Group #34033062 — subgroups

Subgroups — #34033062

Order 18688.

120 conjugacy classes of subgroups; only the 37 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 37 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	#22627025	C₂	1	2	yes	maximal
2	#22626268	C₂	1	1	yes	maximal, characteristic
3	#17989695	—	2	2	no	—
4	C₂×#14556665	C₂²	1	1	yes	characteristic
5	C₂²×#11924031	C₄	1	1	yes	characteristic
6	#17989508	—	2	2	no	—
7	#17988545	C₄	1	1	yes	characteristic
8	#14556681	—	4	2	no	—
9	#14556665	—	4	1	no	—
10	C₂×#11924031	—	2	1	no	—
11	C₂×#11924031	—	4	1	no	—
12	C₂×#11924031	D₄	1	2	yes	—
13	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
14	#14556550	C₈	1	1	yes	characteristic
15	#14556501	—	2	1	no	—
16	C₈×Dic₇₃	—	2	1	no	—
17	C₂×M₁₆×C₇₃	C₈	1	1	yes	characteristic
18	#11924031	—	2	2	no	—
19	#11924031	—	4	1	no	—
20	C₂×#10596831	—	2	1	no	—
21	C₂×#10596831	—	2	1	no	—
22	C₂×#10596831	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×#10596831	M₁₆	1	1	yes	characteristic
24	C₄⋊Dic₇₃	—	2	1	no	—
25	C₂²⋊Dic₇₃	C₁₆	1	2	yes	—
26	C₄×Dic₇₃	—	2	1	no	—
27	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
28	M₁₆×C₇₃	—	4	1	no	—
29	C₂×C₈×C₇₃	—	2	1	no	—
30	C₂³×C₇₃	C₂×C₁₆	1	1	yes	characteristic
31	C₂×C₄×C₇₃	—	2	1	no	—
32	C₂×C₄×C₇₃	—	2	1	no	—
33	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
34	C₂×C₄×C₇₃	M₃₂	1	1	yes	characteristic
35	C₂×Dic₇₃	—	2	2	no	—
36	C₈×C₇₃	—	4	1	no	—
37	#10596831	—	2	1	no	—
38	#10596831	#99	1	1	yes	characteristic
39	#10596831	#101	1	1	yes	characteristic
40	C₂²×C₇₃	—	2	1	no	—
41	C₂²×C₇₃	—	4	1	no	—
42	C₂²×C₇₃	C₂²⋊C₁₆	1	1	yes	characteristic
43	C₄×C₇₃	—	2	1	no	—
44	C₄×C₇₃	#324	1	1	yes	characteristic
45	C₄×C₇₃	#349	1	1	yes	characteristic
46	Dic₇₃	—	4	2	no	—
47	#7365	—	73	1	no	maximal, Sylow
48	C₂×C₇₃	—	2	1	no	—
49	C₂×C₇₃	—	4	1	no	—
50	C₂×C₇₃	#1322	1	1	yes	characteristic
51	#2151	—	73	1	no	—
52	#1400	—	73	2	no	—
53	C₇₃	#7365	1	1	yes	characteristic, complemented, Sylow
54	C₂×M₃₂	—	73	1	no	—
55	C₂²×C₁₆	—	73	1	no	—
56	#405	—	73	1	no	—
57	M₆₄	—	146	2	no	—
58	C₄⋊C₁₆	—	146	2	no	—
59	C₂×M₁₆	#10596842	1	1	yes	characteristic
60	C₂²×C₈	—	73	1	no	—
61	#117	—	73	1	no	—
62	M₃₂	—	292	1	no	—
63	C₂×C₁₆	—	292	1	no	—
64	C₂×C₁₆	—	73	2	no	—
65	C₂×C₁₆	—	146	1	no	—
66	#97	—	146	1	no	—
67	C₄×C₈	—	146	1	no	—
68	C₃₂	—	292	2	no	—
69	C₂²×C₄	C₂×#10596842	1	1	yes	characteristic
70	M₁₆	—	4	1	no	—
71	C₂×C₈	—	2	1	no	—
72	C₂×C₈	—	73	1	no	—
73	C₂×C₈	—	73	1	no	—
74	C₂×C₈	—	146	1	no	—
75	C₂×C₈	—	146	1	no	—
76	C₄⋊C₄	—	146	1	no	—
77	C₂²⋊C₄	—	73	2	no	—
78	C₄²	—	146	1	no	—
79	C₁₆	—	292	1	no	—
80	C₁₆	—	146	2	no	—
81	C₂³	#14556686	1	1	yes	characteristic
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	146	2	no	—
85	C₂×C₄	C₂²⋊#10596842	1	1	yes	characteristic
86	C₂×C₄	#14556688	1	1	yes	characteristic
87	C₈	—	4	1	no	—
88	C₈	—	73	1	no	—
89	C₈	—	73	1	no	—
90	C₈	—	146	1	no	—
91	C₂²	—	2	1	no	—
92	C₂²	—	4	1	no	—
93	C₂²	#17989684	1	1	yes	characteristic
94	C₄	—	2	1	no	—
95	C₄	—	292	2	no	—
96	C₄	#17989673	1	1	yes	characteristic
97	C₄	#17989685	1	1	yes	characteristic
98	C₂	—	2	1	no	—
99	C₂	—	4	1	no	—
100	C₂	#22626965	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22627025

C₂

yes

maximal

#22626268

C₂

yes

maximal, characteristic

#17989695

—

C₂×#14556665

C₂²

yes

characteristic

C₂²×#11924031

C₄

yes

characteristic

#17989508

—

#17988545

C₄

yes

—

—

—

—

D₄

yes

—

C₂²×#10596831

C₂×C₄

yes

characteristic

#14556550

C₈

yes

characteristic

#14556501

—

C₈×Dic₇₃

—

C₂×M₁₆×C₇₃

C₈

yes

—

—

—

—

C₂²⋊C₄

yes

characteristic

C₂×#10596831

M₁₆

yes

characteristic

C₄⋊Dic₇₃

—

C₂²⋊Dic₇₃

C₁₆

yes

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂³×C₇₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₃

M₃₂

yes

characteristic

C₂×Dic₇₃

—

C₈×C₇₃

—

—

yes

yes

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂²⋊C₁₆

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

#7365

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

—

—

C₇₃

#7365

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

M₆₄

—

146

—

C₄⋊C₁₆

—

146

—

C₂×M₁₆

#10596842

yes

characteristic

C₂²×C₈

—

#117

—

M₃₂

—

292

—

C₂×C₁₆

—

292

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

#97

—

146

—

C₄×C₈

—

146

—

C₃₂

—

292

—

C₂²×C₄

C₂×#10596842

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

146

—

C₂×C₈

—

146

—

C₄⋊C₄

—

146

—

C₂²⋊C₄

—

C₄²

—

146

—

C₁₆

—

292

—

C₁₆

—

146

—

C₂³

#14556686

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

146

—

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

—

C₈

—

146

—

C₂²

—

C₂²

—

C₂²

#17989684

yes

characteristic

C₄

—

C₄

—

292

—

C₄

yes

C₄

yes

C₂

—

C₂

—

100

C₂

#22626965

yes

characteristic

	subgroup	G/H
center	C₄	#17989673
commutator, prime radical	C₂³×C₇₃	C₂×C₁₆
Frattini	C₂²×C₄	C₂×#10596842
Fitting	C₂×M₁₆×C₇₃	C₈
socle	C₂×C₇₃	#1322
cosocle	C₂×#14556665	C₂²
hypercenter	C₂×M₁₆	#10596842
hypocenter	C₇₃	#7365
LCS	G ▷ C₂³×C₇₃ ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

subgroup

G/H

center

C₄

#17989673

commutator, prime radical

C₂³×C₇₃

C₂×C₁₆

Frattini

C₂²×C₄

C₂×#10596842

Fitting

C₂×M₁₆×C₇₃

C₈

C₂×C₇₃

C₂²

C₂×M₁₆

C₇₃

G ▷ C₂³×C₇₃ ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
53. C₇₃	#7365	semidirect

H ≅ G/N

type

53. C₇₃

#7365

semidirect

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

subgroup	index
#22627025	2
#22626268	2
#7365	73

subgroup

index

#22627025

#22626268

#7365

Subgroups — #34033062

Conjugacy classes of subgroups

Complements

Maximal subgroups