Subgroups — C₃×#90286 #13060937

Order 1344.

504 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇×#120	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#6516	C₂	1	2	yes	maximal, complemented
3	C₃×#6503	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#6502	C₂	1	2	yes	maximal, complemented
5	C₃×#6501	C₂	1	2	yes	maximal, complemented
6	C₃×#6498	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#6447	C₂	1	2	yes	maximal, complemented
8	C₃×#6446	C₂	1	2	yes	maximal, complemented
9	C₃×#6442	C₂	1	2	yes	maximal, complemented
10	#90286	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
11	C₂⁴×C₃×C₇	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₇×D₄	C₂²	1	2	yes	—
14	C₂×C₃×C₂²⋊D₇	C₂²	1	2	yes	—
15	C₂×C₃×C₂²⋊D₇	C₂²	1	2	yes	—
16	C₂×C₃×C₂²⋊D₇	C₂²	1	2	yes	—
17	C₂×C₃×C₂²⋊D₇	C₂²	1	2	yes	complemented
18	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic, complemented
19	C₂×C₃×D₂₈	C₂²	1	1	yes	characteristic, complemented
20	C₂×C₃×C₄×D₇	C₂²	1	2	yes	complemented
21	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₃×C₇×C₂²⋊C₄	C₂²	1	1	yes	characteristic
23	C₃×C₂²⋊Dic₇	C₂²	1	2	yes	—
24	C₃×C₂²⋊Dic₇	C₂²	1	2	yes	—
25	C₃×C₂²⋊Dic₇	C₂²	1	2	yes	—
26	C₃×C₂²⋊Dic₇	C₂²	1	1	yes	characteristic, complemented
27	C₃×(C₂×D₇)⋊C₄	C₂²	1	2	yes	complemented
28	C₃×C₄⋊Dic₇	C₂²	1	2	yes	complemented
29	C₃×Dic₇⋊C₄	C₂²	1	2	yes	complemented
30	C₃×Dic₇⋊C₄	C₂²	1	2	yes	complemented
31	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic, complemented
32	C₇×#120	C₂×C₃	1	1	yes	characteristic, complemented
33	#6516	C₂×C₃	1	2	yes	complemented
34	#6503	C₂×C₃	1	1	yes	characteristic, complemented
35	#6502	C₂×C₃	1	2	yes	complemented
36	#6501	C₂×C₃	1	2	yes	complemented
37	#6498	C₂×C₃	1	1	yes	characteristic, complemented
38	#6447	C₂×C₃	1	2	yes	complemented
39	#6446	C₂×C₃	1	2	yes	complemented
40	#6442	C₂×C₃	1	2	yes	complemented
41	C₃×#560	—	7	1	no	maximal, Hall
42	C₂³×C₃×C₇	—	4	2	no	—
43	C₂³×C₃×C₇	—	4	1	no	—
44	C₂³×C₃×C₇	C₂³	1	2	yes	—
45	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
46	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
47	C₂²×C₃×D₇	C₂³	1	2	yes	—
48	C₃×C₇×D₄	—	4	2	no	—
49	C₃×C₇×D₄	—	4	1	no	—
50	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
51	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
52	C₃×C₂²⋊D₇	—	4	2	no	—
53	C₃×C₂²⋊D₇	—	4	2	no	—
54	C₃×C₂²⋊D₇	—	4	2	no	—
55	C₃×C₂²⋊D₇	—	4	2	no	—
56	C₂×C₃×Dic₇	—	4	1	no	—
57	C₂×C₃×Dic₇	C₂³	1	2	yes	—
58	C₂×C₃×Dic₇	C₂³	1	2	yes	—
59	C₂×C₃×Dic₇	C₂³	1	2	yes	—
60	C₃×D₂₈	—	4	1	no	—
61	C₃×C₄×D₇	—	4	2	no	—
62	C₂⁴×C₇	C₂²×C₃	1	1	yes	characteristic
63	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
64	C₂×C₇×D₄	C₂²×C₃	1	2	yes	—
65	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
66	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
67	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	—
68	C₂×C₂²⋊D₇	C₂²×C₃	1	2	yes	complemented
69	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic, complemented
70	C₂×D₂₈	C₂²×C₃	1	1	yes	characteristic, complemented
71	C₂×C₄×D₇	C₂²×C₃	1	2	yes	complemented
72	C₇×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
73	C₇×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
74	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
75	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
76	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
77	C₂²⋊Dic₇	C₂²×C₃	1	1	yes	characteristic, complemented
78	(C₂×D₇)⋊C₄	C₂²×C₃	1	2	yes	complemented
79	C₄⋊Dic₇	C₂²×C₃	1	2	yes	complemented
80	Dic₇⋊C₄	C₂²×C₃	1	2	yes	complemented
81	Dic₇⋊C₄	C₂²×C₃	1	2	yes	complemented
82	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic, complemented
83	C₃×#127	—	7	1	no	—
84	C₃×#126	—	7	2	no	—
85	C₃×#123	—	7	1	no	—
86	C₃×#123	—	7	2	no	—
87	C₃×#121	—	7	2	no	—
88	C₃×#121	—	7	2	no	—
89	C₃×#121	—	7	2	no	—
90	C₃×#120	—	7	2	no	—
91	C₃×#120	D₇	1	1	yes	characteristic, complemented
92	C₂²×C₃×C₇	—	2	1	no	—
93	C₂²×C₃×C₇	—	2	2	no	—
94	C₂²×C₃×C₇	—	2	2	no	—
95	C₂²×C₃×C₇	—	2	2	no	—
96	C₂²×C₃×C₇	—	2	2	no	—
97	C₂²×C₃×C₇	—	2	2	no	—
98	C₂²×C₃×C₇	—	2	1	no	—
99	C₂²×C₃×C₇	—	4	4	no	—
100	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
101	C₂×C₃×D₇	—	2	2	no	—
102	C₂×C₃×D₇	—	2	2	no	—
103	C₂×C₃×D₇	—	2	2	no	—
104	C₃×C₄×C₇	—	2	2	no	—
105	C₃×C₄×C₇	—	2	1	no	—
106	C₃×Dic₇	—	2	2	no	—
107	C₃×Dic₇	—	2	2	no	—
108	C₃×Dic₇	—	2	2	no	—
109	#560	—	7	1	no	Sylow
110	C₂³×C₇	—	4	2	no	—
111	C₂³×C₇	—	4	1	no	—
112	C₂³×C₇	C₂³×C₃	1	2	yes	—
113	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
114	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
115	C₂²×D₇	C₂³×C₃	1	2	yes	—
116	C₇×D₄	—	4	2	no	—
117	C₇×D₄	—	4	1	no	—
118	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
119	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
120	C₂²⋊D₇	—	4	2	no	—
121	C₂²⋊D₇	—	4	2	no	—
122	C₂²⋊D₇	—	4	2	no	—
123	C₂²⋊D₇	—	4	2	no	—
124	C₂×Dic₇	—	4	1	no	—
125	C₂×Dic₇	C₂³×C₃	1	2	yes	—
126	C₂×Dic₇	C₂³×C₃	1	2	yes	—
127	C₂×Dic₇	C₂³×C₃	1	2	yes	—
128	D₂₈	—	4	1	no	—
129	C₄×D₇	—	4	2	no	—
130	C₂⁴×C₃	C₂×D₇	1	1	yes	characteristic
131	C₂×C₃×D₄	—	7	2	no	—
132	C₂×C₃×D₄	—	7	2	no	—
133	C₂×C₃×D₄	—	7	2	no	—
134	C₂×C₃×D₄	—	7	2	no	—
135	C₂×C₃×D₄	—	7	1	no	—
136	C₂×C₃×D₄	C₂×D₇	1	2	yes	—
137	C₂×C₃×D₄	C₂×D₇	1	1	yes	characteristic
138	C₂²×C₃×C₄	—	7	1	no	—
139	C₂²×C₃×C₄	—	7	2	no	—
140	C₃×C₄⋊C₄	—	7	2	no	—
141	C₃×C₄⋊C₄	—	7	2	no	—
142	C₃×C₄⋊C₄	—	7	2	no	—
143	C₃×C₂²⋊C₄	—	7	2	no	—
144	C₃×C₂²⋊C₄	—	7	2	no	—
145	C₃×C₂²⋊C₄	—	7	2	no	—
146	C₃×C₂²⋊C₄	—	7	1	no	—
147	C₃×C₂²⋊C₄	—	7	2	no	—
148	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	—
149	C₃×C₂²⋊C₄	C₂×D₇	1	1	yes	characteristic
150	C₃×C₄²	—	7	1	no	—
151	C₂×C₃×C₇	—	4	1	no	—
152	C₂×C₃×C₇	—	4	2	no	—
153	C₂×C₃×C₇	—	4	1	no	—
154	C₂×C₃×C₇	H(5,𝔽₂)	1	2	yes	—
155	C₂×C₃×C₇	H(5,𝔽₂)	1	1	yes	characteristic
156	C₃×D₇	—	4	2	no	—
157	#127	—	7	1	no	—
158	#126	—	7	2	no	—
159	#123	—	7	1	no	—
160	#123	—	7	2	no	—
161	#121	—	7	2	no	—
162	#121	—	7	2	no	—
163	#121	—	7	2	no	—
164	#120	—	7	2	no	—
165	#120	C₃×D₇	1	1	yes	characteristic, complemented
166	C₂²×C₇	—	2	1	no	—
167	C₂²×C₇	—	2	2	no	—
168	C₂²×C₇	—	2	2	no	—
169	C₂²×C₇	—	2	2	no	—
170	C₂²×C₇	—	2	2	no	—
171	C₂²×C₇	—	2	2	no	—
172	C₂²×C₇	—	2	1	no	—
173	C₂²×C₇	—	4	4	no	—
174	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
175	C₂×D₇	—	2	2	no	—
176	C₂×D₇	—	2	2	no	—
177	C₂×D₇	—	2	2	no	—
178	C₄×C₇	—	2	2	no	—
179	C₄×C₇	—	2	1	no	—
180	Dic₇	—	2	2	no	—
181	Dic₇	—	2	2	no	—
182	Dic₇	—	2	2	no	—
183	C₂³×C₃	—	4	2	no	—
184	C₂³×C₃	—	4	1	no	—
185	C₂³×C₃	—	7	2	no	—
186	C₂³×C₃	C₂²×D₇	1	2	yes	—
187	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
188	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
189	C₃×D₄	—	4	2	no	—
190	C₃×D₄	—	4	1	no	—
191	C₃×D₄	—	28	1	no	—
192	C₃×D₄	—	28	2	no	—
193	C₃×D₄	—	28	2	no	—
194	C₃×D₄	—	28	2	no	—
195	C₃×D₄	—	28	2	no	—
196	C₂×C₃×C₄	—	7	2	no	—
197	C₂×C₃×C₄	—	7	2	no	—
198	C₂×C₃×C₄	—	7	2	no	—
199	C₂×C₃×C₄	—	28	2	no	—
200	C₂×C₃×C₄	—	28	1	no	—
201	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
202	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
203	C₃×C₇	#560	1	1	yes	characteristic, complemented, Hall
204	C₂⁴	C₂×C₃×D₇	1	1	yes	characteristic
205	C₂×D₄	—	7	2	no	—
206	C₂×D₄	—	7	2	no	—
207	C₂×D₄	—	7	2	no	—
208	C₂×D₄	—	7	2	no	—
209	C₂×D₄	—	7	1	no	—
210	C₂×D₄	C₂×C₃×D₇	1	2	yes	—
211	C₂×D₄	C₂×C₃×D₇	1	1	yes	characteristic
212	C₂²×C₄	—	7	1	no	—
213	C₂²×C₄	—	7	2	no	—
214	C₄⋊C₄	—	7	2	no	—
215	C₄⋊C₄	—	7	2	no	—
216	C₄⋊C₄	—	7	2	no	—
217	C₂²⋊C₄	—	7	2	no	—
218	C₂²⋊C₄	—	7	2	no	—
219	C₂²⋊C₄	—	7	2	no	—
220	C₂²⋊C₄	—	7	1	no	—
221	C₂²⋊C₄	—	7	2	no	—
222	C₂²⋊C₄	C₂×C₃×D₇	1	2	yes	—
223	C₂²⋊C₄	C₂×C₃×D₇	1	1	yes	characteristic
224	C₄²	—	7	1	no	—
225	C₂×C₇	—	4	2	no	—
226	C₂×C₇	—	4	1	no	—
227	C₂×C₇	—	4	1	no	—
228	C₂×C₇	C₃×H(5,𝔽₂)	1	2	yes	—
229	C₂×C₇	C₃×H(5,𝔽₂)	1	1	yes	characteristic
230	D₇	—	4	2	no	—
231	C₂²×C₃	—	2	1	no	—
232	C₂²×C₃	—	2	2	no	—
233	C₂²×C₃	—	2	2	no	—
234	C₂²×C₃	—	2	1	no	—
235	C₂²×C₃	—	2	2	no	—
236	C₂²×C₃	—	2	2	no	—
237	C₂²×C₃	—	2	2	no	—
238	C₂²×C₃	—	4	4	no	—
239	C₂²×C₃	—	14	2	no	—
240	C₂²×C₃	—	14	2	no	—
241	C₂²×C₃	—	14	2	no	—
242	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
243	C₃×C₄	—	2	2	no	—
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	—	14	2	no	—
246	C₃×C₄	—	14	2	no	—
247	C₃×C₄	—	14	2	no	—
248	C₂³	—	4	2	no	—
249	C₂³	—	4	1	no	—
250	C₂³	—	7	2	no	—
251	C₂³	C₂²×C₃×D₇	1	2	yes	—
252	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
253	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
254	D₄	—	4	2	no	—
255	D₄	—	4	1	no	—
256	D₄	—	28	1	no	—
257	D₄	—	28	2	no	—
258	D₄	—	28	2	no	—
259	D₄	—	28	2	no	—
260	D₄	—	28	2	no	—
261	C₂×C₄	—	7	2	no	—
262	C₂×C₄	—	7	2	no	—
263	C₂×C₄	—	7	2	no	—
264	C₂×C₄	—	28	2	no	—
265	C₂×C₄	—	28	1	no	—
266	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
267	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
268	C₇	C₃×#560	1	1	yes	characteristic, complemented, Sylow
269	C₂×C₃	—	4	2	no	—
270	C₂×C₃	—	4	1	no	—
271	C₂×C₃	—	4	1	no	—
272	C₂×C₃	—	28	2	no	—
273	C₂×C₃	#6548	1	2	yes	—
274	C₂×C₃	#6548	1	1	yes	characteristic
275	C₂²	—	2	1	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	4	4	no	—
283	C₂²	—	14	2	no	—
284	C₂²	—	14	2	no	—
285	C₂²	—	14	2	no	—
286	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	2	1	no	—
289	C₄	—	14	2	no	—
290	C₄	—	14	2	no	—
291	C₄	—	14	2	no	—
292	C₃	#90286	1	1	yes	characteristic, direct product factor, Sylow
293	C₂	—	4	2	no	—
294	C₂	—	4	1	no	—
295	C₂	—	4	1	no	—
296	C₂	—	28	2	no	—
297	C₂	C₃×#6548	1	2	yes	—
298	C₂	C₃×#6548	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₇
commutator, cosocle	C₂²×C₇	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₇
Fitting	C₃×C₇×#120	C₂
socle	C₂²×C₃×C₇	C₂⁴
hypercenter	C₃×#120	D₇
hypocenter	C₇	C₃×#560
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#120

Complements

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

N	H ≅ G/N	type
1. C₃×C₇×#120	C₂	semidirect
2. C₃×#6516	C₂	semidirect
3. C₃×#6503	C₂	semidirect
4. C₃×#6502	C₂	semidirect
5. C₃×#6501	C₂	semidirect
6. C₃×#6498	C₂	semidirect
7. C₃×#6447	C₂	semidirect
8. C₃×#6446	C₂	semidirect
9. C₃×#6442	C₂	semidirect
10. #90286	C₃	direct (N × H)
17. C₂×C₃×C₂²⋊D₇	C₂²	semidirect
18. C₂²×C₃×Dic₇	C₂²	semidirect
19. C₂×C₃×D₂₈	C₂²	semidirect
20. C₂×C₃×C₄×D₇	C₂²	semidirect
26. C₃×C₂²⋊Dic₇	C₂²	semidirect
27. C₃×(C₂×D₇)⋊C₄	C₂²	semidirect
28. C₃×C₄⋊Dic₇	C₂²	semidirect
29. C₃×Dic₇⋊C₄	C₂²	semidirect
30. C₃×Dic₇⋊C₄	C₂²	semidirect
31. C₃×C₄×Dic₇	C₂²	semidirect
32. C₇×#120	C₂×C₃	semidirect
33. #6516	C₂×C₃	semidirect
34. #6503	C₂×C₃	semidirect
35. #6502	C₂×C₃	semidirect
36. #6501	C₂×C₃	semidirect
37. #6498	C₂×C₃	semidirect
38. #6447	C₂×C₃	semidirect
39. #6446	C₂×C₃	semidirect
40. #6442	C₂×C₃	semidirect
68. C₂×C₂²⋊D₇	C₂²×C₃	semidirect
69. C₂²×Dic₇	C₂²×C₃	semidirect
70. C₂×D₂₈	C₂²×C₃	semidirect
71. C₂×C₄×D₇	C₂²×C₃	semidirect
77. C₂²⋊Dic₇	C₂²×C₃	semidirect
78. (C₂×D₇)⋊C₄	C₂²×C₃	semidirect
79. C₄⋊Dic₇	C₂²×C₃	semidirect
80. Dic₇⋊C₄	C₂²×C₃	semidirect
81. Dic₇⋊C₄	C₂²×C₃	semidirect
82. C₄×Dic₇	C₂²×C₃	semidirect
91. C₃×#120	D₇	semidirect
165. #120	C₃×D₇	semidirect
203. C₃×C₇	#560	semidirect
268. C₇	C₃×#560	semidirect
292. C₃	#90286	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×#120	2
C₃×#6516	2
C₃×#6503	2
C₃×#6502	2
C₃×#6501	2
C₃×#6498	2
C₃×#6447	2
C₃×#6446	2
C₃×#6442	2
#90286	3
C₃×#560	7

Subgroups — C3×#90286 #13060937

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90286 #13060937