Subgroups — #13056332

Order 1344.

324 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₆₈	C₂	1	4	yes	maximal, complemented
2	#10621357	C₂	1	2	yes	maximal, complemented
3	C₃×C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#89467	—	3	1	no	maximal, Hall
5	C₂×D₈₄	—	2	4	no	—
6	C₂×D₈₄	C₂²	1	4	yes	—
7	C₂×C₃×C₇×C₈	C₂²	1	2	yes	—
8	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
9	D₁₆₈	—	2	8	no	—
10	C₂×D₅₆	—	3	4	no	—
11	#6437	—	3	2	no	—
12	C₄×C₇×C₈	D₃	1	1	yes	characteristic, complemented
13	#4695	—	7	1	no	maximal, Hall
14	C₂²×D₂₁	—	2	4	no	—
15	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
16	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
17	D₈₄	—	2	8	no	—
18	D₈₄	—	4	8	no	—
19	C₃×C₇×C₈	D₄	1	4	yes	complemented
20	C₂×D₂₈	—	3	4	no	—
21	C₂×D₂₈	—	6	4	no	—
22	C₂×C₇×C₈	C₂×D₃	1	2	yes	—
23	C₄²×C₇	C₂×D₃	1	1	yes	characteristic
24	D₅₆	—	6	8	no	—
25	C₂×D₂₄	—	7	4	no	—
26	#874	—	7	2	no	—
27	C₃×C₄×C₈	D₇	1	1	yes	characteristic, complemented
28	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
29	C₂×D₂₁	—	4	4	no	—
30	C₂×D₂₁	—	4	8	no	—
31	C₃×C₄×C₇	D₈	1	4	yes	complemented
32	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
33	#493	—	21	1	no	Sylow
34	C₂²×D₇	—	6	4	no	—
35	C₂×C₄×C₇	D₁₂	1	2	yes	—
36	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
37	D₂₈	—	6	8	no	—
38	D₂₈	—	12	8	no	—
39	C₇×C₈	D₁₂	1	4	yes	complemented
40	C₂×D₁₂	—	7	4	no	—
41	C₂×D₁₂	—	14	4	no	—
42	C₂×C₃×C₈	C₂×D₇	1	2	yes	—
43	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
44	D₂₄	—	14	8	no	—
45	C₂×C₃×C₇	#127	1	1	yes	characteristic
46	C₂×C₃×C₇	C₂×D₈	1	2	yes	—
47	D₂₁	—	8	4	no	—
48	C₂×D₈	—	21	4	no	—
49	#127	—	21	2	no	—
50	C₄×C₈	D₂₁	1	1	yes	characteristic, complemented
51	C₂²×C₇	C₂×D₁₂	1	1	yes	characteristic
52	C₂×D₇	—	12	4	no	—
53	C₂×D₇	—	12	8	no	—
54	C₄×C₇	D₂₄	1	4	yes	complemented
55	C₄×C₇	C₂×D₁₂	1	2	yes	—
56	C₂²×D₃	—	14	4	no	—
57	C₂×C₃×C₄	D₂₈	1	2	yes	—
58	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
59	D₁₂	—	14	8	no	—
60	D₁₂	—	28	8	no	—
61	C₃×C₈	D₂₈	1	4	yes	complemented
62	C₃×C₇	#493	1	1	yes	characteristic, complemented, Hall
63	C₂×D₄	—	21	4	no	—
64	C₂×D₄	—	42	4	no	—
65	D₈	—	42	8	no	—
66	C₂×C₈	C₂×D₂₁	1	2	yes	—
67	C₄²	C₂×D₂₁	1	1	yes	characteristic
68	C₂×C₇	#874	1	1	yes	characteristic
69	C₂×C₇	C₂×D₂₄	1	2	yes	—
70	D₇	—	24	4	no	—
71	C₂²×C₃	C₂×D₂₈	1	1	yes	characteristic
72	C₂×D₃	—	28	4	no	—
73	C₂×D₃	—	28	8	no	—
74	C₃×C₄	D₅₆	1	4	yes	complemented
75	C₃×C₄	C₂×D₂₈	1	2	yes	—
76	C₂³	—	42	4	no	—
77	D₄	—	42	8	no	—
78	D₄	—	84	8	no	—
79	C₂×C₄	D₈₄	1	2	yes	—
80	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
81	C₈	D₈₄	1	4	yes	complemented
82	C₇	#4695	1	1	yes	characteristic, complemented, Sylow
83	C₂×C₃	#6437	1	1	yes	characteristic
84	C₂×C₃	C₂×D₅₆	1	2	yes	—
85	D₃	—	56	4	no	—
86	C₂²	—	84	4	no	—
87	C₂²	—	84	8	no	—
88	C₂²	C₂×D₈₄	1	1	yes	characteristic
89	C₄	D₁₆₈	1	4	yes	complemented
90	C₄	C₂×D₈₄	1	2	yes	—
91	C₃	#89467	1	1	yes	characteristic, complemented, Sylow
92	C₂	—	168	4	no	—
93	C₂	#10621357	1	1	yes	characteristic
94	C₂	C₂×D₁₆₈	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈₄
commutator, cosocle	C₂×C₃×C₄×C₇	C₂³
Frattini	C₂×C₄	C₂²×D₂₁
Fitting	C₃×C₄×C₇×C₈	C₂
socle	C₂²×C₃×C₇	C₂×D₄
hypercenter	C₄×C₈	D₂₁
hypocenter	C₃×C₇	#493
LCS	G ▷ C₂×C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
1. C₂×D₁₆₈	C₂	semidirect
2. #10621357	C₂	semidirect
3. C₃×C₄×C₇×C₈	C₂	semidirect
12. C₄×C₇×C₈	D₃	semidirect
19. C₃×C₇×C₈	D₄	semidirect
27. C₃×C₄×C₈	D₇	semidirect
31. C₃×C₄×C₇	D₈	semidirect
39. C₇×C₈	D₁₂	semidirect
50. C₄×C₈	D₂₁	semidirect
54. C₄×C₇	D₂₄	semidirect
61. C₃×C₈	D₂₈	semidirect
62. C₃×C₇	#493	semidirect
74. C₃×C₄	D₅₆	semidirect
81. C₈	D₈₄	semidirect
82. C₇	#4695	semidirect
89. C₄	D₁₆₈	semidirect
91. C₃	#89467	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₆₈	2
#10621357	2
C₃×C₄×C₇×C₈	2
#89467	3
#4695	7