Group #21445442 — subgroups

Subgroups — #21445442

Order 7808.

198 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 65 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	#16840158	C₂	1	1	yes	maximal, characteristic, complemented
2	#16840136	C₂	1	1	yes	maximal, characteristic
3	#16840135	C₂	1	1	yes	maximal, characteristic
4	#16840112	C₂	1	1	yes	maximal, characteristic, complemented
5	#16840110	C₂	1	1	yes	maximal, characteristic
6	#16839483	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#500	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11758190	—	2	2	no	—
9	C₂×#11758201	—	2	2	no	—
10	#14519777	C₂²	1	1	yes	characteristic
11	#14519774	C₂²	1	1	yes	characteristic
12	#14519708	—	2	2	no	—
13	#14519701	—	2	2	no	—
14	#14519697	—	2	2	no	—
15	C₄×#92418	C₂²	1	1	yes	characteristic
16	C₄×#92418	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
19	#107×C₆₁	—	2	2	no	—
20	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₆₁	—	2	1	no	—
22	C₂×Q₈×C₆₁	—	2	1	no	—
23	C₂×D₂₄₄	—	2	1	no	—
24	C₂×D₂₄₄	—	2	1	no	—
25	C₂×D₂₄₄	—	2	1	no	—
26	C₂×Dic₁₂₂	—	2	1	no	—
27	C₂×C₈×C₆₁	D₄	1	2	yes	—
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	#11758201	—	4	4	no	—
34	C₄⋊Dic₆₁	—	2	1	no	—
35	C₄⋊Dic₆₁	—	2	1	no	—
36	C₂×#92418	D₄	1	2	yes	—
37	C₂×#92418	D₄	1	2	yes	—
38	#11758190	—	4	4	no	—
39	C₂²×D₆₁	—	4	1	no	—
40	Q₈×C₆₁	—	4	2	no	—
41	Q₈×C₆₁	—	4	2	no	—
42	C₂×C₄×C₆₁	—	4	1	no	—
43	C₂×C₄×C₆₁	—	4	1	no	—
44	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
47	C₂×Dic₆₁	—	4	1	no	—
48	D₂₄₄	—	4	2	no	—
49	D₂₄₄	—	4	2	no	—
50	D₂₄₄	—	4	2	no	—
51	Dic₁₂₂	—	4	2	no	—
52	C₈×C₆₁	SD₁₆	1	4	yes	complemented
53	#92418	—	2	2	no	—
54	#92418	—	2	2	no	—
55	C₂²×C₆₁	#127	1	1	yes	characteristic
56	C₂×D₆₁	—	8	1	no	—
57	C₂×D₆₁	—	8	1	no	—
58	C₂×D₆₁	—	8	1	no	—
59	C₄×C₆₁	—	8	1	no	—
60	C₄×C₆₁	—	8	1	no	—
61	C₄×C₆₁	C₂×SD₁₆	1	2	yes	—
62	C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
63	C₄×C₆₁	#137	1	2	yes	—
64	Dic₆₁	—	8	1	no	—
65	#1714	—	61	1	no	maximal, Sylow
66	C₂×C₆₁	#492	1	1	yes	characteristic
67	C₂×C₆₁	#496	1	1	yes	characteristic
68	C₂×C₆₁	#497	1	1	yes	characteristic
69	D₆₁	—	16	1	no	—
70	#500	D₆₁	1	1	yes	characteristic, complemented
71	#498	—	61	1	no	—
72	#492	—	61	1	no	—
73	#492	—	61	1	no	—
74	#487	—	61	1	no	—
75	#486	—	61	1	no	—
76	#322	—	61	1	no	—
77	C₆₁	#1714	1	1	yes	characteristic, complemented, Sylow
78	C₂×SD₁₆	—	122	2	no	—
79	C₂×SD₁₆	—	122	2	no	—
80	C₄⋊Q₈	—	61	1	no	—
81	C₄⋊Q₈	C₂×D₆₁	1	1	yes	characteristic
82	C₄⋊Q₈	C₂×D₆₁	1	1	yes	characteristic
83	#127	—	61	1	no	—
84	#107	—	2	2	no	—
85	#106	—	122	2	no	—
86	#103	—	122	2	no	—
87	#102	—	122	2	no	—
88	C₄×C₈	—	61	1	no	—
89	C₄×C₈	—	61	1	no	—
90	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	2	1	no	—
93	C₂×Q₈	—	122	1	no	—
94	C₂×D₄	—	122	1	no	—
95	C₂×D₄	—	122	1	no	—
96	C₂×D₄	—	122	1	no	—
97	SD₁₆	—	244	4	no	—
98	SD₁₆	—	244	4	no	—
99	C₂×C₈	—	61	2	no	—
100	C₂×C₈	—	61	2	no	—
101	C₂×C₈	C₂²⋊D₆₁	1	2	yes	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	122	1	no	—
107	C₄⋊C₄	—	122	1	no	—
108	C₄²	C₂²×D₆₁	1	1	yes	characteristic
109	C₂³	—	244	1	no	—
110	Q₈	—	4	2	no	—
111	Q₈	—	4	2	no	—
112	Q₈	—	244	2	no	—
113	D₄	—	244	2	no	—
114	D₄	—	244	2	no	—
115	D₄	—	244	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	244	1	no	—
119	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
120	C₂×C₄	D₄×D₆₁	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
122	C₈	—	122	2	no	—
123	C₈	—	122	2	no	—
124	C₈	#11758201	1	4	yes	complemented
125	C₂²	—	488	1	no	—
126	C₂²	—	488	1	no	—
127	C₂²	—	488	1	no	—
128	C₂²	#14519773	1	1	yes	characteristic
129	C₄	—	8	1	no	—
130	C₄	—	8	1	no	—
131	C₄	—	488	1	no	—
132	C₄	C₂×#11758201	1	2	yes	—
133	C₄	#14519801	1	2	yes	—
134	C₄	#14519809	1	2	yes	—
135	C₂	—	976	1	no	—
136	C₂	#16840136	1	1	yes	characteristic
137	C₂	#16840150	1	1	yes	characteristic
138	C₂	#16840155	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840158

C₂

yes

maximal, characteristic, complemented

#16840136

C₂

yes

maximal, characteristic

#16840135

C₂

yes

maximal, characteristic

#16840112

C₂

yes

maximal, characteristic, complemented

#16840110

C₂

yes

maximal, characteristic

#16839483

C₂

yes

maximal, characteristic, complemented

C₆₁×#500

C₂

yes

maximal, characteristic, complemented

C₂×#11758190

—

C₂×#11758201

—

#14519777

C₂²

yes

characteristic

#14519774

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₄×#92418

C₂²

yes

characteristic

C₄⋊Q₈×C₆₁

C₂²

yes

characteristic

C₄⋊Q₈×C₆₁

C₂²

yes

characteristic

#107×C₆₁

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

D₄

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂³

yes

characteristic

#11758201

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₂×#92418

D₄

yes

—

C₂×#92418

D₄

yes

—

#11758190

—

C₂²×D₆₁

—

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

C₂×D₄

yes

characteristic

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

Dic₁₂₂

—

C₈×C₆₁

SD₁₆

yes

complemented

#92418

—

#92418

—

C₂²×C₆₁

#127

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×SD₁₆

yes

—

C₄×C₆₁

AGL(1,ℤ/8)

yes

—

C₄×C₆₁

#137

yes

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

#500

D₆₁

yes

characteristic, complemented

—

—

—

—

—

—

C₆₁

#1714

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

122

—

C₂×SD₁₆

—

122

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₆₁

yes

characteristic

C₄⋊Q₈

C₂×D₆₁

yes

—

—

—

122

—

#103

—

122

—

#102

—

122

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₆₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

122

—

C₂×D₄

—

122

—

C₂×D₄

—

122

—

C₂×D₄

—

122

—

SD₁₆

—

244

—

SD₁₆

—

244

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₂²⋊D₆₁

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

122

—

107

C₄⋊C₄

—

122

—

108

C₄²

C₂²×D₆₁

yes

characteristic

109

C₂³

—

244

—

110

Q₈

—

111

Q₈

—

112

Q₈

—

244

—

113

D₄

—

244

—

114

D₄

—

244

—

115

D₄

—

244

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

244

—

119

C₂×C₄

D₄×D₆₁

yes

characteristic

120

C₂×C₄

D₄×D₆₁

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

122

C₈

—

122

—

123

C₈

—

122

—

124

C₈

#11758201

yes

complemented

125

C₂²

—

488

—

126

C₂²

—

488

—

127

C₂²

—

488

—

128

C₂²

#14519773

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

488

—

132

C₄

C₂×#11758201

yes

—

133

C₄

#14519801

yes

—

134

C₄

#14519809

yes

—

135

C₂

—

976

—

136

C₂

#16840136

yes

characteristic

137

C₂

#16840150

yes

characteristic

138

C₂

#16840155

yes

characteristic

	subgroup	G/H
center	C₂²	#14519773
commutator	C₄²×C₆₁	C₂³
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×#500	C₂
socle	C₂²×C₆₁	#127
hypercenter	#500	D₆₁
hypocenter	C₆₁	#1714
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

#14519773

commutator

C₄²×C₆₁

C₂³

Frattini

C₄²

C₂²×D₆₁

Fitting

C₆₁×#500

C₂

C₂²×C₆₁

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
1. #16840158	C₂	semidirect
4. #16840112	C₂	semidirect
6. #16839483	C₂	semidirect
7. C₆₁×#500	C₂	semidirect
52. C₈×C₆₁	SD₁₆	semidirect
70. #500	D₆₁	semidirect
77. C₆₁	#1714	semidirect
124. C₈	#11758201	semidirect

H ≅ G/N

type

1. #16840158

C₂

semidirect

4. #16840112

C₂

semidirect

6. #16839483

C₂

semidirect

7. C₆₁×#500

C₂

semidirect

52. C₈×C₆₁

SD₁₆

semidirect

70. #500

D₆₁

semidirect

77. C₆₁

#1714

semidirect

124. C₈

#11758201

semidirect

Subgroups — #21445442

Conjugacy classes of subgroups

Complements