Group #33419 — subgroups

Subgroups — #33419

Order 256.

295 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 295 conjugacy classes of subgroups — too many to draw. Only its 199 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×#405	C₂	1	1	yes	maximal, characteristic
2	#2164	C₂	1	8	yes	maximal, complemented
3	#2150	C₂	1	2	yes	maximal, complemented
4	#2112	C₂	1	2	yes	maximal
5	C₄×M₃₂	C₂	1	2	yes	maximal, complemented
6	C₂²×M₁₆	C₄	1	1	yes	characteristic
7	C₂³×C₈	C₂²	1	1	yes	characteristic
8	C₂×#117	C₄	1	1	yes	characteristic
9	C₂×M₃₂	C₂²	1	4	yes	—
10	#405	C₄	1	4	yes	—
11	#405	C₂²	1	4	yes	—
12	C₂×#97	C₄	1	1	yes	characteristic
13	C₂×#97	C₄	1	1	yes	characteristic
14	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₄⋊C₁₆	C₂²	1	8	yes	complemented
17	C₂²⋊C₁₆	C₂²	1	8	yes	—
18	#346	C₂²	1	4	yes	complemented
19	C₄×C₁₆	C₂²	1	4	yes	complemented
20	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
21	C₂×M₁₆	—	2	4	no	—
22	C₂×M₁₆	C₂×C₄	1	4	yes	—
23	C₂²×C₈	—	2	4	no	—
24	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈	C₂×C₄	1	2	yes	—
26	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈	C₂³	1	2	yes	—
29	C₂²×C₈	C₂³	1	1	yes	characteristic
30	#117	C₈	1	4	yes	—
31	#117	C₂×C₄	1	4	yes	—
32	C₂×C₄⋊C₄	C₈	1	2	yes	—
33	C₂×C₂²⋊C₄	C₈	1	2	yes	—
34	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
36	M₃₂	—	2	8	no	—
37	C₂×C₁₆	C₂³	1	8	yes	—
38	#97	C₂×C₄	1	4	yes	—
39	#97	C₂×C₄	1	4	yes	—
40	C₄×C₈	C₂×C₄	1	2	yes	—
41	C₄×C₈	C₂×C₄	1	2	yes	—
42	C₄×C₈	C₂³	1	2	yes	—
43	C₄×C₈	C₂³	1	2	yes	—
44	C₂⁴	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄	—	2	2	no	—
46	C₂²×C₄	—	2	2	no	—
47	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
48	C₂²×C₄	C₂×C₈	1	2	yes	—
49	C₂²×C₄	C₂×C₈	1	2	yes	—
50	C₂²×C₄	C₂×C₈	1	2	yes	—
51	C₂²×C₄	C₂²×C₄	1	2	yes	—
52	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
53	M₁₆	—	2	8	no	—
54	C₂×C₈	—	2	2	no	—
55	C₂×C₈	—	2	2	no	—
56	C₂×C₈	—	2	4	no	—
57	C₂×C₈	—	2	4	no	—
58	C₂×C₈	M₁₆	1	4	yes	—
59	C₂×C₈	C₂²×C₄	1	2	yes	—
60	C₂×C₈	C₂²×C₄	1	2	yes	—
61	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈	C₂²×C₄	1	2	yes	—
63	C₂×C₈	#41	1	4	yes	—
64	C₂×C₈	C₂⁴	1	1	yes	characteristic
65	C₄⋊C₄	C₂×C₈	1	8	yes	—
66	C₂²⋊C₄	C₂×C₈	1	8	yes	—
67	C₄²	C₂×C₈	1	2	yes	—
68	C₄²	C₂×C₈	1	2	yes	—
69	C₄²	C₂²×C₄	1	2	yes	—
70	C₄²	C₂²×C₄	1	2	yes	—
71	C₁₆	—	2	8	no	—
72	C₂³	—	2	2	no	—
73	C₂³	—	4	2	no	—
74	C₂³	C₂²×C₈	1	2	yes	—
75	C₂³	C₂²×C₈	1	1	yes	characteristic
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	—	2	4	no	—
80	C₂×C₄	—	4	2	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	#117	1	2	yes	—
83	C₂×C₄	C₂²×C₈	1	4	yes	—
84	C₂×C₄	C₂²×C₈	1	4	yes	—
85	C₂×C₄	C₂²×C₈	1	2	yes	—
86	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄	C₂×M₁₆	1	2	yes	—
88	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
89	C₈	—	2	2	no	—
90	C₈	—	2	2	no	—
91	C₈	#117	1	2	yes	—
92	C₈	C₂×M₁₆	1	4	yes	—
93	C₈	C₂×#41	1	2	yes	—
94	C₂²	—	2	2	no	—
95	C₂²	—	2	4	no	—
96	C₂²	—	2	2	no	—
97	C₂²	—	4	4	no	—
98	C₂²	#431	1	2	yes	—
99	C₂²	C₂³×C₈	1	1	yes	characteristic
100	C₄	—	2	4	no	—
101	C₄	—	2	4	no	—
102	C₄	—	2	2	no	—
103	C₄	#431	1	2	yes	—
104	C₄	C₂×#117	1	1	yes	characteristic
105	C₄	C₂²×M₁₆	1	1	yes	characteristic
106	C₂	—	2	2	no	—
107	C₂	—	4	2	no	—
108	C₂	C₂×#431	1	1	yes	characteristic
109	C₂	#3407	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#405

C₂

yes

maximal, characteristic

#2164

C₂

yes

maximal, complemented

#2150

C₂

yes

maximal, complemented

#2112

C₂

yes

maximal

C₄×M₃₂

C₂

yes

maximal, complemented

C₂²×M₁₆

C₄

yes

characteristic

C₂³×C₈

C₂²

yes

characteristic

C₂×#117

C₄

yes

characteristic

C₂×M₃₂

C₂²

yes

—

#405

C₄

yes

—

#405

C₂²

yes

—

C₂×#97

C₄

yes

characteristic

C₂×#97

C₄

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₄⋊C₁₆

C₂²

yes

complemented

C₂²⋊C₁₆

C₂²

yes

—

#346

C₂²

yes

complemented

C₄×C₁₆

C₂²

yes

complemented

C₂³×C₄

C₂×C₄

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×C₄

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂³

yes

—

C₂²×C₈

C₂³

yes

characteristic

#117

C₈

yes

—

#117

C₂×C₄

yes

—

C₂×C₄⋊C₄

C₈

yes

—

C₂×C₂²⋊C₄

C₈

yes

—

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

characteristic

M₃₂

—

C₂×C₁₆

C₂³

yes

—

#97

C₂×C₄

yes

—

#97

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

—

C₂⁴

C₂×C₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₈

yes

characteristic

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

C₂×C₈

yes

—

C₂²×C₄

yes

—

C₂²×C₄

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

M₁₆

yes

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

#41

yes

—

C₂×C₈

C₂⁴

yes

characteristic

C₄⋊C₄

C₂×C₈

yes

—

C₂²⋊C₄

C₂×C₈

yes

—

C₄²

C₂×C₈

yes

—

C₄²

C₂×C₈

yes

—

C₄²

C₂²×C₄

yes

—

C₄²

C₂²×C₄

yes

—

C₁₆

—

C₂³

—

C₂³

—

C₂³

C₂²×C₈

yes

—

C₂³

C₂²×C₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#117

yes

—

C₂×C₄

C₂²×C₈

yes

—

C₂×C₄

C₂²×C₈

yes

—

C₂×C₄

C₂²×C₈

yes

—

C₂×C₄

C₂²×C₈

yes

characteristic

C₂×C₄

C₂×M₁₆

yes

—

C₂×C₄

C₂³×C₄

yes

characteristic

C₈

—

C₈

—

C₈

#117

yes

—

C₈

C₂×M₁₆

yes

—

C₈

C₂×#41

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#431

yes

—

C₂²

C₂³×C₈

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

#431

yes

—

104

C₄

C₂×#117

yes

characteristic

105

C₄

C₂²×M₁₆

yes

characteristic

106

C₂

—

107

C₂

—

108

C₂

C₂×#431

yes

characteristic

109

C₂

#3407

yes

—

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂⁴
commutator, socle	C₂²	C₂³×C₈
prime radical	C₂⁴	C₂×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, Frattini, cosocle

C₂×C₈

C₂⁴

commutator, socle

C₂²

C₂³×C₈

prime radical

C₂⁴

C₂×C₈

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₈ ◁ G

LECS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. #2164	C₂	semidirect
3. #2150	C₂	semidirect
5. C₄×M₃₂	C₂	semidirect
16. C₄⋊C₁₆	C₂²	semidirect
18. #346	C₂²	semidirect
19. C₄×C₁₆	C₂²	semidirect

H ≅ G/N

type

2. #2164

C₂

semidirect

3. #2150

C₂

semidirect

5. C₄×M₃₂

C₂

semidirect

16. C₄⋊C₁₆

C₂²

semidirect

18. #346

C₂²

semidirect

19. C₄×C₁₆

C₂²

semidirect

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

subgroup	index
C₂×#405	2
#2164	2
#2150	2
#2112	2
C₄×M₃₂	2

subgroup

index

C₄×M₃₂

Subgroups — #33419

Conjugacy classes of subgroups

Complements

Maximal subgroups