Subgroups — C₃×#90470 #13061121

Order 1344.

496 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#6554	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#6551	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₇×#137	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#6513	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃×#1144	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#6481	C₂	1	2	yes	maximal, complemented
7	C₃×D₇×Q₁₆	C₂	1	2	yes	maximal, complemented
8	C₃×#6479	C₂	1	2	yes	maximal, complemented
9	C₃×#6478	C₂	1	2	yes	maximal, complemented
10	C₃×#6472	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃×#6470	C₂	1	1	yes	maximal, characteristic, complemented
12	#90470	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
13	C₃×C₇×#41	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₇×Q₈	C₂²	1	1	yes	characteristic
15	C₃×#1161	—	2	1	no	—
16	C₃×#1161	C₂²	1	2	yes	—
17	C₃×Q₈×D₇	—	2	1	no	—
18	C₃×Q₈×D₇	C₂²	1	2	yes	—
19	C₃×Q₈×D₇	C₂²	1	2	yes	—
20	C₃×#1159	—	2	2	no	—
21	C₃×#1159	C₂²	1	2	yes	—
22	C₃×#1157	—	2	2	no	—
23	C₃×#1157	—	2	1	no	—
24	C₃×#1157	C₂²	1	1	yes	characteristic
25	C₃×#1157	C₂²	1	1	yes	characteristic
26	C₂×C₃×Dic₁₄	—	2	1	no	—
27	C₂×C₃×Dic₁₄	C₂²	1	1	yes	characteristic
28	C₃×C₇×Q₁₆	C₂²	1	2	yes	—
29	C₃×C₇×SD₁₆	C₂²	1	2	yes	—
30	C₃×C₇×M₁₆	C₂²	1	1	yes	characteristic
31	C₃×#1144	C₂²	1	2	yes	—
32	C₃×#1144	C₂²	1	2	yes	—
33	C₃×#1144	C₂²	1	1	yes	characteristic
34	C₃×#1143	C₂²	1	1	yes	characteristic
35	C₃×#1142	C₂²	1	1	yes	characteristic
36	C₃×#1141	C₂²	1	1	yes	characteristic
37	C₂×C₃×C₇⋊C₈	C₂²	1	1	yes	characteristic
38	C₃×Dic₂₈	C₂²	1	2	yes	—
39	C₃×#1132	C₂²	1	2	yes	—
40	C₃×#1131	C₂²	1	2	yes	—
41	C₃×C₈×D₇	C₂²	1	2	yes	—
42	#6554	C₂×C₃	1	1	yes	characteristic, complemented
43	#6551	C₂×C₃	1	1	yes	characteristic, complemented
44	C₇×#137	C₂×C₃	1	1	yes	characteristic, complemented
45	#6513	C₂×C₃	1	1	yes	characteristic, complemented
46	C₂×#1144	C₂×C₃	1	1	yes	characteristic, complemented
47	#6481	C₂×C₃	1	2	yes	complemented
48	D₇×Q₁₆	C₂×C₃	1	2	yes	complemented
49	#6479	C₂×C₃	1	2	yes	complemented
50	#6478	C₂×C₃	1	2	yes	complemented
51	#6472	C₂×C₃	1	1	yes	characteristic, complemented
52	#6470	C₂×C₃	1	1	yes	characteristic, complemented
53	C₃×#578	—	7	1	no	maximal, Hall
54	C₃×C₇×Q₈	—	2	1	no	—
55	C₃×C₇×Q₈	C₂³	1	1	yes	characteristic
56	C₃×C₇×Q₈	C₂³	1	2	yes	—
57	C₃×C₇×D₄	—	2	1	no	—
58	C₃×C₇×D₄	C₂³	1	1	yes	characteristic
59	C₂×C₃×C₄×C₇	—	2	1	no	—
60	C₂×C₃×C₄×C₇	—	2	1	no	—
61	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
62	C₃×C₂²⋊D₇	—	2	1	no	—
63	C₃×C₂²⋊D₇	—	2	2	no	—
64	C₃×C₂²⋊D₇	D₄	1	2	yes	—
65	C₂×C₃×Dic₇	—	2	1	no	—
66	C₂×C₃×Dic₇	—	2	2	no	—
67	C₃×D₂₈	—	2	2	no	—
68	C₃×D₂₈	D₄	1	1	yes	characteristic
69	C₃×D₂₈	C₂³	1	1	yes	characteristic
70	C₃×C₄×D₇	—	2	2	no	—
71	C₃×C₄×D₇	—	2	2	no	—
72	C₃×C₄×D₇	—	2	1	no	—
73	C₃×C₄×D₇	—	2	2	no	—
74	C₃×C₄×D₇	C₂³	1	2	yes	—
75	C₃×Dic₁₄	—	2	1	no	—
76	C₃×Dic₁₄	—	2	2	no	—
77	C₃×Dic₁₄	—	2	2	no	—
78	C₃×Dic₁₄	D₄	1	1	yes	characteristic
79	C₃×Dic₁₄	C₂³	1	2	yes	—
80	C₃×Dic₁₄	C₂³	1	1	yes	characteristic
81	C₃×C₇×C₈	C₂³	1	2	yes	—
82	C₃×C₇⋊C₈	C₂³	1	1	yes	characteristic
83	C₃×C₇⋊C₈	C₂³	1	1	yes	characteristic
84	C₇×#41	C₂²×C₃	1	1	yes	characteristic
85	C₂×C₇×Q₈	C₂²×C₃	1	1	yes	characteristic
86	#1161	—	2	1	no	—
87	#1161	C₂²×C₃	1	2	yes	—
88	Q₈×D₇	—	2	1	no	—
89	Q₈×D₇	C₂²×C₃	1	2	yes	—
90	Q₈×D₇	C₂²×C₃	1	2	yes	—
91	#1159	—	2	2	no	—
92	#1159	C₂²×C₃	1	2	yes	—
93	#1157	—	2	2	no	—
94	#1157	—	2	1	no	—
95	#1157	C₂²×C₃	1	1	yes	characteristic
96	#1157	C₂²×C₃	1	1	yes	characteristic
97	C₂×Dic₁₄	—	2	1	no	—
98	C₂×Dic₁₄	C₂²×C₃	1	1	yes	characteristic
99	C₇×Q₁₆	C₂²×C₃	1	2	yes	—
100	C₇×SD₁₆	C₂²×C₃	1	2	yes	—
101	C₇×M₁₆	C₂²×C₃	1	1	yes	characteristic
102	#1144	C₂²×C₃	1	2	yes	—
103	#1144	C₂²×C₃	1	2	yes	—
104	#1144	C₂²×C₃	1	1	yes	characteristic
105	#1143	C₂²×C₃	1	1	yes	characteristic
106	#1142	C₂²×C₃	1	1	yes	characteristic
107	#1141	C₂²×C₃	1	1	yes	characteristic
108	C₂×C₇⋊C₈	C₂²×C₃	1	1	yes	characteristic
109	Dic₂₈	C₂²×C₃	1	2	yes	—
110	#1132	C₂²×C₃	1	2	yes	—
111	#1131	C₂²×C₃	1	2	yes	—
112	C₈×D₇	C₂²×C₃	1	2	yes	—
113	C₃×2₋¹⁺⁴	—	7	1	no	—
114	C₃×2₋¹⁺⁴	—	7	1	no	—
115	C₃×#137	—	7	2	no	—
116	C₃×#137	—	7	2	no	—
117	C₃×#137	—	7	1	no	—
118	C₃×#137	D₇	1	1	yes	characteristic, complemented
119	C₃×#135	—	7	1	no	—
120	C₃×#135	—	7	2	no	—
121	C₂×C₃×Q₁₆	—	7	1	no	—
122	C₂×C₃×Q₁₆	—	7	2	no	—
123	C₃×#131	—	7	1	no	—
124	C₂²×C₃×C₇	—	2	1	no	—
125	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
126	C₂×C₃×D₇	—	2	1	no	—
127	C₂×C₃×D₇	C₂×D₄	1	2	yes	—
128	C₃×C₄×C₇	—	2	1	no	—
129	C₃×C₄×C₇	—	2	2	no	—
130	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
131	C₃×C₄×C₇	C₂⁴	1	1	yes	characteristic
132	C₃×Dic₇	—	2	1	no	—
133	C₃×Dic₇	—	2	2	no	—
134	C₃×Dic₇	C₂×D₄	1	2	yes	—
135	#578	—	7	1	no	Sylow
136	C₇×Q₈	—	2	1	no	—
137	C₇×Q₈	C₂³×C₃	1	1	yes	characteristic
138	C₇×Q₈	C₂³×C₃	1	2	yes	—
139	C₇×D₄	—	2	1	no	—
140	C₇×D₄	C₂³×C₃	1	1	yes	characteristic
141	C₂×C₄×C₇	—	2	1	no	—
142	C₂×C₄×C₇	—	2	1	no	—
143	C₂×C₄×C₇	C₂³×C₃	1	1	yes	characteristic
144	C₂²⋊D₇	—	2	1	no	—
145	C₂²⋊D₇	—	2	2	no	—
146	C₂²⋊D₇	C₃×D₄	1	2	yes	—
147	C₂×Dic₇	—	2	1	no	—
148	C₂×Dic₇	—	2	2	no	—
149	D₂₈	—	2	2	no	—
150	D₂₈	C₃×D₄	1	1	yes	characteristic
151	D₂₈	C₂³×C₃	1	1	yes	characteristic
152	C₄×D₇	—	2	2	no	—
153	C₄×D₇	—	2	2	no	—
154	C₄×D₇	—	2	1	no	—
155	C₄×D₇	—	2	2	no	—
156	C₄×D₇	C₂³×C₃	1	2	yes	—
157	Dic₁₄	—	2	1	no	—
158	Dic₁₄	—	2	2	no	—
159	Dic₁₄	—	2	2	no	—
160	Dic₁₄	C₃×D₄	1	1	yes	characteristic
161	Dic₁₄	C₂³×C₃	1	2	yes	—
162	Dic₁₄	C₂³×C₃	1	1	yes	characteristic
163	C₇×C₈	C₂³×C₃	1	2	yes	—
164	C₇⋊C₈	C₂³×C₃	1	1	yes	characteristic
165	C₇⋊C₈	C₂³×C₃	1	1	yes	characteristic
166	C₃×#41	—	7	1	no	—
167	C₃×#41	—	7	2	no	—
168	C₃×#41	—	7	2	no	—
169	C₃×#41	—	7	1	no	—
170	C₃×#41	—	14	2	no	—
171	C₃×#41	—	14	1	no	—
172	C₃×#41	—	14	2	no	—
173	C₃×#41	—	14	1	no	—
174	C₃×#41	C₂×D₇	1	1	yes	characteristic
175	C₂×C₃×Q₈	—	7	2	no	—
176	C₂×C₃×Q₈	—	7	2	no	—
177	C₂×C₃×Q₈	—	7	1	no	—
178	C₂×C₃×Q₈	—	14	1	no	—
179	C₂×C₃×Q₈	—	14	1	no	—
180	C₂×C₃×Q₈	C₂×D₇	1	1	yes	characteristic
181	C₃×Q₁₆	—	7	1	no	—
182	C₃×Q₁₆	—	7	2	no	—
183	C₃×Q₁₆	—	7	2	no	—
184	C₃×Q₁₆	—	7	2	no	—
185	C₃×Q₁₆	C₂×D₇	1	2	yes	—
186	C₃×SD₁₆	—	7	1	no	—
187	C₃×SD₁₆	—	7	1	no	—
188	C₃×SD₁₆	—	7	2	no	—
189	C₃×SD₁₆	C₂×D₇	1	2	yes	—
190	C₃×D₈	—	7	1	no	—
191	C₃×M₁₆	—	7	2	no	—
192	C₃×M₁₆	C₂×D₇	1	1	yes	characteristic
193	C₂×C₃×C₈	—	7	1	no	—
194	C₂×C₃×C₈	—	7	2	no	—
195	C₂×C₃×C₇	—	2	1	no	—
196	C₂×C₃×C₇	—	4	1	no	—
197	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
198	C₃×D₇	—	2	2	no	—
199	C₃×D₇	—	4	1	no	—
200	2₋¹⁺⁴	—	7	1	no	—
201	2₋¹⁺⁴	—	7	1	no	—
202	#137	—	7	2	no	—
203	#137	—	7	2	no	—
204	#137	—	7	1	no	—
205	#137	C₃×D₇	1	1	yes	characteristic, complemented
206	#135	—	7	1	no	—
207	#135	—	7	2	no	—
208	C₂×Q₁₆	—	7	1	no	—
209	C₂×Q₁₆	—	7	2	no	—
210	#131	—	7	1	no	—
211	C₂²×C₇	—	2	1	no	—
212	C₂²×C₇	C₂×C₃×D₄	1	1	yes	characteristic
213	C₂×D₇	—	2	1	no	—
214	C₂×D₇	C₂×C₃×D₄	1	2	yes	—
215	C₄×C₇	—	2	1	no	—
216	C₄×C₇	—	2	2	no	—
217	C₄×C₇	C₂×C₃×D₄	1	1	yes	characteristic
218	C₄×C₇	C₂⁴×C₃	1	1	yes	characteristic
219	Dic₇	—	2	1	no	—
220	Dic₇	—	2	2	no	—
221	Dic₇	C₂×C₃×D₄	1	2	yes	—
222	C₃×Q₈	—	2	1	no	—
223	C₃×Q₈	—	7	1	no	—
224	C₃×Q₈	—	7	2	no	—
225	C₃×Q₈	—	7	1	no	—
226	C₃×Q₈	—	14	2	no	—
227	C₃×Q₈	—	14	2	no	—
228	C₃×Q₈	—	14	1	no	—
229	C₃×Q₈	C₂²×D₇	1	1	yes	characteristic
230	C₃×Q₈	C₂²×D₇	1	2	yes	—
231	C₃×D₄	—	2	1	no	—
232	C₃×D₄	—	7	2	no	—
233	C₃×D₄	—	7	1	no	—
234	C₃×D₄	—	7	1	no	—
235	C₃×D₄	—	14	2	no	—
236	C₃×D₄	—	14	2	no	—
237	C₃×D₄	—	14	1	no	—
238	C₃×D₄	C₂²×D₇	1	1	yes	characteristic
239	C₂×C₃×C₄	—	2	1	no	—
240	C₂×C₃×C₄	—	2	1	no	—
241	C₂×C₃×C₄	—	7	2	no	—
242	C₂×C₃×C₄	—	14	2	no	—
243	C₂×C₃×C₄	—	14	2	no	—
244	C₂×C₃×C₄	—	14	2	no	—
245	C₂×C₃×C₄	—	14	2	no	—
246	C₂×C₃×C₄	—	14	1	no	—
247	C₂×C₃×C₄	—	14	1	no	—
248	C₂×C₃×C₄	C₂²×D₇	1	1	yes	characteristic
249	C₃×C₈	—	7	1	no	—
250	C₃×C₈	—	7	1	no	—
251	C₃×C₈	C₂²×D₇	1	2	yes	—
252	C₃×C₇	#578	1	1	yes	characteristic, complemented, Hall
253	#41	—	7	1	no	—
254	#41	—	7	2	no	—
255	#41	—	7	2	no	—
256	#41	—	7	1	no	—
257	#41	—	14	2	no	—
258	#41	—	14	1	no	—
259	#41	—	14	2	no	—
260	#41	—	14	1	no	—
261	#41	C₂×C₃×D₇	1	1	yes	characteristic
262	C₂×Q₈	—	7	2	no	—
263	C₂×Q₈	—	7	2	no	—
264	C₂×Q₈	—	7	1	no	—
265	C₂×Q₈	—	14	1	no	—
266	C₂×Q₈	—	14	1	no	—
267	C₂×Q₈	C₂×C₃×D₇	1	1	yes	characteristic
268	Q₁₆	—	7	1	no	—
269	Q₁₆	—	7	2	no	—
270	Q₁₆	—	7	2	no	—
271	Q₁₆	—	7	2	no	—
272	Q₁₆	C₂×C₃×D₇	1	2	yes	—
273	SD₁₆	—	7	1	no	—
274	SD₁₆	—	7	1	no	—
275	SD₁₆	—	7	2	no	—
276	SD₁₆	C₂×C₃×D₇	1	2	yes	—
277	D₈	—	7	1	no	—
278	M₁₆	—	7	2	no	—
279	M₁₆	C₂×C₃×D₇	1	1	yes	characteristic
280	C₂×C₈	—	7	1	no	—
281	C₂×C₈	—	7	2	no	—
282	C₂×C₇	—	2	1	no	—
283	C₂×C₇	—	4	1	no	—
284	C₂×C₇	C₂²×C₃×D₄	1	1	yes	characteristic
285	D₇	—	2	2	no	—
286	D₇	—	4	1	no	—
287	C₂²×C₃	—	2	1	no	—
288	C₂²×C₃	—	7	2	no	—
289	C₂²×C₃	—	14	1	no	—
290	C₂²×C₃	D₄×D₇	1	1	yes	characteristic
291	C₃×C₄	—	2	1	no	—
292	C₃×C₄	—	2	2	no	—
293	C₃×C₄	—	7	2	no	—
294	C₃×C₄	—	14	1	no	—
295	C₃×C₄	—	14	2	no	—
296	C₃×C₄	D₄×D₇	1	1	yes	characteristic
297	C₃×C₄	C₂³×D₇	1	1	yes	characteristic
298	Q₈	—	2	1	no	—
299	Q₈	—	7	1	no	—
300	Q₈	—	7	2	no	—
301	Q₈	—	7	1	no	—
302	Q₈	—	14	2	no	—
303	Q₈	—	14	2	no	—
304	Q₈	—	14	1	no	—
305	Q₈	C₂²×C₃×D₇	1	1	yes	characteristic
306	Q₈	C₂²×C₃×D₇	1	2	yes	—
307	D₄	—	2	1	no	—
308	D₄	—	7	2	no	—
309	D₄	—	7	1	no	—
310	D₄	—	7	1	no	—
311	D₄	—	14	2	no	—
312	D₄	—	14	2	no	—
313	D₄	—	14	1	no	—
314	D₄	C₂²×C₃×D₇	1	1	yes	characteristic
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	7	2	no	—
318	C₂×C₄	—	14	2	no	—
319	C₂×C₄	—	14	2	no	—
320	C₂×C₄	—	14	2	no	—
321	C₂×C₄	—	14	2	no	—
322	C₂×C₄	—	14	1	no	—
323	C₂×C₄	—	14	1	no	—
324	C₂×C₄	C₂²×C₃×D₇	1	1	yes	characteristic
325	C₈	—	7	1	no	—
326	C₈	—	7	1	no	—
327	C₈	C₂²×C₃×D₇	1	2	yes	—
328	C₇	C₃×#578	1	1	yes	characteristic, complemented, Sylow
329	C₂×C₃	—	2	1	no	—
330	C₂×C₃	—	4	1	no	—
331	C₂×C₃	—	14	2	no	—
332	C₂×C₃	—	28	1	no	—
333	C₂×C₃	C₂×D₄×D₇	1	1	yes	characteristic
334	C₂²	—	2	1	no	—
335	C₂²	—	7	2	no	—
336	C₂²	—	14	1	no	—
337	C₂²	C₃×D₄×D₇	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	2	2	no	—
340	C₄	—	7	2	no	—
341	C₄	—	14	1	no	—
342	C₄	—	14	2	no	—
343	C₄	C₃×D₄×D₇	1	1	yes	characteristic
344	C₄	C₂³×C₃×D₇	1	1	yes	characteristic
345	C₃	#90470	1	1	yes	characteristic, direct product factor, Sylow
346	C₂	—	2	1	no	—
347	C₂	—	4	1	no	—
348	C₂	—	14	2	no	—
349	C₂	—	28	1	no	—
350	C₂	C₂×C₃×D₄×D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃	C₂×D₄×D₇
commutator, cosocle	C₄×C₇	C₂⁴×C₃
Frattini	C₄	C₂³×C₃×D₇
Fitting	C₃×C₇×#137	C₂
socle	C₂×C₃×C₇	C₂²×D₄
hypercenter	C₃×#137	D₇
hypocenter	C₇	C₃×#578
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₃×#137

Complements

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

N	H ≅ G/N	type
1. C₃×#6554	C₂	semidirect
2. C₃×#6551	C₂	semidirect
3. C₃×C₇×#137	C₂	semidirect
4. C₃×#6513	C₂	semidirect
5. C₂×C₃×#1144	C₂	semidirect
6. C₃×#6481	C₂	semidirect
7. C₃×D₇×Q₁₆	C₂	semidirect
8. C₃×#6479	C₂	semidirect
9. C₃×#6478	C₂	semidirect
10. C₃×#6472	C₂	semidirect
11. C₃×#6470	C₂	semidirect
12. #90470	C₃	direct (N × H)
42. #6554	C₂×C₃	semidirect
43. #6551	C₂×C₃	semidirect
44. C₇×#137	C₂×C₃	semidirect
45. #6513	C₂×C₃	semidirect
46. C₂×#1144	C₂×C₃	semidirect
47. #6481	C₂×C₃	semidirect
48. D₇×Q₁₆	C₂×C₃	semidirect
49. #6479	C₂×C₃	semidirect
50. #6478	C₂×C₃	semidirect
51. #6472	C₂×C₃	semidirect
52. #6470	C₂×C₃	semidirect
118. C₃×#137	D₇	semidirect
205. #137	C₃×D₇	semidirect
252. C₃×C₇	#578	semidirect
328. C₇	C₃×#578	semidirect
345. C₃	#90470	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×#6554	2
C₃×#6551	2
C₃×C₇×#137	2
C₃×#6513	2
C₂×C₃×#1144	2
C₃×#6481	2
C₃×D₇×Q₁₆	2
C₃×#6479	2
C₃×#6478	2
C₃×#6472	2
C₃×#6470	2
#90470	3
C₃×#578	7

Subgroups — C3×#90470 #13061121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#90470 #13061121