Subgroups — #13056319

Order 1344.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10621352	C₂	1	2	yes	maximal
2	C₃×C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic
3	#10620657	C₂	1	4	yes	maximal
4	#89454	—	3	1	no	maximal, Hall
5	C₂×Dic₄₂	—	2	2	no	—
6	C₂×C₃×C₇×C₈	C₂²	1	2	yes	—
7	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₂₁	—	2	2	no	—
9	C₄⋊Dic₂₁	C₂²	1	4	yes	—
10	#6432	—	3	2	no	—
11	C₄×C₇×C₈	D₃	1	1	yes	characteristic
12	#6392	—	3	4	no	—
13	#4682	—	7	1	no	maximal, Hall
14	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
15	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
16	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
17	C₂×Dic₂₁	—	2	4	no	—
18	Dic₄₂	—	4	4	no	—
19	C₃×C₇×C₈	Q₈	1	4	yes	complemented
20	C₂×Dic₁₄	—	6	2	no	—
21	C₂×C₇×C₈	C₂×D₃	1	2	yes	—
22	C₄²×C₇	C₂×D₃	1	1	yes	characteristic
23	C₄⋊Dic₇	—	3	4	no	—
24	C₄⋊Dic₇	—	6	2	no	—
25	#869	—	7	2	no	—
26	C₃×C₄×C₈	D₇	1	1	yes	characteristic
27	#818	—	7	4	no	—
28	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
29	C₃×C₄×C₇	D₈	1	2	yes	—
30	C₃×C₄×C₇	Q₁₆	1	2	yes	—
31	C₃×C₄×C₇	C₂×Q₈	1	2	yes	—
32	Dic₂₁	—	4	4	no	—
33	#500	—	21	1	no	Sylow
34	C₂×C₄×C₇	D₁₂	1	1	yes	characteristic
35	C₂×C₄×C₇	D₁₂	1	1	yes	characteristic
36	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
37	C₂×Dic₇	—	6	4	no	—
38	Dic₁₄	—	12	4	no	—
39	C₇×C₈	Dic₆	1	4	yes	complemented
40	C₂×Dic₆	—	14	2	no	—
41	C₂×C₃×C₈	C₂×D₇	1	2	yes	—
42	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
43	C₄⋊Dic₃	—	7	4	no	—
44	C₄⋊Dic₃	—	14	2	no	—
45	C₂×C₃×C₇	C₄⋊Q₈	1	1	yes	characteristic
46	C₂×C₃×C₇	C₂×D₈	1	1	yes	characteristic
47	C₂×C₃×C₇	C₂×Q₁₆	1	1	yes	characteristic
48	C₄⋊Q₈	—	21	2	no	—
49	#107	—	21	4	no	—
50	C₄×C₈	D₂₁	1	1	yes	characteristic
51	C₂²×C₇	C₂×D₁₂	1	1	yes	characteristic
52	C₄×C₇	D₂₄	1	2	yes	—
53	C₄×C₇	Dic₁₂	1	2	yes	—
54	C₄×C₇	C₂×Dic₆	1	2	yes	—
55	Dic₇	—	12	4	no	—
56	C₂×C₃×C₄	D₂₈	1	1	yes	characteristic
57	C₂×C₃×C₄	D₂₈	1	1	yes	characteristic
58	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
59	C₂×Dic₃	—	14	4	no	—
60	Dic₆	—	28	4	no	—
61	C₃×C₈	Dic₁₄	1	4	yes	complemented
62	C₃×C₇	#500	1	1	yes	characteristic, complemented, Hall
63	C₂×Q₈	—	42	2	no	—
64	C₂×C₈	C₂×D₂₁	1	2	yes	—
65	C₄⋊C₄	—	21	4	no	—
66	C₄⋊C₄	—	42	2	no	—
67	C₄²	C₂×D₂₁	1	1	yes	characteristic
68	C₂×C₇	#869	1	1	yes	characteristic
69	C₂×C₇	C₂×D₂₄	1	1	yes	characteristic
70	C₂×C₇	C₂×Dic₁₂	1	1	yes	characteristic
71	C₂²×C₃	C₂×D₂₈	1	1	yes	characteristic
72	C₃×C₄	D₅₆	1	2	yes	—
73	C₃×C₄	Dic₂₈	1	2	yes	—
74	C₃×C₄	C₂×Dic₁₄	1	2	yes	—
75	Dic₃	—	28	4	no	—
76	Q₈	—	84	4	no	—
77	C₂×C₄	—	42	4	no	—
78	C₂×C₄	D₈₄	1	1	yes	characteristic
79	C₂×C₄	D₈₄	1	1	yes	characteristic
80	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
81	C₈	Dic₄₂	1	4	yes	complemented
82	C₇	#4682	1	1	yes	characteristic, complemented, Sylow
83	C₂×C₃	#6432	1	1	yes	characteristic
84	C₂×C₃	C₂×D₅₆	1	1	yes	characteristic
85	C₂×C₃	C₂×Dic₂₈	1	1	yes	characteristic
86	C₂²	C₂×D₈₄	1	1	yes	characteristic
87	C₄	—	84	4	no	—
88	C₄	D₁₆₈	1	2	yes	—
89	C₄	Dic₈₄	1	2	yes	—
90	C₄	C₂×Dic₄₂	1	2	yes	—
91	C₃	#89454	1	1	yes	characteristic, complemented, Sylow
92	C₂	#10621352	1	1	yes	characteristic
93	C₂	C₂×D₁₆₈	1	1	yes	characteristic
94	C₂	C₂×Dic₈₄	1	1	yes	characteristic

	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, prime radical	C₂²×C₃×C₇	C₂×D₄
hypercenter	C₄×C₈	D₂₁
hypocenter	C₃×C₇	#500
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
19. C₃×C₇×C₈	Q₈	semidirect
39. C₇×C₈	Dic₆	semidirect
61. C₃×C₈	Dic₁₄	semidirect
62. C₃×C₇	#500	semidirect
81. C₈	Dic₄₂	semidirect
82. C₇	#4682	semidirect
91. C₃	#89454	semidirect

Maximal subgroups

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

subgroup	index
#10621352	2
C₃×C₄×C₇×C₈	2
#10620657	2
#89454	3
#4682	7