Subgroups — C₂×C₇₁×#3261 #31881784

Order 18176.

1754 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#541×C₇₁	C₂	1	2	yes	maximal
2	C₇₁×#3261	C₂	1	16	yes	maximal, direct product factor
3	C₂×#479×C₇₁	C₂	1	4	yes	maximal, complemented
4	C₂×#470×C₇₁	C₂	1	2	yes	maximal
5	C₂×#467×C₇₁	C₂	1	2	yes	maximal
6	C₂×#462×C₇₁	C₂	1	4	yes	maximal, complemented
7	C₂×#424×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₁₆×C₇₁	C₂²	1	2	yes	—
9	C₂²×Q₁₆×C₇₁	C₂²	1	2	yes	—
10	C₂²×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₇₁	C₂²	1	1	yes	characteristic
12	#541×C₇₁	—	2	8	no	—
13	#541×C₇₁	C₂²	1	16	yes	—
14	C₂×C₄⋊Q₈×C₇₁	C₂²	1	2	yes	—
15	C₂×#126×C₇₁	—	2	4	no	—
16	C₂×#125×C₇₁	—	2	4	no	—
17	C₂×#125×C₇₁	C₂²	1	2	yes	—
18	C₂×C₂²⋊Q₈×C₇₁	C₂²	1	4	yes	—
19	C₂×C₄×Q₈×C₇₁	C₂²	1	4	yes	—
20	C₂×#117×C₇₁	C₂²	1	1	yes	characteristic
21	#479×C₇₁	C₂²	1	32	yes	complemented
22	#470×C₇₁	C₂²	1	16	yes	—
23	#467×C₇₁	C₂²	1	16	yes	—
24	#462×C₇₁	C₂²	1	32	yes	complemented
25	C₂×#107×C₇₁	C₂²	1	2	yes	—
26	C₂×#106×C₇₁	C₂²	1	2	yes	—
27	#424×C₇₁	C₂²	1	8	yes	—
28	C₂×C₄⋊C₈×C₇₁	C₂²	1	4	yes	—
29	C₂×#103×C₇₁	C₂²	1	4	yes	—
30	C₂×#103×C₇₁	C₂²	1	4	yes	—
31	C₂²×Q₈×C₇₁	C₂³	1	4	yes	—
32	C₂³×C₄×C₇₁	C₂³	1	1	yes	characteristic
33	C₂×Q₁₆×C₇₁	—	4	4	no	—
34	C₂×Q₁₆×C₇₁	—	4	4	no	—
35	C₂×Q₁₆×C₇₁	C₂³	1	8	yes	—
36	C₂×Q₁₆×C₇₁	C₂³	1	8	yes	—
37	C₂×M₁₆×C₇₁	—	4	2	no	—
38	C₂×M₁₆×C₇₁	C₂³	1	4	yes	—
39	C₂²×C₈×C₇₁	—	4	2	no	—
40	C₂²×C₈×C₇₁	C₂³	1	2	yes	—
41	C₂²×C₈×C₇₁	C₂³	1	2	yes	—
42	C₂²×C₈×C₇₁	C₂³	1	4	yes	—
43	C₄⋊Q₈×C₇₁	—	2	4	no	—
44	C₄⋊Q₈×C₇₁	C₂³	1	8	yes	—
45	#126×C₇₁	—	2	32	no	—
46	#125×C₇₁	—	2	32	no	—
47	#125×C₇₁	—	2	4	no	—
48	#125×C₇₁	C₂³	1	8	yes	—
49	C₂²⋊Q₈×C₇₁	—	2	8	no	—
50	C₂²⋊Q₈×C₇₁	C₂³	1	16	yes	—
51	C₄×Q₈×C₇₁	—	2	8	no	—
52	C₄×Q₈×C₇₁	C₂³	1	16	yes	—
53	#117×C₇₁	D₄	1	4	yes	—
54	#117×C₇₁	C₂³	1	4	yes	—
55	C₂×C₄⋊C₄×C₇₁	—	2	4	no	—
56	C₂×C₄⋊C₄×C₇₁	—	2	4	no	—
57	C₂×C₄⋊C₄×C₇₁	—	2	4	no	—
58	C₂×C₄⋊C₄×C₇₁	—	2	4	no	—
59	C₂×C₄⋊C₄×C₇₁	D₄	1	2	yes	—
60	C₂×C₄⋊C₄×C₇₁	C₂³	1	4	yes	—
61	C₂×C₂²⋊C₄×C₇₁	—	2	4	no	—
62	C₂×C₂²⋊C₄×C₇₁	D₄	1	2	yes	—
63	C₂×C₄²×C₇₁	—	2	4	no	—
64	C₂×C₄²×C₇₁	C₂³	1	2	yes	—
65	#107×C₇₁	C₂³	1	8	yes	—
66	#106×C₇₁	C₂³	1	8	yes	—
67	C₄⋊C₈×C₇₁	C₂³	1	16	yes	—
68	#103×C₇₁	C₂³	1	16	yes	—
69	#103×C₇₁	C₂³	1	16	yes	—
70	C₂⁴×C₇₁	C₂×D₄	1	1	yes	characteristic
71	C₂×Q₈×C₇₁	—	2	4	no	—
72	C₂×Q₈×C₇₁	—	2	8	no	—
73	C₂×Q₈×C₇₁	—	2	8	no	—
74	C₂×Q₈×C₇₁	C₂⁴	1	8	yes	—
75	C₂²×C₄×C₇₁	—	2	2	no	—
76	C₂²×C₄×C₇₁	—	2	2	no	—
77	C₂²×C₄×C₇₁	—	2	4	no	—
78	C₂²×C₄×C₇₁	—	2	4	no	—
79	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₇₁	C₂×D₄	1	4	yes	—
81	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₇₁	C₂⁴	1	2	yes	—
83	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
84	Q₁₆×C₇₁	—	4	8	no	—
85	Q₁₆×C₇₁	—	4	8	no	—
86	M₁₆×C₇₁	—	4	4	no	—
87	C₂×C₈×C₇₁	—	2	4	no	—
88	C₂×C₈×C₇₁	—	2	4	no	—
89	C₂×C₈×C₇₁	—	4	4	no	—
90	C₂×C₈×C₇₁	C₂⁴	1	4	yes	—
91	C₂×C₈×C₇₁	C₂⁴	1	4	yes	—
92	C₄⋊C₄×C₇₁	—	2	16	no	—
93	C₄⋊C₄×C₇₁	—	2	16	no	—
94	C₄⋊C₄×C₇₁	—	2	16	no	—
95	C₄⋊C₄×C₇₁	—	2	16	no	—
96	C₄⋊C₄×C₇₁	—	2	4	no	—
97	C₄⋊C₄×C₇₁	C₂×D₄	1	8	yes	—
98	C₄⋊C₄×C₇₁	C₂⁴	1	8	yes	—
99	C₂²⋊C₄×C₇₁	—	2	16	no	—
100	C₂²⋊C₄×C₇₁	C₂×D₄	1	8	yes	—
101	C₄²×C₇₁	—	2	16	no	—
102	C₄²×C₇₁	C₂×D₄	1	4	yes	—
103	C₄²×C₇₁	C₂⁴	1	4	yes	—
104	C₂³×C₇₁	—	2	2	no	—
105	C₂³×C₇₁	—	2	4	no	—
106	C₂³×C₇₁	C₂²×D₄	1	2	yes	—
107	C₂³×C₇₁	C₂²×D₄	1	1	yes	characteristic
108	Q₈×C₇₁	—	2	8	no	—
109	Q₈×C₇₁	—	2	8	no	—
110	Q₈×C₇₁	—	4	8	no	—
111	C₂×C₄×C₇₁	—	2	2	no	—
112	C₂×C₄×C₇₁	—	2	8	no	—
113	C₂×C₄×C₇₁	—	2	4	no	—
114	C₂×C₄×C₇₁	—	2	2	no	—
115	C₂×C₄×C₇₁	—	2	2	no	—
116	C₂×C₄×C₇₁	—	2	8	no	—
117	C₂×C₄×C₇₁	—	2	8	no	—
118	C₂×C₄×C₇₁	—	4	8	no	—
119	C₂×C₄×C₇₁	—	4	8	no	—
120	C₂×C₄×C₇₁	C₂²×D₄	1	8	yes	—
121	C₂×C₄×C₇₁	C₂²×D₄	1	2	yes	—
122	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₇₁	H(5,𝔽₂)	1	2	yes	—
124	C₂×C₄×C₇₁	2₋¹⁺⁴	1	2	yes	—
125	C₂×C₄×C₇₁	C₂⁵	1	1	yes	characteristic
126	C₈×C₇₁	—	2	4	no	—
127	C₈×C₇₁	—	2	4	no	—
128	C₂²×C₇₁	—	2	2	no	—
129	C₂²×C₇₁	—	2	4	no	—
130	C₂²×C₇₁	—	4	4	no	—
131	C₂²×C₇₁	#537	1	2	yes	—
132	C₂²×C₇₁	#578	1	4	yes	—
133	C₂²×C₇₁	C₂³×D₄	1	1	yes	characteristic
134	C₄×C₇₁	—	2	2	no	—
135	C₄×C₇₁	—	2	8	no	—
136	C₄×C₇₁	—	4	8	no	—
137	C₄×C₇₁	—	4	8	no	—
138	C₄×C₇₁	#537	1	2	yes	—
139	C₄×C₇₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
140	C₄×C₇₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
141	C₂×#3261	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
142	C₂×C₇₁	—	4	2	no	—
143	C₂×C₇₁	#3261	1	4	yes	direct product factor
144	C₂×C₇₁	C₂×#537	1	1	yes	characteristic
145	C₂×C₇₁	C₂×#578	1	2	yes	—
146	C₂×#541	C₂×C₇₁	1	2	yes	—
147	#3261	C₂×C₇₁	1	16	yes	direct product factor
148	C₂×#479	C₂×C₇₁	1	4	yes	complemented
149	C₂×#470	C₂×C₇₁	1	2	yes	—
150	C₂×#467	C₂×C₇₁	1	2	yes	—
151	C₂×#462	C₂×C₇₁	1	4	yes	complemented
152	C₂×#424	C₂×C₇₁	1	1	yes	characteristic
153	C₇₁	C₂×#3261	1	1	yes	characteristic, direct product factor, Sylow
154	C₂²×Q₁₆	C₂²×C₇₁	1	2	yes	—
155	C₂²×Q₁₆	C₂²×C₇₁	1	2	yes	—
156	C₂²×M₁₆	C₂²×C₇₁	1	1	yes	characteristic
157	C₂³×C₈	C₂²×C₇₁	1	1	yes	characteristic
158	#541	—	2	8	no	—
159	#541	C₂²×C₇₁	1	16	yes	—
160	C₂×C₄⋊Q₈	C₂²×C₇₁	1	2	yes	—
161	C₂×#126	—	2	4	no	—
162	C₂×#125	—	2	4	no	—
163	C₂×#125	C₂²×C₇₁	1	2	yes	—
164	C₂×C₂²⋊Q₈	C₂²×C₇₁	1	4	yes	—
165	C₂×C₄×Q₈	C₂²×C₇₁	1	4	yes	—
166	C₂×#117	C₂²×C₇₁	1	1	yes	characteristic
167	#479	C₂²×C₇₁	1	32	yes	complemented
168	#470	C₂²×C₇₁	1	16	yes	—
169	#467	C₂²×C₇₁	1	16	yes	—
170	#462	C₂²×C₇₁	1	32	yes	complemented
171	C₂×#107	C₂²×C₇₁	1	2	yes	—
172	C₂×#106	C₂²×C₇₁	1	2	yes	—
173	#424	C₂²×C₇₁	1	8	yes	—
174	C₂×C₄⋊C₈	C₂²×C₇₁	1	4	yes	—
175	C₂×#103	C₂²×C₇₁	1	4	yes	—
176	C₂×#103	C₂²×C₇₁	1	4	yes	—
177	C₂²×Q₈	C₂³×C₇₁	1	4	yes	—
178	C₂³×C₄	C₂³×C₇₁	1	1	yes	characteristic
179	C₂×Q₁₆	—	4	4	no	—
180	C₂×Q₁₆	—	4	4	no	—
181	C₂×Q₁₆	C₂³×C₇₁	1	8	yes	—
182	C₂×Q₁₆	C₂³×C₇₁	1	8	yes	—
183	C₂×M₁₆	—	4	2	no	—
184	C₂×M₁₆	C₂³×C₇₁	1	4	yes	—
185	C₂²×C₈	—	4	2	no	—
186	C₂²×C₈	C₂³×C₇₁	1	2	yes	—
187	C₂²×C₈	C₂³×C₇₁	1	2	yes	—
188	C₂²×C₈	C₂³×C₇₁	1	4	yes	—
189	C₄⋊Q₈	—	2	4	no	—
190	C₄⋊Q₈	C₂³×C₇₁	1	8	yes	—
191	#126	—	2	32	no	—
192	#125	—	2	32	no	—
193	#125	—	2	4	no	—
194	#125	C₂³×C₇₁	1	8	yes	—
195	C₂²⋊Q₈	—	2	8	no	—
196	C₂²⋊Q₈	C₂³×C₇₁	1	16	yes	—
197	C₄×Q₈	—	2	8	no	—
198	C₄×Q₈	C₂³×C₇₁	1	16	yes	—
199	#117	D₄×C₇₁	1	4	yes	—
200	#117	C₂³×C₇₁	1	4	yes	—
201	C₂×C₄⋊C₄	—	2	4	no	—
202	C₂×C₄⋊C₄	—	2	4	no	—
203	C₂×C₄⋊C₄	—	2	4	no	—
204	C₂×C₄⋊C₄	—	2	4	no	—
205	C₂×C₄⋊C₄	D₄×C₇₁	1	2	yes	—
206	C₂×C₄⋊C₄	C₂³×C₇₁	1	4	yes	—
207	C₂×C₂²⋊C₄	—	2	4	no	—
208	C₂×C₂²⋊C₄	D₄×C₇₁	1	2	yes	—
209	C₂×C₄²	—	2	4	no	—
210	C₂×C₄²	C₂³×C₇₁	1	2	yes	—
211	#107	C₂³×C₇₁	1	8	yes	—
212	#106	C₂³×C₇₁	1	8	yes	—
213	C₄⋊C₈	C₂³×C₇₁	1	16	yes	—
214	#103	C₂³×C₇₁	1	16	yes	—
215	#103	C₂³×C₇₁	1	16	yes	—
216	C₂⁴	C₂×D₄×C₇₁	1	1	yes	characteristic
217	C₂×Q₈	—	2	4	no	—
218	C₂×Q₈	—	2	8	no	—
219	C₂×Q₈	—	2	8	no	—
220	C₂×Q₈	C₂⁴×C₇₁	1	8	yes	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	C₂×D₄×C₇₁	1	4	yes	—
226	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
228	C₂²×C₄	C₂⁴×C₇₁	1	2	yes	—
229	C₂²×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
230	Q₁₆	—	4	8	no	—
231	Q₁₆	—	4	8	no	—
232	M₁₆	—	4	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	4	4	no	—
236	C₂×C₈	C₂⁴×C₇₁	1	4	yes	—
237	C₂×C₈	C₂⁴×C₇₁	1	4	yes	—
238	C₄⋊C₄	—	2	16	no	—
239	C₄⋊C₄	—	2	16	no	—
240	C₄⋊C₄	—	2	16	no	—
241	C₄⋊C₄	—	2	16	no	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	C₂×D₄×C₇₁	1	8	yes	—
244	C₄⋊C₄	C₂⁴×C₇₁	1	8	yes	—
245	C₂²⋊C₄	—	2	16	no	—
246	C₂²⋊C₄	C₂×D₄×C₇₁	1	8	yes	—
247	C₄²	—	2	16	no	—
248	C₄²	C₂×D₄×C₇₁	1	4	yes	—
249	C₄²	C₂⁴×C₇₁	1	4	yes	—
250	C₂³	—	2	2	no	—
251	C₂³	—	2	4	no	—
252	C₂³	C₂²×D₄×C₇₁	1	2	yes	—
253	C₂³	C₂²×D₄×C₇₁	1	1	yes	characteristic
254	Q₈	—	2	8	no	—
255	Q₈	—	2	8	no	—
256	Q₈	—	4	8	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	4	8	no	—
265	C₂×C₄	—	4	8	no	—
266	C₂×C₄	C₂²×D₄×C₇₁	1	8	yes	—
267	C₂×C₄	C₂²×D₄×C₇₁	1	2	yes	—
268	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
269	C₂×C₄	H(5,𝔽₂)×C₇₁	1	2	yes	—
270	C₂×C₄	2₋¹⁺⁴×C₇₁	1	2	yes	—
271	C₂×C₄	C₂⁵×C₇₁	1	1	yes	characteristic
272	C₈	—	2	4	no	—
273	C₈	—	2	4	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	4	4	no	—
277	C₂²	#537×C₇₁	1	2	yes	—
278	C₂²	#578×C₇₁	1	4	yes	—
279	C₂²	C₂³×D₄×C₇₁	1	1	yes	characteristic
280	C₄	—	2	8	no	—
281	C₄	—	2	2	no	—
282	C₄	—	4	8	no	—
283	C₄	—	4	8	no	—
284	C₄	#537×C₇₁	1	2	yes	—
285	C₄	C₂×H(5,𝔽₂)×C₇₁	1	1	yes	characteristic
286	C₄	C₂×2₋¹⁺⁴×C₇₁	1	1	yes	characteristic
287	C₂	—	4	2	no	—
288	C₂	C₇₁×#3261	1	4	yes	direct product factor
289	C₂	C₂×#537×C₇₁	1	1	yes	characteristic
290	C₂	C₂×#578×C₇₁	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₇₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₇₁
prime radical	C₂⁴×C₇₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇₁ ◁ C₂×#117×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇₁×#3261	C₂	direct (N × H)
3. C₂×#479×C₇₁	C₂	semidirect
6. C₂×#462×C₇₁	C₂	semidirect
21. #479×C₇₁	C₂²	semidirect
24. #462×C₇₁	C₂²	semidirect
141. C₂×#3261	C₇₁	direct (N × H)
143. C₂×C₇₁	#3261	direct (N × H)
147. #3261	C₂×C₇₁	direct (N × H)
148. C₂×#479	C₂×C₇₁	semidirect
151. C₂×#462	C₂×C₇₁	semidirect
153. C₇₁	C₂×#3261	direct (N × H)
167. #479	C₂²×C₇₁	semidirect
170. #462	C₂²×C₇₁	semidirect
288. C₂	C₇₁×#3261	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×#541×C₇₁	2
C₇₁×#3261	2
C₂×#479×C₇₁	2
C₂×#470×C₇₁	2
C₂×#467×C₇₁	2
C₂×#462×C₇₁	2
C₂×#424×C₇₁	2
C₂×#3261	71

Subgroups — C2×C71×#3261 #31881784

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇₁×#3261 #31881784