Subgroups — C₂³×#217 #87300

Order 384.

4632 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁶×C₃	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×Dic₃	C₂	1	2	yes	maximal, complemented
3	C₂²×#217	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₂²⋊C₄	—	3	1	no	maximal, Sylow
5	C₂⁵×C₃	—	2	16	no	—
6	C₂⁵×C₃	C₄	1	16	yes	complemented
7	C₂⁵×C₃	C₂²	1	14	yes	—
8	C₂⁵×C₃	C₂²	1	1	yes	characteristic
9	C₂³×Dic₃	—	2	16	no	—
10	C₂³×Dic₃	C₂²	1	28	yes	complemented
11	C₂×#217	C₂²	1	112	yes	direct product factor
12	C₂⁶	D₃	1	1	yes	characteristic
13	C₂⁴×C₄	—	3	2	no	—
14	C₂²×C₂²⋊C₄	—	3	28	no	—
15	C₂⁴×C₃	—	2	128	no	—
16	C₂⁴×C₃	—	2	112	no	—
17	C₂⁴×C₃	C₂×C₄	1	8	yes	—
18	C₂⁴×C₃	C₂×C₄	1	112	yes	complemented
19	C₂⁴×C₃	D₄	1	16	yes	—
20	C₂⁴×C₃	C₂³	1	28	yes	—
21	C₂⁴×C₃	C₂³	1	7	yes	—
22	C₂²×Dic₃	—	2	112	no	—
23	C₂²×Dic₃	C₂³	1	56	yes	complemented
24	#217	C₂³	1	64	yes	direct product factor
25	C₂⁵	—	2	16	no	—
26	C₂⁵	Dic₃	1	16	yes	complemented
27	C₂⁵	C₂×D₃	1	14	yes	—
28	C₂⁵	C₂×D₃	1	1	yes	characteristic
29	C₂³×C₄	—	3	28	no	—
30	C₂³×C₄	—	6	16	no	—
31	C₂×C₂²⋊C₄	—	3	112	no	—
32	C₂³×C₃	—	2	448	no	—
33	C₂³×C₃	—	2	112	no	—
34	C₂³×C₃	C₂²⋊C₄	1	64	yes	complemented
35	C₂³×C₃	C₂²×C₄	1	28	yes	—
36	C₂³×C₃	C₂²×C₄	1	112	yes	complemented
37	C₂³×C₃	C₂×D₄	1	56	yes	—
38	C₂³×C₃	C₂⁴	1	8	yes	—
39	C₂³×C₃	C₂⁴	1	7	yes	—
40	C₂×Dic₃	—	2	112	no	—
41	C₂×Dic₃	C₂⁴	1	16	yes	complemented
42	C₂⁴	—	2	128	no	—
43	C₂⁴	—	2	112	no	—
44	C₂⁴	C₂×Dic₃	1	8	yes	—
45	C₂⁴	C₂×Dic₃	1	112	yes	complemented
46	C₂⁴	#67	1	16	yes	—
47	C₂⁴	C₂²×D₃	1	28	yes	—
48	C₂⁴	C₂²×D₃	1	7	yes	—
49	C₂²×C₄	—	3	56	no	—
50	C₂²×C₄	—	6	112	no	—
51	C₂²⋊C₄	—	3	64	no	—
52	C₂²×C₃	—	2	224	no	—
53	C₂²×C₃	—	2	16	no	—
54	C₂²×C₃	C₂×C₂²⋊C₄	1	112	yes	complemented
55	C₂²×C₃	C₂³×C₄	1	14	yes	—
56	C₂²×C₃	C₂³×C₄	1	16	yes	complemented
57	C₂²×C₃	C₂²×D₄	1	28	yes	—
58	C₂²×C₃	C₂⁵	1	1	yes	characteristic
59	Dic₃	—	2	16	no	—
60	C₂³	—	2	448	no	—
61	C₂³	—	2	112	no	—
62	C₂³	#217	1	64	yes	direct product factor
63	C₂³	C₂²×Dic₃	1	28	yes	—
64	C₂³	C₂²×Dic₃	1	112	yes	complemented
65	C₂³	C₂×#67	1	56	yes	—
66	C₂³	C₂³×D₃	1	8	yes	—
67	C₂³	C₂³×D₃	1	7	yes	—
68	C₂×C₄	—	3	16	no	—
69	C₂×C₄	—	6	112	no	—
70	C₂×C₃	—	2	16	no	—
71	C₂×C₃	C₂²×C₂²⋊C₄	1	28	yes	complemented
72	C₂×C₃	C₂⁴×C₄	1	1	yes	characteristic
73	C₂×C₃	C₂³×D₄	1	2	yes	—
74	C₂²	—	2	16	no	—
75	C₂²	—	2	224	no	—
76	C₂²	C₂×#217	1	112	yes	direct product factor
77	C₂²	C₂³×Dic₃	1	14	yes	—
78	C₂²	C₂³×Dic₃	1	16	yes	complemented
79	C₂²	C₂²×#67	1	28	yes	—
80	C₂²	C₂⁴×D₃	1	1	yes	characteristic
81	C₄	—	6	16	no	—
82	C₃	C₂³×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
83	C₂	—	2	16	no	—
84	C₂	C₂²×#217	1	28	yes	direct product factor
85	C₂	C₂⁴×Dic₃	1	1	yes	characteristic
86	C₂	C₂³×#67	1	2	yes	—

	subgroup	G/H
center	C₂⁵	C₂×D₃
commutator	C₂×C₃	C₂⁴×C₄
Frattini	C₂²	C₂⁴×D₃
Fitting, prime radical	C₂⁶×C₃	C₂
socle	C₂⁵×C₃	C₂²
cosocle	C₂²×C₃	C₂⁵
hypercenter	C₂⁶	D₃
hypocenter	C₃	C₂³×C₂²⋊C₄
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂⁵ ◁ C₂⁶

Complements

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

N	H ≅ G/N	type
2. C₂⁴×Dic₃	C₂	semidirect
3. C₂²×#217	C₂	direct (N × H)
6. C₂⁵×C₃	C₄	semidirect
10. C₂³×Dic₃	C₂²	semidirect
11. C₂×#217	C₂²	direct (N × H)
18. C₂⁴×C₃	C₂×C₄	semidirect
23. C₂²×Dic₃	C₂³	semidirect
24. #217	C₂³	direct (N × H)
26. C₂⁵	Dic₃	semidirect
34. C₂³×C₃	C₂²⋊C₄	semidirect
36. C₂³×C₃	C₂²×C₄	semidirect
41. C₂×Dic₃	C₂⁴	semidirect
45. C₂⁴	C₂×Dic₃	semidirect
54. C₂²×C₃	C₂×C₂²⋊C₄	semidirect
56. C₂²×C₃	C₂³×C₄	semidirect
62. C₂³	#217	direct (N × H)
64. C₂³	C₂²×Dic₃	semidirect
71. C₂×C₃	C₂²×C₂²⋊C₄	semidirect
76. C₂²	C₂×#217	direct (N × H)
78. C₂²	C₂³×Dic₃	semidirect
82. C₃	C₂³×C₂²⋊C₄	semidirect
84. C₂	C₂²×#217	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁶×C₃	2
C₂⁴×Dic₃	2
C₂²×#217	2
C₂³×C₂²⋊C₄	3

Subgroups — C23×#217 #87300

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#217 #87300