Group #17321 — subgroups

Subgroups — C₂×#1453 #17321

Order 256.

433 conjugacy classes of subgroups; only the 303 normal subgroups are shown.

normal subgroups only

This group has 433 conjugacy classes of subgroups — too many to draw. Only its 303 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×#431	C₂	1	2	yes	maximal
2	C₂×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
3	#1453	C₂	1	8	yes	maximal, direct product factor
4	C₂×#322	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#322	C₂	1	2	yes	maximal, complemented
6	C₂×C₈²	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×M₁₆	C₄	1	2	yes	—
8	C₂²×C₄²	C₂²	1	1	yes	characteristic
9	#431	C₄	1	8	yes	—
10	#431	C₂²	1	8	yes	—
11	C₂×C₄⋊C₈	C₄	1	4	yes	—
12	C₂×C₂²⋊C₈	C₄	1	4	yes	—
13	C₄×M₁₆	C₄	1	4	yes	—
14	C₄×M₁₆	C₂²	1	4	yes	—
15	C₂×#97	C₄	1	2	yes	—
16	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈	C₂²	1	2	yes	—
19	C₂×C₄×C₈	C₂²	1	2	yes	—
20	#322	C₂²	1	4	yes	complemented
21	#322	C₂²	1	8	yes	complemented
22	C₈²	C₂²	1	4	yes	complemented
23	C₂³×C₄	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄	C₂×C₄	1	2	yes	—
25	C₂×M₁₆	—	2	8	no	—
26	C₂×M₁₆	C₂×C₄	1	8	yes	—
27	C₂²×C₈	C₂×C₄	1	2	yes	—
28	C₂²×C₈	C₂×C₄	1	4	yes	—
29	C₂²×C₈	C₂×C₄	1	4	yes	—
30	C₂²×C₈	C₂×C₄	1	2	yes	—
31	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²	C₂×C₄	1	2	yes	—
33	C₂×C₄²	C₂×C₄	1	2	yes	—
34	C₂×C₄²	C₂×C₄	1	4	yes	—
35	C₂×C₄²	C₂³	1	2	yes	—
36	C₂×C₄²	C₂³	1	1	yes	characteristic
37	C₄⋊C₈	C₂×C₄	1	16	yes	—
38	C₂²⋊C₈	C₂×C₄	1	16	yes	—
39	#97	C₂×C₄	1	8	yes	—
40	C₄×C₈	C₂×C₄	1	2	yes	—
41	C₄×C₈	C₂×C₄	1	2	yes	—
42	C₄×C₈	C₂×C₄	1	4	yes	—
43	C₄×C₈	C₂×C₄	1	4	yes	—
44	C₄×C₈	C₂³	1	2	yes	—
45	C₄×C₈	C₂³	1	4	yes	—
46	C₄×C₈	C₂³	1	2	yes	—
47	C₄×C₈	C₂³	1	4	yes	—
48	C₂⁴	C₄²	1	1	yes	characteristic
49	C₂²×C₄	—	2	4	no	—
50	C₂²×C₄	—	2	8	no	—
51	C₂²×C₄	C₄²	1	1	yes	characteristic
52	C₂²×C₄	C₄²	1	2	yes	—
53	C₂²×C₄	C₄²	1	2	yes	—
54	C₂²×C₄	C₄²	1	4	yes	—
55	C₂²×C₄	C₂²×C₄	1	4	yes	—
56	C₂²×C₄	C₂²×C₄	1	2	yes	—
57	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄	C₂²×C₄	1	2	yes	—
59	M₁₆	—	2	16	no	—
60	C₂×C₈	—	2	4	no	—
61	C₂×C₈	—	2	8	no	—
62	C₂×C₈	—	2	8	no	—
63	C₂×C₈	M₁₆	1	8	yes	—
64	C₂×C₈	C₂²×C₄	1	4	yes	—
65	C₂×C₈	C₂²×C₄	1	8	yes	—
66	C₂×C₈	C₂²×C₄	1	8	yes	—
67	C₂×C₈	C₂²×C₄	1	4	yes	—
68	C₄²	C₄²	1	2	yes	—
69	C₄²	C₄²	1	4	yes	—
70	C₄²	C₂²×C₄	1	2	yes	—
71	C₄²	C₂²×C₄	1	1	yes	characteristic
72	C₄²	C₂²×C₄	1	2	yes	—
73	C₄²	C₂²×C₄	1	4	yes	—
74	C₄²	C₂⁴	1	1	yes	characteristic
75	C₂³	—	2	2	no	—
76	C₂³	—	2	4	no	—
77	C₂³	C₂×C₄²	1	2	yes	—
78	C₂³	C₂×C₄²	1	1	yes	characteristic
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	4	no	—
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	2	8	no	—
83	C₂×C₄	—	2	4	no	—
84	C₂×C₄	—	2	8	no	—
85	C₂×C₄	C₂×C₄²	1	2	yes	—
86	C₂×C₄	C₂×C₄²	1	4	yes	—
87	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
88	C₂×C₄	C₂×C₄²	1	2	yes	—
89	C₂×C₄	C₂×M₁₆	1	4	yes	—
90	C₂×C₄	#131	1	8	yes	—
91	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
92	C₂×C₄	C₂³×C₄	1	2	yes	—
93	C₈	—	2	4	no	—
94	C₈	—	2	8	no	—
95	C₈	—	2	8	no	—
96	C₈	C₂×M₁₆	1	8	yes	—
97	C₂²	—	2	2	no	—
98	C₂²	—	2	4	no	—
99	C₂²	—	4	4	no	—
100	C₂²	C₄×M₁₆	1	2	yes	—
101	C₂²	#405	1	4	yes	—
102	C₂²	C₂²×C₄²	1	1	yes	characteristic
103	C₄	—	2	2	no	—
104	C₄	—	2	4	no	—
105	C₄	C₄×M₁₆	1	2	yes	—
106	C₄	#405	1	4	yes	—
107	C₄	C₂²×M₁₆	1	2	yes	—
108	C₄	C₂×#131	1	4	yes	—
109	C₂	—	4	2	no	—
110	C₂	#1453	1	4	yes	direct product factor
111	C₂	C₂×C₄×M₁₆	1	1	yes	characteristic
112	C₂	C₂×#405	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#431

C₂

yes

maximal

C₂×C₄×M₁₆

C₂

yes

maximal, characteristic

#1453

C₂

yes

maximal, direct product factor

C₂×#322

C₂

yes

maximal, characteristic, complemented

C₂×#322

C₂

yes

maximal, complemented

C₂×C₈²

C₂

yes

maximal, characteristic, complemented

C₂²×M₁₆

C₄

yes

—

C₂²×C₄²

C₂²

yes

characteristic

#431

C₄

yes

—

#431

C₂²

yes

—

C₂×C₄⋊C₈

C₄

yes

—

C₂×C₂²⋊C₈

C₄

yes

—

C₄×M₁₆

C₄

yes

—

C₄×M₁₆

C₂²

yes

—

C₂×#97

C₄

yes

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

C₂×C₄×C₈

C₂²

yes

—

#322

C₂²

yes

complemented

#322

C₂²

yes

complemented

C₈²

C₂²

yes

complemented

C₂³×C₄

C₂×C₄

yes

characteristic

C₂³×C₄

C₂×C₄

yes

—

C₂×M₁₆

—

C₂×M₁₆

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

—

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

—

C₂×C₄²

C₂×C₄

yes

—

C₂×C₄²

C₂×C₄

yes

—

C₂×C₄²

C₂³

yes

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

C₂×C₄

yes

—

C₂²⋊C₈

C₂×C₄

yes

—

#97

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

—

C₂⁴

C₄²

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄²

yes

characteristic

C₂²×C₄

C₄²

yes

—

C₂²×C₄

C₄²

yes

—

C₂²×C₄

C₄²

yes

—

C₂²×C₄

yes

—

C₂²×C₄

yes

—

C₂²×C₄

yes

characteristic

C₂²×C₄

yes

—

M₁₆

—

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

—

C₂×C₈

C₂²×C₄

yes

—

C₄²

yes

—

C₄²

yes

—

C₄²

C₂²×C₄

yes

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂²×C₄

yes

—

C₄²

C₂²×C₄

yes

—

C₄²

C₂⁴

yes

characteristic

C₂³

—

C₂³

—

C₂³

C₂×C₄²

yes

—

C₂³

C₂×C₄²

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄²

yes

—

C₂×C₄

C₂×C₄²

yes

—

C₂×C₄

C₂×C₄²

yes

characteristic

C₂×C₄

C₂×C₄²

yes

—

C₂×C₄

C₂×M₁₆

yes

—

C₂×C₄

#131

yes

—

C₂×C₄

C₂³×C₄

yes

characteristic

C₂×C₄

C₂³×C₄

yes

—

C₈

—

C₈

—

C₈

—

C₈

C₂×M₁₆

yes

—

C₂²

—

C₂²

—

C₂²

—

100

C₂²

C₄×M₁₆

yes

—

101

C₂²

#405

yes

—

102

C₂²

C₂²×C₄²

yes

characteristic

103

C₄

—

104

C₄

—

105

C₄

C₄×M₁₆

yes

—

106

C₄

#405

yes

—

107

C₄

C₂²×M₁₆

yes

—

108

C₄

C₂×#131

yes

—

109

C₂

—

110

C₂

#1453

yes

direct product factor

111

C₂

C₂×C₄×M₁₆

yes

characteristic

112

C₂

C₂×#405

yes

—

	subgroup	G/H
center	C₂×C₄²	C₂³
commutator	C₂²	C₂²×C₄²
Frattini, cosocle	C₄²	C₂⁴
socle	C₂³	C₂×C₄²
prime radical	C₂⁴	C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂×C₄²

C₂³

commutator

C₂²

C₂²×C₄²

Frattini, cosocle

C₄²

C₂⁴

socle

C₂³

C₂×C₄²

prime radical

C₂⁴

C₄²

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
3. #1453	C₂	direct (N × H)
4. C₂×#322	C₂	semidirect
5. C₂×#322	C₂	semidirect
6. C₂×C₈²	C₂	semidirect
20. #322	C₂²	semidirect
21. #322	C₂²	semidirect
22. C₈²	C₂²	semidirect
110. C₂	#1453	direct (N × H)

H ≅ G/N

type

3. #1453

C₂

direct (N × H)

4. C₂×#322

C₂

semidirect

5. C₂×#322

C₂

semidirect

6. C₂×C₈²

C₂

semidirect

20. #322

C₂²

semidirect

21. #322

C₂²

semidirect

22. C₈²

C₂²

semidirect

110. C₂

#1453

direct (N × H)

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

subgroup	index
C₂×#431	2
C₂×C₄×M₁₆	2
#1453	2
C₂×#322	2
C₂×C₈²	2

subgroup

index

C₂×#431

C₂×C₄×M₁₆

#1453

C₂×#322

C₂×C₈²

Subgroups — C₂×#1453 #17321

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#1453 #17321

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#1453 #17321