Subgroups — C₂×C₇₁×#1550 #31838579

Order 18176.

1074 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#533×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#423×C₇₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄⋊C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₇₁×#1550	C₂	1	8	yes	maximal, direct product factor
5	C₂×#333×C₇₁	C₂	1	2	yes	maximal, complemented
6	C₂×#332×C₇₁	C₂	1	2	yes	maximal, complemented
7	C₂²×M₁₆×C₇₁	—	2	1	no	—
8	C₂³×C₈×C₇₁	—	2	1	no	—
9	#533×C₇₁	—	2	2	no	—
10	#533×C₇₁	—	2	2	no	—
11	#533×C₇₁	C₄	1	4	yes	—
12	#533×C₇₁	C₂²	1	4	yes	—
13	C₂×C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×#125×C₇₁	—	2	1	no	—
16	C₂×C₂²⋊Q₈×C₇₁	—	2	1	no	—
17	C₂×C₂²⋊Q₈×C₇₁	C₄	1	2	yes	—
18	C₂×C₄×Q₈×C₇₁	—	2	1	no	—
19	C₂×C₄×Q₈×C₇₁	C₄	1	2	yes	—
20	C₂×#117×C₇₁	—	2	1	no	—
21	C₂²×C₄²×C₇₁	C₂²	1	1	yes	characteristic
22	#423×C₇₁	—	2	2	no	—
23	#423×C₇₁	C₂²	1	4	yes	—
24	C₂×C₄⋊C₈×C₇₁	—	2	1	no	—
25	C₂×C₄⋊C₈×C₇₁	—	2	2	no	—
26	C₂×C₄⋊C₈×C₇₁	—	2	1	no	—
27	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
28	C₂×C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
29	C₂×C₄⋊C₈×C₇₁	C₂²	1	4	yes	—
30	#333×C₇₁	C₂²	1	8	yes	complemented
31	#332×C₇₁	C₂²	1	8	yes	complemented
32	C₂²×Q₈×C₇₁	—	2	1	no	—
33	C₂²×Q₈×C₇₁	C₂×C₄	1	2	yes	—
34	C₂³×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
35	C₂³×C₄×C₇₁	D₄	1	1	yes	characteristic
36	C₂³×C₄×C₇₁	D₄	1	1	yes	characteristic
37	C₂×M₁₆×C₇₁	—	2	4	no	—
38	C₂×M₁₆×C₇₁	—	4	4	no	—
39	C₂²×C₈×C₇₁	—	2	4	no	—
40	C₂²×C₈×C₇₁	—	2	2	no	—
41	C₂²×C₈×C₇₁	—	2	2	no	—
42	C₂²×C₈×C₇₁	—	4	4	no	—
43	C₄⋊Q₈×C₇₁	—	2	1	no	—
44	C₄⋊Q₈×C₇₁	—	2	1	no	—
45	C₄⋊Q₈×C₇₁	—	2	1	no	—
46	C₄⋊Q₈×C₇₁	—	2	1	no	—
47	C₄⋊Q₈×C₇₁	C₂×C₄	1	2	yes	—
48	C₄⋊Q₈×C₇₁	C₂×C₄	1	2	yes	—
49	C₄⋊Q₈×C₇₁	C₂³	1	2	yes	—
50	C₄⋊Q₈×C₇₁	C₂³	1	2	yes	—
51	#125×C₇₁	—	2	4	no	—
52	#125×C₇₁	—	2	4	no	—
53	C₂²⋊Q₈×C₇₁	—	2	4	no	—
54	C₂²⋊Q₈×C₇₁	—	2	4	no	—
55	C₂²⋊Q₈×C₇₁	—	2	4	no	—
56	C₂²⋊Q₈×C₇₁	C₂×C₄	1	8	yes	—
57	C₄×Q₈×C₇₁	—	2	4	no	—
58	C₄×Q₈×C₇₁	—	2	4	no	—
59	C₄×Q₈×C₇₁	—	2	4	no	—
60	C₄×Q₈×C₇₁	C₂×C₄	1	8	yes	—
61	#117×C₇₁	—	2	4	no	—
62	#117×C₇₁	—	2	4	no	—
63	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
64	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
65	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
66	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
67	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
68	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	2	yes	—
69	C₂×C₂²⋊C₄×C₇₁	—	2	2	no	—
70	C₂×C₄²×C₇₁	—	2	2	no	—
71	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
72	C₂×C₄²×C₇₁	C₂×C₄	1	2	yes	—
73	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
74	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
75	C₂×C₄²×C₇₁	D₄	1	2	yes	—
76	C₂×C₄²×C₇₁	D₄	1	2	yes	—
77	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₇₁	C₂³	1	2	yes	—
79	C₄⋊C₈×C₇₁	—	2	4	no	—
80	C₄⋊C₈×C₇₁	—	2	2	no	—
81	C₄⋊C₈×C₇₁	—	2	4	no	—
82	C₄⋊C₈×C₇₁	—	2	2	no	—
83	C₄⋊C₈×C₇₁	C₂³	1	4	yes	—
84	C₄⋊C₈×C₇₁	C₂³	1	4	yes	—
85	C₂⁴×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
86	C₂×Q₈×C₇₁	—	2	2	no	—
87	C₂×Q₈×C₇₁	—	2	2	no	—
88	C₂×Q₈×C₇₁	—	2	2	no	—
89	C₂×Q₈×C₇₁	—	4	4	no	—
90	C₂×Q₈×C₇₁	—	4	2	no	—
91	C₂×Q₈×C₇₁	—	4	2	no	—
92	C₂×Q₈×C₇₁	C₂²×C₄	1	4	yes	—
93	C₂²×C₄×C₇₁	—	2	2	no	—
94	C₂²×C₄×C₇₁	—	2	2	no	—
95	C₂²×C₄×C₇₁	—	2	2	no	—
96	C₂²×C₄×C₇₁	—	2	2	no	—
97	C₂²×C₄×C₇₁	—	4	2	no	—
98	C₂²×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
99	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
100	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
101	C₂²×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
102	C₂²×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
103	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
104	C₂²×C₄×C₇₁	SD₁₆	1	4	yes	—
105	C₂²×C₄×C₇₁	Q₁₆	1	4	yes	—
106	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₇₁	C₂²×C₄	1	2	yes	—
108	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
109	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
112	M₁₆×C₇₁	—	4	8	no	—
113	C₂×C₈×C₇₁	—	2	4	no	—
114	C₂×C₈×C₇₁	—	2	4	no	—
115	C₂×C₈×C₇₁	—	4	8	no	—
116	C₂×C₈×C₇₁	—	4	4	no	—
117	C₂×C₈×C₇₁	—	4	4	no	—
118	C₄⋊C₄×C₇₁	—	2	2	no	—
119	C₄⋊C₄×C₇₁	—	2	2	no	—
120	C₄⋊C₄×C₇₁	—	2	2	no	—
121	C₄⋊C₄×C₇₁	—	2	8	no	—
122	C₄⋊C₄×C₇₁	—	2	8	no	—
123	C₄⋊C₄×C₇₁	—	2	2	no	—
124	C₄⋊C₄×C₇₁	—	2	2	no	—
125	C₄⋊C₄×C₇₁	—	2	2	no	—
126	C₄⋊C₄×C₇₁	—	2	2	no	—
127	C₄⋊C₄×C₇₁	C₂²×C₄	1	4	yes	—
128	C₂²⋊C₄×C₇₁	—	2	8	no	—
129	C₄²×C₇₁	—	2	8	no	—
130	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
131	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
132	C₄²×C₇₁	C₂²⋊C₄	1	2	yes	—
133	C₄²×C₇₁	C₂²×C₄	1	2	yes	—
134	C₄²×C₇₁	C₂²×C₄	1	1	yes	characteristic
135	C₄²×C₇₁	C₂×D₄	1	2	yes	—
136	C₄²×C₇₁	C₂×D₄	1	2	yes	—
137	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
138	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
139	C₄²×C₇₁	C₂⁴	1	1	yes	characteristic
140	C₂³×C₇₁	—	2	2	no	—
141	C₂³×C₇₁	—	2	2	no	—
142	C₂³×C₇₁	#103	1	4	yes	—
143	C₂³×C₇₁	C₂×C₂²⋊C₄	1	2	yes	—
144	C₂³×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	Q₈×C₇₁	—	4	4	no	—
146	Q₈×C₇₁	—	4	4	no	—
147	Q₈×C₇₁	—	4	2	no	—
148	Q₈×C₇₁	—	4	2	no	—
149	Q₈×C₇₁	—	4	4	no	—
150	C₂×C₄×C₇₁	—	2	2	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	4	no	—
156	C₂×C₄×C₇₁	—	2	4	no	—
157	C₂×C₄×C₇₁	—	2	4	no	—
158	C₂×C₄×C₇₁	—	4	2	no	—
159	C₂×C₄×C₇₁	—	4	2	no	—
160	C₂×C₄×C₇₁	—	4	4	no	—
161	C₂×C₄×C₇₁	—	4	4	no	—
162	C₂×C₄×C₇₁	#103	1	8	yes	—
163	C₂×C₄×C₇₁	#103	1	4	yes	—
164	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	2	yes	—
165	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	2	yes	—
166	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
167	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	2	yes	—
168	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
169	C₂×C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
170	C₂×C₄×C₇₁	C₂×SD₁₆	1	2	yes	—
171	C₂×C₄×C₇₁	C₂×SD₁₆	1	4	yes	—
172	C₂×C₄×C₇₁	C₂×Q₁₆	1	4	yes	—
173	C₂×C₄×C₇₁	C₂×Q₁₆	1	2	yes	—
174	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
175	C₂×C₄×C₇₁	#137	1	2	yes	—
176	C₂×C₄×C₇₁	C₂³×C₄	1	1	yes	characteristic
177	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
179	C₈×C₇₁	—	4	4	no	—
180	C₈×C₇₁	—	4	4	no	—
181	C₂²×C₇₁	—	2	8	no	—
182	C₂²×C₇₁	—	2	2	no	—
183	C₂²×C₇₁	—	2	2	no	—
184	C₂²×C₇₁	#413	1	2	yes	—
185	C₂²×C₇₁	C₂×#103	1	4	yes	—
186	C₂²×C₇₁	C₂×#103	1	2	yes	—
187	C₂²×C₇₁	#417	1	2	yes	—
188	C₂²×C₇₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
189	C₄×C₇₁	—	2	2	no	—
190	C₄×C₇₁	—	2	1	no	—
191	C₄×C₇₁	—	2	1	no	—
192	C₄×C₇₁	—	2	2	no	—
193	C₄×C₇₁	—	4	4	no	—
194	C₄×C₇₁	—	4	4	no	—
195	C₄×C₇₁	C₂×#103	1	4	yes	—
196	C₄×C₇₁	C₂×#103	1	2	yes	—
197	C₄×C₇₁	#417	1	2	yes	—
198	C₄×C₇₁	C₂²×SD₁₆	1	1	yes	characteristic
199	C₄×C₇₁	C₂²×Q₁₆	1	1	yes	characteristic
200	C₄×C₇₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
201	C₄×C₇₁	C₂×#137	1	1	yes	characteristic
202	C₂×#1550	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
203	C₂×C₇₁	—	2	4	no	—
204	C₂×C₇₁	#1550	1	4	yes	direct product factor
205	C₂×C₇₁	C₂×#413	1	1	yes	characteristic
206	C₂×C₇₁	C₂²×#103	1	1	yes	characteristic
207	C₂×C₇₁	C₂×#417	1	1	yes	characteristic
208	C₂×#533	C₂×C₇₁	1	1	yes	characteristic
209	C₂×#423	C₂×C₇₁	1	1	yes	characteristic
210	C₂²×C₄⋊C₈	C₂×C₇₁	1	1	yes	characteristic
211	#1550	C₂×C₇₁	1	8	yes	direct product factor
212	C₂×#333	C₂×C₇₁	1	2	yes	complemented
213	C₂×#332	C₂×C₇₁	1	2	yes	complemented
214	C₇₁	C₂×#1550	1	1	yes	characteristic, direct product factor, Sylow
215	C₂²×M₁₆	—	2	1	no	—
216	C₂³×C₈	—	2	1	no	—
217	#533	—	2	2	no	—
218	#533	—	2	2	no	—
219	#533	C₄×C₇₁	1	4	yes	—
220	#533	C₂²×C₇₁	1	4	yes	—
221	C₂×C₄⋊Q₈	C₂²×C₇₁	1	1	yes	characteristic
222	C₂×C₄⋊Q₈	C₂²×C₇₁	1	1	yes	characteristic
223	C₂×#125	—	2	1	no	—
224	C₂×C₂²⋊Q₈	—	2	1	no	—
225	C₂×C₂²⋊Q₈	C₄×C₇₁	1	2	yes	—
226	C₂×C₄×Q₈	—	2	1	no	—
227	C₂×C₄×Q₈	C₄×C₇₁	1	2	yes	—
228	C₂×#117	—	2	1	no	—
229	C₂²×C₄²	C₂²×C₇₁	1	1	yes	characteristic
230	#423	—	2	2	no	—
231	#423	C₂²×C₇₁	1	4	yes	—
232	C₂×C₄⋊C₈	—	2	1	no	—
233	C₂×C₄⋊C₈	—	2	2	no	—
234	C₂×C₄⋊C₈	—	2	1	no	—
235	C₂×C₄⋊C₈	C₂²×C₇₁	1	2	yes	—
236	C₂×C₄⋊C₈	C₂²×C₇₁	1	2	yes	—
237	C₂×C₄⋊C₈	C₂²×C₇₁	1	4	yes	—
238	#333	C₂²×C₇₁	1	8	yes	complemented
239	#332	C₂²×C₇₁	1	8	yes	complemented
240	C₂²×Q₈	—	2	1	no	—
241	C₂²×Q₈	C₂×C₄×C₇₁	1	2	yes	—
242	C₂³×C₄	C₂×C₄×C₇₁	1	1	yes	characteristic
243	C₂³×C₄	D₄×C₇₁	1	1	yes	characteristic
244	C₂³×C₄	D₄×C₇₁	1	1	yes	characteristic
245	C₂×M₁₆	—	2	4	no	—
246	C₂×M₁₆	—	4	4	no	—
247	C₂²×C₈	—	2	4	no	—
248	C₂²×C₈	—	2	2	no	—
249	C₂²×C₈	—	2	2	no	—
250	C₂²×C₈	—	4	4	no	—
251	C₄⋊Q₈	—	2	1	no	—
252	C₄⋊Q₈	—	2	1	no	—
253	C₄⋊Q₈	—	2	1	no	—
254	C₄⋊Q₈	—	2	1	no	—
255	C₄⋊Q₈	C₂×C₄×C₇₁	1	2	yes	—
256	C₄⋊Q₈	C₂×C₄×C₇₁	1	2	yes	—
257	C₄⋊Q₈	C₂³×C₇₁	1	2	yes	—
258	C₄⋊Q₈	C₂³×C₇₁	1	2	yes	—
259	#125	—	2	4	no	—
260	#125	—	2	4	no	—
261	C₂²⋊Q₈	—	2	4	no	—
262	C₂²⋊Q₈	—	2	4	no	—
263	C₂²⋊Q₈	—	2	4	no	—
264	C₂²⋊Q₈	C₂×C₄×C₇₁	1	8	yes	—
265	C₄×Q₈	—	2	4	no	—
266	C₄×Q₈	—	2	4	no	—
267	C₄×Q₈	—	2	4	no	—
268	C₄×Q₈	C₂×C₄×C₇₁	1	8	yes	—
269	#117	—	2	4	no	—
270	#117	—	2	4	no	—
271	C₂×C₄⋊C₄	—	2	2	no	—
272	C₂×C₄⋊C₄	—	2	2	no	—
273	C₂×C₄⋊C₄	—	2	1	no	—
274	C₂×C₄⋊C₄	—	2	1	no	—
275	C₂×C₄⋊C₄	—	2	1	no	—
276	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	2	yes	—
277	C₂×C₂²⋊C₄	—	2	2	no	—
278	C₂×C₄²	—	2	2	no	—
279	C₂×C₄²	C₂×C₄×C₇₁	1	1	yes	characteristic
280	C₂×C₄²	C₂×C₄×C₇₁	1	2	yes	—
281	C₂×C₄²	D₄×C₇₁	1	1	yes	characteristic
282	C₂×C₄²	D₄×C₇₁	1	1	yes	characteristic
283	C₂×C₄²	D₄×C₇₁	1	2	yes	—
284	C₂×C₄²	D₄×C₇₁	1	2	yes	—
285	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
286	C₂×C₄²	C₂³×C₇₁	1	2	yes	—
287	C₄⋊C₈	—	2	4	no	—
288	C₄⋊C₈	—	2	2	no	—
289	C₄⋊C₈	—	2	4	no	—
290	C₄⋊C₈	—	2	2	no	—
291	C₄⋊C₈	C₂³×C₇₁	1	4	yes	—
292	C₄⋊C₈	C₂³×C₇₁	1	4	yes	—
293	C₂⁴	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
294	C₂×Q₈	—	2	2	no	—
295	C₂×Q₈	—	2	2	no	—
296	C₂×Q₈	—	2	2	no	—
297	C₂×Q₈	—	4	4	no	—
298	C₂×Q₈	—	4	2	no	—
299	C₂×Q₈	—	4	2	no	—
300	C₂×Q₈	C₂²×C₄×C₇₁	1	4	yes	—
301	C₂²×C₄	—	2	2	no	—
302	C₂²×C₄	—	2	2	no	—
303	C₂²×C₄	—	2	2	no	—
304	C₂²×C₄	—	2	2	no	—
305	C₂²×C₄	—	4	2	no	—
306	C₂²×C₄	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
307	C₂²×C₄	C₂²⋊C₄×C₇₁	1	2	yes	—
308	C₂²×C₄	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
309	C₂²×C₄	C₂²⋊C₄×C₇₁	1	2	yes	—
310	C₂²×C₄	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
311	C₂²×C₄	C₂²⋊C₄×C₇₁	1	2	yes	—
312	C₂²×C₄	SD₁₆×C₇₁	1	4	yes	—
313	C₂²×C₄	Q₁₆×C₇₁	1	4	yes	—
314	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
315	C₂²×C₄	C₂²×C₄×C₇₁	1	2	yes	—
316	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
317	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
318	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
319	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
320	M₁₆	—	4	8	no	—
321	C₂×C₈	—	2	4	no	—
322	C₂×C₈	—	2	4	no	—
323	C₂×C₈	—	4	8	no	—
324	C₂×C₈	—	4	4	no	—
325	C₂×C₈	—	4	4	no	—
326	C₄⋊C₄	—	2	2	no	—
327	C₄⋊C₄	—	2	2	no	—
328	C₄⋊C₄	—	2	2	no	—
329	C₄⋊C₄	—	2	8	no	—
330	C₄⋊C₄	—	2	8	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	—	2	2	no	—
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	C₂²×C₄×C₇₁	1	4	yes	—
336	C₂²⋊C₄	—	2	8	no	—
337	C₄²	—	2	8	no	—
338	C₄²	C₂²⋊C₄×C₇₁	1	2	yes	—
339	C₄²	C₂²⋊C₄×C₇₁	1	2	yes	—
340	C₄²	C₂²⋊C₄×C₇₁	1	2	yes	—
341	C₄²	C₂²×C₄×C₇₁	1	2	yes	—
342	C₄²	C₂²×C₄×C₇₁	1	1	yes	characteristic
343	C₄²	C₂×D₄×C₇₁	1	2	yes	—
344	C₄²	C₂×D₄×C₇₁	1	2	yes	—
345	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
346	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
347	C₄²	C₂⁴×C₇₁	1	1	yes	characteristic
348	C₂³	—	2	2	no	—
349	C₂³	—	2	2	no	—
350	C₂³	#103×C₇₁	1	4	yes	—
351	C₂³	C₂×C₂²⋊C₄×C₇₁	1	2	yes	—
352	C₂³	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
353	Q₈	—	4	4	no	—
354	Q₈	—	4	4	no	—
355	Q₈	—	4	4	no	—
356	Q₈	—	4	2	no	—
357	Q₈	—	4	2	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	4	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	4	no	—
365	C₂×C₄	—	2	4	no	—
366	C₂×C₄	—	4	2	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	4	4	no	—
369	C₂×C₄	—	4	4	no	—
370	C₂×C₄	#103×C₇₁	1	8	yes	—
371	C₂×C₄	#103×C₇₁	1	4	yes	—
372	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	2	yes	—
373	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	2	yes	—
374	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
375	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	2	yes	—
376	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
377	C₂×C₄	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
378	C₂×C₄	C₂×SD₁₆×C₇₁	1	2	yes	—
379	C₂×C₄	C₂×SD₁₆×C₇₁	1	4	yes	—
380	C₂×C₄	C₂×Q₁₆×C₇₁	1	4	yes	—
381	C₂×C₄	C₂×Q₁₆×C₇₁	1	2	yes	—
382	C₂×C₄	AGL(1,ℤ/8)×C₇₁	1	2	yes	—
383	C₂×C₄	#137×C₇₁	1	2	yes	—
384	C₂×C₄	C₂³×C₄×C₇₁	1	1	yes	characteristic
385	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
386	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
387	C₈	—	4	4	no	—
388	C₈	—	4	4	no	—
389	C₂²	—	2	8	no	—
390	C₂²	—	2	2	no	—
391	C₂²	—	2	2	no	—
392	C₂²	#413×C₇₁	1	2	yes	—
393	C₂²	C₂×#103×C₇₁	1	4	yes	—
394	C₂²	C₂×#103×C₇₁	1	2	yes	—
395	C₂²	#417×C₇₁	1	2	yes	—
396	C₂²	C₂²×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
397	C₄	—	2	1	no	—
398	C₄	—	2	2	no	—
399	C₄	—	2	1	no	—
400	C₄	—	2	2	no	—
401	C₄	—	4	4	no	—
402	C₄	—	4	4	no	—
403	C₄	C₂×#103×C₇₁	1	4	yes	—
404	C₄	C₂×#103×C₇₁	1	2	yes	—
405	C₄	#417×C₇₁	1	2	yes	—
406	C₄	C₂²×SD₁₆×C₇₁	1	1	yes	characteristic
407	C₄	C₂²×Q₁₆×C₇₁	1	1	yes	characteristic
408	C₄	C₂×AGL(1,ℤ/8)×C₇₁	1	1	yes	characteristic
409	C₄	C₂×#137×C₇₁	1	1	yes	characteristic
410	C₂	—	2	4	no	—
411	C₂	C₇₁×#1550	1	4	yes	direct product factor
412	C₂	C₂×#413×C₇₁	1	1	yes	characteristic
413	C₂	C₂²×#103×C₇₁	1	1	yes	characteristic
414	C₂	C₂×#417×C₇₁	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇₁	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₇₁
Frattini, cosocle	C₄²	C₂⁴×C₇₁
prime radical	C₂⁴×C₇₁	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇₁ ◁ C₂²×C₄²×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₇₁×#1550	C₂	direct (N × H)
5. C₂×#333×C₇₁	C₂	semidirect
6. C₂×#332×C₇₁	C₂	semidirect
30. #333×C₇₁	C₂²	semidirect
31. #332×C₇₁	C₂²	semidirect
202. C₂×#1550	C₇₁	direct (N × H)
204. C₂×C₇₁	#1550	direct (N × H)
211. #1550	C₂×C₇₁	direct (N × H)
212. C₂×#333	C₂×C₇₁	semidirect
213. C₂×#332	C₂×C₇₁	semidirect
214. C₇₁	C₂×#1550	direct (N × H)
238. #333	C₂²×C₇₁	semidirect
239. #332	C₂²×C₇₁	semidirect
411. C₂	C₇₁×#1550	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#533×C₇₁	2
C₂×#423×C₇₁	2
C₂²×C₄⋊C₈×C₇₁	2
C₇₁×#1550	2
C₂×#333×C₇₁	2
C₂×#332×C₇₁	2
C₂×#1550	71

Subgroups — C2×C71×#1550 #31838579

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇₁×#1550 #31838579