Group #21445672 — subgroups

Subgroups — #21445672

Order 7808.

292 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 165 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×C₄⋊#92418	C₂	1	2	yes	maximal
2	C₂×#14519688	C₂	1	2	yes	maximal
3	#16839517	C₂	1	2	yes	maximal
4	C₄³×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×#92418	C₂²	1	4	yes	—
6	C₄⋊#92418	C₄	1	8	yes	complemented
7	#14519688	—	2	4	no	—
8	C₂×C₄²×C₆₁	C₄	1	2	yes	—
9	C₂×C₄²×C₆₁	C₄	1	2	yes	—
10	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₆₁	C₂×C₄	1	2	yes	—
15	C₂²×C₄×C₆₁	C₂×C₄	1	2	yes	—
16	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
18	C₄²×C₆₁	—	2	4	no	—
19	C₄²×C₆₁	—	2	4	no	—
20	C₄²×C₆₁	—	2	2	no	—
21	C₄²×C₆₁	C₂×C₄	1	4	yes	—
22	C₄²×C₆₁	D₄	1	2	yes	—
23	C₄²×C₆₁	Q₈	1	2	yes	—
24	C₂×#92418	—	2	4	no	—
25	C₂×#92418	—	2	4	no	—
26	C₂×#92418	C₂×C₄	1	8	yes	—
27	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄×C₆₁	—	2	2	no	—
29	C₂×C₄×C₆₁	—	2	2	no	—
30	C₂×C₄×C₆₁	—	2	2	no	—
31	C₂×C₄×C₆₁	—	2	2	no	—
32	C₂×C₄×C₆₁	—	2	2	no	—
33	C₂×C₄×C₆₁	C₄²	1	2	yes	—
34	C₂×C₄×C₆₁	C₄²	1	2	yes	—
35	C₂×C₄×C₆₁	C₄⋊C₄	1	4	yes	—
36	C₂×C₄×C₆₁	M₁₆	1	4	yes	—
37	C₂×C₄×C₆₁	M₁₆	1	4	yes	—
38	C₂×C₄×C₆₁	C₂²×C₄	1	2	yes	—
39	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₆₁	#41	1	2	yes	—
42	#92418	—	2	8	no	—
43	C₂²×C₆₁	C₂×C₄²	1	1	yes	characteristic
44	C₂²×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₆₁	#117	1	1	yes	characteristic
46	C₂²×C₆₁	C₂×M₁₆	1	2	yes	—
47	C₂²×C₆₁	C₂×M₁₆	1	2	yes	—
48	C₄×C₆₁	—	2	2	no	—
49	C₄×C₆₁	—	2	4	no	—
50	C₄×C₆₁	—	2	4	no	—
51	C₄×C₆₁	#97	1	4	yes	complemented
52	C₄×C₆₁	C₄×D₄	1	2	yes	—
53	C₄×C₆₁	C₄×Q₈	1	2	yes	—
54	#1767	—	61	1	no	maximal, Sylow
55	C₂×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₆₁	C₂×#97	1	1	yes	characteristic
57	C₂×C₆₁	C₄×M₁₆	1	1	yes	characteristic
58	C₂×C₆₁	#423	1	2	yes	—
59	C₂×C₆₁	#432	1	2	yes	—
60	C₂×C₄⋊C₈	—	61	2	no	—
61	C₂×#97	—	61	2	no	—
62	C₄³	D₆₁	1	1	yes	characteristic
63	#336	—	61	2	no	—
64	C₆₁	#1767	1	1	yes	characteristic, complemented, Sylow
65	C₂²×C₈	—	61	4	no	—
66	C₂×C₄²	Dic₆₁	1	2	yes	—
67	C₂×C₄²	Dic₆₁	1	2	yes	—
68	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
69	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
70	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
71	C₄⋊C₈	—	61	8	no	—
72	#97	—	122	4	no	—
73	C₂²×C₄	C₂×Dic₆₁	1	2	yes	—
74	C₂²×C₄	C₂×Dic₆₁	1	1	yes	characteristic
75	C₂²×C₄	C₂×Dic₆₁	1	2	yes	—
76	C₂²×C₄	C₂×Dic₆₁	1	1	yes	characteristic
77	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
78	C₂×C₈	—	61	8	no	—
79	C₂×C₈	—	122	4	no	—
80	C₂×C₈	—	122	4	no	—
81	C₄²	—	2	4	no	—
82	C₄²	—	2	4	no	—
83	C₄²	—	2	2	no	—
84	C₄²	Dic₁₂₂	1	2	yes	—
85	C₄²	C₄×D₆₁	1	4	yes	—
86	C₄²	D₂₄₄	1	2	yes	—
87	C₂³	C₂²×Dic₆₁	1	1	yes	characteristic
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	#11758194	1	4	yes	—
94	C₂×C₄	#11758194	1	4	yes	—
95	C₂×C₄	C₄×Dic₆₁	1	2	yes	—
96	C₂×C₄	C₄×Dic₆₁	1	2	yes	—
97	C₂×C₄	C₄⋊Dic₆₁	1	4	yes	—
98	C₂×C₄	C₂×Dic₁₂₂	1	1	yes	characteristic
99	C₂×C₄	C₂×C₄×D₆₁	1	2	yes	—
100	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
101	C₂×C₄	#11758222	1	2	yes	—
102	C₈	—	122	8	no	—
103	C₂²	C₂×C₄×Dic₆₁	1	1	yes	characteristic
104	C₂²	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
105	C₂²	#14519736	1	1	yes	characteristic
106	C₂²	C₂×#11758194	1	2	yes	—
107	C₂²	C₂×#11758194	1	2	yes	—
108	C₄	—	2	2	no	—
109	C₄	—	2	4	no	—
110	C₄	—	2	4	no	—
111	C₄	#14519688	1	4	yes	complemented
112	C₄	C₄×D₂₄₄	1	2	yes	—
113	C₄	C₄×Dic₁₂₂	1	2	yes	—
114	C₂	C₄×C₄⋊Dic₆₁	1	1	yes	characteristic
115	C₂	C₂×#14519688	1	1	yes	characteristic
116	C₂	C₄×#11758194	1	1	yes	characteristic
117	C₂	#16839786	1	2	yes	—
118	C₂	#16839819	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#92418

C₂

yes

maximal

C₂×#14519688

C₂

yes

maximal

#16839517

C₂

yes

maximal

C₄³×C₆₁

C₂

yes

maximal, characteristic

C₂²×#92418

C₂²

yes

—

C₄⋊#92418

C₄

yes

complemented

#14519688

—

C₂×C₄²×C₆₁

C₄

yes

—

C₂×C₄²×C₆₁

C₄

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

D₄

yes

—

C₄²×C₆₁

Q₈

yes

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

C₂×C₄

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

—

C₂×C₄×C₆₁

C₄²

yes

—

C₂×C₄×C₆₁

C₄⋊C₄

yes

—

C₂×C₄×C₆₁

M₁₆

yes

—

C₂×C₄×C₆₁

M₁₆

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₁

#41

yes

—

#92418

—

C₂²×C₆₁

C₂×C₄²

yes

characteristic

C₂²×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₆₁

#117

yes

characteristic

C₂²×C₆₁

C₂×M₁₆

yes

—

C₂²×C₆₁

C₂×M₁₆

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#97

yes

complemented

C₄×C₆₁

C₄×D₄

yes

—

C₄×C₆₁

C₄×Q₈

yes

—

#1767

—

maximal, Sylow

C₂×C₆₁

C₄×C₄⋊C₄

yes

characteristic

C₂×C₆₁

C₂×#97

yes

characteristic

C₂×C₆₁

C₄×M₁₆

yes

characteristic

C₂×C₆₁

#423

yes

—

C₂×C₆₁

#432

yes

—

C₂×C₄⋊C₈

—

C₂×#97

—

C₄³

D₆₁

yes

characteristic

#336

—

C₆₁

#1767

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄²

Dic₆₁

yes

—

C₂×C₄²

Dic₆₁

yes

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₄⋊C₈

—

#97

—

122

—

C₂²×C₄

C₂×Dic₆₁

yes

—

C₂²×C₄

C₂×Dic₆₁

yes

characteristic

C₂²×C₄

C₂×Dic₆₁

yes

—

C₂²×C₄

C₂×Dic₆₁

yes

characteristic

C₂²×C₄

C₂²×D₆₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

C₄²

—

C₄²

—

C₄²

—

C₄²

Dic₁₂₂

yes

—

C₄²

C₄×D₆₁

yes

—

C₄²

D₂₄₄

yes

—

C₂³

C₂²×Dic₆₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#11758194

yes

—

C₂×C₄

#11758194

yes

—

C₂×C₄

C₄×Dic₆₁

yes

—

C₂×C₄

C₄×Dic₆₁

yes

—

C₂×C₄

C₄⋊Dic₆₁

yes

—

C₂×C₄

C₂×Dic₁₂₂

yes

characteristic

C₂×C₄

C₂×C₄×D₆₁

yes

—

100

C₂×C₄

C₂×D₂₄₄

yes

characteristic

101

C₂×C₄

#11758222

yes

—

102

C₈

—

122

—

103

C₂²

C₂×C₄×Dic₆₁

yes

characteristic

104

C₂²

C₂×C₄⋊Dic₆₁

yes

characteristic

105

C₂²

#14519736

yes

characteristic

106

C₂²

C₂×#11758194

yes

—

107

C₂²

C₂×#11758194

yes

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

#14519688

yes

complemented

112

C₄

C₄×D₂₄₄

yes

—

113

C₄

C₄×Dic₁₂₂

yes

—

114

C₂

C₄×C₄⋊Dic₆₁

yes

characteristic

115

C₂

C₂×#14519688

yes

characteristic

116

C₂

C₄×#11758194

yes

characteristic

117

C₂

#16839786

yes

—

118

C₂

#16839819

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₆₁
commutator	C₂²×C₆₁	C₂×C₄²
Fitting	C₄³×C₆₁	C₂
socle	C₂³×C₆₁	C₂²×C₄
hypercenter	C₄³	D₆₁
hypocenter	C₆₁	#1767
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₆₁

commutator

C₂²×C₆₁

C₂×C₄²

Fitting

C₄³×C₆₁

C₂

socle

C₂³×C₆₁

C₂²×C₄

hypercenter

C₄³

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂²×C₄ ◁ C₄³

N	H ≅ G/N	type
6. C₄⋊#92418	C₄	semidirect
51. C₄×C₆₁	#97	semidirect
64. C₆₁	#1767	semidirect
111. C₄	#14519688	semidirect

H ≅ G/N

type

6. C₄⋊#92418

C₄

semidirect

51. C₄×C₆₁

#97

semidirect

64. C₆₁

#1767

semidirect

111. C₄

#14519688

semidirect

Subgroups — #21445672

Conjugacy classes of subgroups

Complements