Subgroups — C₃₁×#33940 #21490699

Order 7936.

520 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3396	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2243	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2240	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2226	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2225	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2210	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2206	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#2183	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2177	C₂	1	2	yes	maximal, complemented
11	C₄×C₃₁×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#559	—	2	1	no	—
13	D₄×Q₈×C₃₁	—	2	1	no	—
14	C₂×C₃₁×SD₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×D₁₆	C₂²	1	2	yes	—
16	C₃₁×#500	C₂²	1	1	yes	characteristic
17	C₃₁×#500	C₂²	1	2	yes	—
18	C₃₁×#495	C₂²	1	2	yes	—
19	C₃₁×#493	C₂²	1	1	yes	characteristic
20	C₃₁×#474	—	2	1	no	—
21	C₃₁×#474	—	2	2	no	—
22	C₃₁×#463	—	2	2	no	—
23	C₃₁×#460	—	2	1	no	—
24	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
26	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₄×D₈×C₃₁	C₂²	1	2	yes	—
28	C₄×D₈×C₃₁	C₂²	1	2	yes	—
29	C₃₁×#367	C₂²	1	1	yes	characteristic
30	C₃₁×#367	C₂²	1	2	yes	—
31	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
32	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
33	C₃₁×#358	C₂²	1	1	yes	characteristic
34	C₃₁×#358	C₂²	1	2	yes	—
35	C₃₁×#357	C₂²	1	1	yes	characteristic
36	C₃₁×#357	C₂²	1	2	yes	—
37	C₃₁×#357	C₂²	1	2	yes	—
38	C₃₁×#357	C₂²	1	2	yes	—
39	C₃₁×#346	C₂²	1	2	yes	—
40	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
41	C₂²×Q₈×C₃₁	—	4	1	no	—
42	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
43	C₂×SD₁₆×C₃₁	—	2	2	no	—
44	C₂×D₈×C₃₁	—	2	1	no	—
45	C₂×D₈×C₃₁	C₂³	1	2	yes	—
46	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₃₁×C₄⋊Q₈	—	2	2	no	—
48	C₃₁×C₄⋊Q₈	—	2	1	no	—
49	C₃₁×#127	—	2	1	no	—
50	C₃₁×#124	—	2	2	no	—
51	C₃₁×#124	—	4	1	no	—
52	C₃₁×#123	—	4	2	no	—
53	C₃₁×C₂²⋊Q₈	—	4	1	no	—
54	C₃₁×C₂²⋊Q₈	—	4	2	no	—
55	C₄×Q₈×C₃₁	—	2	1	no	—
56	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
57	C₄×D₄×C₃₁	—	2	2	no	—
58	C₄×D₄×C₃₁	—	2	2	no	—
59	C₄×D₄×C₃₁	—	2	1	no	—
60	C₄×D₄×C₃₁	—	4	1	no	—
61	C₃₁×SD₃₂	—	2	2	no	—
62	C₃₁×D₁₆	—	2	4	no	—
63	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
64	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
65	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
66	C₃₁×#107	—	2	2	no	—
67	C₃₁×#107	—	2	1	no	—
68	C₃₁×#107	C₂³	1	1	yes	characteristic
69	C₃₁×#107	C₂³	1	1	yes	characteristic
70	C₃₁×#107	C₂³	1	2	yes	—
71	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
72	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
73	C₃₁×#103	—	2	1	no	—
74	C₃₁×#102	—	2	2	no	—
75	C₃₁×#102	—	2	2	no	—
76	C₃₁×#102	—	2	1	no	—
77	C₄×C₈×C₃₁	C₂³	1	2	yes	—
78	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
79	C₂×Q₈×C₃₁	—	2	1	no	—
80	C₂×Q₈×C₃₁	—	4	2	no	—
81	C₂×Q₈×C₃₁	—	4	1	no	—
82	C₂×Q₈×C₃₁	—	8	4	no	—
83	C₂×Q₈×C₃₁	D₈	1	1	yes	characteristic
84	C₂×D₄×C₃₁	—	2	1	no	—
85	C₂×D₄×C₃₁	—	2	2	no	—
86	C₂×D₄×C₃₁	—	4	1	no	—
87	C₂×D₄×C₃₁	—	4	1	no	—
88	C₂²×C₄×C₃₁	—	4	2	no	—
89	C₂²×C₄×C₃₁	—	4	1	no	—
90	C₂²×C₄×C₃₁	—	4	2	no	—
91	Q₁₆×C₃₁	—	2	1	no	—
92	Q₁₆×C₃₁	—	2	1	no	—
93	SD₁₆×C₃₁	—	8	2	no	—
94	D₈×C₃₁	—	2	2	no	—
95	D₈×C₃₁	—	2	1	no	—
96	D₈×C₃₁	—	2	2	no	—
97	D₈×C₃₁	—	8	1	no	—
98	D₈×C₃₁	#41	1	2	yes	—
99	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
100	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
102	C₄⋊C₄×C₃₁	—	2	1	no	—
103	C₄⋊C₄×C₃₁	—	2	2	no	—
104	C₄⋊C₄×C₃₁	—	2	1	no	—
105	C₄⋊C₄×C₃₁	—	4	1	no	—
106	C₄⋊C₄×C₃₁	—	4	2	no	—
107	C₄⋊C₄×C₃₁	—	4	1	no	—
108	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
109	C₄⋊C₄×C₃₁	D₈	1	2	yes	—
110	C₂²⋊C₄×C₃₁	—	4	2	no	—
111	C₂²⋊C₄×C₃₁	—	4	1	no	—
112	C₂²⋊C₄×C₃₁	—	4	2	no	—
113	C₂²⋊C₄×C₃₁	—	4	2	no	—
114	C₂²⋊C₄×C₃₁	—	4	2	no	—
115	C₄²×C₃₁	—	4	1	no	—
116	C₄²×C₃₁	C₂×D₄	1	2	yes	—
117	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₁₆×C₃₁	—	2	2	no	—
119	C₁₆×C₃₁	—	2	1	no	—
120	C₁₆×C₃₁	—	2	1	no	—
121	#33940	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
122	C₂³×C₃₁	—	4	1	no	—
123	C₂³×C₃₁	—	4	2	no	—
124	Q₈×C₃₁	—	2	2	no	—
125	Q₈×C₃₁	—	4	1	no	—
126	Q₈×C₃₁	—	4	1	no	—
127	Q₈×C₃₁	—	8	4	no	—
128	Q₈×C₃₁	—	8	2	no	—
129	D₄×C₃₁	—	2	2	no	—
130	D₄×C₃₁	—	4	2	no	—
131	D₄×C₃₁	—	4	1	no	—
132	D₄×C₃₁	—	4	2	no	—
133	D₄×C₃₁	—	8	2	no	—
134	D₄×C₃₁	—	16	1	no	—
135	C₂×C₄×C₃₁	—	4	1	no	—
136	C₂×C₄×C₃₁	—	4	1	no	—
137	C₂×C₄×C₃₁	—	4	2	no	—
138	C₂×C₄×C₃₁	—	4	1	no	—
139	C₂×C₄×C₃₁	—	8	4	no	—
140	C₂×C₄×C₃₁	—	8	2	no	—
141	C₂×C₄×C₃₁	—	16	2	no	—
142	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
143	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
144	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
147	C₈×C₃₁	—	2	1	no	—
148	C₈×C₃₁	—	2	2	no	—
149	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
150	C₈×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
151	#3396	C₂×C₃₁	1	1	yes	characteristic, complemented
152	Q₈×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
153	#2243	C₂×C₃₁	1	2	yes	complemented
154	#2240	C₂×C₃₁	1	1	yes	characteristic, complemented
155	#2226	C₂×C₃₁	1	2	yes	complemented
156	#2225	C₂×C₃₁	1	1	yes	characteristic, complemented
157	#2210	C₂×C₃₁	1	2	yes	complemented
158	#2206	C₂×C₃₁	1	1	yes	characteristic
159	#2183	C₂×C₃₁	1	1	yes	characteristic, complemented
160	#2177	C₂×C₃₁	1	2	yes	complemented
161	C₄×SD₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
162	C₂²×C₃₁	—	4	2	no	—
163	C₂²×C₃₁	—	8	1	no	—
164	C₂²×C₃₁	—	8	1	no	—
165	C₂²×C₃₁	—	8	2	no	—
166	C₂²×C₃₁	—	8	2	no	—
167	C₂²×C₃₁	—	8	2	no	—
168	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
169	C₄×C₃₁	—	2	2	no	—
170	C₄×C₃₁	—	2	2	no	—
171	C₄×C₃₁	—	2	1	no	—
172	C₄×C₃₁	—	8	1	no	—
173	C₄×C₃₁	—	8	2	no	—
174	C₄×C₃₁	—	8	1	no	—
175	C₄×C₃₁	—	8	1	no	—
176	C₄×C₃₁	#509	1	2	yes	—
177	C₄×C₃₁	#548	1	1	yes	characteristic
178	C₄×C₃₁	#577	1	1	yes	characteristic
179	#559	—	2	1	no	—
180	D₄×Q₈	—	2	1	no	—
181	C₂×SD₃₂	C₂²×C₃₁	1	1	yes	characteristic
182	C₂×D₁₆	C₂²×C₃₁	1	2	yes	—
183	#500	C₂²×C₃₁	1	1	yes	characteristic
184	#500	C₂²×C₃₁	1	2	yes	—
185	#495	C₂²×C₃₁	1	2	yes	—
186	#493	C₂²×C₃₁	1	1	yes	characteristic
187	#474	—	2	1	no	—
188	#474	—	2	2	no	—
189	#463	—	2	2	no	—
190	#460	—	2	1	no	—
191	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
192	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
193	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
194	C₄×D₈	C₂²×C₃₁	1	2	yes	—
195	C₄×D₈	C₂²×C₃₁	1	2	yes	—
196	#367	C₂²×C₃₁	1	1	yes	characteristic
197	#367	C₂²×C₃₁	1	2	yes	—
198	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
199	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
200	#358	C₂²×C₃₁	1	1	yes	characteristic
201	#358	C₂²×C₃₁	1	2	yes	—
202	#357	C₂²×C₃₁	1	1	yes	characteristic
203	#357	C₂²×C₃₁	1	2	yes	—
204	#357	C₂²×C₃₁	1	2	yes	—
205	#357	C₂²×C₃₁	1	2	yes	—
206	#346	C₂²×C₃₁	1	2	yes	—
207	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
208	C₂×C₃₁	—	8	2	no	—
209	C₂×C₃₁	—	16	2	no	—
210	C₂×C₃₁	#3358	1	1	yes	characteristic
211	C₂×C₃₁	C₂×#509	1	1	yes	characteristic
212	C₂×C₃₁	#3416	1	1	yes	characteristic
213	C₂²×Q₈	—	4	1	no	—
214	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
215	C₂×SD₁₆	—	2	2	no	—
216	C₂×D₈	—	2	1	no	—
217	C₂×D₈	C₂³×C₃₁	1	2	yes	—
218	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
219	C₄⋊Q₈	—	2	2	no	—
220	C₄⋊Q₈	—	2	1	no	—
221	#127	—	2	1	no	—
222	#124	—	2	2	no	—
223	#124	—	4	1	no	—
224	#123	—	4	2	no	—
225	C₂²⋊Q₈	—	4	1	no	—
226	C₂²⋊Q₈	—	4	2	no	—
227	C₄×Q₈	—	2	1	no	—
228	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
229	C₄×D₄	—	2	2	no	—
230	C₄×D₄	—	2	1	no	—
231	C₄×D₄	—	2	2	no	—
232	C₄×D₄	—	4	1	no	—
233	SD₃₂	—	2	2	no	—
234	D₁₆	—	2	4	no	—
235	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
236	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
237	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
238	#107	—	2	2	no	—
239	#107	—	2	1	no	—
240	#107	C₂³×C₃₁	1	2	yes	—
241	#107	C₂³×C₃₁	1	1	yes	characteristic
242	#107	C₂³×C₃₁	1	1	yes	characteristic
243	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
244	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
245	#103	—	2	1	no	—
246	#102	—	2	2	no	—
247	#102	—	2	2	no	—
248	#102	—	2	1	no	—
249	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
250	C₄×C₈	C₂³×C₃₁	1	2	yes	—
251	C₃₁	#33940	1	1	yes	characteristic, direct product factor, Sylow
252	C₂×Q₈	—	2	1	no	—
253	C₂×Q₈	—	4	2	no	—
254	C₂×Q₈	—	4	1	no	—
255	C₂×Q₈	—	8	4	no	—
256	C₂×Q₈	D₈×C₃₁	1	1	yes	characteristic
257	C₂×D₄	—	2	1	no	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	4	1	no	—
260	C₂×D₄	—	4	1	no	—
261	C₂²×C₄	—	4	2	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	4	2	no	—
264	Q₁₆	—	2	1	no	—
265	Q₁₆	—	2	1	no	—
266	SD₁₆	—	8	2	no	—
267	D₈	—	2	2	no	—
268	D₈	—	2	1	no	—
269	D₈	—	2	2	no	—
270	D₈	—	8	1	no	—
271	D₈	#41×C₃₁	1	2	yes	—
272	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
273	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
274	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₄⋊C₄	—	4	2	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
282	C₄⋊C₄	D₈×C₃₁	1	2	yes	—
283	C₂²⋊C₄	—	4	2	no	—
284	C₂²⋊C₄	—	4	1	no	—
285	C₂²⋊C₄	—	4	2	no	—
286	C₂²⋊C₄	—	4	2	no	—
287	C₂²⋊C₄	—	4	2	no	—
288	C₄²	—	4	1	no	—
289	C₄²	C₂×D₄×C₃₁	1	2	yes	—
290	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
291	C₁₆	—	2	2	no	—
292	C₁₆	—	2	1	no	—
293	C₁₆	—	2	1	no	—
294	C₂³	—	4	1	no	—
295	C₂³	—	4	2	no	—
296	Q₈	—	2	2	no	—
297	Q₈	—	4	1	no	—
298	Q₈	—	4	1	no	—
299	Q₈	—	8	4	no	—
300	Q₈	—	8	2	no	—
301	D₄	—	2	2	no	—
302	D₄	—	4	2	no	—
303	D₄	—	4	1	no	—
304	D₄	—	4	2	no	—
305	D₄	—	8	2	no	—
306	D₄	—	16	1	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	8	4	no	—
312	C₂×C₄	—	8	2	no	—
313	C₂×C₄	—	16	2	no	—
314	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
315	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
316	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
317	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
318	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
319	C₈	—	2	1	no	—
320	C₈	—	2	2	no	—
321	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
322	C₈	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
323	C₂²	—	4	2	no	—
324	C₂²	—	8	1	no	—
325	C₂²	—	8	1	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	8	2	no	—
329	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	2	no	—
332	C₄	—	2	1	no	—
333	C₄	—	8	1	no	—
334	C₄	—	8	2	no	—
335	C₄	—	8	1	no	—
336	C₄	—	8	1	no	—
337	C₄	C₃₁×#509	1	2	yes	—
338	C₄	C₃₁×#548	1	1	yes	characteristic
339	C₄	C₃₁×#577	1	1	yes	characteristic
340	C₂	—	8	2	no	—
341	C₂	—	16	2	no	—
342	C₂	C₃₁×#3358	1	1	yes	characteristic
343	C₂	C₂×C₃₁×#509	1	1	yes	characteristic
344	C₂	C₃₁×#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
prime radical	C₃₁×#2206	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×Q₈×C₃₁ ◁ C₈×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3396	C₂	semidirect
2. Q₈×D₈×C₃₁	C₂	semidirect
3. C₃₁×#2243	C₂	semidirect
4. C₃₁×#2240	C₂	semidirect
5. C₃₁×#2226	C₂	semidirect
6. C₃₁×#2225	C₂	semidirect
7. C₃₁×#2210	C₂	semidirect
9. C₃₁×#2183	C₂	semidirect
10. C₃₁×#2177	C₂	semidirect
11. C₄×C₃₁×SD₃₂	C₂	semidirect
121. #33940	C₃₁	direct (N × H)
151. #3396	C₂×C₃₁	semidirect
152. Q₈×D₈	C₂×C₃₁	semidirect
153. #2243	C₂×C₃₁	semidirect
154. #2240	C₂×C₃₁	semidirect
155. #2226	C₂×C₃₁	semidirect
156. #2225	C₂×C₃₁	semidirect
157. #2210	C₂×C₃₁	semidirect
159. #2183	C₂×C₃₁	semidirect
160. #2177	C₂×C₃₁	semidirect
161. C₄×SD₃₂	C₂×C₃₁	semidirect
251. C₃₁	#33940	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×#3396	2
Q₈×D₈×C₃₁	2
C₃₁×#2243	2
C₃₁×#2240	2
C₃₁×#2226	2
C₃₁×#2225	2
C₃₁×#2210	2
C₃₁×#2206	2
C₃₁×#2183	2
C₃₁×#2177	2
C₄×C₃₁×SD₃₂	2
#33940	31

Subgroups — C31×#33940 #21490699

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33940 #21490699