Subgroups — C₇₁×#24294 #31845166

Order 18176.

1358 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#576×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#472×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₁×#3060	C₂	1	1	yes	maximal, characteristic, complemented
4	C₇₁×#3012	C₂	1	2	yes	maximal, complemented
5	C₂×#444×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₇₁×#2051	C₂	1	2	yes	maximal, complemented
7	C₇₁×#2006	C₂	1	2	yes	maximal, complemented
8	C₇₁×#1975	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₁×#1972	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇₁×#1815	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₁×#1792	C₂	1	2	yes	maximal, complemented
12	C₂×H(5,𝔽₂)×C₇₁	C₂²	1	2	yes	—
13	#576×C₇₁	—	2	4	no	—
14	#576×C₇₁	—	2	4	no	—
15	C₂×AGL(1,ℤ/8)×C₇₁	—	2	2	no	—
16	C₂×AGL(1,ℤ/8)×C₇₁	C₂²	1	2	yes	—
17	C₂×AGL(1,ℤ/8)×C₇₁	C₂²	1	2	yes	complemented
18	C₂×#135×C₇₁	—	2	1	no	—
19	C₂×#135×C₇₁	C₂²	1	1	yes	characteristic
20	C₂×#135×C₇₁	C₂²	1	1	yes	characteristic
21	C₂²×D₈×C₇₁	—	2	1	no	—
22	C₂²×D₈×C₇₁	C₂²	1	1	yes	characteristic
23	C₂×#131×C₇₁	C₂²	1	1	yes	characteristic
24	C₂×#131×C₇₁	C₂²	1	1	yes	characteristic, complemented
25	#537×C₇₁	C₂²	1	2	yes	complemented
26	#472×C₇₁	—	2	4	no	—
27	#470×C₇₁	—	2	1	no	—
28	#469×C₇₁	—	2	1	no	—
29	#465×C₇₁	—	2	2	no	—
30	#452×C₇₁	—	2	2	no	—
31	#450×C₇₁	—	2	2	no	—
32	#449×C₇₁	—	2	2	no	—
33	#447×C₇₁	—	2	2	no	—
34	#444×C₇₁	—	2	4	no	—
35	C₂×#108×C₇₁	C₂²	1	1	yes	characteristic, complemented
36	C₂×#108×C₇₁	C₂²	1	1	yes	characteristic, complemented
37	C₂×#106×C₇₁	C₂²	1	1	yes	characteristic, complemented
38	C₂×#104×C₇₁	C₂²	1	1	yes	characteristic
39	C₂×#104×C₇₁	C₂²	1	1	yes	characteristic, complemented
40	C₂×#104×C₇₁	C₂²	1	2	yes	complemented
41	#417×C₇₁	C₂²	1	2	yes	—
42	#417×C₇₁	C₂²	1	2	yes	complemented
43	C₂×#102×C₇₁	C₂²	1	2	yes	complemented
44	C₂×#100×C₇₁	C₂²	1	2	yes	complemented
45	#408×C₇₁	C₂²	1	2	yes	complemented
46	C₂×#97×C₇₁	C₂²	1	1	yes	characteristic, complemented
47	#390×C₇₁	C₂²	1	2	yes	complemented
48	#343×C₇₁	C₂²	1	2	yes	complemented
49	H(5,𝔽₂)×C₇₁	—	2	4	no	—
50	H(5,𝔽₂)×C₇₁	—	2	4	no	—
51	H(5,𝔽₂)×C₇₁	—	2	4	no	—
52	H(5,𝔽₂)×C₇₁	—	4	2	no	—
53	C₂×#41×C₇₁	—	2	2	no	—
54	C₂×#41×C₇₁	—	2	2	no	—
55	C₂×#41×C₇₁	C₂³	1	2	yes	—
56	C₂×#41×C₇₁	C₂³	1	1	yes	characteristic
57	C₂×#41×C₇₁	C₂³	1	1	yes	characteristic
58	C₂²×D₄×C₇₁	—	2	2	no	—
59	C₂²×D₄×C₇₁	—	2	2	no	—
60	C₂²×D₄×C₇₁	—	4	2	no	—
61	C₂²×D₄×C₇₁	C₂³	1	2	yes	—
62	AGL(1,ℤ/8)×C₇₁	—	2	16	no	—
63	AGL(1,ℤ/8)×C₇₁	—	2	4	no	—
64	AGL(1,ℤ/8)×C₇₁	—	2	4	no	—
65	AGL(1,ℤ/8)×C₇₁	—	2	4	no	—
66	AGL(1,ℤ/8)×C₇₁	—	4	2	no	—
67	#135×C₇₁	—	2	8	no	—
68	#135×C₇₁	—	2	2	no	—
69	#135×C₇₁	—	2	2	no	—
70	#135×C₇₁	—	2	2	no	—
71	#135×C₇₁	—	2	2	no	—
72	C₂×Q₁₆×C₇₁	D₄	1	1	yes	characteristic
73	C₂×Q₁₆×C₇₁	D₄	1	1	yes	characteristic
74	C₂×SD₁₆×C₇₁	—	2	2	no	—
75	C₂×SD₁₆×C₇₁	—	2	2	no	—
76	C₂×SD₁₆×C₇₁	D₄	1	1	yes	characteristic
77	C₂×SD₁₆×C₇₁	D₄	1	1	yes	characteristic
78	C₂×SD₁₆×C₇₁	D₄	1	2	yes	—
79	C₂×D₈×C₇₁	—	2	8	no	—
80	C₂×D₈×C₇₁	—	2	2	no	—
81	C₂×D₈×C₇₁	—	2	2	no	—
82	C₂×D₈×C₇₁	—	2	1	no	—
83	C₂×D₈×C₇₁	—	2	1	no	—
84	C₂×D₈×C₇₁	—	2	2	no	—
85	C₂×D₈×C₇₁	D₄	1	1	yes	characteristic
86	C₂×D₈×C₇₁	D₄	1	1	yes	characteristic
87	C₂×D₈×C₇₁	D₄	1	4	yes	complemented
88	#131×C₇₁	—	2	2	no	—
89	#131×C₇₁	—	2	2	no	—
90	#131×C₇₁	—	2	2	no	—
91	#131×C₇₁	D₄	1	4	yes	complemented
92	C₂×M₁₆×C₇₁	—	2	1	no	—
93	C₂×M₁₆×C₇₁	—	2	1	no	—
94	C₂×M₁₆×C₇₁	C₂³	1	2	yes	—
95	C₂×M₁₆×C₇₁	C₂³	1	1	yes	characteristic
96	C₂×M₁₆×C₇₁	C₂³	1	1	yes	characteristic
97	C₂²×C₈×C₇₁	—	2	1	no	—
98	C₂²×C₈×C₇₁	—	2	1	no	—
99	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
100	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
101	C₂²⋊Q₈×C₇₁	—	2	2	no	—
102	C₂²⋊Q₈×C₇₁	—	2	2	no	—
103	#121×C₇₁	—	2	2	no	—
104	#121×C₇₁	—	2	2	no	—
105	#121×C₇₁	—	4	2	no	—
106	C₂×C₄⋊C₄×C₇₁	C₂³	1	2	yes	—
107	C₂×C₂²⋊C₄×C₇₁	—	4	2	no	—
108	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
109	#108×C₇₁	—	2	2	no	—
110	#108×C₇₁	—	2	2	no	—
111	#106×C₇₁	—	2	1	no	—
112	#106×C₇₁	—	2	1	no	—
113	#104×C₇₁	—	2	2	no	—
114	#104×C₇₁	—	2	2	no	—
115	#104×C₇₁	—	4	2	no	—
116	#103×C₇₁	—	2	2	no	—
117	#103×C₇₁	—	2	2	no	—
118	#102×C₇₁	—	2	2	no	—
119	#102×C₇₁	—	2	2	no	—
120	#102×C₇₁	—	2	2	no	—
121	#102×C₇₁	—	2	2	no	—
122	#100×C₇₁	—	2	4	no	—
123	C₂²⋊C₈×C₇₁	—	2	2	no	—
124	C₂²⋊C₈×C₇₁	—	2	2	no	—
125	#97×C₇₁	—	2	2	no	—
126	C₂⁴×C₇₁	—	2	2	no	—
127	C₂⁴×C₇₁	—	4	2	no	—
128	#41×C₇₁	—	2	1	no	—
129	#41×C₇₁	—	2	1	no	—
130	#41×C₇₁	—	2	8	no	—
131	#41×C₇₁	—	2	1	no	—
132	#41×C₇₁	—	2	1	no	—
133	#41×C₇₁	—	2	2	no	—
134	#41×C₇₁	—	2	2	no	—
135	#41×C₇₁	—	2	1	no	—
136	#41×C₇₁	—	2	2	no	—
137	#41×C₇₁	—	2	2	no	—
138	#41×C₇₁	—	2	1	no	—
139	#41×C₇₁	—	4	4	no	—
140	#41×C₇₁	—	4	4	no	—
141	#41×C₇₁	—	4	4	no	—
142	#41×C₇₁	C₂×D₄	1	2	yes	—
143	#41×C₇₁	#41	1	2	yes	—
144	C₂×Q₈×C₇₁	C₂×D₄	1	2	yes	—
145	C₂×Q₈×C₇₁	C₂×D₄	1	1	yes	characteristic
146	C₂×Q₈×C₇₁	C₂×D₄	1	1	yes	characteristic
147	C₂×D₄×C₇₁	—	2	1	no	—
148	C₂×D₄×C₇₁	—	2	8	no	—
149	C₂×D₄×C₇₁	—	2	1	no	—
150	C₂×D₄×C₇₁	—	2	2	no	—
151	C₂×D₄×C₇₁	—	2	2	no	—
152	C₂×D₄×C₇₁	—	2	2	no	—
153	C₂×D₄×C₇₁	—	2	2	no	—
154	C₂×D₄×C₇₁	—	2	2	no	—
155	C₂×D₄×C₇₁	—	2	2	no	—
156	C₂×D₄×C₇₁	—	4	4	no	—
157	C₂×D₄×C₇₁	—	4	2	no	—
158	C₂×D₄×C₇₁	—	4	2	no	—
159	C₂×D₄×C₇₁	—	4	4	no	—
160	C₂×D₄×C₇₁	—	4	8	no	—
161	C₂×D₄×C₇₁	—	4	8	no	—
162	C₂×D₄×C₇₁	—	4	4	no	—
163	C₂×D₄×C₇₁	—	4	2	no	—
164	C₂×D₄×C₇₁	C₂×D₄	1	4	yes	—
165	C₂×D₄×C₇₁	C₂×D₄	1	2	yes	—
166	C₂×D₄×C₇₁	C₂×D₄	1	1	yes	characteristic
167	C₂×D₄×C₇₁	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₇₁	—	2	1	no	—
169	C₂²×C₄×C₇₁	—	2	2	no	—
170	C₂²×C₄×C₇₁	—	2	1	no	—
171	C₂²×C₄×C₇₁	—	2	1	no	—
172	C₂²×C₄×C₇₁	—	2	2	no	—
173	C₂²×C₄×C₇₁	—	4	2	no	—
174	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
175	Q₁₆×C₇₁	—	2	2	no	—
176	Q₁₆×C₇₁	—	2	2	no	—
177	SD₁₆×C₇₁	—	2	8	no	—
178	SD₁₆×C₇₁	—	2	2	no	—
179	SD₁₆×C₇₁	—	2	2	no	—
180	SD₁₆×C₇₁	—	2	4	no	—
181	SD₁₆×C₇₁	—	4	4	no	—
182	D₈×C₇₁	—	2	8	no	—
183	D₈×C₇₁	—	2	2	no	—
184	D₈×C₇₁	—	2	2	no	—
185	D₈×C₇₁	—	2	4	no	—
186	D₈×C₇₁	—	2	4	no	—
187	D₈×C₇₁	—	4	4	no	—
188	M₁₆×C₇₁	—	2	1	no	—
189	M₁₆×C₇₁	—	2	4	no	—
190	M₁₆×C₇₁	—	2	1	no	—
191	M₁₆×C₇₁	—	2	2	no	—
192	M₁₆×C₇₁	—	2	1	no	—
193	M₁₆×C₇₁	—	2	2	no	—
194	M₁₆×C₇₁	—	4	2	no	—
195	M₁₆×C₇₁	C₂×D₄	1	2	yes	—
196	C₂×C₈×C₇₁	—	2	1	no	—
197	C₂×C₈×C₇₁	—	2	1	no	—
198	C₂×C₈×C₇₁	—	2	1	no	—
199	C₂×C₈×C₇₁	—	2	4	no	—
200	C₂×C₈×C₇₁	—	2	2	no	—
201	C₂×C₈×C₇₁	—	2	1	no	—
202	C₂×C₈×C₇₁	—	4	2	no	—
203	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
204	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
205	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
206	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
207	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
208	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
209	C₄⋊C₄×C₇₁	—	2	2	no	—
210	C₄⋊C₄×C₇₁	—	2	2	no	—
211	C₄⋊C₄×C₇₁	—	4	2	no	—
212	C₂²⋊C₄×C₇₁	—	4	2	no	—
213	C₂²⋊C₄×C₇₁	—	4	2	no	—
214	C₂²⋊C₄×C₇₁	—	8	4	no	—
215	C₄²×C₇₁	—	2	1	no	—
216	C₄²×C₇₁	—	2	1	no	—
217	C₂³×C₇₁	—	2	1	no	—
218	C₂³×C₇₁	—	2	1	no	—
219	C₂³×C₇₁	—	2	2	no	—
220	C₂³×C₇₁	—	2	4	no	—
221	C₂³×C₇₁	—	4	8	no	—
222	C₂³×C₇₁	—	4	2	no	—
223	C₂³×C₇₁	—	4	2	no	—
224	C₂³×C₇₁	—	8	2	no	—
225	C₂³×C₇₁	—	8	2	no	—
226	C₂³×C₇₁	—	16	2	no	—
227	C₂³×C₇₁	—	16	2	no	—
228	C₂³×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
229	Q₈×C₇₁	—	2	1	no	—
230	Q₈×C₇₁	—	2	2	no	—
231	Q₈×C₇₁	—	2	1	no	—
232	Q₈×C₇₁	—	2	1	no	—
233	Q₈×C₇₁	—	2	2	no	—
234	Q₈×C₇₁	#121	1	2	yes	—
235	D₄×C₇₁	—	2	2	no	—
236	D₄×C₇₁	—	2	2	no	—
237	D₄×C₇₁	—	2	2	no	—
238	D₄×C₇₁	—	2	4	no	—
239	D₄×C₇₁	—	2	1	no	—
240	D₄×C₇₁	—	2	1	no	—
241	D₄×C₇₁	—	2	1	no	—
242	D₄×C₇₁	—	2	4	no	—
243	D₄×C₇₁	—	2	2	no	—
244	D₄×C₇₁	—	4	8	no	—
245	D₄×C₇₁	—	4	8	no	—
246	D₄×C₇₁	—	4	2	no	—
247	D₄×C₇₁	—	4	1	no	—
248	D₄×C₇₁	—	4	2	no	—
249	D₄×C₇₁	—	4	1	no	—
250	D₄×C₇₁	—	4	2	no	—
251	D₄×C₇₁	—	4	2	no	—
252	D₄×C₇₁	—	16	2	no	—
253	D₄×C₇₁	#121	1	2	yes	—
254	C₂×C₄×C₇₁	—	2	1	no	—
255	C₂×C₄×C₇₁	—	2	2	no	—
256	C₂×C₄×C₇₁	—	2	2	no	—
257	C₂×C₄×C₇₁	—	2	1	no	—
258	C₂×C₄×C₇₁	—	2	2	no	—
259	C₂×C₄×C₇₁	—	4	8	no	—
260	C₂×C₄×C₇₁	—	4	1	no	—
261	C₂×C₄×C₇₁	—	4	2	no	—
262	C₂×C₄×C₇₁	—	4	2	no	—
263	C₂×C₄×C₇₁	—	4	1	no	—
264	C₂×C₄×C₇₁	—	4	2	no	—
265	C₂×C₄×C₇₁	—	4	1	no	—
266	C₂×C₄×C₇₁	—	4	2	no	—
267	C₂×C₄×C₇₁	—	8	2	no	—
268	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
269	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
270	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
271	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
272	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
273	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
274	C₈×C₇₁	—	2	2	no	—
275	C₈×C₇₁	—	2	1	no	—
276	C₈×C₇₁	—	2	1	no	—
277	C₈×C₇₁	—	4	1	no	—
278	C₈×C₇₁	—	4	2	no	—
279	C₈×C₇₁	#120	1	2	yes	—
280	C₈×C₇₁	#120	1	2	yes	—
281	C₂²×C₇₁	—	2	2	no	—
282	C₂²×C₇₁	—	2	2	no	—
283	C₂²×C₇₁	—	4	2	no	—
284	C₂²×C₇₁	—	4	2	no	—
285	C₂²×C₇₁	—	4	4	no	—
286	C₂²×C₇₁	—	4	1	no	—
287	C₂²×C₇₁	—	4	2	no	—
288	C₂²×C₇₁	—	4	2	no	—
289	C₂²×C₇₁	—	4	2	no	—
290	C₂²×C₇₁	—	8	4	no	—
291	C₂²×C₇₁	—	8	2	no	—
292	C₂²×C₇₁	—	8	2	no	—
293	C₂²×C₇₁	—	16	2	no	—
294	C₂²×C₇₁	—	16	8	no	—
295	C₂²×C₇₁	#535	1	1	yes	characteristic
296	C₂²×C₇₁	D₄²	1	1	yes	characteristic
297	C₂²×C₇₁	#546	1	1	yes	characteristic
298	C₄×C₇₁	—	2	2	no	—
299	C₄×C₇₁	—	4	2	no	—
300	C₄×C₇₁	—	4	1	no	—
301	C₄×C₇₁	—	8	1	no	—
302	C₄×C₇₁	—	8	2	no	—
303	C₄×C₇₁	C₂×#120	1	1	yes	characteristic
304	C₄×C₇₁	C₂×#121	1	1	yes	characteristic
305	C₄×C₇₁	D₄²	1	1	yes	characteristic
306	C₄×C₇₁	#546	1	1	yes	characteristic
307	#24294	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
308	C₂×C₇₁	—	2	1	no	—
309	C₂×C₇₁	—	2	1	no	—
310	C₂×C₇₁	—	4	2	no	—
311	C₂×C₇₁	—	8	1	no	—
312	C₂×C₇₁	—	8	2	no	—
313	C₂×C₇₁	—	8	4	no	—
314	C₂×C₇₁	#2408	1	1	yes	characteristic
315	C₂×C₇₁	#3288	1	2	yes	—
316	C₂×#576	C₂×C₇₁	1	1	yes	characteristic, complemented
317	C₂×#472	C₂×C₇₁	1	1	yes	characteristic, complemented
318	#3060	C₂×C₇₁	1	1	yes	characteristic, complemented
319	#3012	C₂×C₇₁	1	2	yes	complemented
320	C₂×#444	C₂×C₇₁	1	1	yes	characteristic, complemented
321	#2051	C₂×C₇₁	1	2	yes	complemented
322	#2006	C₂×C₇₁	1	2	yes	complemented
323	#1975	C₂×C₇₁	1	1	yes	characteristic, complemented
324	#1972	C₂×C₇₁	1	1	yes	characteristic, complemented
325	#1815	C₂×C₇₁	1	1	yes	characteristic, complemented
326	#1792	C₂×C₇₁	1	2	yes	complemented
327	C₇₁	#24294	1	1	yes	characteristic, direct product factor, Sylow
328	C₂×H(5,𝔽₂)	C₂²×C₇₁	1	2	yes	—
329	#576	—	2	4	no	—
330	#576	—	2	4	no	—
331	C₂×AGL(1,ℤ/8)	—	2	2	no	—
332	C₂×AGL(1,ℤ/8)	C₂²×C₇₁	1	2	yes	—
333	C₂×AGL(1,ℤ/8)	C₂²×C₇₁	1	2	yes	complemented
334	C₂×#135	—	2	1	no	—
335	C₂×#135	C₂²×C₇₁	1	1	yes	characteristic
336	C₂×#135	C₂²×C₇₁	1	1	yes	characteristic
337	C₂²×D₈	—	2	1	no	—
338	C₂²×D₈	C₂²×C₇₁	1	1	yes	characteristic
339	C₂×#131	C₂²×C₇₁	1	1	yes	characteristic
340	C₂×#131	C₂²×C₇₁	1	1	yes	characteristic, complemented
341	#537	C₂²×C₇₁	1	2	yes	complemented
342	#472	—	2	4	no	—
343	#470	—	2	1	no	—
344	#469	—	2	1	no	—
345	#465	—	2	2	no	—
346	#452	—	2	2	no	—
347	#450	—	2	2	no	—
348	#449	—	2	2	no	—
349	#447	—	2	2	no	—
350	#444	—	2	4	no	—
351	C₂×#108	C₂²×C₇₁	1	1	yes	characteristic, complemented
352	C₂×#108	C₂²×C₇₁	1	1	yes	characteristic, complemented
353	C₂×#106	C₂²×C₇₁	1	1	yes	characteristic, complemented
354	C₂×#104	C₂²×C₇₁	1	1	yes	characteristic
355	C₂×#104	C₂²×C₇₁	1	1	yes	characteristic, complemented
356	C₂×#104	C₂²×C₇₁	1	2	yes	complemented
357	#417	C₂²×C₇₁	1	2	yes	—
358	#417	C₂²×C₇₁	1	2	yes	complemented
359	C₂×#102	C₂²×C₇₁	1	2	yes	complemented
360	C₂×#100	C₂²×C₇₁	1	2	yes	complemented
361	#408	C₂²×C₇₁	1	2	yes	complemented
362	C₂×#97	C₂²×C₇₁	1	1	yes	characteristic, complemented
363	#390	C₂²×C₇₁	1	2	yes	complemented
364	#343	C₂²×C₇₁	1	2	yes	complemented
365	H(5,𝔽₂)	—	2	4	no	—
366	H(5,𝔽₂)	—	2	4	no	—
367	H(5,𝔽₂)	—	2	4	no	—
368	H(5,𝔽₂)	—	4	2	no	—
369	C₂×#41	—	2	2	no	—
370	C₂×#41	—	2	2	no	—
371	C₂×#41	C₂³×C₇₁	1	1	yes	characteristic
372	C₂×#41	C₂³×C₇₁	1	2	yes	—
373	C₂×#41	C₂³×C₇₁	1	1	yes	characteristic
374	C₂²×D₄	—	2	2	no	—
375	C₂²×D₄	—	2	2	no	—
376	C₂²×D₄	—	4	2	no	—
377	C₂²×D₄	C₂³×C₇₁	1	2	yes	—
378	AGL(1,ℤ/8)	—	2	16	no	—
379	AGL(1,ℤ/8)	—	2	4	no	—
380	AGL(1,ℤ/8)	—	2	4	no	—
381	AGL(1,ℤ/8)	—	2	4	no	—
382	AGL(1,ℤ/8)	—	4	2	no	—
383	#135	—	2	8	no	—
384	#135	—	2	2	no	—
385	#135	—	2	2	no	—
386	#135	—	2	2	no	—
387	#135	—	2	2	no	—
388	C₂×Q₁₆	D₄×C₇₁	1	1	yes	characteristic
389	C₂×Q₁₆	D₄×C₇₁	1	1	yes	characteristic
390	C₂×SD₁₆	—	2	2	no	—
391	C₂×SD₁₆	—	2	2	no	—
392	C₂×SD₁₆	D₄×C₇₁	1	2	yes	—
393	C₂×SD₁₆	D₄×C₇₁	1	1	yes	characteristic
394	C₂×SD₁₆	D₄×C₇₁	1	1	yes	characteristic
395	C₂×D₈	—	2	8	no	—
396	C₂×D₈	—	2	2	no	—
397	C₂×D₈	—	2	2	no	—
398	C₂×D₈	—	2	1	no	—
399	C₂×D₈	—	2	1	no	—
400	C₂×D₈	—	2	2	no	—
401	C₂×D₈	D₄×C₇₁	1	1	yes	characteristic
402	C₂×D₈	D₄×C₇₁	1	1	yes	characteristic
403	C₂×D₈	D₄×C₇₁	1	4	yes	complemented
404	#131	—	2	2	no	—
405	#131	—	2	2	no	—
406	#131	—	2	2	no	—
407	#131	D₄×C₇₁	1	4	yes	complemented
408	C₂×M₁₆	—	2	1	no	—
409	C₂×M₁₆	—	2	1	no	—
410	C₂×M₁₆	C₂³×C₇₁	1	1	yes	characteristic
411	C₂×M₁₆	C₂³×C₇₁	1	2	yes	—
412	C₂×M₁₆	C₂³×C₇₁	1	1	yes	characteristic
413	C₂²×C₈	—	2	1	no	—
414	C₂²×C₈	—	2	1	no	—
415	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
416	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
417	C₂²⋊Q₈	—	2	2	no	—
418	C₂²⋊Q₈	—	2	2	no	—
419	#121	—	2	2	no	—
420	#121	—	2	2	no	—
421	#121	—	4	2	no	—
422	C₂×C₄⋊C₄	C₂³×C₇₁	1	2	yes	—
423	C₂×C₂²⋊C₄	—	4	2	no	—
424	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
425	#108	—	2	2	no	—
426	#108	—	2	2	no	—
427	#106	—	2	1	no	—
428	#106	—	2	1	no	—
429	#104	—	2	2	no	—
430	#104	—	2	2	no	—
431	#104	—	4	2	no	—
432	#103	—	2	2	no	—
433	#103	—	2	2	no	—
434	#102	—	2	2	no	—
435	#102	—	2	2	no	—
436	#102	—	2	2	no	—
437	#102	—	2	2	no	—
438	#100	—	2	4	no	—
439	C₂²⋊C₈	—	2	2	no	—
440	C₂²⋊C₈	—	2	2	no	—
441	#97	—	2	2	no	—
442	C₂⁴	—	2	2	no	—
443	C₂⁴	—	4	2	no	—
444	#41	—	2	1	no	—
445	#41	—	2	1	no	—
446	#41	—	2	8	no	—
447	#41	—	2	1	no	—
448	#41	—	2	1	no	—
449	#41	—	2	2	no	—
450	#41	—	2	2	no	—
451	#41	—	2	1	no	—
452	#41	—	2	2	no	—
453	#41	—	2	2	no	—
454	#41	—	2	1	no	—
455	#41	—	4	4	no	—
456	#41	—	4	4	no	—
457	#41	—	4	4	no	—
458	#41	C₂×D₄×C₇₁	1	2	yes	—
459	#41	#41×C₇₁	1	2	yes	—
460	C₂×Q₈	C₂×D₄×C₇₁	1	2	yes	—
461	C₂×Q₈	C₂×D₄×C₇₁	1	1	yes	characteristic
462	C₂×Q₈	C₂×D₄×C₇₁	1	1	yes	characteristic
463	C₂×D₄	—	2	1	no	—
464	C₂×D₄	—	2	8	no	—
465	C₂×D₄	—	2	1	no	—
466	C₂×D₄	—	2	2	no	—
467	C₂×D₄	—	2	2	no	—
468	C₂×D₄	—	2	2	no	—
469	C₂×D₄	—	2	2	no	—
470	C₂×D₄	—	2	2	no	—
471	C₂×D₄	—	2	2	no	—
472	C₂×D₄	—	4	4	no	—
473	C₂×D₄	—	4	2	no	—
474	C₂×D₄	—	4	2	no	—
475	C₂×D₄	—	4	4	no	—
476	C₂×D₄	—	4	8	no	—
477	C₂×D₄	—	4	8	no	—
478	C₂×D₄	—	4	4	no	—
479	C₂×D₄	—	4	2	no	—
480	C₂×D₄	C₂×D₄×C₇₁	1	4	yes	—
481	C₂×D₄	C₂×D₄×C₇₁	1	2	yes	—
482	C₂×D₄	C₂×D₄×C₇₁	1	1	yes	characteristic
483	C₂×D₄	C₂×D₄×C₇₁	1	1	yes	characteristic
484	C₂²×C₄	—	2	1	no	—
485	C₂²×C₄	—	2	2	no	—
486	C₂²×C₄	—	2	1	no	—
487	C₂²×C₄	—	2	1	no	—
488	C₂²×C₄	—	2	2	no	—
489	C₂²×C₄	—	4	2	no	—
490	C₂²×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
491	Q₁₆	—	2	2	no	—
492	Q₁₆	—	2	2	no	—
493	SD₁₆	—	2	8	no	—
494	SD₁₆	—	2	2	no	—
495	SD₁₆	—	2	2	no	—
496	SD₁₆	—	2	4	no	—
497	SD₁₆	—	4	4	no	—
498	D₈	—	2	8	no	—
499	D₈	—	2	2	no	—
500	D₈	—	2	2	no	—
501	D₈	—	2	4	no	—
502	D₈	—	2	4	no	—
503	D₈	—	4	4	no	—
504	M₁₆	—	2	1	no	—
505	M₁₆	—	2	4	no	—
506	M₁₆	—	2	1	no	—
507	M₁₆	—	2	2	no	—
508	M₁₆	—	2	1	no	—
509	M₁₆	—	2	2	no	—
510	M₁₆	—	4	2	no	—
511	M₁₆	C₂×D₄×C₇₁	1	2	yes	—
512	C₂×C₈	—	2	1	no	—
513	C₂×C₈	—	2	1	no	—
514	C₂×C₈	—	2	1	no	—
515	C₂×C₈	—	2	4	no	—
516	C₂×C₈	—	2	2	no	—
517	C₂×C₈	—	2	1	no	—
518	C₂×C₈	—	4	2	no	—
519	C₂×C₈	C₂×D₄×C₇₁	1	1	yes	characteristic
520	C₂×C₈	C₂×D₄×C₇₁	1	1	yes	characteristic
521	C₂×C₈	C₂×D₄×C₇₁	1	2	yes	—
522	C₂×C₈	C₂×D₄×C₇₁	1	1	yes	characteristic
523	C₂×C₈	C₂×D₄×C₇₁	1	1	yes	characteristic
524	C₂×C₈	C₂×D₄×C₇₁	1	2	yes	—
525	C₄⋊C₄	—	2	2	no	—
526	C₄⋊C₄	—	2	2	no	—
527	C₄⋊C₄	—	4	2	no	—
528	C₂²⋊C₄	—	4	2	no	—
529	C₂²⋊C₄	—	4	2	no	—
530	C₂²⋊C₄	—	8	4	no	—
531	C₄²	—	2	1	no	—
532	C₄²	—	2	1	no	—
533	C₂³	—	2	1	no	—
534	C₂³	—	2	1	no	—
535	C₂³	—	2	2	no	—
536	C₂³	—	2	4	no	—
537	C₂³	—	4	8	no	—
538	C₂³	—	4	2	no	—
539	C₂³	—	4	2	no	—
540	C₂³	—	8	2	no	—
541	C₂³	—	8	2	no	—
542	C₂³	—	16	2	no	—
543	C₂³	—	16	2	no	—
544	C₂³	H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
545	Q₈	—	2	1	no	—
546	Q₈	—	2	2	no	—
547	Q₈	—	2	1	no	—
548	Q₈	—	2	1	no	—
549	Q₈	—	2	2	no	—
550	Q₈	#121×C₇₁	1	2	yes	—
551	D₄	—	2	2	no	—
552	D₄	—	2	2	no	—
553	D₄	—	2	2	no	—
554	D₄	—	2	4	no	—
555	D₄	—	2	1	no	—
556	D₄	—	2	1	no	—
557	D₄	—	2	1	no	—
558	D₄	—	2	4	no	—
559	D₄	—	2	2	no	—
560	D₄	—	4	8	no	—
561	D₄	—	4	8	no	—
562	D₄	—	4	2	no	—
563	D₄	—	4	1	no	—
564	D₄	—	4	2	no	—
565	D₄	—	4	1	no	—
566	D₄	—	4	2	no	—
567	D₄	—	4	2	no	—
568	D₄	—	16	2	no	—
569	D₄	#121×C₇₁	1	2	yes	—
570	C₂×C₄	—	2	1	no	—
571	C₂×C₄	—	2	2	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	1	no	—
574	C₂×C₄	—	2	2	no	—
575	C₂×C₄	—	4	8	no	—
576	C₂×C₄	—	4	1	no	—
577	C₂×C₄	—	4	2	no	—
578	C₂×C₄	—	4	2	no	—
579	C₂×C₄	—	4	1	no	—
580	C₂×C₄	—	4	2	no	—
581	C₂×C₄	—	4	1	no	—
582	C₂×C₄	—	4	2	no	—
583	C₂×C₄	—	8	2	no	—
584	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
585	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
586	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
587	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
588	C₂×C₄	C₂×#41×C₇₁	1	1	yes	characteristic
589	C₂×C₄	H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
590	C₈	—	2	2	no	—
591	C₈	—	2	1	no	—
592	C₈	—	2	1	no	—
593	C₈	—	4	1	no	—
594	C₈	—	4	2	no	—
595	C₈	#120×C₇₁	1	2	yes	—
596	C₈	#120×C₇₁	1	2	yes	—
597	C₂²	—	2	2	no	—
598	C₂²	—	2	2	no	—
599	C₂²	—	4	2	no	—
600	C₂²	—	4	2	no	—
601	C₂²	—	4	4	no	—
602	C₂²	—	4	1	no	—
603	C₂²	—	4	2	no	—
604	C₂²	—	4	2	no	—
605	C₂²	—	4	2	no	—
606	C₂²	—	8	4	no	—
607	C₂²	—	8	2	no	—
608	C₂²	—	8	2	no	—
609	C₂²	—	16	2	no	—
610	C₂²	—	16	8	no	—
611	C₂²	#535×C₇₁	1	1	yes	characteristic
612	C₂²	D₄²×C₇₁	1	1	yes	characteristic
613	C₂²	#546×C₇₁	1	1	yes	characteristic
614	C₄	—	2	2	no	—
615	C₄	—	4	2	no	—
616	C₄	—	4	1	no	—
617	C₄	—	8	1	no	—
618	C₄	—	8	2	no	—
619	C₄	C₂×#120×C₇₁	1	1	yes	characteristic
620	C₄	C₂×#121×C₇₁	1	1	yes	characteristic
621	C₄	D₄²×C₇₁	1	1	yes	characteristic
622	C₄	#546×C₇₁	1	1	yes	characteristic
623	C₂	—	2	1	no	—
624	C₂	—	2	1	no	—
625	C₂	—	4	2	no	—
626	C₂	—	8	2	no	—
627	C₂	—	8	1	no	—
628	C₂	—	8	4	no	—
629	C₂	C₇₁×#2408	1	1	yes	characteristic
630	C₂	C₇₁×#3288	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇₁	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ C₂²×C₄×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#576×C₇₁	C₂	semidirect
2. C₂×#472×C₇₁	C₂	semidirect
3. C₇₁×#3060	C₂	semidirect
4. C₇₁×#3012	C₂	semidirect
5. C₂×#444×C₇₁	C₂	semidirect
6. C₇₁×#2051	C₂	semidirect
7. C₇₁×#2006	C₂	semidirect
8. C₇₁×#1975	C₂	semidirect
9. C₇₁×#1972	C₂	semidirect
10. C₇₁×#1815	C₂	semidirect
11. C₇₁×#1792	C₂	semidirect
17. C₂×AGL(1,ℤ/8)×C₇₁	C₂²	semidirect
24. C₂×#131×C₇₁	C₂²	semidirect
25. #537×C₇₁	C₂²	semidirect
35. C₂×#108×C₇₁	C₂²	semidirect
36. C₂×#108×C₇₁	C₂²	semidirect
37. C₂×#106×C₇₁	C₂²	semidirect
39. C₂×#104×C₇₁	C₂²	semidirect
40. C₂×#104×C₇₁	C₂²	semidirect
42. #417×C₇₁	C₂²	semidirect
43. C₂×#102×C₇₁	C₂²	semidirect
44. C₂×#100×C₇₁	C₂²	semidirect
45. #408×C₇₁	C₂²	semidirect
46. C₂×#97×C₇₁	C₂²	semidirect
47. #390×C₇₁	C₂²	semidirect
48. #343×C₇₁	C₂²	semidirect
87. C₂×D₈×C₇₁	D₄	semidirect
91. #131×C₇₁	D₄	semidirect
307. #24294	C₇₁	direct (N × H)
316. C₂×#576	C₂×C₇₁	semidirect
317. C₂×#472	C₂×C₇₁	semidirect
318. #3060	C₂×C₇₁	semidirect
319. #3012	C₂×C₇₁	semidirect
320. C₂×#444	C₂×C₇₁	semidirect
321. #2051	C₂×C₇₁	semidirect
322. #2006	C₂×C₇₁	semidirect
323. #1975	C₂×C₇₁	semidirect
324. #1972	C₂×C₇₁	semidirect
325. #1815	C₂×C₇₁	semidirect
326. #1792	C₂×C₇₁	semidirect
327. C₇₁	#24294	direct (N × H)
333. C₂×AGL(1,ℤ/8)	C₂²×C₇₁	semidirect
340. C₂×#131	C₂²×C₇₁	semidirect
341. #537	C₂²×C₇₁	semidirect
351. C₂×#108	C₂²×C₇₁	semidirect
352. C₂×#108	C₂²×C₇₁	semidirect
353. C₂×#106	C₂²×C₇₁	semidirect
355. C₂×#104	C₂²×C₇₁	semidirect
356. C₂×#104	C₂²×C₇₁	semidirect
358. #417	C₂²×C₇₁	semidirect
359. C₂×#102	C₂²×C₇₁	semidirect
360. C₂×#100	C₂²×C₇₁	semidirect
361. #408	C₂²×C₇₁	semidirect
362. C₂×#97	C₂²×C₇₁	semidirect
363. #390	C₂²×C₇₁	semidirect
364. #343	C₂²×C₇₁	semidirect
403. C₂×D₈	D₄×C₇₁	semidirect
407. #131	D₄×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#576×C₇₁	2
C₂×#472×C₇₁	2
C₇₁×#3060	2
C₇₁×#3012	2
C₂×#444×C₇₁	2
C₇₁×#2051	2
C₇₁×#2006	2
C₇₁×#1975	2
C₇₁×#1972	2
C₇₁×#1815	2
C₇₁×#1792	2
#24294	71

Subgroups — C71×#24294 #31845166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#24294 #31845166