Subgroups — C₂×C₇₁×#3185 #31881564

Order 18176.

1954 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#533×C₇₁	C₂	1	2	yes	maximal
2	C₇₁×#3185	C₂	1	16	yes	maximal, direct product factor
3	C₂×#498×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#494×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#487×C₇₁	C₂	1	2	yes	maximal, complemented
6	C₂×#470×C₇₁	C₂	1	4	yes	maximal
7	C₂×#441×C₇₁	C₂	1	4	yes	maximal, complemented
8	C₂×C₄×M₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₇₁	C₂²	1	4	yes	—
10	C₂²×M₁₆×C₇₁	C₂²	1	2	yes	—
11	#533×C₇₁	—	2	8	no	—
12	#533×C₇₁	C₂²	1	16	yes	—
13	C₂×C₄⋊Q₈×C₇₁	C₂²	1	2	yes	—
14	C₂×C₄⋊Q₈×C₇₁	C₂²	1	2	yes	—
15	C₂×#125×C₇₁	—	2	2	no	—
16	C₂×C₂²⋊Q₈×C₇₁	—	2	2	no	—
17	C₂×C₂²⋊Q₈×C₇₁	C₂²	1	4	yes	—
18	C₂×C₄×Q₈×C₇₁	—	2	2	no	—
19	C₂×C₄×Q₈×C₇₁	C₂²	1	4	yes	—
20	C₂×#117×C₇₁	—	2	2	no	—
21	C₂²×C₄²×C₇₁	C₂²	1	1	yes	characteristic
22	#498×C₇₁	C₂²	1	8	yes	complemented
23	#494×C₇₁	C₂²	1	8	yes	complemented
24	#487×C₇₁	C₂²	1	16	yes	complemented
25	#470×C₇₁	C₂²	1	32	yes	—
26	#441×C₇₁	C₂²	1	32	yes	complemented
27	C₂×#106×C₇₁	C₂²	1	4	yes	—
28	C₂×#103×C₇₁	C₂²	1	8	yes	—
29	C₄×M₁₆×C₇₁	C₂²	1	8	yes	—
30	C₂×#97×C₇₁	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
32	C₂×C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
33	C₂²×Q₈×C₇₁	—	2	2	no	—
34	C₂²×Q₈×C₇₁	C₂³	1	4	yes	—
35	C₂³×C₄×C₇₁	D₄	1	2	yes	—
36	C₂³×C₄×C₇₁	C₂³	1	1	yes	characteristic
37	C₂×Q₁₆×C₇₁	—	2	16	no	—
38	C₂×Q₁₆×C₇₁	C₂³	1	16	yes	—
39	C₂×M₁₆×C₇₁	—	2	8	no	—
40	C₂×M₁₆×C₇₁	C₂³	1	8	yes	—
41	C₂²×C₈×C₇₁	C₂³	1	2	yes	—
42	C₂²×C₈×C₇₁	C₂³	1	2	yes	—
43	C₄⋊Q₈×C₇₁	—	2	4	no	—
44	C₄⋊Q₈×C₇₁	—	2	4	no	—
45	C₄⋊Q₈×C₇₁	C₂³	1	8	yes	—
46	C₄⋊Q₈×C₇₁	C₂³	1	8	yes	—
47	#125×C₇₁	—	2	16	no	—
48	C₂²⋊Q₈×C₇₁	—	2	16	no	—
49	C₂²⋊Q₈×C₇₁	—	2	8	no	—
50	C₂²⋊Q₈×C₇₁	C₂³	1	16	yes	—
51	C₄×Q₈×C₇₁	—	2	16	no	—
52	C₄×Q₈×C₇₁	—	2	8	no	—
53	C₄×Q₈×C₇₁	C₂³	1	16	yes	—
54	#117×C₇₁	—	2	16	no	—
55	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
56	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
57	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
58	C₂×C₄⋊C₄×C₇₁	—	2	4	no	—
59	C₂×C₄⋊C₄×C₇₁	—	2	4	no	—
60	C₂×C₄⋊C₄×C₇₁	C₂³	1	4	yes	—
61	C₂×C₂²⋊C₄×C₇₁	—	2	4	no	—
62	C₂×C₄²×C₇₁	—	2	4	no	—
63	C₂×C₄²×C₇₁	D₄	1	2	yes	—
64	C₂×C₄²×C₇₁	D₄	1	4	yes	—
65	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₇₁	C₂³	1	4	yes	—
68	#106×C₇₁	C₂³	1	16	yes	—
69	#103×C₇₁	C₂³	1	32	yes	—
70	#97×C₇₁	C₂³	1	8	yes	—
71	C₄×C₈×C₇₁	C₂³	1	4	yes	—
72	C₄×C₈×C₇₁	C₂³	1	4	yes	—
73	C₂⁴×C₇₁	C₂×D₄	1	1	yes	characteristic
74	C₂×Q₈×C₇₁	—	2	4	no	—
75	C₂×Q₈×C₇₁	—	2	8	no	—
76	C₂×Q₈×C₇₁	—	2	16	no	—
77	C₂×Q₈×C₇₁	—	4	8	no	—
78	C₂×Q₈×C₇₁	C₂⁴	1	8	yes	—
79	C₂²×C₄×C₇₁	—	2	8	no	—
80	C₂²×C₄×C₇₁	—	2	4	no	—
81	C₂²×C₄×C₇₁	—	2	4	no	—
82	C₂²×C₄×C₇₁	—	2	4	no	—
83	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₇₁	C₂×D₄	1	8	yes	—
85	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₇₁	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₇₁	C₂⁴	1	2	yes	—
89	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
90	Q₁₆×C₇₁	—	2	32	no	—
91	M₁₆×C₇₁	—	2	16	no	—
92	C₂×C₈×C₇₁	—	2	4	no	—
93	C₂×C₈×C₇₁	#41	1	8	yes	—
94	C₂×C₈×C₇₁	C₂⁴	1	4	yes	—
95	C₂×C₈×C₇₁	C₂⁴	1	4	yes	—
96	C₄⋊C₄×C₇₁	—	2	8	no	—
97	C₄⋊C₄×C₇₁	—	2	8	no	—
98	C₄⋊C₄×C₇₁	—	2	8	no	—
99	C₄⋊C₄×C₇₁	—	2	16	no	—
100	C₄⋊C₄×C₇₁	—	2	16	no	—
101	C₄⋊C₄×C₇₁	—	2	4	no	—
102	C₄⋊C₄×C₇₁	C₂⁴	1	8	yes	—
103	C₂²⋊C₄×C₇₁	—	2	16	no	—
104	C₄²×C₇₁	—	2	16	no	—
105	C₄²×C₇₁	C₂×D₄	1	2	yes	—
106	C₄²×C₇₁	C₂×D₄	1	8	yes	—
107	C₄²×C₇₁	C₂×D₄	1	2	yes	—
108	C₄²×C₇₁	C₂⁴	1	2	yes	—
109	C₄²×C₇₁	C₂⁴	1	2	yes	—
110	C₂³×C₇₁	—	2	2	no	—
111	C₂³×C₇₁	—	2	4	no	—
112	C₂³×C₇₁	C₂²×D₄	1	2	yes	—
113	C₂³×C₇₁	C₂²×D₄	1	1	yes	characteristic
114	Q₈×C₇₁	—	2	16	no	—
115	Q₈×C₇₁	—	4	8	no	—
116	Q₈×C₇₁	—	4	16	no	—
117	C₂×C₄×C₇₁	—	2	2	no	—
118	C₂×C₄×C₇₁	—	2	4	no	—
119	C₂×C₄×C₇₁	—	2	4	no	—
120	C₂×C₄×C₇₁	—	2	16	no	—
121	C₂×C₄×C₇₁	—	2	4	no	—
122	C₂×C₄×C₇₁	—	2	8	no	—
123	C₂×C₄×C₇₁	—	2	8	no	—
124	C₂×C₄×C₇₁	—	4	8	no	—
125	C₂×C₄×C₇₁	—	4	8	no	—
126	C₂×C₄×C₇₁	#137	1	8	yes	—
127	C₂×C₄×C₇₁	C₂²×D₄	1	4	yes	—
128	C₂×C₄×C₇₁	C₂²×D₄	1	2	yes	—
129	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₇₁	C₂²×D₄	1	4	yes	—
131	C₂×C₄×C₇₁	C₂×#41	1	4	yes	—
132	C₂×C₄×C₇₁	C₂⁵	1	1	yes	characteristic
133	C₈×C₇₁	—	2	4	no	—
134	C₈×C₇₁	C₂×#41	1	8	yes	—
135	C₂²×C₇₁	—	2	2	no	—
136	C₂²×C₇₁	—	2	4	no	—
137	C₂²×C₇₁	—	4	4	no	—
138	C₂²×C₇₁	#532	1	2	yes	—
139	C₂²×C₇₁	C₂×#137	1	4	yes	—
140	C₂²×C₇₁	C₂³×D₄	1	1	yes	characteristic
141	C₄×C₇₁	—	2	2	no	—
142	C₄×C₇₁	—	2	4	no	—
143	C₄×C₇₁	—	4	8	no	—
144	C₄×C₇₁	—	4	8	no	—
145	C₄×C₇₁	#532	1	2	yes	—
146	C₄×C₇₁	C₂×#137	1	8	yes	—
147	C₄×C₇₁	C₂²×#41	1	2	yes	—
148	C₂×#3185	C₇₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
149	C₂×C₇₁	—	4	2	no	—
150	C₂×C₇₁	#3185	1	4	yes	direct product factor
151	C₂×C₇₁	C₂×#532	1	1	yes	characteristic
152	C₂×C₇₁	C₂²×#137	1	2	yes	—
153	C₂×#533	C₂×C₇₁	1	2	yes	—
154	#3185	C₂×C₇₁	1	16	yes	direct product factor
155	C₂×#498	C₂×C₇₁	1	1	yes	characteristic, complemented
156	C₂×#494	C₂×C₇₁	1	1	yes	characteristic, complemented
157	C₂×#487	C₂×C₇₁	1	2	yes	complemented
158	C₂×#470	C₂×C₇₁	1	4	yes	—
159	C₂×#441	C₂×C₇₁	1	4	yes	complemented
160	C₂×C₄×M₁₆	C₂×C₇₁	1	1	yes	characteristic
161	C₇₁	C₂×#3185	1	1	yes	characteristic, direct product factor, Sylow
162	C₂²×Q₁₆	C₂²×C₇₁	1	4	yes	—
163	C₂²×M₁₆	C₂²×C₇₁	1	2	yes	—
164	#533	—	2	8	no	—
165	#533	C₂²×C₇₁	1	16	yes	—
166	C₂×C₄⋊Q₈	C₂²×C₇₁	1	2	yes	—
167	C₂×C₄⋊Q₈	C₂²×C₇₁	1	2	yes	—
168	C₂×#125	—	2	2	no	—
169	C₂×C₂²⋊Q₈	—	2	2	no	—
170	C₂×C₂²⋊Q₈	C₂²×C₇₁	1	4	yes	—
171	C₂×C₄×Q₈	—	2	2	no	—
172	C₂×C₄×Q₈	C₂²×C₇₁	1	4	yes	—
173	C₂×#117	—	2	2	no	—
174	C₂²×C₄²	C₂²×C₇₁	1	1	yes	characteristic
175	#498	C₂²×C₇₁	1	8	yes	complemented
176	#494	C₂²×C₇₁	1	8	yes	complemented
177	#487	C₂²×C₇₁	1	16	yes	complemented
178	#470	C₂²×C₇₁	1	32	yes	—
179	#441	C₂²×C₇₁	1	32	yes	complemented
180	C₂×#106	C₂²×C₇₁	1	4	yes	—
181	C₂×#103	C₂²×C₇₁	1	8	yes	—
182	C₄×M₁₆	C₂²×C₇₁	1	8	yes	—
183	C₂×#97	C₂²×C₇₁	1	2	yes	—
184	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
185	C₂×C₄×C₈	C₂²×C₇₁	1	1	yes	characteristic
186	C₂²×Q₈	—	2	2	no	—
187	C₂²×Q₈	C₂³×C₇₁	1	4	yes	—
188	C₂³×C₄	D₄×C₇₁	1	2	yes	—
189	C₂³×C₄	C₂³×C₇₁	1	1	yes	characteristic
190	C₂×Q₁₆	—	2	16	no	—
191	C₂×Q₁₆	C₂³×C₇₁	1	16	yes	—
192	C₂×M₁₆	—	2	8	no	—
193	C₂×M₁₆	C₂³×C₇₁	1	8	yes	—
194	C₂²×C₈	C₂³×C₇₁	1	2	yes	—
195	C₂²×C₈	C₂³×C₇₁	1	2	yes	—
196	C₄⋊Q₈	—	2	4	no	—
197	C₄⋊Q₈	—	2	4	no	—
198	C₄⋊Q₈	C₂³×C₇₁	1	8	yes	—
199	C₄⋊Q₈	C₂³×C₇₁	1	8	yes	—
200	#125	—	2	16	no	—
201	C₂²⋊Q₈	—	2	16	no	—
202	C₂²⋊Q₈	—	2	8	no	—
203	C₂²⋊Q₈	C₂³×C₇₁	1	16	yes	—
204	C₄×Q₈	—	2	16	no	—
205	C₄×Q₈	—	2	8	no	—
206	C₄×Q₈	C₂³×C₇₁	1	16	yes	—
207	#117	—	2	16	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	—	2	4	no	—
212	C₂×C₄⋊C₄	—	2	4	no	—
213	C₂×C₄⋊C₄	C₂³×C₇₁	1	4	yes	—
214	C₂×C₂²⋊C₄	—	2	4	no	—
215	C₂×C₄²	—	2	4	no	—
216	C₂×C₄²	D₄×C₇₁	1	2	yes	—
217	C₂×C₄²	D₄×C₇₁	1	4	yes	—
218	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
219	C₂×C₄²	C₂³×C₇₁	1	1	yes	characteristic
220	C₂×C₄²	C₂³×C₇₁	1	4	yes	—
221	#106	C₂³×C₇₁	1	16	yes	—
222	#103	C₂³×C₇₁	1	32	yes	—
223	#97	C₂³×C₇₁	1	8	yes	—
224	C₄×C₈	C₂³×C₇₁	1	4	yes	—
225	C₄×C₈	C₂³×C₇₁	1	4	yes	—
226	C₂⁴	C₂×D₄×C₇₁	1	1	yes	characteristic
227	C₂×Q₈	—	2	4	no	—
228	C₂×Q₈	—	2	8	no	—
229	C₂×Q₈	—	2	16	no	—
230	C₂×Q₈	—	4	8	no	—
231	C₂×Q₈	C₂⁴×C₇₁	1	8	yes	—
232	C₂²×C₄	—	2	8	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	—	2	4	no	—
235	C₂²×C₄	—	2	4	no	—
236	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
237	C₂²×C₄	C₂×D₄×C₇₁	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
239	C₂²×C₄	C₂×D₄×C₇₁	1	8	yes	—
240	C₂²×C₄	C₂×D₄×C₇₁	1	2	yes	—
241	C₂²×C₄	C₂⁴×C₇₁	1	2	yes	—
242	C₂²×C₄	C₂⁴×C₇₁	1	1	yes	characteristic
243	Q₁₆	—	2	32	no	—
244	M₁₆	—	2	16	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	#41×C₇₁	1	8	yes	—
247	C₂×C₈	C₂⁴×C₇₁	1	4	yes	—
248	C₂×C₈	C₂⁴×C₇₁	1	4	yes	—
249	C₄⋊C₄	—	2	8	no	—
250	C₄⋊C₄	—	2	8	no	—
251	C₄⋊C₄	—	2	8	no	—
252	C₄⋊C₄	—	2	16	no	—
253	C₄⋊C₄	—	2	16	no	—
254	C₄⋊C₄	—	2	4	no	—
255	C₄⋊C₄	C₂⁴×C₇₁	1	8	yes	—
256	C₂²⋊C₄	—	2	16	no	—
257	C₄²	—	2	16	no	—
258	C₄²	C₂×D₄×C₇₁	1	2	yes	—
259	C₄²	C₂×D₄×C₇₁	1	8	yes	—
260	C₄²	C₂×D₄×C₇₁	1	2	yes	—
261	C₄²	C₂⁴×C₇₁	1	2	yes	—
262	C₄²	C₂⁴×C₇₁	1	2	yes	—
263	C₂³	—	2	2	no	—
264	C₂³	—	2	4	no	—
265	C₂³	C₂²×D₄×C₇₁	1	2	yes	—
266	C₂³	C₂²×D₄×C₇₁	1	1	yes	characteristic
267	Q₈	—	2	16	no	—
268	Q₈	—	4	8	no	—
269	Q₈	—	4	16	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	16	no	—
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	2	8	no	—
276	C₂×C₄	—	2	8	no	—
277	C₂×C₄	—	4	8	no	—
278	C₂×C₄	—	4	8	no	—
279	C₂×C₄	#137×C₇₁	1	8	yes	—
280	C₂×C₄	C₂²×D₄×C₇₁	1	4	yes	—
281	C₂×C₄	C₂²×D₄×C₇₁	1	2	yes	—
282	C₂×C₄	C₂²×D₄×C₇₁	1	1	yes	characteristic
283	C₂×C₄	C₂²×D₄×C₇₁	1	4	yes	—
284	C₂×C₄	C₂×#41×C₇₁	1	4	yes	—
285	C₂×C₄	C₂⁵×C₇₁	1	1	yes	characteristic
286	C₈	—	2	4	no	—
287	C₈	C₂×#41×C₇₁	1	8	yes	—
288	C₂²	—	2	4	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	4	4	no	—
291	C₂²	#532×C₇₁	1	2	yes	—
292	C₂²	C₂×#137×C₇₁	1	4	yes	—
293	C₂²	C₂³×D₄×C₇₁	1	1	yes	characteristic
294	C₄	—	2	4	no	—
295	C₄	—	2	2	no	—
296	C₄	—	4	8	no	—
297	C₄	—	4	8	no	—
298	C₄	#532×C₇₁	1	2	yes	—
299	C₄	C₂×#137×C₇₁	1	8	yes	—
300	C₄	C₂²×#41×C₇₁	1	2	yes	—
301	C₂	—	4	2	no	—
302	C₂	C₇₁×#3185	1	4	yes	direct product factor
303	C₂	C₂×#532×C₇₁	1	1	yes	characteristic
304	C₂	C₂²×#137×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 ◁ C₂²×C₄²×C₇₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₇₁×#3185	C₂	direct (N × H)
3. C₂×#498×C₇₁	C₂	semidirect
4. C₂×#494×C₇₁	C₂	semidirect
5. C₂×#487×C₇₁	C₂	semidirect
7. C₂×#441×C₇₁	C₂	semidirect
22. #498×C₇₁	C₂²	semidirect
23. #494×C₇₁	C₂²	semidirect
24. #487×C₇₁	C₂²	semidirect
26. #441×C₇₁	C₂²	semidirect
148. C₂×#3185	C₇₁	direct (N × H)
150. C₂×C₇₁	#3185	direct (N × H)
154. #3185	C₂×C₇₁	direct (N × H)
155. C₂×#498	C₂×C₇₁	semidirect
156. C₂×#494	C₂×C₇₁	semidirect
157. C₂×#487	C₂×C₇₁	semidirect
159. C₂×#441	C₂×C₇₁	semidirect
161. C₇₁	C₂×#3185	direct (N × H)
175. #498	C₂²×C₇₁	semidirect
176. #494	C₂²×C₇₁	semidirect
177. #487	C₂²×C₇₁	semidirect
179. #441	C₂²×C₇₁	semidirect
302. C₂	C₇₁×#3185	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×#533×C₇₁	2
C₇₁×#3185	2
C₂×#498×C₇₁	2
C₂×#494×C₇₁	2
C₂×#487×C₇₁	2
C₂×#470×C₇₁	2
C₂×#441×C₇₁	2
C₂×C₄×M₁₆×C₇₁	2
C₂×#3185	71

Subgroups — C2×C71×#3185 #31881564

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇₁×#3185 #31881564