Group #21459446 — subgroups

Subgroups — #21459446

Order 7808.

1202 conjugacy classes of subgroups; only the 491 normal subgroups are shown.

normal subgroups only

This group has 1202 conjugacy classes of subgroups — too many to draw. Only its 491 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	#16840478	C₂	1	9	yes	maximal, complemented
2	#16840434	C₂	1	6	yes	maximal, complemented
3	#16840313	C₂	1	9	yes	maximal, complemented
4	C₄×#11758226	C₂	1	6	yes	maximal, complemented
5	Q₈²×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11758226	C₂²	1	6	yes	—
7	C₂×#11758226	C₂²	1	6	yes	—
8	#14519774	C₂²	1	9	yes	—
9	#14519766	C₂²	1	18	yes	—
10	#14519752	C₂²	1	36	yes	—
11	Q₈×Dic₆₁	C₂²	1	2	yes	—
12	C₄×D₂₄₄	C₂²	1	18	yes	—
13	#14519739	C₂²	1	18	yes	—
14	#14519735	C₂²	1	18	yes	—
15	C₄²×D₆₁	C₂²	1	9	yes	—
16	C₄⋊Q₈×C₆₁	C₂²	1	9	yes	—
17	C₄×Q₈×C₆₁	C₂²	1	6	yes	—
18	C₂×Q₈×C₆₁	C₂³	1	6	yes	—
19	C₂×Q₈×C₆₁	C₂³	1	2	yes	—
20	#11758226	—	2	24	no	—
21	#11758226	—	4	12	no	—
22	C₂×D₂₄₄	C₂³	1	18	yes	—
23	C₂×D₂₄₄	C₂³	1	18	yes	—
24	C₂×C₄×D₆₁	C₂³	1	9	yes	—
25	C₂×C₄×D₆₁	C₂³	1	18	yes	—
26	C₄⋊C₄×C₆₁	C₂³	1	18	yes	—
27	C₄²×C₆₁	C₂³	1	9	yes	—
28	#11758198	C₂³	1	36	yes	—
29	C₄⋊Dic₆₁	C₂³	1	6	yes	—
30	C₄×Dic₆₁	C₂³	1	6	yes	—
31	C₄×Dic₆₁	C₂³	1	9	yes	—
32	C₂²×D₆₁	C₂⁴	1	9	yes	—
33	Q₈×C₆₁	—	4	6	no	—
34	Q₈×C₆₁	#41	1	8	yes	complemented
35	C₂×C₄×C₆₁	C₂⁴	1	9	yes	—
36	C₂×C₄×C₆₁	C₂⁴	1	6	yes	—
37	C₂×Dic₆₁	C₂⁴	1	6	yes	—
38	C₂×Dic₆₁	C₂⁴	1	1	yes	characteristic
39	D₂₄₄	—	2	36	no	—
40	D₂₄₄	—	4	18	no	—
41	C₄×D₆₁	—	2	36	no	—
42	C₄×D₆₁	—	2	18	no	—
43	C₂²×C₆₁	C₂⁵	1	1	yes	characteristic
44	C₂×D₆₁	—	2	18	no	—
45	C₂×D₆₁	—	2	9	no	—
46	C₄×C₆₁	—	2	9	no	—
47	C₄×C₆₁	C₂×#41	1	12	yes	—
48	Dic₆₁	—	2	6	no	—
49	Dic₆₁	H(5,𝔽₂)	1	2	yes	—
50	#3483	—	61	1	no	maximal, Sylow
51	C₂×C₆₁	C₂²×#41	1	2	yes	—
52	C₂×C₆₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
53	D₆₁	—	4	9	no	—
54	Q₈²	D₆₁	1	1	yes	characteristic, complemented
55	#555	—	61	9	no	—
56	#550	—	61	6	no	—
57	#532	—	61	9	no	—
58	C₄×#41	—	61	6	no	—
59	C₆₁	#3483	1	1	yes	characteristic, complemented, Sylow
60	C₂×#41	—	61	6	no	—
61	C₂×#41	—	61	6	no	—
62	C₄⋊Q₈	C₂×D₆₁	1	9	yes	—
63	#127	—	61	9	no	—
64	#124	—	61	18	no	—
65	#121	—	61	36	no	—
66	C₄×Q₈	—	61	2	no	—
67	C₄×Q₈	C₂×D₆₁	1	6	yes	—
68	C₄×D₄	—	61	18	no	—
69	C₄×D₄	—	61	18	no	—
70	#117	—	61	18	no	—
71	C₂×C₄²	—	61	9	no	—
72	#41	—	122	24	no	—
73	#41	—	244	12	no	—
74	C₂×Q₈	C₂²×D₆₁	1	6	yes	—
75	C₂×Q₈	C₂²×D₆₁	1	2	yes	—
76	C₂×D₄	—	61	18	no	—
77	C₂×D₄	—	61	18	no	—
78	C₂²×C₄	—	61	18	no	—
79	C₂²×C₄	—	61	9	no	—
80	C₄⋊C₄	—	61	6	no	—
81	C₄⋊C₄	C₂²×D₆₁	1	18	yes	—
82	C₂²⋊C₄	—	61	36	no	—
83	C₄²	—	61	6	no	—
84	C₄²	—	61	9	no	—
85	C₄²	C₂²×D₆₁	1	9	yes	—
86	C₂³	—	61	9	no	—
87	Q₈	—	4	6	no	—
88	Q₈	#11758226	1	8	yes	complemented
89	D₄	—	122	36	no	—
90	D₄	—	244	18	no	—
91	C₂×C₄	—	61	1	no	—
92	C₂×C₄	—	61	6	no	—
93	C₂×C₄	—	122	18	no	—
94	C₂×C₄	—	122	36	no	—
95	C₂×C₄	C₂³×D₆₁	1	9	yes	—
96	C₂×C₄	C₂³×D₆₁	1	6	yes	—
97	C₂²	—	122	18	no	—
98	C₂²	—	122	9	no	—
99	C₂²	C₂⁴×D₆₁	1	1	yes	characteristic
100	C₄	—	2	9	no	—
101	C₄	—	61	2	no	—
102	C₄	—	122	6	no	—
103	C₄	C₂×#11758226	1	12	yes	—
104	C₂	—	244	9	no	—
105	C₂	C₂²×#11758226	1	2	yes	—
106	C₂	H(5,𝔽₂)×D₆₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840478

C₂

yes

maximal, complemented

#16840434

C₂

yes

maximal, complemented

#16840313

C₂

yes

maximal, complemented

C₄×#11758226

C₂

yes

maximal, complemented

Q₈²×C₆₁

C₂

yes

maximal, characteristic, complemented

C₂×#11758226

C₂²

yes

—

C₂×#11758226

C₂²

yes

—

#14519774

C₂²

yes

—

#14519766

C₂²

yes

—

#14519752

C₂²

yes

—

Q₈×Dic₆₁

C₂²

yes

—

C₄×D₂₄₄

C₂²

yes

—

#14519739

C₂²

yes

—

#14519735

C₂²

yes

—

C₄²×D₆₁

C₂²

yes

—

C₄⋊Q₈×C₆₁

C₂²

yes

—

C₄×Q₈×C₆₁

C₂²

yes

—

C₂×Q₈×C₆₁

C₂³

yes

—

C₂×Q₈×C₆₁

C₂³

yes

—

#11758226

—

#11758226

—

C₂×D₂₄₄

C₂³

yes

—

C₂×D₂₄₄

C₂³

yes

—

C₂×C₄×D₆₁

C₂³

yes

—

C₂×C₄×D₆₁

C₂³

yes

—

C₄⋊C₄×C₆₁

C₂³

yes

—

C₄²×C₆₁

C₂³

yes

—

#11758198

C₂³

yes

—

C₄⋊Dic₆₁

C₂³

yes

—

C₄×Dic₆₁

C₂³

yes

—

C₄×Dic₆₁

C₂³

yes

—

C₂²×D₆₁

C₂⁴

yes

—

Q₈×C₆₁

—

Q₈×C₆₁

#41

yes

complemented

C₂×C₄×C₆₁

C₂⁴

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

—

C₂×Dic₆₁

C₂⁴

yes

—

C₂×Dic₆₁

C₂⁴

yes

characteristic

D₂₄₄

—

D₂₄₄

—

C₄×D₆₁

—

C₄×D₆₁

—

C₂²×C₆₁

C₂⁵

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×#41

yes

—

Dic₆₁

—

Dic₆₁

H(5,𝔽₂)

yes

—

#3483

—

maximal, Sylow

C₂×C₆₁

C₂²×#41

yes

—

C₂×C₆₁

C₂×H(5,𝔽₂)

yes

characteristic

D₆₁

—

Q₈²

D₆₁

yes

characteristic, complemented

#555

—

#550

—

#532

—

C₄×#41

—

C₆₁

#3483

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×#41

—

C₄⋊Q₈

C₂×D₆₁

yes

—

#127

—

#124

—

#121

—

C₄×Q₈

—

C₄×Q₈

C₂×D₆₁

yes

—

C₄×D₄

—

C₄×D₄

—

#117

—

C₂×C₄²

—

#41

—

122

—

#41

—

244

—

C₂×Q₈

C₂²×D₆₁

yes

—

C₂×Q₈

C₂²×D₆₁

yes

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₆₁

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²×D₆₁

yes

—

C₂³

—

Q₈

—

Q₈

#11758226

yes

complemented

D₄

—

122

—

D₄

—

244

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

122

—

C₂×C₄

—

122

—

C₂×C₄

C₂³×D₆₁

yes

—

C₂×C₄

C₂³×D₆₁

yes

—

C₂²

—

122

—

C₂²

—

122

—

C₂²

C₂⁴×D₆₁

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

122

—

103

C₄

C₂×#11758226

yes

—

104

C₂

—

244

—

105

C₂

C₂²×#11758226

yes

—

106

C₂

H(5,𝔽₂)×D₆₁

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₆₁
commutator, socle	C₂²×C₆₁	C₂⁵
Fitting	Q₈²×C₆₁	C₂
hypercenter	Q₈²	D₆₁
hypocenter	C₆₁	#3483
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ Q₈²

subgroup

G/H

center, Frattini

C₂²

C₂⁴×D₆₁

commutator, socle

C₂²×C₆₁

C₂⁵

Fitting

Q₈²×C₆₁

C₂

hypercenter

Q₈²

D₆₁

C₆₁

G ▷ C₂²×C₆₁ ▷ C₆₁

C₁ ◁ C₂² ◁ Q₈²

N	H ≅ G/N	type
1. #16840478	C₂	semidirect
2. #16840434	C₂	semidirect
3. #16840313	C₂	semidirect
4. C₄×#11758226	C₂	semidirect
5. Q₈²×C₆₁	C₂	semidirect
34. Q₈×C₆₁	#41	semidirect
54. Q₈²	D₆₁	semidirect
59. C₆₁	#3483	semidirect
88. Q₈	#11758226	semidirect

H ≅ G/N

type

1. #16840478

C₂

semidirect

2. #16840434

C₂

semidirect

3. #16840313

C₂

semidirect

4. C₄×#11758226

C₂

semidirect

5. Q₈²×C₆₁

C₂

semidirect

34. Q₈×C₆₁

#41

semidirect

54. Q₈²

D₆₁

semidirect

59. C₆₁

#3483

semidirect

88. Q₈

#11758226

semidirect

Subgroups — #21459446

Conjugacy classes of subgroups

Complements