Subgroups — C₃×#90291 #13060942

Order 1344.

468 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×#121	C₂	1	1	yes	maximal, characteristic
2	C₃×#6499	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#6498	C₂	1	2	yes	maximal, complemented
4	C₃×D₄×Dic₇	C₂	1	2	yes	maximal, complemented
5	C₃×D₄×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#6487	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃×C₄×Dic₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#6453	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#6450	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃×#6442	C₂	1	2	yes	maximal, complemented
11	C₃×#6440	C₂	1	2	yes	maximal, complemented
12	#90291	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₂×C₃×C₇×D₄	C₂²	1	2	yes	—
14	C₂×C₃×C₇×D₄	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
16	C₂²×C₃×Dic₇	C₂²	1	2	yes	—
17	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
18	C₂²×C₃×Dic₇	C₂²	1	1	yes	characteristic
19	C₂×C₃×Dic₁₄	C₂²	1	1	yes	characteristic
20	C₃×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₃×C₇×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₃×C₂²⋊Dic₇	C₂²	1	2	yes	—
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×C₄⋊Dic₇	C₂²	1	2	yes	—
27	C₃×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
28	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
29	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
30	C₃×Dic₇⋊C₄	C₂²	1	2	yes	—
31	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
32	C₃×C₄×Dic₇	C₂²	1	2	yes	—
33	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
34	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
35	C₃×C₄×Dic₇	C₂²	1	1	yes	characteristic
36	C₇×#121	C₂×C₃	1	1	yes	characteristic
37	#6499	C₂×C₃	1	1	yes	characteristic, complemented
38	#6498	C₂×C₃	1	2	yes	complemented
39	D₄×Dic₇	C₂×C₃	1	2	yes	complemented
40	D₄×Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
41	#6487	C₂×C₃	1	1	yes	characteristic, complemented
42	C₂×C₄×Dic₇	C₂×C₃	1	1	yes	characteristic, complemented
43	#6453	C₂×C₃	1	1	yes	characteristic, complemented
44	#6450	C₂×C₃	1	1	yes	characteristic, complemented
45	#6442	C₂×C₃	1	2	yes	complemented
46	#6440	C₂×C₃	1	2	yes	complemented
47	C₃×#529	—	7	1	no	maximal, Hall
48	C₂³×C₃×C₇	C₂³	1	2	yes	—
49	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
50	C₃×C₇×D₄	—	2	4	no	—
51	C₃×C₇×D₄	—	2	2	no	—
52	C₂×C₃×C₄×C₇	—	2	2	no	—
53	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
54	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
55	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
56	C₂×C₃×Dic₇	—	2	2	no	—
57	C₂×C₃×Dic₇	—	2	2	no	—
58	C₂×C₃×Dic₇	—	2	4	no	—
59	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
60	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
61	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
62	C₂×C₃×Dic₇	C₂³	1	2	yes	—
63	C₂×C₃×Dic₇	C₂³	1	2	yes	—
64	C₂×C₃×Dic₇	C₂³	1	1	yes	characteristic
65	C₃×Dic₁₄	—	2	2	no	—
66	C₂×C₇×D₄	C₂²×C₃	1	2	yes	—
67	C₂×C₇×D₄	C₂²×C₃	1	1	yes	characteristic
68	C₂²×C₄×C₇	C₂²×C₃	1	1	yes	characteristic
69	C₂²×Dic₇	C₂²×C₃	1	2	yes	—
70	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
71	C₂²×Dic₇	C₂²×C₃	1	1	yes	characteristic
72	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
73	C₇×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
74	C₇×C₂²⋊C₄	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	2	yes	—
78	C₂²⋊Dic₇	C₂²×C₃	1	2	yes	—
79	C₄⋊Dic₇	C₂²×C₃	1	2	yes	—
80	C₄⋊Dic₇	C₂²×C₃	1	1	yes	characteristic
81	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
82	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
83	Dic₇⋊C₄	C₂²×C₃	1	2	yes	—
84	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
85	C₄×Dic₇	C₂²×C₃	1	2	yes	—
86	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
87	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
88	C₄×Dic₇	C₂²×C₃	1	1	yes	characteristic
89	C₃×#126	—	7	2	no	—
90	C₃×#125	—	7	1	no	—
91	C₃×#124	—	7	1	no	—
92	C₃×#123	—	7	2	no	—
93	C₃×C₂²⋊Q₈	—	7	1	no	—
94	C₃×#121	D₇	1	1	yes	characteristic
95	C₃×C₄×Q₈	—	7	1	no	—
96	C₃×C₄×D₄	—	7	1	no	—
97	C₃×C₄×D₄	—	7	2	no	—
98	C₃×#117	—	7	2	no	—
99	C₂×C₃×C₄²	—	7	1	no	—
100	C₂²×C₃×C₇	—	2	1	no	—
101	C₂²×C₃×C₇	—	2	1	no	—
102	C₂²×C₃×C₇	—	2	2	no	—
103	C₂²×C₃×C₇	—	2	2	no	—
104	C₂²×C₃×C₇	—	2	2	no	—
105	C₂²×C₃×C₇	#41	1	2	yes	—
106	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
107	C₃×C₄×C₇	—	2	1	no	—
108	C₃×C₄×C₇	—	2	2	no	—
109	C₃×C₄×C₇	#41	1	2	yes	—
110	C₃×Dic₇	—	2	2	no	—
111	C₃×Dic₇	—	2	1	no	—
112	C₃×Dic₇	—	2	1	no	—
113	C₃×Dic₇	—	2	1	no	—
114	C₃×Dic₇	—	2	2	no	—
115	C₃×Dic₇	#41	1	2	yes	—
116	#529	—	7	1	no	Sylow
117	C₂³×C₇	C₂³×C₃	1	2	yes	—
118	C₂³×C₇	C₂³×C₃	1	1	yes	characteristic
119	C₇×D₄	—	2	4	no	—
120	C₇×D₄	—	2	2	no	—
121	C₂×C₄×C₇	—	2	2	no	—
122	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
123	C₂×C₄×C₇	C₂³×C₃	1	2	yes	—
124	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
125	C₂×Dic₇	—	2	2	no	—
126	C₂×Dic₇	—	2	2	no	—
127	C₂×Dic₇	—	2	4	no	—
128	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
129	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
130	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
131	C₂×Dic₇	C₂³×C₃	1	2	yes	—
132	C₂×Dic₇	C₂³×C₃	1	2	yes	—
133	C₂×Dic₇	C₂³×C₃	1	1	yes	characteristic
134	Dic₁₄	—	2	2	no	—
135	C₂×C₃×Q₈	—	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	1	no	—
140	C₂²×C₃×C₄	—	7	2	no	—
141	C₂²×C₃×C₄	C₂×D₇	1	1	yes	characteristic
142	C₃×C₄⋊C₄	—	7	2	no	—
143	C₃×C₄⋊C₄	—	7	2	no	—
144	C₃×C₄⋊C₄	—	7	2	no	—
145	C₃×C₄⋊C₄	—	7	1	no	—
146	C₃×C₄⋊C₄	—	7	2	no	—
147	C₃×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
148	C₃×C₂²⋊C₄	—	7	2	no	—
149	C₃×C₂²⋊C₄	—	7	2	no	—
150	C₃×C₂²⋊C₄	—	7	2	no	—
151	C₃×C₂²⋊C₄	—	7	2	no	—
152	C₃×C₂²⋊C₄	C₂×D₇	1	2	yes	—
153	C₃×C₄²	—	7	1	no	—
154	C₃×C₄²	—	7	1	no	—
155	C₃×C₄²	—	7	1	no	—
156	C₃×C₄²	—	7	1	no	—
157	C₃×C₄²	—	7	2	no	—
158	C₂×C₃×C₇	—	2	2	no	—
159	C₂×C₃×C₇	—	4	2	no	—
160	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
161	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
162	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
163	#126	—	7	2	no	—
164	#125	—	7	1	no	—
165	#124	—	7	1	no	—
166	#123	—	7	2	no	—
167	C₂²⋊Q₈	—	7	1	no	—
168	#121	C₃×D₇	1	1	yes	characteristic
169	C₄×Q₈	—	7	1	no	—
170	C₄×D₄	—	7	1	no	—
171	C₄×D₄	—	7	2	no	—
172	#117	—	7	2	no	—
173	C₂×C₄²	—	7	1	no	—
174	C₂²×C₇	—	2	1	no	—
175	C₂²×C₇	—	2	1	no	—
176	C₂²×C₇	—	2	2	no	—
177	C₂²×C₇	—	2	2	no	—
178	C₂²×C₇	—	2	2	no	—
179	C₂²×C₇	C₃×#41	1	2	yes	—
180	C₂²×C₇	C₂⁴×C₃	1	1	yes	characteristic
181	C₄×C₇	—	2	1	no	—
182	C₄×C₇	—	2	2	no	—
183	C₄×C₇	C₃×#41	1	2	yes	—
184	Dic₇	—	2	2	no	—
185	Dic₇	—	2	1	no	—
186	Dic₇	—	2	1	no	—
187	Dic₇	—	2	1	no	—
188	Dic₇	—	2	2	no	—
189	Dic₇	C₃×#41	1	2	yes	—
190	C₂³×C₃	C₂²×D₇	1	2	yes	—
191	C₂³×C₃	C₂²×D₇	1	1	yes	characteristic
192	C₃×Q₈	—	14	2	no	—
193	C₃×D₄	—	2	4	no	—
194	C₃×D₄	—	2	2	no	—
195	C₂×C₃×C₄	—	2	2	no	—
196	C₂×C₃×C₄	—	7	1	no	—
197	C₂×C₃×C₄	—	7	1	no	—
198	C₂×C₃×C₄	—	7	1	no	—
199	C₂×C₃×C₄	—	7	1	no	—
200	C₂×C₃×C₄	—	7	2	no	—
201	C₂×C₃×C₄	—	7	2	no	—
202	C₂×C₃×C₄	—	14	2	no	—
203	C₂×C₃×C₄	—	14	4	no	—
204	C₂×C₃×C₄	—	14	2	no	—
205	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
206	C₂×C₃×C₄	C₂²×D₇	1	2	yes	—
207	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
208	C₃×C₇	#529	1	1	yes	characteristic, complemented, Hall
209	C₂×Q₈	—	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	1	no	—
214	C₂²×C₄	—	7	2	no	—
215	C₂²×C₄	C₂×C₃×D₇	1	1	yes	characteristic
216	C₄⋊C₄	—	7	2	no	—
217	C₄⋊C₄	—	7	2	no	—
218	C₄⋊C₄	—	7	2	no	—
219	C₄⋊C₄	—	7	1	no	—
220	C₄⋊C₄	—	7	2	no	—
221	C₄⋊C₄	C₂×C₃×D₇	1	1	yes	characteristic
222	C₂²⋊C₄	—	7	2	no	—
223	C₂²⋊C₄	—	7	2	no	—
224	C₂²⋊C₄	—	7	2	no	—
225	C₂²⋊C₄	—	7	2	no	—
226	C₂²⋊C₄	C₂×C₃×D₇	1	2	yes	—
227	C₄²	—	7	1	no	—
228	C₄²	—	7	1	no	—
229	C₄²	—	7	1	no	—
230	C₄²	—	7	1	no	—
231	C₄²	—	7	2	no	—
232	C₂×C₇	—	2	2	no	—
233	C₂×C₇	—	4	2	no	—
234	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
235	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
236	C₂×C₇	C₂×C₃×#41	1	1	yes	characteristic
237	C₂²×C₃	—	2	1	no	—
238	C₂²×C₃	—	2	1	no	—
239	C₂²×C₃	—	2	2	no	—
240	C₂²×C₃	—	2	2	no	—
241	C₂²×C₃	—	2	2	no	—
242	C₂²×C₃	#1159	1	2	yes	—
243	C₂²×C₃	C₂³×D₇	1	1	yes	characteristic
244	C₃×C₄	—	2	1	no	—
245	C₃×C₄	—	2	2	no	—
246	C₃×C₄	—	7	2	no	—
247	C₃×C₄	—	14	1	no	—
248	C₃×C₄	—	14	1	no	—
249	C₃×C₄	—	14	2	no	—
250	C₃×C₄	—	14	1	no	—
251	C₃×C₄	—	14	2	no	—
252	C₃×C₄	#1159	1	2	yes	—
253	C₂³	C₂²×C₃×D₇	1	2	yes	—
254	C₂³	C₂²×C₃×D₇	1	1	yes	characteristic
255	Q₈	—	14	2	no	—
256	D₄	—	2	4	no	—
257	D₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	7	1	no	—
260	C₂×C₄	—	7	1	no	—
261	C₂×C₄	—	7	1	no	—
262	C₂×C₄	—	7	1	no	—
263	C₂×C₄	—	7	2	no	—
264	C₂×C₄	—	7	2	no	—
265	C₂×C₄	—	14	2	no	—
266	C₂×C₄	—	14	4	no	—
267	C₂×C₄	—	14	2	no	—
268	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
269	C₂×C₄	C₂²×C₃×D₇	1	2	yes	—
270	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
271	C₇	C₃×#529	1	1	yes	characteristic, complemented, Sylow
272	C₂×C₃	—	2	2	no	—
273	C₂×C₃	—	4	2	no	—
274	C₂×C₃	C₂×#1159	1	1	yes	characteristic
275	C₂×C₃	C₂×#1159	1	1	yes	characteristic
276	C₂×C₃	D₇×#41	1	1	yes	characteristic
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	C₃×#1159	1	2	yes	—
283	C₂²	C₂³×C₃×D₇	1	1	yes	characteristic
284	C₄	—	2	1	no	—
285	C₄	—	2	2	no	—
286	C₄	—	7	2	no	—
287	C₄	—	14	1	no	—
288	C₄	—	14	1	no	—
289	C₄	—	14	2	no	—
290	C₄	—	14	1	no	—
291	C₄	—	14	2	no	—
292	C₄	C₃×#1159	1	2	yes	—
293	C₃	#90291	1	1	yes	characteristic, direct product factor, Sylow
294	C₂	—	2	2	no	—
295	C₂	—	4	2	no	—
296	C₂	C₂×C₃×#1159	1	1	yes	characteristic
297	C₂	C₂×C₃×#1159	1	1	yes	characteristic
298	C₂	C₃×D₇×#41	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	C₂³×D₇
commutator, cosocle	C₂²×C₇	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₇
Fitting, prime radical	C₃×C₇×#121	C₂
socle	C₂²×C₃×C₇	C₂⁴
hypercenter	C₃×#121	D₇
hypocenter	C₇	C₃×#529
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#121

Complements

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

N	H ≅ G/N	type
2. C₃×#6499	C₂	semidirect
3. C₃×#6498	C₂	semidirect
4. C₃×D₄×Dic₇	C₂	semidirect
5. C₃×D₄×Dic₇	C₂	semidirect
6. C₃×#6487	C₂	semidirect
7. C₂×C₃×C₄×Dic₇	C₂	semidirect
8. C₃×#6453	C₂	semidirect
9. C₃×#6450	C₂	semidirect
10. C₃×#6442	C₂	semidirect
11. C₃×#6440	C₂	semidirect
12. #90291	C₃	direct (N × H)
37. #6499	C₂×C₃	semidirect
38. #6498	C₂×C₃	semidirect
39. D₄×Dic₇	C₂×C₃	semidirect
40. D₄×Dic₇	C₂×C₃	semidirect
41. #6487	C₂×C₃	semidirect
42. C₂×C₄×Dic₇	C₂×C₃	semidirect
43. #6453	C₂×C₃	semidirect
44. #6450	C₂×C₃	semidirect
45. #6442	C₂×C₃	semidirect
46. #6440	C₂×C₃	semidirect
208. C₃×C₇	#529	semidirect
271. C₇	C₃×#529	semidirect
293. C₃	#90291	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×#121	2
C₃×#6499	2
C₃×#6498	2
C₃×D₄×Dic₇	2
C₃×#6487	2
C₂×C₃×C₄×Dic₇	2
C₃×#6453	2
C₃×#6450	2
C₃×#6442	2
C₃×#6440	2
#90291	3
C₃×#529	7

Subgroups — C3×#90291 #13060942

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90291 #13060942