Group #72234379 — subgroups

Subgroups — #72234379

Order 1024.

220 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 179 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×#7511	C₂	1	2	yes	maximal
2	#149496	C₂	1	1	yes	maximal, characteristic
3	#94506	C₂	1	4	yes	maximal, complemented
4	C₂×M₁₂₈	C₄	1	4	yes	—
5	C₄×M₆₄	C₄	1	1	yes	characteristic
6	C₂×#1397	C₂²	1	1	yes	characteristic
7	#11399	C₄	1	1	yes	characteristic
8	#7511	C₄	1	4	yes	—
9	#7511	C₂²	1	4	yes	—
10	#7330	C₄	1	2	yes	—
11	#7330	C₂²	1	2	yes	—
12	C₂×M₆₄	C₈	1	2	yes	—
13	C₂²×C₃₂	C₂×C₄	1	2	yes	—
14	C₄×M₃₂	C₈	1	1	yes	characteristic
15	C₂×C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
16	#1450	C₈	1	1	yes	characteristic
17	M₁₂₈	C₈	1	8	yes	complemented
18	M₁₂₈	C₂×C₄	1	8	yes	—
19	C₂×C₆₄	C₈	1	8	yes	complemented
20	#1397	C₂×C₄	1	2	yes	—
21	#1397	C₂×C₄	1	1	yes	characteristic
22	#1397	C₂×C₄	1	2	yes	—
23	#1397	C₂³	1	1	yes	characteristic
24	C₄×C₃₂	C₈	1	2	yes	—
25	#1311	C₂×C₄	1	2	yes	—
26	C₈×C₁₆	C₈	1	2	yes	—
27	C₂×M₃₂	—	2	1	no	—
28	C₂²×C₁₆	C₄²	1	1	yes	characteristic
29	C₂²×C₁₆	C₂×C₈	1	1	yes	characteristic
30	#431	—	2	1	no	—
31	C₂×C₄×C₈	C₂×C₈	1	1	yes	characteristic
32	M₆₄	—	2	4	no	—
33	C₂×C₃₂	C₄²	1	2	yes	—
34	C₂×C₃₂	C₂×C₈	1	4	yes	—
35	C₂×C₃₂	C₂×C₈	1	4	yes	—
36	C₂×C₃₂	C₂×C₈	1	4	yes	—
37	C₂×C₃₂	C₂²×C₄	1	2	yes	—
38	#346	—	2	1	no	—
39	C₄×C₁₆	C₄²	1	1	yes	characteristic
40	C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
41	C₄×C₁₆	C₂×C₈	1	2	yes	—
42	C₄×C₁₆	C₂×C₈	1	1	yes	characteristic
43	C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
44	#322	—	2	1	no	—
45	C₈²	C₂×C₈	1	2	yes	—
46	C₆₄	C₂×C₈	1	16	yes	—
47	C₂²×C₈	C₄×C₈	1	1	yes	characteristic
48	C₂²×C₈	M₃₂	1	1	yes	characteristic
49	C₂×C₄²	M₃₂	1	1	yes	characteristic
50	M₃₂	—	4	2	no	—
51	C₂×C₁₆	—	2	2	no	—
52	C₂×C₁₆	C₄×C₈	1	1	yes	characteristic
53	C₂×C₁₆	C₄×C₈	1	2	yes	—
54	C₂×C₁₆	C₄×C₈	1	2	yes	—
55	C₂×C₁₆	C₄×C₈	1	1	yes	characteristic
56	C₂×C₁₆	M₃₂	1	4	yes	—
57	C₂×C₁₆	C₂×C₄²	1	1	yes	characteristic
58	C₂×C₁₆	C₂²×C₈	1	1	yes	characteristic
59	C₄⋊C₈	—	4	1	no	—
60	C₂²⋊C₈	—	4	1	no	—
61	C₄×C₈	—	2	2	no	—
62	C₄×C₈	C₄×C₈	1	1	yes	characteristic
63	C₄×C₈	M₃₂	1	1	yes	characteristic
64	C₄×C₈	M₃₂	1	1	yes	characteristic
65	C₄×C₈	C₂²×C₈	1	1	yes	characteristic
66	C₃₂	—	2	4	no	—
67	C₃₂	C₄×C₈	1	4	yes	—
68	C₃₂	C₂²×C₈	1	4	yes	—
69	C₂²×C₄	—	2	1	no	—
70	C₂²×C₄	#346	1	1	yes	characteristic
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	4	2	no	—
73	C₂×C₈	C₈²	1	2	yes	—
74	C₂×C₈	#346	1	1	yes	characteristic
75	C₂×C₈	#346	1	1	yes	characteristic
76	C₂×C₈	#346	1	2	yes	—
77	C₂×C₈	C₂×C₄×C₈	1	1	yes	characteristic
78	C₂×C₈	C₂×M₃₂	1	1	yes	characteristic
79	C₄²	—	2	1	no	—
80	C₄²	#346	1	1	yes	characteristic
81	C₄²	C₂×M₃₂	1	1	yes	characteristic
82	C₁₆	—	4	2	no	—
83	C₁₆	C₈²	1	1	yes	characteristic
84	C₁₆	C₈²	1	1	yes	characteristic
85	C₁₆	#346	1	2	yes	—
86	C₁₆	C₂×C₄×C₈	1	2	yes	—
87	C₁₆	C₂×M₃₂	1	2	yes	—
88	C₂³	#1398	1	1	yes	characteristic
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	8	1	no	—
92	C₂×C₄	#1311	1	2	yes	—
93	C₂×C₄	#1398	1	1	yes	characteristic
94	C₂×C₄	#1398	1	2	yes	—
95	C₂×C₄	C₂×#346	1	1	yes	characteristic
96	C₈	—	2	1	no	—
97	C₈	—	2	1	no	—
98	C₈	—	8	2	no	—
99	C₈	#1311	1	2	yes	—
100	C₈	C₂×C₈²	1	1	yes	characteristic
101	C₈	C₂×#346	1	1	yes	characteristic
102	C₂²	—	4	1	no	—
103	C₂²	#7330	1	2	yes	—
104	C₂²	C₂×#1398	1	1	yes	characteristic
105	C₄	—	4	1	no	—
106	C₄	—	4	1	no	—
107	C₄	#7330	1	2	yes	—
108	C₄	C₂×#1311	1	1	yes	characteristic
109	C₄	C₂×#1398	1	1	yes	characteristic
110	C₂	—	2	1	no	—
111	C₂	—	2	2	no	—
112	C₂	C₂×#7330	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#7511

C₂

yes

maximal

#149496

C₂

yes

maximal, characteristic

#94506

C₂

yes

maximal, complemented

C₂×M₁₂₈

C₄

yes

—

C₄×M₆₄

C₄

yes

characteristic

C₂×#1397

C₂²

yes

characteristic

#11399

C₄

yes

characteristic

#7511

C₄

yes

—

#7511

C₂²

yes

—

#7330

C₄

yes

—

#7330

C₂²

yes

—

C₂×M₆₄

C₈

yes

—

C₂²×C₃₂

C₂×C₄

yes

—

C₄×M₃₂

C₈

yes

characteristic

C₂×C₄×C₁₆

C₂×C₄

yes

characteristic

#1450

C₈

yes

characteristic

M₁₂₈

C₈

yes

complemented

M₁₂₈

C₂×C₄

yes

—

C₂×C₆₄

C₈

yes

complemented

#1397

C₂×C₄

yes

—

#1397

C₂×C₄

yes

characteristic

#1397

C₂×C₄

yes

—

#1397

C₂³

yes

characteristic

C₄×C₃₂

C₈

yes

—

#1311

C₂×C₄

yes

—

C₈×C₁₆

C₈

yes

—

C₂×M₃₂

—

C₂²×C₁₆

C₄²

yes

characteristic

C₂²×C₁₆

C₂×C₈

yes

characteristic

#431

—

C₂×C₄×C₈

C₂×C₈

yes

characteristic

M₆₄

—

C₂×C₃₂

C₄²

yes

—

C₂×C₃₂

C₂×C₈

yes

—

C₂×C₃₂

C₂×C₈

yes

—

C₂×C₃₂

C₂×C₈

yes

—

C₂×C₃₂

C₂²×C₄

yes

—

#346

—

C₄×C₁₆

C₄²

yes

characteristic

C₄×C₁₆

C₂×C₈

yes

characteristic

C₄×C₁₆

C₂×C₈

yes

—

C₄×C₁₆

C₂×C₈

yes

characteristic

C₄×C₁₆

C₂²×C₄

yes

characteristic

#322

—

C₈²

C₂×C₈

yes

—

C₆₄

C₂×C₈

yes

—

C₂²×C₈

C₄×C₈

yes

characteristic

C₂²×C₈

M₃₂

yes

characteristic

C₂×C₄²

M₃₂

yes

characteristic

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₄×C₈

yes

characteristic

C₂×C₁₆

C₄×C₈

yes

—

C₂×C₁₆

C₄×C₈

yes

—

C₂×C₁₆

C₄×C₈

yes

characteristic

C₂×C₁₆

M₃₂

yes

—

C₂×C₁₆

C₂×C₄²

yes

characteristic

C₂×C₁₆

C₂²×C₈

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

yes

characteristic

C₄×C₈

M₃₂

yes

characteristic

C₄×C₈

M₃₂

yes

characteristic

C₄×C₈

C₂²×C₈

yes

characteristic

C₃₂

—

C₃₂

C₄×C₈

yes

—

C₃₂

C₂²×C₈

yes

—

C₂²×C₄

—

C₂²×C₄

#346

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₈²

yes

—

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

—

C₂×C₈

C₂×C₄×C₈

yes

characteristic

C₂×C₈

C₂×M₃₂

yes

characteristic

C₄²

—

C₄²

#346

yes

characteristic

C₄²

C₂×M₃₂

yes

characteristic

C₁₆

—

C₁₆

C₈²

yes

characteristic

C₁₆

C₈²

yes

characteristic

C₁₆

#346

yes

—

C₁₆

C₂×C₄×C₈

yes

—

C₁₆

C₂×M₃₂

yes

—

C₂³

#1398

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#1311

yes

—

C₂×C₄

#1398

yes

characteristic

C₂×C₄

#1398

yes

—

C₂×C₄

C₂×#346

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

#1311

yes

—

100

C₈

C₂×C₈²

yes

characteristic

101

C₈

C₂×#346

yes

characteristic

102

C₂²

—

103

C₂²

#7330

yes

—

104

C₂²

C₂×#1398

yes

characteristic

105

C₄

—

106

C₄

—

107

C₄

#7330

yes

—

108

C₄

C₂×#1311

yes

characteristic

109

C₄

C₂×#1398

yes

characteristic

110

C₂

—

111

C₂

—

112

C₂

C₂×#7330

yes

characteristic

	subgroup	G/H
center	C₁₆	C₈²
commutator	C₈	C₂×C₈²
Frattini, cosocle	#1397	C₂³
socle	C₂	C₂×#7330
prime radical	C₂³	#1398
LCS	G ▷ C₈ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ G
LECS	G ▷ #1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₁₆

C₈²

commutator

C₈

C₂×C₈²

C₂³

C₂

C₂³

G ▷ C₈ ▷ C₁

C₁ ◁ C₁₆ ◁ G

G ▷ #1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
3. #94506	C₂	semidirect
17. M₁₂₈	C₈	semidirect
19. C₂×C₆₄	C₈	semidirect

H ≅ G/N

type

3. #94506

C₂

semidirect

17. M₁₂₈

C₈

semidirect

19. C₂×C₆₄

C₈

semidirect

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

subgroup	index
C₂×#7511	2
#149496	2
#94506	2

subgroup

index

C₂×#7511

#149496

#94506

Subgroups — #72234379

Conjugacy classes of subgroups

Complements

Maximal subgroups