Group #34014529 — subgroups

Subgroups — #34014529

Order 18688.

186 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 92 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	#22626354	C₂	1	2	yes	maximal, complemented
2	#22625090	C₂	1	1	yes	maximal, characteristic, complemented
3	#22625048	C₂	1	2	yes	maximal, complemented
4	C₄×#14556641	C₂	1	1	yes	maximal, characteristic, complemented
5	#22611768	C₂	1	1	yes	maximal, characteristic
6	#17989519	—	2	2	no	—
7	#17989519	C₂²	1	2	yes	—
8	C₂×#14556641	—	2	2	no	—
9	C₂×#14556641	C₂²	1	1	yes	characteristic
10	C₂×#14556641	C₂²	1	1	yes	characteristic
11	#17989004	C₂²	1	2	yes	—
12	#17988408	C₄	1	1	yes	characteristic
13	#17988371	C₄	1	2	yes	—
14	C₄×#11924003	C₂²	1	1	yes	characteristic
15	#443×C₇₃	C₄	1	1	yes	characteristic
16	#14556641	—	2	2	no	—
17	#14556641	—	2	1	no	—
18	#14556641	D₄	1	1	yes	characteristic
19	#14556641	D₄	1	1	yes	characteristic
20	#14556531	—	2	2	no	—
21	#14556531	C₂×C₄	1	2	yes	—
22	C₂×#11924003	—	2	2	no	—
23	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
24	C₂×#11924003	C₂³	1	1	yes	characteristic
25	#135×C₇₃	C₈	1	1	yes	characteristic
26	#131×C₇₃	C₂×C₄	1	2	yes	—
27	#108×C₇₃	C₈	1	1	yes	characteristic
28	#104×C₇₃	C₈	1	2	yes	—
29	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
30	#11924003	—	2	2	no	—
31	#11924003	—	2	1	no	—
32	#11924003	C₂×D₄	1	1	yes	characteristic
33	#11924003	#41	1	1	yes	characteristic
34	#41×C₇₃	C₂×C₈	1	2	yes	—
35	Q₁₆×C₇₃	C₁₆	1	1	yes	characteristic
36	SD₁₆×C₇₃	C₁₆	1	2	yes	—
37	D₈×C₇₃	C₁₆	1	1	yes	characteristic
38	M₁₆×C₇₃	—	2	2	no	—
39	M₁₆×C₇₃	C₂×C₈	1	2	yes	—
40	C₂×C₈×C₇₃	—	2	2	no	—
41	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
42	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
43	C₄²×C₇₃	C₂×C₈	1	1	yes	characteristic
44	Q₈×C₇₃	C₂×C₁₆	1	2	yes	—
45	D₄×C₇₃	—	2	2	no	—
46	D₄×C₇₃	C₂×C₁₆	1	2	yes	—
47	C₂×C₄×C₇₃	—	2	1	no	—
48	C₂×C₄×C₇₃	—	2	2	no	—
49	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
50	C₈×C₇₃	—	2	2	no	—
51	C₈×C₇₃	C₂×C₁₆	1	1	yes	characteristic
52	C₈×C₇₃	C₂×C₁₆	1	1	yes	characteristic
53	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
54	C₈×C₇₃	#131	1	1	yes	characteristic
55	C₂²×C₇₃	—	2	2	no	—
56	C₂²×C₇₃	#504	1	1	yes	characteristic
57	C₄×C₇₃	—	2	2	no	—
58	C₄×C₇₃	—	2	2	no	—
59	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
60	C₄×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
61	#13647	—	73	1	no	maximal, Sylow
62	C₂×C₇₃	—	2	1	no	—
63	C₂×C₇₃	—	4	2	no	—
64	C₂×C₇₃	D₄×C₁₆	1	1	yes	characteristic
65	#2258	—	73	2	no	—
66	#2170	—	73	1	no	—
67	#1422	—	73	1	no	—
68	#1401	—	73	2	no	—
69	C₄×C₃₂	—	73	1	no	—
70	C₇₃	#13647	1	1	yes	characteristic, complemented, Sylow
71	#504	—	73	2	no	—
72	#443	#64408	1	1	yes	characteristic
73	M₆₄	—	73	2	no	—
74	M₆₄	—	146	2	no	—
75	C₂×C₃₂	—	73	1	no	—
76	C₂×C₃₂	—	73	1	no	—
77	C₂×C₃₂	—	146	2	no	—
78	#364	—	73	1	no	—
79	#350	—	73	2	no	—
80	C₄×C₁₆	—	73	1	no	—
81	#135	#10596833	1	1	yes	characteristic
82	#131	C₂×#64408	1	2	yes	—
83	M₃₂	—	73	2	no	—
84	M₃₂	—	146	2	no	—
85	C₂×C₁₆	—	73	1	no	—
86	C₂×C₁₆	—	73	1	no	—
87	C₂×C₁₆	—	146	2	no	—
88	#108	#10596833	1	1	yes	characteristic
89	#104	#10596833	1	2	yes	—
90	C₄×C₈	C₂×#64408	1	1	yes	characteristic
91	C₃₂	—	73	1	no	—
92	C₃₂	—	73	1	no	—
93	C₃₂	—	146	1	no	—
94	C₃₂	—	146	2	no	—
95	#41	C₂×#10596833	1	2	yes	—
96	Q₁₆	#11924031	1	1	yes	characteristic
97	SD₁₆	#11924031	1	2	yes	—
98	D₈	#11924031	1	1	yes	characteristic
99	M₁₆	—	2	2	no	—
100	M₁₆	C₂×#10596833	1	2	yes	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	C₂×#10596833	1	1	yes	characteristic
103	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
104	C₄²	C₂×#10596833	1	1	yes	characteristic
105	C₁₆	—	73	1	no	—
106	C₁₆	—	73	1	no	—
107	C₁₆	—	146	1	no	—
108	C₁₆	—	146	2	no	—
109	Q₈	C₂×#11924031	1	2	yes	—
110	D₄	—	2	2	no	—
111	D₄	C₂×#11924031	1	2	yes	—
112	C₂×C₄	—	2	1	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
115	C₈	—	2	2	no	—
116	C₈	C₂×#11924031	1	1	yes	characteristic
117	C₈	C₂×#11924031	1	1	yes	characteristic
118	C₈	D₄×#64408	1	1	yes	characteristic
119	C₈	#14556678	1	1	yes	characteristic
120	C₂²	—	2	2	no	—
121	C₂²	#17989645	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	D₄×#10596833	1	1	yes	characteristic
125	C₄	C₂²×#11924031	1	1	yes	characteristic
126	C₂	—	2	1	no	—
127	C₂	—	4	2	no	—
128	C₂	D₄×#11924031	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626354

C₂

yes

maximal, complemented

#22625090

C₂

yes

maximal, characteristic, complemented

#22625048

C₂

yes

maximal, complemented

C₄×#14556641

C₂

yes

maximal, characteristic, complemented

#22611768

C₂

yes

maximal, characteristic

#17989519

—

#17989519

C₂²

yes

—

C₂×#14556641

—

C₂×#14556641

C₂²

yes

characteristic

C₂×#14556641

C₂²

yes

characteristic

#17989004

C₂²

yes

—

#17988408

C₄

yes

characteristic

#17988371

C₄

yes

—

C₄×#11924003

C₂²

yes

characteristic

#443×C₇₃

C₄

yes

—

—

D₄

yes

characteristic

#14556641

D₄

yes

characteristic

#14556531

—

#14556531

C₂×C₄

yes

—

C₂×#11924003

—

C₂×#11924003

C₂×C₄

yes

characteristic

C₂×#11924003

C₂³

yes

characteristic

#135×C₇₃

C₈

yes

characteristic

#131×C₇₃

C₂×C₄

yes

—

#108×C₇₃

C₈

yes

characteristic

#104×C₇₃

C₈

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

—

C₂×D₄

yes

yes

#41×C₇₃

C₂×C₈

yes

—

Q₁₆×C₇₃

C₁₆

yes

characteristic

SD₁₆×C₇₃

C₁₆

yes

—

D₈×C₇₃

C₁₆

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

C₂×C₈

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₂×C₈

yes

characteristic

Q₈×C₇₃

C₂×C₁₆

yes

—

D₄×C₇₃

—

D₄×C₇₃

C₂×C₁₆

yes

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

C₂×C₁₆

yes

characteristic

C₈×C₇₃

C₂×C₁₆

yes

characteristic

C₈×C₇₃

C₄×D₄

yes

characteristic

C₈×C₇₃

#131

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

#504

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₈×D₄

yes

characteristic

C₄×C₇₃

C₂²×C₁₆

yes

characteristic

#13647

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

D₄×C₁₆

yes

—

—

—

—

C₄×C₃₂

—

C₇₃

#13647

yes

characteristic, complemented, Sylow

—

yes

M₆₄

—

M₆₄

—

146

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

146

—

#364

—

#350

—

C₄×C₁₆

—

yes

yes

—

M₃₂

—

M₃₂

—

146

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

yes

yes

—

C₄×C₈

C₂×#64408

yes

characteristic

C₃₂

—

C₃₂

—

C₃₂

—

146

—

C₃₂

—

146

—

#41

C₂×#10596833

yes

—

Q₁₆

#11924031

yes

characteristic

SD₁₆

#11924031

yes

—

D₈

#11924031

yes

characteristic

M₁₆

—

100

M₁₆

C₂×#10596833

yes

—

101

C₂×C₈

—

102

C₂×C₈

C₂×#10596833

yes

characteristic

103

C₂×C₈

C₂²×#64408

yes

characteristic

104

C₄²

C₂×#10596833

yes

characteristic

105

C₁₆

—

106

C₁₆

—

107

C₁₆

—

146

—

108

C₁₆

—

146

—

109

Q₈

C₂×#11924031

yes

—

110

D₄

—

111

D₄

C₂×#11924031

yes

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂²×#10596833

yes

characteristic

115

C₈

—

116

C₈

C₂×#11924031

yes

characteristic

117

C₈

C₂×#11924031

yes

characteristic

118

C₈

D₄×#64408

yes

characteristic

119

C₈

#14556678

yes

characteristic

120

C₂²

—

121

C₂²

#17989645

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

D₄×#10596833

yes

characteristic

125

C₄

C₂²×#11924031

yes

characteristic

126

C₂

—

127

C₂

—

128

C₂

D₄×#11924031

yes

characteristic

	subgroup	G/H
center	C₈	D₄×#64408
commutator	C₄×C₇₃	C₂²×C₁₆
Frattini	C₂×C₈	C₂²×#64408
Fitting	#443×C₇₃	C₄
socle	C₂×C₇₃	D₄×C₁₆
prime radical	#135×C₇₃	C₈
cosocle	C₂×#11924003	C₂³
hypercenter	#443	#64408
hypocenter	C₇₃	#13647
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #443

subgroup

G/H

center

C₈

D₄×#64408

commutator

C₄×C₇₃

C₂²×C₁₆

Frattini

C₂×C₈

C₂²×#64408

Fitting

#443×C₇₃

C₄

socle

C₂×C₇₃

D₄×C₁₆

prime radical

#135×C₇₃

C₈

C₂³

C₇₃

G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ #443

N	H ≅ G/N	type
1. #22626354	C₂	semidirect
2. #22625090	C₂	semidirect
3. #22625048	C₂	semidirect
4. C₄×#14556641	C₂	semidirect
70. C₇₃	#13647	semidirect

H ≅ G/N

type

1. #22626354

C₂

semidirect

2. #22625090

C₂

semidirect

3. #22625048

C₂

semidirect

4. C₄×#14556641

C₂

semidirect

70. C₇₃

#13647

semidirect

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

subgroup	index
#22626354	2
#22625090	2
#22625048	2
C₄×#14556641	2
#22611768	2
#13647	73

subgroup

index

Subgroups — #34014529

Conjugacy classes of subgroups

Complements

Maximal subgroups