Subgroups — C₂×C₃₁×#1971 #21477984

Order 7936.

1574 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×#106	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃₁×#103	C₂	1	2	yes	maximal
4	C₂²×C₃₁×#97	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃₁×#391	C₂	1	2	yes	maximal
6	C₃₁×#1971	C₂	1	8	yes	maximal, direct product factor
7	C₂³×Q₈×C₃₁	C₂²	1	2	yes	—
8	C₂²×Q₁₆×C₃₁	—	2	4	no	—
9	C₂²×Q₁₆×C₃₁	—	2	4	no	—
10	C₂²×Q₁₆×C₃₁	C₄	1	8	yes	complemented
11	C₂²×Q₁₆×C₃₁	C₂²	1	4	yes	—
12	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
15	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#106	—	2	4	no	—
17	C₂×C₃₁×#106	C₂²	1	4	yes	—
18	C₂×C₃₁×#103	—	2	8	no	—
19	C₂×C₃₁×#103	C₂²	1	8	yes	—
20	C₂×C₃₁×#97	—	2	4	no	—
21	C₂×C₃₁×#97	C₂²	1	4	yes	—
22	C₃₁×#391	—	2	4	no	—
23	C₃₁×#391	C₂²	1	8	yes	—
24	C₂×C₃₁×#95	—	2	4	no	—
25	C₂²×Q₈×C₃₁	—	2	2	no	—
26	C₂²×Q₈×C₃₁	—	2	4	no	—
27	C₂²×Q₈×C₃₁	—	2	4	no	—
28	C₂²×Q₈×C₃₁	—	2	4	no	—
29	C₂²×Q₈×C₃₁	—	2	4	no	—
30	C₂²×Q₈×C₃₁	—	2	4	no	—
31	C₂²×Q₈×C₃₁	C₂×C₄	1	8	yes	—
32	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
33	C₂³×C₄×C₃₁	—	2	2	no	—
34	C₂³×C₄×C₃₁	—	2	2	no	—
35	C₂³×C₄×C₃₁	D₄	1	2	yes	—
36	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
37	C₂×Q₁₆×C₃₁	—	2	32	no	—
38	C₂×Q₁₆×C₃₁	—	2	8	no	—
39	C₂×Q₁₆×C₃₁	—	2	8	no	—
40	C₂×Q₁₆×C₃₁	C₂×C₄	1	16	yes	complemented
41	C₂²×C₈×C₃₁	—	2	2	no	—
42	C₂²×C₈×C₃₁	—	2	2	no	—
43	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
44	C₂²×C₈×C₃₁	D₄	1	4	yes	—
45	C₂²×C₈×C₃₁	D₄	1	4	yes	—
46	C₂²×C₈×C₃₁	D₄	1	4	yes	—
47	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
48	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
50	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
51	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
52	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
53	C₂×C₄²×C₃₁	—	2	2	no	—
54	C₂×C₄²×C₃₁	—	2	2	no	—
55	C₂×C₄²×C₃₁	D₄	1	2	yes	—
56	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
57	C₃₁×#106	—	2	8	no	—
58	C₃₁×#103	—	2	16	no	—
59	C₃₁×#97	—	2	8	no	—
60	C₃₁×#95	—	2	16	no	—
61	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
62	C₂×Q₈×C₃₁	—	2	8	no	—
63	C₂×Q₈×C₃₁	—	2	4	no	—
64	C₂×Q₈×C₃₁	—	2	16	no	—
65	C₂×Q₈×C₃₁	—	2	16	no	—
66	C₂×Q₈×C₃₁	—	2	4	no	—
67	C₂×Q₈×C₃₁	—	2	4	no	—
68	C₂×Q₈×C₃₁	—	2	4	no	—
69	C₂×Q₈×C₃₁	—	2	4	no	—
70	C₂×Q₈×C₃₁	—	4	4	no	—
71	C₂×Q₈×C₃₁	—	4	8	no	—
72	C₂×Q₈×C₃₁	—	4	8	no	—
73	C₂×Q₈×C₃₁	—	4	4	no	—
74	C₂×Q₈×C₃₁	C₂²×C₄	1	8	yes	—
75	C₂²×C₄×C₃₁	—	2	8	no	—
76	C₂²×C₄×C₃₁	—	2	4	no	—
77	C₂²×C₄×C₃₁	—	2	8	no	—
78	C₂²×C₄×C₃₁	—	2	4	no	—
79	C₂²×C₄×C₃₁	—	2	4	no	—
80	C₂²×C₄×C₃₁	—	4	4	no	—
81	C₂²×C₄×C₃₁	—	4	4	no	—
82	C₂²×C₄×C₃₁	—	4	4	no	—
83	C₂²×C₄×C₃₁	—	4	8	no	—
84	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
86	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₃₁	#41	1	2	yes	—
91	C₂²×C₄×C₃₁	#41	1	2	yes	—
92	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
93	Q₁₆×C₃₁	—	2	32	no	—
94	C₂×C₈×C₃₁	—	2	8	no	—
95	C₂×C₈×C₃₁	—	2	2	no	—
96	C₂×C₈×C₃₁	—	2	2	no	—
97	C₂×C₈×C₃₁	C₂²⋊C₄	1	16	yes	—
98	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
99	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
100	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
101	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
102	C₄⋊C₄×C₃₁	—	2	4	no	—
103	C₄⋊C₄×C₃₁	—	2	4	no	—
104	C₄⋊C₄×C₃₁	—	4	4	no	—
105	C₄²×C₃₁	—	2	4	no	—
106	C₄²×C₃₁	—	2	2	no	—
107	C₄²×C₃₁	—	2	2	no	—
108	C₂×#1971	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂³×C₃₁	C₄×D₄	1	2	yes	—
110	C₂³×C₃₁	#121	1	2	yes	—
111	C₂³×C₃₁	#127	1	2	yes	—
112	C₂³×C₃₁	#137	1	8	yes	—
113	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
114	Q₈×C₃₁	—	2	8	no	—
115	Q₈×C₃₁	—	2	8	no	—
116	Q₈×C₃₁	—	4	4	no	—
117	Q₈×C₃₁	—	4	4	no	—
118	Q₈×C₃₁	—	4	16	no	—
119	C₂×C₄×C₃₁	—	2	8	no	—
120	C₂×C₄×C₃₁	—	2	4	no	—
121	C₂×C₄×C₃₁	—	2	8	no	—
122	C₂×C₄×C₃₁	—	4	4	no	—
123	C₂×C₄×C₃₁	—	4	4	no	—
124	C₂×C₄×C₃₁	—	4	16	no	—
125	C₂×C₄×C₃₁	—	4	8	no	—
126	C₂×C₄×C₃₁	—	4	4	no	—
127	C₂×C₄×C₃₁	—	4	8	no	—
128	C₂×C₄×C₃₁	—	8	8	no	—
129	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
130	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
131	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
132	C₂×C₄×C₃₁	#121	1	4	yes	—
133	C₂×C₄×C₃₁	#121	1	2	yes	—
134	C₂×C₄×C₃₁	#124	1	4	yes	—
135	C₂×C₄×C₃₁	#127	1	2	yes	—
136	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
142	C₈×C₃₁	—	2	4	no	—
143	C₈×C₃₁	C₂×C₂²⋊C₄	1	8	yes	—
144	C₂³×Q₁₆	C₂×C₃₁	1	1	yes	characteristic
145	C₂²×#106	C₂×C₃₁	1	1	yes	characteristic
146	C₂²×#103	C₂×C₃₁	1	2	yes	—
147	C₂²×#97	C₂×C₃₁	1	1	yes	characteristic
148	C₂×#391	C₂×C₃₁	1	2	yes	—
149	#1971	C₂×C₃₁	1	8	yes	direct product factor
150	C₂²×C₃₁	#390	1	4	yes	—
151	C₂²×C₃₁	#441	1	8	yes	—
152	C₂²×C₃₁	#470	1	8	yes	—
153	C₂²×C₃₁	#497	1	8	yes	—
154	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
155	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
156	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
157	C₂²×C₃₁	C₂×#137	1	4	yes	—
158	C₄×C₃₁	—	4	4	no	—
159	C₄×C₃₁	—	4	8	no	—
160	C₄×C₃₁	—	8	4	no	—
161	C₄×C₃₁	#390	1	4	yes	—
162	C₄×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
163	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
164	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
165	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
166	C₂³×Q₈	C₂²×C₃₁	1	2	yes	—
167	C₂²×Q₁₆	—	2	4	no	—
168	C₂²×Q₁₆	—	2	4	no	—
169	C₂²×Q₁₆	C₄×C₃₁	1	8	yes	complemented
170	C₂²×Q₁₆	C₂²×C₃₁	1	4	yes	—
171	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
172	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
173	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
174	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
175	C₂×#106	—	2	4	no	—
176	C₂×#106	C₂²×C₃₁	1	4	yes	—
177	C₂×#103	—	2	8	no	—
178	C₂×#103	C₂²×C₃₁	1	8	yes	—
179	C₂×#97	—	2	4	no	—
180	C₂×#97	C₂²×C₃₁	1	4	yes	—
181	#391	—	2	4	no	—
182	#391	C₂²×C₃₁	1	8	yes	—
183	C₂×#95	—	2	4	no	—
184	C₂×C₃₁	#1971	1	8	yes	direct product factor
185	C₂×C₃₁	C₂×#390	1	1	yes	characteristic
186	C₂×C₃₁	C₂×#441	1	2	yes	—
187	C₂×C₃₁	C₂×#470	1	2	yes	—
188	C₂×C₃₁	C₂×#497	1	2	yes	—
189	C₂²×Q₈	—	2	2	no	—
190	C₂²×Q₈	—	2	4	no	—
191	C₂²×Q₈	—	2	4	no	—
192	C₂²×Q₈	—	2	4	no	—
193	C₂²×Q₈	—	2	4	no	—
194	C₂²×Q₈	—	2	4	no	—
195	C₂²×Q₈	C₂×C₄×C₃₁	1	8	yes	—
196	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
197	C₂³×C₄	—	2	2	no	—
198	C₂³×C₄	—	2	2	no	—
199	C₂³×C₄	D₄×C₃₁	1	2	yes	—
200	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
201	C₂×Q₁₆	—	2	32	no	—
202	C₂×Q₁₆	—	2	8	no	—
203	C₂×Q₁₆	—	2	8	no	—
204	C₂×Q₁₆	C₂×C₄×C₃₁	1	16	yes	complemented
205	C₂²×C₈	—	2	2	no	—
206	C₂²×C₈	—	2	2	no	—
207	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
208	C₂²×C₈	D₄×C₃₁	1	4	yes	—
209	C₂²×C₈	D₄×C₃₁	1	4	yes	—
210	C₂²×C₈	D₄×C₃₁	1	4	yes	—
211	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
212	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
213	C₂×C₄⋊C₄	—	2	4	no	—
214	C₂×C₄⋊C₄	—	2	4	no	—
215	C₂×C₄⋊C₄	—	2	2	no	—
216	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
217	C₂×C₄²	—	2	2	no	—
218	C₂×C₄²	—	2	2	no	—
219	C₂×C₄²	D₄×C₃₁	1	2	yes	—
220	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
221	#106	—	2	8	no	—
222	#103	—	2	16	no	—
223	#97	—	2	8	no	—
224	#95	—	2	16	no	—
225	C₃₁	C₂×#1971	1	1	yes	characteristic, direct product factor, Sylow
226	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
227	C₂×Q₈	—	2	8	no	—
228	C₂×Q₈	—	2	4	no	—
229	C₂×Q₈	—	2	16	no	—
230	C₂×Q₈	—	2	16	no	—
231	C₂×Q₈	—	2	4	no	—
232	C₂×Q₈	—	2	4	no	—
233	C₂×Q₈	—	2	4	no	—
234	C₂×Q₈	—	2	4	no	—
235	C₂×Q₈	—	4	4	no	—
236	C₂×Q₈	—	4	8	no	—
237	C₂×Q₈	—	4	8	no	—
238	C₂×Q₈	—	4	4	no	—
239	C₂×Q₈	C₂²×C₄×C₃₁	1	8	yes	—
240	C₂²×C₄	—	2	8	no	—
241	C₂²×C₄	—	2	4	no	—
242	C₂²×C₄	—	2	8	no	—
243	C₂²×C₄	—	2	4	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	4	4	no	—
246	C₂²×C₄	—	4	4	no	—
247	C₂²×C₄	—	4	4	no	—
248	C₂²×C₄	—	4	8	no	—
249	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
250	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
251	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
252	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
253	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
254	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
255	C₂²×C₄	#41×C₃₁	1	2	yes	—
256	C₂²×C₄	#41×C₃₁	1	2	yes	—
257	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
258	Q₁₆	—	2	32	no	—
259	C₂×C₈	—	2	8	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	C₂²⋊C₄×C₃₁	1	16	yes	—
263	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
264	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
265	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
266	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	4	4	no	—
270	C₄²	—	2	4	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₂³	C₄×D₄×C₃₁	1	2	yes	—
274	C₂³	C₃₁×#121	1	2	yes	—
275	C₂³	C₃₁×#127	1	2	yes	—
276	C₂³	C₃₁×#137	1	8	yes	—
277	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
278	Q₈	—	2	8	no	—
279	Q₈	—	2	8	no	—
280	Q₈	—	4	4	no	—
281	Q₈	—	4	4	no	—
282	Q₈	—	4	16	no	—
283	C₂×C₄	—	2	8	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	8	no	—
286	C₂×C₄	—	4	8	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	4	16	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	4	8	no	—
292	C₂×C₄	—	8	8	no	—
293	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
294	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
295	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
296	C₂×C₄	C₃₁×#121	1	4	yes	—
297	C₂×C₄	C₃₁×#121	1	2	yes	—
298	C₂×C₄	C₃₁×#124	1	4	yes	—
299	C₂×C₄	C₃₁×#127	1	2	yes	—
300	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
301	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
302	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
303	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
304	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
305	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
306	C₈	—	2	4	no	—
307	C₈	C₂×C₂²⋊C₄×C₃₁	1	8	yes	—
308	C₂²	C₃₁×#390	1	4	yes	—
309	C₂²	C₃₁×#441	1	8	yes	—
310	C₂²	C₃₁×#470	1	8	yes	—
311	C₂²	C₃₁×#497	1	8	yes	—
312	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
313	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
314	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
315	C₂²	C₂×C₃₁×#137	1	4	yes	—
316	C₄	—	4	4	no	—
317	C₄	—	4	8	no	—
318	C₄	—	8	4	no	—
319	C₄	C₃₁×#390	1	4	yes	—
320	C₄	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
321	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
322	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
323	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
324	C₂	C₃₁×#1971	1	8	yes	direct product factor
325	C₂	C₂×C₃₁×#390	1	1	yes	characteristic
326	C₂	C₂×C₃₁×#441	1	2	yes	—
327	C₂	C₂×C₃₁×#470	1	2	yes	—
328	C₂	C₂×C₃₁×#497	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₃₁×#1971	C₂	direct (N × H)
10. C₂²×Q₁₆×C₃₁	C₄	semidirect
40. C₂×Q₁₆×C₃₁	C₂×C₄	semidirect
108. C₂×#1971	C₃₁	direct (N × H)
149. #1971	C₂×C₃₁	direct (N × H)
169. C₂²×Q₁₆	C₄×C₃₁	semidirect
184. C₂×C₃₁	#1971	direct (N × H)
204. C₂×Q₁₆	C₂×C₄×C₃₁	semidirect
225. C₃₁	C₂×#1971	direct (N × H)
324. C₂	C₃₁×#1971	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×Q₁₆×C₃₁	2
C₂²×C₃₁×#106	2
C₂²×C₃₁×#103	2
C₂²×C₃₁×#97	2
C₂×C₃₁×#391	2
C₃₁×#1971	2
C₂×#1971	31

Subgroups — C2×C31×#1971 #21477984

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1971 #21477984