Subgroups — C₃×#90503 #13061151

Order 1344.

532 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₃×Q₈×Dic₇	C₂	1	8	yes	maximal, complemented
3	C₃×#6489	C₂	1	6	yes	maximal
4	#90503	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
5	C₂×C₃×C₇×Q₈	C₄	1	8	yes	complemented
6	C₂×C₃×C₇×Q₈	C₂²	1	4	yes	—
7	C₂²×C₃×C₄×C₇	C₂²	1	3	yes	—
8	C₃×C₂²⋊Dic₇	C₂²	1	4	yes	—
9	C₃×C₄⋊Dic₇	C₂²	1	12	yes	—
10	C₃×C₄×Dic₇	C₂²	1	12	yes	—
11	C₂²×C₇×Q₈	C₂×C₃	1	1	yes	characteristic
12	Q₈×Dic₇	C₂×C₃	1	8	yes	complemented
13	#6489	C₂×C₃	1	6	yes	—
14	C₃×#519	—	7	1	no	maximal, Hall
15	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
16	C₃×C₇×Q₈	C₂×C₄	1	16	yes	—
17	C₂×C₃×C₄×C₇	C₂×C₄	1	12	yes	—
18	C₂×C₃×C₄×C₇	C₂³	1	6	yes	—
19	C₂×C₃×Dic₇	C₂³	1	8	yes	—
20	C₂×C₇×Q₈	C₃×C₄	1	8	yes	complemented
21	C₂×C₇×Q₈	C₂²×C₃	1	4	yes	—
22	C₂²×C₄×C₇	C₂²×C₃	1	3	yes	—
23	C₂²⋊Dic₇	C₂²×C₃	1	4	yes	—
24	C₄⋊Dic₇	C₂²×C₃	1	12	yes	—
25	C₄×Dic₇	C₂²×C₃	1	12	yes	—
26	C₂²×C₃×Q₈	D₇	1	1	yes	characteristic
27	C₃×C₄×Q₈	—	7	8	no	—
28	C₃×#117	—	7	6	no	—
29	C₂²×C₃×C₇	—	2	2	no	—
30	C₂²×C₃×C₇	C₂²×C₄	1	2	yes	—
31	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
32	C₃×C₄×C₇	C₂²×C₄	1	12	yes	—
33	C₃×Dic₇	—	2	8	no	—
34	#519	—	7	1	no	Sylow
35	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
36	C₇×Q₈	C₂×C₃×C₄	1	16	yes	—
37	C₂×C₄×C₇	C₂×C₃×C₄	1	12	yes	—
38	C₂×C₄×C₇	C₂³×C₃	1	6	yes	—
39	C₂×Dic₇	C₂³×C₃	1	8	yes	—
40	C₂×C₃×Q₈	Dic₇	1	8	yes	complemented
41	C₂×C₃×Q₈	C₂×D₇	1	4	yes	—
42	C₂²×C₃×C₄	C₂×D₇	1	3	yes	—
43	C₃×C₄⋊C₄	—	7	12	no	—
44	C₃×C₂²⋊C₄	—	7	4	no	—
45	C₃×C₄²	—	7	12	no	—
46	C₂×C₃×C₇	—	2	2	no	—
47	C₂×C₃×C₇	C₂³×C₄	1	1	yes	characteristic
48	C₂×C₃×C₇	2₋¹⁺⁴	1	2	yes	—
49	C₂²×Q₈	C₃×D₇	1	1	yes	characteristic
50	C₄×Q₈	—	7	8	no	—
51	#117	—	7	6	no	—
52	C₂²×C₇	—	2	2	no	—
53	C₂²×C₇	C₂²×C₃×C₄	1	2	yes	—
54	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
55	C₄×C₇	C₂²×C₃×C₄	1	12	yes	—
56	Dic₇	—	2	8	no	—
57	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
58	C₃×Q₈	C₂×Dic₇	1	16	yes	—
59	C₂×C₃×C₄	—	7	8	no	—
60	C₂×C₃×C₄	C₂×Dic₇	1	12	yes	—
61	C₂×C₃×C₄	C₂²×D₇	1	6	yes	—
62	C₃×C₇	#519	1	1	yes	characteristic, complemented, Hall
63	C₂×Q₈	C₃×Dic₇	1	8	yes	complemented
64	C₂×Q₈	C₂×C₃×D₇	1	4	yes	—
65	C₂²×C₄	C₂×C₃×D₇	1	3	yes	—
66	C₄⋊C₄	—	7	12	no	—
67	C₂²⋊C₄	—	7	4	no	—
68	C₄²	—	7	12	no	—
69	C₂×C₇	—	2	2	no	—
70	C₂×C₇	C₂³×C₃×C₄	1	1	yes	characteristic
71	C₂×C₇	C₃×2₋¹⁺⁴	1	2	yes	—
72	C₂²×C₃	—	2	2	no	—
73	C₂²×C₃	C₂²×Dic₇	1	2	yes	—
74	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
75	C₃×C₄	—	14	8	no	—
76	C₃×C₄	C₂²×Dic₇	1	12	yes	—
77	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
78	Q₈	C₂×C₃×Dic₇	1	16	yes	—
79	C₂×C₄	—	7	8	no	—
80	C₂×C₄	C₂×C₃×Dic₇	1	12	yes	—
81	C₂×C₄	C₂²×C₃×D₇	1	6	yes	—
82	C₇	C₃×#519	1	1	yes	characteristic, complemented, Sylow
83	C₂×C₃	—	2	2	no	—
84	C₂×C₃	#6551	1	2	yes	—
85	C₂×C₃	C₂³×Dic₇	1	1	yes	characteristic
86	C₂²	—	2	2	no	—
87	C₂²	C₂²×C₃×Dic₇	1	2	yes	—
88	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
89	C₄	—	14	8	no	—
90	C₄	C₂²×C₃×Dic₇	1	12	yes	—
91	C₃	#90503	1	1	yes	characteristic, direct product factor, Sylow
92	C₂	—	2	2	no	—
93	C₂	C₃×#6551	1	2	yes	—
94	C₂	C₂³×C₃×Dic₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₇
commutator	C₂×C₇	C₂³×C₃×C₄
Frattini	C₂²	C₂³×C₃×D₇
Fitting	C₂²×C₃×C₇×Q₈	C₂
socle	C₂²×C₃×C₇	C₂⁴
prime radical	C₂³×C₃×C₇	C₂³
cosocle	C₂²×C₇	C₂⁴×C₃
hypercenter	C₂²×C₃×Q₈	D₇
hypocenter	C₇	C₃×#519
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×Q₈

Complements

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

N	H ≅ G/N	type
2. C₃×Q₈×Dic₇	C₂	semidirect
4. #90503	C₃	direct (N × H)
5. C₂×C₃×C₇×Q₈	C₄	semidirect
12. Q₈×Dic₇	C₂×C₃	semidirect
20. C₂×C₇×Q₈	C₃×C₄	semidirect
40. C₂×C₃×Q₈	Dic₇	semidirect
62. C₃×C₇	#519	semidirect
63. C₂×Q₈	C₃×Dic₇	semidirect
82. C₇	C₃×#519	semidirect
91. C₃	#90503	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×C₃×C₇×Q₈	2
C₃×Q₈×Dic₇	2
C₃×#6489	2
#90503	3
C₃×#519	7

Subgroups — C3×#90503 #13061151

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90503 #13061151