Group #34009762 — subgroups

Subgroups — #34009762

Order 18688.

228 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 92 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	#22626294	C₂	1	2	yes	maximal
2	#22624849	C₂	1	4	yes	maximal, complemented
3	#22608213	C₂	1	1	yes	maximal, characteristic
4	#17989508	C₄	1	4	yes	—
5	#17989482	C₄	1	4	yes	—
6	#17989475	C₂²	1	4	yes	—
7	#17988638	C₄	1	2	yes	—
8	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
9	#17988309	C₂²	1	2	yes	—
10	C₈×#10596831	C₄	1	1	yes	characteristic
11	C₈²×C₇₃	C₄	1	1	yes	characteristic
12	C₂×#11924031	C₂×C₄	1	8	yes	—
13	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
14	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
15	#14556520	—	2	2	no	—
16	#14556503	—	2	2	no	—
17	C₄×#10596831	C₂×C₄	1	2	yes	—
18	C₄×#10596831	C₂³	1	1	yes	characteristic
19	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
20	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
22	#11924031	—	2	8	no	—
23	C₂×C₄×D₇₃	C₄²	1	1	yes	characteristic
24	C₂×C₄×D₇₃	M₁₆	1	2	yes	—
25	C₂×#10596831	—	2	2	no	—
26	C₂×#10596831	C₂²×C₄	1	2	yes	—
27	C₈×D₇₃	—	2	4	no	—
28	C₄×Dic₇₃	C₄²	1	1	yes	characteristic
29	C₄×Dic₇₃	M₁₆	1	2	yes	—
30	C₂×C₈×C₇₃	—	2	2	no	—
31	C₂×C₈×C₇₃	C₄²	1	2	yes	—
32	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
33	C₂²×D₇₃	#97	1	1	yes	characteristic
34	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
35	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
36	C₂×Dic₇₃	#97	1	2	yes	—
37	C₂×Dic₇₃	#97	1	1	yes	characteristic
38	C₄×D₇₃	—	2	2	no	—
39	C₄×D₇₃	—	4	2	no	—
40	C₈×C₇₃	—	2	2	no	—
41	C₈×C₇₃	—	4	2	no	—
42	#10596831	—	4	2	no	—
43	#10596831	#131	1	4	yes	—
44	C₂²×C₇₃	C₂×#97	1	1	yes	characteristic
45	C₂×D₇₃	—	2	2	no	—
46	C₂×D₇₃	—	2	1	no	—
47	C₄×C₇₃	—	2	2	no	—
48	C₄×C₇₃	#405	1	2	yes	—
49	Dic₇₃	—	2	2	no	—
50	Dic₇₃	—	2	1	no	—
51	#11416	—	73	1	no	maximal, Sylow
52	C₂×C₇₃	#1452	1	1	yes	characteristic
53	C₂×C₇₃	#2110	1	2	yes	—
54	D₇₃	—	4	1	no	—
55	#2163	—	73	2	no	—
56	#1450	—	73	1	no	—
57	#1313	—	73	4	no	—
58	C₇₃	#11416	1	1	yes	characteristic, complemented, Sylow
59	#431	—	73	2	no	—
60	C₂×C₄×C₈	—	73	1	no	—
61	C₄⋊C₁₆	—	73	4	no	—
62	C₂²⋊C₁₆	—	73	4	no	—
63	#346	—	73	4	no	—
64	#322	—	73	2	no	—
65	C₈²	—	73	1	no	—
66	C₈²	#64408	1	1	yes	characteristic
67	C₂²×C₈	—	73	2	no	—
68	C₂×C₄²	—	73	1	no	—
69	C₂×C₁₆	—	73	8	no	—
70	C₄⋊C₈	—	146	2	no	—
71	C₂²⋊C₈	—	146	2	no	—
72	C₄×C₈	—	73	2	no	—
73	C₄×C₈	—	73	1	no	—
74	C₄×C₈	—	73	2	no	—
75	C₄×C₈	C₂×#64408	1	2	yes	—
76	C₄×C₈	C₂×#64408	1	1	yes	characteristic
77	C₂²×C₄	—	73	2	no	—
78	C₂²×C₄	—	73	1	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	73	2	no	—
81	C₂×C₈	—	146	4	no	—
82	C₂×C₈	—	146	2	no	—
83	C₂×C₈	C₄×#64408	1	2	yes	—
84	C₄²	—	73	2	no	—
85	C₄²	—	73	1	no	—
86	C₄²	C₂²×#64408	1	1	yes	characteristic
87	C₁₆	—	146	8	no	—
88	C₂³	—	73	1	no	—
89	C₂×C₄	—	292	2	no	—
90	C₂×C₄	—	73	2	no	—
91	C₂×C₄	—	73	1	no	—
92	C₂×C₄	—	146	2	no	—
93	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
94	C₂×C₄	D₇₃⋊M₁₆	1	2	yes	—
95	C₈	—	2	2	no	—
96	C₈	—	4	2	no	—
97	C₈	—	292	2	no	—
98	C₈	—	73	4	no	—
99	C₂²	—	146	2	no	—
100	C₂²	—	146	1	no	—
101	C₂²	#17989546	1	1	yes	characteristic
102	C₄	—	2	2	no	—
103	C₄	—	146	2	no	—
104	C₄	—	146	1	no	—
105	C₄	#17989550	1	2	yes	—
106	C₂	—	292	1	no	—
107	C₂	#22625145	1	1	yes	characteristic
108	C₂	#22626153	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22626294

C₂

yes

maximal

#22624849

C₂

yes

maximal, complemented

#22608213

C₂

yes

maximal, characteristic

#17989508

C₄

yes

—

#17989482

C₄

yes

—

#17989475

C₂²

yes

—

#17988638

C₄

yes

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

#17988309

C₂²

yes

—

C₈×#10596831

C₄

yes

characteristic

C₈²×C₇₃

C₄

yes

characteristic

C₂×#11924031

C₂×C₄

yes

—

C₂×C₈×D₇₃

C₂×C₄

yes

—

C₄²×D₇₃

C₂×C₄

yes

—

—

C₂×C₄

yes

—

C₄×#10596831

C₂³

yes

characteristic

C₈×Dic₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

#11924031

—

C₂×C₄×D₇₃

C₄²

yes

characteristic

C₂×C₄×D₇₃

M₁₆

yes

—

C₂×#10596831

—

C₂×#10596831

C₂²×C₄

yes

—

C₈×D₇₃

—

C₄×Dic₇₃

C₄²

yes

characteristic

C₄×Dic₇₃

M₁₆

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄²

yes

—

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₂²×D₇₃

#97

yes

characteristic

C₂×C₄×C₇₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₃

C₂×M₁₆

yes

—

C₂×Dic₇₃

#97

yes

—

C₂×Dic₇₃

#97

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

#10596831

—

#10596831

#131

yes

—

C₂²×C₇₃

C₂×#97

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#405

yes

—

Dic₇₃

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

—

#2163

—

#1450

—

#1313

—

C₇₃

#11416

yes

characteristic, complemented, Sylow

#431

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

#346

—

#322

—

C₈²

—

C₈²

#64408

yes

characteristic

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

C₄⋊C₈

—

146

—

C₂²⋊C₈

—

146

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#64408

yes

—

C₄×C₈

C₂×#64408

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

146

—

C₂×C₈

—

146

—

C₂×C₈

C₄×#64408

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×#64408

yes

characteristic

C₁₆

—

146

—

C₂³

—

C₂×C₄

—

292

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

146

—

C₂×C₄

C₂×C₄×#64408

yes

characteristic

C₂×C₄

D₇₃⋊M₁₆

yes

—

C₈

—

C₈

—

C₈

—

292

—

C₈

—

C₂²

—

146

—

100

C₂²

—

146

—

101

C₂²

#17989546

yes

characteristic

102

C₄

—

103

C₄

—

146

—

104

C₄

—

146

—

105

C₄

#17989550

yes

—

106

C₂

—

292

—

107

C₂

#22625145

yes

characteristic

108

C₂

#22626153

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#64408
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₄²	C₂²×#64408
Fitting	C₈²×C₇₃	C₄
socle	C₂²×C₇₃	C₂×#97
prime radical	C₂²×D₇₃	#97
cosocle	C₄×#10596831	C₂³
hypercenter	C₈²	#64408
hypocenter	C₇₃	#11416
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₈²

subgroup

G/H

center

C₂×C₄

C₂×C₄×#64408

commutator

C₂×C₄×C₇₃

C₂×C₄²

Frattini

C₄²

C₂²×#64408

Fitting

C₈²×C₇₃

C₄

socle

C₂²×C₇₃

C₂×#97

C₂²×D₇₃

C₂³

C₈²

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₄ ◁ C₈²

N	H ≅ G/N	type
2. #22624849	C₂	semidirect
58. C₇₃	#11416	semidirect

H ≅ G/N

type

2. #22624849

C₂

semidirect

58. C₇₃

#11416

semidirect

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

subgroup	index
#22626294	2
#22624849	2
#22608213	2
#11416	73

subgroup

index

Subgroups — #34009762

Conjugacy classes of subgroups

Complements

Maximal subgroups