Group #22611719 — subgroups

Subgroups — #22611719

Order 9344.

156 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 93 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	#17988625	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556525	C₂	1	1	yes	maximal, characteristic
3	#17988415	C₂	1	2	yes	maximal, complemented
4	#17988413	C₂	1	2	yes	maximal, complemented
5	C₂×M₃₂×C₇₃	C₂	1	1	yes	maximal, characteristic
6	#14556550	—	2	1	no	—
7	C₂×C₄⋊Dic₇₃	—	2	1	no	—
8	#14556525	—	2	1	no	—
9	#14556525	C₂²	1	2	yes	—
10	#14556525	C₂²	1	1	yes	characteristic
11	#14556525	C₂²	1	1	yes	characteristic
12	#14556523	—	2	1	no	—
13	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
14	M₃₂×C₇₃	C₄	1	4	yes	complemented
15	C₂×C₁₆×C₇₃	C₂²	1	2	yes	—
16	C₂²×Dic₇₃	—	4	1	no	—
17	C₄⋊Dic₇₃	—	2	1	no	—
18	C₄⋊Dic₇₃	—	2	1	no	—
19	C₄⋊Dic₇₃	—	2	2	no	—
20	C₄⋊Dic₇₃	—	4	1	no	—
21	C₂²⋊Dic₇₃	—	4	1	no	—
22	C₄×Dic₇₃	—	4	1	no	—
23	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
24	C₂×C₈×C₇₃	C₂×C₄	1	2	yes	—
25	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
26	C₂×C₈×C₇₃	Q₈	1	2	yes	—
27	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
28	C₁₆×C₇₃	C₂×C₄	1	4	yes	complemented
29	C₂³×C₇₃	D₈	1	1	yes	characteristic
30	C₂×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
31	C₂×C₄×C₇₃	D₈	1	1	yes	characteristic
32	C₂×C₄×C₇₃	Q₁₆	1	2	yes	—
33	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
34	C₂×Dic₇₃	—	4	1	no	—
35	C₂×Dic₇₃	—	4	1	no	—
36	C₂×Dic₇₃	—	4	2	no	—
37	C₂×Dic₇₃	—	8	2	no	—
38	C₈×C₇₃	C₄⋊C₄	1	2	yes	—
39	C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
40	C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
41	C₂²×C₇₃	—	2	1	no	—
42	C₂²×C₇₃	—	2	1	no	—
43	C₂²×C₇₃	#107	1	2	yes	—
44	C₂²×C₇₃	C₂×D₈	1	1	yes	characteristic
45	C₄×C₇₃	#107	1	2	yes	—
46	C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
48	Dic₇₃	—	8	1	no	—
49	Dic₇₃	—	8	2	no	—
50	Dic₇₃	—	8	1	no	—
51	C₂×C₇₃	—	2	2	no	—
52	C₂×C₇₃	C₂×#107	1	1	yes	characteristic
53	C₂×C₇₃	#509	1	1	yes	characteristic
54	C₂×C₇₃	#510	1	1	yes	characteristic
55	#2159	—	73	1	no	maximal, Sylow
56	C₇₃	#2159	1	1	yes	characteristic, complemented, Sylow
57	C₂×M₃₂	D₇₃	1	1	yes	characteristic
58	#427	—	73	1	no	—
59	C₂×#107	—	73	1	no	—
60	#367	—	73	2	no	—
61	#366	—	73	2	no	—
62	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
63	#117	—	146	1	no	—
64	C₂×C₄⋊C₄	—	146	1	no	—
65	M₃₂	Dic₇₃	1	4	yes	complemented
66	C₂×C₁₆	C₂×D₇₃	1	2	yes	—
67	#107	—	73	1	no	—
68	#107	—	73	2	no	—
69	#107	—	73	1	no	—
70	#107	—	146	1	no	—
71	#106	—	146	1	no	—
72	C₂²×C₄	—	292	1	no	—
73	C₂²×C₄	D₂₉₂	1	1	yes	characteristic
74	C₂×C₈	Dic₁₄₆	1	2	yes	—
75	C₂×C₈	D₂₉₂	1	1	yes	characteristic
76	C₂×C₈	C₂×Dic₇₃	1	2	yes	—
77	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
78	C₄⋊C₄	—	292	1	no	—
79	C₄⋊C₄	—	146	1	no	—
80	C₄⋊C₄	—	146	1	no	—
81	C₄⋊C₄	—	146	2	no	—
82	C₂²⋊C₄	—	292	1	no	—
83	C₄²	—	292	1	no	—
84	C₁₆	C₂×Dic₇₃	1	4	yes	complemented
85	C₂³	D₅₈₄	1	1	yes	characteristic
86	C₂×C₄	—	292	1	no	—
87	C₂×C₄	—	292	1	no	—
88	C₂×C₄	—	292	2	no	—
89	C₂×C₄	—	584	2	no	—
90	C₂×C₄	C₄⋊Dic₇₃	1	2	yes	—
91	C₂×C₄	D₅₈₄	1	1	yes	characteristic
92	C₂×C₄	Dic₂₉₂	1	2	yes	—
93	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
94	C₈	C₄⋊Dic₇₃	1	2	yes	—
95	C₈	C₂²×Dic₇₃	1	1	yes	characteristic
96	C₈	C₂×Dic₁₄₆	1	1	yes	characteristic
97	C₂²	—	2	1	no	—
98	C₂²	—	2	1	no	—
99	C₂²	#14556525	1	2	yes	—
100	C₂²	C₂×D₅₈₄	1	1	yes	characteristic
101	C₄	—	584	2	no	—
102	C₄	—	584	1	no	—
103	C₄	—	584	1	no	—
104	C₄	#14556525	1	2	yes	—
105	C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
106	C₄	C₂×Dic₂₉₂	1	1	yes	characteristic
107	C₂	—	2	2	no	—
108	C₂	C₂×#14556525	1	1	yes	characteristic
109	C₂	#17989025	1	1	yes	characteristic
110	C₂	#17989030	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988625

C₂

yes

maximal, characteristic

C₂×#14556525

C₂

yes

maximal, characteristic

#17988415

C₂

yes

maximal, complemented

#17988413

C₂

yes

maximal, complemented

C₂×M₃₂×C₇₃

C₂

yes

maximal, characteristic

#14556550

—

C₂×C₄⋊Dic₇₃

—

#14556525

—

#14556525

C₂²

yes

—

#14556525

C₂²

yes

characteristic

#14556525

C₂²

yes

characteristic

#14556523

—

C₂²×C₈×C₇₃

C₂²

yes

characteristic

M₃₂×C₇₃

C₄

yes

complemented

C₂×C₁₆×C₇₃

C₂²

yes

—

C₂²×Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×C₄

yes

—

C₂×C₈×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

Q₈

yes

—

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₁₆×C₇₃

C₂×C₄

yes

complemented

C₂³×C₇₃

D₈

yes

characteristic

C₂×C₄×C₇₃

C₄⋊C₄

yes

—

C₂×C₄×C₇₃

D₈

yes

characteristic

C₂×C₄×C₇₃

Q₁₆

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

C₄⋊C₄

yes

—

C₈×C₇₃

C₂²×C₄

yes

characteristic

C₈×C₇₃

C₂×Q₈

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#107

yes

—

C₂²×C₇₃

C₂×D₈

yes

characteristic

C₄×C₇₃

#107

yes

—

C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₃

C₂×Q₁₆

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×M₃₂

D₇₃

yes

—

—

—

—

C₂²×C₈

C₂×D₇₃

yes

characteristic

#117

—

146

—

C₂×C₄⋊C₄

—

146

—

M₃₂

Dic₇₃

yes

complemented

C₂×C₁₆

C₂×D₇₃

yes

—

—

—

—

—

146

—

#106

—

146

—

C₂²×C₄

—

292

—

C₂²×C₄

D₂₉₂

yes

characteristic

C₂×C₈

Dic₁₄₆

yes

—

C₂×C₈

D₂₉₂

yes

characteristic

C₂×C₈

C₂×Dic₇₃

yes

—

C₂×C₈

C₂²×D₇₃

yes

characteristic

C₄⋊C₄

—

292

—

C₄⋊C₄

—

146

—

C₄⋊C₄

—

146

—

C₄⋊C₄

—

146

—

C₂²⋊C₄

—

292

—

C₄²

—

292

—

C₁₆

C₂×Dic₇₃

yes

complemented

C₂³

D₅₈₄

yes

characteristic

C₂×C₄

—

292

—

C₂×C₄

—

292

—

C₂×C₄

—

292

—

C₂×C₄

—

584

—

C₂×C₄

C₄⋊Dic₇₃

yes

—

C₂×C₄

D₅₈₄

yes

characteristic

C₂×C₄

Dic₂₉₂

yes

—

C₂×C₄

C₂×D₂₉₂

yes

characteristic

C₈

C₄⋊Dic₇₃

yes

—

C₈

C₂²×Dic₇₃

yes

characteristic

C₈

C₂×Dic₁₄₆

yes

characteristic

C₂²

—

C₂²

—

C₂²

#14556525

yes

—

100

C₂²

C₂×D₅₈₄

yes

characteristic

101

C₄

—

584

—

102

C₄

—

584

—

103

C₄

—

584

—

104

C₄

#14556525

yes

—

105

C₄

C₂×C₄⋊Dic₇₃

yes

characteristic

106

C₄

C₂×Dic₂₉₂

yes

characteristic

107

C₂

—

108

C₂

C₂×#14556525

yes

characteristic

109

C₂

#17989025

yes

characteristic

110

C₂

#17989030

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₅₈₄
commutator	C₈×C₇₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₇₃
Fitting	C₂×M₃₂×C₇₃	C₂
socle	C₂²×C₇₃	C₂×D₈
hypercenter	C₂×M₃₂	D₇₃
hypocenter	C₇₃	#2159
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₂²

C₂×D₅₈₄

commutator

C₈×C₇₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₇₃

Fitting

C₂×M₃₂×C₇₃

C₂

socle

C₂²×C₇₃

C₂×D₈

hypercenter

C₂×M₃₂

D₇₃

hypocenter

C₇₃

#2159

LCS

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
3. #17988415	C₂	semidirect
4. #17988413	C₂	semidirect
14. M₃₂×C₇₃	C₄	semidirect
28. C₁₆×C₇₃	C₂×C₄	semidirect
56. C₇₃	#2159	semidirect
65. M₃₂	Dic₇₃	semidirect
84. C₁₆	C₂×Dic₇₃	semidirect

H ≅ G/N

type

3. #17988415

C₂

semidirect

4. #17988413

C₂

semidirect

14. M₃₂×C₇₃

C₄

semidirect

28. C₁₆×C₇₃

C₂×C₄

semidirect

56. C₇₃

#2159

semidirect

65. M₃₂

Dic₇₃

semidirect

84. C₁₆

C₂×Dic₇₃

semidirect

Subgroups — #22611719

Conjugacy classes of subgroups

Complements