Subgroups — #31891749

Order 18176.

438 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22596909	C₂	1	1	yes	maximal, characteristic
2	#22588357	C₂	1	1	yes	maximal, characteristic
3	#22587116	C₂	1	1	yes	maximal, characteristic, complemented
4	#22586723	C₂	1	1	yes	maximal, characteristic
5	#22586241	C₂	1	1	yes	maximal, characteristic, complemented
6	#22586180	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₁×#1567	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14556136	—	2	1	no	—
9	C₄×#11765865	C₂²	1	1	yes	characteristic
10	#17986052	C₂²	1	1	yes	characteristic
11	#17986027	—	2	1	no	—
12	#17986024	—	2	2	no	—
13	#17986021	—	2	1	no	—
14	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×D₇₁	—	2	1	no	—
16	C₂×#14556054	—	2	1	no	—
17	C₄×C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
18	#17985760	—	2	1	no	—
19	#17985751	—	2	1	no	—
20	#17985751	—	2	1	no	—
21	#17985731	C₄	1	1	yes	characteristic, complemented
22	#17985731	C₄	1	1	yes	characteristic, complemented
23	#17985728	C₄	1	1	yes	characteristic, complemented
24	#17985728	C₄	1	1	yes	characteristic, complemented
25	#423×C₇₁	C₂²	1	1	yes	characteristic
26	#423×C₇₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
28	#335×C₇₁	—	2	2	no	—
29	C₂×#11765865	D₄	1	1	yes	characteristic
30	#14556136	—	4	2	no	—
31	#14556136	—	4	2	no	—
32	C₂×#11765848	—	2	2	no	—
33	C₂²×C₇₁⋊C₈	—	2	1	no	—
34	C₂×C₈×D₇₁	—	2	2	no	—
35	C₄×Dic₁₄₂	C₂×C₄	1	1	yes	characteristic
36	C₄×Dic₁₄₂	Q₈	1	1	yes	characteristic
37	C₄×D₂₈₄	C₂×C₄	1	1	yes	characteristic
38	C₄×D₂₈₄	Q₈	1	1	yes	characteristic
39	C₄×C₂²⋊D₇₁	—	2	1	no	—
40	C₄×C₂²⋊D₇₁	—	2	1	no	—
41	#14556087	D₄	1	1	yes	characteristic
42	#14556085	—	2	1	no	—
43	C₂×C₄⋊Dic₇₁	—	2	1	no	—
44	C₂×C₄×Dic₇₁	—	2	1	no	—
45	C₄²×D₇₁	—	2	1	no	—
46	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
47	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
48	C₄⋊(C₇₁⋊C₈)	C₂×C₄	1	1	yes	characteristic
49	C₄⋊(C₇₁⋊C₈)	D₄	1	1	yes	characteristic
50	C₄⋊(C₇₁⋊C₈)	D₄	1	1	yes	characteristic
51	Dic₇₁⋊C₈	—	2	2	no	—
52	Dic₇₁⋊C₈	—	2	2	no	—
53	#14556054	—	4	2	no	—
54	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
55	C₄×C₇₁⋊C₈	—	2	1	no	—
56	C₇₁⋊#95	—	2	1	no	—
57	C₂×M₁₆×C₇₁	—	2	1	no	—
58	C₂×M₁₆×C₇₁	—	2	1	no	—
59	C₂²×C₈×C₇₁	—	2	1	no	—
60	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
61	C₄⋊C₈×C₇₁	—	2	1	no	—
62	C₄⋊C₈×C₇₁	—	2	1	no	—
63	C₄⋊C₈×C₇₁	—	2	1	no	—
64	C₄⋊C₈×C₇₁	D₄	1	2	yes	—
65	C₄×C₈×C₇₁	—	2	1	no	—
66	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
67	C₄×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
68	#11765865	—	2	1	no	—
69	#11765865	—	2	1	no	—
70	#11765865	D₈	1	2	yes	—
71	#11765865	Q₁₆	1	2	yes	—
72	C₂×Dic₁₄₂	C₄⋊C₄	1	1	yes	characteristic
73	C₂×D₂₈₄	C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₂²⋊D₇₁	—	2	1	no	—
75	C₂²×Dic₇₁	—	4	1	no	—
76	C₂×C₄×D₇₁	—	2	1	no	—
77	C₂×C₄×D₇₁	—	2	1	no	—
78	C₂×C₄×D₇₁	—	2	1	no	—
79	#11765848	—	4	4	no	—
80	C₂×C₇₁⋊C₈	—	2	1	no	—
81	C₂×C₇₁⋊C₈	—	2	1	no	—
82	C₂×C₇₁⋊C₈	—	2	1	no	—
83	C₂×C₇₁⋊C₈	—	2	1	no	—
84	C₂×C₇₁⋊C₈	—	4	1	no	—
85	C₂×C₇₁⋊C₈	—	4	1	no	—
86	C₈×D₇₁	—	4	4	no	—
87	C₄⋊Dic₇₁	—	2	2	no	—
88	C₄⋊Dic₇₁	—	4	1	no	—
89	C₄⋊Dic₇₁	C₄⋊C₄	1	1	yes	characteristic
90	C₄⋊Dic₇₁	C₄⋊C₄	1	1	yes	characteristic
91	Dic₇₁⋊C₄	—	2	1	no	—
92	Dic₇₁⋊C₄	—	2	1	no	—
93	C₂²⋊Dic₇₁	—	2	1	no	—
94	(C₂×D₇₁)⋊C₄	—	2	1	no	—
95	(C₂×D₇₁)⋊C₄	—	2	1	no	—
96	C₄×Dic₇₁	—	2	1	no	—
97	C₄×Dic₇₁	—	2	1	no	—
98	C₄×Dic₇₁	—	2	1	no	—
99	C₄×Dic₇₁	—	4	1	no	—
100	C₄×Dic₇₁	—	4	1	no	—
101	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
103	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
104	M₁₆×C₇₁	—	4	1	no	—
105	M₁₆×C₇₁	—	4	1	no	—
106	M₁₆×C₇₁	—	4	1	no	—
107	M₁₆×C₇₁	—	4	1	no	—
108	C₂×C₈×C₇₁	—	2	1	no	—
109	C₂×C₈×C₇₁	—	2	1	no	—
110	C₂×C₈×C₇₁	—	2	2	no	—
111	C₂×C₈×C₇₁	—	4	2	no	—
112	C₂×C₈×C₇₁	—	4	1	no	—
113	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
117	C₂³×C₇₁	#123	1	1	yes	characteristic
118	C₂²×D₇₁	—	2	1	no	—
119	C₂×C₄×C₇₁	—	2	1	no	—
120	C₂×C₄×C₇₁	—	2	1	no	—
121	C₂×C₄×C₇₁	—	4	1	no	—
122	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₇₁	#120	1	1	yes	characteristic
126	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
129	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
130	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
132	C₂²⋊D₇₁	—	2	2	no	—
133	C₂²⋊D₇₁	—	2	2	no	—
134	C₂×Dic₇₁	—	2	1	no	—
135	C₂×Dic₇₁	—	2	1	no	—
136	C₂×Dic₇₁	—	2	1	no	—
137	C₂×Dic₇₁	—	4	2	no	—
138	C₂×Dic₇₁	—	8	1	no	—
139	C₂×Dic₇₁	—	8	1	no	—
140	D₂₈₄	—	2	1	no	—
141	D₂₈₄	#107	1	2	yes	—
142	C₄×D₇₁	—	2	2	no	—
143	C₄×D₇₁	—	2	2	no	—
144	C₄×D₇₁	—	4	1	no	—
145	C₄×D₇₁	—	4	1	no	—
146	C₄×D₇₁	—	8	1	no	—
147	Dic₁₄₂	—	2	1	no	—
148	Dic₁₄₂	#107	1	2	yes	—
149	C₈×C₇₁	—	2	2	no	—
150	C₈×C₇₁	—	4	1	no	—
151	C₈×C₇₁	—	4	2	no	—
152	C₈×C₇₁	—	8	1	no	—
153	C₇₁⋊C₈	—	4	1	no	—
154	C₇₁⋊C₈	—	4	1	no	—
155	C₇₁⋊C₈	—	4	1	no	—
156	C₇₁⋊C₈	—	4	1	no	—
157	C₂²×C₇₁	—	2	1	no	—
158	C₂²×C₇₁	—	2	1	no	—
159	C₂²×C₇₁	#385	1	1	yes	characteristic
160	C₂²×C₇₁	#480	1	1	yes	characteristic
161	C₂²×C₇₁	#484	1	1	yes	characteristic
162	C₂×D₇₁	—	2	2	no	—
163	C₂×D₇₁	—	4	1	no	—
164	C₂×D₇₁	—	4	1	no	—
165	C₄×C₇₁	—	2	1	no	—
166	C₄×C₇₁	—	2	1	no	—
167	C₄×C₇₁	—	2	1	no	—
168	C₄×C₇₁	C₂×#107	1	1	yes	characteristic
169	C₄×C₇₁	#428	1	1	yes	characteristic
170	C₄×C₇₁	#443	1	1	yes	characteristic
171	C₄×C₇₁	#444	1	1	yes	characteristic
172	C₄×C₇₁	#447	1	1	yes	characteristic
173	C₄×C₇₁	#451	1	1	yes	characteristic
174	Dic₇₁	—	2	2	no	—
175	Dic₇₁	—	4	1	no	—
176	Dic₇₁	—	4	1	no	—
177	Dic₇₁	—	8	2	no	—
178	#8902	—	71	1	no	maximal, Sylow
179	C₂×C₇₁	—	2	1	no	—
180	C₂×C₇₁	—	2	1	no	—
181	C₂×C₇₁	#1852	1	1	yes	characteristic
182	C₂×C₇₁	#1861	1	1	yes	characteristic
183	C₂×C₇₁	#1866	1	1	yes	characteristic
184	D₇₁	—	4	2	no	—
185	#2906	—	71	1	no	—
186	#1806	—	71	1	no	—
187	#1571	—	71	1	no	—
188	#1567	D₇₁	1	1	yes	characteristic, complemented
189	#1476	—	71	1	no	—
190	#1303	—	71	1	no	—
191	#1279	—	71	1	no	—
192	C₇₁	#8902	1	1	yes	characteristic, complemented, Sylow
193	C₂×#131	—	142	1	no	—
194	C₄×#41	—	71	1	no	—
195	#431	—	71	1	no	—
196	#424	—	142	1	no	—
197	#424	—	142	2	no	—
198	#423	—	142	1	no	—
199	#423	C₂×D₇₁	1	1	yes	characteristic
200	#423	C₂×D₇₁	1	1	yes	characteristic
201	C₂×C₄⋊C₈	—	71	1	no	—
202	C₂×C₄⋊C₈	—	142	1	no	—
203	C₂×#104	—	142	1	no	—
204	C₂×C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
205	C₄×C₄⋊C₄	—	71	1	no	—
206	#339	—	142	1	no	—
207	#335	—	2	2	no	—
208	#335	—	142	1	no	—
209	#335	—	142	1	no	—
210	Q₈⋊C₈	—	71	1	no	—
211	Q₈⋊C₈	—	71	1	no	—
212	D₄⋊C₈	—	71	1	no	—
213	D₄⋊C₈	—	71	1	no	—
214	C₂×#41	—	71	1	no	—
215	#131	—	284	2	no	—
216	#131	—	284	2	no	—
217	C₂×M₁₆	—	2	1	no	—
218	C₂×M₁₆	—	2	1	no	—
219	C₂×M₁₆	—	142	2	no	—
220	C₂²×C₈	—	2	1	no	—
221	C₂²×C₈	—	142	2	no	—
222	C₂²×C₈	—	142	1	no	—
223	C₄×Q₈	—	71	1	no	—
224	C₄×Q₈	—	71	1	no	—
225	C₄×D₄	—	71	1	no	—
226	C₄×D₄	—	71	1	no	—
227	C₄×D₄	—	142	1	no	—
228	C₄×D₄	—	142	1	no	—
229	#117	—	71	1	no	—
230	#117	—	142	1	no	—
231	C₂×C₄⋊C₄	—	142	1	no	—
232	C₂×C₄²	—	142	1	no	—
233	C₂×C₄²	—	142	1	no	—
234	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
235	C₄⋊C₈	—	2	1	no	—
236	C₄⋊C₈	—	2	1	no	—
237	C₄⋊C₈	—	2	1	no	—
238	C₄⋊C₈	—	71	1	no	—
239	C₄⋊C₈	—	71	1	no	—
240	C₄⋊C₈	—	71	1	no	—
241	C₄⋊C₈	—	71	1	no	—
242	C₄⋊C₈	—	142	2	no	—
243	C₄⋊C₈	—	142	2	no	—
244	C₄⋊C₈	—	142	1	no	—
245	C₄⋊C₈	C₂²⋊D₇₁	1	2	yes	—
246	#104	—	284	2	no	—
247	C₂²⋊C₈	—	142	1	no	—
248	C₄×C₈	—	2	1	no	—
249	C₄×C₈	—	142	1	no	—
250	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
251	C₄×C₈	C₄×D₇₁	1	1	yes	characteristic
252	#95	—	142	1	no	—
253	#41	—	71	2	no	—
254	#41	—	71	2	no	—
255	#41	—	142	1	no	—
256	#41	—	142	1	no	—
257	C₂×Q₈	—	71	1	no	—
258	C₂×D₄	—	71	1	no	—
259	C₂×D₄	—	142	1	no	—
260	C₂²×C₄	—	284	1	no	—
261	C₂²×C₄	—	142	1	no	—
262	C₂²×C₄	—	142	1	no	—
263	C₂²×C₄	—	142	1	no	—
264	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
265	C₂²×C₄	#11765865	1	1	yes	characteristic
266	C₂²×C₄	#11765866	1	1	yes	characteristic
267	M₁₆	—	4	1	no	—
268	M₁₆	—	4	1	no	—
269	M₁₆	—	4	1	no	—
270	M₁₆	—	4	1	no	—
271	M₁₆	—	284	4	no	—
272	C₂×C₈	—	2	1	no	—
273	C₂×C₈	—	2	1	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	4	2	no	—
276	C₂×C₈	—	4	1	no	—
277	C₂×C₈	—	284	4	no	—
278	C₂×C₈	—	284	1	no	—
279	C₂×C₈	—	284	1	no	—
280	C₂×C₈	—	142	1	no	—
281	C₂×C₈	—	142	1	no	—
282	C₂×C₈	—	142	1	no	—
283	C₂×C₈	—	142	1	no	—
284	C₄⋊C₄	—	284	1	no	—
285	C₄⋊C₄	—	71	1	no	—
286	C₄⋊C₄	—	71	1	no	—
287	C₄⋊C₄	—	142	2	no	—
288	C₄⋊C₄	—	142	1	no	—
289	C₄⋊C₄	—	142	1	no	—
290	C₂²⋊C₄	—	142	1	no	—
291	C₂²⋊C₄	—	142	1	no	—
292	C₂²⋊C₄	—	142	1	no	—
293	C₄²	—	284	1	no	—
294	C₄²	—	284	1	no	—
295	C₄²	—	142	1	no	—
296	C₄²	—	142	1	no	—
297	C₄²	—	142	1	no	—
298	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
299	C₄²	D₄×D₇₁	1	1	yes	characteristic
300	C₄²	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
301	C₄²	Q₈×D₇₁	1	1	yes	characteristic
302	C₂³	—	142	1	no	—
303	C₂³	#14556113	1	1	yes	characteristic
304	Q₈	—	71	2	no	—
305	Q₈	—	142	1	no	—
306	D₄	—	71	2	no	—
307	D₄	—	142	2	no	—
308	D₄	—	142	2	no	—
309	D₄	—	142	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	4	1	no	—
313	C₂×C₄	—	284	1	no	—
314	C₂×C₄	—	284	1	no	—
315	C₂×C₄	—	284	2	no	—
316	C₂×C₄	—	568	1	no	—
317	C₂×C₄	—	568	1	no	—
318	C₂×C₄	—	568	1	no	—
319	C₂×C₄	—	142	1	no	—
320	C₂×C₄	—	142	1	no	—
321	C₂×C₄	—	142	1	no	—
322	C₂×C₄	—	142	2	no	—
323	C₂×C₄	—	142	2	no	—
324	C₂×C₄	C₄⋊C₄×D₇₁	1	1	yes	characteristic
325	C₂×C₄	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
326	C₂×C₄	#14556090	1	1	yes	characteristic
327	C₂×C₄	#14556096	1	1	yes	characteristic
328	C₂×C₄	#14556104	1	1	yes	characteristic
329	C₂×C₄	#14556106	1	1	yes	characteristic
330	C₂×C₄	D₈×D₇₁	1	1	yes	characteristic
331	C₂×C₄	Q₁₆×D₇₁	1	1	yes	characteristic
332	C₂×C₄	#14556152	1	1	yes	characteristic
333	C₂×C₄	#14556160	1	1	yes	characteristic
334	C₈	—	2	2	no	—
335	C₈	—	4	1	no	—
336	C₈	—	4	2	no	—
337	C₈	—	8	1	no	—
338	C₈	—	284	1	no	—
339	C₈	—	284	1	no	—
340	C₈	—	284	1	no	—
341	C₈	—	284	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	284	1	no	—
345	C₂²	—	284	1	no	—
346	C₂²	—	142	2	no	—
347	C₂²	#17985870	1	1	yes	characteristic
348	C₂²	#17986314	1	1	yes	characteristic
349	C₂²	#17986336	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	284	1	no	—
354	C₄	—	284	1	no	—
355	C₄	—	568	2	no	—
356	C₄	—	142	2	no	—
357	C₄	#107×D₇₁	1	1	yes	characteristic
358	C₄	#17986040	1	1	yes	characteristic
359	C₄	#17986115	1	1	yes	characteristic
360	C₄	#17986120	1	1	yes	characteristic
361	C₄	#17986131	1	1	yes	characteristic
362	C₄	#17986153	1	1	yes	characteristic
363	C₂	—	2	1	no	—
364	C₂	—	2	1	no	—
365	C₂	—	284	2	no	—
366	C₂	#22588526	1	1	yes	characteristic
367	C₂	#22588579	1	1	yes	characteristic
368	C₂	#22588612	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556096
commutator	C₄²×C₇₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₇₁×#1567	C₂
socle	C₂²×C₇₁	#385
prime radical	C₂×#11765865	D₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	#1567	D₇₁
hypocenter	C₇₁	#8902
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
3. #22587116	C₂	semidirect
5. #22586241	C₂	semidirect
6. #22586180	C₂	semidirect
7. C₇₁×#1567	C₂	semidirect
21. #17985731	C₄	semidirect
22. #17985731	C₄	semidirect
23. #17985728	C₄	semidirect
24. #17985728	C₄	semidirect
188. #1567	D₇₁	semidirect
192. C₇₁	#8902	semidirect

Maximal subgroups

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

subgroup	index
#22596909	2
#22588357	2
#22587116	2
#22586723	2
#22586241	2
#22586180	2
C₇₁×#1567	2
#8902	71