Subgroups — C₂×#22553033 #30664236

Order 16768.

248 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₃₁⋊C₈	C₂	1	2	yes	maximal
2	#22553033	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×C₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₃₁⋊C₈	C₄	1	4	yes	—
5	C₄×C₁₃₁⋊C₈	C₄	1	4	yes	—
6	C₄×C₁₃₁⋊C₈	C₂²	1	4	yes	—
7	C₂²×C₈×C₁₃₁	C₄	1	2	yes	—
8	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
9	C₄×C₈×C₁₃₁	C₄	1	2	yes	—
10	C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
11	C₂×C₁₃₁⋊C₈	—	2	8	no	—
12	C₂×C₁₃₁⋊C₈	C₂×C₄	1	8	yes	—
13	C₂²×C₄×C₁₃₁	C₂×C₄	1	2	yes	—
14	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×C₁₃₁	C₈	1	8	yes	complemented
16	C₂×C₈×C₁₃₁	C₂×C₄	1	4	yes	—
17	C₄²×C₁₃₁	C₂×C₄	1	2	yes	—
18	C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
19	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
20	C₁₃₁⋊C₈	—	2	8	no	—
21	C₂³×C₁₃₁	C₄²	1	1	yes	characteristic
22	C₂×C₄×C₁₃₁	C₄²	1	1	yes	characteristic
23	C₂×C₄×C₁₃₁	C₄²	1	2	yes	—
24	C₂×C₄×C₁₃₁	C₂×C₈	1	4	yes	—
25	C₂×C₄×C₁₃₁	M₁₆	1	8	yes	—
26	C₂×C₄×C₁₃₁	C₂²×C₄	1	2	yes	—
27	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
28	C₈×C₁₃₁	C₂×C₈	1	8	yes	complemented
29	C₂²×C₁₃₁	C₄×C₈	1	2	yes	—
30	C₂²×C₁₃₁	#97	1	4	yes	—
31	C₂²×C₁₃₁	C₂×C₄²	1	1	yes	characteristic
32	C₄×C₁₃₁	C₄×C₈	1	2	yes	—
33	C₄×C₁₃₁	#97	1	4	yes	—
34	C₄×C₁₃₁	C₂²×C₈	1	2	yes	—
35	C₄×C₁₃₁	C₂×M₁₆	1	4	yes	—
36	C₂×C₁₃₁	#322	1	4	yes	complemented
37	C₂×C₁₃₁	C₂×C₄×C₈	1	1	yes	characteristic
38	C₂×C₁₃₁	C₂×#97	1	2	yes	—
39	C₁₃₁	C₂×#322	1	1	yes	characteristic, complemented, Sylow
40	C₂×#322	—	131	1	no	maximal, Sylow
41	C₂×C₄×C₈	—	131	2	no	—
42	C₂×C₄×C₈	D₁₃₁	1	1	yes	characteristic
43	#322	—	131	4	no	—
44	C₂²×C₈	—	131	4	no	—
45	C₂²×C₈	Dic₁₃₁	1	2	yes	—
46	C₂×C₄²	C₂×D₁₃₁	1	1	yes	characteristic
47	C₄×C₈	—	131	4	no	—
48	C₄×C₈	—	131	4	no	—
49	C₄×C₈	Dic₁₃₁	1	2	yes	—
50	C₄×C₈	C₂×D₁₃₁	1	2	yes	—
51	C₂²×C₄	C₄×D₁₃₁	1	2	yes	—
52	C₂²×C₄	C₂×Dic₁₃₁	1	1	yes	characteristic
53	C₂×C₈	—	262	8	no	—
54	C₂×C₈	—	131	8	no	—
55	C₂×C₈	C₁₃₁⋊C₈	1	8	yes	complemented
56	C₂×C₈	C₂×Dic₁₃₁	1	4	yes	—
57	C₄²	C₄×D₁₃₁	1	2	yes	—
58	C₄²	C₂×Dic₁₃₁	1	1	yes	characteristic
59	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
60	C₂³	C₄×Dic₁₃₁	1	1	yes	characteristic
61	C₂×C₄	C₄×Dic₁₃₁	1	1	yes	characteristic
62	C₂×C₄	C₄×Dic₁₃₁	1	2	yes	—
63	C₂×C₄	C₂×C₁₃₁⋊C₈	1	4	yes	—
64	C₂×C₄	#14523496	1	8	yes	—
65	C₂×C₄	C₂×C₄×D₁₃₁	1	2	yes	—
66	C₂×C₄	C₂²×Dic₁₃₁	1	1	yes	characteristic
67	C₈	—	262	8	no	—
68	C₈	C₂×C₁₃₁⋊C₈	1	8	yes	complemented
69	C₂²	C₄×C₁₃₁⋊C₈	1	2	yes	—
70	C₂²	#16862156	1	4	yes	—
71	C₂²	C₂×C₄×Dic₁₃₁	1	1	yes	characteristic
72	C₄	C₄×C₁₃₁⋊C₈	1	2	yes	—
73	C₄	#16862156	1	4	yes	—
74	C₄	C₂²×C₁₃₁⋊C₈	1	2	yes	—
75	C₄	C₂×#14523496	1	4	yes	—
76	C₂	#22553033	1	4	yes	direct product factor
77	C₂	C₂×C₄×C₁₃₁⋊C₈	1	1	yes	characteristic
78	C₂	C₂×#16862156	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂×D₁₃₁
commutator	C₂×C₁₃₁	C₂×C₄×C₈
Frattini	C₄²	C₂²×D₁₃₁
Fitting	C₂×C₄×C₈×C₁₃₁	C₂
socle, prime radical	C₂³×C₁₃₁	C₄²
cosocle	C₄²×C₁₃₁	C₂³
hypercenter	C₂×C₄×C₈	D₁₃₁
hypocenter	C₁₃₁	C₂×#322
LCS	G ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. #22553033	C₂	direct (N × H)
15. C₂×C₈×C₁₃₁	C₈	semidirect
28. C₈×C₁₃₁	C₂×C₈	semidirect
36. C₂×C₁₃₁	#322	semidirect
39. C₁₃₁	C₂×#322	semidirect
55. C₂×C₈	C₁₃₁⋊C₈	semidirect
68. C₈	C₂×C₁₃₁⋊C₈	semidirect
76. C₂	#22553033	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×C₄×C₁₃₁⋊C₈	2
#22553033	2
C₂×C₄×C₈×C₁₃₁	2
C₂×#322	131

Subgroups — C2×#22553033 #30664236

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22553033 #30664236