Group #34011800 — subgroups

Subgroups — #34011800

Order 18688.

216 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 84 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	#22625492	C₂	1	2	yes	maximal
2	#22624957	C₂	1	4	yes	maximal, complemented
3	C₄×#14556530	C₂	1	1	yes	maximal, characteristic
4	#17989609	—	2	2	no	—
5	#17989450	—	2	2	no	—
6	#17989448	C₂²	1	4	yes	—
7	C₂×#14556530	C₄	1	2	yes	—
8	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
9	#17988362	C₂²	1	2	yes	—
10	C₄×#11924003	C₄	1	1	yes	characteristic
11	C₄×C₁₆×C₇₃	C₄	1	1	yes	characteristic
12	C₄⋊#10596833	—	2	4	no	—
13	#14556648	—	4	2	no	—
14	#14556646	—	4	2	no	—
15	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×D₇₃	Q₈	1	1	yes	characteristic
17	C₄²×D₇₃	D₄	1	1	yes	characteristic
18	#14556530	—	2	4	no	—
19	C₂×#11924003	C₂×C₄	1	2	yes	—
20	C₄×#10596831	Q₈	1	1	yes	characteristic
21	C₈×Dic₇₃	D₄	1	1	yes	characteristic
22	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
23	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
25	C₂×#10596833	—	4	4	no	—
26	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
27	C₂×C₄×D₇₃	SD₁₆	1	2	yes	—
28	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
29	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
30	C₈×D₇₃	—	2	2	no	—
31	C₈×D₇₃	—	2	2	no	—
32	C₄×Dic₇₃	SD₁₆	1	2	yes	—
33	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
34	#11924003	M₁₆	1	4	yes	—
35	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
37	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
38	C₁₆×C₇₃	—	2	2	no	—
39	C₂²×D₇₃	#106	1	1	yes	characteristic
40	C₂×C₄×C₇₃	#106	1	2	yes	—
41	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂×Dic₇₃	#106	1	1	yes	characteristic
43	C₂×Dic₇₃	C₂×SD₁₆	1	2	yes	—
44	C₄×D₇₃	—	2	1	no	—
45	C₄×D₇₃	—	2	1	no	—
46	C₄×D₇₃	—	4	2	no	—
47	#10596833	—	8	4	no	—
48	C₈×C₇₃	#108	1	2	yes	—
49	C₈×C₇₃	C₂×M₁₆	1	2	yes	—
50	#10596831	—	2	1	no	—
51	#10596831	—	2	1	no	—
52	C₂²×C₇₃	C₂×#106	1	1	yes	characteristic
53	C₂×D₇₃	—	2	1	no	—
54	C₂×D₇₃	—	2	2	no	—
55	C₄×C₇₃	—	2	2	no	—
56	C₄×C₇₃	#423	1	1	yes	characteristic
57	C₄×C₇₃	C₂×#108	1	1	yes	characteristic
58	Dic₇₃	—	2	1	no	—
59	Dic₇₃	—	2	2	no	—
60	#12021	—	73	1	no	maximal, Sylow
61	C₂×C₇₃	#1566	1	1	yes	characteristic
62	C₂×C₇₃	#2155	1	2	yes	—
63	D₇₃	—	4	1	no	—
64	C₄×M₃₂	—	73	1	no	—
65	#1568	—	73	2	no	—
66	#1368	—	73	4	no	—
67	C₇₃	#12021	1	1	yes	characteristic, complemented, Sylow
68	C₂×M₃₂	—	73	2	no	—
69	#432	—	146	2	no	—
70	C₂×C₄×C₈	—	73	1	no	—
71	#346	—	73	2	no	—
72	C₄×C₁₆	—	73	1	no	—
73	C₄×C₁₆	#64408	1	1	yes	characteristic
74	#335	—	146	2	no	—
75	#334	—	73	4	no	—
76	C₂²×C₈	—	73	1	no	—
77	C₂²×C₈	—	73	1	no	—
78	C₂×C₄²	—	73	1	no	—
79	M₃₂	—	146	4	no	—
80	C₂×C₁₆	—	73	2	no	—
81	C₂×C₁₆	C₂×#64408	1	2	yes	—
82	C₄⋊C₈	—	146	4	no	—
83	C₂²⋊C₈	—	292	2	no	—
84	#97	—	292	2	no	—
85	C₄×C₈	—	73	1	no	—
86	C₄×C₈	—	73	1	no	—
87	C₄×C₈	—	73	1	no	—
88	C₄×C₈	C₂×#64408	1	1	yes	characteristic
89	C₂²×C₄	—	73	2	no	—
90	C₂²×C₄	—	73	1	no	—
91	C₂×C₈	—	292	4	no	—
92	C₂×C₈	—	73	1	no	—
93	C₂×C₈	—	73	1	no	—
94	C₂×C₈	—	146	2	no	—
95	C₂×C₈	—	146	2	no	—
96	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
97	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
98	C₄²	—	73	2	no	—
99	C₄²	—	73	1	no	—
100	C₄²	C₄⋊#64408	1	1	yes	characteristic
101	C₁₆	—	2	2	no	—
102	C₁₆	—	73	4	no	—
103	C₂³	—	73	1	no	—
104	C₂×C₄	—	292	2	no	—
105	C₂×C₄	—	73	2	no	—
106	C₂×C₄	—	73	1	no	—
107	C₂×C₄	—	146	1	no	—
108	C₂×C₄	—	146	1	no	—
109	C₂×C₄	#14556659	1	2	yes	—
110	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
111	C₈	—	584	4	no	—
112	C₈	—	146	1	no	—
113	C₈	—	146	1	no	—
114	C₈	#14556662	1	2	yes	—
115	C₈	D₇₃⋊M₁₆	1	2	yes	—
116	C₂²	—	146	1	no	—
117	C₂²	—	146	2	no	—
118	C₂²	C₂×#14556659	1	1	yes	characteristic
119	C₄	—	2	2	no	—
120	C₄	—	146	1	no	—
121	C₄	—	146	2	no	—
122	C₄	#17989592	1	1	yes	characteristic
123	C₄	C₂×#14556662	1	1	yes	characteristic
124	C₂	—	292	1	no	—
125	C₂	#22625487	1	1	yes	characteristic
126	C₂	#22626278	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22625492

C₂

yes

maximal

#22624957

C₂

yes

maximal, complemented

C₄×#14556530

C₂

yes

maximal, characteristic

#17989609

—

#17989450

—

#17989448

C₂²

yes

—

C₂×#14556530

C₄

yes

—

C₄×C₈×D₇₃

C₂²

yes

characteristic

#17988362

C₂²

yes

—

C₄×#11924003

C₄

yes

characteristic

C₄×C₁₆×C₇₃

C₄

yes

—

—

—

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

C₂×C₈×D₇₃

Q₈

yes

characteristic

C₄²×D₇₃

D₄

yes

characteristic

#14556530

—

C₂×#11924003

C₂×C₄

yes

—

C₄×#10596831

Q₈

yes

characteristic

C₈×Dic₇₃

D₄

yes

characteristic

C₈×Dic₇₃

C₂³

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×#10596833

—

C₂×C₄×D₇₃

C₄⋊C₄

yes

characteristic

C₂×C₄×D₇₃

SD₁₆

yes

—

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₂×#10596831

C₂×Q₈

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

C₄×Dic₇₃

SD₁₆

yes

—

C₄×Dic₇₃

C₂×D₄

yes

characteristic

#11924003

M₁₆

yes

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₄⋊C₄

yes

characteristic

C₁₆×C₇₃

—

C₂²×D₇₃

#106

yes

characteristic

C₂×C₄×C₇₃

#106

yes

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₇₃

#106

yes

characteristic

C₂×Dic₇₃

C₂×SD₁₆

yes

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

#10596833

—

C₈×C₇₃

#108

yes

—

C₈×C₇₃

C₂×M₁₆

yes

—

#10596831

—

#10596831

—

C₂²×C₇₃

C₂×#106

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

—

C₄×M₃₂

—

#1568

—

#1368

—

C₇₃

#12021

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

#432

—

146

—

C₂×C₄×C₈

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

#64408

yes

characteristic

#335

—

146

—

#334

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

146

—

C₂×C₁₆

—

C₂×C₁₆

C₂×#64408

yes

—

C₄⋊C₈

—

146

—

C₂²⋊C₈

—

292

—

#97

—

292

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#64408

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

292

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

146

—

C₂×C₈

—

146

—

C₂×C₈

C₄⋊#64408

yes

characteristic

C₂×C₈

C₂²×#64408

yes

characteristic

C₄²

—

C₄²

—

100

C₄²

C₄⋊#64408

yes

characteristic

101

C₁₆

—

102

C₁₆

—

103

C₂³

—

104

C₂×C₄

—

292

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

146

—

108

C₂×C₄

—

146

—

109

C₂×C₄

#14556659

yes

—

110

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

111

C₈

—

584

—

112

C₈

—

146

—

113

C₈

—

146

—

114

C₈

#14556662

yes

—

115

C₈

D₇₃⋊M₁₆

yes

—

116

C₂²

—

146

—

117

C₂²

—

146

—

118

C₂²

C₂×#14556659

yes

characteristic

119

C₄

—

120

C₄

—

146

—

121

C₄

—

146

—

122

C₄

#17989592

yes

characteristic

123

C₄

C₂×#14556662

yes

characteristic

124

C₂

—

292

—

125

C₂

#22625487

yes

characteristic

126

C₂

#22626278

yes

—

	subgroup	G/H
center	C₂²	C₂×#14556659
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×#64408
Fitting	C₄×C₁₆×C₇₃	C₄
socle	C₂²×C₇₃	C₂×#106
prime radical	C₂²×D₇₃	#106
cosocle	C₈×Dic₇₃	C₂³
hypercenter	C₄×C₁₆	#64408
hypocenter	C₇₃	#12021
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

subgroup

G/H

center

C₂²

C₂×#14556659

commutator

C₂×C₈×C₇₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×#64408

Fitting

C₄×C₁₆×C₇₃

C₄

C₂²×C₇₃

C₂²×D₇₃

C₈×Dic₇₃

C₂³

C₄×C₁₆

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

N	H ≅ G/N	type
2. #22624957	C₂	semidirect
67. C₇₃	#12021	semidirect

H ≅ G/N

type

2. #22624957

C₂

semidirect

67. C₇₃

#12021

semidirect

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

subgroup	index
#22625492	2
#22624957	2
C₄×#14556530	2
#12021	73

subgroup

index

Subgroups — #34011800

Conjugacy classes of subgroups

Complements

Maximal subgroups