Subgroups — C₂×#10621357 #13061565

Order 1344.

996 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₈₄	C₂	1	6	yes	maximal, complemented
2	C₂×C₃×C₄²×C₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#10621357	C₂	1	8	yes	maximal, direct product factor
4	C₂×#6437	—	3	1	no	maximal, Hall
5	C₂³×D₂₁	C₂²	1	4	yes	—
6	C₂²×C₃×C₄×C₇	C₂²	1	3	yes	—
7	C₂×D₈₄	—	2	24	no	—
8	C₂×D₈₄	C₂²	1	24	yes	complemented
9	C₃×C₄²×C₇	C₂²	1	4	yes	complemented
10	C₂²×D₂₈	—	3	6	no	—
11	C₂×C₄²×C₇	D₃	1	1	yes	characteristic, complemented
12	#6437	—	3	8	no	—
13	C₂×#874	—	7	1	no	maximal, Hall
14	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
15	C₂²×D₂₁	—	2	24	no	—
16	C₂²×D₂₁	C₂³	1	8	yes	—
17	C₂×C₃×C₄×C₇	D₄	1	12	yes	complemented
18	C₂×C₃×C₄×C₇	C₂³	1	6	yes	—
19	D₈₄	—	2	48	no	—
20	C₂³×D₇	—	3	4	no	—
21	C₂²×C₄×C₇	C₂×D₃	1	3	yes	—
22	C₂×D₂₈	—	3	24	no	—
23	C₂×D₂₈	—	6	24	no	—
24	C₄²×C₇	C₂×D₃	1	4	yes	complemented
25	C₂²×D₁₂	—	7	6	no	—
26	C₂×C₃×C₄²	D₇	1	1	yes	characteristic, complemented
27	#874	—	7	8	no	—
28	C₂²×C₃×C₇	C₂×D₄	1	6	yes	—
29	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
30	C₂×D₂₁	—	2	24	no	—
31	C₂×D₂₁	—	4	16	no	—
32	C₃×C₄×C₇	C₂×D₄	1	12	yes	complemented
33	C₂×#127	—	21	1	no	Sylow
34	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
35	C₂²×D₇	—	3	8	no	—
36	C₂²×D₇	—	6	24	no	—
37	C₂×C₄×C₇	D₁₂	1	12	yes	complemented
38	C₂×C₄×C₇	C₂²×D₃	1	6	yes	—
39	D₂₈	—	6	48	no	—
40	C₂³×D₃	—	7	4	no	—
41	C₂²×C₃×C₄	C₂×D₇	1	3	yes	—
42	C₂×D₁₂	—	7	24	no	—
43	C₂×D₁₂	—	14	24	no	—
44	C₃×C₄²	C₂×D₇	1	4	yes	complemented
45	C₂×C₃×C₇	#127	1	4	yes	complemented
46	C₂×C₃×C₇	C₂²×D₄	1	3	yes	—
47	D₂₁	—	4	8	no	—
48	C₂²×D₄	—	21	6	no	—
49	#127	—	21	8	no	—
50	C₂×C₄²	D₂₁	1	1	yes	characteristic, complemented
51	C₂²×C₇	C₂×D₁₂	1	6	yes	—
52	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
53	C₂×D₇	—	6	24	no	—
54	C₂×D₇	—	12	16	no	—
55	C₄×C₇	C₂×D₁₂	1	12	yes	complemented
56	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
57	C₂²×D₃	—	7	8	no	—
58	C₂²×D₃	—	14	24	no	—
59	C₂×C₃×C₄	D₂₈	1	12	yes	complemented
60	C₂×C₃×C₄	C₂²×D₇	1	6	yes	—
61	D₁₂	—	14	48	no	—
62	C₃×C₇	C₂×#127	1	1	yes	characteristic, complemented, Hall
63	C₂⁴	—	21	4	no	—
64	C₂×D₄	—	21	24	no	—
65	C₂×D₄	—	42	24	no	—
66	C₂²×C₄	C₂×D₂₁	1	3	yes	—
67	C₄²	C₂×D₂₁	1	4	yes	complemented
68	C₂×C₇	#874	1	4	yes	complemented
69	C₂×C₇	C₂²×D₁₂	1	3	yes	—
70	D₇	—	12	8	no	—
71	C₂²×C₃	C₂×D₂₈	1	6	yes	—
72	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
73	C₂×D₃	—	14	24	no	—
74	C₂×D₃	—	28	16	no	—
75	C₃×C₄	C₂×D₂₈	1	12	yes	complemented
76	C₂³	—	21	8	no	—
77	C₂³	—	42	24	no	—
78	C₂³	C₂²×D₂₁	1	1	yes	characteristic
79	D₄	—	42	48	no	—
80	C₂×C₄	D₈₄	1	12	yes	complemented
81	C₂×C₄	C₂²×D₂₁	1	6	yes	—
82	C₇	C₂×#874	1	1	yes	characteristic, complemented, Sylow
83	C₂×C₃	#6437	1	4	yes	complemented
84	C₂×C₃	C₂²×D₂₈	1	3	yes	—
85	D₃	—	28	8	no	—
86	C₂²	—	42	24	no	—
87	C₂²	—	84	16	no	—
88	C₂²	C₂×D₈₄	1	6	yes	—
89	C₂²	C₂³×D₂₁	1	1	yes	characteristic
90	C₄	C₂×D₈₄	1	12	yes	complemented
91	C₃	C₂×#6437	1	1	yes	characteristic, complemented, Sylow
92	C₂	—	84	8	no	—
93	C₂	#10621357	1	4	yes	direct product factor
94	C₂	C₂²×D₈₄	1	3	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₂₁
commutator, cosocle	C₂²×C₃×C₇	C₂⁴
Frattini	C₂²	C₂³×D₂₁
Fitting	C₂×C₃×C₄²×C₇	C₂
socle	C₂³×C₃×C₇	C₂³
hypercenter	C₂×C₄²	D₂₁
hypocenter	C₃×C₇	C₂×#127
LCS	G ▷ C₂²×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂²×D₈₄	C₂	semidirect
2. C₂×C₃×C₄²×C₇	C₂	semidirect
3. #10621357	C₂	direct (N × H)
8. C₂×D₈₄	C₂²	semidirect
9. C₃×C₄²×C₇	C₂²	semidirect
11. C₂×C₄²×C₇	D₃	semidirect
17. C₂×C₃×C₄×C₇	D₄	semidirect
24. C₄²×C₇	C₂×D₃	semidirect
26. C₂×C₃×C₄²	D₇	semidirect
32. C₃×C₄×C₇	C₂×D₄	semidirect
37. C₂×C₄×C₇	D₁₂	semidirect
44. C₃×C₄²	C₂×D₇	semidirect
45. C₂×C₃×C₇	#127	semidirect
50. C₂×C₄²	D₂₁	semidirect
55. C₄×C₇	C₂×D₁₂	semidirect
59. C₂×C₃×C₄	D₂₈	semidirect
62. C₃×C₇	C₂×#127	semidirect
67. C₄²	C₂×D₂₁	semidirect
68. C₂×C₇	#874	semidirect
75. C₃×C₄	C₂×D₂₈	semidirect
80. C₂×C₄	D₈₄	semidirect
82. C₇	C₂×#874	semidirect
83. C₂×C₃	#6437	semidirect
90. C₄	C₂×D₈₄	semidirect
91. C₃	C₂×#6437	semidirect
93. C₂	#10621357	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×D₈₄	2
C₂×C₃×C₄²×C₇	2
#10621357	2
C₂×#6437	3
C₂×#874	7

Subgroups — C2×#10621357 #13061565

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10621357 #13061565