Subgroups — C₃×C₇×#427 #13056938

Order 1344.

228 conjugacy classes of subgroups.

The subgroup lattice has 124 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₇×C₈	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇×#117	C₂	1	2	yes	maximal
3	C₃×C₇×#107	C₂	1	2	yes	maximal, complemented
4	C₃×C₇×#106	C₂	1	2	yes	maximal, complemented
5	C₇×#427	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₇×C₈	C₄	1	4	yes	complemented
8	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₃×C₇×C₄⋊C₄	C₂²	1	4	yes	—
11	C₃×C₇×C₂²⋊C₄	—	2	2	no	—
12	C₃×C₄²×C₇	—	2	2	no	—
13	C₂²×C₇×C₈	C₂×C₃	1	1	yes	characteristic
14	C₇×#117	C₂×C₃	1	2	yes	—
15	C₇×#107	C₂×C₃	1	2	yes	complemented
16	C₇×#106	C₂×C₃	1	2	yes	complemented
17	C₃×#427	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
18	C₂³×C₃×C₇	D₄	1	1	yes	characteristic
19	C₂×C₃×C₄×C₇	—	2	4	no	—
20	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
21	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
22	C₂×C₃×C₄×C₇	Q₈	1	2	yes	—
23	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
24	C₃×C₇×C₈	C₂×C₄	1	2	yes	—
25	C₃×C₇×C₈	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
27	C₂×C₇×C₈	C₃×C₄	1	4	yes	complemented
28	C₂×C₇×C₈	C₂²×C₃	1	1	yes	characteristic
29	C₂×C₇×C₈	C₂²×C₃	1	1	yes	characteristic
30	C₇×C₄⋊C₄	C₂²×C₃	1	4	yes	—
31	C₇×C₂²⋊C₄	—	2	2	no	—
32	C₄²×C₇	—	2	2	no	—
33	C₂²×C₃×C₈	C₂×C₇	1	1	yes	characteristic
34	C₃×#117	C₂×C₇	1	2	yes	—
35	C₃×#107	C₂×C₇	1	2	yes	complemented
36	C₃×#106	C₂×C₇	1	2	yes	complemented
37	C₂²×C₃×C₇	—	2	2	no	—
38	C₂²×C₃×C₇	C₄⋊C₄	1	2	yes	—
39	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
40	C₃×C₄×C₇	—	4	4	no	—
41	C₃×C₄×C₇	C₄⋊C₄	1	2	yes	—
42	C₃×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
43	C₃×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
44	#427	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
45	C₂³×C₇	C₃×D₄	1	1	yes	characteristic
46	C₂×C₄×C₇	—	2	4	no	—
47	C₂×C₄×C₇	C₂×C₃×C₄	1	2	yes	—
48	C₂×C₄×C₇	C₃×D₄	1	1	yes	characteristic
49	C₂×C₄×C₇	C₃×Q₈	1	2	yes	—
50	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
51	C₇×C₈	C₂×C₃×C₄	1	2	yes	—
52	C₇×C₈	C₂×C₃×C₄	1	2	yes	—
53	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
54	C₂×C₃×C₈	C₄×C₇	1	4	yes	complemented
55	C₂×C₃×C₈	C₂²×C₇	1	1	yes	characteristic
56	C₂×C₃×C₈	C₂²×C₇	1	1	yes	characteristic
57	C₃×C₄⋊C₄	C₂²×C₇	1	4	yes	—
58	C₃×C₂²⋊C₄	—	2	2	no	—
59	C₃×C₄²	—	2	2	no	—
60	C₂×C₃×C₇	—	2	2	no	—
61	C₂×C₃×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₃×C₇	#135	1	2	yes	—
63	C₂²×C₈	C₂×C₃×C₇	1	1	yes	characteristic
64	#117	C₂×C₃×C₇	1	2	yes	—
65	#107	C₂×C₃×C₇	1	2	yes	complemented
66	#106	C₂×C₃×C₇	1	2	yes	complemented
67	C₂²×C₇	—	2	2	no	—
68	C₂²×C₇	C₃×C₄⋊C₄	1	2	yes	—
69	C₂²×C₇	C₂×C₃×D₄	1	1	yes	characteristic
70	C₄×C₇	—	4	4	no	—
71	C₄×C₇	C₃×C₄⋊C₄	1	2	yes	—
72	C₄×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
73	C₄×C₇	C₂×C₃×Q₈	1	1	yes	characteristic
74	C₂³×C₃	C₇×D₄	1	1	yes	characteristic
75	C₂×C₃×C₄	—	2	4	no	—
76	C₂×C₃×C₄	C₂×C₄×C₇	1	2	yes	—
77	C₂×C₃×C₄	C₇×D₄	1	1	yes	characteristic
78	C₂×C₃×C₄	C₇×Q₈	1	2	yes	—
79	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
80	C₃×C₈	C₂×C₄×C₇	1	2	yes	—
81	C₃×C₈	C₂×C₄×C₇	1	2	yes	—
82	C₃×C₇	#427	1	1	yes	characteristic, direct product factor, Hall
83	C₂²×C₄	C₂²×C₃×C₇	1	1	yes	characteristic
84	C₂×C₈	C₃×C₄×C₇	1	4	yes	complemented
85	C₂×C₈	C₂²×C₃×C₇	1	1	yes	characteristic
86	C₂×C₈	C₂²×C₃×C₇	1	1	yes	characteristic
87	C₄⋊C₄	C₂²×C₃×C₇	1	4	yes	—
88	C₂²⋊C₄	—	2	2	no	—
89	C₄²	—	2	2	no	—
90	C₂×C₇	—	2	2	no	—
91	C₂×C₇	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₇	C₃×#135	1	2	yes	—
93	C₂²×C₃	—	2	2	no	—
94	C₂²×C₃	C₇×C₄⋊C₄	1	2	yes	—
95	C₂²×C₃	C₂×C₇×D₄	1	1	yes	characteristic
96	C₃×C₄	—	4	4	no	—
97	C₃×C₄	C₇×C₄⋊C₄	1	2	yes	—
98	C₃×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
99	C₃×C₄	C₂×C₇×Q₈	1	1	yes	characteristic
100	C₂³	C₃×C₇×D₄	1	1	yes	characteristic
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	C₂×C₃×C₄×C₇	1	2	yes	—
103	C₂×C₄	C₃×C₇×D₄	1	1	yes	characteristic
104	C₂×C₄	C₃×C₇×Q₈	1	2	yes	—
105	C₂×C₄	C₂³×C₃×C₇	1	1	yes	characteristic
106	C₈	C₂×C₃×C₄×C₇	1	2	yes	—
107	C₈	C₂×C₃×C₄×C₇	1	2	yes	—
108	C₇	C₃×#427	1	1	yes	characteristic, direct product factor, Sylow
109	C₂×C₃	—	2	2	no	—
110	C₂×C₃	C₂×C₇×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₃	C₇×#135	1	2	yes	—
112	C₂²	—	2	2	no	—
113	C₂²	C₃×C₇×C₄⋊C₄	1	2	yes	—
114	C₂²	C₂×C₃×C₇×D₄	1	1	yes	characteristic
115	C₄	—	4	4	no	—
116	C₄	C₃×C₇×C₄⋊C₄	1	2	yes	—
117	C₄	C₂²×C₃×C₄×C₇	1	1	yes	characteristic
118	C₄	C₂×C₃×C₇×Q₈	1	1	yes	characteristic
119	C₃	C₇×#427	1	1	yes	characteristic, direct product factor, Sylow
120	C₂	—	2	2	no	—
121	C₂	C₂×C₃×C₇×C₄⋊C₄	1	1	yes	characteristic
122	C₂	C₃×C₇×#135	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₄×C₇	D₄
commutator	C₄	C₂²×C₃×C₄×C₇
Frattini, cosocle	C₂×C₄	C₂³×C₃×C₇
socle	C₂²×C₃×C₇	C₂×D₄
prime radical	C₂³×C₃×C₇	D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄×C₇ ◁ C₂²×C₃×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×C₇×#107	C₂	semidirect
4. C₃×C₇×#106	C₂	semidirect
5. C₇×#427	C₃	direct (N × H)
7. C₂×C₃×C₇×C₈	C₄	semidirect
15. C₇×#107	C₂×C₃	semidirect
16. C₇×#106	C₂×C₃	semidirect
17. C₃×#427	C₇	direct (N × H)
27. C₂×C₇×C₈	C₃×C₄	semidirect
35. C₃×#107	C₂×C₇	semidirect
36. C₃×#106	C₂×C₇	semidirect
44. #427	C₃×C₇	direct (N × H)
54. C₂×C₃×C₈	C₄×C₇	semidirect
65. #107	C₂×C₃×C₇	semidirect
66. #106	C₂×C₃×C₇	semidirect
82. C₃×C₇	#427	direct (N × H)
84. C₂×C₈	C₃×C₄×C₇	semidirect
108. C₇	C₃×#427	direct (N × H)
119. C₃	C₇×#427	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×C₈	2
C₃×C₇×#117	2
C₃×C₇×#107	2
C₃×C₇×#106	2
C₇×#427	3
C₃×#427	7

Subgroups — C3×C7×#427 #13056938

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇×#427 #13056938