Subgroups — #31967478

Order 18176.

624 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22598418	C₂	1	1	yes	maximal, characteristic
2	#22597312	C₂	1	1	yes	maximal, characteristic
3	C₈×Q₈×D₇₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#22587189	C₂	1	1	yes	maximal, characteristic, complemented
5	#22587187	C₂	1	1	yes	maximal, characteristic, complemented
6	#22587186	C₂	1	1	yes	maximal, characteristic, complemented
7	#22587184	C₂	1	1	yes	maximal, characteristic, complemented
8	#22587160	C₂	1	1	yes	maximal, characteristic, complemented
9	#22587157	C₂	1	1	yes	maximal, characteristic, complemented
10	#22587152	C₂	1	1	yes	maximal, characteristic, complemented
11	#22587124	C₂	1	1	yes	maximal, characteristic, complemented
12	#22586785	C₂	1	1	yes	maximal, characteristic, complemented
13	#22586750	C₂	1	1	yes	maximal, characteristic
14	#22586628	C₂	1	1	yes	maximal, characteristic, complemented
15	C₇₁×SD₁₆⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
16	#17986531	C₂²	1	1	yes	characteristic
17	C₄×Q₈×D₇₁	C₂²	1	1	yes	characteristic
18	#17986331	C₄	1	1	yes	characteristic
19	#17986311	C₄	1	1	yes	characteristic
20	#17986309	C₄	1	1	yes	characteristic
21	#17986308	C₄	1	1	yes	characteristic
22	#17986220	C₄	1	1	yes	characteristic
23	#17986219	C₄	1	1	yes	characteristic
24	#17986156	C₄	1	1	yes	characteristic
25	#17986136	C₄	1	1	yes	characteristic
26	#17986126	C₂²	1	1	yes	characteristic
27	Q₈×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
28	C₈×Dic₁₄₂	—	2	1	no	—
29	C₈×Dic₁₄₂	C₂²	1	1	yes	characteristic
30	C₈×Dic₁₄₂	C₂²	1	1	yes	characteristic
31	#17986123	—	2	1	no	—
32	#17986123	C₂²	1	1	yes	characteristic
33	C₄×Dic₂₈₄	C₂²	1	1	yes	characteristic
34	C₄×#11765855	C₂²	1	1	yes	characteristic
35	C₄×#11765853	C₂²	1	1	yes	characteristic
36	#17986077	C₂²	1	1	yes	characteristic
37	#17986073	—	2	1	no	—
38	C₈×C₂²⋊D₇₁	—	2	1	no	—
39	#17986057	—	2	1	no	—
40	#17986055	C₂²	1	1	yes	characteristic
41	#17986050	C₂²	1	1	yes	characteristic
42	C₄⋊C₈×D₇₁	—	2	1	no	—
43	C₄⋊C₈×D₇₁	C₂²	1	1	yes	characteristic
44	#17985954	—	2	1	no	—
45	C₄×#11765848	C₂²	1	1	yes	characteristic
46	C₄×C₈×D₇₁	—	2	1	no	—
47	C₄×C₈×D₇₁	C₂²	1	1	yes	characteristic
48	#17985751	C₂²	1	1	yes	characteristic
49	#17985750	C₂²	1	1	yes	characteristic
50	#17985749	C₂²	1	1	yes	characteristic
51	#17985731	C₂²	1	1	yes	characteristic
52	#17985731	C₂²	1	1	yes	characteristic
53	Q₈⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
54	#17985729	C₂²	1	1	yes	characteristic
55	#17985729	C₂²	1	1	yes	characteristic
56	D₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
57	#17985721	C₂²	1	1	yes	characteristic
58	#17985721	C₂²	1	1	yes	characteristic
59	#17985721	C₂²	1	1	yes	characteristic
60	C₈×Q₈×C₇₁	C₂²	1	1	yes	characteristic
61	C₄×SD₁₆×C₇₁	C₂²	1	1	yes	characteristic
62	C₈×D₄×C₇₁	C₂²	1	1	yes	characteristic
63	#335×C₇₁	C₂²	1	1	yes	characteristic
64	Q₈⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
65	D₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
66	#322×C₇₁	C₂²	1	1	yes	characteristic
67	C₂×Q₈×D₇₁	C₂×C₄	1	1	yes	characteristic
68	C₂×Dic₂₈₄	C₂×C₄	1	1	yes	characteristic
69	C₂×#11765855	C₂×C₄	1	1	yes	characteristic
70	C₂×#11765853	C₂×C₄	1	1	yes	characteristic
71	C₂×#11765849	—	2	1	no	—
72	C₂×#11765848	—	2	1	no	—
73	C₂×C₈×D₇₁	—	2	1	no	—
74	C₂×C₈×D₇₁	D₄	1	1	yes	characteristic
75	C₂×C₈×D₇₁	D₄	1	1	yes	characteristic
76	#14556121	—	2	1	no	—
77	#14556119	C₂×C₄	1	1	yes	characteristic
78	#14556115	C₂×C₄	1	1	yes	characteristic
79	#14556110	—	2	1	no	—
80	#14556108	C₂×C₄	1	1	yes	characteristic
81	#14556099	C₂×C₄	1	1	yes	characteristic
82	Q₈×Dic₇₁	C₂×C₄	1	1	yes	characteristic
83	C₄×Dic₁₄₂	—	2	1	no	—
84	C₄×Dic₁₄₂	C₂³	1	1	yes	characteristic
85	C₄×Dic₁₄₂	C₂³	1	1	yes	characteristic
86	#14556093	—	2	1	no	—
87	#14556093	C₂×C₄	1	1	yes	characteristic
88	C₄×C₂²⋊D₇₁	—	2	1	no	—
89	#14556087	—	2	1	no	—
90	C₄⋊C₄×D₇₁	—	2	1	no	—
91	C₄⋊C₄×D₇₁	C₂×C₄	1	1	yes	characteristic
92	C₄²×D₇₁	—	2	1	no	—
93	C₄²×D₇₁	C₂³	1	1	yes	characteristic
94	#14556062	C₂×C₄	1	1	yes	characteristic
95	#14556061	C₂×C₄	1	1	yes	characteristic
96	#14556059	C₂×C₄	1	1	yes	characteristic
97	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
98	C₄⋊(C₇₁⋊C₈)	C₂³	1	1	yes	characteristic
99	C₄⋊(C₇₁⋊C₈)	C₂³	1	1	yes	characteristic
100	Dic₇₁⋊C₈	—	2	1	no	—
101	Dic₇₁⋊C₈	—	2	1	no	—
102	Dic₇₁⋊C₈	—	2	1	no	—
103	Dic₇₁⋊C₈	—	2	1	no	—
104	Dic₇₁⋊C₈	—	2	1	no	—
105	Dic₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
106	Dic₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
107	#14556053	C₂×C₄	1	1	yes	characteristic
108	#14556053	C₂×C₄	1	1	yes	characteristic
109	#14556052	C₂×C₄	1	1	yes	characteristic
110	#14556051	C₂×C₄	1	1	yes	characteristic
111	#14556051	C₂×C₄	1	1	yes	characteristic
112	#14556049	C₂×C₄	1	1	yes	characteristic
113	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
114	(C₂×D₇₁)⋊C₈	—	2	1	no	—
115	(C₂×D₇₁)⋊C₈	C₂×C₄	1	1	yes	characteristic
116	(C₂×D₇₁)⋊C₈	C₂×C₄	1	1	yes	characteristic
117	#14556038	—	2	1	no	—
118	#14556038	—	2	1	no	—
119	#14556038	—	2	1	no	—
120	C₄×C₇₁⋊C₈	—	2	1	no	—
121	C₄×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
122	C₄×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
123	C₄×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
124	C₈×Dic₇₁	—	2	1	no	—
125	C₈×Dic₇₁	—	2	1	no	—
126	C₈×Dic₇₁	—	2	1	no	—
127	C₈×Dic₇₁	D₄	1	1	yes	characteristic
128	C₈×Dic₇₁	D₄	1	1	yes	characteristic
129	C₂×SD₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
130	C₂²×C₈×C₇₁	—	2	1	no	—
131	C₄×Q₈×C₇₁	C₂³	1	1	yes	characteristic
132	C₄×D₄×C₇₁	C₂³	1	1	yes	characteristic
133	#106×C₇₁	C₂×C₄	1	1	yes	characteristic
134	C₄⋊C₈×C₇₁	—	2	1	no	—
135	C₄⋊C₈×C₇₁	C₂³	1	1	yes	characteristic
136	C₄⋊C₈×C₇₁	C₂³	1	1	yes	characteristic
137	#103×C₇₁	C₂×C₄	1	1	yes	characteristic
138	#102×C₇₁	C₂×C₄	1	1	yes	characteristic
139	C₂²⋊C₈×C₇₁	—	2	1	no	—
140	C₄×C₈×C₇₁	—	2	1	no	—
141	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
142	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
143	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
144	C₂×Dic₁₄₂	—	2	1	no	—
145	C₂×Dic₁₄₂	C₂²×C₄	1	1	yes	characteristic
146	C₂×Dic₁₄₂	C₂²×C₄	1	1	yes	characteristic
147	Q₈×D₇₁	—	2	2	no	—
148	Q₈×D₇₁	—	2	2	no	—
149	C₂×C₂²⋊D₇₁	—	2	1	no	—
150	C₂×C₄×D₇₁	—	2	1	no	—
151	C₂×C₄×D₇₁	—	2	1	no	—
152	C₂×C₄×D₇₁	C₂²×C₄	1	1	yes	characteristic
153	C₂×C₄×D₇₁	C₂×D₄	1	1	yes	characteristic
154	C₂×C₄×D₇₁	#41	1	1	yes	characteristic
155	Dic₂₈₄	—	2	2	no	—
156	#11765855	—	2	2	no	—
157	#11765853	—	2	2	no	—
158	#11765849	—	4	2	no	—
159	#11765848	—	4	2	no	—
160	C₂×C₇₁⋊C₈	—	2	1	no	—
161	C₂×C₇₁⋊C₈	—	2	1	no	—
162	C₂×C₇₁⋊C₈	—	2	1	no	—
163	C₂×C₇₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
164	C₂×C₇₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
165	C₂×C₇₁⋊C₈	C₂×D₄	1	1	yes	characteristic
166	C₂×C₇₁⋊C₈	C₂×D₄	1	1	yes	characteristic
167	C₈×D₇₁	—	2	2	no	—
168	C₈×D₇₁	—	4	1	no	—
169	C₈×D₇₁	—	4	2	no	—
170	C₄⋊Dic₇₁	—	2	1	no	—
171	C₄⋊Dic₇₁	—	2	1	no	—
172	C₄⋊Dic₇₁	—	2	1	no	—
173	C₄⋊Dic₇₁	C₂²×C₄	1	1	yes	characteristic
174	C₄⋊Dic₇₁	C₂²×C₄	1	1	yes	characteristic
175	Dic₇₁⋊C₄	—	2	1	no	—
176	Dic₇₁⋊C₄	—	2	1	no	—
177	Dic₇₁⋊C₄	—	2	1	no	—
178	Dic₇₁⋊C₄	—	2	1	no	—
179	Dic₇₁⋊C₄	—	2	1	no	—
180	C₂²⋊Dic₇₁	—	2	1	no	—
181	C₂²⋊Dic₇₁	—	2	1	no	—
182	C₂²⋊Dic₇₁	—	2	1	no	—
183	(C₂×D₇₁)⋊C₄	—	2	1	no	—
184	C₄×Dic₇₁	—	2	1	no	—
185	C₄×Dic₇₁	—	2	1	no	—
186	C₄×Dic₇₁	—	2	1	no	—
187	C₄×Dic₇₁	—	2	1	no	—
188	C₄×Dic₇₁	C₂²×C₄	1	1	yes	characteristic
189	C₄×Dic₇₁	C₂×D₄	1	1	yes	characteristic
190	C₄×Dic₇₁	#41	1	1	yes	characteristic
191	C₂×Q₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
192	C₂×D₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
193	C₂²×C₄×C₇₁	—	2	1	no	—
194	SD₁₆×C₇₁	—	2	2	no	—
195	C₂×C₈×C₇₁	—	2	1	no	—
196	C₂×C₈×C₇₁	—	2	1	no	—
197	C₂×C₈×C₇₁	—	2	1	no	—
198	C₂×C₈×C₇₁	—	4	2	no	—
199	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
200	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
201	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
202	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
203	C₄⋊C₄×C₇₁	—	2	1	no	—
204	C₄⋊C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
205	C₄⋊C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
206	C₂²⋊C₄×C₇₁	—	2	1	no	—
207	C₄²×C₇₁	—	2	1	no	—
208	C₄²×C₇₁	C₂⁴	1	1	yes	characteristic
209	C₂³×C₇₁	—	2	1	no	—
210	C₂²×D₇₁	C₄×D₄	1	1	yes	characteristic
211	Q₈×C₇₁	—	2	1	no	—
212	Q₈×C₇₁	#131	1	2	yes	—
213	D₄×C₇₁	—	2	1	no	—
214	D₄×C₇₁	—	2	1	no	—
215	C₂×C₄×C₇₁	—	2	1	no	—
216	C₂×C₄×C₇₁	—	2	1	no	—
217	C₂×C₄×C₇₁	—	2	1	no	—
218	C₂×C₄×C₇₁	—	4	1	no	—
219	C₂×C₄×C₇₁	—	4	1	no	—
220	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
221	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
223	C₂²⋊D₇₁	—	4	2	no	—
224	C₂×Dic₇₁	—	2	1	no	—
225	C₂×Dic₇₁	—	2	1	no	—
226	C₂×Dic₇₁	—	2	1	no	—
227	C₂×Dic₇₁	—	2	1	no	—
228	C₂×Dic₇₁	C₄×D₄	1	1	yes	characteristic
229	C₂×Dic₇₁	C₄×D₄	1	1	yes	characteristic
230	C₂×Dic₇₁	C₄×D₄	1	1	yes	characteristic
231	C₄×D₇₁	—	2	4	no	—
232	C₄×D₇₁	—	2	2	no	—
233	C₄×D₇₁	—	2	1	no	—
234	C₄×D₇₁	—	2	1	no	—
235	C₄×D₇₁	—	2	1	no	—
236	C₄×D₇₁	—	2	1	no	—
237	C₄×D₇₁	—	4	2	no	—
238	Dic₁₄₂	—	2	4	no	—
239	Dic₁₄₂	—	2	1	no	—
240	Dic₁₄₂	—	2	1	no	—
241	Dic₁₄₂	—	2	1	no	—
242	Dic₁₄₂	#131	1	2	yes	—
243	C₈×C₇₁	—	2	1	no	—
244	C₈×C₇₁	—	2	1	no	—
245	C₈×C₇₁	—	2	1	no	—
246	C₈×C₇₁	—	2	1	no	—
247	C₈×C₇₁	—	4	1	no	—
248	C₈×C₇₁	—	4	1	no	—
249	C₈×C₇₁	—	4	1	no	—
250	C₇₁⋊C₈	—	2	1	no	—
251	C₇₁⋊C₈	—	2	1	no	—
252	C₇₁⋊C₈	—	2	1	no	—
253	C₇₁⋊C₈	—	4	1	no	—
254	C₇₁⋊C₈	—	4	1	no	—
255	C₇₁⋊C₈	—	4	1	no	—
256	C₇₁⋊C₈	#137	1	2	yes	—
257	C₂²×C₇₁	—	4	1	no	—
258	C₂²×C₇₁	—	4	1	no	—
259	C₂²×C₇₁	—	4	1	no	—
260	C₂²×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
261	C₂×D₇₁	—	2	1	no	—
262	C₂×D₇₁	—	2	1	no	—
263	C₂×D₇₁	#444	1	2	yes	—
264	C₄×C₇₁	—	2	2	no	—
265	C₄×C₇₁	—	2	1	no	—
266	C₄×C₇₁	—	4	1	no	—
267	C₄×C₇₁	—	4	1	no	—
268	C₄×C₇₁	C₂×#131	1	1	yes	characteristic
269	C₄×C₇₁	C₂×#131	1	1	yes	characteristic
270	C₄×C₇₁	C₂×#137	1	1	yes	characteristic
271	C₄×C₇₁	#575	1	1	yes	characteristic
272	Dic₇₁	—	2	1	no	—
273	Dic₇₁	—	2	2	no	—
274	Dic₇₁	—	2	1	no	—
275	Dic₇₁	—	2	1	no	—
276	Dic₇₁	—	4	1	no	—
277	Dic₇₁	—	4	1	no	—
278	Dic₇₁	—	4	1	no	—
279	#17915	—	71	1	no	maximal, Sylow
280	C₂×C₇₁	—	8	1	no	—
281	C₂×C₇₁	#2929	1	1	yes	characteristic
282	C₂×C₇₁	C₄×#137	1	1	yes	characteristic
283	C₂×C₇₁	C₂×#444	1	1	yes	characteristic
284	D₇₁	—	2	2	no	—
285	#3108	—	71	1	no	—
286	#2966	—	71	1	no	—
287	C₂×C₈×Q₈	—	71	1	no	—
288	#1584	—	71	1	no	—
289	#1584	—	71	1	no	—
290	#1583	—	71	1	no	—
291	#1583	—	71	1	no	—
292	#1579	—	71	1	no	—
293	#1579	—	71	1	no	—
294	SD₁₆⋊C₈	—	71	1	no	—
295	SD₁₆⋊C₈	D₇₁	1	1	yes	characteristic, complemented
296	#1573	—	71	1	no	—
297	#1491	—	71	1	no	—
298	#1483	—	71	1	no	—
299	#1451	—	71	1	no	—
300	C₇₁	#17915	1	1	yes	characteristic, complemented, Sylow
301	#529	—	71	1	no	—
302	C₂×C₄×Q₈	—	71	1	no	—
303	#483	—	71	1	no	—
304	#480	—	71	1	no	—
305	#479	—	71	1	no	—
306	#479	—	71	1	no	—
307	#464	—	71	1	no	—
308	#464	—	71	1	no	—
309	#451	—	71	1	no	—
310	#448	—	71	1	no	—
311	#446	—	71	1	no	—
312	C₈×Q₈	—	71	1	no	—
313	C₈×Q₈	—	71	1	no	—
314	C₈×Q₈	—	71	1	no	—
315	C₈×Q₈	—	71	1	no	—
316	C₈×Q₈	—	142	1	no	—
317	C₈×Q₈	—	142	1	no	—
318	C₈×Q₈	C₂×D₇₁	1	1	yes	characteristic
319	C₄×Q₁₆	—	71	1	no	—
320	C₄×Q₁₆	—	71	1	no	—
321	C₄×SD₁₆	—	71	1	no	—
322	C₄×SD₁₆	C₂×D₇₁	1	1	yes	characteristic
323	#436	—	71	1	no	—
324	#435	—	142	1	no	—
325	C₈×D₄	—	142	1	no	—
326	C₈×D₄	C₂×D₇₁	1	1	yes	characteristic
327	#433	—	142	1	no	—
328	#432	—	71	1	no	—
329	#431	—	71	1	no	—
330	C₂×C₄⋊C₈	—	71	1	no	—
331	C₂×C₄⋊C₈	—	142	1	no	—
332	#405	—	142	1	no	—
333	C₄×M₁₆	—	71	1	no	—
334	C₂×C₄×C₈	—	71	1	no	—
335	C₂×C₄×C₈	—	142	1	no	—
336	#335	—	71	1	no	—
337	#335	C₂×D₇₁	1	1	yes	characteristic
338	#334	—	71	1	no	—
339	#334	—	71	1	no	—
340	Q₈⋊C₈	—	71	1	no	—
341	Q₈⋊C₈	—	71	1	no	—
342	Q₈⋊C₈	—	71	1	no	—
343	Q₈⋊C₈	—	71	1	no	—
344	Q₈⋊C₈	—	71	1	no	—
345	Q₈⋊C₈	C₂×D₇₁	1	1	yes	characteristic
346	D₄⋊C₈	—	71	1	no	—
347	D₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
348	#322	—	71	1	no	—
349	#322	—	71	1	no	—
350	#322	—	71	1	no	—
351	#322	C₂×D₇₁	1	1	yes	characteristic
352	C₂²×Q₈	—	71	1	no	—
353	C₂×Q₁₆	—	71	1	no	—
354	C₂×Q₁₆	—	71	1	no	—
355	C₂×SD₁₆	—	71	1	no	—
356	C₂×SD₁₆	C₄×D₇₁	1	1	yes	characteristic
357	C₂×M₁₆	—	142	1	no	—
358	C₂×M₁₆	—	142	1	no	—
359	C₂²×C₈	—	2	1	no	—
360	C₂²×C₈	—	71	1	no	—
361	C₂²×C₈	—	71	1	no	—
362	C₂²×C₈	—	142	1	no	—
363	#126	—	142	1	no	—
364	#125	—	71	1	no	—
365	#124	—	71	1	no	—
366	#123	—	142	1	no	—
367	C₂²⋊Q₈	—	71	1	no	—
368	#121	—	71	1	no	—
369	C₄×Q₈	—	71	1	no	—
370	C₄×Q₈	—	71	1	no	—
371	C₄×Q₈	—	71	1	no	—
372	C₄×Q₈	—	71	1	no	—
373	C₄×Q₈	—	142	1	no	—
374	C₄×Q₈	—	142	1	no	—
375	C₄×Q₈	C₂²×D₇₁	1	1	yes	characteristic
376	C₄×D₄	—	142	1	no	—
377	C₄×D₄	C₂²×D₇₁	1	1	yes	characteristic
378	#117	—	142	1	no	—
379	C₂×C₄⋊C₄	—	71	1	no	—
380	C₂×C₄⋊C₄	—	142	1	no	—
381	C₂×C₄²	—	71	1	no	—
382	C₂×C₄²	—	142	1	no	—
383	#107	—	71	1	no	—
384	#107	—	71	1	no	—
385	#106	—	71	1	no	—
386	#106	C₄×D₇₁	1	1	yes	characteristic
387	C₄⋊C₈	—	2	1	no	—
388	C₄⋊C₈	—	71	1	no	—
389	C₄⋊C₈	—	71	1	no	—
390	C₄⋊C₈	—	71	1	no	—
391	C₄⋊C₈	—	71	1	no	—
392	C₄⋊C₈	—	142	1	no	—
393	C₄⋊C₈	—	142	1	no	—
394	C₄⋊C₈	—	142	1	no	—
395	C₄⋊C₈	—	142	1	no	—
396	C₄⋊C₈	—	142	1	no	—
397	C₄⋊C₈	—	142	1	no	—
398	C₄⋊C₈	C₂²×D₇₁	1	1	yes	characteristic
399	C₄⋊C₈	C₂²×D₇₁	1	1	yes	characteristic
400	#103	—	71	1	no	—
401	#103	—	71	1	no	—
402	#103	—	71	1	no	—
403	#103	—	71	1	no	—
404	#103	—	71	1	no	—
405	#103	C₄×D₇₁	1	1	yes	characteristic
406	#102	—	71	1	no	—
407	#102	C₄×D₇₁	1	1	yes	characteristic
408	C₂²⋊C₈	—	2	1	no	—
409	C₂²⋊C₈	—	71	1	no	—
410	C₂²⋊C₈	—	71	1	no	—
411	C₂²⋊C₈	—	142	1	no	—
412	C₂²⋊C₈	—	142	1	no	—
413	#97	—	142	1	no	—
414	#97	—	142	1	no	—
415	#97	—	142	1	no	—
416	C₄×C₈	—	2	1	no	—
417	C₄×C₈	—	71	1	no	—
418	C₄×C₈	—	71	1	no	—
419	C₄×C₈	—	71	1	no	—
420	C₄×C₈	—	71	1	no	—
421	C₄×C₈	—	71	1	no	—
422	C₄×C₈	—	142	1	no	—
423	C₄×C₈	—	142	1	no	—
424	C₄×C₈	—	142	1	no	—
425	C₄×C₈	—	142	1	no	—
426	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
427	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
428	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
429	C₂×Q₈	—	71	1	no	—
430	C₂×Q₈	—	71	1	no	—
431	C₂×Q₈	—	142	1	no	—
432	C₂×Q₈	—	142	2	no	—
433	C₂×Q₈	—	142	2	no	—
434	C₂×Q₈	C₂×C₄×D₇₁	1	1	yes	characteristic
435	C₂×D₄	—	142	1	no	—
436	C₂×D₄	C₂×C₄×D₇₁	1	1	yes	characteristic
437	C₂²×C₄	—	2	1	no	—
438	C₂²×C₄	—	71	1	no	—
439	C₂²×C₄	—	71	1	no	—
440	C₂²×C₄	—	71	1	no	—
441	C₂²×C₄	—	142	1	no	—
442	C₂²×C₄	—	142	1	no	—
443	Q₁₆	—	142	2	no	—
444	Q₁₆	—	142	2	no	—
445	SD₁₆	—	2	2	no	—
446	SD₁₆	—	142	2	no	—
447	M₁₆	—	284	2	no	—
448	M₁₆	—	284	2	no	—
449	C₂×C₈	—	2	1	no	—
450	C₂×C₈	—	2	1	no	—
451	C₂×C₈	—	2	1	no	—
452	C₂×C₈	—	4	2	no	—
453	C₂×C₈	—	284	1	no	—
454	C₂×C₈	—	284	2	no	—
455	C₂×C₈	—	71	1	no	—
456	C₂×C₈	—	71	1	no	—
457	C₂×C₈	—	71	1	no	—
458	C₂×C₈	—	71	1	no	—
459	C₂×C₈	—	142	2	no	—
460	C₂×C₈	—	142	1	no	—
461	C₂×C₈	—	142	1	no	—
462	C₂×C₈	—	142	1	no	—
463	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
464	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
465	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
466	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	71	1	no	—
469	C₄⋊C₄	—	71	1	no	—
470	C₄⋊C₄	—	142	1	no	—
471	C₄⋊C₄	—	142	1	no	—
472	C₄⋊C₄	—	142	1	no	—
473	C₄⋊C₄	—	142	1	no	—
474	C₄⋊C₄	—	142	1	no	—
475	C₄⋊C₄	—	142	1	no	—
476	C₄⋊C₄	—	142	1	no	—
477	C₄⋊C₄	—	142	1	no	—
478	C₄⋊C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
479	C₄⋊C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
480	C₂²⋊C₄	—	2	1	no	—
481	C₂²⋊C₄	—	142	1	no	—
482	C₂²⋊C₄	—	142	1	no	—
483	C₂²⋊C₄	—	142	1	no	—
484	C₂²⋊C₄	—	142	1	no	—
485	C₄²	—	2	1	no	—
486	C₄²	—	71	1	no	—
487	C₄²	—	71	1	no	—
488	C₄²	—	71	1	no	—
489	C₄²	—	142	1	no	—
490	C₄²	—	142	1	no	—
491	C₄²	—	142	1	no	—
492	C₄²	—	142	1	no	—
493	C₄²	C₂³×D₇₁	1	1	yes	characteristic
494	C₂³	—	2	1	no	—
495	C₂³	—	71	1	no	—
496	Q₈	—	2	1	no	—
497	Q₈	—	71	2	no	—
498	Q₈	—	142	4	no	—
499	Q₈	—	142	1	no	—
500	Q₈	—	142	1	no	—
501	Q₈	—	142	1	no	—
502	Q₈	#14556134	1	2	yes	—
503	D₄	—	2	1	no	—
504	D₄	—	2	1	no	—
505	D₄	—	284	2	no	—
506	C₂×C₄	—	2	1	no	—
507	C₂×C₄	—	2	1	no	—
508	C₂×C₄	—	2	1	no	—
509	C₂×C₄	—	4	1	no	—
510	C₂×C₄	—	4	1	no	—
511	C₂×C₄	—	284	2	no	—
512	C₂×C₄	—	71	1	no	—
513	C₂×C₄	—	71	1	no	—
514	C₂×C₄	—	71	1	no	—
515	C₂×C₄	—	142	2	no	—
516	C₂×C₄	—	142	4	no	—
517	C₂×C₄	—	142	1	no	—
518	C₂×C₄	—	142	1	no	—
519	C₂×C₄	—	142	1	no	—
520	C₂×C₄	—	142	1	no	—
521	C₂×C₄	—	142	1	no	—
522	C₂×C₄	—	142	1	no	—
523	C₂×C₄	—	142	1	no	—
524	C₂×C₄	—	142	1	no	—
525	C₂×C₄	C₂²×C₄×D₇₁	1	1	yes	characteristic
526	C₂×C₄	C₂×D₄×D₇₁	1	1	yes	characteristic
527	C₂×C₄	#41×D₇₁	1	1	yes	characteristic
528	C₈	—	2	1	no	—
529	C₈	—	2	1	no	—
530	C₈	—	2	1	no	—
531	C₈	—	2	1	no	—
532	C₈	—	4	1	no	—
533	C₈	—	4	1	no	—
534	C₈	—	4	1	no	—
535	C₈	—	284	1	no	—
536	C₈	—	284	1	no	—
537	C₈	—	284	1	no	—
538	C₈	—	71	2	no	—
539	C₈	—	142	1	no	—
540	C₈	—	142	1	no	—
541	C₈	—	142	1	no	—
542	C₂²	—	4	1	no	—
543	C₂²	—	4	1	no	—
544	C₂²	—	4	1	no	—
545	C₂²	—	71	2	no	—
546	C₂²	—	142	1	no	—
547	C₂²	—	142	1	no	—
548	C₂²	C₄×D₄×D₇₁	1	1	yes	characteristic
549	C₄	—	2	1	no	—
550	C₄	—	2	2	no	—
551	C₄	—	4	1	no	—
552	C₄	—	4	1	no	—
553	C₄	—	284	1	no	—
554	C₄	—	284	1	no	—
555	C₄	—	284	1	no	—
556	C₄	—	142	1	no	—
557	C₄	—	142	1	no	—
558	C₄	—	142	1	no	—
559	C₄	—	142	2	no	—
560	C₄	#131×D₇₁	1	1	yes	characteristic
561	C₄	C₂×#14556134	1	1	yes	characteristic
562	C₄	#137×D₇₁	1	1	yes	characteristic
563	C₄	#17986795	1	1	yes	characteristic
564	C₂	—	8	1	no	—
565	C₂	—	142	2	no	—
566	C₂	#22597029	1	1	yes	characteristic
567	C₂	#22597162	1	1	yes	characteristic
568	C₂	#444×D₇₁	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×SD₁₆⋊C₈	C₂
socle	C₂²×C₇₁	C₂×C₄×D₄
prime radical	#14556099	C₂×C₄
cosocle	C₄²×C₇₁	C₂⁴
hypercenter	SD₁₆⋊C₈	D₇₁
hypocenter	C₇₁	#17915
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ SD₁₆⋊C₈

Complements

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

N	H ≅ G/N	type
3. C₈×Q₈×D₇₁	C₂	semidirect
4. #22587189	C₂	semidirect
5. #22587187	C₂	semidirect
6. #22587186	C₂	semidirect
7. #22587184	C₂	semidirect
8. #22587160	C₂	semidirect
9. #22587157	C₂	semidirect
10. #22587152	C₂	semidirect
11. #22587124	C₂	semidirect
12. #22586785	C₂	semidirect
14. #22586628	C₂	semidirect
15. C₇₁×SD₁₆⋊C₈	C₂	semidirect
295. SD₁₆⋊C₈	D₇₁	semidirect
300. C₇₁	#17915	semidirect

Maximal subgroups

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

subgroup	index
#22598418	2
#22597312	2
C₈×Q₈×D₇₁	2
#22587189	2
#22587187	2
#22587186	2
#22587184	2
#22587160	2
#22587157	2
#22587152	2
#22587124	2
#22586785	2
#22586750	2
#22586628	2
C₇₁×SD₁₆⋊C₈	2
#17915	71