Group #34011727 — subgroups

Subgroups — #34011727

Order 18688.

234 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 112 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	#22625478	C₂	1	2	yes	maximal
2	#22624957	C₂	1	4	yes	maximal, complemented
3	C₂×#17988362	C₂	1	1	yes	maximal, characteristic
4	#17989603	—	2	2	no	—
5	#17989450	—	2	2	no	—
6	#17989448	C₂²	1	4	yes	—
7	C₂²×#11924003	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
9	#17988362	C₄	1	2	yes	—
10	#17988362	C₂²	1	2	yes	—
11	C₂²×C₁₆×C₇₃	C₄	1	1	yes	characteristic
12	C₄⋊#10596833	—	2	4	no	—
13	C₂²⋊#10596833	—	4	2	no	—
14	C₄×#10596833	—	4	2	no	—
15	C₂²×#10596831	Q₈	1	1	yes	characteristic
16	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
17	C₂×#11924003	C₈	1	4	yes	complemented
18	C₂×#11924003	C₂×C₄	1	2	yes	—
19	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
20	C₈×Dic₇₃	D₄	1	1	yes	characteristic
21	C₈×Dic₇₃	Q₈	1	1	yes	characteristic
22	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
23	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₆×C₇₃	—	2	2	no	—
25	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
26	C₂×#10596833	—	4	4	no	—
27	C₂²×Dic₇₃	SD₁₆	1	2	yes	—
28	C₂×#10596831	—	2	2	no	—
29	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
30	C₂×#10596831	C₂×Q₈	1	1	yes	characteristic
31	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
32	C₄×Dic₇₃	SD₁₆	1	2	yes	—
33	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
34	#11924003	C₂×C₈	1	4	yes	—
35	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
38	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
39	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
40	C₁₆×C₇₃	—	2	2	no	—
41	C₂³×C₇₃	#106	1	1	yes	characteristic
42	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
43	C₂×C₄×C₇₃	#106	1	1	yes	characteristic
44	C₂×C₄×C₇₃	#108	1	2	yes	—
45	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×Dic₇₃	—	4	2	no	—
47	C₂×Dic₇₃	#106	1	2	yes	—
48	C₂×Dic₇₃	C₂×SD₁₆	1	2	yes	—
49	#10596833	—	8	4	no	—
50	C₈×C₇₃	C₄⋊C₈	1	2	yes	—
51	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
52	C₈×C₇₃	C₂×M₁₆	1	1	yes	characteristic
53	#10596831	—	2	1	no	—
54	#10596831	—	2	1	no	—
55	C₂²×C₇₃	—	2	2	no	—
56	C₂²×C₇₃	#334	1	2	yes	—
57	C₂²×C₇₃	C₂×#106	1	1	yes	characteristic
58	C₄×C₇₃	#334	1	2	yes	—
59	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
60	C₄×C₇₃	C₂×#108	1	1	yes	characteristic
61	Dic₇₃	—	2	2	no	—
62	Dic₇₃	—	2	2	no	—
63	#11999	—	73	1	no	maximal, Sylow
64	C₂×C₇₃	—	2	2	no	—
65	C₂×C₇₃	C₂×#334	1	1	yes	characteristic
66	C₂×C₇₃	#2155	1	2	yes	—
67	C₂×#346	—	73	1	no	—
68	#1564	—	73	2	no	—
69	#1368	—	73	4	no	—
70	C₇₃	#11999	1	1	yes	characteristic, complemented, Sylow
71	C₂²×C₁₆	—	73	1	no	—
72	C₂²×C₁₆	#64408	1	1	yes	characteristic
73	#431	—	146	2	no	—
74	C₂×C₄×C₈	—	73	1	no	—
75	#346	—	73	2	no	—
76	#346	—	73	2	no	—
77	#335	—	146	2	no	—
78	#334	—	73	4	no	—
79	C₂²×C₈	—	73	1	no	—
80	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
81	C₂×C₄²	—	73	1	no	—
82	C₂×C₁₆	—	2	2	no	—
83	C₂×C₁₆	—	73	4	no	—
84	C₂×C₁₆	—	73	2	no	—
85	C₂×C₁₆	C₂×#64408	1	2	yes	—
86	C₄⋊C₈	—	146	4	no	—
87	C₂²⋊C₈	—	292	2	no	—
88	C₄×C₈	—	292	2	no	—
89	C₄×C₈	—	73	1	no	—
90	C₄×C₈	—	73	1	no	—
91	C₄×C₈	—	73	1	no	—
92	C₄×C₈	—	73	1	no	—
93	C₂²×C₄	—	73	2	no	—
94	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
95	C₂×C₈	—	292	4	no	—
96	C₂×C₈	—	73	1	no	—
97	C₂×C₈	—	73	1	no	—
98	C₂×C₈	—	146	2	no	—
99	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
100	C₂×C₈	#11924036	1	2	yes	—
101	C₂×C₈	#11924037	1	2	yes	—
102	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
103	C₄²	—	73	2	no	—
104	C₄²	—	73	1	no	—
105	C₄²	—	73	1	no	—
106	C₁₆	—	2	2	no	—
107	C₁₆	—	73	4	no	—
108	C₂³	#14556659	1	1	yes	characteristic
109	C₂×C₄	—	292	2	no	—
110	C₂×C₄	—	73	2	no	—
111	C₂×C₄	—	73	2	no	—
112	C₂×C₄	#14556658	1	2	yes	—
113	C₂×C₄	#14556659	1	1	yes	characteristic
114	C₂×C₄	#14556662	1	2	yes	—
115	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
116	C₈	—	584	4	no	—
117	C₈	—	146	1	no	—
118	C₈	—	146	1	no	—
119	C₈	#14556658	1	2	yes	—
120	C₈	C₂×#11924036	1	1	yes	characteristic
121	C₈	C₂×#11924037	1	1	yes	characteristic
122	C₂²	—	2	2	no	—
123	C₂²	#17989449	1	2	yes	—
124	C₂²	C₂×#14556659	1	1	yes	characteristic
125	C₄	—	146	2	no	—
126	C₄	—	146	2	no	—
127	C₄	#17989449	1	2	yes	—
128	C₄	C₂×#14556658	1	1	yes	characteristic
129	C₄	C₂×#14556662	1	1	yes	characteristic
130	C₂	—	2	2	no	—
131	C₂	C₂×#17989449	1	1	yes	characteristic
132	C₂	#22626278	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22625478

C₂

yes

maximal

#22624957

C₂

yes

maximal, complemented

C₂×#17988362

C₂

yes

maximal, characteristic

#17989603

—

#17989450

—

#17989448

C₂²

yes

—

C₂²×#11924003

C₄

yes

characteristic

C₂×C₈×Dic₇₃

C₂²

yes

characteristic

#17988362

C₄

yes

—

#17988362

C₂²

yes

—

C₂²×C₁₆×C₇₃

C₄

yes

—

—

—

Q₈

yes

characteristic

C₂×C₄×Dic₇₃

D₄

yes

characteristic

C₂×#11924003

C₈

yes

complemented

C₂×#11924003

C₂×C₄

yes

—

C₈×Dic₇₃

C₂×C₄

yes

characteristic

C₈×Dic₇₃

D₄

yes

characteristic

C₈×Dic₇₃

Q₈

yes

characteristic

C₈×Dic₇₃

C₂³

yes

characteristic

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

—

C₂×#10596833

—

C₂²×Dic₇₃

SD₁₆

yes

—

C₂×#10596831

—

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₂×#10596831

C₂×Q₈

yes

characteristic

C₄×Dic₇₃

C₄⋊C₄

yes

characteristic

C₄×Dic₇₃

SD₁₆

yes

—

C₄×Dic₇₃

C₂×D₄

yes

characteristic

#11924003

C₂×C₈

yes

—

C₂²×C₄×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×C₈

yes

—

C₂×C₈×C₇₃

M₁₆

yes

—

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₁₆×C₇₃

—

C₂³×C₇₃

#106

yes

characteristic

C₂×C₄×C₇₃

C₄⋊C₈

yes

—

C₂×C₄×C₇₃

#106

yes

characteristic

C₂×C₄×C₇₃

#108

yes

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

#106

yes

—

C₂×Dic₇₃

C₂×SD₁₆

yes

—

#10596833

—

C₈×C₇₃

C₄⋊C₈

yes

—

C₈×C₇₃

C₂²×C₈

yes

characteristic

C₈×C₇₃

C₂×M₁₆

yes

characteristic

#10596831

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

#334

yes

—

C₂²×C₇₃

C₂×#106

yes

characteristic

C₄×C₇₃

#334

yes

—

C₄×C₇₃

C₂×C₄⋊C₈

yes

characteristic

C₄×C₇₃

C₂×#108

yes

characteristic

Dic₇₃

—

Dic₇₃

—

#11999

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

C₂×#334

yes

characteristic

C₂×C₇₃

#2155

yes

—

C₂×#346

—

#1564

—

#1368

—

C₇₃

#11999

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

#64408

yes

characteristic

#431

—

146

—

C₂×C₄×C₈

—

#346

—

#346

—

#335

—

146

—

#334

—

C₂²×C₈

—

C₂²×C₈

C₂×#64408

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×#64408

yes

—

C₄⋊C₈

—

146

—

C₂²⋊C₈

—

292

—

C₄×C₈

—

292

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

C₄⋊#64408

yes

characteristic

C₂×C₈

—

292

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

146

—

C₂×C₈

C₄⋊#64408

yes

characteristic

100

C₂×C₈

#11924036

yes

—

101

C₂×C₈

#11924037

yes

—

102

C₂×C₈

C₂²×#64408

yes

characteristic

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₁₆

—

107

C₁₆

—

108

C₂³

#14556659

yes

characteristic

109

C₂×C₄

—

292

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

#14556658

yes

—

113

C₂×C₄

#14556659

yes

characteristic

114

C₂×C₄

#14556662

yes

—

115

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

116

C₈

—

584

—

117

C₈

—

146

—

118

C₈

—

146

—

119

C₈

#14556658

yes

—

120

C₈

C₂×#11924036

yes

characteristic

121

C₈

C₂×#11924037

yes

characteristic

122

C₂²

—

123

C₂²

#17989449

yes

—

124

C₂²

C₂×#14556659

yes

characteristic

125

C₄

—

146

—

126

C₄

—

146

—

127

C₄

#17989449

yes

—

128

C₄

C₂×#14556658

yes

characteristic

129

C₄

C₂×#14556662

yes

characteristic

130

C₂

—

131

C₂

C₂×#17989449

yes

characteristic

132

C₂

#22626278

yes

—

	subgroup	G/H
center	C₂×C₄	#14556659
commutator	C₈×C₇₃	C₂²×C₈
Frattini	C₂×C₈	C₂²×#64408
Fitting	C₂²×C₁₆×C₇₃	C₄
socle	C₂²×C₇₃	C₂×#106
prime radical	C₂³×C₇₃	#106
cosocle	C₈×Dic₇₃	C₂³
hypercenter	C₂²×C₁₆	#64408
hypocenter	C₇₃	#11999
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂×C₄

#14556659

commutator

C₈×C₇₃

C₂²×C₈

Frattini

C₂×C₈

C₂²×#64408

Fitting

C₂²×C₁₆×C₇₃

C₄

C₂²×C₇₃

C₂³×C₇₃

C₈×Dic₇₃

C₂³

C₂²×C₁₆

C₇₃

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
2. #22624957	C₂	semidirect
17. C₂×#11924003	C₈	semidirect
70. C₇₃	#11999	semidirect

H ≅ G/N

type

2. #22624957

C₂

semidirect

17. C₂×#11924003

C₈

semidirect

70. C₇₃

#11999

semidirect

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

subgroup	index
#22625478	2
#22624957	2
C₂×#17988362	2
#11999	73

subgroup

index

Subgroups — #34011727

Conjugacy classes of subgroups

Complements

Maximal subgroups