Subgroups — C₇₁×#34313 #31855185

Order 18176.

3276 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#550×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#520×C₇₁	C₂	1	2	yes	maximal, complemented
3	C₇₁×#2366	C₂	1	3	yes	maximal, complemented
4	C₇₁×#2361	C₂	1	6	yes	maximal, complemented
5	Q₈×C₄⋊C₄×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄×C₄⋊C₄×C₇₁	C₂	1	3	yes	maximal, complemented
7	C₇₁×#2341	C₂	1	2	yes	maximal, complemented
8	C₇₁×#2324	C₂	1	6	yes	maximal, complemented
9	C₄×#127×C₇₁	C₂	1	3	yes	maximal, complemented
10	C₇₁×#2293	C₂	1	3	yes	maximal, complemented
11	C₄²×Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₇₁	C₂²	1	2	yes	—
13	#550×C₇₁	C₄	1	16	yes	complemented
14	C₂×#127×C₇₁	C₂²	1	3	yes	—
15	C₂×#121×C₇₁	C₂²	1	6	yes	—
16	#520×C₇₁	—	2	16	no	—
17	C₂×C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₇₁	C₂²	1	2	yes	—
19	C₂×C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₇₁	C₂²	1	6	yes	—
22	C₂×C₄×D₄×C₇₁	C₂²	1	6	yes	—
23	C₂×C₄×D₄×C₇₁	C₂²	1	6	yes	—
24	C₂×C₄×D₄×C₇₁	C₂²	1	6	yes	—
25	C₂×C₄×D₄×C₇₁	C₂²	1	3	yes	—
26	C₂×#117×C₇₁	C₂²	1	6	yes	—
27	C₂²×C₄⋊C₄×C₇₁	C₂²	1	6	yes	—
28	#390×C₇₁	C₂²	1	3	yes	—
29	#390×C₇₁	C₂²	1	3	yes	—
30	#390×C₇₁	C₂²	1	6	yes	—
31	#390×C₇₁	C₂²	1	6	yes	—
32	#390×C₇₁	C₂²	1	6	yes	—
33	#389×C₇₁	C₂²	1	1	yes	characteristic
34	#389×C₇₁	C₂²	1	3	yes	—
35	#389×C₇₁	C₂²	1	6	yes	—
36	#388×C₇₁	C₂²	1	6	yes	—
37	#388×C₇₁	C₂²	1	6	yes	—
38	#386×C₇₁	C₂²	1	12	yes	—
39	#385×C₇₁	C₂²	1	12	yes	—
40	#384×C₇₁	C₂²	1	3	yes	—
41	#380×C₇₁	C₂²	1	6	yes	—
42	C₄×C₄⋊C₄×C₇₁	C₂²	1	3	yes	—
43	C₄×C₄⋊C₄×C₇₁	C₂²	1	3	yes	—
44	C₄×C₄⋊C₄×C₇₁	C₂²	1	3	yes	—
45	C₄×C₄⋊C₄×C₇₁	C₂²	1	3	yes	—
46	C₄×C₂²⋊C₄×C₇₁	C₂²	1	6	yes	—
47	C₄³×C₇₁	C₂²	1	3	yes	—
48	C₂×#41×C₇₁	—	2	8	no	—
49	C₂×#41×C₇₁	—	2	8	no	—
50	C₂×#41×C₇₁	C₂×C₄	1	16	yes	complemented
51	C₂²×Q₈×C₇₁	C₂³	1	1	yes	characteristic
52	C₂²×D₄×C₇₁	C₂³	1	3	yes	—
53	C₂²×D₄×C₇₁	C₂³	1	6	yes	—
54	C₂²×D₄×C₇₁	C₂³	1	6	yes	—
55	C₂³×C₄×C₇₁	C₂³	1	6	yes	—
56	C₂³×C₄×C₇₁	C₂³	1	12	yes	—
57	#127×C₇₁	C₂×C₄	1	24	yes	complemented
58	#121×C₇₁	C₂×C₄	1	48	yes	complemented
59	C₄×Q₈×C₇₁	—	2	8	no	—
60	C₄×Q₈×C₇₁	—	2	4	no	—
61	C₄×Q₈×C₇₁	C₂×C₄	1	8	yes	complemented
62	C₄×Q₈×C₇₁	D₄	1	8	yes	complemented
63	C₄×D₄×C₇₁	—	2	24	no	—
64	C₄×D₄×C₇₁	—	2	24	no	—
65	C₄×D₄×C₇₁	—	4	12	no	—
66	C₄×D₄×C₇₁	—	4	12	no	—
67	C₄×D₄×C₇₁	C₂×C₄	1	24	yes	complemented
68	#117×C₇₁	—	2	24	no	—
69	C₂×C₄⋊C₄×C₇₁	—	2	24	no	—
70	C₂×C₄⋊C₄×C₇₁	C₂³	1	6	yes	—
71	C₂×C₄⋊C₄×C₇₁	C₂³	1	6	yes	—
72	C₂×C₄⋊C₄×C₇₁	C₂³	1	2	yes	—
73	C₂×C₄⋊C₄×C₇₁	C₂³	1	3	yes	—
74	C₂×C₄⋊C₄×C₇₁	C₂³	1	3	yes	—
75	C₂×C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
76	C₂×C₄⋊C₄×C₇₁	C₂³	1	6	yes	—
77	C₂×C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₇₁	C₂³	1	3	yes	—
79	C₂×C₄⋊C₄×C₇₁	C₂³	1	3	yes	—
80	C₂×C₂²⋊C₄×C₇₁	C₂³	1	6	yes	—
81	C₂×C₂²⋊C₄×C₇₁	C₂³	1	6	yes	—
82	C₂×C₂²⋊C₄×C₇₁	C₂³	1	6	yes	—
83	C₂×C₂²⋊C₄×C₇₁	C₂³	1	12	yes	—
84	C₂×C₂²⋊C₄×C₇₁	C₂³	1	6	yes	—
85	C₂×C₂²⋊C₄×C₇₁	C₂³	1	6	yes	—
86	C₂×C₄²×C₇₁	C₂³	1	6	yes	—
87	C₂×C₄²×C₇₁	C₂³	1	3	yes	—
88	C₂×C₄²×C₇₁	C₂³	1	3	yes	—
89	C₂×C₄²×C₇₁	C₂³	1	3	yes	—
90	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
91	C₂×C₄²×C₇₁	C₂³	1	3	yes	—
92	C₂×C₄²×C₇₁	C₂³	1	3	yes	—
93	C₂×C₄²×C₇₁	C₂³	1	3	yes	—
94	#95×C₇₁	C₂³	1	2	yes	—
95	#95×C₇₁	C₂³	1	6	yes	—
96	#95×C₇₁	C₂³	1	6	yes	—
97	#95×C₇₁	C₂³	1	6	yes	—
98	C₂⁴×C₇₁	C₂⁴	1	6	yes	—
99	#41×C₇₁	—	2	64	no	—
100	C₂×Q₈×C₇₁	C₂²×C₄	1	4	yes	—
101	C₂×Q₈×C₇₁	C₂×D₄	1	4	yes	—
102	C₂×Q₈×C₇₁	#41	1	4	yes	—
103	C₂×D₄×C₇₁	—	2	12	no	—
104	C₂×D₄×C₇₁	—	2	12	no	—
105	C₂×D₄×C₇₁	—	2	6	no	—
106	C₂×D₄×C₇₁	—	2	6	no	—
107	C₂×D₄×C₇₁	—	4	6	no	—
108	C₂×D₄×C₇₁	—	4	6	no	—
109	C₂×D₄×C₇₁	C₂²×C₄	1	12	yes	—
110	C₂×D₄×C₇₁	C₂²×C₄	1	24	yes	—
111	C₂×D₄×C₇₁	C₂²×C₄	1	24	yes	—
112	C₂²×C₄×C₇₁	—	2	12	no	—
113	C₂²×C₄×C₇₁	—	2	12	no	—
114	C₂²×C₄×C₇₁	—	2	24	no	—
115	C₂²×C₄×C₇₁	—	4	12	no	—
116	C₂²×C₄×C₇₁	—	4	12	no	—
117	C₂²×C₄×C₇₁	C₂²×C₄	1	24	yes	—
118	C₂²×C₄×C₇₁	C₂⁴	1	2	yes	—
119	C₂²×C₄×C₇₁	C₂⁴	1	3	yes	—
120	C₂²×C₄×C₇₁	C₂⁴	1	6	yes	—
121	C₂²×C₄×C₇₁	C₂⁴	1	3	yes	—
122	C₂²×C₄×C₇₁	C₂⁴	1	2	yes	—
123	C₂²×C₄×C₇₁	C₂⁴	1	3	yes	—
124	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
125	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
126	C₂²×C₄×C₇₁	C₂⁴	1	3	yes	—
127	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
128	C₄⋊C₄×C₇₁	—	2	12	no	—
129	C₄⋊C₄×C₇₁	—	2	12	no	—
130	C₄⋊C₄×C₇₁	—	2	4	no	—
131	C₄⋊C₄×C₇₁	—	2	6	no	—
132	C₄⋊C₄×C₇₁	—	2	6	no	—
133	C₄⋊C₄×C₇₁	—	2	2	no	—
134	C₄⋊C₄×C₇₁	—	4	6	no	—
135	C₄⋊C₄×C₇₁	C₂²×C₄	1	4	yes	—
136	C₄⋊C₄×C₇₁	C₂²×C₄	1	12	yes	—
137	C₄⋊C₄×C₇₁	C₂×D₄	1	12	yes	—
138	C₂²⋊C₄×C₇₁	—	2	12	no	—
139	C₂²⋊C₄×C₇₁	—	2	12	no	—
140	C₂²⋊C₄×C₇₁	—	2	12	no	—
141	C₂²⋊C₄×C₇₁	—	4	6	no	—
142	C₂²⋊C₄×C₇₁	—	4	12	no	—
143	C₂²⋊C₄×C₇₁	C₂²×C₄	1	24	yes	—
144	C₄²×C₇₁	—	2	12	no	—
145	C₄²×C₇₁	—	2	6	no	—
146	C₄²×C₇₁	—	2	6	no	—
147	C₄²×C₇₁	—	2	6	no	—
148	C₄²×C₇₁	—	2	2	no	—
149	C₄²×C₇₁	—	4	3	no	—
150	C₄²×C₇₁	C₂²×C₄	1	12	yes	—
151	C₄²×C₇₁	C₂×D₄	1	12	yes	—
152	C₂³×C₇₁	—	2	6	no	—
153	C₂³×C₇₁	—	2	6	no	—
154	C₂³×C₇₁	—	2	6	no	—
155	C₂³×C₇₁	—	2	6	no	—
156	C₂³×C₇₁	—	2	6	no	—
157	C₂³×C₇₁	—	2	6	no	—
158	C₂³×C₇₁	C₂³×C₄	1	12	yes	—
159	C₂³×C₇₁	C₂⁵	1	1	yes	characteristic
160	Q₈×C₇₁	C₄×D₄	1	16	yes	complemented
161	D₄×C₇₁	—	2	48	no	—
162	D₄×C₇₁	—	2	24	no	—
163	D₄×C₇₁	—	4	24	no	—
164	C₂×C₄×C₇₁	—	2	2	no	—
165	C₂×C₄×C₇₁	—	2	1	no	—
166	C₂×C₄×C₇₁	—	2	2	no	—
167	C₂×C₄×C₇₁	—	2	3	no	—
168	C₂×C₄×C₇₁	—	2	3	no	—
169	C₂×C₄×C₇₁	—	2	3	no	—
170	C₂×C₄×C₇₁	—	2	1	no	—
171	C₂×C₄×C₇₁	—	2	1	no	—
172	C₂×C₄×C₇₁	—	2	1	no	—
173	C₂×C₄×C₇₁	—	2	3	no	—
174	C₂×C₄×C₇₁	—	2	48	no	—
175	C₂×C₄×C₇₁	—	2	2	no	—
176	C₂×C₄×C₇₁	—	2	6	no	—
177	C₂×C₄×C₇₁	—	2	3	no	—
178	C₂×C₄×C₇₁	—	2	3	no	—
179	C₂×C₄×C₇₁	—	2	6	no	—
180	C₂×C₄×C₇₁	—	2	6	no	—
181	C₂×C₄×C₇₁	—	2	2	no	—
182	C₂×C₄×C₇₁	—	2	3	no	—
183	C₂×C₄×C₇₁	—	2	2	no	—
184	C₂×C₄×C₇₁	—	8	24	no	—
185	C₂×C₄×C₇₁	C₂³×C₄	1	6	yes	—
186	C₂×C₄×C₇₁	C₂³×C₄	1	4	yes	—
187	C₂×C₄×C₇₁	C₂³×C₄	1	6	yes	—
188	C₂×C₄×C₇₁	C₂³×C₄	1	2	yes	—
189	C₂×C₄×C₇₁	C₂²×D₄	1	2	yes	—
190	C₂×C₄×C₇₁	C₂²×D₄	1	6	yes	—
191	C₂×C₄×C₇₁	C₂²×D₄	1	6	yes	—
192	C₂×C₄×C₇₁	C₂×#41	1	6	yes	—
193	C₂×C₄×C₇₁	H(5,𝔽₂)	1	2	yes	—
194	C₂×C₄×C₇₁	H(5,𝔽₂)	1	2	yes	—
195	C₂×C₄×C₇₁	2₋¹⁺⁴	1	2	yes	—
196	C₂²×C₇₁	—	2	12	no	—
197	C₂²×C₇₁	—	2	12	no	—
198	C₂²×C₇₁	—	2	12	no	—
199	C₂²×C₇₁	—	4	6	no	—
200	C₂²×C₇₁	—	4	6	no	—
201	C₂²×C₇₁	—	4	6	no	—
202	C₂²×C₇₁	—	4	6	no	—
203	C₂²×C₇₁	C₂⁴×C₄	1	1	yes	characteristic
204	C₂²×C₇₁	C₂³×D₄	1	1	yes	characteristic
205	C₂²×C₇₁	C₂²×#41	1	1	yes	characteristic
206	C₂²×C₇₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
207	C₂²×C₇₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
208	C₂²×C₇₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
209	C₂²×C₇₁	#585	1	1	yes	characteristic
210	C₄×C₇₁	—	2	4	no	—
211	C₄×C₇₁	—	2	6	no	—
212	C₄×C₇₁	—	2	2	no	—
213	C₄×C₇₁	—	2	6	no	—
214	C₄×C₇₁	—	2	2	no	—
215	C₄×C₇₁	—	4	2	no	—
216	C₄×C₇₁	—	4	6	no	—
217	C₄×C₇₁	—	4	3	no	—
218	C₄×C₇₁	C₂×C₄×D₄	1	12	yes	—
219	C₄×C₇₁	#520	1	4	yes	complemented
220	#34313	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
221	C₂×C₇₁	—	4	12	no	—
222	C₂×C₇₁	C₂²×C₄×D₄	1	1	yes	characteristic
223	C₂×C₇₁	C₂×#520	1	1	yes	characteristic
224	C₂×C₇₁	C₄×H(5,𝔽₂)	1	1	yes	characteristic
225	C₂×C₇₁	#3497	1	1	yes	characteristic
226	C₂×C₇₁	#3498	1	1	yes	characteristic
227	C₂×C₇₁	#3517	1	1	yes	characteristic
228	C₂×C₇₁	#3522	1	1	yes	characteristic
229	C₂×#550	C₂×C₇₁	1	1	yes	characteristic
230	C₂×#520	C₂×C₇₁	1	2	yes	complemented
231	#2366	C₂×C₇₁	1	3	yes	complemented
232	#2361	C₂×C₇₁	1	6	yes	complemented
233	Q₈×C₄⋊C₄	C₂×C₇₁	1	1	yes	characteristic, complemented
234	D₄×C₄⋊C₄	C₂×C₇₁	1	3	yes	complemented
235	#2341	C₂×C₇₁	1	2	yes	complemented
236	#2324	C₂×C₇₁	1	6	yes	complemented
237	C₄×#127	C₂×C₇₁	1	3	yes	complemented
238	#2293	C₂×C₇₁	1	3	yes	complemented
239	C₄²×Q₈	C₂×C₇₁	1	1	yes	characteristic, complemented
240	C₇₁	#34313	1	1	yes	characteristic, direct product factor, Sylow
241	C₂²×#41	C₂²×C₇₁	1	2	yes	—
242	#550	C₄×C₇₁	1	16	yes	complemented
243	C₂×#127	C₂²×C₇₁	1	3	yes	—
244	C₂×#121	C₂²×C₇₁	1	6	yes	—
245	#520	—	2	16	no	—
246	C₂×C₄×Q₈	C₂²×C₇₁	1	1	yes	characteristic
247	C₂×C₄×Q₈	C₂²×C₇₁	1	2	yes	—
248	C₂×C₄×Q₈	C₂²×C₇₁	1	1	yes	characteristic
249	C₂×C₄×Q₈	C₂²×C₇₁	1	1	yes	characteristic
250	C₂×C₄×D₄	C₂²×C₇₁	1	6	yes	—
251	C₂×C₄×D₄	C₂²×C₇₁	1	6	yes	—
252	C₂×C₄×D₄	C₂²×C₇₁	1	6	yes	—
253	C₂×C₄×D₄	C₂²×C₇₁	1	6	yes	—
254	C₂×C₄×D₄	C₂²×C₇₁	1	3	yes	—
255	C₂×#117	C₂²×C₇₁	1	6	yes	—
256	C₂²×C₄⋊C₄	C₂²×C₇₁	1	6	yes	—
257	#390	C₂²×C₇₁	1	3	yes	—
258	#390	C₂²×C₇₁	1	6	yes	—
259	#390	C₂²×C₇₁	1	6	yes	—
260	#390	C₂²×C₇₁	1	6	yes	—
261	#390	C₂²×C₇₁	1	3	yes	—
262	#389	C₂²×C₇₁	1	6	yes	—
263	#389	C₂²×C₇₁	1	3	yes	—
264	#389	C₂²×C₇₁	1	1	yes	characteristic
265	#388	C₂²×C₇₁	1	6	yes	—
266	#388	C₂²×C₇₁	1	6	yes	—
267	#386	C₂²×C₇₁	1	12	yes	—
268	#385	C₂²×C₇₁	1	12	yes	—
269	#384	C₂²×C₇₁	1	3	yes	—
270	#380	C₂²×C₇₁	1	6	yes	—
271	C₄×C₄⋊C₄	C₂²×C₇₁	1	3	yes	—
272	C₄×C₄⋊C₄	C₂²×C₇₁	1	3	yes	—
273	C₄×C₄⋊C₄	C₂²×C₇₁	1	3	yes	—
274	C₄×C₄⋊C₄	C₂²×C₇₁	1	3	yes	—
275	C₄×C₂²⋊C₄	C₂²×C₇₁	1	6	yes	—
276	C₄³	C₂²×C₇₁	1	3	yes	—
277	C₂×#41	—	2	8	no	—
278	C₂×#41	—	2	8	no	—
279	C₂×#41	C₂×C₄×C₇₁	1	16	yes	complemented
280	C₂²×Q₈	C₂³×C₇₁	1	1	yes	characteristic
281	C₂²×D₄	C₂³×C₇₁	1	3	yes	—
282	C₂²×D₄	C₂³×C₇₁	1	6	yes	—
283	C₂²×D₄	C₂³×C₇₁	1	6	yes	—
284	C₂³×C₄	C₂³×C₇₁	1	6	yes	—
285	C₂³×C₄	C₂³×C₇₁	1	12	yes	—
286	#127	C₂×C₄×C₇₁	1	24	yes	complemented
287	#121	C₂×C₄×C₇₁	1	48	yes	complemented
288	C₄×Q₈	—	2	8	no	—
289	C₄×Q₈	—	2	4	no	—
290	C₄×Q₈	C₂×C₄×C₇₁	1	8	yes	complemented
291	C₄×Q₈	D₄×C₇₁	1	8	yes	complemented
292	C₄×D₄	—	2	24	no	—
293	C₄×D₄	—	2	24	no	—
294	C₄×D₄	—	4	12	no	—
295	C₄×D₄	—	4	12	no	—
296	C₄×D₄	C₂×C₄×C₇₁	1	24	yes	complemented
297	#117	—	2	24	no	—
298	C₂×C₄⋊C₄	—	2	24	no	—
299	C₂×C₄⋊C₄	C₂³×C₇₁	1	6	yes	—
300	C₂×C₄⋊C₄	C₂³×C₇₁	1	6	yes	—
301	C₂×C₄⋊C₄	C₂³×C₇₁	1	2	yes	—
302	C₂×C₄⋊C₄	C₂³×C₇₁	1	3	yes	—
303	C₂×C₄⋊C₄	C₂³×C₇₁	1	3	yes	—
304	C₂×C₄⋊C₄	C₂³×C₇₁	1	1	yes	characteristic
305	C₂×C₄⋊C₄	C₂³×C₇₁	1	6	yes	—
306	C₂×C₄⋊C₄	C₂³×C₇₁	1	1	yes	characteristic
307	C₂×C₄⋊C₄	C₂³×C₇₁	1	3	yes	—
308	C₂×C₄⋊C₄	C₂³×C₇₁	1	3	yes	—
309	C₂×C₂²⋊C₄	C₂³×C₇₁	1	6	yes	—
310	C₂×C₂²⋊C₄	C₂³×C₇₁	1	6	yes	—
311	C₂×C₂²⋊C₄	C₂³×C₇₁	1	6	yes	—
312	C₂×C₂²⋊C₄	C₂³×C₇₁	1	12	yes	—
313	C₂×C₂²⋊C₄	C₂³×C₇₁	1	6	yes	—
314	C₂×C₂²⋊C₄	C₂³×C₇₁	1	6	yes	—
315	C₂×C₄²	C₂³×C₇₁	1	6	yes	—
316	C₂×C₄²	C₂³×C₇₁	1	3	yes	—
317	C₂×C₄²	C₂³×C₇₁	1	3	yes	—
318	C₂×C₄²	C₂³×C₇₁	1	3	yes	—
319	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
320	C₂×C₄²	C₂³×C₇₁	1	3	yes	—
321	C₂×C₄²	C₂³×C₇₁	1	3	yes	—
322	C₂×C₄²	C₂³×C₇₁	1	3	yes	—
323	#95	C₂³×C₇₁	1	2	yes	—
324	#95	C₂³×C₇₁	1	6	yes	—
325	#95	C₂³×C₇₁	1	6	yes	—
326	#95	C₂³×C₇₁	1	6	yes	—
327	C₂⁴	C₂⁴×C₇₁	1	6	yes	—
328	#41	—	2	64	no	—
329	C₂×Q₈	C₂²×C₄×C₇₁	1	4	yes	—
330	C₂×Q₈	C₂×D₄×C₇₁	1	4	yes	—
331	C₂×Q₈	#41×C₇₁	1	4	yes	—
332	C₂×D₄	—	2	12	no	—
333	C₂×D₄	—	2	12	no	—
334	C₂×D₄	—	2	6	no	—
335	C₂×D₄	—	2	6	no	—
336	C₂×D₄	—	4	6	no	—
337	C₂×D₄	—	4	6	no	—
338	C₂×D₄	C₂²×C₄×C₇₁	1	12	yes	—
339	C₂×D₄	C₂²×C₄×C₇₁	1	24	yes	—
340	C₂×D₄	C₂²×C₄×C₇₁	1	24	yes	—
341	C₂²×C₄	—	2	12	no	—
342	C₂²×C₄	—	2	12	no	—
343	C₂²×C₄	—	2	24	no	—
344	C₂²×C₄	—	4	12	no	—
345	C₂²×C₄	—	4	12	no	—
346	C₂²×C₄	C₂²×C₄×C₇₁	1	24	yes	—
347	C₂²×C₄	C₂⁴×C₇₁	1	2	yes	—
348	C₂²×C₄	C₂⁴×C₇₁	1	3	yes	—
349	C₂²×C₄	C₂⁴×C₇₁	1	6	yes	—
350	C₂²×C₄	C₂⁴×C₇₁	1	3	yes	—
351	C₂²×C₄	C₂⁴×C₇₁	1	2	yes	—
352	C₂²×C₄	C₂⁴×C₇₁	1	3	yes	—
353	C₂²×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
354	C₂²×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
355	C₂²×C₄	C₂⁴×C₇₁	1	3	yes	—
356	C₂²×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
357	C₄⋊C₄	—	2	12	no	—
358	C₄⋊C₄	—	2	12	no	—
359	C₄⋊C₄	—	2	4	no	—
360	C₄⋊C₄	—	2	6	no	—
361	C₄⋊C₄	—	2	6	no	—
362	C₄⋊C₄	—	2	2	no	—
363	C₄⋊C₄	—	4	6	no	—
364	C₄⋊C₄	C₂²×C₄×C₇₁	1	4	yes	—
365	C₄⋊C₄	C₂²×C₄×C₇₁	1	12	yes	—
366	C₄⋊C₄	C₂×D₄×C₇₁	1	12	yes	—
367	C₂²⋊C₄	—	2	12	no	—
368	C₂²⋊C₄	—	2	12	no	—
369	C₂²⋊C₄	—	2	12	no	—
370	C₂²⋊C₄	—	4	6	no	—
371	C₂²⋊C₄	—	4	12	no	—
372	C₂²⋊C₄	C₂²×C₄×C₇₁	1	24	yes	—
373	C₄²	—	2	12	no	—
374	C₄²	—	2	6	no	—
375	C₄²	—	2	6	no	—
376	C₄²	—	2	6	no	—
377	C₄²	—	2	2	no	—
378	C₄²	—	4	3	no	—
379	C₄²	C₂²×C₄×C₇₁	1	12	yes	—
380	C₄²	C₂×D₄×C₇₁	1	12	yes	—
381	C₂³	—	2	6	no	—
382	C₂³	—	2	6	no	—
383	C₂³	—	2	6	no	—
384	C₂³	—	2	6	no	—
385	C₂³	—	2	6	no	—
386	C₂³	—	2	6	no	—
387	C₂³	C₂³×C₄×C₇₁	1	12	yes	—
388	C₂³	C₂⁵×C₇₁	1	1	yes	characteristic
389	Q₈	C₄×D₄×C₇₁	1	16	yes	complemented
390	D₄	—	2	48	no	—
391	D₄	—	2	24	no	—
392	D₄	—	4	24	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	1	no	—
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	2	3	no	—
397	C₂×C₄	—	2	3	no	—
398	C₂×C₄	—	2	3	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	2	1	no	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	2	3	no	—
403	C₂×C₄	—	2	48	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	6	no	—
406	C₂×C₄	—	2	3	no	—
407	C₂×C₄	—	2	3	no	—
408	C₂×C₄	—	2	6	no	—
409	C₂×C₄	—	2	6	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	3	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	8	24	no	—
414	C₂×C₄	C₂³×C₄×C₇₁	1	6	yes	—
415	C₂×C₄	C₂³×C₄×C₇₁	1	4	yes	—
416	C₂×C₄	C₂³×C₄×C₇₁	1	6	yes	—
417	C₂×C₄	C₂³×C₄×C₇₁	1	2	yes	—
418	C₂×C₄	C₂²×D₄×C₇₁	1	2	yes	—
419	C₂×C₄	C₂²×D₄×C₇₁	1	6	yes	—
420	C₂×C₄	C₂²×D₄×C₇₁	1	6	yes	—
421	C₂×C₄	C₂×#41×C₇₁	1	6	yes	—
422	C₂×C₄	H(5,𝔽₂)×C₇₁	1	2	yes	—
423	C₂×C₄	H(5,𝔽₂)×C₇₁	1	2	yes	—
424	C₂×C₄	2₋¹⁺⁴×C₇₁	1	2	yes	—
425	C₂²	—	2	12	no	—
426	C₂²	—	2	12	no	—
427	C₂²	—	2	12	no	—
428	C₂²	—	4	6	no	—
429	C₂²	—	4	6	no	—
430	C₂²	—	4	6	no	—
431	C₂²	—	4	6	no	—
432	C₂²	C₂⁴×C₄×C₇₁	1	1	yes	characteristic
433	C₂²	C₂³×D₄×C₇₁	1	1	yes	characteristic
434	C₂²	C₂²×#41×C₇₁	1	1	yes	characteristic
435	C₂²	C₂×H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
436	C₂²	C₂×H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
437	C₂²	C₂×2₋¹⁺⁴×C₇₁	1	1	yes	characteristic
438	C₂²	#585×C₇₁	1	1	yes	characteristic
439	C₄	—	2	4	no	—
440	C₄	—	2	6	no	—
441	C₄	—	2	2	no	—
442	C₄	—	2	6	no	—
443	C₄	—	2	2	no	—
444	C₄	—	4	2	no	—
445	C₄	—	4	6	no	—
446	C₄	—	4	3	no	—
447	C₄	C₂×C₄×D₄×C₇₁	1	12	yes	—
448	C₄	#520×C₇₁	1	4	yes	complemented
449	C₂	—	4	12	no	—
450	C₂	C₂²×C₄×D₄×C₇₁	1	1	yes	characteristic
451	C₂	C₂×#520×C₇₁	1	1	yes	characteristic
452	C₂	C₄×H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
453	C₂	C₇₁×#3497	1	1	yes	characteristic
454	C₂	C₇₁×#3498	1	1	yes	characteristic
455	C₂	C₇₁×#3517	1	1	yes	characteristic
456	C₂	C₇₁×#3522	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₇₁	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₇₁
Frattini, cosocle	C₂³	C₂⁵×C₇₁
prime radical	C₂×#550×C₇₁	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#520×C₇₁	C₂	semidirect
3. C₇₁×#2366	C₂	semidirect
4. C₇₁×#2361	C₂	semidirect
5. Q₈×C₄⋊C₄×C₇₁	C₂	semidirect
6. D₄×C₄⋊C₄×C₇₁	C₂	semidirect
7. C₇₁×#2341	C₂	semidirect
8. C₇₁×#2324	C₂	semidirect
9. C₄×#127×C₇₁	C₂	semidirect
10. C₇₁×#2293	C₂	semidirect
11. C₄²×Q₈×C₇₁	C₂	semidirect
13. #550×C₇₁	C₄	semidirect
50. C₂×#41×C₇₁	C₂×C₄	semidirect
57. #127×C₇₁	C₂×C₄	semidirect
58. #121×C₇₁	C₂×C₄	semidirect
61. C₄×Q₈×C₇₁	C₂×C₄	semidirect
62. C₄×Q₈×C₇₁	D₄	semidirect
67. C₄×D₄×C₇₁	C₂×C₄	semidirect
160. Q₈×C₇₁	C₄×D₄	semidirect
219. C₄×C₇₁	#520	semidirect
220. #34313	C₇₁	direct (N × H)
230. C₂×#520	C₂×C₇₁	semidirect
231. #2366	C₂×C₇₁	semidirect
232. #2361	C₂×C₇₁	semidirect
233. Q₈×C₄⋊C₄	C₂×C₇₁	semidirect
234. D₄×C₄⋊C₄	C₂×C₇₁	semidirect
235. #2341	C₂×C₇₁	semidirect
236. #2324	C₂×C₇₁	semidirect
237. C₄×#127	C₂×C₇₁	semidirect
238. #2293	C₂×C₇₁	semidirect
239. C₄²×Q₈	C₂×C₇₁	semidirect
240. C₇₁	#34313	direct (N × H)
242. #550	C₄×C₇₁	semidirect
279. C₂×#41	C₂×C₄×C₇₁	semidirect
286. #127	C₂×C₄×C₇₁	semidirect
287. #121	C₂×C₄×C₇₁	semidirect
290. C₄×Q₈	C₂×C₄×C₇₁	semidirect
291. C₄×Q₈	D₄×C₇₁	semidirect
296. C₄×D₄	C₂×C₄×C₇₁	semidirect
389. Q₈	C₄×D₄×C₇₁	semidirect
448. C₄	#520×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#550×C₇₁	2
C₂×#520×C₇₁	2
C₇₁×#2366	2
C₇₁×#2361	2
Q₈×C₄⋊C₄×C₇₁	2
D₄×C₄⋊C₄×C₇₁	2
C₇₁×#2341	2
C₇₁×#2324	2
C₄×#127×C₇₁	2
C₇₁×#2293	2
C₄²×Q₈×C₇₁	2
#34313	71

Subgroups — C71×#34313 #31855185

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#34313 #31855185