Subgroups — C₇₁×#23646 #31844518

Order 18176.

906 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#427×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇₁×#2896	C₂	1	2	yes	maximal
3	C₂×#408×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₇₁×#2079	C₂	1	4	yes	maximal, complemented
5	C₇₁×#2078	C₂	1	4	yes	maximal, complemented
6	C₇₁×#1742	C₂	1	2	yes	maximal, complemented
7	C₂×#341×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#41×C₇₁	C₂²	1	1	yes	characteristic
9	C₂²×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₇₁	C₂²	1	2	yes	—
11	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
12	C₂×#117×C₇₁	C₂²	1	2	yes	—
13	#427×C₇₁	—	2	2	no	—
14	#427×C₇₁	—	2	2	no	—
15	C₂×#107×C₇₁	C₂²	1	2	yes	complemented
16	C₂×#106×C₇₁	C₂²	1	2	yes	complemented
17	#419×C₇₁	—	2	2	no	—
18	#418×C₇₁	—	2	2	no	—
19	#417×C₇₁	C₂²	1	8	yes	—
20	#416×C₇₁	—	2	4	no	—
21	#408×C₇₁	—	2	2	no	—
22	#408×C₇₁	C₂²	1	2	yes	—
23	#408×C₇₁	C₂²	1	2	yes	—
24	C₂×C₂²⋊C₈×C₇₁	—	2	2	no	—
25	#343×C₇₁	C₂²	1	8	yes	complemented
26	#341×C₇₁	C₂²	1	2	yes	complemented
27	#341×C₇₁	C₂²	1	2	yes	complemented
28	C₂×#41×C₇₁	—	2	1	no	—
29	C₂×#41×C₇₁	—	2	1	no	—
30	C₂×#41×C₇₁	—	2	4	no	—
31	C₂×#41×C₇₁	—	2	4	no	—
32	C₂×#41×C₇₁	D₄	1	2	yes	—
33	C₂×#41×C₇₁	C₂³	1	2	yes	—
34	C₂²×Q₈×C₇₁	D₄	1	1	yes	characteristic
35	C₂²×D₄×C₇₁	—	2	1	no	—
36	C₂²×D₄×C₇₁	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₇₁	—	2	1	no	—
38	C₂³×C₄×C₇₁	C₂³	1	1	yes	characteristic
39	C₂×M₁₆×C₇₁	—	2	1	no	—
40	C₂×M₁₆×C₇₁	—	2	2	no	—
41	C₂×M₁₆×C₇₁	—	2	4	no	—
42	C₂×M₁₆×C₇₁	—	2	4	no	—
43	C₂×M₁₆×C₇₁	—	2	4	no	—
44	C₂×M₁₆×C₇₁	C₂³	1	2	yes	—
45	C₂×M₁₆×C₇₁	C₂³	1	4	yes	—
46	C₂²×C₈×C₇₁	—	2	4	no	—
47	C₂²×C₈×C₇₁	—	2	1	no	—
48	C₂²×C₈×C₇₁	—	2	2	no	—
49	C₂²×C₈×C₇₁	D₄	1	2	yes	—
50	C₂²×C₈×C₇₁	D₄	1	2	yes	—
51	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
53	#117×C₇₁	—	2	2	no	—
54	#117×C₇₁	—	2	2	no	—
55	#117×C₇₁	—	2	2	no	—
56	#117×C₇₁	—	2	2	no	—
57	C₂×C₄⋊C₄×C₇₁	C₂³	1	4	yes	—
58	C₂×C₂²⋊C₄×C₇₁	—	2	2	no	—
59	C₂×C₄²×C₇₁	—	2	2	no	—
60	#107×C₇₁	—	2	4	no	—
61	#106×C₇₁	—	2	4	no	—
62	#103×C₇₁	—	2	8	no	—
63	#102×C₇₁	—	2	8	no	—
64	C₂²⋊C₈×C₇₁	—	2	4	no	—
65	C₂²⋊C₈×C₇₁	—	2	4	no	—
66	C₂⁴×C₇₁	—	2	1	no	—
67	C₂⁴×C₇₁	#41	1	1	yes	characteristic
68	#41×C₇₁	—	2	8	no	—
69	#41×C₇₁	—	2	4	no	—
70	#41×C₇₁	—	2	4	no	—
71	#41×C₇₁	—	4	2	no	—
72	#41×C₇₁	—	4	2	no	—
73	#41×C₇₁	—	4	4	no	—
74	C₂×Q₈×C₇₁	—	2	1	no	—
75	C₂×Q₈×C₇₁	—	2	4	no	—
76	C₂×Q₈×C₇₁	C₂×D₄	1	2	yes	—
77	C₂×D₄×C₇₁	—	2	2	no	—
78	C₂×D₄×C₇₁	—	2	2	no	—
79	C₂×D₄×C₇₁	—	2	1	no	—
80	C₂×D₄×C₇₁	—	2	4	no	—
81	C₂×D₄×C₇₁	—	4	4	no	—
82	C₂×D₄×C₇₁	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₇₁	—	2	1	no	—
84	C₂²×C₄×C₇₁	—	2	2	no	—
85	C₂²×C₄×C₇₁	—	2	2	no	—
86	C₂²×C₄×C₇₁	—	2	4	no	—
87	C₂²×C₄×C₇₁	—	4	2	no	—
88	C₂²×C₄×C₇₁	—	4	2	no	—
89	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
94	C₂²×C₄×C₇₁	#41	1	4	yes	—
95	C₂²×C₄×C₇₁	#41	1	4	yes	—
96	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
97	M₁₆×C₇₁	—	2	4	no	—
98	M₁₆×C₇₁	—	2	4	no	—
99	M₁₆×C₇₁	—	2	4	no	—
100	M₁₆×C₇₁	—	4	4	no	—
101	M₁₆×C₇₁	—	4	2	no	—
102	C₂×C₈×C₇₁	—	2	4	no	—
103	C₂×C₈×C₇₁	—	2	4	no	—
104	C₂×C₈×C₇₁	—	2	2	no	—
105	C₂×C₈×C₇₁	—	2	2	no	—
106	C₂×C₈×C₇₁	—	2	2	no	—
107	C₂×C₈×C₇₁	—	2	4	no	—
108	C₂×C₈×C₇₁	—	4	4	no	—
109	C₂×C₈×C₇₁	—	4	2	no	—
110	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
111	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
112	C₄⋊C₄×C₇₁	—	2	4	no	—
113	C₄⋊C₄×C₇₁	—	2	4	no	—
114	C₂²⋊C₄×C₇₁	—	4	2	no	—
115	C₂²⋊C₄×C₇₁	—	4	2	no	—
116	C₄²×C₇₁	—	4	2	no	—
117	C₄²×C₇₁	—	4	2	no	—
118	C₂³×C₇₁	—	2	2	no	—
119	C₂³×C₇₁	—	4	2	no	—
120	C₂³×C₇₁	—	4	4	no	—
121	C₂³×C₇₁	—	4	2	no	—
122	C₂³×C₇₁	#121	1	1	yes	characteristic
123	C₂³×C₇₁	#121	1	1	yes	characteristic
124	C₂³×C₇₁	#126	1	4	yes	—
125	C₂³×C₇₁	C₂×#41	1	1	yes	characteristic
126	Q₈×C₇₁	—	2	4	no	—
127	Q₈×C₇₁	—	4	2	no	—
128	D₄×C₇₁	—	2	4	no	—
129	D₄×C₇₁	—	4	4	no	—
130	D₄×C₇₁	—	4	4	no	—
131	D₄×C₇₁	—	4	2	no	—
132	C₂×C₄×C₇₁	—	2	2	no	—
133	C₂×C₄×C₇₁	—	4	2	no	—
134	C₂×C₄×C₇₁	—	4	2	no	—
135	C₂×C₄×C₇₁	—	4	2	no	—
136	C₂×C₄×C₇₁	—	4	4	no	—
137	C₂×C₄×C₇₁	—	4	4	no	—
138	C₂×C₄×C₇₁	—	8	4	no	—
139	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
140	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
141	C₂×C₄×C₇₁	#123	1	2	yes	—
142	C₂×C₄×C₇₁	#123	1	2	yes	—
143	C₂×C₄×C₇₁	#123	1	4	yes	—
144	C₂×C₄×C₇₁	#123	1	4	yes	—
145	C₂×C₄×C₇₁	#124	1	2	yes	—
146	C₂×C₄×C₇₁	#124	1	2	yes	—
147	C₂×C₄×C₇₁	#126	1	4	yes	—
148	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₇₁	C₂×#41	1	2	yes	—
151	C₂×C₄×C₇₁	C₂×#41	1	2	yes	—
152	C₈×C₇₁	—	2	2	no	—
153	C₈×C₇₁	—	2	2	no	—
154	C₈×C₇₁	—	4	4	no	—
155	C₈×C₇₁	—	4	2	no	—
156	C₂²×C₇₁	—	2	2	no	—
157	C₂²×C₇₁	—	2	2	no	—
158	C₂²×C₇₁	—	2	2	no	—
159	C₂²×C₇₁	—	4	4	no	—
160	C₂²×C₇₁	—	4	2	no	—
161	C₂²×C₇₁	—	4	4	no	—
162	C₂²×C₇₁	—	8	4	no	—
163	C₂²×C₇₁	#397	1	2	yes	—
164	C₂²×C₇₁	#397	1	2	yes	—
165	C₂²×C₇₁	C₂×#121	1	1	yes	characteristic
166	C₂²×C₇₁	C₂×#126	1	2	yes	—
167	C₄×C₇₁	—	4	2	no	—
168	C₄×C₇₁	—	8	4	no	—
169	C₄×C₇₁	#397	1	2	yes	—
170	C₄×C₇₁	#397	1	2	yes	—
171	C₄×C₇₁	C₂×#123	1	1	yes	characteristic
172	C₄×C₇₁	C₂×#123	1	2	yes	—
173	C₄×C₇₁	C₂×#124	1	1	yes	characteristic
174	#23646	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
175	C₂×C₇₁	—	2	2	no	—
176	C₂×C₇₁	—	2	2	no	—
177	C₂×C₇₁	—	2	2	no	—
178	C₂×C₇₁	—	8	2	no	—
179	C₂×C₇₁	C₂×#397	1	1	yes	characteristic
180	C₂×C₇₁	#3067	1	2	yes	—
181	C₂×#427	C₂×C₇₁	1	1	yes	characteristic, complemented
182	#2896	C₂×C₇₁	1	2	yes	—
183	C₂×#408	C₂×C₇₁	1	1	yes	characteristic
184	#2079	C₂×C₇₁	1	4	yes	complemented
185	#2078	C₂×C₇₁	1	4	yes	complemented
186	#1742	C₂×C₇₁	1	2	yes	complemented
187	C₂×#341	C₂×C₇₁	1	1	yes	characteristic, complemented
188	C₇₁	#23646	1	1	yes	characteristic, direct product factor, Sylow
189	C₂²×#41	C₂²×C₇₁	1	1	yes	characteristic
190	C₂²×M₁₆	C₂²×C₇₁	1	1	yes	characteristic
191	C₂²×M₁₆	C₂²×C₇₁	1	2	yes	—
192	C₂³×C₈	C₂²×C₇₁	1	1	yes	characteristic
193	C₂×#117	C₂²×C₇₁	1	2	yes	—
194	#427	—	2	2	no	—
195	#427	—	2	2	no	—
196	C₂×#107	C₂²×C₇₁	1	2	yes	complemented
197	C₂×#106	C₂²×C₇₁	1	2	yes	complemented
198	#419	—	2	2	no	—
199	#418	—	2	2	no	—
200	#417	C₂²×C₇₁	1	8	yes	—
201	#416	—	2	4	no	—
202	#408	—	2	2	no	—
203	#408	C₂²×C₇₁	1	2	yes	—
204	#408	C₂²×C₇₁	1	2	yes	—
205	C₂×C₂²⋊C₈	—	2	2	no	—
206	#343	C₂²×C₇₁	1	8	yes	complemented
207	#341	C₂²×C₇₁	1	2	yes	complemented
208	#341	C₂²×C₇₁	1	2	yes	complemented
209	C₂×#41	—	2	1	no	—
210	C₂×#41	—	2	1	no	—
211	C₂×#41	—	2	4	no	—
212	C₂×#41	—	2	4	no	—
213	C₂×#41	D₄×C₇₁	1	2	yes	—
214	C₂×#41	C₂³×C₇₁	1	2	yes	—
215	C₂²×Q₈	D₄×C₇₁	1	1	yes	characteristic
216	C₂²×D₄	—	2	1	no	—
217	C₂²×D₄	D₄×C₇₁	1	1	yes	characteristic
218	C₂³×C₄	—	2	1	no	—
219	C₂³×C₄	C₂³×C₇₁	1	1	yes	characteristic
220	C₂×M₁₆	—	2	1	no	—
221	C₂×M₁₆	—	2	2	no	—
222	C₂×M₁₆	—	2	4	no	—
223	C₂×M₁₆	—	2	4	no	—
224	C₂×M₁₆	—	2	4	no	—
225	C₂×M₁₆	C₂³×C₇₁	1	4	yes	—
226	C₂×M₁₆	C₂³×C₇₁	1	2	yes	—
227	C₂²×C₈	—	2	4	no	—
228	C₂²×C₈	—	2	1	no	—
229	C₂²×C₈	—	2	2	no	—
230	C₂²×C₈	D₄×C₇₁	1	2	yes	—
231	C₂²×C₈	D₄×C₇₁	1	2	yes	—
232	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
233	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
234	#117	—	2	2	no	—
235	#117	—	2	2	no	—
236	#117	—	2	2	no	—
237	#117	—	2	2	no	—
238	C₂×C₄⋊C₄	C₂³×C₇₁	1	4	yes	—
239	C₂×C₂²⋊C₄	—	2	2	no	—
240	C₂×C₄²	—	2	2	no	—
241	#107	—	2	4	no	—
242	#106	—	2	4	no	—
243	#103	—	2	8	no	—
244	#102	—	2	8	no	—
245	C₂²⋊C₈	—	2	4	no	—
246	C₂²⋊C₈	—	2	4	no	—
247	C₂⁴	—	2	1	no	—
248	C₂⁴	#41×C₇₁	1	1	yes	characteristic
249	#41	—	2	8	no	—
250	#41	—	2	4	no	—
251	#41	—	2	4	no	—
252	#41	—	4	2	no	—
253	#41	—	4	2	no	—
254	#41	—	4	4	no	—
255	C₂×Q₈	—	2	1	no	—
256	C₂×Q₈	—	2	4	no	—
257	C₂×Q₈	C₂×D₄×C₇₁	1	2	yes	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	2	1	no	—
261	C₂×D₄	—	2	4	no	—
262	C₂×D₄	—	4	4	no	—
263	C₂×D₄	C₂×D₄×C₇₁	1	2	yes	—
264	C₂²×C₄	—	2	1	no	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	2	4	no	—
268	C₂²×C₄	—	4	2	no	—
269	C₂²×C₄	—	4	2	no	—
270	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
271	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
272	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
273	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
274	C₂²×C₄	#41×C₇₁	1	1	yes	characteristic
275	C₂²×C₄	#41×C₇₁	1	4	yes	—
276	C₂²×C₄	#41×C₇₁	1	4	yes	—
277	C₂²×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
278	M₁₆	—	2	4	no	—
279	M₁₆	—	2	4	no	—
280	M₁₆	—	2	4	no	—
281	M₁₆	—	4	4	no	—
282	M₁₆	—	4	2	no	—
283	C₂×C₈	—	2	4	no	—
284	C₂×C₈	—	2	4	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	4	no	—
289	C₂×C₈	—	4	4	no	—
290	C₂×C₈	—	4	2	no	—
291	C₂×C₈	C₂×D₄×C₇₁	1	2	yes	—
292	C₂×C₈	C₂×D₄×C₇₁	1	2	yes	—
293	C₄⋊C₄	—	2	4	no	—
294	C₄⋊C₄	—	2	4	no	—
295	C₂²⋊C₄	—	4	2	no	—
296	C₂²⋊C₄	—	4	2	no	—
297	C₄²	—	4	2	no	—
298	C₄²	—	4	2	no	—
299	C₂³	—	2	2	no	—
300	C₂³	—	4	2	no	—
301	C₂³	—	4	4	no	—
302	C₂³	—	4	2	no	—
303	C₂³	#121×C₇₁	1	1	yes	characteristic
304	C₂³	#121×C₇₁	1	1	yes	characteristic
305	C₂³	#126×C₇₁	1	4	yes	—
306	C₂³	C₂×#41×C₇₁	1	1	yes	characteristic
307	Q₈	—	2	4	no	—
308	Q₈	—	4	2	no	—
309	D₄	—	2	4	no	—
310	D₄	—	4	4	no	—
311	D₄	—	4	4	no	—
312	D₄	—	4	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	4	no	—
318	C₂×C₄	—	4	4	no	—
319	C₂×C₄	—	8	4	no	—
320	C₂×C₄	#121×C₇₁	1	1	yes	characteristic
321	C₂×C₄	#121×C₇₁	1	1	yes	characteristic
322	C₂×C₄	#123×C₇₁	1	2	yes	—
323	C₂×C₄	#123×C₇₁	1	2	yes	—
324	C₂×C₄	#123×C₇₁	1	4	yes	—
325	C₂×C₄	#123×C₇₁	1	4	yes	—
326	C₂×C₄	#124×C₇₁	1	2	yes	—
327	C₂×C₄	#124×C₇₁	1	2	yes	—
328	C₂×C₄	#126×C₇₁	1	4	yes	—
329	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
330	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
331	C₂×C₄	C₂×#41×C₇₁	1	2	yes	—
332	C₂×C₄	C₂×#41×C₇₁	1	2	yes	—
333	C₈	—	2	2	no	—
334	C₈	—	2	2	no	—
335	C₈	—	4	4	no	—
336	C₈	—	4	2	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	4	4	no	—
341	C₂²	—	4	4	no	—
342	C₂²	—	4	2	no	—
343	C₂²	—	8	4	no	—
344	C₂²	#397×C₇₁	1	2	yes	—
345	C₂²	#397×C₇₁	1	2	yes	—
346	C₂²	C₂×#121×C₇₁	1	1	yes	characteristic
347	C₂²	C₂×#126×C₇₁	1	2	yes	—
348	C₄	—	4	2	no	—
349	C₄	—	8	4	no	—
350	C₄	#397×C₇₁	1	2	yes	—
351	C₄	#397×C₇₁	1	2	yes	—
352	C₄	C₂×#123×C₇₁	1	1	yes	characteristic
353	C₄	C₂×#123×C₇₁	1	2	yes	—
354	C₄	C₂×#124×C₇₁	1	1	yes	characteristic
355	C₂	—	2	2	no	—
356	C₂	—	2	2	no	—
357	C₂	—	2	2	no	—
358	C₂	—	8	2	no	—
359	C₂	C₂×#397×C₇₁	1	1	yes	characteristic
360	C₂	C₇₁×#3067	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₇₁	#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₁
socle	C₂²×C₇₁	C₂×#121
prime radical	C₂²×#41×C₇₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₁ ◁ C₂³×C₄×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#427×C₇₁	C₂	semidirect
4. C₇₁×#2079	C₂	semidirect
5. C₇₁×#2078	C₂	semidirect
6. C₇₁×#1742	C₂	semidirect
7. C₂×#341×C₇₁	C₂	semidirect
15. C₂×#107×C₇₁	C₂²	semidirect
16. C₂×#106×C₇₁	C₂²	semidirect
25. #343×C₇₁	C₂²	semidirect
26. #341×C₇₁	C₂²	semidirect
27. #341×C₇₁	C₂²	semidirect
174. #23646	C₇₁	direct (N × H)
181. C₂×#427	C₂×C₇₁	semidirect
184. #2079	C₂×C₇₁	semidirect
185. #2078	C₂×C₇₁	semidirect
186. #1742	C₂×C₇₁	semidirect
187. C₂×#341	C₂×C₇₁	semidirect
188. C₇₁	#23646	direct (N × H)
196. C₂×#107	C₂²×C₇₁	semidirect
197. C₂×#106	C₂²×C₇₁	semidirect
206. #343	C₂²×C₇₁	semidirect
207. #341	C₂²×C₇₁	semidirect
208. #341	C₂²×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#427×C₇₁	2
C₇₁×#2896	2
C₂×#408×C₇₁	2
C₇₁×#2079	2
C₇₁×#2078	2
C₇₁×#1742	2
C₂×#341×C₇₁	2
#23646	71

Subgroups — C71×#23646 #31844518

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#23646 #31844518