Subgroups — C₇₁×#23203 #31844075

Order 18176.

892 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#497×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₇₁×#2950	C₂	1	2	yes	maximal, complemented
3	C₇₁×#2949	C₂	1	2	yes	maximal
4	C₇₁×#1979	C₂	1	2	yes	maximal, complemented
5	C₇₁×#1977	C₂	1	2	yes	maximal
6	C₇₁×#1888	C₂	1	1	yes	maximal, characteristic
7	C₇₁×#1886	C₂	1	2	yes	maximal, complemented
8	C₇₁×#1885	C₂	1	1	yes	maximal, characteristic
9	C₇₁×#1773	C₂	1	2	yes	maximal, complemented
10	C₂²×Q₁₆×C₇₁	C₂²	1	2	yes	—
11	C₂²×SD₁₆×C₇₁	C₂²	1	2	yes	—
12	C₂×C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
13	C₂×#124×C₇₁	C₂²	1	1	yes	characteristic
14	#519×C₇₁	C₂²	1	2	yes	—
15	#519×C₇₁	C₂²	1	1	yes	characteristic
16	#518×C₇₁	C₂²	1	1	yes	characteristic
17	#497×C₇₁	C₄	1	8	yes	complemented
18	#441×C₇₁	—	2	4	no	—
19	#440×C₇₁	—	2	4	no	—
20	C₄×Q₁₆×C₇₁	—	2	4	no	—
21	C₄×SD₁₆×C₇₁	—	2	4	no	—
22	C₂×#108×C₇₁	C₂²	1	2	yes	—
23	#427×C₇₁	C₂²	1	2	yes	—
24	#419×C₇₁	C₂²	1	2	yes	—
25	#419×C₇₁	C₂²	1	2	yes	—
26	#419×C₇₁	C₂²	1	2	yes	—
27	#418×C₇₁	C₂²	1	2	yes	—
28	C₂×#101×C₇₁	C₂²	1	2	yes	—
29	C₂×#101×C₇₁	C₂²	1	1	yes	characteristic
30	C₂×#100×C₇₁	C₂²	1	1	yes	characteristic
31	#405×C₇₁	C₂²	1	2	yes	—
32	#405×C₇₁	C₂²	1	2	yes	—
33	C₂×#97×C₇₁	C₂²	1	1	yes	characteristic
34	#339×C₇₁	C₂²	1	2	yes	—
35	#339×C₇₁	C₂²	1	2	yes	—
36	C₂²×Q₈×C₇₁	C₂³	1	1	yes	characteristic
37	C₂²×Q₈×C₇₁	C₂³	1	2	yes	—
38	C₂²×D₄×C₇₁	C₂³	1	1	yes	characteristic
39	C₂×Q₁₆×C₇₁	—	2	2	no	—
40	C₂×Q₁₆×C₇₁	—	2	2	no	—
41	C₂×Q₁₆×C₇₁	—	2	4	no	—
42	C₂×Q₁₆×C₇₁	C₂×C₄	1	8	yes	complemented
43	C₂×SD₁₆×C₇₁	—	2	2	no	—
44	C₂×SD₁₆×C₇₁	—	2	2	no	—
45	C₂×SD₁₆×C₇₁	—	2	4	no	—
46	C₂×SD₁₆×C₇₁	C₂×C₄	1	8	yes	—
47	C₂×M₁₆×C₇₁	C₂³	1	2	yes	—
48	C₂×M₁₆×C₇₁	C₂³	1	2	yes	—
49	C₂²×C₈×C₇₁	C₂³	1	2	yes	—
50	C₄⋊Q₈×C₇₁	—	4	1	no	—
51	C₄⋊Q₈×C₇₁	C₂×C₄	1	4	yes	complemented
52	#124×C₇₁	—	4	1	no	—
53	#124×C₇₁	C₂×C₄	1	4	yes	—
54	C₄×Q₈×C₇₁	—	2	4	no	—
55	C₄×Q₈×C₇₁	—	2	4	no	—
56	C₄×Q₈×C₇₁	—	2	4	no	—
57	C₄×D₄×C₇₁	—	2	4	no	—
58	#117×C₇₁	—	4	2	no	—
59	#117×C₇₁	—	4	1	no	—
60	#117×C₇₁	C₂³	1	2	yes	—
61	#117×C₇₁	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₇₁	—	4	1	no	—
63	C₂×C₂²⋊C₄×C₇₁	—	4	1	no	—
64	C₂×C₂²⋊C₄×C₇₁	—	4	1	no	—
65	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
66	#108×C₇₁	—	2	4	no	—
67	#107×C₇₁	D₄	1	4	yes	—
68	#106×C₇₁	D₄	1	4	yes	—
69	#103×C₇₁	—	2	4	no	—
70	#103×C₇₁	—	2	4	no	—
71	#103×C₇₁	—	2	4	no	—
72	#102×C₇₁	—	2	4	no	—
73	#101×C₇₁	—	4	2	no	—
74	#101×C₇₁	—	4	1	no	—
75	#100×C₇₁	—	4	1	no	—
76	#97×C₇₁	—	2	2	no	—
77	#97×C₇₁	—	2	2	no	—
78	#97×C₇₁	C₂×C₄	1	4	yes	complemented
79	C₄×C₈×C₇₁	—	2	2	no	—
80	C₄×C₈×C₇₁	—	2	2	no	—
81	C₂⁴×C₇₁	—	2	1	no	—
82	C₂×Q₈×C₇₁	—	2	1	no	—
83	C₂×Q₈×C₇₁	—	2	2	no	—
84	C₂×Q₈×C₇₁	—	2	2	no	—
85	C₂×Q₈×C₇₁	—	2	2	no	—
86	C₂×Q₈×C₇₁	—	2	2	no	—
87	C₂×Q₈×C₇₁	—	2	2	no	—
88	C₂×Q₈×C₇₁	—	2	2	no	—
89	C₂×Q₈×C₇₁	—	2	2	no	—
90	C₂×Q₈×C₇₁	C₂²×C₄	1	2	yes	—
91	C₂×Q₈×C₇₁	C₂²×C₄	1	4	yes	—
92	C₂×D₄×C₇₁	—	2	2	no	—
93	C₂×D₄×C₇₁	—	2	2	no	—
94	C₂×D₄×C₇₁	—	2	1	no	—
95	C₂×D₄×C₇₁	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₇₁	—	2	1	no	—
97	C₂²×C₄×C₇₁	—	2	1	no	—
98	C₂²×C₄×C₇₁	—	2	2	no	—
99	C₂²×C₄×C₇₁	—	4	2	no	—
100	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
101	Q₁₆×C₇₁	—	2	8	no	—
102	Q₁₆×C₇₁	—	2	8	no	—
103	SD₁₆×C₇₁	—	2	8	no	—
104	SD₁₆×C₇₁	—	2	8	no	—
105	M₁₆×C₇₁	—	2	2	no	—
106	M₁₆×C₇₁	—	2	2	no	—
107	M₁₆×C₇₁	—	2	2	no	—
108	M₁₆×C₇₁	—	2	2	no	—
109	C₂×C₈×C₇₁	—	2	2	no	—
110	C₂×C₈×C₇₁	—	2	2	no	—
111	C₂×C₈×C₇₁	C₂²×C₄	1	4	yes	—
112	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
113	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
114	C₂×C₈×C₇₁	#41	1	4	yes	—
115	C₄⋊C₄×C₇₁	—	4	4	no	—
116	C₄⋊C₄×C₇₁	—	4	2	no	—
117	C₄⋊C₄×C₇₁	—	4	4	no	—
118	C₄⋊C₄×C₇₁	C₂×D₄	1	4	yes	—
119	C₄⋊C₄×C₇₁	C₂×D₄	1	4	yes	—
120	C₂²⋊C₄×C₇₁	—	2	2	no	—
121	C₂²⋊C₄×C₇₁	—	2	2	no	—
122	C₂²⋊C₄×C₇₁	—	4	4	no	—
123	C₂²⋊C₄×C₇₁	—	4	2	no	—
124	C₄²×C₇₁	—	2	2	no	—
125	C₄²×C₇₁	—	2	2	no	—
126	C₄²×C₇₁	—	2	1	no	—
127	C₄²×C₇₁	—	4	2	no	—
128	C₄²×C₇₁	—	4	4	no	—
129	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
130	C₂³×C₇₁	—	2	2	no	—
131	C₂³×C₇₁	—	4	1	no	—
132	C₂³×C₇₁	—	4	1	no	—
133	C₂³×C₇₁	—	8	2	no	—
134	C₂³×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
135	Q₈×C₇₁	—	2	4	no	—
136	Q₈×C₇₁	—	2	4	no	—
137	Q₈×C₇₁	—	2	4	no	—
138	Q₈×C₇₁	—	4	2	no	—
139	Q₈×C₇₁	—	4	2	no	—
140	Q₈×C₇₁	—	4	2	no	—
141	D₄×C₇₁	—	2	4	no	—
142	D₄×C₇₁	—	4	2	no	—
143	C₂×C₄×C₇₁	—	2	2	no	—
144	C₂×C₄×C₇₁	—	2	2	no	—
145	C₂×C₄×C₇₁	—	2	4	no	—
146	C₂×C₄×C₇₁	—	2	4	no	—
147	C₂×C₄×C₇₁	—	2	4	no	—
148	C₂×C₄×C₇₁	—	4	2	no	—
149	C₂×C₄×C₇₁	—	4	1	no	—
150	C₂×C₄×C₇₁	—	4	2	no	—
151	C₂×C₄×C₇₁	—	4	1	no	—
152	C₂×C₄×C₇₁	—	4	2	no	—
153	C₂×C₄×C₇₁	—	16	2	no	—
154	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
155	C₂×C₄×C₇₁	C₂²×D₄	1	2	yes	—
156	C₂×C₄×C₇₁	C₂×#41	1	2	yes	—
157	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
158	C₈×C₇₁	—	4	2	no	—
159	C₈×C₇₁	—	4	2	no	—
160	C₈×C₇₁	C₄×D₄	1	4	yes	—
161	C₈×C₇₁	C₄×D₄	1	4	yes	—
162	C₂²×C₇₁	—	2	2	no	—
163	C₂²×C₇₁	—	4	2	no	—
164	C₂²×C₇₁	—	8	2	no	—
165	C₂²×C₇₁	—	8	2	no	—
166	C₂²×C₇₁	—	16	1	no	—
167	C₂²×C₇₁	#518	1	1	yes	characteristic
168	C₂²×C₇₁	D₄²	1	1	yes	characteristic
169	C₂²×C₇₁	#559	1	1	yes	characteristic
170	C₄×C₇₁	—	4	2	no	—
171	C₄×C₇₁	—	4	2	no	—
172	C₄×C₇₁	—	4	4	no	—
173	C₄×C₇₁	—	4	4	no	—
174	C₄×C₇₁	—	4	4	no	—
175	C₄×C₇₁	C₂×C₄×D₄	1	2	yes	—
176	C₄×C₇₁	#550	1	2	yes	—
177	#23203	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
178	C₂×C₇₁	—	2	1	no	—
179	C₂×C₇₁	—	2	1	no	—
180	C₂×C₇₁	—	8	2	no	—
181	C₂×C₇₁	#2366	1	1	yes	characteristic
182	C₂×C₇₁	#3289	1	2	yes	—
183	C₂×#497	C₂×C₇₁	1	1	yes	characteristic
184	#2950	C₂×C₇₁	1	2	yes	complemented
185	#2949	C₂×C₇₁	1	2	yes	—
186	#1979	C₂×C₇₁	1	2	yes	complemented
187	#1977	C₂×C₇₁	1	2	yes	—
188	#1888	C₂×C₇₁	1	1	yes	characteristic
189	#1886	C₂×C₇₁	1	2	yes	complemented
190	#1885	C₂×C₇₁	1	1	yes	characteristic
191	#1773	C₂×C₇₁	1	2	yes	complemented
192	C₇₁	#23203	1	1	yes	characteristic, direct product factor, Sylow
193	C₂²×Q₁₆	C₂²×C₇₁	1	2	yes	—
194	C₂²×SD₁₆	C₂²×C₇₁	1	2	yes	—
195	C₂×C₄⋊Q₈	C₂²×C₇₁	1	1	yes	characteristic
196	C₂×#124	C₂²×C₇₁	1	1	yes	characteristic
197	#519	C₂²×C₇₁	1	2	yes	—
198	#519	C₂²×C₇₁	1	1	yes	characteristic
199	#518	C₂²×C₇₁	1	1	yes	characteristic
200	#497	C₄×C₇₁	1	8	yes	complemented
201	#441	—	2	4	no	—
202	#440	—	2	4	no	—
203	C₄×Q₁₆	—	2	4	no	—
204	C₄×SD₁₆	—	2	4	no	—
205	C₂×#108	C₂²×C₇₁	1	2	yes	—
206	#427	C₂²×C₇₁	1	2	yes	—
207	#419	C₂²×C₇₁	1	2	yes	—
208	#419	C₂²×C₇₁	1	2	yes	—
209	#419	C₂²×C₇₁	1	2	yes	—
210	#418	C₂²×C₇₁	1	2	yes	—
211	C₂×#101	C₂²×C₇₁	1	2	yes	—
212	C₂×#101	C₂²×C₇₁	1	1	yes	characteristic
213	C₂×#100	C₂²×C₇₁	1	1	yes	characteristic
214	#405	C₂²×C₇₁	1	2	yes	—
215	#405	C₂²×C₇₁	1	2	yes	—
216	C₂×#97	C₂²×C₇₁	1	1	yes	characteristic
217	#339	C₂²×C₇₁	1	2	yes	—
218	#339	C₂²×C₇₁	1	2	yes	—
219	C₂²×Q₈	C₂³×C₇₁	1	1	yes	characteristic
220	C₂²×Q₈	C₂³×C₇₁	1	2	yes	—
221	C₂²×D₄	C₂³×C₇₁	1	1	yes	characteristic
222	C₂×Q₁₆	—	2	2	no	—
223	C₂×Q₁₆	—	2	2	no	—
224	C₂×Q₁₆	—	2	4	no	—
225	C₂×Q₁₆	C₂×C₄×C₇₁	1	8	yes	complemented
226	C₂×SD₁₆	—	2	2	no	—
227	C₂×SD₁₆	—	2	2	no	—
228	C₂×SD₁₆	—	2	4	no	—
229	C₂×SD₁₆	C₂×C₄×C₇₁	1	8	yes	—
230	C₂×M₁₆	C₂³×C₇₁	1	2	yes	—
231	C₂×M₁₆	C₂³×C₇₁	1	2	yes	—
232	C₂²×C₈	C₂³×C₇₁	1	2	yes	—
233	C₄⋊Q₈	—	4	1	no	—
234	C₄⋊Q₈	C₂×C₄×C₇₁	1	4	yes	complemented
235	#124	—	4	1	no	—
236	#124	C₂×C₄×C₇₁	1	4	yes	—
237	C₄×Q₈	—	2	4	no	—
238	C₄×Q₈	—	2	4	no	—
239	C₄×Q₈	—	2	4	no	—
240	C₄×D₄	—	2	4	no	—
241	#117	—	4	2	no	—
242	#117	—	4	1	no	—
243	#117	C₂³×C₇₁	1	2	yes	—
244	#117	C₂³×C₇₁	1	2	yes	—
245	C₂×C₄⋊C₄	—	4	1	no	—
246	C₂×C₂²⋊C₄	—	4	1	no	—
247	C₂×C₂²⋊C₄	—	4	1	no	—
248	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
249	#108	—	2	4	no	—
250	#107	D₄×C₇₁	1	4	yes	—
251	#106	D₄×C₇₁	1	4	yes	—
252	#103	—	2	4	no	—
253	#103	—	2	4	no	—
254	#103	—	2	4	no	—
255	#102	—	2	4	no	—
256	#101	—	4	2	no	—
257	#101	—	4	1	no	—
258	#100	—	4	1	no	—
259	#97	—	2	2	no	—
260	#97	—	2	2	no	—
261	#97	C₂×C₄×C₇₁	1	4	yes	complemented
262	C₄×C₈	—	2	2	no	—
263	C₄×C₈	—	2	2	no	—
264	C₂⁴	—	2	1	no	—
265	C₂×Q₈	—	2	1	no	—
266	C₂×Q₈	—	2	2	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	—	2	2	no	—
269	C₂×Q₈	—	2	2	no	—
270	C₂×Q₈	—	2	2	no	—
271	C₂×Q₈	—	2	2	no	—
272	C₂×Q₈	—	2	2	no	—
273	C₂×Q₈	C₂²×C₄×C₇₁	1	2	yes	—
274	C₂×Q₈	C₂²×C₄×C₇₁	1	4	yes	—
275	C₂×D₄	—	2	2	no	—
276	C₂×D₄	—	2	2	no	—
277	C₂×D₄	—	2	1	no	—
278	C₂×D₄	C₂²×C₄×C₇₁	1	2	yes	—
279	C₂²×C₄	—	2	1	no	—
280	C₂²×C₄	—	2	1	no	—
281	C₂²×C₄	—	2	2	no	—
282	C₂²×C₄	—	4	2	no	—
283	C₂²×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
284	Q₁₆	—	2	8	no	—
285	Q₁₆	—	2	8	no	—
286	SD₁₆	—	2	8	no	—
287	SD₁₆	—	2	8	no	—
288	M₁₆	—	2	2	no	—
289	M₁₆	—	2	2	no	—
290	M₁₆	—	2	2	no	—
291	M₁₆	—	2	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	C₂²×C₄×C₇₁	1	4	yes	—
295	C₂×C₈	C₂×D₄×C₇₁	1	2	yes	—
296	C₂×C₈	C₂×D₄×C₇₁	1	2	yes	—
297	C₂×C₈	#41×C₇₁	1	4	yes	—
298	C₄⋊C₄	—	4	4	no	—
299	C₄⋊C₄	—	4	2	no	—
300	C₄⋊C₄	—	4	4	no	—
301	C₄⋊C₄	C₂×D₄×C₇₁	1	4	yes	—
302	C₄⋊C₄	C₂×D₄×C₇₁	1	4	yes	—
303	C₂²⋊C₄	—	2	2	no	—
304	C₂²⋊C₄	—	2	2	no	—
305	C₂²⋊C₄	—	4	4	no	—
306	C₂²⋊C₄	—	4	2	no	—
307	C₄²	—	2	2	no	—
308	C₄²	—	2	2	no	—
309	C₄²	—	2	1	no	—
310	C₄²	—	4	2	no	—
311	C₄²	—	4	4	no	—
312	C₄²	C₂²×C₄×C₇₁	1	2	yes	—
313	C₂³	—	2	2	no	—
314	C₂³	—	4	1	no	—
315	C₂³	—	4	1	no	—
316	C₂³	—	8	2	no	—
317	C₂³	H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
318	Q₈	—	2	4	no	—
319	Q₈	—	2	4	no	—
320	Q₈	—	2	4	no	—
321	Q₈	—	4	2	no	—
322	Q₈	—	4	2	no	—
323	Q₈	—	4	2	no	—
324	D₄	—	2	4	no	—
325	D₄	—	4	2	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	4	no	—
329	C₂×C₄	—	2	4	no	—
330	C₂×C₄	—	2	4	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	16	2	no	—
337	C₂×C₄	C₂³×C₄×C₇₁	1	1	yes	characteristic
338	C₂×C₄	C₂²×D₄×C₇₁	1	2	yes	—
339	C₂×C₄	C₂×#41×C₇₁	1	2	yes	—
340	C₂×C₄	H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
341	C₈	—	4	2	no	—
342	C₈	—	4	2	no	—
343	C₈	C₄×D₄×C₇₁	1	4	yes	—
344	C₈	C₄×D₄×C₇₁	1	4	yes	—
345	C₂²	—	2	2	no	—
346	C₂²	—	4	2	no	—
347	C₂²	—	8	2	no	—
348	C₂²	—	8	2	no	—
349	C₂²	—	16	1	no	—
350	C₂²	#518×C₇₁	1	1	yes	characteristic
351	C₂²	D₄²×C₇₁	1	1	yes	characteristic
352	C₂²	#559×C₇₁	1	1	yes	characteristic
353	C₄	—	4	2	no	—
354	C₄	—	4	2	no	—
355	C₄	—	4	4	no	—
356	C₄	—	4	4	no	—
357	C₄	—	4	4	no	—
358	C₄	C₂×C₄×D₄×C₇₁	1	2	yes	—
359	C₄	#550×C₇₁	1	2	yes	—
360	C₂	—	2	1	no	—
361	C₂	—	2	1	no	—
362	C₂	—	8	2	no	—
363	C₂	C₇₁×#2366	1	1	yes	characteristic
364	C₂	C₇₁×#3289	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇₁	D₄²
commutator	C₂×C₄	C₂³×C₄×C₇₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₁
prime radical	C₂²×D₄×C₇₁	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₂²×C₄×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇₁×#2950	C₂	semidirect
4. C₇₁×#1979	C₂	semidirect
7. C₇₁×#1886	C₂	semidirect
9. C₇₁×#1773	C₂	semidirect
17. #497×C₇₁	C₄	semidirect
42. C₂×Q₁₆×C₇₁	C₂×C₄	semidirect
51. C₄⋊Q₈×C₇₁	C₂×C₄	semidirect
78. #97×C₇₁	C₂×C₄	semidirect
177. #23203	C₇₁	direct (N × H)
184. #2950	C₂×C₇₁	semidirect
186. #1979	C₂×C₇₁	semidirect
189. #1886	C₂×C₇₁	semidirect
191. #1773	C₂×C₇₁	semidirect
192. C₇₁	#23203	direct (N × H)
200. #497	C₄×C₇₁	semidirect
225. C₂×Q₁₆	C₂×C₄×C₇₁	semidirect
234. C₄⋊Q₈	C₂×C₄×C₇₁	semidirect
261. #97	C₂×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×#497×C₇₁	2
C₇₁×#2950	2
C₇₁×#2949	2
C₇₁×#1979	2
C₇₁×#1977	2
C₇₁×#1888	2
C₇₁×#1886	2
C₇₁×#1885	2
C₇₁×#1773	2
#23203	71

Subgroups — C71×#23203 #31844075

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#23203 #31844075