Subgroups — #30667720

Order 16768.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16862186	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₃₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₁₃₁⋊C₁₆)	C₂	1	4	yes	maximal, complemented
4	#405×C₁₃₁	C₂	1	1	yes	maximal, characteristic
5	#16862186	—	2	2	no	—
6	C₂×C₁₃₁⋊C₁₆	—	2	2	no	—
7	C₂×C₁₃₁⋊C₁₆	C₂²	1	2	yes	—
8	C₂×C₁₃₁⋊C₁₆	C₂²	1	2	yes	—
9	C₂×M₁₆×C₁₃₁	C₄	1	1	yes	characteristic
10	C₂²×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
11	#117×C₁₃₁	C₄	1	1	yes	characteristic
12	#97×C₁₃₁	C₄	1	2	yes	—
13	C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
14	C₁₃₁⋊C₁₆	—	2	2	no	—
15	C₁₃₁⋊C₁₆	—	2	2	no	—
16	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
17	M₁₆×C₁₃₁	—	2	2	no	—
18	C₂×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
19	C₂×C₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₁₃₁	D₄	1	2	yes	—
21	C₂×C₈×C₁₃₁	Q₈	1	2	yes	—
22	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
23	C₄⋊C₄×C₁₃₁	C₈	1	2	yes	—
24	C₂²⋊C₄×C₁₃₁	C₈	1	2	yes	—
25	C₄²×C₁₃₁	C₂×C₄	1	2	yes	—
26	C₂³×C₁₃₁	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄×C₁₃₁	C₄⋊C₄	1	2	yes	—
28	C₂×C₄×C₁₃₁	C₂×C₈	1	4	yes	—
29	C₂×C₄×C₁₃₁	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₁₃₁	M₁₆	1	2	yes	—
31	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
32	C₈×C₁₃₁	—	2	2	no	—
33	C₈×C₁₃₁	C₄⋊C₄	1	2	yes	—
34	C₈×C₁₃₁	C₂×D₄	1	1	yes	characteristic
35	C₈×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₁₃₁	—	2	2	no	—
37	C₂²×C₁₃₁	C₄⋊C₈	1	2	yes	—
38	C₂²×C₁₃₁	C₂²×C₈	1	1	yes	characteristic
39	C₄×C₁₃₁	—	2	4	no	—
40	C₄×C₁₃₁	C₄⋊C₈	1	2	yes	—
41	C₄×C₁₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₄×C₁₃₁	C₂×M₁₆	1	1	yes	characteristic
43	C₂×C₁₃₁	—	2	2	no	—
44	C₂×C₁₃₁	C₂×C₄⋊C₈	1	1	yes	characteristic
45	C₂×C₁₃₁	#504	1	2	yes	—
46	C₁₃₁	#2151	1	1	yes	characteristic, complemented, Sylow
47	#2151	—	131	1	no	maximal, Sylow
48	C₂×M₃₂	—	131	1	no	—
49	C₂²×C₁₆	—	131	1	no	—
50	#405	D₁₃₁	1	1	yes	characteristic
51	C₄⋊C₁₆	—	131	4	no	—
52	C₂×M₁₆	Dic₁₃₁	1	1	yes	characteristic
53	C₂²×C₈	C₂×D₁₃₁	1	1	yes	characteristic
54	#117	Dic₁₃₁	1	1	yes	characteristic
55	M₃₂	—	262	2	no	—
56	C₂×C₁₆	—	262	2	no	—
57	C₂×C₁₆	—	131	2	no	—
58	C₂×C₁₆	—	131	2	no	—
59	#97	Dic₁₃₁	1	2	yes	—
60	C₄×C₈	C₂×D₁₃₁	1	2	yes	—
61	C₂²×C₄	C₂×Dic₁₃₁	1	1	yes	characteristic
62	M₁₆	—	2	2	no	—
63	C₂×C₈	C₂×Dic₁₃₁	1	2	yes	—
64	C₂×C₈	C₂×Dic₁₃₁	1	1	yes	characteristic
65	C₂×C₈	D₅₂₄	1	2	yes	—
66	C₂×C₈	Dic₂₆₂	1	2	yes	—
67	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
68	C₄⋊C₄	C₁₃₁⋊C₈	1	2	yes	—
69	C₂²⋊C₄	C₁₃₁⋊C₈	1	2	yes	—
70	C₄²	C₂×Dic₁₃₁	1	2	yes	—
71	C₁₆	—	262	2	no	—
72	C₁₆	—	262	2	no	—
73	C₂³	C₂×C₁₃₁⋊C₈	1	1	yes	characteristic
74	C₂×C₄	C₄⋊Dic₁₃₁	1	2	yes	—
75	C₂×C₄	C₂×C₁₃₁⋊C₈	1	4	yes	—
76	C₂×C₄	C₂×C₁₃₁⋊C₈	1	1	yes	characteristic
77	C₂×C₄	#14523497	1	2	yes	—
78	C₂×C₄	C₂²×Dic₁₃₁	1	1	yes	characteristic
79	C₈	—	2	2	no	—
80	C₈	C₄⋊Dic₁₃₁	1	2	yes	—
81	C₈	C₂×D₅₂₄	1	1	yes	characteristic
82	C₈	C₂×Dic₂₆₂	1	1	yes	characteristic
83	C₂²	—	2	2	no	—
84	C₂²	C₄⋊(C₁₃₁⋊C₈)	1	2	yes	—
85	C₂²	C₂²×C₁₃₁⋊C₈	1	1	yes	characteristic
86	C₄	—	2	4	no	—
87	C₄	C₄⋊(C₁₃₁⋊C₈)	1	2	yes	—
88	C₄	C₂×C₄⋊Dic₁₃₁	1	1	yes	characteristic
89	C₄	C₂×#14523497	1	1	yes	characteristic
90	C₂	—	2	2	no	—
91	C₂	C₂×C₄⋊(C₁₃₁⋊C₈)	1	1	yes	characteristic
92	C₂	#22553729	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₃₁
commutator, socle	C₂²×C₁₃₁	C₂²×C₈
Fitting	#405×C₁₃₁	C₂
prime radical	C₂³×C₁₃₁	C₂×C₈
cosocle	C₂×C₈×C₁₃₁	C₂³
hypercenter	#405	D₁₃₁
hypocenter	C₁₃₁	#2151
LCS	G ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂×C₈ ◁ #405

Complements

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

N	H ≅ G/N	type
3. C₄⋊(C₁₃₁⋊C₁₆)	C₂	semidirect
46. C₁₃₁	#2151	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16862186	2
C₂²×C₁₃₁⋊C₁₆	2
C₄⋊(C₁₃₁⋊C₁₆)	2
#405×C₁₃₁	2
#2151	131