Group #21454215 — subgroups

Subgroups — #21454215

Order 7808.

648 conjugacy classes of subgroups; only the 245 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 245 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×#14519777	C₂	1	1	yes	maximal, characteristic
2	C₂×#14519775	C₂	1	4	yes	maximal
3	#16839631	C₂	1	8	yes	maximal
4	C₄²×Dic₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₂₂	C₂²	1	2	yes	—
7	#14519777	—	2	4	no	—
8	#14519775	—	2	16	no	—
9	C₂×C₄⋊Dic₆₁	C₂²	1	4	yes	—
10	C₂×#11758196	C₂²	1	16	yes	—
11	C₂×C₄×Dic₆₁	C₂²	1	2	yes	—
12	C₂×C₄×Dic₆₁	C₂²	1	4	yes	—
13	C₂²×Q₈×C₆₁	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₆₁	—	2	4	no	—
15	C₂×C₄⋊C₄×C₆₁	C₂²	1	4	yes	—
16	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₆₁	—	2	4	no	—
18	C₂×Q₈×C₆₁	—	2	8	no	—
19	C₂²×C₄×C₆₁	C₂³	1	4	yes	—
20	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
21	C₂²×C₄×C₆₁	C₂³	1	2	yes	—
22	C₂²×Dic₆₁	C₂³	1	4	yes	—
23	C₂²×Dic₆₁	C₂³	1	4	yes	—
24	C₂×Dic₁₂₂	—	2	8	no	—
25	C₂×Dic₁₂₂	—	2	4	no	—
26	C₄⋊C₄×C₆₁	—	2	8	no	—
27	C₄²×C₆₁	D₄	1	4	yes	—
28	C₄⋊Dic₆₁	—	2	8	no	—
29	#11758196	—	2	32	no	—
30	C₄×Dic₆₁	D₄	1	8	yes	—
31	C₄×Dic₆₁	Q₈	1	16	yes	complemented
32	C₂³×C₆₁	C₂⁴	1	1	yes	characteristic
33	Q₈×C₆₁	—	4	8	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×D₄	1	2	yes	—
38	C₂×C₄×C₆₁	C₂×D₄	1	4	yes	—
39	C₂×C₄×C₆₁	C₂×Q₈	1	8	yes	—
40	C₂×Dic₆₁	—	2	8	no	—
41	C₂×Dic₆₁	—	2	4	no	—
42	C₂×Dic₆₁	C₂×D₄	1	8	yes	—
43	C₂×Dic₆₁	C₂×Q₈	1	16	yes	—
44	Dic₁₂₂	—	4	8	no	—
45	C₂²×C₆₁	C₂²×D₄	1	2	yes	—
46	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
47	C₂²×C₆₁	C₂²×Q₈	1	4	yes	—
48	C₄×C₆₁	—	4	4	no	—
49	C₄×C₆₁	#127	1	4	yes	—
50	C₄×C₆₁	C₄⋊Q₈	1	8	yes	complemented
51	Dic₆₁	—	4	4	no	—
52	Dic₆₁	C₄⋊Q₈	1	16	yes	complemented
53	#2868	—	61	1	no	maximal, Sylow
54	C₂×C₆₁	C₂×#127	1	1	yes	characteristic
55	C₂×C₆₁	C₂×C₄⋊Q₈	1	4	yes	—
56	C₂×C₆₁	C₂×C₄⋊Q₈	1	2	yes	—
57	C₂×C₄⋊Q₈	—	61	4	no	—
58	C₂×C₄⋊Q₈	—	61	1	no	—
59	C₂×C₄⋊Q₈	D₆₁	1	1	yes	characteristic
60	#384	—	61	8	no	—
61	C₄³	—	61	1	no	—
62	C₆₁	#2868	1	1	yes	characteristic, complemented, Sylow
63	C₂²×Q₈	—	61	2	no	—
64	C₂²×Q₈	C₂×D₆₁	1	2	yes	—
65	C₄⋊Q₈	—	2	4	no	—
66	C₄⋊Q₈	—	122	16	no	—
67	C₄⋊Q₈	—	122	4	no	—
68	C₂×C₄⋊C₄	—	61	16	no	—
69	C₂×C₄⋊C₄	—	61	4	no	—
70	C₂×C₄⋊C₄	C₂×D₆₁	1	4	yes	—
71	C₂×C₄²	—	61	2	no	—
72	C₂×C₄²	—	61	4	no	—
73	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
74	C₂×Q₈	—	2	8	no	—
75	C₂×Q₈	—	2	4	no	—
76	C₂×Q₈	—	122	8	no	—
77	C₂×Q₈	—	122	4	no	—
78	C₂²×C₄	—	61	4	no	—
79	C₂²×C₄	—	61	4	no	—
80	C₂²×C₄	C₂²×D₆₁	1	4	yes	—
81	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
82	C₂²×C₄	C₂²×D₆₁	1	2	yes	—
83	C₄⋊C₄	—	2	8	no	—
84	C₄⋊C₄	—	122	32	no	—
85	C₄⋊C₄	—	122	8	no	—
86	C₄²	—	61	8	no	—
87	C₄²	—	61	16	no	—
88	C₄²	C₂²⋊D₆₁	1	4	yes	—
89	C₂³	C₂³×D₆₁	1	1	yes	characteristic
90	Q₈	—	4	8	no	—
91	Q₈	—	244	8	no	—
92	C₂×C₄	—	2	8	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	61	8	no	—
95	C₂×C₄	—	61	16	no	—
96	C₂×C₄	—	122	8	no	—
97	C₂×C₄	—	122	4	no	—
98	C₂×C₄	D₄×D₆₁	1	4	yes	—
99	C₂×C₄	Q₈×D₆₁	1	8	yes	—
100	C₂×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
101	C₂×C₄	C₂×C₂²⋊D₆₁	1	4	yes	—
102	C₂²	C₂×D₄×D₆₁	1	2	yes	—
103	C₂²	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
104	C₂²	C₂×Q₈×D₆₁	1	4	yes	—
105	C₄	—	4	4	no	—
106	C₄	—	61	16	no	—
107	C₄	—	244	4	no	—
108	C₄	#14519773	1	4	yes	—
109	C₄	#14519775	1	8	yes	complemented
110	C₂	C₂×#14519773	1	1	yes	characteristic
111	C₂	C₄⋊Q₈×D₆₁	1	4	yes	—
112	C₂	C₂×#14519775	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519777

C₂

yes

maximal, characteristic

C₂×#14519775

C₂

yes

maximal

#16839631

C₂

yes

maximal

C₄²×Dic₆₁

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₆₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₂₂

C₂²

yes

—

#14519777

—

#14519775

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×#11758196

C₂²

yes

—

C₂×C₄×Dic₆₁

C₂²

yes

—

C₂×C₄×Dic₆₁

C₂²

yes

—

C₂²×Q₈×C₆₁

C₂²

yes

—

C₄⋊Q₈×C₆₁

—

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

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

—

C₄²×C₆₁

D₄

yes

—

C₄⋊Dic₆₁

—

#11758196

—

C₄×Dic₆₁

D₄

yes

—

C₄×Dic₆₁

Q₈

yes

complemented

C₂³×C₆₁

C₂⁴

yes

characteristic

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂×D₄

yes

—

C₂×Dic₆₁

C₂×Q₈

yes

—

Dic₁₂₂

—

C₂²×C₆₁

C₂²×D₄

yes

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₂²×C₆₁

C₂²×Q₈

yes

—

C₄×C₆₁

—

C₄×C₆₁

#127

yes

—

C₄×C₆₁

C₄⋊Q₈

yes

complemented

Dic₆₁

—

Dic₆₁

C₄⋊Q₈

yes

—

C₂×C₆₁

yes

C₂×C₆₁

C₂×C₄⋊Q₈

yes

—

C₂×C₆₁

C₂×C₄⋊Q₈

yes

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

D₆₁

yes

characteristic

#384

—

C₄³

—

C₆₁

#2868

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

C₂×D₆₁

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

122

—

C₄⋊Q₈

—

122

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

122

—

C₂×Q₈

—

122

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

C₂²×C₄

C₂²×D₆₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄²

—

C₄²

—

C₄²

C₂²⋊D₆₁

yes

—

C₂³

C₂³×D₆₁

yes

characteristic

Q₈

—

Q₈

—

244

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

122

—

C₂×C₄

—

122

—

C₂×C₄

D₄×D₆₁

yes

—

C₂×C₄

Q₈×D₆₁

yes

—

100

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

101

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

102

C₂²

C₂×D₄×D₆₁

yes

—

103

C₂²

C₂²×C₂²⋊D₆₁

yes

characteristic

104

C₂²

C₂×Q₈×D₆₁

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

244

—

108

C₄

#14519773

yes

—

109

C₄

#14519775

yes

complemented

110

C₂

C₂×#14519773

yes

characteristic

111

C₂

C₄⋊Q₈×D₆₁

yes

—

112

C₂

C₂×#14519775

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₁
commutator, socle	C₂³×C₆₁	C₂⁴
Fitting	C₂×C₄⋊Q₈×C₆₁	C₂
hypercenter	C₂×C₄⋊Q₈	D₆₁
hypocenter	C₆₁	#2868
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₆₁

commutator, socle

C₂³×C₆₁

C₂⁴

Fitting

C₂×C₄⋊Q₈×C₆₁

C₂

hypercenter

C₂×C₄⋊Q₈

D₆₁

C₆₁

G ▷ C₂³×C₆₁ ▷ C₆₁

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
31. C₄×Dic₆₁	Q₈	semidirect
50. C₄×C₆₁	C₄⋊Q₈	semidirect
52. Dic₆₁	C₄⋊Q₈	semidirect
62. C₆₁	#2868	semidirect
109. C₄	#14519775	semidirect

H ≅ G/N

type

31. C₄×Dic₆₁

Q₈

semidirect

50. C₄×C₆₁

C₄⋊Q₈

semidirect

52. Dic₆₁

C₄⋊Q₈

semidirect

62. C₆₁

#2868

semidirect

109. C₄

#14519775

semidirect

Subgroups — #21454215

Conjugacy classes of subgroups

Complements