Subgroups — C₃₁×#19933 #21476692

Order 7936.

624 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3367	C₂	1	2	yes	maximal
2	C₃₁×#3342	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1707	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1700	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1673	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1669	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1652	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1651	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1591	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₁₆×C₃₁	—	2	1	no	—
13	C₃₁×#556	C₂²	1	2	yes	—
14	D₄×Q₈×C₃₁	C₂²	1	2	yes	—
15	C₃₁×#500	C₂²	1	1	yes	characteristic
16	C₃₁×#498	C₂²	1	2	yes	—
17	C₃₁×#494	C₂²	1	1	yes	characteristic
18	C₃₁×#489	—	2	2	no	—
19	C₃₁×#487	C₂²	1	2	yes	—
20	C₃₁×#485	—	2	2	no	—
21	C₃₁×#477	—	2	2	no	—
22	C₃₁×#475	C₂²	1	2	yes	—
23	C₃₁×#474	C₂²	1	2	yes	—
24	C₃₁×#467	—	2	1	no	—
25	C₃₁×#467	—	2	1	no	—
26	C₃₁×#463	—	2	2	no	—
27	C₃₁×#462	C₂²	1	2	yes	—
28	C₃₁×#461	C₂²	1	2	yes	—
29	C₃₁×#451	—	2	2	no	—
30	C₃₁×#446	C₂²	1	2	yes	—
31	C₃₁×#440	—	2	2	no	—
32	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
33	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
34	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
36	C₃₁×#427	—	2	1	no	—
37	C₃₁×#335	C₂²	1	1	yes	characteristic
38	C₃₁×#335	C₂²	1	1	yes	characteristic
39	C₃₁×#332	C₂²	1	2	yes	—
40	C₃₁×#332	C₂²	1	2	yes	—
41	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
42	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
43	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
44	C₈²×C₃₁	C₂²	1	1	yes	characteristic
45	C₂²×Q₈×C₃₁	—	2	2	no	—
46	C₂×Q₁₆×C₃₁	—	2	2	no	—
47	C₂×Q₁₆×C₃₁	—	2	1	no	—
48	C₂×Q₁₆×C₃₁	—	2	1	no	—
49	C₂×Q₁₆×C₃₁	—	4	4	no	—
50	C₂×Q₁₆×C₃₁	D₄	1	1	yes	characteristic
51	C₂×SD₁₆×C₃₁	—	2	2	no	—
52	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₃₁	—	2	1	no	—
54	C₂²×C₈×C₃₁	—	2	1	no	—
55	C₃₁×C₄⋊Q₈	—	2	2	no	—
56	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
57	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
58	C₃₁×#126	—	4	2	no	—
59	C₃₁×#124	—	2	2	no	—
60	C₃₁×C₂²⋊Q₈	—	2	2	no	—
61	C₃₁×C₂²⋊Q₈	—	2	2	no	—
62	C₃₁×C₂²⋊Q₈	—	4	2	no	—
63	C₄×Q₈×C₃₁	—	2	2	no	—
64	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
65	C₄×D₄×C₃₁	—	2	2	no	—
66	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₃₁×#117	—	2	2	no	—
69	C₃₁×#107	—	2	1	no	—
70	C₃₁×#107	—	2	1	no	—
71	C₃₁×#107	—	2	2	no	—
72	C₃₁×#107	D₄	1	1	yes	characteristic
73	C₃₁×#107	D₄	1	1	yes	characteristic
74	C₃₁×#106	—	2	2	no	—
75	C₃₁×#106	—	4	2	no	—
76	C₃₁×C₄⋊C₈	—	2	2	no	—
77	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
78	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
79	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
80	C₃₁×#103	—	2	2	no	—
81	C₃₁×#103	—	2	2	no	—
82	C₃₁×#103	—	2	2	no	—
83	C₃₁×#103	—	2	2	no	—
84	C₃₁×#103	—	2	2	no	—
85	C₃₁×#103	D₄	1	2	yes	—
86	C₃₁×#102	—	2	2	no	—
87	C₃₁×#102	D₄	1	1	yes	characteristic
88	C₃₁×#102	D₄	1	1	yes	characteristic
89	C₃₁×C₂²⋊C₈	—	2	1	no	—
90	C₃₁×C₂²⋊C₈	—	2	1	no	—
91	C₃₁×#97	—	2	2	no	—
92	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₃₁	—	2	2	no	—
96	C₂×Q₈×C₃₁	—	2	2	no	—
97	C₂×Q₈×C₃₁	—	4	2	no	—
98	C₂×Q₈×C₃₁	—	4	4	no	—
99	C₂×Q₈×C₃₁	—	8	2	no	—
100	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
101	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₃₁	—	2	1	no	—
104	C₂²×C₄×C₃₁	—	2	1	no	—
105	C₂²×C₄×C₃₁	—	4	2	no	—
106	Q₁₆×C₃₁	—	2	2	no	—
107	Q₁₆×C₃₁	—	4	4	no	—
108	Q₁₆×C₃₁	—	4	2	no	—
109	Q₁₆×C₃₁	—	8	1	no	—
110	SD₁₆×C₃₁	—	4	4	no	—
111	D₈×C₃₁	—	2	2	no	—
112	C₂×C₈×C₃₁	—	2	1	no	—
113	C₂×C₈×C₃₁	—	2	2	no	—
114	C₂×C₈×C₃₁	—	2	1	no	—
115	C₂×C₈×C₃₁	—	2	1	no	—
116	C₂×C₈×C₃₁	—	4	2	no	—
117	C₂×C₈×C₃₁	—	4	2	no	—
118	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
122	C₄⋊C₄×C₃₁	—	2	2	no	—
123	C₄⋊C₄×C₃₁	—	2	2	no	—
124	C₄⋊C₄×C₃₁	—	2	2	no	—
125	C₄⋊C₄×C₃₁	—	2	2	no	—
126	C₄⋊C₄×C₃₁	—	2	2	no	—
127	C₄⋊C₄×C₃₁	—	4	2	no	—
128	C₄⋊C₄×C₃₁	—	4	2	no	—
129	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
130	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
131	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
132	C₂²⋊C₄×C₃₁	—	2	1	no	—
133	C₂²⋊C₄×C₃₁	—	2	1	no	—
134	C₂²⋊C₄×C₃₁	—	4	2	no	—
135	C₂²⋊C₄×C₃₁	—	4	2	no	—
136	C₂²⋊C₄×C₃₁	—	4	2	no	—
137	C₄²×C₃₁	—	2	2	no	—
138	C₄²×C₃₁	—	4	2	no	—
139	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
140	#19933	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂³×C₃₁	—	2	1	no	—
142	C₂³×C₃₁	—	2	1	no	—
143	Q₈×C₃₁	—	2	2	no	—
144	Q₈×C₃₁	—	2	2	no	—
145	Q₈×C₃₁	—	4	2	no	—
146	Q₈×C₃₁	—	4	2	no	—
147	Q₈×C₃₁	—	4	2	no	—
148	Q₈×C₃₁	—	4	2	no	—
149	Q₈×C₃₁	—	8	4	no	—
150	D₄×C₃₁	—	2	1	no	—
151	D₄×C₃₁	—	2	1	no	—
152	D₄×C₃₁	—	2	1	no	—
153	D₄×C₃₁	#135	1	2	yes	—
154	C₂×C₄×C₃₁	—	2	2	no	—
155	C₂×C₄×C₃₁	—	2	2	no	—
156	C₂×C₄×C₃₁	—	2	1	no	—
157	C₂×C₄×C₃₁	—	2	1	no	—
158	C₂×C₄×C₃₁	—	4	1	no	—
159	C₂×C₄×C₃₁	—	4	1	no	—
160	C₂×C₄×C₃₁	—	4	1	no	—
161	C₂×C₄×C₃₁	—	4	1	no	—
162	C₂×C₄×C₃₁	—	4	2	no	—
163	C₂×C₄×C₃₁	—	4	2	no	—
164	C₂×C₄×C₃₁	—	8	4	no	—
165	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
166	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
167	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
168	C₈×C₃₁	—	2	2	no	—
169	C₈×C₃₁	—	2	1	no	—
170	C₈×C₃₁	—	2	1	no	—
171	C₈×C₃₁	—	2	1	no	—
172	C₈×C₃₁	—	4	2	no	—
173	C₈×C₃₁	—	4	1	no	—
174	C₈×C₃₁	—	4	1	no	—
175	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
176	#3367	C₂×C₃₁	1	2	yes	—
177	#3342	C₂×C₃₁	1	1	yes	characteristic, complemented
178	D₄×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
179	#1707	C₂×C₃₁	1	1	yes	characteristic, complemented
180	#1700	C₂×C₃₁	1	2	yes	complemented
181	#1673	C₂×C₃₁	1	1	yes	characteristic, complemented
182	#1669	C₂×C₃₁	1	1	yes	characteristic, complemented
183	#1652	C₂×C₃₁	1	2	yes	complemented
184	#1651	C₂×C₃₁	1	2	yes	complemented
185	#1591	C₂×C₃₁	1	1	yes	characteristic, complemented
186	C₈×D₈	C₂×C₃₁	1	1	yes	characteristic
187	C₂²×C₃₁	—	2	2	no	—
188	C₂²×C₃₁	—	4	1	no	—
189	C₂²×C₃₁	—	4	1	no	—
190	C₂²×C₃₁	—	4	1	no	—
191	C₂²×C₃₁	—	4	1	no	—
192	C₂²×C₃₁	—	4	1	no	—
193	C₂²×C₃₁	D₄²	1	1	yes	characteristic
194	C₄×C₃₁	—	2	1	no	—
195	C₄×C₃₁	—	4	1	no	—
196	C₄×C₃₁	—	4	1	no	—
197	C₄×C₃₁	—	4	2	no	—
198	C₄×C₃₁	—	4	2	no	—
199	C₄×C₃₁	—	8	2	no	—
200	C₄×C₃₁	—	8	2	no	—
201	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
202	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
203	C₄×C₃₁	#577	1	1	yes	characteristic
204	C₄×C₃₁	#578	1	1	yes	characteristic
205	C₂²×Q₁₆	—	2	1	no	—
206	#556	C₂²×C₃₁	1	2	yes	—
207	D₄×Q₈	C₂²×C₃₁	1	2	yes	—
208	#500	C₂²×C₃₁	1	1	yes	characteristic
209	#498	C₂²×C₃₁	1	2	yes	—
210	#494	C₂²×C₃₁	1	1	yes	characteristic
211	#489	—	2	2	no	—
212	#487	C₂²×C₃₁	1	2	yes	—
213	#485	—	2	2	no	—
214	#477	—	2	2	no	—
215	#475	C₂²×C₃₁	1	2	yes	—
216	#474	C₂²×C₃₁	1	2	yes	—
217	#467	—	2	1	no	—
218	#467	—	2	1	no	—
219	#463	—	2	2	no	—
220	#462	C₂²×C₃₁	1	2	yes	—
221	#461	C₂²×C₃₁	1	2	yes	—
222	#451	—	2	2	no	—
223	#446	C₂²×C₃₁	1	2	yes	—
224	#440	—	2	2	no	—
225	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
226	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
227	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
228	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
229	#427	—	2	1	no	—
230	#335	C₂²×C₃₁	1	1	yes	characteristic
231	#335	C₂²×C₃₁	1	1	yes	characteristic
232	#332	C₂²×C₃₁	1	2	yes	—
233	#332	C₂²×C₃₁	1	2	yes	—
234	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
235	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
236	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
237	C₈²	C₂²×C₃₁	1	1	yes	characteristic
238	C₂×C₃₁	—	4	2	no	—
239	C₂×C₃₁	—	8	1	no	—
240	C₂×C₃₁	#3272	1	1	yes	characteristic
241	C₂×C₃₁	#3285	1	1	yes	characteristic
242	C₂×C₃₁	#3293	1	1	yes	characteristic
243	C₂²×Q₈	—	2	2	no	—
244	C₂×Q₁₆	—	2	2	no	—
245	C₂×Q₁₆	—	2	1	no	—
246	C₂×Q₁₆	—	2	1	no	—
247	C₂×Q₁₆	—	4	4	no	—
248	C₂×Q₁₆	D₄×C₃₁	1	1	yes	characteristic
249	C₂×SD₁₆	—	2	2	no	—
250	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
251	C₂²×C₈	—	2	1	no	—
252	C₂²×C₈	—	2	1	no	—
253	C₄⋊Q₈	—	2	2	no	—
254	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
255	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
256	#126	—	4	2	no	—
257	#124	—	2	2	no	—
258	C₂²⋊Q₈	—	2	2	no	—
259	C₂²⋊Q₈	—	2	2	no	—
260	C₂²⋊Q₈	—	4	2	no	—
261	C₄×Q₈	—	2	2	no	—
262	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
263	C₄×D₄	—	2	2	no	—
264	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
265	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
266	#117	—	2	2	no	—
267	#107	—	2	1	no	—
268	#107	—	2	1	no	—
269	#107	—	2	2	no	—
270	#107	D₄×C₃₁	1	1	yes	characteristic
271	#107	D₄×C₃₁	1	1	yes	characteristic
272	#106	—	2	2	no	—
273	#106	—	4	2	no	—
274	C₄⋊C₈	—	2	2	no	—
275	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
276	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
277	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
278	#103	—	2	2	no	—
279	#103	—	2	2	no	—
280	#103	—	2	2	no	—
281	#103	—	2	2	no	—
282	#103	—	2	2	no	—
283	#103	D₄×C₃₁	1	2	yes	—
284	#102	—	2	2	no	—
285	#102	D₄×C₃₁	1	1	yes	characteristic
286	#102	D₄×C₃₁	1	1	yes	characteristic
287	C₂²⋊C₈	—	2	1	no	—
288	C₂²⋊C₈	—	2	1	no	—
289	#97	—	2	2	no	—
290	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
291	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
292	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
293	C₃₁	#19933	1	1	yes	characteristic, direct product factor, Sylow
294	C₂×Q₈	—	2	2	no	—
295	C₂×Q₈	—	2	2	no	—
296	C₂×Q₈	—	4	2	no	—
297	C₂×Q₈	—	4	4	no	—
298	C₂×Q₈	—	8	2	no	—
299	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
300	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
301	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
302	C₂²×C₄	—	2	1	no	—
303	C₂²×C₄	—	2	1	no	—
304	C₂²×C₄	—	4	2	no	—
305	Q₁₆	—	2	2	no	—
306	Q₁₆	—	4	4	no	—
307	Q₁₆	—	4	2	no	—
308	Q₁₆	—	8	1	no	—
309	SD₁₆	—	4	4	no	—
310	D₈	—	2	2	no	—
311	C₂×C₈	—	2	1	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	1	no	—
314	C₂×C₈	—	2	1	no	—
315	C₂×C₈	—	4	2	no	—
316	C₂×C₈	—	4	2	no	—
317	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
318	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
319	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
320	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	2	2	no	—
323	C₄⋊C₄	—	2	2	no	—
324	C₄⋊C₄	—	2	2	no	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	4	2	no	—
327	C₄⋊C₄	—	4	2	no	—
328	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
329	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
330	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
331	C₂²⋊C₄	—	2	1	no	—
332	C₂²⋊C₄	—	2	1	no	—
333	C₂²⋊C₄	—	4	2	no	—
334	C₂²⋊C₄	—	4	2	no	—
335	C₂²⋊C₄	—	4	2	no	—
336	C₄²	—	2	2	no	—
337	C₄²	—	4	2	no	—
338	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
339	C₂³	—	2	1	no	—
340	C₂³	—	2	1	no	—
341	Q₈	—	2	2	no	—
342	Q₈	—	2	2	no	—
343	Q₈	—	4	2	no	—
344	Q₈	—	4	2	no	—
345	Q₈	—	4	2	no	—
346	Q₈	—	4	2	no	—
347	Q₈	—	8	4	no	—
348	D₄	—	2	1	no	—
349	D₄	—	2	1	no	—
350	D₄	—	2	1	no	—
351	D₄	C₃₁×#135	1	2	yes	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	4	1	no	—
357	C₂×C₄	—	4	1	no	—
358	C₂×C₄	—	4	1	no	—
359	C₂×C₄	—	4	1	no	—
360	C₂×C₄	—	4	2	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	8	4	no	—
363	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
364	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
365	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
366	C₈	—	2	2	no	—
367	C₈	—	2	1	no	—
368	C₈	—	2	1	no	—
369	C₈	—	2	1	no	—
370	C₈	—	4	2	no	—
371	C₈	—	4	1	no	—
372	C₈	—	4	1	no	—
373	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
374	C₂²	—	2	2	no	—
375	C₂²	—	4	1	no	—
376	C₂²	—	4	1	no	—
377	C₂²	—	4	1	no	—
378	C₂²	—	4	1	no	—
379	C₂²	—	4	1	no	—
380	C₂²	D₄²×C₃₁	1	1	yes	characteristic
381	C₄	—	2	1	no	—
382	C₄	—	4	1	no	—
383	C₄	—	4	1	no	—
384	C₄	—	4	2	no	—
385	C₄	—	4	2	no	—
386	C₄	—	8	2	no	—
387	C₄	—	8	2	no	—
388	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
389	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
390	C₄	C₃₁×#577	1	1	yes	characteristic
391	C₄	C₃₁×#578	1	1	yes	characteristic
392	C₂	—	4	2	no	—
393	C₂	—	8	1	no	—
394	C₂	C₃₁×#3272	1	1	yes	characteristic
395	C₂	C₃₁×#3285	1	1	yes	characteristic
396	C₂	C₃₁×#3293	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₂×D₈×C₃₁	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#3342	C₂	semidirect
3. D₄×Q₁₆×C₃₁	C₂	semidirect
4. C₃₁×#1707	C₂	semidirect
5. C₃₁×#1700	C₂	semidirect
6. C₃₁×#1673	C₂	semidirect
7. C₃₁×#1669	C₂	semidirect
8. C₃₁×#1652	C₂	semidirect
9. C₃₁×#1651	C₂	semidirect
10. C₃₁×#1591	C₂	semidirect
140. #19933	C₃₁	direct (N × H)
177. #3342	C₂×C₃₁	semidirect
178. D₄×Q₁₆	C₂×C₃₁	semidirect
179. #1707	C₂×C₃₁	semidirect
180. #1700	C₂×C₃₁	semidirect
181. #1673	C₂×C₃₁	semidirect
182. #1669	C₂×C₃₁	semidirect
183. #1652	C₂×C₃₁	semidirect
184. #1651	C₂×C₃₁	semidirect
185. #1591	C₂×C₃₁	semidirect
293. C₃₁	#19933	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×#3367	2
C₃₁×#3342	2
D₄×Q₁₆×C₃₁	2
C₃₁×#1707	2
C₃₁×#1700	2
C₃₁×#1673	2
C₃₁×#1669	2
C₃₁×#1652	2
C₃₁×#1651	2
C₃₁×#1591	2
C₈×D₈×C₃₁	2
#19933	31

Subgroups — C31×#19933 #21476692

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19933 #21476692