Subgroups — C₇₁×#61054 #31881926

Order 18176.

2710 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#578×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#135×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₇₁×#3487	C₂	1	2	yes	maximal, complemented
4	C₇₁×#3287	C₂	1	4	yes	maximal, complemented
5	C₇₁×#3285	C₂	1	4	yes	maximal, complemented
6	C₇₁×#3271	C₂	1	8	yes	maximal, complemented
7	C₇₁×#3155	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇₁×#3058	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₁×#3050	C₂	1	1	yes	maximal, characteristic, complemented
10	C₇₁×#3043	C₂	1	2	yes	maximal, complemented
11	C₇₁×#3010	C₂	1	2	yes	maximal, complemented
12	C₇₁×#3005	C₂	1	2	yes	maximal, complemented
13	C₇₁×#2950	C₂	1	1	yes	maximal, characteristic, complemented
14	C₇₁×#2929	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×2₋¹⁺⁴×C₇₁	C₂²	1	2	yes	—
16	C₂²×#41×C₇₁	C₂²	1	2	yes	—
17	#578×C₇₁	—	2	8	no	—
18	C₂×#137×C₇₁	C₂²	1	2	yes	—
19	C₂×#137×C₇₁	C₂²	1	4	yes	complemented
20	C₂×AGL(1,ℤ/8)×C₇₁	C₂²	1	2	yes	—
21	C₂×#135×C₇₁	—	2	8	no	—
22	C₂×#135×C₇₁	C₂²	1	1	yes	characteristic
23	C₂×#135×C₇₁	C₂²	1	4	yes	—
24	C₂×#135×C₇₁	C₂²	1	4	yes	—
25	C₂×#135×C₇₁	C₂²	1	2	yes	complemented
26	C₂²×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
27	C₂²×Q₁₆×C₇₁	C₂²	1	2	yes	complemented
28	C₂²×SD₁₆×C₇₁	C₂²	1	2	yes	—
29	C₂²×D₈×C₇₁	C₂²	1	1	yes	characteristic
30	C₂×#131×C₇₁	C₂²	1	1	yes	characteristic
31	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
32	#550×C₇₁	—	2	4	no	—
33	#550×C₇₁	C₂²	1	8	yes	complemented
34	#548×C₇₁	—	2	4	no	—
35	#548×C₇₁	C₂²	1	8	yes	complemented
36	#535×C₇₁	—	2	4	no	—
37	#535×C₇₁	C₂²	1	2	yes	—
38	#535×C₇₁	C₂²	1	2	yes	—
39	#525×C₇₁	—	2	4	no	—
40	#525×C₇₁	C₂²	1	2	yes	—
41	#525×C₇₁	C₂²	1	1	yes	characteristic
42	#519×C₇₁	C₂²	1	2	yes	—
43	#496×C₇₁	C₂²	1	4	yes	complemented
44	#495×C₇₁	C₂²	1	2	yes	complemented
45	#493×C₇₁	C₂²	1	2	yes	complemented
46	#469×C₇₁	C₂²	1	4	yes	complemented
47	#467×C₇₁	C₂²	1	2	yes	complemented
48	#466×C₇₁	C₂²	1	2	yes	complemented
49	#466×C₇₁	C₂²	1	4	yes	complemented
50	#465×C₇₁	C₂²	1	4	yes	complemented
51	#464×C₇₁	C₂²	1	8	yes	complemented
52	#460×C₇₁	C₂²	1	8	yes	complemented
53	#452×C₇₁	C₂²	1	8	yes	complemented
54	#449×C₇₁	C₂²	1	8	yes	complemented
55	#449×C₇₁	C₂²	1	8	yes	complemented
56	#448×C₇₁	C₂²	1	8	yes	complemented
57	#441×C₇₁	C₂²	1	4	yes	complemented
58	C₄×Q₁₆×C₇₁	C₂²	1	4	yes	complemented
59	#435×C₇₁	C₂²	1	4	yes	complemented
60	C₈×D₄×C₇₁	C₂²	1	4	yes	complemented
61	#427×C₇₁	C₂²	1	1	yes	characteristic
62	#424×C₇₁	C₂²	1	1	yes	characteristic
63	#419×C₇₁	C₂²	1	2	yes	—
64	#418×C₇₁	C₂²	1	2	yes	—
65	#416×C₇₁	C₂²	1	2	yes	—
66	#408×C₇₁	C₂²	1	1	yes	characteristic
67	#407×C₇₁	C₂²	1	1	yes	characteristic
68	#405×C₇₁	C₂²	1	1	yes	characteristic
69	2₋¹⁺⁴×C₇₁	—	2	8	no	—
70	2₋¹⁺⁴×C₇₁	—	4	4	no	—
71	C₂×#41×C₇₁	—	2	4	no	—
72	C₂×#41×C₇₁	—	2	2	no	—
73	C₂×#41×C₇₁	—	2	4	no	—
74	C₂×#41×C₇₁	—	2	8	no	—
75	C₂×#41×C₇₁	—	4	4	no	—
76	C₂×#41×C₇₁	C₂³	1	2	yes	—
77	C₂×#41×C₇₁	C₂³	1	4	yes	—
78	C₂×#41×C₇₁	C₂³	1	8	yes	—
79	C₂×#41×C₇₁	C₂³	1	1	yes	characteristic
80	C₂²×Q₈×C₇₁	—	2	2	no	—
81	C₂²×Q₈×C₇₁	C₂³	1	4	yes	—
82	C₂²×Q₈×C₇₁	C₂³	1	2	yes	—
83	C₂²×D₄×C₇₁	—	2	2	no	—
84	C₂²×D₄×C₇₁	C₂³	1	2	yes	—
85	C₂³×C₄×C₇₁	—	2	2	no	—
86	C₂³×C₄×C₇₁	C₂³	1	1	yes	characteristic
87	#137×C₇₁	—	2	8	no	—
88	#137×C₇₁	—	2	16	no	—
89	AGL(1,ℤ/8)×C₇₁	—	4	4	no	—
90	#135×C₇₁	—	2	16	no	—
91	#135×C₇₁	—	2	16	no	—
92	#135×C₇₁	—	2	8	no	—
93	#135×C₇₁	—	2	4	no	—
94	C₂×Q₁₆×C₇₁	—	2	8	no	—
95	C₂×Q₁₆×C₇₁	D₄	1	8	yes	complemented
96	C₂×Q₁₆×C₇₁	C₂³	1	4	yes	—
97	C₂×Q₁₆×C₇₁	C₂³	1	1	yes	characteristic
98	C₂×Q₁₆×C₇₁	C₂³	1	2	yes	—
99	C₂×Q₁₆×C₇₁	C₂³	1	2	yes	—
100	C₂×SD₁₆×C₇₁	—	2	8	no	—
101	C₂×SD₁₆×C₇₁	C₂³	1	8	yes	—
102	C₂×SD₁₆×C₇₁	C₂³	1	4	yes	—
103	C₂×SD₁₆×C₇₁	C₂³	1	2	yes	—
104	C₂×SD₁₆×C₇₁	C₂³	1	4	yes	—
105	C₂×D₈×C₇₁	—	2	4	no	—
106	C₂×D₈×C₇₁	C₂³	1	1	yes	characteristic
107	C₂×D₈×C₇₁	C₂³	1	2	yes	—
108	C₂×D₈×C₇₁	C₂³	1	2	yes	—
109	C₂×D₈×C₇₁	C₂³	1	4	yes	—
110	#131×C₇₁	—	2	4	no	—
111	C₂×M₁₆×C₇₁	C₂³	1	1	yes	characteristic
112	C₂×M₁₆×C₇₁	C₂³	1	1	yes	characteristic
113	C₂×M₁₆×C₇₁	C₂³	1	2	yes	—
114	C₂²×C₈×C₇₁	—	2	4	no	—
115	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
116	C₂²×C₈×C₇₁	C₂³	1	2	yes	—
117	C₂²×C₈×C₇₁	C₂³	1	2	yes	—
118	C₂²×C₈×C₇₁	C₂³	1	2	yes	—
119	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
120	#127×C₇₁	—	2	8	no	—
121	#127×C₇₁	—	2	2	no	—
122	#127×C₇₁	C₂³	1	4	yes	—
123	#124×C₇₁	—	2	2	no	—
124	#124×C₇₁	—	2	8	no	—
125	#124×C₇₁	C₂³	1	4	yes	—
126	#123×C₇₁	—	2	2	no	—
127	#123×C₇₁	D₄	1	2	yes	—
128	C₂²⋊Q₈×C₇₁	—	2	8	no	—
129	C₂²⋊Q₈×C₇₁	D₄	1	2	yes	—
130	C₂²⋊Q₈×C₇₁	C₂³	1	4	yes	—
131	#121×C₇₁	—	2	8	no	—
132	#121×C₇₁	—	2	4	no	—
133	#121×C₇₁	—	2	8	no	—
134	#121×C₇₁	—	2	8	no	—
135	#121×C₇₁	D₄	1	2	yes	—
136	#121×C₇₁	C₂³	1	8	yes	—
137	#121×C₇₁	C₂³	1	4	yes	—
138	#120×C₇₁	—	2	2	no	—
139	#120×C₇₁	—	2	4	no	—
140	#120×C₇₁	—	2	4	no	—
141	#120×C₇₁	D₄	1	2	yes	—
142	C₄×Q₈×C₇₁	—	2	4	no	—
143	C₄×Q₈×C₇₁	C₂³	1	8	yes	—
144	C₄×D₄×C₇₁	—	2	4	no	—
145	C₄×D₄×C₇₁	—	2	8	no	—
146	C₄×D₄×C₇₁	—	2	8	no	—
147	C₄×D₄×C₇₁	C₂³	1	4	yes	—
148	#117×C₇₁	—	2	4	no	—
149	#117×C₇₁	C₂³	1	2	yes	—
150	#117×C₇₁	C₂³	1	1	yes	characteristic
151	#107×C₇₁	C₂³	1	2	yes	—
152	#106×C₇₁	C₂³	1	2	yes	—
153	C₄⋊C₈×C₇₁	C₂³	1	4	yes	—
154	#103×C₇₁	C₂³	1	4	yes	—
155	#103×C₇₁	C₂³	1	8	yes	—
156	#102×C₇₁	C₂³	1	4	yes	—
157	#102×C₇₁	C₂³	1	8	yes	—
158	C₂²⋊C₈×C₇₁	C₂³	1	4	yes	—
159	C₂²⋊C₈×C₇₁	C₂³	1	4	yes	—
160	#97×C₇₁	C₂³	1	2	yes	—
161	C₄×C₈×C₇₁	C₂³	1	2	yes	—
162	C₂⁴×C₇₁	—	2	2	no	—
163	C₂⁴×C₇₁	C₂×D₄	1	1	yes	characteristic
164	#41×C₇₁	—	2	2	no	—
165	#41×C₇₁	—	2	2	no	—
166	#41×C₇₁	—	2	16	no	—
167	#41×C₇₁	—	2	8	no	—
168	#41×C₇₁	—	2	4	no	—
169	#41×C₇₁	—	2	4	no	—
170	#41×C₇₁	—	4	8	no	—
171	#41×C₇₁	—	4	4	no	—
172	#41×C₇₁	—	4	16	no	—
173	#41×C₇₁	—	4	16	no	—
174	#41×C₇₁	—	4	8	no	—
175	C₂×Q₈×C₇₁	—	2	2	no	—
176	C₂×Q₈×C₇₁	—	2	4	no	—
177	C₂×Q₈×C₇₁	—	2	4	no	—
178	C₂×Q₈×C₇₁	—	2	8	no	—
179	C₂×Q₈×C₇₁	—	4	4	no	—
180	C₂×Q₈×C₇₁	—	4	8	no	—
181	C₂×Q₈×C₇₁	C₂×D₄	1	8	yes	—
182	C₂×Q₈×C₇₁	C₂×D₄	1	1	yes	characteristic
183	C₂×Q₈×C₇₁	C₂⁴	1	2	yes	—
184	C₂×Q₈×C₇₁	C₂⁴	1	4	yes	—
185	C₂×D₄×C₇₁	—	2	2	no	—
186	C₂×D₄×C₇₁	—	2	8	no	—
187	C₂×D₄×C₇₁	—	2	4	no	—
188	C₂×D₄×C₇₁	—	2	8	no	—
189	C₂×D₄×C₇₁	—	2	4	no	—
190	C₂×D₄×C₇₁	—	2	4	no	—
191	C₂×D₄×C₇₁	—	2	2	no	—
192	C₂×D₄×C₇₁	—	2	8	no	—
193	C₂×D₄×C₇₁	—	2	4	no	—
194	C₂×D₄×C₇₁	—	4	8	no	—
195	C₂×D₄×C₇₁	C₂×D₄	1	2	yes	—
196	C₂×D₄×C₇₁	C₂×D₄	1	1	yes	characteristic
197	C₂×D₄×C₇₁	C₂×D₄	1	4	yes	—
198	C₂×D₄×C₇₁	C₂⁴	1	4	yes	—
199	C₂×D₄×C₇₁	C₂⁴	1	2	yes	—
200	C₂²×C₄×C₇₁	—	2	2	no	—
201	C₂²×C₄×C₇₁	—	2	2	no	—
202	C₂²×C₄×C₇₁	—	2	2	no	—
203	C₂²×C₄×C₇₁	—	2	4	no	—
204	C₂²×C₄×C₇₁	—	2	4	no	—
205	C₂²×C₄×C₇₁	—	2	2	no	—
206	C₂²×C₄×C₇₁	—	2	8	no	—
207	C₂²×C₄×C₇₁	—	4	8	no	—
208	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
209	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
210	C₂²×C₄×C₇₁	C₂⁴	1	2	yes	—
211	C₂²×C₄×C₇₁	C₂⁴	1	2	yes	—
212	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
213	Q₁₆×C₇₁	—	2	8	no	—
214	Q₁₆×C₇₁	—	2	2	no	—
215	Q₁₆×C₇₁	C₂×D₄	1	8	yes	—
216	Q₁₆×C₇₁	C₂×D₄	1	8	yes	—
217	SD₁₆×C₇₁	—	2	16	no	—
218	SD₁₆×C₇₁	—	2	8	no	—
219	SD₁₆×C₇₁	—	2	4	no	—
220	SD₁₆×C₇₁	—	4	4	no	—
221	D₈×C₇₁	—	2	4	no	—
222	D₈×C₇₁	—	2	4	no	—
223	D₈×C₇₁	—	2	2	no	—
224	D₈×C₇₁	—	4	4	no	—
225	M₁₆×C₇₁	—	2	2	no	—
226	M₁₆×C₇₁	—	2	2	no	—
227	M₁₆×C₇₁	—	2	4	no	—
228	C₂×C₈×C₇₁	—	2	2	no	—
229	C₂×C₈×C₇₁	—	2	4	no	—
230	C₂×C₈×C₇₁	—	2	4	no	—
231	C₂×C₈×C₇₁	—	2	4	no	—
232	C₂×C₈×C₇₁	C₂×D₄	1	4	yes	—
233	C₂×C₈×C₇₁	C₂⁴	1	2	yes	—
234	C₂×C₈×C₇₁	C₂⁴	1	2	yes	—
235	C₂×C₈×C₇₁	C₂⁴	1	2	yes	—
236	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
237	C₂×C₈×C₇₁	C₂⁴	1	1	yes	characteristic
238	C₄⋊C₄×C₇₁	—	2	8	no	—
239	C₄⋊C₄×C₇₁	—	2	4	no	—
240	C₄⋊C₄×C₇₁	C₂×D₄	1	2	yes	—
241	C₄⋊C₄×C₇₁	C₂×D₄	1	4	yes	—
242	C₄⋊C₄×C₇₁	C₂⁴	1	4	yes	—
243	C₂²⋊C₄×C₇₁	—	2	2	no	—
244	C₂²⋊C₄×C₇₁	—	2	4	no	—
245	C₂²⋊C₄×C₇₁	—	2	4	no	—
246	C₂²⋊C₄×C₇₁	—	2	8	no	—
247	C₂²⋊C₄×C₇₁	—	2	8	no	—
248	C₂²⋊C₄×C₇₁	C₂×D₄	1	2	yes	—
249	C₂²⋊C₄×C₇₁	C₂×D₄	1	4	yes	—
250	C₂²⋊C₄×C₇₁	C₂×D₄	1	4	yes	—
251	C₄²×C₇₁	—	2	8	no	—
252	C₄²×C₇₁	—	2	2	no	—
253	C₄²×C₇₁	C₂⁴	1	2	yes	—
254	C₂³×C₇₁	—	2	2	no	—
255	C₂³×C₇₁	—	2	4	no	—
256	C₂³×C₇₁	—	2	2	no	—
257	C₂³×C₇₁	—	4	2	no	—
258	C₂³×C₇₁	—	4	4	no	—
259	C₂³×C₇₁	—	8	4	no	—
260	C₂³×C₇₁	C₂²×D₄	1	2	yes	—
261	C₂³×C₇₁	C₂²×D₄	1	2	yes	—
262	C₂³×C₇₁	C₂²×D₄	1	1	yes	characteristic
263	Q₈×C₇₁	—	2	8	no	—
264	Q₈×C₇₁	—	2	2	no	—
265	Q₈×C₇₁	—	2	4	no	—
266	Q₈×C₇₁	—	2	2	no	—
267	Q₈×C₇₁	—	2	2	no	—
268	Q₈×C₇₁	—	4	8	no	—
269	Q₈×C₇₁	C₂²×D₄	1	8	yes	—
270	D₄×C₇₁	—	2	4	no	—
271	D₄×C₇₁	—	2	2	no	—
272	D₄×C₇₁	—	2	4	no	—
273	D₄×C₇₁	—	2	2	no	—
274	D₄×C₇₁	—	2	2	no	—
275	D₄×C₇₁	—	2	4	no	—
276	D₄×C₇₁	—	4	4	no	—
277	D₄×C₇₁	—	4	16	no	—
278	D₄×C₇₁	—	4	8	no	—
279	D₄×C₇₁	—	4	4	no	—
280	D₄×C₇₁	—	4	4	no	—
281	D₄×C₇₁	—	8	8	no	—
282	D₄×C₇₁	—	8	8	no	—
283	D₄×C₇₁	—	8	4	no	—
284	C₂×C₄×C₇₁	—	2	2	no	—
285	C₂×C₄×C₇₁	—	2	8	no	—
286	C₂×C₄×C₇₁	—	2	4	no	—
287	C₂×C₄×C₇₁	—	2	4	no	—
288	C₂×C₄×C₇₁	—	2	2	no	—
289	C₂×C₄×C₇₁	—	2	2	no	—
290	C₂×C₄×C₇₁	—	2	2	no	—
291	C₂×C₄×C₇₁	—	2	2	no	—
292	C₂×C₄×C₇₁	—	2	4	no	—
293	C₂×C₄×C₇₁	—	2	2	no	—
294	C₂×C₄×C₇₁	—	4	16	no	—
295	C₂×C₄×C₇₁	—	4	8	no	—
296	C₂×C₄×C₇₁	—	4	8	no	—
297	C₂×C₄×C₇₁	—	4	4	no	—
298	C₂×C₄×C₇₁	C₂²×D₄	1	4	yes	—
299	C₂×C₄×C₇₁	C₂²×D₄	1	2	yes	—
300	C₂×C₄×C₇₁	C₂²×D₄	1	2	yes	—
301	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
302	C₂×C₄×C₇₁	C₂²×D₄	1	2	yes	—
303	C₂×C₄×C₇₁	H(5,𝔽₂)	1	2	yes	—
304	C₂×C₄×C₇₁	C₂⁵	1	1	yes	characteristic
305	C₈×C₇₁	—	2	2	no	—
306	C₈×C₇₁	—	2	2	no	—
307	C₈×C₇₁	—	2	2	no	—
308	C₈×C₇₁	C₂²×D₄	1	2	yes	—
309	C₈×C₇₁	C₂²×D₄	1	2	yes	—
310	C₂²×C₇₁	—	2	2	no	—
311	C₂²×C₇₁	—	2	2	no	—
312	C₂²×C₇₁	—	2	4	no	—
313	C₂²×C₇₁	—	2	2	no	—
314	C₂²×C₇₁	—	2	2	no	—
315	C₂²×C₇₁	—	2	2	no	—
316	C₂²×C₇₁	—	4	4	no	—
317	C₂²×C₇₁	—	4	4	no	—
318	C₂²×C₇₁	—	4	8	no	—
319	C₂²×C₇₁	—	4	2	no	—
320	C₂²×C₇₁	—	4	4	no	—
321	C₂²×C₇₁	—	8	8	no	—
322	C₂²×C₇₁	D₄²	1	2	yes	—
323	C₂²×C₇₁	C₂³×D₄	1	1	yes	characteristic
324	C₄×C₇₁	—	2	4	no	—
325	C₄×C₇₁	—	2	2	no	—
326	C₄×C₇₁	—	2	8	no	—
327	C₄×C₇₁	—	2	2	no	—
328	C₄×C₇₁	—	4	4	no	—
329	C₄×C₇₁	—	4	2	no	—
330	C₄×C₇₁	D₄²	1	2	yes	—
331	C₄×C₇₁	C₂³×D₄	1	1	yes	characteristic
332	C₄×C₇₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
333	#61054	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
334	C₂×C₇₁	—	2	2	no	—
335	C₂×C₇₁	—	4	2	no	—
336	C₂×C₇₁	—	4	2	no	—
337	C₂×C₇₁	—	8	2	no	—
338	C₂×C₇₁	—	8	4	no	—
339	C₂×C₇₁	C₂×D₄²	1	1	yes	characteristic
340	C₂×C₇₁	#3584	1	2	yes	—
341	C₂×#578	C₂×C₇₁	1	1	yes	characteristic, complemented
342	C₂²×#135	C₂×C₇₁	1	1	yes	characteristic, complemented
343	#3487	C₂×C₇₁	1	2	yes	complemented
344	#3287	C₂×C₇₁	1	4	yes	complemented
345	#3285	C₂×C₇₁	1	4	yes	complemented
346	#3271	C₂×C₇₁	1	8	yes	complemented
347	#3155	C₂×C₇₁	1	1	yes	characteristic, complemented
348	#3058	C₂×C₇₁	1	1	yes	characteristic, complemented
349	#3050	C₂×C₇₁	1	1	yes	characteristic, complemented
350	#3043	C₂×C₇₁	1	2	yes	complemented
351	#3010	C₂×C₇₁	1	2	yes	complemented
352	#3005	C₂×C₇₁	1	2	yes	complemented
353	#2950	C₂×C₇₁	1	1	yes	characteristic, complemented
354	#2929	C₂×C₇₁	1	1	yes	characteristic, complemented
355	C₇₁	#61054	1	1	yes	characteristic, direct product factor, Sylow
356	C₂×2₋¹⁺⁴	C₂²×C₇₁	1	2	yes	—
357	C₂²×#41	C₂²×C₇₁	1	2	yes	—
358	#578	—	2	8	no	—
359	C₂×#137	C₂²×C₇₁	1	2	yes	—
360	C₂×#137	C₂²×C₇₁	1	4	yes	complemented
361	C₂×AGL(1,ℤ/8)	C₂²×C₇₁	1	2	yes	—
362	C₂×#135	—	2	8	no	—
363	C₂×#135	C₂²×C₇₁	1	1	yes	characteristic
364	C₂×#135	C₂²×C₇₁	1	4	yes	—
365	C₂×#135	C₂²×C₇₁	1	4	yes	—
366	C₂×#135	C₂²×C₇₁	1	2	yes	complemented
367	C₂²×Q₁₆	C₂²×C₇₁	1	1	yes	characteristic
368	C₂²×Q₁₆	C₂²×C₇₁	1	2	yes	complemented
369	C₂²×SD₁₆	C₂²×C₇₁	1	2	yes	—
370	C₂²×D₈	C₂²×C₇₁	1	1	yes	characteristic
371	C₂×#131	C₂²×C₇₁	1	1	yes	characteristic
372	C₂³×C₈	C₂²×C₇₁	1	1	yes	characteristic
373	#550	—	2	4	no	—
374	#550	C₂²×C₇₁	1	8	yes	complemented
375	#548	—	2	4	no	—
376	#548	C₂²×C₇₁	1	8	yes	complemented
377	#535	—	2	4	no	—
378	#535	C₂²×C₇₁	1	2	yes	—
379	#535	C₂²×C₇₁	1	2	yes	—
380	#525	—	2	4	no	—
381	#525	C₂²×C₇₁	1	2	yes	—
382	#525	C₂²×C₇₁	1	1	yes	characteristic
383	#519	C₂²×C₇₁	1	2	yes	—
384	#496	C₂²×C₇₁	1	4	yes	complemented
385	#495	C₂²×C₇₁	1	2	yes	complemented
386	#493	C₂²×C₇₁	1	2	yes	complemented
387	#469	C₂²×C₇₁	1	4	yes	complemented
388	#467	C₂²×C₇₁	1	2	yes	complemented
389	#466	C₂²×C₇₁	1	4	yes	complemented
390	#466	C₂²×C₇₁	1	2	yes	complemented
391	#465	C₂²×C₇₁	1	4	yes	complemented
392	#464	C₂²×C₇₁	1	8	yes	complemented
393	#460	C₂²×C₇₁	1	8	yes	complemented
394	#452	C₂²×C₇₁	1	8	yes	complemented
395	#449	C₂²×C₇₁	1	8	yes	complemented
396	#449	C₂²×C₇₁	1	8	yes	complemented
397	#448	C₂²×C₇₁	1	8	yes	complemented
398	#441	C₂²×C₇₁	1	4	yes	complemented
399	C₄×Q₁₆	C₂²×C₇₁	1	4	yes	complemented
400	#435	C₂²×C₇₁	1	4	yes	complemented
401	C₈×D₄	C₂²×C₇₁	1	4	yes	complemented
402	#427	C₂²×C₇₁	1	1	yes	characteristic
403	#424	C₂²×C₇₁	1	1	yes	characteristic
404	#419	C₂²×C₇₁	1	2	yes	—
405	#418	C₂²×C₇₁	1	2	yes	—
406	#416	C₂²×C₇₁	1	2	yes	—
407	#408	C₂²×C₇₁	1	1	yes	characteristic
408	#407	C₂²×C₇₁	1	1	yes	characteristic
409	#405	C₂²×C₇₁	1	1	yes	characteristic
410	2₋¹⁺⁴	—	2	8	no	—
411	2₋¹⁺⁴	—	4	4	no	—
412	C₂×#41	—	2	4	no	—
413	C₂×#41	—	2	2	no	—
414	C₂×#41	—	2	4	no	—
415	C₂×#41	—	2	8	no	—
416	C₂×#41	—	4	4	no	—
417	C₂×#41	C₂³×C₇₁	1	2	yes	—
418	C₂×#41	C₂³×C₇₁	1	8	yes	—
419	C₂×#41	C₂³×C₇₁	1	4	yes	—
420	C₂×#41	C₂³×C₇₁	1	1	yes	characteristic
421	C₂²×Q₈	—	2	2	no	—
422	C₂²×Q₈	C₂³×C₇₁	1	4	yes	—
423	C₂²×Q₈	C₂³×C₇₁	1	2	yes	—
424	C₂²×D₄	—	2	2	no	—
425	C₂²×D₄	C₂³×C₇₁	1	2	yes	—
426	C₂³×C₄	—	2	2	no	—
427	C₂³×C₄	C₂³×C₇₁	1	1	yes	characteristic
428	#137	—	2	8	no	—
429	#137	—	2	16	no	—
430	AGL(1,ℤ/8)	—	4	4	no	—
431	#135	—	2	16	no	—
432	#135	—	2	16	no	—
433	#135	—	2	8	no	—
434	#135	—	2	4	no	—
435	C₂×Q₁₆	—	2	8	no	—
436	C₂×Q₁₆	D₄×C₇₁	1	8	yes	complemented
437	C₂×Q₁₆	C₂³×C₇₁	1	1	yes	characteristic
438	C₂×Q₁₆	C₂³×C₇₁	1	4	yes	—
439	C₂×Q₁₆	C₂³×C₇₁	1	2	yes	—
440	C₂×Q₁₆	C₂³×C₇₁	1	2	yes	—
441	C₂×SD₁₆	—	2	8	no	—
442	C₂×SD₁₆	C₂³×C₇₁	1	8	yes	—
443	C₂×SD₁₆	C₂³×C₇₁	1	2	yes	—
444	C₂×SD₁₆	C₂³×C₇₁	1	4	yes	—
445	C₂×SD₁₆	C₂³×C₇₁	1	4	yes	—
446	C₂×D₈	—	2	4	no	—
447	C₂×D₈	C₂³×C₇₁	1	2	yes	—
448	C₂×D₈	C₂³×C₇₁	1	2	yes	—
449	C₂×D₈	C₂³×C₇₁	1	1	yes	characteristic
450	C₂×D₈	C₂³×C₇₁	1	4	yes	—
451	#131	—	2	4	no	—
452	C₂×M₁₆	C₂³×C₇₁	1	1	yes	characteristic
453	C₂×M₁₆	C₂³×C₇₁	1	2	yes	—
454	C₂×M₁₆	C₂³×C₇₁	1	1	yes	characteristic
455	C₂²×C₈	—	2	4	no	—
456	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
457	C₂²×C₈	C₂³×C₇₁	1	2	yes	—
458	C₂²×C₈	C₂³×C₇₁	1	2	yes	—
459	C₂²×C₈	C₂³×C₇₁	1	2	yes	—
460	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
461	#127	—	2	8	no	—
462	#127	—	2	2	no	—
463	#127	C₂³×C₇₁	1	4	yes	—
464	#124	—	2	2	no	—
465	#124	—	2	8	no	—
466	#124	C₂³×C₇₁	1	4	yes	—
467	#123	—	2	2	no	—
468	#123	D₄×C₇₁	1	2	yes	—
469	C₂²⋊Q₈	—	2	8	no	—
470	C₂²⋊Q₈	D₄×C₇₁	1	2	yes	—
471	C₂²⋊Q₈	C₂³×C₇₁	1	4	yes	—
472	#121	—	2	8	no	—
473	#121	—	2	4	no	—
474	#121	—	2	8	no	—
475	#121	—	2	8	no	—
476	#121	D₄×C₇₁	1	2	yes	—
477	#121	C₂³×C₇₁	1	4	yes	—
478	#121	C₂³×C₇₁	1	8	yes	—
479	#120	—	2	2	no	—
480	#120	—	2	4	no	—
481	#120	—	2	4	no	—
482	#120	D₄×C₇₁	1	2	yes	—
483	C₄×Q₈	—	2	4	no	—
484	C₄×Q₈	C₂³×C₇₁	1	8	yes	—
485	C₄×D₄	—	2	4	no	—
486	C₄×D₄	—	2	8	no	—
487	C₄×D₄	—	2	8	no	—
488	C₄×D₄	C₂³×C₇₁	1	4	yes	—
489	#117	—	2	4	no	—
490	#117	C₂³×C₇₁	1	2	yes	—
491	#117	C₂³×C₇₁	1	1	yes	characteristic
492	#107	C₂³×C₇₁	1	2	yes	—
493	#106	C₂³×C₇₁	1	2	yes	—
494	C₄⋊C₈	C₂³×C₇₁	1	4	yes	—
495	#103	C₂³×C₇₁	1	4	yes	—
496	#103	C₂³×C₇₁	1	8	yes	—
497	#102	C₂³×C₇₁	1	4	yes	—
498	#102	C₂³×C₇₁	1	8	yes	—
499	C₂²⋊C₈	C₂³×C₇₁	1	4	yes	—
500	C₂²⋊C₈	C₂³×C₇₁	1	4	yes	—
501	#97	C₂³×C₇₁	1	2	yes	—
502	C₄×C₈	C₂³×C₇₁	1	2	yes	—
503	C₂⁴	—	2	2	no	—
504	C₂⁴	C₂×D₄×C₇₁	1	1	yes	characteristic
505	#41	—	2	2	no	—
506	#41	—	2	2	no	—
507	#41	—	2	16	no	—
508	#41	—	2	8	no	—
509	#41	—	2	4	no	—
510	#41	—	2	4	no	—
511	#41	—	4	8	no	—
512	#41	—	4	4	no	—
513	#41	—	4	16	no	—
514	#41	—	4	16	no	—
515	#41	—	4	8	no	—
516	C₂×Q₈	—	2	2	no	—
517	C₂×Q₈	—	2	4	no	—
518	C₂×Q₈	—	2	4	no	—
519	C₂×Q₈	—	2	8	no	—
520	C₂×Q₈	—	4	4	no	—
521	C₂×Q₈	—	4	8	no	—
522	C₂×Q₈	C₂×D₄×C₇₁	1	8	yes	—
523	C₂×Q₈	C₂×D₄×C₇₁	1	1	yes	characteristic
524	C₂×Q₈	C₂⁴×C₇₁	1	2	yes	—
525	C₂×Q₈	C₂⁴×C₇₁	1	4	yes	—
526	C₂×D₄	—	2	2	no	—
527	C₂×D₄	—	2	8	no	—
528	C₂×D₄	—	2	4	no	—
529	C₂×D₄	—	2	8	no	—
530	C₂×D₄	—	2	4	no	—
531	C₂×D₄	—	2	4	no	—
532	C₂×D₄	—	2	2	no	—
533	C₂×D₄	—	2	8	no	—
534	C₂×D₄	—	2	4	no	—
535	C₂×D₄	—	4	8	no	—
536	C₂×D₄	C₂×D₄×C₇₁	1	2	yes	—
537	C₂×D₄	C₂×D₄×C₇₁	1	1	yes	characteristic
538	C₂×D₄	C₂×D₄×C₇₁	1	4	yes	—
539	C₂×D₄	C₂⁴×C₇₁	1	4	yes	—
540	C₂×D₄	C₂⁴×C₇₁	1	2	yes	—
541	C₂²×C₄	—	2	2	no	—
542	C₂²×C₄	—	2	2	no	—
543	C₂²×C₄	—	2	2	no	—
544	C₂²×C₄	—	2	4	no	—
545	C₂²×C₄	—	2	4	no	—
546	C₂²×C₄	—	2	2	no	—
547	C₂²×C₄	—	2	8	no	—
548	C₂²×C₄	—	4	8	no	—
549	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
550	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
551	C₂²×C₄	C₂⁴×C₇₁	1	2	yes	—
552	C₂²×C₄	C₂⁴×C₇₁	1	2	yes	—
553	C₂²×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
554	Q₁₆	—	2	8	no	—
555	Q₁₆	—	2	2	no	—
556	Q₁₆	C₂×D₄×C₇₁	1	8	yes	—
557	Q₁₆	C₂×D₄×C₇₁	1	8	yes	—
558	SD₁₆	—	2	16	no	—
559	SD₁₆	—	2	8	no	—
560	SD₁₆	—	2	4	no	—
561	SD₁₆	—	4	4	no	—
562	D₈	—	2	4	no	—
563	D₈	—	2	4	no	—
564	D₈	—	2	2	no	—
565	D₈	—	4	4	no	—
566	M₁₆	—	2	2	no	—
567	M₁₆	—	2	2	no	—
568	M₁₆	—	2	4	no	—
569	C₂×C₈	—	2	2	no	—
570	C₂×C₈	—	2	4	no	—
571	C₂×C₈	—	2	4	no	—
572	C₂×C₈	—	2	4	no	—
573	C₂×C₈	C₂×D₄×C₇₁	1	4	yes	—
574	C₂×C₈	C₂⁴×C₇₁	1	2	yes	—
575	C₂×C₈	C₂⁴×C₇₁	1	2	yes	—
576	C₂×C₈	C₂⁴×C₇₁	1	2	yes	—
577	C₂×C₈	C₂⁴×C₇₁	1	1	yes	characteristic
578	C₂×C₈	C₂⁴×C₇₁	1	1	yes	characteristic
579	C₄⋊C₄	—	2	8	no	—
580	C₄⋊C₄	—	2	4	no	—
581	C₄⋊C₄	C₂×D₄×C₇₁	1	2	yes	—
582	C₄⋊C₄	C₂×D₄×C₇₁	1	4	yes	—
583	C₄⋊C₄	C₂⁴×C₇₁	1	4	yes	—
584	C₂²⋊C₄	—	2	2	no	—
585	C₂²⋊C₄	—	2	4	no	—
586	C₂²⋊C₄	—	2	4	no	—
587	C₂²⋊C₄	—	2	8	no	—
588	C₂²⋊C₄	—	2	8	no	—
589	C₂²⋊C₄	C₂×D₄×C₇₁	1	2	yes	—
590	C₂²⋊C₄	C₂×D₄×C₇₁	1	4	yes	—
591	C₂²⋊C₄	C₂×D₄×C₇₁	1	4	yes	—
592	C₄²	—	2	8	no	—
593	C₄²	—	2	2	no	—
594	C₄²	C₂⁴×C₇₁	1	2	yes	—
595	C₂³	—	2	2	no	—
596	C₂³	—	2	4	no	—
597	C₂³	—	2	2	no	—
598	C₂³	—	4	2	no	—
599	C₂³	—	4	4	no	—
600	C₂³	—	8	4	no	—
601	C₂³	C₂²×D₄×C₇₁	1	2	yes	—
602	C₂³	C₂²×D₄×C₇₁	1	2	yes	—
603	C₂³	C₂²×D₄×C₇₁	1	1	yes	characteristic
604	Q₈	—	2	8	no	—
605	Q₈	—	2	2	no	—
606	Q₈	—	2	4	no	—
607	Q₈	—	2	2	no	—
608	Q₈	—	2	2	no	—
609	Q₈	—	4	8	no	—
610	Q₈	C₂²×D₄×C₇₁	1	8	yes	—
611	D₄	—	2	4	no	—
612	D₄	—	2	2	no	—
613	D₄	—	2	4	no	—
614	D₄	—	2	2	no	—
615	D₄	—	2	2	no	—
616	D₄	—	2	4	no	—
617	D₄	—	4	4	no	—
618	D₄	—	4	16	no	—
619	D₄	—	4	8	no	—
620	D₄	—	4	4	no	—
621	D₄	—	4	4	no	—
622	D₄	—	8	8	no	—
623	D₄	—	8	8	no	—
624	D₄	—	8	4	no	—
625	C₂×C₄	—	2	2	no	—
626	C₂×C₄	—	2	8	no	—
627	C₂×C₄	—	2	4	no	—
628	C₂×C₄	—	2	4	no	—
629	C₂×C₄	—	2	2	no	—
630	C₂×C₄	—	2	2	no	—
631	C₂×C₄	—	2	2	no	—
632	C₂×C₄	—	2	2	no	—
633	C₂×C₄	—	2	4	no	—
634	C₂×C₄	—	2	2	no	—
635	C₂×C₄	—	4	16	no	—
636	C₂×C₄	—	4	8	no	—
637	C₂×C₄	—	4	8	no	—
638	C₂×C₄	—	4	4	no	—
639	C₂×C₄	C₂²×D₄×C₇₁	1	4	yes	—
640	C₂×C₄	C₂²×D₄×C₇₁	1	2	yes	—
641	C₂×C₄	C₂²×D₄×C₇₁	1	2	yes	—
642	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
643	C₂×C₄	C₂²×D₄×C₇₁	1	2	yes	—
644	C₂×C₄	H(5,𝔽₂)×C₇₁	1	2	yes	—
645	C₂×C₄	C₂⁵×C₇₁	1	1	yes	characteristic
646	C₈	—	2	2	no	—
647	C₈	—	2	2	no	—
648	C₈	—	2	2	no	—
649	C₈	C₂²×D₄×C₇₁	1	2	yes	—
650	C₈	C₂²×D₄×C₇₁	1	2	yes	—
651	C₂²	—	2	2	no	—
652	C₂²	—	2	2	no	—
653	C₂²	—	2	4	no	—
654	C₂²	—	2	2	no	—
655	C₂²	—	2	2	no	—
656	C₂²	—	2	2	no	—
657	C₂²	—	4	4	no	—
658	C₂²	—	4	4	no	—
659	C₂²	—	4	8	no	—
660	C₂²	—	4	2	no	—
661	C₂²	—	4	4	no	—
662	C₂²	—	8	8	no	—
663	C₂²	D₄²×C₇₁	1	2	yes	—
664	C₂²	C₂³×D₄×C₇₁	1	1	yes	characteristic
665	C₄	—	2	4	no	—
666	C₄	—	2	2	no	—
667	C₄	—	2	8	no	—
668	C₄	—	2	2	no	—
669	C₄	—	4	4	no	—
670	C₄	—	4	2	no	—
671	C₄	D₄²×C₇₁	1	2	yes	—
672	C₄	C₂³×D₄×C₇₁	1	1	yes	characteristic
673	C₄	C₂×H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
674	C₂	—	2	2	no	—
675	C₂	—	4	2	no	—
676	C₂	—	4	2	no	—
677	C₂	—	8	4	no	—
678	C₂	—	8	2	no	—
679	C₂	C₂×D₄²×C₇₁	1	1	yes	characteristic
680	C₂	C₇₁×#3584	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₇₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₇₁ ◁ #525×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#578×C₇₁	C₂	semidirect
2. C₂²×#135×C₇₁	C₂	semidirect
3. C₇₁×#3487	C₂	semidirect
4. C₇₁×#3287	C₂	semidirect
5. C₇₁×#3285	C₂	semidirect
6. C₇₁×#3271	C₂	semidirect
7. C₇₁×#3155	C₂	semidirect
8. C₇₁×#3058	C₂	semidirect
9. C₇₁×#3050	C₂	semidirect
10. C₇₁×#3043	C₂	semidirect
11. C₇₁×#3010	C₂	semidirect
12. C₇₁×#3005	C₂	semidirect
13. C₇₁×#2950	C₂	semidirect
14. C₇₁×#2929	C₂	semidirect
19. C₂×#137×C₇₁	C₂²	semidirect
25. C₂×#135×C₇₁	C₂²	semidirect
27. C₂²×Q₁₆×C₇₁	C₂²	semidirect
33. #550×C₇₁	C₂²	semidirect
35. #548×C₇₁	C₂²	semidirect
43. #496×C₇₁	C₂²	semidirect
44. #495×C₇₁	C₂²	semidirect
45. #493×C₇₁	C₂²	semidirect
46. #469×C₇₁	C₂²	semidirect
47. #467×C₇₁	C₂²	semidirect
48. #466×C₇₁	C₂²	semidirect
49. #466×C₇₁	C₂²	semidirect
50. #465×C₇₁	C₂²	semidirect
51. #464×C₇₁	C₂²	semidirect
52. #460×C₇₁	C₂²	semidirect
53. #452×C₇₁	C₂²	semidirect
54. #449×C₇₁	C₂²	semidirect
55. #449×C₇₁	C₂²	semidirect
56. #448×C₇₁	C₂²	semidirect
57. #441×C₇₁	C₂²	semidirect
58. C₄×Q₁₆×C₇₁	C₂²	semidirect
59. #435×C₇₁	C₂²	semidirect
60. C₈×D₄×C₇₁	C₂²	semidirect
95. C₂×Q₁₆×C₇₁	D₄	semidirect
333. #61054	C₇₁	direct (N × H)
341. C₂×#578	C₂×C₇₁	semidirect
342. C₂²×#135	C₂×C₇₁	semidirect
343. #3487	C₂×C₇₁	semidirect
344. #3287	C₂×C₇₁	semidirect
345. #3285	C₂×C₇₁	semidirect
346. #3271	C₂×C₇₁	semidirect
347. #3155	C₂×C₇₁	semidirect
348. #3058	C₂×C₇₁	semidirect
349. #3050	C₂×C₇₁	semidirect
350. #3043	C₂×C₇₁	semidirect
351. #3010	C₂×C₇₁	semidirect
352. #3005	C₂×C₇₁	semidirect
353. #2950	C₂×C₇₁	semidirect
354. #2929	C₂×C₇₁	semidirect
355. C₇₁	#61054	direct (N × H)
360. C₂×#137	C₂²×C₇₁	semidirect
366. C₂×#135	C₂²×C₇₁	semidirect
368. C₂²×Q₁₆	C₂²×C₇₁	semidirect
374. #550	C₂²×C₇₁	semidirect
376. #548	C₂²×C₇₁	semidirect
384. #496	C₂²×C₇₁	semidirect
385. #495	C₂²×C₇₁	semidirect
386. #493	C₂²×C₇₁	semidirect
387. #469	C₂²×C₇₁	semidirect
388. #467	C₂²×C₇₁	semidirect
389. #466	C₂²×C₇₁	semidirect
390. #466	C₂²×C₇₁	semidirect
391. #465	C₂²×C₇₁	semidirect
392. #464	C₂²×C₇₁	semidirect
393. #460	C₂²×C₇₁	semidirect
394. #452	C₂²×C₇₁	semidirect
395. #449	C₂²×C₇₁	semidirect
396. #449	C₂²×C₇₁	semidirect
397. #448	C₂²×C₇₁	semidirect
398. #441	C₂²×C₇₁	semidirect
399. C₄×Q₁₆	C₂²×C₇₁	semidirect
400. #435	C₂²×C₇₁	semidirect
401. C₈×D₄	C₂²×C₇₁	semidirect
436. C₂×Q₁₆	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×#578×C₇₁	2
C₂²×#135×C₇₁	2
C₇₁×#3487	2
C₇₁×#3287	2
C₇₁×#3285	2
C₇₁×#3271	2
C₇₁×#3155	2
C₇₁×#3058	2
C₇₁×#3050	2
C₇₁×#3043	2
C₇₁×#3010	2
C₇₁×#3005	2
C₇₁×#2950	2
C₇₁×#2929	2
#61054	71

Subgroups — C71×#61054 #31881926

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#61054 #31881926