Subgroups — C₇₁×#33600 #31854472

Order 18176.

710 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#508×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₁×#3050	C₂	1	2	yes	maximal, complemented
4	C₇₁×#2217	C₂	1	4	yes	maximal, complemented
5	C₇₁×#2216	C₂	1	2	yes	maximal, complemented
6	C₇₁×#2215	C₂	1	2	yes	maximal, complemented
7	C₇₁×#2158	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇₁×#2138	C₂	1	2	yes	maximal, complemented
9	C₂×#135×C₇₁	C₂²	1	2	yes	—
10	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
11	#525×C₇₁	—	2	2	no	—
12	#508×C₇₁	—	2	4	no	—
13	C₂×Q₃₂×C₇₁	C₂²	1	1	yes	characteristic, complemented
14	C₂×SD₃₂×C₇₁	C₂²	1	2	yes	complemented
15	C₂×D₁₆×C₇₁	C₂²	1	1	yes	characteristic, complemented
16	C₂²×C₁₆×C₇₁	—	2	4	no	—
17	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₇₁	C₂²	1	2	yes	—
19	C₂²×C₁₆×C₇₁	C₂²	1	2	yes	—
20	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
21	#467×C₇₁	C₂²	1	4	yes	—
22	#466×C₇₁	C₂²	1	4	yes	—
23	#465×C₇₁	—	2	4	no	—
24	#427×C₇₁	C₂²	1	2	yes	—
25	#416×C₇₁	—	2	2	no	—
26	#367×C₇₁	C₂²	1	2	yes	complemented
27	#366×C₇₁	C₂²	1	2	yes	complemented
28	#358×C₇₁	C₂²	1	4	yes	complemented
29	#357×C₇₁	C₂²	1	4	yes	complemented
30	C₂×#41×C₇₁	—	2	2	no	—
31	C₂³×C₄×C₇₁	D₄	1	1	yes	characteristic
32	#135×C₇₁	—	2	4	no	—
33	#135×C₇₁	—	2	4	no	—
34	C₂×Q₁₆×C₇₁	C₂³	1	2	yes	—
35	C₂×SD₁₆×C₇₁	—	2	2	no	—
36	C₂×D₈×C₇₁	C₂³	1	2	yes	—
37	C₂²×C₈×C₇₁	—	2	2	no	—
38	C₂²×C₈×C₇₁	—	2	2	no	—
39	C₂²×C₈×C₇₁	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₇₁	D₄	1	2	yes	—
41	C₂²×C₈×C₇₁	C₂³	1	2	yes	—
42	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
43	#123×C₇₁	—	4	2	no	—
44	C₂²⋊Q₈×C₇₁	—	2	4	no	—
45	#121×C₇₁	—	2	4	no	—
46	#120×C₇₁	—	4	2	no	—
47	C₄×D₄×C₇₁	—	4	4	no	—
48	#117×C₇₁	—	2	2	no	—
49	Q₃₂×C₇₁	—	2	2	no	—
50	SD₃₂×C₇₁	—	2	4	no	—
51	D₁₆×C₇₁	—	2	2	no	—
52	C₂×C₁₆×C₇₁	—	2	2	no	—
53	C₂×C₁₆×C₇₁	—	2	4	no	—
54	C₂×C₁₆×C₇₁	—	2	4	no	—
55	C₂×C₁₆×C₇₁	D₄	1	4	yes	complemented
56	C₂×C₁₆×C₇₁	C₂³	1	2	yes	—
57	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
58	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
59	#107×C₇₁	C₂³	1	4	yes	—
60	#106×C₇₁	—	2	2	no	—
61	#103×C₇₁	—	2	4	no	—
62	#102×C₇₁	—	2	4	no	—
63	C₂⁴×C₇₁	D₈	1	1	yes	characteristic
64	#41×C₇₁	—	4	4	no	—
65	#41×C₇₁	—	4	4	no	—
66	C₂×Q₈×C₇₁	—	2	2	no	—
67	C₂×D₄×C₇₁	—	2	2	no	—
68	C₂×D₄×C₇₁	—	4	4	no	—
69	C₂×D₄×C₇₁	—	4	2	no	—
70	C₂²×C₄×C₇₁	—	2	2	no	—
71	C₂²×C₄×C₇₁	—	2	2	no	—
72	C₂²×C₄×C₇₁	—	4	2	no	—
73	C₂²×C₄×C₇₁	D₈	1	1	yes	characteristic
74	C₂²×C₄×C₇₁	D₈	1	2	yes	—
75	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
77	Q₁₆×C₇₁	—	2	2	no	—
78	Q₁₆×C₇₁	—	2	2	no	—
79	SD₁₆×C₇₁	—	4	4	no	—
80	D₈×C₇₁	—	2	2	no	—
81	D₈×C₇₁	—	2	2	no	—
82	C₂×C₈×C₇₁	—	2	2	no	—
83	C₂×C₈×C₇₁	—	2	4	no	—
84	C₂×C₈×C₇₁	—	2	2	no	—
85	C₂×C₈×C₇₁	—	2	2	no	—
86	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₇₁	#41	1	2	yes	—
90	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
91	C₄⋊C₄×C₇₁	—	2	4	no	—
92	C₄⋊C₄×C₇₁	—	4	4	no	—
93	C₂²⋊C₄×C₇₁	—	4	2	no	—
94	C₂²⋊C₄×C₇₁	—	4	4	no	—
95	C₂²⋊C₄×C₇₁	—	4	4	no	—
96	C₄²×C₇₁	—	4	2	no	—
97	C₁₆×C₇₁	—	2	2	no	—
98	C₁₆×C₇₁	C₂×D₄	1	2	yes	—
99	C₁₆×C₇₁	C₂×D₄	1	2	yes	—
100	C₂³×C₇₁	—	2	2	no	—
101	C₂³×C₇₁	—	2	4	no	—
102	C₂³×C₇₁	—	4	2	no	—
103	C₂³×C₇₁	C₂×D₈	1	1	yes	characteristic
104	C₂³×C₇₁	C₂×D₈	1	2	yes	—
105	Q₈×C₇₁	—	4	2	no	—
106	Q₈×C₇₁	—	4	2	no	—
107	D₄×C₇₁	—	4	2	no	—
108	D₄×C₇₁	—	4	2	no	—
109	D₄×C₇₁	—	8	4	no	—
110	D₄×C₇₁	—	8	8	no	—
111	C₂×C₄×C₇₁	—	2	2	no	—
112	C₂×C₄×C₇₁	—	2	2	no	—
113	C₂×C₄×C₇₁	—	2	2	no	—
114	C₂×C₄×C₇₁	—	2	4	no	—
115	C₂×C₄×C₇₁	—	4	4	no	—
116	C₂×C₄×C₇₁	—	4	2	no	—
117	C₂×C₄×C₇₁	—	8	4	no	—
118	C₂×C₄×C₇₁	#121	1	2	yes	—
119	C₂×C₄×C₇₁	C₂×D₈	1	2	yes	—
120	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₇₁	#135	1	2	yes	—
122	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
123	C₈×C₇₁	—	2	2	no	—
124	C₈×C₇₁	#121	1	2	yes	—
125	C₈×C₇₁	C₂²×D₄	1	1	yes	characteristic
126	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
127	C₂²×C₇₁	—	2	8	no	—
128	C₂²×C₇₁	—	2	2	no	—
129	C₂²×C₇₁	—	2	2	no	—
130	C₂²×C₇₁	—	2	2	no	—
131	C₂²×C₇₁	—	8	4	no	—
132	C₂²×C₇₁	—	8	2	no	—
133	C₂²×C₇₁	#466	1	2	yes	—
134	C₂²×C₇₁	#508	1	4	yes	complemented
135	C₂²×C₇₁	C₂²×D₈	1	1	yes	characteristic
136	C₄×C₇₁	—	2	2	no	—
137	C₄×C₇₁	—	8	4	no	—
138	C₄×C₇₁	—	8	2	no	—
139	C₄×C₇₁	#466	1	2	yes	—
140	C₄×C₇₁	C₂×#121	1	1	yes	characteristic
141	C₄×C₇₁	C₂×#135	1	1	yes	characteristic
142	#33600	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂×C₇₁	—	2	2	no	—
144	C₂×C₇₁	—	2	4	no	—
145	C₂×C₇₁	—	16	2	no	—
146	C₂×C₇₁	C₂×#466	1	1	yes	characteristic
147	C₂×C₇₁	C₂×#508	1	2	yes	—
148	C₂×#508	C₂×C₇₁	1	1	yes	characteristic, complemented
149	C₂³×C₁₆	C₂×C₇₁	1	1	yes	characteristic, complemented
150	#3050	C₂×C₇₁	1	2	yes	complemented
151	#2217	C₂×C₇₁	1	4	yes	complemented
152	#2216	C₂×C₇₁	1	2	yes	complemented
153	#2215	C₂×C₇₁	1	2	yes	complemented
154	#2158	C₂×C₇₁	1	1	yes	characteristic, complemented
155	#2138	C₂×C₇₁	1	2	yes	complemented
156	C₇₁	#33600	1	1	yes	characteristic, direct product factor, Sylow
157	C₂×#135	C₂²×C₇₁	1	2	yes	—
158	C₂³×C₈	C₂²×C₇₁	1	1	yes	characteristic
159	#525	—	2	2	no	—
160	#508	—	2	4	no	—
161	C₂×Q₃₂	C₂²×C₇₁	1	1	yes	characteristic, complemented
162	C₂×SD₃₂	C₂²×C₇₁	1	2	yes	complemented
163	C₂×D₁₆	C₂²×C₇₁	1	1	yes	characteristic, complemented
164	C₂²×C₁₆	—	2	4	no	—
165	C₂²×C₁₆	C₂²×C₇₁	1	1	yes	characteristic
166	C₂²×C₁₆	C₂²×C₇₁	1	2	yes	—
167	C₂²×C₁₆	C₂²×C₇₁	1	2	yes	—
168	C₂²×C₁₆	C₂²×C₇₁	1	1	yes	characteristic
169	#467	C₂²×C₇₁	1	4	yes	—
170	#466	C₂²×C₇₁	1	4	yes	—
171	#465	—	2	4	no	—
172	#427	C₂²×C₇₁	1	2	yes	—
173	#416	—	2	2	no	—
174	#367	C₂²×C₇₁	1	2	yes	complemented
175	#366	C₂²×C₇₁	1	2	yes	complemented
176	#358	C₂²×C₇₁	1	4	yes	complemented
177	#357	C₂²×C₇₁	1	4	yes	complemented
178	C₂×#41	—	2	2	no	—
179	C₂³×C₄	D₄×C₇₁	1	1	yes	characteristic
180	#135	—	2	4	no	—
181	#135	—	2	4	no	—
182	C₂×Q₁₆	C₂³×C₇₁	1	2	yes	—
183	C₂×SD₁₆	—	2	2	no	—
184	C₂×D₈	C₂³×C₇₁	1	2	yes	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	D₄×C₇₁	1	1	yes	characteristic
188	C₂²×C₈	D₄×C₇₁	1	2	yes	—
189	C₂²×C₈	C₂³×C₇₁	1	2	yes	—
190	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
191	#123	—	4	2	no	—
192	C₂²⋊Q₈	—	2	4	no	—
193	#121	—	2	4	no	—
194	#120	—	4	2	no	—
195	C₄×D₄	—	4	4	no	—
196	#117	—	2	2	no	—
197	Q₃₂	—	2	2	no	—
198	SD₃₂	—	2	4	no	—
199	D₁₆	—	2	2	no	—
200	C₂×C₁₆	—	2	2	no	—
201	C₂×C₁₆	—	2	4	no	—
202	C₂×C₁₆	—	2	4	no	—
203	C₂×C₁₆	D₄×C₇₁	1	4	yes	complemented
204	C₂×C₁₆	C₂³×C₇₁	1	2	yes	—
205	C₂×C₁₆	C₂³×C₇₁	1	1	yes	characteristic
206	C₂×C₁₆	C₂³×C₇₁	1	1	yes	characteristic
207	#107	C₂³×C₇₁	1	4	yes	—
208	#106	—	2	2	no	—
209	#103	—	2	4	no	—
210	#102	—	2	4	no	—
211	C₂⁴	D₈×C₇₁	1	1	yes	characteristic
212	#41	—	4	4	no	—
213	#41	—	4	4	no	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	—	4	4	no	—
217	C₂×D₄	—	4	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	4	2	no	—
221	C₂²×C₄	D₈×C₇₁	1	2	yes	—
222	C₂²×C₄	D₈×C₇₁	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
224	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
225	Q₁₆	—	2	2	no	—
226	Q₁₆	—	2	2	no	—
227	SD₁₆	—	4	4	no	—
228	D₈	—	2	2	no	—
229	D₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	C₂×D₄×C₇₁	1	2	yes	—
235	C₂×C₈	C₂×D₄×C₇₁	1	1	yes	characteristic
236	C₂×C₈	C₂×D₄×C₇₁	1	2	yes	—
237	C₂×C₈	#41×C₇₁	1	2	yes	—
238	C₂×C₈	C₂⁴×C₇₁	1	1	yes	characteristic
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	4	4	no	—
241	C₂²⋊C₄	—	4	2	no	—
242	C₂²⋊C₄	—	4	4	no	—
243	C₂²⋊C₄	—	4	4	no	—
244	C₄²	—	4	2	no	—
245	C₁₆	—	2	2	no	—
246	C₁₆	C₂×D₄×C₇₁	1	2	yes	—
247	C₁₆	C₂×D₄×C₇₁	1	2	yes	—
248	C₂³	—	2	2	no	—
249	C₂³	—	2	4	no	—
250	C₂³	—	4	2	no	—
251	C₂³	C₂×D₈×C₇₁	1	1	yes	characteristic
252	C₂³	C₂×D₈×C₇₁	1	2	yes	—
253	Q₈	—	4	2	no	—
254	Q₈	—	4	2	no	—
255	D₄	—	4	2	no	—
256	D₄	—	4	2	no	—
257	D₄	—	8	4	no	—
258	D₄	—	8	8	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	8	4	no	—
266	C₂×C₄	#121×C₇₁	1	2	yes	—
267	C₂×C₄	C₂×D₈×C₇₁	1	2	yes	—
268	C₂×C₄	C₂×D₈×C₇₁	1	1	yes	characteristic
269	C₂×C₄	#135×C₇₁	1	2	yes	—
270	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
271	C₈	—	2	2	no	—
272	C₈	#121×C₇₁	1	2	yes	—
273	C₈	C₂²×D₄×C₇₁	1	1	yes	characteristic
274	C₈	C₂×#41×C₇₁	1	1	yes	characteristic
275	C₂²	—	2	8	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	8	2	no	—
280	C₂²	—	8	4	no	—
281	C₂²	#466×C₇₁	1	2	yes	—
282	C₂²	#508×C₇₁	1	4	yes	complemented
283	C₂²	C₂²×D₈×C₇₁	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	8	2	no	—
286	C₄	—	8	4	no	—
287	C₄	#466×C₇₁	1	2	yes	—
288	C₄	C₂×#121×C₇₁	1	1	yes	characteristic
289	C₄	C₂×#135×C₇₁	1	1	yes	characteristic
290	C₂	—	2	2	no	—
291	C₂	—	2	4	no	—
292	C₂	—	16	2	no	—
293	C₂	C₂×#466×C₇₁	1	1	yes	characteristic
294	C₂	C₂×#508×C₇₁	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₇₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₇₁
socle	C₂²×C₇₁	C₂²×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇₁ ◁ C₂³×C₄×C₇₁ ◁ C₂³×C₈×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#508×C₇₁	C₂	semidirect
2. C₂³×C₁₆×C₇₁	C₂	semidirect
3. C₇₁×#3050	C₂	semidirect
4. C₇₁×#2217	C₂	semidirect
5. C₇₁×#2216	C₂	semidirect
6. C₇₁×#2215	C₂	semidirect
7. C₇₁×#2158	C₂	semidirect
8. C₇₁×#2138	C₂	semidirect
13. C₂×Q₃₂×C₇₁	C₂²	semidirect
14. C₂×SD₃₂×C₇₁	C₂²	semidirect
15. C₂×D₁₆×C₇₁	C₂²	semidirect
26. #367×C₇₁	C₂²	semidirect
27. #366×C₇₁	C₂²	semidirect
28. #358×C₇₁	C₂²	semidirect
29. #357×C₇₁	C₂²	semidirect
55. C₂×C₁₆×C₇₁	D₄	semidirect
134. C₂²×C₇₁	#508	semidirect
142. #33600	C₇₁	direct (N × H)
148. C₂×#508	C₂×C₇₁	semidirect
149. C₂³×C₁₆	C₂×C₇₁	semidirect
150. #3050	C₂×C₇₁	semidirect
151. #2217	C₂×C₇₁	semidirect
152. #2216	C₂×C₇₁	semidirect
153. #2215	C₂×C₇₁	semidirect
154. #2158	C₂×C₇₁	semidirect
155. #2138	C₂×C₇₁	semidirect
156. C₇₁	#33600	direct (N × H)
161. C₂×Q₃₂	C₂²×C₇₁	semidirect
162. C₂×SD₃₂	C₂²×C₇₁	semidirect
163. C₂×D₁₆	C₂²×C₇₁	semidirect
174. #367	C₂²×C₇₁	semidirect
175. #366	C₂²×C₇₁	semidirect
176. #358	C₂²×C₇₁	semidirect
177. #357	C₂²×C₇₁	semidirect
203. C₂×C₁₆	D₄×C₇₁	semidirect
282. C₂²	#508×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#508×C₇₁	2
C₂³×C₁₆×C₇₁	2
C₇₁×#3050	2
C₇₁×#2217	2
C₇₁×#2216	2
C₇₁×#2215	2
C₇₁×#2158	2
C₇₁×#2138	2
#33600	71

Subgroups — C71×#33600 #31854472

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#33600 #31854472