Subgroups — C₂³×#14523497 #30680663

Order 16768.

1720 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523497	C₂	1	28	yes	maximal, direct product factor
2	C₂³×C₁₃₁⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₂⁴×C₄×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#14523497	C₂²	1	112	yes	direct product factor
5	C₂²×C₁₃₁⋊C₈	C₂²	1	28	yes	complemented
6	C₂⁵×C₁₃₁	C₄	1	1	yes	characteristic
7	C₂³×C₄×C₁₃₁	C₄	1	14	yes	—
8	C₂³×C₄×C₁₃₁	C₄	1	1	yes	characteristic
9	C₂³×C₄×C₁₃₁	C₂²	1	14	yes	—
10	C₂³×C₄×C₁₃₁	C₂²	1	1	yes	characteristic
11	#14523497	C₂³	1	64	yes	direct product factor
12	C₂×C₁₃₁⋊C₈	C₂³	1	56	yes	complemented
13	C₂⁴×C₁₃₁	—	2	8	no	—
14	C₂⁴×C₁₃₁	C₂×C₄	1	14	yes	—
15	C₂⁴×C₁₃₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₁₃₁	C₂×C₄	1	84	yes	—
17	C₂²×C₄×C₁₃₁	C₂×C₄	1	14	yes	—
18	C₂²×C₄×C₁₃₁	C₂×C₄	1	7	yes	—
19	C₂²×C₄×C₁₃₁	C₂³	1	7	yes	—
20	C₂²×C₄×C₁₃₁	C₂³	1	28	yes	—
21	C₁₃₁⋊C₈	C₂⁴	1	16	yes	complemented
22	C₂³×C₁₃₁	—	2	56	no	—
23	C₂³×C₁₃₁	M₁₆	1	8	yes	complemented
24	C₂³×C₁₃₁	C₂²×C₄	1	28	yes	—
25	C₂³×C₁₃₁	C₂²×C₄	1	7	yes	—
26	C₂×C₄×C₁₃₁	C₂²×C₄	1	56	yes	—
27	C₂×C₄×C₁₃₁	C₂²×C₄	1	21	yes	—
28	C₂×C₄×C₁₃₁	C₂²×C₄	1	28	yes	—
29	C₂×C₄×C₁₃₁	C₂⁴	1	7	yes	—
30	C₂×C₄×C₁₃₁	C₂⁴	1	8	yes	—
31	C₂²×C₁₃₁	—	2	56	no	—
32	C₂²×C₁₃₁	C₂×M₁₆	1	28	yes	complemented
33	C₂²×C₁₃₁	C₂³×C₄	1	8	yes	—
34	C₂²×C₁₃₁	C₂³×C₄	1	7	yes	—
35	C₄×C₁₃₁	C₂³×C₄	1	7	yes	—
36	C₄×C₁₃₁	C₂³×C₄	1	8	yes	—
37	C₄×C₁₃₁	C₂⁵	1	1	yes	characteristic
38	C₂×C₁₃₁	—	2	8	no	—
39	C₂×C₁₃₁	C₂²×M₁₆	1	14	yes	complemented
40	C₂×C₁₃₁	C₂⁴×C₄	1	1	yes	characteristic
41	C₁₃₁	C₂³×M₁₆	1	1	yes	characteristic, complemented, Sylow
42	C₂³×M₁₆	—	131	1	no	maximal, Sylow
43	C₂⁴×C₄	D₁₃₁	1	1	yes	characteristic
44	C₂²×M₁₆	—	131	28	no	—
45	C₂³×C₈	—	131	2	no	—
46	C₂⁵	Dic₁₃₁	1	1	yes	characteristic
47	C₂³×C₄	Dic₁₃₁	1	1	yes	characteristic
48	C₂³×C₄	Dic₁₃₁	1	14	yes	—
49	C₂³×C₄	C₂×D₁₃₁	1	14	yes	—
50	C₂³×C₄	C₂×D₁₃₁	1	1	yes	characteristic
51	C₂×M₁₆	—	131	112	no	—
52	C₂²×C₈	—	131	28	no	—
53	C₂⁴	—	2	8	no	—
54	C₂⁴	C₂×Dic₁₃₁	1	14	yes	—
55	C₂⁴	C₂×Dic₁₃₁	1	1	yes	characteristic
56	C₂²×C₄	C₂×Dic₁₃₁	1	84	yes	—
57	C₂²×C₄	C₂×Dic₁₃₁	1	14	yes	—
58	C₂²×C₄	C₂×Dic₁₃₁	1	7	yes	—
59	C₂²×C₄	C₂²×D₁₃₁	1	7	yes	—
60	C₂²×C₄	C₂²×D₁₃₁	1	28	yes	—
61	M₁₆	—	131	64	no	—
62	C₂×C₈	—	131	56	no	—
63	C₂³	—	2	56	no	—
64	C₂³	#14523497	1	8	yes	direct product factor
65	C₂³	C₂²×Dic₁₃₁	1	28	yes	—
66	C₂³	C₂²×Dic₁₃₁	1	7	yes	—
67	C₂×C₄	C₂²×Dic₁₃₁	1	56	yes	—
68	C₂×C₄	C₂²×Dic₁₃₁	1	28	yes	—
69	C₂×C₄	C₂²×Dic₁₃₁	1	21	yes	—
70	C₂×C₄	C₂³×D₁₃₁	1	7	yes	—
71	C₂×C₄	C₂³×D₁₃₁	1	8	yes	—
72	C₈	—	131	16	no	—
73	C₂²	—	2	56	no	—
74	C₂²	C₂×#14523497	1	28	yes	direct product factor
75	C₂²	C₂³×Dic₁₃₁	1	8	yes	—
76	C₂²	C₂³×Dic₁₃₁	1	7	yes	—
77	C₄	C₂³×Dic₁₃₁	1	7	yes	—
78	C₄	C₂³×Dic₁₃₁	1	8	yes	—
79	C₄	C₂⁴×D₁₃₁	1	1	yes	characteristic
80	C₂	—	2	8	no	—
81	C₂	C₂²×#14523497	1	14	yes	direct product factor
82	C₂	C₂⁴×Dic₁₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₄	C₂×D₁₃₁
commutator	C₂×C₁₃₁	C₂⁴×C₄
Frattini	C₄	C₂⁴×D₁₃₁
Fitting	C₂⁴×C₄×C₁₃₁	C₂
socle	C₂⁴×C₁₃₁	C₂×C₄
prime radical	C₂⁵×C₁₃₁	C₄
cosocle	C₄×C₁₃₁	C₂⁵
hypercenter	C₂⁴×C₄	D₁₃₁
hypocenter	C₁₃₁	C₂³×M₁₆
LCS	G ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂⁴×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#14523497	C₂	direct (N × H)
2. C₂³×C₁₃₁⋊C₈	C₂	semidirect
4. C₂×#14523497	C₂²	direct (N × H)
5. C₂²×C₁₃₁⋊C₈	C₂²	semidirect
11. #14523497	C₂³	direct (N × H)
12. C₂×C₁₃₁⋊C₈	C₂³	semidirect
21. C₁₃₁⋊C₈	C₂⁴	semidirect
23. C₂³×C₁₃₁	M₁₆	semidirect
32. C₂²×C₁₃₁	C₂×M₁₆	semidirect
39. C₂×C₁₃₁	C₂²×M₁₆	semidirect
41. C₁₃₁	C₂³×M₁₆	semidirect
64. C₂³	#14523497	direct (N × H)
74. C₂²	C₂×#14523497	direct (N × H)
81. C₂	C₂²×#14523497	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×#14523497	2
C₂³×C₁₃₁⋊C₈	2
C₂⁴×C₄×C₁₃₁	2
C₂³×M₁₆	131

Subgroups — C23×#14523497 #30680663

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14523497 #30680663