Subgroups — C₇₁×#23753 #31844625

Order 18176.

648 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#430×C₇₁	C₂	1	1	yes	maximal, characteristic
2	C₇₁×#2905	C₂	1	1	yes	maximal, characteristic
3	C₇₁×#2103	C₂	1	8	yes	maximal, complemented
4	C₂×#344×C₇₁	C₂	1	2	yes	maximal
5	C₇₁×#1738	C₂	1	2	yes	maximal
6	C₂×#338×C₇₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×M₁₆×C₇₁	C₂²	1	2	yes	—
8	C₂²×M₁₆×C₇₁	C₂²	1	2	yes	—
9	C₂²×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
11	C₂×#117×C₇₁	C₂²	1	1	yes	characteristic
12	#430×C₇₁	—	2	2	no	—
13	#430×C₇₁	C₂²	1	4	yes	—
14	C₂×#108×C₇₁	—	2	2	no	—
15	#424×C₇₁	—	2	2	no	—
16	#424×C₇₁	C₂²	1	4	yes	—
17	#423×C₇₁	—	2	2	no	—
18	#344×C₇₁	C₂²	1	8	yes	—
19	#341×C₇₁	C₂²	1	8	yes	—
20	#338×C₇₁	C₂²	1	4	yes	—
21	C₂³×C₄×C₇₁	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₇₁	—	2	2	no	—
23	C₂×M₁₆×C₇₁	—	2	2	no	—
24	C₂×M₁₆×C₇₁	—	2	1	no	—
25	C₂×M₁₆×C₇₁	—	2	1	no	—
26	C₂×M₁₆×C₇₁	—	2	2	no	—
27	C₂×M₁₆×C₇₁	—	2	2	no	—
28	C₂×M₁₆×C₇₁	—	2	2	no	—
29	C₂×M₁₆×C₇₁	—	2	2	no	—
30	C₂×M₁₆×C₇₁	—	2	2	no	—
31	C₂×M₁₆×C₇₁	—	2	2	no	—
32	C₂×M₁₆×C₇₁	—	2	2	no	—
33	C₂×M₁₆×C₇₁	—	2	2	no	—
34	C₂×M₁₆×C₇₁	—	2	2	no	—
35	C₂×M₁₆×C₇₁	—	2	2	no	—
36	C₂×M₁₆×C₇₁	—	2	2	no	—
37	C₂×M₁₆×C₇₁	—	2	2	no	—
38	C₂×M₁₆×C₇₁	D₄	1	2	yes	—
39	C₂×M₁₆×C₇₁	C₂³	1	4	yes	—
40	C₂×M₁₆×C₇₁	C₂³	1	4	yes	—
41	C₂×M₁₆×C₇₁	C₂³	1	2	yes	—
42	C₂²×C₈×C₇₁	—	2	2	no	—
43	C₂²×C₈×C₇₁	—	2	2	no	—
44	C₂²×C₈×C₇₁	D₄	1	2	yes	—
45	C₂²×C₈×C₇₁	C₂³	1	2	yes	—
46	#117×C₇₁	—	2	1	no	—
47	#117×C₇₁	—	2	1	no	—
48	#117×C₇₁	Q₈	1	2	yes	—
49	#117×C₇₁	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
51	C₂×C₂²⋊C₄×C₇₁	Q₈	1	2	yes	—
52	C₂×C₄²×C₇₁	—	2	1	no	—
53	#108×C₇₁	—	2	8	no	—
54	C₄⋊C₈×C₇₁	—	2	8	no	—
55	C₂⁴×C₇₁	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₇₁	—	2	2	no	—
57	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
61	C₂²×C₄×C₇₁	#41	1	1	yes	characteristic
62	C₂²×C₄×C₇₁	#41	1	2	yes	—
63	C₂²×C₄×C₇₁	#41	1	2	yes	—
64	C₂²×C₄×C₇₁	#41	1	2	yes	—
65	C₂²×C₄×C₇₁	#41	1	2	yes	—
66	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
67	M₁₆×C₇₁	—	2	4	no	—
68	M₁₆×C₇₁	—	2	4	no	—
69	M₁₆×C₇₁	—	2	4	no	—
70	M₁₆×C₇₁	—	2	4	no	—
71	M₁₆×C₇₁	—	2	4	no	—
72	M₁₆×C₇₁	—	2	4	no	—
73	M₁₆×C₇₁	—	4	1	no	—
74	M₁₆×C₇₁	—	4	1	no	—
75	M₁₆×C₇₁	—	4	2	no	—
76	M₁₆×C₇₁	—	4	2	no	—
77	M₁₆×C₇₁	—	4	2	no	—
78	M₁₆×C₇₁	—	4	2	no	—
79	M₁₆×C₇₁	—	4	2	no	—
80	C₂×C₈×C₇₁	—	2	2	no	—
81	C₂×C₈×C₇₁	—	2	2	no	—
82	C₂×C₈×C₇₁	—	2	2	no	—
83	C₂×C₈×C₇₁	—	2	4	no	—
84	C₂×C₈×C₇₁	—	2	4	no	—
85	C₂×C₈×C₇₁	—	2	4	no	—
86	C₂×C₈×C₇₁	—	4	2	no	—
87	C₂×C₈×C₇₁	—	4	2	no	—
88	C₂×C₈×C₇₁	—	4	2	no	—
89	C₂×C₈×C₇₁	—	4	2	no	—
90	C₂×C₈×C₇₁	C₂×D₄	1	4	yes	—
91	C₄⋊C₄×C₇₁	—	2	2	no	—
92	C₄⋊C₄×C₇₁	—	2	2	no	—
93	C₂²⋊C₄×C₇₁	—	2	2	no	—
94	C₂²⋊C₄×C₇₁	C₂×Q₈	1	4	yes	—
95	C₄²×C₇₁	—	2	2	no	—
96	C₄²×C₇₁	—	2	2	no	—
97	C₂³×C₇₁	—	8	1	no	—
98	C₂³×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂³×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂³×C₇₁	#125	1	2	yes	—
101	C₂³×C₇₁	#125	1	2	yes	—
102	C₂³×C₇₁	C₂²×Q₈	1	1	yes	characteristic
103	C₂×C₄×C₇₁	—	2	4	no	—
104	C₂×C₄×C₇₁	—	8	2	no	—
105	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₇₁	#123	1	1	yes	characteristic
108	C₂×C₄×C₇₁	#123	1	1	yes	characteristic
109	C₂×C₄×C₇₁	#123	1	2	yes	—
110	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
111	C₂×C₄×C₇₁	#124	1	2	yes	—
112	C₂×C₄×C₇₁	#124	1	1	yes	characteristic
113	C₂×C₄×C₇₁	#125	1	2	yes	—
114	C₂×C₄×C₇₁	#125	1	2	yes	—
115	C₂×C₄×C₇₁	#126	1	2	yes	—
116	C₂×C₄×C₇₁	#126	1	2	yes	—
117	C₂×C₄×C₇₁	#126	1	2	yes	—
118	C₂×C₄×C₇₁	#126	1	2	yes	—
119	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₇₁	C₂×#41	1	2	yes	—
121	C₂×C₄×C₇₁	C₂×#41	1	2	yes	—
122	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
123	C₈×C₇₁	—	2	4	no	—
124	C₈×C₇₁	—	4	2	no	—
125	C₈×C₇₁	—	4	4	no	—
126	C₈×C₇₁	—	4	4	no	—
127	C₂²×C₇₁	—	4	1	no	—
128	C₂²×C₇₁	—	4	1	no	—
129	C₂²×C₇₁	—	4	2	no	—
130	C₂²×C₇₁	—	4	2	no	—
131	C₂²×C₇₁	—	4	1	no	—
132	C₂²×C₇₁	#400	1	1	yes	characteristic
133	C₂²×C₇₁	#400	1	1	yes	characteristic
134	C₂²×C₇₁	#400	1	2	yes	—
135	C₂²×C₇₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
136	C₂²×C₇₁	C₂×#125	1	2	yes	—
137	C₄×C₇₁	—	4	4	no	—
138	C₄×C₇₁	#400	1	2	yes	—
139	C₄×C₇₁	#400	1	1	yes	characteristic
140	C₄×C₇₁	#400	1	1	yes	characteristic
141	C₄×C₇₁	C₂×#123	1	1	yes	characteristic
142	C₄×C₇₁	C₂×#124	1	1	yes	characteristic
143	C₄×C₇₁	C₂×#126	1	2	yes	—
144	#23753	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₂×C₇₁	—	2	1	no	—
146	C₂×C₇₁	—	2	2	no	—
147	C₂×C₇₁	—	2	2	no	—
148	C₂×C₇₁	—	2	1	no	—
149	C₂×C₇₁	—	2	1	no	—
150	C₂×C₇₁	C₂×#400	1	1	yes	characteristic
151	C₂×#430	C₂×C₇₁	1	1	yes	characteristic
152	#2905	C₂×C₇₁	1	1	yes	characteristic
153	#2103	C₂×C₇₁	1	8	yes	complemented
154	C₂×#344	C₂×C₇₁	1	2	yes	—
155	#1738	C₂×C₇₁	1	2	yes	—
156	C₂×#338	C₂×C₇₁	1	1	yes	characteristic
157	C₇₁	#23753	1	1	yes	characteristic, direct product factor, Sylow
158	C₂²×M₁₆	C₂²×C₇₁	1	2	yes	—
159	C₂²×M₁₆	C₂²×C₇₁	1	2	yes	—
160	C₂²×M₁₆	C₂²×C₇₁	1	1	yes	characteristic
161	C₂²×M₁₆	C₂²×C₇₁	1	1	yes	characteristic
162	C₂×#117	C₂²×C₇₁	1	1	yes	characteristic
163	#430	—	2	2	no	—
164	#430	C₂²×C₇₁	1	4	yes	—
165	C₂×#108	—	2	2	no	—
166	#424	—	2	2	no	—
167	#424	C₂²×C₇₁	1	4	yes	—
168	#423	—	2	2	no	—
169	#344	C₂²×C₇₁	1	8	yes	—
170	#341	C₂²×C₇₁	1	8	yes	—
171	#338	C₂²×C₇₁	1	4	yes	—
172	C₂³×C₄	C₂³×C₇₁	1	1	yes	characteristic
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	1	no	—
176	C₂×M₁₆	—	2	1	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	2	2	no	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	D₄×C₇₁	1	2	yes	—
190	C₂×M₁₆	C₂³×C₇₁	1	4	yes	—
191	C₂×M₁₆	C₂³×C₇₁	1	4	yes	—
192	C₂×M₁₆	C₂³×C₇₁	1	2	yes	—
193	C₂²×C₈	—	2	2	no	—
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	D₄×C₇₁	1	2	yes	—
196	C₂²×C₈	C₂³×C₇₁	1	2	yes	—
197	#117	—	2	1	no	—
198	#117	—	2	1	no	—
199	#117	Q₈×C₇₁	1	2	yes	—
200	#117	C₂³×C₇₁	1	2	yes	—
201	C₂×C₄⋊C₄	—	2	1	no	—
202	C₂×C₂²⋊C₄	Q₈×C₇₁	1	2	yes	—
203	C₂×C₄²	—	2	1	no	—
204	#108	—	2	8	no	—
205	C₄⋊C₈	—	2	8	no	—
206	C₂⁴	C₂×Q₈×C₇₁	1	1	yes	characteristic
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
210	C₂²×C₄	C₂×Q₈×C₇₁	1	1	yes	characteristic
211	C₂²×C₄	#41×C₇₁	1	1	yes	characteristic
212	C₂²×C₄	#41×C₇₁	1	1	yes	characteristic
213	C₂²×C₄	#41×C₇₁	1	2	yes	—
214	C₂²×C₄	#41×C₇₁	1	2	yes	—
215	C₂²×C₄	#41×C₇₁	1	2	yes	—
216	C₂²×C₄	#41×C₇₁	1	2	yes	—
217	C₂²×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
218	M₁₆	—	2	4	no	—
219	M₁₆	—	2	4	no	—
220	M₁₆	—	2	4	no	—
221	M₁₆	—	2	4	no	—
222	M₁₆	—	2	4	no	—
223	M₁₆	—	2	4	no	—
224	M₁₆	—	4	1	no	—
225	M₁₆	—	4	1	no	—
226	M₁₆	—	4	2	no	—
227	M₁₆	—	4	2	no	—
228	M₁₆	—	4	2	no	—
229	M₁₆	—	4	2	no	—
230	M₁₆	—	4	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	4	2	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	C₂×D₄×C₇₁	1	4	yes	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₂²⋊C₄	—	2	2	no	—
245	C₂²⋊C₄	C₂×Q₈×C₇₁	1	4	yes	—
246	C₄²	—	2	2	no	—
247	C₄²	—	2	2	no	—
248	C₂³	—	8	1	no	—
249	C₂³	C₂²⋊Q₈×C₇₁	1	1	yes	characteristic
250	C₂³	C₂²⋊Q₈×C₇₁	1	1	yes	characteristic
251	C₂³	#125×C₇₁	1	2	yes	—
252	C₂³	#125×C₇₁	1	2	yes	—
253	C₂³	C₂²×Q₈×C₇₁	1	1	yes	characteristic
254	C₂×C₄	—	2	4	no	—
255	C₂×C₄	—	8	2	no	—
256	C₂×C₄	C₂²⋊Q₈×C₇₁	1	1	yes	characteristic
257	C₂×C₄	C₂²⋊Q₈×C₇₁	1	1	yes	characteristic
258	C₂×C₄	#123×C₇₁	1	1	yes	characteristic
259	C₂×C₄	#123×C₇₁	1	1	yes	characteristic
260	C₂×C₄	#123×C₇₁	1	2	yes	—
261	C₂×C₄	#124×C₇₁	1	1	yes	characteristic
262	C₂×C₄	#124×C₇₁	1	2	yes	—
263	C₂×C₄	#124×C₇₁	1	1	yes	characteristic
264	C₂×C₄	#125×C₇₁	1	2	yes	—
265	C₂×C₄	#125×C₇₁	1	2	yes	—
266	C₂×C₄	#126×C₇₁	1	2	yes	—
267	C₂×C₄	#126×C₇₁	1	2	yes	—
268	C₂×C₄	#126×C₇₁	1	2	yes	—
269	C₂×C₄	#126×C₇₁	1	2	yes	—
270	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
271	C₂×C₄	C₂×#41×C₇₁	1	2	yes	—
272	C₂×C₄	C₂×#41×C₇₁	1	2	yes	—
273	C₂×C₄	C₂×#41×C₇₁	1	1	yes	characteristic
274	C₈	—	2	4	no	—
275	C₈	—	4	2	no	—
276	C₈	—	4	4	no	—
277	C₈	—	4	4	no	—
278	C₂²	—	4	1	no	—
279	C₂²	—	4	1	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	—	4	1	no	—
283	C₂²	#400×C₇₁	1	1	yes	characteristic
284	C₂²	#400×C₇₁	1	1	yes	characteristic
285	C₂²	#400×C₇₁	1	2	yes	—
286	C₂²	C₂×C₂²⋊Q₈×C₇₁	1	1	yes	characteristic
287	C₂²	C₂×#125×C₇₁	1	2	yes	—
288	C₄	—	4	4	no	—
289	C₄	#400×C₇₁	1	2	yes	—
290	C₄	#400×C₇₁	1	1	yes	characteristic
291	C₄	#400×C₇₁	1	1	yes	characteristic
292	C₄	C₂×#123×C₇₁	1	1	yes	characteristic
293	C₄	C₂×#124×C₇₁	1	1	yes	characteristic
294	C₄	C₂×#126×C₇₁	1	2	yes	—
295	C₂	—	2	1	no	—
296	C₂	—	2	2	no	—
297	C₂	—	2	2	no	—
298	C₂	—	2	1	no	—
299	C₂	—	2	1	no	—
300	C₂	C₂×#400×C₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₇₁	C₂×#124
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇₁
socle	C₂×C₇₁	C₂×#400
prime radical	C₂⁴×C₇₁	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₇₁ ◁ C₂³×C₄×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇₁×#2103	C₂	semidirect
144. #23753	C₇₁	direct (N × H)
153. #2103	C₂×C₇₁	semidirect
157. C₇₁	#23753	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×#430×C₇₁	2
C₇₁×#2905	2
C₇₁×#2103	2
C₂×#344×C₇₁	2
C₇₁×#1738	2
C₂×#338×C₇₁	2
#23753	71

Subgroups — C71×#23753 #31844625

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇₁×#23753 #31844625