Subgroups — C₇₁×#9735 #31830607

Order 18176.

522 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	C₂×#486×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₇₁×#1987	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₁×#1982	C₂	1	1	yes	maximal, characteristic
4	C₈×C₄⋊C₄×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#334×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#333×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#331×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
9	C₂×#127×C₇₁	C₂²	1	1	yes	characteristic
10	#486×C₇₁	—	2	2	no	—
11	#486×C₇₁	C₄	1	4	yes	complemented
12	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×#102×C₇₁	—	2	1	no	—
16	C₂×#102×C₇₁	—	2	1	no	—
17	C₂×#102×C₇₁	—	2	2	no	—
18	C₂×C₄×C₈×C₇₁	—	2	1	no	—
19	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
20	C₄×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
21	#340×C₇₁	—	2	2	no	—
22	#336×C₇₁	—	2	1	no	—
23	#334×C₇₁	—	2	2	no	—
24	#333×C₇₁	—	2	2	no	—
25	#331×C₇₁	—	2	2	no	—
26	C₂²×Q₈×C₇₁	—	2	1	no	—
27	C₂²×D₄×C₇₁	—	2	1	no	—
28	C₂²×D₄×C₇₁	—	2	1	no	—
29	C₂²×D₄×C₇₁	—	2	1	no	—
30	C₂²×C₈×C₇₁	—	2	1	no	—
31	C₂²×C₈×C₇₁	—	2	1	no	—
32	C₂²×C₈×C₇₁	—	2	1	no	—
33	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
35	C₄⋊Q₈×C₇₁	—	2	1	no	—
36	C₄⋊Q₈×C₇₁	—	2	1	no	—
37	C₄⋊Q₈×C₇₁	—	2	1	no	—
38	C₄⋊Q₈×C₇₁	C₂×C₄	1	2	yes	—
39	#127×C₇₁	—	2	1	no	—
40	#127×C₇₁	—	2	1	no	—
41	#127×C₇₁	—	2	1	no	—
42	#127×C₇₁	C₂×C₄	1	2	yes	—
43	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
44	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
45	C₂×C₄⋊C₄×C₇₁	D₄	1	2	yes	—
46	C₂×C₄²×C₇₁	—	2	1	no	—
47	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₇₁	—	2	1	no	—
49	C₄⋊C₈×C₇₁	—	2	1	no	—
50	C₄⋊C₈×C₇₁	—	2	1	no	—
51	C₄⋊C₈×C₇₁	—	2	1	no	—
52	C₄⋊C₈×C₇₁	—	2	1	no	—
53	C₄⋊C₈×C₇₁	D₄	1	2	yes	—
54	#102×C₇₁	—	2	4	no	—
55	#102×C₇₁	—	2	4	no	—
56	#102×C₇₁	—	4	4	no	—
57	C₄×C₈×C₇₁	—	4	2	no	—
58	C₄×C₈×C₇₁	C₂×C₄	1	2	yes	—
59	C₄×C₈×C₇₁	D₄	1	2	yes	—
60	#95×C₇₁	—	2	1	no	—
61	C₂⁴×C₇₁	—	4	1	no	—
62	C₂×Q₈×C₇₁	—	2	2	no	—
63	C₂×Q₈×C₇₁	—	4	1	no	—
64	C₂×Q₈×C₇₁	—	8	2	no	—
65	C₂×D₄×C₇₁	—	2	2	no	—
66	C₂×D₄×C₇₁	—	2	2	no	—
67	C₂×D₄×C₇₁	—	2	2	no	—
68	C₂×D₄×C₇₁	—	4	1	no	—
69	C₂×D₄×C₇₁	—	4	1	no	—
70	C₂×D₄×C₇₁	—	4	1	no	—
71	C₂×D₄×C₇₁	—	4	2	no	—
72	C₂×D₄×C₇₁	—	4	1	no	—
73	C₂×D₄×C₇₁	—	4	1	no	—
74	C₂×D₄×C₇₁	—	4	2	no	—
75	C₂×D₄×C₇₁	—	4	1	no	—
76	C₂×D₄×C₇₁	—	4	1	no	—
77	C₂×D₄×C₇₁	—	8	2	no	—
78	C₂²×C₄×C₇₁	—	2	2	no	—
79	C₂²×C₄×C₇₁	—	4	1	no	—
80	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
83	C₂×C₈×C₇₁	—	2	1	no	—
84	C₂×C₈×C₇₁	—	2	1	no	—
85	C₂×C₈×C₇₁	—	2	1	no	—
86	C₂×C₈×C₇₁	—	2	1	no	—
87	C₂×C₈×C₇₁	—	2	2	no	—
88	C₂×C₈×C₇₁	—	4	1	no	—
89	C₂×C₈×C₇₁	—	4	1	no	—
90	C₂×C₈×C₇₁	—	4	1	no	—
91	C₂×C₈×C₇₁	—	4	1	no	—
92	C₂×C₈×C₇₁	—	4	2	no	—
93	C₂×C₈×C₇₁	—	4	2	no	—
94	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
95	C₂×C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
96	C₄⋊C₄×C₇₁	—	2	2	no	—
97	C₄⋊C₄×C₇₁	—	2	2	no	—
98	C₄⋊C₄×C₇₁	—	2	2	no	—
99	C₄⋊C₄×C₇₁	—	4	1	no	—
100	C₄⋊C₄×C₇₁	—	4	1	no	—
101	C₄⋊C₄×C₇₁	SD₁₆	1	4	yes	—
102	C₄²×C₇₁	—	4	1	no	—
103	C₄²×C₇₁	—	4	1	no	—
104	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
105	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₇₁	#41	1	1	yes	characteristic
108	C₂³×C₇₁	—	4	2	no	—
109	C₂³×C₇₁	—	8	1	no	—
110	C₂³×C₇₁	—	8	1	no	—
111	C₂³×C₇₁	—	8	2	no	—
112	C₂³×C₇₁	—	8	2	no	—
113	C₂³×C₇₁	#121	1	1	yes	characteristic
114	Q₈×C₇₁	—	8	2	no	—
115	Q₈×C₇₁	—	8	2	no	—
116	D₄×C₇₁	—	4	2	no	—
117	D₄×C₇₁	—	4	2	no	—
118	D₄×C₇₁	—	4	4	no	—
119	D₄×C₇₁	—	8	2	no	—
120	D₄×C₇₁	—	8	2	no	—
121	D₄×C₇₁	—	8	1	no	—
122	D₄×C₇₁	—	8	1	no	—
123	D₄×C₇₁	—	8	1	no	—
124	D₄×C₇₁	—	8	1	no	—
125	C₂×C₄×C₇₁	—	2	2	no	—
126	C₂×C₄×C₇₁	—	2	4	no	—
127	C₂×C₄×C₇₁	—	4	2	no	—
128	C₂×C₄×C₇₁	—	4	2	no	—
129	C₂×C₄×C₇₁	—	4	2	no	—
130	C₂×C₄×C₇₁	—	8	2	no	—
131	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
135	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
136	C₂×C₄×C₇₁	#127	1	1	yes	characteristic
137	C₂×C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
138	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
139	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
140	C₂×C₄×C₇₁	#135	1	2	yes	—
141	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
143	C₈×C₇₁	—	2	2	no	—
144	C₈×C₇₁	—	2	2	no	—
145	C₈×C₇₁	—	4	2	no	—
146	C₈×C₇₁	—	8	1	no	—
147	C₈×C₇₁	—	8	1	no	—
148	C₂²×C₇₁	—	8	2	no	—
149	C₂²×C₇₁	—	8	2	no	—
150	C₂²×C₇₁	—	8	2	no	—
151	C₂²×C₇₁	—	16	2	no	—
152	C₂²×C₇₁	—	16	2	no	—
153	C₂²×C₇₁	#390	1	1	yes	characteristic
154	C₂²×C₇₁	#463	1	1	yes	characteristic
155	C₂²×C₇₁	#464	1	1	yes	characteristic
156	C₂²×C₇₁	#465	1	2	yes	—
157	C₂²×C₇₁	#472	1	2	yes	—
158	C₄×C₇₁	—	2	1	no	—
159	C₄×C₇₁	—	2	1	no	—
160	C₄×C₇₁	—	4	4	no	—
161	C₄×C₇₁	—	8	2	no	—
162	C₄×C₇₁	#416	1	1	yes	characteristic
163	C₄×C₇₁	#417	1	1	yes	characteristic
164	C₄×C₇₁	C₄×SD₁₆	1	2	yes	—
165	C₄×C₇₁	#463	1	1	yes	characteristic
166	C₄×C₇₁	#464	1	1	yes	characteristic
167	C₄×C₇₁	#492	1	1	yes	characteristic
168	C₄×C₇₁	#495	1	1	yes	characteristic
169	#9735	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
170	C₂×C₇₁	—	16	2	no	—
171	C₂×C₇₁	#1677	1	2	yes	—
172	C₂×C₇₁	#1678	1	2	yes	—
173	C₂×C₇₁	#1968	1	1	yes	characteristic
174	C₂×C₇₁	#1978	1	1	yes	characteristic
175	C₂×C₇₁	#1983	1	1	yes	characteristic
176	C₂×#486	C₂×C₇₁	1	1	yes	characteristic
177	#1987	C₂×C₇₁	1	1	yes	characteristic, complemented
178	#1982	C₂×C₇₁	1	1	yes	characteristic
179	C₈×C₄⋊C₄	C₂×C₇₁	1	1	yes	characteristic, complemented
180	C₂×#334	C₂×C₇₁	1	1	yes	characteristic, complemented
181	C₂×#333	C₂×C₇₁	1	1	yes	characteristic, complemented
182	C₂×#331	C₂×C₇₁	1	1	yes	characteristic
183	C₇₁	#9735	1	1	yes	characteristic, direct product factor, Sylow
184	C₂×C₄⋊Q₈	C₂²×C₇₁	1	1	yes	characteristic
185	C₂×#127	C₂²×C₇₁	1	1	yes	characteristic
186	#486	—	2	2	no	—
187	#486	C₄×C₇₁	1	4	yes	complemented
188	C₂×C₄⋊C₈	C₂²×C₇₁	1	1	yes	characteristic
189	C₂×C₄⋊C₈	C₂²×C₇₁	1	1	yes	characteristic
190	C₂×C₄⋊C₈	C₂²×C₇₁	1	1	yes	characteristic
191	C₂×#102	—	2	1	no	—
192	C₂×#102	—	2	1	no	—
193	C₂×#102	—	2	2	no	—
194	C₂×C₄×C₈	—	2	1	no	—
195	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
196	C₄×C₄⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
197	#340	—	2	2	no	—
198	#336	—	2	1	no	—
199	#334	—	2	2	no	—
200	#333	—	2	2	no	—
201	#331	—	2	2	no	—
202	C₂²×Q₈	—	2	1	no	—
203	C₂²×D₄	—	2	1	no	—
204	C₂²×D₄	—	2	1	no	—
205	C₂²×D₄	—	2	1	no	—
206	C₂²×C₈	—	2	1	no	—
207	C₂²×C₈	—	2	1	no	—
208	C₂²×C₈	—	2	1	no	—
209	C₂²×C₈	D₄×C₇₁	1	1	yes	characteristic
210	C₂²×C₈	D₄×C₇₁	1	1	yes	characteristic
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	C₂×C₄×C₇₁	1	2	yes	—
215	#127	—	2	1	no	—
216	#127	—	2	1	no	—
217	#127	—	2	1	no	—
218	#127	C₂×C₄×C₇₁	1	2	yes	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	D₄×C₇₁	1	2	yes	—
222	C₂×C₄²	—	2	1	no	—
223	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	—	2	1	no	—
227	C₄⋊C₈	—	2	1	no	—
228	C₄⋊C₈	—	2	1	no	—
229	C₄⋊C₈	D₄×C₇₁	1	2	yes	—
230	#102	—	2	4	no	—
231	#102	—	2	4	no	—
232	#102	—	4	4	no	—
233	C₄×C₈	—	4	2	no	—
234	C₄×C₈	C₂×C₄×C₇₁	1	2	yes	—
235	C₄×C₈	D₄×C₇₁	1	2	yes	—
236	#95	—	2	1	no	—
237	C₂⁴	—	4	1	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×Q₈	—	8	2	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	2	2	no	—
243	C₂×D₄	—	2	2	no	—
244	C₂×D₄	—	4	1	no	—
245	C₂×D₄	—	4	1	no	—
246	C₂×D₄	—	4	1	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂×D₄	—	4	1	no	—
249	C₂×D₄	—	4	1	no	—
250	C₂×D₄	—	4	2	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	—	8	2	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
257	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
258	C₂²×C₄	#41×C₇₁	1	1	yes	characteristic
259	C₂×C₈	—	2	1	no	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	4	1	no	—
265	C₂×C₈	—	4	1	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	4	1	no	—
268	C₂×C₈	—	4	2	no	—
269	C₂×C₈	—	4	2	no	—
270	C₂×C₈	C₂²⋊C₄×C₇₁	1	2	yes	—
271	C₂×C₈	C₂²⋊C₄×C₇₁	1	2	yes	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	SD₁₆×C₇₁	1	4	yes	—
278	C₄²	—	4	1	no	—
279	C₄²	—	4	1	no	—
280	C₄²	C₂²×C₄×C₇₁	1	1	yes	characteristic
281	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
282	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
283	C₄²	#41×C₇₁	1	1	yes	characteristic
284	C₂³	—	4	2	no	—
285	C₂³	—	8	1	no	—
286	C₂³	—	8	1	no	—
287	C₂³	—	8	2	no	—
288	C₂³	—	8	2	no	—
289	C₂³	#121×C₇₁	1	1	yes	characteristic
290	Q₈	—	8	2	no	—
291	Q₈	—	8	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	4	2	no	—
294	D₄	—	4	4	no	—
295	D₄	—	8	2	no	—
296	D₄	—	8	2	no	—
297	D₄	—	8	1	no	—
298	D₄	—	8	1	no	—
299	D₄	—	8	1	no	—
300	D₄	—	8	1	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	4	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	8	2	no	—
307	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
308	C₂×C₄	C₄×D₄×C₇₁	1	1	yes	characteristic
309	C₂×C₄	C₄×D₄×C₇₁	1	1	yes	characteristic
310	C₂×C₄	#121×C₇₁	1	1	yes	characteristic
311	C₂×C₄	#124×C₇₁	1	1	yes	characteristic
312	C₂×C₄	#127×C₇₁	1	1	yes	characteristic
313	C₂×C₄	C₂×SD₁₆×C₇₁	1	2	yes	—
314	C₂×C₄	#135×C₇₁	1	1	yes	characteristic
315	C₂×C₄	#135×C₇₁	1	1	yes	characteristic
316	C₂×C₄	#135×C₇₁	1	2	yes	—
317	C₂×C₄	AGL(1,ℤ/8)×C₇₁	1	1	yes	characteristic
318	C₂×C₄	#137×C₇₁	1	1	yes	characteristic
319	C₈	—	2	2	no	—
320	C₈	—	2	2	no	—
321	C₈	—	4	2	no	—
322	C₈	—	8	1	no	—
323	C₈	—	8	1	no	—
324	C₂²	—	8	2	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	16	2	no	—
328	C₂²	—	16	2	no	—
329	C₂²	#390×C₇₁	1	1	yes	characteristic
330	C₂²	#463×C₇₁	1	1	yes	characteristic
331	C₂²	#464×C₇₁	1	1	yes	characteristic
332	C₂²	#465×C₇₁	1	2	yes	—
333	C₂²	#472×C₇₁	1	2	yes	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	4	4	no	—
337	C₄	—	8	2	no	—
338	C₄	#416×C₇₁	1	1	yes	characteristic
339	C₄	#417×C₇₁	1	1	yes	characteristic
340	C₄	C₄×SD₁₆×C₇₁	1	2	yes	—
341	C₄	#463×C₇₁	1	1	yes	characteristic
342	C₄	#464×C₇₁	1	1	yes	characteristic
343	C₄	#492×C₇₁	1	1	yes	characteristic
344	C₄	#495×C₇₁	1	1	yes	characteristic
345	C₂	—	16	2	no	—
346	C₂	C₇₁×#1677	1	2	yes	—
347	C₂	C₇₁×#1678	1	2	yes	—
348	C₂	C₇₁×#1968	1	1	yes	characteristic
349	C₂	C₇₁×#1978	1	1	yes	characteristic
350	C₂	C₇₁×#1983	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇₁	#121
commutator	C₄²	C₂²×C₄×C₇₁
Frattini, cosocle	C₂×C₄²	C₂³×C₇₁
prime radical	C₂×#127×C₇₁	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇₁ ◁ C₂×C₄²×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇₁×#1987	C₂	semidirect
4. C₈×C₄⋊C₄×C₇₁	C₂	semidirect
5. C₂×#334×C₇₁	C₂	semidirect
6. C₂×#333×C₇₁	C₂	semidirect
11. #486×C₇₁	C₄	semidirect
169. #9735	C₇₁	direct (N × H)
177. #1987	C₂×C₇₁	semidirect
179. C₈×C₄⋊C₄	C₂×C₇₁	semidirect
180. C₂×#334	C₂×C₇₁	semidirect
181. C₂×#333	C₂×C₇₁	semidirect
183. C₇₁	#9735	direct (N × H)
187. #486	C₄×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#486×C₇₁	2
C₇₁×#1987	2
C₇₁×#1982	2
C₈×C₄⋊C₄×C₇₁	2
C₂×#334×C₇₁	2
C₂×#333×C₇₁	2
C₂×#331×C₇₁	2
#9735	71

Subgroups — C71×#9735 #31830607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#9735 #31830607