Subgroups — C₇₁×#23151 #31844023

Order 18176.

772 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#446×C₇₁	C₂	1	2	yes	maximal
2	C₄×C₄⋊Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#1985	C₂	1	2	yes	maximal
4	C₇₁×#1939	C₂	1	1	yes	maximal, characteristic
5	C₇₁×#1939	C₂	1	1	yes	maximal, characteristic
6	C₇₁×#1917	C₂	1	2	yes	maximal
7	C₇₁×#1916	C₂	1	2	yes	maximal
8	C₇₁×#1770	C₂	1	1	yes	maximal, characteristic
9	C₈×C₄⋊C₄×C₇₁	C₂	1	1	yes	maximal, characteristic
10	C₇₁×#1767	C₂	1	1	yes	maximal, characteristic
11	C₄×C₄⋊C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊Q₈×C₇₁	C₄	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₇₁	C₂²	1	2	yes	—
14	#446×C₇₁	—	2	8	no	—
15	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
21	C₂×#97×C₇₁	C₂²	1	2	yes	—
22	C₂×#97×C₇₁	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₇₁	C₂²	1	2	yes	—
26	#391×C₇₁	C₄	1	2	yes	—
27	#389×C₇₁	C₄	1	1	yes	characteristic
28	#389×C₇₁	C₄	1	2	yes	—
29	#389×C₇₁	C₄	1	1	yes	characteristic
30	#384×C₇₁	C₄	1	1	yes	characteristic
31	C₄×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
33	C₄×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
34	C₄³×C₇₁	C₂²	1	1	yes	characteristic
35	#336×C₇₁	C₂²	1	1	yes	characteristic
36	#336×C₇₁	C₂²	1	1	yes	characteristic
37	#336×C₇₁	C₂²	1	2	yes	—
38	#336×C₇₁	C₂²	1	2	yes	—
39	#336×C₇₁	C₂²	1	2	yes	—
40	#336×C₇₁	C₂²	1	2	yes	—
41	#336×C₇₁	C₂²	1	2	yes	—
42	C₂²×Q₈×C₇₁	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₇₁	C₂³	1	2	yes	—
44	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
47	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₇₁	C₂³	1	2	yes	—
49	C₄⋊Q₈×C₇₁	—	2	4	no	—
50	C₄×Q₈×C₇₁	—	2	4	no	—
51	C₄×Q₈×C₇₁	—	2	4	no	—
52	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
53	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
54	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
55	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
57	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
58	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₇₁	C₂³	1	2	yes	—
66	C₂×C₄²×C₇₁	C₂³	1	2	yes	—
67	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
70	C₄⋊C₈×C₇₁	—	2	4	no	—
71	C₄⋊C₈×C₇₁	—	2	4	no	—
72	C₄⋊C₈×C₇₁	—	2	2	no	—
73	C₄⋊C₈×C₇₁	—	2	2	no	—
74	C₄⋊C₈×C₇₁	D₄	1	4	yes	—
75	C₄⋊C₈×C₇₁	Q₈	1	4	yes	complemented
76	#97×C₇₁	—	2	4	no	—
77	#97×C₇₁	—	2	4	no	—
78	C₄×C₈×C₇₁	—	2	4	no	—
79	C₄×C₈×C₇₁	—	2	2	no	—
80	C₄×C₈×C₇₁	—	2	2	no	—
81	#95×C₇₁	C₂×C₄	1	2	yes	—
82	#95×C₇₁	C₂×C₄	1	2	yes	—
83	#95×C₇₁	C₂×C₄	1	2	yes	—
84	#95×C₇₁	C₂×C₄	1	2	yes	—
85	C₂×Q₈×C₇₁	—	2	4	no	—
86	C₂×Q₈×C₇₁	—	2	4	no	—
87	C₂×Q₈×C₇₁	—	4	2	no	—
88	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
99	C₂×C₈×C₇₁	—	2	2	no	—
100	C₂×C₈×C₇₁	—	2	1	no	—
101	C₂×C₈×C₇₁	—	2	1	no	—
102	C₂×C₈×C₇₁	—	2	1	no	—
103	C₂×C₈×C₇₁	—	2	1	no	—
104	C₂×C₈×C₇₁	—	2	2	no	—
105	C₂×C₈×C₇₁	—	2	2	no	—
106	C₂×C₈×C₇₁	—	2	1	no	—
107	C₂×C₈×C₇₁	—	2	1	no	—
108	C₂×C₈×C₇₁	—	2	2	no	—
109	C₂×C₈×C₇₁	—	2	1	no	—
110	C₂×C₈×C₇₁	—	2	1	no	—
111	C₂×C₈×C₇₁	—	2	2	no	—
112	C₂×C₈×C₇₁	—	2	1	no	—
113	C₂×C₈×C₇₁	—	2	1	no	—
114	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
115	C₂×C₈×C₇₁	C₂×D₄	1	2	yes	—
116	C₂×C₈×C₇₁	C₂×Q₈	1	4	yes	—
117	C₄⋊C₄×C₇₁	—	2	4	no	—
118	C₄⋊C₄×C₇₁	—	2	4	no	—
119	C₄⋊C₄×C₇₁	—	2	4	no	—
120	C₄⋊C₄×C₇₁	—	2	4	no	—
121	C₄⋊C₄×C₇₁	—	2	2	no	—
122	C₄⋊C₄×C₇₁	—	2	4	no	—
123	C₄⋊C₄×C₇₁	—	2	2	no	—
124	C₄⋊C₄×C₇₁	—	2	2	no	—
125	C₄⋊C₄×C₇₁	M₁₆	1	4	yes	—
126	C₄²×C₇₁	—	2	2	no	—
127	C₄²×C₇₁	—	2	2	no	—
128	C₄²×C₇₁	—	2	2	no	—
129	C₄²×C₇₁	—	2	2	no	—
130	C₄²×C₇₁	—	2	2	no	—
131	C₄²×C₇₁	—	2	2	no	—
132	C₄²×C₇₁	—	2	2	no	—
133	C₄²×C₇₁	—	2	1	no	—
134	C₄²×C₇₁	—	2	1	no	—
135	C₄²×C₇₁	—	4	1	no	—
136	C₄²×C₇₁	C₂×D₄	1	2	yes	—
137	C₄²×C₇₁	C₂×Q₈	1	2	yes	—
138	C₄²×C₇₁	#41	1	2	yes	—
139	C₄²×C₇₁	#41	1	2	yes	—
140	C₂³×C₇₁	C₂³×C₄	1	1	yes	characteristic
141	Q₈×C₇₁	—	4	8	no	—
142	C₂×C₄×C₇₁	—	2	2	no	—
143	C₂×C₄×C₇₁	—	2	1	no	—
144	C₂×C₄×C₇₁	—	2	1	no	—
145	C₂×C₄×C₇₁	—	2	1	no	—
146	C₂×C₄×C₇₁	—	2	1	no	—
147	C₂×C₄×C₇₁	—	2	1	no	—
148	C₂×C₄×C₇₁	—	2	1	no	—
149	C₂×C₄×C₇₁	—	2	1	no	—
150	C₂×C₄×C₇₁	—	2	1	no	—
151	C₂×C₄×C₇₁	—	2	2	no	—
152	C₂×C₄×C₇₁	—	2	2	no	—
153	C₂×C₄×C₇₁	—	2	2	no	—
154	C₂×C₄×C₇₁	—	2	2	no	—
155	C₂×C₄×C₇₁	—	2	1	no	—
156	C₂×C₄×C₇₁	—	2	2	no	—
157	C₂×C₄×C₇₁	—	2	1	no	—
158	C₂×C₄×C₇₁	—	2	2	no	—
159	C₂×C₄×C₇₁	—	2	2	no	—
160	C₂×C₄×C₇₁	—	2	2	no	—
161	C₂×C₄×C₇₁	—	2	2	no	—
162	C₂×C₄×C₇₁	—	2	1	no	—
163	C₂×C₄×C₇₁	—	2	2	no	—
164	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
165	C₂×C₄×C₇₁	C₄×D₄	1	2	yes	—
166	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
167	C₂×C₄×C₇₁	C₄×Q₈	1	2	yes	—
168	C₂×C₄×C₇₁	C₂×M₁₆	1	4	yes	—
169	C₂×C₄×C₇₁	C₂×M₁₆	1	2	yes	—
170	C₂×C₄×C₇₁	#131	1	2	yes	—
171	C₂×C₄×C₇₁	#131	1	2	yes	—
172	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
173	C₂×C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
174	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
175	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
176	C₂×C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
177	C₂×C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
178	C₈×C₇₁	—	2	2	no	—
179	C₈×C₇₁	—	2	2	no	—
180	C₈×C₇₁	—	2	4	no	—
181	C₈×C₇₁	—	4	2	no	—
182	C₈×C₇₁	—	4	1	no	—
183	C₈×C₇₁	—	4	1	no	—
184	C₂²×C₇₁	C₂×C₄×D₄	1	1	yes	characteristic
185	C₂²×C₇₁	C₂×C₄×Q₈	1	1	yes	characteristic
186	C₂²×C₇₁	#519	1	1	yes	characteristic
187	C₂²×C₇₁	C₂²×M₁₆	1	1	yes	characteristic
188	C₂²×C₇₁	C₂×#131	1	1	yes	characteristic
189	C₂²×C₇₁	C₂×#131	1	1	yes	characteristic
190	C₂²×C₇₁	#568	1	1	yes	characteristic
191	C₄×C₇₁	—	2	2	no	—
192	C₄×C₇₁	—	2	2	no	—
193	C₄×C₇₁	—	2	4	no	—
194	C₄×C₇₁	—	2	2	no	—
195	C₄×C₇₁	—	2	2	no	—
196	C₄×C₇₁	—	4	2	no	—
197	C₄×C₇₁	—	4	2	no	—
198	C₄×C₇₁	—	4	2	no	—
199	C₄×C₇₁	—	4	1	no	—
200	C₄×C₇₁	#446	1	4	yes	complemented
201	C₄×C₇₁	#547	1	1	yes	characteristic
202	C₄×C₇₁	D₄×Q₈	1	1	yes	characteristic
203	C₄×C₇₁	#556	1	1	yes	characteristic
204	C₄×C₇₁	#557	1	1	yes	characteristic
205	#23151	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
206	C₂×C₇₁	#2365	1	1	yes	characteristic
207	C₂×C₇₁	#2931	1	1	yes	characteristic
208	C₂×C₇₁	D₄×M₁₆	1	1	yes	characteristic
209	C₂×C₇₁	C₂×#446	1	1	yes	characteristic
210	C₂×C₇₁	#2962	1	1	yes	characteristic
211	C₂×C₇₁	#2984	1	1	yes	characteristic
212	C₂×C₇₁	#2985	1	1	yes	characteristic
213	C₂×#446	C₂×C₇₁	1	2	yes	—
214	C₄×C₄⋊Q₈	C₂×C₇₁	1	1	yes	characteristic
215	#1985	C₂×C₇₁	1	2	yes	—
216	#1939	C₂×C₇₁	1	1	yes	characteristic
217	#1939	C₂×C₇₁	1	1	yes	characteristic
218	#1917	C₂×C₇₁	1	2	yes	—
219	#1916	C₂×C₇₁	1	2	yes	—
220	#1770	C₂×C₇₁	1	1	yes	characteristic
221	C₈×C₄⋊C₄	C₂×C₇₁	1	1	yes	characteristic
222	#1767	C₂×C₇₁	1	1	yes	characteristic
223	C₄×C₄⋊C₈	C₂×C₇₁	1	1	yes	characteristic
224	C₇₁	#23151	1	1	yes	characteristic, direct product factor, Sylow
225	C₂×C₄⋊Q₈	C₄×C₇₁	1	1	yes	characteristic
226	C₂×C₄×Q₈	C₂²×C₇₁	1	2	yes	—
227	#446	—	2	8	no	—
228	C₂×C₄⋊C₈	C₂²×C₇₁	1	2	yes	—
229	C₂×C₄⋊C₈	C₂²×C₇₁	1	2	yes	—
230	C₂×C₄⋊C₈	C₂²×C₇₁	1	1	yes	characteristic
231	C₂×C₄⋊C₈	C₂²×C₇₁	1	1	yes	characteristic
232	C₂×C₄⋊C₈	C₂²×C₇₁	1	1	yes	characteristic
233	C₂×C₄⋊C₈	C₂²×C₇₁	1	1	yes	characteristic
234	C₂×#97	C₂²×C₇₁	1	2	yes	—
235	C₂×#97	C₂²×C₇₁	1	2	yes	—
236	C₂×C₄×C₈	C₂²×C₇₁	1	2	yes	—
237	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
238	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
239	#391	C₄×C₇₁	1	2	yes	—
240	#389	C₄×C₇₁	1	1	yes	characteristic
241	#389	C₄×C₇₁	1	2	yes	—
242	#389	C₄×C₇₁	1	1	yes	characteristic
243	#384	C₄×C₇₁	1	1	yes	characteristic
244	C₄×C₄⋊C₄	C₂²×C₇₁	1	2	yes	—
245	C₄×C₄⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
246	C₄×C₄⋊C₄	C₂²×C₇₁	1	1	yes	characteristic
247	C₄³	C₂²×C₇₁	1	1	yes	characteristic
248	#336	C₂²×C₇₁	1	1	yes	characteristic
249	#336	C₂²×C₇₁	1	2	yes	—
250	#336	C₂²×C₇₁	1	2	yes	—
251	#336	C₂²×C₇₁	1	2	yes	—
252	#336	C₂²×C₇₁	1	1	yes	characteristic
253	#336	C₂²×C₇₁	1	2	yes	—
254	#336	C₂²×C₇₁	1	2	yes	—
255	C₂²×Q₈	C₂×C₄×C₇₁	1	2	yes	—
256	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
257	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
258	C₂²×C₈	C₂³×C₇₁	1	2	yes	—
259	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
260	C₂²×C₈	C₂³×C₇₁	1	1	yes	characteristic
261	C₂²×C₈	C₂³×C₇₁	1	2	yes	—
262	C₄⋊Q₈	—	2	4	no	—
263	C₄×Q₈	—	2	4	no	—
264	C₄×Q₈	—	2	4	no	—
265	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	2	yes	—
266	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	2	yes	—
267	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	2	yes	—
268	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	1	yes	characteristic
269	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	2	yes	—
270	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	2	yes	—
271	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	1	yes	characteristic
272	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	1	yes	characteristic
273	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	1	yes	characteristic
274	C₂×C₄²	C₂×C₄×C₇₁	1	1	yes	characteristic
275	C₂×C₄²	C₂×C₄×C₇₁	1	1	yes	characteristic
276	C₂×C₄²	C₂×C₄×C₇₁	1	1	yes	characteristic
277	C₂×C₄²	C₂×C₄×C₇₁	1	1	yes	characteristic
278	C₂×C₄²	C₂³×C₇₁	1	2	yes	—
279	C₂×C₄²	C₂³×C₇₁	1	2	yes	—
280	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
281	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
282	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
283	C₄⋊C₈	—	2	4	no	—
284	C₄⋊C₈	—	2	4	no	—
285	C₄⋊C₈	—	2	2	no	—
286	C₄⋊C₈	—	2	2	no	—
287	C₄⋊C₈	D₄×C₇₁	1	4	yes	—
288	C₄⋊C₈	Q₈×C₇₁	1	4	yes	complemented
289	#97	—	2	4	no	—
290	#97	—	2	4	no	—
291	C₄×C₈	—	2	4	no	—
292	C₄×C₈	—	2	2	no	—
293	C₄×C₈	—	2	2	no	—
294	#95	C₂×C₄×C₇₁	1	2	yes	—
295	#95	C₂×C₄×C₇₁	1	2	yes	—
296	#95	C₂×C₄×C₇₁	1	2	yes	—
297	#95	C₂×C₄×C₇₁	1	2	yes	—
298	C₂×Q₈	—	2	4	no	—
299	C₂×Q₈	—	2	4	no	—
300	C₂×Q₈	—	4	2	no	—
301	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
302	C₂²×C₄	C₂²×C₄×C₇₁	1	2	yes	—
303	C₂²×C₄	C₂²×C₄×C₇₁	1	2	yes	—
304	C₂²×C₄	C₂²×C₄×C₇₁	1	2	yes	—
305	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
306	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
307	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
308	C₂²×C₄	C₂²×C₄×C₇₁	1	2	yes	—
309	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
310	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
311	C₂²×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	1	no	—
314	C₂×C₈	—	2	1	no	—
315	C₂×C₈	—	2	1	no	—
316	C₂×C₈	—	2	1	no	—
317	C₂×C₈	—	2	2	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	2	1	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	1	no	—
323	C₂×C₈	—	2	1	no	—
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	2	1	no	—
327	C₂×C₈	C₂×D₄×C₇₁	1	2	yes	—
328	C₂×C₈	C₂×D₄×C₇₁	1	2	yes	—
329	C₂×C₈	C₂×Q₈×C₇₁	1	4	yes	—
330	C₄⋊C₄	—	2	4	no	—
331	C₄⋊C₄	—	2	4	no	—
332	C₄⋊C₄	—	2	4	no	—
333	C₄⋊C₄	—	2	4	no	—
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	—	2	4	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	2	2	no	—
338	C₄⋊C₄	M₁₆×C₇₁	1	4	yes	—
339	C₄²	—	2	2	no	—
340	C₄²	—	2	2	no	—
341	C₄²	—	2	2	no	—
342	C₄²	—	2	2	no	—
343	C₄²	—	2	2	no	—
344	C₄²	—	2	2	no	—
345	C₄²	—	2	2	no	—
346	C₄²	—	2	1	no	—
347	C₄²	—	2	1	no	—
348	C₄²	—	4	1	no	—
349	C₄²	C₂×D₄×C₇₁	1	2	yes	—
350	C₄²	C₂×Q₈×C₇₁	1	2	yes	—
351	C₄²	#41×C₇₁	1	2	yes	—
352	C₄²	#41×C₇₁	1	2	yes	—
353	C₂³	C₂³×C₄×C₇₁	1	1	yes	characteristic
354	Q₈	—	4	8	no	—
355	C₂×C₄	—	2	2	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	1	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	C₄×D₄×C₇₁	1	2	yes	—
378	C₂×C₄	C₄×D₄×C₇₁	1	2	yes	—
379	C₂×C₄	C₄×Q₈×C₇₁	1	2	yes	—
380	C₂×C₄	C₄×Q₈×C₇₁	1	2	yes	—
381	C₂×C₄	C₂×M₁₆×C₇₁	1	4	yes	—
382	C₂×C₄	C₂×M₁₆×C₇₁	1	2	yes	—
383	C₂×C₄	#131×C₇₁	1	2	yes	—
384	C₂×C₄	#131×C₇₁	1	2	yes	—
385	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
386	C₂×C₄	C₂²×Q₈×C₇₁	1	1	yes	characteristic
387	C₂×C₄	C₂×#41×C₇₁	1	1	yes	characteristic
388	C₂×C₄	C₂×#41×C₇₁	1	1	yes	characteristic
389	C₂×C₄	2₋¹⁺⁴×C₇₁	1	1	yes	characteristic
390	C₂×C₄	2₋¹⁺⁴×C₇₁	1	1	yes	characteristic
391	C₈	—	2	2	no	—
392	C₈	—	2	2	no	—
393	C₈	—	2	4	no	—
394	C₈	—	4	2	no	—
395	C₈	—	4	1	no	—
396	C₈	—	4	1	no	—
397	C₂²	C₂×C₄×D₄×C₇₁	1	1	yes	characteristic
398	C₂²	C₂×C₄×Q₈×C₇₁	1	1	yes	characteristic
399	C₂²	#519×C₇₁	1	1	yes	characteristic
400	C₂²	C₂²×M₁₆×C₇₁	1	1	yes	characteristic
401	C₂²	C₂×#131×C₇₁	1	1	yes	characteristic
402	C₂²	C₂×#131×C₇₁	1	1	yes	characteristic
403	C₂²	#568×C₇₁	1	1	yes	characteristic
404	C₄	—	2	2	no	—
405	C₄	—	2	2	no	—
406	C₄	—	2	4	no	—
407	C₄	—	2	2	no	—
408	C₄	—	2	2	no	—
409	C₄	—	4	2	no	—
410	C₄	—	4	2	no	—
411	C₄	—	4	2	no	—
412	C₄	—	4	1	no	—
413	C₄	#446×C₇₁	1	4	yes	complemented
414	C₄	#547×C₇₁	1	1	yes	characteristic
415	C₄	D₄×Q₈×C₇₁	1	1	yes	characteristic
416	C₄	#556×C₇₁	1	1	yes	characteristic
417	C₄	#557×C₇₁	1	1	yes	characteristic
418	C₂	C₇₁×#2365	1	1	yes	characteristic
419	C₂	C₇₁×#2931	1	1	yes	characteristic
420	C₂	D₄×M₁₆×C₇₁	1	1	yes	characteristic
421	C₂	C₂×#446×C₇₁	1	1	yes	characteristic
422	C₂	C₇₁×#2962	1	1	yes	characteristic
423	C₂	C₇₁×#2984	1	1	yes	characteristic
424	C₂	C₇₁×#2985	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇₁	C₂⁴
commutator	C₂³	C₂³×C₄×C₇₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₁
socle, prime radical	C₂³×C₇₁	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
75. C₄⋊C₈×C₇₁	Q₈	semidirect
200. C₄×C₇₁	#446	semidirect
205. #23151	C₇₁	direct (N × H)
224. C₇₁	#23151	direct (N × H)
288. C₄⋊C₈	Q₈×C₇₁	semidirect
413. C₄	#446×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#446×C₇₁	2
C₄×C₄⋊Q₈×C₇₁	2
C₇₁×#1985	2
C₇₁×#1939	2
C₇₁×#1917	2
C₇₁×#1916	2
C₇₁×#1770	2
C₈×C₄⋊C₄×C₇₁	2
C₇₁×#1767	2
C₄×C₄⋊C₈×C₇₁	2
#23151	71

Subgroups — C71×#23151 #31844023

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#23151 #31844023