1	1	yes	characteristic
137	C₄	#14519808	1	1	yes	characteristic
138	C₂	#16839969	1	1	yes	characteristic
139	C₂	#16839988	1	1	yes	characteristic
140	C₂	#16840022	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840054

C₂

yes

maximal, characteristic

#16839969

C₂

yes

maximal, characteristic

C₈×Dic₁₂₂

C₂

yes

maximal, characteristic

#16839513

C₂

yes

maximal, characteristic

#16839511

C₂

yes

maximal, characteristic

#16839508

C₂

yes

maximal, characteristic

C₆₁×#498

C₂

yes

maximal, characteristic

C₂×#11758202

—

C₄×Dic₁₂₂

C₂²

yes

characteristic

#14519710

—

C₄⋊#92418

C₂²

yes

characteristic

C₄⋊#92418

C₂²

yes

characteristic

C₄⋊#92418

C₂²

yes

—

—

—

C₈×Dic₆₁

—

C₄⋊Q₈×C₆₁

C₂²

yes

characteristic

C₄⋊Q₈×C₆₁

C₂²

yes

characteristic

#106×C₆₁

—

#106×C₆₁

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×Dic₁₂₂

D₄

yes

characteristic

C₂×C₈×C₆₁

D₄

yes

characteristic

C₂×C₈×C₆₁

D₄

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂³

yes

characteristic

#11758202

—

C₄⋊Dic₆₁

D₄

yes

characteristic

#11758196

—

C₄×Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

—

Q₈×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

SD₁₆

yes

—

C₈×C₆₁

—

C₈×C₆₁

Q₁₆

yes

—

—

—

C₂²×C₆₁

#121

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×SD₁₆

yes

characteristic

C₄×C₆₁

C₂×Q₁₆

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

yes

characteristic

#477

—

#462

—

C₈×Q₈

—

#334

—

#333

—

#332

—

C₆₁

#1672

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

122

—

C₄⋊Q₈

C₂×D₆₁

yes

characteristic

C₄⋊Q₈

C₂×D₆₁

yes

characteristic

C₄×Q₈

—

#107

—

122

—

#106

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

122

—

#103

—

122

—

#103

—

122

—

C₄×C₈

—

122

—

C₄×C₈

C₂×D₆₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

244

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

100

C₂×C₈

—

122

—

101

C₂×C₈

C₂²⋊D₆₁

yes

characteristic

102

C₂×C₈

C₂²⋊D₆₁

yes

characteristic

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

122

—

109

C₄²

—

122

—

110

C₄²

C₂²×D₆₁

yes

characteristic

111

Q₈

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

Q₈

—

122

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

122

—

119

C₂×C₄

—

122

—

120

C₂×C₄

D₄×D₆₁

yes

characteristic

121

C₂×C₄

#11758224

yes

characteristic

122

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

123

C₈

—

124

C₈

—

244

—

125

C₈

—

244

—

126

C₈

—

244

—

127

C₈

#11758202

yes

complemented

128

C₂²

#14519748

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

122

—

133

C₄

—

244

—

134

C₄

SD₁₆×D₆₁

yes

characteristic

135

C₄

C₂×#11758202

yes

characteristic

136

C₄

#14519798

yes

characteristic

137

C₄

#14519808

yes

characteristic

138

C₂

#16839969

yes

characteristic

139

C₂

#16839988

yes

characteristic

140

C₂

#16840022

yes

characteristic

	subgroup	G/H
center	C₂²	#14519748
commutator	C₄²×C₆₁	C₂³
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×#498	C₂
socle	C₂²×C₆₁	#121
hypercenter	#498	D₆₁
hypocenter	C₆₁	#1672
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#14519748

commutator

C₄²×C₆₁

C₂³

Frattini

C₄²

C₂²×D₆₁

Fitting

C₆₁×#498

C₂

C₂²×C₆₁

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
53. C₈×C₆₁	Q₁₆	semidirect
78. C₆₁	#1672	semidirect
127. C₈	#11758202	semidirect

H ≅ G/N

type

53. C₈×C₆₁

Q₁₆

semidirect

78. C₆₁

#1672

semidirect

127. C₈

#11758202

semidirect

Subgroups — #21445198

Conjugacy classes of subgroups

Complements