Subgroups — C₃×C₇×#559 #13061979

Order 1344.

472 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×#127	C₂	1	1	yes	maximal, characteristic
2	C₃×C₇×#124	C₂	1	4	yes	maximal, complemented
3	C₃×C₇×#123	C₂	1	4	yes	maximal, complemented
4	C₃×C₄×C₇×Q₈	C₂	1	2	yes	maximal, complemented
5	C₃×C₄×C₇×D₄	C₂	1	4	yes	maximal, complemented
6	C₇×#559	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₇×Q₈	C₂²	1	2	yes	—
8	C₂×C₃×C₇×D₄	C₂²	1	4	yes	—
9	C₂×C₃×C₇×D₄	C₂²	1	2	yes	—
10	C₂²×C₃×C₄×C₇	C₂²	1	4	yes	—
11	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	—
12	C₃×C₇×C₄⋊C₄	C₂²	1	4	yes	—
13	C₃×C₇×C₂²⋊C₄	C₂²	1	8	yes	—
14	C₃×C₇×C₂²⋊C₄	C₂²	1	4	yes	—
15	C₃×C₄²×C₇	C₂²	1	4	yes	—
16	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
17	C₇×#127	C₂×C₃	1	1	yes	characteristic
18	C₇×#124	C₂×C₃	1	4	yes	complemented
19	C₇×#123	C₂×C₃	1	4	yes	complemented
20	C₄×C₇×Q₈	C₂×C₃	1	2	yes	complemented
21	C₄×C₇×D₄	C₂×C₃	1	4	yes	complemented
22	C₃×#559	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
23	C₂³×C₃×C₇	C₂³	1	4	yes	—
24	C₃×C₇×Q₈	—	2	4	no	—
25	C₃×C₇×D₄	—	2	8	no	—
26	C₃×C₇×D₄	—	4	2	no	—
27	C₂×C₃×C₄×C₇	—	4	4	no	—
28	C₂×C₃×C₄×C₇	C₂³	1	4	yes	—
29	C₂×C₃×C₄×C₇	C₂³	1	4	yes	—
30	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
31	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
32	C₂×C₇×Q₈	C₂²×C₃	1	2	yes	—
33	C₂×C₇×D₄	C₂²×C₃	1	4	yes	—
34	C₂×C₇×D₄	C₂²×C₃	1	2	yes	—
35	C₂²×C₄×C₇	C₂²×C₃	1	4	yes	—
36	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
37	C₇×C₄⋊C₄	C₂²×C₃	1	4	yes	—
38	C₇×C₂²⋊C₄	C₂²×C₃	1	8	yes	—
39	C₇×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
40	C₄²×C₇	C₂²×C₃	1	4	yes	—
41	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
42	C₃×#127	C₂×C₇	1	1	yes	characteristic
43	C₃×#124	C₂×C₇	1	4	yes	complemented
44	C₃×#123	C₂×C₇	1	4	yes	complemented
45	C₃×C₄×Q₈	C₂×C₇	1	2	yes	complemented
46	C₃×C₄×D₄	C₂×C₇	1	4	yes	complemented
47	C₂²×C₃×C₇	—	2	8	no	—
48	C₂²×C₃×C₇	—	2	4	no	—
49	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
50	C₃×C₄×C₇	—	2	4	no	—
51	C₃×C₄×C₇	—	2	4	no	—
52	C₃×C₄×C₇	—	2	1	no	—
53	C₃×C₄×C₇	#41	1	4	yes	—
54	#559	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
55	C₂³×C₇	C₂³×C₃	1	4	yes	—
56	C₇×Q₈	—	2	4	no	—
57	C₇×D₄	—	2	8	no	—
58	C₇×D₄	—	4	2	no	—
59	C₂×C₄×C₇	—	4	4	no	—
60	C₂×C₄×C₇	C₂³×C₃	1	4	yes	—
61	C₂×C₄×C₇	C₂³×C₃	1	4	yes	—
62	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
63	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
64	C₂×C₃×Q₈	C₂²×C₇	1	2	yes	—
65	C₂×C₃×D₄	C₂²×C₇	1	4	yes	—
66	C₂×C₃×D₄	C₂²×C₇	1	2	yes	—
67	C₂²×C₃×C₄	C₂²×C₇	1	4	yes	—
68	C₃×C₄⋊C₄	C₂²×C₇	1	2	yes	—
69	C₃×C₄⋊C₄	C₂²×C₇	1	4	yes	—
70	C₃×C₂²⋊C₄	C₂²×C₇	1	8	yes	—
71	C₃×C₂²⋊C₄	C₂²×C₇	1	4	yes	—
72	C₃×C₄²	C₂²×C₇	1	4	yes	—
73	C₃×C₄²	C₂²×C₇	1	1	yes	characteristic
74	C₂×C₃×C₇	—	4	4	no	—
75	C₂×C₃×C₇	C₂×#41	1	2	yes	—
76	C₂×C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic
77	#127	C₂×C₃×C₇	1	1	yes	characteristic
78	#124	C₂×C₃×C₇	1	4	yes	complemented
79	#123	C₂×C₃×C₇	1	4	yes	complemented
80	C₄×Q₈	C₂×C₃×C₇	1	2	yes	complemented
81	C₄×D₄	C₂×C₃×C₇	1	4	yes	complemented
82	C₂²×C₇	—	2	8	no	—
83	C₂²×C₇	—	2	4	no	—
84	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
85	C₄×C₇	—	2	4	no	—
86	C₄×C₇	—	2	4	no	—
87	C₄×C₇	—	2	1	no	—
88	C₄×C₇	C₃×#41	1	4	yes	—
89	C₂³×C₃	C₂³×C₇	1	4	yes	—
90	C₃×Q₈	—	2	4	no	—
91	C₃×D₄	—	2	8	no	—
92	C₃×D₄	—	4	2	no	—
93	C₂×C₃×C₄	—	4	4	no	—
94	C₂×C₃×C₄	C₂³×C₇	1	4	yes	—
95	C₂×C₃×C₄	C₂³×C₇	1	4	yes	—
96	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
97	C₂×C₃×C₄	C₂³×C₇	1	2	yes	—
98	C₃×C₇	#559	1	1	yes	characteristic, direct product factor, Hall
99	C₂×Q₈	C₂²×C₃×C₇	1	2	yes	—
100	C₂×D₄	C₂²×C₃×C₇	1	4	yes	—
101	C₂×D₄	C₂²×C₃×C₇	1	2	yes	—
102	C₂²×C₄	C₂²×C₃×C₇	1	4	yes	—
103	C₄⋊C₄	C₂²×C₃×C₇	1	2	yes	—
104	C₄⋊C₄	C₂²×C₃×C₇	1	4	yes	—
105	C₂²⋊C₄	C₂²×C₃×C₇	1	8	yes	—
106	C₂²⋊C₄	C₂²×C₃×C₇	1	4	yes	—
107	C₄²	C₂²×C₃×C₇	1	4	yes	—
108	C₄²	C₂²×C₃×C₇	1	1	yes	characteristic
109	C₂×C₇	—	4	4	no	—
110	C₂×C₇	C₂×C₃×#41	1	2	yes	—
111	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
112	C₂²×C₃	—	2	8	no	—
113	C₂²×C₃	—	2	4	no	—
114	C₂²×C₃	C₂⁴×C₇	1	1	yes	characteristic
115	C₃×C₄	—	2	4	no	—
116	C₃×C₄	—	2	4	no	—
117	C₃×C₄	—	2	1	no	—
118	C₃×C₄	C₇×#41	1	4	yes	—
119	C₂³	C₂³×C₃×C₇	1	4	yes	—
120	Q₈	—	2	4	no	—
121	D₄	—	2	8	no	—
122	D₄	—	4	2	no	—
123	C₂×C₄	—	4	4	no	—
124	C₂×C₄	C₂³×C₃×C₇	1	4	yes	—
125	C₂×C₄	C₂³×C₃×C₇	1	4	yes	—
126	C₂×C₄	C₂³×C₃×C₇	1	1	yes	characteristic
127	C₂×C₄	C₂³×C₃×C₇	1	2	yes	—
128	C₇	C₃×#559	1	1	yes	characteristic, direct product factor, Sylow
129	C₂×C₃	—	4	4	no	—
130	C₂×C₃	C₂×C₇×#41	1	2	yes	—
131	C₂×C₃	C₇×H(5,𝔽₂)	1	1	yes	characteristic
132	C₂²	—	2	8	no	—
133	C₂²	—	2	4	no	—
134	C₂²	C₂⁴×C₃×C₇	1	1	yes	characteristic
135	C₄	—	2	4	no	—
136	C₄	—	2	4	no	—
137	C₄	—	2	1	no	—
138	C₄	C₃×C₇×#41	1	4	yes	—
139	C₃	C₇×#559	1	1	yes	characteristic, direct product factor, Sylow
140	C₂	—	4	4	no	—
141	C₂	C₂×C₃×C₇×#41	1	2	yes	—
142	C₂	C₃×C₇×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃×C₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃×C₇
prime radical	C₃×C₇×#127	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×C₇×#124	C₂	semidirect
3. C₃×C₇×#123	C₂	semidirect
4. C₃×C₄×C₇×Q₈	C₂	semidirect
5. C₃×C₄×C₇×D₄	C₂	semidirect
6. C₇×#559	C₃	direct (N × H)
18. C₇×#124	C₂×C₃	semidirect
19. C₇×#123	C₂×C₃	semidirect
20. C₄×C₇×Q₈	C₂×C₃	semidirect
21. C₄×C₇×D₄	C₂×C₃	semidirect
22. C₃×#559	C₇	direct (N × H)
43. C₃×#124	C₂×C₇	semidirect
44. C₃×#123	C₂×C₇	semidirect
45. C₃×C₄×Q₈	C₂×C₇	semidirect
46. C₃×C₄×D₄	C₂×C₇	semidirect
54. #559	C₃×C₇	direct (N × H)
78. #124	C₂×C₃×C₇	semidirect
79. #123	C₂×C₃×C₇	semidirect
80. C₄×Q₈	C₂×C₃×C₇	semidirect
81. C₄×D₄	C₂×C₃×C₇	semidirect
98. C₃×C₇	#559	direct (N × H)
128. C₇	C₃×#559	direct (N × H)
139. C₃	C₇×#559	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×#127	2
C₃×C₇×#124	2
C₃×C₇×#123	2
C₃×C₄×C₇×Q₈	2
C₃×C₄×C₇×D₄	2
C₇×#559	3
C₃×#559	7

Subgroups — C3×C7×#559 #13061979

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇×#559 #13061979