Subgroups — C₃₁×#19434 #21476193

Order 7936.

720 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3205	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3037	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#460	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2983	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1702	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1688	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1570	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1545	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1540	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#137	—	2	1	no	—
11	C₂²×SD₁₆×C₃₁	—	2	1	no	—
12	C₃₁×#532	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#493	C₂²	1	2	yes	—
16	C₃₁×#486	C₂²	1	2	yes	—
17	C₃₁×#464	—	2	2	no	—
18	C₃₁×#462	C₂²	1	2	yes	complemented
19	C₃₁×#460	—	2	2	no	—
20	C₃₁×#460	C₂²	1	4	yes	—
21	C₃₁×#460	C₂²	1	2	yes	complemented
22	C₃₁×#459	—	2	2	no	—
23	C₃₁×#447	—	2	2	no	—
24	C₃₁×#446	—	2	2	no	—
25	C₃₁×#446	C₂²	1	4	yes	complemented
26	C₃₁×#432	—	2	2	no	—
27	C₃₁×#431	C₂²	1	1	yes	characteristic
28	C₃₁×#431	C₂²	1	1	yes	characteristic
29	C₃₁×#423	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
31	C₃₁×#417	—	2	1	no	—
32	C₂×C₃₁×#102	—	2	1	no	—
33	C₃₁×#334	C₂²	1	4	yes	complemented
34	C₃₁×#332	C₂²	1	2	yes	complemented
35	C₃₁×#331	C₂²	1	4	yes	—
36	C₃₁×#331	C₂²	1	2	yes	complemented
37	C₂×#41×C₃₁	—	2	1	no	—
38	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
39	C₂²×D₄×C₃₁	—	2	1	no	—
40	C₂²×D₄×C₃₁	—	2	1	no	—
41	C₂²×D₄×C₃₁	—	2	1	no	—
42	C₃₁×#137	—	4	4	no	—
43	C₂×Q₁₆×C₃₁	—	2	2	no	—
44	C₂×SD₁₆×C₃₁	—	2	4	no	—
45	C₂×SD₁₆×C₃₁	—	2	2	no	—
46	C₂×SD₁₆×C₃₁	—	4	4	no	—
47	C₂×D₈×C₃₁	—	2	4	no	—
48	C₂×M₁₆×C₃₁	—	2	1	no	—
49	C₂²×C₈×C₃₁	—	2	1	no	—
50	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
51	C₃₁×#127	—	2	1	no	—
52	C₃₁×#127	—	2	1	no	—
53	C₃₁×#127	—	2	1	no	—
54	C₃₁×#127	C₂³	1	2	yes	—
55	C₃₁×#127	C₂³	1	1	yes	characteristic
56	C₃₁×#124	—	2	1	no	—
57	C₃₁×#121	—	2	1	no	—
58	C₃₁×#121	—	2	1	no	—
59	C₄×Q₈×C₃₁	—	2	1	no	—
60	C₄×Q₈×C₃₁	—	2	1	no	—
61	C₄×Q₈×C₃₁	D₄	1	2	yes	—
62	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
63	C₄×D₄×C₃₁	—	2	1	no	—
64	C₄×D₄×C₃₁	—	2	1	no	—
65	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
66	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
67	C₂×C₄²×C₃₁	—	2	1	no	—
68	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
69	C₃₁×C₄⋊C₈	—	2	4	no	—
70	C₃₁×C₄⋊C₈	—	2	1	no	—
71	C₃₁×C₄⋊C₈	—	2	1	no	—
72	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
73	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
74	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
75	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
76	C₃₁×#103	—	2	2	no	—
77	C₃₁×#102	—	2	4	no	—
78	C₃₁×#102	—	2	4	no	—
79	C₃₁×#102	—	2	2	no	—
80	C₃₁×C₂²⋊C₈	—	2	1	no	—
81	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
82	C₃₁×#97	—	2	4	no	—
83	C₄×C₈×C₃₁	—	2	1	no	—
84	C₄×C₈×C₃₁	C₂³	1	2	yes	—
85	C₂⁴×C₃₁	—	4	1	no	—
86	#41×C₃₁	—	4	2	no	—
87	#41×C₃₁	—	4	2	no	—
88	C₂×Q₈×C₃₁	—	2	1	no	—
89	C₂×Q₈×C₃₁	—	2	2	no	—
90	C₂×Q₈×C₃₁	—	2	2	no	—
91	C₂×Q₈×C₃₁	—	2	1	no	—
92	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
93	C₂×D₄×C₃₁	—	2	1	no	—
94	C₂×D₄×C₃₁	—	2	2	no	—
95	C₂×D₄×C₃₁	—	2	2	no	—
96	C₂×D₄×C₃₁	—	2	1	no	—
97	C₂×D₄×C₃₁	—	2	1	no	—
98	C₂×D₄×C₃₁	—	2	2	no	—
99	C₂×D₄×C₃₁	—	4	1	no	—
100	C₂×D₄×C₃₁	—	4	1	no	—
101	C₂×D₄×C₃₁	—	4	1	no	—
102	C₂×D₄×C₃₁	—	4	1	no	—
103	C₂×D₄×C₃₁	—	4	2	no	—
104	C₂×D₄×C₃₁	—	4	2	no	—
105	C₂×D₄×C₃₁	—	8	2	no	—
106	C₂×D₄×C₃₁	—	8	2	no	—
107	C₂²×C₄×C₃₁	—	2	1	no	—
108	C₂²×C₄×C₃₁	—	2	1	no	—
109	C₂²×C₄×C₃₁	—	4	1	no	—
110	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
111	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
113	Q₁₆×C₃₁	—	4	4	no	—
114	SD₁₆×C₃₁	—	4	8	no	—
115	SD₁₆×C₃₁	—	4	4	no	—
116	D₈×C₃₁	—	8	4	no	—
117	M₁₆×C₃₁	—	4	2	no	—
118	C₂×C₈×C₃₁	—	2	2	no	—
119	C₂×C₈×C₃₁	—	2	2	no	—
120	C₂×C₈×C₃₁	—	2	2	no	—
121	C₂×C₈×C₃₁	—	4	2	no	—
122	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
123	C₄⋊C₄×C₃₁	—	2	2	no	—
124	C₄⋊C₄×C₃₁	—	2	1	no	—
125	C₄⋊C₄×C₃₁	—	2	2	no	—
126	C₄⋊C₄×C₃₁	—	2	1	no	—
127	C₄⋊C₄×C₃₁	—	2	1	no	—
128	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
129	C₂²⋊C₄×C₃₁	—	4	1	no	—
130	C₂²⋊C₄×C₃₁	—	4	1	no	—
131	C₄²×C₃₁	—	2	2	no	—
132	C₄²×C₃₁	—	2	2	no	—
133	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
134	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₄²×C₃₁	#41	1	1	yes	characteristic
136	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
137	#19434	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
138	C₂³×C₃₁	—	4	1	no	—
139	C₂³×C₃₁	—	4	2	no	—
140	C₂³×C₃₁	—	8	2	no	—
141	C₂³×C₃₁	—	8	2	no	—
142	C₂³×C₃₁	—	8	2	no	—
143	C₂³×C₃₁	#121	1	1	yes	characteristic
144	Q₈×C₃₁	—	2	2	no	—
145	Q₈×C₃₁	—	2	2	no	—
146	Q₈×C₃₁	—	4	2	no	—
147	Q₈×C₃₁	—	4	1	no	—
148	Q₈×C₃₁	—	4	1	no	—
149	D₄×C₃₁	—	4	2	no	—
150	D₄×C₃₁	—	4	1	no	—
151	D₄×C₃₁	—	4	2	no	—
152	D₄×C₃₁	—	4	1	no	—
153	D₄×C₃₁	—	4	2	no	—
154	D₄×C₃₁	—	4	4	no	—
155	D₄×C₃₁	—	8	1	no	—
156	D₄×C₃₁	—	8	2	no	—
157	D₄×C₃₁	—	8	2	no	—
158	D₄×C₃₁	—	8	2	no	—
159	D₄×C₃₁	—	8	2	no	—
160	D₄×C₃₁	—	8	2	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	2	2	no	—
163	C₂×C₄×C₃₁	—	2	2	no	—
164	C₂×C₄×C₃₁	—	4	1	no	—
165	C₂×C₄×C₃₁	—	4	2	no	—
166	C₂×C₄×C₃₁	—	4	1	no	—
167	C₂×C₄×C₃₁	—	8	1	no	—
168	C₂×C₄×C₃₁	—	8	2	no	—
169	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
170	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
171	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
172	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
173	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
174	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
175	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
177	C₈×C₃₁	—	2	4	no	—
178	C₈×C₃₁	—	4	2	no	—
179	C₈×C₃₁	—	4	2	no	—
180	C₈×C₃₁	—	4	2	no	—
181	#3205	C₂×C₃₁	1	1	yes	characteristic
182	#3037	C₂×C₃₁	1	1	yes	characteristic, complemented
183	C₂×#460	C₂×C₃₁	1	1	yes	characteristic, complemented
184	#2983	C₂×C₃₁	1	1	yes	characteristic, complemented
185	#1702	C₂×C₃₁	1	4	yes	complemented
186	#1688	C₂×C₃₁	1	4	yes	complemented
187	#1570	C₂×C₃₁	1	1	yes	characteristic, complemented
188	#1545	C₂×C₃₁	1	1	yes	characteristic, complemented
189	#1540	C₂×C₃₁	1	1	yes	characteristic, complemented
190	C₂²×C₃₁	—	2	1	no	—
191	C₂²×C₃₁	—	2	1	no	—
192	C₂²×C₃₁	—	8	2	no	—
193	C₂²×C₃₁	—	8	1	no	—
194	C₂²×C₃₁	—	8	1	no	—
195	C₂²×C₃₁	—	8	2	no	—
196	C₂²×C₃₁	—	8	2	no	—
197	C₂²×C₃₁	—	8	1	no	—
198	C₂²×C₃₁	—	16	4	no	—
199	C₂²×C₃₁	#469	1	2	yes	—
200	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
201	C₄×C₃₁	—	2	1	no	—
202	C₄×C₃₁	—	2	1	no	—
203	C₄×C₃₁	—	2	1	no	—
204	C₄×C₃₁	—	4	2	no	—
205	C₄×C₃₁	—	4	2	no	—
206	C₄×C₃₁	—	8	1	no	—
207	C₄×C₃₁	#469	1	2	yes	—
208	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
209	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
210	C₄×C₃₁	#577	1	2	yes	—
211	C₂×#137	—	2	1	no	—
212	C₂²×SD₁₆	—	2	1	no	—
213	#532	C₂²×C₃₁	1	1	yes	characteristic
214	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
215	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
216	#493	C₂²×C₃₁	1	2	yes	—
217	#486	C₂²×C₃₁	1	2	yes	—
218	#464	—	2	2	no	—
219	#462	C₂²×C₃₁	1	2	yes	complemented
220	#460	—	2	2	no	—
221	#460	C₂²×C₃₁	1	4	yes	—
222	#460	C₂²×C₃₁	1	2	yes	complemented
223	#459	—	2	2	no	—
224	#447	—	2	2	no	—
225	#446	—	2	2	no	—
226	#446	C₂²×C₃₁	1	4	yes	complemented
227	#432	—	2	2	no	—
228	#431	C₂²×C₃₁	1	1	yes	characteristic
229	#431	C₂²×C₃₁	1	1	yes	characteristic
230	#423	C₂²×C₃₁	1	1	yes	characteristic
231	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
232	#417	—	2	1	no	—
233	C₂×#102	—	2	1	no	—
234	#334	C₂²×C₃₁	1	4	yes	complemented
235	#332	C₂²×C₃₁	1	2	yes	complemented
236	#331	C₂²×C₃₁	1	4	yes	—
237	#331	C₂²×C₃₁	1	2	yes	complemented
238	C₂×C₃₁	—	2	2	no	—
239	C₂×C₃₁	—	16	1	no	—
240	C₂×C₃₁	—	16	2	no	—
241	C₂×C₃₁	#3043	1	1	yes	characteristic
242	C₂×C₃₁	C₂×#469	1	1	yes	characteristic
243	C₂×C₃₁	#3068	1	1	yes	characteristic
244	C₂×#41	—	2	1	no	—
245	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
246	C₂²×D₄	—	2	1	no	—
247	C₂²×D₄	—	2	1	no	—
248	C₂²×D₄	—	2	1	no	—
249	#137	—	4	4	no	—
250	C₂×Q₁₆	—	2	2	no	—
251	C₂×SD₁₆	—	2	4	no	—
252	C₂×SD₁₆	—	2	2	no	—
253	C₂×SD₁₆	—	4	4	no	—
254	C₂×D₈	—	2	4	no	—
255	C₂×M₁₆	—	2	1	no	—
256	C₂²×C₈	—	2	1	no	—
257	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
258	#127	—	2	1	no	—
259	#127	—	2	1	no	—
260	#127	—	2	1	no	—
261	#127	C₂³×C₃₁	1	2	yes	—
262	#127	C₂³×C₃₁	1	1	yes	characteristic
263	#124	—	2	1	no	—
264	#121	—	2	1	no	—
265	#121	—	2	1	no	—
266	C₄×Q₈	—	2	1	no	—
267	C₄×Q₈	—	2	1	no	—
268	C₄×Q₈	D₄×C₃₁	1	2	yes	—
269	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
270	C₄×D₄	—	2	1	no	—
271	C₄×D₄	—	2	1	no	—
272	C₂×C₄⋊C₄	—	2	1	no	—
273	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
274	C₂×C₄²	—	2	1	no	—
275	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
276	C₄⋊C₈	—	2	4	no	—
277	C₄⋊C₈	—	2	1	no	—
278	C₄⋊C₈	—	2	1	no	—
279	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
280	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
281	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
282	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
283	#103	—	2	2	no	—
284	#102	—	2	4	no	—
285	#102	—	2	4	no	—
286	#102	—	2	2	no	—
287	C₂²⋊C₈	—	2	1	no	—
288	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
289	#97	—	2	4	no	—
290	C₄×C₈	—	2	1	no	—
291	C₄×C₈	C₂³×C₃₁	1	2	yes	—
292	C₃₁	#19434	1	1	yes	characteristic, direct product factor, Sylow
293	C₂⁴	—	4	1	no	—
294	#41	—	4	2	no	—
295	#41	—	4	2	no	—
296	C₂×Q₈	—	2	1	no	—
297	C₂×Q₈	—	2	2	no	—
298	C₂×Q₈	—	2	2	no	—
299	C₂×Q₈	—	2	1	no	—
300	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
301	C₂×D₄	—	2	1	no	—
302	C₂×D₄	—	2	2	no	—
303	C₂×D₄	—	2	2	no	—
304	C₂×D₄	—	2	1	no	—
305	C₂×D₄	—	2	1	no	—
306	C₂×D₄	—	2	2	no	—
307	C₂×D₄	—	4	1	no	—
308	C₂×D₄	—	4	1	no	—
309	C₂×D₄	—	4	1	no	—
310	C₂×D₄	—	4	1	no	—
311	C₂×D₄	—	4	2	no	—
312	C₂×D₄	—	4	2	no	—
313	C₂×D₄	—	8	2	no	—
314	C₂×D₄	—	8	2	no	—
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	2	1	no	—
317	C₂²×C₄	—	4	1	no	—
318	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
319	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
320	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
321	Q₁₆	—	4	4	no	—
322	SD₁₆	—	4	8	no	—
323	SD₁₆	—	4	4	no	—
324	D₈	—	8	4	no	—
325	M₁₆	—	4	2	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	4	2	no	—
330	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	2	1	no	—
333	C₄⋊C₄	—	2	2	no	—
334	C₄⋊C₄	—	2	1	no	—
335	C₄⋊C₄	—	2	1	no	—
336	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
337	C₂²⋊C₄	—	4	1	no	—
338	C₂²⋊C₄	—	4	1	no	—
339	C₄²	—	2	2	no	—
340	C₄²	—	2	2	no	—
341	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
342	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
343	C₄²	#41×C₃₁	1	1	yes	characteristic
344	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
345	C₂³	—	4	1	no	—
346	C₂³	—	4	2	no	—
347	C₂³	—	8	2	no	—
348	C₂³	—	8	2	no	—
349	C₂³	—	8	2	no	—
350	C₂³	C₃₁×#121	1	1	yes	characteristic
351	Q₈	—	2	2	no	—
352	Q₈	—	2	2	no	—
353	Q₈	—	4	2	no	—
354	Q₈	—	4	1	no	—
355	Q₈	—	4	1	no	—
356	D₄	—	4	2	no	—
357	D₄	—	4	1	no	—
358	D₄	—	4	2	no	—
359	D₄	—	4	1	no	—
360	D₄	—	4	2	no	—
361	D₄	—	4	4	no	—
362	D₄	—	8	1	no	—
363	D₄	—	8	2	no	—
364	D₄	—	8	2	no	—
365	D₄	—	8	2	no	—
366	D₄	—	8	2	no	—
367	D₄	—	8	2	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	4	1	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	—	4	1	no	—
374	C₂×C₄	—	8	1	no	—
375	C₂×C₄	—	8	2	no	—
376	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
377	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
378	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
379	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
380	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
381	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
382	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
383	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
384	C₈	—	2	4	no	—
385	C₈	—	4	2	no	—
386	C₈	—	4	2	no	—
387	C₈	—	4	2	no	—
388	C₂²	—	2	1	no	—
389	C₂²	—	2	1	no	—
390	C₂²	—	8	2	no	—
391	C₂²	—	8	1	no	—
392	C₂²	—	8	1	no	—
393	C₂²	—	8	2	no	—
394	C₂²	—	8	2	no	—
395	C₂²	—	8	1	no	—
396	C₂²	—	16	4	no	—
397	C₂²	C₃₁×#469	1	2	yes	—
398	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
399	C₄	—	2	1	no	—
400	C₄	—	2	1	no	—
401	C₄	—	2	1	no	—
402	C₄	—	4	2	no	—
403	C₄	—	4	2	no	—
404	C₄	—	8	1	no	—
405	C₄	C₃₁×#469	1	2	yes	—
406	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
407	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
408	C₄	C₃₁×#577	1	2	yes	—
409	C₂	—	2	2	no	—
410	C₂	—	16	2	no	—
411	C₂	—	16	1	no	—
412	C₂	C₃₁×#3043	1	1	yes	characteristic
413	C₂	C₂×C₃₁×#469	1	1	yes	characteristic
414	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×#3205	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
2. C₃₁×#3037	C₂	semidirect
3. C₂×C₃₁×#460	C₂	semidirect
4. C₃₁×#2983	C₂	semidirect
5. C₃₁×#1702	C₂	semidirect
6. C₃₁×#1688	C₂	semidirect
7. C₃₁×#1570	C₂	semidirect
8. C₃₁×#1545	C₂	semidirect
9. C₃₁×#1540	C₂	semidirect
18. C₃₁×#462	C₂²	semidirect
21. C₃₁×#460	C₂²	semidirect
25. C₃₁×#446	C₂²	semidirect
33. C₃₁×#334	C₂²	semidirect
34. C₃₁×#332	C₂²	semidirect
36. C₃₁×#331	C₂²	semidirect
137. #19434	C₃₁	direct (N × H)
182. #3037	C₂×C₃₁	semidirect
183. C₂×#460	C₂×C₃₁	semidirect
184. #2983	C₂×C₃₁	semidirect
185. #1702	C₂×C₃₁	semidirect
186. #1688	C₂×C₃₁	semidirect
187. #1570	C₂×C₃₁	semidirect
188. #1545	C₂×C₃₁	semidirect
189. #1540	C₂×C₃₁	semidirect
219. #462	C₂²×C₃₁	semidirect
222. #460	C₂²×C₃₁	semidirect
226. #446	C₂²×C₃₁	semidirect
234. #334	C₂²×C₃₁	semidirect
235. #332	C₂²×C₃₁	semidirect
237. #331	C₂²×C₃₁	semidirect
292. C₃₁	#19434	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×#3205	2
C₃₁×#3037	2
C₂×C₃₁×#460	2
C₃₁×#2983	2
C₃₁×#1702	2
C₃₁×#1688	2
C₃₁×#1570	2
C₃₁×#1545	2
C₃₁×#1540	2
#19434	31

Subgroups — C31×#19434 #21476193

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19434 #21476193