Subgroups — C₂×C₇₁×#2615 #31855477

Order 18176.

6166 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	C₂³×#41×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#121×C₇₁	C₂	1	3	yes	maximal, complemented
3	C₇₁×#2615	C₂	1	16	yes	maximal, direct product factor
4	C₂×#393×C₇₁	C₂	1	2	yes	maximal, complemented
5	C₂×#392×C₇₁	C₂	1	6	yes	maximal, complemented
6	C₂×#380×C₇₁	C₂	1	3	yes	maximal, complemented
7	C₂²×#41×C₇₁	—	2	24	no	—
8	C₂²×#41×C₇₁	C₂²	1	8	yes	complemented
9	C₂³×Q₈×C₇₁	C₂²	1	1	yes	characteristic
10	C₂³×D₄×C₇₁	C₂²	1	3	yes	—
11	C₂³×D₄×C₇₁	C₂²	1	1	yes	characteristic
12	C₂³×D₄×C₇₁	C₂²	1	6	yes	—
13	C₂⁴×C₄×C₇₁	C₂²	1	3	yes	—
14	C₂×#121×C₇₁	—	2	24	no	—
15	C₂×#121×C₇₁	C₂²	1	24	yes	complemented
16	C₂²×C₄⋊C₄×C₇₁	C₂²	1	3	yes	—
17	C₂²×C₂²⋊C₄×C₇₁	C₂²	1	6	yes	—
18	C₂²×C₂²⋊C₄×C₇₁	C₂²	1	6	yes	—
19	#393×C₇₁	C₂²	1	16	yes	complemented
20	#392×C₇₁	C₂²	1	48	yes	complemented
21	#380×C₇₁	C₂²	1	24	yes	complemented
22	C₂×#95×C₇₁	C₂²	1	6	yes	—
23	C₂⁵×C₇₁	C₂³	1	2	yes	—
24	C₂⁵×C₇₁	C₂³	1	3	yes	—
25	C₂×#41×C₇₁	—	2	128	no	—
26	C₂×#41×C₇₁	—	2	96	no	—
27	C₂²×Q₈×C₇₁	—	2	12	no	—
28	C₂²×Q₈×C₇₁	C₂³	1	4	yes	—
29	C₂²×D₄×C₇₁	—	2	24	no	—
30	C₂²×D₄×C₇₁	—	2	24	no	—
31	C₂²×D₄×C₇₁	—	2	12	no	—
32	C₂²×D₄×C₇₁	—	2	12	no	—
33	C₂²×D₄×C₇₁	—	4	24	no	—
34	C₂²×D₄×C₇₁	C₂³	1	4	yes	—
35	C₂²×D₄×C₇₁	C₂³	1	12	yes	—
36	C₂²×D₄×C₇₁	C₂³	1	24	yes	—
37	C₂³×C₄×C₇₁	—	2	24	no	—
38	C₂³×C₄×C₇₁	D₄	1	24	yes	complemented
39	C₂³×C₄×C₇₁	C₂³	1	6	yes	—
40	C₂³×C₄×C₇₁	C₂³	1	3	yes	—
41	C₂³×C₄×C₇₁	C₂³	1	12	yes	—
42	C₂³×C₄×C₇₁	C₂³	1	1	yes	characteristic
43	#121×C₇₁	—	2	96	no	—
44	C₂×C₄⋊C₄×C₇₁	—	2	12	no	—
45	C₂×C₄⋊C₄×C₇₁	C₂³	1	12	yes	—
46	C₂×C₂²⋊C₄×C₇₁	—	2	24	no	—
47	C₂×C₂²⋊C₄×C₇₁	—	4	12	no	—
48	C₂×C₂²⋊C₄×C₇₁	C₂³	1	24	yes	—
49	C₂×C₂²⋊C₄×C₇₁	C₂³	1	24	yes	—
50	#95×C₇₁	C₂³	1	24	yes	—
51	C₂⁴×C₇₁	—	2	12	no	—
52	C₂⁴×C₇₁	—	2	6	no	—
53	C₂⁴×C₇₁	—	2	12	no	—
54	C₂⁴×C₇₁	—	2	6	no	—
55	C₂⁴×C₇₁	—	2	12	no	—
56	C₂⁴×C₇₁	—	2	12	no	—
57	C₂⁴×C₇₁	—	2	4	no	—
58	C₂⁴×C₇₁	C₂×D₄	1	12	yes	—
59	C₂⁴×C₇₁	C₂⁴	1	4	yes	—
60	C₂⁴×C₇₁	C₂⁴	1	6	yes	—
61	C₂⁴×C₇₁	C₂⁴	1	1	yes	characteristic
62	#41×C₇₁	—	2	256	no	—
63	C₂×Q₈×C₇₁	—	2	32	no	—
64	C₂×Q₈×C₇₁	—	2	24	no	—
65	C₂×D₄×C₇₁	—	2	96	no	—
66	C₂×D₄×C₇₁	—	2	48	no	—
67	C₂×D₄×C₇₁	—	2	48	no	—
68	C₂×D₄×C₇₁	—	2	24	no	—
69	C₂×D₄×C₇₁	—	2	24	no	—
70	C₂×D₄×C₇₁	—	4	48	no	—
71	C₂×D₄×C₇₁	—	4	96	no	—
72	C₂×D₄×C₇₁	—	4	16	no	—
73	C₂²×C₄×C₇₁	—	2	12	no	—
74	C₂²×C₄×C₇₁	—	2	96	no	—
75	C₂²×C₄×C₇₁	—	2	48	no	—
76	C₂²×C₄×C₇₁	—	2	12	no	—
77	C₂²×C₄×C₇₁	—	2	24	no	—
78	C₂²×C₄×C₇₁	C₂×D₄	1	12	yes	—
79	C₂²×C₄×C₇₁	C₂×D₄	1	12	yes	—
80	C₂²×C₄×C₇₁	C₂×D₄	1	48	yes	complemented
81	C₂²×C₄×C₇₁	C₂⁴	1	12	yes	—
82	C₂²×C₄×C₇₁	C₂⁴	1	6	yes	—
83	C₂²×C₄×C₇₁	C₂⁴	1	2	yes	—
84	C₄⋊C₄×C₇₁	—	2	24	no	—
85	C₂²⋊C₄×C₇₁	—	2	48	no	—
86	C₂²⋊C₄×C₇₁	—	4	24	no	—
87	C₂³×C₇₁	—	2	24	no	—
88	C₂³×C₇₁	—	2	24	no	—
89	C₂³×C₇₁	—	2	24	no	—
90	C₂³×C₇₁	—	2	12	no	—
91	C₂³×C₇₁	—	2	6	no	—
92	C₂³×C₇₁	—	2	12	no	—
93	C₂³×C₇₁	—	2	6	no	—
94	C₂³×C₇₁	—	2	12	no	—
95	C₂³×C₇₁	—	2	12	no	—
96	C₂³×C₇₁	—	2	4	no	—
97	C₂³×C₇₁	—	4	24	no	—
98	C₂³×C₇₁	—	4	24	no	—
99	C₂³×C₇₁	—	4	8	no	—
100	C₂³×C₇₁	—	4	24	no	—
101	C₂³×C₇₁	—	4	24	no	—
102	C₂³×C₇₁	C₂²×D₄	1	12	yes	—
103	C₂³×C₇₁	C₂²×D₄	1	6	yes	—
104	C₂³×C₇₁	H(5,𝔽₂)	1	6	yes	—
105	C₂³×C₇₁	2₋¹⁺⁴	1	2	yes	—
106	C₂³×C₇₁	C₂⁵	1	1	yes	characteristic
107	Q₈×C₇₁	—	2	32	no	—
108	D₄×C₇₁	—	2	96	no	—
109	D₄×C₇₁	—	4	96	no	—
110	D₄×C₇₁	—	4	16	no	—
111	C₂×C₄×C₇₁	—	2	24	no	—
112	C₂×C₄×C₇₁	—	2	12	no	—
113	C₂×C₄×C₇₁	—	2	96	no	—
114	C₂×C₄×C₇₁	—	2	12	no	—
115	C₂×C₄×C₇₁	—	2	24	no	—
116	C₂×C₄×C₇₁	—	4	24	no	—
117	C₂×C₄×C₇₁	#120	1	16	yes	—
118	C₂×C₄×C₇₁	C₂²×D₄	1	12	yes	—
119	C₂×C₄×C₇₁	C₂²×D₄	1	12	yes	—
120	C₂²×C₇₁	—	2	12	no	—
121	C₂²×C₇₁	—	2	12	no	—
122	C₂²×C₇₁	—	2	12	no	—
123	C₂²×C₇₁	—	4	48	no	—
124	C₂²×C₇₁	—	4	12	no	—
125	C₂²×C₇₁	—	4	12	no	—
126	C₂²×C₇₁	—	4	4	no	—
127	C₂²×C₇₁	—	4	12	no	—
128	C₂²×C₇₁	—	4	12	no	—
129	C₂²×C₇₁	—	8	16	no	—
130	C₂²×C₇₁	C₂×#120	1	4	yes	—
131	C₂²×C₇₁	#535	1	12	yes	—
132	C₂²×C₇₁	#537	1	12	yes	—
133	C₂²×C₇₁	C₂³×D₄	1	3	yes	—
134	C₂²×C₇₁	C₂×H(5,𝔽₂)	1	3	yes	—
135	C₂²×C₇₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
136	C₄×C₇₁	—	2	12	no	—
137	C₄×C₇₁	—	4	12	no	—
138	C₄×C₇₁	C₂×#120	1	8	yes	—
139	C₂×#2615	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₂×C₇₁	—	4	12	no	—
141	C₂×C₇₁	—	8	4	no	—
142	C₂×C₇₁	#2615	1	8	yes	direct product factor
143	C₂×C₇₁	C₂²×#120	1	1	yes	characteristic
144	C₂×C₇₁	C₂×#535	1	3	yes	—
145	C₂×C₇₁	C₂×#537	1	3	yes	—
146	C₂³×#41	C₂×C₇₁	1	1	yes	characteristic, complemented
147	C₂²×#121	C₂×C₇₁	1	3	yes	complemented
148	#2615	C₂×C₇₁	1	16	yes	direct product factor
149	C₂×#393	C₂×C₇₁	1	2	yes	complemented
150	C₂×#392	C₂×C₇₁	1	6	yes	complemented
151	C₂×#380	C₂×C₇₁	1	3	yes	complemented
152	C₇₁	C₂×#2615	1	1	yes	characteristic, direct product factor, Sylow
153	C₂²×#41	—	2	24	no	—
154	C₂²×#41	C₂²×C₇₁	1	8	yes	complemented
155	C₂³×Q₈	C₂²×C₇₁	1	1	yes	characteristic
156	C₂³×D₄	C₂²×C₇₁	1	3	yes	—
157	C₂³×D₄	C₂²×C₇₁	1	1	yes	characteristic
158	C₂³×D₄	C₂²×C₇₁	1	6	yes	—
159	C₂⁴×C₄	C₂²×C₇₁	1	3	yes	—
160	C₂×#121	—	2	24	no	—
161	C₂×#121	C₂²×C₇₁	1	24	yes	complemented
162	C₂²×C₄⋊C₄	C₂²×C₇₁	1	3	yes	—
163	C₂²×C₂²⋊C₄	C₂²×C₇₁	1	6	yes	—
164	C₂²×C₂²⋊C₄	C₂²×C₇₁	1	6	yes	—
165	#393	C₂²×C₇₁	1	16	yes	complemented
166	#392	C₂²×C₇₁	1	48	yes	complemented
167	#380	C₂²×C₇₁	1	24	yes	complemented
168	C₂×#95	C₂²×C₇₁	1	6	yes	—
169	C₂⁵	C₂³×C₇₁	1	2	yes	—
170	C₂⁵	C₂³×C₇₁	1	3	yes	—
171	C₂×#41	—	2	128	no	—
172	C₂×#41	—	2	96	no	—
173	C₂²×Q₈	—	2	12	no	—
174	C₂²×Q₈	C₂³×C₇₁	1	4	yes	—
175	C₂²×D₄	—	2	24	no	—
176	C₂²×D₄	—	2	24	no	—
177	C₂²×D₄	—	2	12	no	—
178	C₂²×D₄	—	2	12	no	—
179	C₂²×D₄	—	4	24	no	—
180	C₂²×D₄	C₂³×C₇₁	1	4	yes	—
181	C₂²×D₄	C₂³×C₇₁	1	12	yes	—
182	C₂²×D₄	C₂³×C₇₁	1	24	yes	—
183	C₂³×C₄	—	2	24	no	—
184	C₂³×C₄	D₄×C₇₁	1	24	yes	complemented
185	C₂³×C₄	C₂³×C₇₁	1	6	yes	—
186	C₂³×C₄	C₂³×C₇₁	1	3	yes	—
187	C₂³×C₄	C₂³×C₇₁	1	12	yes	—
188	C₂³×C₄	C₂³×C₇₁	1	1	yes	characteristic
189	#121	—	2	96	no	—
190	C₂×C₄⋊C₄	—	2	12	no	—
191	C₂×C₄⋊C₄	C₂³×C₇₁	1	12	yes	—
192	C₂×C₂²⋊C₄	—	2	24	no	—
193	C₂×C₂²⋊C₄	—	4	12	no	—
194	C₂×C₂²⋊C₄	C₂³×C₇₁	1	24	yes	—
195	C₂×C₂²⋊C₄	C₂³×C₇₁	1	24	yes	—
196	#95	C₂³×C₇₁	1	24	yes	—
197	C₂⁴	—	2	12	no	—
198	C₂⁴	—	2	6	no	—
199	C₂⁴	—	2	12	no	—
200	C₂⁴	—	2	6	no	—
201	C₂⁴	—	2	12	no	—
202	C₂⁴	—	2	12	no	—
203	C₂⁴	—	2	4	no	—
204	C₂⁴	C₂×D₄×C₇₁	1	12	yes	—
205	C₂⁴	C₂⁴×C₇₁	1	4	yes	—
206	C₂⁴	C₂⁴×C₇₁	1	6	yes	—
207	C₂⁴	C₂⁴×C₇₁	1	1	yes	characteristic
208	#41	—	2	256	no	—
209	C₂×Q₈	—	2	32	no	—
210	C₂×Q₈	—	2	24	no	—
211	C₂×D₄	—	2	96	no	—
212	C₂×D₄	—	2	48	no	—
213	C₂×D₄	—	2	48	no	—
214	C₂×D₄	—	2	24	no	—
215	C₂×D₄	—	2	24	no	—
216	C₂×D₄	—	4	48	no	—
217	C₂×D₄	—	4	96	no	—
218	C₂×D₄	—	4	16	no	—
219	C₂²×C₄	—	2	12	no	—
220	C₂²×C₄	—	2	96	no	—
221	C₂²×C₄	—	2	48	no	—
222	C₂²×C₄	—	2	12	no	—
223	C₂²×C₄	—	2	24	no	—
224	C₂²×C₄	C₂×D₄×C₇₁	1	12	yes	—
225	C₂²×C₄	C₂×D₄×C₇₁	1	12	yes	—
226	C₂²×C₄	C₂×D₄×C₇₁	1	48	yes	complemented
227	C₂²×C₄	C₂⁴×C₇₁	1	12	yes	—
228	C₂²×C₄	C₂⁴×C₇₁	1	6	yes	—
229	C₂²×C₄	C₂⁴×C₇₁	1	2	yes	—
230	C₄⋊C₄	—	2	24	no	—
231	C₂²⋊C₄	—	2	48	no	—
232	C₂²⋊C₄	—	4	24	no	—
233	C₂³	—	2	24	no	—
234	C₂³	—	2	24	no	—
235	C₂³	—	2	24	no	—
236	C₂³	—	2	12	no	—
237	C₂³	—	2	6	no	—
238	C₂³	—	2	12	no	—
239	C₂³	—	2	6	no	—
240	C₂³	—	2	12	no	—
241	C₂³	—	2	12	no	—
242	C₂³	—	2	4	no	—
243	C₂³	—	4	24	no	—
244	C₂³	—	4	24	no	—
245	C₂³	—	4	8	no	—
246	C₂³	—	4	24	no	—
247	C₂³	—	4	24	no	—
248	C₂³	C₂²×D₄×C₇₁	1	12	yes	—
249	C₂³	C₂²×D₄×C₇₁	1	6	yes	—
250	C₂³	H(5,𝔽₂)×C₇₁	1	6	yes	—
251	C₂³	2₋¹⁺⁴×C₇₁	1	2	yes	—
252	C₂³	C₂⁵×C₇₁	1	1	yes	characteristic
253	Q₈	—	2	32	no	—
254	D₄	—	2	96	no	—
255	D₄	—	4	96	no	—
256	D₄	—	4	16	no	—
257	C₂×C₄	—	2	24	no	—
258	C₂×C₄	—	2	12	no	—
259	C₂×C₄	—	2	96	no	—
260	C₂×C₄	—	2	12	no	—
261	C₂×C₄	—	2	24	no	—
262	C₂×C₄	—	4	24	no	—
263	C₂×C₄	#120×C₇₁	1	16	yes	—
264	C₂×C₄	C₂²×D₄×C₇₁	1	12	yes	—
265	C₂×C₄	C₂²×D₄×C₇₁	1	12	yes	—
266	C₂²	—	2	12	no	—
267	C₂²	—	2	12	no	—
268	C₂²	—	2	12	no	—
269	C₂²	—	4	48	no	—
270	C₂²	—	4	12	no	—
271	C₂²	—	4	12	no	—
272	C₂²	—	4	4	no	—
273	C₂²	—	4	12	no	—
274	C₂²	—	4	12	no	—
275	C₂²	—	8	16	no	—
276	C₂²	C₂×#120×C₇₁	1	4	yes	—
277	C₂²	#535×C₇₁	1	12	yes	—
278	C₂²	#537×C₇₁	1	12	yes	—
279	C₂²	C₂³×D₄×C₇₁	1	3	yes	—
280	C₂²	C₂×H(5,𝔽₂)×C₇₁	1	3	yes	—
281	C₂²	C₂×2₋¹⁺⁴×C₇₁	1	1	yes	characteristic
282	C₄	—	2	12	no	—
283	C₄	—	4	12	no	—
284	C₄	C₂×#120×C₇₁	1	8	yes	—
285	C₂	—	4	12	no	—
286	C₂	—	8	4	no	—
287	C₂	C₇₁×#2615	1	8	yes	direct product factor
288	C₂	C₂²×#120×C₇₁	1	1	yes	characteristic
289	C₂	C₂×#535×C₇₁	1	3	yes	—
290	C₂	C₂×#537×C₇₁	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₇₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₇₁
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×#41×C₇₁	C₂	semidirect
2. C₂²×#121×C₇₁	C₂	semidirect
3. C₇₁×#2615	C₂	direct (N × H)
4. C₂×#393×C₇₁	C₂	semidirect
5. C₂×#392×C₇₁	C₂	semidirect
6. C₂×#380×C₇₁	C₂	semidirect
8. C₂²×#41×C₇₁	C₂²	semidirect
15. C₂×#121×C₇₁	C₂²	semidirect
19. #393×C₇₁	C₂²	semidirect
20. #392×C₇₁	C₂²	semidirect
21. #380×C₇₁	C₂²	semidirect
38. C₂³×C₄×C₇₁	D₄	semidirect
80. C₂²×C₄×C₇₁	C₂×D₄	semidirect
139. C₂×#2615	C₇₁	direct (N × H)
142. C₂×C₇₁	#2615	direct (N × H)
146. C₂³×#41	C₂×C₇₁	semidirect
147. C₂²×#121	C₂×C₇₁	semidirect
148. #2615	C₂×C₇₁	direct (N × H)
149. C₂×#393	C₂×C₇₁	semidirect
150. C₂×#392	C₂×C₇₁	semidirect
151. C₂×#380	C₂×C₇₁	semidirect
152. C₇₁	C₂×#2615	direct (N × H)
154. C₂²×#41	C₂²×C₇₁	semidirect
161. C₂×#121	C₂²×C₇₁	semidirect
165. #393	C₂²×C₇₁	semidirect
166. #392	C₂²×C₇₁	semidirect
167. #380	C₂²×C₇₁	semidirect
184. C₂³×C₄	D₄×C₇₁	semidirect
226. C₂²×C₄	C₂×D₄×C₇₁	semidirect
287. C₂	C₇₁×#2615	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#41×C₇₁	2
C₂²×#121×C₇₁	2
C₇₁×#2615	2
C₂×#393×C₇₁	2
C₂×#392×C₇₁	2
C₂×#380×C₇₁	2
C₂×#2615	71

Subgroups — C2×C71×#2615 #31855477

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇₁×#2615 #31855477