Group #17989142 — subgroups

Subgroups — #17989142

Order 4672.

222 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 111 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	#14556591	C₂	1	1	yes	maximal, characteristic, complemented
2	#14556582	C₂	1	2	yes	maximal, complemented
3	#14556571	C₂	1	2	yes	maximal
4	#14556567	C₂	1	2	yes	maximal
5	Q₈×Dic₇₃	C₂	1	2	yes	maximal, complemented
6	#14556556	C₂	1	2	yes	maximal, complemented
7	#14556549	C₂	1	2	yes	maximal
8	C₄²×D₇₃	C₂	1	1	yes	maximal, characteristic
9	C₄⋊Q₈×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₁₄₆	C₂²	1	2	yes	—
11	C₂×C₄×D₇₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₇₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₇₃	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₇₃	C₂²	1	2	yes	—
15	C₄⋊Dic₇₃	C₂²	1	2	yes	—
16	C₄⋊Dic₇₃	C₂²	1	4	yes	—
17	#11924007	C₂²	1	4	yes	—
18	#11924005	C₂²	1	4	yes	—
19	C₄×Dic₇₃	C₂²	1	1	yes	characteristic
20	C₄×Dic₇₃	C₂²	1	1	yes	characteristic
21	C₄×Dic₇₃	C₂²	1	1	yes	characteristic
22	C₄×Dic₇₃	C₂²	1	4	yes	—
23	C₂×Q₈×C₇₃	C₂²	1	2	yes	—
24	C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
25	C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
26	C₄²×C₇₃	C₂²	1	1	yes	characteristic
27	C₂²×D₇₃	C₂³	1	1	yes	characteristic
28	Q₈×C₇₃	—	2	4	no	—
29	C₂×C₄×C₇₃	C₂³	1	4	yes	—
30	C₂×C₄×C₇₃	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₇₃	C₂³	1	1	yes	characteristic
32	C₂×C₄×C₇₃	C₂³	1	1	yes	characteristic
33	C₂×Dic₇₃	C₂³	1	1	yes	characteristic
34	C₂×Dic₇₃	C₂³	1	1	yes	characteristic
35	C₂×Dic₇₃	C₂³	1	4	yes	—
36	C₂×Dic₇₃	C₂³	1	1	yes	characteristic
37	C₄×D₇₃	—	2	2	no	—
38	C₄×D₇₃	—	2	2	no	—
39	C₄×D₇₃	Q₈	1	4	yes	complemented
40	Dic₁₄₆	—	2	4	no	—
41	C₂²×C₇₃	C₂⁴	1	1	yes	characteristic
42	C₂×D₇₃	—	2	1	no	—
43	C₂×D₇₃	—	2	1	no	—
44	C₂×D₇₃	C₂×Q₈	1	2	yes	—
45	C₄×C₇₃	—	2	4	no	—
46	C₄×C₇₃	C₂×Q₈	1	2	yes	—
47	C₄×C₇₃	#41	1	2	yes	—
48	C₄×C₇₃	#41	1	2	yes	—
49	Dic₇₃	—	2	4	no	—
50	Dic₇₃	—	2	1	no	—
51	Dic₇₃	—	2	1	no	—
52	Dic₇₃	C₂×Q₈	1	2	yes	—
53	C₂×C₇₃	C₂²×Q₈	1	1	yes	characteristic
54	C₂×C₇₃	C₂×#41	1	1	yes	characteristic
55	C₂×C₇₃	C₂×#41	1	1	yes	characteristic
56	D₇₃	—	2	2	no	—
57	C₇₃	#533	1	1	yes	characteristic, complemented, Sylow
58	#533	—	73	1	no	maximal, Sylow
59	C₄⋊Q₈	—	73	1	no	—
60	C₄⋊Q₈	D₇₃	1	1	yes	characteristic, complemented
61	#125	—	73	2	no	—
62	C₂²⋊Q₈	—	73	2	no	—
63	C₂²⋊Q₈	—	73	2	no	—
64	C₄×Q₈	—	73	2	no	—
65	C₄×Q₈	—	73	2	no	—
66	#117	—	73	2	no	—
67	C₂×C₄²	—	73	1	no	—
68	C₂×Q₈	—	73	2	no	—
69	C₂×Q₈	C₂×D₇₃	1	2	yes	—
70	C₂²×C₄	—	73	1	no	—
71	C₂²×C₄	—	73	1	no	—
72	C₂²×C₄	—	73	1	no	—
73	C₄⋊C₄	—	73	4	no	—
74	C₄⋊C₄	—	73	2	no	—
75	C₄⋊C₄	—	73	4	no	—
76	C₄⋊C₄	—	73	2	no	—
77	C₄⋊C₄	C₂×D₇₃	1	2	yes	—
78	C₄⋊C₄	C₂×D₇₃	1	2	yes	—
79	C₂²⋊C₄	—	73	4	no	—
80	C₄²	—	73	1	no	—
81	C₄²	—	73	1	no	—
82	C₄²	—	73	1	no	—
83	C₄²	—	73	4	no	—
84	C₄²	C₂×D₇₃	1	1	yes	characteristic
85	C₂³	—	73	1	no	—
86	Q₈	—	2	4	no	—
87	Q₈	—	146	4	no	—
88	C₂×C₄	—	73	1	no	—
89	C₂×C₄	—	73	4	no	—
90	C₂×C₄	—	73	1	no	—
91	C₂×C₄	—	73	1	no	—
92	C₂×C₄	—	73	4	no	—
93	C₂×C₄	—	146	2	no	—
94	C₂×C₄	—	146	2	no	—
95	C₂×C₄	C₂²×D₇₃	1	4	yes	—
96	C₂×C₄	C₂²×D₇₃	1	1	yes	characteristic
97	C₂×C₄	C₂²×D₇₃	1	1	yes	characteristic
98	C₂×C₄	C₂²×D₇₃	1	1	yes	characteristic
99	C₂²	—	73	2	no	—
100	C₂²	—	146	1	no	—
101	C₂²	—	146	1	no	—
102	C₂²	C₂³×D₇₃	1	1	yes	characteristic
103	C₄	—	2	4	no	—
104	C₄	—	73	2	no	—
105	C₄	—	146	1	no	—
106	C₄	—	146	1	no	—
107	C₄	—	146	4	no	—
108	C₄	Q₈×D₇₃	1	2	yes	—
109	C₄	#11924027	1	2	yes	—
110	C₄	#11924029	1	2	yes	—
111	C₂	—	146	2	no	—
112	C₂	C₂×Q₈×D₇₃	1	1	yes	characteristic
113	C₂	C₂×#11924027	1	1	yes	characteristic
114	C₂	C₂×#11924029	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14556591

C₂

yes

maximal, characteristic, complemented

#14556582

C₂

yes

maximal, complemented

#14556571

C₂

yes

maximal

#14556567

C₂

yes

maximal

Q₈×Dic₇₃

C₂

yes

maximal, complemented

#14556556

C₂

yes

maximal, complemented

#14556549

C₂

yes

maximal

C₄²×D₇₃

C₂

yes

maximal, characteristic

C₄⋊Q₈×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×Dic₁₄₆

C₂²

yes

—

C₂×C₄×D₇₃

C₂²

yes

characteristic

C₂×C₄×D₇₃

C₂²

yes

characteristic

C₂×C₄×D₇₃

C₂²

yes

characteristic

C₄⋊Dic₇₃

C₂²

yes

—

C₄⋊Dic₇₃

C₂²

yes

—

C₄⋊Dic₇₃

C₂²

yes

—

#11924007

C₂²

yes

—

#11924005

C₂²

yes

—

C₄×Dic₇₃

C₂²

yes

characteristic

C₄×Dic₇₃

C₂²

yes

characteristic

C₄×Dic₇₃

C₂²

yes

characteristic

C₄×Dic₇₃

C₂²

yes

—

C₂×Q₈×C₇₃

C₂²

yes

—

C₄⋊C₄×C₇₃

C₂²

yes

—

C₄⋊C₄×C₇₃

C₂²

yes

—

C₄²×C₇₃

C₂²

yes

characteristic

C₂²×D₇₃

C₂³

yes

characteristic

Q₈×C₇₃

—

C₂×C₄×C₇₃

C₂³

yes

—

C₂×C₄×C₇₃

C₂³

yes

characteristic

C₂×C₄×C₇₃

C₂³

yes

characteristic

C₂×C₄×C₇₃

C₂³

yes

characteristic

C₂×Dic₇₃

C₂³

yes

characteristic

C₂×Dic₇₃

C₂³

yes

characteristic

C₂×Dic₇₃

C₂³

yes

—

C₂×Dic₇₃

C₂³

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

Q₈

yes

complemented

Dic₁₄₆

—

C₂²×C₇₃

C₂⁴

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

C₂×Q₈

yes

—

C₄×C₇₃

—

C₄×C₇₃

C₂×Q₈

yes

—

C₄×C₇₃

#41

yes

—

C₄×C₇₃

#41

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

C₂×Q₈

yes

—

C₂×C₇₃

C₂²×Q₈

yes

characteristic

C₂×C₇₃

C₂×#41

yes

characteristic

C₂×C₇₃

C₂×#41

yes

characteristic

D₇₃

—

C₇₃

#533

yes

characteristic, complemented, Sylow

#533

—

maximal, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

D₇₃

yes

characteristic, complemented

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₂×Q₈

—

C₂×Q₈

C₂×D₇₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₇₃

yes

—

C₄⋊C₄

C₂×D₇₃

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×D₇₃

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

146

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

146

—

C₂×C₄

—

146

—

C₂×C₄

C₂²×D₇₃

yes

—

C₂×C₄

C₂²×D₇₃

yes

characteristic

C₂×C₄

C₂²×D₇₃

yes

characteristic

C₂×C₄

C₂²×D₇₃

yes

characteristic

C₂²

—

100

C₂²

—

146

—

101

C₂²

—

146

—

102

C₂²

C₂³×D₇₃

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

—

146

—

106

C₄

—

146

—

107

C₄

—

146

—

108

C₄

Q₈×D₇₃

yes

—

109

C₄

#11924027

yes

—

110

C₄

#11924029

yes

—

111

C₂

—

146

—

112

C₂

C₂×Q₈×D₇₃

yes

characteristic

113

C₂

C₂×#11924027

yes

characteristic

114

C₂

C₂×#11924029

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₇₃
commutator, socle, cosocle	C₂²×C₇₃	C₂⁴
Fitting	C₄⋊Q₈×C₇₃	C₂
prime radical	C₂²×D₇₃	C₂³
hypercenter	C₄⋊Q₈	D₇₃
hypocenter	C₇₃	#533
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₇₃

commutator, socle, cosocle

C₂²×C₇₃

C₂⁴

Fitting

C₄⋊Q₈×C₇₃

C₂

prime radical

C₂²×D₇₃

C₂³

hypercenter

C₄⋊Q₈

D₇₃

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. #14556591	C₂	semidirect
2. #14556582	C₂	semidirect
5. Q₈×Dic₇₃	C₂	semidirect
6. #14556556	C₂	semidirect
9. C₄⋊Q₈×C₇₃	C₂	semidirect
39. C₄×D₇₃	Q₈	semidirect
57. C₇₃	#533	semidirect
60. C₄⋊Q₈	D₇₃	semidirect

H ≅ G/N

type

1. #14556591

C₂

semidirect

2. #14556582

C₂

semidirect

5. Q₈×Dic₇₃

C₂

semidirect

6. #14556556

C₂

semidirect

9. C₄⋊Q₈×C₇₃

C₂

semidirect

39. C₄×D₇₃

Q₈

semidirect

57. C₇₃

#533

semidirect

60. C₄⋊Q₈

D₇₃

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#14556591	2
#14556582	2
#14556571	2
#14556567	2
Q₈×Dic₇₃	2
#14556556	2
#14556549	2
C₄²×D₇₃	2
C₄⋊Q₈×C₇₃	2
#533	73

subgroup

index

Q₈×Dic₇₃

C₄²×D₇₃

C₄⋊Q₈×C₇₃

#533

Subgroups — #17989142

Conjugacy classes of subgroups

Complements

Maximal subgroups