Subgroups — #30666132

Order 16768.

220 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16862181	C₂	1	4	yes	maximal
2	#22553336	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₁₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#423×C₁₃₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523497	C₂²	1	2	yes	—
6	C₂²×C₁₃₁⋊C₈	C₂²	1	2	yes	—
7	#16862181	—	2	8	no	—
8	C₄⋊(C₁₃₁⋊C₈)	—	2	2	no	—
9	C₄×C₁₃₁⋊C₈	C₄	1	2	yes	—
10	C₄×C₁₃₁⋊C₈	C₄	1	2	yes	—
11	C₂×M₁₆×C₁₃₁	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₁₃₁	—	2	2	no	—
14	#14523497	—	2	4	no	—
15	C₂×C₁₃₁⋊C₈	—	2	2	no	—
16	C₂×C₁₃₁⋊C₈	C₂×C₄	1	4	yes	—
17	C₂×C₁₃₁⋊C₈	D₄	1	4	yes	—
18	C₂×C₁₃₁⋊C₈	Q₈	1	4	yes	—
19	C₂²×C₄×C₁₃₁	Q₈	1	2	yes	—
20	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
21	M₁₆×C₁₃₁	—	2	4	no	—
22	C₂×C₈×C₁₃₁	—	2	2	no	—
23	C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
25	C₄²×C₁₃₁	D₄	1	2	yes	—
26	C₁₃₁⋊C₈	—	4	2	no	—
27	C₁₃₁⋊C₈	C₄⋊C₄	1	8	yes	—
28	C₂³×C₁₃₁	C₂×Q₈	1	1	yes	characteristic
29	C₂×C₄×C₁₃₁	—	2	4	no	—
30	C₂×C₄×C₁₃₁	C₄⋊C₄	1	2	yes	—
31	C₂×C₄×C₁₃₁	C₄⋊C₄	1	2	yes	—
32	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₃₁	C₂×D₄	1	2	yes	—
34	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₃₁	C₂×Q₈	1	2	yes	—
36	C₈×C₁₃₁	—	4	2	no	—
37	C₂²×C₁₃₁	—	2	2	no	—
38	C₂²×C₁₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₁₃₁	C₄⋊Q₈	1	2	yes	—
40	C₄×C₁₃₁	—	2	2	no	—
41	C₄×C₁₃₁	#108	1	4	yes	complemented
42	C₄×C₁₃₁	C₂×C₄⋊C₄	1	2	yes	—
43	C₄×C₁₃₁	#127	1	1	yes	characteristic
44	C₄×C₁₃₁	C₄⋊Q₈	1	1	yes	characteristic
45	C₂×C₁₃₁	—	2	2	no	—
46	C₂×C₁₃₁	#384	1	1	yes	characteristic
47	C₂×C₁₃₁	C₂×#108	1	2	yes	—
48	C₁₃₁	#1848	1	1	yes	characteristic, complemented, Sylow
49	#1848	—	131	1	no	maximal, Sylow
50	C₂×#108	—	131	4	no	—
51	#423	—	131	1	no	—
52	#423	D₁₃₁	1	1	yes	characteristic
53	C₂×C₄×C₈	—	131	1	no	—
54	C₂×M₁₆	—	131	2	no	—
55	C₂×M₁₆	C₂×D₁₃₁	1	2	yes	—
56	C₂²×C₈	—	131	2	no	—
57	C₂×C₄²	C₂×D₁₃₁	1	1	yes	characteristic
58	#108	—	262	8	no	—
59	C₄⋊C₈	—	2	2	no	—
60	C₄⋊C₈	—	262	2	no	—
61	C₄×C₈	—	131	2	no	—
62	C₄×C₈	—	131	2	no	—
63	C₂²×C₄	Dic₂₆₂	1	2	yes	—
64	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
65	M₁₆	—	2	4	no	—
66	M₁₆	—	262	4	no	—
67	C₂×C₈	—	2	2	no	—
68	C₂×C₈	—	262	2	no	—
69	C₂×C₈	—	131	4	no	—
70	C₂×C₈	—	131	4	no	—
71	C₂×C₈	—	131	4	no	—
72	C₄²	C₄×D₁₃₁	1	1	yes	characteristic
73	C₄²	C₄×D₁₃₁	1	1	yes	characteristic
74	C₄²	C₂²⋊D₁₃₁	1	2	yes	—
75	C₂³	C₂×Dic₂₆₂	1	1	yes	characteristic
76	C₂×C₄	—	2	4	no	—
77	C₂×C₄	Dic₁₃₁⋊C₄	1	2	yes	—
78	C₂×C₄	Dic₁₃₁⋊C₄	1	2	yes	—
79	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
80	C₂×C₄	D₄×D₁₃₁	1	2	yes	—
81	C₂×C₄	C₂×C₂²⋊D₁₃₁	1	1	yes	characteristic
82	C₂×C₄	Q₈×D₁₃₁	1	2	yes	—
83	C₈	—	4	2	no	—
84	C₈	—	524	2	no	—
85	C₈	—	131	8	no	—
86	C₂²	—	2	2	no	—
87	C₂²	C₂×Dic₁₃₁⋊C₄	1	1	yes	characteristic
88	C₂²	#16862245	1	2	yes	—
89	C₄	—	2	2	no	—
90	C₄	#16862181	1	4	yes	complemented
91	C₄	C₄⋊C₄×D₁₃₁	1	2	yes	—
92	C₄	#16862242	1	1	yes	characteristic
93	C₄	#16862244	1	1	yes	characteristic
94	C₂	—	2	2	no	—
95	C₂	#22553181	1	1	yes	characteristic
96	C₂	C₂×#16862181	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₃₁
commutator	C₂×C₄×C₁₃₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₁
Fitting	#423×C₁₃₁	C₂
socle	C₂²×C₁₃₁	C₂×C₄⋊C₄
prime radical	C₂³×C₁₃₁	C₂×Q₈
cosocle	C₂²×C₄×C₁₃₁	C₂³
hypercenter	#423	D₁₃₁
hypocenter	C₁₃₁	#1848
LCS	G ▷ C₂×C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

Complements

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

N	H ≅ G/N	type
41. C₄×C₁₃₁	#108	semidirect
48. C₁₃₁	#1848	semidirect
90. C₄	#16862181	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16862181	2
#22553336	2
C₂×C₄×C₁₃₁⋊C₈	2
#423×C₁₃₁	2
#1848	131