Subgroups — C₇₁×#29140 #31850012

Order 18176.

754 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₁×#2859	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#124×C₇₁	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#2026	C₂	1	2	yes	maximal, complemented
4	C₇₁×#2025	C₂	1	2	yes	maximal
5	C₇₁×#1958	C₂	1	2	yes	maximal
6	C₇₁×#1835	C₂	1	2	yes	maximal, complemented
7	C₇₁×#1803	C₂	1	2	yes	maximal, complemented
8	C₇₁×#1802	C₂	1	2	yes	maximal
9	C₄×C₄⋊C₈×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#125×C₇₁	C₂²	1	2	yes	—
11	C₂×#124×C₇₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
15	C₂×#103×C₇₁	C₂²	1	4	yes	—
16	C₂×#102×C₇₁	C₂²	1	4	yes	—
17	C₂×C₄×C₈×C₇₁	C₂²	1	2	yes	—
18	#391×C₇₁	C₂²	1	1	yes	characteristic
19	#390×C₇₁	C₂²	1	1	yes	characteristic
20	#389×C₇₁	—	2	2	no	—
21	#389×C₇₁	—	2	2	no	—
22	#388×C₇₁	—	2	2	no	—
23	#384×C₇₁	C₂²	1	2	yes	—
24	#382×C₇₁	C₂²	1	1	yes	characteristic
25	#382×C₇₁	C₂²	1	2	yes	—
26	C₄×C₂²⋊C₄×C₇₁	—	2	2	no	—
27	C₄³×C₇₁	C₂²	1	1	yes	characteristic
28	#340×C₇₁	C₂²	1	4	yes	—
29	#340×C₇₁	C₂²	1	4	yes	—
30	#336×C₇₁	C₂²	1	2	yes	—
31	C₂²×Q₈×C₇₁	C₂³	1	1	yes	characteristic
32	C₂²×D₄×C₇₁	C₂³	1	1	yes	characteristic
33	C₂³×C₄×C₇₁	—	2	1	no	—
34	C₂²×C₈×C₇₁	C₂³	1	4	yes	—
35	#125×C₇₁	—	4	4	no	—
36	#124×C₇₁	—	2	2	no	—
37	#124×C₇₁	Q₈	1	4	yes	—
38	C₄×Q₈×C₇₁	—	2	4	no	—
39	C₄×D₄×C₇₁	—	2	4	no	—
40	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
41	C₂×C₄⋊C₄×C₇₁	—	2	4	no	—
42	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₇₁	C₂³	1	4	yes	—
45	C₂×C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
47	C₂×C₂²⋊C₄×C₇₁	—	2	2	no	—
48	C₂×C₂²⋊C₄×C₇₁	—	2	1	no	—
49	C₂×C₂²⋊C₄×C₇₁	—	2	2	no	—
50	C₂×C₄²×C₇₁	—	2	1	no	—
51	C₂×C₄²×C₇₁	D₄	1	2	yes	—
52	C₂×C₄²×C₇₁	D₄	1	2	yes	—
53	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₇₁	—	2	4	no	—
57	#103×C₇₁	—	4	4	no	—
58	#102×C₇₁	—	4	4	no	—
59	C₄×C₈×C₇₁	—	2	4	no	—
60	#95×C₇₁	—	2	2	no	—
61	#95×C₇₁	—	2	2	no	—
62	#95×C₇₁	—	2	4	no	—
63	C₂⁴×C₇₁	—	2	1	no	—
64	C₂×Q₈×C₇₁	—	2	1	no	—
65	C₂×Q₈×C₇₁	—	2	2	no	—
66	C₂×Q₈×C₇₁	—	2	2	no	—
67	C₂×Q₈×C₇₁	C₂×Q₈	1	2	yes	—
68	C₂×D₄×C₇₁	—	2	2	no	—
69	C₂×D₄×C₇₁	—	2	1	no	—
70	C₂×D₄×C₇₁	—	2	2	no	—
71	C₂×D₄×C₇₁	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₇₁	—	2	1	no	—
73	C₂²×C₄×C₇₁	—	2	1	no	—
74	C₂²×C₄×C₇₁	—	2	1	no	—
75	C₂²×C₄×C₇₁	—	2	4	no	—
76	C₂²×C₄×C₇₁	—	4	2	no	—
77	C₂²×C₄×C₇₁	—	4	4	no	—
78	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
83	C₂×C₈×C₇₁	—	2	4	no	—
84	C₂×C₈×C₇₁	—	2	4	no	—
85	C₂×C₈×C₇₁	—	2	4	no	—
86	C₄⋊C₄×C₇₁	—	2	2	no	—
87	C₄⋊C₄×C₇₁	—	2	2	no	—
88	C₄⋊C₄×C₇₁	—	2	4	no	—
89	C₄⋊C₄×C₇₁	—	2	4	no	—
90	C₄⋊C₄×C₇₁	—	4	4	no	—
91	C₄⋊C₄×C₇₁	—	4	8	no	—
92	C₄⋊C₄×C₇₁	—	4	2	no	—
93	C₄⋊C₄×C₇₁	—	4	4	no	—
94	C₂²⋊C₄×C₇₁	—	2	8	no	—
95	C₂²⋊C₄×C₇₁	—	4	4	no	—
96	C₂²⋊C₄×C₇₁	—	4	2	no	—
97	C₄²×C₇₁	—	2	4	no	—
98	C₄²×C₇₁	—	2	2	no	—
99	C₄²×C₇₁	—	4	2	no	—
100	C₄²×C₇₁	—	4	2	no	—
101	C₄²×C₇₁	C₂×Q₈	1	2	yes	—
102	C₄²×C₇₁	#41	1	2	yes	—
103	C₄²×C₇₁	#41	1	4	yes	—
104	C₂³×C₇₁	—	2	2	no	—
105	C₂³×C₇₁	—	4	1	no	—
106	C₂³×C₇₁	—	4	1	no	—
107	C₂³×C₇₁	—	4	2	no	—
108	C₂³×C₇₁	—	4	2	no	—
109	C₂³×C₇₁	C₂²×D₄	1	1	yes	characteristic
110	Q₈×C₇₁	—	2	4	no	—
111	Q₈×C₇₁	—	4	2	no	—
112	D₄×C₇₁	—	2	4	no	—
113	D₄×C₇₁	—	4	2	no	—
114	C₂×C₄×C₇₁	—	2	2	no	—
115	C₂×C₄×C₇₁	—	2	2	no	—
116	C₂×C₄×C₇₁	—	2	2	no	—
117	C₂×C₄×C₇₁	—	2	2	no	—
118	C₂×C₄×C₇₁	—	2	2	no	—
119	C₂×C₄×C₇₁	—	2	2	no	—
120	C₂×C₄×C₇₁	—	2	2	no	—
121	C₂×C₄×C₇₁	—	4	1	no	—
122	C₂×C₄×C₇₁	—	4	1	no	—
123	C₂×C₄×C₇₁	—	4	1	no	—
124	C₂×C₄×C₇₁	—	4	1	no	—
125	C₂×C₄×C₇₁	—	4	4	no	—
126	C₂×C₄×C₇₁	—	4	4	no	—
127	C₂×C₄×C₇₁	—	4	2	no	—
128	C₂×C₄×C₇₁	—	4	2	no	—
129	C₂×C₄×C₇₁	—	4	4	no	—
130	C₂×C₄×C₇₁	—	8	4	no	—
131	C₂×C₄×C₇₁	C₂²⋊Q₈	1	4	yes	—
132	C₂×C₄×C₇₁	#124	1	2	yes	—
133	C₂×C₄×C₇₁	#124	1	2	yes	—
134	C₂×C₄×C₇₁	#135	1	4	yes	—
135	C₂×C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
136	C₂×C₄×C₇₁	C₂×#41	1	2	yes	—
137	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₇₁	H(5,𝔽₂)	1	2	yes	—
139	C₈×C₇₁	—	4	4	no	—
140	C₂²×C₇₁	—	4	2	no	—
141	C₂²×C₇₁	—	4	4	no	—
142	C₂²×C₇₁	—	8	1	no	—
143	C₂²×C₇₁	—	8	1	no	—
144	C₂²×C₇₁	—	8	2	no	—
145	C₂²×C₇₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
146	C₂²×C₇₁	C₂×#124	1	1	yes	characteristic
147	C₂²×C₇₁	#535	1	1	yes	characteristic
148	C₂²×C₇₁	C₂×#135	1	2	yes	—
149	C₂²×C₇₁	#575	1	2	yes	—
150	C₄×C₇₁	—	2	4	no	—
151	C₄×C₇₁	—	2	2	no	—
152	C₄×C₇₁	—	4	2	no	—
153	C₄×C₇₁	—	4	2	no	—
154	C₄×C₇₁	—	8	1	no	—
155	C₄×C₇₁	—	8	1	no	—
156	C₄×C₇₁	—	8	4	no	—
157	C₄×C₇₁	#488	1	4	yes	—
158	C₄×C₇₁	#554	1	2	yes	—
159	C₄×C₇₁	#555	1	2	yes	—
160	#29140	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₂×C₇₁	—	8	2	no	—
162	C₂×C₇₁	#2563	1	1	yes	characteristic
163	C₂×C₇₁	#3076	1	2	yes	—
164	C₂×C₇₁	C₂×#488	1	1	yes	characteristic
165	C₂×C₇₁	#3142	1	1	yes	characteristic
166	C₂×C₇₁	#3212	1	2	yes	—
167	#2859	C₂×C₇₁	1	1	yes	characteristic, complemented
168	C₄×#124	C₂×C₇₁	1	1	yes	characteristic
169	#2026	C₂×C₇₁	1	2	yes	complemented
170	#2025	C₂×C₇₁	1	2	yes	—
171	#1958	C₂×C₇₁	1	2	yes	—
172	#1835	C₂×C₇₁	1	2	yes	complemented
173	#1803	C₂×C₇₁	1	2	yes	complemented
174	#1802	C₂×C₇₁	1	2	yes	—
175	C₄×C₄⋊C₈	C₂×C₇₁	1	1	yes	characteristic, complemented
176	C₇₁	#29140	1	1	yes	characteristic, direct product factor, Sylow
177	C₂×#125	C₂²×C₇₁	1	2	yes	—
178	C₂×#124	C₂²×C₇₁	1	1	yes	characteristic
179	C₂×C₄×Q₈	C₂²×C₇₁	1	1	yes	characteristic
180	C₂×C₄×D₄	C₂²×C₇₁	1	1	yes	characteristic
181	C₂×C₄⋊C₈	C₂²×C₇₁	1	2	yes	—
182	C₂×#103	C₂²×C₇₁	1	4	yes	—
183	C₂×#102	C₂²×C₇₁	1	4	yes	—
184	C₂×C₄×C₈	C₂²×C₇₁	1	2	yes	—
185	#391	C₂²×C₇₁	1	1	yes	characteristic
186	#390	C₂²×C₇₁	1	1	yes	characteristic
187	#389	—	2	2	no	—
188	#389	—	2	2	no	—
189	#388	—	2	2	no	—
190	#384	C₂²×C₇₁	1	2	yes	—
191	#382	C₂²×C₇₁	1	1	yes	characteristic
192	#382	C₂²×C₇₁	1	2	yes	—
193	C₄×C₂²⋊C₄	—	2	2	no	—
194	C₄³	C₂²×C₇₁	1	1	yes	characteristic
195	#340	C₂²×C₇₁	1	4	yes	—
196	#340	C₂²×C₇₁	1	4	yes	—
197	#336	C₂²×C₇₁	1	2	yes	—
198	C₂²×Q₈	C₂³×C₇₁	1	1	yes	characteristic
199	C₂²×D₄	C₂³×C₇₁	1	1	yes	characteristic
200	C₂³×C₄	—	2	1	no	—
201	C₂²×C₈	C₂³×C₇₁	1	4	yes	—
202	#125	—	4	4	no	—
203	#124	—	2	2	no	—
204	#124	Q₈×C₇₁	1	4	yes	—
205	C₄×Q₈	—	2	4	no	—
206	C₄×D₄	—	2	4	no	—
207	C₂×C₄⋊C₄	—	2	1	no	—
208	C₂×C₄⋊C₄	—	2	4	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	C₂³×C₇₁	1	1	yes	characteristic
212	C₂×C₄⋊C₄	C₂³×C₇₁	1	1	yes	characteristic
213	C₂×C₄⋊C₄	C₂³×C₇₁	1	4	yes	—
214	C₂×C₂²⋊C₄	—	2	2	no	—
215	C₂×C₂²⋊C₄	—	2	1	no	—
216	C₂×C₂²⋊C₄	—	2	2	no	—
217	C₂×C₄²	—	2	1	no	—
218	C₂×C₄²	D₄×C₇₁	1	2	yes	—
219	C₂×C₄²	D₄×C₇₁	1	2	yes	—
220	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
221	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
222	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
223	C₄⋊C₈	—	2	4	no	—
224	#103	—	4	4	no	—
225	#102	—	4	4	no	—
226	C₄×C₈	—	2	4	no	—
227	#95	—	2	2	no	—
228	#95	—	2	2	no	—
229	#95	—	2	4	no	—
230	C₂⁴	—	2	1	no	—
231	C₂×Q₈	—	2	1	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂×Q₈	C₂×Q₈×C₇₁	1	2	yes	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	2	1	no	—
237	C₂×D₄	—	2	2	no	—
238	C₂×D₄	C₂×Q₈×C₇₁	1	2	yes	—
239	C₂²×C₄	—	2	1	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	2	1	no	—
242	C₂²×C₄	—	2	4	no	—
243	C₂²×C₄	—	4	2	no	—
244	C₂²×C₄	—	4	4	no	—
245	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
246	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
247	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
249	C₂²×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	2	4	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	4	4	no	—
258	C₄⋊C₄	—	4	8	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	—	4	4	no	—
261	C₂²⋊C₄	—	2	8	no	—
262	C₂²⋊C₄	—	4	4	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₄²	—	2	4	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	4	2	no	—
267	C₄²	—	4	2	no	—
268	C₄²	C₂×Q₈×C₇₁	1	2	yes	—
269	C₄²	#41×C₇₁	1	2	yes	—
270	C₄²	#41×C₇₁	1	4	yes	—
271	C₂³	—	2	2	no	—
272	C₂³	—	4	1	no	—
273	C₂³	—	4	1	no	—
274	C₂³	—	4	2	no	—
275	C₂³	—	4	2	no	—
276	C₂³	C₂²×D₄×C₇₁	1	1	yes	characteristic
277	Q₈	—	2	4	no	—
278	Q₈	—	4	2	no	—
279	D₄	—	2	4	no	—
280	D₄	—	4	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	4	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	4	no	—
297	C₂×C₄	—	8	4	no	—
298	C₂×C₄	C₂²⋊Q₈×C₇₁	1	4	yes	—
299	C₂×C₄	#124×C₇₁	1	2	yes	—
300	C₂×C₄	#124×C₇₁	1	2	yes	—
301	C₂×C₄	#135×C₇₁	1	4	yes	—
302	C₂×C₄	C₂²×Q₈×C₇₁	1	1	yes	characteristic
303	C₂×C₄	C₂×#41×C₇₁	1	2	yes	—
304	C₂×C₄	C₂×#41×C₇₁	1	1	yes	characteristic
305	C₂×C₄	H(5,𝔽₂)×C₇₁	1	2	yes	—
306	C₈	—	4	4	no	—
307	C₂²	—	4	2	no	—
308	C₂²	—	4	4	no	—
309	C₂²	—	8	1	no	—
310	C₂²	—	8	1	no	—
311	C₂²	—	8	2	no	—
312	C₂²	C₂×C₂²⋊Q₈×C₇₁	1	1	yes	characteristic
313	C₂²	C₂×#124×C₇₁	1	1	yes	characteristic
314	C₂²	#535×C₇₁	1	1	yes	characteristic
315	C₂²	C₂×#135×C₇₁	1	2	yes	—
316	C₂²	#575×C₇₁	1	2	yes	—
317	C₄	—	2	4	no	—
318	C₄	—	2	2	no	—
319	C₄	—	4	2	no	—
320	C₄	—	4	2	no	—
321	C₄	—	8	1	no	—
322	C₄	—	8	1	no	—
323	C₄	—	8	4	no	—
324	C₄	#488×C₇₁	1	4	yes	—
325	C₄	#554×C₇₁	1	2	yes	—
326	C₄	#555×C₇₁	1	2	yes	—
327	C₂	—	8	2	no	—
328	C₂	C₇₁×#2563	1	1	yes	characteristic
329	C₂	C₇₁×#3076	1	2	yes	—
330	C₂	C₂×#488×C₇₁	1	1	yes	characteristic
331	C₂	C₇₁×#3142	1	1	yes	characteristic
332	C₂	C₇₁×#3212	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₇₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₁
prime radical	C₂²×D₄×C₇₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇₁ ◁ C₄³×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇₁×#2859	C₂	semidirect
3. C₇₁×#2026	C₂	semidirect
6. C₇₁×#1835	C₂	semidirect
7. C₇₁×#1803	C₂	semidirect
9. C₄×C₄⋊C₈×C₇₁	C₂	semidirect
160. #29140	C₇₁	direct (N × H)
167. #2859	C₂×C₇₁	semidirect
169. #2026	C₂×C₇₁	semidirect
172. #1835	C₂×C₇₁	semidirect
173. #1803	C₂×C₇₁	semidirect
175. C₄×C₄⋊C₈	C₂×C₇₁	semidirect
176. C₇₁	#29140	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×#2859	2
C₄×#124×C₇₁	2
C₇₁×#2026	2
C₇₁×#2025	2
C₇₁×#1958	2
C₇₁×#1835	2
C₇₁×#1803	2
C₇₁×#1802	2
C₄×C₄⋊C₈×C₇₁	2
#29140	71

Subgroups — C71×#29140 #31850012

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#29140 #31850012