Subgroups — #32150690

Order 18176.

646 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#504×D₇₁	C₂	1	2	yes	maximal, complemented
2	#22597238	C₂	1	1	yes	maximal, characteristic
3	#22590183	C₂	1	1	yes	maximal, characteristic, complemented
4	#22590180	C₂	1	2	yes	maximal, complemented
5	#22590177	C₂	1	1	yes	maximal, characteristic, complemented
6	#22590176	C₂	1	1	yes	maximal, characteristic, complemented
7	#22590174	C₂	1	2	yes	maximal, complemented
8	#22590104	C₂	1	1	yes	maximal, characteristic, complemented
9	#22589966	C₂	1	2	yes	maximal, complemented
10	#22589938	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₁×#2171	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986796	C₄	1	1	yes	characteristic
13	#131×D₇₁	C₂²	1	2	yes	—
14	#17986417	—	2	2	no	—
15	#17986417	C₂²	1	2	yes	complemented
16	#17986416	C₂²	1	2	yes	complemented
17	#17986415	—	2	2	no	—
18	#17986415	C₂²	1	2	yes	complemented
19	C₂×#14556067	C₂²	1	1	yes	characteristic
20	M₃₂×D₇₁	—	2	2	no	—
21	M₃₂×D₇₁	C₂²	1	2	yes	—
22	C₂×C₁₆×D₇₁	—	2	2	no	—
23	C₂×C₁₆×D₇₁	C₂²	1	1	yes	characteristic
24	#17986121	C₄	1	1	yes	characteristic
25	#17986120	C₄	1	1	yes	characteristic
26	#17986118	C₄	1	2	yes	—
27	#17986117	C₂²	1	1	yes	characteristic
28	#17986114	C₂²	1	2	yes	—
29	#17986046	C₄	1	1	yes	characteristic
30	#17986014	C₄	1	2	yes	—
31	#17985953	C₂²	1	1	yes	characteristic
32	#17985820	C₂²	1	1	yes	characteristic, complemented
33	#17985819	C₂²	1	2	yes	complemented
34	#17985784	C₂²	1	2	yes	complemented
35	#17985783	C₂²	1	2	yes	complemented
36	#17985782	C₂²	1	2	yes	complemented
37	#17985774	C₂²	1	1	yes	characteristic, complemented
38	C₄×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
39	C₁₆×Dic₇₁	C₂²	1	1	yes	characteristic, complemented
40	#504×C₇₁	C₂²	1	2	yes	complemented
41	#443×C₇₁	C₂²	1	1	yes	characteristic
42	#364×C₇₁	C₂²	1	1	yes	characteristic, complemented
43	#350×C₇₁	C₂²	1	2	yes	complemented
44	#346×C₇₁	C₂²	1	1	yes	characteristic, complemented
45	#41×D₇₁	C₂×C₄	1	2	yes	—
46	#14556160	C₈	1	2	yes	—
47	#14556157	C₈	1	2	yes	—
48	#14556155	C₈	1	2	yes	—
49	#14556152	C₈	1	2	yes	—
50	#14556150	C₂×C₄	1	1	yes	characteristic
51	#14556148	C₂×C₄	1	2	yes	—
52	#14556136	—	2	2	no	—
53	#14556136	C₂×C₄	1	2	yes	—
54	#14556135	C₂×C₄	1	2	yes	—
55	#14556134	—	2	2	no	—
56	#14556134	C₂³	1	2	yes	—
57	C₂×#11765848	C₂×C₄	1	1	yes	characteristic
58	M₁₆×D₇₁	—	2	2	no	—
59	M₁₆×D₇₁	C₂×C₄	1	2	yes	—
60	C₂×C₈×D₇₁	—	2	2	no	—
61	C₂×C₈×D₇₁	C₂³	1	1	yes	characteristic
62	#14556085	C₂×C₄	1	1	yes	characteristic
63	#14556068	—	2	2	no	—
64	#14556068	C₂³	1	2	yes	—
65	#14556067	—	2	2	no	—
66	#14556067	—	2	2	no	—
67	#14556067	—	2	2	no	—
68	#14556067	—	2	2	no	—
69	C₂×C₇₁⋊C₁₆	—	2	2	no	—
70	C₂×C₇₁⋊C₁₆	C₂³	1	1	yes	characteristic
71	C₂×C₇₁⋊C₁₆	C₂³	1	1	yes	characteristic
72	C₁₆×D₇₁	—	2	2	no	—
73	C₁₆×D₇₁	—	2	2	no	—
74	C₁₆×D₇₁	—	2	2	no	—
75	C₁₆×D₇₁	D₄	1	2	yes	—
76	C₁₆×D₇₁	D₄	1	2	yes	—
77	#14556064	C₂×C₄	1	1	yes	characteristic
78	#14556063	C₂×C₄	1	2	yes	—
79	#14556056	C₂×C₄	1	2	yes	—
80	#14556055	C₂×C₄	1	2	yes	—
81	#14556054	C₂×C₄	1	2	yes	—
82	#14556039	C₂×C₄	1	1	yes	characteristic
83	#14556038	C₂×C₄	1	1	yes	characteristic
84	C₈×Dic₇₁	C₂³	1	1	yes	characteristic
85	#135×C₇₁	C₂×C₄	1	1	yes	characteristic
86	#131×C₇₁	C₂³	1	2	yes	—
87	M₃₂×C₇₁	—	2	2	no	—
88	M₃₂×C₇₁	C₂³	1	2	yes	—
89	C₂×C₁₆×C₇₁	—	2	2	no	—
90	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
91	C₂×C₁₆×C₇₁	C₂³	1	1	yes	characteristic
92	#108×C₇₁	C₂×C₄	1	1	yes	characteristic
93	#104×C₇₁	C₂×C₄	1	2	yes	—
94	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
95	#11765866	C₂×C₈	1	2	yes	—
96	#11765865	—	2	2	no	—
97	#11765865	C₂²×C₄	1	2	yes	—
98	#11765864	—	2	2	no	—
99	#11765864	C₂×C₈	1	2	yes	—
100	Q₈×D₇₁	C₂×C₈	1	2	yes	—
101	D₄×D₇₁	—	2	2	no	—
102	D₄×D₇₁	C₂×C₈	1	2	yes	—
103	C₂×C₄×D₇₁	—	2	2	no	—
104	C₂×C₄×D₇₁	C₂²×C₄	1	1	yes	characteristic
105	Dic₂₈₄	C₂×C₈	1	1	yes	characteristic
106	#11765855	C₂×C₈	1	2	yes	—
107	#11765854	C₂×C₈	1	2	yes	—
108	#11765853	C₂×C₈	1	2	yes	—
109	#11765852	C₂×C₈	1	2	yes	—
110	D₅₆₈	C₂×C₈	1	1	yes	characteristic
111	#11765850	C₂×C₈	1	2	yes	—
112	#11765849	—	2	2	no	—
113	#11765849	C₂²×C₄	1	2	yes	—
114	#11765848	—	2	2	no	—
115	#11765848	—	2	2	no	—
116	#11765848	—	2	2	no	—
117	#11765848	C₂×C₈	1	2	yes	—
118	#11765848	C₂×C₈	1	2	yes	—
119	C₂×C₇₁⋊C₈	—	2	2	no	—
120	C₂×C₇₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₇₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
122	C₈×D₇₁	—	2	2	no	—
123	C₈×D₇₁	—	2	2	no	—
124	C₈×D₇₁	—	2	2	no	—
125	C₈×D₇₁	C₂×D₄	1	2	yes	—
126	C₈×D₇₁	#41	1	2	yes	—
127	Dic₇₁⋊C₄	—	2	1	no	—
128	(C₂×D₇₁)⋊C₄	—	2	1	no	—
129	C₄×Dic₇₁	C₂²×C₄	1	1	yes	characteristic
130	C₇₁⋊C₁₆	—	2	2	no	—
131	C₇₁⋊C₁₆	—	2	1	no	—
132	C₇₁⋊C₁₆	C₂×D₄	1	1	yes	characteristic
133	C₇₁⋊C₁₆	C₂×D₄	1	1	yes	characteristic
134	#41×C₇₁	C₂²×C₄	1	2	yes	—
135	Q₁₆×C₇₁	C₂×C₈	1	1	yes	characteristic
136	SD₁₆×C₇₁	C₂×C₈	1	2	yes	—
137	D₈×C₇₁	C₂×C₈	1	1	yes	characteristic
138	M₁₆×C₇₁	—	2	2	no	—
139	M₁₆×C₇₁	C₂²×C₄	1	2	yes	—
140	C₂×C₈×C₇₁	—	2	2	no	—
141	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
142	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
143	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
144	C₁₆×C₇₁	—	2	2	no	—
145	C₁₆×C₇₁	—	2	1	no	—
146	C₁₆×C₇₁	C₂×D₄	1	2	yes	—
147	C₂²×D₇₁	—	2	2	no	—
148	C₂²×D₇₁	#131	1	1	yes	characteristic
149	Q₈×C₇₁	C₂²×C₈	1	2	yes	—
150	D₄×C₇₁	—	2	2	no	—
151	D₄×C₇₁	C₂²×C₈	1	2	yes	—
152	C₂×C₄×C₇₁	—	2	2	no	—
153	C₂×C₄×C₇₁	—	2	1	no	—
154	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
155	C₂²⋊D₇₁	—	2	2	no	—
156	C₂²⋊D₇₁	—	2	2	no	—
157	C₂²⋊D₇₁	—	2	2	no	—
158	C₂×Dic₇₁	—	2	2	no	—
159	C₂×Dic₇₁	—	2	1	no	—
160	C₂×Dic₇₁	#131	1	1	yes	characteristic
161	D₂₈₄	—	2	2	no	—
162	D₂₈₄	C₂²×C₈	1	2	yes	—
163	C₄×D₇₁	—	2	2	no	—
164	C₄×D₇₁	—	2	2	no	—
165	C₄×D₇₁	—	2	2	no	—
166	C₄×D₇₁	C₄×D₄	1	2	yes	—
167	C₄×D₇₁	C₂²×C₈	1	2	yes	—
168	Dic₁₄₂	—	2	2	no	—
169	Dic₁₄₂	C₂²×C₈	1	2	yes	—
170	C₈×C₇₁	—	2	2	no	—
171	C₈×C₇₁	C₂²×C₈	1	1	yes	characteristic
172	C₈×C₇₁	C₂²×C₈	1	1	yes	characteristic
173	C₈×C₇₁	C₂²×D₄	1	1	yes	characteristic
174	C₈×C₇₁	C₂×#41	1	1	yes	characteristic
175	C₇₁⋊C₈	—	2	2	no	—
176	C₇₁⋊C₈	C₄×D₄	1	2	yes	—
177	C₇₁⋊C₈	C₂²×C₈	1	2	yes	—
178	C₂²×C₇₁	—	2	2	no	—
179	C₂²×C₇₁	C₂×#131	1	1	yes	characteristic
180	C₂×D₇₁	—	2	2	no	—
181	C₂×D₇₁	—	4	1	no	—
182	C₂×D₇₁	—	4	4	no	—
183	C₂×D₇₁	C₈×D₄	1	2	yes	—
184	C₄×C₇₁	—	2	2	no	—
185	C₄×C₇₁	—	4	1	no	—
186	C₄×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
187	C₄×C₇₁	C₂³×C₈	1	1	yes	characteristic
188	Dic₇₁	—	2	2	no	—
189	Dic₇₁	—	4	1	no	—
190	Dic₇₁	C₈×D₄	1	2	yes	—
191	#33441	—	71	1	no	maximal, Sylow
192	C₂×C₇₁	—	2	1	no	—
193	C₂×C₇₁	—	4	2	no	—
194	C₂×C₇₁	C₂×C₈×D₄	1	1	yes	characteristic
195	D₇₁	—	2	2	no	—
196	D₇₁	—	4	2	no	—
197	C₂×#504	—	71	2	no	—
198	#2955	—	71	1	no	—
199	#2171	—	71	2	no	—
200	#2171	—	71	1	no	—
201	#2171	D₇₁	1	1	yes	characteristic, complemented
202	#2170	—	71	1	no	—
203	#2170	—	71	1	no	—
204	#2170	—	71	2	no	—
205	#2153	—	71	1	no	—
206	#2117	—	71	2	no	—
207	#2108	—	71	1	no	—
208	C₇₁	#33441	1	1	yes	characteristic, complemented, Sylow
209	#575	—	71	1	no	—
210	C₂×#131	—	71	2	no	—
211	#504	—	71	2	no	—
212	#504	—	71	2	no	—
213	#504	—	71	2	no	—
214	#504	—	142	2	no	—
215	#504	—	142	2	no	—
216	#504	C₂×D₇₁	1	2	yes	complemented
217	C₂×M₃₂	—	71	1	no	—
218	C₂×M₃₂	—	71	2	no	—
219	C₂×M₃₂	—	142	2	no	—
220	C₂²×C₁₆	—	71	1	no	—
221	C₂²×C₁₆	—	142	2	no	—
222	#444	—	71	1	no	—
223	#444	—	71	1	no	—
224	#444	—	71	2	no	—
225	#443	—	71	2	no	—
226	#443	—	71	1	no	—
227	#443	C₂×D₇₁	1	1	yes	characteristic
228	#430	—	71	1	no	—
229	#421	—	71	2	no	—
230	#405	—	71	1	no	—
231	#364	—	71	2	no	—
232	#364	—	71	1	no	—
233	#364	C₂×D₇₁	1	1	yes	characteristic, complemented
234	#350	—	71	2	no	—
235	#350	—	71	2	no	—
236	#350	—	71	2	no	—
237	#350	C₂×D₇₁	1	2	yes	complemented
238	#346	—	71	1	no	—
239	#346	C₂×D₇₁	1	1	yes	characteristic, complemented
240	C₄×C₁₆	—	71	1	no	—
241	C₄×C₁₆	—	71	1	no	—
242	C₂×#41	—	71	2	no	—
243	#137	—	71	2	no	—
244	#137	—	71	2	no	—
245	AGL(1,ℤ/8)	—	71	2	no	—
246	AGL(1,ℤ/8)	—	71	2	no	—
247	#135	—	71	2	no	—
248	#135	—	71	1	no	—
249	#135	C₄×D₇₁	1	1	yes	characteristic
250	#131	—	71	2	no	—
251	#131	—	71	2	no	—
252	#131	—	71	2	no	—
253	#131	—	142	2	no	—
254	#131	—	142	2	no	—
255	#131	C₂²×D₇₁	1	2	yes	—
256	C₂×M₁₆	—	71	1	no	—
257	C₂×M₁₆	—	71	2	no	—
258	C₂×M₁₆	—	142	2	no	—
259	C₂²×C₈	—	71	1	no	—
260	C₂²×C₈	—	142	2	no	—
261	#117	—	71	1	no	—
262	M₃₂	—	2	2	no	—
263	M₃₂	—	71	2	no	—
264	M₃₂	—	142	2	no	—
265	M₃₂	—	142	2	no	—
266	M₃₂	—	142	2	no	—
267	M₃₂	—	142	2	no	—
268	M₃₂	—	142	2	no	—
269	M₃₂	C₂²×D₇₁	1	2	yes	—
270	C₂×C₁₆	—	2	2	no	—
271	C₂×C₁₆	—	71	1	no	—
272	C₂×C₁₆	—	71	2	no	—
273	C₂×C₁₆	—	71	2	no	—
274	C₂×C₁₆	—	71	1	no	—
275	C₂×C₁₆	—	142	2	no	—
276	C₂×C₁₆	—	142	2	no	—
277	C₂×C₁₆	—	142	2	no	—
278	C₂×C₁₆	—	142	2	no	—
279	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
280	C₂×C₁₆	C₂²×D₇₁	1	1	yes	characteristic
281	#108	—	71	2	no	—
282	#108	—	71	1	no	—
283	#108	C₄×D₇₁	1	1	yes	characteristic
284	#104	—	71	2	no	—
285	#104	—	71	2	no	—
286	#104	—	71	2	no	—
287	#104	C₄×D₇₁	1	2	yes	—
288	#97	—	71	1	no	—
289	#97	—	71	1	no	—
290	C₄×C₈	—	71	1	no	—
291	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
292	#41	—	71	2	no	—
293	#41	—	71	2	no	—
294	#41	—	71	2	no	—
295	#41	—	142	2	no	—
296	#41	—	142	2	no	—
297	#41	C₂×C₄×D₇₁	1	2	yes	—
298	C₂×Q₈	—	71	2	no	—
299	C₂×D₄	—	71	2	no	—
300	C₂×D₄	—	142	2	no	—
301	C₂²×C₄	—	71	1	no	—
302	C₂²×C₄	—	142	2	no	—
303	Q₁₆	—	71	2	no	—
304	Q₁₆	—	71	1	no	—
305	Q₁₆	C₈×D₇₁	1	1	yes	characteristic
306	SD₁₆	—	71	2	no	—
307	SD₁₆	—	71	2	no	—
308	SD₁₆	—	71	2	no	—
309	SD₁₆	C₈×D₇₁	1	2	yes	—
310	D₈	—	71	2	no	—
311	D₈	—	71	1	no	—
312	D₈	C₈×D₇₁	1	1	yes	characteristic
313	M₁₆	—	2	2	no	—
314	M₁₆	—	71	2	no	—
315	M₁₆	—	71	2	no	—
316	M₁₆	—	71	2	no	—
317	M₁₆	—	142	2	no	—
318	M₁₆	—	142	2	no	—
319	M₁₆	—	142	2	no	—
320	M₁₆	—	142	2	no	—
321	M₁₆	C₂×C₄×D₇₁	1	2	yes	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	71	1	no	—
324	C₂×C₈	—	71	1	no	—
325	C₂×C₈	—	71	2	no	—
326	C₂×C₈	—	71	2	no	—
327	C₂×C₈	—	142	2	no	—
328	C₂×C₈	—	142	2	no	—
329	C₂×C₈	—	142	2	no	—
330	C₂×C₈	—	142	2	no	—
331	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
332	C₂×C₈	C₂³×D₇₁	1	1	yes	characteristic
333	C₄⋊C₄	—	142	1	no	—
334	C₂²⋊C₄	—	142	1	no	—
335	C₄²	—	71	1	no	—
336	C₄²	C₂×C₄×D₇₁	1	1	yes	characteristic
337	C₁₆	—	2	2	no	—
338	C₁₆	—	2	1	no	—
339	C₁₆	—	71	1	no	—
340	C₁₆	—	71	1	no	—
341	C₁₆	—	142	2	no	—
342	C₁₆	—	142	1	no	—
343	C₁₆	D₄×D₇₁	1	2	yes	—
344	C₂³	—	71	1	no	—
345	C₂³	—	142	2	no	—
346	Q₈	—	71	2	no	—
347	Q₈	—	142	2	no	—
348	Q₈	C₂×C₈×D₇₁	1	2	yes	—
349	D₄	—	2	2	no	—
350	D₄	—	71	2	no	—
351	D₄	—	142	2	no	—
352	D₄	—	142	2	no	—
353	D₄	—	142	2	no	—
354	D₄	—	142	2	no	—
355	D₄	C₂×C₈×D₇₁	1	2	yes	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	71	1	no	—
359	C₂×C₄	—	71	2	no	—
360	C₂×C₄	—	71	2	no	—
361	C₂×C₄	—	142	2	no	—
362	C₂×C₄	—	142	2	no	—
363	C₂×C₄	—	142	1	no	—
364	C₂×C₄	—	142	2	no	—
365	C₂×C₄	—	142	2	no	—
366	C₂×C₄	C₂²×C₄×D₇₁	1	1	yes	characteristic
367	C₈	—	2	2	no	—
368	C₈	—	71	2	no	—
369	C₈	—	71	2	no	—
370	C₈	—	142	2	no	—
371	C₈	C₂×C₈×D₇₁	1	1	yes	characteristic
372	C₈	C₂×C₈×D₇₁	1	1	yes	characteristic
373	C₈	C₂×D₄×D₇₁	1	1	yes	characteristic
374	C₈	#41×D₇₁	1	1	yes	characteristic
375	C₂²	—	2	2	no	—
376	C₂²	—	284	1	no	—
377	C₂²	—	284	4	no	—
378	C₂²	—	71	2	no	—
379	C₂²	—	142	2	no	—
380	C₂²	#131×D₇₁	1	1	yes	characteristic
381	C₄	—	2	2	no	—
382	C₄	—	4	1	no	—
383	C₄	—	284	1	no	—
384	C₄	—	71	2	no	—
385	C₄	—	142	2	no	—
386	C₄	C₄×D₄×D₇₁	1	1	yes	characteristic
387	C₄	C₂²×C₈×D₇₁	1	1	yes	characteristic
388	C₂	—	2	1	no	—
389	C₂	—	4	2	no	—
390	C₂	—	284	2	no	—
391	C₂	—	142	2	no	—
392	C₂	C₈×D₄×D₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×D₄×D₇₁
commutator	C₄×C₇₁	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₇₁
Fitting	C₇₁×#2171	C₂
socle	C₂×C₇₁	C₂×C₈×D₄
prime radical	#17986796	C₄
cosocle	C₂×C₈×C₇₁	C₂⁴
hypercenter	#2171	D₇₁
hypocenter	C₇₁	#33441
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ #2171

Complements

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

N	H ≅ G/N	type
1. #504×D₇₁	C₂	semidirect
3. #22590183	C₂	semidirect
4. #22590180	C₂	semidirect
5. #22590177	C₂	semidirect
6. #22590176	C₂	semidirect
7. #22590174	C₂	semidirect
8. #22590104	C₂	semidirect
9. #22589966	C₂	semidirect
10. #22589938	C₂	semidirect
11. C₇₁×#2171	C₂	semidirect
15. #17986417	C₂²	semidirect
16. #17986416	C₂²	semidirect
18. #17986415	C₂²	semidirect
32. #17985820	C₂²	semidirect
33. #17985819	C₂²	semidirect
34. #17985784	C₂²	semidirect
35. #17985783	C₂²	semidirect
36. #17985782	C₂²	semidirect
37. #17985774	C₂²	semidirect
38. C₄×C₇₁⋊C₁₆	C₂²	semidirect
39. C₁₆×Dic₇₁	C₂²	semidirect
40. #504×C₇₁	C₂²	semidirect
42. #364×C₇₁	C₂²	semidirect
43. #350×C₇₁	C₂²	semidirect
44. #346×C₇₁	C₂²	semidirect
201. #2171	D₇₁	semidirect
208. C₇₁	#33441	semidirect
216. #504	C₂×D₇₁	semidirect
233. #364	C₂×D₇₁	semidirect
237. #350	C₂×D₇₁	semidirect
239. #346	C₂×D₇₁	semidirect

Maximal subgroups

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

subgroup	index
#504×D₇₁	2
#22597238	2
#22590183	2
#22590180	2
#22590177	2
#22590176	2
#22590174	2
#22590104	2
#22589966	2
#22589938	2
C₇₁×#2171	2
#33441	71