Group #21461353 — subgroups

Subgroups — #21461353

Order 7808.

228 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 60 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	#16840785	C₂	1	2	yes	maximal
2	#16840622	C₂	1	4	yes	maximal, complemented
3	C₄⋊Q₈×D₆₁	C₂	1	1	yes	maximal, characteristic
4	#14519844	C₂²	1	4	yes	—
5	#14519834	—	2	2	no	—
6	#14519832	—	2	2	no	—
7	C₂×Q₈×D₆₁	C₄	1	2	yes	—
8	#14519777	C₄	1	1	yes	characteristic
9	#14519776	—	2	2	no	—
10	#14519775	C₂²	1	2	yes	—
11	C₄⋊C₄×D₆₁	—	2	2	no	—
12	C₄²×D₆₁	C₂²	1	1	yes	characteristic
13	C₄⋊Q₈×C₆₁	C₄	1	1	yes	characteristic
14	C₂×Q₈×C₆₁	C₂×C₄	1	2	yes	—
15	Q₈×D₆₁	—	4	4	no	—
16	C₂×C₄×D₆₁	—	2	2	no	—
17	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×D₆₁	D₄	1	2	yes	—
19	C₂×Dic₁₂₂	—	2	2	no	—
20	C₂×Dic₁₂₂	C₂×C₄	1	2	yes	—
21	C₂×#92420	—	2	4	no	—
22	C₄⋊C₄×C₆₁	—	2	2	no	—
23	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
24	C₄⋊Dic₆₁	—	2	2	no	—
25	#11758196	—	2	4	no	—
26	C₄×Dic₆₁	D₄	1	2	yes	—
27	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
28	C₂²×D₆₁	C₂²⋊C₄	1	1	yes	characteristic
29	Q₈×C₆₁	—	4	2	no	—
30	C₂×C₄×C₆₁	—	2	2	no	—
31	C₂×C₄×C₆₁	C₂²⋊C₄	1	2	yes	—
32	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
33	C₂×Dic₆₁	—	2	2	no	—
34	C₂×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×Dic₆₁	C₂×D₄	1	2	yes	—
36	C₄×D₆₁	—	2	2	no	—
37	C₄×D₆₁	—	2	4	no	—
38	C₄×D₆₁	—	4	4	no	—
39	Dic₁₂₂	—	4	4	no	—
40	Dic₁₂₂	—	4	2	no	—
41	#92420	—	4	4	no	—
42	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
43	C₂×D₆₁	—	2	2	no	—
44	C₂×D₆₁	#100	1	2	yes	—
45	C₄×C₆₁	—	2	1	no	—
46	C₄×C₆₁	—	2	2	no	—
47	C₄×C₆₁	—	4	2	no	—
48	Dic₆₁	—	2	1	no	—
49	Dic₆₁	—	4	2	no	—
50	Dic₆₁	#137	1	4	yes	—
51	#1558	—	61	1	no	maximal, Sylow
52	C₂×C₆₁	C₂×#100	1	1	yes	characteristic
53	C₂×C₆₁	#419	1	2	yes	—
54	D₆₁	—	2	2	no	—
55	C₂×C₄⋊Q₈	—	61	1	no	—
56	#432	—	61	2	no	—
57	#333	—	61	4	no	—
58	C₆₁	#1558	1	1	yes	characteristic, complemented, Sylow
59	C₂²×Q₈	—	61	2	no	—
60	C₄⋊Q₈	—	61	2	no	—
61	C₄⋊Q₈	—	61	1	no	—
62	C₄⋊Q₈	—	122	2	no	—
63	C₄⋊Q₈	#6923	1	1	yes	characteristic
64	C₂×C₄⋊C₄	—	122	2	no	—
65	C₂×C₄²	—	61	1	no	—
66	C₄⋊C₈	—	61	4	no	—
67	C₂²⋊C₈	—	122	2	no	—
68	#97	—	122	2	no	—
69	C₂×Q₈	—	61	2	no	—
70	C₂×Q₈	—	122	2	no	—
71	C₂×Q₈	—	244	4	no	—
72	C₂×Q₈	C₂×#6923	1	2	yes	—
73	C₂²×C₄	—	61	2	no	—
74	C₂²×C₄	—	61	1	no	—
75	C₂²×C₄	—	122	2	no	—
76	C₂×C₈	—	122	4	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄⋊C₄	—	122	4	no	—
79	C₄⋊C₄	—	122	2	no	—
80	C₄²	—	61	2	no	—
81	C₄²	—	61	1	no	—
82	C₄²	C₂×#6923	1	1	yes	characteristic
83	C₂³	—	61	1	no	—
84	Q₈	—	4	2	no	—
85	Q₈	—	244	2	no	—
86	Q₈	—	244	4	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	61	2	no	—
89	C₂×C₄	—	61	1	no	—
90	C₂×C₄	—	122	4	no	—
91	C₂×C₄	—	122	2	no	—
92	C₂×C₄	—	122	2	no	—
93	C₂×C₄	—	244	4	no	—
94	C₂×C₄	C₂²⋊#6923	1	2	yes	—
95	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
96	C₈	—	244	4	no	—
97	C₂²	—	61	2	no	—
98	C₂²	—	122	2	no	—
99	C₂²	C₂×C₂²⋊#6923	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	2	2	no	—
102	C₄	—	4	2	no	—
103	C₄	—	61	4	no	—
104	C₄	—	122	1	no	—
105	C₄	—	244	2	no	—
106	C₂	—	122	2	no	—
107	C₂	D₆₁⋊#100	1	1	yes	characteristic
108	C₂	#16840755	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16840785

C₂

yes

maximal

#16840622

C₂

yes

maximal, complemented

C₄⋊Q₈×D₆₁

C₂

yes

maximal, characteristic

#14519844

C₂²

yes

—

#14519834

—

#14519832

—

C₂×Q₈×D₆₁

C₄

yes

—

#14519777

C₄

yes

characteristic

#14519776

—

#14519775

C₂²

yes

—

C₄⋊C₄×D₆₁

—

C₄²×D₆₁

C₂²

yes

characteristic

C₄⋊Q₈×C₆₁

C₄

yes

characteristic

C₂×Q₈×C₆₁

C₂×C₄

yes

—

Q₈×D₆₁

—

C₂×C₄×D₆₁

—

C₂×C₄×D₆₁

C₂×C₄

yes

characteristic

C₂×C₄×D₆₁

D₄

yes

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

C₂×C₄

yes

—

C₂×#92420

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄⋊Dic₆₁

—

#11758196

—

C₄×Dic₆₁

D₄

yes

—

C₄×Dic₆₁

C₂³

yes

characteristic

C₂²×D₆₁

C₂²⋊C₄

yes

characteristic

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₆₁

C₂×D₄

yes

—

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

#92420

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

#100

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

#137

yes

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

—

D₆₁

—

C₂×C₄⋊Q₈

—

#432

—

#333

—

C₆₁

#1558

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

122

—

C₄⋊Q₈

#6923

yes

characteristic

C₂×C₄⋊C₄

—

122

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

122

—

#97

—

122

—

C₂×Q₈

—

C₂×Q₈

—

122

—

C₂×Q₈

—

244

—

C₂×Q₈

C₂×#6923

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

122

—

C₂×C₈

—

122

—

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄²

—

C₄²

—

C₄²

C₂×#6923

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

244

—

Q₈

—

244

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

122

—

C₂×C₄

—

122

—

C₂×C₄

—

122

—

C₂×C₄

—

244

—

C₂×C₄

C₂²⋊#6923

yes

—

C₂×C₄

C₂²×#6923

yes

characteristic

C₈

—

244

—

C₂²

—

C₂²

—

122

—

C₂²

C₂×C₂²⋊#6923

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

—

122

—

105

C₄

—

244

—

106

C₂

—

122

—

107

C₂

D₆₁⋊#100

yes

characteristic

108

C₂

#16840755

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#6923
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6923
Fitting	C₄⋊Q₈×C₆₁	C₄
socle	C₂²×C₆₁	C₂×C₂²⋊C₄
hypercenter	C₄⋊Q₈	#6923
hypocenter	C₆₁	#1558
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#6923

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6923

Fitting

C₄⋊Q₈×C₆₁

C₄

socle

C₂²×C₆₁

C₂×C₂²⋊C₄

C₄⋊Q₈

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
2. #16840622	C₂	semidirect
58. C₆₁	#1558	semidirect

H ≅ G/N

type

2. #16840622

C₂

semidirect

58. C₆₁

#1558

semidirect

Subgroups — #21461353

Conjugacy classes of subgroups

Complements