Group #22627147 — subgroups

Subgroups — #22627147

Order 9344.

162 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 61 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×#14556687	C₂	1	2	yes	maximal, complemented
2	C₄⋊#11924040	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊#11924040	C₂	1	2	yes	maximal, complemented
4	C₄×#11924040	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄×#10596833	C₂	1	1	yes	maximal, characteristic
6	#14556687	—	2	4	no	—
7	C₂×#11924040	C₂²	1	1	yes	characteristic
8	C₂×#11924040	C₂²	1	2	yes	—
9	C₂×#11924040	C₂²	1	1	yes	characteristic
10	C₂²×#10596833	C₂²	1	2	yes	—
11	C₄⋊#10596833	C₄	1	1	yes	characteristic
12	C₂²⋊#10596833	C₄	1	2	yes	—
13	C₄×#10596833	C₂²	1	1	yes	characteristic
14	D₄×Dic₇₃	C₄	1	1	yes	characteristic
15	#11924040	—	2	2	no	—
16	#11924040	—	2	1	no	—
17	#11924040	D₄	1	2	yes	complemented
18	C₂×#10596833	—	2	2	no	—
19	C₂×#10596833	—	2	2	no	—
20	C₂×#10596833	C₂×C₄	1	2	yes	—
21	C₂×#10596833	C₂×C₄	1	1	yes	characteristic
22	C₂×#10596833	C₂³	1	1	yes	characteristic
23	C₂²×Dic₇₃	C₂×C₄	1	2	yes	—
24	C₄⋊Dic₇₃	C₈	1	1	yes	characteristic
25	C₂²⋊Dic₇₃	C₈	1	2	yes	—
26	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂×D₄×C₇₃	C₈	1	1	yes	characteristic
28	C₂³×C₇₃	C₂×C₈	1	2	yes	—
29	D₄×C₇₃	—	2	2	no	—
30	C₂×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
31	C₂×Dic₇₃	—	2	2	no	—
32	C₂×Dic₇₃	—	2	2	no	—
33	C₂×Dic₇₃	C₂×C₈	1	1	yes	characteristic
34	C₂×Dic₇₃	C₂×C₈	1	2	yes	—
35	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
36	#10596833	—	2	2	no	—
37	#10596833	—	2	1	no	—
38	#10596833	C₂×D₄	1	1	yes	characteristic
39	#10596833	#41	1	1	yes	characteristic
40	C₂²×C₇₃	—	2	2	no	—
41	C₂²×C₇₃	—	2	2	no	—
42	C₂²×C₇₃	—	2	2	no	—
43	C₂²×C₇₃	C₂²×C₈	1	1	yes	characteristic
44	C₄×C₇₃	M₃₂	1	2	yes	complemented
45	Dic₇₃	—	2	2	no	—
46	Dic₇₃	—	2	1	no	—
47	Dic₇₃	C₄×D₄	1	1	yes	characteristic
48	Dic₇₃	#131	1	1	yes	characteristic
49	C₂×C₇₃	—	4	2	no	—
50	C₂×C₇₃	C₈×D₄	1	1	yes	characteristic
51	C₂×C₇₃	C₂×M₃₂	1	1	yes	characteristic
52	C₂×C₇₃	#504	1	1	yes	characteristic
53	#2169	—	73	1	no	maximal, Sylow
54	C₇₃	#2169	1	1	yes	characteristic, complemented, Sylow
55	C₂×M₃₂	—	73	2	no	—
56	C₈×D₄	—	73	1	no	—
57	C₄⋊C₁₆	—	73	1	no	—
58	C₂²⋊C₁₆	—	73	2	no	—
59	C₄×C₁₆	—	73	1	no	—
60	C₂²×C₈	—	73	2	no	—
61	C₄×D₄	—	73	1	no	—
62	M₃₂	—	146	4	no	—
63	C₂×C₁₆	—	73	1	no	—
64	C₂×C₁₆	—	73	1	no	—
65	C₂×C₁₆	—	73	2	no	—
66	C₄⋊C₈	—	73	1	no	—
67	C₂²⋊C₈	—	73	2	no	—
68	C₄×C₈	—	73	1	no	—
69	C₂×D₄	#10596842	1	1	yes	characteristic
70	C₂²×C₄	—	73	2	no	—
71	C₂×C₈	—	73	1	no	—
72	C₂×C₈	—	73	1	no	—
73	C₂×C₈	—	73	2	no	—
74	C₂×C₈	—	146	2	no	—
75	C₂×C₈	—	146	2	no	—
76	C₄⋊C₄	—	73	1	no	—
77	C₂²⋊C₄	—	73	2	no	—
78	C₄²	—	73	1	no	—
79	C₁₆	—	73	2	no	—
80	C₁₆	—	146	1	no	—
81	C₁₆	—	146	2	no	—
82	C₂³	C₂×#10596842	1	2	yes	—
83	D₄	—	2	2	no	—
84	C₂×C₄	—	73	1	no	—
85	C₂×C₄	—	73	1	no	—
86	C₂×C₄	—	73	2	no	—
87	C₂×C₄	—	146	2	no	—
88	C₂×C₄	—	146	2	no	—
89	C₂×C₄	C₂×#10596842	1	1	yes	characteristic
90	C₈	—	73	1	no	—
91	C₈	—	73	1	no	—
92	C₈	—	146	2	no	—
93	C₈	—	146	1	no	—
94	C₂²	—	2	2	no	—
95	C₂²	—	2	2	no	—
96	C₂²	—	2	2	no	—
97	C₂²	C₂²×#10596842	1	1	yes	characteristic
98	C₄	—	73	1	no	—
99	C₄	—	73	1	no	—
100	C₄	—	146	2	no	—
101	C₄	—	146	1	no	—
102	C₄	#14556687	1	2	yes	complemented
103	C₂	—	4	2	no	—
104	C₂	D₄×#10596842	1	1	yes	characteristic
105	C₂	C₂×#14556687	1	1	yes	characteristic
106	C₂	#17989708	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556687

C₂

yes

maximal, complemented

C₄⋊#11924040

C₂

yes

maximal, characteristic, complemented

C₂²⋊#11924040

C₂

yes

maximal, complemented

C₄×#11924040

C₂

yes

maximal, characteristic, complemented

D₄×#10596833

C₂

yes

maximal, characteristic

#14556687

—

C₂×#11924040

C₂²

yes

characteristic

C₂×#11924040

C₂²

yes

—

C₂×#11924040

C₂²

yes

characteristic

C₂²×#10596833

C₂²

yes

—

C₄⋊#10596833

C₄

yes

characteristic

C₂²⋊#10596833

C₄

yes

—

C₄×#10596833

C₂²

yes

characteristic

D₄×Dic₇₃

C₄

yes

—

—

D₄

yes

—

—

C₂×C₄

yes

—

C₂×#10596833

C₂×C₄

yes

characteristic

C₂×#10596833

C₂³

yes

characteristic

C₂²×Dic₇₃

C₂×C₄

yes

—

C₄⋊Dic₇₃

C₈

yes

characteristic

C₂²⋊Dic₇₃

C₈

yes

—

C₄×Dic₇₃

C₂×C₄

yes

characteristic

C₂×D₄×C₇₃

C₈

yes

characteristic

C₂³×C₇₃

C₂×C₈

yes

—

D₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂×C₈

yes

characteristic

C₂×Dic₇₃

C₂×C₈

yes

—

C₂×Dic₇₃

C₂²×C₄

yes

—

—

C₂×D₄

yes

yes

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂²×C₈

yes

characteristic

C₄×C₇₃

M₃₂

yes

complemented

Dic₇₃

—

Dic₇₃

—

Dic₇₃

C₄×D₄

yes

characteristic

Dic₇₃

#131

yes

characteristic

C₂×C₇₃

—

C₂×C₇₃

C₈×D₄

yes

characteristic

C₂×C₇₃

C₂×M₃₂

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₈×D₄

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₄×C₁₆

—

C₂²×C₈

—

C₄×D₄

—

M₃₂

—

146

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₂×D₄

#10596842

yes

characteristic

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

146

—

C₂×C₈

—

146

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

C₁₆

—

146

—

C₁₆

—

146

—

C₂³

C₂×#10596842

yes

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

146

—

C₂×C₄

—

146

—

C₂×C₄

C₂×#10596842

yes

characteristic

C₈

—

C₈

—

C₈

—

146

—

C₈

—

146

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂²×#10596842

yes

characteristic

C₄

—

C₄

—

100

C₄

—

146

—

101

C₄

—

146

—

102

C₄

#14556687

yes

complemented

103

C₂

—

104

C₂

D₄×#10596842

yes

characteristic

105

C₂

C₂×#14556687

yes

characteristic

106

C₂

#17989708

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#10596842
commutator, socle	C₂²×C₇₃	C₂²×C₈
Fitting	C₂×D₄×C₇₃	C₈
hypercenter	C₂×D₄	#10596842
hypocenter	C₇₃	#2169
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×D₄

subgroup

G/H

center, Frattini

C₂²

C₂²×#10596842

commutator, socle

C₂²×C₇₃

C₂²×C₈

Fitting

C₂×D₄×C₇₃

C₈

C₂×D₄

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂² ◁ C₂×D₄

N	H ≅ G/N	type
1. C₂×#14556687	C₂	semidirect
2. C₄⋊#11924040	C₂	semidirect
3. C₂²⋊#11924040	C₂	semidirect
4. C₄×#11924040	C₂	semidirect
17. #11924040	D₄	semidirect
44. C₄×C₇₃	M₃₂	semidirect
54. C₇₃	#2169	semidirect
102. C₄	#14556687	semidirect

H ≅ G/N

type

1. C₂×#14556687

C₂

semidirect

2. C₄⋊#11924040

C₂

semidirect

3. C₂²⋊#11924040

C₂

semidirect

4. C₄×#11924040

C₂

semidirect

17. #11924040

D₄

semidirect

44. C₄×C₇₃

M₃₂

semidirect

54. C₇₃

#2169

semidirect

102. C₄

#14556687

semidirect

Subgroups — #22627147

Conjugacy classes of subgroups

Complements