Subgroups — C₇₁×#8579 #31829451

Order 18176.

540 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#124×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₇₁×#1767	C₂	1	2	yes	maximal, complemented
3	C₂×#329×C₇₁	C₂	1	2	yes	maximal
4	C₇₁×#1295	C₂	1	2	yes	maximal, complemented
5	C₂×#124×C₇₁	C₂²	1	1	yes	characteristic
6	C₂×C₄×Q₈×C₇₁	C₄	1	1	yes	characteristic
7	C₂×C₄×D₄×C₇₁	C₄	1	1	yes	characteristic
8	C₂×C₄⋊C₈×C₇₁	—	2	2	no	—
9	C₂×#97×C₇₁	C₂²	1	4	yes	—
10	#391×C₇₁	C₄	1	1	yes	characteristic
11	#390×C₇₁	C₄	1	1	yes	characteristic
12	#388×C₇₁	—	2	2	no	—
13	#382×C₇₁	C₂²	1	1	yes	characteristic
14	C₄×C₂²⋊C₄×C₇₁	—	2	2	no	—
15	C₄³×C₇₁	C₂²	1	1	yes	characteristic
16	#336×C₇₁	—	2	2	no	—
17	#329×C₇₁	—	2	4	no	—
18	C₂²×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×D₄×C₇₁	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₇₁	—	2	1	no	—
21	C₂²×C₈×C₇₁	—	2	4	no	—
22	#124×C₇₁	—	2	2	no	—
23	#124×C₇₁	D₄	1	2	yes	—
24	#124×C₇₁	Q₈	1	2	yes	—
25	C₄×Q₈×C₇₁	—	4	2	no	—
26	C₄×D₄×C₇₁	—	4	2	no	—
27	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
28	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₂²⋊C₄×C₇₁	—	2	2	no	—
31	C₂×C₂²⋊C₄×C₇₁	—	2	1	no	—
32	C₂×C₂²⋊C₄×C₇₁	—	2	2	no	—
33	C₂×C₄²×C₇₁	—	2	1	no	—
34	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₇₁	D₄	1	2	yes	—
37	C₂×C₄²×C₇₁	D₄	1	2	yes	—
38	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₇₁	—	2	8	no	—
40	#97×C₇₁	—	2	4	no	—
41	#97×C₇₁	—	2	4	no	—
42	#95×C₇₁	—	2	2	no	—
43	#95×C₇₁	—	2	2	no	—
44	C₂⁴×C₇₁	—	2	1	no	—
45	C₂×Q₈×C₇₁	—	2	1	no	—
46	C₂×Q₈×C₇₁	—	4	1	no	—
47	C₂×Q₈×C₇₁	—	4	1	no	—
48	C₂×Q₈×C₇₁	C₄⋊C₄	1	2	yes	—
49	C₂×D₄×C₇₁	—	2	1	no	—
50	C₂×D₄×C₇₁	—	4	1	no	—
51	C₂×D₄×C₇₁	—	4	1	no	—
52	C₂×D₄×C₇₁	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₇₁	—	2	1	no	—
54	C₂²×C₄×C₇₁	—	2	1	no	—
55	C₂²×C₄×C₇₁	—	2	1	no	—
56	C₂²×C₄×C₇₁	—	4	2	no	—
57	C₂²×C₄×C₇₁	—	4	4	no	—
58	C₂²×C₄×C₇₁	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
63	C₂×C₈×C₇₁	—	2	8	no	—
64	C₂×C₈×C₇₁	—	4	4	no	—
65	C₂×C₈×C₇₁	—	4	4	no	—
66	C₄⋊C₄×C₇₁	—	4	1	no	—
67	C₄⋊C₄×C₇₁	—	4	1	no	—
68	C₄⋊C₄×C₇₁	—	4	2	no	—
69	C₂²⋊C₄×C₇₁	—	2	4	no	—
70	C₂²⋊C₄×C₇₁	—	2	4	no	—
71	C₂²⋊C₄×C₇₁	—	4	4	no	—
72	C₂²⋊C₄×C₇₁	—	4	2	no	—
73	C₄²×C₇₁	—	2	2	no	—
74	C₄²×C₇₁	—	2	4	no	—
75	C₄²×C₇₁	—	4	1	no	—
76	C₄²×C₇₁	—	4	2	no	—
77	C₄²×C₇₁	—	4	2	no	—
78	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
80	C₄²×C₇₁	#41	1	2	yes	—
81	C₂³×C₇₁	—	2	2	no	—
82	C₂³×C₇₁	—	4	1	no	—
83	C₂³×C₇₁	—	4	1	no	—
84	C₂³×C₇₁	—	4	2	no	—
85	C₂³×C₇₁	—	4	2	no	—
86	C₂³×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	Q₈×C₇₁	—	4	2	no	—
88	Q₈×C₇₁	—	4	2	no	—
89	D₄×C₇₁	—	4	2	no	—
90	D₄×C₇₁	—	4	2	no	—
91	C₂×C₄×C₇₁	—	2	1	no	—
92	C₂×C₄×C₇₁	—	2	1	no	—
93	C₂×C₄×C₇₁	—	2	2	no	—
94	C₂×C₄×C₇₁	—	2	2	no	—
95	C₂×C₄×C₇₁	—	2	2	no	—
96	C₂×C₄×C₇₁	—	2	1	no	—
97	C₂×C₄×C₇₁	—	2	2	no	—
98	C₂×C₄×C₇₁	—	2	2	no	—
99	C₂×C₄×C₇₁	—	2	2	no	—
100	C₂×C₄×C₇₁	—	2	2	no	—
101	C₂×C₄×C₇₁	—	4	1	no	—
102	C₂×C₄×C₇₁	—	4	1	no	—
103	C₂×C₄×C₇₁	—	4	1	no	—
104	C₂×C₄×C₇₁	—	4	1	no	—
105	C₂×C₄×C₇₁	—	4	2	no	—
106	C₂×C₄×C₇₁	—	4	2	no	—
107	C₂×C₄×C₇₁	—	8	4	no	—
108	C₂×C₄×C₇₁	#100	1	2	yes	—
109	C₂×C₄×C₇₁	#104	1	4	yes	—
110	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
112	C₂×C₄×C₇₁	#121	1	2	yes	—
113	C₂×C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
114	C₈×C₇₁	—	4	8	no	—
115	C₂²×C₇₁	—	4	2	no	—
116	C₂²×C₇₁	—	4	4	no	—
117	C₂²×C₇₁	—	8	1	no	—
118	C₂²×C₇₁	—	8	1	no	—
119	C₂²×C₇₁	—	8	2	no	—
120	C₂²×C₇₁	#380	1	1	yes	characteristic
121	C₂²×C₇₁	C₂×#100	1	1	yes	characteristic
122	C₂²×C₇₁	#413	1	1	yes	characteristic
123	C₂²×C₇₁	C₂×#104	1	2	yes	—
124	C₂²×C₇₁	#421	1	2	yes	—
125	C₄×C₇₁	—	2	1	no	—
126	C₄×C₇₁	—	2	1	no	—
127	C₄×C₇₁	—	2	2	no	—
128	C₄×C₇₁	—	2	2	no	—
129	C₄×C₇₁	—	4	1	no	—
130	C₄×C₇₁	—	4	2	no	—
131	C₄×C₇₁	—	4	2	no	—
132	C₄×C₇₁	—	8	1	no	—
133	C₄×C₇₁	—	8	1	no	—
134	C₄×C₇₁	#329	1	4	yes	complemented
135	#8579	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₂×C₇₁	—	8	2	no	—
137	C₂×C₇₁	C₂×#329	1	1	yes	characteristic
138	C₂×C₇₁	#1529	1	2	yes	—
139	C₂×C₇₁	#1536	1	1	yes	characteristic
140	C₂×C₇₁	#1800	1	1	yes	characteristic
141	C₂×C₇₁	#1806	1	2	yes	—
142	C₄×#124	C₂×C₇₁	1	1	yes	characteristic
143	#1767	C₂×C₇₁	1	2	yes	complemented
144	C₂×#329	C₂×C₇₁	1	2	yes	—
145	#1295	C₂×C₇₁	1	2	yes	complemented
146	C₇₁	#8579	1	1	yes	characteristic, direct product factor, Sylow
147	C₂×#124	C₂²×C₇₁	1	1	yes	characteristic
148	C₂×C₄×Q₈	C₄×C₇₁	1	1	yes	characteristic
149	C₂×C₄×D₄	C₄×C₇₁	1	1	yes	characteristic
150	C₂×C₄⋊C₈	—	2	2	no	—
151	C₂×#97	C₂²×C₇₁	1	4	yes	—
152	#391	C₄×C₇₁	1	1	yes	characteristic
153	#390	C₄×C₇₁	1	1	yes	characteristic
154	#388	—	2	2	no	—
155	#382	C₂²×C₇₁	1	1	yes	characteristic
156	C₄×C₂²⋊C₄	—	2	2	no	—
157	C₄³	C₂²×C₇₁	1	1	yes	characteristic
158	#336	—	2	2	no	—
159	#329	—	2	4	no	—
160	C₂²×Q₈	C₂×C₄×C₇₁	1	1	yes	characteristic
161	C₂²×D₄	C₂×C₄×C₇₁	1	1	yes	characteristic
162	C₂³×C₄	—	2	1	no	—
163	C₂²×C₈	—	2	4	no	—
164	#124	—	2	2	no	—
165	#124	D₄×C₇₁	1	2	yes	—
166	#124	Q₈×C₇₁	1	2	yes	—
167	C₄×Q₈	—	4	2	no	—
168	C₄×D₄	—	4	2	no	—
169	C₂×C₄⋊C₄	—	2	1	no	—
170	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	1	yes	characteristic
171	C₂×C₄⋊C₄	C₂×C₄×C₇₁	1	1	yes	characteristic
172	C₂×C₂²⋊C₄	—	2	2	no	—
173	C₂×C₂²⋊C₄	—	2	1	no	—
174	C₂×C₂²⋊C₄	—	2	2	no	—
175	C₂×C₄²	—	2	1	no	—
176	C₂×C₄²	C₂×C₄×C₇₁	1	1	yes	characteristic
177	C₂×C₄²	C₂×C₄×C₇₁	1	1	yes	characteristic
178	C₂×C₄²	D₄×C₇₁	1	2	yes	—
179	C₂×C₄²	D₄×C₇₁	1	2	yes	—
180	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
181	C₄⋊C₈	—	2	8	no	—
182	#97	—	2	4	no	—
183	#97	—	2	4	no	—
184	#95	—	2	2	no	—
185	#95	—	2	2	no	—
186	C₂⁴	—	2	1	no	—
187	C₂×Q₈	—	2	1	no	—
188	C₂×Q₈	—	4	1	no	—
189	C₂×Q₈	—	4	1	no	—
190	C₂×Q₈	C₄⋊C₄×C₇₁	1	2	yes	—
191	C₂×D₄	—	2	1	no	—
192	C₂×D₄	—	4	1	no	—
193	C₂×D₄	—	4	1	no	—
194	C₂×D₄	C₄⋊C₄×C₇₁	1	2	yes	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	—	4	4	no	—
200	C₂²×C₄	C₂²⋊C₄×C₇₁	1	2	yes	—
201	C₂²×C₄	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
202	C₂²×C₄	C₂²⋊C₄×C₇₁	1	1	yes	characteristic
203	C₂²×C₄	C₂²×C₄×C₇₁	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
205	C₂×C₈	—	2	8	no	—
206	C₂×C₈	—	4	4	no	—
207	C₂×C₈	—	4	4	no	—
208	C₄⋊C₄	—	4	1	no	—
209	C₄⋊C₄	—	4	1	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₂²⋊C₄	—	2	4	no	—
212	C₂²⋊C₄	—	2	4	no	—
213	C₂²⋊C₄	—	4	4	no	—
214	C₂²⋊C₄	—	4	2	no	—
215	C₄²	—	2	2	no	—
216	C₄²	—	2	4	no	—
217	C₄²	—	4	1	no	—
218	C₄²	—	4	2	no	—
219	C₄²	—	4	2	no	—
220	C₄²	C₂×D₄×C₇₁	1	1	yes	characteristic
221	C₄²	C₂×Q₈×C₇₁	1	1	yes	characteristic
222	C₄²	#41×C₇₁	1	2	yes	—
223	C₂³	—	2	2	no	—
224	C₂³	—	4	1	no	—
225	C₂³	—	4	1	no	—
226	C₂³	—	4	2	no	—
227	C₂³	—	4	2	no	—
228	C₂³	C₂×C₂²⋊C₄×C₇₁	1	1	yes	characteristic
229	Q₈	—	4	2	no	—
230	Q₈	—	4	2	no	—
231	D₄	—	4	2	no	—
232	D₄	—	4	2	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	8	4	no	—
250	C₂×C₄	#100×C₇₁	1	2	yes	—
251	C₂×C₄	#104×C₇₁	1	4	yes	—
252	C₂×C₄	C₂×C₄⋊C₄×C₇₁	1	1	yes	characteristic
253	C₂×C₄	#117×C₇₁	1	1	yes	characteristic
254	C₂×C₄	#121×C₇₁	1	2	yes	—
255	C₂×C₄	C₂²⋊Q₈×C₇₁	1	2	yes	—
256	C₈	—	4	8	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	4	no	—
259	C₂²	—	8	1	no	—
260	C₂²	—	8	2	no	—
261	C₂²	—	8	1	no	—
262	C₂²	#380×C₇₁	1	1	yes	characteristic
263	C₂²	C₂×#100×C₇₁	1	1	yes	characteristic
264	C₂²	#413×C₇₁	1	1	yes	characteristic
265	C₂²	C₂×#104×C₇₁	1	2	yes	—
266	C₂²	#421×C₇₁	1	2	yes	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	4	2	no	—
272	C₄	—	4	1	no	—
273	C₄	—	4	2	no	—
274	C₄	—	8	1	no	—
275	C₄	—	8	1	no	—
276	C₄	#329×C₇₁	1	4	yes	complemented
277	C₂	—	8	2	no	—
278	C₂	C₂×#329×C₇₁	1	1	yes	characteristic
279	C₂	C₇₁×#1529	1	2	yes	—
280	C₂	C₇₁×#1536	1	1	yes	characteristic
281	C₂	C₇₁×#1800	1	1	yes	characteristic
282	C₂	C₇₁×#1806	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₇₁	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₇₁
Frattini, cosocle	C₂×C₄²	C₂³×C₇₁
prime radical	C₂²×D₄×C₇₁	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇₁ ◁ C₄³×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇₁×#1767	C₂	semidirect
4. C₇₁×#1295	C₂	semidirect
134. C₄×C₇₁	#329	semidirect
135. #8579	C₇₁	direct (N × H)
143. #1767	C₂×C₇₁	semidirect
145. #1295	C₂×C₇₁	semidirect
146. C₇₁	#8579	direct (N × H)
276. C₄	#329×C₇₁	semidirect

Maximal subgroups

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

subgroup	index
C₄×#124×C₇₁	2
C₇₁×#1767	2
C₂×#329×C₇₁	2
C₇₁×#1295	2
#8579	71

Subgroups — C71×#8579 #31829451

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#8579 #31829451