Group #32979669 — subgroups

Subgroups — #32979669

Order 18688.

262 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 115 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	#22610069	C₂	1	1	yes	maximal, characteristic
2	#22610062	C₂	1	1	yes	maximal, characteristic
3	#22607790	C₂	1	4	yes	maximal
4	C₇₃×#1450	C₂	1	1	yes	maximal, characteristic
5	#17988611	C₂²	1	2	yes	—
6	#17988453	C₂²	1	2	yes	—
7	#17988352	—	2	2	no	—
8	#17988350	—	2	2	no	—
9	#431×C₇₃	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
11	#322×C₇₃	C₄	1	2	yes	complemented
12	C₈²×C₇₃	C₄	1	2	yes	complemented
13	C₂×#11924012	—	2	2	no	—
14	C₂×C₄⋊Dic₇₃	—	2	2	no	—
15	C₄⋊#10596831	—	2	4	no	—
16	#14556498	—	2	4	no	—
17	C₂²×C₈×C₇₃	D₄	1	2	yes	—
18	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
19	C₄⋊C₈×C₇₃	—	2	2	no	—
20	C₂²⋊C₈×C₇₃	—	2	2	no	—
21	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	—
22	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₇₃	Q₈	1	2	yes	—
25	C₂²×Dic₇₃	—	4	2	no	—
26	#11924012	—	4	4	no	—
27	C₂×#10596831	—	4	2	no	—
28	C₄⋊Dic₇₃	—	4	4	no	—
29	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
30	C₂²×C₄×C₇₃	#41	1	2	yes	—
31	C₂×C₈×C₇₃	—	2	4	no	—
32	C₂×C₈×C₇₃	—	2	4	no	—
33	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
34	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
35	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
36	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
37	C₄²×C₇₃	#41	1	2	yes	—
38	C₂³×C₇₃	#124	1	1	yes	characteristic
39	C₂×C₄×C₇₃	—	4	2	no	—
40	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₇₃	#117	1	2	yes	—
42	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
43	C₂×C₄×C₇₃	#125	1	2	yes	—
44	C₂×C₄×C₇₃	#135	1	4	yes	—
45	C₂×Dic₇₃	—	8	4	no	—
46	C₂×Dic₇₃	—	8	2	no	—
47	C₈×C₇₃	—	2	2	no	—
48	C₈×C₇₃	—	2	2	no	—
49	C₈×C₇₃	—	4	4	no	—
50	#10596831	—	8	2	no	—
51	C₂²×C₇₃	—	2	2	no	—
52	C₂²×C₇₃	#382	1	1	yes	characteristic
53	C₂²×C₇₃	#488	1	2	yes	—
54	C₄×C₇₃	—	2	2	no	—
55	C₄×C₇₃	—	2	2	no	—
56	C₄×C₇₃	#427	1	2	yes	—
57	C₄×C₇₃	#488	1	2	yes	—
58	Dic₇₃	—	16	2	no	—
59	#8791	—	73	1	no	maximal, Sylow
60	C₂×C₇₃	—	2	2	no	—
61	C₂×C₇₃	#1833	1	2	yes	—
62	C₂×C₇₃	#1835	1	1	yes	characteristic
63	#1837	—	73	1	no	—
64	#1835	—	73	1	no	—
65	#1450	D₇₃	1	1	yes	characteristic
66	#1286	—	73	4	no	—
67	C₇₃	#8791	1	1	yes	characteristic, complemented, Sylow
68	#431	C₂×D₇₃	1	2	yes	—
69	#423	—	73	2	no	—
70	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
71	#382	—	73	2	no	—
72	#341	—	146	2	no	—
73	#340	—	146	2	no	—
74	#322	Dic₇₃	1	2	yes	complemented
75	C₈²	Dic₇₃	1	2	yes	complemented
76	C₂×M₁₆	—	146	2	no	—
77	C₂²×C₈	D₂₉₂	1	2	yes	—
78	C₂×C₄⋊C₄	—	146	2	no	—
79	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
80	C₄⋊C₈	—	2	2	no	—
81	C₄⋊C₈	—	146	4	no	—
82	C₂²⋊C₈	—	2	2	no	—
83	C₄×C₈	Dic₁₄₆	1	2	yes	—
84	C₄×C₈	C₂×Dic₇₃	1	4	yes	—
85	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
86	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
87	#95	—	146	4	no	—
88	C₂²×C₄	—	292	2	no	—
89	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
90	C₂²×C₄	#11924028	1	2	yes	—
91	M₁₆	—	292	4	no	—
92	C₂×C₈	—	2	4	no	—
93	C₂×C₈	—	2	4	no	—
94	C₂×C₈	—	292	2	no	—
95	C₂×C₈	C₄⋊Dic₇₃	1	2	yes	—
96	C₂×C₈	C₄⋊Dic₇₃	1	2	yes	—
97	C₄⋊C₄	—	292	4	no	—
98	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
99	C₄²	C₂×Dic₁₄₆	1	1	yes	characteristic
100	C₄²	#11924028	1	2	yes	—
101	C₂³	#14556579	1	1	yes	characteristic
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	584	4	no	—
104	C₂×C₄	—	584	2	no	—
105	C₂×C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
106	C₂×C₄	#14556550	1	2	yes	—
107	C₂×C₄	#14556579	1	1	yes	characteristic
108	C₂×C₄	#14556583	1	2	yes	—
109	C₂×C₄	#14556613	1	4	yes	—
110	C₈	—	2	2	no	—
111	C₈	—	2	2	no	—
112	C₈	—	4	4	no	—
113	C₈	—	584	2	no	—
114	C₂²	—	2	2	no	—
115	C₂²	#17988453	1	1	yes	characteristic
116	C₂²	#17988942	1	2	yes	—
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	1168	2	no	—
120	C₄	#17988625	1	2	yes	—
121	C₄	#17988942	1	2	yes	—
122	C₂	—	2	2	no	—
123	C₂	#22610055	1	2	yes	—
124	C₂	#22610062	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22610069

C₂

yes

maximal, characteristic

#22610062

C₂

yes

maximal, characteristic

#22607790

C₂

yes

maximal

C₇₃×#1450

C₂

yes

maximal, characteristic

#17988611

C₂²

yes

—

#17988453

C₂²

yes

—

#17988352

—

#17988350

—

#431×C₇₃

C₂²

yes

—

C₂×C₄×C₈×C₇₃

C₂²

yes

characteristic

#322×C₇₃

C₄

yes

complemented

C₈²×C₇₃

C₄

yes

complemented

C₂×#11924012

—

C₂×C₄⋊Dic₇₃

—

C₄⋊#10596831

—

#14556498

—

C₂²×C₈×C₇₃

D₄

yes

—

C₂×C₄²×C₇₃

C₂³

yes

characteristic

C₄⋊C₈×C₇₃

—

C₂²⋊C₈×C₇₃

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

Q₈

yes

—

C₂²×Dic₇₃

—

#11924012

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₂²×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₇₃

#41

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₂×Q₈

yes

characteristic

C₄²×C₇₃

#41

yes

—

C₂³×C₇₃

#124

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

#117

yes

—

C₂×C₄×C₇₃

#124

yes

characteristic

C₂×C₄×C₇₃

#125

yes

—

C₂×C₄×C₇₃

#135

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

#382

yes

characteristic

C₂²×C₇₃

#488

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#427

yes

—

C₄×C₇₃

#488

yes

—

Dic₇₃

—

#8791

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#1833

yes

—

C₂×C₇₃

yes

—

—

D₇₃

yes

characteristic

#1286

—

C₇₃

#8791

yes

characteristic, complemented, Sylow

#431

C₂×D₇₃

yes

—

#423

—

C₂×C₄×C₈

C₂×D₇₃

yes

characteristic

#382

—

#341

—

146

—

#340

—

146

—

#322

Dic₇₃

yes

complemented

C₈²

Dic₇₃

yes

complemented

C₂×M₁₆

—

146

—

C₂²×C₈

D₂₉₂

yes

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄²

C₂²×D₇₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

146

—

C₂²⋊C₈

—

C₄×C₈

Dic₁₄₆

yes

—

C₄×C₈

C₂×Dic₇₃

yes

—

C₄×C₈

C₂×Dic₇₃

yes

characteristic

C₄×C₈

C₂×Dic₇₃

yes

characteristic

#95

—

146

—

C₂²×C₄

—

292

—

C₂²×C₄

C₂×D₂₉₂

yes

characteristic

C₂²×C₄

#11924028

yes

—

M₁₆

—

292

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

292

—

C₂×C₈

C₄⋊Dic₇₃

yes

—

C₂×C₈

C₄⋊Dic₇₃

yes

—

C₄⋊C₄

—

292

—

C₄²

C₂²×Dic₇₃

yes

characteristic

C₄²

C₂×Dic₁₄₆

yes

characteristic

100

C₄²

#11924028

yes

—

101

C₂³

#14556579

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

584

—

104

C₂×C₄

—

584

—

105

C₂×C₄

C₂×C₄⋊Dic₇₃

yes

characteristic

106

C₂×C₄

#14556550

yes

—

107

C₂×C₄

#14556579

yes

characteristic

108

C₂×C₄

#14556583

yes

—

109

C₂×C₄

#14556613

yes

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

584

—

114

C₂²

—

115

C₂²

#17988453

yes

characteristic

116

C₂²

#17988942

yes

—

117

C₄

—

118

C₄

—

119

C₄

—

1168

—

120

C₄

#17988625

yes

—

121

C₄

#17988942

yes

—

122

C₂

—

123

C₂

#22610055

yes

—

124

C₂

#22610062

yes

characteristic

	subgroup	G/H
center	C₂²	#17988453
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1450	C₂
socle	C₂²×C₇₃	#382
prime radical	C₂³×C₇₃	#124
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1450	D₇₃
hypocenter	C₇₃	#8791
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

subgroup

G/H

center

C₂²

#17988453

commutator

C₄²×C₇₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₇₃

Fitting

C₇₃×#1450

C₂

C₂²×C₇₃

C₂³×C₇₃

C₂×C₄²×C₇₃

C₂³

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450

N	H ≅ G/N	type
11. #322×C₇₃	C₄	semidirect
12. C₈²×C₇₃	C₄	semidirect
67. C₇₃	#8791	semidirect
74. #322	Dic₇₃	semidirect
75. C₈²	Dic₇₃	semidirect

H ≅ G/N

type

11. #322×C₇₃

C₄

semidirect

12. C₈²×C₇₃

C₄

semidirect

67. C₇₃

#8791

semidirect

74. #322

Dic₇₃

semidirect

75. C₈²

Dic₇₃

semidirect

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

subgroup	index
#22610069	2
#22610062	2
#22607790	2
C₇₃×#1450	2
#8791	73

subgroup

index

C₇₃×#1450

Subgroups — #32979669

Conjugacy classes of subgroups

Complements

Maximal subgroups