Subgroups — #1332×D₇₁ #31907185

Order 18176.

406 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₁₆×D₇₁	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₆×D₇₁	C₂	1	1	yes	maximal, characteristic
3	#335×D₇₁	C₂	1	1	yes	maximal, characteristic
4	#22586305	C₂	1	1	yes	maximal, characteristic, complemented
5	#22586304	C₂	1	1	yes	maximal, characteristic, complemented
6	#22586303	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₁×#1332	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₁₆×D₇₁	C₄	1	1	yes	characteristic
9	C₄×Q₈×D₇₁	—	2	1	no	—
10	C₂×C₁₆×D₇₁	—	2	1	no	—
11	C₄×Dic₂₈₄	C₂²	1	1	yes	characteristic
12	Q₁₆×Dic₇₁	C₄	1	1	yes	characteristic
13	#17986093	C₄	1	1	yes	characteristic
14	C₄×#11765855	—	2	1	no	—
15	#17986091	—	2	1	no	—
16	#107×D₇₁	C₄	1	1	yes	characteristic
17	C₄⋊C₈×D₇₁	—	2	1	no	—
18	#103×D₇₁	—	2	1	no	—
19	C₄×C₈×D₇₁	C₂²	1	1	yes	characteristic
20	C₄×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
21	C₁₆×Dic₇₁	—	2	1	no	—
22	#17985752	C₂²	1	1	yes	characteristic
23	#17985751	—	2	1	no	—
24	C₄×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
25	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
26	#335×C₇₁	C₂²	1	1	yes	characteristic
27	C₂×Q₈×D₇₁	—	2	1	no	—
28	C₂×Dic₂₈₄	C₂×C₄	1	1	yes	characteristic
29	C₂×#11765855	—	2	1	no	—
30	Q₁₆×D₇₁	—	2	2	no	—
31	Q₁₆×D₇₁	C₈	1	4	yes	complemented
32	C₂×C₈×D₇₁	—	4	1	no	—
33	C₂×C₈×D₇₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₈×D₇₁	D₄	1	1	yes	characteristic
35	Q₈×Dic₇₁	—	2	1	no	—
36	C₄×Dic₁₄₂	—	2	1	no	—
37	C₄×Dic₁₄₂	—	4	1	no	—
38	#14556093	—	2	1	no	—
39	#14556093	—	4	1	no	—
40	C₄⋊C₄×D₇₁	—	2	1	no	—
41	C₄⋊C₄×D₇₁	—	4	1	no	—
42	C₄²×D₇₁	—	4	1	no	—
43	C₄²×D₇₁	D₄	1	1	yes	characteristic
44	C₂×C₇₁⋊C₁₆	—	2	1	no	—
45	C₁₆×D₇₁	—	2	4	no	—
46	#14556062	C₂×C₄	1	1	yes	characteristic
47	#14556061	—	2	1	no	—
48	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
49	Dic₇₁⋊C₈	—	4	1	no	—
50	#14556053	—	2	1	no	—
51	#14556052	—	2	1	no	—
52	#14556051	—	2	1	no	—
53	C₄×C₇₁⋊C₈	D₄	1	1	yes	characteristic
54	C₈×Dic₇₁	C₂×C₄	1	1	yes	characteristic
55	C₈×Dic₇₁	D₄	1	1	yes	characteristic
56	C₂×Q₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
57	C₄×Q₈×C₇₁	—	2	1	no	—
58	C₂×C₁₆×C₇₁	—	2	1	no	—
59	#107×C₇₁	C₂×C₄	1	1	yes	characteristic
60	C₄⋊C₈×C₇₁	—	2	1	no	—
61	#103×C₇₁	—	2	1	no	—
62	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
63	C₂×Dic₁₄₂	—	2	1	no	—
64	C₂×Dic₁₄₂	—	4	1	no	—
65	Q₈×D₇₁	—	2	4	no	—
66	Q₈×D₇₁	—	4	2	no	—
67	C₂×C₄×D₇₁	—	4	1	no	—
68	C₂×C₄×D₇₁	—	4	1	no	—
69	C₂×C₄×D₇₁	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₄×D₇₁	D₈	1	1	yes	characteristic
71	C₂×C₄×D₇₁	SD₁₆	1	1	yes	characteristic
72	Dic₂₈₄	—	2	1	no	—
73	Dic₂₈₄	C₂×C₈	1	2	yes	complemented
74	#11765855	—	2	4	no	—
75	C₂×C₇₁⋊C₈	—	4	1	no	—
76	C₂×C₇₁⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₇₁⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₈×D₇₁	—	2	2	no	—
79	C₈×D₇₁	—	8	2	no	—
80	C₈×D₇₁	C₂×C₈	1	2	yes	—
81	C₈×D₇₁	M₁₆	1	2	yes	—
82	C₄⋊Dic₇₁	—	2	1	no	—
83	C₄⋊Dic₇₁	—	4	1	no	—
84	Dic₇₁⋊C₄	—	4	1	no	—
85	Dic₇₁⋊C₄	—	4	1	no	—
86	Dic₇₁⋊C₄	—	4	1	no	—
87	C₄×Dic₇₁	—	4	1	no	—
88	C₄×Dic₇₁	—	4	1	no	—
89	C₄×Dic₇₁	—	4	1	no	—
90	C₄×Dic₇₁	C₂²⋊C₄	1	1	yes	characteristic
91	C₄×Dic₇₁	D₈	1	1	yes	characteristic
92	C₄×Dic₇₁	SD₁₆	1	1	yes	characteristic
93	C₇₁⋊C₁₆	—	2	2	no	—
94	C₂×Q₈×C₇₁	—	2	1	no	—
95	Q₁₆×C₇₁	—	2	1	no	—
96	Q₁₆×C₇₁	C₂×C₈	1	2	yes	complemented
97	C₂×C₈×C₇₁	—	4	1	no	—
98	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
100	C₄⋊C₄×C₇₁	—	2	1	no	—
101	C₄⋊C₄×C₇₁	—	4	1	no	—
102	C₄²×C₇₁	—	4	1	no	—
103	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
104	C₁₆×C₇₁	—	2	2	no	—
105	C₂²×D₇₁	#102	1	1	yes	characteristic
106	Q₈×C₇₁	—	2	2	no	—
107	Q₈×C₇₁	—	4	1	no	—
108	C₂×C₄×C₇₁	—	4	1	no	—
109	C₂×C₄×C₇₁	—	4	1	no	—
110	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×Dic₇₁	—	4	1	no	—
114	C₂×Dic₇₁	—	4	1	no	—
115	C₂×Dic₇₁	#102	1	1	yes	characteristic
116	C₂×Dic₇₁	#102	1	1	yes	characteristic
117	C₂×Dic₇₁	#102	1	1	yes	characteristic
118	C₄×D₇₁	—	2	2	no	—
119	C₄×D₇₁	—	4	4	no	—
120	C₄×D₇₁	—	8	2	no	—
121	C₄×D₇₁	C₂²⋊C₈	1	2	yes	—
122	C₄×D₇₁	#104	1	2	yes	—
123	C₄×D₇₁	SD₃₂	1	2	yes	—
124	C₄×D₇₁	Q₃₂	1	2	yes	—
125	Dic₁₄₂	—	2	2	no	—
126	Dic₁₄₂	—	4	1	no	—
127	Dic₁₄₂	—	4	4	no	—
128	C₈×C₇₁	—	2	1	no	—
129	C₈×C₇₁	—	8	1	no	—
130	C₈×C₇₁	C₂²×C₈	1	1	yes	characteristic
131	C₈×C₇₁	C₂×M₁₆	1	1	yes	characteristic
132	C₇₁⋊C₈	—	2	1	no	—
133	C₇₁⋊C₈	—	8	1	no	—
134	C₇₁⋊C₈	C₂²⋊C₈	1	2	yes	—
135	C₂²×C₇₁	C₂×#102	1	1	yes	characteristic
136	C₂×D₇₁	D₄⋊C₈	1	2	yes	—
137	C₂×D₇₁	#358	1	2	yes	—
138	C₂×D₇₁	#359	1	2	yes	—
139	C₄×C₇₁	—	2	1	no	—
140	C₄×C₇₁	—	4	2	no	—
141	C₄×C₇₁	—	8	1	no	—
142	C₄×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
143	C₄×C₇₁	C₂×#104	1	1	yes	characteristic
144	C₄×C₇₁	C₂×SD₃₂	1	1	yes	characteristic
145	C₄×C₇₁	C₂×Q₃₂	1	1	yes	characteristic
146	Dic₇₁	—	2	1	no	—
147	Dic₇₁	—	4	2	no	—
148	Dic₇₁	—	8	1	no	—
149	Dic₇₁	D₄⋊C₈	1	2	yes	—
150	Dic₇₁	#358	1	2	yes	—
151	C₂×#1332	—	71	1	no	maximal, Sylow
152	C₂×C₇₁	C₂×D₄⋊C₈	1	1	yes	characteristic
153	C₂×C₇₁	C₂×#358	1	1	yes	characteristic
154	C₂×C₇₁	C₂×#359	1	1	yes	characteristic
155	D₇₁	#1332	1	4	yes	direct product factor
156	C₂×C₄×Q₁₆	—	71	1	no	—
157	C₂×C₄×C₁₆	—	71	1	no	—
158	C₂×#335	—	71	1	no	—
159	#1332	—	71	1	no	—
160	#1332	—	71	1	no	—
161	#1332	—	71	1	no	—
162	#1332	D₇₁	1	1	yes	characteristic, direct product factor
163	C₇₁	C₂×#1332	1	1	yes	characteristic, complemented, Sylow
164	C₂²×Q₁₆	—	71	1	no	—
165	C₂×C₄×Q₈	—	142	1	no	—
166	C₂²×C₁₆	—	142	1	no	—
167	C₄×Q₁₆	—	71	1	no	—
168	C₄×Q₁₆	—	71	1	no	—
169	C₄×Q₁₆	—	71	1	no	—
170	C₄×Q₁₆	—	142	1	no	—
171	C₄×Q₁₆	—	142	1	no	—
172	C₄×Q₁₆	C₂×D₇₁	1	1	yes	characteristic
173	C₂×#107	—	71	1	no	—
174	C₂×C₄⋊C₈	—	142	1	no	—
175	C₂×#103	—	142	1	no	—
176	C₂×C₄×C₈	—	71	1	no	—
177	C₄×C₁₆	—	71	1	no	—
178	C₄×C₁₆	—	142	1	no	—
179	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
180	#335	—	71	1	no	—
181	#335	—	142	1	no	—
182	#335	C₂×D₇₁	1	1	yes	characteristic
183	C₂²×Q₈	—	142	1	no	—
184	C₂×Q₁₆	—	71	4	no	—
185	C₂×Q₁₆	—	71	1	no	—
186	C₂×Q₁₆	—	142	2	no	—
187	C₂×Q₁₆	—	142	1	no	—
188	C₂×Q₁₆	C₄×D₇₁	1	1	yes	characteristic
189	C₂²×C₈	—	284	1	no	—
190	C₂²×C₈	—	71	1	no	—
191	C₂²×C₈	—	71	1	no	—
192	C₄×Q₈	—	2	1	no	—
193	C₄×Q₈	—	284	1	no	—
194	C₄×Q₈	—	284	1	no	—
195	C₄×Q₈	—	142	1	no	—
196	C₄×Q₈	—	142	1	no	—
197	C₄×Q₈	—	142	1	no	—
198	C₂×C₄⋊C₄	—	284	1	no	—
199	C₂×C₄⋊C₄	—	142	1	no	—
200	C₂×C₄²	—	284	1	no	—
201	C₂×C₄²	—	71	1	no	—
202	C₂×C₁₆	—	2	1	no	—
203	C₂×C₁₆	—	142	4	no	—
204	C₂×C₁₆	—	142	1	no	—
205	#107	—	71	1	no	—
206	#107	—	142	1	no	—
207	#107	C₄×D₇₁	1	1	yes	characteristic
208	C₄⋊C₈	—	2	1	no	—
209	C₄⋊C₈	—	284	1	no	—
210	C₄⋊C₈	—	142	1	no	—
211	#103	—	2	1	no	—
212	#103	—	142	1	no	—
213	#103	—	142	1	no	—
214	#103	—	142	1	no	—
215	C₄×C₈	—	71	1	no	—
216	C₄×C₈	—	71	1	no	—
217	C₄×C₈	—	71	1	no	—
218	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
219	C₂×Q₈	—	2	1	no	—
220	C₂×Q₈	—	284	1	no	—
221	C₂×Q₈	—	284	2	no	—
222	C₂×Q₈	—	142	4	no	—
223	C₂×Q₈	—	142	1	no	—
224	C₂²×C₄	—	284	1	no	—
225	C₂²×C₄	—	284	1	no	—
226	C₂²×C₄	—	71	1	no	—
227	C₂²×C₄	—	71	1	no	—
228	C₂²×C₄	—	71	1	no	—
229	Q₁₆	—	2	1	no	—
230	Q₁₆	—	71	2	no	—
231	Q₁₆	—	142	4	no	—
232	Q₁₆	—	142	1	no	—
233	Q₁₆	C₈×D₇₁	1	2	yes	complemented
234	C₂×C₈	—	4	1	no	—
235	C₂×C₈	—	284	1	no	—
236	C₂×C₈	—	568	2	no	—
237	C₂×C₈	—	71	1	no	—
238	C₂×C₈	—	71	2	no	—
239	C₂×C₈	—	71	2	no	—
240	C₂×C₈	—	71	1	no	—
241	C₂×C₈	—	142	2	no	—
242	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
243	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	4	1	no	—
246	C₄⋊C₄	—	284	1	no	—
247	C₄⋊C₄	—	284	1	no	—
248	C₄⋊C₄	—	284	1	no	—
249	C₄⋊C₄	—	284	1	no	—
250	C₄⋊C₄	—	142	1	no	—
251	C₄²	—	4	1	no	—
252	C₄²	—	284	1	no	—
253	C₄²	—	284	1	no	—
254	C₄²	—	284	1	no	—
255	C₄²	—	71	1	no	—
256	C₄²	—	71	1	no	—
257	C₄²	—	71	1	no	—
258	C₄²	D₄×D₇₁	1	1	yes	characteristic
259	C₁₆	—	2	2	no	—
260	C₁₆	—	142	2	no	—
261	C₂³	—	71	1	no	—
262	Q₈	—	2	2	no	—
263	Q₈	—	4	1	no	—
264	Q₈	—	284	4	no	—
265	Q₈	—	284	1	no	—
266	Q₈	—	142	2	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	1	no	—
269	C₂×C₄	—	284	4	no	—
270	C₂×C₄	—	284	1	no	—
271	C₂×C₄	—	284	1	no	—
272	C₂×C₄	—	568	2	no	—
273	C₂×C₄	—	71	1	no	—
274	C₂×C₄	—	71	1	no	—
275	C₂×C₄	—	71	2	no	—
276	C₂×C₄	—	71	2	no	—
277	C₂×C₄	—	71	2	no	—
278	C₂×C₄	—	71	2	no	—
279	C₂×C₄	—	71	1	no	—
280	C₂×C₄	—	142	2	no	—
281	C₂×C₄	C₂²⋊C₄×D₇₁	1	1	yes	characteristic
282	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
283	C₂×C₄	SD₁₆×D₇₁	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	8	1	no	—
286	C₈	—	568	1	no	—
287	C₈	—	71	2	no	—
288	C₈	—	142	1	no	—
289	C₈	C₂×C₈×D₇₁	1	1	yes	characteristic
290	C₈	M₁₆×D₇₁	1	1	yes	characteristic
291	C₂²	—	71	2	no	—
292	C₂²	—	71	2	no	—
293	C₂²	—	71	2	no	—
294	C₂²	#102×D₇₁	1	1	yes	characteristic
295	C₄	—	2	1	no	—
296	C₄	—	4	2	no	—
297	C₄	—	8	1	no	—
298	C₄	—	284	2	no	—
299	C₄	—	568	1	no	—
300	C₄	—	71	2	no	—
301	C₄	—	71	2	no	—
302	C₄	—	142	1	no	—
303	C₄	C₂²⋊C₈×D₇₁	1	1	yes	characteristic
304	C₄	#104×D₇₁	1	1	yes	characteristic
305	C₄	SD₃₂×D₇₁	1	1	yes	characteristic
306	C₄	Q₃₂×D₇₁	1	1	yes	characteristic
307	C₂	—	71	4	no	—
308	C₂	D₄⋊C₈×D₇₁	1	1	yes	characteristic
309	C₂	#358×D₇₁	1	1	yes	characteristic
310	C₂	#359×D₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₇₁
commutator	C₈×C₇₁	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₇₁
Fitting	C₇₁×#1332	C₂
socle	C₂²×C₇₁	C₂×#102
prime radical	C₂²×D₇₁	#102
cosocle	C₄×C₈×C₇₁	C₂³
hypercenter	#1332	D₇₁
hypocenter	C₇₁	C₂×#1332
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #1332

Complements

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

N	H ≅ G/N	type
4. #22586305	C₂	semidirect
5. #22586304	C₂	semidirect
6. #22586303	C₂	semidirect
7. C₇₁×#1332	C₂	semidirect
31. Q₁₆×D₇₁	C₈	semidirect
73. Dic₂₈₄	C₂×C₈	semidirect
96. Q₁₆×C₇₁	C₂×C₈	semidirect
155. D₇₁	#1332	direct (N × H)
162. #1332	D₇₁	direct (N × H)
163. C₇₁	C₂×#1332	semidirect
233. Q₁₆	C₈×D₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×Q₁₆×D₇₁	2
C₄×C₁₆×D₇₁	2
#335×D₇₁	2
#22586305	2
#22586304	2
#22586303	2
C₇₁×#1332	2
C₂×#1332	71

Subgroups — #1332×D71 #31907185

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1332×D₇₁ #31907185