Group #33007015 — subgroups

Subgroups — #33007015

Order 18688.

150 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 75 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	#22612210	C₂	1	1	yes	maximal, characteristic
2	#22612206	C₂	1	1	yes	maximal, characteristic
3	#22608133	C₂	1	2	yes	maximal
4	#22608131	C₂	1	2	yes	maximal
5	C₇₃×#1397	C₂	1	1	yes	maximal, characteristic
6	#17988992	—	2	1	no	—
7	#17988989	—	2	1	no	—
8	#17988415	—	2	1	no	—
9	#17988413	—	2	1	no	—
10	#17988413	C₂²	1	1	yes	characteristic
11	#17988413	C₂²	1	2	yes	—
12	#17988413	C₂²	1	1	yes	characteristic
13	C₂×C₃₂×C₇₃	C₂²	1	2	yes	—
14	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
15	#14556591	—	4	1	no	—
16	#14556583	—	4	1	no	—
17	#14556525	—	2	2	no	—
18	#14556525	—	2	1	no	—
19	#14556525	—	2	1	no	—
20	#14556525	—	4	1	no	—
21	#14556523	—	4	1	no	—
22	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
23	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
24	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
25	C₃₂×C₇₃	Q₈	1	4	yes	complemented
26	C₂×Dic₁₄₆	—	8	1	no	—
27	C₄⋊Dic₇₃	—	4	1	no	—
28	C₄⋊Dic₇₃	—	4	1	no	—
29	C₄⋊Dic₇₃	—	4	2	no	—
30	C₄⋊Dic₇₃	—	8	1	no	—
31	#11924007	—	8	1	no	—
32	#11924007	—	8	1	no	—
33	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
34	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
35	C₄²×C₇₃	D₈	1	1	yes	characteristic
36	C₁₆×C₇₃	Q₁₆	1	2	yes	—
37	C₁₆×C₇₃	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₇₃	D₁₆	1	1	yes	characteristic
39	C₂×C₄×C₇₃	D₁₆	1	1	yes	characteristic
40	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
41	C₂×Dic₇₃	—	8	1	no	—
42	C₂×Dic₇₃	—	8	1	no	—
43	C₂×Dic₇₃	—	8	2	no	—
44	Dic₁₄₆	—	16	2	no	—
45	C₈×C₇₃	Q₃₂	1	2	yes	—
46	C₈×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
47	C₈×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
48	C₂²×C₇₃	C₂×D₁₆	1	1	yes	characteristic
49	C₄×C₇₃	—	2	1	no	—
50	C₄×C₇₃	—	2	1	no	—
51	C₄×C₇₃	#500	1	1	yes	characteristic
52	C₄×C₇₃	C₂×Q₃₂	1	1	yes	characteristic
53	Dic₇₃	—	16	2	no	—
54	Dic₇₃	—	16	1	no	—
55	Dic₇₃	—	16	1	no	—
56	#13734	—	73	1	no	maximal, Sylow
57	C₂×C₇₃	#2252	1	1	yes	characteristic
58	C₂×C₇₃	#2263	1	1	yes	characteristic
59	C₂×C₇₃	#2264	1	1	yes	characteristic
60	#2253	—	73	1	no	—
61	#2252	—	73	1	no	—
62	#1424	—	73	2	no	—
63	#1423	—	73	2	no	—
64	#1397	D₇₃	1	1	yes	characteristic
65	C₇₃	#13734	1	1	yes	characteristic, complemented, Sylow
66	#500	—	146	1	no	—
67	#499	—	146	1	no	—
68	C₂×C₃₂	C₂×D₇₃	1	2	yes	—
69	#367	—	146	1	no	—
70	#366	—	73	1	no	—
71	#366	—	73	2	no	—
72	#366	—	73	1	no	—
73	#366	—	146	1	no	—
74	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
75	C₄⋊Q₈	—	292	1	no	—
76	#125	—	292	1	no	—
77	C₂×C₁₆	D₂₉₂	1	1	yes	characteristic
78	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
79	#107	—	292	1	no	—
80	#107	—	146	2	no	—
81	#107	—	146	1	no	—
82	#107	—	146	1	no	—
83	#106	—	292	1	no	—
84	C₄×C₈	D₂₉₂	1	1	yes	characteristic
85	C₃₂	Dic₁₄₆	1	4	yes	complemented
86	C₂×Q₈	—	584	1	no	—
87	C₂×C₈	D₅₈₄	1	1	yes	characteristic
88	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
89	C₄⋊C₄	—	292	1	no	—
90	C₄⋊C₄	—	292	1	no	—
91	C₄⋊C₄	—	292	2	no	—
92	C₄⋊C₄	—	584	1	no	—
93	C₄⋊C₄	—	584	1	no	—
94	C₄⋊C₄	—	584	1	no	—
95	C₄²	D₅₈₄	1	1	yes	characteristic
96	C₁₆	Dic₂₉₂	1	2	yes	—
97	C₁₆	C₂×Dic₁₄₆	1	2	yes	—
98	Q₈	—	1168	2	no	—
99	C₂×C₄	—	584	1	no	—
100	C₂×C₄	—	584	1	no	—
101	C₂×C₄	—	584	2	no	—
102	C₂×C₄	D₁₁₆₈	1	1	yes	characteristic
103	C₂×C₄	D₁₁₆₈	1	1	yes	characteristic
104	C₂×C₄	C₂×D₅₈₄	1	1	yes	characteristic
105	C₈	Dic₅₈₄	1	2	yes	—
106	C₈	#14556591	1	1	yes	characteristic
107	C₈	C₂×Dic₂₉₂	1	1	yes	characteristic
108	C₂²	C₂×D₁₁₆₈	1	1	yes	characteristic
109	C₄	—	2	1	no	—
110	C₄	—	2	1	no	—
111	C₄	—	1168	2	no	—
112	C₄	—	1168	1	no	—
113	C₄	—	1168	1	no	—
114	C₄	#17988992	1	1	yes	characteristic
115	C₄	C₂×Dic₅₈₄	1	1	yes	characteristic
116	C₂	#22612206	1	1	yes	characteristic
117	C₂	#22612246	1	1	yes	characteristic
118	C₂	#22612251	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22612210

C₂

yes

maximal, characteristic

#22612206

C₂

yes

maximal, characteristic

#22608133

C₂

yes

maximal

#22608131

C₂

yes

maximal

C₇₃×#1397

C₂

yes

maximal, characteristic

—

—

—

—

C₂²

yes

characteristic

#17988413

C₂²

yes

—

#17988413

C₂²

yes

characteristic

C₂×C₃₂×C₇₃

C₂²

yes

—

C₄×C₁₆×C₇₃

C₂²

yes

—

—

—

—

—

—

—

C₂×C₁₆×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂³

yes

characteristic

C₄×C₈×C₇₃

D₄

yes

characteristic

C₃₂×C₇₃

Q₈

yes

complemented

C₂×Dic₁₄₆

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#11924007

—

#11924007

—

C₂×C₈×C₇₃

D₈

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₄²×C₇₃

D₈

yes

characteristic

C₁₆×C₇₃

Q₁₆

yes

—

C₁₆×C₇₃

C₂×Q₈

yes

—

C₂×C₄×C₇₃

D₁₆

yes

characteristic

C₂×C₄×C₇₃

D₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

C₈×C₇₃

Q₃₂

yes

—

C₈×C₇₃

C₄⋊Q₈

yes

characteristic

C₈×C₇₃

C₂×Q₁₆

yes

characteristic

C₂²×C₇₃

C₂×D₁₆

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#500

yes

characteristic

C₄×C₇₃

C₂×Q₃₂

yes

characteristic

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

—

—

—

D₇₃

yes

characteristic

C₇₃

#13734

yes

characteristic, complemented, Sylow

#500

—

146

—

#499

—

146

—

C₂×C₃₂

C₂×D₇₃

yes

—

#367

—

146

—

—

—

—

—

146

—

C₄×C₁₆

C₂×D₇₃

yes

characteristic

C₄⋊Q₈

—

292

—

#125

—

292

—

C₂×C₁₆

D₂₉₂

yes

characteristic

C₂×C₁₆

C₂²×D₇₃

yes

characteristic

#107

—

292

—

#107

—

146

—

#107

—

146

—

#107

—

146

—

#106

—

292

—

C₄×C₈

D₂₉₂

yes

characteristic

C₃₂

Dic₁₄₆

yes

complemented

C₂×Q₈

—

584

—

C₂×C₈

D₅₈₄

yes

characteristic

C₂×C₈

C₂×D₂₉₂

yes

characteristic

C₄⋊C₄

—

292

—

C₄⋊C₄

—

292

—

C₄⋊C₄

—

292

—

C₄⋊C₄

—

584

—

C₄⋊C₄

—

584

—

C₄⋊C₄

—

584

—

C₄²

D₅₈₄

yes

characteristic

C₁₆

Dic₂₉₂

yes

—

C₁₆

C₂×Dic₁₄₆

yes

—

Q₈

—

1168

—

C₂×C₄

—

584

—

100

C₂×C₄

—

584

—

101

C₂×C₄

—

584

—

102

C₂×C₄

D₁₁₆₈

yes

characteristic

103

C₂×C₄

D₁₁₆₈

yes

characteristic

104

C₂×C₄

C₂×D₅₈₄

yes

characteristic

105

C₈

Dic₅₈₄

yes

—

106

C₈

#14556591

yes

characteristic

107

C₈

C₂×Dic₂₉₂

yes

characteristic

108

C₂²

C₂×D₁₁₆₈

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

1168

—

112

C₄

—

1168

—

113

C₄

—

1168

—

114

C₄

#17988992

yes

characteristic

115

C₄

C₂×Dic₅₈₄

yes

characteristic

116

C₂

#22612206

yes

characteristic

117

C₂

#22612246

yes

characteristic

118

C₂

#22612251

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₁₆₈
commutator, cosocle	C₂×C₁₆×C₇₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₃
Fitting	C₇₃×#1397	C₂
socle, prime radical	C₂²×C₇₃	C₂×D₁₆
hypercenter	#1397	D₇₃
hypocenter	C₇₃	#13734
LCS	G ▷ C₂×C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1397

subgroup

G/H

center

C₂²

C₂×D₁₁₆₈

commutator, cosocle

C₂×C₁₆×C₇₃

C₂³

Frattini

C₂×C₁₆

C₂²×D₇₃

Fitting

C₇₃×#1397

C₂

socle, prime radical

C₂²×C₇₃

C₂×D₁₆

D₇₃

C₇₃

G ▷ C₂×C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #1397

N	H ≅ G/N	type
25. C₃₂×C₇₃	Q₈	semidirect
65. C₇₃	#13734	semidirect
85. C₃₂	Dic₁₄₆	semidirect

H ≅ G/N

type

25. C₃₂×C₇₃

Q₈

semidirect

65. C₇₃

#13734

semidirect

85. C₃₂

Dic₁₄₆

semidirect

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

subgroup	index
#22612210	2
#22612206	2
#22608133	2
#22608131	2
C₇₃×#1397	2
#13734	73

subgroup

index

C₇₃×#1397

Subgroups — #33007015

Conjugacy classes of subgroups

Complements

Maximal subgroups