Group #21445298 — subgroups

Subgroups — #21445298

Order 7808.

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

normal subgroups only

This group has 198 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	#16840139	C₂	1	1	yes	maximal, characteristic, complemented
2	#16839957	C₂	1	2	yes	maximal, complemented
3	#16839513	C₂	1	1	yes	maximal, characteristic, complemented
4	#16839505	C₂	1	2	yes	maximal, complemented
5	C₆₁×#436	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄₈₈	—	2	1	no	—
7	C₂×D₄₈₈	—	2	1	no	—
8	C₂×#11758199	—	2	2	no	—
9	#14519774	C₂²	1	2	yes	—
10	C₄⋊#92418	C₂²	1	2	yes	—
11	#14519697	—	2	2	no	—
12	C₂×M₁₆×C₆₁	—	2	1	no	—
13	C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂²⋊C₈×C₆₁	—	2	1	no	—
16	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×D₄×C₆₁	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₆₁	—	2	1	no	—
19	C₂×D₂₄₄	—	2	2	no	—
20	C₂×D₂₄₄	—	2	2	no	—
21	C₂×D₂₄₄	—	2	2	no	—
22	M₁₆×C₆₁	—	4	2	no	—
23	C₂×C₈×C₆₁	—	2	1	no	—
24	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
25	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
26	C₄⋊C₄×C₆₁	D₄	1	1	yes	characteristic
27	C₂²⋊C₄×C₆₁	—	2	1	no	—
28	C₄²×C₆₁	C₂³	1	1	yes	characteristic
29	#11758199	—	4	4	no	—
30	C₂×#92418	—	2	2	no	—
31	D₄₈₈	—	4	2	no	—
32	D₄₈₈	—	4	2	no	—
33	C₂³×C₆₁	—	2	1	no	—
34	C₂²×D₆₁	—	4	2	no	—
35	D₄×C₆₁	—	2	2	no	—
36	C₂×C₄×C₆₁	—	2	1	no	—
37	C₂×C₄×C₆₁	—	4	1	no	—
38	C₂×C₄×C₆₁	—	4	1	no	—
39	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
42	D₂₄₄	—	4	2	no	—
43	D₂₄₄	—	4	2	no	—
44	D₂₄₄	—	4	2	no	—
45	D₂₄₄	—	4	2	no	—
46	D₂₄₄	—	8	2	no	—
47	C₈×C₆₁	—	2	1	no	—
48	C₈×C₆₁	—	4	1	no	—
49	C₈×C₆₁	D₈	1	2	yes	complemented
50	#92418	—	4	2	no	—
51	C₂²×C₆₁	—	4	1	no	—
52	C₂²×C₆₁	—	4	1	no	—
53	C₂²×C₆₁	—	4	1	no	—
54	C₂²×C₆₁	#121	1	1	yes	characteristic
55	C₂×D₆₁	—	8	2	no	—
56	C₂×D₆₁	—	8	2	no	—
57	C₂×D₆₁	—	8	2	no	—
58	C₄×C₆₁	—	2	1	no	—
59	C₄×C₆₁	—	4	1	no	—
60	C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
61	C₄×C₆₁	AGL(1,ℤ/8)	1	1	yes	characteristic
62	C₄×C₆₁	AGL(1,ℤ/8)	1	2	yes	—
63	#1687	—	61	1	no	maximal, Sylow
64	C₂×C₆₁	—	8	1	no	—
65	C₂×C₆₁	#459	1	1	yes	characteristic
66	C₂×C₆₁	#469	1	1	yes	characteristic
67	C₂×C₆₁	#472	1	1	yes	characteristic
68	D₆₁	—	16	2	no	—
69	#493	—	61	1	no	—
70	#459	—	61	2	no	—
71	#436	D₆₁	1	1	yes	characteristic, complemented
72	#334	—	61	1	no	—
73	#331	—	61	2	no	—
74	C₆₁	#1687	1	1	yes	characteristic, complemented, Sylow
75	C₂×D₈	—	122	2	no	—
76	C₂×D₈	—	122	1	no	—
77	C₂×D₈	—	122	1	no	—
78	C₂×M₁₆	—	2	1	no	—
79	#127	—	61	2	no	—
80	C₄×D₄	C₂×D₆₁	1	1	yes	characteristic
81	C₄⋊C₈	—	61	2	no	—
82	C₄⋊C₈	C₂×D₆₁	1	1	yes	characteristic
83	#102	—	122	2	no	—
84	C₂²⋊C₈	—	2	1	no	—
85	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
86	C₂×D₄	—	122	2	no	—
87	C₂×D₄	—	122	2	no	—
88	C₂×D₄	—	122	2	no	—
89	C₂×D₄	D₂₄₄	1	1	yes	characteristic
90	C₂²×C₄	—	2	1	no	—
91	D₈	—	244	2	no	—
92	D₈	—	244	4	no	—
93	D₈	—	244	2	no	—
94	M₁₆	—	4	2	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	122	2	no	—
97	C₂×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
98	C₂×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
99	C₄⋊C₄	D₂₄₄	1	1	yes	characteristic
100	C₂²⋊C₄	—	2	1	no	—
101	C₄²	C₂²×D₆₁	1	1	yes	characteristic
102	C₂³	—	2	1	no	—
103	C₂³	—	244	2	no	—
104	D₄	—	2	2	no	—
105	D₄	—	488	2	no	—
106	D₄	—	244	2	no	—
107	D₄	—	244	2	no	—
108	D₄	—	244	2	no	—
109	D₄	—	244	2	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
114	C₂×C₄	#11758222	1	1	yes	characteristic
115	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
116	C₈	—	2	1	no	—
117	C₈	—	4	1	no	—
118	C₈	—	244	2	no	—
119	C₈	#11758199	1	2	yes	complemented
120	C₂²	—	4	1	no	—
121	C₂²	—	4	1	no	—
122	C₂²	—	4	1	no	—
123	C₂²	—	488	2	no	—
124	C₂²	—	488	2	no	—
125	C₂²	—	488	2	no	—
126	C₂²	#14519751	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	4	1	no	—
129	C₄	C₂×#11758199	1	1	yes	characteristic
130	C₄	#14519802	1	1	yes	characteristic
131	C₄	#14519805	1	2	yes	—
132	C₂	—	8	1	no	—
133	C₂	—	976	2	no	—
134	C₂	#16839957	1	1	yes	characteristic
135	C₂	#16840015	1	1	yes	characteristic
136	C₂	#16840032	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840139

C₂

yes

maximal, characteristic, complemented

#16839957

C₂

yes

maximal, complemented

#16839513

C₂

yes

maximal, characteristic, complemented

#16839505

C₂

yes

maximal, complemented

C₆₁×#436

C₂

yes

maximal, characteristic, complemented

C₂×D₄₈₈

—

C₂×D₄₈₈

—

C₂×#11758199

—

#14519774

C₂²

yes

—

C₄⋊#92418

C₂²

yes

—

#14519697

—

C₂×M₁₆×C₆₁

—

C₄×D₄×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

C₂²

yes

characteristic

C₂²⋊C₈×C₆₁

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×D₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

D₄

yes

characteristic

C₂×C₈×C₆₁

D₄

yes

characteristic

C₄⋊C₄×C₆₁

D₄

yes

characteristic

C₂²⋊C₄×C₆₁

—

C₄²×C₆₁

C₂³

yes

characteristic

#11758199

—

C₂×#92418

—

D₄₈₈

—

D₄₈₈

—

C₂³×C₆₁

—

C₂²×D₆₁

—

D₄×C₆₁

—

C₂×C₄×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

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

D₈

yes

complemented

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#121

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×D₈

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

#1687

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

#493

—

#459

—

#436

D₆₁

yes

characteristic, complemented

#334

—

#331

—

C₆₁

#1687

yes

characteristic, complemented, Sylow

C₂×D₈

—

122

—

C₂×D₈

—

122

—

C₂×D₈

—

122

—

C₂×M₁₆

—

#127

—

C₄×D₄

C₂×D₆₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂×D₆₁

yes

characteristic

#102

—

122

—

C₂²⋊C₈

—

C₄×C₈

C₂×D₆₁

yes

characteristic

C₂×D₄

—

122

—

C₂×D₄

—

122

—

C₂×D₄

—

122

—

C₂×D₄

D₂₄₄

yes

characteristic

C₂²×C₄

—

D₈

—

244

—

D₈

—

244

—

D₈

—

244

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

C₂²⋊D₆₁

yes

characteristic

C₂×C₈

C₂²⋊D₆₁

yes

characteristic

C₄⋊C₄

D₂₄₄

yes

characteristic

100

C₂²⋊C₄

—

101

C₄²

C₂²×D₆₁

yes

characteristic

102

C₂³

—

103

C₂³

—

244

—

104

D₄

—

105

D₄

—

488

—

106

D₄

—

244

—

107

D₄

—

244

—

108

D₄

—

244

—

109

D₄

—

244

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂×D₂₄₄

yes

characteristic

114

C₂×C₄

#11758222

yes

characteristic

115

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

116

C₈

—

117

C₈

—

118

C₈

—

244

—

119

C₈

#11758199

yes

complemented

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

488

—

124

C₂²

—

488

—

125

C₂²

—

488

—

126

C₂²

#14519751

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

C₂×#11758199

yes

characteristic

130

C₄

#14519802

yes

characteristic

131

C₄

#14519805

yes

—

132

C₂

—

133

C₂

—

976

—

134

C₂

#16839957

yes

characteristic

135

C₂

#16840015

yes

characteristic

136

C₂

#16840032

yes

characteristic

	subgroup	G/H
center	C₂²	#14519751
commutator	C₄²×C₆₁	C₂³
Frattini	C₄²	C₂²×D₆₁
Fitting	C₆₁×#436	C₂
socle	C₂²×C₆₁	#121
hypercenter	#436	D₆₁
hypocenter	C₆₁	#1687
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #436

subgroup

G/H

center

C₂²

#14519751

commutator

C₄²×C₆₁

C₂³

Frattini

C₄²

C₂²×D₆₁

Fitting

C₆₁×#436

C₂

C₂²×C₆₁

D₆₁

C₆₁

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #436

N	H ≅ G/N	type
1. #16840139	C₂	semidirect
2. #16839957	C₂	semidirect
3. #16839513	C₂	semidirect
4. #16839505	C₂	semidirect
5. C₆₁×#436	C₂	semidirect
49. C₈×C₆₁	D₈	semidirect
71. #436	D₆₁	semidirect
74. C₆₁	#1687	semidirect
119. C₈	#11758199	semidirect

H ≅ G/N

type

1. #16840139

C₂

semidirect

2. #16839957

C₂

semidirect

3. #16839513

C₂

semidirect

4. #16839505

C₂

semidirect

5. C₆₁×#436

C₂

semidirect

49. C₈×C₆₁

D₈

semidirect

71. #436

D₆₁

semidirect

74. C₆₁

#1687

semidirect

119. C₈

#11758199

semidirect

Subgroups — #21445298

Conjugacy classes of subgroups

Complements