Group #21447526 — subgroups

Subgroups — #21447526

Order 7808.

228 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₂²×#11758185	C₂	1	1	yes	maximal, characteristic, complemented
2	#16839800	C₂	1	1	yes	maximal, characteristic, complemented
3	#16839566	C₂	1	2	yes	maximal, complemented
4	#16839563	C₂	1	2	yes	maximal, complemented
5	C₂×#135×C₆₁	C₂	1	1	yes	maximal, characteristic
6	#14519736	—	2	1	no	—
7	C₂×#11758185	—	2	2	no	—
8	C₂×#11758185	C₂²	1	2	yes	—
9	#14519711	C₂²	1	2	yes	—
10	#14519709	—	2	1	no	—
11	C₂×#41×C₆₁	—	2	1	no	—
12	#135×C₆₁	—	2	2	no	—
13	#135×C₆₁	C₄	1	4	yes	complemented
14	C₂×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×SD₁₆×C₆₁	—	2	1	no	—
16	C₂×D₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
18	#41×C₆₁	—	2	4	no	—
19	#41×C₆₁	—	4	2	no	—
20	C₂×Q₈×C₆₁	—	2	1	no	—
21	C₂×D₄×C₆₁	—	2	1	no	—
22	C₂×D₄×C₆₁	—	4	1	no	—
23	C₂²×C₄×C₆₁	—	4	1	no	—
24	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
25	Q₁₆×C₆₁	—	2	1	no	—
26	Q₁₆×C₆₁	C₂×C₄	1	2	yes	—
27	SD₁₆×C₆₁	—	2	4	no	—
28	D₈×C₆₁	—	2	1	no	—
29	D₈×C₆₁	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₆₁	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
32	C₂×C₈×C₆₁	D₄	1	2	yes	—
33	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
34	C₂²⋊Dic₆₁	—	4	1	no	—
35	C₄⋊Dic₆₁	—	2	2	no	—
36	C₄×Dic₆₁	—	4	1	no	—
37	#11758185	—	2	2	no	—
38	C₂³×C₆₁	—	4	1	no	—
39	C₂³×C₆₁	D₈	1	1	yes	characteristic
40	Q₈×C₆₁	—	2	2	no	—
41	Q₈×C₆₁	—	4	1	no	—
42	D₄×C₆₁	—	2	2	no	—
43	D₄×C₆₁	—	4	4	no	—
44	D₄×C₆₁	—	4	1	no	—
45	C₂×C₄×C₆₁	—	4	1	no	—
46	C₂×C₄×C₆₁	—	4	4	no	—
47	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
48	C₂×C₄×C₆₁	D₈	1	1	yes	characteristic
49	C₂×C₄×C₆₁	SD₁₆	1	2	yes	—
50	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
51	C₂×Dic₆₁	—	4	2	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	—	4	2	no	—
57	C₂²×C₆₁	—	8	2	no	—
58	C₂²×C₆₁	#102	1	2	yes	—
59	C₂²×C₆₁	C₂×D₈	1	1	yes	characteristic
60	C₄×C₆₁	—	4	2	no	—
61	C₄×C₆₁	#102	1	2	yes	—
62	C₄×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₄×C₆₁	C₂×SD₁₆	1	1	yes	characteristic
64	Dic₆₁	—	8	2	no	—
65	#2138	—	61	1	no	maximal, Sylow
66	C₂×C₆₁	—	2	2	no	—
67	C₂×C₆₁	—	8	2	no	—
68	C₂×C₆₁	C₂×#102	1	1	yes	characteristic
69	C₂×C₆₁	#508	1	2	yes	—
70	C₂×#135	D₆₁	1	1	yes	characteristic
71	C₂²×C₁₆	—	61	1	no	—
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	—	2	1	no	—
77	#135	—	2	2	no	—
78	#135	Dic₆₁	1	4	yes	complemented
79	C₂×Q₁₆	C₂×D₆₁	1	1	yes	characteristic
80	C₂×SD₁₆	—	2	1	no	—
81	C₂×D₈	C₂×D₆₁	1	1	yes	characteristic
82	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
83	#117	—	122	1	no	—
84	C₂×C₁₆	—	61	2	no	—
85	C₂×C₁₆	—	122	2	no	—
86	#107	—	61	2	no	—
87	#106	—	122	1	no	—
88	#41	—	2	4	no	—
89	#41	—	4	2	no	—
90	C₂×Q₈	—	2	1	no	—
91	C₂×D₄	—	2	1	no	—
92	C₂×D₄	—	4	1	no	—
93	C₂²×C₄	—	4	1	no	—
94	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
95	Q₁₆	—	2	1	no	—
96	Q₁₆	C₂×Dic₆₁	1	2	yes	—
97	SD₁₆	—	2	4	no	—
98	D₈	—	2	1	no	—
99	D₈	C₂×Dic₆₁	1	2	yes	—
100	C₂×C₈	C₂×Dic₆₁	1	2	yes	—
101	C₂×C₈	C₂²⋊D₆₁	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	—	122	2	no	—
108	C₂³	—	4	1	no	—
109	C₂³	#11758199	1	1	yes	characteristic
110	Q₈	—	2	2	no	—
111	Q₈	—	4	1	no	—
112	D₄	—	2	2	no	—
113	D₄	—	4	4	no	—
114	D₄	—	4	1	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	—	4	4	no	—
117	C₂×C₄	—	244	2	no	—
118	C₂×C₄	#11758199	1	1	yes	characteristic
119	C₂×C₄	#11758200	1	2	yes	—
120	C₂×C₄	C₂²⋊Dic₆₁	1	2	yes	—
121	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
122	C₈	C₂²⋊Dic₆₁	1	2	yes	—
123	C₈	C₂²×Dic₆₁	1	1	yes	characteristic
124	C₈	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
125	C₂²	—	2	2	no	—
126	C₂²	—	4	2	no	—
127	C₂²	—	8	2	no	—
128	C₂²	#14519698	1	2	yes	—
129	C₂²	C₂×#11758199	1	1	yes	characteristic
130	C₄	—	4	2	no	—
131	C₄	—	488	2	no	—
132	C₄	#14519698	1	2	yes	—
133	C₄	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
134	C₄	C₂×#11758200	1	1	yes	characteristic
135	C₂	—	2	2	no	—
136	C₂	—	8	2	no	—
137	C₂	C₂×#14519698	1	1	yes	characteristic
138	C₂	#16840192	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11758185

C₂

yes

maximal, characteristic, complemented

#16839800

C₂

yes

maximal, characteristic, complemented

#16839566

C₂

yes

maximal, complemented

#16839563

C₂

yes

maximal, complemented

C₂×#135×C₆₁

C₂

yes

maximal, characteristic

#14519736

—

C₂×#11758185

—

C₂×#11758185

C₂²

yes

—

#14519711

C₂²

yes

—

#14519709

—

C₂×#41×C₆₁

—

#135×C₆₁

—

#135×C₆₁

C₄

yes

complemented

C₂×Q₁₆×C₆₁

C₂²

yes

characteristic

C₂×SD₁₆×C₆₁

—

C₂×D₈×C₆₁

C₂²

yes

characteristic

C₂²×C₈×C₆₁

C₂²

yes

characteristic

#41×C₆₁

—

#41×C₆₁

—

C₂×Q₈×C₆₁

—

C₂×D₄×C₆₁

—

C₂×D₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

D₄

yes

characteristic

Q₁₆×C₆₁

—

Q₁₆×C₆₁

C₂×C₄

yes

—

SD₁₆×C₆₁

—

D₈×C₆₁

—

D₈×C₆₁

C₂×C₄

yes

—

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

—

#11758185

—

C₂³×C₆₁

—

C₂³×C₆₁

D₈

yes

characteristic

Q₈×C₆₁

—

Q₈×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

D₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂×C₄×C₆₁

D₈

yes

characteristic

C₂×C₄×C₆₁

SD₁₆

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

—

C₈×C₆₁

C₂²⋊C₄

yes

—

C₈×C₆₁

C₂²×C₄

yes

characteristic

C₈×C₆₁

C₂×D₄

yes

characteristic

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#102

yes

—

C₂²×C₆₁

C₂×D₈

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

#102

yes

—

C₄×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₆₁

C₂×SD₁₆

yes

characteristic

Dic₆₁

—

#2138

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

C₂×#102

yes

characteristic

C₂×C₆₁

#508

yes

—

C₂×#135

D₆₁

yes

characteristic

C₂²×C₁₆

—

#427

—

#358

—

#357

—

C₆₁

#2138

yes

characteristic, complemented, Sylow

C₂×#41

—

#135

—

#135

Dic₆₁

yes

complemented

C₂×Q₁₆

C₂×D₆₁

yes

characteristic

C₂×SD₁₆

—

C₂×D₈

C₂×D₆₁

yes

characteristic

C₂²×C₈

C₂×D₆₁

yes

characteristic

#117

—

122

—

C₂×C₁₆

—

C₂×C₁₆

—

122

—

#107

—

#106

—

122

—

#41

—

#41

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₆₁

yes

characteristic

Q₁₆

—

Q₁₆

C₂×Dic₆₁

yes

—

SD₁₆

—

D₈

—

D₈

C₂×Dic₆₁

yes

—

100

C₂×C₈

C₂×Dic₆₁

yes

—

101

C₂×C₈

C₂²⋊D₆₁

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

—

122

—

108

C₂³

—

109

C₂³

#11758199

yes

characteristic

110

Q₈

—

111

Q₈

—

112

D₄

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

244

—

118

C₂×C₄

#11758199

yes

characteristic

119

C₂×C₄

#11758200

yes

—

120

C₂×C₄

C₂²⋊Dic₆₁

yes

—

121

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

122

C₈

C₂²⋊Dic₆₁

yes

—

123

C₈

C₂²×Dic₆₁

yes

characteristic

124

C₈

C₂×C₂²⋊D₆₁

yes

characteristic

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

#14519698

yes

—

129

C₂²

C₂×#11758199

yes

characteristic

130

C₄

—

131

C₄

—

488

—

132

C₄

#14519698

yes

—

133

C₄

C₂×C₂²⋊Dic₆₁

yes

characteristic

134

C₄

C₂×#11758200

yes

characteristic

135

C₂

—

136

C₂

—

137

C₂

C₂×#14519698

yes

characteristic

138

C₂

#16840192

yes

—

	subgroup	G/H
center	C₂×C₄	#11758199
commutator	C₈×C₆₁	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₆₁
Fitting	C₂×#135×C₆₁	C₂
socle	C₂²×C₆₁	C₂×D₈
hypercenter	C₂×#135	D₆₁
hypocenter	C₆₁	#2138
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#135

subgroup

G/H

center

C₂×C₄

#11758199

commutator

C₈×C₆₁

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₆₁

Fitting

C₂×#135×C₆₁

C₂

socle

C₂²×C₆₁

C₂×D₈

D₆₁

C₆₁

G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#135

N	H ≅ G/N	type
1. C₂²×#11758185	C₂	semidirect
2. #16839800	C₂	semidirect
3. #16839566	C₂	semidirect
4. #16839563	C₂	semidirect
13. #135×C₆₁	C₄	semidirect
75. C₆₁	#2138	semidirect
78. #135	Dic₆₁	semidirect

H ≅ G/N

type

1. C₂²×#11758185

C₂

semidirect

2. #16839800

C₂

semidirect

3. #16839566

C₂

semidirect

4. #16839563

C₂

semidirect

13. #135×C₆₁

C₄

semidirect

75. C₆₁

#2138

semidirect

78. #135

Dic₆₁

semidirect

Subgroups — #21447526

Conjugacy classes of subgroups

Complements