Subgroups — C₃₁×#8609 #21465368

Order 7936.

550 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	C₂×C₈×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1808	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1799	C₂	1	2	yes	maximal
4	C₃₁×#1289	C₂	1	2	yes	maximal, complemented
5	C₂³×C₈×C₃₁	C₄	1	1	yes	characteristic, complemented
6	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
8	C₈×D₄×C₃₁	—	2	2	no	—
9	C₈×D₄×C₃₁	—	2	2	no	—
10	C₂×C₃₁×#107	—	2	2	no	—
11	C₂×C₃₁×C₄⋊C₈	C₄	1	1	yes	characteristic, complemented
12	C₂×C₃₁×#99	—	2	2	no	—
13	C₂×C₃₁×C₂²⋊C₈	C₄	1	1	yes	characteristic, complemented
14	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₃₁	C₄	1	1	yes	characteristic, complemented
16	C₃₁×#380	C₂²	1	4	yes	—
17	C₃₁×#342	—	2	2	no	—
18	C₃₁×#340	—	2	2	no	—
19	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
20	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
21	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
22	C₂²×C₈×C₃₁	—	2	1	no	—
23	C₂²×C₈×C₃₁	—	2	1	no	—
24	C₂²×C₈×C₃₁	—	4	2	no	—
25	C₂²×C₈×C₃₁	—	4	2	no	—
26	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₃₁	D₄	1	2	yes	—
31	C₂²×C₈×C₃₁	D₄	1	2	yes	—
32	C₄×D₄×C₃₁	—	2	2	no	—
33	C₄×D₄×C₃₁	—	2	2	no	—
34	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
35	C₂×C₄⋊C₄×C₃₁	Q₈	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
37	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
38	C₂×C₂²⋊C₄×C₃₁	Q₈	1	1	yes	characteristic
39	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
40	C₃₁×#107	—	4	4	no	—
41	C₃₁×C₄⋊C₈	—	2	2	no	—
42	C₃₁×#99	—	4	4	no	—
43	C₃₁×C₂²⋊C₈	—	2	2	no	—
44	C₃₁×C₂²⋊C₈	—	2	2	no	—
45	C₄×C₈×C₃₁	—	2	1	no	—
46	C₄×C₈×C₃₁	—	2	1	no	—
47	C₃₁×#95	—	2	4	no	—
48	C₂⁴×C₃₁	C₄⋊C₄	1	1	yes	characteristic
49	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
50	C₂×D₄×C₃₁	—	2	2	no	—
51	C₂×D₄×C₃₁	—	2	2	no	—
52	C₂×D₄×C₃₁	D₈	1	2	yes	—
53	C₂×D₄×C₃₁	Q₁₆	1	2	yes	—
54	C₂²×C₄×C₃₁	—	2	1	no	—
55	C₂²×C₄×C₃₁	—	2	1	no	—
56	C₂²×C₄×C₃₁	—	4	1	no	—
57	C₂²×C₄×C₃₁	—	4	1	no	—
58	C₂²×C₄×C₃₁	—	4	1	no	—
59	C₂²×C₄×C₃₁	—	4	1	no	—
60	C₂²×C₄×C₃₁	—	4	4	no	—
61	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	#41	1	2	yes	—
68	C₂×C₈×C₃₁	—	2	4	no	—
69	C₂×C₈×C₃₁	—	2	2	no	—
70	C₂×C₈×C₃₁	—	2	1	no	—
71	C₂×C₈×C₃₁	—	2	1	no	—
72	C₂×C₈×C₃₁	—	4	1	no	—
73	C₂×C₈×C₃₁	—	4	2	no	—
74	C₂×C₈×C₃₁	—	4	2	no	—
75	C₂×C₈×C₃₁	—	4	2	no	—
76	C₂×C₈×C₃₁	—	4	1	no	—
77	C₂×C₈×C₃₁	—	4	1	no	—
78	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
80	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
81	C₄⋊C₄×C₃₁	—	2	2	no	—
82	C₄⋊C₄×C₃₁	—	4	4	no	—
83	C₄⋊C₄×C₃₁	—	4	4	no	—
84	C₂²⋊C₄×C₃₁	—	2	2	no	—
85	C₂²⋊C₄×C₃₁	—	2	2	no	—
86	C₂²⋊C₄×C₃₁	—	4	4	no	—
87	C₂²⋊C₄×C₃₁	—	4	4	no	—
88	C₄²×C₃₁	—	2	1	no	—
89	C₄²×C₃₁	—	2	1	no	—
90	#8609	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₃₁	—	2	1	no	—
92	C₂³×C₃₁	—	2	1	no	—
93	C₂³×C₃₁	—	4	1	no	—
94	C₂³×C₃₁	—	4	1	no	—
95	C₂³×C₃₁	—	4	1	no	—
96	C₂³×C₃₁	—	4	1	no	—
97	C₂³×C₃₁	#107	1	2	yes	complemented
98	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂³×C₃₁	#121	1	2	yes	—
100	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
101	C₂³×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
102	C₂³×C₃₁	#135	1	2	yes	—
103	D₄×C₃₁	—	2	4	no	—
104	D₄×C₃₁	—	2	4	no	—
105	C₂×C₄×C₃₁	—	2	1	no	—
106	C₂×C₄×C₃₁	—	2	1	no	—
107	C₂×C₄×C₃₁	—	2	2	no	—
108	C₂×C₄×C₃₁	—	2	1	no	—
109	C₂×C₄×C₃₁	—	2	2	no	—
110	C₂×C₄×C₃₁	—	2	1	no	—
111	C₂×C₄×C₃₁	—	2	1	no	—
112	C₂×C₄×C₃₁	—	2	1	no	—
113	C₂×C₄×C₃₁	—	4	1	no	—
114	C₂×C₄×C₃₁	—	4	2	no	—
115	C₂×C₄×C₃₁	—	4	2	no	—
116	C₂×C₄×C₃₁	—	4	1	no	—
117	C₂×C₄×C₃₁	—	4	1	no	—
118	C₂×C₄×C₃₁	—	4	1	no	—
119	C₂×C₄×C₃₁	—	4	1	no	—
120	C₂×C₄×C₃₁	—	8	4	no	—
121	C₂×C₄×C₃₁	—	8	4	no	—
122	C₂×C₄×C₃₁	—	8	4	no	—
123	C₂×C₄×C₃₁	#107	1	2	yes	complemented
124	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
126	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
127	C₈×C₃₁	—	2	1	no	—
128	C₈×C₃₁	—	2	2	no	—
129	C₈×C₃₁	—	4	1	no	—
130	C₈×C₃₁	—	4	1	no	—
131	C₈×C₃₁	#99	1	2	yes	complemented
132	C₂×C₈×D₄	C₂×C₃₁	1	1	yes	characteristic
133	#1808	C₂×C₃₁	1	2	yes	complemented
134	#1799	C₂×C₃₁	1	2	yes	—
135	#1289	C₂×C₃₁	1	2	yes	complemented
136	C₂²×C₃₁	—	2	1	no	—
137	C₂²×C₃₁	—	2	2	no	—
138	C₂²×C₃₁	—	2	1	no	—
139	C₂²×C₃₁	—	2	2	no	—
140	C₂²×C₃₁	—	2	2	no	—
141	C₂²×C₃₁	—	4	1	no	—
142	C₂²×C₃₁	—	4	1	no	—
143	C₂²×C₃₁	—	4	2	no	—
144	C₂²×C₃₁	—	4	2	no	—
145	C₂²×C₃₁	—	4	2	no	—
146	C₂²×C₃₁	—	4	1	no	—
147	C₂²×C₃₁	—	4	1	no	—
148	C₂²×C₃₁	#380	1	1	yes	characteristic
149	C₂²×C₃₁	C₂×#107	1	1	yes	characteristic
150	C₂²×C₃₁	#427	1	1	yes	characteristic
151	C₂²×C₃₁	#466	1	2	yes	—
152	C₂²×C₃₁	#467	1	2	yes	—
153	C₄×C₃₁	—	2	1	no	—
154	C₄×C₃₁	—	2	2	no	—
155	C₄×C₃₁	—	2	2	no	—
156	C₄×C₃₁	—	4	1	no	—
157	C₄×C₃₁	—	4	1	no	—
158	C₄×C₃₁	—	16	4	no	—
159	C₄×C₃₁	C₂×#99	1	1	yes	characteristic
160	C₄×C₃₁	#410	1	1	yes	characteristic
161	C₂³×C₈	C₄×C₃₁	1	1	yes	characteristic, complemented
162	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
163	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
164	C₈×D₄	—	2	2	no	—
165	C₈×D₄	—	2	2	no	—
166	C₂×#107	—	2	2	no	—
167	C₂×C₄⋊C₈	C₄×C₃₁	1	1	yes	characteristic, complemented
168	C₂×#99	—	2	2	no	—
169	C₂×C₂²⋊C₈	C₄×C₃₁	1	1	yes	characteristic, complemented
170	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
171	C₂×C₄×C₈	C₄×C₃₁	1	1	yes	characteristic, complemented
172	#380	C₂²×C₃₁	1	4	yes	—
173	#342	—	2	2	no	—
174	#340	—	2	2	no	—
175	C₂×C₃₁	—	2	2	no	—
176	C₂×C₃₁	—	2	1	no	—
177	C₂×C₃₁	—	2	1	no	—
178	C₂×C₃₁	—	2	2	no	—
179	C₂×C₃₁	—	4	2	no	—
180	C₂×C₃₁	#1799	1	1	yes	characteristic
181	C₂×C₃₁	#1808	1	1	yes	characteristic
182	C₂×C₃₁	#1812	1	1	yes	characteristic
183	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
184	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
185	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
186	C₂²×C₈	—	2	1	no	—
187	C₂²×C₈	—	2	1	no	—
188	C₂²×C₈	—	4	2	no	—
189	C₂²×C₈	—	4	2	no	—
190	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
191	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
192	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
193	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
194	C₂²×C₈	D₄×C₃₁	1	2	yes	—
195	C₂²×C₈	D₄×C₃₁	1	2	yes	—
196	C₄×D₄	—	2	2	no	—
197	C₄×D₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	4	no	—
199	C₂×C₄⋊C₄	Q₈×C₃₁	1	1	yes	characteristic
200	C₂×C₂²⋊C₄	—	2	4	no	—
201	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
202	C₂×C₂²⋊C₄	Q₈×C₃₁	1	1	yes	characteristic
203	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
204	#107	—	4	4	no	—
205	C₄⋊C₈	—	2	2	no	—
206	#99	—	4	4	no	—
207	C₂²⋊C₈	—	2	2	no	—
208	C₂²⋊C₈	—	2	2	no	—
209	C₄×C₈	—	2	1	no	—
210	C₄×C₈	—	2	1	no	—
211	#95	—	2	4	no	—
212	C₃₁	#8609	1	1	yes	characteristic, direct product factor, Sylow
213	C₂⁴	C₄⋊C₄×C₃₁	1	1	yes	characteristic
214	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	—	2	2	no	—
217	C₂×D₄	D₈×C₃₁	1	2	yes	—
218	C₂×D₄	Q₁₆×C₃₁	1	2	yes	—
219	C₂²×C₄	—	2	1	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	—	4	1	no	—
224	C₂²×C₄	—	4	1	no	—
225	C₂²×C₄	—	4	4	no	—
226	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
227	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
228	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
229	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
230	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
231	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
232	C₂²×C₄	#41×C₃₁	1	2	yes	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	—	4	2	no	—
241	C₂×C₈	—	4	1	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
244	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
245	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	4	4	no	—
248	C₄⋊C₄	—	4	4	no	—
249	C₂²⋊C₄	—	2	2	no	—
250	C₂²⋊C₄	—	2	2	no	—
251	C₂²⋊C₄	—	4	4	no	—
252	C₂²⋊C₄	—	4	4	no	—
253	C₄²	—	2	1	no	—
254	C₄²	—	2	1	no	—
255	C₂³	—	2	1	no	—
256	C₂³	—	2	1	no	—
257	C₂³	—	4	1	no	—
258	C₂³	—	4	1	no	—
259	C₂³	—	4	1	no	—
260	C₂³	—	4	1	no	—
261	C₂³	C₃₁×#107	1	2	yes	complemented
262	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
263	C₂³	C₃₁×#121	1	2	yes	—
264	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
265	C₂³	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
266	C₂³	C₃₁×#135	1	2	yes	—
267	D₄	—	2	4	no	—
268	D₄	—	2	4	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	8	4	no	—
285	C₂×C₄	—	8	4	no	—
286	C₂×C₄	—	8	4	no	—
287	C₂×C₄	C₃₁×#107	1	2	yes	complemented
288	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
289	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
290	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
291	C₈	—	2	1	no	—
292	C₈	—	2	2	no	—
293	C₈	—	4	1	no	—
294	C₈	—	4	1	no	—
295	C₈	C₃₁×#99	1	2	yes	complemented
296	C₂²	—	2	1	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	1	no	—
303	C₂²	—	4	2	no	—
304	C₂²	—	4	2	no	—
305	C₂²	—	4	2	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	C₃₁×#380	1	1	yes	characteristic
309	C₂²	C₂×C₃₁×#107	1	1	yes	characteristic
310	C₂²	C₃₁×#427	1	1	yes	characteristic
311	C₂²	C₃₁×#466	1	2	yes	—
312	C₂²	C₃₁×#467	1	2	yes	—
313	C₄	—	2	1	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	16	4	no	—
319	C₄	C₂×C₃₁×#99	1	1	yes	characteristic
320	C₄	C₃₁×#410	1	1	yes	characteristic
321	C₂	—	2	2	no	—
322	C₂	—	2	1	no	—
323	C₂	—	2	1	no	—
324	C₂	—	2	2	no	—
325	C₂	—	4	2	no	—
326	C₂	C₃₁×#1799	1	1	yes	characteristic
327	C₂	C₃₁×#1808	1	1	yes	characteristic
328	C₂	C₃₁×#1812	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	#380
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂³×C₄	C₂³×C₃₁
prime radical	C₂²×D₄×C₃₁	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1808	C₂	semidirect
4. C₃₁×#1289	C₂	semidirect
5. C₂³×C₈×C₃₁	C₄	semidirect
11. C₂×C₃₁×C₄⋊C₈	C₄	semidirect
13. C₂×C₃₁×C₂²⋊C₈	C₄	semidirect
15. C₂×C₄×C₈×C₃₁	C₄	semidirect
90. #8609	C₃₁	direct (N × H)
97. C₂³×C₃₁	#107	semidirect
123. C₂×C₄×C₃₁	#107	semidirect
131. C₈×C₃₁	#99	semidirect
133. #1808	C₂×C₃₁	semidirect
135. #1289	C₂×C₃₁	semidirect
161. C₂³×C₈	C₄×C₃₁	semidirect
167. C₂×C₄⋊C₈	C₄×C₃₁	semidirect
169. C₂×C₂²⋊C₈	C₄×C₃₁	semidirect
171. C₂×C₄×C₈	C₄×C₃₁	semidirect
212. C₃₁	#8609	direct (N × H)
261. C₂³	C₃₁×#107	semidirect
287. C₂×C₄	C₃₁×#107	semidirect
295. C₈	C₃₁×#99	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₄×C₃₁	2
C₃₁×#1808	2
C₃₁×#1799	2
C₃₁×#1289	2
#8609	31

Subgroups — C31×#8609 #21465368

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8609 #21465368