Subgroups — #31968909

Order 18176.

638 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	#22597292	C₂	1	1	yes	maximal, characteristic
2	Q₈×#11765848	C₂	1	1	yes	maximal, characteristic
3	C₄×Q₁₆×D₇₁	C₂	1	1	yes	maximal, characteristic
4	#22587235	C₂	1	1	yes	maximal, characteristic, complemented
5	#22587234	C₂	1	1	yes	maximal, characteristic, complemented
6	#22587192	C₂	1	1	yes	maximal, characteristic, complemented
7	#22587191	C₂	1	1	yes	maximal, characteristic, complemented
8	#22587189	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈×#11765855	C₂	1	1	yes	maximal, characteristic, complemented
10	#22587144	C₂	1	1	yes	maximal, characteristic, complemented
11	#22587097	C₂	1	1	yes	maximal, characteristic
12	#22586767	C₂	1	1	yes	maximal, characteristic
13	#22586767	C₂	1	1	yes	maximal, characteristic
14	#22586619	C₂	1	1	yes	maximal, characteristic
15	C₇₁×#1584	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₁₆×D₇₁	C₄	1	1	yes	characteristic
17	C₄×Q₈×D₇₁	C₂²	1	1	yes	characteristic
18	C₄×Q₈×D₇₁	C₂²	1	1	yes	characteristic
19	#17986130	C₂²	1	1	yes	characteristic
20	#17986129	—	2	1	no	—
21	#17986129	C₂²	1	1	yes	characteristic
22	#17986128	—	2	1	no	—
23	#17986127	—	2	1	no	—
24	#17986126	—	2	1	no	—
25	#17986126	C₂²	1	1	yes	characteristic
26	Q₈×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
27	C₈×Dic₁₄₂	C₂²	1	1	yes	characteristic
28	#17986123	C₂²	1	1	yes	characteristic
29	C₄×Dic₂₈₄	C₂²	1	1	yes	characteristic
30	Q₁₆×Dic₇₁	C₄	1	1	yes	characteristic
31	#17986093	C₄	1	1	yes	characteristic
32	C₄×#11765855	C₂²	1	1	yes	characteristic
33	C₄×#11765855	C₂²	1	1	yes	characteristic
34	#17986091	C₄	1	1	yes	characteristic
35	#17986091	C₄	1	1	yes	characteristic
36	#17986050	C₂²	1	1	yes	characteristic
37	#107×D₇₁	C₄	1	1	yes	characteristic
38	#17986021	—	2	1	no	—
39	#17986021	—	2	1	no	—
40	#17986021	C₂²	1	1	yes	characteristic
41	#17986021	C₂²	1	1	yes	characteristic
42	#103×D₇₁	C₄	1	1	yes	characteristic
43	#103×D₇₁	C₄	1	1	yes	characteristic
44	#17985951	—	2	1	no	—
45	C₄×#11765848	—	2	1	no	—
46	C₄×#11765848	C₂²	1	1	yes	characteristic
47	C₄×C₈×D₇₁	C₂²	1	1	yes	characteristic
48	#17985752	C₂²	1	1	yes	characteristic
49	#17985751	C₂²	1	1	yes	characteristic
50	#17985751	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	Q₈⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
55	#17985729	C₂²	1	1	yes	characteristic
56	#17985729	C₂²	1	1	yes	characteristic
57	#17985721	C₂²	1	1	yes	characteristic
58	C₈×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
59	C₈×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
60	#446×C₇₁	C₂²	1	1	yes	characteristic
61	C₈×Q₈×C₇₁	C₂²	1	1	yes	characteristic
62	C₄×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
63	#335×C₇₁	C₂²	1	1	yes	characteristic
64	Q₈⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
65	Q₈⋊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×Q₈×D₇₁	C₂×C₄	1	1	yes	characteristic
69	C₂×Dic₂₈₄	C₂×C₄	1	1	yes	characteristic
70	C₂×#11765855	C₂×C₄	1	1	yes	characteristic
71	C₂×#11765855	C₂×C₄	1	1	yes	characteristic
72	Q₁₆×D₇₁	—	2	4	no	—
73	C₂×#11765848	—	2	1	no	—
74	C₂×#11765848	—	2	1	no	—
75	C₂×#11765848	D₄	1	1	yes	characteristic
76	C₂×#11765848	D₄	1	1	yes	characteristic
77	C₂×C₈×D₇₁	C₂×C₄	1	1	yes	characteristic
78	C₂×C₈×D₇₁	C₂×C₄	1	1	yes	characteristic
79	Q₈×Dic₇₁	C₂×C₄	1	1	yes	characteristic
80	Q₈×Dic₇₁	C₂×C₄	1	1	yes	characteristic
81	C₄×Dic₁₄₂	—	2	1	no	—
82	C₄×Dic₁₄₂	—	2	1	no	—
83	C₄×Dic₁₄₂	C₂³	1	1	yes	characteristic
84	C₄×Dic₁₄₂	C₂³	1	1	yes	characteristic
85	#14556093	—	2	1	no	—
86	#14556093	—	2	1	no	—
87	#14556093	C₂×C₄	1	1	yes	characteristic
88	#14556093	C₂×C₄	1	1	yes	characteristic
89	C₄⋊C₄×D₇₁	—	2	1	no	—
90	C₄⋊C₄×D₇₁	—	2	1	no	—
91	C₄⋊C₄×D₇₁	C₂×C₄	1	1	yes	characteristic
92	C₄⋊C₄×D₇₁	C₂×C₄	1	1	yes	characteristic
93	C₄²×D₇₁	—	2	1	no	—
94	C₄²×D₇₁	—	2	1	no	—
95	C₄²×D₇₁	C₂³	1	1	yes	characteristic
96	#14556062	C₂×C₄	1	1	yes	characteristic
97	#14556061	C₂×C₄	1	1	yes	characteristic
98	#14556061	C₂×C₄	1	1	yes	characteristic
99	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
100	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
101	C₄⋊(C₇₁⋊C₈)	C₂³	1	1	yes	characteristic
102	C₄⋊(C₇₁⋊C₈)	C₂³	1	1	yes	characteristic
103	Dic₇₁⋊C₈	—	2	1	no	—
104	Dic₇₁⋊C₈	—	2	1	no	—
105	Dic₇₁⋊C₈	—	2	1	no	—
106	Dic₇₁⋊C₈	—	2	1	no	—
107	Dic₇₁⋊C₈	—	2	1	no	—
108	Dic₇₁⋊C₈	—	2	1	no	—
109	Dic₇₁⋊C₈	—	2	1	no	—
110	#14556053	C₂×C₄	1	1	yes	characteristic
111	#14556053	C₂×C₄	1	1	yes	characteristic
112	#14556052	C₂×C₄	1	1	yes	characteristic
113	#14556052	C₂×C₄	1	1	yes	characteristic
114	#14556051	C₂×C₄	1	1	yes	characteristic
115	#14556051	C₂×C₄	1	1	yes	characteristic
116	(C₂×D₇₁)⋊C₈	—	2	1	no	—
117	#14556039	—	2	1	no	—
118	#14556038	—	2	1	no	—
119	#14556038	—	2	1	no	—
120	#14556038	D₄	1	1	yes	characteristic
121	#14556038	D₄	1	1	yes	characteristic
122	C₄×C₇₁⋊C₈	—	2	1	no	—
123	C₄×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
124	C₄×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
125	C₄×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
126	C₈×Dic₇₁	—	2	1	no	—
127	C₈×Dic₇₁	—	2	1	no	—
128	C₈×Dic₇₁	C₂×C₄	1	1	yes	characteristic
129	C₈×Dic₇₁	C₂×C₄	1	1	yes	characteristic
130	C₂×Q₁₆×C₇₁	C₂×C₄	1	1	yes	characteristic
131	C₄×Q₈×C₇₁	C₂³	1	1	yes	characteristic
132	C₄×Q₈×C₇₁	C₂³	1	1	yes	characteristic
133	#107×C₇₁	C₂×C₄	1	1	yes	characteristic
134	C₄⋊C₈×C₇₁	—	2	1	no	—
135	C₄⋊C₈×C₇₁	—	2	1	no	—
136	C₄⋊C₈×C₇₁	C₂³	1	1	yes	characteristic
137	C₄⋊C₈×C₇₁	C₂³	1	1	yes	characteristic
138	#103×C₇₁	C₂×C₄	1	1	yes	characteristic
139	#103×C₇₁	C₂×C₄	1	1	yes	characteristic
140	#97×C₇₁	—	2	1	no	—
141	C₄×C₈×C₇₁	—	2	1	no	—
142	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
143	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
144	C₄×C₈×C₇₁	C₂³	1	1	yes	characteristic
145	C₂×Dic₁₄₂	—	2	1	no	—
146	C₂×Dic₁₄₂	—	2	1	no	—
147	C₂×Dic₁₄₂	C₂²×C₄	1	1	yes	characteristic
148	C₂×Dic₁₄₂	C₂²×C₄	1	1	yes	characteristic
149	Q₈×D₇₁	—	2	2	no	—
150	Q₈×D₇₁	—	2	2	no	—
151	Q₈×D₇₁	—	4	1	no	—
152	Q₈×D₇₁	—	4	1	no	—
153	C₂×C₄×D₇₁	—	2	1	no	—
154	C₂×C₄×D₇₁	—	2	1	no	—
155	C₂×C₄×D₇₁	—	2	1	no	—
156	C₂×C₄×D₇₁	—	2	1	no	—
157	C₂×C₄×D₇₁	C₂²×C₄	1	1	yes	characteristic
158	C₂×C₄×D₇₁	C₂×D₄	1	1	yes	characteristic
159	C₂×C₄×D₇₁	#41	1	1	yes	characteristic
160	Dic₂₈₄	—	2	2	no	—
161	#11765855	—	2	2	no	—
162	#11765855	M₁₆	1	4	yes	complemented
163	#11765848	—	2	2	no	—
164	#11765848	—	4	2	no	—
165	#11765848	—	4	2	no	—
166	#11765848	Q₁₆	1	4	yes	complemented
167	C₂×C₇₁⋊C₈	—	2	1	no	—
168	C₂×C₇₁⋊C₈	—	2	1	no	—
169	C₂×C₇₁⋊C₈	—	2	1	no	—
170	C₂×C₇₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
171	C₂×C₇₁⋊C₈	C₂²×C₄	1	1	yes	characteristic
172	C₂×C₇₁⋊C₈	C₂×D₄	1	1	yes	characteristic
173	C₂×C₇₁⋊C₈	C₂×D₄	1	1	yes	characteristic
174	C₈×D₇₁	—	2	2	no	—
175	C₈×D₇₁	—	4	1	no	—
176	C₄⋊Dic₇₁	—	2	1	no	—
177	C₄⋊Dic₇₁	—	2	1	no	—
178	C₄⋊Dic₇₁	C₂²×C₄	1	1	yes	characteristic
179	C₄⋊Dic₇₁	C₂²×C₄	1	1	yes	characteristic
180	Dic₇₁⋊C₄	—	2	1	no	—
181	Dic₇₁⋊C₄	—	2	1	no	—
182	Dic₇₁⋊C₄	—	2	1	no	—
183	Dic₇₁⋊C₄	—	2	1	no	—
184	Dic₇₁⋊C₄	—	2	1	no	—
185	Dic₇₁⋊C₄	—	2	1	no	—
186	C₄×Dic₇₁	—	2	1	no	—
187	C₄×Dic₇₁	—	2	1	no	—
188	C₄×Dic₇₁	—	2	1	no	—
189	C₄×Dic₇₁	—	2	1	no	—
190	C₄×Dic₇₁	—	2	1	no	—
191	C₄×Dic₇₁	—	2	1	no	—
192	C₄×Dic₇₁	C₂²×C₄	1	1	yes	characteristic
193	C₄×Dic₇₁	C₂×D₄	1	1	yes	characteristic
194	C₄×Dic₇₁	#41	1	1	yes	characteristic
195	C₂×Q₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
196	C₂×Q₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
197	Q₁₆×C₇₁	—	2	2	no	—
198	C₂×C₈×C₇₁	—	2	1	no	—
199	C₂×C₈×C₇₁	—	2	1	no	—
200	C₂×C₈×C₇₁	—	2	1	no	—
201	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
202	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
203	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
204	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
205	C₄⋊C₄×C₇₁	—	2	1	no	—
206	C₄⋊C₄×C₇₁	—	2	1	no	—
207	C₄⋊C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
208	C₄⋊C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
209	C₄²×C₇₁	—	2	1	no	—
210	C₄²×C₇₁	—	2	1	no	—
211	C₄²×C₇₁	C₂⁴	1	1	yes	characteristic
212	C₂²×D₇₁	C₄×D₄	1	1	yes	characteristic
213	Q₈×C₇₁	—	2	1	no	—
214	Q₈×C₇₁	—	2	1	no	—
215	Q₈×C₇₁	—	2	1	no	—
216	Q₈×C₇₁	C₂×M₁₆	1	2	yes	—
217	C₂×C₄×C₇₁	—	2	1	no	—
218	C₂×C₄×C₇₁	—	2	1	no	—
219	C₂×C₄×C₇₁	—	2	1	no	—
220	C₂×C₄×C₇₁	—	2	1	no	—
221	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
222	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
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	1	no	—
232	C₄×D₇₁	—	2	1	no	—
233	C₄×D₇₁	—	4	1	no	—
234	C₄×D₇₁	—	4	2	no	—
235	C₄×D₇₁	—	4	2	no	—
236	C₄×D₇₁	—	8	1	no	—
237	C₄×D₇₁	—	8	1	no	—
238	C₄×D₇₁	C₂×Q₁₆	1	2	yes	—
239	C₄×D₇₁	#135	1	2	yes	—
240	Dic₁₄₂	—	2	1	no	—
241	Dic₁₄₂	—	2	1	no	—
242	Dic₁₄₂	—	2	1	no	—
243	Dic₁₄₂	—	4	2	no	—
244	Dic₁₄₂	—	4	2	no	—
245	Dic₁₄₂	C₂×M₁₆	1	2	yes	—
246	C₈×C₇₁	—	2	1	no	—
247	C₈×C₇₁	—	2	1	no	—
248	C₈×C₇₁	—	2	1	no	—
249	C₈×C₇₁	—	4	1	no	—
250	C₈×C₇₁	—	4	1	no	—
251	C₈×C₇₁	—	4	1	no	—
252	C₈×C₇₁	C₂×Q₁₆	1	2	yes	complemented
253	C₇₁⋊C₈	—	2	1	no	—
254	C₇₁⋊C₈	—	2	1	no	—
255	C₇₁⋊C₈	—	4	1	no	—
256	C₇₁⋊C₈	—	4	1	no	—
257	C₇₁⋊C₈	—	4	1	no	—
258	C₇₁⋊C₈	C₂×M₁₆	1	2	yes	—
259	C₇₁⋊C₈	C₂×Q₁₆	1	2	yes	complemented
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	C₄×Q₁₆	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	—	4	1	no	—
269	C₄×C₇₁	C₂²×M₁₆	1	1	yes	characteristic
270	C₄×C₇₁	#568	1	1	yes	characteristic
271	C₄×C₇₁	C₂²×Q₁₆	1	1	yes	characteristic
272	C₄×C₇₁	C₂×#135	1	1	yes	characteristic
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	Dic₇₁	C₄×Q₁₆	1	2	yes	—
280	M₁₆×Q₁₆	—	71	1	no	maximal, Sylow
281	C₂×C₇₁	D₄×M₁₆	1	1	yes	characteristic
282	C₂×C₇₁	C₂×C₄×Q₁₆	1	1	yes	characteristic
283	C₂×C₇₁	#2957	1	1	yes	characteristic
284	D₇₁	—	2	2	no	—
285	Q₈×M₁₆	—	71	1	no	—
286	Q₈×M₁₆	—	71	1	no	—
287	C₂×C₄×Q₁₆	—	71	1	no	—
288	#1591	—	71	1	no	—
289	#1591	—	71	1	no	—
290	#1584	—	71	1	no	—
291	#1584	—	71	1	no	—
292	#1584	—	71	1	no	—
293	#1584	D₇₁	1	1	yes	characteristic, complemented
294	C₈×Q₁₆	—	71	1	no	—
295	C₈×Q₁₆	—	71	1	no	—
296	#1567	—	71	1	no	—
297	#1487	—	71	1	no	—
298	#1487	—	71	1	no	—
299	C₈×M₁₆	—	71	1	no	—
300	C₇₁	M₁₆×Q₁₆	1	1	yes	characteristic, complemented, Sylow
301	C₂²×Q₁₆	—	71	1	no	—
302	C₂×C₄×Q₈	—	71	1	no	—
303	C₂×C₄×Q₈	—	71	1	no	—
304	#446	—	71	1	no	—
305	#446	—	71	1	no	—
306	#446	—	71	1	no	—
307	#446	—	142	1	no	—
308	#446	—	142	1	no	—
309	#446	—	142	1	no	—
310	#446	—	142	1	no	—
311	#446	C₂×D₇₁	1	1	yes	characteristic
312	C₈×Q₈	—	71	1	no	—
313	C₈×Q₈	—	71	1	no	—
314	C₈×Q₈	—	71	1	no	—
315	C₈×Q₈	C₂×D₇₁	1	1	yes	characteristic
316	C₄×Q₁₆	—	71	1	no	—
317	C₄×Q₁₆	—	71	1	no	—
318	C₄×Q₁₆	—	71	1	no	—
319	C₄×Q₁₆	—	71	1	no	—
320	C₄×Q₁₆	—	71	1	no	—
321	C₄×Q₁₆	—	71	1	no	—
322	C₄×Q₁₆	—	71	1	no	—
323	C₄×Q₁₆	C₂×D₇₁	1	1	yes	characteristic
324	#431	—	71	1	no	—
325	C₂×#107	—	71	1	no	—
326	#423	—	71	1	no	—
327	#423	—	71	1	no	—
328	#423	—	142	1	no	—
329	#423	—	142	1	no	—
330	C₂×#103	—	71	1	no	—
331	C₂×#103	—	71	1	no	—
332	C₄×M₁₆	—	71	1	no	—
333	C₄×M₁₆	—	142	1	no	—
334	C₄×M₁₆	—	142	1	no	—
335	C₂×C₄×C₈	—	71	1	no	—
336	#335	—	71	1	no	—
337	#335	—	71	1	no	—
338	#335	—	71	1	no	—
339	#335	C₂×D₇₁	1	1	yes	characteristic
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	—	71	1	no	—
346	Q₈⋊C₈	C₂×D₇₁	1	1	yes	characteristic
347	Q₈⋊C₈	C₂×D₇₁	1	1	yes	characteristic
348	#322	—	71	1	no	—
349	#322	C₂×D₇₁	1	1	yes	characteristic
350	C₈²	—	71	1	no	—
351	C₈²	—	71	1	no	—
352	C₂²×Q₈	—	71	1	no	—
353	C₂²×Q₈	—	71	1	no	—
354	C₂×Q₁₆	—	71	1	no	—
355	C₂×Q₁₆	—	71	1	no	—
356	C₂×Q₁₆	—	71	1	no	—
357	C₂×Q₁₆	—	142	4	no	—
358	C₂×Q₁₆	C₄×D₇₁	1	1	yes	characteristic
359	C₂×M₁₆	—	71	1	no	—
360	C₂×M₁₆	—	71	1	no	—
361	C₂×M₁₆	—	142	1	no	—
362	C₂×M₁₆	—	142	1	no	—
363	C₂²×C₈	—	71	1	no	—
364	C₂²×C₈	—	71	1	no	—
365	C₄×Q₈	—	71	1	no	—
366	C₄×Q₈	—	71	1	no	—
367	C₄×Q₈	—	71	1	no	—
368	C₄×Q₈	—	71	1	no	—
369	C₄×Q₈	—	71	1	no	—
370	C₄×Q₈	—	71	1	no	—
371	C₄×Q₈	—	142	1	no	—
372	C₄×Q₈	—	142	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×Q₈	C₂²×D₇₁	1	1	yes	characteristic
377	C₂×C₄⋊C₄	—	71	1	no	—
378	C₂×C₄⋊C₄	—	71	1	no	—
379	C₂×C₄⋊C₄	—	142	1	no	—
380	C₂×C₄⋊C₄	—	142	1	no	—
381	C₂×C₄²	—	71	1	no	—
382	C₂×C₄²	—	142	1	no	—
383	C₂×C₄²	—	142	1	no	—
384	#107	—	71	1	no	—
385	#107	—	71	1	no	—
386	#107	—	71	1	no	—
387	#107	C₄×D₇₁	1	1	yes	characteristic
388	C₄⋊C₈	—	2	1	no	—
389	C₄⋊C₈	—	2	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	—	142	1	no	—
399	C₄⋊C₈	—	142	1	no	—
400	C₄⋊C₈	—	142	1	no	—
401	C₄⋊C₈	C₂²×D₇₁	1	1	yes	characteristic
402	C₄⋊C₈	C₂²×D₇₁	1	1	yes	characteristic
403	#103	—	71	1	no	—
404	#103	—	71	1	no	—
405	#103	—	71	1	no	—
406	#103	—	71	1	no	—
407	#103	—	71	1	no	—
408	#103	—	71	1	no	—
409	#103	C₄×D₇₁	1	1	yes	characteristic
410	#103	C₄×D₇₁	1	1	yes	characteristic
411	C₂²⋊C₈	—	142	1	no	—
412	#97	—	2	1	no	—
413	#97	—	71	1	no	—
414	#97	—	71	1	no	—
415	#97	—	142	1	no	—
416	#97	—	142	1	no	—
417	#97	—	142	1	no	—
418	C₄×C₈	—	2	1	no	—
419	C₄×C₈	—	71	1	no	—
420	C₄×C₈	—	71	1	no	—
421	C₄×C₈	—	71	1	no	—
422	C₄×C₈	—	71	1	no	—
423	C₄×C₈	—	71	1	no	—
424	C₄×C₈	—	142	1	no	—
425	C₄×C₈	—	142	1	no	—
426	C₄×C₈	—	142	1	no	—
427	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
428	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
429	C₄×C₈	C₂²×D₇₁	1	1	yes	characteristic
430	C₂×Q₈	—	284	1	no	—
431	C₂×Q₈	—	284	1	no	—
432	C₂×Q₈	—	71	1	no	—
433	C₂×Q₈	—	71	1	no	—
434	C₂×Q₈	—	142	1	no	—
435	C₂×Q₈	—	142	1	no	—
436	C₂×Q₈	—	142	2	no	—
437	C₂×Q₈	—	142	2	no	—
438	C₂×Q₈	C₂×C₄×D₇₁	1	1	yes	characteristic
439	C₂×Q₈	C₂×C₄×D₇₁	1	1	yes	characteristic
440	C₂²×C₄	—	71	1	no	—
441	C₂²×C₄	—	71	1	no	—
442	C₂²×C₄	—	71	1	no	—
443	C₂²×C₄	—	142	1	no	—
444	C₂²×C₄	—	142	1	no	—
445	C₂²×C₄	—	142	1	no	—
446	C₂²×C₄	—	142	1	no	—
447	Q₁₆	—	2	2	no	—
448	Q₁₆	—	71	4	no	—
449	Q₁₆	—	142	2	no	—
450	Q₁₆	—	142	2	no	—
451	M₁₆	—	284	2	no	—
452	M₁₆	—	284	2	no	—
453	M₁₆	—	71	4	no	—
454	M₁₆	—	142	2	no	—
455	C₂×C₈	—	2	1	no	—
456	C₂×C₈	—	2	1	no	—
457	C₂×C₈	—	2	1	no	—
458	C₂×C₈	—	284	1	no	—
459	C₂×C₈	—	71	1	no	—
460	C₂×C₈	—	71	1	no	—
461	C₂×C₈	—	71	1	no	—
462	C₂×C₈	—	71	1	no	—
463	C₂×C₈	—	142	2	no	—
464	C₂×C₈	—	142	1	no	—
465	C₂×C₈	—	142	1	no	—
466	C₂×C₈	—	142	1	no	—
467	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
468	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
469	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
470	C₂×C₈	D₄×D₇₁	1	1	yes	characteristic
471	C₄⋊C₄	—	2	1	no	—
472	C₄⋊C₄	—	2	1	no	—
473	C₄⋊C₄	—	71	1	no	—
474	C₄⋊C₄	—	71	1	no	—
475	C₄⋊C₄	—	142	1	no	—
476	C₄⋊C₄	—	142	1	no	—
477	C₄⋊C₄	—	142	1	no	—
478	C₄⋊C₄	—	142	1	no	—
479	C₄⋊C₄	—	142	1	no	—
480	C₄⋊C₄	—	142	1	no	—
481	C₄⋊C₄	—	142	1	no	—
482	C₄⋊C₄	—	142	1	no	—
483	C₄⋊C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
484	C₄⋊C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
485	C₄²	—	2	1	no	—
486	C₄²	—	2	1	no	—
487	C₄²	—	71	1	no	—
488	C₄²	—	71	1	no	—
489	C₄²	—	71	1	no	—
490	C₄²	—	142	1	no	—
491	C₄²	—	142	1	no	—
492	C₄²	—	142	1	no	—
493	C₄²	—	142	1	no	—
494	C₄²	—	142	1	no	—
495	C₄²	—	142	1	no	—
496	C₄²	C₂³×D₇₁	1	1	yes	characteristic
497	C₂³	—	71	1	no	—
498	Q₈	—	2	1	no	—
499	Q₈	—	2	1	no	—
500	Q₈	—	2	1	no	—
501	Q₈	—	284	2	no	—
502	Q₈	—	284	2	no	—
503	Q₈	—	71	2	no	—
504	Q₈	—	142	1	no	—
505	Q₈	—	142	1	no	—
506	Q₈	—	142	1	no	—
507	Q₈	M₁₆×D₇₁	1	2	yes	—
508	C₂×C₄	—	2	1	no	—
509	C₂×C₄	—	2	1	no	—
510	C₂×C₄	—	2	1	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	284	2	no	—
513	C₂×C₄	—	284	1	no	—
514	C₂×C₄	—	284	2	no	—
515	C₂×C₄	—	568	1	no	—
516	C₂×C₄	—	568	1	no	—
517	C₂×C₄	—	71	1	no	—
518	C₂×C₄	—	71	2	no	—
519	C₂×C₄	—	71	2	no	—
520	C₂×C₄	—	71	1	no	—
521	C₂×C₄	—	71	1	no	—
522	C₂×C₄	—	142	1	no	—
523	C₂×C₄	—	142	1	no	—
524	C₂×C₄	—	142	1	no	—
525	C₂×C₄	—	142	1	no	—
526	C₂×C₄	—	142	1	no	—
527	C₂×C₄	—	142	1	no	—
528	C₂×C₄	C₂²×C₄×D₇₁	1	1	yes	characteristic
529	C₂×C₄	C₂×D₄×D₇₁	1	1	yes	characteristic
530	C₂×C₄	#41×D₇₁	1	1	yes	characteristic
531	C₈	—	2	1	no	—
532	C₈	—	2	1	no	—
533	C₈	—	2	1	no	—
534	C₈	—	4	1	no	—
535	C₈	—	4	1	no	—
536	C₈	—	4	1	no	—
537	C₈	—	284	1	no	—
538	C₈	—	284	1	no	—
539	C₈	—	284	1	no	—
540	C₈	—	71	2	no	—
541	C₈	—	71	2	no	—
542	C₈	—	142	1	no	—
543	C₈	—	142	1	no	—
544	C₈	Q₁₆×D₇₁	1	2	yes	complemented
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	—	4	1	no	—
554	C₄	—	284	1	no	—
555	C₄	—	284	1	no	—
556	C₄	—	284	1	no	—
557	C₄	—	71	2	no	—
558	C₄	—	142	1	no	—
559	C₄	—	142	2	no	—
560	C₄	—	142	1	no	—
561	C₄	C₂×M₁₆×D₇₁	1	1	yes	characteristic
562	C₄	#17986764	1	1	yes	characteristic
563	C₄	C₂×Q₁₆×D₇₁	1	1	yes	characteristic
564	C₄	#135×D₇₁	1	1	yes	characteristic
565	C₂	—	142	2	no	—
566	C₂	#22597073	1	1	yes	characteristic
567	C₂	C₄×Q₁₆×D₇₁	1	1	yes	characteristic
568	C₂	#22597254	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×#1584	C₂
socle	C₂²×C₇₁	C₂×C₄×D₄
prime radical	C₂²×D₇₁	C₄×D₄
cosocle	C₄²×C₇₁	C₂⁴
hypercenter	#1584	D₇₁
hypocenter	C₇₁	M₁₆×Q₁₆
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1584

Complements

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

N	H ≅ G/N	type
4. #22587235	C₂	semidirect
5. #22587234	C₂	semidirect
6. #22587192	C₂	semidirect
7. #22587191	C₂	semidirect
8. #22587189	C₂	semidirect
9. C₈×#11765855	C₂	semidirect
10. #22587144	C₂	semidirect
15. C₇₁×#1584	C₂	semidirect
162. #11765855	M₁₆	semidirect
166. #11765848	Q₁₆	semidirect
252. C₈×C₇₁	C₂×Q₁₆	semidirect
259. C₇₁⋊C₈	C₂×Q₁₆	semidirect
293. #1584	D₇₁	semidirect
300. C₇₁	M₁₆×Q₁₆	semidirect
544. C₈	Q₁₆×D₇₁	semidirect

Maximal subgroups

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

subgroup	index
#22597292	2
Q₈×#11765848	2
C₄×Q₁₆×D₇₁	2
#22587235	2
#22587234	2
#22587192	2
#22587191	2
#22587189	2
C₈×#11765855	2
#22587144	2
#22587097	2
#22586767	2
#22586619	2
C₇₁×#1584	2
M₁₆×Q₁₆	71