Subgroups — C₂×C₄×#15667732 #36378742

Order 22592.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#15667732	C₂	1	2	yes	maximal
2	C₄×#15667732	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×Dic₃₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#15667732	C₄	1	8	yes	direct product factor
5	C₂×#15667732	C₂²	1	4	yes	—
6	C₂²×Dic₃₅₃	C₄	1	1	yes	characteristic
7	C₂²×Dic₃₅₃	C₂²	1	1	yes	characteristic
8	C₄×Dic₃₅₃	C₄	1	2	yes	—
9	C₄×Dic₃₅₃	C₂²	1	2	yes	—
10	C₂²×C₄×C₃₅₃	C₄	1	1	yes	characteristic
11	#15667732	C₂×C₄	1	8	yes	direct product factor
12	C₂×Dic₃₅₃	C₈	1	4	yes	complemented
13	C₂×Dic₃₅₃	C₂×C₄	1	1	yes	characteristic
14	C₂×Dic₃₅₃	C₂×C₄	1	4	yes	—
15	C₂×Dic₃₅₃	C₂×C₄	1	2	yes	—
16	C₂×Dic₃₅₃	C₂³	1	1	yes	characteristic
17	C₂³×C₃₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₃₅₃	C₈	1	4	yes	complemented
19	C₂×C₄×C₃₅₃	C₂×C₄	1	2	yes	—
20	C₂²×C₃₅₃	C₄²	1	2	yes	—
21	C₂²×C₃₅₃	C₂×C₈	1	2	yes	—
22	C₂²×C₃₅₃	C₂×C₈	1	2	yes	—
23	C₂²×C₃₅₃	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₃₅₃	C₂×C₈	1	4	yes	complemented
25	Dic₃₅₃	C₄²	1	2	yes	—
26	Dic₃₅₃	C₂×C₈	1	4	yes	complemented
27	Dic₃₅₃	C₂²×C₄	1	2	yes	—
28	C₂×C₃₅₃	C₄×C₈	1	4	yes	complemented
29	C₂×C₃₅₃	C₂×C₄²	1	1	yes	characteristic
30	C₂×C₃₅₃	C₂²×C₈	1	1	yes	characteristic
31	C₂×C₃₅₃	C₂²×C₈	1	1	yes	characteristic
32	C₃₅₃	C₂×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
33	C₂×C₄×C₈	—	353	1	no	maximal, Sylow
34	C₂²×C₈	—	353	2	no	—
35	C₂×C₄²	—	353	1	no	—
36	C₄×C₈	—	353	4	no	—
37	C₂²×C₄	—	353	1	no	—
38	C₂²×C₄	—	353	1	no	—
39	C₂²×C₄	#13083443	1	1	yes	characteristic
40	C₂×C₈	—	353	4	no	—
41	C₂×C₈	—	353	8	no	—
42	C₄²	—	353	2	no	—
43	C₄²	—	353	2	no	—
44	C₂³	C₂×#13083443	1	1	yes	characteristic
45	C₂×C₄	—	353	4	no	—
46	C₂×C₄	—	353	4	no	—
47	C₂×C₄	—	353	1	no	—
48	C₂×C₄	—	353	2	no	—
49	C₂×C₄	—	353	1	no	—
50	C₂×C₄	#15667732	1	4	yes	direct product factor
51	C₂×C₄	C₂×#13083443	1	2	yes	—
52	C₈	—	353	8	no	—
53	C₂²	C₄×#13083443	1	2	yes	—
54	C₂²	C₂×#15667732	1	2	yes	—
55	C₂²	#19150745	1	2	yes	—
56	C₂²	C₂²×#13083443	1	1	yes	characteristic
57	C₄	—	353	2	no	—
58	C₄	—	353	2	no	—
59	C₄	—	353	4	no	—
60	C₄	C₂×#15667732	1	4	yes	direct product factor
61	C₂	C₄×#15667732	1	4	yes	direct product factor
62	C₂	C₂×C₄×#13083443	1	1	yes	characteristic
63	C₂	C₂²×#15667732	1	1	yes	characteristic
64	C₂	C₂×#19150745	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂²×C₄	#13083443
commutator, hypocenter	C₃₅₃	C₂×C₄×C₈
Frattini	C₂²	C₂²×#13083443
Fitting	C₂²×C₄×C₃₅₃	C₄
socle, prime radical	C₂³×C₃₅₃	C₂×C₄
cosocle	C₂×Dic₃₅₃	C₂³
LCS	G ▷ C₃₅₃
UCS	C₁ ◁ C₂²×C₄

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₄×#15667732	C₂	direct (N × H)
4. C₂×#15667732	C₄	direct (N × H)
11. #15667732	C₂×C₄	direct (N × H)
12. C₂×Dic₃₅₃	C₈	semidirect
18. C₂×C₄×C₃₅₃	C₈	semidirect
24. C₄×C₃₅₃	C₂×C₈	semidirect
26. Dic₃₅₃	C₂×C₈	semidirect
28. C₂×C₃₅₃	C₄×C₈	semidirect
32. C₃₅₃	C₂×C₄×C₈	semidirect
50. C₂×C₄	#15667732	direct (N × H)
60. C₄	C₂×#15667732	direct (N × H)
61. C₂	C₄×#15667732	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#15667732	2
C₄×#15667732	2
C₂×C₄×Dic₃₅₃	2
C₂×C₄×C₈	353

Subgroups — C2×C4×#15667732 #36378742

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#15667732 #36378742