Subgroups — C₃×C₇×#540 #13061960

Order 1344.

480 conjugacy classes of subgroups.

The subgroup lattice has 196 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, complemented
2	C₃×C₇×#125	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₇×#123	C₂	1	4	yes	maximal, complemented
4	C₃×C₇×#121	C₂	1	4	yes	maximal, complemented
5	C₃×C₇×#121	C₂	1	2	yes	maximal, complemented
6	C₃×C₄×C₇×D₄	C₂	1	2	yes	maximal, complemented
7	C₃×C₇×#117	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇×#540	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
9	C₂×C₃×C₇×D₄	C₂²	1	2	yes	—
10	C₂×C₃×C₇×D₄	C₂²	1	4	yes	—
11	C₂×C₃×C₇×D₄	C₂²	1	4	yes	—
12	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₄×C₇	C₂²	1	4	yes	—
14	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	—
15	C₃×C₇×C₄⋊C₄	C₂²	1	2	yes	—
16	C₃×C₇×C₄⋊C₄	C₂²	1	4	yes	—
17	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
18	C₃×C₇×C₂²⋊C₄	C₂²	1	4	yes	—
19	C₃×C₇×C₂²⋊C₄	C₂²	1	4	yes	—
20	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
21	C₃×C₄²×C₇	C₂²	1	1	yes	characteristic
22	C₇×#127	C₂×C₃	1	1	yes	characteristic, complemented
23	C₇×#125	C₂×C₃	1	1	yes	characteristic, complemented
24	C₇×#123	C₂×C₃	1	4	yes	complemented
25	C₇×#121	C₂×C₃	1	4	yes	complemented
26	C₇×#121	C₂×C₃	1	2	yes	complemented
27	C₄×C₇×D₄	C₂×C₃	1	2	yes	complemented
28	C₇×#117	C₂×C₃	1	1	yes	characteristic, complemented
29	C₃×#540	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
30	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
31	C₂³×C₃×C₇	C₂³	1	4	yes	—
32	C₃×C₇×D₄	—	2	4	no	—
33	C₃×C₇×D₄	—	4	4	no	—
34	C₃×C₇×D₄	—	4	4	no	—
35	C₂×C₃×C₄×C₇	—	2	2	no	—
36	C₂×C₃×C₄×C₇	—	4	4	no	—
37	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
38	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
39	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
40	C₂×C₃×C₄×C₇	C₂³	1	4	yes	—
41	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
42	C₂×C₇×D₄	C₂²×C₃	1	2	yes	—
43	C₂×C₇×D₄	C₂²×C₃	1	4	yes	—
44	C₂×C₇×D₄	C₂²×C₃	1	4	yes	—
45	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
46	C₂²×C₄×C₇	C₂²×C₃	1	4	yes	—
47	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
48	C₇×C₄⋊C₄	C₂²×C₃	1	2	yes	—
49	C₇×C₄⋊C₄	C₂²×C₃	1	4	yes	—
50	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
51	C₇×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
52	C₇×C₂²⋊C₄	C₂²×C₃	1	4	yes	—
53	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
54	C₄²×C₇	C₂²×C₃	1	1	yes	characteristic
55	C₃×#127	C₂×C₇	1	1	yes	characteristic, complemented
56	C₃×#125	C₂×C₇	1	1	yes	characteristic, complemented
57	C₃×#123	C₂×C₇	1	4	yes	complemented
58	C₃×#121	C₂×C₇	1	4	yes	complemented
59	C₃×#121	C₂×C₇	1	2	yes	complemented
60	C₃×C₄×D₄	C₂×C₇	1	2	yes	complemented
61	C₃×#117	C₂×C₇	1	1	yes	characteristic, complemented
62	C₂²×C₃×C₇	—	2	1	no	—
63	C₂²×C₃×C₇	—	2	4	no	—
64	C₂²×C₃×C₇	—	2	4	no	—
65	C₂²×C₃×C₇	—	2	2	no	—
66	C₂²×C₃×C₇	—	2	4	no	—
67	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
68	C₃×C₄×C₇	—	2	1	no	—
69	C₃×C₄×C₇	—	2	2	no	—
70	C₃×C₄×C₇	—	2	4	no	—
71	C₃×C₄×C₇	—	2	2	no	—
72	C₃×C₄×C₇	#41	1	2	yes	—
73	#540	C₃×C₇	1	1	yes	characteristic, direct product factor, Sylow
74	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
75	C₂³×C₇	C₂³×C₃	1	4	yes	—
76	C₇×D₄	—	2	4	no	—
77	C₇×D₄	—	4	4	no	—
78	C₇×D₄	—	4	4	no	—
79	C₂×C₄×C₇	—	2	2	no	—
80	C₂×C₄×C₇	—	4	4	no	—
81	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
82	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
83	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
84	C₂×C₄×C₇	C₂³×C₃	1	4	yes	—
85	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
86	C₂×C₃×D₄	C₂²×C₇	1	2	yes	—
87	C₂×C₃×D₄	C₂²×C₇	1	4	yes	—
88	C₂×C₃×D₄	C₂²×C₇	1	4	yes	—
89	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
90	C₂²×C₃×C₄	C₂²×C₇	1	4	yes	—
91	C₃×C₄⋊C₄	C₂²×C₇	1	2	yes	—
92	C₃×C₄⋊C₄	C₂²×C₇	1	2	yes	—
93	C₃×C₄⋊C₄	C₂²×C₇	1	4	yes	—
94	C₃×C₂²⋊C₄	C₂²×C₇	1	2	yes	—
95	C₃×C₂²⋊C₄	C₂²×C₇	1	4	yes	—
96	C₃×C₂²⋊C₄	C₂²×C₇	1	4	yes	—
97	C₃×C₄²	C₂²×C₇	1	1	yes	characteristic
98	C₃×C₄²	C₂²×C₇	1	1	yes	characteristic
99	C₂×C₃×C₇	—	4	1	no	—
100	C₂×C₃×C₇	—	4	4	no	—
101	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
102	C₂×C₃×C₇	H(5,𝔽₂)	1	2	yes	—
103	#127	C₂×C₃×C₇	1	1	yes	characteristic, complemented
104	#125	C₂×C₃×C₇	1	1	yes	characteristic, complemented
105	#123	C₂×C₃×C₇	1	4	yes	complemented
106	#121	C₂×C₃×C₇	1	4	yes	complemented
107	#121	C₂×C₃×C₇	1	2	yes	complemented
108	C₄×D₄	C₂×C₃×C₇	1	2	yes	complemented
109	#117	C₂×C₃×C₇	1	1	yes	characteristic, complemented
110	C₂²×C₇	—	2	1	no	—
111	C₂²×C₇	—	2	4	no	—
112	C₂²×C₇	—	2	4	no	—
113	C₂²×C₇	—	2	2	no	—
114	C₂²×C₇	—	2	4	no	—
115	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
116	C₄×C₇	—	2	1	no	—
117	C₄×C₇	—	2	2	no	—
118	C₄×C₇	—	2	4	no	—
119	C₄×C₇	—	2	2	no	—
120	C₄×C₇	C₃×#41	1	2	yes	—
121	C₂³×C₃	C₂³×C₇	1	1	yes	characteristic
122	C₂³×C₃	C₂³×C₇	1	4	yes	—
123	C₃×D₄	—	2	4	no	—
124	C₃×D₄	—	4	4	no	—
125	C₃×D₄	—	4	4	no	—
126	C₂×C₃×C₄	—	2	2	no	—
127	C₂×C₃×C₄	—	4	4	no	—
128	C₂×C₃×C₄	C₂³×C₇	1	2	yes	—
129	C₂×C₃×C₄	C₂³×C₇	1	2	yes	—
130	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
131	C₂×C₃×C₄	C₂³×C₇	1	4	yes	—
132	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
133	C₃×C₇	#540	1	1	yes	characteristic, direct product factor, Hall
134	C₂×D₄	C₂²×C₃×C₇	1	2	yes	—
135	C₂×D₄	C₂²×C₃×C₇	1	4	yes	—
136	C₂×D₄	C₂²×C₃×C₇	1	4	yes	—
137	C₂²×C₄	C₂²×C₃×C₇	1	1	yes	characteristic
138	C₂²×C₄	C₂²×C₃×C₇	1	4	yes	—
139	C₄⋊C₄	C₂²×C₃×C₇	1	2	yes	—
140	C₄⋊C₄	C₂²×C₃×C₇	1	2	yes	—
141	C₄⋊C₄	C₂²×C₃×C₇	1	4	yes	—
142	C₂²⋊C₄	C₂²×C₃×C₇	1	2	yes	—
143	C₂²⋊C₄	C₂²×C₃×C₇	1	4	yes	—
144	C₂²⋊C₄	C₂²×C₃×C₇	1	4	yes	—
145	C₄²	C₂²×C₃×C₇	1	1	yes	characteristic
146	C₄²	C₂²×C₃×C₇	1	1	yes	characteristic
147	C₂×C₇	—	4	1	no	—
148	C₂×C₇	—	4	4	no	—
149	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
150	C₂×C₇	C₃×H(5,𝔽₂)	1	2	yes	—
151	C₂²×C₃	—	2	1	no	—
152	C₂²×C₃	—	2	4	no	—
153	C₂²×C₃	—	2	4	no	—
154	C₂²×C₃	—	2	2	no	—
155	C₂²×C₃	—	2	4	no	—
156	C₂²×C₃	C₂⁴×C₇	1	1	yes	characteristic
157	C₃×C₄	—	2	1	no	—
158	C₃×C₄	—	2	2	no	—
159	C₃×C₄	—	2	4	no	—
160	C₃×C₄	—	2	2	no	—
161	C₃×C₄	C₇×#41	1	2	yes	—
162	C₂³	C₂³×C₃×C₇	1	1	yes	characteristic
163	C₂³	C₂³×C₃×C₇	1	4	yes	—
164	D₄	—	2	4	no	—
165	D₄	—	4	4	no	—
166	D₄	—	4	4	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	4	4	no	—
169	C₂×C₄	C₂³×C₃×C₇	1	2	yes	—
170	C₂×C₄	C₂³×C₃×C₇	1	2	yes	—
171	C₂×C₄	C₂³×C₃×C₇	1	1	yes	characteristic
172	C₂×C₄	C₂³×C₃×C₇	1	4	yes	—
173	C₂×C₄	C₂³×C₃×C₇	1	1	yes	characteristic
174	C₇	C₃×#540	1	1	yes	characteristic, direct product factor, Sylow
175	C₂×C₃	—	4	1	no	—
176	C₂×C₃	—	4	4	no	—
177	C₂×C₃	C₂×C₇×#41	1	1	yes	characteristic
178	C₂×C₃	C₇×H(5,𝔽₂)	1	2	yes	—
179	C₂²	—	2	1	no	—
180	C₂²	—	2	4	no	—
181	C₂²	—	2	4	no	—
182	C₂²	—	2	2	no	—
183	C₂²	—	2	4	no	—
184	C₂²	C₂⁴×C₃×C₇	1	1	yes	characteristic
185	C₄	—	2	1	no	—
186	C₄	—	2	2	no	—
187	C₄	—	2	4	no	—
188	C₄	—	2	2	no	—
189	C₄	C₃×C₇×#41	1	2	yes	—
190	C₃	C₇×#540	1	1	yes	characteristic, direct product factor, Sylow
191	C₂	—	4	1	no	—
192	C₂	—	4	4	no	—
193	C₂	C₂×C₃×C₇×#41	1	1	yes	characteristic
194	C₂	C₃×C₇×H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃×C₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃×C₇
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
1. C₃×C₇×#127	C₂	semidirect
2. C₃×C₇×#125	C₂	semidirect
3. C₃×C₇×#123	C₂	semidirect
4. C₃×C₇×#121	C₂	semidirect
5. C₃×C₇×#121	C₂	semidirect
6. C₃×C₄×C₇×D₄	C₂	semidirect
7. C₃×C₇×#117	C₂	semidirect
8. C₇×#540	C₃	direct (N × H)
22. C₇×#127	C₂×C₃	semidirect
23. C₇×#125	C₂×C₃	semidirect
24. C₇×#123	C₂×C₃	semidirect
25. C₇×#121	C₂×C₃	semidirect
26. C₇×#121	C₂×C₃	semidirect
27. C₄×C₇×D₄	C₂×C₃	semidirect
28. C₇×#117	C₂×C₃	semidirect
29. C₃×#540	C₇	direct (N × H)
55. C₃×#127	C₂×C₇	semidirect
56. C₃×#125	C₂×C₇	semidirect
57. C₃×#123	C₂×C₇	semidirect
58. C₃×#121	C₂×C₇	semidirect
59. C₃×#121	C₂×C₇	semidirect
60. C₃×C₄×D₄	C₂×C₇	semidirect
61. C₃×#117	C₂×C₇	semidirect
73. #540	C₃×C₇	direct (N × H)
103. #127	C₂×C₃×C₇	semidirect
104. #125	C₂×C₃×C₇	semidirect
105. #123	C₂×C₃×C₇	semidirect
106. #121	C₂×C₃×C₇	semidirect
107. #121	C₂×C₃×C₇	semidirect
108. C₄×D₄	C₂×C₃×C₇	semidirect
109. #117	C₂×C₃×C₇	semidirect
133. C₃×C₇	#540	direct (N × H)
174. C₇	C₃×#540	direct (N × H)
190. C₃	C₇×#540	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×#125	2
C₃×C₇×#123	2
C₃×C₇×#121	2
C₃×C₄×C₇×D₄	2
C₃×C₇×#117	2
C₇×#540	3
C₃×#540	7

Subgroups — C3×C7×#540 #13061960

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₇×#540 #13061960