Group #45688507 — subgroups

Subgroups — C₂×#30659795 #45688507

Order 33344.

266 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 140 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	#30659795	C₂	1	8	yes	maximal, direct product factor
2	C₂×#22552374	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#22552373	C₂	1	2	yes	maximal, complemented
4	C₂×#22552372	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#16861967	C₂	1	2	yes	maximal, complemented
6	C₂×#22552363	C₂	1	1	yes	maximal, characteristic
7	#22552374	C₂²	1	4	yes	complemented
8	#22552373	C₂²	1	8	yes	complemented
9	#22552372	C₂²	1	4	yes	complemented
10	C₂×#16861967	C₂²	1	1	yes	characteristic
11	C₂×#16861967	C₂²	1	2	yes	—
12	C₂×#16861967	C₂²	1	1	yes	characteristic
13	C₂×#16861967	C₂²	1	8	yes	complemented
14	#22552363	C₄	1	4	yes	—
15	#22552363	C₂²	1	4	yes	—
16	C₂×Dic₁₀₄₂	C₄	1	1	yes	characteristic
17	C₂×C₂²⋊D₅₂₁	C₄	1	2	yes	—
18	C₂²×Dic₅₂₁	C₂²	1	2	yes	—
19	C₂×C₄×D₅₂₁	C₂²	1	1	yes	characteristic
20	C₂×D₄×C₅₂₁	C₄	1	1	yes	characteristic
21	#16861967	C₂³	1	2	yes	—
22	#16861967	C₂³	1	4	yes	—
23	#16861967	C₂³	1	2	yes	—
24	C₂²×D₅₂₁	C₂×C₄	1	1	yes	characteristic
25	Dic₁₀₄₂	C₂×C₄	1	4	yes	—
26	C₂²⋊D₅₂₁	C₂×C₄	1	8	yes	—
27	C₂×Dic₅₂₁	C₂×C₄	1	2	yes	—
28	C₂×Dic₅₂₁	C₂×C₄	1	4	yes	—
29	C₂×Dic₅₂₁	C₂³	1	1	yes	characteristic
30	C₂×Dic₅₂₁	C₂³	1	4	yes	—
31	C₄×D₅₂₁	C₂×C₄	1	2	yes	—
32	C₄×D₅₂₁	C₂³	1	2	yes	—
33	C₂³×C₅₂₁	C₂×C₄	1	2	yes	—
34	D₄×C₅₂₁	C₂×C₄	1	4	yes	—
35	C₂×C₄×C₅₂₁	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₅₂₁	—	2	4	no	—
37	C₂²×C₅₂₁	C₂²×C₄	1	4	yes	—
38	C₂²×C₅₂₁	C₂²×C₄	1	1	yes	characteristic
39	C₂×D₅₂₁	—	2	2	no	—
40	C₂×D₅₂₁	C₂²×C₄	1	2	yes	—
41	C₄×C₅₂₁	C₂²×C₄	1	2	yes	—
42	Dic₅₂₁	C₂²×C₄	1	1	yes	characteristic
43	Dic₅₂₁	C₂²×C₄	1	4	yes	—
44	Dic₅₂₁	C₂⁴	1	1	yes	characteristic
45	C₂×C₅₂₁	—	2	4	no	—
46	C₂×C₅₂₁	#131	1	2	yes	complemented
47	C₂×C₅₂₁	C₂³×C₄	1	1	yes	characteristic
48	D₅₂₁	—	2	2	no	—
49	C₅₂₁	C₂×#131	1	1	yes	characteristic, complemented, Sylow
50	C₂×#131	—	521	1	no	maximal, Sylow
51	C₂×#41	—	521	1	no	—
52	#131	—	521	8	no	—
53	C₂×M₁₆	—	521	2	no	—
54	C₂×M₁₆	—	521	1	no	—
55	C₂²×C₈	—	521	2	no	—
56	C₂²×C₈	—	521	1	no	—
57	#41	—	521	4	no	—
58	#41	—	521	4	no	—
59	C₂×Q₈	—	521	1	no	—
60	C₂×D₄	—	521	2	no	—
61	C₂×D₄	#14523438	1	1	yes	characteristic
62	C₂²×C₄	—	521	2	no	—
63	C₂²×C₄	—	521	1	no	—
64	M₁₆	—	521	8	no	—
65	M₁₆	—	521	4	no	—
66	C₂×C₈	—	521	1	no	—
67	C₂×C₈	—	521	2	no	—
68	C₂×C₈	—	521	1	no	—
69	C₂×C₈	—	521	8	no	—
70	C₂×C₈	—	521	4	no	—
71	C₂³	—	521	1	no	—
72	C₂³	C₂×#14523438	1	2	yes	—
73	Q₈	—	521	4	no	—
74	D₄	—	521	8	no	—
75	D₄	C₂×#14523438	1	4	yes	—
76	C₂×C₄	—	521	4	no	—
77	C₂×C₄	—	521	4	no	—
78	C₂×C₄	—	521	2	no	—
79	C₂×C₄	—	521	1	no	—
80	C₂×C₄	—	521	2	no	—
81	C₂×C₄	—	521	2	no	—
82	C₂×C₄	C₂×#14523438	1	1	yes	characteristic
83	C₈	—	521	2	no	—
84	C₈	—	521	4	no	—
85	C₈	—	521	2	no	—
86	C₂²	—	2	4	no	—
87	C₂²	—	521	2	no	—
88	C₂²	—	1042	2	no	—
89	C₂²	C₂²×#14523438	1	4	yes	—
90	C₂²	C₂²×#14523438	1	1	yes	characteristic
91	C₄	—	521	1	no	—
92	C₄	—	521	1	no	—
93	C₄	—	521	4	no	—
94	C₄	C₂²×#14523438	1	2	yes	—
95	C₂	—	2	4	no	—
96	C₂	—	1042	2	no	—
97	C₂	#30659795	1	2	yes	direct product factor
98	C₂	C₂³×#14523438	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#30659795

C₂

yes

maximal, direct product factor

C₂×#22552374

C₂

yes

maximal, characteristic, complemented

C₂×#22552373

C₂

yes

maximal, complemented

C₂×#22552372

C₂

yes

maximal, characteristic, complemented

C₂²×#16861967

C₂

yes

maximal, complemented

C₂×#22552363

C₂

yes

maximal, characteristic

#22552374

C₂²

yes

complemented

#22552373

C₂²

yes

complemented

#22552372

C₂²

yes

complemented

C₂×#16861967

C₂²

yes

characteristic

C₂×#16861967

C₂²

yes

—

C₂×#16861967

C₂²

yes

characteristic

C₂×#16861967

C₂²

yes

complemented

#22552363

C₄

yes

—

#22552363

C₂²

yes

—

C₂×Dic₁₀₄₂

C₄

yes

characteristic

C₂×C₂²⋊D₅₂₁

C₄

yes

—

C₂²×Dic₅₂₁

C₂²

yes

—

C₂×C₄×D₅₂₁

C₂²

yes

characteristic

C₂×D₄×C₅₂₁

C₄

yes

characteristic

#16861967

C₂³

yes

—

#16861967

C₂³

yes

—

#16861967

C₂³

yes

—

C₂²×D₅₂₁

C₂×C₄

yes

characteristic

Dic₁₀₄₂

C₂×C₄

yes

—

C₂²⋊D₅₂₁

C₂×C₄

yes

—

C₂×Dic₅₂₁

C₂×C₄

yes

—

C₂×Dic₅₂₁

C₂×C₄

yes

—

C₂×Dic₅₂₁

C₂³

yes

characteristic

C₂×Dic₅₂₁

C₂³

yes

—

C₄×D₅₂₁

C₂×C₄

yes

—

C₄×D₅₂₁

C₂³

yes

—

C₂³×C₅₂₁

C₂×C₄

yes

—

D₄×C₅₂₁

C₂×C₄

yes

—

C₂×C₄×C₅₂₁

C₂×C₄

yes

characteristic

C₂²×C₅₂₁

—

C₂²×C₅₂₁

C₂²×C₄

yes

—

C₂²×C₅₂₁

C₂²×C₄

yes

characteristic

C₂×D₅₂₁

—

C₂×D₅₂₁

C₂²×C₄

yes

—

C₄×C₅₂₁

C₂²×C₄

yes

—

Dic₅₂₁

C₂²×C₄

yes

characteristic

Dic₅₂₁

C₂²×C₄

yes

—

Dic₅₂₁

C₂⁴

yes

characteristic

C₂×C₅₂₁

—

C₂×C₅₂₁

#131

yes

complemented

C₂×C₅₂₁

C₂³×C₄

yes

characteristic

D₅₂₁

—

C₅₂₁

C₂×#131

yes

characteristic, complemented, Sylow

C₂×#131

—

521

maximal, Sylow

C₂×#41

—

521

—

#131

—

521

—

C₂×M₁₆

—

521

—

C₂×M₁₆

—

521

—

C₂²×C₈

—

521

—

C₂²×C₈

—

521

—

#41

—

521

—

#41

—

521

—

C₂×Q₈

—

521

—

C₂×D₄

—

521

—

C₂×D₄

#14523438

yes

characteristic

C₂²×C₄

—

521

—

C₂²×C₄

—

521

—

M₁₆

—

521

—

M₁₆

—

521

—

C₂×C₈

—

521

—

C₂×C₈

—

521

—

C₂×C₈

—

521

—

C₂×C₈

—

521

—

C₂×C₈

—

521

—

C₂³

—

521

—

C₂³

C₂×#14523438

yes

—

Q₈

—

521

—

D₄

—

521

—

D₄

C₂×#14523438

yes

—

C₂×C₄

—

521

—

C₂×C₄

—

521

—

C₂×C₄

—

521

—

C₂×C₄

—

521

—

C₂×C₄

—

521

—

C₂×C₄

—

521

—

C₂×C₄

C₂×#14523438

yes

characteristic

C₈

—

521

—

C₈

—

521

—

C₈

—

521

—

C₂²

—

C₂²

—

521

—

C₂²

—

1042

—

C₂²

C₂²×#14523438

yes

—

C₂²

C₂²×#14523438

yes

characteristic

C₄

—

521

—

C₄

—

521

—

C₄

—

521

—

C₄

C₂²×#14523438

yes

—

C₂

—

C₂

—

1042

—

C₂

#30659795

yes

direct product factor

C₂

C₂³×#14523438

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×#14523438
commutator	C₂×C₅₂₁	C₂³×C₄
Frattini	C₂	C₂³×#14523438
Fitting	C₂×D₄×C₅₂₁	C₄
socle	C₂²×C₅₂₁	C₂²×C₄
prime radical	C₂×#22552363	C₂
cosocle	Dic₅₂₁	C₂⁴
hypercenter	C₂×D₄	#14523438
hypocenter	C₅₂₁	C₂×#131
LCS	G ▷ C₂×C₅₂₁ ▷ C₅₂₁
UCS	C₁ ◁ C₂² ◁ C₂×D₄

subgroup

G/H

center

C₂²

C₂²×#14523438

commutator

C₂×C₅₂₁

C₂³×C₄

Frattini

C₂

C₂³×#14523438

Fitting

C₂×D₄×C₅₂₁

C₄

socle

C₂²×C₅₂₁

C₂²×C₄

prime radical

C₂×#22552363

C₂

cosocle

Dic₅₂₁

C₂⁴

C₂×D₄

C₅₂₁

G ▷ C₂×C₅₂₁ ▷ C₅₂₁

C₁ ◁ C₂² ◁ C₂×D₄

N	H ≅ G/N	type
1. #30659795	C₂	direct (N × H)
2. C₂×#22552374	C₂	semidirect
3. C₂×#22552373	C₂	semidirect
4. C₂×#22552372	C₂	semidirect
5. C₂²×#16861967	C₂	semidirect
7. #22552374	C₂²	semidirect
8. #22552373	C₂²	semidirect
9. #22552372	C₂²	semidirect
13. C₂×#16861967	C₂²	semidirect
46. C₂×C₅₂₁	#131	semidirect
49. C₅₂₁	C₂×#131	semidirect
97. C₂	#30659795	direct (N × H)

H ≅ G/N

type

1. #30659795

C₂

direct (N × H)

2. C₂×#22552374

C₂

semidirect

3. C₂×#22552373

C₂

semidirect

4. C₂×#22552372

C₂

semidirect

5. C₂²×#16861967

C₂

semidirect

7. #22552374

C₂²

semidirect

8. #22552373

C₂²

semidirect

9. #22552372

C₂²

semidirect

13. C₂×#16861967

C₂²

semidirect

46. C₂×C₅₂₁

#131

semidirect

49. C₅₂₁

C₂×#131

semidirect

97. C₂

#30659795

direct (N × H)

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

subgroup	index
#30659795	2
C₂×#22552374	2
C₂×#22552373	2
C₂×#22552372	2
C₂²×#16861967	2
C₂×#22552363	2
C₂×#131	521

subgroup

index

521

Subgroups — C₂×#30659795 #45688507

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#30659795 #45688507

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#30659795 #45688507