Group #21445513 — subgroups

Subgroups — #21445513

Order 7808.

480 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 221 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×#11758194	C₂	1	2	yes	maximal
2	#16839517	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄²×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#11758194	—	2	4	no	—
5	C₂×#11758194	—	2	4	no	—
6	C₂×#11758194	C₄	1	8	yes	complemented
7	C₂²×#92418	C₂²	1	4	yes	—
8	C₂³×C₄×C₆₁	C₄	1	2	yes	—
9	C₂³×C₄×C₆₁	C₂²	1	1	yes	characteristic
10	C₂×C₄²×C₆₁	—	2	4	no	—
11	C₂×C₄²×C₆₁	C₄	1	2	yes	—
12	C₂×C₄²×C₆₁	C₂²	1	2	yes	—
13	C₂⁴×C₆₁	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₆₁	—	2	4	no	—
15	C₂²×C₄×C₆₁	—	2	8	no	—
16	C₂²×C₄×C₆₁	C₂×C₄	1	4	yes	—
17	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₆₁	C₂×C₄	1	4	yes	—
19	C₂²×C₄×C₆₁	D₄	1	2	yes	—
20	C₂²×C₄×C₆₁	D₄	1	4	yes	—
21	C₂²×C₄×C₆₁	Q₈	1	2	yes	—
22	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
23	C₄²×C₆₁	—	2	4	no	—
24	C₄²×C₆₁	—	2	4	no	—
25	#11758194	—	2	16	no	—
26	C₂×#92418	—	2	4	no	—
27	C₂×#92418	—	2	4	no	—
28	C₂×#92418	C₂×C₄	1	8	yes	complemented
29	C₂³×C₆₁	—	2	4	no	—
30	C₂³×C₆₁	C₄²	1	2	yes	—
31	C₂³×C₆₁	C₂²⋊C₄	1	2	yes	—
32	C₂³×C₆₁	C₄⋊C₄	1	2	yes	—
33	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₆₁	—	2	16	no	—
35	C₂×C₄×C₆₁	—	2	4	no	—
36	C₂×C₄×C₆₁	—	2	4	no	—
37	C₂×C₄×C₆₁	C₄²	1	2	yes	—
38	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₆₁	C₂²⋊C₄	1	8	yes	—
40	C₂×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
41	C₂×C₄×C₆₁	C₄⋊C₄	1	8	yes	—
42	C₂×C₄×C₆₁	M₁₆	1	8	yes	complemented
43	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
47	#92418	—	2	8	no	—
48	C₂²×C₆₁	—	2	4	no	—
49	C₂²×C₆₁	—	2	4	no	—
50	C₂²×C₆₁	—	2	4	no	—
51	C₂²×C₆₁	#95	1	4	yes	—
52	C₂²×C₆₁	C₂×C₄²	1	1	yes	characteristic
53	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₆₁	C₂×M₁₆	1	4	yes	—
56	C₄×C₆₁	—	2	8	no	—
57	C₄×C₆₁	#95	1	4	yes	—
58	C₄×C₆₁	C₂×C₂²⋊C₄	1	2	yes	—
59	C₄×C₆₁	C₂×C₄⋊C₄	1	2	yes	—
60	#1728	—	61	1	no	maximal, Sylow
61	C₂×C₆₁	—	2	4	no	—
62	C₂×C₆₁	C₂×#95	1	1	yes	characteristic
63	C₂×C₆₁	C₄×M₁₆	1	2	yes	—
64	C₂×C₆₁	#407	1	2	yes	—
65	C₂×C₆₁	#423	1	2	yes	—
66	C₂²×M₁₆	—	61	2	no	—
67	C₂²×C₄²	D₆₁	1	1	yes	characteristic
68	#336	—	61	4	no	—
69	C₆₁	#1728	1	1	yes	characteristic, complemented, Sylow
70	C₂³×C₄	Dic₆₁	1	2	yes	—
71	C₂³×C₄	C₂×D₆₁	1	1	yes	characteristic
72	C₂×M₁₆	—	61	8	no	—
73	C₂×M₁₆	—	122	4	no	—
74	C₂×M₁₆	—	122	4	no	—
75	C₂²×C₈	—	61	4	no	—
76	C₂×C₄²	—	2	4	no	—
77	C₂×C₄²	Dic₆₁	1	2	yes	—
78	C₂×C₄²	C₂×D₆₁	1	2	yes	—
79	C₂⁴	C₂×Dic₆₁	1	1	yes	characteristic
80	C₂²×C₄	—	2	4	no	—
81	C₂²×C₄	—	2	8	no	—
82	C₂²×C₄	Dic₁₂₂	1	2	yes	—
83	C₂²×C₄	C₄×D₆₁	1	4	yes	—
84	C₂²×C₄	D₂₄₄	1	2	yes	—
85	C₂²×C₄	C₂×Dic₆₁	1	4	yes	—
86	C₂²×C₄	C₂×Dic₆₁	1	1	yes	characteristic
87	C₂²×C₄	C₂²⋊D₆₁	1	4	yes	—
88	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
89	M₁₆	—	122	16	no	—
90	C₂×C₈	—	61	8	no	—
91	C₂×C₈	—	122	4	no	—
92	C₂×C₈	—	122	4	no	—
93	C₄²	—	2	4	no	—
94	C₄²	—	2	4	no	—
95	C₂³	—	2	4	no	—
96	C₂³	C₄×Dic₆₁	1	2	yes	—
97	C₂³	C₄⋊Dic₆₁	1	2	yes	—
98	C₂³	C₂²⋊Dic₆₁	1	2	yes	—
99	C₂³	C₂²×Dic₆₁	1	1	yes	characteristic
100	C₂×C₄	—	2	16	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	#11758194	1	8	yes	complemented
104	C₂×C₄	C₄×Dic₆₁	1	2	yes	—
105	C₂×C₄	#11758196	1	8	yes	—
106	C₂×C₄	C₄⋊Dic₆₁	1	2	yes	—
107	C₂×C₄	#11758198	1	8	yes	—
108	C₂×C₄	C₂²⋊Dic₆₁	1	2	yes	—
109	C₂×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
110	C₂×C₄	C₂×C₄×D₆₁	1	2	yes	—
111	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
112	C₂×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
113	C₈	—	122	8	no	—
114	C₂²	—	2	4	no	—
115	C₂²	—	2	4	no	—
116	C₂²	—	2	4	no	—
117	C₂²	#14519684	1	4	yes	—
118	C₂²	C₂×C₄×Dic₆₁	1	1	yes	characteristic
119	C₂²	C₂×C₂²⋊Dic₆₁	1	1	yes	characteristic
120	C₂²	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
121	C₂²	C₂×#11758194	1	4	yes	—
122	C₄	—	2	8	no	—
123	C₄	#14519684	1	4	yes	—
124	C₄	C₂×#11758198	1	2	yes	—
125	C₄	C₂×#11758196	1	2	yes	—
126	C₂	—	2	4	no	—
127	C₂	C₂×#14519684	1	1	yes	characteristic
128	C₂	C₄×#11758194	1	2	yes	—
129	C₂	#16839726	1	2	yes	—
130	C₂	#16839786	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11758194

C₂

yes

maximal

#16839517

C₂

yes

maximal, complemented

C₂²×C₄²×C₆₁

C₂

yes

maximal, characteristic

C₂×#11758194

—

C₂×#11758194

—

C₂×#11758194

C₄

yes

complemented

C₂²×#92418

C₂²

yes

—

C₂³×C₄×C₆₁

C₄

yes

—

C₂³×C₄×C₆₁

C₂²

yes

characteristic

C₂×C₄²×C₆₁

—

C₂×C₄²×C₆₁

C₄

yes

—

C₂×C₄²×C₆₁

C₂²

yes

—

C₂⁴×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

Q₈

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₄²×C₆₁

—

C₄²×C₆₁

—

—

—

—

C₂×C₄

yes

complemented

C₂³×C₆₁

—

C₂³×C₆₁

C₄²

yes

—

C₂³×C₆₁

C₂²⋊C₄

yes

—

C₂³×C₆₁

C₄⋊C₄

yes

—

C₂³×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄²

yes

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂×C₄×C₆₁

C₄⋊C₄

yes

—

C₂×C₄×C₆₁

C₄⋊C₄

yes

—

C₂×C₄×C₆₁

M₁₆

yes

complemented

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

characteristic

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

#95

yes

—

C₂²×C₆₁

C₂×C₄²

yes

characteristic

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₆₁

C₂×M₁₆

yes

—

C₄×C₆₁

—

C₄×C₆₁

#95

yes

—

C₄×C₆₁

C₂×C₂²⋊C₄

yes

—

C₄×C₆₁

C₂×C₄⋊C₄

yes

—

#1728

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

C₂×#95

yes

characteristic

C₂×C₆₁

C₄×M₁₆

yes

—

C₂×C₆₁

#407

yes

—

C₂×C₆₁

#423

yes

—

C₂²×M₁₆

—

C₂²×C₄²

D₆₁

yes

characteristic

#336

—

C₆₁

#1728

yes

characteristic, complemented, Sylow

C₂³×C₄

Dic₆₁

yes

—

C₂³×C₄

C₂×D₆₁

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

122

—

C₂×M₁₆

—

122

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

Dic₆₁

yes

—

C₂×C₄²

C₂×D₆₁

yes

—

C₂⁴

C₂×Dic₆₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₁₂₂

yes

—

C₂²×C₄

C₄×D₆₁

yes

—

C₂²×C₄

D₂₄₄

yes

—

C₂²×C₄

C₂×Dic₆₁

yes

—

C₂²×C₄

C₂×Dic₆₁

yes

characteristic

C₂²×C₄

C₂²⋊D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

M₁₆

—

122

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

C₄²

—

C₄²

—

C₂³

—

C₂³

C₄×Dic₆₁

yes

—

C₂³

C₄⋊Dic₆₁

yes

—

C₂³

C₂²⋊Dic₆₁

yes

—

C₂³

C₂²×Dic₆₁

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

#11758194

yes

complemented

104

C₂×C₄

C₄×Dic₆₁

yes

—

105

C₂×C₄

#11758196

yes

—

106

C₂×C₄

C₄⋊Dic₆₁

yes

—

107

C₂×C₄

#11758198

yes

—

108

C₂×C₄

C₂²⋊Dic₆₁

yes

—

109

C₂×C₄

C₂×Dic₁₂₂

yes

characteristic

110

C₂×C₄

C₂×C₄×D₆₁

yes

—

111

C₂×C₄

C₂×D₂₄₄

yes

characteristic

112

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

113

C₈

—

122

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

#14519684

yes

—

118

C₂²

C₂×C₄×Dic₆₁

yes

characteristic

119

C₂²

C₂×C₂²⋊Dic₆₁

yes

characteristic

120

C₂²

C₂×C₄⋊Dic₆₁

yes

characteristic

121

C₂²

C₂×#11758194

yes

—

122

C₄

—

123

C₄

#14519684

yes

—

124

C₄

C₂×#11758198

yes

—

125

C₄

C₂×#11758196

yes

—

126

C₂

—

127

C₂

C₂×#14519684

yes

characteristic

128

C₂

C₄×#11758194

yes

—

129

C₂

#16839726

yes

—

130

C₂

#16839786

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₆₁
commutator	C₂²×C₆₁	C₂×C₄²
Fitting	C₂²×C₄²×C₆₁	C₂
socle	C₂³×C₆₁	C₂²×C₄
hypercenter	C₂²×C₄²	D₆₁
hypocenter	C₆₁	#1728
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₆₁

commutator

C₂²×C₆₁

C₂×C₄²

Fitting

C₂²×C₄²×C₆₁

C₂

socle

C₂³×C₆₁

C₂²×C₄

hypercenter

C₂²×C₄²

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂²×C₄ ◁ C₂²×C₄²

N	H ≅ G/N	type
2. #16839517	C₂	semidirect
6. C₂×#11758194	C₄	semidirect
28. C₂×#92418	C₂×C₄	semidirect
42. C₂×C₄×C₆₁	M₁₆	semidirect
69. C₆₁	#1728	semidirect
103. C₂×C₄	#11758194	semidirect

H ≅ G/N

type

2. #16839517

C₂

semidirect

6. C₂×#11758194

C₄

semidirect

28. C₂×#92418

C₂×C₄

semidirect

42. C₂×C₄×C₆₁

M₁₆

semidirect

69. C₆₁

#1728

semidirect

103. C₂×C₄

#11758194

semidirect

Subgroups — #21445513

Conjugacy classes of subgroups

Complements