Subgroups — C₃₁×#33123 #21489882

Order 7936.

808 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3094	C₂	1	2	yes	maximal, complemented
2	C₂×C₃₁×#466	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#405	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1978	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1952	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1931	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1900	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1810	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#121	C₂²	1	2	yes	—
13	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
14	C₃₁×#482	—	2	4	no	—
15	C₃₁×#481	—	2	2	no	—
16	C₃₁×#480	—	2	2	no	—
17	C₃₁×#466	—	2	1	no	—
18	C₃₁×#466	—	2	1	no	—
19	C₃₁×#466	—	2	1	no	—
20	C₃₁×#466	—	2	1	no	—
21	C₂×C₃₁×#108	C₂²	1	2	yes	—
22	C₃₁×#428	C₂²	1	4	yes	—
23	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
24	C₃₁×#418	C₂²	1	4	yes	—
25	C₂×C₃₁×#102	C₂²	1	2	yes	—
26	C₂×C₃₁×#100	C₂²	1	4	yes	—
27	C₃₁×#407	C₂²	1	2	yes	—
28	C₃₁×#405	—	2	1	no	—
29	C₃₁×#405	—	2	1	no	—
30	C₃₁×#405	C₂²	1	2	yes	—
31	C₃₁×#405	C₂²	1	2	yes	—
32	C₂×C₃₁×#97	—	2	1	no	—
33	C₂×C₄×C₈×C₃₁	—	2	1	no	—
34	C₃₁×#343	C₂²	1	4	yes	—
35	C₃₁×#341	C₂²	1	2	yes	—
36	C₂²×D₄×C₃₁	—	2	2	no	—
37	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
38	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
39	C₂×D₈×C₃₁	—	2	1	no	—
40	C₂×D₈×C₃₁	—	2	1	no	—
41	C₂×D₈×C₃₁	—	2	4	no	—
42	C₂×M₁₆×C₃₁	—	2	1	no	—
43	C₂×M₁₆×C₃₁	—	2	4	no	—
44	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
45	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
46	C₂²×C₈×C₃₁	—	2	1	no	—
47	C₂²×C₈×C₃₁	—	2	4	no	—
48	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₃₁×#121	—	2	2	no	—
55	C₃₁×#121	—	2	2	no	—
56	C₃₁×#121	—	2	2	no	—
57	C₃₁×#121	—	2	2	no	—
58	C₃₁×#117	—	2	1	no	—
59	C₃₁×#117	—	2	1	no	—
60	C₃₁×#117	D₄	1	2	yes	—
61	C₃₁×#117	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
65	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
66	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
67	C₂×C₄²×C₃₁	—	2	1	no	—
68	C₃₁×#108	—	2	4	no	—
69	C₃₁×#107	—	2	1	no	—
70	C₃₁×#107	—	2	1	no	—
71	C₃₁×#107	—	2	4	no	—
72	C₃₁×#106	—	2	4	no	—
73	C₃₁×#102	—	2	2	no	—
74	C₃₁×#102	—	2	2	no	—
75	C₃₁×#102	—	2	4	no	—
76	C₃₁×#102	—	2	4	no	—
77	C₃₁×#100	—	2	8	no	—
78	C₃₁×C₂²⋊C₈	—	2	2	no	—
79	C₃₁×C₂²⋊C₈	—	2	2	no	—
80	C₃₁×#97	—	2	2	no	—
81	C₃₁×#97	—	2	2	no	—
82	C₄×C₈×C₃₁	—	2	2	no	—
83	C₄×C₈×C₃₁	—	2	2	no	—
84	C₂⁴×C₃₁	—	2	2	no	—
85	C₂⁴×C₃₁	#41	1	1	yes	characteristic
86	C₂×D₄×C₃₁	—	2	2	no	—
87	C₂×D₄×C₃₁	—	2	2	no	—
88	C₂×D₄×C₃₁	—	2	4	no	—
89	C₂×D₄×C₃₁	—	2	4	no	—
90	C₂×D₄×C₃₁	—	4	2	no	—
91	C₂×D₄×C₃₁	—	4	2	no	—
92	C₂×D₄×C₃₁	—	8	4	no	—
93	C₂²×C₄×C₃₁	—	2	2	no	—
94	C₂²×C₄×C₃₁	—	2	2	no	—
95	C₂²×C₄×C₃₁	—	2	1	no	—
96	C₂²×C₄×C₃₁	—	2	1	no	—
97	C₂²×C₄×C₃₁	—	2	2	no	—
98	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
103	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
104	D₈×C₃₁	—	4	2	no	—
105	D₈×C₃₁	—	4	2	no	—
106	M₁₆×C₃₁	—	2	4	no	—
107	M₁₆×C₃₁	—	2	2	no	—
108	M₁₆×C₃₁	—	2	4	no	—
109	M₁₆×C₃₁	—	2	4	no	—
110	M₁₆×C₃₁	#41	1	4	yes	—
111	C₂×C₈×C₃₁	—	2	4	no	—
112	C₂×C₈×C₃₁	—	2	2	no	—
113	C₂×C₈×C₃₁	—	2	2	no	—
114	C₂×C₈×C₃₁	—	2	2	no	—
115	C₂×C₈×C₃₁	—	2	2	no	—
116	C₂×C₈×C₃₁	—	2	2	no	—
117	C₂×C₈×C₃₁	—	2	2	no	—
118	C₂×C₈×C₃₁	—	2	4	no	—
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	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₃₁	#41	1	4	yes	—
124	C₄⋊C₄×C₃₁	—	2	1	no	—
125	C₄⋊C₄×C₃₁	—	2	1	no	—
126	C₄⋊C₄×C₃₁	—	2	2	no	—
127	C₄⋊C₄×C₃₁	—	2	2	no	—
128	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
129	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
130	C₂²⋊C₄×C₃₁	—	2	2	no	—
131	C₂²⋊C₄×C₃₁	—	2	2	no	—
132	C₂²⋊C₄×C₃₁	—	4	2	no	—
133	C₂²⋊C₄×C₃₁	—	4	2	no	—
134	C₄²×C₃₁	—	2	2	no	—
135	C₄²×C₃₁	—	2	2	no	—
136	#33123	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂³×C₃₁	—	2	2	no	—
138	C₂³×C₃₁	—	2	2	no	—
139	C₂³×C₃₁	—	4	1	no	—
140	C₂³×C₃₁	—	4	2	no	—
141	C₂³×C₃₁	—	4	4	no	—
142	C₂³×C₃₁	—	8	2	no	—
143	C₂³×C₃₁	—	8	2	no	—
144	C₂³×C₃₁	#121	1	1	yes	characteristic
145	C₂³×C₃₁	#121	1	1	yes	characteristic
146	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
147	D₄×C₃₁	—	4	2	no	—
148	D₄×C₃₁	—	4	2	no	—
149	D₄×C₃₁	—	4	4	no	—
150	D₄×C₃₁	—	8	4	no	—
151	D₄×C₃₁	—	8	4	no	—
152	C₂×C₄×C₃₁	—	2	2	no	—
153	C₂×C₄×C₃₁	—	2	1	no	—
154	C₂×C₄×C₃₁	—	2	1	no	—
155	C₂×C₄×C₃₁	—	2	1	no	—
156	C₂×C₄×C₃₁	—	2	1	no	—
157	C₂×C₄×C₃₁	—	2	2	no	—
158	C₂×C₄×C₃₁	—	2	2	no	—
159	C₂×C₄×C₃₁	—	2	2	no	—
160	C₂×C₄×C₃₁	—	2	2	no	—
161	C₂×C₄×C₃₁	—	4	2	no	—
162	C₂×C₄×C₃₁	—	4	2	no	—
163	C₂×C₄×C₃₁	—	4	2	no	—
164	C₂×C₄×C₃₁	—	8	2	no	—
165	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
166	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
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	2	yes	—
170	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
171	C₈×C₃₁	—	2	1	no	—
172	C₈×C₃₁	—	2	1	no	—
173	C₈×C₃₁	—	2	4	no	—
174	C₈×C₃₁	—	4	4	no	—
175	C₈×C₃₁	#123	1	2	yes	—
176	C₈×C₃₁	#123	1	2	yes	—
177	#3094	C₂×C₃₁	1	2	yes	complemented
178	C₂×#466	C₂×C₃₁	1	1	yes	characteristic
179	C₂×#405	C₂×C₃₁	1	1	yes	characteristic, complemented
180	#1978	C₂×C₃₁	1	2	yes	complemented
181	#1952	C₂×C₃₁	1	2	yes	complemented
182	#1931	C₂×C₃₁	1	4	yes	complemented
183	#1900	C₂×C₃₁	1	2	yes	complemented
184	#1810	C₂×C₃₁	1	1	yes	characteristic, complemented
185	C₂²×C₃₁	—	2	1	no	—
186	C₂²×C₃₁	—	2	1	no	—
187	C₂²×C₃₁	—	2	1	no	—
188	C₂²×C₃₁	—	2	1	no	—
189	C₂²×C₃₁	—	2	1	no	—
190	C₂²×C₃₁	—	2	1	no	—
191	C₂²×C₃₁	—	2	1	no	—
192	C₂²×C₃₁	—	4	4	no	—
193	C₂²×C₃₁	—	8	2	no	—
194	C₂²×C₃₁	—	8	2	no	—
195	C₂²×C₃₁	—	8	2	no	—
196	C₂²×C₃₁	—	16	4	no	—
197	C₂²×C₃₁	#472	1	2	yes	—
198	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
199	C₂²×C₃₁	#529	1	2	yes	—
200	C₄×C₃₁	—	2	1	no	—
201	C₄×C₃₁	—	2	1	no	—
202	C₄×C₃₁	—	4	2	no	—
203	C₄×C₃₁	—	4	2	no	—
204	C₄×C₃₁	—	8	2	no	—
205	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
206	C₄×C₃₁	#525	1	1	yes	characteristic
207	C₄×C₃₁	#532	1	2	yes	—
208	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
209	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
210	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
211	C₂×#121	C₂²×C₃₁	1	2	yes	—
212	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
213	#482	—	2	4	no	—
214	#481	—	2	2	no	—
215	#480	—	2	2	no	—
216	#466	—	2	1	no	—
217	#466	—	2	1	no	—
218	#466	—	2	1	no	—
219	#466	—	2	1	no	—
220	C₂×#108	C₂²×C₃₁	1	2	yes	—
221	#428	C₂²×C₃₁	1	4	yes	—
222	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
223	#418	C₂²×C₃₁	1	4	yes	—
224	C₂×#102	C₂²×C₃₁	1	2	yes	—
225	C₂×#100	C₂²×C₃₁	1	4	yes	—
226	#407	C₂²×C₃₁	1	2	yes	—
227	#405	—	2	1	no	—
228	#405	—	2	1	no	—
229	#405	C₂²×C₃₁	1	2	yes	—
230	#405	C₂²×C₃₁	1	2	yes	—
231	C₂×#97	—	2	1	no	—
232	C₂×C₄×C₈	—	2	1	no	—
233	#343	C₂²×C₃₁	1	4	yes	—
234	#341	C₂²×C₃₁	1	2	yes	—
235	C₂×C₃₁	—	2	2	no	—
236	C₂×C₃₁	—	2	2	no	—
237	C₂×C₃₁	—	4	1	no	—
238	C₂×C₃₁	—	16	2	no	—
239	C₂×C₃₁	#2854	1	1	yes	characteristic
240	C₂×C₃₁	C₂×#472	1	1	yes	characteristic
241	C₂×C₃₁	#3067	1	1	yes	characteristic
242	C₂²×D₄	—	2	2	no	—
243	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
244	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
245	C₂×D₈	—	2	1	no	—
246	C₂×D₈	—	2	1	no	—
247	C₂×D₈	—	2	4	no	—
248	C₂×M₁₆	—	2	1	no	—
249	C₂×M₁₆	—	2	4	no	—
250	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
251	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
252	C₂²×C₈	—	2	1	no	—
253	C₂²×C₈	—	2	4	no	—
254	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
255	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
256	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
257	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
258	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
259	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
260	#121	—	2	2	no	—
261	#121	—	2	2	no	—
262	#121	—	2	2	no	—
263	#121	—	2	2	no	—
264	#117	—	2	1	no	—
265	#117	—	2	1	no	—
266	#117	D₄×C₃₁	1	2	yes	—
267	#117	C₂³×C₃₁	1	2	yes	—
268	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
269	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
270	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
271	C₂×C₂²⋊C₄	—	2	1	no	—
272	C₂×C₂²⋊C₄	—	2	2	no	—
273	C₂×C₄²	—	2	1	no	—
274	#108	—	2	4	no	—
275	#107	—	2	1	no	—
276	#107	—	2	1	no	—
277	#107	—	2	4	no	—
278	#106	—	2	4	no	—
279	#102	—	2	2	no	—
280	#102	—	2	2	no	—
281	#102	—	2	4	no	—
282	#102	—	2	4	no	—
283	#100	—	2	8	no	—
284	C₂²⋊C₈	—	2	2	no	—
285	C₂²⋊C₈	—	2	2	no	—
286	#97	—	2	2	no	—
287	#97	—	2	2	no	—
288	C₄×C₈	—	2	2	no	—
289	C₄×C₈	—	2	2	no	—
290	C₃₁	#33123	1	1	yes	characteristic, direct product factor, Sylow
291	C₂⁴	—	2	2	no	—
292	C₂⁴	#41×C₃₁	1	1	yes	characteristic
293	C₂×D₄	—	2	2	no	—
294	C₂×D₄	—	2	2	no	—
295	C₂×D₄	—	2	4	no	—
296	C₂×D₄	—	2	4	no	—
297	C₂×D₄	—	4	2	no	—
298	C₂×D₄	—	4	2	no	—
299	C₂×D₄	—	8	4	no	—
300	C₂²×C₄	—	2	2	no	—
301	C₂²×C₄	—	2	2	no	—
302	C₂²×C₄	—	2	1	no	—
303	C₂²×C₄	—	2	1	no	—
304	C₂²×C₄	—	2	2	no	—
305	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
306	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
307	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
308	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
309	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
310	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
311	D₈	—	4	2	no	—
312	D₈	—	4	2	no	—
313	M₁₆	—	2	4	no	—
314	M₁₆	—	2	2	no	—
315	M₁₆	—	2	4	no	—
316	M₁₆	—	2	4	no	—
317	M₁₆	#41×C₃₁	1	4	yes	—
318	C₂×C₈	—	2	4	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	2	2	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	—	2	4	no	—
326	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
327	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
328	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
329	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
330	C₂×C₈	#41×C₃₁	1	4	yes	—
331	C₄⋊C₄	—	2	1	no	—
332	C₄⋊C₄	—	2	1	no	—
333	C₄⋊C₄	—	2	2	no	—
334	C₄⋊C₄	—	2	2	no	—
335	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
336	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
337	C₂²⋊C₄	—	2	2	no	—
338	C₂²⋊C₄	—	2	2	no	—
339	C₂²⋊C₄	—	4	2	no	—
340	C₂²⋊C₄	—	4	2	no	—
341	C₄²	—	2	2	no	—
342	C₄²	—	2	2	no	—
343	C₂³	—	2	2	no	—
344	C₂³	—	2	2	no	—
345	C₂³	—	4	1	no	—
346	C₂³	—	4	2	no	—
347	C₂³	—	4	4	no	—
348	C₂³	—	8	2	no	—
349	C₂³	—	8	2	no	—
350	C₂³	C₃₁×#121	1	1	yes	characteristic
351	C₂³	C₃₁×#121	1	1	yes	characteristic
352	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
353	D₄	—	4	2	no	—
354	D₄	—	4	2	no	—
355	D₄	—	4	4	no	—
356	D₄	—	8	4	no	—
357	D₄	—	8	4	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	—	8	2	no	—
371	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
372	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
373	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
374	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
375	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
376	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
377	C₈	—	2	1	no	—
378	C₈	—	2	1	no	—
379	C₈	—	2	4	no	—
380	C₈	—	4	4	no	—
381	C₈	C₃₁×#123	1	2	yes	—
382	C₈	C₃₁×#123	1	2	yes	—
383	C₂²	—	2	1	no	—
384	C₂²	—	2	1	no	—
385	C₂²	—	2	1	no	—
386	C₂²	—	2	1	no	—
387	C₂²	—	2	1	no	—
388	C₂²	—	2	1	no	—
389	C₂²	—	2	1	no	—
390	C₂²	—	4	4	no	—
391	C₂²	—	8	2	no	—
392	C₂²	—	8	2	no	—
393	C₂²	—	8	2	no	—
394	C₂²	—	16	4	no	—
395	C₂²	C₃₁×#472	1	2	yes	—
396	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
397	C₂²	C₃₁×#529	1	2	yes	—
398	C₄	—	2	1	no	—
399	C₄	—	2	1	no	—
400	C₄	—	4	2	no	—
401	C₄	—	4	2	no	—
402	C₄	—	8	2	no	—
403	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
404	C₄	C₃₁×#525	1	1	yes	characteristic
405	C₄	C₃₁×#532	1	2	yes	—
406	C₂	—	2	2	no	—
407	C₂	—	2	2	no	—
408	C₂	—	4	1	no	—
409	C₂	—	16	2	no	—
410	C₂	C₃₁×#2854	1	1	yes	characteristic
411	C₂	C₂×C₃₁×#472	1	1	yes	characteristic
412	C₂	C₃₁×#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#466	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3094	C₂	semidirect
3. C₂×C₃₁×#405	C₂	semidirect
4. C₃₁×#1978	C₂	semidirect
5. C₃₁×#1952	C₂	semidirect
6. C₃₁×#1931	C₂	semidirect
7. C₃₁×#1900	C₂	semidirect
8. C₃₁×#1810	C₂	semidirect
136. #33123	C₃₁	direct (N × H)
177. #3094	C₂×C₃₁	semidirect
179. C₂×#405	C₂×C₃₁	semidirect
180. #1978	C₂×C₃₁	semidirect
181. #1952	C₂×C₃₁	semidirect
182. #1931	C₂×C₃₁	semidirect
183. #1900	C₂×C₃₁	semidirect
184. #1810	C₂×C₃₁	semidirect
290. C₃₁	#33123	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×#3094	2
C₂×C₃₁×#466	2
C₂×C₃₁×#405	2
C₃₁×#1978	2
C₃₁×#1952	2
C₃₁×#1931	2
C₃₁×#1900	2
C₃₁×#1810	2
#33123	31

Subgroups — C31×#33123 #21489882

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33123 #21489882