Subgroups — C₃×C₇×#578 #13061998

Order 1344.

496 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×C₇×2₋¹⁺⁴	C₂	1	2	yes	maximal, complemented
2	C₃×C₇×#137	C₂	1	6	yes	maximal, complemented
3	C₃×C₇×#135	C₂	1	3	yes	maximal, complemented
4	C₂×C₃×C₇×Q₁₆	C₂	1	3	yes	maximal, complemented
5	C₃×C₇×#131	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇×#578	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₃×C₇×#41	—	2	6	no	—
8	C₃×C₇×#41	C₂²	1	6	yes	—
9	C₃×C₇×#41	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₇×Q₈	—	2	2	no	—
11	C₂×C₃×C₇×Q₈	C₂²	1	6	yes	—
12	C₃×C₇×Q₁₆	C₂²	1	3	yes	—
13	C₃×C₇×Q₁₆	C₂²	1	6	yes	—
14	C₃×C₇×SD₁₆	C₂²	1	6	yes	—
15	C₃×C₇×D₈	C₂²	1	1	yes	characteristic
16	C₃×C₇×M₁₆	C₂²	1	3	yes	—
17	C₂×C₃×C₇×C₈	C₂²	1	3	yes	—
18	C₇×2₋¹⁺⁴	C₂×C₃	1	2	yes	complemented
19	C₇×#137	C₂×C₃	1	6	yes	complemented
20	C₇×#135	C₂×C₃	1	3	yes	complemented
21	C₂×C₇×Q₁₆	C₂×C₃	1	3	yes	complemented
22	C₇×#131	C₂×C₃	1	1	yes	characteristic, complemented
23	C₃×#578	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
24	C₃×C₇×Q₈	—	2	6	no	—
25	C₃×C₇×Q₈	D₄	1	1	yes	characteristic
26	C₃×C₇×Q₈	C₂³	1	6	yes	—
27	C₃×C₇×D₄	—	2	6	no	—
28	C₃×C₇×D₄	D₄	1	3	yes	—
29	C₃×C₇×D₄	C₂³	1	2	yes	—
30	C₂×C₃×C₄×C₇	—	2	6	no	—
31	C₂×C₃×C₄×C₇	—	2	6	no	—
32	C₂×C₃×C₄×C₇	C₂³	1	3	yes	—
33	C₃×C₇×C₈	C₂³	1	1	yes	characteristic
34	C₃×C₇×C₈	C₂³	1	3	yes	—
35	C₇×#41	—	2	6	no	—
36	C₇×#41	C₂²×C₃	1	6	yes	—
37	C₇×#41	C₂²×C₃	1	1	yes	characteristic
38	C₂×C₇×Q₈	—	2	2	no	—
39	C₂×C₇×Q₈	C₂²×C₃	1	6	yes	—
40	C₇×Q₁₆	C₂²×C₃	1	3	yes	—
41	C₇×Q₁₆	C₂²×C₃	1	6	yes	—
42	C₇×SD₁₆	C₂²×C₃	1	6	yes	—
43	C₇×D₈	C₂²×C₃	1	1	yes	characteristic
44	C₇×M₁₆	C₂²×C₃	1	3	yes	—
45	C₂×C₇×C₈	C₂²×C₃	1	3	yes	—
46	C₃×2₋¹⁺⁴	C₂×C₇	1	2	yes	complemented
47	C₃×#137	C₂×C₇	1	6	yes	complemented
48	C₃×#135	C₂×C₇	1	3	yes	complemented
49	C₂×C₃×Q₁₆	C₂×C₇	1	3	yes	complemented
50	C₃×#131	C₂×C₇	1	1	yes	characteristic, complemented
51	C₂²×C₃×C₇	—	2	2	no	—
52	C₂²×C₃×C₇	C₂×D₄	1	3	yes	—
53	C₃×C₄×C₇	—	2	6	no	—
54	C₃×C₄×C₇	C₂×D₄	1	3	yes	—
55	C₃×C₄×C₇	C₂⁴	1	1	yes	characteristic
56	#578	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
57	C₇×Q₈	—	2	6	no	—
58	C₇×Q₈	C₃×D₄	1	1	yes	characteristic
59	C₇×Q₈	C₂³×C₃	1	6	yes	—
60	C₇×D₄	—	2	6	no	—
61	C₇×D₄	C₃×D₄	1	3	yes	—
62	C₇×D₄	C₂³×C₃	1	2	yes	—
63	C₂×C₄×C₇	—	2	6	no	—
64	C₂×C₄×C₇	—	2	6	no	—
65	C₂×C₄×C₇	C₂³×C₃	1	3	yes	—
66	C₇×C₈	C₂³×C₃	1	1	yes	characteristic
67	C₇×C₈	C₂³×C₃	1	3	yes	—
68	C₃×#41	—	2	6	no	—
69	C₃×#41	C₂²×C₇	1	6	yes	—
70	C₃×#41	C₂²×C₇	1	1	yes	characteristic
71	C₂×C₃×Q₈	—	2	2	no	—
72	C₂×C₃×Q₈	C₂²×C₇	1	6	yes	—
73	C₃×Q₁₆	C₂²×C₇	1	3	yes	—
74	C₃×Q₁₆	C₂²×C₇	1	6	yes	—
75	C₃×SD₁₆	C₂²×C₇	1	6	yes	—
76	C₃×D₈	C₂²×C₇	1	1	yes	characteristic
77	C₃×M₁₆	C₂²×C₇	1	3	yes	—
78	C₂×C₃×C₈	C₂²×C₇	1	3	yes	—
79	C₂×C₃×C₇	—	2	3	no	—
80	C₂×C₃×C₇	—	4	2	no	—
81	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
82	2₋¹⁺⁴	C₂×C₃×C₇	1	2	yes	complemented
83	#137	C₂×C₃×C₇	1	6	yes	complemented
84	#135	C₂×C₃×C₇	1	3	yes	complemented
85	C₂×Q₁₆	C₂×C₃×C₇	1	3	yes	complemented
86	#131	C₂×C₃×C₇	1	1	yes	characteristic, complemented
87	C₂²×C₇	—	2	2	no	—
88	C₂²×C₇	C₂×C₃×D₄	1	3	yes	—
89	C₄×C₇	—	2	6	no	—
90	C₄×C₇	C₂×C₃×D₄	1	3	yes	—
91	C₄×C₇	C₂⁴×C₃	1	1	yes	characteristic
92	C₃×Q₈	—	2	6	no	—
93	C₃×Q₈	C₇×D₄	1	1	yes	characteristic
94	C₃×Q₈	C₂³×C₇	1	6	yes	—
95	C₃×D₄	—	2	6	no	—
96	C₃×D₄	C₇×D₄	1	3	yes	—
97	C₃×D₄	C₂³×C₇	1	2	yes	—
98	C₂×C₃×C₄	—	2	6	no	—
99	C₂×C₃×C₄	—	2	6	no	—
100	C₂×C₃×C₄	C₂³×C₇	1	3	yes	—
101	C₃×C₈	C₂³×C₇	1	1	yes	characteristic
102	C₃×C₈	C₂³×C₇	1	3	yes	—
103	C₃×C₇	#578	1	1	yes	characteristic, direct product factor, Hall
104	#41	—	2	6	no	—
105	#41	C₂²×C₃×C₇	1	6	yes	—
106	#41	C₂²×C₃×C₇	1	1	yes	characteristic
107	C₂×Q₈	—	2	2	no	—
108	C₂×Q₈	C₂²×C₃×C₇	1	6	yes	—
109	Q₁₆	C₂²×C₃×C₇	1	3	yes	—
110	Q₁₆	C₂²×C₃×C₇	1	6	yes	—
111	SD₁₆	C₂²×C₃×C₇	1	6	yes	—
112	D₈	C₂²×C₃×C₇	1	1	yes	characteristic
113	M₁₆	C₂²×C₃×C₇	1	3	yes	—
114	C₂×C₈	C₂²×C₃×C₇	1	3	yes	—
115	C₂×C₇	—	2	3	no	—
116	C₂×C₇	—	4	2	no	—
117	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
118	C₂²×C₃	—	2	2	no	—
119	C₂²×C₃	C₂×C₇×D₄	1	3	yes	—
120	C₃×C₄	—	2	6	no	—
121	C₃×C₄	C₂×C₇×D₄	1	3	yes	—
122	C₃×C₄	C₂⁴×C₇	1	1	yes	characteristic
123	Q₈	—	2	6	no	—
124	Q₈	C₃×C₇×D₄	1	1	yes	characteristic
125	Q₈	C₂³×C₃×C₇	1	6	yes	—
126	D₄	—	2	6	no	—
127	D₄	C₃×C₇×D₄	1	3	yes	—
128	D₄	C₂³×C₃×C₇	1	2	yes	—
129	C₂×C₄	—	2	6	no	—
130	C₂×C₄	—	2	6	no	—
131	C₂×C₄	C₂³×C₃×C₇	1	3	yes	—
132	C₈	C₂³×C₃×C₇	1	1	yes	characteristic
133	C₈	C₂³×C₃×C₇	1	3	yes	—
134	C₇	C₃×#578	1	1	yes	characteristic, direct product factor, Sylow
135	C₂×C₃	—	2	3	no	—
136	C₂×C₃	—	4	2	no	—
137	C₂×C₃	C₂²×C₇×D₄	1	1	yes	characteristic
138	C₂²	—	2	2	no	—
139	C₂²	C₂×C₃×C₇×D₄	1	3	yes	—
140	C₄	—	2	6	no	—
141	C₄	C₂×C₃×C₇×D₄	1	3	yes	—
142	C₄	C₂⁴×C₃×C₇	1	1	yes	characteristic
143	C₃	C₇×#578	1	1	yes	characteristic, direct product factor, Sylow
144	C₂	—	2	3	no	—
145	C₂	—	4	2	no	—
146	C₂	C₂²×C₃×C₇×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃×C₇	C₂²×D₄
commutator, Frattini, cosocle	C₄	C₂⁴×C₃×C₇
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₇ ◁ C₃×C₇×#41 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃×C₇×2₋¹⁺⁴	C₂	semidirect
2. C₃×C₇×#137	C₂	semidirect
3. C₃×C₇×#135	C₂	semidirect
4. C₂×C₃×C₇×Q₁₆	C₂	semidirect
5. C₃×C₇×#131	C₂	semidirect
6. C₇×#578	C₃	direct (N × H)
18. C₇×2₋¹⁺⁴	C₂×C₃	semidirect
19. C₇×#137	C₂×C₃	semidirect
20. C₇×#135	C₂×C₃	semidirect
21. C₂×C₇×Q₁₆	C₂×C₃	semidirect
22. C₇×#131	C₂×C₃	semidirect
23. C₃×#578	C₇	direct (N × H)
46. C₃×2₋¹⁺⁴	C₂×C₇	semidirect
47. C₃×#137	C₂×C₇	semidirect
48. C₃×#135	C₂×C₇	semidirect
49. C₂×C₃×Q₁₆	C₂×C₇	semidirect
50. C₃×#131	C₂×C₇	semidirect
56. #578	C₃×C₇	direct (N × H)
82. 2₋¹⁺⁴	C₂×C₃×C₇	semidirect
83. #137	C₂×C₃×C₇	semidirect
84. #135	C₂×C₃×C₇	semidirect
85. C₂×Q₁₆	C₂×C₃×C₇	semidirect
86. #131	C₂×C₃×C₇	semidirect
103. C₃×C₇	#578	direct (N × H)
134. C₇	C₃×#578	direct (N × H)
143. C₃	C₇×#578	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×C₇×2₋¹⁺⁴	2
C₃×C₇×#137	2
C₃×C₇×#135	2
C₂×C₃×C₇×Q₁₆	2
C₃×C₇×#131	2
C₇×#578	3
C₃×#578	7

Subgroups — C3×C7×#578 #13061998

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇×#578 #13061998