Group #34005498 — subgroups

Subgroups — #34005498

Order 18688.

354 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 136 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	#22625870	C₂	1	2	yes	maximal
2	#22625487	C₂	1	4	yes	maximal
3	C₈²×D₇₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556661	—	2	4	no	—
5	#17989591	C₂²	1	4	yes	—
6	#17989449	—	2	8	no	—
7	C₄×C₈×D₇₃	C₂²	1	2	yes	—
8	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
9	C₈×#10596831	C₄	1	2	yes	—
10	C₈×#10596831	C₄	1	1	yes	characteristic
11	C₈²×C₇₃	C₄	1	1	yes	characteristic
12	C₂×#11924038	—	2	4	no	—
13	#14556661	—	4	8	no	—
14	C₄⋊#10596833	—	2	8	no	—
15	C₂×C₈×D₇₃	D₄	1	2	yes	—
16	C₂×C₈×D₇₃	Q₈	1	4	yes	—
17	C₄²×D₇₃	C₂³	1	1	yes	characteristic
18	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
19	C₄×#10596831	C₂×C₄	1	2	yes	—
20	C₈×Dic₇₃	D₄	1	4	yes	—
21	C₈×Dic₇₃	Q₈	1	2	yes	—
22	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
24	#11924038	—	4	8	no	—
25	C₂×#10596833	—	4	4	no	—
26	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
27	C₂×C₄×D₇₃	C₂×Q₈	1	2	yes	—
28	C₂×#10596831	C₄⋊C₄	1	2	yes	—
29	C₂×#10596831	C₄⋊C₄	1	4	yes	—
30	C₈×D₇₃	—	2	8	no	—
31	C₈×D₇₃	SD₁₆	1	8	yes	—
32	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
33	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₈×C₇₃	C₄⋊C₄	1	4	yes	—
35	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
36	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²×D₇₃	C₄⋊Q₈	1	1	yes	characteristic
38	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
40	C₂×Dic₇₃	#127	1	1	yes	characteristic
41	C₂×Dic₇₃	C₄⋊Q₈	1	2	yes	—
42	C₄×D₇₃	—	2	4	no	—
43	C₄×D₇₃	C₂×SD₁₆	1	4	yes	—
44	#10596833	—	8	4	no	—
45	C₈×C₇₃	—	2	4	no	—
46	C₈×C₇₃	#106	1	4	yes	—
47	#10596831	#106	1	4	yes	—
48	#10596831	#108	1	8	yes	complemented
49	C₂²×C₇₃	#384	1	1	yes	characteristic
50	C₂×D₇₃	—	2	2	no	—
51	C₂×D₇₃	#498	1	2	yes	—
52	C₄×C₇₃	C₂×#106	1	2	yes	—
53	C₄×C₇₃	C₂×#108	1	4	yes	—
54	Dic₇₃	—	2	2	no	—
55	Dic₇₃	#492	1	2	yes	—
56	#8822	—	73	1	no	maximal, Sylow
57	C₂×C₇₃	#1844	1	1	yes	characteristic
58	C₂×C₇₃	#1848	1	2	yes	—
59	D₇₃	—	2	2	no	—
60	#1848	—	73	2	no	—
61	#1566	—	73	4	no	—
62	C₂×C₈²	—	73	1	no	—
63	C₇₃	#8822	1	1	yes	characteristic, complemented, Sylow
64	C₂×#108	—	146	4	no	—
65	#423	—	73	4	no	—
66	C₂×C₄×C₈	—	73	2	no	—
67	C₂×C₄×C₈	—	73	1	no	—
68	#334	—	146	8	no	—
69	C₈²	—	73	1	no	—
70	C₈²	—	73	2	no	—
71	C₈²	#64408	1	1	yes	characteristic
72	C₂×M₁₆	—	146	4	no	—
73	C₂²×C₈	—	73	2	no	—
74	C₂²×C₈	—	73	4	no	—
75	C₂×C₄²	—	73	1	no	—
76	#108	—	292	8	no	—
77	C₄⋊C₈	—	146	8	no	—
78	C₄×C₈	—	73	4	no	—
79	C₄×C₈	—	73	2	no	—
80	C₄×C₈	—	73	2	no	—
81	C₄×C₈	—	73	1	no	—
82	C₄×C₈	C₂×#64408	1	2	yes	—
83	C₄×C₈	C₂×#64408	1	1	yes	characteristic
84	C₂²×C₄	—	73	1	no	—
85	C₂²×C₄	—	73	2	no	—
86	M₁₆	—	292	8	no	—
87	C₂×C₈	—	292	4	no	—
88	C₂×C₈	—	73	2	no	—
89	C₂×C₈	—	73	4	no	—
90	C₂×C₈	—	73	8	no	—
91	C₂×C₈	—	146	8	no	—
92	C₂×C₈	C₄⋊#64408	1	4	yes	—
93	C₂×C₈	C₄⋊#64408	1	2	yes	—
94	C₄²	—	73	2	no	—
95	C₄²	—	73	1	no	—
96	C₄²	C₂²×#64408	1	1	yes	characteristic
97	C₂³	—	73	1	no	—
98	C₂×C₄	—	73	1	no	—
99	C₂×C₄	—	73	4	no	—
100	C₂×C₄	—	73	2	no	—
101	C₂×C₄	—	146	4	no	—
102	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
103	C₂×C₄	C₂×C₄⋊#64408	1	2	yes	—
104	C₈	—	2	4	no	—
105	C₈	—	584	4	no	—
106	C₈	—	73	4	no	—
107	C₈	—	73	8	no	—
108	C₈	#14556659	1	4	yes	—
109	C₂²	—	73	2	no	—
110	C₂²	—	146	2	no	—
111	C₂²	#17989533	1	1	yes	characteristic
112	C₄	—	73	2	no	—
113	C₄	—	146	2	no	—
114	C₄	C₂×#14556659	1	2	yes	—
115	C₄	D₇₃⋊#108	1	4	yes	—
116	C₂	—	146	2	no	—
117	C₂	#22625866	1	1	yes	characteristic
118	C₂	#22625871	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22625870

C₂

yes

maximal

#22625487

C₂

yes

maximal

C₈²×D₇₃

C₂

yes

maximal, characteristic

C₂×#14556661

—

#17989591

C₂²

yes

—

#17989449

—

C₄×C₈×D₇₃

C₂²

yes

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

C₈×#10596831

C₄

yes

—

C₈×#10596831

C₄

yes

characteristic

C₈²×C₇₃

C₄

yes

—

—

—

C₂×C₈×D₇₃

D₄

yes

—

C₂×C₈×D₇₃

Q₈

yes

—

C₄²×D₇₃

C₂³

yes

characteristic

C₄×#10596831

C₂×C₄

yes

characteristic

C₄×#10596831

C₂×C₄

yes

—

C₈×Dic₇₃

D₄

yes

—

C₈×Dic₇₃

Q₈

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

#11924038

—

C₂×#10596833

—

C₂×C₄×D₇₃

C₂×D₄

yes

characteristic

C₂×C₄×D₇₃

C₂×Q₈

yes

—

C₂×#10596831

C₄⋊C₄

yes

—

C₂×#10596831

C₄⋊C₄

yes

—

C₈×D₇₃

—

C₈×D₇₃

SD₁₆

yes

—

C₄×Dic₇₃

C₂×D₄

yes

—

C₄×Dic₇₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₄²×C₇₃

C₂²×C₄

yes

characteristic

C₂²×D₇₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

—

C₂×Dic₇₃

#127

yes

characteristic

C₂×Dic₇₃

C₄⋊Q₈

yes

—

C₄×D₇₃

—

C₄×D₇₃

C₂×SD₁₆

yes

—

#10596833

—

C₈×C₇₃

—

C₈×C₇₃

#106

yes

—

#10596831

#106

yes

—

yes

C₂²×C₇₃

yes

C₂×D₇₃

—

C₂×D₇₃

#498

yes

—

C₄×C₇₃

C₂×#106

yes

—

C₄×C₇₃

C₂×#108

yes

—

Dic₇₃

—

Dic₇₃

#492

yes

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

—

#1848

—

#1566

—

C₂×C₈²

—

C₇₃

#8822

yes

characteristic, complemented, Sylow

C₂×#108

—

146

—

#423

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#334

—

146

—

C₈²

—

C₈²

—

C₈²

#64408

yes

characteristic

C₂×M₁₆

—

146

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

#108

—

292

—

C₄⋊C₈

—

146

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#64408

yes

—

C₄×C₈

C₂×#64408

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

292

—

C₂×C₈

—

292

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

146

—

C₂×C₈

C₄⋊#64408

yes

—

C₂×C₈

C₄⋊#64408

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×#64408

yes

characteristic

C₂³

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

146

—

102

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

103

C₂×C₄

C₂×C₄⋊#64408

yes

—

104

C₈

—

105

C₈

—

584

—

106

C₈

—

107

C₈

—

108

C₈

#14556659

yes

—

109

C₂²

—

110

C₂²

—

146

—

111

C₂²

#17989533

yes

characteristic

112

C₄

—

113

C₄

—

146

—

114

C₄

C₂×#14556659

yes

—

115

C₄

D₇₃⋊#108

yes

—

116

C₂

—

146

—

117

C₂

#22625866

yes

characteristic

118

C₂

#22625871

yes

—

	subgroup	G/H
center	C₂²	#17989533
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₄²	C₂²×#64408
Fitting	C₈²×C₇₃	C₄
socle	C₂²×C₇₃	#384
prime radical	C₂²×D₇₃	C₄⋊Q₈
cosocle	C₄²×D₇₃	C₂³
hypercenter	C₈²	#64408
hypocenter	C₇₃	#8822
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

subgroup

G/H

center

C₂²

#17989533

commutator

C₄²×C₇₃

C₂²×C₄

Frattini

C₄²

C₂²×#64408

Fitting

C₈²×C₇₃

C₄

socle

C₂²×C₇₃

#384

prime radical

C₂²×D₇₃

C₄⋊Q₈

cosocle

C₄²×D₇₃

C₂³

C₈²

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈²

N	H ≅ G/N	type
48. #10596831	#108	semidirect
63. C₇₃	#8822	semidirect

H ≅ G/N

type

48. #10596831

#108

semidirect

63. C₇₃

#8822

semidirect

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

subgroup	index
#22625870	2
#22625487	2
C₈²×D₇₃	2
#8822	73

subgroup

index

#22625870

#22625487

C₈²×D₇₃

#8822

Subgroups — #34005498

Conjugacy classes of subgroups

Complements

Maximal subgroups