Subgroups — C₃×C₇×#405 #13056916

Order 1344.

260 conjugacy classes of subgroups.

The subgroup lattice has 140 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₇×C₈	C₂	1	1	yes	maximal, characteristic
3	C₃×C₇×#117	C₂	1	1	yes	maximal, characteristic
4	C₃×C₇×#97	C₂	1	2	yes	maximal, complemented
5	C₃×C₄×C₇×C₈	C₂	1	2	yes	maximal, complemented
6	C₇×#405	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
8	C₃×C₇×M₁₆	C₄	1	4	yes	complemented
9	C₂×C₃×C₇×C₈	C₄	1	4	yes	complemented
10	C₂×C₃×C₇×C₈	C₂²	1	2	yes	—
11	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
13	C₃×C₇×C₄⋊C₄	C₄	1	2	yes	—
14	C₃×C₇×C₂²⋊C₄	C₄	1	2	yes	—
15	C₃×C₄²×C₇	C₂²	1	2	yes	—
16	C₂×C₇×M₁₆	C₂×C₃	1	1	yes	characteristic
17	C₂²×C₇×C₈	C₂×C₃	1	1	yes	characteristic
18	C₇×#117	C₂×C₃	1	1	yes	characteristic
19	C₇×#97	C₂×C₃	1	2	yes	complemented
20	C₄×C₇×C₈	C₂×C₃	1	2	yes	complemented
21	C₃×#405	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
22	C₂³×C₃×C₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₃×C₄×C₇	C₂×C₄	1	4	yes	—
24	C₂×C₃×C₄×C₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
26	C₂×C₃×C₄×C₇	C₂×C₄	1	2	yes	—
27	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
28	C₃×C₇×C₈	C₂×C₄	1	2	yes	—
29	C₃×C₇×C₈	C₂×C₄	1	2	yes	—
30	C₃×C₇×C₈	C₂×C₄	1	4	yes	—
31	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
32	C₇×M₁₆	C₃×C₄	1	4	yes	complemented
33	C₂×C₇×C₈	C₃×C₄	1	4	yes	complemented
34	C₂×C₇×C₈	C₂²×C₃	1	1	yes	characteristic
35	C₂×C₇×C₈	C₂²×C₃	1	2	yes	—
36	C₂×C₇×C₈	C₂²×C₃	1	1	yes	characteristic
37	C₇×C₄⋊C₄	C₃×C₄	1	2	yes	—
38	C₇×C₂²⋊C₄	C₃×C₄	1	2	yes	—
39	C₄²×C₇	C₂²×C₃	1	2	yes	—
40	C₂×C₃×M₁₆	C₂×C₇	1	1	yes	characteristic
41	C₂²×C₃×C₈	C₂×C₇	1	1	yes	characteristic
42	C₃×#117	C₂×C₇	1	1	yes	characteristic
43	C₃×#97	C₂×C₇	1	2	yes	complemented
44	C₃×C₄×C₈	C₂×C₇	1	2	yes	complemented
45	C₂²×C₃×C₇	—	2	2	no	—
46	C₂²×C₃×C₇	C₄²	1	2	yes	—
47	C₂²×C₃×C₇	C₂²×C₄	1	1	yes	characteristic
48	C₃×C₄×C₇	—	2	4	no	—
49	C₃×C₄×C₇	C₄²	1	2	yes	—
50	C₃×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
51	C₃×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
52	#405	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
53	C₂³×C₇	C₂×C₃×C₄	1	1	yes	characteristic
54	C₂×C₄×C₇	C₂×C₃×C₄	1	4	yes	—
55	C₂×C₄×C₇	C₂×C₃×C₄	1	1	yes	characteristic
56	C₂×C₄×C₇	C₂×C₃×C₄	1	2	yes	—
57	C₂×C₄×C₇	C₂×C₃×C₄	1	2	yes	—
58	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
59	C₇×C₈	C₂×C₃×C₄	1	2	yes	—
60	C₇×C₈	C₂×C₃×C₄	1	2	yes	—
61	C₇×C₈	C₂×C₃×C₄	1	4	yes	—
62	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
63	C₃×M₁₆	C₄×C₇	1	4	yes	complemented
64	C₂×C₃×C₈	C₄×C₇	1	4	yes	complemented
65	C₂×C₃×C₈	C₂²×C₇	1	1	yes	characteristic
66	C₂×C₃×C₈	C₂²×C₇	1	2	yes	—
67	C₂×C₃×C₈	C₂²×C₇	1	1	yes	characteristic
68	C₃×C₄⋊C₄	C₄×C₇	1	2	yes	—
69	C₃×C₂²⋊C₄	C₄×C₇	1	2	yes	—
70	C₃×C₄²	C₂²×C₇	1	2	yes	—
71	C₂×C₃×C₇	—	2	2	no	—
72	C₂×C₃×C₇	C₂×C₄²	1	1	yes	characteristic
73	C₂×C₃×C₇	#131	1	2	yes	—
74	C₂×M₁₆	C₂×C₃×C₇	1	1	yes	characteristic
75	C₂²×C₈	C₂×C₃×C₇	1	1	yes	characteristic
76	#117	C₂×C₃×C₇	1	1	yes	characteristic
77	#97	C₂×C₃×C₇	1	2	yes	complemented
78	C₄×C₈	C₂×C₃×C₇	1	2	yes	complemented
79	C₂²×C₇	—	2	2	no	—
80	C₂²×C₇	C₃×C₄²	1	2	yes	—
81	C₂²×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
82	C₄×C₇	—	2	4	no	—
83	C₄×C₇	C₃×C₄²	1	2	yes	—
84	C₄×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
85	C₄×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
86	C₂³×C₃	C₂×C₄×C₇	1	1	yes	characteristic
87	C₂×C₃×C₄	C₂×C₄×C₇	1	4	yes	—
88	C₂×C₃×C₄	C₂×C₄×C₇	1	1	yes	characteristic
89	C₂×C₃×C₄	C₂×C₄×C₇	1	2	yes	—
90	C₂×C₃×C₄	C₂×C₄×C₇	1	2	yes	—
91	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
92	C₃×C₈	C₂×C₄×C₇	1	2	yes	—
93	C₃×C₈	C₂×C₄×C₇	1	2	yes	—
94	C₃×C₈	C₂×C₄×C₇	1	4	yes	—
95	C₃×C₇	#405	1	1	yes	characteristic, direct product factor, Hall
96	C₂²×C₄	C₂²×C₃×C₇	1	1	yes	characteristic
97	M₁₆	C₃×C₄×C₇	1	4	yes	complemented
98	C₂×C₈	C₃×C₄×C₇	1	4	yes	complemented
99	C₂×C₈	C₂²×C₃×C₇	1	1	yes	characteristic
100	C₂×C₈	C₂²×C₃×C₇	1	2	yes	—
101	C₂×C₈	C₂²×C₃×C₇	1	1	yes	characteristic
102	C₄⋊C₄	C₃×C₄×C₇	1	2	yes	—
103	C₂²⋊C₄	C₃×C₄×C₇	1	2	yes	—
104	C₄²	C₂²×C₃×C₇	1	2	yes	—
105	C₂×C₇	—	2	2	no	—
106	C₂×C₇	C₂×C₃×C₄²	1	1	yes	characteristic
107	C₂×C₇	C₃×#131	1	2	yes	—
108	C₂²×C₃	—	2	2	no	—
109	C₂²×C₃	C₄²×C₇	1	2	yes	—
110	C₂²×C₃	C₂²×C₄×C₇	1	1	yes	characteristic
111	C₃×C₄	—	2	4	no	—
112	C₃×C₄	C₄²×C₇	1	2	yes	—
113	C₃×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
114	C₃×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
115	C₂³	C₂×C₃×C₄×C₇	1	1	yes	characteristic
116	C₂×C₄	C₂×C₃×C₄×C₇	1	4	yes	—
117	C₂×C₄	C₂×C₃×C₄×C₇	1	1	yes	characteristic
118	C₂×C₄	C₂×C₃×C₄×C₇	1	2	yes	—
119	C₂×C₄	C₂×C₃×C₄×C₇	1	2	yes	—
120	C₂×C₄	C₂³×C₃×C₇	1	1	yes	characteristic
121	C₈	C₂×C₃×C₄×C₇	1	2	yes	—
122	C₈	C₂×C₃×C₄×C₇	1	2	yes	—
123	C₈	C₂×C₃×C₄×C₇	1	4	yes	—
124	C₇	C₃×#405	1	1	yes	characteristic, direct product factor, Sylow
125	C₂×C₃	—	2	2	no	—
126	C₂×C₃	C₂×C₄²×C₇	1	1	yes	characteristic
127	C₂×C₃	C₇×#131	1	2	yes	—
128	C₂²	—	2	2	no	—
129	C₂²	C₃×C₄²×C₇	1	2	yes	—
130	C₂²	C₂²×C₃×C₄×C₇	1	1	yes	characteristic
131	C₄	—	2	4	no	—
132	C₄	C₃×C₄²×C₇	1	2	yes	—
133	C₄	C₂²×C₃×C₄×C₇	1	1	yes	characteristic
134	C₄	C₂²×C₃×C₄×C₇	1	1	yes	characteristic
135	C₃	C₇×#405	1	1	yes	characteristic, direct product factor, Sylow
136	C₂	—	2	2	no	—
137	C₂	C₂×C₃×C₄²×C₇	1	1	yes	characteristic
138	C₂	C₃×C₇×#131	1	2	yes	—

	subgroup	G/H
center	C₂×C₃×C₇×C₈	C₂²
commutator	C₂	C₂×C₃×C₄²×C₇
Frattini, cosocle	C₂×C₄	C₂³×C₃×C₇
socle	C₂²×C₃×C₇	C₂²×C₄
prime radical	C₂³×C₃×C₇	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃×C₇×#97	C₂	semidirect
5. C₃×C₄×C₇×C₈	C₂	semidirect
6. C₇×#405	C₃	direct (N × H)
8. C₃×C₇×M₁₆	C₄	semidirect
9. C₂×C₃×C₇×C₈	C₄	semidirect
19. C₇×#97	C₂×C₃	semidirect
20. C₄×C₇×C₈	C₂×C₃	semidirect
21. C₃×#405	C₇	direct (N × H)
32. C₇×M₁₆	C₃×C₄	semidirect
33. C₂×C₇×C₈	C₃×C₄	semidirect
43. C₃×#97	C₂×C₇	semidirect
44. C₃×C₄×C₈	C₂×C₇	semidirect
52. #405	C₃×C₇	direct (N × H)
63. C₃×M₁₆	C₄×C₇	semidirect
64. C₂×C₃×C₈	C₄×C₇	semidirect
77. #97	C₂×C₃×C₇	semidirect
78. C₄×C₈	C₂×C₃×C₇	semidirect
95. C₃×C₇	#405	direct (N × H)
97. M₁₆	C₃×C₄×C₇	semidirect
98. C₂×C₈	C₃×C₄×C₇	semidirect
124. C₇	C₃×#405	direct (N × H)
135. C₃	C₇×#405	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₇×M₁₆	2
C₂²×C₃×C₇×C₈	2
C₃×C₇×#117	2
C₃×C₇×#97	2
C₃×C₄×C₇×C₈	2
C₇×#405	3
C₃×#405	7

Subgroups — C3×C7×#405 #13056916

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇×#405 #13056916