Subgroups — #30664041

Order 16768.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16862186	C₂	1	2	yes	maximal
2	C₂×#97×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#16862186	—	2	2	no	—
4	#16862186	—	2	2	no	—
5	C₂×C₁₃₁⋊C₁₆	C₄	1	4	yes	complemented
6	C₂²×C₈×C₁₃₁	C₄	1	1	yes	characteristic
7	C₂²×C₈×C₁₃₁	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₁₃₁	C₄	1	1	yes	characteristic
9	#97×C₁₃₁	—	2	2	no	—
10	C₁₃₁⋊C₁₆	—	2	4	no	—
11	C₂²×C₄×C₁₃₁	—	2	1	no	—
12	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
13	C₂×C₈×C₁₃₁	—	2	1	no	—
14	C₂×C₈×C₁₃₁	—	2	1	no	—
15	C₂×C₈×C₁₃₁	C₈	1	2	yes	—
16	C₂×C₈×C₁₃₁	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₁₃₁	D₄	1	2	yes	—
18	C₂×C₈×C₁₃₁	D₄	1	1	yes	characteristic
19	C₂×C₈×C₁₃₁	Q₈	1	1	yes	characteristic
20	C₄²×C₁₃₁	—	2	1	no	—
21	C₄²×C₁₃₁	C₈	1	2	yes	—
22	C₂³×C₁₃₁	M₁₆	1	1	yes	characteristic
23	C₂×C₄×C₁₃₁	—	2	2	no	—
24	C₂×C₄×C₁₃₁	—	4	2	no	—
25	C₂×C₄×C₁₃₁	C₄²	1	1	yes	characteristic
26	C₂×C₄×C₁₃₁	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₃₁	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₁₃₁	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₁₃₁	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₁₃₁	M₁₆	1	1	yes	characteristic
31	C₈×C₁₃₁	—	2	2	no	—
32	C₈×C₁₃₁	C₂²⋊C₄	1	2	yes	—
33	C₈×C₁₃₁	C₄⋊C₄	1	2	yes	—
34	C₂²×C₁₃₁	—	2	2	no	—
35	C₂²×C₁₃₁	#97	1	1	yes	characteristic
36	C₂²×C₁₃₁	C₂²⋊C₈	1	1	yes	characteristic
37	C₂²×C₁₃₁	C₄⋊C₈	1	1	yes	characteristic
38	C₄×C₁₃₁	—	4	2	no	—
39	C₄×C₁₃₁	#95	1	1	yes	characteristic
40	C₄×C₁₃₁	C₄×C₈	1	1	yes	characteristic
41	C₄×C₁₃₁	C₂²⋊C₈	1	1	yes	characteristic
42	C₄×C₁₃₁	C₄⋊C₈	1	1	yes	characteristic
43	C₂×C₁₃₁	—	2	1	no	—
44	C₂×C₁₃₁	—	2	1	no	—
45	C₂×C₁₃₁	#336	1	1	yes	characteristic
46	C₂×C₁₃₁	#347	1	2	yes	—
47	C₁₃₁	#1375	1	1	yes	characteristic, complemented, Sylow
48	#1375	—	131	1	no	maximal, Sylow
49	C₂×M₃₂	—	131	2	no	—
50	C₂×#97	D₁₃₁	1	1	yes	characteristic
51	C₂²×C₈	Dic₁₃₁	1	1	yes	characteristic
52	C₂²×C₈	C₂×D₁₃₁	1	1	yes	characteristic
53	C₂×C₄²	Dic₁₃₁	1	1	yes	characteristic
54	M₃₂	—	262	2	no	—
55	M₃₂	—	262	2	no	—
56	C₂×C₁₆	—	131	4	no	—
57	#97	—	2	2	no	—
58	C₂²×C₄	—	2	1	no	—
59	C₂²×C₄	C₂×Dic₁₃₁	1	1	yes	characteristic
60	C₂×C₈	—	2	1	no	—
61	C₂×C₈	—	2	1	no	—
62	C₂×C₈	C₁₃₁⋊C₈	1	2	yes	—
63	C₂×C₈	C₄×D₁₃₁	1	2	yes	—
64	C₂×C₈	C₂²⋊D₁₃₁	1	2	yes	—
65	C₂×C₈	D₅₂₄	1	1	yes	characteristic
66	C₂×C₈	Dic₂₆₂	1	1	yes	characteristic
67	C₄²	—	2	1	no	—
68	C₄²	C₁₃₁⋊C₈	1	2	yes	—
69	C₁₆	—	262	4	no	—
70	C₂³	#14523497	1	1	yes	characteristic
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	4	2	no	—
73	C₂×C₄	C₄×Dic₁₃₁	1	1	yes	characteristic
74	C₂×C₄	C₂²⋊Dic₁₃₁	1	1	yes	characteristic
75	C₂×C₄	C₄⋊Dic₁₃₁	1	1	yes	characteristic
76	C₂×C₄	C₂×C₁₃₁⋊C₈	1	1	yes	characteristic
77	C₂×C₄	C₂×C₁₃₁⋊C₈	1	1	yes	characteristic
78	C₂×C₄	#14523497	1	1	yes	characteristic
79	C₈	—	2	2	no	—
80	C₈	(C₂×D₁₃₁)⋊C₄	1	2	yes	—
81	C₈	Dic₁₃₁⋊C₄	1	2	yes	—
82	C₂²	—	2	2	no	—
83	C₂²	#16862157	1	1	yes	characteristic
84	C₂²	C₂²⋊(C₁₃₁⋊C₈)	1	1	yes	characteristic
85	C₂²	C₄⋊(C₁₃₁⋊C₈)	1	1	yes	characteristic
86	C₄	—	4	2	no	—
87	C₄	C₁₃₁⋊#95	1	1	yes	characteristic
88	C₄	C₄×C₁₃₁⋊C₈	1	1	yes	characteristic
89	C₄	C₂²⋊(C₁₃₁⋊C₈)	1	1	yes	characteristic
90	C₄	C₄⋊(C₁₃₁⋊C₈)	1	1	yes	characteristic
91	C₂	—	2	1	no	—
92	C₂	—	2	1	no	—
93	C₂	#22553067	1	1	yes	characteristic
94	C₂	#22553090	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₄×Dic₁₃₁
commutator	C₄×C₁₃₁	C₄×C₈
Frattini	C₂²×C₈	C₂×D₁₃₁
Fitting	C₂×#97×C₁₃₁	C₂
socle	C₂²×C₁₃₁	#97
prime radical	C₂³×C₁₃₁	M₁₆
cosocle	C₂²×C₈×C₁₃₁	C₂²
hypercenter	C₂×#97	D₁₃₁
hypocenter	C₁₃₁	#1375
LCS	G ▷ C₄×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
5. C₂×C₁₃₁⋊C₁₆	C₄	semidirect
47. C₁₃₁	#1375	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
C₂×#97×C₁₃₁	2
#1375	131