Subgroups — C₇₁×#23833 #31844705

Order 18176.

616 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇₁×#2960	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇₁×#2955	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#2103	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₁×#2102	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇₁×#2080	C₂	1	2	yes	maximal, complemented
6	C₇₁×#2053	C₂	1	1	yes	maximal, characteristic
7	C₇₁×#2050	C₂	1	1	yes	maximal, characteristic
8	C₇₁×#2031	C₂	1	2	yes	maximal, complemented
9	C₇₁×#1764	C₂	1	2	yes	maximal, complemented
10	C₇₁×#1757	C₂	1	2	yes	maximal, complemented
11	C₇₁×#1747	C₂	1	1	yes	maximal, characteristic, complemented
12	#575×C₇₁	C₂²	1	1	yes	characteristic
13	C₂×#131×C₇₁	C₂²	1	2	yes	—
14	#533×C₇₁	C₂²	1	1	yes	characteristic
15	#446×C₇₁	—	2	2	no	—
16	C₈×Q₈×C₇₁	—	2	2	no	—
17	#444×C₇₁	—	2	2	no	—
18	#443×C₇₁	—	2	2	no	—
19	#430×C₇₁	C₂²	1	1	yes	characteristic
20	#424×C₇₁	C₂²	1	1	yes	characteristic
21	#424×C₇₁	C₂²	1	1	yes	characteristic
22	#423×C₇₁	C₂²	1	1	yes	characteristic
23	#421×C₇₁	C₂²	1	2	yes	—
24	C₂×#104×C₇₁	C₂²	1	2	yes	—
25	#416×C₇₁	C₂²	1	2	yes	—
26	#413×C₇₁	C₂²	1	2	yes	—
27	#410×C₇₁	C₂²	1	2	yes	—
28	#405×C₇₁	C₂²	1	2	yes	—
29	#405×C₇₁	C₂²	1	1	yes	characteristic
30	#405×C₇₁	C₂²	1	1	yes	characteristic
31	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
32	#344×C₇₁	C₂²	1	2	yes	—
33	#344×C₇₁	C₂²	1	2	yes	—
34	#341×C₇₁	C₂²	1	2	yes	—
35	#339×C₇₁	C₂²	1	2	yes	—
36	#338×C₇₁	C₂²	1	1	yes	characteristic
37	H(3,ℤ/4)×C₇₁	C₂²	1	2	yes	—
38	H(3,ℤ/4)×C₇₁	C₂²	1	1	yes	characteristic
39	C₂×#41×C₇₁	C₂³	1	2	yes	—
40	#137×C₇₁	—	2	2	no	—
41	AGL(1,ℤ/8)×C₇₁	—	2	2	no	—
42	#135×C₇₁	—	2	2	no	—
43	#131×C₇₁	—	2	2	no	—
44	#131×C₇₁	—	2	2	no	—
45	#131×C₇₁	—	2	2	no	—
46	#131×C₇₁	—	2	2	no	—
47	C₂×M₁₆×C₇₁	—	2	2	no	—
48	C₂×M₁₆×C₇₁	C₂³	1	1	yes	characteristic
49	C₂×M₁₆×C₇₁	C₂³	1	2	yes	—
50	C₂×M₁₆×C₇₁	C₂³	1	2	yes	—
51	C₂×M₁₆×C₇₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₇₁	—	2	2	no	—
53	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
55	C₄⋊Q₈×C₇₁	—	2	1	no	—
56	#125×C₇₁	—	2	1	no	—
57	C₂²⋊Q₈×C₇₁	—	2	2	no	—
58	C₄×Q₈×C₇₁	—	2	2	no	—
59	#117×C₇₁	C₂³	1	2	yes	—
60	#117×C₇₁	C₂³	1	1	yes	characteristic
61	#117×C₇₁	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
63	#108×C₇₁	—	2	2	no	—
64	C₄⋊C₈×C₇₁	—	2	2	no	—
65	C₄⋊C₈×C₇₁	—	2	2	no	—
66	C₄⋊C₈×C₇₁	—	2	2	no	—
67	#104×C₇₁	—	2	4	no	—
68	#104×C₇₁	—	2	2	no	—
69	#104×C₇₁	—	2	2	no	—
70	#103×C₇₁	—	2	2	no	—
71	#102×C₇₁	—	2	2	no	—
72	#101×C₇₁	—	2	2	no	—
73	#100×C₇₁	—	2	2	no	—
74	#99×C₇₁	—	2	4	no	—
75	#97×C₇₁	—	2	1	no	—
76	#97×C₇₁	—	2	2	no	—
77	#97×C₇₁	D₄	1	2	yes	—
78	C₄×C₈×C₇₁	—	2	1	no	—
79	C₄×C₈×C₇₁	—	2	1	no	—
80	C₄×C₈×C₇₁	—	2	1	no	—
81	C₄×C₈×C₇₁	—	2	2	no	—
82	C₄×C₈×C₇₁	D₄	1	2	yes	—
83	#41×C₇₁	—	2	2	no	—
84	#41×C₇₁	—	2	2	no	—
85	#41×C₇₁	—	2	2	no	—
86	#41×C₇₁	—	2	2	no	—
87	C₂×Q₈×C₇₁	—	2	2	no	—
88	C₂×Q₈×C₇₁	#41	1	2	yes	—
89	C₂×D₄×C₇₁	—	2	2	no	—
90	C₂×D₄×C₇₁	#41	1	2	yes	—
91	C₂²×C₄×C₇₁	—	2	1	no	—
92	C₂²×C₄×C₇₁	—	2	2	no	—
93	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
94	Q₁₆×C₇₁	—	2	2	no	—
95	SD₁₆×C₇₁	—	2	4	no	—
96	D₈×C₇₁	—	2	2	no	—
97	M₁₆×C₇₁	—	2	2	no	—
98	M₁₆×C₇₁	—	2	2	no	—
99	M₁₆×C₇₁	—	2	2	no	—
100	M₁₆×C₇₁	—	2	2	no	—
101	M₁₆×C₇₁	—	2	2	no	—
102	M₁₆×C₇₁	—	2	2	no	—
103	M₁₆×C₇₁	—	4	2	no	—
104	M₁₆×C₇₁	—	4	2	no	—
105	C₂×C₈×C₇₁	—	2	1	no	—
106	C₂×C₈×C₇₁	—	2	1	no	—
107	C₂×C₈×C₇₁	—	2	2	no	—
108	C₂×C₈×C₇₁	—	2	2	no	—
109	C₂×C₈×C₇₁	—	2	2	no	—
110	C₂×C₈×C₇₁	—	4	2	no	—
111	C₂×C₈×C₇₁	—	4	2	no	—
112	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
113	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₇₁	#41	1	2	yes	—
116	C₂×C₈×C₇₁	#41	1	2	yes	—
117	C₄⋊C₄×C₇₁	—	2	4	no	—
118	C₄⋊C₄×C₇₁	—	2	2	no	—
119	C₄⋊C₄×C₇₁	—	2	1	no	—
120	C₄⋊C₄×C₇₁	—	2	1	no	—
121	C₄⋊C₄×C₇₁	—	2	2	no	—
122	C₂²⋊C₄×C₇₁	—	2	1	no	—
123	C₂²⋊C₄×C₇₁	—	2	2	no	—
124	C₂²⋊C₄×C₇₁	#41	1	2	yes	—
125	C₄²×C₇₁	—	2	1	no	—
126	C₄²×C₇₁	—	2	1	no	—
127	C₄²×C₇₁	—	2	2	no	—
128	C₄²×C₇₁	C₂×D₄	1	2	yes	—
129	C₄²×C₇₁	#41	1	1	yes	characteristic
130	C₄²×C₇₁	#41	1	1	yes	characteristic
131	C₂³×C₇₁	—	2	2	no	—
132	C₂³×C₇₁	C₂×#41	1	1	yes	characteristic
133	Q₈×C₇₁	—	2	2	no	—
134	Q₈×C₇₁	—	2	2	no	—
135	Q₈×C₇₁	—	4	4	no	—
136	D₄×C₇₁	—	2	2	no	—
137	D₄×C₇₁	—	2	2	no	—
138	D₄×C₇₁	—	4	2	no	—
139	D₄×C₇₁	—	4	2	no	—
140	C₂×C₄×C₇₁	—	2	1	no	—
141	C₂×C₄×C₇₁	—	2	1	no	—
142	C₂×C₄×C₇₁	—	2	2	no	—
143	C₂×C₄×C₇₁	—	2	2	no	—
144	C₂×C₄×C₇₁	—	2	2	no	—
145	C₂×C₄×C₇₁	—	4	2	no	—
146	C₂×C₄×C₇₁	—	4	2	no	—
147	C₂×C₄×C₇₁	—	4	2	no	—
148	C₂×C₄×C₇₁	—	4	2	no	—
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	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
153	C₈×C₇₁	—	2	2	no	—
154	C₈×C₇₁	—	4	1	no	—
155	C₈×C₇₁	—	4	1	no	—
156	C₈×C₇₁	—	4	2	no	—
157	C₈×C₇₁	—	4	2	no	—
158	C₈×C₇₁	#124	1	2	yes	—
159	C₈×C₇₁	#124	1	1	yes	characteristic
160	C₈×C₇₁	#124	1	1	yes	characteristic
161	C₂²×C₇₁	—	4	1	no	—
162	C₂²×C₇₁	—	4	2	no	—
163	C₂²×C₇₁	—	8	2	no	—
164	C₂²×C₇₁	#525	1	1	yes	characteristic
165	C₂²×C₇₁	#529	1	2	yes	—
166	C₄×C₇₁	—	2	2	no	—
167	C₄×C₇₁	—	4	1	no	—
168	C₄×C₇₁	—	4	2	no	—
169	C₄×C₇₁	—	4	2	no	—
170	C₄×C₇₁	—	4	2	no	—
171	C₄×C₇₁	—	8	2	no	—
172	C₄×C₇₁	#525	1	1	yes	characteristic
173	C₄×C₇₁	C₂×#124	1	1	yes	characteristic
174	C₄×C₇₁	#529	1	2	yes	—
175	#23833	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
176	C₂×C₇₁	—	2	1	no	—
177	C₂×C₇₁	—	2	2	no	—
178	C₂×C₇₁	—	8	2	no	—
179	C₂×C₇₁	#2398	1	1	yes	characteristic
180	#2960	C₂×C₇₁	1	1	yes	characteristic, complemented
181	#2955	C₂×C₇₁	1	1	yes	characteristic
182	#2103	C₂×C₇₁	1	1	yes	characteristic, complemented
183	#2102	C₂×C₇₁	1	1	yes	characteristic, complemented
184	#2080	C₂×C₇₁	1	2	yes	complemented
185	#2053	C₂×C₇₁	1	1	yes	characteristic
186	#2050	C₂×C₇₁	1	1	yes	characteristic
187	#2031	C₂×C₇₁	1	2	yes	complemented
188	#1764	C₂×C₇₁	1	2	yes	complemented
189	#1757	C₂×C₇₁	1	2	yes	complemented
190	#1747	C₂×C₇₁	1	1	yes	characteristic, complemented
191	C₇₁	#23833	1	1	yes	characteristic, direct product factor, Sylow
192	#575	C₂²×C₇₁	1	1	yes	characteristic
193	C₂×#131	C₂²×C₇₁	1	2	yes	—
194	#533	C₂²×C₇₁	1	1	yes	characteristic
195	#446	—	2	2	no	—
196	C₈×Q₈	—	2	2	no	—
197	#444	—	2	2	no	—
198	#443	—	2	2	no	—
199	#430	C₂²×C₇₁	1	1	yes	characteristic
200	#424	C₂²×C₇₁	1	1	yes	characteristic
201	#424	C₂²×C₇₁	1	1	yes	characteristic
202	#423	C₂²×C₇₁	1	1	yes	characteristic
203	#421	C₂²×C₇₁	1	2	yes	—
204	C₂×#104	C₂²×C₇₁	1	2	yes	—
205	#416	C₂²×C₇₁	1	2	yes	—
206	#413	C₂²×C₇₁	1	2	yes	—
207	#410	C₂²×C₇₁	1	2	yes	—
208	#405	C₂²×C₇₁	1	1	yes	characteristic
209	#405	C₂²×C₇₁	1	2	yes	—
210	#405	C₂²×C₇₁	1	1	yes	characteristic
211	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
212	#344	C₂²×C₇₁	1	2	yes	—
213	#344	C₂²×C₇₁	1	2	yes	—
214	#341	C₂²×C₇₁	1	2	yes	—
215	#339	C₂²×C₇₁	1	2	yes	—
216	#338	C₂²×C₇₁	1	1	yes	characteristic
217	H(3,ℤ/4)	C₂²×C₇₁	1	1	yes	characteristic
218	H(3,ℤ/4)	C₂²×C₇₁	1	2	yes	—
219	C₂×#41	C₂³×C₇₁	1	2	yes	—
220	#137	—	2	2	no	—
221	AGL(1,ℤ/8)	—	2	2	no	—
222	#135	—	2	2	no	—
223	#131	—	2	2	no	—
224	#131	—	2	2	no	—
225	#131	—	2	2	no	—
226	#131	—	2	2	no	—
227	C₂×M₁₆	—	2	2	no	—
228	C₂×M₁₆	C₂³×C₇₁	1	2	yes	—
229	C₂×M₁₆	C₂³×C₇₁	1	2	yes	—
230	C₂×M₁₆	C₂³×C₇₁	1	1	yes	characteristic
231	C₂×M₁₆	C₂³×C₇₁	1	1	yes	characteristic
232	C₂²×C₈	—	2	2	no	—
233	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
234	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
235	C₄⋊Q₈	—	2	1	no	—
236	#125	—	2	1	no	—
237	C₂²⋊Q₈	—	2	2	no	—
238	C₄×Q₈	—	2	2	no	—
239	#117	C₂³×C₇₁	1	2	yes	—
240	#117	C₂³×C₇₁	1	1	yes	characteristic
241	#117	C₂³×C₇₁	1	1	yes	characteristic
242	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
243	#108	—	2	2	no	—
244	C₄⋊C₈	—	2	2	no	—
245	C₄⋊C₈	—	2	2	no	—
246	C₄⋊C₈	—	2	2	no	—
247	#104	—	2	4	no	—
248	#104	—	2	2	no	—
249	#104	—	2	2	no	—
250	#103	—	2	2	no	—
251	#102	—	2	2	no	—
252	#101	—	2	2	no	—
253	#100	—	2	2	no	—
254	#99	—	2	4	no	—
255	#97	—	2	1	no	—
256	#97	—	2	2	no	—
257	#97	D₄×C₇₁	1	2	yes	—
258	C₄×C₈	—	2	1	no	—
259	C₄×C₈	—	2	1	no	—
260	C₄×C₈	—	2	1	no	—
261	C₄×C₈	—	2	2	no	—
262	C₄×C₈	D₄×C₇₁	1	2	yes	—
263	#41	—	2	2	no	—
264	#41	—	2	2	no	—
265	#41	—	2	2	no	—
266	#41	—	2	2	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	#41×C₇₁	1	2	yes	—
269	C₂×D₄	—	2	2	no	—
270	C₂×D₄	#41×C₇₁	1	2	yes	—
271	C₂²×C₄	—	2	1	no	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
274	Q₁₆	—	2	2	no	—
275	SD₁₆	—	2	4	no	—
276	D₈	—	2	2	no	—
277	M₁₆	—	2	2	no	—
278	M₁₆	—	2	2	no	—
279	M₁₆	—	2	2	no	—
280	M₁₆	—	2	2	no	—
281	M₁₆	—	2	2	no	—
282	M₁₆	—	2	2	no	—
283	M₁₆	—	4	2	no	—
284	M₁₆	—	4	2	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	4	2	no	—
291	C₂×C₈	—	4	2	no	—
292	C₂×C₈	C₂×D₄×C₇₁	1	2	yes	—
293	C₂×C₈	C₂×D₄×C₇₁	1	1	yes	characteristic
294	C₂×C₈	C₂×D₄×C₇₁	1	1	yes	characteristic
295	C₂×C₈	#41×C₇₁	1	2	yes	—
296	C₂×C₈	#41×C₇₁	1	2	yes	—
297	C₄⋊C₄	—	2	4	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	1	no	—
300	C₄⋊C₄	—	2	1	no	—
301	C₄⋊C₄	—	2	2	no	—
302	C₂²⋊C₄	—	2	1	no	—
303	C₂²⋊C₄	—	2	2	no	—
304	C₂²⋊C₄	#41×C₇₁	1	2	yes	—
305	C₄²	—	2	1	no	—
306	C₄²	—	2	1	no	—
307	C₄²	—	2	2	no	—
308	C₄²	C₂×D₄×C₇₁	1	2	yes	—
309	C₄²	#41×C₇₁	1	1	yes	characteristic
310	C₄²	#41×C₇₁	1	1	yes	characteristic
311	C₂³	—	2	2	no	—
312	C₂³	C₂×#41×C₇₁	1	1	yes	characteristic
313	Q₈	—	2	2	no	—
314	Q₈	—	2	2	no	—
315	Q₈	—	4	4	no	—
316	D₄	—	2	2	no	—
317	D₄	—	2	2	no	—
318	D₄	—	4	2	no	—
319	D₄	—	4	2	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
330	C₂×C₄	C₂×#41×C₇₁	1	2	yes	—
331	C₂×C₄	C₂×#41×C₇₁	1	2	yes	—
332	C₂×C₄	C₂×#41×C₇₁	1	1	yes	characteristic
333	C₈	—	2	2	no	—
334	C₈	—	4	1	no	—
335	C₈	—	4	1	no	—
336	C₈	—	4	2	no	—
337	C₈	—	4	2	no	—
338	C₈	#124×C₇₁	1	1	yes	characteristic
339	C₈	#124×C₇₁	1	2	yes	—
340	C₈	#124×C₇₁	1	1	yes	characteristic
341	C₂²	—	4	1	no	—
342	C₂²	—	4	2	no	—
343	C₂²	—	8	2	no	—
344	C₂²	#525×C₇₁	1	1	yes	characteristic
345	C₂²	#529×C₇₁	1	2	yes	—
346	C₄	—	2	2	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	2	no	—
349	C₄	—	4	2	no	—
350	C₄	—	4	2	no	—
351	C₄	—	8	2	no	—
352	C₄	#525×C₇₁	1	1	yes	characteristic
353	C₄	C₂×#124×C₇₁	1	1	yes	characteristic
354	C₄	#529×C₇₁	1	2	yes	—
355	C₂	—	2	1	no	—
356	C₂	—	2	2	no	—
357	C₂	—	8	2	no	—
358	C₂	C₇₁×#2398	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₇₁	#124
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₁
socle	C₂×C₇₁	#2398
prime radical	#575×C₇₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₇₁ ◁ C₂²×C₈×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₇₁×#2960	C₂	semidirect
3. C₇₁×#2103	C₂	semidirect
4. C₇₁×#2102	C₂	semidirect
5. C₇₁×#2080	C₂	semidirect
8. C₇₁×#2031	C₂	semidirect
9. C₇₁×#1764	C₂	semidirect
10. C₇₁×#1757	C₂	semidirect
11. C₇₁×#1747	C₂	semidirect
175. #23833	C₇₁	direct (N × H)
180. #2960	C₂×C₇₁	semidirect
182. #2103	C₂×C₇₁	semidirect
183. #2102	C₂×C₇₁	semidirect
184. #2080	C₂×C₇₁	semidirect
187. #2031	C₂×C₇₁	semidirect
188. #1764	C₂×C₇₁	semidirect
189. #1757	C₂×C₇₁	semidirect
190. #1747	C₂×C₇₁	semidirect
191. C₇₁	#23833	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×#2960	2
C₇₁×#2955	2
C₇₁×#2103	2
C₇₁×#2102	2
C₇₁×#2080	2
C₇₁×#2053	2
C₇₁×#2050	2
C₇₁×#2031	2
C₇₁×#1764	2
C₇₁×#1757	2
C₇₁×#1747	2
#23833	71

Subgroups — C71×#23833 #31844705

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#23833 #31844705