Subgroups — C₃×C₇×#443 #13056954

Order 1344.

212 conjugacy classes of subgroups.

The subgroup lattice has 148 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×#135	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇×#131	C₂	1	2	yes	maximal, complemented
3	C₃×C₇×#108	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₇×#104	C₂	1	2	yes	maximal, complemented
5	C₃×C₄×C₇×C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#443	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×C₇×#41	C₂²	1	2	yes	—
8	C₃×C₇×Q₁₆	C₄	1	1	yes	characteristic, complemented
9	C₃×C₇×SD₁₆	C₄	1	2	yes	complemented
10	C₃×C₇×D₈	C₄	1	1	yes	characteristic, complemented
11	C₃×C₇×M₁₆	—	2	2	no	—
12	C₃×C₇×M₁₆	C₂²	1	2	yes	—
13	C₂×C₃×C₇×C₈	—	2	2	no	—
14	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃×C₇×C₈	C₂²	1	1	yes	characteristic
16	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
17	C₇×#135	C₂×C₃	1	1	yes	characteristic
18	C₇×#131	C₂×C₃	1	2	yes	complemented
19	C₇×#108	C₂×C₃	1	1	yes	characteristic, complemented
20	C₇×#104	C₂×C₃	1	2	yes	complemented
21	C₄×C₇×C₈	C₂×C₃	1	1	yes	characteristic, complemented
22	C₃×#443	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₃×C₇×Q₈	C₂×C₄	1	2	yes	—
24	C₃×C₇×D₄	—	2	2	no	—
25	C₃×C₇×D₄	C₂×C₄	1	2	yes	—
26	C₂×C₃×C₄×C₇	—	2	1	no	—
27	C₂×C₃×C₄×C₇	—	2	2	no	—
28	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
29	C₃×C₇×C₈	—	2	2	no	—
30	C₃×C₇×C₈	C₂×C₄	1	1	yes	characteristic
31	C₃×C₇×C₈	C₂×C₄	1	1	yes	characteristic
32	C₃×C₇×C₈	D₄	1	1	yes	characteristic
33	C₃×C₇×C₈	D₄	1	1	yes	characteristic
34	C₇×#41	C₂²×C₃	1	2	yes	—
35	C₇×Q₁₆	C₃×C₄	1	1	yes	characteristic, complemented
36	C₇×SD₁₆	C₃×C₄	1	2	yes	complemented
37	C₇×D₈	C₃×C₄	1	1	yes	characteristic, complemented
38	C₇×M₁₆	—	2	2	no	—
39	C₇×M₁₆	C₂²×C₃	1	2	yes	—
40	C₂×C₇×C₈	—	2	2	no	—
41	C₂×C₇×C₈	C₂²×C₃	1	1	yes	characteristic
42	C₂×C₇×C₈	C₂²×C₃	1	1	yes	characteristic
43	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
44	C₃×#135	C₂×C₇	1	1	yes	characteristic
45	C₃×#131	C₂×C₇	1	2	yes	complemented
46	C₃×#108	C₂×C₇	1	1	yes	characteristic, complemented
47	C₃×#104	C₂×C₇	1	2	yes	complemented
48	C₃×C₄×C₈	C₂×C₇	1	1	yes	characteristic, complemented
49	C₂²×C₃×C₇	—	2	2	no	—
50	C₂²×C₃×C₇	#41	1	1	yes	characteristic
51	C₃×C₄×C₇	—	2	2	no	—
52	C₃×C₄×C₇	—	2	2	no	—
53	C₃×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
54	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
55	#443	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
56	C₇×Q₈	C₂×C₃×C₄	1	2	yes	—
57	C₇×D₄	—	2	2	no	—
58	C₇×D₄	C₂×C₃×C₄	1	2	yes	—
59	C₂×C₄×C₇	—	2	1	no	—
60	C₂×C₄×C₇	—	2	2	no	—
61	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
62	C₇×C₈	—	2	2	no	—
63	C₇×C₈	C₂×C₃×C₄	1	1	yes	characteristic
64	C₇×C₈	C₂×C₃×C₄	1	1	yes	characteristic
65	C₇×C₈	C₃×D₄	1	1	yes	characteristic
66	C₇×C₈	C₃×D₄	1	1	yes	characteristic
67	C₃×#41	C₂²×C₇	1	2	yes	—
68	C₃×Q₁₆	C₄×C₇	1	1	yes	characteristic, complemented
69	C₃×SD₁₆	C₄×C₇	1	2	yes	complemented
70	C₃×D₈	C₄×C₇	1	1	yes	characteristic, complemented
71	C₃×M₁₆	—	2	2	no	—
72	C₃×M₁₆	C₂²×C₇	1	2	yes	—
73	C₂×C₃×C₈	—	2	2	no	—
74	C₂×C₃×C₈	C₂²×C₇	1	1	yes	characteristic
75	C₂×C₃×C₈	C₂²×C₇	1	1	yes	characteristic
76	C₃×C₄²	C₂²×C₇	1	1	yes	characteristic
77	C₂×C₃×C₇	—	2	1	no	—
78	C₂×C₃×C₇	—	4	2	no	—
79	C₂×C₃×C₇	C₄×D₄	1	1	yes	characteristic
80	#135	C₂×C₃×C₇	1	1	yes	characteristic
81	#131	C₂×C₃×C₇	1	2	yes	complemented
82	#108	C₂×C₃×C₇	1	1	yes	characteristic, complemented
83	#104	C₂×C₃×C₇	1	2	yes	complemented
84	C₄×C₈	C₂×C₃×C₇	1	1	yes	characteristic, complemented
85	C₂²×C₇	—	2	2	no	—
86	C₂²×C₇	C₃×#41	1	1	yes	characteristic
87	C₄×C₇	—	2	2	no	—
88	C₄×C₇	—	2	2	no	—
89	C₄×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
90	C₄×C₇	C₂×C₃×D₄	1	1	yes	characteristic
91	C₃×Q₈	C₂×C₄×C₇	1	2	yes	—
92	C₃×D₄	—	2	2	no	—
93	C₃×D₄	C₂×C₄×C₇	1	2	yes	—
94	C₂×C₃×C₄	—	2	1	no	—
95	C₂×C₃×C₄	—	2	2	no	—
96	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
97	C₃×C₈	—	2	2	no	—
98	C₃×C₈	C₂×C₄×C₇	1	1	yes	characteristic
99	C₃×C₈	C₂×C₄×C₇	1	1	yes	characteristic
100	C₃×C₈	C₇×D₄	1	1	yes	characteristic
101	C₃×C₈	C₇×D₄	1	1	yes	characteristic
102	C₃×C₇	#443	1	1	yes	characteristic, direct product factor, Hall
103	#41	C₂²×C₃×C₇	1	2	yes	—
104	Q₁₆	C₃×C₄×C₇	1	1	yes	characteristic, complemented
105	SD₁₆	C₃×C₄×C₇	1	2	yes	complemented
106	D₈	C₃×C₄×C₇	1	1	yes	characteristic, complemented
107	M₁₆	—	2	2	no	—
108	M₁₆	C₂²×C₃×C₇	1	2	yes	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	C₂²×C₃×C₇	1	1	yes	characteristic
111	C₂×C₈	C₂²×C₃×C₇	1	1	yes	characteristic
112	C₄²	C₂²×C₃×C₇	1	1	yes	characteristic
113	C₂×C₇	—	2	1	no	—
114	C₂×C₇	—	4	2	no	—
115	C₂×C₇	C₃×C₄×D₄	1	1	yes	characteristic
116	C₂²×C₃	—	2	2	no	—
117	C₂²×C₃	C₇×#41	1	1	yes	characteristic
118	C₃×C₄	—	2	2	no	—
119	C₃×C₄	—	2	2	no	—
120	C₃×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
121	C₃×C₄	C₂×C₇×D₄	1	1	yes	characteristic
122	Q₈	C₂×C₃×C₄×C₇	1	2	yes	—
123	D₄	—	2	2	no	—
124	D₄	C₂×C₃×C₄×C₇	1	2	yes	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	C₂³×C₃×C₇	1	1	yes	characteristic
128	C₈	—	2	2	no	—
129	C₈	C₂×C₃×C₄×C₇	1	1	yes	characteristic
130	C₈	C₂×C₃×C₄×C₇	1	1	yes	characteristic
131	C₈	C₃×C₇×D₄	1	1	yes	characteristic
132	C₈	C₃×C₇×D₄	1	1	yes	characteristic
133	C₇	C₃×#443	1	1	yes	characteristic, direct product factor, Sylow
134	C₂×C₃	—	2	1	no	—
135	C₂×C₃	—	4	2	no	—
136	C₂×C₃	C₄×C₇×D₄	1	1	yes	characteristic
137	C₂²	—	2	2	no	—
138	C₂²	C₃×C₇×#41	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	2	2	no	—
141	C₄	C₂²×C₃×C₄×C₇	1	1	yes	characteristic
142	C₄	C₂×C₃×C₇×D₄	1	1	yes	characteristic
143	C₃	C₇×#443	1	1	yes	characteristic, direct product factor, Sylow
144	C₂	—	2	1	no	—
145	C₂	—	4	2	no	—
146	C₂	C₃×C₄×C₇×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₇×C₈	D₄
commutator	C₄	C₂²×C₃×C₄×C₇
Frattini, cosocle	C₂×C₄	C₂³×C₃×C₇
socle	C₂×C₃×C₇	C₄×D₄
prime radical	C₃×C₇×#135	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₇×C₈ ◁ C₂×C₃×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₇×#131	C₂	semidirect
3. C₃×C₇×#108	C₂	semidirect
4. C₃×C₇×#104	C₂	semidirect
5. C₃×C₄×C₇×C₈	C₂	semidirect
6. C₇×#443	C₃	direct (N × H)
8. C₃×C₇×Q₁₆	C₄	semidirect
9. C₃×C₇×SD₁₆	C₄	semidirect
10. C₃×C₇×D₈	C₄	semidirect
18. C₇×#131	C₂×C₃	semidirect
19. C₇×#108	C₂×C₃	semidirect
20. C₇×#104	C₂×C₃	semidirect
21. C₄×C₇×C₈	C₂×C₃	semidirect
22. C₃×#443	C₇	direct (N × H)
35. C₇×Q₁₆	C₃×C₄	semidirect
36. C₇×SD₁₆	C₃×C₄	semidirect
37. C₇×D₈	C₃×C₄	semidirect
45. C₃×#131	C₂×C₇	semidirect
46. C₃×#108	C₂×C₇	semidirect
47. C₃×#104	C₂×C₇	semidirect
48. C₃×C₄×C₈	C₂×C₇	semidirect
55. #443	C₃×C₇	direct (N × H)
68. C₃×Q₁₆	C₄×C₇	semidirect
69. C₃×SD₁₆	C₄×C₇	semidirect
70. C₃×D₈	C₄×C₇	semidirect
81. #131	C₂×C₃×C₇	semidirect
82. #108	C₂×C₃×C₇	semidirect
83. #104	C₂×C₃×C₇	semidirect
84. C₄×C₈	C₂×C₃×C₇	semidirect
102. C₃×C₇	#443	direct (N × H)
104. Q₁₆	C₃×C₄×C₇	semidirect
105. SD₁₆	C₃×C₄×C₇	semidirect
106. D₈	C₃×C₄×C₇	semidirect
133. C₇	C₃×#443	direct (N × H)
143. C₃	C₇×#443	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₇×#135	2
C₃×C₇×#131	2
C₃×C₇×#108	2
C₃×C₇×#104	2
C₃×C₄×C₇×C₈	2
C₇×#443	3
C₃×#443	7

Subgroups — C3×C7×#443 #13056954

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇×#443 #13056954