Subgroups — #31967491

Order 18176.

624 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#22597275	C₂	1	1	yes	maximal, characteristic, complemented
3	#22587191	C₂	1	2	yes	maximal, complemented
4	#22587177	C₂	1	2	yes	maximal, complemented
5	#22587159	C₂	1	2	yes	maximal, complemented
6	SD₁₆⋊(C₇₁⋊C₈)	C₂	1	2	yes	maximal, complemented
7	#22587129	C₂	1	1	yes	maximal, characteristic, complemented
8	#22586779	C₂	1	1	yes	maximal, characteristic
9	#22586761	C₂	1	1	yes	maximal, characteristic, complemented
10	#22586630	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₁×#3107	C₂	1	1	yes	maximal, characteristic
12	#17986130	—	2	1	no	—
13	#17986130	C₂²	1	2	yes	—
14	Q₈×C₇₁⋊C₈	—	2	1	no	—
15	Q₈×C₇₁⋊C₈	C₂²	1	2	yes	—
16	Q₈×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
17	Q₈×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
18	D₄×C₇₁⋊C₈	—	2	1	no	—
19	D₄×C₇₁⋊C₈	—	2	1	no	—
20	D₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
21	D₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
22	#17986052	—	2	1	no	—
23	#17986052	C₂²	1	1	yes	characteristic
24	#17986026	—	2	1	no	—
25	#17986023	C₂²	1	1	yes	characteristic
26	#17985955	—	2	1	no	—
27	C₄×#11765849	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₇₁⋊C₈	—	2	1	no	—
29	C₂×C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
30	#17985752	C₂²	1	2	yes	—
31	#17985750	C₂²	1	2	yes	—
32	Q₈⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
33	Q₈⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
34	Q₈⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
35	D₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
36	#17985721	C₂²	1	2	yes	—
37	#17985720	C₂²	1	2	yes	—
38	#533×C₇₁	C₂²	1	1	yes	characteristic
39	C₄×#41×C₇₁	C₂²	1	1	yes	characteristic
40	#485×C₇₁	C₄	1	2	yes	—
41	#475×C₇₁	C₄	1	1	yes	characteristic
42	#474×C₇₁	C₄	1	1	yes	characteristic
43	#462×C₇₁	C₄	1	1	yes	characteristic
44	#461×C₇₁	C₄	1	1	yes	characteristic
45	#452×C₇₁	C₄	1	2	yes	—
46	C₄×Q₁₆×C₇₁	C₂²	1	2	yes	—
47	C₄×SD₁₆×C₇₁	C₂²	1	2	yes	—
48	#431×C₇₁	C₂²	1	1	yes	characteristic
49	C₂×#11765849	—	2	1	no	—
50	C₂×#11765849	—	2	1	no	—
51	C₂²×C₇₁⋊C₈	—	2	1	no	—
52	C₂²×C₇₁⋊C₈	—	2	1	no	—
53	C₂²×C₇₁⋊C₈	D₄	1	1	yes	characteristic
54	C₂²×C₇₁⋊C₈	D₄	1	1	yes	characteristic
55	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
56	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
57	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
58	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
59	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
60	C₄⋊(C₇₁⋊C₈)	C₂³	1	1	yes	characteristic
61	C₄⋊(C₇₁⋊C₈)	C₂³	1	1	yes	characteristic
62	C₄⋊(C₇₁⋊C₈)	C₂³	1	2	yes	—
63	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
64	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
65	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
66	#14556039	—	2	2	no	—
67	#14556039	—	2	1	no	—
68	C₄×C₇₁⋊C₈	—	2	1	no	—
69	C₄×C₇₁⋊C₈	—	2	1	no	—
70	C₄×C₇₁⋊C₈	—	2	2	no	—
71	C₄×C₇₁⋊C₈	—	2	1	no	—
72	C₄×C₇₁⋊C₈	D₄	1	1	yes	characteristic
73	C₄×C₇₁⋊C₈	D₄	1	1	yes	characteristic
74	C₄×C₇₁⋊C₈	C₂³	1	2	yes	—
75	C₄×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
76	C₄×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
77	C₂×#41×C₇₁	C₂×C₄	1	1	yes	characteristic
78	C₂×Q₁₆×C₇₁	C₂×C₄	1	2	yes	—
79	C₂×SD₁₆×C₇₁	C₂×C₄	1	2	yes	—
80	C₄⋊Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
81	C₄⋊Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
82	#125×C₇₁	—	2	1	no	—
83	C₂²⋊Q₈×C₇₁	—	2	1	no	—
84	C₂²⋊Q₈×C₇₁	C₂×C₄	1	2	yes	—
85	C₄×Q₈×C₇₁	—	2	1	no	—
86	C₄×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
87	C₄×Q₈×C₇₁	C₂³	1	2	yes	—
88	C₄×Q₈×C₇₁	C₂³	1	1	yes	characteristic
89	C₄×D₄×C₇₁	—	2	1	no	—
90	C₄×D₄×C₇₁	—	2	1	no	—
91	C₄×D₄×C₇₁	C₂×C₄	1	1	yes	characteristic
92	C₄×D₄×C₇₁	C₂³	1	1	yes	characteristic
93	#117×C₇₁	—	2	1	no	—
94	#117×C₇₁	—	2	1	no	—
95	#117×C₇₁	C₂×C₄	1	1	yes	characteristic
96	C₂×C₄²×C₇₁	—	2	1	no	—
97	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
98	#107×C₇₁	C₂×C₄	1	2	yes	—
99	#106×C₇₁	C₂×C₄	1	2	yes	—
100	C₄⋊C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
101	C₄⋊C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
102	#103×C₇₁	C₂×C₄	1	2	yes	—
103	#103×C₇₁	C₂×C₄	1	2	yes	—
104	#103×C₇₁	C₂×C₄	1	2	yes	—
105	#102×C₇₁	C₂×C₄	1	2	yes	—
106	C₂²⋊C₈×C₇₁	C₂×C₄	1	2	yes	—
107	C₄×C₈×C₇₁	C₂³	1	2	yes	—
108	#11765849	—	4	2	no	—
109	#11765849	—	4	2	no	—
110	C₂×C₇₁⋊C₈	—	2	2	no	—
111	C₂×C₇₁⋊C₈	—	2	1	no	—
112	C₂×C₇₁⋊C₈	—	2	1	no	—
113	C₂×C₇₁⋊C₈	—	2	2	no	—
114	C₂×C₇₁⋊C₈	—	2	2	no	—
115	C₂×C₇₁⋊C₈	—	4	1	no	—
116	C₂×C₇₁⋊C₈	—	4	2	no	—
117	C₂×C₇₁⋊C₈	—	4	2	no	—
118	C₂×C₇₁⋊C₈	C₂×D₄	1	1	yes	characteristic
119	C₂×C₇₁⋊C₈	C₂×D₄	1	1	yes	characteristic
120	C₂×C₇₁⋊C₈	C₂×D₄	1	1	yes	characteristic
121	C₂×C₇₁⋊C₈	C₂×D₄	1	1	yes	characteristic
122	#41×C₇₁	—	2	2	no	—
123	#41×C₇₁	—	2	2	no	—
124	C₂×Q₈×C₇₁	—	2	1	no	—
125	C₂×Q₈×C₇₁	C₂²×C₄	1	2	yes	—
126	C₂×Q₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
127	C₂×D₄×C₇₁	—	2	1	no	—
128	C₂×D₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄×C₇₁	—	2	1	no	—
130	C₂²×C₄×C₇₁	—	2	1	no	—
131	C₂²×C₄×C₇₁	—	2	1	no	—
132	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
133	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
135	Q₁₆×C₇₁	—	2	4	no	—
136	SD₁₆×C₇₁	—	2	4	no	—
137	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
138	C₂×C₈×C₇₁	C₂²×C₄	1	2	yes	—
139	C₄⋊C₄×C₇₁	—	2	1	no	—
140	C₄⋊C₄×C₇₁	—	2	1	no	—
141	C₄⋊C₄×C₇₁	—	2	1	no	—
142	C₄⋊C₄×C₇₁	—	2	1	no	—
143	C₄⋊C₄×C₇₁	—	2	2	no	—
144	C₄⋊C₄×C₇₁	—	2	1	no	—
145	C₄⋊C₄×C₇₁	—	2	2	no	—
146	C₄⋊C₄×C₇₁	C₂²×C₄	1	2	yes	—
147	C₄⋊C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
148	C₄⋊C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
149	C₂²⋊C₄×C₇₁	—	2	2	no	—
150	C₂²⋊C₄×C₇₁	—	2	1	no	—
151	C₂²⋊C₄×C₇₁	—	2	1	no	—
152	C₂²⋊C₄×C₇₁	—	2	1	no	—
153	C₄²×C₇₁	—	2	1	no	—
154	C₄²×C₇₁	—	2	1	no	—
155	C₄²×C₇₁	—	2	1	no	—
156	C₄²×C₇₁	—	2	2	no	—
157	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
158	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
159	C₄²×C₇₁	#41	1	1	yes	characteristic
160	C₄²×C₇₁	C₂⁴	1	1	yes	characteristic
161	C₂³×C₇₁	—	2	1	no	—
162	C₂³×C₇₁	C₄×D₄	1	1	yes	characteristic
163	Q₈×C₇₁	—	2	1	no	—
164	Q₈×C₇₁	—	2	2	no	—
165	Q₈×C₇₁	—	2	2	no	—
166	Q₈×C₇₁	—	4	2	no	—
167	Q₈×C₇₁	#131	1	2	yes	—
168	D₄×C₇₁	—	2	4	no	—
169	D₄×C₇₁	—	2	1	no	—
170	D₄×C₇₁	#131	1	2	yes	—
171	C₂×C₄×C₇₁	—	2	4	no	—
172	C₂×C₄×C₇₁	—	2	1	no	—
173	C₂×C₄×C₇₁	—	2	1	no	—
174	C₂×C₄×C₇₁	—	2	1	no	—
175	C₂×C₄×C₇₁	—	2	1	no	—
176	C₂×C₄×C₇₁	—	2	1	no	—
177	C₂×C₄×C₇₁	—	2	2	no	—
178	C₂×C₄×C₇₁	—	2	1	no	—
179	C₂×C₄×C₇₁	—	2	1	no	—
180	C₂×C₄×C₇₁	—	2	2	no	—
181	C₂×C₄×C₇₁	—	2	2	no	—
182	C₂×C₄×C₇₁	—	4	1	no	—
183	C₂×C₄×C₇₁	—	4	1	no	—
184	C₂×C₄×C₇₁	—	4	2	no	—
185	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
186	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
187	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
188	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
189	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
191	C₈×C₇₁	—	2	2	no	—
192	C₈×C₇₁	—	2	2	no	—
193	C₇₁⋊C₈	—	2	1	no	—
194	C₇₁⋊C₈	—	2	1	no	—
195	C₇₁⋊C₈	—	2	1	no	—
196	C₇₁⋊C₈	—	4	2	no	—
197	C₇₁⋊C₈	—	4	1	no	—
198	C₇₁⋊C₈	—	4	1	no	—
199	C₇₁⋊C₈	—	4	2	no	—
200	C₇₁⋊C₈	#137	1	2	yes	—
201	C₂²×C₇₁	—	2	1	no	—
202	C₂²×C₇₁	—	2	2	no	—
203	C₂²×C₇₁	—	2	1	no	—
204	C₂²×C₇₁	—	2	1	no	—
205	C₂²×C₇₁	—	4	1	no	—
206	C₂²×C₇₁	—	4	1	no	—
207	C₂²×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
208	C₄×C₇₁	—	2	1	no	—
209	C₄×C₇₁	—	2	2	no	—
210	C₄×C₇₁	—	2	1	no	—
211	C₄×C₇₁	—	2	1	no	—
212	C₄×C₇₁	—	4	2	no	—
213	C₄×C₇₁	—	4	1	no	—
214	C₄×C₇₁	—	4	1	no	—
215	C₄×C₇₁	—	4	2	no	—
216	C₄×C₇₁	#444	1	2	yes	—
217	C₄×C₇₁	C₂×#131	1	1	yes	characteristic
218	C₄×C₇₁	C₂×#131	1	1	yes	characteristic
219	C₄×C₇₁	C₂×#137	1	1	yes	characteristic
220	C₄×C₇₁	#575	1	1	yes	characteristic
221	#17916	—	71	1	no	maximal, Sylow
222	C₂×C₇₁	—	4	1	no	—
223	C₂×C₇₁	—	4	2	no	—
224	C₂×C₇₁	#2929	1	1	yes	characteristic
225	C₂×C₇₁	C₄×#137	1	1	yes	characteristic
226	C₂×C₇₁	C₂×#444	1	1	yes	characteristic
227	#3107	D₇₁	1	1	yes	characteristic
228	C₈×#41	—	71	1	no	—
229	#2960	—	71	1	no	—
230	#1584	—	71	2	no	—
231	#1582	—	71	2	no	—
232	#1579	—	71	2	no	—
233	SD₁₆⋊C₈	—	71	2	no	—
234	#1574	—	71	1	no	—
235	#1489	—	71	1	no	—
236	#1485	—	71	1	no	—
237	#1451	—	71	1	no	—
238	C₇₁	#17916	1	1	yes	characteristic, complemented, Sylow
239	#533	C₂×D₇₁	1	1	yes	characteristic
240	C₄×#41	C₂×D₇₁	1	1	yes	characteristic
241	#485	Dic₇₁	1	2	yes	—
242	#475	Dic₇₁	1	1	yes	characteristic
243	#474	Dic₇₁	1	1	yes	characteristic
244	#462	Dic₇₁	1	1	yes	characteristic
245	#461	Dic₇₁	1	1	yes	characteristic
246	#452	Dic₇₁	1	2	yes	—
247	#446	—	71	2	no	—
248	#446	—	142	1	no	—
249	C₈×Q₈	—	71	2	no	—
250	C₈×Q₈	—	71	1	no	—
251	C₈×Q₈	—	71	1	no	—
252	C₈×Q₈	—	142	1	no	—
253	C₄×Q₁₆	C₂×D₇₁	1	2	yes	—
254	C₄×SD₁₆	C₂×D₇₁	1	2	yes	—
255	C₈×D₄	—	71	1	no	—
256	C₈×D₄	—	71	1	no	—
257	C₈×D₄	—	142	1	no	—
258	C₈×D₄	—	142	1	no	—
259	#431	—	71	1	no	—
260	#431	—	142	1	no	—
261	#431	C₂×D₇₁	1	1	yes	characteristic
262	#424	—	142	1	no	—
263	#423	—	71	1	no	—
264	#405	—	142	1	no	—
265	C₄×M₁₆	—	71	1	no	—
266	C₂×C₄×C₈	—	71	1	no	—
267	C₂×C₄×C₈	—	142	1	no	—
268	#335	—	71	2	no	—
269	#334	—	71	2	no	—
270	Q₈⋊C₈	—	71	2	no	—
271	Q₈⋊C₈	—	71	2	no	—
272	Q₈⋊C₈	—	71	2	no	—
273	D₄⋊C₈	—	71	2	no	—
274	#322	—	71	2	no	—
275	#322	—	71	2	no	—
276	C₂×#41	C₂×Dic₇₁	1	1	yes	characteristic
277	C₂×Q₁₆	C₂×Dic₇₁	1	2	yes	—
278	C₂×SD₁₆	C₂×Dic₇₁	1	2	yes	—
279	C₂×M₁₆	—	142	1	no	—
280	C₂×M₁₆	—	142	1	no	—
281	C₂²×C₈	—	71	1	no	—
282	C₂²×C₈	—	71	1	no	—
283	C₂²×C₈	—	142	1	no	—
284	C₂²×C₈	—	142	1	no	—
285	C₄⋊Q₈	C₂×Dic₇₁	1	1	yes	characteristic
286	C₄⋊Q₈	C₂×Dic₇₁	1	1	yes	characteristic
287	#125	—	2	1	no	—
288	C₂²⋊Q₈	—	2	1	no	—
289	C₂²⋊Q₈	C₂×Dic₇₁	1	2	yes	—
290	C₄×Q₈	—	2	1	no	—
291	C₄×Q₈	C₂×Dic₇₁	1	1	yes	characteristic
292	C₄×Q₈	C₂²×D₇₁	1	1	yes	characteristic
293	C₄×Q₈	C₂²×D₇₁	1	2	yes	—
294	C₄×D₄	—	2	1	no	—
295	C₄×D₄	—	2	1	no	—
296	C₄×D₄	C₂×Dic₇₁	1	1	yes	characteristic
297	C₄×D₄	C₂²×D₇₁	1	1	yes	characteristic
298	#117	—	2	1	no	—
299	#117	—	2	1	no	—
300	#117	C₂×Dic₇₁	1	1	yes	characteristic
301	C₂×C₄²	—	2	1	no	—
302	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
303	#107	C₂×Dic₇₁	1	2	yes	—
304	#106	C₂×Dic₇₁	1	2	yes	—
305	C₄⋊C₈	—	71	2	no	—
306	C₄⋊C₈	—	71	1	no	—
307	C₄⋊C₈	—	71	1	no	—
308	C₄⋊C₈	—	142	2	no	—
309	C₄⋊C₈	—	142	2	no	—
310	C₄⋊C₈	—	142	1	no	—
311	C₄⋊C₈	—	142	1	no	—
312	C₄⋊C₈	—	142	1	no	—
313	C₄⋊C₈	C₂×Dic₇₁	1	1	yes	characteristic
314	C₄⋊C₈	C₂×Dic₇₁	1	1	yes	characteristic
315	#103	C₂×Dic₇₁	1	2	yes	—
316	#103	C₂×Dic₇₁	1	2	yes	—
317	#103	C₂×Dic₇₁	1	2	yes	—
318	#102	C₂×Dic₇₁	1	2	yes	—
319	C₂²⋊C₈	—	142	1	no	—
320	C₂²⋊C₈	—	142	1	no	—
321	C₂²⋊C₈	—	142	1	no	—
322	C₂²⋊C₈	C₂×Dic₇₁	1	2	yes	—
323	#97	—	142	2	no	—
324	#97	—	142	1	no	—
325	C₄×C₈	—	71	1	no	—
326	C₄×C₈	—	71	1	no	—
327	C₄×C₈	—	71	1	no	—
328	C₄×C₈	—	71	1	no	—
329	C₄×C₈	—	71	2	no	—
330	C₄×C₈	—	142	1	no	—
331	C₄×C₈	—	142	1	no	—
332	C₄×C₈	—	142	2	no	—
333	C₄×C₈	—	142	1	no	—
334	C₄×C₈	C₂²×D₇₁	1	2	yes	—
335	#41	—	2	2	no	—
336	#41	—	2	2	no	—
337	C₂×Q₈	—	2	1	no	—
338	C₂×Q₈	C₂²×Dic₇₁	1	2	yes	—
339	C₂×Q₈	C₂²×Dic₇₁	1	1	yes	characteristic
340	C₂×D₄	—	2	1	no	—
341	C₂×D₄	C₂²×Dic₇₁	1	1	yes	characteristic
342	C₂²×C₄	—	2	1	no	—
343	C₂²×C₄	—	2	1	no	—
344	C₂²×C₄	—	2	1	no	—
345	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
346	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
347	C₂²×C₄	#11765864	1	1	yes	characteristic
348	Q₁₆	—	2	4	no	—
349	SD₁₆	—	2	4	no	—
350	M₁₆	—	284	2	no	—
351	M₁₆	—	284	2	no	—
352	C₂×C₈	—	284	2	no	—
353	C₂×C₈	—	284	2	no	—
354	C₂×C₈	—	284	1	no	—
355	C₂×C₈	—	71	1	no	—
356	C₂×C₈	—	71	1	no	—
357	C₂×C₈	—	71	1	no	—
358	C₂×C₈	—	71	1	no	—
359	C₂×C₈	—	142	1	no	—
360	C₂×C₈	—	142	1	no	—
361	C₂×C₈	—	142	2	no	—
362	C₂×C₈	—	142	2	no	—
363	C₂×C₈	—	142	2	no	—
364	C₂×C₈	C₂²×Dic₇₁	1	2	yes	—
365	C₂×C₈	C₂²×Dic₇₁	1	2	yes	—
366	C₄⋊C₄	—	2	1	no	—
367	C₄⋊C₄	—	2	2	no	—
368	C₄⋊C₄	—	2	1	no	—
369	C₄⋊C₄	—	2	1	no	—
370	C₄⋊C₄	—	2	1	no	—
371	C₄⋊C₄	—	2	1	no	—
372	C₄⋊C₄	—	2	2	no	—
373	C₄⋊C₄	C₂²×Dic₇₁	1	2	yes	—
374	C₄⋊C₄	C₂²×Dic₇₁	1	1	yes	characteristic
375	C₄⋊C₄	C₂²×Dic₇₁	1	1	yes	characteristic
376	C₂²⋊C₄	—	2	2	no	—
377	C₂²⋊C₄	—	2	1	no	—
378	C₂²⋊C₄	—	2	1	no	—
379	C₂²⋊C₄	—	2	1	no	—
380	C₄²	—	2	1	no	—
381	C₄²	—	2	2	no	—
382	C₄²	—	2	1	no	—
383	C₄²	—	2	1	no	—
384	C₄²	C₂²×Dic₇₁	1	1	yes	characteristic
385	C₄²	D₄×D₇₁	1	1	yes	characteristic
386	C₄²	#11765864	1	1	yes	characteristic
387	C₄²	C₂³×D₇₁	1	1	yes	characteristic
388	C₂³	—	2	1	no	—
389	C₂³	D₄×Dic₇₁	1	1	yes	characteristic
390	Q₈	—	2	1	no	—
391	Q₈	—	2	2	no	—
392	Q₈	—	2	2	no	—
393	Q₈	—	4	2	no	—
394	Q₈	#14556135	1	2	yes	—
395	D₄	—	2	4	no	—
396	D₄	—	2	1	no	—
397	D₄	#14556135	1	2	yes	—
398	C₂×C₄	—	2	4	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	2	1	no	—
403	C₂×C₄	—	2	1	no	—
404	C₂×C₄	—	2	1	no	—
405	C₂×C₄	—	2	1	no	—
406	C₂×C₄	—	2	1	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	4	2	no	—
410	C₂×C₄	—	4	1	no	—
411	C₂×C₄	—	4	1	no	—
412	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
413	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
414	C₂×C₄	D₄×Dic₇₁	1	1	yes	characteristic
415	C₂×C₄	C₂³×Dic₇₁	1	1	yes	characteristic
416	C₂×C₄	C₂×D₄×D₇₁	1	1	yes	characteristic
417	C₂×C₄	C₂×#11765864	1	1	yes	characteristic
418	C₈	—	2	2	no	—
419	C₈	—	2	2	no	—
420	C₈	—	284	2	no	—
421	C₈	—	284	1	no	—
422	C₈	—	284	1	no	—
423	C₈	—	284	2	no	—
424	C₈	—	71	2	no	—
425	C₈	—	142	1	no	—
426	C₈	—	142	1	no	—
427	C₈	—	142	1	no	—
428	C₂²	—	2	1	no	—
429	C₂²	—	2	1	no	—
430	C₂²	—	2	1	no	—
431	C₂²	—	2	2	no	—
432	C₂²	—	4	1	no	—
433	C₂²	—	4	1	no	—
434	C₂²	C₂×D₄×Dic₇₁	1	1	yes	characteristic
435	C₄	—	2	1	no	—
436	C₄	—	2	1	no	—
437	C₄	—	2	1	no	—
438	C₄	—	2	2	no	—
439	C₄	—	4	1	no	—
440	C₄	—	4	1	no	—
441	C₄	—	4	2	no	—
442	C₄	—	4	2	no	—
443	C₄	#17986121	1	2	yes	—
444	C₄	C₂×#14556135	1	1	yes	characteristic
445	C₄	C₂×#14556135	1	1	yes	characteristic
446	C₄	#137×D₇₁	1	1	yes	characteristic
447	C₄	#17986798	1	1	yes	characteristic
448	C₂	—	4	1	no	—
449	C₂	—	4	2	no	—
450	C₂	#22597035	1	1	yes	characteristic
451	C₂	#137×Dic₇₁	1	1	yes	characteristic
452	C₂	C₂×#17986121	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂³×C₄
Frattini	C₄²	C₂³×D₇₁
Fitting	C₇₁×#3107	C₂
socle	C₂²×C₇₁	C₂×C₄×D₄
prime radical	C₂×#41×C₇₁	C₂×C₄
cosocle	C₄²×C₇₁	C₂⁴
hypercenter	#3107	D₇₁
hypocenter	C₇₁	#17916
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3107

Complements

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

N	H ≅ G/N	type
2. #22597275	C₂	semidirect
3. #22587191	C₂	semidirect
4. #22587177	C₂	semidirect
5. #22587159	C₂	semidirect
6. SD₁₆⋊(C₇₁⋊C₈)	C₂	semidirect
7. #22587129	C₂	semidirect
9. #22586761	C₂	semidirect
10. #22586630	C₂	semidirect
238. C₇₁	#17916	semidirect

Maximal subgroups

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

subgroup	index
#41×C₇₁⋊C₈	2
#22597275	2
#22587191	2
#22587177	2
#22587159	2
SD₁₆⋊(C₇₁⋊C₈)	2
#22587129	2
#22586779	2
#22586761	2
#22586630	2
C₇₁×#3107	2
#17916	71