Group #22612073 — subgroups

Subgroups — #22612073

Order 9344.

130 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 130 conjugacy classes of subgroups — too many to draw. Only its 57 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	#17988992	C₂	1	1	yes	maximal, characteristic
2	#17988414	C₂	1	2	yes	maximal
3	C₄⋊#11924003	C₂	1	1	yes	maximal, characteristic
4	#17988391	C₂	1	2	yes	maximal
5	C₄×Q₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
6	#14556591	—	2	1	no	—
7	C₂×#11924003	C₂²	1	2	yes	—
8	#14556525	—	2	1	no	—
9	#14556525	C₂²	1	2	yes	—
10	C₂×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
11	C₄×Q₈×C₇₃	—	2	1	no	—
12	#107×C₇₃	C₂²	1	1	yes	characteristic
13	#103×C₇₃	—	2	1	no	—
14	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₄₆	—	4	1	no	—
16	C₄⋊Dic₇₃	—	2	2	no	—
17	C₄⋊Dic₇₃	—	4	1	no	—
18	#11924003	—	2	2	no	—
19	C₂×Q₈×C₇₃	—	2	1	no	—
20	Q₁₆×C₇₃	—	2	1	no	—
21	Q₁₆×C₇₃	Q₈	1	2	yes	complemented
22	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
23	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₇₃	—	2	1	no	—
25	C₄⋊C₄×C₇₃	—	4	1	no	—
26	C₄²×C₇₃	—	4	1	no	—
27	C₄²×C₇₃	D₄	1	1	yes	characteristic
28	Q₈×C₇₃	—	2	2	no	—
29	Q₈×C₇₃	—	4	1	no	—
30	C₂×C₄×C₇₃	—	4	1	no	—
31	C₂×C₄×C₇₃	—	4	1	no	—
32	C₂×C₄×C₇₃	D₈	1	1	yes	characteristic
33	C₂×C₄×C₇₃	D₈	1	1	yes	characteristic
34	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
35	C₂×Dic₇₃	—	4	2	no	—
36	Dic₁₄₆	—	8	2	no	—
37	C₈×C₇₃	—	2	1	no	—
38	C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
39	C₈×C₇₃	#41	1	1	yes	characteristic
40	C₂²×C₇₃	C₂×D₈	1	1	yes	characteristic
41	C₄×C₇₃	—	2	1	no	—
42	C₄×C₇₃	—	4	2	no	—
43	C₄×C₇₃	—	8	1	no	—
44	C₄×C₇₃	SD₃₂	1	2	yes	—
45	C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
46	C₄×C₇₃	#137	1	1	yes	characteristic
47	Dic₇₃	—	8	2	no	—
48	C₂×C₇₃	#474	1	1	yes	characteristic
49	C₂×C₇₃	C₂×SD₃₂	1	1	yes	characteristic
50	C₂×C₇₃	#510	1	1	yes	characteristic
51	#2225	—	73	1	no	maximal, Sylow
52	C₇₃	#2225	1	1	yes	characteristic, complemented, Sylow
53	#500	—	73	1	no	—
54	C₄×Q₁₆	D₇₃	1	1	yes	characteristic
55	#367	—	73	2	no	—
56	C₄⋊C₁₆	—	73	1	no	—
57	#358	—	73	2	no	—
58	C₂×Q₁₆	C₂×D₇₃	1	1	yes	characteristic
59	C₄⋊Q₈	—	146	1	no	—
60	C₄×Q₈	—	2	1	no	—
61	C₂×C₁₆	—	73	2	no	—
62	#107	—	73	2	no	—
63	#107	—	146	1	no	—
64	#107	C₂×D₇₃	1	1	yes	characteristic
65	#103	—	2	1	no	—
66	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
67	C₂×Q₈	—	2	1	no	—
68	C₂×Q₈	—	292	1	no	—
69	Q₁₆	—	2	1	no	—
70	Q₁₆	Dic₁₄₆	1	2	yes	complemented
71	C₂×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
72	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
73	C₄⋊C₄	—	2	1	no	—
74	C₄⋊C₄	—	4	1	no	—
75	C₄⋊C₄	—	292	1	no	—
76	C₄⋊C₄	—	146	2	no	—
77	C₄²	—	4	1	no	—
78	C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
79	C₁₆	—	146	2	no	—
80	Q₈	—	2	2	no	—
81	Q₈	—	4	1	no	—
82	Q₈	—	584	2	no	—
83	C₂×C₄	—	4	1	no	—
84	C₂×C₄	—	4	1	no	—
85	C₂×C₄	—	292	2	no	—
86	C₂×C₄	#11924013	1	1	yes	characteristic
87	C₂×C₄	#11924013	1	1	yes	characteristic
88	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
89	C₈	—	2	1	no	—
90	C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
91	C₈	#11924028	1	1	yes	characteristic
92	C₂²	C₂×#11924013	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	4	2	no	—
95	C₄	—	8	1	no	—
96	C₄	—	584	2	no	—
97	C₄	#14556534	1	2	yes	—
98	C₄	#14556570	1	1	yes	characteristic
99	C₄	#14556621	1	1	yes	characteristic
100	C₂	#17988867	1	1	yes	characteristic
101	C₂	C₂×#14556534	1	1	yes	characteristic
102	C₂	#17989027	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988992

C₂

yes

maximal, characteristic

#17988414

C₂

yes

maximal

C₄⋊#11924003

C₂

yes

maximal, characteristic

#17988391

C₂

yes

maximal

C₄×Q₁₆×C₇₃

C₂

yes

maximal, characteristic

#14556591

—

C₂×#11924003

C₂²

yes

—

#14556525

—

#14556525

C₂²

yes

—

C₂×Q₁₆×C₇₃

C₂²

yes

characteristic

C₄×Q₈×C₇₃

—

#107×C₇₃

C₂²

yes

characteristic

#103×C₇₃

—

C₄×C₈×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924003

—

C₂×Q₈×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

Q₈

yes

complemented

C₂×C₈×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

D₄

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

D₈

yes

characteristic

C₂×C₄×C₇₃

D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

C₂×Q₈

yes

characteristic

C₈×C₇₃

#41

yes

characteristic

C₂²×C₇₃

C₂×D₈

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

SD₃₂

yes

—

C₄×C₇₃

C₂²⋊Q₈

yes

characteristic

C₄×C₇₃

#137

yes

characteristic

Dic₇₃

—

C₂×C₇₃

#474

yes

characteristic

C₂×C₇₃

C₂×SD₃₂

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

#500

—

C₄×Q₁₆

D₇₃

yes

characteristic

#367

—

C₄⋊C₁₆

—

#358

—

C₂×Q₁₆

C₂×D₇₃

yes

characteristic

C₄⋊Q₈

—

146

—

C₄×Q₈

—

C₂×C₁₆

—

#107

—

#107

—

146

—

#107

C₂×D₇₃

yes

characteristic

#103

—

C₄×C₈

C₂×D₇₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

292

—

Q₁₆

—

Q₁₆

Dic₁₄₆

yes

complemented

C₂×C₈

C₂²⋊D₇₃

yes

characteristic

C₂×C₈

C₂²×D₇₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

292

—

C₄⋊C₄

—

146

—

C₄²

—

C₄²

C₂²⋊D₇₃

yes

characteristic

C₁₆

—

146

—

Q₈

—

Q₈

—

Q₈

—

584

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

292

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

C₈

—

C₈

C₂×Dic₁₄₆

yes

C₈

yes

C₂²

yes

C₄

—

C₄

—

C₄

—

C₄

—

584

—

C₄

#14556534

yes

—

C₄

yes

C₄

yes

100

C₂

#17988867

yes

characteristic

101

C₂

C₂×#14556534

yes

characteristic

102

C₂

#17989027

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11924013
commutator	C₂×C₈×C₇₃	C₂³
Frattini	C₂×C₈	C₂²×D₇₃
Fitting	C₄×Q₁₆×C₇₃	C₂
socle	C₂²×C₇₃	C₂×D₈
hypercenter	C₄×Q₁₆	D₇₃
hypocenter	C₇₃	#2225
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₁₆

subgroup

G/H

center

C₂²

C₂×#11924013

commutator

C₂×C₈×C₇₃

C₂³

Frattini

C₂×C₈

C₂²×D₇₃

Fitting

C₄×Q₁₆×C₇₃

C₂

socle

C₂²×C₇₃

C₂×D₈

hypercenter

C₄×Q₁₆

D₇₃

hypocenter

C₇₃

#2225

LCS

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₁₆

N	H ≅ G/N	type
21. Q₁₆×C₇₃	Q₈	semidirect
52. C₇₃	#2225	semidirect
70. Q₁₆	Dic₁₄₆	semidirect

H ≅ G/N

type

21. Q₁₆×C₇₃

Q₈

semidirect

52. C₇₃

#2225

semidirect

70. Q₁₆

Dic₁₄₆

semidirect

Subgroups — #22612073

Conjugacy classes of subgroups

Complements