Group #36378862 — subgroups

Subgroups — C₂×#26015079 #36378862

Order 22592.

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	#26015079	C₂	1	8	yes	maximal, direct product factor
2	C₂×#19150747	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19150746	C₂	1	2	yes	maximal, complemented
4	C₂×#19150745	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×#15667732	C₂	1	2	yes	maximal, complemented
6	C₂×#19150736	C₂	1	1	yes	maximal, characteristic
7	#19150747	C₂²	1	4	yes	complemented
8	#19150746	C₂²	1	8	yes	complemented
9	#19150745	C₂²	1	4	yes	complemented
10	C₂×#15667732	C₂²	1	1	yes	characteristic
11	C₂×#15667732	C₂²	1	2	yes	—
12	C₂×#15667732	C₂²	1	1	yes	characteristic
13	C₂×#15667732	C₂²	1	8	yes	complemented
14	#19150736	C₄	1	4	yes	—
15	#19150736	C₂²	1	4	yes	—
16	C₂×Dic₇₀₆	C₄	1	1	yes	characteristic
17	C₂×C₂²⋊D₃₅₃	C₄	1	2	yes	—
18	C₂²×Dic₃₅₃	C₂²	1	2	yes	—
19	C₂×C₄×D₃₅₃	C₂²	1	1	yes	characteristic
20	C₂×D₄×C₃₅₃	C₄	1	1	yes	characteristic
21	#15667732	C₂³	1	2	yes	—
22	#15667732	C₂³	1	4	yes	—
23	#15667732	C₂³	1	2	yes	—
24	C₂²×D₃₅₃	C₂×C₄	1	1	yes	characteristic
25	Dic₇₀₆	C₂×C₄	1	4	yes	—
26	C₂²⋊D₃₅₃	C₂×C₄	1	8	yes	—
27	C₂×Dic₃₅₃	C₂×C₄	1	2	yes	—
28	C₂×Dic₃₅₃	C₂×C₄	1	4	yes	—
29	C₂×Dic₃₅₃	C₂³	1	1	yes	characteristic
30	C₂×Dic₃₅₃	C₂³	1	4	yes	—
31	C₄×D₃₅₃	C₂×C₄	1	2	yes	—
32	C₄×D₃₅₃	C₂³	1	2	yes	—
33	C₂³×C₃₅₃	C₂×C₄	1	2	yes	—
34	D₄×C₃₅₃	C₂×C₄	1	4	yes	—
35	C₂×C₄×C₃₅₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₃₅₃	—	2	4	no	—
37	C₂²×C₃₅₃	C₂²×C₄	1	4	yes	—
38	C₂²×C₃₅₃	C₂²×C₄	1	1	yes	characteristic
39	C₂×D₃₅₃	—	2	2	no	—
40	C₂×D₃₅₃	C₂²×C₄	1	2	yes	—
41	C₄×C₃₅₃	C₂²×C₄	1	2	yes	—
42	Dic₃₅₃	C₂²×C₄	1	1	yes	characteristic
43	Dic₃₅₃	C₂²×C₄	1	4	yes	—
44	Dic₃₅₃	C₂⁴	1	1	yes	characteristic
45	C₂×C₃₅₃	—	2	4	no	—
46	C₂×C₃₅₃	#131	1	2	yes	complemented
47	C₂×C₃₅₃	C₂³×C₄	1	1	yes	characteristic
48	D₃₅₃	—	2	2	no	—
49	C₃₅₃	C₂×#131	1	1	yes	characteristic, complemented, Sylow
50	C₂×#131	—	353	1	no	maximal, Sylow
51	C₂×#41	—	353	1	no	—
52	#131	—	353	8	no	—
53	C₂×M₁₆	—	353	2	no	—
54	C₂×M₁₆	—	353	1	no	—
55	C₂²×C₈	—	353	2	no	—
56	C₂²×C₈	—	353	1	no	—
57	#41	—	353	4	no	—
58	#41	—	353	4	no	—
59	C₂×Q₈	—	353	1	no	—
60	C₂×D₄	—	353	2	no	—
61	C₂×D₄	#13083443	1	1	yes	characteristic
62	C₂²×C₄	—	353	2	no	—
63	C₂²×C₄	—	353	1	no	—
64	M₁₆	—	353	8	no	—
65	M₁₆	—	353	4	no	—
66	C₂×C₈	—	353	1	no	—
67	C₂×C₈	—	353	2	no	—
68	C₂×C₈	—	353	1	no	—
69	C₂×C₈	—	353	8	no	—
70	C₂×C₈	—	353	4	no	—
71	C₂³	—	353	1	no	—
72	C₂³	C₂×#13083443	1	2	yes	—
73	Q₈	—	353	4	no	—
74	D₄	—	353	8	no	—
75	D₄	C₂×#13083443	1	4	yes	—
76	C₂×C₄	—	353	4	no	—
77	C₂×C₄	—	353	4	no	—
78	C₂×C₄	—	353	2	no	—
79	C₂×C₄	—	353	1	no	—
80	C₂×C₄	—	353	2	no	—
81	C₂×C₄	—	353	2	no	—
82	C₂×C₄	C₂×#13083443	1	1	yes	characteristic
83	C₈	—	353	2	no	—
84	C₈	—	353	4	no	—
85	C₈	—	353	2	no	—
86	C₂²	—	2	4	no	—
87	C₂²	—	353	2	no	—
88	C₂²	—	706	2	no	—
89	C₂²	C₂²×#13083443	1	4	yes	—
90	C₂²	C₂²×#13083443	1	1	yes	characteristic
91	C₄	—	353	1	no	—
92	C₄	—	353	1	no	—
93	C₄	—	353	4	no	—
94	C₄	C₂²×#13083443	1	2	yes	—
95	C₂	—	2	4	no	—
96	C₂	—	706	2	no	—
97	C₂	#26015079	1	2	yes	direct product factor
98	C₂	C₂³×#13083443	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#26015079

C₂

yes

maximal, direct product factor

C₂×#19150747

C₂

yes

maximal, characteristic, complemented

C₂×#19150746

C₂

yes

maximal, complemented

C₂×#19150745

C₂

yes

maximal, characteristic, complemented

C₂²×#15667732

C₂

yes

maximal, complemented

C₂×#19150736

C₂

yes

maximal, characteristic

#19150747

C₂²

yes

complemented

#19150746

C₂²

yes

complemented

#19150745

C₂²

yes

complemented

C₂×#15667732

C₂²

yes

characteristic

C₂×#15667732

C₂²

yes

—

C₂×#15667732

C₂²

yes

characteristic

C₂×#15667732

C₂²

yes

complemented

#19150736

C₄

yes

—

#19150736

C₂²

yes

—

C₂×Dic₇₀₆

C₄

yes

characteristic

C₂×C₂²⋊D₃₅₃

C₄

yes

—

C₂²×Dic₃₅₃

C₂²

yes

—

C₂×C₄×D₃₅₃

C₂²

yes

characteristic

C₂×D₄×C₃₅₃

C₄

yes

characteristic

#15667732

C₂³

yes

—

#15667732

C₂³

yes

—

#15667732

C₂³

yes

—

C₂²×D₃₅₃

C₂×C₄

yes

characteristic

Dic₇₀₆

C₂×C₄

yes

—

C₂²⋊D₃₅₃

C₂×C₄

yes

—

C₂×Dic₃₅₃

C₂×C₄

yes

—

C₂×Dic₃₅₃

C₂×C₄

yes

—

C₂×Dic₃₅₃

C₂³

yes

characteristic

C₂×Dic₃₅₃

C₂³

yes

—

C₄×D₃₅₃

C₂×C₄

yes

—

C₄×D₃₅₃

C₂³

yes

—

C₂³×C₃₅₃

C₂×C₄

yes

—

D₄×C₃₅₃

C₂×C₄

yes

—

C₂×C₄×C₃₅₃

C₂×C₄

yes

characteristic

C₂²×C₃₅₃

—

C₂²×C₃₅₃

C₂²×C₄

yes

—

C₂²×C₃₅₃

C₂²×C₄

yes

characteristic

C₂×D₃₅₃

—

C₂×D₃₅₃

C₂²×C₄

yes

—

C₄×C₃₅₃

C₂²×C₄

yes

—

Dic₃₅₃

C₂²×C₄

yes

characteristic

Dic₃₅₃

C₂²×C₄

yes

—

Dic₃₅₃

C₂⁴

yes

characteristic

C₂×C₃₅₃

—

C₂×C₃₅₃

#131

yes

complemented

C₂×C₃₅₃

C₂³×C₄

yes

characteristic

D₃₅₃

—

C₃₅₃

C₂×#131

yes

characteristic, complemented, Sylow

C₂×#131

—

353

maximal, Sylow

C₂×#41

—

353

—

#131

—

353

—

C₂×M₁₆

—

353

—

C₂×M₁₆

—

353

—

C₂²×C₈

—

353

—

C₂²×C₈

—

353

—

#41

—

353

—

#41

—

353

—

C₂×Q₈

—

353

—

C₂×D₄

—

353

—

C₂×D₄

#13083443

yes

characteristic

C₂²×C₄

—

353

—

C₂²×C₄

—

353

—

M₁₆

—

353

—

M₁₆

—

353

—

C₂×C₈

—

353

—

C₂×C₈

—

353

—

C₂×C₈

—

353

—

C₂×C₈

—

353

—

C₂×C₈

—

353

—

C₂³

—

353

—

C₂³

C₂×#13083443

yes

—

Q₈

—

353

—

D₄

—

353

—

D₄

C₂×#13083443

yes

—

C₂×C₄

—

353

—

C₂×C₄

—

353

—

C₂×C₄

—

353

—

C₂×C₄

—

353

—

C₂×C₄

—

353

—

C₂×C₄

—

353

—

C₂×C₄

C₂×#13083443

yes

characteristic

C₈

—

353

—

C₈

—

353

—

C₈

—

353

—

C₂²

—

C₂²

—

353

—

C₂²

—

706

—

C₂²

C₂²×#13083443

yes

—

C₂²

C₂²×#13083443

yes

characteristic

C₄

—

353

—

C₄

—

353

—

C₄

—

353

—

C₄

C₂²×#13083443

yes

—

C₂

—

C₂

—

706

—

C₂

#26015079

yes

direct product factor

C₂

C₂³×#13083443

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×#13083443
commutator	C₂×C₃₅₃	C₂³×C₄
Frattini	C₂	C₂³×#13083443
Fitting	C₂×D₄×C₃₅₃	C₄
socle	C₂²×C₃₅₃	C₂²×C₄
prime radical	C₂×#19150736	C₂
cosocle	Dic₃₅₃	C₂⁴
hypercenter	C₂×D₄	#13083443
hypocenter	C₃₅₃	C₂×#131
LCS	G ▷ C₂×C₃₅₃ ▷ C₃₅₃
UCS	C₁ ◁ C₂² ◁ C₂×D₄

subgroup

G/H

center

C₂²

C₂²×#13083443

commutator

C₂×C₃₅₃

C₂³×C₄

Frattini

C₂

C₂³×#13083443

Fitting

C₂×D₄×C₃₅₃

C₄

socle

C₂²×C₃₅₃

C₂²×C₄

prime radical

C₂×#19150736

C₂

cosocle

Dic₃₅₃

C₂⁴

C₂×D₄

C₃₅₃

G ▷ C₂×C₃₅₃ ▷ C₃₅₃

C₁ ◁ C₂² ◁ C₂×D₄

N	H ≅ G/N	type
1. #26015079	C₂	direct (N × H)
2. C₂×#19150747	C₂	semidirect
3. C₂×#19150746	C₂	semidirect
4. C₂×#19150745	C₂	semidirect
5. C₂²×#15667732	C₂	semidirect
7. #19150747	C₂²	semidirect
8. #19150746	C₂²	semidirect
9. #19150745	C₂²	semidirect
13. C₂×#15667732	C₂²	semidirect
46. C₂×C₃₅₃	#131	semidirect
49. C₃₅₃	C₂×#131	semidirect
97. C₂	#26015079	direct (N × H)

H ≅ G/N

type

1. #26015079

C₂

direct (N × H)

2. C₂×#19150747

C₂

semidirect

3. C₂×#19150746

C₂

semidirect

4. C₂×#19150745

C₂

semidirect

5. C₂²×#15667732

C₂

semidirect

7. #19150747

C₂²

semidirect

8. #19150746

C₂²

semidirect

9. #19150745

C₂²

semidirect

13. C₂×#15667732

C₂²

semidirect

46. C₂×C₃₅₃

#131

semidirect

49. C₃₅₃

C₂×#131

semidirect

97. C₂

#26015079

direct (N × H)

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

subgroup	index
#26015079	2
C₂×#19150747	2
C₂×#19150746	2
C₂×#19150745	2
C₂²×#15667732	2
C₂×#19150736	2
C₂×#131	353

subgroup

index

353

Subgroups — C₂×#26015079 #36378862

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#26015079 #36378862

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#26015079 #36378862