Group #22612019 — subgroups

Subgroups — #22612019

Order 9344.

154 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 51 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×C₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic
2	#17988825	C₂	1	2	yes	maximal
3	#17988413	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988391	C₂	1	2	yes	maximal, complemented
5	C₂×Q₃₂×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₇₃	C₂²	1	1	yes	characteristic
7	#14556567	—	2	2	no	—
8	C₂×#11924003	C₂²	1	1	yes	characteristic
9	C₁₆×D₇₃	—	2	2	no	—
10	#14556525	C₂²	1	2	yes	—
11	#14556515	—	2	2	no	—
12	C₂×Q₁₆×C₇₃	C₂²	1	2	yes	—
13	Q₃₂×C₇₃	—	2	2	no	—
14	C₂×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
16	C₂×#10596831	D₄	1	1	yes	characteristic
17	C₈×D₇₃	—	2	1	no	—
18	C₈×D₇₃	—	2	1	no	—
19	C₄⋊Dic₇₃	—	2	2	no	—
20	#11924007	—	4	2	no	—
21	#11924005	—	4	2	no	—
22	#11924003	—	2	1	no	—
23	C₂×Q₈×C₇₃	—	2	2	no	—
24	Q₁₆×C₇₃	—	2	2	no	—
25	Q₁₆×C₇₃	—	2	2	no	—
26	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
27	C₁₆×C₇₃	D₄	1	2	yes	—
28	C₂²×D₇₃	D₈	1	1	yes	characteristic
29	Q₈×C₇₃	—	4	2	no	—
30	Q₈×C₇₃	—	4	2	no	—
31	C₂×C₄×C₇₃	—	4	2	no	—
32	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
33	C₂×Dic₇₃	—	4	2	no	—
34	C₂×Dic₇₃	D₈	1	1	yes	characteristic
35	C₄×D₇₃	—	2	1	no	—
36	C₄×D₇₃	—	2	1	no	—
37	C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
38	C₈×C₇₃	#41	1	1	yes	characteristic
39	#10596831	—	2	1	no	—
40	C₂²×C₇₃	C₂×D₈	1	1	yes	characteristic
41	C₂×D₇₃	—	2	1	no	—
42	C₂×D₇₃	—	2	1	no	—
43	C₂×D₇₃	Q₃₂	1	2	yes	complemented
44	C₄×C₇₃	—	8	2	no	—
45	C₄×C₇₃	#121	1	1	yes	characteristic
46	C₄×C₇₃	#135	1	1	yes	characteristic
47	Dic₇₃	—	2	1	no	—
48	Dic₇₃	—	8	2	no	—
49	C₂×C₇₃	#466	1	1	yes	characteristic
50	C₂×C₇₃	C₂×Q₃₂	1	1	yes	characteristic
51	C₂×C₇₃	#508	1	1	yes	characteristic
52	D₇₃	—	2	2	no	—
53	#2216	—	73	1	no	maximal, Sylow
54	C₇₃	#2216	1	1	yes	characteristic, complemented, Sylow
55	C₂×Q₃₂	D₇₃	1	1	yes	characteristic, complemented
56	C₂²×C₁₆	—	73	1	no	—
57	#467	—	73	2	no	—
58	#366	—	73	1	no	—
59	#358	—	73	2	no	—
60	C₂×Q₁₆	C₂×D₇₃	1	2	yes	—
61	C₂²×C₈	—	73	1	no	—
62	C₂²⋊Q₈	—	146	2	no	—
63	Q₃₂	—	2	2	no	—
64	C₂×C₁₆	—	73	1	no	—
65	C₂×C₁₆	—	146	2	no	—
66	C₂×C₁₆	C₂×D₇₃	1	1	yes	characteristic
67	#107	—	73	2	no	—
68	#103	—	146	2	no	—
69	C₂×Q₈	—	2	2	no	—
70	C₂²×C₄	—	73	1	no	—
71	Q₁₆	—	2	2	no	—
72	Q₁₆	—	2	2	no	—
73	C₂×C₈	—	73	1	no	—
74	C₂×C₈	—	146	1	no	—
75	C₂×C₈	—	146	1	no	—
76	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
77	C₄⋊C₄	—	292	2	no	—
78	C₄⋊C₄	—	146	2	no	—
79	C₂²⋊C₄	—	292	2	no	—
80	C₁₆	—	146	1	no	—
81	C₁₆	C₂²⋊D₇₃	1	2	yes	—
82	C₂³	—	73	1	no	—
83	Q₈	—	4	2	no	—
84	Q₈	—	4	2	no	—
85	C₂×C₄	—	4	2	no	—
86	C₂×C₄	—	292	2	no	—
87	C₂×C₄	—	73	1	no	—
88	C₂×C₄	—	146	1	no	—
89	C₂×C₄	—	146	1	no	—
90	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
91	C₈	—	146	1	no	—
92	C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
93	C₈	#11924027	1	1	yes	characteristic
94	C₂²	—	73	2	no	—
95	C₂²	—	146	1	no	—
96	C₂²	—	146	1	no	—
97	C₂²	D₈×D₇₃	1	1	yes	characteristic
98	C₄	—	8	2	no	—
99	C₄	—	584	2	no	—
100	C₄	—	146	1	no	—
101	C₄	#14556562	1	1	yes	characteristic
102	C₄	#14556610	1	1	yes	characteristic
103	C₂	—	146	2	no	—
104	C₂	#17988820	1	1	yes	characteristic
105	C₂	Q₃₂×D₇₃	1	1	yes	characteristic
106	C₂	#17989020	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₆×D₇₃

C₂

yes

maximal, characteristic

#17988825

C₂

yes

maximal

#17988413

C₂

yes

maximal, characteristic, complemented

#17988391

C₂

yes

maximal, complemented

C₂×Q₃₂×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₇₃

C₂²

yes

characteristic

#14556567

—

C₂×#11924003

C₂²

yes

characteristic

C₁₆×D₇₃

—

#14556525

C₂²

yes

—

#14556515

—

C₂×Q₁₆×C₇₃

C₂²

yes

—

Q₃₂×C₇₃

—

C₂×C₁₆×C₇₃

C₂²

yes

characteristic

C₂×C₄×D₇₃

D₄

yes

characteristic

C₂×#10596831

D₄

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

C₄⋊Dic₇₃

—

#11924007

—

#11924005

—

#11924003

—

C₂×Q₈×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₁₆×C₇₃

D₄

yes

—

C₂²×D₇₃

D₈

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

D₈

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₈×C₇₃

C₂×D₄

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

C₂×D₈

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

Q₃₂

yes

complemented

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

#466

yes

characteristic

C₂×C₇₃

C₂×Q₃₂

yes

characteristic

C₂×C₇₃

#508

yes

characteristic

D₇₃

—

#2216

—

maximal, Sylow

C₇₃

#2216

yes

characteristic, complemented, Sylow

C₂×Q₃₂

D₇₃

yes

characteristic, complemented

C₂²×C₁₆

—

#467

—

#366

—

#358

—

C₂×Q₁₆

C₂×D₇₃

yes

—

C₂²×C₈

—

C₂²⋊Q₈

—

146

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₂×C₁₆

C₂×D₇₃

yes

characteristic

#107

—

#103

—

146

—

C₂×Q₈

—

C₂²×C₄

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

146

—

C₂×C₈

—

146

—

C₂×C₈

C₂²×D₇₃

yes

characteristic

C₄⋊C₄

—

292

—

C₄⋊C₄

—

146

—

C₂²⋊C₄

—

292

—

C₁₆

—

146

—

C₁₆

C₂²⋊D₇₃

yes

—

C₂³

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

292

—

C₂×C₄

—

C₂×C₄

—

146

—

C₂×C₄

—

146

—

C₂×C₄

D₄×D₇₃

yes

characteristic

C₈

—

146

—

C₈

C₂×C₂²⋊D₇₃

yes

characteristic

C₈

#11924027

yes

characteristic

C₂²

—

C₂²

—

146

—

C₂²

—

146

—

C₂²

D₈×D₇₃

yes

characteristic

C₄

—

C₄

—

584

—

100

C₄

—

146

—

101

C₄

#14556562

yes

characteristic

102

C₄

#14556610

yes

characteristic

103

C₂

—

146

—

104

C₂

#17988820

yes

characteristic

105

C₂

Q₃₂×D₇₃

yes

characteristic

106

C₂

#17989020

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₇₃
commutator	C₂×C₈×C₇₃	C₂³
Frattini	C₂×C₈	C₂²×D₇₃
Fitting	C₂×Q₃₂×C₇₃	C₂
socle	C₂²×C₇₃	C₂×D₈
hypercenter	C₂×Q₃₂	D₇₃
hypocenter	C₇₃	#2216
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

subgroup

G/H

center

C₂²

D₈×D₇₃

commutator

C₂×C₈×C₇₃

C₂³

Frattini

C₂×C₈

C₂²×D₇₃

Fitting

C₂×Q₃₂×C₇₃

C₂

socle

C₂²×C₇₃

C₂×D₈

hypercenter

C₂×Q₃₂

D₇₃

hypocenter

C₇₃

#2216

LCS

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

N	H ≅ G/N	type
3. #17988413	C₂	semidirect
4. #17988391	C₂	semidirect
5. C₂×Q₃₂×C₇₃	C₂	semidirect
43. C₂×D₇₃	Q₃₂	semidirect
54. C₇₃	#2216	semidirect
55. C₂×Q₃₂	D₇₃	semidirect

H ≅ G/N

type

3. #17988413

C₂

semidirect

4. #17988391

C₂

semidirect

5. C₂×Q₃₂×C₇₃

C₂

semidirect

43. C₂×D₇₃

Q₃₂

semidirect

54. C₇₃

#2216

semidirect

55. C₂×Q₃₂

D₇₃

semidirect

Subgroups — #22612019

Conjugacy classes of subgroups

Complements