Subgroups — C₃₁×#29263 #21486022

Order 7936.

798 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2331	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#2328	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2092	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2023	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1982	C₂	1	2	yes	maximal
6	C₃₁×#1949	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1864	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1862	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1832	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
13	C₂×C₃₁×#107	C₂²	1	2	yes	—
14	C₂×C₃₁×#106	C₂²	1	2	yes	—
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃₁×#102	C₂²	1	2	yes	—
17	C₂×C₃₁×#102	C₂²	1	2	yes	—
18	C₂×C₃₁×#102	C₂²	1	2	yes	—
19	C₂×C₃₁×#102	C₂²	1	2	yes	—
20	C₃₁×#391	—	2	2	no	—
21	C₃₁×#390	C₂²	1	2	yes	—
22	C₃₁×#389	C₂²	1	2	yes	—
23	C₃₁×#387	—	2	2	no	—
24	C₃₁×#386	—	2	2	no	—
25	C₃₁×#386	—	2	2	no	—
26	C₃₁×#382	C₂²	1	1	yes	characteristic
27	C₃₁×#382	C₂²	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
29	C₃₁×#376	C₂²	1	1	yes	characteristic
30	C₃₁×#340	C₂²	1	2	yes	—
31	C₃₁×#340	C₂²	1	2	yes	—
32	C₃₁×#336	C₂²	1	2	yes	—
33	C₃₁×#336	C₂²	1	1	yes	characteristic
34	C₃₁×#336	C₂²	1	1	yes	characteristic
35	C₂²×Q₈×C₃₁	—	2	1	no	—
36	C₂²×D₄×C₃₁	—	2	2	no	—
37	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
38	C₂³×C₄×C₃₁	—	2	2	no	—
39	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
40	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
41	C₃₁×C₄⋊Q₈	—	2	4	no	—
42	C₃₁×#127	—	2	2	no	—
43	C₃₁×#127	Q₈	1	4	yes	complemented
44	C₄×D₄×C₃₁	—	4	2	no	—
45	C₄×D₄×C₃₁	—	4	2	no	—
46	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
47	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
48	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
51	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
52	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
53	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
54	C₂×C₄²×C₃₁	—	2	2	no	—
55	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
58	C₃₁×#107	—	4	2	no	—
59	C₃₁×#106	—	4	2	no	—
60	C₃₁×C₄⋊C₈	—	2	4	no	—
61	C₃₁×#102	—	2	4	no	—
62	C₃₁×#102	—	2	4	no	—
63	C₃₁×#102	—	4	2	no	—
64	C₃₁×#102	—	4	2	no	—
65	C₃₁×#95	—	2	2	no	—
66	C₃₁×#95	—	2	2	no	—
67	C₃₁×#95	—	2	1	no	—
68	C₃₁×#95	—	2	1	no	—
69	C₃₁×#95	—	2	2	no	—
70	C₃₁×#95	D₄	1	2	yes	—
71	C₃₁×#95	D₄	1	2	yes	—
72	C₂⁴×C₃₁	—	2	2	no	—
73	C₂×Q₈×C₃₁	—	2	4	no	—
74	C₂×Q₈×C₃₁	—	8	2	no	—
75	C₂×D₄×C₃₁	—	2	8	no	—
76	C₂×D₄×C₃₁	—	2	2	no	—
77	C₂×D₄×C₃₁	—	2	2	no	—
78	C₂×D₄×C₃₁	—	2	2	no	—
79	C₂×D₄×C₃₁	—	2	2	no	—
80	C₂×D₄×C₃₁	—	2	2	no	—
81	C₂×D₄×C₃₁	—	8	2	no	—
82	C₂×D₄×C₃₁	—	8	2	no	—
83	C₂×D₄×C₃₁	C₂×Q₈	1	4	yes	—
84	C₂²×C₄×C₃₁	—	2	2	no	—
85	C₂²×C₄×C₃₁	—	2	2	no	—
86	C₂²×C₄×C₃₁	—	2	2	no	—
87	C₂²×C₄×C₃₁	—	4	4	no	—
88	C₂²×C₄×C₃₁	—	8	2	no	—
89	C₂²×C₄×C₃₁	—	8	2	no	—
90	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
93	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
95	C₂×C₈×C₃₁	—	2	2	no	—
96	C₂×C₈×C₃₁	—	2	2	no	—
97	C₂×C₈×C₃₁	—	2	2	no	—
98	C₂×C₈×C₃₁	—	2	2	no	—
99	C₂×C₈×C₃₁	—	2	2	no	—
100	C₂×C₈×C₃₁	—	2	2	no	—
101	C₄⋊C₄×C₃₁	—	2	4	no	—
102	C₄⋊C₄×C₃₁	—	2	2	no	—
103	C₄⋊C₄×C₃₁	—	2	2	no	—
104	C₄⋊C₄×C₃₁	—	2	2	no	—
105	C₄⋊C₄×C₃₁	—	2	2	no	—
106	C₄⋊C₄×C₃₁	—	2	2	no	—
107	C₄⋊C₄×C₃₁	—	8	2	no	—
108	C₄⋊C₄×C₃₁	#41	1	4	yes	—
109	C₂²⋊C₄×C₃₁	—	4	4	no	—
110	C₂²⋊C₄×C₃₁	—	8	2	no	—
111	C₂²⋊C₄×C₃₁	—	8	2	no	—
112	C₄²×C₃₁	—	2	2	no	—
113	C₄²×C₃₁	—	2	2	no	—
114	C₄²×C₃₁	—	8	2	no	—
115	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
116	C₄²×C₃₁	#41	1	2	yes	—
117	#29263	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂³×C₃₁	—	2	4	no	—
119	C₂³×C₃₁	—	4	2	no	—
120	C₂³×C₃₁	—	4	2	no	—
121	C₂³×C₃₁	—	4	2	no	—
122	C₂³×C₃₁	—	4	2	no	—
123	C₂³×C₃₁	—	4	2	no	—
124	C₂³×C₃₁	—	4	2	no	—
125	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
126	Q₈×C₃₁	—	8	4	no	—
127	D₄×C₃₁	—	2	4	no	—
128	D₄×C₃₁	—	2	4	no	—
129	D₄×C₃₁	—	4	2	no	—
130	D₄×C₃₁	—	4	2	no	—
131	D₄×C₃₁	—	8	8	no	—
132	C₂×C₄×C₃₁	—	2	2	no	—
133	C₂×C₄×C₃₁	—	2	2	no	—
134	C₂×C₄×C₃₁	—	2	1	no	—
135	C₂×C₄×C₃₁	—	2	1	no	—
136	C₂×C₄×C₃₁	—	2	4	no	—
137	C₂×C₄×C₃₁	—	2	2	no	—
138	C₂×C₄×C₃₁	—	2	2	no	—
139	C₂×C₄×C₃₁	—	2	2	no	—
140	C₂×C₄×C₃₁	—	2	2	no	—
141	C₂×C₄×C₃₁	—	2	2	no	—
142	C₂×C₄×C₃₁	—	4	2	no	—
143	C₂×C₄×C₃₁	—	4	2	no	—
144	C₂×C₄×C₃₁	—	4	2	no	—
145	C₂×C₄×C₃₁	—	4	2	no	—
146	C₂×C₄×C₃₁	—	4	2	no	—
147	C₂×C₄×C₃₁	—	4	2	no	—
148	C₂×C₄×C₃₁	—	4	2	no	—
149	C₂×C₄×C₃₁	—	4	2	no	—
150	C₂×C₄×C₃₁	—	8	8	no	—
151	C₂×C₄×C₃₁	C₂²⋊Q₈	1	4	yes	—
152	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
153	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
154	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
155	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
156	C₈×C₃₁	—	4	2	no	—
157	C₈×C₃₁	—	4	2	no	—
158	#2331	C₂×C₃₁	1	1	yes	characteristic, complemented
159	#2328	C₂×C₃₁	1	1	yes	characteristic
160	#2092	C₂×C₃₁	1	2	yes	complemented
161	#2023	C₂×C₃₁	1	2	yes	complemented
162	#1982	C₂×C₃₁	1	2	yes	—
163	#1949	C₂×C₃₁	1	2	yes	complemented
164	#1864	C₂×C₃₁	1	2	yes	complemented
165	#1862	C₂×C₃₁	1	2	yes	complemented
166	#1832	C₂×C₃₁	1	1	yes	characteristic, complemented
167	C₂²×C₃₁	—	4	4	no	—
168	C₂²×C₃₁	—	4	4	no	—
169	C₂²×C₃₁	—	4	4	no	—
170	C₂²×C₃₁	—	8	2	no	—
171	C₂²×C₃₁	—	8	2	no	—
172	C₂²×C₃₁	—	8	2	no	—
173	C₂²×C₃₁	—	8	2	no	—
174	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
175	C₂²×C₃₁	#525	1	1	yes	characteristic
176	C₂²×C₃₁	#535	1	1	yes	characteristic
177	C₂²×C₃₁	#576	1	2	yes	—
178	C₂²×C₃₁	#577	1	2	yes	—
179	C₄×C₃₁	—	2	4	no	—
180	C₄×C₃₁	—	4	1	no	—
181	C₄×C₃₁	—	4	4	no	—
182	C₄×C₃₁	—	4	1	no	—
183	C₄×C₃₁	—	4	2	no	—
184	C₄×C₃₁	—	8	2	no	—
185	C₄×C₃₁	—	8	2	no	—
186	C₄×C₃₁	#554	1	2	yes	—
187	C₄×C₃₁	#559	1	2	yes	—
188	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
189	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
190	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
191	C₂×#107	C₂²×C₃₁	1	2	yes	—
192	C₂×#106	C₂²×C₃₁	1	2	yes	—
193	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
194	C₂×#102	C₂²×C₃₁	1	2	yes	—
195	C₂×#102	C₂²×C₃₁	1	2	yes	—
196	C₂×#102	C₂²×C₃₁	1	2	yes	—
197	C₂×#102	C₂²×C₃₁	1	2	yes	—
198	#391	—	2	2	no	—
199	#390	C₂²×C₃₁	1	2	yes	—
200	#389	C₂²×C₃₁	1	2	yes	—
201	#387	—	2	2	no	—
202	#386	—	2	2	no	—
203	#386	—	2	2	no	—
204	#382	C₂²×C₃₁	1	1	yes	characteristic
205	#382	C₂²×C₃₁	1	1	yes	characteristic
206	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
207	#376	C₂²×C₃₁	1	1	yes	characteristic
208	#340	C₂²×C₃₁	1	2	yes	—
209	#340	C₂²×C₃₁	1	2	yes	—
210	#336	C₂²×C₃₁	1	2	yes	—
211	#336	C₂²×C₃₁	1	1	yes	characteristic
212	#336	C₂²×C₃₁	1	1	yes	characteristic
213	C₂×C₃₁	—	8	4	no	—
214	C₂×C₃₁	#2580	1	1	yes	characteristic
215	C₂×C₃₁	#3081	1	2	yes	—
216	C₂×C₃₁	#3122	1	1	yes	characteristic
217	C₂×C₃₁	#3126	1	1	yes	characteristic
218	C₂×C₃₁	#3224	1	2	yes	—
219	C₂²×Q₈	—	2	1	no	—
220	C₂²×D₄	—	2	2	no	—
221	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
222	C₂³×C₄	—	2	2	no	—
223	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
224	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
225	C₄⋊Q₈	—	2	4	no	—
226	#127	—	2	2	no	—
227	#127	Q₈×C₃₁	1	4	yes	complemented
228	C₄×D₄	—	4	2	no	—
229	C₄×D₄	—	4	2	no	—
230	C₂×C₄⋊C₄	—	2	1	no	—
231	C₂×C₄⋊C₄	—	2	2	no	—
232	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
233	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
234	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
235	C₂×C₂²⋊C₄	—	2	2	no	—
236	C₂×C₂²⋊C₄	—	2	2	no	—
237	C₂×C₂²⋊C₄	—	2	2	no	—
238	C₂×C₄²	—	2	2	no	—
239	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
240	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
241	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
242	#107	—	4	2	no	—
243	#106	—	4	2	no	—
244	C₄⋊C₈	—	2	4	no	—
245	#102	—	2	4	no	—
246	#102	—	2	4	no	—
247	#102	—	4	2	no	—
248	#102	—	4	2	no	—
249	#95	—	2	2	no	—
250	#95	—	2	2	no	—
251	#95	—	2	1	no	—
252	#95	—	2	1	no	—
253	#95	—	2	2	no	—
254	#95	D₄×C₃₁	1	2	yes	—
255	#95	D₄×C₃₁	1	2	yes	—
256	C₃₁	#29263	1	1	yes	characteristic, direct product factor, Sylow
257	C₂⁴	—	2	2	no	—
258	C₂×Q₈	—	2	4	no	—
259	C₂×Q₈	—	8	2	no	—
260	C₂×D₄	—	2	8	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	2	2	no	—
263	C₂×D₄	—	2	2	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	2	2	no	—
266	C₂×D₄	—	8	2	no	—
267	C₂×D₄	—	8	2	no	—
268	C₂×D₄	C₂×Q₈×C₃₁	1	4	yes	—
269	C₂²×C₄	—	2	2	no	—
270	C₂²×C₄	—	2	2	no	—
271	C₂²×C₄	—	2	2	no	—
272	C₂²×C₄	—	4	4	no	—
273	C₂²×C₄	—	8	2	no	—
274	C₂²×C₄	—	8	2	no	—
275	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
276	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
277	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
278	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
279	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₄⋊C₄	—	2	4	no	—
287	C₄⋊C₄	—	2	2	no	—
288	C₄⋊C₄	—	2	2	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	8	2	no	—
293	C₄⋊C₄	#41×C₃₁	1	4	yes	—
294	C₂²⋊C₄	—	4	4	no	—
295	C₂²⋊C₄	—	8	2	no	—
296	C₂²⋊C₄	—	8	2	no	—
297	C₄²	—	2	2	no	—
298	C₄²	—	2	2	no	—
299	C₄²	—	8	2	no	—
300	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
301	C₄²	#41×C₃₁	1	2	yes	—
302	C₂³	—	2	4	no	—
303	C₂³	—	4	2	no	—
304	C₂³	—	4	2	no	—
305	C₂³	—	4	2	no	—
306	C₂³	—	4	2	no	—
307	C₂³	—	4	2	no	—
308	C₂³	—	4	2	no	—
309	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
310	Q₈	—	8	4	no	—
311	D₄	—	2	4	no	—
312	D₄	—	2	4	no	—
313	D₄	—	4	2	no	—
314	D₄	—	4	2	no	—
315	D₄	—	8	8	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	4	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	8	8	no	—
335	C₂×C₄	C₃₁×C₂²⋊Q₈	1	4	yes	—
336	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
337	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
338	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
339	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
340	C₈	—	4	2	no	—
341	C₈	—	4	2	no	—
342	C₂²	—	4	4	no	—
343	C₂²	—	4	4	no	—
344	C₂²	—	4	4	no	—
345	C₂²	—	8	2	no	—
346	C₂²	—	8	2	no	—
347	C₂²	—	8	2	no	—
348	C₂²	—	8	2	no	—
349	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
350	C₂²	C₃₁×#525	1	1	yes	characteristic
351	C₂²	C₃₁×#535	1	1	yes	characteristic
352	C₂²	C₃₁×#576	1	2	yes	—
353	C₂²	C₃₁×#577	1	2	yes	—
354	C₄	—	2	4	no	—
355	C₄	—	4	1	no	—
356	C₄	—	4	4	no	—
357	C₄	—	4	1	no	—
358	C₄	—	4	2	no	—
359	C₄	—	8	2	no	—
360	C₄	—	8	2	no	—
361	C₄	C₃₁×#554	1	2	yes	—
362	C₄	C₃₁×#559	1	2	yes	—
363	C₂	—	8	4	no	—
364	C₂	C₃₁×#2580	1	1	yes	characteristic
365	C₂	C₃₁×#3081	1	2	yes	—
366	C₂	C₃₁×#3122	1	1	yes	characteristic
367	C₂	C₃₁×#3126	1	1	yes	characteristic
368	C₂	C₃₁×#3224	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₃₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#2331	C₂	semidirect
3. C₃₁×#2092	C₂	semidirect
4. C₃₁×#2023	C₂	semidirect
6. C₃₁×#1949	C₂	semidirect
7. C₃₁×#1864	C₂	semidirect
8. C₃₁×#1862	C₂	semidirect
9. C₃₁×#1832	C₂	semidirect
43. C₃₁×#127	Q₈	semidirect
117. #29263	C₃₁	direct (N × H)
158. #2331	C₂×C₃₁	semidirect
160. #2092	C₂×C₃₁	semidirect
161. #2023	C₂×C₃₁	semidirect
163. #1949	C₂×C₃₁	semidirect
164. #1864	C₂×C₃₁	semidirect
165. #1862	C₂×C₃₁	semidirect
166. #1832	C₂×C₃₁	semidirect
227. #127	Q₈×C₃₁	semidirect
256. C₃₁	#29263	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#2331	2
C₃₁×#2328	2
C₃₁×#2092	2
C₃₁×#2023	2
C₃₁×#1982	2
C₃₁×#1949	2
C₃₁×#1864	2
C₃₁×#1862	2
C₃₁×#1832	2
#29263	31

Subgroups — C31×#29263 #21486022

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#29263 #21486022