Subgroups — C₄×Q₃₂×D₇₁ #32150723

Order 18176.

678 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₃₂×D₇₁	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₁₆×D₇₁	C₂	1	2	yes	maximal
3	C₄×Dic₅₆₈	C₂	1	1	yes	maximal, characteristic, complemented
4	Q₃₂×Dic₇₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#22590198	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#14556074	C₂	1	2	yes	maximal, complemented
7	#22590196	C₂	1	2	yes	maximal, complemented
8	#366×D₇₁	C₂	1	1	yes	maximal, characteristic
9	#358×D₇₁	C₂	1	2	yes	maximal
10	C₄×C₁₆×D₇₁	C₂	1	1	yes	maximal, characteristic
11	C₄×Q₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₇₁	C₂²	1	2	yes	—
13	C₄×Q₈×D₇₁	—	2	2	no	—
14	C₂×Dic₅₆₈	C₂²	1	1	yes	characteristic
15	C₂×#14556074	C₂²	1	2	yes	—
16	Q₃₂×D₇₁	C₄	1	8	yes	direct product factor
17	C₂×C₁₆×D₇₁	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₇₁	C₂²	1	1	yes	characteristic
19	C₄×Dic₂₈₄	C₂²	1	2	yes	—
20	Q₁₆×Dic₇₁	C₂²	1	2	yes	—
21	#17986093	C₂²	1	2	yes	—
22	C₄×#11765855	—	2	2	no	—
23	#17986091	—	2	2	no	—
24	#107×D₇₁	C₂²	1	2	yes	—
25	#103×D₇₁	—	2	2	no	—
26	C₄×C₈×D₇₁	C₂²	1	1	yes	characteristic
27	#17985825	C₂²	1	1	yes	characteristic
28	#17985824	C₂²	1	2	yes	—
29	#17985804	C₂²	1	2	yes	—
30	#17985803	C₂²	1	2	yes	—
31	#17985802	C₂²	1	2	yes	—
32	C₄×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₇₁	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₇₁	C₂²	1	1	yes	characteristic
35	C₂×Q₃₂×C₇₁	C₂²	1	1	yes	characteristic
36	C₄×Q₁₆×C₇₁	C₂²	1	2	yes	—
37	#366×C₇₁	C₂²	1	1	yes	characteristic
38	#358×C₇₁	C₂²	1	2	yes	—
39	C₄×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
40	C₂×Q₈×D₇₁	—	2	2	no	—
41	C₂×Dic₂₈₄	C₂³	1	2	yes	—
42	C₂×#11765855	—	2	2	no	—
43	Q₁₆×D₇₁	—	2	4	no	—
44	Q₁₆×D₇₁	C₂×C₄	1	8	yes	—
45	C₂×C₈×D₇₁	D₄	1	1	yes	characteristic
46	C₂×C₈×D₇₁	C₂³	1	1	yes	characteristic
47	Q₈×Dic₇₁	—	2	2	no	—
48	C₄×Dic₁₄₂	—	2	2	no	—
49	C₄×Dic₁₄₂	—	4	2	no	—
50	#14556093	—	2	2	no	—
51	#14556093	—	4	2	no	—
52	C₄⋊C₄×D₇₁	—	2	2	no	—
53	C₄⋊C₄×D₇₁	—	4	2	no	—
54	C₄²×D₇₁	—	4	2	no	—
55	C₄²×D₇₁	D₄	1	1	yes	characteristic
56	Dic₅₆₈	C₂×C₄	1	4	yes	complemented
57	#14556074	C₂×C₄	1	8	yes	complemented
58	C₂×C₇₁⋊C₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₇₁⋊C₁₆	C₂³	1	1	yes	characteristic
60	C₁₆×D₇₁	—	2	2	no	—
61	C₁₆×D₇₁	C₂×C₄	1	4	yes	—
62	#14556062	C₂³	1	2	yes	—
63	#14556061	—	2	2	no	—
64	#14556053	—	2	2	no	—
65	#14556052	—	2	2	no	—
66	#14556051	—	2	2	no	—
67	C₄×C₇₁⋊C₈	D₄	1	1	yes	characteristic
68	C₈×Dic₇₁	D₄	1	1	yes	characteristic
69	C₈×Dic₇₁	C₂³	1	1	yes	characteristic
70	C₂×Q₁₆×C₇₁	C₂³	1	2	yes	—
71	C₄×Q₈×C₇₁	—	2	2	no	—
72	Q₃₂×C₇₁	C₂×C₄	1	4	yes	complemented
73	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
75	#107×C₇₁	C₂³	1	2	yes	—
76	#103×C₇₁	—	2	2	no	—
77	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
78	C₂×Dic₁₄₂	—	2	2	no	—
79	C₂×Dic₁₄₂	—	4	2	no	—
80	Q₈×D₇₁	—	2	8	no	—
81	Q₈×D₇₁	—	4	4	no	—
82	C₂×C₄×D₇₁	—	4	2	no	—
83	C₂×C₄×D₇₁	—	4	2	no	—
84	C₂×C₄×D₇₁	D₈	1	1	yes	characteristic
85	C₂×C₄×D₇₁	D₈	1	1	yes	characteristic
86	C₂×C₄×D₇₁	C₂×D₄	1	1	yes	characteristic
87	Dic₂₈₄	—	2	2	no	—
88	Dic₂₈₄	C₂²×C₄	1	4	yes	—
89	#11765855	—	2	8	no	—
90	C₂×C₇₁⋊C₈	C₂×D₄	1	1	yes	characteristic
91	C₂×C₇₁⋊C₈	C₂×D₄	1	1	yes	characteristic
92	C₈×D₇₁	—	2	2	no	—
93	C₈×D₇₁	C₂²×C₄	1	2	yes	—
94	C₈×D₇₁	#41	1	2	yes	—
95	C₄⋊Dic₇₁	—	2	2	no	—
96	C₄⋊Dic₇₁	—	4	2	no	—
97	Dic₇₁⋊C₄	—	4	2	no	—
98	Dic₇₁⋊C₄	—	4	2	no	—
99	Dic₇₁⋊C₄	—	4	2	no	—
100	C₄×Dic₇₁	—	4	2	no	—
101	C₄×Dic₇₁	—	4	2	no	—
102	C₄×Dic₇₁	—	4	2	no	—
103	C₄×Dic₇₁	D₈	1	1	yes	characteristic
104	C₄×Dic₇₁	D₈	1	1	yes	characteristic
105	C₄×Dic₇₁	C₂×D₄	1	1	yes	characteristic
106	C₇₁⋊C₁₆	—	2	1	no	—
107	C₇₁⋊C₁₆	C₂²×C₄	1	2	yes	—
108	C₂×Q₈×C₇₁	—	2	2	no	—
109	Q₁₆×C₇₁	—	2	2	no	—
110	Q₁₆×C₇₁	C₂²×C₄	1	4	yes	—
111	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₇₁	—	2	2	no	—
114	C₄⋊C₄×C₇₁	—	4	2	no	—
115	C₄²×C₇₁	—	4	2	no	—
116	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
117	C₁₆×C₇₁	—	2	1	no	—
118	C₁₆×C₇₁	C₂²×C₄	1	2	yes	—
119	C₂²×D₇₁	C₂×D₈	1	1	yes	characteristic
120	Q₈×C₇₁	—	2	4	no	—
121	Q₈×C₇₁	—	4	2	no	—
122	C₂×C₄×C₇₁	—	4	2	no	—
123	C₂×C₄×C₇₁	—	4	2	no	—
124	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
127	C₂×Dic₇₁	—	4	2	no	—
128	C₂×Dic₇₁	—	4	2	no	—
129	C₂×Dic₇₁	C₂×D₈	1	1	yes	characteristic
130	C₂×Dic₇₁	C₂×D₈	1	1	yes	characteristic
131	C₂×Dic₇₁	C₂×D₈	1	1	yes	characteristic
132	C₄×D₇₁	—	2	2	no	—
133	C₄×D₇₁	—	4	8	no	—
134	C₄×D₇₁	—	8	4	no	—
135	C₄×D₇₁	Q₃₂	1	4	yes	direct product factor
136	C₄×D₇₁	C₄×D₄	1	2	yes	—
137	C₄×D₇₁	#135	1	2	yes	—
138	Dic₁₄₂	—	2	4	no	—
139	Dic₁₄₂	—	4	2	no	—
140	Dic₁₄₂	—	4	8	no	—
141	C₈×C₇₁	—	2	1	no	—
142	C₈×C₇₁	C₂³×C₄	1	1	yes	characteristic
143	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
144	C₇₁⋊C₈	—	2	1	no	—
145	C₇₁⋊C₈	C₄×D₄	1	2	yes	—
146	C₂²×C₇₁	C₂²×D₈	1	1	yes	characteristic
147	C₂×D₇₁	C₄×D₈	1	2	yes	—
148	C₂×D₇₁	C₂×Q₃₂	1	2	yes	—
149	C₂×D₇₁	#508	1	2	yes	—
150	C₄×C₇₁	—	2	1	no	—
151	C₄×C₇₁	—	4	4	no	—
152	C₄×C₇₁	—	8	2	no	—
153	C₄×C₇₁	C₂×Q₃₂	1	2	yes	complemented
154	C₄×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
155	C₄×C₇₁	C₂×#135	1	1	yes	characteristic
156	Dic₇₁	—	2	1	no	—
157	Dic₇₁	—	4	4	no	—
158	Dic₇₁	—	8	2	no	—
159	Dic₇₁	C₄×D₈	1	2	yes	—
160	Dic₇₁	C₂×Q₃₂	1	2	yes	complemented
161	C₂×C₄×Q₃₂	—	71	1	no	maximal, Sylow
162	C₂×C₇₁	C₂×C₄×D₈	1	1	yes	characteristic
163	C₂×C₇₁	C₂²×Q₃₂	1	1	yes	characteristic
164	C₂×C₇₁	C₂×#508	1	1	yes	characteristic
165	D₇₁	C₄×Q₃₂	1	4	yes	direct product factor
166	C₂²×Q₃₂	—	71	1	no	—
167	C₂×C₄×Q₁₆	—	71	2	no	—
168	C₄×Q₃₂	—	71	1	no	—
169	C₄×Q₃₂	—	71	2	no	—
170	C₄×Q₃₂	—	71	1	no	—
171	C₄×Q₃₂	—	71	2	no	—
172	C₄×Q₃₂	—	71	1	no	—
173	C₄×Q₃₂	D₇₁	1	1	yes	characteristic, direct product factor
174	C₂×#366	—	71	1	no	—
175	C₂×#358	—	71	2	no	—
176	C₂×C₄×C₁₆	—	71	1	no	—
177	C₇₁	C₂×C₄×Q₃₂	1	1	yes	characteristic, complemented, Sylow
178	C₂²×Q₁₆	—	71	2	no	—
179	C₂×C₄×Q₈	—	142	2	no	—
180	C₂×Q₃₂	—	71	2	no	—
181	C₂×Q₃₂	—	71	8	no	—
182	C₂×Q₃₂	—	71	1	no	—
183	C₂×Q₃₂	C₂×D₇₁	1	1	yes	characteristic
184	C₂²×C₁₆	—	71	1	no	—
185	C₂²×C₁₆	—	71	1	no	—
186	C₄×Q₁₆	—	71	2	no	—
187	C₄×Q₁₆	—	71	2	no	—
188	C₄×Q₁₆	—	71	2	no	—
189	C₄×Q₁₆	—	142	2	no	—
190	C₄×Q₁₆	—	142	2	no	—
191	C₄×Q₁₆	C₂×D₇₁	1	2	yes	—
192	C₂×#107	—	71	2	no	—
193	C₂×#103	—	142	2	no	—
194	C₂×C₄×C₈	—	71	1	no	—
195	#366	—	71	1	no	—
196	#366	—	71	2	no	—
197	#366	C₂×D₇₁	1	1	yes	characteristic
198	#358	—	71	2	no	—
199	#358	—	71	2	no	—
200	#358	—	71	2	no	—
201	#358	C₂×D₇₁	1	2	yes	—
202	C₄×C₁₆	—	71	1	no	—
203	C₄×C₁₆	—	71	1	no	—
204	C₄×C₁₆	—	71	1	no	—
205	C₄×C₁₆	C₂×D₇₁	1	1	yes	characteristic
206	C₂²×Q₈	—	142	2	no	—
207	C₂×Q₁₆	—	71	8	no	—
208	C₂×Q₁₆	—	71	2	no	—
209	C₂×Q₁₆	—	142	4	no	—
210	C₂×Q₁₆	—	142	2	no	—
211	C₂×Q₁₆	C₂²×D₇₁	1	2	yes	—
212	C₂²×C₈	—	71	1	no	—
213	C₂²×C₈	—	71	1	no	—
214	C₄×Q₈	—	2	2	no	—
215	C₄×Q₈	—	284	2	no	—
216	C₄×Q₈	—	284	2	no	—
217	C₄×Q₈	—	142	2	no	—
218	C₄×Q₈	—	142	2	no	—
219	C₄×Q₈	—	142	2	no	—
220	C₂×C₄⋊C₄	—	284	2	no	—
221	C₂×C₄⋊C₄	—	142	2	no	—
222	C₂×C₄²	—	284	2	no	—
223	C₂×C₄²	—	71	1	no	—
224	Q₃₂	—	71	8	no	—
225	Q₃₂	—	71	4	no	—
226	Q₃₂	C₄×D₇₁	1	4	yes	direct product factor
227	C₂×C₁₆	—	71	1	no	—
228	C₂×C₁₆	—	71	4	no	—
229	C₂×C₁₆	—	71	1	no	—
230	C₂×C₁₆	—	142	2	no	—
231	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
232	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
233	#107	—	71	2	no	—
234	#107	—	142	2	no	—
235	#107	C₂²×D₇₁	1	2	yes	—
236	#103	—	2	2	no	—
237	#103	—	142	2	no	—
238	#103	—	142	2	no	—
239	#103	—	142	2	no	—
240	C₄×C₈	—	71	1	no	—
241	C₄×C₈	—	71	1	no	—
242	C₄×C₈	—	71	1	no	—
243	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	284	2	no	—
246	C₂×Q₈	—	284	4	no	—
247	C₂×Q₈	—	142	8	no	—
248	C₂×Q₈	—	142	2	no	—
249	C₂²×C₄	—	284	2	no	—
250	C₂²×C₄	—	284	2	no	—
251	C₂²×C₄	—	71	1	no	—
252	C₂²×C₄	—	71	1	no	—
253	C₂²×C₄	—	71	1	no	—
254	Q₁₆	—	2	2	no	—
255	Q₁₆	—	71	4	no	—
256	Q₁₆	—	142	8	no	—
257	Q₁₆	—	142	2	no	—
258	Q₁₆	C₂×C₄×D₇₁	1	4	yes	—
259	C₂×C₈	—	71	1	no	—
260	C₂×C₈	—	71	2	no	—
261	C₂×C₈	—	71	2	no	—
262	C₂×C₈	—	71	1	no	—
263	C₂×C₈	—	142	2	no	—
264	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
265	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	284	2	no	—
269	C₄⋊C₄	—	284	2	no	—
270	C₄⋊C₄	—	284	2	no	—
271	C₄⋊C₄	—	284	2	no	—
272	C₄⋊C₄	—	142	2	no	—
273	C₄²	—	4	2	no	—
274	C₄²	—	284	2	no	—
275	C₄²	—	284	2	no	—
276	C₄²	—	284	2	no	—
277	C₄²	—	71	1	no	—
278	C₄²	—	71	1	no	—
279	C₄²	—	71	1	no	—
280	C₄²	D₄×D₇₁	1	1	yes	characteristic
281	C₁₆	—	2	1	no	—
282	C₁₆	—	71	2	no	—
283	C₁₆	—	142	1	no	—
284	C₁₆	C₂×C₄×D₇₁	1	2	yes	—
285	C₂³	—	71	1	no	—
286	Q₈	—	2	4	no	—
287	Q₈	—	4	2	no	—
288	Q₈	—	284	8	no	—
289	Q₈	—	284	2	no	—
290	Q₈	—	142	4	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	284	8	no	—
294	C₂×C₄	—	284	2	no	—
295	C₂×C₄	—	284	2	no	—
296	C₂×C₄	—	568	4	no	—
297	C₂×C₄	—	71	1	no	—
298	C₂×C₄	—	71	1	no	—
299	C₂×C₄	—	71	4	no	—
300	C₂×C₄	—	71	2	no	—
301	C₂×C₄	—	71	2	no	—
302	C₂×C₄	—	71	1	no	—
303	C₂×C₄	—	142	2	no	—
304	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
305	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
306	C₂×C₄	C₂×D₄×D₇₁	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	71	2	no	—
309	C₈	—	142	1	no	—
310	C₈	C₂²×C₄×D₇₁	1	1	yes	characteristic
311	C₈	#41×D₇₁	1	1	yes	characteristic
312	C₂²	—	71	2	no	—
313	C₂²	—	71	2	no	—
314	C₂²	—	71	2	no	—
315	C₂²	C₂×D₈×D₇₁	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	4	4	no	—
318	C₄	—	8	2	no	—
319	C₄	—	284	4	no	—
320	C₄	—	568	2	no	—
321	C₄	—	71	2	no	—
322	C₄	—	71	2	no	—
323	C₄	—	142	1	no	—
324	C₄	Q₃₂×D₇₁	1	2	yes	direct product factor
325	C₄	C₄×D₄×D₇₁	1	1	yes	characteristic
326	C₄	#135×D₇₁	1	1	yes	characteristic
327	C₂	—	71	4	no	—
328	C₂	C₄×D₈×D₇₁	1	1	yes	characteristic
329	C₂	C₂×Q₃₂×D₇₁	1	1	yes	characteristic
330	C₂	#508×D₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₇₁
commutator	C₈×C₇₁	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₄×Q₃₂×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₈
prime radical	C₂²×D₇₁	C₂×D₈
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	C₄×Q₃₂	D₇₁
hypocenter	C₇₁	C₂×C₄×Q₃₂
LCS	G ▷ C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
3. C₄×Dic₅₆₈	C₂	semidirect
4. Q₃₂×Dic₇₁	C₂	semidirect
5. #22590198	C₂	semidirect
6. C₄×#14556074	C₂	semidirect
7. #22590196	C₂	semidirect
11. C₄×Q₃₂×C₇₁	C₂	semidirect
16. Q₃₂×D₇₁	C₄	direct (N × H)
56. Dic₅₆₈	C₂×C₄	semidirect
57. #14556074	C₂×C₄	semidirect
72. Q₃₂×C₇₁	C₂×C₄	semidirect
135. C₄×D₇₁	Q₃₂	direct (N × H)
153. C₄×C₇₁	C₂×Q₃₂	semidirect
160. Dic₇₁	C₂×Q₃₂	semidirect
165. D₇₁	C₄×Q₃₂	direct (N × H)
173. C₄×Q₃₂	D₇₁	direct (N × H)
177. C₇₁	C₂×C₄×Q₃₂	semidirect
226. Q₃₂	C₄×D₇₁	direct (N × H)
324. C₄	Q₃₂×D₇₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×Q₃₂×D₇₁	2
C₄×Q₁₆×D₇₁	2
C₄×Dic₅₆₈	2
Q₃₂×Dic₇₁	2
#22590198	2
C₄×#14556074	2
#22590196	2
#366×D₇₁	2
#358×D₇₁	2
C₄×C₁₆×D₇₁	2
C₄×Q₃₂×C₇₁	2
C₂×C₄×Q₃₂	71

Subgroups — C4×Q32×D71 #32150723

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×Q₃₂×D₇₁ #32150723