Group #21457213 — subgroups

Subgroups — #21457213

Order 7808.

342 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 125 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	#16840319	C₂	1	2	yes	maximal
2	#16840048	C₂	1	4	yes	maximal, complemented
3	#16840020	C₂	1	4	yes	maximal
4	#16839973	C₂	1	4	yes	maximal, complemented
5	C₆₁×#423	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₄₄	C₂²	1	4	yes	—
7	#14519777	C₂²	1	2	yes	—
8	#14519777	C₂²	1	2	yes	—
9	#14519769	—	2	2	no	—
10	#14519756	—	2	2	no	—
11	#14519756	C₂²	1	4	yes	—
12	C₄×Dic₁₂₂	—	2	2	no	—
13	C₄×Dic₁₂₂	C₂²	1	4	yes	—
14	#14519736	—	2	2	no	—
15	#14519709	C₂²	1	4	yes	—
16	#14519702	C₂²	1	8	yes	—
17	C₂×M₁₆×C₆₁	C₂²	1	2	yes	—
18	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
19	C₄⋊C₈×C₆₁	C₂²	1	4	yes	—
20	C₂²×C₄×C₆₁	D₄	1	2	yes	—
21	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
22	C₂×Dic₁₂₂	—	2	2	no	—
23	C₂×Dic₁₂₂	C₂³	1	4	yes	—
24	M₁₆×C₆₁	—	4	2	no	—
25	C₂×C₈×C₆₁	C₂³	1	4	yes	—
26	C₄²×C₆₁	D₄	1	1	yes	characteristic
27	C₄²×C₆₁	D₄	1	1	yes	characteristic
28	C₄²×C₆₁	C₂³	1	2	yes	—
29	C₂²⋊Dic₆₁	—	2	4	no	—
30	C₄⋊Dic₆₁	—	2	2	no	—
31	C₄⋊Dic₆₁	—	2	2	no	—
32	C₄⋊Dic₆₁	—	2	2	no	—
33	C₄⋊Dic₆₁	C₂³	1	4	yes	—
34	#11758196	—	2	4	no	—
35	#11758196	—	2	4	no	—
36	C₄×Dic₆₁	—	2	4	no	—
37	Dic₂₄₄	—	4	4	no	—
38	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₆₁	—	2	4	no	—
40	C₂×C₄×C₆₁	—	2	1	no	—
41	C₂×C₄×C₆₁	—	2	1	no	—
42	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₆₁	—	2	4	no	—
47	C₂×Dic₆₁	—	2	4	no	—
48	Dic₁₂₂	—	2	4	no	—
49	Dic₁₂₂	—	2	4	no	—
50	Dic₁₂₂	—	4	4	no	—
51	C₈×C₆₁	—	2	4	no	—
52	C₂²×C₆₁	—	2	1	no	—
53	C₂²×C₆₁	—	2	2	no	—
54	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
55	C₄×C₆₁	—	2	1	no	—
56	C₄×C₆₁	—	2	2	no	—
57	C₄×C₆₁	#137	1	4	yes	—
58	C₄×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
59	C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
60	Dic₆₁	—	4	4	no	—
61	Dic₆₁	—	4	4	no	—
62	#3255	—	61	1	no	maximal, Sylow
63	C₂×C₆₁	—	4	1	no	—
64	C₂×C₆₁	#537	1	1	yes	characteristic
65	C₂×C₆₁	C₂×#137	1	2	yes	—
66	#533	—	61	2	no	—
67	#475	—	61	4	no	—
68	#470	—	61	4	no	—
69	#462	—	61	4	no	—
70	#423	D₆₁	1	1	yes	characteristic
71	C₆₁	#3255	1	1	yes	characteristic, complemented, Sylow
72	C₂×Q₁₆	—	61	4	no	—
73	C₂×M₁₆	C₂×D₆₁	1	2	yes	—
74	C₄⋊Q₈	—	61	2	no	—
75	C₄⋊Q₈	—	61	2	no	—
76	#125	—	122	2	no	—
77	C₂²⋊Q₈	—	61	4	no	—
78	C₂²⋊Q₈	—	122	2	no	—
79	C₄×Q₈	—	61	4	no	—
80	C₄×Q₈	—	122	2	no	—
81	#117	—	122	2	no	—
82	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
83	#106	—	61	4	no	—
84	C₄⋊C₈	C₂×D₆₁	1	4	yes	—
85	#103	—	61	8	no	—
86	C₂×Q₈	—	61	4	no	—
87	C₂×Q₈	—	122	2	no	—
88	C₂²×C₄	D₂₄₄	1	2	yes	—
89	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
90	Q₁₆	—	244	4	no	—
91	M₁₆	—	4	2	no	—
92	C₂×C₈	C₂²×D₆₁	1	4	yes	—
93	C₄⋊C₄	—	61	4	no	—
94	C₄⋊C₄	—	122	4	no	—
95	C₄⋊C₄	—	122	4	no	—
96	C₄⋊C₄	—	122	2	no	—
97	C₄⋊C₄	—	122	2	no	—
98	C₄⋊C₄	—	122	2	no	—
99	C₂²⋊C₄	—	122	4	no	—
100	C₄²	—	122	4	no	—
101	C₄²	D₂₄₄	1	1	yes	characteristic
102	C₄²	D₂₄₄	1	1	yes	characteristic
103	C₄²	C₂²×D₆₁	1	2	yes	—
104	C₂³	C₂×D₂₄₄	1	1	yes	characteristic
105	Q₈	—	122	4	no	—
106	Q₈	—	122	4	no	—
107	Q₈	—	244	4	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	122	4	no	—
112	C₂×C₄	—	122	4	no	—
113	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
114	C₂×C₄	C₂×D₂₄₄	1	2	yes	—
115	C₂×C₄	C₂×D₂₄₄	1	2	yes	—
116	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
117	C₈	—	2	4	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	2	1	no	—
120	C₂²	C₂²×D₂₄₄	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	2	1	no	—
123	C₄	—	244	4	no	—
124	C₄	—	244	4	no	—
125	C₄	#14519810	1	4	yes	—
126	C₄	#14519822	1	1	yes	characteristic
127	C₄	#14519825	1	1	yes	characteristic
128	C₂	—	4	1	no	—
129	C₂	#16840343	1	1	yes	characteristic
130	C₂	C₂×#14519810	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16840319

C₂

yes

maximal

#16840048

C₂

yes

maximal, complemented

#16840020

C₂

yes

maximal

#16839973

C₂

yes

maximal, complemented

C₆₁×#423

C₂

yes

maximal, characteristic

C₂×Dic₂₄₄

C₂²

yes

—

#14519777

C₂²

yes

—

#14519777

C₂²

yes

—

#14519769

—

#14519756

—

#14519756

C₂²

yes

—

C₄×Dic₁₂₂

—

C₄×Dic₁₂₂

C₂²

yes

—

#14519736

—

#14519709

C₂²

yes

—

#14519702

C₂²

yes

—

C₂×M₁₆×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

C₂²

yes

—

C₂²×C₄×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

C₂³

yes

—

M₁₆×C₆₁

—

C₂×C₈×C₆₁

C₂³

yes

—

C₄²×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

C₂³

yes

—

C₂²⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂³

yes

—

#11758196

—

#11758196

—

C₄×Dic₆₁

—

Dic₂₄₄

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#137

yes

—

C₄×C₆₁

H(5,𝔽₂)

yes

characteristic

C₄×C₆₁

2₋¹⁺⁴

yes

characteristic

Dic₆₁

—

Dic₆₁

—

#3255

—

maximal, Sylow

C₂×C₆₁

—

C₂×C₆₁

#537

yes

characteristic

C₂×C₆₁

C₂×#137

yes

—

—

—

—

—

D₆₁

yes

characteristic

C₆₁

#3255

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×M₁₆

C₂×D₆₁

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

122

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

122

—

C₄×Q₈

—

C₄×Q₈

—

122

—

#117

—

122

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

#106

—

C₄⋊C₈

C₂×D₆₁

yes

—

#103

—

C₂×Q₈

—

C₂×Q₈

—

122

—

C₂²×C₄

D₂₄₄

yes

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

Q₁₆

—

244

—

M₁₆

—

C₂×C₈

C₂²×D₆₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₂²⋊C₄

—

122

—

100

C₄²

—

122

—

101

C₄²

D₂₄₄

yes

characteristic

102

C₄²

D₂₄₄

yes

characteristic

103

C₄²

C₂²×D₆₁

yes

—

104

C₂³

C₂×D₂₄₄

yes

characteristic

105

Q₈

—

122

—

106

Q₈

—

122

—

107

Q₈

—

244

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

122

—

112

C₂×C₄

—

122

—

113

C₂×C₄

C₂×D₂₄₄

yes

characteristic

114

C₂×C₄

C₂×D₂₄₄

yes

—

115

C₂×C₄

C₂×D₂₄₄

yes

—

116

C₂×C₄

C₂³×D₆₁

yes

characteristic

117

C₈

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂²×D₂₄₄

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

244

—

124

C₄

—

244

—

125

C₄

#14519810

yes

—

126

C₄

#14519822

yes

characteristic

127

C₄

#14519825

yes

characteristic

128

C₂

—

129

C₂

#16840343

yes

characteristic

130

C₂

C₂×#14519810

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₆₁×#423	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	#423	D₆₁
hypocenter	C₆₁	#3255
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂²

C₂²×D₂₄₄

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₆₁×#423

C₂

socle

C₂²×C₆₁

C₂²×D₄

D₆₁

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
2. #16840048	C₂	semidirect
4. #16839973	C₂	semidirect
71. C₆₁	#3255	semidirect

H ≅ G/N

type

2. #16840048

C₂

semidirect

4. #16839973

C₂

semidirect

71. C₆₁

#3255

semidirect

Subgroups — #21457213

Conjugacy classes of subgroups

Complements