Group #34000489 — subgroups

Subgroups — #34000489

Order 18688.

148 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 50 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	#22626305	C₂	1	2	yes	maximal
2	#22611689	C₂	1	1	yes	maximal, characteristic
3	#17989509	—	2	2	no	—
4	#17989483	—	2	2	no	—
5	#17989480	C₄	1	4	yes	—
6	C₂×#14556531	C₄	1	1	yes	characteristic
7	#17988542	C₂²	1	1	yes	characteristic
8	#17988405	—	2	2	no	—
9	C₂²×C₁₆×C₇₃	C₄	1	1	yes	characteristic
10	#14556665	—	2	4	no	—
11	C₂×#11924012	D₄	1	1	yes	characteristic
12	#14556550	Q₈	1	1	yes	characteristic
13	#14556531	—	4	1	no	—
14	C₂×#11924003	—	2	1	no	—
15	#14556501	C₂×C₄	1	2	yes	—
16	C₈×Dic₇₃	D₄	1	2	yes	—
17	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₆×C₇₃	—	2	1	no	—
19	C₂×C₁₆×C₇₃	—	4	1	no	—
20	#11924031	—	4	4	no	—
21	#11924012	—	4	1	no	—
22	C₂×#10596831	C₂²⋊C₄	1	2	yes	—
23	C₄⋊Dic₇₃	—	2	1	no	—
24	C₂²⋊Dic₇₃	—	2	1	no	—
25	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
26	#11924003	—	4	1	no	—
27	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₈×C₇₃	—	2	1	no	—
29	C₂×C₈×C₇₃	—	2	1	no	—
30	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
31	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₁₆×C₇₃	—	4	1	no	—
33	C₂³×C₇₃	#108	1	1	yes	characteristic
34	C₂×C₄×C₇₃	—	2	1	no	—
35	C₂×C₄×C₇₃	—	2	1	no	—
36	C₂×C₄×C₇₃	#95	1	1	yes	characteristic
37	C₂×C₄×C₇₃	#108	1	1	yes	characteristic
38	C₂×Dic₇₃	—	2	2	no	—
39	C₈×C₇₃	—	2	1	no	—
40	C₈×C₇₃	#100	1	1	yes	characteristic
41	C₈×C₇₃	#101	1	1	yes	characteristic
42	#10596831	—	2	2	no	—
43	C₂²×C₇₃	—	2	1	no	—
44	C₂²×C₇₃	—	4	1	no	—
45	C₂²×C₇₃	#341	1	1	yes	characteristic
46	C₄×C₇₃	—	2	1	no	—
47	C₄×C₇₃	H(3,ℤ/4)	1	1	yes	characteristic
48	C₄×C₇₃	#343	1	1	yes	characteristic
49	Dic₇₃	—	4	2	no	—
50	#7240	—	73	1	no	maximal, Sylow
51	C₂×C₇₃	—	2	1	no	—
52	C₂×C₇₃	—	4	1	no	—
53	C₂×C₇₃	#1290	1	1	yes	characteristic
54	#2166	—	73	2	no	—
55	#2151	—	73	1	no	—
56	C₇₃	#7240	1	1	yes	characteristic, complemented, Sylow
57	C₂×M₃₂	—	73	1	no	—
58	C₂²×C₁₆	#64408	1	1	yes	characteristic
59	#405	—	73	1	no	—
60	#364	—	146	2	no	—
61	C₄⋊C₁₆	—	146	2	no	—
62	#349	—	146	2	no	—
63	#347	—	73	4	no	—
64	C₂×M₁₆	—	73	1	no	—
65	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
66	#117	—	73	1	no	—
67	M₃₂	—	292	1	no	—
68	M₃₂	—	146	4	no	—
69	C₂×C₁₆	—	2	1	no	—
70	C₂×C₁₆	—	4	1	no	—
71	C₂×C₁₆	—	146	1	no	—
72	#97	—	73	2	no	—
73	C₄×C₈	—	73	2	no	—
74	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
75	M₁₆	—	292	1	no	—
76	C₂×C₈	—	2	1	no	—
77	C₂×C₈	—	2	1	no	—
78	C₂×C₈	—	73	2	no	—
79	C₂×C₈	C₄×#64408	1	1	yes	characteristic
80	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
81	C₄⋊C₄	—	146	1	no	—
82	C₂²⋊C₄	—	146	1	no	—
83	C₄²	—	73	2	no	—
84	C₁₆	—	4	1	no	—
85	C₁₆	—	292	4	no	—
86	C₁₆	—	292	1	no	—
87	C₂³	#14556661	1	1	yes	characteristic
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	146	2	no	—
91	C₂×C₄	(C₂×#64408)⋊C₄	1	1	yes	characteristic
92	C₂×C₄	#14556662	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	146	2	no	—
95	C₈	#14556651	1	1	yes	characteristic
96	C₈	#14556652	1	1	yes	characteristic
97	C₂²	—	2	1	no	—
98	C₂²	—	4	1	no	—
99	C₂²	#17989463	1	1	yes	characteristic
100	C₄	—	2	1	no	—
101	C₄	—	292	2	no	—
102	C₄	#17989455	1	1	yes	characteristic
103	C₄	#17989467	1	1	yes	characteristic
104	C₂	—	2	1	no	—
105	C₂	—	4	1	no	—
106	C₂	#22624796	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626305

C₂

yes

maximal

#22611689

C₂

yes

maximal, characteristic

#17989509

—

#17989483

—

#17989480

C₄

yes

—

C₂×#14556531

C₄

yes

characteristic

#17988542

C₂²

yes

characteristic

#17988405

—

C₂²×C₁₆×C₇₃

C₄

yes

characteristic

#14556665

—

C₂×#11924012

D₄

yes

characteristic

#14556550

Q₈

yes

—

—

C₂×C₄

yes

—

C₈×Dic₇₃

D₄

yes

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

—

#11924031

—

#11924012

—

C₂×#10596831

C₂²⋊C₄

yes

—

C₄⋊Dic₇₃

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

C₄⋊C₄

yes

—

#11924003

—

C₂²×C₄×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₄²

yes

characteristic

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₁₆×C₇₃

—

C₂³×C₇₃

#108

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

C₂×Dic₇₃

—

C₈×C₇₃

—

C₈×C₇₃

yes

C₈×C₇₃

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#341

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

#7240

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

—

—

C₇₃

#7240

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂²×C₁₆

yes

—

—

146

—

C₄⋊C₁₆

—

146

—

#349

—

146

—

#347

—

C₂×M₁₆

—

C₂²×C₈

C₂×#64408

yes

characteristic

#117

—

M₃₂

—

292

—

M₃₂

—

146

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

#97

—

C₄×C₈

—

C₂²×C₄

C₄⋊#64408

yes

characteristic

M₁₆

—

292

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

146

—

C₂²⋊C₄

—

146

—

C₄²

—

C₁₆

—

C₁₆

—

292

—

C₁₆

—

292

—

C₂³

#14556661

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

146

—

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

146

—

C₈

yes

C₈

yes

C₂²

—

C₂²

—

C₂²

#17989463

yes

characteristic

100

C₄

—

101

C₄

—

292

—

102

C₄

#17989455

yes

characteristic

103

C₄

#17989467

yes

characteristic

104

C₂

—

105

C₂

—

106

C₂

#22624796

yes

characteristic

	subgroup	G/H
center	C₄	#17989455
commutator	C₂×C₈×C₇₃	C₄²
Frattini	C₂²×C₈	C₂×#64408
Fitting	C₂²×C₁₆×C₇₃	C₄
socle	C₂×C₇₃	#1290
prime radical	C₂³×C₇₃	#108
cosocle	#17988542	C₂²
hypercenter	C₂²×C₁₆	#64408
hypocenter	C₇₃	#7240
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₄

#17989455

commutator

C₂×C₈×C₇₃

C₄²

Frattini

C₂²×C₈

C₂×#64408

Fitting

C₂²×C₁₆×C₇₃

C₄

C₂×C₇₃

C₂³×C₇₃

C₂²

C₂²×C₁₆

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
56. C₇₃	#7240	semidirect

H ≅ G/N

type

56. C₇₃

#7240

semidirect

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

subgroup	index
#22626305	2
#22611689	2
#7240	73

subgroup

index

#22626305

#22611689

#7240

Subgroups — #34000489

Conjugacy classes of subgroups

Complements

Maximal subgroups