Group #21444959 — subgroups

Subgroups — #21444959

Order 7808.

206 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 65 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	#16840141	C₂	1	1	yes	maximal, characteristic
2	#16840057	C₂	1	2	yes	maximal
3	#16839511	C₂	1	1	yes	maximal, characteristic
4	Q₈⋊#92418	C₂	1	2	yes	maximal
5	Q₈²×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#11758202	—	2	2	no	—
7	#14519777	C₂²	1	1	yes	characteristic
8	#14519710	—	2	2	no	—
9	C₄⋊#92418	C₂²	1	2	yes	—
10	#14519701	—	2	2	no	—
11	C₄×#92418	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₆₁	—	2	2	no	—
13	C₄⋊Q₈×C₆₁	—	2	2	no	—
14	C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₄×Q₈×C₆₁	—	2	2	no	—
16	C₄×Q₈×C₆₁	C₂²	1	2	yes	—
17	C₂×Q₈×C₆₁	—	2	2	no	—
18	C₂×Q₈×C₆₁	D₄	1	2	yes	—
19	C₂×Q₈×C₆₁	D₄	1	2	yes	—
20	C₂×Dic₁₂₂	—	2	1	no	—
21	C₄⋊C₄×C₆₁	—	2	4	no	—
22	C₄⋊C₄×C₆₁	—	2	2	no	—
23	C₄⋊C₄×C₆₁	—	2	2	no	—
24	C₄⋊C₄×C₆₁	D₄	1	2	yes	—
25	C₄²×C₆₁	—	2	2	no	—
26	C₄²×C₆₁	—	2	2	no	—
27	C₄²×C₆₁	C₂³	1	1	yes	characteristic
28	#11758202	—	4	4	no	—
29	C₄⋊Dic₆₁	—	2	2	no	—
30	C₂×#92418	—	2	1	no	—
31	C₂×#92418	—	2	2	no	—
32	Q₈×C₆₁	—	2	2	no	—
33	Q₈×C₆₁	—	4	2	no	—
34	Q₈×C₆₁	—	8	2	no	—
35	Q₈×C₆₁	Q₁₆	1	4	yes	complemented
36	C₂×C₄×C₆₁	—	2	2	no	—
37	C₂×C₄×C₆₁	—	2	2	no	—
38	C₂×C₄×C₆₁	—	2	2	no	—
39	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
41	C₂×Dic₆₁	—	4	1	no	—
42	Dic₁₂₂	—	4	2	no	—
43	#92418	—	2	2	no	—
44	#92418	—	4	2	no	—
45	C₂²×C₆₁	#120	1	1	yes	characteristic
46	C₄×C₆₁	—	2	4	no	—
47	C₄×C₆₁	—	2	1	no	—
48	C₄×C₆₁	—	4	2	no	—
49	C₄×C₆₁	—	4	2	no	—
50	C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
51	C₄×C₆₁	#137	1	2	yes	—
52	Dic₆₁	—	8	1	no	—
53	#1633	—	61	1	no	maximal, Sylow
54	C₂×C₆₁	#451	1	2	yes	—
55	C₂×C₆₁	#453	1	1	yes	characteristic
56	Q₈²	D₆₁	1	1	yes	characteristic
57	#494	—	61	1	no	—
58	#477	—	61	2	no	—
59	#333	—	61	1	no	—
60	Q₈⋊C₈	—	61	2	no	—
61	C₆₁	#1633	1	1	yes	characteristic, complemented, Sylow
62	C₂×Q₁₆	—	122	2	no	—
63	C₄⋊Q₈	—	2	2	no	—
64	C₄⋊Q₈	—	2	2	no	—
65	C₄⋊Q₈	—	61	1	no	—
66	C₄⋊Q₈	C₂×D₆₁	1	1	yes	characteristic
67	C₄×Q₈	—	2	2	no	—
68	C₄×Q₈	C₂×D₆₁	1	2	yes	—
69	#107	—	122	2	no	—
70	C₄⋊C₈	—	61	2	no	—
71	#103	—	122	2	no	—
72	C₄×C₈	—	61	1	no	—
73	C₂×Q₈	—	2	2	no	—
74	C₂×Q₈	—	122	1	no	—
75	C₂×Q₈	C₂²⋊D₆₁	1	2	yes	—
76	C₂×Q₈	C₂²⋊D₆₁	1	2	yes	—
77	Q₁₆	—	244	4	no	—
78	C₂×C₈	—	122	1	no	—
79	C₂×C₈	—	122	2	no	—
80	C₄⋊C₄	—	2	4	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	122	2	no	—
84	C₄⋊C₄	C₂²⋊D₆₁	1	2	yes	—
85	C₄²	—	2	2	no	—
86	C₄²	—	2	2	no	—
87	C₄²	C₂²×D₆₁	1	1	yes	characteristic
88	Q₈	—	2	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	8	2	no	—
91	Q₈	—	244	2	no	—
92	Q₈	#11758202	1	4	yes	complemented
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	244	1	no	—
97	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
99	C₈	—	122	2	no	—
100	C₈	—	244	2	no	—
101	C₂²	#14519746	1	1	yes	characteristic
102	C₄	—	2	4	no	—
103	C₄	—	2	1	no	—
104	C₄	—	4	2	no	—
105	C₄	—	4	2	no	—
106	C₄	—	488	1	no	—
107	C₄	C₂×#11758202	1	2	yes	—
108	C₄	#14519808	1	2	yes	—
109	C₂	#16839918	1	2	yes	—
110	C₂	#16839930	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840141

C₂

yes

maximal, characteristic

#16840057

C₂

yes

maximal

#16839511

C₂

yes

maximal, characteristic

Q₈⋊#92418

C₂

yes

maximal

Q₈²×C₆₁

C₂

yes

maximal, characteristic

C₂×#11758202

—

#14519777

C₂²

yes

characteristic

#14519710

—

C₄⋊#92418

C₂²

yes

—

#14519701

—

C₄×#92418

C₂²

yes

characteristic

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

C₂²

yes

characteristic

C₄×Q₈×C₆₁

—

C₄×Q₈×C₆₁

C₂²

yes

—

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

D₄

yes

—

C₂×Q₈×C₆₁

D₄

yes

—

C₂×Dic₁₂₂

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

D₄

yes

—

C₄²×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂³

yes

characteristic

#11758202

—

C₄⋊Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

Q₈×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

Q₁₆

yes

complemented

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×Dic₆₁

—

Dic₁₂₂

—

#92418

—

#92418

—

C₂²×C₆₁

#120

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×Q₁₆

yes

—

C₄×C₆₁

#137

yes

—

Dic₆₁

—

#1633

—

maximal, Sylow

C₂×C₆₁

#451

yes

—

C₂×C₆₁

#453

yes

characteristic

Q₈²

D₆₁

yes

—

—

—

Q₈⋊C₈

—

C₆₁

#1633

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

122

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₆₁

yes

characteristic

C₄×Q₈

—

C₄×Q₈

C₂×D₆₁

yes

—

#107

—

122

—

C₄⋊C₈

—

#103

—

122

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

122

—

C₂×Q₈

C₂²⋊D₆₁

yes

—

C₂×Q₈

C₂²⋊D₆₁

yes

—

Q₁₆

—

244

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄⋊C₄

C₂²⋊D₆₁

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×D₆₁

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

244

—

Q₈

#11758202

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

244

—

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

C₈

—

122

—

100

C₈

—

244

—

101

C₂²

#14519746

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

488

—

107

C₄

C₂×#11758202

yes

—

108

C₄

#14519808

yes

—

109

C₂

#16839918

yes

—

110

C₂

#16839930

yes

characteristic

	subgroup	G/H
center	C₂²	#14519746
commutator	C₄²×C₆₁	C₂³
Frattini	C₄²	C₂²×D₆₁
Fitting	Q₈²×C₆₁	C₂
socle	C₂²×C₆₁	#120
hypercenter	Q₈²	D₆₁
hypocenter	C₆₁	#1633
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

subgroup

G/H

center

C₂²

#14519746

commutator

C₄²×C₆₁

C₂³

Frattini

C₄²

C₂²×D₆₁

Fitting

Q₈²×C₆₁

C₂

socle

C₂²×C₆₁

#120

hypercenter

Q₈²

D₆₁

hypocenter

C₆₁

#1633

LCS

G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

N	H ≅ G/N	type
35. Q₈×C₆₁	Q₁₆	semidirect
61. C₆₁	#1633	semidirect
92. Q₈	#11758202	semidirect

H ≅ G/N

type

35. Q₈×C₆₁

Q₁₆

semidirect

61. C₆₁

#1633

semidirect

92. Q₈

#11758202

semidirect

Subgroups — #21444959

Conjugacy classes of subgroups

Complements