Group #33006613 — subgroups

Subgroups — #33006613

Order 18688.

166 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₁₆×Dic₁₄₆	C₂	1	1	yes	maximal, characteristic
2	#22608126	C₂	1	2	yes	maximal
3	C₄×#14556497	C₂	1	1	yes	maximal, characteristic
4	C₃₂×Dic₇₃	C₂	1	2	yes	maximal
5	C₇₃×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic
6	C₈×Dic₁₄₆	C₄	1	1	yes	characteristic
7	C₂×#14556497	C₂²	1	1	yes	characteristic
8	C₂×#14556497	C₂²	1	1	yes	characteristic
9	C₄⋊#11924003	C₄	1	1	yes	characteristic
10	#17988405	C₄	1	2	yes	—
11	C₁₆×Dic₇₃	C₂²	1	2	yes	—
12	C₂×C₃₂×C₇₃	C₂²	1	2	yes	—
13	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
14	C₄×Dic₁₄₆	C₈	1	1	yes	characteristic
15	C₂×#11924003	C₂×C₄	1	2	yes	—
16	C₄⋊#10596831	C₈	1	1	yes	characteristic
17	#14556520	C₈	1	2	yes	—
18	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
19	#14556497	—	2	1	no	—
20	#14556497	Q₈	1	2	yes	complemented
21	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
23	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
24	C₃₂×C₇₃	—	2	2	no	—
25	C₂×Dic₁₄₆	C₁₆	1	1	yes	characteristic
26	C₂×#10596831	C₂×C₈	1	2	yes	—
27	C₄⋊Dic₇₃	C₁₆	1	1	yes	characteristic
28	#11924007	C₁₆	1	2	yes	—
29	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
30	#11924003	—	2	2	no	—
31	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
33	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
34	C₁₆×C₇₃	—	2	1	no	—
35	C₁₆×C₇₃	C₂×Q₈	1	1	yes	characteristic
36	C₁₆×C₇₃	#41	1	1	yes	characteristic
37	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
38	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
39	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
40	C₂×Dic₇₃	C₂×C₁₆	1	2	yes	—
41	C₂×Dic₇₃	C₂×C₁₆	1	2	yes	—
42	Dic₁₄₆	C₃₂	1	4	yes	complemented
43	C₈×C₇₃	—	2	1	no	—
44	C₈×C₇₃	C₄×Q₈	1	1	yes	characteristic
45	C₈×C₇₃	#131	1	1	yes	characteristic
46	#10596831	—	2	2	no	—
47	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
48	C₄×C₇₃	—	2	1	no	—
49	C₄×C₇₃	C₂×C₃₂	1	2	yes	—
50	C₄×C₇₃	C₈×Q₈	1	1	yes	characteristic
51	C₄×C₇₃	#504	1	1	yes	characteristic
52	Dic₇₃	—	2	2	no	—
53	Dic₇₃	C₂×C₃₂	1	4	yes	—
54	Q₈×C₃₂	—	73	1	no	maximal, Sylow
55	C₂×C₇₃	Q₈×C₁₆	1	1	yes	characteristic
56	C₂×C₇₃	C₂²×C₃₂	1	1	yes	characteristic
57	C₂×C₇₃	#2258	1	1	yes	characteristic
58	Q₈×C₁₆	—	73	1	no	—
59	C₄⋊C₃₂	—	73	2	no	—
60	C₄⋊C₃₂	D₇₃	1	1	yes	characteristic
61	C₄×C₃₂	—	73	2	no	—
62	C₄×C₃₂	—	73	1	no	—
63	C₇₃	Q₈×C₃₂	1	1	yes	characteristic, complemented, Sylow
64	C₈×Q₈	—	73	1	no	—
65	C₂×C₃₂	—	73	1	no	—
66	C₂×C₃₂	—	73	1	no	—
67	C₂×C₃₂	C₂×D₇₃	1	2	yes	—
68	C₄⋊C₁₆	—	73	2	no	—
69	C₄⋊C₁₆	—	73	1	no	—
70	C₄×C₁₆	—	73	2	no	—
71	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
72	C₄×Q₈	—	73	1	no	—
73	C₂×C₁₆	—	73	2	no	—
74	C₂×C₁₆	C₄×D₇₃	1	1	yes	characteristic
75	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
76	C₄⋊C₈	—	73	2	no	—
77	C₄⋊C₈	—	73	1	no	—
78	C₄×C₈	—	73	2	no	—
79	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
80	C₃₂	—	2	2	no	—
81	C₃₂	—	73	2	no	—
82	C₃₂	—	146	1	no	—
83	C₂×Q₈	—	73	1	no	—
84	C₂×C₈	—	73	2	no	—
85	C₂×C₈	C₈×D₇₃	1	1	yes	characteristic
86	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
87	C₄⋊C₄	—	73	2	no	—
88	C₄⋊C₄	—	73	1	no	—
89	C₄²	—	73	2	no	—
90	C₄²	C₈×D₇₃	1	1	yes	characteristic
91	C₁₆	—	2	1	no	—
92	C₁₆	—	146	2	no	—
93	C₁₆	Q₈×D₇₃	1	1	yes	characteristic
94	C₁₆	#11924027	1	1	yes	characteristic
95	Q₈	—	73	4	no	—
96	C₂×C₄	—	73	2	no	—
97	C₂×C₄	—	73	2	no	—
98	C₂×C₄	C₁₆×D₇₃	1	1	yes	characteristic
99	C₂×C₄	C₁₆×D₇₃	1	1	yes	characteristic
100	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
101	C₈	—	2	1	no	—
102	C₈	—	146	2	no	—
103	C₈	#14556556	1	1	yes	characteristic
104	C₈	#14556599	1	1	yes	characteristic
105	C₂²	C₂×C₁₆×D₇₃	1	1	yes	characteristic
106	C₄	—	2	1	no	—
107	C₄	—	73	4	no	—
108	C₄	—	146	2	no	—
109	C₄	C₃₂×D₇₃	1	2	yes	—
110	C₄	#17988711	1	1	yes	characteristic
111	C₄	#17989005	1	1	yes	characteristic
112	C₂	#22611833	1	1	yes	characteristic
113	C₂	C₂×C₃₂×D₇₃	1	1	yes	characteristic
114	C₂	#22612226	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₆×Dic₁₄₆

C₂

yes

maximal, characteristic

#22608126

C₂

yes

maximal

C₄×#14556497

C₂

yes

maximal, characteristic

C₃₂×Dic₇₃

C₂

yes

maximal

C₇₃×C₄⋊C₃₂

C₂

yes

maximal, characteristic

C₈×Dic₁₄₆

C₄

yes

characteristic

C₂×#14556497

C₂²

yes

characteristic

C₂×#14556497

C₂²

yes

characteristic

C₄⋊#11924003

C₄

yes

characteristic

#17988405

C₄

yes

—

C₁₆×Dic₇₃

C₂²

yes

—

C₂×C₃₂×C₇₃

C₂²

yes

—

C₄×C₁₆×C₇₃

C₂²

yes

characteristic

C₄×Dic₁₄₆

C₈

yes

characteristic

C₂×#11924003

C₂×C₄

yes

—

C₄⋊#10596831

C₈

yes

characteristic

#14556520

C₈

yes

—

C₈×Dic₇₃

C₂×C₄

yes

—

#14556497

—

#14556497

Q₈

yes

complemented

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂³

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₃₂×C₇₃

—

C₂×Dic₁₄₆

C₁₆

yes

characteristic

C₂×#10596831

C₂×C₈

yes

—

C₄⋊Dic₇₃

C₁₆

yes

characteristic

#11924007

C₁₆

yes

—

C₄×Dic₇₃

C₂×C₈

yes

—

#11924003

—

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₂×C₈

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

C₂×Q₈

yes

characteristic

C₁₆×C₇₃

#41

yes

characteristic

C₂×C₄×C₇₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×Dic₇₃

C₂×C₁₆

yes

—

C₂×Dic₇₃

C₂×C₁₆

yes

—

Dic₁₄₆

C₃₂

yes

complemented

C₈×C₇₃

—

C₈×C₇₃

C₄×Q₈

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

C₂²×C₁₆

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

C₂×C₃₂

yes

—

C₄×C₇₃

C₈×Q₈

yes

characteristic

C₄×C₇₃

#504

yes

characteristic

Dic₇₃

—

Dic₇₃

C₂×C₃₂

yes

—

Q₈×C₃₂

—

maximal, Sylow

C₂×C₇₃

Q₈×C₁₆

yes

characteristic

C₂×C₇₃

C₂²×C₃₂

yes

characteristic

C₂×C₇₃

#2258

yes

characteristic

Q₈×C₁₆

—

C₄⋊C₃₂

—

C₄⋊C₃₂

D₇₃

yes

characteristic

C₄×C₃₂

—

C₄×C₃₂

—

C₇₃

Q₈×C₃₂

yes

characteristic, complemented, Sylow

C₈×Q₈

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₇₃

yes

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₃

yes

characteristic

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₇₃

yes

characteristic

C₂×C₁₆

C₂²×D₇₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₇₃

yes

characteristic

C₃₂

—

C₃₂

—

C₃₂

—

146

—

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

C₈×D₇₃

yes

characteristic

C₂×C₈

C₂×C₄×D₇₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₈×D₇₃

yes

characteristic

C₁₆

—

C₁₆

—

146

—

C₁₆

Q₈×D₇₃

yes

characteristic

C₁₆

#11924027

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₁₆×D₇₃

yes

characteristic

C₂×C₄

C₁₆×D₇₃

yes

characteristic

100

C₂×C₄

C₂×C₈×D₇₃

yes

characteristic

101

C₈

—

102

C₈

—

146

—

103

C₈

#14556556

yes

characteristic

104

C₈

#14556599

yes

characteristic

105

C₂²

C₂×C₁₆×D₇₃

yes

characteristic

106

C₄

—

107

C₄

—

108

C₄

—

146

—

109

C₄

C₃₂×D₇₃

yes

—

110

C₄

#17988711

yes

characteristic

111

C₄

#17989005

yes

characteristic

112

C₂

#22611833

yes

characteristic

113

C₂

C₂×C₃₂×D₇₃

yes

characteristic

114

C₂

#22612226

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₇₃
commutator	C₂×C₇₃	C₂²×C₃₂
Fitting	C₇₃×C₄⋊C₃₂	C₂
socle, prime radical	C₂²×C₇₃	C₂²×C₁₆
cosocle	C₂×C₁₆×C₇₃	C₂³
hypercenter	C₄⋊C₃₂	D₇₃
hypocenter	C₇₃	Q₈×C₃₂
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄⋊C₃₂

subgroup

G/H

center, Frattini

C₂×C₁₆

C₂²×D₇₃

commutator

C₂×C₇₃

C₂²×C₃₂

Fitting

C₇₃×C₄⋊C₃₂

C₂

socle, prime radical

C₂²×C₇₃

C₂²×C₁₆

cosocle

C₂×C₁₆×C₇₃

C₂³

hypercenter

C₄⋊C₃₂

D₇₃

hypocenter

C₇₃

Q₈×C₃₂

LCS

G ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₁₆ ◁ C₄⋊C₃₂

N	H ≅ G/N	type
20. #14556497	Q₈	semidirect
42. Dic₁₄₆	C₃₂	semidirect
63. C₇₃	Q₈×C₃₂	semidirect

H ≅ G/N

type

20. #14556497

Q₈

semidirect

42. Dic₁₄₆

C₃₂

semidirect

63. C₇₃

Q₈×C₃₂

semidirect

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

subgroup	index
C₁₆×Dic₁₄₆	2
#22608126	2
C₄×#14556497	2
C₃₂×Dic₇₃	2
C₇₃×C₄⋊C₃₂	2
Q₈×C₃₂	73

subgroup

index

C₁₆×Dic₁₄₆

#22608126

C₄×#14556497

C₃₂×Dic₇₃

C₇₃×C₄⋊C₃₂

Q₈×C₃₂

Subgroups — #33006613

Conjugacy classes of subgroups

Complements

Maximal subgroups