Subgroups — C₇₁×#61322 #31882194

Order 18176.

1782 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#41×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#557×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₁×#3374	C₂	1	6	yes	maximal, complemented
4	C₇₁×#3372	C₂	1	6	yes	maximal, complemented
5	C₇₁×#3363	C₂	1	2	yes	maximal, complemented
6	C₇₁×#3359	C₂	1	2	yes	maximal, complemented
7	C₇₁×#3139	C₂	1	3	yes	maximal, complemented
8	C₇₁×#3078	C₂	1	3	yes	maximal, complemented
9	C₇₁×#3039	C₂	1	3	yes	maximal
10	Q₈×M₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#137×C₇₁	C₂	1	3	yes	maximal
12	C₂×#137×C₇₁	C₂²	1	3	yes	—
13	Q₈²×C₇₁	C₂²	1	1	yes	characteristic
14	#557×C₇₁	—	2	1	no	—
15	#557×C₇₁	—	2	3	no	—
16	#557×C₇₁	C₂²	1	2	yes	—
17	#557×C₇₁	C₂²	1	6	yes	—
18	#557×C₇₁	C₂²	1	3	yes	—
19	#554×C₇₁	—	2	3	no	—
20	#554×C₇₁	C₂²	1	3	yes	—
21	D₄×Q₈×C₇₁	—	2	1	no	—
22	D₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
23	#533×C₇₁	—	2	3	no	—
24	#533×C₇₁	C₂²	1	3	yes	—
25	C₂×C₄⋊Q₈×C₇₁	C₂²	1	3	yes	—
26	C₂×#125×C₇₁	—	2	3	no	—
27	C₄×#41×C₇₁	C₂²	1	3	yes	—
28	C₂×C₄×Q₈×C₇₁	—	2	1	no	—
29	C₂×C₄×Q₈×C₇₁	—	2	1	no	—
30	C₂×C₄×Q₈×C₇₁	C₂²	1	3	yes	—
31	C₂×C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
32	#491×C₇₁	C₂²	1	12	yes	—
33	#487×C₇₁	C₂²	1	6	yes	—
34	#487×C₇₁	C₂²	1	6	yes	—
35	#479×C₇₁	C₂²	1	12	yes	—
36	#477×C₇₁	C₂²	1	6	yes	—
37	#475×C₇₁	C₂²	1	6	yes	—
38	#462×C₇₁	C₂²	1	6	yes	—
39	#462×C₇₁	C₂²	1	6	yes	—
40	#461×C₇₁	C₂²	1	6	yes	—
41	#461×C₇₁	C₂²	1	6	yes	—
42	#446×C₇₁	C₂²	1	6	yes	—
43	C₈×Q₈×C₇₁	C₂²	1	2	yes	—
44	#441×C₇₁	C₂²	1	6	yes	—
45	#440×C₇₁	C₂²	1	6	yes	—
46	C₄×Q₁₆×C₇₁	C₂²	1	6	yes	—
47	C₄×SD₁₆×C₇₁	C₂²	1	6	yes	—
48	#428×C₇₁	C₂²	1	3	yes	—
49	#423×C₇₁	C₂²	1	3	yes	—
50	#419×C₇₁	C₂²	1	6	yes	—
51	#419×C₇₁	C₂²	1	3	yes	—
52	#417×C₇₁	C₂²	1	1	yes	characteristic
53	C₄×M₁₆×C₇₁	C₂²	1	3	yes	—
54	C₂×#41×C₇₁	C₂³	1	1	yes	characteristic
55	C₂²×Q₈×C₇₁	—	2	1	no	—
56	C₂²×Q₈×C₇₁	D₄	1	1	yes	characteristic
57	C₂²×Q₈×C₇₁	C₂³	1	3	yes	—
58	#137×C₇₁	—	2	12	no	—
59	C₂×Q₁₆×C₇₁	C₂³	1	6	yes	—
60	C₂×SD₁₆×C₇₁	C₂³	1	6	yes	—
61	C₂×M₁₆×C₇₁	C₂³	1	1	yes	characteristic
62	C₂×M₁₆×C₇₁	C₂³	1	3	yes	—
63	C₄⋊Q₈×C₇₁	—	2	3	no	—
64	C₄⋊Q₈×C₇₁	—	2	3	no	—
65	C₄⋊Q₈×C₇₁	—	2	3	no	—
66	C₄⋊Q₈×C₇₁	—	2	3	no	—
67	C₄⋊Q₈×C₇₁	C₂³	1	6	yes	—
68	C₄⋊Q₈×C₇₁	C₂³	1	6	yes	—
69	C₄⋊Q₈×C₇₁	C₂³	1	3	yes	—
70	C₄⋊Q₈×C₇₁	C₂³	1	3	yes	—
71	#125×C₇₁	—	2	6	no	—
72	#125×C₇₁	—	2	12	no	—
73	#125×C₇₁	—	2	6	no	—
74	#125×C₇₁	—	2	6	no	—
75	#125×C₇₁	C₂³	1	6	yes	—
76	C₂²⋊Q₈×C₇₁	—	2	3	no	—
77	C₂²⋊Q₈×C₇₁	—	2	3	no	—
78	C₂²⋊Q₈×C₇₁	—	2	3	no	—
79	C₂²⋊Q₈×C₇₁	—	2	3	no	—
80	C₂²⋊Q₈×C₇₁	—	2	6	no	—
81	C₄×Q₈×C₇₁	—	2	1	no	—
82	C₄×Q₈×C₇₁	—	2	6	no	—
83	C₄×Q₈×C₇₁	—	2	2	no	—
84	C₄×Q₈×C₇₁	—	2	3	no	—
85	C₄×Q₈×C₇₁	—	2	3	no	—
86	C₄×Q₈×C₇₁	—	2	3	no	—
87	C₄×Q₈×C₇₁	—	2	3	no	—
88	C₄×Q₈×C₇₁	—	2	3	no	—
89	C₄×Q₈×C₇₁	D₄	1	1	yes	characteristic
90	C₄×Q₈×C₇₁	D₄	1	3	yes	—
91	C₄×Q₈×C₇₁	C₂³	1	6	yes	—
92	C₄×Q₈×C₇₁	C₂³	1	3	yes	—
93	C₄×Q₈×C₇₁	C₂³	1	6	yes	—
94	C₄×Q₈×C₇₁	C₂³	1	3	yes	—
95	C₄×Q₈×C₇₁	C₂³	1	1	yes	characteristic
96	C₄×Q₈×C₇₁	C₂³	1	3	yes	—
97	C₄×D₄×C₇₁	—	2	3	no	—
98	C₄×D₄×C₇₁	—	2	3	no	—
99	C₄×D₄×C₇₁	C₂³	1	3	yes	—
100	C₄×D₄×C₇₁	C₂³	1	3	yes	—
101	#117×C₇₁	—	2	3	no	—
102	#117×C₇₁	C₂³	1	3	yes	—
103	C₂×C₄⋊C₄×C₇₁	—	2	3	no	—
104	C₂×C₄⋊C₄×C₇₁	—	2	3	no	—
105	C₂×C₄⋊C₄×C₇₁	—	2	3	no	—
106	C₂×C₄⋊C₄×C₇₁	—	2	3	no	—
107	C₂×C₄⋊C₄×C₇₁	D₄	1	3	yes	—
108	C₂×C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
109	C₂×C₄²×C₇₁	—	2	3	no	—
110	C₂×C₄²×C₇₁	—	2	3	no	—
111	C₂×C₄²×C₇₁	C₂³	1	3	yes	—
112	#107×C₇₁	C₂³	1	6	yes	—
113	#106×C₇₁	C₂³	1	6	yes	—
114	C₄⋊C₈×C₇₁	C₂³	1	6	yes	—
115	C₄⋊C₈×C₇₁	C₂³	1	6	yes	—
116	#103×C₇₁	C₂³	1	2	yes	—
117	#103×C₇₁	C₂³	1	6	yes	—
118	#103×C₇₁	C₂³	1	6	yes	—
119	#103×C₇₁	C₂³	1	12	yes	—
120	#103×C₇₁	C₂³	1	12	yes	—
121	#102×C₇₁	C₂³	1	2	yes	—
122	#97×C₇₁	C₂³	1	6	yes	—
123	C₄×C₈×C₇₁	C₂³	1	6	yes	—
124	#41×C₇₁	—	2	2	no	—
125	#41×C₇₁	#41	1	4	yes	—
126	C₂×Q₈×C₇₁	—	2	4	no	—
127	C₂×Q₈×C₇₁	—	2	3	no	—
128	C₂×Q₈×C₇₁	—	2	3	no	—
129	C₂×Q₈×C₇₁	—	2	3	no	—
130	C₂×Q₈×C₇₁	—	2	6	no	—
131	C₂×Q₈×C₇₁	—	2	6	no	—
132	C₂×Q₈×C₇₁	—	4	4	no	—
133	C₂×Q₈×C₇₁	C₂×D₄	1	3	yes	—
134	C₂×Q₈×C₇₁	C₂×D₄	1	1	yes	characteristic
135	C₂×Q₈×C₇₁	C₂⁴	1	6	yes	—
136	C₂×Q₈×C₇₁	C₂⁴	1	1	yes	characteristic
137	C₂×D₄×C₇₁	—	2	1	no	—
138	C₂×D₄×C₇₁	C₂⁴	1	1	yes	characteristic
139	C₂²×C₄×C₇₁	—	2	1	no	—
140	C₂²×C₄×C₇₁	—	2	1	no	—
141	C₂²×C₄×C₇₁	—	2	3	no	—
142	C₂²×C₄×C₇₁	—	2	3	no	—
143	C₂²×C₄×C₇₁	—	2	3	no	—
144	C₂²×C₄×C₇₁	C₂×D₄	1	3	yes	—
145	C₂²×C₄×C₇₁	C₂×D₄	1	3	yes	—
146	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
147	Q₁₆×C₇₁	—	2	12	no	—
148	SD₁₆×C₇₁	—	2	12	no	—
149	M₁₆×C₇₁	—	2	6	no	—
150	M₁₆×C₇₁	—	2	2	no	—
151	C₂×C₈×C₇₁	C₂⁴	1	2	yes	—
152	C₂×C₈×C₇₁	C₂⁴	1	6	yes	—
153	C₄⋊C₄×C₇₁	—	2	6	no	—
154	C₄⋊C₄×C₇₁	—	2	3	no	—
155	C₄⋊C₄×C₇₁	—	2	6	no	—
156	C₄⋊C₄×C₇₁	—	2	6	no	—
157	C₄⋊C₄×C₇₁	—	2	6	no	—
158	C₄⋊C₄×C₇₁	—	2	6	no	—
159	C₄⋊C₄×C₇₁	—	2	6	no	—
160	C₄⋊C₄×C₇₁	—	2	6	no	—
161	C₄⋊C₄×C₇₁	—	2	1	no	—
162	C₄⋊C₄×C₇₁	—	2	3	no	—
163	C₄⋊C₄×C₇₁	—	2	3	no	—
164	C₄⋊C₄×C₇₁	—	2	3	no	—
165	C₄⋊C₄×C₇₁	—	2	6	no	—
166	C₄⋊C₄×C₇₁	C₂×D₄	1	3	yes	—
167	C₄⋊C₄×C₇₁	C₂×D₄	1	6	yes	—
168	C₄⋊C₄×C₇₁	C₂×D₄	1	3	yes	—
169	C₄⋊C₄×C₇₁	C₂⁴	1	2	yes	—
170	C₄⋊C₄×C₇₁	C₂⁴	1	3	yes	—
171	C₄⋊C₄×C₇₁	C₂⁴	1	3	yes	—
172	C₂²⋊C₄×C₇₁	—	2	3	no	—
173	C₂²⋊C₄×C₇₁	—	2	3	no	—
174	C₂²⋊C₄×C₇₁	—	2	3	no	—
175	C₄²×C₇₁	—	2	3	no	—
176	C₄²×C₇₁	—	2	3	no	—
177	C₄²×C₇₁	—	2	3	no	—
178	C₄²×C₇₁	—	2	6	no	—
179	C₄²×C₇₁	—	2	6	no	—
180	C₄²×C₇₁	C₂×D₄	1	3	yes	—
181	C₄²×C₇₁	C₂×D₄	1	3	yes	—
182	C₄²×C₇₁	C₂⁴	1	3	yes	—
183	C₄²×C₇₁	C₂⁴	1	3	yes	—
184	C₂³×C₇₁	—	2	1	no	—
185	C₂³×C₇₁	C₂²×D₄	1	1	yes	characteristic
186	Q₈×C₇₁	—	2	6	no	—
187	Q₈×C₇₁	—	2	1	no	—
188	Q₈×C₇₁	—	2	6	no	—
189	Q₈×C₇₁	—	2	6	no	—
190	Q₈×C₇₁	—	4	6	no	—
191	Q₈×C₇₁	—	4	6	no	—
192	Q₈×C₇₁	—	8	3	no	—
193	Q₈×C₇₁	#137	1	4	yes	complemented
194	Q₈×C₇₁	C₂×#41	1	2	yes	—
195	D₄×C₇₁	—	2	4	no	—
196	D₄×C₇₁	—	2	1	no	—
197	D₄×C₇₁	C₂×#41	1	2	yes	—
198	C₂×C₄×C₇₁	—	2	1	no	—
199	C₂×C₄×C₇₁	—	2	4	no	—
200	C₂×C₄×C₇₁	—	2	6	no	—
201	C₂×C₄×C₇₁	—	2	2	no	—
202	C₂×C₄×C₇₁	—	2	3	no	—
203	C₂×C₄×C₇₁	—	2	3	no	—
204	C₂×C₄×C₇₁	—	2	6	no	—
205	C₂×C₄×C₇₁	—	4	3	no	—
206	C₂×C₄×C₇₁	—	4	2	no	—
207	C₂×C₄×C₇₁	—	4	6	no	—
208	C₂×C₄×C₇₁	—	4	6	no	—
209	C₂×C₄×C₇₁	—	8	3	no	—
210	C₂×C₄×C₇₁	C₂²×D₄	1	3	yes	—
211	C₂×C₄×C₇₁	C₂²×D₄	1	3	yes	—
212	C₂×C₄×C₇₁	C₂²×D₄	1	3	yes	—
213	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₇₁	C₂²×D₄	1	3	yes	—
215	C₂×C₄×C₇₁	C₂×#41	1	2	yes	—
216	C₂×C₄×C₇₁	2₋¹⁺⁴	1	2	yes	—
217	C₂×C₄×C₇₁	C₂⁵	1	1	yes	characteristic
218	C₈×C₇₁	—	2	6	no	—
219	C₈×C₇₁	—	2	2	no	—
220	C₂²×C₇₁	—	2	1	no	—
221	C₂²×C₇₁	—	2	1	no	—
222	C₂²×C₇₁	—	2	2	no	—
223	C₂²×C₇₁	—	4	1	no	—
224	C₂²×C₇₁	—	4	1	no	—
225	C₂²×C₇₁	#548	1	2	yes	—
226	C₂²×C₇₁	C₂³×D₄	1	1	yes	characteristic
227	C₄×C₇₁	—	2	3	no	—
228	C₄×C₇₁	—	2	3	no	—
229	C₄×C₇₁	—	2	2	no	—
230	C₄×C₇₁	—	2	3	no	—
231	C₄×C₇₁	—	4	2	no	—
232	C₄×C₇₁	—	4	3	no	—
233	C₄×C₇₁	—	4	6	no	—
234	C₄×C₇₁	—	4	3	no	—
235	C₄×C₇₁	#548	1	2	yes	—
236	C₄×C₇₁	C₂×#137	1	6	yes	—
237	C₄×C₇₁	C₂²×#41	1	1	yes	characteristic
238	C₄×C₇₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
239	#61322	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
240	C₂×C₇₁	—	2	2	no	—
241	C₂×C₇₁	—	4	2	no	—
242	C₂×C₇₁	C₂×#548	1	1	yes	characteristic
243	C₂×C₇₁	C₂²×#137	1	1	yes	characteristic
244	C₂×C₇₁	#3586	1	1	yes	characteristic
245	Q₈×#41	C₂×C₇₁	1	1	yes	characteristic
246	C₂×#557	C₂×C₇₁	1	1	yes	characteristic, complemented
247	#3374	C₂×C₇₁	1	6	yes	complemented
248	#3372	C₂×C₇₁	1	6	yes	complemented
249	#3363	C₂×C₇₁	1	2	yes	complemented
250	#3359	C₂×C₇₁	1	2	yes	complemented
251	#3139	C₂×C₇₁	1	3	yes	complemented
252	#3078	C₂×C₇₁	1	3	yes	complemented
253	#3039	C₂×C₇₁	1	3	yes	—
254	Q₈×M₁₆	C₂×C₇₁	1	1	yes	characteristic, complemented
255	C₄×#137	C₂×C₇₁	1	3	yes	—
256	C₇₁	#61322	1	1	yes	characteristic, direct product factor, Sylow
257	C₂×#137	C₂²×C₇₁	1	3	yes	—
258	Q₈²	C₂²×C₇₁	1	1	yes	characteristic
259	#557	—	2	1	no	—
260	#557	—	2	3	no	—
261	#557	C₂²×C₇₁	1	2	yes	—
262	#557	C₂²×C₇₁	1	6	yes	—
263	#557	C₂²×C₇₁	1	3	yes	—
264	#554	—	2	3	no	—
265	#554	C₂²×C₇₁	1	3	yes	—
266	D₄×Q₈	—	2	1	no	—
267	D₄×Q₈	C₂²×C₇₁	1	1	yes	characteristic
268	#533	—	2	3	no	—
269	#533	C₂²×C₇₁	1	3	yes	—
270	C₂×C₄⋊Q₈	C₂²×C₇₁	1	3	yes	—
271	C₂×#125	—	2	3	no	—
272	C₄×#41	C₂²×C₇₁	1	3	yes	—
273	C₂×C₄×Q₈	—	2	1	no	—
274	C₂×C₄×Q₈	—	2	1	no	—
275	C₂×C₄×Q₈	C₂²×C₇₁	1	3	yes	—
276	C₂×C₄×Q₈	C₂²×C₇₁	1	1	yes	characteristic
277	#491	C₂²×C₇₁	1	12	yes	—
278	#487	C₂²×C₇₁	1	6	yes	—
279	#487	C₂²×C₇₁	1	6	yes	—
280	#479	C₂²×C₇₁	1	12	yes	—
281	#477	C₂²×C₇₁	1	6	yes	—
282	#475	C₂²×C₇₁	1	6	yes	—
283	#462	C₂²×C₇₁	1	6	yes	—
284	#462	C₂²×C₇₁	1	6	yes	—
285	#461	C₂²×C₇₁	1	6	yes	—
286	#461	C₂²×C₇₁	1	6	yes	—
287	#446	C₂²×C₇₁	1	6	yes	—
288	C₈×Q₈	C₂²×C₇₁	1	2	yes	—
289	#441	C₂²×C₇₁	1	6	yes	—
290	#440	C₂²×C₇₁	1	6	yes	—
291	C₄×Q₁₆	C₂²×C₇₁	1	6	yes	—
292	C₄×SD₁₆	C₂²×C₇₁	1	6	yes	—
293	#428	C₂²×C₇₁	1	3	yes	—
294	#423	C₂²×C₇₁	1	3	yes	—
295	#419	C₂²×C₇₁	1	6	yes	—
296	#419	C₂²×C₇₁	1	3	yes	—
297	#417	C₂²×C₇₁	1	1	yes	characteristic
298	C₄×M₁₆	C₂²×C₇₁	1	3	yes	—
299	C₂×#41	C₂³×C₇₁	1	1	yes	characteristic
300	C₂²×Q₈	—	2	1	no	—
301	C₂²×Q₈	D₄×C₇₁	1	1	yes	characteristic
302	C₂²×Q₈	C₂³×C₇₁	1	3	yes	—
303	#137	—	2	12	no	—
304	C₂×Q₁₆	C₂³×C₇₁	1	6	yes	—
305	C₂×SD₁₆	C₂³×C₇₁	1	6	yes	—
306	C₂×M₁₆	C₂³×C₇₁	1	3	yes	—
307	C₂×M₁₆	C₂³×C₇₁	1	1	yes	characteristic
308	C₄⋊Q₈	—	2	3	no	—
309	C₄⋊Q₈	—	2	3	no	—
310	C₄⋊Q₈	—	2	3	no	—
311	C₄⋊Q₈	—	2	3	no	—
312	C₄⋊Q₈	C₂³×C₇₁	1	6	yes	—
313	C₄⋊Q₈	C₂³×C₇₁	1	6	yes	—
314	C₄⋊Q₈	C₂³×C₇₁	1	3	yes	—
315	C₄⋊Q₈	C₂³×C₇₁	1	3	yes	—
316	#125	—	2	6	no	—
317	#125	—	2	12	no	—
318	#125	—	2	6	no	—
319	#125	—	2	6	no	—
320	#125	C₂³×C₇₁	1	6	yes	—
321	C₂²⋊Q₈	—	2	3	no	—
322	C₂²⋊Q₈	—	2	3	no	—
323	C₂²⋊Q₈	—	2	3	no	—
324	C₂²⋊Q₈	—	2	3	no	—
325	C₂²⋊Q₈	—	2	6	no	—
326	C₄×Q₈	—	2	1	no	—
327	C₄×Q₈	—	2	6	no	—
328	C₄×Q₈	—	2	2	no	—
329	C₄×Q₈	—	2	3	no	—
330	C₄×Q₈	—	2	3	no	—
331	C₄×Q₈	—	2	3	no	—
332	C₄×Q₈	—	2	3	no	—
333	C₄×Q₈	—	2	3	no	—
334	C₄×Q₈	D₄×C₇₁	1	1	yes	characteristic
335	C₄×Q₈	D₄×C₇₁	1	3	yes	—
336	C₄×Q₈	C₂³×C₇₁	1	6	yes	—
337	C₄×Q₈	C₂³×C₇₁	1	6	yes	—
338	C₄×Q₈	C₂³×C₇₁	1	3	yes	—
339	C₄×Q₈	C₂³×C₇₁	1	3	yes	—
340	C₄×Q₈	C₂³×C₇₁	1	1	yes	characteristic
341	C₄×Q₈	C₂³×C₇₁	1	3	yes	—
342	C₄×D₄	—	2	3	no	—
343	C₄×D₄	—	2	3	no	—
344	C₄×D₄	C₂³×C₇₁	1	3	yes	—
345	C₄×D₄	C₂³×C₇₁	1	3	yes	—
346	#117	—	2	3	no	—
347	#117	C₂³×C₇₁	1	3	yes	—
348	C₂×C₄⋊C₄	—	2	3	no	—
349	C₂×C₄⋊C₄	—	2	3	no	—
350	C₂×C₄⋊C₄	—	2	3	no	—
351	C₂×C₄⋊C₄	—	2	3	no	—
352	C₂×C₄⋊C₄	D₄×C₇₁	1	3	yes	—
353	C₂×C₄⋊C₄	C₂³×C₇₁	1	1	yes	characteristic
354	C₂×C₄²	—	2	3	no	—
355	C₂×C₄²	—	2	3	no	—
356	C₂×C₄²	C₂³×C₇₁	1	3	yes	—
357	#107	C₂³×C₇₁	1	6	yes	—
358	#106	C₂³×C₇₁	1	6	yes	—
359	C₄⋊C₈	C₂³×C₇₁	1	6	yes	—
360	C₄⋊C₈	C₂³×C₇₁	1	6	yes	—
361	#103	C₂³×C₇₁	1	2	yes	—
362	#103	C₂³×C₇₁	1	6	yes	—
363	#103	C₂³×C₇₁	1	6	yes	—
364	#103	C₂³×C₇₁	1	12	yes	—
365	#103	C₂³×C₇₁	1	12	yes	—
366	#102	C₂³×C₇₁	1	2	yes	—
367	#97	C₂³×C₇₁	1	6	yes	—
368	C₄×C₈	C₂³×C₇₁	1	6	yes	—
369	#41	—	2	2	no	—
370	#41	#41×C₇₁	1	4	yes	—
371	C₂×Q₈	—	2	4	no	—
372	C₂×Q₈	—	2	3	no	—
373	C₂×Q₈	—	2	3	no	—
374	C₂×Q₈	—	2	3	no	—
375	C₂×Q₈	—	2	6	no	—
376	C₂×Q₈	—	2	6	no	—
377	C₂×Q₈	—	4	4	no	—
378	C₂×Q₈	C₂×D₄×C₇₁	1	3	yes	—
379	C₂×Q₈	C₂×D₄×C₇₁	1	1	yes	characteristic
380	C₂×Q₈	C₂⁴×C₇₁	1	6	yes	—
381	C₂×Q₈	C₂⁴×C₇₁	1	1	yes	characteristic
382	C₂×D₄	—	2	1	no	—
383	C₂×D₄	C₂⁴×C₇₁	1	1	yes	characteristic
384	C₂²×C₄	—	2	1	no	—
385	C₂²×C₄	—	2	1	no	—
386	C₂²×C₄	—	2	3	no	—
387	C₂²×C₄	—	2	3	no	—
388	C₂²×C₄	—	2	3	no	—
389	C₂²×C₄	C₂×D₄×C₇₁	1	3	yes	—
390	C₂²×C₄	C₂×D₄×C₇₁	1	3	yes	—
391	C₂²×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
392	Q₁₆	—	2	12	no	—
393	SD₁₆	—	2	12	no	—
394	M₁₆	—	2	6	no	—
395	M₁₆	—	2	2	no	—
396	C₂×C₈	C₂⁴×C₇₁	1	2	yes	—
397	C₂×C₈	C₂⁴×C₇₁	1	6	yes	—
398	C₄⋊C₄	—	2	6	no	—
399	C₄⋊C₄	—	2	3	no	—
400	C₄⋊C₄	—	2	6	no	—
401	C₄⋊C₄	—	2	6	no	—
402	C₄⋊C₄	—	2	6	no	—
403	C₄⋊C₄	—	2	6	no	—
404	C₄⋊C₄	—	2	6	no	—
405	C₄⋊C₄	—	2	6	no	—
406	C₄⋊C₄	—	2	1	no	—
407	C₄⋊C₄	—	2	3	no	—
408	C₄⋊C₄	—	2	3	no	—
409	C₄⋊C₄	—	2	3	no	—
410	C₄⋊C₄	—	2	6	no	—
411	C₄⋊C₄	C₂×D₄×C₇₁	1	3	yes	—
412	C₄⋊C₄	C₂×D₄×C₇₁	1	6	yes	—
413	C₄⋊C₄	C₂×D₄×C₇₁	1	3	yes	—
414	C₄⋊C₄	C₂⁴×C₇₁	1	2	yes	—
415	C₄⋊C₄	C₂⁴×C₇₁	1	3	yes	—
416	C₄⋊C₄	C₂⁴×C₇₁	1	3	yes	—
417	C₂²⋊C₄	—	2	3	no	—
418	C₂²⋊C₄	—	2	3	no	—
419	C₂²⋊C₄	—	2	3	no	—
420	C₄²	—	2	3	no	—
421	C₄²	—	2	3	no	—
422	C₄²	—	2	3	no	—
423	C₄²	—	2	6	no	—
424	C₄²	—	2	6	no	—
425	C₄²	C₂×D₄×C₇₁	1	3	yes	—
426	C₄²	C₂×D₄×C₇₁	1	3	yes	—
427	C₄²	C₂⁴×C₇₁	1	3	yes	—
428	C₄²	C₂⁴×C₇₁	1	3	yes	—
429	C₂³	—	2	1	no	—
430	C₂³	C₂²×D₄×C₇₁	1	1	yes	characteristic
431	Q₈	—	2	6	no	—
432	Q₈	—	2	1	no	—
433	Q₈	—	2	6	no	—
434	Q₈	—	2	6	no	—
435	Q₈	—	4	6	no	—
436	Q₈	—	4	6	no	—
437	Q₈	—	8	3	no	—
438	Q₈	#137×C₇₁	1	4	yes	complemented
439	Q₈	C₂×#41×C₇₁	1	2	yes	—
440	D₄	—	2	4	no	—
441	D₄	—	2	1	no	—
442	D₄	C₂×#41×C₇₁	1	2	yes	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	4	no	—
445	C₂×C₄	—	2	6	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	3	no	—
448	C₂×C₄	—	2	3	no	—
449	C₂×C₄	—	2	6	no	—
450	C₂×C₄	—	4	3	no	—
451	C₂×C₄	—	4	2	no	—
452	C₂×C₄	—	4	6	no	—
453	C₂×C₄	—	4	6	no	—
454	C₂×C₄	—	8	3	no	—
455	C₂×C₄	C₂²×D₄×C₇₁	1	3	yes	—
456	C₂×C₄	C₂²×D₄×C₇₁	1	3	yes	—
457	C₂×C₄	C₂²×D₄×C₇₁	1	3	yes	—
458	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
459	C₂×C₄	C₂²×D₄×C₇₁	1	3	yes	—
460	C₂×C₄	C₂×#41×C₇₁	1	2	yes	—
461	C₂×C₄	2₋¹⁺⁴×C₇₁	1	2	yes	—
462	C₂×C₄	C₂⁵×C₇₁	1	1	yes	characteristic
463	C₈	—	2	6	no	—
464	C₈	—	2	2	no	—
465	C₂²	—	2	1	no	—
466	C₂²	—	2	1	no	—
467	C₂²	—	2	2	no	—
468	C₂²	—	4	1	no	—
469	C₂²	—	4	1	no	—
470	C₂²	#548×C₇₁	1	2	yes	—
471	C₂²	C₂³×D₄×C₇₁	1	1	yes	characteristic
472	C₄	—	2	3	no	—
473	C₄	—	2	3	no	—
474	C₄	—	2	3	no	—
475	C₄	—	2	2	no	—
476	C₄	—	4	3	no	—
477	C₄	—	4	2	no	—
478	C₄	—	4	3	no	—
479	C₄	—	4	6	no	—
480	C₄	#548×C₇₁	1	2	yes	—
481	C₄	C₂×#137×C₇₁	1	6	yes	—
482	C₄	C₂²×#41×C₇₁	1	1	yes	characteristic
483	C₄	C₂×2₋¹⁺⁴×C₇₁	1	1	yes	characteristic
484	C₂	—	2	2	no	—
485	C₂	—	4	2	no	—
486	C₂	C₂×#548×C₇₁	1	1	yes	characteristic
487	C₂	C₂²×#137×C₇₁	1	1	yes	characteristic
488	C₂	C₇₁×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₇₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇₁
prime radical	C₂×#41×C₇₁	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₂×C₄×Q₈×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#557×C₇₁	C₂	semidirect
3. C₇₁×#3374	C₂	semidirect
4. C₇₁×#3372	C₂	semidirect
5. C₇₁×#3363	C₂	semidirect
6. C₇₁×#3359	C₂	semidirect
7. C₇₁×#3139	C₂	semidirect
8. C₇₁×#3078	C₂	semidirect
10. Q₈×M₁₆×C₇₁	C₂	semidirect
193. Q₈×C₇₁	#137	semidirect
239. #61322	C₇₁	direct (N × H)
246. C₂×#557	C₂×C₇₁	semidirect
247. #3374	C₂×C₇₁	semidirect
248. #3372	C₂×C₇₁	semidirect
249. #3363	C₂×C₇₁	semidirect
250. #3359	C₂×C₇₁	semidirect
251. #3139	C₂×C₇₁	semidirect
252. #3078	C₂×C₇₁	semidirect
254. Q₈×M₁₆	C₂×C₇₁	semidirect
256. C₇₁	#61322	direct (N × H)
438. Q₈	#137×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#41×C₇₁	2
C₂×#557×C₇₁	2
C₇₁×#3374	2
C₇₁×#3372	2
C₇₁×#3363	2
C₇₁×#3359	2
C₇₁×#3139	2
C₇₁×#3078	2
C₇₁×#3039	2
Q₈×M₁₆×C₇₁	2
C₄×#137×C₇₁	2
#61322	71

Subgroups — C71×#61322 #31882194

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#61322 #31882194