Subgroups — #30664244

Order 16768.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22553365	C₂	1	2	yes	maximal
2	#22553033	C₂	1	2	yes	maximal, complemented
3	C₈×C₁₃₁⋊C₈	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic
5	C₄⋊(C₁₃₁⋊C₈)	C₄	1	4	yes	—
6	C₂²⋊(C₁₃₁⋊C₈)	C₄	1	4	yes	—
7	C₄×C₁₃₁⋊C₈	C₂²	1	4	yes	—
8	C₂²×C₈×C₁₃₁	C₄	1	2	yes	—
9	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
10	C₄×C₈×C₁₃₁	C₄	1	2	yes	—
11	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
12	C₄×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₁₃₁⋊C₈	C₂×C₄	1	8	yes	—
14	C₂²×C₄×C₁₃₁	C₂×C₄	1	2	yes	—
15	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
16	C₂×C₈×C₁₃₁	C₈	1	8	yes	complemented
17	C₂×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
18	C₂×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
19	C₄²×C₁₃₁	C₂×C₄	1	2	yes	—
20	C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
21	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
22	C₁₃₁⋊C₈	—	2	8	no	—
23	C₂³×C₁₃₁	C₄²	1	1	yes	characteristic
24	C₂×C₄×C₁₃₁	—	2	4	no	—
25	C₂×C₄×C₁₃₁	C₄²	1	2	yes	—
26	C₂×C₄×C₁₃₁	C₄²	1	1	yes	characteristic
27	C₂×C₄×C₁₃₁	C₂×C₈	1	4	yes	—
28	C₂×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
29	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
30	C₈×C₁₃₁	C₂×C₈	1	4	yes	—
31	C₈×C₁₃₁	C₂×C₈	1	4	yes	—
32	C₂²×C₁₃₁	—	2	2	no	—
33	C₂²×C₁₃₁	C₄×C₈	1	2	yes	—
34	C₂²×C₁₃₁	C₂×C₄²	1	1	yes	characteristic
35	C₄×C₁₃₁	—	2	2	no	—
36	C₄×C₁₃₁	C₄×C₈	1	2	yes	—
37	C₄×C₁₃₁	C₂²×C₈	1	2	yes	—
38	C₄×C₁₃₁	#131	1	4	yes	—
39	C₂×C₁₃₁	—	2	2	no	—
40	C₂×C₁₃₁	C₂×C₄×C₈	1	1	yes	characteristic
41	C₂×C₁₃₁	#405	1	2	yes	—
42	C₁₃₁	#1450	1	1	yes	characteristic, complemented, Sylow
43	#1450	—	131	1	no	maximal, Sylow
44	#431	—	131	2	no	—
45	C₂×C₄×C₈	D₁₃₁	1	1	yes	characteristic
46	#322	—	131	2	no	—
47	C₈²	—	131	2	no	—
48	C₂²×C₈	Dic₁₃₁	1	2	yes	—
49	C₂×C₄²	C₂×D₁₃₁	1	1	yes	characteristic
50	C₄⋊C₈	—	131	4	no	—
51	C₂²⋊C₈	—	131	4	no	—
52	C₄×C₈	—	131	4	no	—
53	C₄×C₈	Dic₁₃₁	1	2	yes	—
54	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
55	C₄×C₈	C₂×D₁₃₁	1	1	yes	characteristic
56	C₂²×C₄	C₄×D₁₃₁	1	2	yes	—
57	C₂²×C₄	C₂×Dic₁₃₁	1	1	yes	characteristic
58	C₂×C₈	—	131	8	no	—
59	C₂×C₈	C₁₃₁⋊C₈	1	8	yes	complemented
60	C₂×C₈	C₂×Dic₁₃₁	1	2	yes	—
61	C₂×C₈	C₂×Dic₁₃₁	1	2	yes	—
62	C₄²	C₄×D₁₃₁	1	2	yes	—
63	C₄²	C₂×Dic₁₃₁	1	1	yes	characteristic
64	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
65	C₂³	C₄×Dic₁₃₁	1	1	yes	characteristic
66	C₂×C₄	—	2	4	no	—
67	C₂×C₄	C₄×Dic₁₃₁	1	2	yes	—
68	C₂×C₄	C₄×Dic₁₃₁	1	1	yes	characteristic
69	C₂×C₄	C₂×C₁₃₁⋊C₈	1	4	yes	—
70	C₂×C₄	C₂×C₄×D₁₃₁	1	2	yes	—
71	C₂×C₄	C₂²×Dic₁₃₁	1	1	yes	characteristic
72	C₈	—	262	8	no	—
73	C₈	C₂×C₁₃₁⋊C₈	1	4	yes	—
74	C₈	C₂×C₁₃₁⋊C₈	1	4	yes	—
75	C₂²	—	2	2	no	—
76	C₂²	C₄×C₁₃₁⋊C₈	1	2	yes	—
77	C₂²	C₂×C₄×Dic₁₃₁	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	C₄×C₁₃₁⋊C₈	1	2	yes	—
80	C₄	C₂²×C₁₃₁⋊C₈	1	2	yes	—
81	C₄	#16862252	1	4	yes	—
82	C₂	—	2	2	no	—
83	C₂	C₂×C₄×C₁₃₁⋊C₈	1	1	yes	characteristic
84	C₂	#22553267	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂×D₁₃₁
commutator	C₂×C₁₃₁	C₂×C₄×C₈
Frattini	C₄²	C₂²×D₁₃₁
Fitting	C₂×C₄×C₈×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂×C₄²
prime radical	C₂³×C₁₃₁	C₄²
cosocle	C₄²×C₁₃₁	C₂³
hypercenter	C₂×C₄×C₈	D₁₃₁
hypocenter	C₁₃₁	#1450
LCS	G ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #22553033	C₂	semidirect
3. C₈×C₁₃₁⋊C₈	C₂	semidirect
16. C₂×C₈×C₁₃₁	C₈	semidirect
42. C₁₃₁	#1450	semidirect
59. C₂×C₈	C₁₃₁⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22553365	2
#22553033	2
C₈×C₁₃₁⋊C₈	2
C₂×C₄×C₈×C₁₃₁	2
#1450	131