Group #34034219 — subgroups

Subgroups — #34034219

Order 18688.

370 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 141 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	C₂×#17989690	C₂	1	3	yes	maximal
2	C₂×C₄⋊#11924040	C₂	1	3	yes	maximal
3	#22625588	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11924040	C₂²	1	3	yes	—
5	C₂²×#11924040	C₂²	1	1	yes	characteristic
6	#17989690	—	2	6	no	—
7	C₄⋊#11924040	—	2	6	no	—
8	C₂×#14556646	C₄	1	3	yes	—
9	C₂×C₄×#10596833	C₂²	1	3	yes	—
10	#17988432	C₄	1	1	yes	characteristic
11	C₂×#11924040	—	2	6	no	—
12	C₂×#11924040	—	2	3	no	—
13	C₂×#11924040	—	2	3	no	—
14	C₂²×#10596833	C₂×C₄	1	3	yes	—
15	C₂²×#10596833	C₂³	1	1	yes	characteristic
16	#14556646	—	2	6	no	—
17	C₄×#10596833	D₄	1	6	yes	—
18	C₄×#10596833	Q₈	1	6	yes	—
19	C₂×C₄×Dic₇₃	C₂×C₄	1	3	yes	—
20	#14556498	C₈	1	3	yes	—
21	#95×C₇₃	C₈	1	1	yes	characteristic
22	#11924040	—	4	3	no	—
23	#11924040	—	4	1	no	—
24	C₂×#10596833	—	2	6	no	—
25	C₂×#10596833	C₄⋊C₄	1	6	yes	—
26	C₂×#10596833	C₂×D₄	1	3	yes	—
27	C₂×#10596833	C₂×Q₈	1	3	yes	—
28	C₂²×Dic₇₃	C₂×C₈	1	3	yes	—
29	C₂²×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
30	C₄×Dic₇₃	C₄⋊C₄	1	6	yes	—
31	C₄×Dic₇₃	M₁₆	1	6	yes	—
32	C₂²×C₄×C₇₃	C₂×C₈	1	3	yes	—
33	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
34	C₂×C₄×C₇₃	—	2	6	no	—
35	C₂×C₄×C₇₃	C₄⋊C₈	1	6	yes	—
36	C₂×Dic₇₃	—	2	3	no	—
37	C₂×Dic₇₃	—	2	3	no	—
38	C₂×Dic₇₃	C₄⋊C₈	1	6	yes	—
39	C₂×Dic₇₃	C₂×C₄⋊C₄	1	3	yes	—
40	C₂×Dic₇₃	C₂×M₁₆	1	3	yes	—
41	#10596833	—	2	6	no	—
42	#10596833	#127	1	1	yes	characteristic
43	#10596833	C₄⋊Q₈	1	3	yes	—
44	C₂²×C₇₃	C₂×C₄⋊C₈	1	3	yes	—
45	C₂²×C₇₃	#504	1	3	yes	—
46	C₂²×C₇₃	#504	1	1	yes	characteristic
47	C₄×C₇₃	—	2	6	no	—
48	Dic₇₃	—	2	6	no	—
49	Dic₇₃	#384	1	1	yes	characteristic
50	Dic₇₃	#423	1	3	yes	—
51	#12899	—	73	1	no	maximal, Sylow
52	C₂×C₇₃	#1842	1	1	yes	characteristic
53	C₂×C₇₃	#2151	1	3	yes	—
54	C₂×C₇₃	#2151	1	3	yes	—
55	C₂×C₄⋊C₁₆	—	73	3	no	—
56	C₂×C₄⋊C₁₆	—	73	3	no	—
57	#1726	—	73	1	no	—
58	C₇₃	#12899	1	1	yes	characteristic, complemented, Sylow
59	C₂²×C₁₆	—	73	3	no	—
60	C₂²×C₁₆	—	73	1	no	—
61	C₂×#97	—	73	3	no	—
62	C₂×C₄×C₈	—	73	3	no	—
63	#376	—	73	1	no	—
64	C₄⋊C₁₆	—	146	6	no	—
65	C₄⋊C₁₆	—	146	6	no	—
66	C₂²×C₈	—	73	1	no	—
67	C₂²×C₈	—	73	3	no	—
68	C₂×C₄²	—	73	3	no	—
69	C₂×C₁₆	—	146	6	no	—
70	C₂×C₁₆	—	146	3	no	—
71	C₂×C₁₆	—	146	3	no	—
72	#97	—	146	6	no	—
73	C₄×C₈	—	73	6	no	—
74	C₄×C₈	—	73	6	no	—
75	#95	—	73	3	no	—
76	#95	#10596842	1	1	yes	characteristic
77	C₂²×C₄	—	73	1	no	—
78	C₂²×C₄	—	73	3	no	—
79	C₂²×C₄	C₂×#10596842	1	3	yes	—
80	C₂×C₈	—	73	3	no	—
81	C₂×C₈	—	73	3	no	—
82	C₂×C₈	—	73	6	no	—
83	C₂×C₈	—	146	6	no	—
84	C₄²	—	73	6	no	—
85	C₄²	—	73	6	no	—
86	C₁₆	—	292	3	no	—
87	C₁₆	—	292	1	no	—
88	C₂³	C₂²×#10596842	1	1	yes	characteristic
89	C₂×C₄	—	2	6	no	—
90	C₂×C₄	—	73	3	no	—
91	C₂×C₄	—	73	3	no	—
92	C₂×C₄	—	73	6	no	—
93	C₂×C₄	—	146	3	no	—
94	C₂×C₄	—	146	3	no	—
95	C₂×C₄	C₄⋊#10596842	1	6	yes	—
96	C₈	—	73	1	no	—
97	C₈	—	73	3	no	—
98	C₈	—	146	6	no	—
99	C₂²	C₂×C₄⋊#10596842	1	3	yes	—
100	C₂²	#17989708	1	3	yes	—
101	C₂²	#17989709	1	1	yes	characteristic
102	C₄	—	2	6	no	—
103	C₄	—	73	1	no	—
104	C₄	—	73	3	no	—
105	C₄	—	146	6	no	—
106	C₂	#22627075	1	1	yes	characteristic
107	C₂	#22627122	1	3	yes	—
108	C₂	#22627123	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17989690

C₂

yes

maximal

C₂×C₄⋊#11924040

C₂

yes

maximal

#22625588

C₂

yes

maximal, characteristic

C₂²×#11924040

C₂²

yes

—

C₂²×#11924040

C₂²

yes

—

—

C₄

yes

—

C₂×C₄×#10596833

C₂²

yes

—

#17988432

C₄

yes

—

—

—

C₂×C₄

yes

—

C₂²×#10596833

C₂³

yes

characteristic

#14556646

—

C₄×#10596833

D₄

yes

—

C₄×#10596833

Q₈

yes

—

C₂×C₄×Dic₇₃

C₂×C₄

yes

—

#14556498

C₈

yes

—

#95×C₇₃

C₈

yes

—

—

—

C₄⋊C₄

yes

—

C₂×#10596833

C₂×D₄

yes

—

C₂×#10596833

C₂×Q₈

yes

—

C₂²×Dic₇₃

C₂×C₈

yes

—

C₂²×Dic₇₃

C₂²×C₄

yes

characteristic

C₄×Dic₇₃

C₄⋊C₄

yes

—

C₄×Dic₇₃

M₁₆

yes

—

C₂²×C₄×C₇₃

C₂×C₈

yes

—

C₂³×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄⋊C₈

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₄⋊C₈

yes

—

C₂×Dic₇₃

C₂×C₄⋊C₄

yes

—

C₂×Dic₇₃

C₂×M₁₆

yes

—

—

yes

C₄⋊Q₈

yes

—

C₂²×C₇₃

C₂×C₄⋊C₈

yes

—

C₂²×C₇₃

#504

yes

—

C₂²×C₇₃

#504

yes

characteristic

C₄×C₇₃

—

Dic₇₃

—

Dic₇₃

#384

yes

characteristic

Dic₇₃

#423

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₂×C₇₃

#2151

yes

—

C₂×C₄⋊C₁₆

—

C₂×C₄⋊C₁₆

—

#1726

—

C₇₃

#12899

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂×#97

—

C₂×C₄×C₈

—

#376

—

C₄⋊C₁₆

—

146

—

C₄⋊C₁₆

—

146

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₁₆

—

146

—

C₂×C₁₆

—

146

—

C₂×C₁₆

—

146

—

#97

—

146

—

C₄×C₈

—

C₄×C₈

—

—

yes

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#10596842

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

146

—

C₄²

—

C₄²

—

C₁₆

—

292

—

C₁₆

—

292

—

C₂³

C₂²×#10596842

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

146

—

C₂×C₄

—

146

—

C₂×C₄

C₄⋊#10596842

yes

—

C₈

—

C₈

—

C₈

—

146

—

C₂²

C₂×C₄⋊#10596842

yes

—

100

C₂²

#17989708

yes

—

101

C₂²

#17989709

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

146

—

106

C₂

#22627075

yes

characteristic

107

C₂

#22627122

yes

—

108

C₂

#22627123

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂²×#10596842
commutator, socle, prime radical	C₂³×C₇₃	C₂²×C₈
Fitting	#95×C₇₃	C₈
cosocle	C₂²×#10596833	C₂³
hypercenter	#95	#10596842
hypocenter	C₇₃	#12899
LCS	G ▷ C₂³×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ #95

subgroup

G/H

center, Frattini

C₂³

C₂²×#10596842

commutator, socle, prime radical

C₂³×C₇₃

C₂²×C₈

Fitting

#95×C₇₃

C₈

C₂³

C₇₃

G ▷ C₂³×C₇₃ ▷ C₇₃

C₁ ◁ C₂³ ◁ #95

N	H ≅ G/N	type
58. C₇₃	#12899	semidirect

H ≅ G/N

type

58. C₇₃

#12899

semidirect

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

subgroup	index
C₂×#17989690	2
C₂×C₄⋊#11924040	2
#22625588	2
#12899	73

subgroup

index

Subgroups — #34034219

Conjugacy classes of subgroups

Complements

Maximal subgroups