Subgroups — #13056320

Order 1344.

172 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#10621353	C₂	1	2	yes	maximal
2	C₃×C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic
3	#10620657	C₂	1	2	yes	maximal
4	#10620656	C₂	1	2	yes	maximal
5	#89455	—	3	1	no	maximal, Hall
6	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₂₁	C₂²	1	4	yes	—
10	#66629	—	2	4	no	—
11	#6433	—	3	2	no	—
12	C₄×C₇×C₈	D₃	1	1	yes	characteristic
13	#6392	—	3	2	no	—
14	#6391	—	3	2	no	—
15	#4683	—	7	1	no	maximal, Hall
16	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
17	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
18	C₂×Dic₂₁	—	2	4	no	—
19	C₃×C₇×C₈	Q₈	1	2	yes	—
20	C₃×C₇×C₈	Q₈	1	2	yes	—
21	C₂×C₇×C₈	C₂×D₃	1	1	yes	characteristic
22	C₂×C₇×C₈	C₂×D₃	1	1	yes	characteristic
23	C₄²×C₇	C₂×D₃	1	1	yes	characteristic
24	C₄⋊Dic₇	—	3	4	no	—
25	Dic₇⋊C₄	—	6	4	no	—
26	#870	—	7	2	no	—
27	C₃×C₄×C₈	D₇	1	1	yes	characteristic
28	#818	—	7	2	no	—
29	#817	—	7	2	no	—
30	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
31	C₃×C₄×C₇	—	2	2	no	—
32	C₃×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
33	C₃×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
34	Dic₂₁	—	4	4	no	—
35	#499	—	21	1	no	Sylow
36	C₂×C₄×C₇	D₁₂	1	2	yes	—
37	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
38	C₂×Dic₇	—	6	4	no	—
39	C₇×C₈	Dic₆	1	2	yes	—
40	C₇×C₈	Dic₆	1	2	yes	—
41	C₂×C₃×C₈	C₂×D₇	1	1	yes	characteristic
42	C₂×C₃×C₈	C₂×D₇	1	1	yes	characteristic
43	C₃×C₄²	C₂×D₇	1	1	yes	characteristic
44	C₄⋊Dic₃	—	7	4	no	—
45	Dic₃⋊C₄	—	14	4	no	—
46	C₂×C₃×C₇	C₄⋊Q₈	1	1	yes	characteristic
47	C₂×C₃×C₇	#135	1	2	yes	—
48	#125	—	21	2	no	—
49	#107	—	21	2	no	—
50	#106	—	21	2	no	—
51	C₄×C₈	D₂₁	1	1	yes	characteristic
52	C₂²×C₇	C₂×D₁₂	1	1	yes	characteristic
53	C₄×C₇	—	2	2	no	—
54	C₄×C₇	C₂×Dic₆	1	1	yes	characteristic
55	C₄×C₇	C₂×Dic₆	1	1	yes	characteristic
56	Dic₇	—	12	4	no	—
57	C₂×C₃×C₄	D₂₈	1	2	yes	—
58	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
59	C₂×Dic₃	—	14	4	no	—
60	C₃×C₈	Dic₁₄	1	2	yes	—
61	C₃×C₈	Dic₁₄	1	2	yes	—
62	C₃×C₇	#499	1	1	yes	characteristic, complemented, Hall
63	C₂×C₈	C₂×D₂₁	1	1	yes	characteristic
64	C₂×C₈	C₂×D₂₁	1	1	yes	characteristic
65	C₄⋊C₄	—	21	4	no	—
66	C₄⋊C₄	—	42	4	no	—
67	C₄²	C₂×D₂₁	1	1	yes	characteristic
68	C₂×C₇	#869	1	1	yes	characteristic
69	C₂×C₇	#904	1	2	yes	—
70	C₂²×C₃	C₂×D₂₈	1	1	yes	characteristic
71	C₃×C₄	—	2	2	no	—
72	C₃×C₄	C₂×Dic₁₄	1	1	yes	characteristic
73	C₃×C₄	C₂×Dic₁₄	1	1	yes	characteristic
74	Dic₃	—	28	4	no	—
75	C₂×C₄	—	42	4	no	—
76	C₂×C₄	D₈₄	1	2	yes	—
77	C₂×C₄	C₂²×D₂₁	1	1	yes	characteristic
78	C₈	Dic₄₂	1	2	yes	—
79	C₈	Dic₄₂	1	2	yes	—
80	C₇	#4683	1	1	yes	characteristic, complemented, Sylow
81	C₂×C₃	#6432	1	1	yes	characteristic
82	C₂×C₃	#6467	1	2	yes	—
83	C₂²	C₂×D₈₄	1	1	yes	characteristic
84	C₄	—	2	2	no	—
85	C₄	—	84	4	no	—
86	C₄	C₂×Dic₄₂	1	1	yes	characteristic
87	C₄	C₂×Dic₄₂	1	1	yes	characteristic
88	C₃	#89455	1	1	yes	characteristic, complemented, Sylow
89	C₂	#10621352	1	1	yes	characteristic
90	C₂	#10621387	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, prime radical	C₂²×C₃×C₇	C₂×D₄
hypercenter	C₄×C₈	D₂₁
hypocenter	C₃×C₇	#499
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
62. C₃×C₇	#499	semidirect
80. C₇	#4683	semidirect
88. C₃	#89455	semidirect

Maximal subgroups

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

subgroup	index
#10621353	2
C₃×C₄×C₇×C₈	2
#10620657	2
#10620656	2
#89455	3
#4683	7