Group #33000727 — subgroups

Subgroups — #33000727

Order 18688.

262 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 95 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	#22612213	C₂	1	1	yes	maximal, characteristic
2	#22612211	C₂	1	2	yes	maximal
3	#22609548	C₂	1	2	yes	maximal
4	C₁₆×#10596831	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#2254	C₂	1	1	yes	maximal, characteristic
6	#17988992	C₂²	1	2	yes	—
7	#17988990	—	2	2	no	—
8	#17988966	—	2	2	no	—
9	#17988415	—	2	2	no	—
10	#17988414	—	2	4	no	—
11	C₄×#11924003	C₂²	1	1	yes	characteristic
12	C₈×#10596831	C₂²	1	1	yes	characteristic
13	#500×C₇₃	C₂²	1	2	yes	—
14	#367×C₇₃	—	2	2	no	—
15	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
16	C₂×#11924018	—	4	4	no	—
17	#14556591	—	2	2	no	—
18	C₂×#11924003	D₄	1	2	yes	—
19	#14556525	—	2	2	no	—
20	#14556525	—	2	2	no	—
21	#14556524	—	4	4	no	—
22	C₄×#10596831	D₄	1	2	yes	—
23	C₄⋊Q₈×C₇₃	—	2	2	no	—
24	C₂×C₁₆×C₇₃	D₄	1	2	yes	—
25	#107×C₇₃	—	2	2	no	—
26	#107×C₇₃	—	2	2	no	—
27	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
28	C₂×Dic₁₄₆	—	4	2	no	—
29	#11924018	—	8	8	no	—
30	C₂×#10596831	D₈	1	4	yes	—
31	C₄⋊Dic₇₃	—	4	2	no	—
32	C₄⋊Dic₇₃	—	4	2	no	—
33	#11924003	Q₁₆	1	4	yes	complemented
34	C₂×Q₈×C₇₃	—	4	2	no	—
35	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
37	C₄⋊C₄×C₇₃	—	4	2	no	—
38	C₄⋊C₄×C₇₃	—	4	2	no	—
39	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
40	C₁₆×C₇₃	Q₁₆	1	4	yes	complemented
41	Q₈×C₇₃	—	8	4	no	—
42	C₂×C₄×C₇₃	—	8	2	no	—
43	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
44	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
45	C₂×Dic₇₃	—	8	2	no	—
46	Dic₁₄₆	—	8	4	no	—
47	C₈×C₇₃	C₂×Q₁₆	1	2	yes	—
48	C₈×C₇₃	C₂×Q₁₆	1	2	yes	—
49	#10596831	SD₃₂	1	8	yes	—
50	C₂²×C₇₃	#493	1	1	yes	characteristic
51	C₄×C₇₃	—	16	2	no	—
52	C₄×C₇₃	#494	1	1	yes	characteristic
53	C₄×C₇₃	#494	1	1	yes	characteristic
54	C₄×C₇₃	C₂×SD₃₂	1	4	yes	—
55	Dic₇₃	—	16	2	no	—
56	#12523	—	73	1	no	maximal, Sylow
57	C₂×C₇₃	#1713	1	1	yes	characteristic
58	C₂×C₇₃	#2248	1	2	yes	—
59	#2254	—	73	2	no	—
60	#2254	—	73	1	no	—
61	#2254	D₇₃	1	1	yes	characteristic
62	#1713	—	73	2	no	—
63	C₈×C₁₆	—	73	1	no	—
64	C₇₃	#12523	1	1	yes	characteristic, complemented, Sylow
65	#500	—	73	2	no	—
66	#500	—	146	2	no	—
67	#500	C₂×D₇₃	1	2	yes	—
68	#494	—	146	2	no	—
69	#367	—	2	2	no	—
70	#367	—	146	4	no	—
71	#367	—	146	2	no	—
72	C₄×C₁₆	—	73	1	no	—
73	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
74	C₈²	—	73	1	no	—
75	C₂×Q₁₆	—	292	4	no	—
76	C₄⋊Q₈	—	2	2	no	—
77	C₄⋊Q₈	—	146	2	no	—
78	C₂×C₁₆	—	73	2	no	—
79	C₂×C₁₆	C₂²⋊D₇₃	1	2	yes	—
80	#107	—	2	2	no	—
81	#107	—	2	2	no	—
82	#107	—	292	4	no	—
83	#107	—	146	2	no	—
84	#107	—	146	2	no	—
85	C₄×C₈	—	73	2	no	—
86	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
87	C₂×Q₈	—	4	2	no	—
88	C₂×Q₈	—	292	2	no	—
89	Q₁₆	—	584	8	no	—
90	C₂×C₈	—	73	4	no	—
91	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
92	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	4	2	no	—
95	C₄⋊C₄	—	292	2	no	—
96	C₄⋊C₄	—	292	2	no	—
97	C₄²	D₄×D₇₃	1	1	yes	characteristic
98	C₁₆	—	73	4	no	—
99	C₁₆	#11924018	1	4	yes	complemented
100	Q₈	—	8	4	no	—
101	Q₈	—	584	4	no	—
102	C₂×C₄	—	8	2	no	—
103	C₂×C₄	—	584	2	no	—
104	C₂×C₄	#14556587	1	1	yes	characteristic
105	C₂×C₄	D₈×D₇₃	1	2	yes	—
106	C₈	—	73	8	no	—
107	C₈	Q₁₆×D₇₃	1	2	yes	—
108	C₈	C₂×#11924018	1	2	yes	—
109	C₂²	#17988962	1	1	yes	characteristic
110	C₄	—	16	2	no	—
111	C₄	—	1168	2	no	—
112	C₄	#17988965	1	1	yes	characteristic
113	C₄	#17988966	1	1	yes	characteristic
114	C₄	SD₃₂×D₇₃	1	4	yes	—
115	C₂	#22609548	1	1	yes	characteristic
116	C₂	#22612185	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22612213

C₂

yes

maximal, characteristic

#22612211

C₂

yes

maximal

#22609548

C₂

yes

maximal

C₁₆×#10596831

C₂

yes

maximal, characteristic

C₇₃×#2254

C₂

yes

maximal, characteristic

#17988992

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₈×#10596831

C₂²

yes

characteristic

#500×C₇₃

C₂²

yes

—

#367×C₇₃

—

C₄×C₁₆×C₇₃

C₂²

yes

—

—

D₄

yes

—

—

—

—

D₄

yes

—

C₄⋊Q₈×C₇₃

—

C₂×C₁₆×C₇₃

D₄

yes

—

#107×C₇₃

—

#107×C₇₃

—

C₄×C₈×C₇₃

C₂³

yes

characteristic

C₂×Dic₁₄₆

—

#11924018

—

C₂×#10596831

D₈

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924003

Q₁₆

yes

complemented

C₂×Q₈×C₇₃

—

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×D₄

yes

characteristic

C₁₆×C₇₃

Q₁₆

yes

complemented

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#127

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

—

C₂×Dic₇₃

—

Dic₁₄₆

—

C₈×C₇₃

C₂×Q₁₆

yes

—

C₈×C₇₃

C₂×Q₁₆

yes

—

#10596831

SD₃₂

yes

—

C₂²×C₇₃

#493

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

C₄×C₇₃

C₂×SD₃₂

yes

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

#2254

—

#2254

—

#2254

D₇₃

yes

characteristic

#1713

—

C₈×C₁₆

—

C₇₃

#12523

yes

characteristic, complemented, Sylow

#500

—

#500

—

146

—

#500

C₂×D₇₃

yes

—

#494

—

146

—

#367

—

#367

—

146

—

#367

—

146

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₇₃

yes

characteristic

C₈²

—

C₂×Q₁₆

—

292

—

C₄⋊Q₈

—

C₄⋊Q₈

—

146

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₇₃

yes

—

#107

—

#107

—

#107

—

292

—

#107

—

146

—

#107

—

146

—

C₄×C₈

—

C₄×C₈

C₂²×D₇₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

292

—

Q₁₆

—

584

—

C₂×C₈

—

C₂×C₈

D₄×D₇₃

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₇₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

292

—

C₄⋊C₄

—

292

—

C₄²

D₄×D₇₃

yes

characteristic

C₁₆

—

C₁₆

#11924018

yes

complemented

100

Q₈

—

101

Q₈

—

584

—

102

C₂×C₄

—

103

C₂×C₄

—

584

—

104

C₂×C₄

#14556587

yes

characteristic

105

C₂×C₄

D₈×D₇₃

yes

—

106

C₈

—

107

C₈

Q₁₆×D₇₃

yes

—

108

C₈

C₂×#11924018

yes

—

109

C₂²

#17988962

yes

characteristic

110

C₄

—

111

C₄

—

1168

—

112

C₄

#17988965

yes

characteristic

113

C₄

#17988966

yes

characteristic

114

C₄

SD₃₂×D₇₃

yes

—

115

C₂

#22609548

yes

characteristic

116

C₂

#22612185

yes

—

	subgroup	G/H
center	C₂²	#17988962
commutator, cosocle	C₄×C₈×C₇₃	C₂³
Frattini	C₄×C₈	C₂²×D₇₃
Fitting	C₇₃×#2254	C₂
socle, prime radical	C₂²×C₇₃	#493
hypercenter	#2254	D₇₃
hypocenter	C₇₃	#12523
LCS	G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

subgroup

G/H

center

C₂²

#17988962

commutator, cosocle

C₄×C₈×C₇₃

C₂³

Frattini

C₄×C₈

C₂²×D₇₃

Fitting

C₇₃×#2254

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₄×C₈×C₇₃ ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

N	H ≅ G/N	type
33. #11924003	Q₁₆	semidirect
40. C₁₆×C₇₃	Q₁₆	semidirect
64. C₇₃	#12523	semidirect
99. C₁₆	#11924018	semidirect

H ≅ G/N

type

33. #11924003

Q₁₆

semidirect

40. C₁₆×C₇₃

Q₁₆

semidirect

64. C₇₃

#12523

semidirect

99. C₁₆

#11924018

semidirect

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

subgroup	index
#22612213	2
#22612211	2
#22609548	2
C₁₆×#10596831	2
C₇₃×#2254	2
#12523	73

subgroup

index

#22612213

#22612211

#22609548

C₁₆×#10596831

C₇₃×#2254

#12523

Subgroups — #33000727

Conjugacy classes of subgroups

Complements

Maximal subgroups