Subgroups — #350×Dic₇₁ #31912502

Order 18176.

490 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#131×Dic₇₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#17985783	C₂	1	1	yes	maximal, characteristic
3	M₃₂×Dic₇₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₆×Dic₇₁	C₂	1	1	yes	maximal, characteristic, complemented
5	#22586430	C₂	1	1	yes	maximal, characteristic, complemented
6	#22586428	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#350×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂×#14556135	C₄	1	1	yes	characteristic
9	#41×Dic₇₁	C₄	1	1	yes	characteristic
10	C₂×#14556068	C₂²	1	1	yes	characteristic
11	C₂²×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
12	#17985957	—	2	1	no	—
13	#17985957	C₄	1	1	yes	characteristic
14	#17985954	—	2	1	no	—
15	#17985954	C₂²	1	1	yes	characteristic
16	M₁₆×Dic₇₁	—	2	1	no	—
17	M₁₆×Dic₇₁	C₄	1	1	yes	characteristic
18	C₂×C₈×Dic₇₁	—	2	1	no	—
19	C₂×C₈×Dic₇₁	C₂²	1	1	yes	characteristic
20	#17985783	C₄	1	4	yes	complemented
21	#17985774	—	2	1	no	—
22	C₁₆×Dic₇₁	—	2	1	no	—
23	C₁₆×Dic₇₁	—	2	1	no	—
24	C₁₆×Dic₇₁	—	2	1	no	—
25	C₂×#131×C₇₁	C₂²	1	1	yes	characteristic
26	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
27	C₂²×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
28	#350×C₇₁	C₄	1	4	yes	complemented
29	#14556135	—	2	2	no	—
30	#14556135	C₈	1	4	yes	—
31	C₂×#11765849	—	2	1	no	—
32	C₂×#11765849	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₇₁⋊C₈	—	2	1	no	—
34	C₂²×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
35	Q₈×Dic₇₁	C₈	1	2	yes	—
36	D₄×Dic₇₁	—	2	2	no	—
37	D₄×Dic₇₁	C₈	1	2	yes	—
38	#14556087	—	2	1	no	—
39	#14556087	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×Dic₇₁	—	2	1	no	—
41	C₂×C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
42	#14556068	—	2	1	no	—
43	#14556068	C₂×C₄	1	2	yes	complemented
44	C₂×C₇₁⋊C₁₆	—	2	1	no	—
45	C₂×C₇₁⋊C₁₆	—	2	1	no	—
46	C₂×C₇₁⋊C₁₆	—	2	1	no	—
47	C₂×C₇₁⋊C₁₆	C₂×C₄	1	2	yes	complemented
48	#14556038	—	2	1	no	—
49	#14556038	—	2	1	no	—
50	#14556038	—	2	1	no	—
51	C₈×Dic₇₁	—	2	1	no	—
52	C₈×Dic₇₁	—	2	1	no	—
53	C₈×Dic₇₁	—	2	1	no	—
54	C₈×Dic₇₁	D₄	1	1	yes	characteristic
55	C₈×Dic₇₁	D₄	1	1	yes	characteristic
56	C₈×Dic₇₁	D₄	1	1	yes	characteristic
57	C₈×Dic₇₁	D₄	1	1	yes	characteristic
58	C₂×#41×C₇₁	C₂×C₄	1	1	yes	characteristic
59	#131×C₇₁	—	2	1	no	—
60	#131×C₇₁	—	2	1	no	—
61	#131×C₇₁	C₂×C₄	1	2	yes	—
62	#131×C₇₁	C₂×C₄	1	2	yes	—
63	C₂×M₁₆×C₇₁	—	2	1	no	—
64	C₂×M₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₈×C₇₁	—	2	1	no	—
66	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
67	M₃₂×C₇₁	—	2	1	no	—
68	M₃₂×C₇₁	C₂×C₄	1	2	yes	complemented
69	C₂×C₁₆×C₇₁	—	2	1	no	—
70	C₂×C₁₆×C₇₁	—	2	1	no	—
71	C₂×C₁₆×C₇₁	—	2	1	no	—
72	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	complemented
73	C₂²×Dic₇₁	—	2	2	no	—
74	C₂²×Dic₇₁	M₁₆	1	2	yes	—
75	#11765849	—	2	1	no	—
76	#11765849	—	2	4	no	—
77	#11765849	C₂×C₈	1	2	yes	—
78	C₂×C₇₁⋊C₈	—	2	1	no	—
79	C₂×C₇₁⋊C₈	—	2	4	no	—
80	C₂×C₇₁⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₇₁⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₇₁⋊C₈	C₂×C₈	1	2	yes	—
83	C₂×C₇₁⋊C₈	M₁₆	1	2	yes	—
84	C₄⋊Dic₇₁	—	2	2	no	—
85	C₄⋊Dic₇₁	C₂×C₈	1	2	yes	—
86	C₂²⋊Dic₇₁	—	2	1	no	—
87	C₂²⋊Dic₇₁	—	2	2	no	—
88	C₄×Dic₇₁	—	2	1	no	—
89	C₄×Dic₇₁	—	2	2	no	—
90	C₄×Dic₇₁	C₂²⋊C₄	1	1	yes	characteristic
91	C₄×Dic₇₁	C₂²⋊C₄	1	1	yes	characteristic
92	C₄×Dic₇₁	C₂×C₈	1	2	yes	—
93	C₇₁⋊C₁₆	—	2	2	no	—
94	C₇₁⋊C₁₆	—	2	2	no	—
95	#41×C₇₁	—	2	1	no	—
96	#41×C₇₁	—	2	1	no	—
97	#41×C₇₁	C₄²	1	2	yes	—
98	#41×C₇₁	C₂×C₈	1	2	yes	—
99	C₂×Q₈×C₇₁	C₂×C₈	1	1	yes	characteristic
100	C₂×D₄×C₇₁	—	2	1	no	—
101	C₂×D₄×C₇₁	C₂×C₈	1	1	yes	characteristic
102	C₂²×C₄×C₇₁	—	2	1	no	—
103	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
104	M₁₆×C₇₁	—	2	1	no	—
105	M₁₆×C₇₁	—	2	2	no	—
106	M₁₆×C₇₁	—	2	2	no	—
107	M₁₆×C₇₁	C₄²	1	2	yes	—
108	C₂×C₈×C₇₁	—	2	1	no	—
109	C₂×C₈×C₇₁	—	2	2	no	—
110	C₂×C₈×C₇₁	—	2	2	no	—
111	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
116	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
117	C₁₆×C₇₁	—	2	2	no	—
118	C₁₆×C₇₁	—	2	2	no	—
119	C₂³×C₇₁	—	2	1	no	—
120	C₂³×C₇₁	C₂×M₁₆	1	1	yes	characteristic
121	Q₈×C₇₁	—	2	1	no	—
122	Q₈×C₇₁	C₄×C₈	1	2	yes	—
123	D₄×C₇₁	—	2	2	no	—
124	D₄×C₇₁	—	2	2	no	—
125	D₄×C₇₁	—	2	1	no	—
126	D₄×C₇₁	C₄×C₈	1	2	yes	—
127	C₂×C₄×C₇₁	—	2	1	no	—
128	C₂×C₄×C₇₁	—	2	2	no	—
129	C₂×C₄×C₇₁	—	2	2	no	—
130	C₂×C₄×C₇₁	C₂×C₄²	1	1	yes	characteristic
131	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
133	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
135	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
136	C₂×Dic₇₁	—	2	4	no	—
137	C₂×Dic₇₁	—	2	2	no	—
138	C₂×Dic₇₁	—	4	4	no	—
139	C₂×Dic₇₁	C₂²⋊C₈	1	2	yes	—
140	C₂×Dic₇₁	C₂²⋊C₈	1	2	yes	—
141	C₈×C₇₁	—	2	2	no	—
142	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
143	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
144	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
145	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
146	C₇₁⋊C₈	—	2	2	no	—
147	C₇₁⋊C₈	C₂²⋊C₈	1	2	yes	—
148	C₇₁⋊C₈	C₂²⋊C₈	1	2	yes	—
149	C₂²×C₇₁	—	2	2	no	—
150	C₂²×C₇₁	—	2	2	no	—
151	C₂²×C₇₁	—	4	2	no	—
152	C₂²×C₇₁	C₄×M₁₆	1	1	yes	characteristic
153	C₂²×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
154	C₂²×C₇₁	#431	1	1	yes	characteristic
155	C₄×C₇₁	—	2	2	no	—
156	C₄×C₇₁	C₄×C₂²⋊C₄	1	1	yes	characteristic
157	C₄×C₇₁	C₂×C₄×C₈	1	1	yes	characteristic
158	C₄×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
159	C₄×C₇₁	#431	1	1	yes	characteristic
160	Dic₇₁	—	2	2	no	—
161	Dic₇₁	—	4	2	no	—
162	Dic₇₁	#350	1	4	yes	direct product factor
163	C₄×#350	—	71	1	no	maximal, Sylow
164	C₂×C₇₁	—	2	1	no	—
165	C₂×C₇₁	—	2	1	no	—
166	C₂×C₇₁	—	4	2	no	—
167	C₂×C₇₁	C₄×C₂²⋊C₈	1	1	yes	characteristic
168	C₂×C₇₁	C₂×#350	1	2	yes	—
169	C₄×#131	—	71	1	no	—
170	C₂×#350	—	71	1	no	—
171	C₂×#350	D₇₁	1	1	yes	characteristic
172	C₄×M₃₂	—	71	1	no	—
173	C₂×C₄×C₁₆	—	71	1	no	—
174	#1379	—	71	1	no	—
175	#1379	—	71	1	no	—
176	C₇₁	C₄×#350	1	1	yes	characteristic, complemented, Sylow
177	C₂×#131	—	71	1	no	—
178	C₂×#131	C₂×D₇₁	1	1	yes	characteristic
179	C₄×#41	—	71	1	no	—
180	C₂×M₃₂	—	71	1	no	—
181	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
182	C₂²×C₁₆	—	71	1	no	—
183	C₂²×C₁₆	C₂×D₇₁	1	1	yes	characteristic
184	#405	—	71	1	no	—
185	#405	—	71	1	no	—
186	#405	—	142	1	no	—
187	#405	—	142	1	no	—
188	C₄×M₁₆	—	71	1	no	—
189	C₄×M₁₆	—	142	1	no	—
190	C₂×C₄×C₈	—	71	1	no	—
191	C₂×C₄×C₈	—	142	1	no	—
192	#350	—	71	4	no	—
193	#350	Dic₇₁	1	4	yes	direct product factor
194	#346	—	142	1	no	—
195	C₄×C₁₆	—	142	1	no	—
196	C₄×C₁₆	—	142	1	no	—
197	C₄×C₁₆	—	142	1	no	—
198	C₂×#41	C₄×D₇₁	1	1	yes	characteristic
199	#131	—	2	1	no	—
200	#131	—	2	1	no	—
201	#131	—	71	4	no	—
202	#131	—	142	2	no	—
203	#131	C₄×D₇₁	1	2	yes	—
204	#131	C₂×Dic₇₁	1	2	yes	—
205	C₂×M₁₆	—	2	1	no	—
206	C₂×M₁₆	—	71	1	no	—
207	C₂×M₁₆	—	142	1	no	—
208	C₂×M₁₆	C₄×D₇₁	1	1	yes	characteristic
209	C₂²×C₈	—	2	1	no	—
210	C₂²×C₈	—	71	1	no	—
211	C₂²×C₈	—	142	1	no	—
212	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
213	C₄×Q₈	—	71	2	no	—
214	C₄×D₄	—	71	2	no	—
215	C₄×D₄	—	142	2	no	—
216	#117	—	71	1	no	—
217	#117	—	142	1	no	—
218	C₂×C₄²	—	71	1	no	—
219	C₂×C₄²	—	142	1	no	—
220	M₃₂	—	2	1	no	—
221	M₃₂	—	71	2	no	—
222	M₃₂	—	142	1	no	—
223	M₃₂	C₂×Dic₇₁	1	2	yes	complemented
224	C₂×C₁₆	—	2	1	no	—
225	C₂×C₁₆	—	2	1	no	—
226	C₂×C₁₆	—	2	1	no	—
227	C₂×C₁₆	—	71	2	no	—
228	C₂×C₁₆	—	142	1	no	—
229	C₂×C₁₆	—	142	1	no	—
230	C₂×C₁₆	—	142	1	no	—
231	C₂×C₁₆	C₂×Dic₇₁	1	2	yes	complemented
232	#97	—	142	1	no	—
233	#97	—	142	1	no	—
234	#97	—	142	1	no	—
235	C₄×C₈	—	71	1	no	—
236	C₄×C₈	—	71	1	no	—
237	C₄×C₈	—	71	1	no	—
238	C₄×C₈	—	71	1	no	—
239	C₄×C₈	—	142	1	no	—
240	C₄×C₈	—	142	1	no	—
241	C₄×C₈	—	142	1	no	—
242	#41	—	2	1	no	—
243	#41	—	2	1	no	—
244	#41	C₄×Dic₇₁	1	2	yes	—
245	#41	C₈×D₇₁	1	2	yes	—
246	C₂×Q₈	C₈×D₇₁	1	1	yes	characteristic
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	C₈×D₇₁	1	1	yes	characteristic
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	71	2	no	—
251	C₂²×C₄	—	142	2	no	—
252	C₂²×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
253	M₁₆	—	2	1	no	—
254	M₁₆	—	2	2	no	—
255	M₁₆	—	2	2	no	—
256	M₁₆	—	71	2	no	—
257	M₁₆	—	142	4	no	—
258	M₁₆	—	142	1	no	—
259	M₁₆	C₄×Dic₇₁	1	2	yes	—
260	C₂×C₈	—	2	1	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	71	1	no	—
264	C₂×C₈	—	71	1	no	—
265	C₂×C₈	—	71	2	no	—
266	C₂×C₈	—	71	2	no	—
267	C₂×C₈	—	142	4	no	—
268	C₂×C₈	—	142	1	no	—
269	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
270	C₂×C₈	C₂²×Dic₇₁	1	1	yes	characteristic
271	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
272	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
273	C₂×C₈	#11765864	1	1	yes	characteristic
274	C₂×C₈	#11765864	1	1	yes	characteristic
275	C₄⋊C₄	—	71	2	no	—
276	C₄⋊C₄	—	142	2	no	—
277	C₂²⋊C₄	—	142	2	no	—
278	C₂²⋊C₄	—	142	1	no	—
279	C₄²	—	71	1	no	—
280	C₄²	—	71	1	no	—
281	C₄²	—	71	2	no	—
282	C₄²	—	142	2	no	—
283	C₄²	—	142	1	no	—
284	C₁₆	—	2	2	no	—
285	C₁₆	—	2	2	no	—
286	C₁₆	—	142	2	no	—
287	C₁₆	—	142	2	no	—
288	C₂³	—	2	1	no	—
289	C₂³	M₁₆×D₇₁	1	1	yes	characteristic
290	Q₈	—	2	1	no	—
291	Q₈	C₈×Dic₇₁	1	2	yes	—
292	D₄	—	2	2	no	—
293	D₄	—	2	2	no	—
294	D₄	—	2	1	no	—
295	D₄	C₈×Dic₇₁	1	2	yes	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	284	4	no	—
300	C₂×C₄	—	71	2	no	—
301	C₂×C₄	—	71	2	no	—
302	C₂×C₄	—	142	2	no	—
303	C₂×C₄	—	142	4	no	—
304	C₂×C₄	C₂×C₄×Dic₇₁	1	1	yes	characteristic
305	C₂×C₄	C₂²⋊C₄×D₇₁	1	1	yes	characteristic
306	C₂×C₄	#14556084	1	1	yes	characteristic
307	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
308	C₂×C₄	C₂×C₈×D₇₁	1	1	yes	characteristic
309	C₂×C₄	M₁₆×D₇₁	1	1	yes	characteristic
310	C₈	—	2	2	no	—
311	C₈	—	71	2	no	—
312	C₈	—	71	2	no	—
313	C₈	—	142	2	no	—
314	C₈	(C₂²⋊D₇₁)⋊C₄	1	1	yes	characteristic
315	C₈	(C₂²⋊D₇₁)⋊C₄	1	1	yes	characteristic
316	C₈	D₄×Dic₇₁	1	1	yes	characteristic
317	C₈	D₄×Dic₇₁	1	1	yes	characteristic
318	C₂²	—	2	2	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	4	2	no	—
321	C₂²	M₁₆×Dic₇₁	1	1	yes	characteristic
322	C₂²	C₂²⋊C₈×D₇₁	1	1	yes	characteristic
323	C₂²	#17986049	1	1	yes	characteristic
324	C₄	—	2	2	no	—
325	C₄	—	284	2	no	—
326	C₄	—	71	4	no	—
327	C₄	—	142	2	no	—
328	C₄	C₂²⋊C₄×Dic₇₁	1	1	yes	characteristic
329	C₄	C₂×C₈×Dic₇₁	1	1	yes	characteristic
330	C₄	C₂²⋊C₈×D₇₁	1	1	yes	characteristic
331	C₄	#17986049	1	1	yes	characteristic
332	C₂	—	2	1	no	—
333	C₂	—	2	1	no	—
334	C₂	—	4	2	no	—
335	C₂	C₂²⋊C₈×Dic₇₁	1	1	yes	characteristic
336	C₂	#350×D₇₁	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	D₄×D₇₁
commutator	C₄×C₇₁	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#350×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₂²⋊C₈
prime radical	C₂×#41×C₇₁	C₂×C₄
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#350	D₇₁
hypocenter	C₇₁	C₄×#350
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#350

Complements

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

N	H ≅ G/N	type
3. M₃₂×Dic₇₁	C₂	semidirect
4. C₂×C₁₆×Dic₇₁	C₂	semidirect
5. #22586430	C₂	semidirect
6. #22586428	C₂	semidirect
20. #17985783	C₄	semidirect
28. #350×C₇₁	C₄	semidirect
43. #14556068	C₂×C₄	semidirect
47. C₂×C₇₁⋊C₁₆	C₂×C₄	semidirect
68. M₃₂×C₇₁	C₂×C₄	semidirect
72. C₂×C₁₆×C₇₁	C₂×C₄	semidirect
162. Dic₇₁	#350	direct (N × H)
176. C₇₁	C₄×#350	semidirect
193. #350	Dic₇₁	direct (N × H)
223. M₃₂	C₂×Dic₇₁	semidirect
231. C₂×C₁₆	C₂×Dic₇₁	semidirect

Maximal subgroups

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

subgroup	index
#131×Dic₇₁	2
C₂×#17985783	2
M₃₂×Dic₇₁	2
C₂×C₁₆×Dic₇₁	2
#22586430	2
#22586428	2
C₂×#350×C₇₁	2
C₄×#350	71

Subgroups — #350×Dic71 #31912502

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #350×Dic₇₁ #31912502