Subgroups — C₃₁×#33961 #21490720

Order 7936.

520 conjugacy classes of subgroups.

The subgroup lattice has 356 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	C₃₁×#3393	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2249	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2248	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2211	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2210	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2209	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2206	C₂	1	1	yes	maximal, characteristic
9	Q₈×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₃₁×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₃₁×SD₃₂	C₂	1	2	yes	maximal, complemented
12	C₃₁×#559	—	2	1	no	—
13	C₃₁×#556	—	2	1	no	—
14	C₂×C₃₁×Q₃₂	C₂²	1	2	yes	—
15	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	—
16	C₂×C₃₁×SD₃₂	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	—
18	C₂×C₃₁×D₁₆	C₂²	1	2	yes	—
19	C₃₁×#495	C₂²	1	2	yes	—
20	C₃₁×#495	C₂²	1	2	yes	—
21	C₃₁×#494	C₂²	1	1	yes	characteristic
22	C₃₁×#493	C₂²	1	1	yes	characteristic
23	C₃₁×#464	—	2	2	no	—
24	C₃₁×#463	—	2	2	no	—
25	C₃₁×#461	—	2	1	no	—
26	C₃₁×#460	—	2	1	no	—
27	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
29	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
30	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
31	C₄×D₈×C₃₁	C₂²	1	2	yes	—
32	C₃₁×#367	C₂²	1	1	yes	characteristic
33	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
34	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
35	C₃₁×#358	C₂²	1	2	yes	—
36	C₃₁×#358	C₂²	1	1	yes	characteristic
37	C₃₁×#357	C₂²	1	2	yes	—
38	C₃₁×#357	C₂²	1	1	yes	characteristic
39	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₃₁	C₂²	1	2	yes	—
41	C₂×Q₁₆×C₃₁	—	2	1	no	—
42	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
43	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
44	C₂×SD₁₆×C₃₁	—	2	2	no	—
45	C₂×SD₁₆×C₃₁	—	2	2	no	—
46	C₂×D₈×C₃₁	—	2	1	no	—
47	C₂×D₈×C₃₁	C₂³	1	2	yes	—
48	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₃₁×C₄⋊Q₈	—	2	1	no	—
50	C₃₁×#127	—	2	1	no	—
51	C₃₁×#126	—	4	2	no	—
52	C₃₁×#124	—	2	2	no	—
53	C₃₁×#124	—	2	2	no	—
54	C₃₁×#124	—	4	1	no	—
55	C₃₁×#123	—	4	2	no	—
56	C₃₁×C₂²⋊Q₈	—	4	1	no	—
57	C₄×Q₈×C₃₁	—	2	1	no	—
58	C₄×Q₈×C₃₁	—	2	2	no	—
59	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
60	C₄×D₄×C₃₁	—	2	1	no	—
61	C₄×D₄×C₃₁	—	2	2	no	—
62	C₄×D₄×C₃₁	—	4	1	no	—
63	C₃₁×#117	—	4	1	no	—
64	C₃₁×Q₃₂	—	4	2	no	—
65	C₃₁×SD₃₂	—	2	4	no	—
66	C₃₁×SD₃₂	—	2	2	no	—
67	C₃₁×SD₃₂	—	4	2	no	—
68	C₃₁×D₁₆	—	4	2	no	—
69	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
70	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
71	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
72	C₃₁×#107	C₂³	1	1	yes	characteristic
73	C₃₁×#107	C₂³	1	1	yes	characteristic
74	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
75	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
76	C₃₁×#103	—	2	2	no	—
77	C₃₁×#103	—	2	1	no	—
78	C₃₁×#102	—	2	2	no	—
79	C₃₁×#102	—	2	1	no	—
80	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₃₁	C₂³	1	2	yes	—
82	C₂×Q₈×C₃₁	—	2	1	no	—
83	C₂×Q₈×C₃₁	—	2	2	no	—
84	C₂×Q₈×C₃₁	D₈	1	1	yes	characteristic
85	C₂×D₄×C₃₁	—	2	1	no	—
86	C₂×D₄×C₃₁	—	2	2	no	—
87	C₂×D₄×C₃₁	—	4	1	no	—
88	C₂×D₄×C₃₁	—	4	1	no	—
89	C₂²×C₄×C₃₁	—	4	1	no	—
90	C₂²×C₄×C₃₁	—	4	2	no	—
91	Q₁₆×C₃₁	—	2	2	no	—
92	Q₁₆×C₃₁	—	2	1	no	—
93	Q₁₆×C₃₁	—	2	2	no	—
94	Q₁₆×C₃₁	—	2	1	no	—
95	Q₁₆×C₃₁	—	8	1	no	—
96	SD₁₆×C₃₁	—	8	2	no	—
97	SD₁₆×C₃₁	—	8	2	no	—
98	D₈×C₃₁	—	2	2	no	—
99	D₈×C₃₁	—	2	1	no	—
100	D₈×C₃₁	—	2	1	no	—
101	D₈×C₃₁	—	2	2	no	—
102	D₈×C₃₁	—	8	1	no	—
103	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
106	C₄⋊C₄×C₃₁	—	2	1	no	—
107	C₄⋊C₄×C₃₁	—	2	1	no	—
108	C₄⋊C₄×C₃₁	—	4	1	no	—
109	C₄⋊C₄×C₃₁	—	4	2	no	—
110	C₄⋊C₄×C₃₁	—	4	1	no	—
111	C₄⋊C₄×C₃₁	—	4	1	no	—
112	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
113	C₄⋊C₄×C₃₁	D₈	1	2	yes	—
114	C₂²⋊C₄×C₃₁	—	4	2	no	—
115	C₂²⋊C₄×C₃₁	—	4	2	no	—
116	C₂²⋊C₄×C₃₁	—	4	1	no	—
117	C₂²⋊C₄×C₃₁	—	4	2	no	—
118	C₂²⋊C₄×C₃₁	—	4	2	no	—
119	C₂²⋊C₄×C₃₁	—	4	2	no	—
120	C₄²×C₃₁	—	4	2	no	—
121	C₄²×C₃₁	—	4	1	no	—
122	C₄²×C₃₁	C₂×D₄	1	2	yes	—
123	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₁₆×C₃₁	—	2	2	no	—
125	C₁₆×C₃₁	—	2	1	no	—
126	C₁₆×C₃₁	#41	1	2	yes	—
127	#33961	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
128	C₂³×C₃₁	—	4	1	no	—
129	C₂³×C₃₁	—	4	2	no	—
130	Q₈×C₃₁	—	2	2	no	—
131	Q₈×C₃₁	—	4	2	no	—
132	Q₈×C₃₁	—	4	1	no	—
133	Q₈×C₃₁	—	4	1	no	—
134	Q₈×C₃₁	—	4	2	no	—
135	D₄×C₃₁	—	4	1	no	—
136	D₄×C₃₁	—	4	2	no	—
137	D₄×C₃₁	—	4	2	no	—
138	D₄×C₃₁	—	4	1	no	—
139	D₄×C₃₁	—	8	2	no	—
140	D₄×C₃₁	—	16	1	no	—
141	C₂×C₄×C₃₁	—	4	1	no	—
142	C₂×C₄×C₃₁	—	4	1	no	—
143	C₂×C₄×C₃₁	—	4	1	no	—
144	C₂×C₄×C₃₁	—	4	2	no	—
145	C₂×C₄×C₃₁	—	8	2	no	—
146	C₂×C₄×C₃₁	—	16	2	no	—
147	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
148	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
149	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
152	C₈×C₃₁	—	2	1	no	—
153	C₈×C₃₁	—	2	2	no	—
154	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
155	C₈×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
156	#3396	C₂×C₃₁	1	1	yes	characteristic, complemented
157	#3393	C₂×C₃₁	1	1	yes	characteristic, complemented
158	#2249	C₂×C₃₁	1	2	yes	complemented
159	#2248	C₂×C₃₁	1	1	yes	characteristic, complemented
160	#2211	C₂×C₃₁	1	2	yes	complemented
161	#2210	C₂×C₃₁	1	2	yes	complemented
162	#2209	C₂×C₃₁	1	1	yes	characteristic, complemented
163	#2206	C₂×C₃₁	1	1	yes	characteristic
164	Q₈×C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
165	C₄×SD₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
166	C₄×SD₃₂	C₂×C₃₁	1	2	yes	complemented
167	C₂²×C₃₁	—	8	2	no	—
168	C₂²×C₃₁	—	8	1	no	—
169	C₂²×C₃₁	—	8	1	no	—
170	C₂²×C₃₁	—	8	2	no	—
171	C₂²×C₃₁	—	8	2	no	—
172	C₂²×C₃₁	—	8	1	no	—
173	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
174	C₄×C₃₁	—	2	2	no	—
175	C₄×C₃₁	—	2	2	no	—
176	C₄×C₃₁	—	2	1	no	—
177	C₄×C₃₁	—	8	1	no	—
178	C₄×C₃₁	—	8	2	no	—
179	C₄×C₃₁	—	8	1	no	—
180	C₄×C₃₁	—	8	1	no	—
181	C₄×C₃₁	#508	1	2	yes	—
182	C₄×C₃₁	#550	1	1	yes	characteristic
183	C₄×C₃₁	#576	1	1	yes	characteristic
184	#559	—	2	1	no	—
185	#556	—	2	1	no	—
186	C₂×Q₃₂	C₂²×C₃₁	1	2	yes	—
187	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	—
188	C₂×SD₃₂	C₂²×C₃₁	1	1	yes	characteristic
189	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	—
190	C₂×D₁₆	C₂²×C₃₁	1	2	yes	—
191	#495	C₂²×C₃₁	1	2	yes	—
192	#495	C₂²×C₃₁	1	2	yes	—
193	#494	C₂²×C₃₁	1	1	yes	characteristic
194	#493	C₂²×C₃₁	1	1	yes	characteristic
195	#464	—	2	2	no	—
196	#463	—	2	2	no	—
197	#461	—	2	1	no	—
198	#460	—	2	1	no	—
199	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
200	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
201	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
202	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
203	C₄×D₈	C₂²×C₃₁	1	2	yes	—
204	#367	C₂²×C₃₁	1	1	yes	characteristic
205	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
206	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
207	#358	C₂²×C₃₁	1	2	yes	—
208	#358	C₂²×C₃₁	1	1	yes	characteristic
209	#357	C₂²×C₃₁	1	2	yes	—
210	#357	C₂²×C₃₁	1	1	yes	characteristic
211	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
212	C₄×C₁₆	C₂²×C₃₁	1	2	yes	—
213	C₂×C₃₁	—	16	1	no	—
214	C₂×C₃₁	—	16	2	no	—
215	C₂×C₃₁	#3391	1	1	yes	characteristic
216	C₂×C₃₁	C₂×#508	1	1	yes	characteristic
217	C₂×C₃₁	#3416	1	1	yes	characteristic
218	C₂×Q₁₆	—	2	1	no	—
219	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
220	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
221	C₂×SD₁₆	—	2	2	no	—
222	C₂×SD₁₆	—	2	2	no	—
223	C₂×D₈	—	2	1	no	—
224	C₂×D₈	C₂³×C₃₁	1	2	yes	—
225	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
226	C₄⋊Q₈	—	2	1	no	—
227	#127	—	2	1	no	—
228	#126	—	4	2	no	—
229	#124	—	2	2	no	—
230	#124	—	2	2	no	—
231	#124	—	4	1	no	—
232	#123	—	4	2	no	—
233	C₂²⋊Q₈	—	4	1	no	—
234	C₄×Q₈	—	2	2	no	—
235	C₄×Q₈	—	2	1	no	—
236	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
237	C₄×D₄	—	2	1	no	—
238	C₄×D₄	—	2	2	no	—
239	C₄×D₄	—	4	1	no	—
240	#117	—	4	1	no	—
241	Q₃₂	—	4	2	no	—
242	SD₃₂	—	2	4	no	—
243	SD₃₂	—	2	2	no	—
244	SD₃₂	—	4	2	no	—
245	D₁₆	—	4	2	no	—
246	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
247	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
248	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
249	#107	C₂³×C₃₁	1	1	yes	characteristic
250	#107	C₂³×C₃₁	1	1	yes	characteristic
251	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
252	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
253	#103	—	2	2	no	—
254	#103	—	2	1	no	—
255	#102	—	2	2	no	—
256	#102	—	2	1	no	—
257	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
258	C₄×C₈	C₂³×C₃₁	1	2	yes	—
259	C₃₁	#33961	1	1	yes	characteristic, direct product factor, Sylow
260	C₂×Q₈	—	2	1	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	D₈×C₃₁	1	1	yes	characteristic
263	C₂×D₄	—	2	1	no	—
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	4	1	no	—
266	C₂×D₄	—	4	1	no	—
267	C₂²×C₄	—	4	1	no	—
268	C₂²×C₄	—	4	2	no	—
269	Q₁₆	—	2	2	no	—
270	Q₁₆	—	2	1	no	—
271	Q₁₆	—	2	2	no	—
272	Q₁₆	—	2	1	no	—
273	Q₁₆	—	8	1	no	—
274	SD₁₆	—	8	2	no	—
275	SD₁₆	—	8	2	no	—
276	D₈	—	2	2	no	—
277	D₈	—	2	1	no	—
278	D₈	—	2	1	no	—
279	D₈	—	2	2	no	—
280	D₈	—	8	1	no	—
281	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
282	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
283	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
284	C₄⋊C₄	—	2	1	no	—
285	C₄⋊C₄	—	2	1	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₄⋊C₄	—	4	2	no	—
288	C₄⋊C₄	—	4	1	no	—
289	C₄⋊C₄	—	4	1	no	—
290	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
291	C₄⋊C₄	D₈×C₃₁	1	2	yes	—
292	C₂²⋊C₄	—	4	2	no	—
293	C₂²⋊C₄	—	4	2	no	—
294	C₂²⋊C₄	—	4	1	no	—
295	C₂²⋊C₄	—	4	2	no	—
296	C₂²⋊C₄	—	4	2	no	—
297	C₂²⋊C₄	—	4	2	no	—
298	C₄²	—	4	2	no	—
299	C₄²	—	4	1	no	—
300	C₄²	C₂×D₄×C₃₁	1	2	yes	—
301	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
302	C₁₆	—	2	2	no	—
303	C₁₆	—	2	1	no	—
304	C₁₆	#41×C₃₁	1	2	yes	—
305	C₂³	—	4	1	no	—
306	C₂³	—	4	2	no	—
307	Q₈	—	2	2	no	—
308	Q₈	—	4	2	no	—
309	Q₈	—	4	1	no	—
310	Q₈	—	4	1	no	—
311	Q₈	—	4	2	no	—
312	D₄	—	4	1	no	—
313	D₄	—	4	2	no	—
314	D₄	—	4	2	no	—
315	D₄	—	4	1	no	—
316	D₄	—	8	2	no	—
317	D₄	—	16	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	8	2	no	—
323	C₂×C₄	—	16	2	no	—
324	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
325	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
326	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
327	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
328	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
329	C₈	—	2	1	no	—
330	C₈	—	2	2	no	—
331	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
332	C₈	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
333	C₂²	—	8	2	no	—
334	C₂²	—	8	1	no	—
335	C₂²	—	8	1	no	—
336	C₂²	—	8	2	no	—
337	C₂²	—	8	2	no	—
338	C₂²	—	8	1	no	—
339	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	1	no	—
343	C₄	—	8	1	no	—
344	C₄	—	8	2	no	—
345	C₄	—	8	1	no	—
346	C₄	—	8	1	no	—
347	C₄	C₃₁×#508	1	2	yes	—
348	C₄	C₃₁×#550	1	1	yes	characteristic
349	C₄	C₃₁×#576	1	1	yes	characteristic
350	C₂	—	16	1	no	—
351	C₂	—	16	2	no	—
352	C₂	C₃₁×#3391	1	1	yes	characteristic
353	C₂	C₂×C₃₁×#508	1	1	yes	characteristic
354	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. C₃₁×#3393	C₂	semidirect
3. C₃₁×#2249	C₂	semidirect
4. C₃₁×#2248	C₂	semidirect
5. C₃₁×#2211	C₂	semidirect
6. C₃₁×#2210	C₂	semidirect
7. C₃₁×#2209	C₂	semidirect
9. Q₈×C₁₆×C₃₁	C₂	semidirect
10. C₄×C₃₁×SD₃₂	C₂	semidirect
11. C₄×C₃₁×SD₃₂	C₂	semidirect
127. #33961	C₃₁	direct (N × H)
156. #3396	C₂×C₃₁	semidirect
157. #3393	C₂×C₃₁	semidirect
158. #2249	C₂×C₃₁	semidirect
159. #2248	C₂×C₃₁	semidirect
160. #2211	C₂×C₃₁	semidirect
161. #2210	C₂×C₃₁	semidirect
162. #2209	C₂×C₃₁	semidirect
164. Q₈×C₁₆	C₂×C₃₁	semidirect
165. C₄×SD₃₂	C₂×C₃₁	semidirect
166. C₄×SD₃₂	C₂×C₃₁	semidirect
259. C₃₁	#33961	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
C₃₁×#3393	2
C₃₁×#2249	2
C₃₁×#2248	2
C₃₁×#2211	2
C₃₁×#2210	2
C₃₁×#2209	2
C₃₁×#2206	2
Q₈×C₁₆×C₃₁	2
C₄×C₃₁×SD₃₂	2
#33961	31

Subgroups — C31×#33961 #21490720

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33961 #21490720