Subgroups — C₃₁×#19391 #21476150

Order 7936.

696 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3227	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#476	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3036	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2972	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1677	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1671	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1570	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1548	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1540	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×AGL(1,ℤ/8)	—	2	1	no	—
11	C₃₁×#532	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#498	C₂²	1	2	yes	complemented
15	C₃₁×#486	C₂²	1	2	yes	complemented
16	C₃₁×#481	—	2	2	no	—
17	C₃₁×#476	—	2	2	no	—
18	C₃₁×#476	—	2	2	no	—
19	C₃₁×#476	C₂²	1	4	yes	—
20	C₃₁×#463	—	2	2	no	—
21	C₃₁×#461	C₂²	1	2	yes	—
22	C₃₁×#459	C₂²	1	2	yes	—
23	C₈×D₄×C₃₁	—	2	2	no	—
24	C₈×D₄×C₃₁	C₂²	1	4	yes	—
25	C₃₁×#431	C₂²	1	1	yes	characteristic
26	C₃₁×#431	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#106	—	2	1	no	—
28	C₃₁×#423	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₃₁×#417	—	2	1	no	—
31	C₂×C₃₁×#102	—	2	1	no	—
32	C₂×C₃₁×C₂²⋊C₈	—	2	2	no	—
33	C₃₁×#334	C₂²	1	4	yes	complemented
34	C₃₁×#333	C₂²	1	4	yes	complemented
35	C₃₁×#332	C₂²	1	2	yes	complemented
36	C₃₁×#331	C₂²	1	2	yes	complemented
37	C₂×#41×C₃₁	—	2	1	no	—
38	C₂²×Q₈×C₃₁	—	2	1	no	—
39	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
40	C₂³×C₄×C₃₁	—	2	1	no	—
41	C₃₁×AGL(1,ℤ/8)	—	4	4	no	—
42	C₂×SD₁₆×C₃₁	—	2	2	no	—
43	C₂×D₈×C₃₁	—	2	2	no	—
44	C₂×M₁₆×C₃₁	—	2	1	no	—
45	C₂²×C₈×C₃₁	—	2	1	no	—
46	C₂²×C₈×C₃₁	—	2	4	no	—
47	C₃₁×C₄⋊Q₈	—	2	1	no	—
48	C₃₁×C₄⋊Q₈	—	2	1	no	—
49	C₃₁×C₄⋊Q₈	—	2	1	no	—
50	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
51	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
52	C₃₁×#127	C₂³	1	1	yes	characteristic
53	C₃₁×#124	—	2	1	no	—
54	C₃₁×#121	—	2	1	no	—
55	C₃₁×#121	—	2	1	no	—
56	C₄×D₄×C₃₁	—	2	1	no	—
57	C₄×D₄×C₃₁	—	2	1	no	—
58	C₄×D₄×C₃₁	—	2	1	no	—
59	C₄×D₄×C₃₁	—	2	1	no	—
60	C₄×D₄×C₃₁	D₄	1	2	yes	—
61	C₄×D₄×C₃₁	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
63	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
64	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
65	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
66	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
67	C₃₁×#106	—	2	4	no	—
68	C₃₁×#106	—	2	4	no	—
69	C₃₁×C₄⋊C₈	—	2	1	no	—
70	C₃₁×C₄⋊C₈	—	2	1	no	—
71	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
72	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
73	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
74	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
75	C₃₁×#103	—	2	2	no	—
76	C₃₁×#102	—	2	4	no	—
77	C₃₁×#102	—	2	4	no	—
78	C₃₁×#102	—	2	2	no	—
79	C₃₁×C₂²⋊C₈	—	2	1	no	—
80	C₃₁×C₂²⋊C₈	—	2	4	no	—
81	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
82	C₄×C₈×C₃₁	—	2	1	no	—
83	C₄×C₈×C₃₁	C₂³	1	2	yes	—
84	C₂⁴×C₃₁	—	2	1	no	—
85	#41×C₃₁	—	4	2	no	—
86	#41×C₃₁	—	4	2	no	—
87	C₂×Q₈×C₃₁	—	2	1	no	—
88	C₂×Q₈×C₃₁	—	2	2	no	—
89	C₂×Q₈×C₃₁	—	4	1	no	—
90	C₂×Q₈×C₃₁	—	8	2	no	—
91	C₂×D₄×C₃₁	—	2	1	no	—
92	C₂×D₄×C₃₁	—	2	2	no	—
93	C₂×D₄×C₃₁	—	2	1	no	—
94	C₂×D₄×C₃₁	—	2	1	no	—
95	C₂×D₄×C₃₁	—	2	1	no	—
96	C₂×D₄×C₃₁	—	4	1	no	—
97	C₂×D₄×C₃₁	SD₁₆	1	4	yes	—
98	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₃₁	—	2	1	no	—
100	C₂²×C₄×C₃₁	—	2	2	no	—
101	C₂²×C₄×C₃₁	—	2	2	no	—
102	C₂²×C₄×C₃₁	—	4	1	no	—
103	C₂²×C₄×C₃₁	—	4	1	no	—
104	C₂²×C₄×C₃₁	—	8	2	no	—
105	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
108	SD₁₆×C₃₁	—	4	4	no	—
109	D₈×C₃₁	—	4	4	no	—
110	M₁₆×C₃₁	—	4	2	no	—
111	C₂×C₈×C₃₁	—	2	2	no	—
112	C₂×C₈×C₃₁	—	2	2	no	—
113	C₂×C₈×C₃₁	—	2	2	no	—
114	C₂×C₈×C₃₁	—	4	2	no	—
115	C₂×C₈×C₃₁	—	8	4	no	—
116	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
117	C₄⋊C₄×C₃₁	—	2	1	no	—
118	C₄⋊C₄×C₃₁	—	2	2	no	—
119	C₄⋊C₄×C₃₁	—	2	2	no	—
120	C₄⋊C₄×C₃₁	—	2	1	no	—
121	C₄⋊C₄×C₃₁	—	2	1	no	—
122	C₄⋊C₄×C₃₁	—	4	1	no	—
123	C₄⋊C₄×C₃₁	—	4	1	no	—
124	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
125	C₂²⋊C₄×C₃₁	—	2	2	no	—
126	C₂²⋊C₄×C₃₁	—	2	2	no	—
127	C₂²⋊C₄×C₃₁	—	4	1	no	—
128	C₂²⋊C₄×C₃₁	—	4	1	no	—
129	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
131	C₄²×C₃₁	#41	1	1	yes	characteristic
132	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
133	#19391	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
134	C₂³×C₃₁	—	2	2	no	—
135	C₂³×C₃₁	—	2	4	no	—
136	C₂³×C₃₁	—	4	1	no	—
137	C₂³×C₃₁	—	8	1	no	—
138	C₂³×C₃₁	—	8	1	no	—
139	C₂³×C₃₁	#121	1	1	yes	characteristic
140	Q₈×C₃₁	—	4	1	no	—
141	Q₈×C₃₁	—	4	1	no	—
142	Q₈×C₃₁	—	8	2	no	—
143	Q₈×C₃₁	—	8	2	no	—
144	D₄×C₃₁	—	2	2	no	—
145	D₄×C₃₁	—	2	4	no	—
146	D₄×C₃₁	—	4	1	no	—
147	D₄×C₃₁	—	4	1	no	—
148	D₄×C₃₁	—	4	2	no	—
149	D₄×C₃₁	—	8	1	no	—
150	D₄×C₃₁	—	8	2	no	—
151	D₄×C₃₁	C₂×SD₁₆	1	4	yes	—
152	C₂×C₄×C₃₁	—	2	2	no	—
153	C₂×C₄×C₃₁	—	2	2	no	—
154	C₂×C₄×C₃₁	—	4	1	no	—
155	C₂×C₄×C₃₁	—	4	4	no	—
156	C₂×C₄×C₃₁	—	4	1	no	—
157	C₂×C₄×C₃₁	—	4	2	no	—
158	C₂×C₄×C₃₁	—	8	2	no	—
159	C₂×C₄×C₃₁	—	8	1	no	—
160	C₂×C₄×C₃₁	—	8	2	no	—
161	C₂×C₄×C₃₁	—	8	2	no	—
162	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
163	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
164	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
165	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
166	C₂×C₄×C₃₁	#135	1	2	yes	—
167	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
170	C₈×C₃₁	—	2	4	no	—
171	C₈×C₃₁	—	4	2	no	—
172	C₈×C₃₁	—	4	2	no	—
173	C₈×C₃₁	—	4	2	no	—
174	#3227	C₂×C₃₁	1	1	yes	characteristic, complemented
175	C₂×#476	C₂×C₃₁	1	1	yes	characteristic, complemented
176	#3036	C₂×C₃₁	1	1	yes	characteristic
177	#2972	C₂×C₃₁	1	1	yes	characteristic, complemented
178	#1677	C₂×C₃₁	1	4	yes	complemented
179	#1671	C₂×C₃₁	1	4	yes	complemented
180	#1570	C₂×C₃₁	1	1	yes	characteristic, complemented
181	#1548	C₂×C₃₁	1	1	yes	characteristic, complemented
182	#1540	C₂×C₃₁	1	1	yes	characteristic, complemented
183	C₂²×C₃₁	—	2	1	no	—
184	C₂²×C₃₁	—	2	1	no	—
185	C₂²×C₃₁	—	2	4	no	—
186	C₂²×C₃₁	—	4	2	no	—
187	C₂²×C₃₁	—	4	2	no	—
188	C₂²×C₃₁	—	8	4	no	—
189	C₂²×C₃₁	—	8	1	no	—
190	C₂²×C₃₁	—	8	1	no	—
191	C₂²×C₃₁	—	8	1	no	—
192	C₂²×C₃₁	#465	1	2	yes	—
193	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
194	C₄×C₃₁	—	2	1	no	—
195	C₄×C₃₁	—	2	1	no	—
196	C₄×C₃₁	—	2	1	no	—
197	C₄×C₃₁	—	4	2	no	—
198	C₄×C₃₁	—	8	1	no	—
199	C₄×C₃₁	—	8	2	no	—
200	C₄×C₃₁	#465	1	2	yes	—
201	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
202	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
203	C₄×C₃₁	#577	1	2	yes	—
204	C₂×AGL(1,ℤ/8)	—	2	1	no	—
205	#532	C₂²×C₃₁	1	1	yes	characteristic
206	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
207	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
208	#498	C₂²×C₃₁	1	2	yes	complemented
209	#486	C₂²×C₃₁	1	2	yes	complemented
210	#481	—	2	2	no	—
211	#476	—	2	2	no	—
212	#476	—	2	2	no	—
213	#476	C₂²×C₃₁	1	4	yes	—
214	#463	—	2	2	no	—
215	#461	C₂²×C₃₁	1	2	yes	—
216	#459	C₂²×C₃₁	1	2	yes	—
217	C₈×D₄	—	2	2	no	—
218	C₈×D₄	C₂²×C₃₁	1	4	yes	—
219	#431	C₂²×C₃₁	1	1	yes	characteristic
220	#431	C₂²×C₃₁	1	1	yes	characteristic
221	C₂×#106	—	2	1	no	—
222	#423	C₂²×C₃₁	1	1	yes	characteristic
223	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
224	#417	—	2	1	no	—
225	C₂×#102	—	2	1	no	—
226	C₂×C₂²⋊C₈	—	2	2	no	—
227	#334	C₂²×C₃₁	1	4	yes	complemented
228	#333	C₂²×C₃₁	1	4	yes	complemented
229	#332	C₂²×C₃₁	1	2	yes	complemented
230	#331	C₂²×C₃₁	1	2	yes	complemented
231	C₂×C₃₁	—	2	2	no	—
232	C₂×C₃₁	—	4	4	no	—
233	C₂×C₃₁	—	16	1	no	—
234	C₂×C₃₁	#3042	1	1	yes	characteristic
235	C₂×C₃₁	C₂×#465	1	1	yes	characteristic
236	C₂×C₃₁	#3068	1	1	yes	characteristic
237	C₂×#41	—	2	1	no	—
238	C₂²×Q₈	—	2	1	no	—
239	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
240	C₂³×C₄	—	2	1	no	—
241	AGL(1,ℤ/8)	—	4	4	no	—
242	C₂×SD₁₆	—	2	2	no	—
243	C₂×D₈	—	2	2	no	—
244	C₂×M₁₆	—	2	1	no	—
245	C₂²×C₈	—	2	1	no	—
246	C₂²×C₈	—	2	4	no	—
247	C₄⋊Q₈	—	2	1	no	—
248	C₄⋊Q₈	—	2	1	no	—
249	C₄⋊Q₈	—	2	1	no	—
250	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
251	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
252	#127	C₂³×C₃₁	1	1	yes	characteristic
253	#124	—	2	1	no	—
254	#121	—	2	1	no	—
255	#121	—	2	1	no	—
256	C₄×D₄	—	2	1	no	—
257	C₄×D₄	—	2	1	no	—
258	C₄×D₄	—	2	1	no	—
259	C₄×D₄	—	2	1	no	—
260	C₄×D₄	D₄×C₃₁	1	2	yes	—
261	C₄×D₄	C₂³×C₃₁	1	2	yes	—
262	C₂×C₄⋊C₄	—	2	1	no	—
263	C₂×C₄⋊C₄	—	2	1	no	—
264	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
265	C₂×C₂²⋊C₄	—	2	1	no	—
266	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
267	#106	—	2	4	no	—
268	#106	—	2	4	no	—
269	C₄⋊C₈	—	2	1	no	—
270	C₄⋊C₈	—	2	1	no	—
271	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
272	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
273	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
274	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
275	#103	—	2	2	no	—
276	#102	—	2	4	no	—
277	#102	—	2	4	no	—
278	#102	—	2	2	no	—
279	C₂²⋊C₈	—	2	1	no	—
280	C₂²⋊C₈	—	2	4	no	—
281	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
282	C₄×C₈	—	2	1	no	—
283	C₄×C₈	C₂³×C₃₁	1	2	yes	—
284	C₃₁	#19391	1	1	yes	characteristic, direct product factor, Sylow
285	C₂⁴	—	2	1	no	—
286	#41	—	4	2	no	—
287	#41	—	4	2	no	—
288	C₂×Q₈	—	2	1	no	—
289	C₂×Q₈	—	2	2	no	—
290	C₂×Q₈	—	4	1	no	—
291	C₂×Q₈	—	8	2	no	—
292	C₂×D₄	—	2	1	no	—
293	C₂×D₄	—	2	2	no	—
294	C₂×D₄	—	2	1	no	—
295	C₂×D₄	—	2	1	no	—
296	C₂×D₄	—	2	1	no	—
297	C₂×D₄	—	4	1	no	—
298	C₂×D₄	SD₁₆×C₃₁	1	4	yes	—
299	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
300	C₂²×C₄	—	2	1	no	—
301	C₂²×C₄	—	2	2	no	—
302	C₂²×C₄	—	2	2	no	—
303	C₂²×C₄	—	4	1	no	—
304	C₂²×C₄	—	4	1	no	—
305	C₂²×C₄	—	8	2	no	—
306	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
307	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
308	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
309	SD₁₆	—	4	4	no	—
310	D₈	—	4	4	no	—
311	M₁₆	—	4	2	no	—
312	C₂×C₈	—	2	2	no	—
313	C₂×C₈	—	2	2	no	—
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	4	2	no	—
316	C₂×C₈	—	8	4	no	—
317	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
318	C₄⋊C₄	—	2	1	no	—
319	C₄⋊C₄	—	2	2	no	—
320	C₄⋊C₄	—	2	2	no	—
321	C₄⋊C₄	—	2	1	no	—
322	C₄⋊C₄	—	2	1	no	—
323	C₄⋊C₄	—	4	1	no	—
324	C₄⋊C₄	—	4	1	no	—
325	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
326	C₂²⋊C₄	—	2	2	no	—
327	C₂²⋊C₄	—	2	2	no	—
328	C₂²⋊C₄	—	4	1	no	—
329	C₂²⋊C₄	—	4	1	no	—
330	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
331	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
332	C₄²	#41×C₃₁	1	1	yes	characteristic
333	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
334	C₂³	—	2	2	no	—
335	C₂³	—	2	4	no	—
336	C₂³	—	4	1	no	—
337	C₂³	—	8	1	no	—
338	C₂³	—	8	1	no	—
339	C₂³	C₃₁×#121	1	1	yes	characteristic
340	Q₈	—	4	1	no	—
341	Q₈	—	4	1	no	—
342	Q₈	—	8	2	no	—
343	Q₈	—	8	2	no	—
344	D₄	—	2	4	no	—
345	D₄	—	2	2	no	—
346	D₄	—	4	1	no	—
347	D₄	—	4	1	no	—
348	D₄	—	4	2	no	—
349	D₄	—	8	1	no	—
350	D₄	—	8	2	no	—
351	D₄	C₂×SD₁₆×C₃₁	1	4	yes	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	4	1	no	—
355	C₂×C₄	—	4	4	no	—
356	C₂×C₄	—	4	1	no	—
357	C₂×C₄	—	4	2	no	—
358	C₂×C₄	—	8	2	no	—
359	C₂×C₄	—	8	1	no	—
360	C₂×C₄	—	8	2	no	—
361	C₂×C₄	—	8	2	no	—
362	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
363	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
364	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
365	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
366	C₂×C₄	C₃₁×#135	1	2	yes	—
367	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
368	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
369	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
370	C₈	—	2	4	no	—
371	C₈	—	4	2	no	—
372	C₈	—	4	2	no	—
373	C₈	—	4	2	no	—
374	C₂²	—	2	1	no	—
375	C₂²	—	2	1	no	—
376	C₂²	—	2	4	no	—
377	C₂²	—	4	2	no	—
378	C₂²	—	4	2	no	—
379	C₂²	—	8	4	no	—
380	C₂²	—	8	1	no	—
381	C₂²	—	8	1	no	—
382	C₂²	—	8	1	no	—
383	C₂²	C₃₁×#465	1	2	yes	—
384	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
385	C₄	—	2	1	no	—
386	C₄	—	2	1	no	—
387	C₄	—	2	1	no	—
388	C₄	—	4	2	no	—
389	C₄	—	8	1	no	—
390	C₄	—	8	2	no	—
391	C₄	C₃₁×#465	1	2	yes	—
392	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
393	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
394	C₄	C₃₁×#577	1	2	yes	—
395	C₂	—	2	2	no	—
396	C₂	—	4	4	no	—
397	C₂	—	16	1	no	—
398	C₂	C₃₁×#3042	1	1	yes	characteristic
399	C₂	C₂×C₃₁×#465	1	1	yes	characteristic
400	C₂	C₃₁×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3036	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3227	C₂	semidirect
2. C₂×C₃₁×#476	C₂	semidirect
4. C₃₁×#2972	C₂	semidirect
5. C₃₁×#1677	C₂	semidirect
6. C₃₁×#1671	C₂	semidirect
7. C₃₁×#1570	C₂	semidirect
8. C₃₁×#1548	C₂	semidirect
9. C₃₁×#1540	C₂	semidirect
14. C₃₁×#498	C₂²	semidirect
15. C₃₁×#486	C₂²	semidirect
33. C₃₁×#334	C₂²	semidirect
34. C₃₁×#333	C₂²	semidirect
35. C₃₁×#332	C₂²	semidirect
36. C₃₁×#331	C₂²	semidirect
133. #19391	C₃₁	direct (N × H)
174. #3227	C₂×C₃₁	semidirect
175. C₂×#476	C₂×C₃₁	semidirect
177. #2972	C₂×C₃₁	semidirect
178. #1677	C₂×C₃₁	semidirect
179. #1671	C₂×C₃₁	semidirect
180. #1570	C₂×C₃₁	semidirect
181. #1548	C₂×C₃₁	semidirect
182. #1540	C₂×C₃₁	semidirect
208. #498	C₂²×C₃₁	semidirect
209. #486	C₂²×C₃₁	semidirect
227. #334	C₂²×C₃₁	semidirect
228. #333	C₂²×C₃₁	semidirect
229. #332	C₂²×C₃₁	semidirect
230. #331	C₂²×C₃₁	semidirect
284. C₃₁	#19391	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×#3227	2
C₂×C₃₁×#476	2
C₃₁×#3036	2
C₃₁×#2972	2
C₃₁×#1677	2
C₃₁×#1671	2
C₃₁×#1570	2
C₃₁×#1548	2
C₃₁×#1540	2
#19391	31

Subgroups — C31×#19391 #21476150

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19391 #21476150