Subgroups — C₃×C₇×#444 #13056955

Order 1344.

208 conjugacy classes of subgroups.

The subgroup lattice has 144 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×C₇×#97	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#444	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	C₇×#97	C₂×C₃	1	1	yes	characteristic, complemented
22	C₃×#444	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	2	no	—
27	C₂×C₃×C₄×C₇	—	2	1	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	2	yes	—
33	C₇×#41	C₂²×C₃	1	2	yes	—
34	C₇×Q₁₆	C₃×C₄	1	1	yes	characteristic, complemented
35	C₇×SD₁₆	C₃×C₄	1	2	yes	complemented
36	C₇×D₈	C₃×C₄	1	1	yes	characteristic, complemented
37	C₇×M₁₆	—	2	2	no	—
38	C₇×M₁₆	C₂²×C₃	1	2	yes	—
39	C₂×C₇×C₈	—	2	2	no	—
40	C₂×C₇×C₈	C₂²×C₃	1	1	yes	characteristic
41	C₂×C₇×C₈	C₂²×C₃	1	1	yes	characteristic
42	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
43	C₃×#135	C₂×C₇	1	1	yes	characteristic
44	C₃×#131	C₂×C₇	1	2	yes	complemented
45	C₃×#108	C₂×C₇	1	1	yes	characteristic, complemented
46	C₃×#104	C₂×C₇	1	2	yes	complemented
47	C₃×#97	C₂×C₇	1	1	yes	characteristic, complemented
48	C₂²×C₃×C₇	—	2	2	no	—
49	C₂²×C₃×C₇	#41	1	1	yes	characteristic
50	C₃×C₄×C₇	—	2	2	no	—
51	C₃×C₄×C₇	—	4	1	no	—
52	C₃×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
53	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
54	#444	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
55	C₇×Q₈	C₂×C₃×C₄	1	2	yes	—
56	C₇×D₄	—	2	2	no	—
57	C₇×D₄	C₂×C₃×C₄	1	2	yes	—
58	C₂×C₄×C₇	—	2	2	no	—
59	C₂×C₄×C₇	—	2	1	no	—
60	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
61	C₇×C₈	—	2	2	no	—
62	C₇×C₈	C₂×C₃×C₄	1	1	yes	characteristic
63	C₇×C₈	C₂×C₃×C₄	1	1	yes	characteristic
64	C₇×C₈	C₃×D₄	1	2	yes	—
65	C₃×#41	C₂²×C₇	1	2	yes	—
66	C₃×Q₁₆	C₄×C₇	1	1	yes	characteristic, complemented
67	C₃×SD₁₆	C₄×C₇	1	2	yes	complemented
68	C₃×D₈	C₄×C₇	1	1	yes	characteristic, complemented
69	C₃×M₁₆	—	2	2	no	—
70	C₃×M₁₆	C₂²×C₇	1	2	yes	—
71	C₂×C₃×C₈	—	2	2	no	—
72	C₂×C₃×C₈	C₂²×C₇	1	1	yes	characteristic
73	C₂×C₃×C₈	C₂²×C₇	1	1	yes	characteristic
74	C₃×C₄²	C₂²×C₇	1	1	yes	characteristic
75	C₂×C₃×C₇	—	2	1	no	—
76	C₂×C₃×C₇	—	4	2	no	—
77	C₂×C₃×C₇	C₄×D₄	1	1	yes	characteristic
78	#135	C₂×C₃×C₇	1	1	yes	characteristic
79	#131	C₂×C₃×C₇	1	2	yes	complemented
80	#108	C₂×C₃×C₇	1	1	yes	characteristic, complemented
81	#104	C₂×C₃×C₇	1	2	yes	complemented
82	#97	C₂×C₃×C₇	1	1	yes	characteristic, complemented
83	C₂²×C₇	—	2	2	no	—
84	C₂²×C₇	C₃×#41	1	1	yes	characteristic
85	C₄×C₇	—	2	2	no	—
86	C₄×C₇	—	4	1	no	—
87	C₄×C₇	C₂²×C₃×C₄	1	1	yes	characteristic
88	C₄×C₇	C₂×C₃×D₄	1	1	yes	characteristic
89	C₃×Q₈	C₂×C₄×C₇	1	2	yes	—
90	C₃×D₄	—	2	2	no	—
91	C₃×D₄	C₂×C₄×C₇	1	2	yes	—
92	C₂×C₃×C₄	—	2	2	no	—
93	C₂×C₃×C₄	—	2	1	no	—
94	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
95	C₃×C₈	—	2	2	no	—
96	C₃×C₈	C₂×C₄×C₇	1	1	yes	characteristic
97	C₃×C₈	C₂×C₄×C₇	1	1	yes	characteristic
98	C₃×C₈	C₇×D₄	1	2	yes	—
99	C₃×C₇	#444	1	1	yes	characteristic, direct product factor, Hall
100	#41	C₂²×C₃×C₇	1	2	yes	—
101	Q₁₆	C₃×C₄×C₇	1	1	yes	characteristic, complemented
102	SD₁₆	C₃×C₄×C₇	1	2	yes	complemented
103	D₈	C₃×C₄×C₇	1	1	yes	characteristic, complemented
104	M₁₆	—	2	2	no	—
105	M₁₆	C₂²×C₃×C₇	1	2	yes	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	C₂²×C₃×C₇	1	1	yes	characteristic
108	C₂×C₈	C₂²×C₃×C₇	1	1	yes	characteristic
109	C₄²	C₂²×C₃×C₇	1	1	yes	characteristic
110	C₂×C₇	—	2	1	no	—
111	C₂×C₇	—	4	2	no	—
112	C₂×C₇	C₃×C₄×D₄	1	1	yes	characteristic
113	C₂²×C₃	—	2	2	no	—
114	C₂²×C₃	C₇×#41	1	1	yes	characteristic
115	C₃×C₄	—	2	2	no	—
116	C₃×C₄	—	4	1	no	—
117	C₃×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
118	C₃×C₄	C₂×C₇×D₄	1	1	yes	characteristic
119	Q₈	C₂×C₃×C₄×C₇	1	2	yes	—
120	D₄	—	2	2	no	—
121	D₄	C₂×C₃×C₄×C₇	1	2	yes	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	C₂³×C₃×C₇	1	1	yes	characteristic
125	C₈	—	2	2	no	—
126	C₈	C₂×C₃×C₄×C₇	1	1	yes	characteristic
127	C₈	C₂×C₃×C₄×C₇	1	1	yes	characteristic
128	C₈	C₃×C₇×D₄	1	2	yes	—
129	C₇	C₃×#444	1	1	yes	characteristic, direct product factor, Sylow
130	C₂×C₃	—	2	1	no	—
131	C₂×C₃	—	4	2	no	—
132	C₂×C₃	C₄×C₇×D₄	1	1	yes	characteristic
133	C₂²	—	2	2	no	—
134	C₂²	C₃×C₇×#41	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	4	1	no	—
137	C₄	C₂²×C₃×C₄×C₇	1	1	yes	characteristic
138	C₄	C₂×C₃×C₇×D₄	1	1	yes	characteristic
139	C₃	C₇×#444	1	1	yes	characteristic, direct product factor, Sylow
140	C₂	—	2	1	no	—
141	C₂	—	4	2	no	—
142	C₂	C₃×C₄×C₇×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₃×C₄×C₇	C₂×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×C₄×C₇ ◁ 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×C₇×#97	C₂	semidirect
6. C₇×#444	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. C₇×#97	C₂×C₃	semidirect
22. C₃×#444	C₇	direct (N × H)
34. C₇×Q₁₆	C₃×C₄	semidirect
35. C₇×SD₁₆	C₃×C₄	semidirect
36. C₇×D₈	C₃×C₄	semidirect
44. C₃×#131	C₂×C₇	semidirect
45. C₃×#108	C₂×C₇	semidirect
46. C₃×#104	C₂×C₇	semidirect
47. C₃×#97	C₂×C₇	semidirect
54. #444	C₃×C₇	direct (N × H)
66. C₃×Q₁₆	C₄×C₇	semidirect
67. C₃×SD₁₆	C₄×C₇	semidirect
68. C₃×D₈	C₄×C₇	semidirect
79. #131	C₂×C₃×C₇	semidirect
80. #108	C₂×C₃×C₇	semidirect
81. #104	C₂×C₃×C₇	semidirect
82. #97	C₂×C₃×C₇	semidirect
99. C₃×C₇	#444	direct (N × H)
101. Q₁₆	C₃×C₄×C₇	semidirect
102. SD₁₆	C₃×C₄×C₇	semidirect
103. D₈	C₃×C₄×C₇	semidirect
129. C₇	C₃×#444	direct (N × H)
139. C₃	C₇×#444	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×C₇×#97	2
C₇×#444	3
C₃×#444	7

Subgroups — C3×C7×#444 #13056955

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇×#444 #13056955