Group #21450848 — subgroups

Subgroups — #21450848

Order 7808.

556 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 197 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×#14519776	C₂	1	4	yes	maximal
2	#16839682	C₂	1	4	yes	maximal
3	#16839658	C₂	1	4	yes	maximal
4	C₄×C₄⋊Dic₆₁	C₂	1	2	yes	maximal
5	C₆₁×#384	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₂₂	C₂²	1	4	yes	—
7	#14519776	—	2	16	no	—
8	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
10	C₂×#11758196	C₂²	1	8	yes	—
11	C₂×C₄×Dic₆₁	C₂²	1	4	yes	—
12	#14519684	C₂²	1	4	yes	—
13	#14519684	C₂²	1	4	yes	—
14	C₂×C₄⋊C₄×C₆₁	C₂²	1	4	yes	—
15	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
16	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₆₁	C₂³	1	4	yes	—
18	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
19	C₂²×C₄×C₆₁	C₂³	1	2	yes	—
20	C₂²×Dic₆₁	C₂³	1	4	yes	—
21	C₂²×Dic₆₁	C₂³	1	4	yes	—
22	C₂×Dic₁₂₂	—	2	16	no	—
23	C₂×Dic₁₂₂	—	4	4	no	—
24	C₄⋊C₄×C₆₁	—	2	8	no	—
25	C₄⋊C₄×C₆₁	—	4	2	no	—
26	C₄²×C₆₁	Q₈	1	4	yes	complemented
27	C₄⋊Dic₆₁	D₄	1	8	yes	—
28	C₄⋊Dic₆₁	Q₈	1	8	yes	complemented
29	#11758196	—	2	16	no	—
30	C₄×Dic₆₁	—	2	8	no	—
31	C₂³×C₆₁	C₂⁴	1	1	yes	characteristic
32	C₂×C₄×C₆₁	—	2	1	no	—
33	C₂×C₄×C₆₁	—	2	1	no	—
34	C₂×C₄×C₆₁	—	2	8	no	—
35	C₂×C₄×C₆₁	—	2	4	no	—
36	C₂×C₄×C₆₁	C₂×D₄	1	4	yes	—
37	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₆₁	C₂×Q₈	1	4	yes	—
39	C₂×C₄×C₆₁	C₂×Q₈	1	4	yes	—
40	C₂×Dic₆₁	—	2	4	no	—
41	C₂×Dic₆₁	—	2	4	no	—
42	C₂×Dic₆₁	—	2	4	no	—
43	C₂×Dic₆₁	—	2	4	no	—
44	C₂×Dic₆₁	C₂×D₄	1	8	yes	—
45	C₂×Dic₆₁	C₂×Q₈	1	8	yes	—
46	Dic₁₂₂	—	4	16	no	—
47	C₂²×C₆₁	C₂²×D₄	1	2	yes	—
48	C₂²×C₆₁	C₂²×Q₈	1	2	yes	—
49	C₂²×C₆₁	C₂²×Q₈	1	1	yes	characteristic
50	C₂²×C₆₁	H(5,𝔽₂)	1	1	yes	characteristic
51	C₂²×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
52	C₄×C₆₁	—	2	2	no	—
53	C₄×C₆₁	—	4	4	no	—
54	C₄×C₆₁	C₄⋊Q₈	1	8	yes	complemented
55	Dic₆₁	—	2	8	no	—
56	Dic₆₁	—	2	8	no	—
57	#2551	—	61	1	no	maximal, Sylow
58	C₂×C₆₁	C₂×C₄⋊Q₈	1	2	yes	—
59	C₂×C₆₁	#544	1	1	yes	characteristic
60	C₂×C₆₁	D₄²	1	1	yes	characteristic
61	C₂×C₆₁	D₄×Q₈	1	2	yes	—
62	C₂×C₆₁	Q₈²	1	1	yes	characteristic
63	C₂×C₄⋊Q₈	—	61	4	no	—
64	#395	—	61	4	no	—
65	#389	—	61	4	no	—
66	#384	D₆₁	1	1	yes	characteristic
67	C₄×C₄⋊C₄	—	61	2	no	—
68	C₆₁	#2551	1	1	yes	characteristic, complemented, Sylow
69	C₂²×Q₈	—	61	4	no	—
70	C₄⋊Q₈	—	122	16	no	—
71	C₂×C₄⋊C₄	—	61	2	no	—
72	C₂×C₄⋊C₄	—	61	2	no	—
73	C₂×C₄⋊C₄	—	61	8	no	—
74	C₂×C₄⋊C₄	C₂×D₆₁	1	4	yes	—
75	C₂×C₄⋊C₄	C₂×D₆₁	1	2	yes	—
76	C₂×C₄²	—	61	4	no	—
77	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
78	#95	—	61	4	no	—
79	#95	—	61	4	no	—
80	C₂×Q₈	—	122	16	no	—
81	C₂×Q₈	—	244	4	no	—
82	C₂²×C₄	—	61	4	no	—
83	C₂²×C₄	—	61	4	no	—
84	C₂²×C₄	C₂²×D₆₁	1	4	yes	—
85	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
86	C₂²×C₄	C₂²×D₆₁	1	2	yes	—
87	C₄⋊C₄	—	2	8	no	—
88	C₄⋊C₄	—	4	2	no	—
89	C₄⋊C₄	—	61	8	no	—
90	C₄⋊C₄	—	61	8	no	—
91	C₄⋊C₄	—	122	16	no	—
92	C₄²	—	122	8	no	—
93	C₄²	Dic₁₂₂	1	4	yes	complemented
94	C₂³	C₂³×D₆₁	1	1	yes	characteristic
95	Q₈	—	244	16	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	2	8	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	61	8	no	—
101	C₂×C₄	—	61	8	no	—
102	C₂×C₄	—	122	4	no	—
103	C₂×C₄	—	122	4	no	—
104	C₂×C₄	—	122	4	no	—
105	C₂×C₄	—	122	4	no	—
106	C₂×C₄	C₂×Dic₁₂₂	1	2	yes	—
107	C₂×C₄	C₂×Dic₁₂₂	1	4	yes	—
108	C₂×C₄	D₄×D₆₁	1	4	yes	—
109	C₂×C₄	Q₈×D₆₁	1	4	yes	—
110	C₂²	C₂×D₄×D₆₁	1	2	yes	—
111	C₂²	C₂×Q₈×D₆₁	1	2	yes	—
112	C₂²	C₂²×Dic₁₂₂	1	1	yes	characteristic
113	C₂²	#14519822	1	1	yes	characteristic
114	C₂²	#14519825	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	4	4	no	—
117	C₄	—	122	8	no	—
118	C₄	—	122	8	no	—
119	C₄	#14519776	1	8	yes	complemented
120	C₂	C₂×#14519776	1	2	yes	—
121	C₂	#16840393	1	1	yes	characteristic
122	C₂	#16840397	1	1	yes	characteristic
123	C₂	#16840427	1	2	yes	—
124	C₂	#16840492	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519776

C₂

yes

maximal

#16839682

C₂

yes

maximal

#16839658

C₂

yes

maximal

C₄×C₄⋊Dic₆₁

C₂

yes

maximal

C₆₁×#384

C₂

yes

maximal, characteristic

C₂²×Dic₁₂₂

C₂²

yes

—

#14519776

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×#11758196

C₂²

yes

—

C₂×C₄×Dic₆₁

C₂²

yes

—

#14519684

C₂²

yes

—

#14519684

C₂²

yes

—

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂²×C₄×C₆₁

C₂³

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×C₄×C₆₁

C₂³

yes

—

C₂²×Dic₆₁

C₂³

yes

—

C₂²×Dic₆₁

C₂³

yes

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

Q₈

yes

complemented

C₄⋊Dic₆₁

D₄

yes

—

C₄⋊Dic₆₁

Q₈

yes

complemented

#11758196

—

C₄×Dic₆₁

—

C₂³×C₆₁

C₂⁴

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂×D₄

yes

—

C₂×Dic₆₁

C₂×Q₈

yes

—

Dic₁₂₂

—

C₂²×C₆₁

C₂²×D₄

yes

—

C₂²×C₆₁

C₂²×Q₈

yes

—

C₂²×C₆₁

C₂²×Q₈

yes

characteristic

C₂²×C₆₁

H(5,𝔽₂)

yes

characteristic

C₂²×C₆₁

2₋¹⁺⁴

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₄⋊Q₈

yes

complemented

Dic₆₁

—

Dic₆₁

—

#2551

—

maximal, Sylow

C₂×C₆₁

C₂×C₄⋊Q₈

yes

—

C₂×C₆₁

#544

yes

characteristic

C₂×C₆₁

D₄²

yes

characteristic

C₂×C₆₁

D₄×Q₈

yes

—

C₂×C₆₁

Q₈²

yes

characteristic

C₂×C₄⋊Q₈

—

#395

—

#389

—

#384

D₆₁

yes

characteristic

C₄×C₄⋊C₄

—

C₆₁

#2551

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₄⋊Q₈

—

122

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

#95

—

#95

—

C₂×Q₈

—

122

—

C₂×Q₈

—

244

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

C₂²×C₄

C₂²×D₆₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄²

—

122

—

C₄²

Dic₁₂₂

yes

complemented

C₂³

C₂³×D₆₁

yes

characteristic

Q₈

—

244

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

122

—

103

C₂×C₄

—

122

—

104

C₂×C₄

—

122

—

105

C₂×C₄

—

122

—

106

C₂×C₄

C₂×Dic₁₂₂

yes

—

107

C₂×C₄

C₂×Dic₁₂₂

yes

—

108

C₂×C₄

D₄×D₆₁

yes

—

109

C₂×C₄

Q₈×D₆₁

yes

—

110

C₂²

C₂×D₄×D₆₁

yes

—

111

C₂²

C₂×Q₈×D₆₁

yes

—

112

C₂²

C₂²×Dic₁₂₂

yes

characteristic

113

C₂²

#14519822

yes

characteristic

114

C₂²

#14519825

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

122

—

118

C₄

—

122

—

119

C₄

#14519776

yes

complemented

120

C₂

C₂×#14519776

yes

—

121

C₂

#16840393

yes

characteristic

122

C₂

#16840397

yes

characteristic

123

C₂

#16840427

yes

—

124

C₂

#16840492

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₁
commutator, socle	C₂³×C₆₁	C₂⁴
Fitting	C₆₁×#384	C₂
hypercenter	#384	D₆₁
hypocenter	C₆₁	#2551
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₆₁

commutator, socle

C₂³×C₆₁

C₂⁴

Fitting

C₆₁×#384

C₂

D₆₁

C₆₁

G ▷ C₂³×C₆₁ ▷ C₆₁

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
26. C₄²×C₆₁	Q₈	semidirect
28. C₄⋊Dic₆₁	Q₈	semidirect
54. C₄×C₆₁	C₄⋊Q₈	semidirect
68. C₆₁	#2551	semidirect
93. C₄²	Dic₁₂₂	semidirect
119. C₄	#14519776	semidirect

H ≅ G/N

type

26. C₄²×C₆₁

Q₈

semidirect

28. C₄⋊Dic₆₁

Q₈

semidirect

54. C₄×C₆₁

C₄⋊Q₈

semidirect

68. C₆₁

#2551

semidirect

93. C₄²

Dic₁₂₂

semidirect

119. C₄

#14519776

semidirect

Subgroups — #21450848

Conjugacy classes of subgroups

Complements