Group #21447527 — subgroups

Subgroups — #21447527

Order 7808.

228 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₂×#14519799	C₂	1	1	yes	maximal, characteristic
2	#16839800	C₂	1	1	yes	maximal, characteristic, complemented
3	#16839567	C₂	1	2	yes	maximal, complemented
4	#16839564	C₂	1	2	yes	maximal, complemented
5	C₂²×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11758222	—	2	1	no	—
7	#14519799	—	2	2	no	—
8	#14519799	C₄	1	4	yes	complemented
9	C₂×Dic₂₄₄	C₂²	1	1	yes	characteristic
10	C₂×#11758190	—	2	1	no	—
11	C₂×D₄₈₈	C₂²	1	1	yes	characteristic
12	#14519736	—	2	1	no	—
13	#14519711	C₂²	1	2	yes	—
14	#14519709	—	2	1	no	—
15	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×C₆₁	—	2	2	no	—
17	C₂×C₁₆×C₆₁	C₂²	1	2	yes	—
18	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
19	C₂×C₂²⋊D₆₁	—	4	1	no	—
20	#11758222	—	2	4	no	—
21	#11758222	—	4	2	no	—
22	C₂×D₂₄₄	—	2	1	no	—
23	C₂×C₄×D₆₁	—	4	1	no	—
24	C₂×Dic₁₂₂	—	2	1	no	—
25	C₂×C₈×C₆₁	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
27	C₂×C₈×C₆₁	D₄	1	2	yes	—
28	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
29	C₂²⋊Dic₆₁	—	4	1	no	—
30	C₄⋊Dic₆₁	—	2	2	no	—
31	C₄×Dic₆₁	—	4	1	no	—
32	Dic₂₄₄	—	2	1	no	—
33	Dic₂₄₄	C₂×C₄	1	2	yes	—
34	D₄₈₈	—	2	1	no	—
35	D₄₈₈	C₂×C₄	1	2	yes	—
36	#11758190	—	2	4	no	—
37	C₁₆×C₆₁	—	2	2	no	—
38	C₂³×C₆₁	D₈	1	1	yes	characteristic
39	C₂²×D₆₁	—	4	1	no	—
40	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
41	C₂×C₄×C₆₁	D₈	1	1	yes	characteristic
42	C₂×C₄×C₆₁	SD₁₆	1	2	yes	—
43	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
44	C₂²⋊D₆₁	—	4	4	no	—
45	C₂×Dic₆₁	—	4	1	no	—
46	C₂×Dic₆₁	—	4	2	no	—
47	D₂₄₄	—	2	2	no	—
48	D₂₄₄	—	4	1	no	—
49	C₄×D₆₁	—	4	4	no	—
50	Dic₁₂₂	—	2	2	no	—
51	Dic₁₂₂	—	4	1	no	—
52	C₈×C₆₁	C₂²⋊C₄	1	2	yes	—
53	C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
54	C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₆₁	—	2	2	no	—
56	C₂²×C₆₁	#102	1	2	yes	—
57	C₂²×C₆₁	C₂×D₈	1	1	yes	characteristic
58	C₂×D₆₁	—	4	2	no	—
59	C₂×D₆₁	—	8	2	no	—
60	C₄×C₆₁	#102	1	2	yes	—
61	C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
63	Dic₆₁	—	4	2	no	—
64	Dic₆₁	—	8	2	no	—
65	#2138	—	61	1	no	maximal, Sylow
66	C₂×C₆₁	—	2	2	no	—
67	C₂×C₆₁	C₂×#102	1	1	yes	characteristic
68	C₂×C₆₁	#508	1	2	yes	—
69	D₆₁	—	8	2	no	—
70	C₂×#135	—	61	1	no	—
71	C₂²×C₁₆	D₆₁	1	1	yes	characteristic, complemented
72	#427	—	61	1	no	—
73	#358	—	61	2	no	—
74	#357	—	61	2	no	—
75	C₆₁	#2138	1	1	yes	characteristic, complemented, Sylow
76	C₂×#41	—	122	1	no	—
77	#135	—	61	4	no	—
78	#135	—	122	2	no	—
79	C₂×Q₁₆	—	61	1	no	—
80	C₂×SD₁₆	—	122	1	no	—
81	C₂×D₈	—	61	1	no	—
82	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
83	#117	—	122	1	no	—
84	C₂×C₁₆	—	2	2	no	—
85	C₂×C₁₆	C₂×D₆₁	1	2	yes	—
86	#107	—	61	2	no	—
87	#106	—	122	1	no	—
88	#41	—	122	4	no	—
89	#41	—	244	2	no	—
90	C₂×Q₈	—	122	1	no	—
91	C₂×D₄	—	122	1	no	—
92	C₂×D₄	—	244	1	no	—
93	C₂²×C₄	—	244	1	no	—
94	C₂²×C₄	D₂₄₄	1	1	yes	characteristic
95	Q₁₆	—	61	2	no	—
96	Q₁₆	—	122	1	no	—
97	SD₁₆	—	122	4	no	—
98	D₈	—	61	2	no	—
99	D₈	—	122	1	no	—
100	C₂×C₈	C₄×D₆₁	1	2	yes	—
101	C₂×C₈	D₂₄₄	1	1	yes	characteristic
102	C₂×C₈	C₂²⋊D₆₁	1	2	yes	—
103	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
104	C₄⋊C₄	—	122	2	no	—
105	C₂²⋊C₄	—	244	1	no	—
106	C₄²	—	244	1	no	—
107	C₁₆	—	2	2	no	—
108	C₂³	—	244	1	no	—
109	C₂³	D₄₈₈	1	1	yes	characteristic
110	Q₈	—	122	2	no	—
111	Q₈	—	244	1	no	—
112	D₄	—	122	2	no	—
113	D₄	—	244	4	no	—
114	D₄	—	244	1	no	—
115	C₂×C₄	—	244	2	no	—
116	C₂×C₄	—	244	1	no	—
117	C₂×C₄	—	244	4	no	—
118	C₂×C₄	#11758190	1	2	yes	—
119	C₂×C₄	D₄₈₈	1	1	yes	characteristic
120	C₂×C₄	#11758198	1	2	yes	—
121	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
122	C₈	#11758198	1	2	yes	—
123	C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
124	C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
125	C₂²	—	2	2	no	—
126	C₂²	—	488	2	no	—
127	C₂²	—	244	2	no	—
128	C₂²	#14519699	1	2	yes	—
129	C₂²	C₂×D₄₈₈	1	1	yes	characteristic
130	C₄	—	488	2	no	—
131	C₄	—	244	2	no	—
132	C₄	#14519699	1	2	yes	—
133	C₄	C₂×#11758198	1	1	yes	characteristic
134	C₄	C₂×#11758190	1	1	yes	characteristic
135	C₂	—	2	2	no	—
136	C₂	—	488	2	no	—
137	C₂	C₂×#14519699	1	1	yes	characteristic
138	C₂	#16840194	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519799

C₂

yes

maximal, characteristic

#16839800

C₂

yes

maximal, characteristic, complemented

#16839567

C₂

yes

maximal, complemented

#16839564

C₂

yes

maximal, complemented

C₂²×C₁₆×C₆₁

C₂

yes

maximal, characteristic, complemented

C₂×#11758222

—

#14519799

—

#14519799

C₄

yes

complemented

C₂×Dic₂₄₄

C₂²

yes

characteristic

C₂×#11758190

—

C₂×D₄₈₈

C₂²

yes

characteristic

#14519736

—

#14519711

C₂²

yes

—

#14519709

—

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂×C₁₆×C₆₁

—

C₂×C₁₆×C₆₁

C₂²

yes

—

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂×C₂²⋊D₆₁

—

#11758222

—

#11758222

—

C₂×D₂₄₄

—

C₂×C₄×D₆₁

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₈×C₆₁

D₄

yes

characteristic

C₂×C₈×C₆₁

D₄

yes

—

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄×Dic₆₁

—

Dic₂₄₄

—

Dic₂₄₄

C₂×C₄

yes

—

D₄₈₈

—

D₄₈₈

C₂×C₄

yes

—

#11758190

—

C₁₆×C₆₁

—

C₂³×C₆₁

D₈

yes

characteristic

C₂²×D₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂×C₄×C₆₁

D₈

yes

characteristic

C₂×C₄×C₆₁

SD₁₆

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂²⋊D₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

—

C₄×D₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

C₂²⋊C₄

yes

—

C₈×C₆₁

C₂²×C₄

yes

characteristic

C₈×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₆₁

—

C₂²×C₆₁

#102

yes

—

C₂²×C₆₁

C₂×D₈

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

#102

yes

—

C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₆₁

C₂×SD₁₆

yes

characteristic

Dic₆₁

—

Dic₆₁

—

#2138

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

C₂×#102

yes

characteristic

C₂×C₆₁

#508

yes

—

D₆₁

—

C₂×#135

—

C₂²×C₁₆

D₆₁

yes

characteristic, complemented

#427

—

#358

—

#357

—

C₆₁

#2138

yes

characteristic, complemented, Sylow

C₂×#41

—

122

—

#135

—

#135

—

122

—

C₂×Q₁₆

—

C₂×SD₁₆

—

122

—

C₂×D₈

—

C₂²×C₈

C₂×D₆₁

yes

characteristic

#117

—

122

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₆₁

yes

—

#107

—

#106

—

122

—

#41

—

122

—

#41

—

244

—

C₂×Q₈

—

122

—

C₂×D₄

—

122

—

C₂×D₄

—

244

—

C₂²×C₄

—

244

—

C₂²×C₄

D₂₄₄

yes

characteristic

Q₁₆

—

Q₁₆

—

122

—

SD₁₆

—

122

—

D₈

—

D₈

—

122

—

100

C₂×C₈

C₄×D₆₁

yes

—

101

C₂×C₈

D₂₄₄

yes

characteristic

102

C₂×C₈

C₂²⋊D₆₁

yes

—

103

C₂×C₈

C₂²×D₆₁

yes

characteristic

104

C₄⋊C₄

—

122

—

105

C₂²⋊C₄

—

244

—

106

C₄²

—

244

—

107

C₁₆

—

108

C₂³

—

244

—

109

C₂³

D₄₈₈

yes

characteristic

110

Q₈

—

122

—

111

Q₈

—

244

—

112

D₄

—

122

—

113

D₄

—

244

—

114

D₄

—

244

—

115

C₂×C₄

—

244

—

116

C₂×C₄

—

244

—

117

C₂×C₄

—

244

—

118

C₂×C₄

#11758190

yes

—

119

C₂×C₄

D₄₈₈

yes

characteristic

120

C₂×C₄

#11758198

yes

—

121

C₂×C₄

C₂×D₂₄₄

yes

characteristic

122

C₈

#11758198

yes

—

123

C₈

C₂×C₄×D₆₁

yes

characteristic

124

C₈

C₂×C₂²⋊D₆₁

yes

characteristic

125

C₂²

—

126

C₂²

—

488

—

127

C₂²

—

244

—

128

C₂²

#14519699

yes

—

129

C₂²

C₂×D₄₈₈

yes

characteristic

130

C₄

—

488

—

131

C₄

—

244

—

132

C₄

#14519699

yes

—

133

C₄

C₂×#11758198

yes

characteristic

134

C₄

C₂×#11758190

yes

characteristic

135

C₂

—

136

C₂

—

488

—

137

C₂

C₂×#14519699

yes

characteristic

138

C₂

#16840194

yes

—

	subgroup	G/H
center	C₂×C₄	D₄₈₈
commutator	C₈×C₆₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₆₁
Fitting	C₂²×C₁₆×C₆₁	C₂
socle	C₂²×C₆₁	C₂×D₈
hypercenter	C₂²×C₁₆	D₆₁
hypocenter	C₆₁	#2138
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂×C₄

D₄₈₈

commutator

C₈×C₆₁

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₆₁

Fitting

C₂²×C₁₆×C₆₁

C₂

socle

C₂²×C₆₁

C₂×D₈

hypercenter

C₂²×C₁₆

D₆₁

hypocenter

C₆₁

#2138

LCS

G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
2. #16839800	C₂	semidirect
3. #16839567	C₂	semidirect
4. #16839564	C₂	semidirect
5. C₂²×C₁₆×C₆₁	C₂	semidirect
8. #14519799	C₄	semidirect
71. C₂²×C₁₆	D₆₁	semidirect
75. C₆₁	#2138	semidirect

H ≅ G/N

type

2. #16839800

C₂

semidirect

3. #16839567

C₂

semidirect

4. #16839564

C₂

semidirect

5. C₂²×C₁₆×C₆₁

C₂

semidirect

8. #14519799

C₄

semidirect

71. C₂²×C₁₆

D₆₁

semidirect

75. C₆₁

#2138

semidirect

Subgroups — #21447527

Conjugacy classes of subgroups

Complements