Subgroups — C₇₁×#33869 #31854741

Order 18176.

546 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₁×#3342	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇₁×#3283	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#2241	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₁×#2237	C₂	1	2	yes	maximal, complemented
5	C₇₁×#2224	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇₁×#2216	C₂	1	2	yes	maximal, complemented
7	C₇₁×#2209	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇₁×#2189	C₂	1	2	yes	maximal, complemented
9	C₄×SD₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
10	D₄×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₁×#2138	C₂	1	2	yes	maximal, complemented
12	C₂×#135×C₇₁	C₂²	1	2	yes	—
13	#556×C₇₁	—	2	1	no	—
14	#547×C₇₁	—	2	1	no	—
15	C₂×Q₃₂×C₇₁	C₂²	1	2	yes	—
16	C₂×SD₃₂×C₇₁	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₇₁	C₂²	1	2	yes	—
18	#500×C₇₁	C₂²	1	1	yes	characteristic
19	#494×C₇₁	C₂²	1	1	yes	characteristic
20	#485×C₇₁	—	2	2	no	—
21	#475×C₇₁	—	2	1	no	—
22	#467×C₇₁	C₂²	1	2	yes	—
23	#467×C₇₁	C₂²	1	2	yes	—
24	#461×C₇₁	—	2	1	no	—
25	#452×C₇₁	—	2	2	no	—
26	C₄×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
27	C₄×D₈×C₇₁	C₂²	1	1	yes	characteristic
28	C₈×D₄×C₇₁	C₂²	1	1	yes	characteristic
29	#427×C₇₁	C₂²	1	2	yes	—
30	#367×C₇₁	C₂²	1	1	yes	characteristic
31	#366×C₇₁	C₂²	1	2	yes	—
32	C₄⋊C₁₆×C₇₁	C₂²	1	1	yes	characteristic
33	#358×C₇₁	C₂²	1	2	yes	—
34	#358×C₇₁	C₂²	1	2	yes	—
35	#358×C₇₁	C₂²	1	1	yes	characteristic
36	#358×C₇₁	C₂²	1	2	yes	—
37	#357×C₇₁	C₂²	1	1	yes	characteristic
38	#357×C₇₁	C₂²	1	2	yes	—
39	C₂²⋊C₁₆×C₇₁	C₂²	1	2	yes	—
40	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
41	C₂×#41×C₇₁	—	2	2	no	—
42	#135×C₇₁	—	2	4	no	—
43	#135×C₇₁	—	4	2	no	—
44	C₂×Q₁₆×C₇₁	—	2	1	no	—
45	C₂×Q₁₆×C₇₁	C₂³	1	1	yes	characteristic
46	C₂×Q₁₆×C₇₁	C₂³	1	2	yes	—
47	C₂×SD₁₆×C₇₁	—	2	2	no	—
48	C₂×D₈×C₇₁	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₇₁	C₂³	1	2	yes	—
50	C₄⋊Q₈×C₇₁	—	2	1	no	—
51	C₄⋊Q₈×C₇₁	—	2	1	no	—
52	#126×C₇₁	—	4	2	no	—
53	#124×C₇₁	—	4	1	no	—
54	#123×C₇₁	—	4	2	no	—
55	C₂²⋊Q₈×C₇₁	—	2	2	no	—
56	C₂²⋊Q₈×C₇₁	—	2	2	no	—
57	#121×C₇₁	—	4	1	no	—
58	C₄×Q₈×C₇₁	—	2	1	no	—
59	C₄×Q₈×C₇₁	—	4	1	no	—
60	C₄×D₄×C₇₁	—	2	1	no	—
61	C₄×D₄×C₇₁	D₄	1	1	yes	characteristic
62	#117×C₇₁	—	2	2	no	—
63	C₂×C₄⋊C₄×C₇₁	—	4	1	no	—
64	Q₃₂×C₇₁	—	4	2	no	—
65	SD₃₂×C₇₁	—	2	2	no	—
66	C₂×C₁₆×C₇₁	—	4	2	no	—
67	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
68	C₂×C₁₆×C₇₁	C₂³	1	2	yes	—
69	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
70	#107×C₇₁	—	2	1	no	—
71	#107×C₇₁	C₂³	1	1	yes	characteristic
72	#107×C₇₁	C₂³	1	2	yes	—
73	#107×C₇₁	C₂³	1	1	yes	characteristic
74	#106×C₇₁	—	2	2	no	—
75	C₄⋊C₈×C₇₁	D₄	1	1	yes	characteristic
76	#103×C₇₁	—	2	1	no	—
77	#103×C₇₁	—	2	2	no	—
78	#103×C₇₁	—	2	2	no	—
79	#102×C₇₁	—	2	1	no	—
80	C₂²⋊C₈×C₇₁	D₄	1	2	yes	—
81	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
82	#41×C₇₁	—	4	4	no	—
83	#41×C₇₁	—	8	2	no	—
84	C₂×Q₈×C₇₁	—	2	1	no	—
85	C₂×Q₈×C₇₁	—	2	2	no	—
86	C₂×Q₈×C₇₁	—	4	1	no	—
87	C₂×D₄×C₇₁	—	2	1	no	—
88	C₂×D₄×C₇₁	—	4	2	no	—
89	C₂×D₄×C₇₁	D₈	1	1	yes	characteristic
90	C₂²×C₄×C₇₁	—	4	1	no	—
91	C₂²×C₄×C₇₁	—	4	2	no	—
92	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
93	Q₁₆×C₇₁	—	2	1	no	—
94	Q₁₆×C₇₁	—	2	2	no	—
95	Q₁₆×C₇₁	—	2	1	no	—
96	Q₁₆×C₇₁	—	2	2	no	—
97	Q₁₆×C₇₁	—	8	1	no	—
98	SD₁₆×C₇₁	—	4	4	no	—
99	D₈×C₇₁	—	2	1	no	—
100	D₈×C₇₁	#41	1	2	yes	—
101	C₂×C₈×C₇₁	—	2	2	no	—
102	C₂×C₈×C₇₁	—	2	2	no	—
103	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
106	C₄⋊C₄×C₇₁	—	2	1	no	—
107	C₄⋊C₄×C₇₁	—	2	1	no	—
108	C₄⋊C₄×C₇₁	—	2	2	no	—
109	C₄⋊C₄×C₇₁	—	4	1	no	—
110	C₄⋊C₄×C₇₁	—	4	2	no	—
111	C₄⋊C₄×C₇₁	—	4	1	no	—
112	C₄⋊C₄×C₇₁	—	4	2	no	—
113	C₄⋊C₄×C₇₁	—	4	1	no	—
114	C₄⋊C₄×C₇₁	—	4	1	no	—
115	C₄⋊C₄×C₇₁	D₈	1	1	yes	characteristic
116	C₂²⋊C₄×C₇₁	—	4	2	no	—
117	C₂²⋊C₄×C₇₁	—	4	2	no	—
118	C₂²⋊C₄×C₇₁	—	4	2	no	—
119	C₂²⋊C₄×C₇₁	—	4	1	no	—
120	C₂²⋊C₄×C₇₁	D₈	1	2	yes	—
121	C₄²×C₇₁	—	4	2	no	—
122	C₄²×C₇₁	—	4	1	no	—
123	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
124	C₁₆×C₇₁	—	2	1	no	—
125	C₁₆×C₇₁	—	2	1	no	—
126	C₁₆×C₇₁	—	2	2	no	—
127	C₂³×C₇₁	—	4	1	no	—
128	C₂³×C₇₁	C₂×D₈	1	2	yes	—
129	Q₈×C₇₁	—	4	2	no	—
130	Q₈×C₇₁	—	4	1	no	—
131	Q₈×C₇₁	—	4	2	no	—
132	Q₈×C₇₁	—	4	1	no	—
133	Q₈×C₇₁	—	8	2	no	—
134	D₄×C₇₁	—	2	2	no	—
135	D₄×C₇₁	—	2	2	no	—
136	D₄×C₇₁	—	4	1	no	—
137	D₄×C₇₁	—	8	4	no	—
138	C₂×C₄×C₇₁	—	2	2	no	—
139	C₂×C₄×C₇₁	—	2	2	no	—
140	C₂×C₄×C₇₁	—	4	1	no	—
141	C₂×C₄×C₇₁	—	4	2	no	—
142	C₂×C₄×C₇₁	—	4	1	no	—
143	C₂×C₄×C₇₁	—	4	1	no	—
144	C₂×C₄×C₇₁	—	4	2	no	—
145	C₂×C₄×C₇₁	—	8	4	no	—
146	C₂×C₄×C₇₁	—	8	2	no	—
147	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
148	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
151	C₈×C₇₁	—	2	2	no	—
152	C₈×C₇₁	—	2	1	no	—
153	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
154	C₈×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
155	C₂²×C₇₁	—	2	2	no	—
156	C₂²×C₇₁	—	2	2	no	—
157	C₂²×C₇₁	—	2	2	no	—
158	C₂²×C₇₁	—	4	2	no	—
159	C₂²×C₇₁	—	8	1	no	—
160	C₂²×C₇₁	—	8	1	no	—
161	C₂²×C₇₁	C₂²×D₈	1	1	yes	characteristic
162	C₄×C₇₁	—	2	1	no	—
163	C₄×C₇₁	—	2	2	no	—
164	C₄×C₇₁	—	8	1	no	—
165	C₄×C₇₁	—	8	2	no	—
166	C₄×C₇₁	—	8	1	no	—
167	C₄×C₇₁	—	8	2	no	—
168	C₄×C₇₁	—	8	1	no	—
169	C₄×C₇₁	#508	1	2	yes	—
170	C₄×C₇₁	#546	1	1	yes	characteristic
171	C₄×C₇₁	#577	1	1	yes	characteristic
172	#33869	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
173	C₂×C₇₁	—	4	2	no	—
174	C₂×C₇₁	—	8	2	no	—
175	C₂×C₇₁	#3294	1	1	yes	characteristic
176	C₂×C₇₁	C₂×#508	1	1	yes	characteristic
177	C₂×C₇₁	#3417	1	1	yes	characteristic
178	#3342	C₂×C₇₁	1	1	yes	characteristic, complemented
179	#3283	C₂×C₇₁	1	1	yes	characteristic
180	#2241	C₂×C₇₁	1	1	yes	characteristic, complemented
181	#2237	C₂×C₇₁	1	2	yes	complemented
182	#2224	C₂×C₇₁	1	1	yes	characteristic, complemented
183	#2216	C₂×C₇₁	1	2	yes	complemented
184	#2209	C₂×C₇₁	1	1	yes	characteristic, complemented
185	#2189	C₂×C₇₁	1	2	yes	complemented
186	C₄×SD₃₂	C₂×C₇₁	1	1	yes	characteristic, complemented
187	D₄×C₁₆	C₂×C₇₁	1	1	yes	characteristic, complemented
188	#2138	C₂×C₇₁	1	2	yes	complemented
189	C₇₁	#33869	1	1	yes	characteristic, direct product factor, Sylow
190	C₂×#135	C₂²×C₇₁	1	2	yes	—
191	#556	—	2	1	no	—
192	#547	—	2	1	no	—
193	C₂×Q₃₂	C₂²×C₇₁	1	2	yes	—
194	C₂×SD₃₂	C₂²×C₇₁	1	1	yes	characteristic
195	C₂²×C₁₆	C₂²×C₇₁	1	2	yes	—
196	#500	C₂²×C₇₁	1	1	yes	characteristic
197	#494	C₂²×C₇₁	1	1	yes	characteristic
198	#485	—	2	2	no	—
199	#475	—	2	1	no	—
200	#467	C₂²×C₇₁	1	2	yes	—
201	#467	C₂²×C₇₁	1	2	yes	—
202	#461	—	2	1	no	—
203	#452	—	2	2	no	—
204	C₄×Q₁₆	C₂²×C₇₁	1	1	yes	characteristic
205	C₄×D₈	C₂²×C₇₁	1	1	yes	characteristic
206	C₈×D₄	C₂²×C₇₁	1	1	yes	characteristic
207	#427	C₂²×C₇₁	1	2	yes	—
208	#367	C₂²×C₇₁	1	1	yes	characteristic
209	#366	C₂²×C₇₁	1	2	yes	—
210	C₄⋊C₁₆	C₂²×C₇₁	1	1	yes	characteristic
211	#358	C₂²×C₇₁	1	2	yes	—
212	#358	C₂²×C₇₁	1	2	yes	—
213	#358	C₂²×C₇₁	1	1	yes	characteristic
214	#358	C₂²×C₇₁	1	2	yes	—
215	#357	C₂²×C₇₁	1	1	yes	characteristic
216	#357	C₂²×C₇₁	1	2	yes	—
217	C₂²⋊C₁₆	C₂²×C₇₁	1	2	yes	—
218	C₄×C₁₆	C₂²×C₇₁	1	1	yes	characteristic
219	C₂×#41	—	2	2	no	—
220	#135	—	2	4	no	—
221	#135	—	4	2	no	—
222	C₂×Q₁₆	—	2	1	no	—
223	C₂×Q₁₆	C₂³×C₇₁	1	2	yes	—
224	C₂×Q₁₆	C₂³×C₇₁	1	1	yes	characteristic
225	C₂×SD₁₆	—	2	2	no	—
226	C₂×D₈	C₂³×C₇₁	1	1	yes	characteristic
227	C₂²×C₈	C₂³×C₇₁	1	2	yes	—
228	C₄⋊Q₈	—	2	1	no	—
229	C₄⋊Q₈	—	2	1	no	—
230	#126	—	4	2	no	—
231	#124	—	4	1	no	—
232	#123	—	4	2	no	—
233	C₂²⋊Q₈	—	2	2	no	—
234	C₂²⋊Q₈	—	2	2	no	—
235	#121	—	4	1	no	—
236	C₄×Q₈	—	2	1	no	—
237	C₄×Q₈	—	4	1	no	—
238	C₄×D₄	—	2	1	no	—
239	C₄×D₄	D₄×C₇₁	1	1	yes	characteristic
240	#117	—	2	2	no	—
241	C₂×C₄⋊C₄	—	4	1	no	—
242	Q₃₂	—	4	2	no	—
243	SD₃₂	—	2	2	no	—
244	C₂×C₁₆	—	4	2	no	—
245	C₂×C₁₆	C₂³×C₇₁	1	1	yes	characteristic
246	C₂×C₁₆	C₂³×C₇₁	1	1	yes	characteristic
247	C₂×C₁₆	C₂³×C₇₁	1	2	yes	—
248	#107	—	2	1	no	—
249	#107	C₂³×C₇₁	1	2	yes	—
250	#107	C₂³×C₇₁	1	1	yes	characteristic
251	#107	C₂³×C₇₁	1	1	yes	characteristic
252	#106	—	2	2	no	—
253	C₄⋊C₈	D₄×C₇₁	1	1	yes	characteristic
254	#103	—	2	1	no	—
255	#103	—	2	2	no	—
256	#103	—	2	2	no	—
257	#102	—	2	1	no	—
258	C₂²⋊C₈	D₄×C₇₁	1	2	yes	—
259	C₄×C₈	C₂³×C₇₁	1	1	yes	characteristic
260	#41	—	4	4	no	—
261	#41	—	8	2	no	—
262	C₂×Q₈	—	2	1	no	—
263	C₂×Q₈	—	2	2	no	—
264	C₂×Q₈	—	4	1	no	—
265	C₂×D₄	—	2	1	no	—
266	C₂×D₄	—	4	2	no	—
267	C₂×D₄	D₈×C₇₁	1	1	yes	characteristic
268	C₂²×C₄	—	4	1	no	—
269	C₂²×C₄	—	4	2	no	—
270	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
271	Q₁₆	—	2	1	no	—
272	Q₁₆	—	2	2	no	—
273	Q₁₆	—	2	1	no	—
274	Q₁₆	—	2	2	no	—
275	Q₁₆	—	8	1	no	—
276	SD₁₆	—	4	4	no	—
277	D₈	—	2	1	no	—
278	D₈	#41×C₇₁	1	2	yes	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	C₂×D₄×C₇₁	1	1	yes	characteristic
282	C₂×C₈	C₂×D₄×C₇₁	1	2	yes	—
283	C₂×C₈	C₂⁴×C₇₁	1	1	yes	characteristic
284	C₄⋊C₄	—	2	1	no	—
285	C₄⋊C₄	—	2	1	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	4	1	no	—
288	C₄⋊C₄	—	4	2	no	—
289	C₄⋊C₄	—	4	1	no	—
290	C₄⋊C₄	—	4	2	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	D₈×C₇₁	1	1	yes	characteristic
294	C₂²⋊C₄	—	4	2	no	—
295	C₂²⋊C₄	—	4	2	no	—
296	C₂²⋊C₄	—	4	2	no	—
297	C₂²⋊C₄	—	4	1	no	—
298	C₂²⋊C₄	D₈×C₇₁	1	2	yes	—
299	C₄²	—	4	2	no	—
300	C₄²	—	4	1	no	—
301	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
302	C₁₆	—	2	1	no	—
303	C₁₆	—	2	1	no	—
304	C₁₆	—	2	2	no	—
305	C₂³	—	4	1	no	—
306	C₂³	C₂×D₈×C₇₁	1	2	yes	—
307	Q₈	—	4	2	no	—
308	Q₈	—	4	1	no	—
309	Q₈	—	4	2	no	—
310	Q₈	—	4	1	no	—
311	Q₈	—	8	2	no	—
312	D₄	—	2	2	no	—
313	D₄	—	2	2	no	—
314	D₄	—	4	1	no	—
315	D₄	—	8	4	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	8	4	no	—
324	C₂×C₄	—	8	2	no	—
325	C₂×C₄	C₂×D₈×C₇₁	1	2	yes	—
326	C₂×C₄	C₂×D₈×C₇₁	1	1	yes	characteristic
327	C₂×C₄	C₂×D₈×C₇₁	1	1	yes	characteristic
328	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
329	C₈	—	2	2	no	—
330	C₈	—	2	1	no	—
331	C₈	C₂×#41×C₇₁	1	1	yes	characteristic
332	C₈	H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
333	C₂²	—	2	2	no	—
334	C₂²	—	2	2	no	—
335	C₂²	—	2	2	no	—
336	C₂²	—	4	2	no	—
337	C₂²	—	8	1	no	—
338	C₂²	—	8	1	no	—
339	C₂²	C₂²×D₈×C₇₁	1	1	yes	characteristic
340	C₄	—	2	2	no	—
341	C₄	—	2	1	no	—
342	C₄	—	8	2	no	—
343	C₄	—	8	1	no	—
344	C₄	—	8	1	no	—
345	C₄	—	8	2	no	—
346	C₄	—	8	1	no	—
347	C₄	#508×C₇₁	1	2	yes	—
348	C₄	#546×C₇₁	1	1	yes	characteristic
349	C₄	#577×C₇₁	1	1	yes	characteristic
350	C₂	—	4	2	no	—
351	C₂	—	8	2	no	—
352	C₂	C₇₁×#3294	1	1	yes	characteristic
353	C₂	C₂×#508×C₇₁	1	1	yes	characteristic
354	C₂	C₇₁×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₇₁
prime radical	C₇₁×#3283	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₄×D₄×C₇₁ ◁ C₈×D₄×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇₁×#3342	C₂	semidirect
3. C₇₁×#2241	C₂	semidirect
4. C₇₁×#2237	C₂	semidirect
5. C₇₁×#2224	C₂	semidirect
6. C₇₁×#2216	C₂	semidirect
7. C₇₁×#2209	C₂	semidirect
8. C₇₁×#2189	C₂	semidirect
9. C₄×SD₃₂×C₇₁	C₂	semidirect
10. D₄×C₁₆×C₇₁	C₂	semidirect
11. C₇₁×#2138	C₂	semidirect
172. #33869	C₇₁	direct (N × H)
178. #3342	C₂×C₇₁	semidirect
180. #2241	C₂×C₇₁	semidirect
181. #2237	C₂×C₇₁	semidirect
182. #2224	C₂×C₇₁	semidirect
183. #2216	C₂×C₇₁	semidirect
184. #2209	C₂×C₇₁	semidirect
185. #2189	C₂×C₇₁	semidirect
186. C₄×SD₃₂	C₂×C₇₁	semidirect
187. D₄×C₁₆	C₂×C₇₁	semidirect
188. #2138	C₂×C₇₁	semidirect
189. C₇₁	#33869	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×#3342	2
C₇₁×#3283	2
C₇₁×#2241	2
C₇₁×#2237	2
C₇₁×#2224	2
C₇₁×#2216	2
C₇₁×#2209	2
C₇₁×#2189	2
C₄×SD₃₂×C₇₁	2
D₄×C₁₆×C₇₁	2
C₇₁×#2138	2
#33869	71

Subgroups — C71×#33869 #31854741

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#33869 #31854741