Group #34014223 — subgroups

Subgroups — #34014223

Order 18688.

214 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 132 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×#17989519	C₂	1	2	yes	maximal
2	#22625042	C₂	1	4	yes	maximal, complemented
3	#22611528	C₂	1	1	yes	maximal, characteristic
4	#17989519	C₄	1	4	yes	complemented
5	#17989519	C₂²	1	4	yes	—
6	C₂×#14556641	C₄	1	4	yes	complemented
7	C₂×#14556531	C₄	1	1	yes	characteristic
8	C₂²×#11924003	C₂²	1	1	yes	characteristic
9	#17988363	C₂²	1	2	yes	—
10	C₄×#11924003	C₄	1	2	yes	—
11	#405×C₇₃	C₄	1	1	yes	characteristic
12	#14556641	C₂×C₄	1	8	yes	—
13	#14556531	C₈	1	4	yes	—
14	C₂×#11924003	C₂×C₄	1	2	yes	—
15	C₂×#11924003	C₂×C₄	1	2	yes	—
16	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
17	C₂×#11924003	C₂×C₄	1	2	yes	—
18	C₂×#11924003	C₂³	1	1	yes	characteristic
19	C₂×M₁₆×C₇₃	C₈	1	1	yes	characteristic
20	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
21	#117×C₇₃	C₈	1	1	yes	characteristic
22	#97×C₇₃	C₈	1	2	yes	—
23	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
24	#11924003	C₄²	1	1	yes	characteristic
25	#11924003	C₄²	1	1	yes	characteristic
26	#11924003	C₂×C₈	1	4	yes	—
27	#11924003	C₂²×C₄	1	2	yes	—
28	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
29	M₁₆×C₇₃	—	2	2	no	—
30	C₂×C₈×C₇₃	C₄²	1	2	yes	—
31	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
32	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
34	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
35	C₄⋊C₄×C₇₃	—	2	1	no	—
36	C₂²⋊C₄×C₇₃	—	2	1	no	—
37	C₄²×C₇₃	C₂×C₈	1	2	yes	—
38	C₂³×C₇₃	M₃₂	1	1	yes	characteristic
39	C₂×C₄×C₇₃	—	2	2	no	—
40	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
41	C₂×C₄×C₇₃	M₃₂	1	1	yes	characteristic
42	C₂×C₄×C₇₃	M₃₂	1	2	yes	—
43	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
44	C₈×C₇₃	—	2	2	no	—
45	C₈×C₇₃	C₄×C₈	1	2	yes	—
46	C₈×C₇₃	C₂×C₄²	1	1	yes	characteristic
47	C₈×C₇₃	C₂²×C₈	1	1	yes	characteristic
48	C₂²×C₇₃	—	4	1	no	—
49	C₂²×C₇₃	#346	1	2	yes	—
50	C₂²×C₇₃	C₂×M₃₂	1	1	yes	characteristic
51	C₄×C₇₃	—	4	2	no	—
52	C₄×C₇₃	#346	1	2	yes	—
53	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
54	C₄×C₇₃	C₂×M₃₂	1	1	yes	characteristic
55	#13551	—	73	1	no	maximal, Sylow
56	C₂×C₇₃	—	2	1	no	—
57	C₂×C₇₃	—	2	2	no	—
58	C₂×C₇₃	C₂×#346	1	1	yes	characteristic
59	C₂×M₆₄	—	73	2	no	—
60	#2108	—	73	1	no	—
61	#1398	—	73	4	no	—
62	C₇₃	#13551	1	1	yes	characteristic, complemented, Sylow
63	C₂×M₃₂	—	73	1	no	—
64	C₂²×C₁₆	—	73	1	no	—
65	#405	#64408	1	1	yes	characteristic
66	M₆₄	—	73	4	no	—
67	M₆₄	—	73	4	no	—
68	C₂×C₃₂	—	73	4	no	—
69	#346	—	73	2	no	—
70	C₄×C₁₆	—	73	2	no	—
71	C₂×M₁₆	#10596833	1	1	yes	characteristic
72	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
73	#117	#10596833	1	1	yes	characteristic
74	M₃₂	—	73	4	no	—
75	C₂×C₁₆	—	73	2	no	—
76	C₂×C₁₆	—	73	2	no	—
77	C₂×C₁₆	—	73	1	no	—
78	C₂×C₁₆	—	73	2	no	—
79	C₂×C₁₆	—	73	1	no	—
80	#97	#10596833	1	2	yes	—
81	C₄×C₈	C₂×#64408	1	2	yes	—
82	C₃₂	—	73	8	no	—
83	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
84	M₁₆	—	2	2	no	—
85	C₂×C₈	C₄×#64408	1	2	yes	—
86	C₂×C₈	C₂×#10596833	1	2	yes	—
87	C₂×C₈	C₂×#10596833	1	1	yes	characteristic
88	C₂×C₈	#11924036	1	2	yes	—
89	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
90	C₄⋊C₄	—	2	1	no	—
91	C₂²⋊C₄	—	2	1	no	—
92	C₄²	C₂×#10596833	1	2	yes	—
93	C₁₆	—	73	4	no	—
94	C₁₆	—	73	1	no	—
95	C₁₆	—	73	1	no	—
96	C₁₆	—	73	2	no	—
97	C₂³	#14556665	1	1	yes	characteristic
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	C₄×#10596833	1	2	yes	—
100	C₂×C₄	#14556665	1	2	yes	—
101	C₂×C₄	#14556665	1	1	yes	characteristic
102	C₂×C₄	C₂²×#10596833	1	1	yes	characteristic
103	C₈	—	2	2	no	—
104	C₈	C₄×#10596833	1	2	yes	—
105	C₈	C₂×C₄×#64408	1	1	yes	characteristic
106	C₈	C₂×#11924036	1	1	yes	characteristic
107	C₂²	—	4	1	no	—
108	C₂²	#17989475	1	2	yes	—
109	C₂²	C₂×#14556665	1	1	yes	characteristic
110	C₄	—	4	2	no	—
111	C₄	#17989475	1	2	yes	—
112	C₄	C₂×C₄×#10596833	1	1	yes	characteristic
113	C₄	C₂×#14556665	1	1	yes	characteristic
114	C₂	—	2	1	no	—
115	C₂	—	2	2	no	—
116	C₂	C₂×#17989475	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17989519

C₂

yes

maximal

#22625042

C₂

yes

maximal, complemented

#22611528

C₂

yes

maximal, characteristic

#17989519

C₄

yes

complemented

#17989519

C₂²

yes

—

C₂×#14556641

C₄

yes

complemented

C₂×#14556531

C₄

yes

characteristic

C₂²×#11924003

C₂²

yes

characteristic

#17988363

C₂²

yes

—

C₄×#11924003

C₄

yes

—

#405×C₇₃

C₄

yes

characteristic

#14556641

C₂×C₄

yes

—

#14556531

C₈

yes

—

C₂×#11924003

C₂×C₄

yes

—

C₂×#11924003

C₂×C₄

yes

—

C₂×#11924003

C₂×C₄

yes

characteristic

C₂×#11924003

C₂×C₄

yes

—

C₂×#11924003

C₂³

yes

characteristic

C₂×M₁₆×C₇₃

C₈

yes

characteristic

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

#117×C₇₃

C₈

yes

characteristic

#97×C₇₃

C₈

yes

—

C₄×C₈×C₇₃

C₂×C₄

yes

—

#11924003

C₄²

yes

characteristic

#11924003

C₄²

yes

characteristic

#11924003

C₂×C₈

yes

—

#11924003

C₂²×C₄

yes

—

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

M₁₆×C₇₃

—

C₂×C₈×C₇₃

C₄²

yes

—

C₂×C₈×C₇₃

C₂×C₈

yes

—

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₂×C₈

yes

—

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₂²⋊C₄×C₇₃

—

C₄²×C₇₃

C₂×C₈

yes

—

C₂³×C₇₃

M₃₂

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₄×C₈

yes

—

C₂×C₄×C₇₃

M₃₂

yes

characteristic

C₂×C₄×C₇₃

M₃₂

yes

—

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₈×C₇₃

—

C₈×C₇₃

C₄×C₈

yes

—

C₈×C₇₃

C₂×C₄²

yes

characteristic

C₈×C₇₃

C₂²×C₈

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

#346

yes

—

C₂²×C₇₃

C₂×M₃₂

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

#346

yes

—

C₄×C₇₃

C₂×C₄×C₈

yes

characteristic

C₄×C₇₃

C₂×M₃₂

yes

characteristic

#13551

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

C₂×#346

yes

characteristic

C₂×M₆₄

—

#2108

—

#1398

—

C₇₃

#13551

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂²×C₁₆

—

#405

#64408

yes

characteristic

M₆₄

—

M₆₄

—

C₂×C₃₂

—

#346

—

C₄×C₁₆

—

C₂×M₁₆

yes

C₂²×C₈

yes

yes

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

#10596833

yes

—

C₄×C₈

C₂×#64408

yes

—

C₃₂

—

C₂²×C₄

C₂×#10596833

yes

characteristic

M₁₆

—

C₂×C₈

C₄×#64408

yes

—

C₂×C₈

C₂×#10596833

yes

—

C₂×C₈

C₂×#10596833

yes

characteristic

C₂×C₈

#11924036

yes

—

C₂×C₈

C₂²×#64408

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×#10596833

yes

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

C₂³

#14556665

yes

characteristic

C₂×C₄

—

C₂×C₄

C₄×#10596833

yes

—

100

C₂×C₄

#14556665

yes

—

101

C₂×C₄

#14556665

yes

characteristic

102

C₂×C₄

C₂²×#10596833

yes

characteristic

103

C₈

—

104

C₈

C₄×#10596833

yes

—

105

C₈

C₂×C₄×#64408

yes

characteristic

106

C₈

C₂×#11924036

yes

characteristic

107

C₂²

—

108

C₂²

#17989475

yes

—

109

C₂²

C₂×#14556665

yes

characteristic

110

C₄

—

111

C₄

#17989475

yes

—

112

C₄

C₂×C₄×#10596833

yes

characteristic

113

C₄

C₂×#14556665

yes

characteristic

114

C₂

—

115

C₂

—

116

C₂

C₂×#17989475

yes

characteristic

	subgroup	G/H
center	C₈	C₂×C₄×#64408
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₂×C₈	C₂²×#64408
Fitting	#405×C₇₃	C₄
socle	C₂×C₇₃	C₂×#346
prime radical	C₂³×C₇₃	M₃₂
cosocle	C₂×#11924003	C₂³
hypercenter	#405	#64408
hypocenter	C₇₃	#13551
LCS	G ▷ C₄×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₈ ◁ #405

subgroup

G/H

center

C₈

C₂×C₄×#64408

commutator

C₄×C₇₃

C₂×C₄×C₈

Frattini

C₂×C₈

C₂²×#64408

Fitting

#405×C₇₃

C₄

socle

C₂×C₇₃

C₂×#346

prime radical

C₂³×C₇₃

M₃₂

C₂³

C₇₃

G ▷ C₄×C₇₃ ▷ C₇₃

C₁ ◁ C₈ ◁ #405

N	H ≅ G/N	type
2. #22625042	C₂	semidirect
4. #17989519	C₄	semidirect
6. C₂×#14556641	C₄	semidirect
62. C₇₃	#13551	semidirect

H ≅ G/N

type

2. #22625042

C₂

semidirect

4. #17989519

C₄

semidirect

6. C₂×#14556641

C₄

semidirect

62. C₇₃

#13551

semidirect

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

subgroup	index
C₂×#17989519	2
#22625042	2
#22611528	2
#13551	73

subgroup

index

Subgroups — #34014223

Conjugacy classes of subgroups

Complements

Maximal subgroups