Subgroups — C₇₁×#19330 #31840202

Order 18176.

616 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	C₇₁×#3235	C₂	1	2	yes	maximal
2	C₇₁×#3170	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₁×#2931	C₂	1	1	yes	maximal, characteristic
4	C₇₁×#1703	C₂	1	4	yes	maximal, complemented
5	C₇₁×#1675	C₂	1	2	yes	maximal, complemented
6	C₇₁×#1669	C₂	1	2	yes	maximal, complemented
7	C₇₁×#1568	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇₁×#1559	C₂	1	2	yes	maximal, complemented
9	C₂×#131×C₇₁	—	2	1	no	—
10	#533×C₇₁	C₂²	1	2	yes	—
11	#532×C₇₁	C₂²	1	1	yes	characteristic
12	#501×C₇₁	—	2	2	no	—
13	#500×C₇₁	C₂²	1	1	yes	characteristic
14	#494×C₇₁	C₂²	1	1	yes	characteristic
15	#490×C₇₁	—	2	2	no	—
16	#487×C₇₁	C₂²	1	2	yes	—
17	#482×C₇₁	—	2	4	no	—
18	#476×C₇₁	C₂²	1	4	yes	—
19	#474×C₇₁	C₂²	1	4	yes	—
20	#467×C₇₁	—	2	1	no	—
21	#467×C₇₁	—	2	1	no	—
22	C₄×Q₁₆×C₇₁	—	2	1	no	—
23	C₄×Q₁₆×C₇₁	—	2	1	no	—
24	#436×C₇₁	—	2	1	no	—
25	#436×C₇₁	C₂²	1	2	yes	—
26	C₈×D₄×C₇₁	—	2	1	no	—
27	C₈×D₄×C₇₁	C₂²	1	2	yes	—
28	#432×C₇₁	C₂²	1	2	yes	—
29	#423×C₇₁	C₂²	1	1	yes	characteristic
30	#408×C₇₁	—	2	1	no	—
31	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
32	#335×C₇₁	C₂²	1	2	yes	—
33	#334×C₇₁	C₂²	1	2	yes	—
34	#333×C₇₁	C₂²	1	4	yes	—
35	#332×C₇₁	C₂²	1	4	yes	—
36	C₂×#41×C₇₁	—	2	1	no	—
37	C₂×Q₁₆×C₇₁	—	2	1	no	—
38	C₂×Q₁₆×C₇₁	—	2	1	no	—
39	#131×C₇₁	—	4	4	no	—
40	C₂×M₁₆×C₇₁	—	2	1	no	—
41	C₂×M₁₆×C₇₁	—	2	2	no	—
42	C₂²×C₈×C₇₁	—	2	2	no	—
43	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
45	C₄⋊Q₈×C₇₁	D₄	1	1	yes	characteristic
46	C₄⋊Q₈×C₇₁	C₂³	1	2	yes	—
47	C₄⋊Q₈×C₇₁	C₂³	1	2	yes	—
48	#127×C₇₁	D₄	1	1	yes	characteristic
49	#125×C₇₁	—	2	2	no	—
50	#124×C₇₁	—	2	1	no	—
51	C₂²⋊Q₈×C₇₁	—	2	2	no	—
52	C₂²⋊Q₈×C₇₁	—	2	2	no	—
53	#121×C₇₁	—	2	1	no	—
54	#121×C₇₁	D₄	1	2	yes	—
55	C₄×Q₈×C₇₁	—	2	2	no	—
56	C₄×Q₈×C₇₁	—	2	2	no	—
57	C₄×D₄×C₇₁	—	2	1	no	—
58	C₄×D₄×C₇₁	C₂³	1	2	yes	—
59	#117×C₇₁	—	2	2	no	—
60	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
61	#107×C₇₁	—	2	1	no	—
62	#107×C₇₁	—	2	1	no	—
63	#107×C₇₁	—	2	4	no	—
64	#106×C₇₁	—	2	4	no	—
65	C₄⋊C₈×C₇₁	—	2	1	no	—
66	C₄⋊C₈×C₇₁	C₂³	1	2	yes	—
67	C₄⋊C₈×C₇₁	C₂³	1	4	yes	—
68	#103×C₇₁	—	2	2	no	—
69	#103×C₇₁	—	2	2	no	—
70	#102×C₇₁	—	2	4	no	—
71	#102×C₇₁	—	2	4	no	—
72	C₂²⋊C₈×C₇₁	—	2	2	no	—
73	C₂²⋊C₈×C₇₁	—	2	2	no	—
74	C₂²⋊C₈×C₇₁	—	2	2	no	—
75	#97×C₇₁	—	2	2	no	—
76	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
77	C₄×C₈×C₇₁	D₄	1	1	yes	characteristic
78	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
80	#41×C₇₁	—	4	2	no	—
81	#41×C₇₁	—	4	2	no	—
82	C₂×Q₈×C₇₁	—	2	1	no	—
83	C₂×Q₈×C₇₁	—	2	2	no	—
84	C₂×Q₈×C₇₁	—	2	2	no	—
85	C₂×D₄×C₇₁	—	2	2	no	—
86	C₂×D₄×C₇₁	—	2	1	no	—
87	C₂×D₄×C₇₁	—	2	1	no	—
88	C₂×D₄×C₇₁	C₂×D₄	1	2	yes	—
89	C₂²×C₄×C₇₁	—	2	2	no	—
90	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
93	Q₁₆×C₇₁	—	4	2	no	—
94	Q₁₆×C₇₁	—	4	2	no	—
95	M₁₆×C₇₁	—	4	4	no	—
96	M₁₆×C₇₁	—	4	2	no	—
97	C₂×C₈×C₇₁	—	2	2	no	—
98	C₂×C₈×C₇₁	—	2	4	no	—
99	C₂×C₈×C₇₁	—	2	2	no	—
100	C₂×C₈×C₇₁	—	4	1	no	—
101	C₂×C₈×C₇₁	—	4	4	no	—
102	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
106	C₄⋊C₄×C₇₁	—	2	1	no	—
107	C₄⋊C₄×C₇₁	—	2	2	no	—
108	C₄⋊C₄×C₇₁	—	2	2	no	—
109	C₄⋊C₄×C₇₁	—	2	2	no	—
110	C₄⋊C₄×C₇₁	—	2	2	no	—
111	C₄⋊C₄×C₇₁	—	4	2	no	—
112	C₄⋊C₄×C₇₁	—	4	2	no	—
113	C₄⋊C₄×C₇₁	C₂×D₄	1	2	yes	—
114	C₂²⋊C₄×C₇₁	—	2	2	no	—
115	C₂²⋊C₄×C₇₁	—	2	2	no	—
116	C₂²⋊C₄×C₇₁	—	4	2	no	—
117	C₄²×C₇₁	—	4	2	no	—
118	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
119	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₇₁	#41	1	1	yes	characteristic
121	C₄²×C₇₁	C₂⁴	1	1	yes	characteristic
122	C₂³×C₇₁	—	2	2	no	—
123	C₂³×C₇₁	#121	1	1	yes	characteristic
124	Q₈×C₇₁	—	4	2	no	—
125	Q₈×C₇₁	—	4	2	no	—
126	Q₈×C₇₁	—	4	2	no	—
127	Q₈×C₇₁	—	8	2	no	—
128	D₄×C₇₁	—	2	4	no	—
129	D₄×C₇₁	—	4	2	no	—
130	D₄×C₇₁	—	4	2	no	—
131	D₄×C₇₁	—	4	4	no	—
132	C₂×C₄×C₇₁	—	2	1	no	—
133	C₂×C₄×C₇₁	—	2	1	no	—
134	C₂×C₄×C₇₁	—	2	2	no	—
135	C₂×C₄×C₇₁	—	2	2	no	—
136	C₂×C₄×C₇₁	—	4	1	no	—
137	C₂×C₄×C₇₁	—	4	2	no	—
138	C₂×C₄×C₇₁	—	4	2	no	—
139	C₂×C₄×C₇₁	—	4	2	no	—
140	C₂×C₄×C₇₁	—	4	2	no	—
141	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
142	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
143	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
144	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
147	C₈×C₇₁	—	2	1	no	—
148	C₈×C₇₁	—	2	1	no	—
149	C₈×C₇₁	—	2	1	no	—
150	C₈×C₇₁	—	4	2	no	—
151	C₈×C₇₁	—	4	4	no	—
152	C₈×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
153	C₂²×C₇₁	—	2	1	no	—
154	C₂²×C₇₁	—	2	1	no	—
155	C₂²×C₇₁	—	2	1	no	—
156	C₂²×C₇₁	—	4	2	no	—
157	C₂²×C₇₁	—	4	2	no	—
158	C₂²×C₇₁	—	4	2	no	—
159	C₂²×C₇₁	C₂×#121	1	1	yes	characteristic
160	C₄×C₇₁	—	2	1	no	—
161	C₄×C₇₁	—	2	1	no	—
162	C₄×C₇₁	—	2	1	no	—
163	C₄×C₇₁	—	2	1	no	—
164	C₄×C₇₁	—	4	2	no	—
165	C₄×C₇₁	—	8	2	no	—
166	C₄×C₇₁	—	8	2	no	—
167	C₄×C₇₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
168	C₄×C₇₁	#575	1	1	yes	characteristic
169	C₄×C₇₁	#578	1	2	yes	—
170	#19330	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
171	C₂×C₇₁	—	4	1	no	—
172	C₂×C₇₁	—	8	2	no	—
173	C₂×C₇₁	#3036	1	1	yes	characteristic
174	C₂×C₇₁	#3059	1	1	yes	characteristic
175	C₂×C₇₁	#3067	1	1	yes	characteristic
176	#3235	C₂×C₇₁	1	2	yes	—
177	#3170	C₂×C₇₁	1	1	yes	characteristic, complemented
178	#2931	C₂×C₇₁	1	1	yes	characteristic
179	#1703	C₂×C₇₁	1	4	yes	complemented
180	#1675	C₂×C₇₁	1	2	yes	complemented
181	#1669	C₂×C₇₁	1	2	yes	complemented
182	#1568	C₂×C₇₁	1	1	yes	characteristic, complemented
183	#1559	C₂×C₇₁	1	2	yes	complemented
184	C₇₁	#19330	1	1	yes	characteristic, direct product factor, Sylow
185	C₂×#131	—	2	1	no	—
186	#533	C₂²×C₇₁	1	2	yes	—
187	#532	C₂²×C₇₁	1	1	yes	characteristic
188	#501	—	2	2	no	—
189	#500	C₂²×C₇₁	1	1	yes	characteristic
190	#494	C₂²×C₇₁	1	1	yes	characteristic
191	#490	—	2	2	no	—
192	#487	C₂²×C₇₁	1	2	yes	—
193	#482	—	2	4	no	—
194	#476	C₂²×C₇₁	1	4	yes	—
195	#474	C₂²×C₇₁	1	4	yes	—
196	#467	—	2	1	no	—
197	#467	—	2	1	no	—
198	C₄×Q₁₆	—	2	1	no	—
199	C₄×Q₁₆	—	2	1	no	—
200	#436	—	2	1	no	—
201	#436	C₂²×C₇₁	1	2	yes	—
202	C₈×D₄	—	2	1	no	—
203	C₈×D₄	C₂²×C₇₁	1	2	yes	—
204	#432	C₂²×C₇₁	1	2	yes	—
205	#423	C₂²×C₇₁	1	1	yes	characteristic
206	#408	—	2	1	no	—
207	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
208	#335	C₂²×C₇₁	1	2	yes	—
209	#334	C₂²×C₇₁	1	2	yes	—
210	#333	C₂²×C₇₁	1	4	yes	—
211	#332	C₂²×C₇₁	1	4	yes	—
212	C₂×#41	—	2	1	no	—
213	C₂×Q₁₆	—	2	1	no	—
214	C₂×Q₁₆	—	2	1	no	—
215	#131	—	4	4	no	—
216	C₂×M₁₆	—	2	1	no	—
217	C₂×M₁₆	—	2	2	no	—
218	C₂²×C₈	—	2	2	no	—
219	C₂²×C₈	D₄×C₇₁	1	1	yes	characteristic
220	C₂²×C₈	D₄×C₇₁	1	1	yes	characteristic
221	C₄⋊Q₈	D₄×C₇₁	1	1	yes	characteristic
222	C₄⋊Q₈	C₂³×C₇₁	1	2	yes	—
223	C₄⋊Q₈	C₂³×C₇₁	1	2	yes	—
224	#127	D₄×C₇₁	1	1	yes	characteristic
225	#125	—	2	2	no	—
226	#124	—	2	1	no	—
227	C₂²⋊Q₈	—	2	2	no	—
228	C₂²⋊Q₈	—	2	2	no	—
229	#121	—	2	1	no	—
230	#121	D₄×C₇₁	1	2	yes	—
231	C₄×Q₈	—	2	2	no	—
232	C₄×Q₈	—	2	2	no	—
233	C₄×D₄	—	2	1	no	—
234	C₄×D₄	C₂³×C₇₁	1	2	yes	—
235	#117	—	2	2	no	—
236	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
237	#107	—	2	1	no	—
238	#107	—	2	1	no	—
239	#107	—	2	4	no	—
240	#106	—	2	4	no	—
241	C₄⋊C₈	—	2	1	no	—
242	C₄⋊C₈	C₂³×C₇₁	1	2	yes	—
243	C₄⋊C₈	C₂³×C₇₁	1	4	yes	—
244	#103	—	2	2	no	—
245	#103	—	2	2	no	—
246	#102	—	2	4	no	—
247	#102	—	2	4	no	—
248	C₂²⋊C₈	—	2	2	no	—
249	C₂²⋊C₈	—	2	2	no	—
250	C₂²⋊C₈	—	2	2	no	—
251	#97	—	2	2	no	—
252	C₄×C₈	D₄×C₇₁	1	1	yes	characteristic
253	C₄×C₈	D₄×C₇₁	1	1	yes	characteristic
254	C₄×C₈	C₂³×C₇₁	1	1	yes	characteristic
255	C₄×C₈	C₂³×C₇₁	1	1	yes	characteristic
256	#41	—	4	2	no	—
257	#41	—	4	2	no	—
258	C₂×Q₈	—	2	1	no	—
259	C₂×Q₈	—	2	2	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	2	1	no	—
263	C₂×D₄	—	2	1	no	—
264	C₂×D₄	C₂×D₄×C₇₁	1	2	yes	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
267	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
268	C₂²×C₄	#41×C₇₁	1	1	yes	characteristic
269	Q₁₆	—	4	2	no	—
270	Q₁₆	—	4	2	no	—
271	M₁₆	—	4	4	no	—
272	M₁₆	—	4	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	4	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	4	1	no	—
277	C₂×C₈	—	4	4	no	—
278	C₂×C₈	C₂×D₄×C₇₁	1	1	yes	characteristic
279	C₂×C₈	C₂×D₄×C₇₁	1	1	yes	characteristic
280	C₂×C₈	C₂×D₄×C₇₁	1	1	yes	characteristic
281	C₂×C₈	C₂×D₄×C₇₁	1	1	yes	characteristic
282	C₄⋊C₄	—	2	1	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	2	2	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	4	2	no	—
288	C₄⋊C₄	—	4	2	no	—
289	C₄⋊C₄	C₂×D₄×C₇₁	1	2	yes	—
290	C₂²⋊C₄	—	2	2	no	—
291	C₂²⋊C₄	—	2	2	no	—
292	C₂²⋊C₄	—	4	2	no	—
293	C₄²	—	4	2	no	—
294	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
295	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
296	C₄²	#41×C₇₁	1	1	yes	characteristic
297	C₄²	C₂⁴×C₇₁	1	1	yes	characteristic
298	C₂³	—	2	2	no	—
299	C₂³	#121×C₇₁	1	1	yes	characteristic
300	Q₈	—	4	2	no	—
301	Q₈	—	4	2	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	8	2	no	—
304	D₄	—	2	4	no	—
305	D₄	—	4	2	no	—
306	D₄	—	4	2	no	—
307	D₄	—	4	4	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	#121×C₇₁	1	1	yes	characteristic
318	C₂×C₄	#121×C₇₁	1	1	yes	characteristic
319	C₂×C₄	#121×C₇₁	1	1	yes	characteristic
320	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
321	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
322	C₂×C₄	C₂×#41×C₇₁	1	1	yes	characteristic
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	2	1	no	—
326	C₈	—	4	2	no	—
327	C₈	—	4	4	no	—
328	C₈	AGL(1,ℤ/8)×C₇₁	1	2	yes	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	4	2	no	—
333	C₂²	—	4	2	no	—
334	C₂²	—	4	2	no	—
335	C₂²	C₂×#121×C₇₁	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	4	2	no	—
341	C₄	—	8	2	no	—
342	C₄	—	8	2	no	—
343	C₄	C₂×AGL(1,ℤ/8)×C₇₁	1	1	yes	characteristic
344	C₄	#575×C₇₁	1	1	yes	characteristic
345	C₄	#578×C₇₁	1	2	yes	—
346	C₂	—	4	1	no	—
347	C₂	—	8	2	no	—
348	C₂	C₇₁×#3036	1	1	yes	characteristic
349	C₂	C₇₁×#3059	1	1	yes	characteristic
350	C₂	C₇₁×#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₇₁
prime radical	#532×C₇₁	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇₁×#3170	C₂	semidirect
4. C₇₁×#1703	C₂	semidirect
5. C₇₁×#1675	C₂	semidirect
6. C₇₁×#1669	C₂	semidirect
7. C₇₁×#1568	C₂	semidirect
8. C₇₁×#1559	C₂	semidirect
170. #19330	C₇₁	direct (N × H)
177. #3170	C₂×C₇₁	semidirect
179. #1703	C₂×C₇₁	semidirect
180. #1675	C₂×C₇₁	semidirect
181. #1669	C₂×C₇₁	semidirect
182. #1568	C₂×C₇₁	semidirect
183. #1559	C₂×C₇₁	semidirect
184. C₇₁	#19330	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₇₁×#3235	2
C₇₁×#3170	2
C₇₁×#2931	2
C₇₁×#1703	2
C₇₁×#1675	2
C₇₁×#1669	2
C₇₁×#1568	2
C₇₁×#1559	2
#19330	71

Subgroups — C71×#19330 #31840202

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#19330 #31840202