Group #21443722 — subgroups

Subgroups — #21443722

Order 7808.

176 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 77 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	#16839815	C₂	1	1	yes	maximal, characteristic
2	#16839786	C₂	1	1	yes	maximal, characteristic
3	C₄×C₄⋊C₄×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#11758194	—	2	1	no	—
5	C₄⋊#92418	—	2	1	no	—
6	C₄⋊#92418	C₄	1	2	yes	complemented
7	C₄⋊#92418	C₄	1	2	yes	complemented
8	C₂²⋊#92418	—	2	1	no	—
9	C₄×#92418	—	2	1	no	—
10	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
11	C₂×C₄²×C₆₁	—	2	1	no	—
12	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
13	#95×C₆₁	C₄	1	2	yes	—
14	C₂²×C₄×C₆₁	—	2	2	no	—
15	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
18	C₄⋊C₄×C₆₁	—	2	4	no	—
19	C₄²×C₆₁	—	4	1	no	—
20	C₄²×C₆₁	—	4	1	no	—
21	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₆₁	D₄	1	1	yes	characteristic
24	C₄²×C₆₁	Q₈	1	1	yes	characteristic
25	#11758194	—	4	1	no	—
26	#11758194	—	4	1	no	—
27	C₂×#92418	—	2	2	no	—
28	C₂×#92418	—	2	2	no	—
29	C₂³×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₆₁	—	2	2	no	—
31	C₂×C₄×C₆₁	—	2	4	no	—
32	C₂×C₄×C₆₁	—	4	1	no	—
33	C₂×C₄×C₆₁	—	4	2	no	—
34	C₂×C₄×C₆₁	—	4	2	no	—
35	C₂×C₄×C₆₁	C₄²	1	1	yes	characteristic
36	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
40	C₂×C₄×C₆₁	C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₆₁	D₈	1	1	yes	characteristic
42	C₂×C₄×C₆₁	SD₁₆	1	1	yes	characteristic
43	C₂×C₄×C₆₁	SD₁₆	1	1	yes	characteristic
44	C₂×C₄×C₆₁	Q₁₆	1	1	yes	characteristic
45	#92418	—	4	2	no	—
46	#92418	—	4	2	no	—
47	C₂²×C₆₁	—	2	1	no	—
48	C₂²×C₆₁	—	2	1	no	—
49	C₂²×C₆₁	#95	1	1	yes	characteristic
50	C₂²×C₆₁	#102	1	1	yes	characteristic
51	C₂²×C₆₁	#103	1	1	yes	characteristic
52	C₄×C₆₁	—	2	1	no	—
53	C₄×C₆₁	—	2	1	no	—
54	C₄×C₆₁	—	2	1	no	—
55	C₄×C₆₁	—	4	4	no	—
56	C₄×C₆₁	#102	1	1	yes	characteristic
57	C₄×C₆₁	#103	1	1	yes	characteristic
58	C₄×C₆₁	#104	1	2	yes	—
59	C₄×C₆₁	#106	1	1	yes	characteristic
60	C₄×C₆₁	#107	1	1	yes	characteristic
61	#1303	—	61	1	no	maximal, Sylow
62	C₂×C₆₁	—	2	1	no	—
63	C₂×C₆₁	—	2	1	no	—
64	C₂×C₆₁	#339	1	1	yes	characteristic
65	C₂×C₆₁	#340	1	1	yes	characteristic
66	C₂×C₆₁	#344	1	1	yes	characteristic
67	#431	—	61	1	no	—
68	#423	—	61	1	no	—
69	C₄×C₄⋊C₄	D₆₁	1	1	yes	characteristic
70	C₆₁	#1303	1	1	yes	characteristic, complemented, Sylow
71	C₂×M₁₆	—	122	1	no	—
72	C₂×C₄⋊C₄	—	2	1	no	—
73	C₂×C₄²	—	2	1	no	—
74	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
75	C₄⋊C₈	—	61	2	no	—
76	C₄⋊C₈	—	61	2	no	—
77	C₄⋊C₈	—	122	1	no	—
78	C₂²⋊C₈	—	122	1	no	—
79	C₄×C₈	—	122	1	no	—
80	#95	Dic₆₁	1	2	yes	—
81	C₂²×C₄	—	2	2	no	—
82	C₂²×C₄	C₂×Dic₆₁	1	1	yes	characteristic
83	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
84	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
85	M₁₆	—	244	1	no	—
86	M₁₆	—	244	1	no	—
87	C₂×C₈	—	122	2	no	—
88	C₂×C₈	—	122	2	no	—
89	C₄⋊C₄	—	2	4	no	—
90	C₄²	—	4	1	no	—
91	C₄²	—	4	1	no	—
92	C₄²	Dic₁₂₂	1	1	yes	characteristic
93	C₄²	C₄×D₆₁	1	1	yes	characteristic
94	C₄²	C₄×D₆₁	1	1	yes	characteristic
95	C₄²	D₂₄₄	1	1	yes	characteristic
96	C₂³	C₂²⋊Dic₆₁	1	1	yes	characteristic
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	4	1	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	C₄×Dic₆₁	1	1	yes	characteristic
103	C₂×C₄	#11758196	1	1	yes	characteristic
104	C₂×C₄	#11758196	1	1	yes	characteristic
105	C₂×C₄	C₄⋊Dic₆₁	1	1	yes	characteristic
106	C₂×C₄	#11758198	1	1	yes	characteristic
107	C₂×C₄	#11758198	1	1	yes	characteristic
108	C₂×C₄	#11758199	1	1	yes	characteristic
109	C₂×C₄	#11758200	1	1	yes	characteristic
110	C₂×C₄	#11758201	1	1	yes	characteristic
111	C₂×C₄	#11758202	1	1	yes	characteristic
112	C₈	—	244	2	no	—
113	C₈	—	244	2	no	—
114	C₂²	—	2	1	no	—
115	C₂²	—	2	1	no	—
116	C₂²	#14519684	1	1	yes	characteristic
117	C₂²	#14519698	1	1	yes	characteristic
118	C₂²	#14519701	1	1	yes	characteristic
119	C₄	—	2	1	no	—
120	C₄	—	2	1	no	—
121	C₄	—	2	1	no	—
122	C₄	—	4	4	no	—
123	C₄	#14519697	1	1	yes	characteristic
124	C₄	#14519700	1	1	yes	characteristic
125	C₄	#14519705	1	2	yes	—
126	C₄	#14519708	1	1	yes	characteristic
127	C₄	#14519710	1	1	yes	characteristic
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#16839522	1	1	yes	characteristic
131	C₂	#16839524	1	1	yes	characteristic
132	C₂	#16839532	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16839815

C₂

yes

maximal, characteristic

#16839786

C₂

yes

maximal, characteristic

C₄×C₄⋊C₄×C₆₁

C₂

yes

maximal, characteristic

C₂×#11758194

—

C₄⋊#92418

—

C₄⋊#92418

C₄

yes

complemented

C₄⋊#92418

C₄

yes

complemented

C₂²⋊#92418

—

C₄×#92418

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄²×C₆₁

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

#95×C₆₁

C₄

yes

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄²×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

Q₈

yes

—

—

—

—

C₂³×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₄²

yes

characteristic

C₂×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₁

D₈

yes

characteristic

C₂×C₄×C₆₁

SD₁₆

yes

characteristic

C₂×C₄×C₆₁

SD₁₆

yes

characteristic

C₂×C₄×C₆₁

Q₁₆

yes

characteristic

#92418

—

#92418

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

yes

C₄×C₆₁

yes

C₄×C₆₁

yes

—

C₄×C₆₁

yes

C₄×C₆₁

yes

—

C₂×C₆₁

—

C₂×C₆₁

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

—

—

C₄×C₄⋊C₄

D₆₁

yes

characteristic

C₆₁

#1303

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

122

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

122

—

C₂²⋊C₈

—

122

—

C₄×C₈

—

122

—

#95

Dic₆₁

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₆₁

yes

characteristic

C₂²×C₄

C₂²⋊D₆₁

yes

characteristic

C₂²×C₄

C₂²⋊D₆₁

yes

characteristic

M₁₆

—

244

—

M₁₆

—

244

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

Dic₁₂₂

yes

characteristic

C₄²

C₄×D₆₁

yes

characteristic

C₄²

C₄×D₆₁

yes

characteristic

C₄²

D₂₄₄

yes

characteristic

C₂³

C₂²⋊Dic₆₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

C₄×Dic₆₁

yes

characteristic

103

C₂×C₄

#11758196

yes

characteristic

104

C₂×C₄

#11758196

yes

characteristic

105

C₂×C₄

C₄⋊Dic₆₁

yes

characteristic

106

C₂×C₄

#11758198

yes

characteristic

107

C₂×C₄

#11758198

yes

characteristic

108

C₂×C₄

#11758199

yes

characteristic

109

C₂×C₄

#11758200

yes

characteristic

110

C₂×C₄

#11758201

yes

characteristic

111

C₂×C₄

#11758202

yes

characteristic

112

C₈

—

244

—

113

C₈

—

244

—

114

C₂²

—

115

C₂²

—

116

C₂²

#14519684

yes

characteristic

117

C₂²

#14519698

yes

characteristic

118

C₂²

#14519701

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

#14519697

yes

characteristic

124

C₄

#14519700

yes

characteristic

125

C₄

#14519705

yes

—

126

C₄

#14519708

yes

characteristic

127

C₄

#14519710

yes

characteristic

128

C₂

—

129

C₂

—

130

C₂

#16839522

yes

characteristic

131

C₂

#16839524

yes

characteristic

132

C₂

#16839532

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#11758198
commutator	C₂×C₄×C₆₁	C₄²
Frattini	C₂×C₄²	C₂×D₆₁
Fitting	C₄×C₄⋊C₄×C₆₁	C₂
socle	C₂²×C₆₁	#95
hypercenter	C₄×C₄⋊C₄	D₆₁
hypocenter	C₆₁	#1303
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂×C₄

#11758198

commutator

C₂×C₄×C₆₁

C₄²

Frattini

C₂×C₄²

C₂×D₆₁

Fitting

C₄×C₄⋊C₄×C₆₁

C₂

socle

C₂²×C₆₁

#95

hypercenter

C₄×C₄⋊C₄

D₆₁

hypocenter

C₆₁

#1303

LCS

G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
6. C₄⋊#92418	C₄	semidirect
7. C₄⋊#92418	C₄	semidirect
70. C₆₁	#1303	semidirect

H ≅ G/N

type

6. C₄⋊#92418

C₄

semidirect

7. C₄⋊#92418

C₄

semidirect

70. C₆₁

#1303

semidirect

Subgroups — #21443722

Conjugacy classes of subgroups

Complements