Subgroups — C₂×#4455 #69722

Order 384.

248 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₃⋊C₈	C₂	1	2	yes	maximal
3	#4455	C₂	1	4	yes	maximal, direct product factor
4	C₂×#322	—	3	1	no	maximal, Sylow
5	C₂²×C₃×C₈	C₄	1	2	yes	—
6	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
7	C₂²×C₃⋊C₈	C₄	1	4	yes	—
8	C₃×C₄×C₈	C₄	1	2	yes	—
9	C₃×C₄×C₈	C₂²	1	2	yes	—
10	C₄×C₃⋊C₈	C₄	1	4	yes	—
11	C₄×C₃⋊C₈	C₂²	1	4	yes	—
12	C₂×C₄×C₈	—	3	2	no	—
13	C₂×C₄×C₈	D₃	1	1	yes	characteristic
14	#322	—	3	4	no	—
15	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
16	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
17	C₂×C₃×C₈	C₈	1	8	yes	complemented
18	C₂×C₃×C₈	C₂×C₄	1	4	yes	—
19	C₃×C₄²	C₂×C₄	1	2	yes	—
20	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
21	C₃×C₄²	C₂³	1	1	yes	characteristic
22	C₂×C₃⋊C₈	—	2	8	no	—
23	C₂×C₃⋊C₈	C₂×C₄	1	8	yes	—
24	C₂²×C₈	—	3	4	no	—
25	C₂²×C₈	Dic₃	1	2	yes	—
26	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
27	C₄×C₈	—	3	4	no	—
28	C₄×C₈	—	3	4	no	—
29	C₄×C₈	Dic₃	1	2	yes	—
30	C₄×C₈	C₂×D₃	1	2	yes	—
31	C₂³×C₃	C₄²	1	1	yes	characteristic
32	C₂×C₃×C₄	C₄²	1	1	yes	characteristic
33	C₂×C₃×C₄	C₄²	1	2	yes	—
34	C₂×C₃×C₄	C₂×C₈	1	4	yes	—
35	C₂×C₃×C₄	M₁₆	1	8	yes	—
36	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
37	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
38	C₃×C₈	C₂×C₈	1	8	yes	complemented
39	C₃⋊C₈	—	2	8	no	—
40	C₂²×C₄	C₄×D₃	1	2	yes	—
41	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
42	C₂×C₈	—	3	8	no	—
43	C₂×C₈	—	6	8	no	—
44	C₂×C₈	C₃⋊C₈	1	8	yes	complemented
45	C₂×C₈	C₂×Dic₃	1	4	yes	—
46	C₄²	C₄×D₃	1	2	yes	—
47	C₄²	C₂×Dic₃	1	1	yes	characteristic
48	C₄²	C₂²×D₃	1	1	yes	characteristic
49	C₂²×C₃	C₄×C₈	1	2	yes	—
50	C₂²×C₃	#97	1	4	yes	—
51	C₂²×C₃	C₂×C₄²	1	1	yes	characteristic
52	C₃×C₄	C₄×C₈	1	2	yes	—
53	C₃×C₄	#97	1	4	yes	—
54	C₃×C₄	C₂²×C₈	1	2	yes	—
55	C₃×C₄	C₂×M₁₆	1	4	yes	—
56	C₂³	C₄×Dic₃	1	1	yes	characteristic
57	C₂×C₄	ΓL(1,𝔽₂₅)	1	8	yes	—
58	C₂×C₄	C₂×C₃⋊C₈	1	4	yes	—
59	C₂×C₄	C₄×Dic₃	1	1	yes	characteristic
60	C₂×C₄	C₄×Dic₃	1	2	yes	—
61	C₂×C₄	C₂×C₄×D₃	1	2	yes	—
62	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
63	C₈	—	6	8	no	—
64	C₈	C₂×C₃⋊C₈	1	8	yes	complemented
65	C₂×C₃	#322	1	4	yes	complemented
66	C₂×C₃	C₂×C₄×C₈	1	1	yes	characteristic
67	C₂×C₃	C₂×#97	1	2	yes	—
68	C₂²	C₄×C₃⋊C₈	1	2	yes	—
69	C₂²	#815	1	4	yes	—
70	C₂²	C₂×C₄×Dic₃	1	1	yes	characteristic
71	C₄	C₄×C₃⋊C₈	1	2	yes	—
72	C₄	#815	1	4	yes	—
73	C₄	C₂×ΓL(1,𝔽₂₅)	1	4	yes	—
74	C₄	C₂²×C₃⋊C₈	1	2	yes	—
75	C₃	C₂×#322	1	1	yes	characteristic, complemented, Sylow
76	C₂	#4455	1	4	yes	direct product factor
77	C₂	C₂×C₄×C₃⋊C₈	1	1	yes	characteristic
78	C₂	C₂×#815	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂×D₃
commutator	C₂×C₃	C₂×C₄×C₈
Frattini	C₄²	C₂²×D₃
Fitting	C₂×C₃×C₄×C₈	C₂
socle, prime radical	C₂³×C₃	C₄²
cosocle	C₃×C₄²	C₂³
hypercenter	C₂×C₄×C₈	D₃
hypocenter	C₃	C₂×#322
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #4455	C₂	direct (N × H)
17. C₂×C₃×C₈	C₈	semidirect
38. C₃×C₈	C₂×C₈	semidirect
44. C₂×C₈	C₃⋊C₈	semidirect
64. C₈	C₂×C₃⋊C₈	semidirect
65. C₂×C₃	#322	semidirect
75. C₃	C₂×#322	semidirect
76. C₂	#4455	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₈	2
C₂×C₄×C₃⋊C₈	2
#4455	2
C₂×#322	3

Subgroups — C2×#4455 #69722

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#4455 #69722