Group #34035033 — subgroups

Subgroups — #34035033

Order 18688.

462 conjugacy classes of subgroups; only the 289 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 289 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	#22627164	C₂	1	8	yes	maximal, complemented
2	C₂×#17989695	C₂	1	6	yes	maximal, complemented
3	C₂×#17989645	C₂	1	1	yes	maximal, characteristic
4	#17989695	C₂²	1	4	yes	—
5	#17989695	C₂²	1	12	yes	complemented
6	C₂×#14556681	C₂²	1	12	yes	complemented
7	#17989645	C₄	1	4	yes	—
8	#17989645	C₂²	1	4	yes	—
9	C₂×#14556665	C₄	1	3	yes	—
10	C₂²×#11924031	C₂²	1	3	yes	—
11	C₂×#14556598	C₄	1	1	yes	characteristic
12	#14556681	C₂³	1	8	yes	—
13	#14556665	C₂×C₄	1	12	yes	—
14	C₂×#11924031	C₂×C₄	1	3	yes	—
15	C₂×#11924031	C₂×C₄	1	6	yes	—
16	C₂×#11924031	C₂³	1	1	yes	characteristic
17	C₂×#11924031	C₂³	1	6	yes	—
18	#14556598	C₈	1	4	yes	—
19	#14556598	C₂×C₄	1	4	yes	—
20	C₂×#11924012	C₈	1	3	yes	—
21	C₂²×#10596831	C₂×C₄	1	3	yes	—
22	C₂×#41×C₇₃	C₈	1	1	yes	characteristic
23	#11924031	C₂²×C₄	1	1	yes	characteristic
24	#11924031	C₂²×C₄	1	6	yes	—
25	#11924031	C₂⁴	1	1	yes	characteristic
26	#11924012	C₂×C₈	1	12	yes	—
27	C₂×#10596831	C₂×C₈	1	3	yes	—
28	C₂×#10596831	C₂×C₈	1	6	yes	—
29	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
30	C₂×#10596831	C₂²×C₄	1	6	yes	—
31	#41×C₇₃	C₁₆	1	4	yes	—
32	#41×C₇₃	C₂×C₈	1	4	yes	—
33	C₂×Q₈×C₇₃	C₁₆	1	1	yes	characteristic
34	C₂×D₄×C₇₃	C₁₆	1	3	yes	—
35	C₂²×C₄×C₇₃	C₂×C₈	1	3	yes	—
36	C₂³×C₇₃	C₂×C₁₆	1	3	yes	—
37	Q₈×C₇₃	C₂×C₁₆	1	4	yes	—
38	D₄×C₇₃	C₂×C₁₆	1	12	yes	—
39	C₂×C₄×C₇₃	C₂×C₁₆	1	3	yes	—
40	C₂×C₄×C₇₃	C₂×C₁₆	1	6	yes	—
41	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
42	C₂×C₄×C₇₃	C₂²×C₈	1	6	yes	—
43	#10596831	C₂²×C₈	1	1	yes	characteristic
44	#10596831	C₂²×C₈	1	6	yes	—
45	#10596831	C₂³×C₄	1	1	yes	characteristic
46	C₂²×C₇₃	—	4	3	no	—
47	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
48	C₂²×C₇₃	C₂²×C₁₆	1	6	yes	—
49	C₄×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
50	C₄×C₇₃	C₂²×C₁₆	1	6	yes	—
51	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
52	#33974	—	73	1	no	maximal, Sylow
53	C₂×C₇₃	—	2	1	no	—
54	C₂×C₇₃	—	2	6	no	—
55	C₂×C₇₃	C₂³×C₁₆	1	1	yes	characteristic
56	C₂×#504	—	73	1	no	—
57	#2258	—	73	8	no	—
58	C₂×M₆₄	—	73	6	no	—
59	C₇₃	#33974	1	1	yes	characteristic, complemented, Sylow
60	C₂×#131	—	73	1	no	—
61	#504	—	73	4	no	—
62	#504	—	73	4	no	—
63	C₂×M₃₂	—	73	3	no	—
64	C₂²×C₁₆	—	73	3	no	—
65	M₆₄	—	73	4	no	—
66	M₆₄	—	73	12	no	—
67	C₂×C₃₂	—	73	12	no	—
68	C₂×#41	#10596842	1	1	yes	characteristic
69	#131	—	73	4	no	—
70	#131	—	73	4	no	—
71	C₂×M₁₆	—	73	3	no	—
72	C₂²×C₈	—	73	3	no	—
73	M₃₂	—	73	12	no	—
74	C₂×C₁₆	—	73	6	no	—
75	C₂×C₁₆	—	73	1	no	—
76	C₂×C₁₆	—	73	3	no	—
77	C₂×C₁₆	—	73	6	no	—
78	C₃₂	—	73	8	no	—
79	#41	#11924040	1	4	yes	—
80	#41	C₂×#10596842	1	4	yes	—
81	C₂×Q₈	#11924040	1	1	yes	characteristic
82	C₂×D₄	#11924040	1	3	yes	—
83	C₂²×C₄	C₂×#10596842	1	3	yes	—
84	M₁₆	—	73	12	no	—
85	C₂×C₈	—	73	3	no	—
86	C₂×C₈	—	73	6	no	—
87	C₂×C₈	—	73	1	no	—
88	C₂×C₈	—	73	6	no	—
89	C₁₆	—	73	1	no	—
90	C₁₆	—	73	6	no	—
91	C₁₆	—	73	1	no	—
92	C₂³	C₂×#11924040	1	3	yes	—
93	Q₈	C₂×#11924040	1	4	yes	—
94	D₄	C₂×#11924040	1	12	yes	—
95	C₂×C₄	C₂×#11924040	1	3	yes	—
96	C₂×C₄	C₂×#11924040	1	6	yes	—
97	C₂×C₄	C₂²×#10596842	1	1	yes	characteristic
98	C₂×C₄	C₂²×#10596842	1	6	yes	—
99	C₈	—	73	1	no	—
100	C₈	—	73	1	no	—
101	C₈	—	73	6	no	—
102	C₂²	—	4	3	no	—
103	C₂²	C₂²×#11924040	1	1	yes	characteristic
104	C₂²	C₂²×#11924040	1	6	yes	—
105	C₄	C₂²×#11924040	1	1	yes	characteristic
106	C₄	C₂²×#11924040	1	6	yes	—
107	C₄	C₂³×#10596842	1	1	yes	characteristic
108	C₂	—	2	1	no	—
109	C₂	—	2	6	no	—
110	C₂	C₂³×#11924040	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22627164

C₂

yes

maximal, complemented

C₂×#17989695

C₂

yes

maximal, complemented

C₂×#17989645

C₂

yes

maximal, characteristic

#17989695

C₂²

yes

—

#17989695

C₂²

yes

complemented

C₂×#14556681

C₂²

yes

complemented

#17989645

C₄

yes

—

#17989645

C₂²

yes

—

C₂×#14556665

C₄

yes

—

C₂²×#11924031

C₂²

yes

—

C₂×#14556598

C₄

yes

characteristic

#14556681

C₂³

yes

—

#14556665

C₂×C₄

yes

—

C₂×#11924031

C₂×C₄

yes

—

C₂×#11924031

C₂×C₄

yes

—

C₂×#11924031

C₂³

yes

characteristic

C₂×#11924031

C₂³

yes

—

#14556598

C₈

yes

—

#14556598

C₂×C₄

yes

—

C₂×#11924012

C₈

yes

—

C₂²×#10596831

C₂×C₄

yes

—

C₂×#41×C₇₃

C₈

yes

characteristic

#11924031

C₂²×C₄

yes

characteristic

#11924031

C₂²×C₄

yes

—

#11924031

C₂⁴

yes

characteristic

#11924012

C₂×C₈

yes

—

C₂×#10596831

C₂×C₈

yes

—

C₂×#10596831

C₂×C₈

yes

—

C₂×#10596831

C₂²×C₄

yes

characteristic

C₂×#10596831

C₂²×C₄

yes

—

#41×C₇₃

C₁₆

yes

—

#41×C₇₃

C₂×C₈

yes

—

C₂×Q₈×C₇₃

C₁₆

yes

characteristic

C₂×D₄×C₇₃

C₁₆

yes

—

C₂²×C₄×C₇₃

C₂×C₈

yes

—

C₂³×C₇₃

C₂×C₁₆

yes

—

Q₈×C₇₃

C₂×C₁₆

yes

—

D₄×C₇₃

C₂×C₁₆

yes

—

C₂×C₄×C₇₃

C₂×C₁₆

yes

—

C₂×C₄×C₇₃

C₂×C₁₆

yes

—

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₈

yes

—

#10596831

C₂²×C₈

yes

characteristic

#10596831

C₂²×C₈

yes

—

#10596831

C₂³×C₄

yes

characteristic

C₂²×C₇₃

—

C₂²×C₇₃

C₂²×C₁₆

yes

characteristic

C₂²×C₇₃

C₂²×C₁₆

yes

—

C₄×C₇₃

C₂²×C₁₆

yes

characteristic

C₄×C₇₃

C₂²×C₁₆

yes

—

C₄×C₇₃

C₂³×C₈

yes

characteristic

#33974

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

C₂³×C₁₆

yes

characteristic

C₂×#504

—

#2258

—

C₂×M₆₄

—

C₇₃

#33974

yes

characteristic, complemented, Sylow

C₂×#131

—

#504

—

#504

—

C₂×M₃₂

—

C₂²×C₁₆

—

M₆₄

—

M₆₄

—

C₂×C₃₂

—

C₂×#41

yes

—

—

C₂×M₁₆

—

C₂²×C₈

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₃₂

—

#41

#11924040

yes

—

#41

C₂×#10596842

yes

—

C₂×Q₈

#11924040

yes

characteristic

C₂×D₄

#11924040

yes

—

C₂²×C₄

C₂×#10596842

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₁₆

—

C₁₆

—

C₁₆

—

C₂³

C₂×#11924040

yes

—

Q₈

C₂×#11924040

yes

—

D₄

C₂×#11924040

yes

—

C₂×C₄

C₂×#11924040

yes

—

C₂×C₄

C₂×#11924040

yes

—

C₂×C₄

C₂²×#10596842

yes

characteristic

C₂×C₄

C₂²×#10596842

yes

—

C₈

—

100

C₈

—

101

C₈

—

102

C₂²

—

103

C₂²

C₂²×#11924040

yes

characteristic

104

C₂²

C₂²×#11924040

yes

—

105

C₄

C₂²×#11924040

yes

characteristic

106

C₄

C₂²×#11924040

yes

—

107

C₄

C₂³×#10596842

yes

characteristic

108

C₂

—

109

C₂

—

110

C₂

C₂³×#11924040

yes

characteristic

	subgroup	G/H
center, Frattini	C₄	C₂³×#10596842
commutator, socle	C₂×C₇₃	C₂³×C₁₆
Fitting, prime radical	C₂×#41×C₇₃	C₈
cosocle	#11924031	C₂⁴
hypercenter	C₂×#41	#10596842
hypocenter	C₇₃	#33974
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×#41

subgroup

G/H

center, Frattini

C₄

C₂³×#10596842

commutator, socle

C₂×C₇₃

C₂³×C₁₆

Fitting, prime radical

C₂×#41×C₇₃

C₈

C₂⁴

C₂×#41

C₇₃

G ▷ C₂×C₇₃ ▷ C₇₃

C₁ ◁ C₄ ◁ C₂×#41

N	H ≅ G/N	type
1. #22627164	C₂	semidirect
2. C₂×#17989695	C₂	semidirect
5. #17989695	C₂²	semidirect
6. C₂×#14556681	C₂²	semidirect
59. C₇₃	#33974	semidirect

H ≅ G/N

type

1. #22627164

C₂

semidirect

2. C₂×#17989695

C₂

semidirect

5. #17989695

C₂²

semidirect

6. C₂×#14556681

C₂²

semidirect

59. C₇₃

#33974

semidirect

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

subgroup	index
#22627164	2
C₂×#17989695	2
C₂×#17989645	2
#33974	73

subgroup

index

Subgroups — #34035033

Conjugacy classes of subgroups

Complements

Maximal subgroups