Group #34000829 — subgroups

Subgroups — #34000829

Order 18688.

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

normal subgroups only

This group has 124 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	C₂²⋊#14556641	C₂	1	2	yes	maximal
2	#22611690	C₂	1	1	yes	maximal, characteristic
3	C₂×#14556641	—	2	2	no	—
4	C₂×#14556531	C₄	1	1	yes	characteristic
5	C₂²×#11924003	C₂²	1	1	yes	characteristic
6	C₄⋊#11924003	—	2	2	no	—
7	#405×C₇₃	C₄	1	1	yes	characteristic
8	#14556641	—	4	2	no	—
9	#14556531	—	4	1	no	—
10	C₂×#11924003	—	2	1	no	—
11	C₂×#11924003	—	4	1	no	—
12	C₂×#11924003	D₄	1	2	yes	—
13	C₂×M₁₆×C₇₃	C₈	1	1	yes	characteristic
14	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
15	#117×C₇₃	C₈	1	1	yes	characteristic
16	#97×C₇₃	—	2	1	no	—
17	C₄×C₈×C₇₃	—	2	1	no	—
18	#11924003	—	2	2	no	—
19	#11924003	—	4	1	no	—
20	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
21	M₁₆×C₇₃	—	4	1	no	—
22	C₂×C₈×C₇₃	—	2	1	no	—
23	C₂×C₈×C₇₃	—	2	1	no	—
24	C₂×C₈×C₇₃	—	2	1	no	—
25	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₈×C₇₃	M₁₆	1	1	yes	characteristic
27	C₄⋊C₄×C₇₃	C₁₆	1	2	yes	—
28	C₂²⋊C₄×C₇₃	—	2	1	no	—
29	C₄²×C₇₃	—	2	1	no	—
30	C₂³×C₇₃	M₃₂	1	1	yes	characteristic
31	C₂×C₄×C₇₃	—	2	1	no	—
32	C₂×C₄×C₇₃	—	2	1	no	—
33	C₂×C₄×C₇₃	—	2	2	no	—
34	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
35	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
36	C₈×C₇₃	—	2	1	no	—
37	C₈×C₇₃	—	4	1	no	—
38	C₈×C₇₃	#99	1	1	yes	characteristic
39	C₈×C₇₃	#101	1	1	yes	characteristic
40	C₂²×C₇₃	—	2	1	no	—
41	C₂²×C₇₃	—	2	2	no	—
42	C₂²×C₇₃	C₂²⋊C₁₆	1	1	yes	characteristic
43	C₄×C₇₃	—	2	1	no	—
44	C₄×C₇₃	—	4	2	no	—
45	C₄×C₇₃	#324	1	1	yes	characteristic
46	C₄×C₇₃	#349	1	1	yes	characteristic
47	#7363	—	73	1	no	maximal, Sylow
48	C₂×C₇₃	—	4	1	no	—
49	C₂×C₇₃	#1322	1	1	yes	characteristic
50	C₂×C₇₃	#1401	1	2	yes	—
51	#2151	—	73	1	no	—
52	C₂²⋊C₃₂	—	73	2	no	—
53	C₇₃	#7363	1	1	yes	characteristic, complemented, Sylow
54	C₂×M₃₂	—	73	1	no	—
55	C₂²×C₁₆	—	73	1	no	—
56	#405	#64408	1	1	yes	characteristic
57	C₂×C₃₂	—	146	2	no	—
58	C₄⋊C₁₆	—	146	2	no	—
59	C₂×M₁₆	#10596833	1	1	yes	characteristic
60	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
61	#117	#10596833	1	1	yes	characteristic
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	—	2	1	no	—
67	C₄×C₈	—	2	1	no	—
68	C₃₂	—	292	2	no	—
69	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
70	M₁₆	—	4	1	no	—
71	C₂×C₈	—	2	1	no	—
72	C₂×C₈	—	2	1	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	C₂²⋊#64408	1	1	yes	characteristic
75	C₂×C₈	#11924037	1	1	yes	characteristic
76	C₄⋊C₄	#11924031	1	2	yes	—
77	C₂²⋊C₄	—	2	1	no	—
78	C₄²	—	2	1	no	—
79	C₁₆	—	292	1	no	—
80	C₁₆	—	146	2	no	—
81	C₂³	#14556665	1	1	yes	characteristic
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	C₂²⋊#10596833	1	1	yes	characteristic
86	C₂×C₄	C₂×#11924031	1	1	yes	characteristic
87	C₈	—	2	1	no	—
88	C₈	—	4	1	no	—
89	C₈	#14556650	1	1	yes	characteristic
90	C₈	#14556652	1	1	yes	characteristic
91	C₂²	—	2	2	no	—
92	C₂²	—	2	1	no	—
93	C₂²	C₂²⋊#11924031	1	1	yes	characteristic
94	C₄	—	2	1	no	—
95	C₄	—	4	2	no	—
96	C₄	#17989432	1	1	yes	characteristic
97	C₄	#17989483	1	1	yes	characteristic
98	C₂	—	4	1	no	—
99	C₂	#22624867	1	1	yes	characteristic
100	C₂	#22625048	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²⋊#14556641

C₂

yes

maximal

#22611690

C₂

yes

maximal, characteristic

C₂×#14556641

—

C₂×#14556531

C₄

yes

characteristic

C₂²×#11924003

C₂²

yes

characteristic

C₄⋊#11924003

—

#405×C₇₃

C₄

yes

—

—

—

—

D₄

yes

—

C₂×M₁₆×C₇₃

C₈

yes

characteristic

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

#117×C₇₃

C₈

yes

characteristic

#97×C₇₃

—

C₄×C₈×C₇₃

—

#11924003

—

#11924003

—

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₇₃

M₁₆

yes

characteristic

C₄⋊C₄×C₇₃

C₁₆

yes

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

M₃₂

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₈

yes

characteristic

C₂×C₄×C₇₃

C₂×C₁₆

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

yes

C₈×C₇₃

yes

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂²⋊C₁₆

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

#1322

yes

characteristic

C₂×C₇₃

#1401

yes

—

#2151

—

C₂²⋊C₃₂

—

C₇₃

#7363

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂²×C₁₆

—

#405

#64408

yes

characteristic

C₂×C₃₂

—

146

—

C₄⋊C₁₆

—

146

—

C₂×M₁₆

yes

C₂²×C₈

yes

yes

M₃₂

—

292

—

C₂×C₁₆

—

292

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

#97

—

C₄×C₈

—

C₃₂

—

292

—

C₂²×C₄

C₂×#10596833

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

yes

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

292

—

C₁₆

—

146

—

C₂³

#14556665

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

C₈

yes

C₈

yes

C₂²

—

C₂²

—

C₂²

C₂²⋊#11924031

yes

characteristic

C₄

—

C₄

—

C₄

yes

C₄

yes

C₂

—

C₂

#22624867

yes

characteristic

100

C₂

#22625048

yes

—

	subgroup	G/H
center	C₂×C₈	C₂²⋊#64408
commutator	C₂×C₄×C₇₃	C₂×C₁₆
Frattini	C₂²×C₈	C₂×#64408
Fitting	#405×C₇₃	C₄
socle	C₂²×C₇₃	C₂²⋊C₁₆
prime radical	C₂³×C₇₃	M₃₂
cosocle	C₂²×#11924003	C₂²
hypercenter	#405	#64408
hypocenter	C₇₃	#7363
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #405

subgroup

G/H

center

C₂×C₈

C₂²⋊#64408

commutator

C₂×C₄×C₇₃

C₂×C₁₆

Frattini

C₂²×C₈

C₂×#64408

Fitting

#405×C₇₃

C₄

socle

C₂²×C₇₃

C₂²⋊C₁₆

prime radical

C₂³×C₇₃

M₃₂

C₂²

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #405

N	H ≅ G/N	type
53. C₇₃	#7363	semidirect

H ≅ G/N

type

53. C₇₃

#7363

semidirect

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

subgroup	index
C₂²⋊#14556641	2
#22611690	2
#7363	73

subgroup

index

C₂²⋊#14556641

#22611690

#7363

Subgroups — #34000829

Conjugacy classes of subgroups

Complements

Maximal subgroups