Subgroups — C₃₁×#9318 #21466077

Order 7936.

580 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3038	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2988	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1982	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1806	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1545	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1303	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1292	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×AGL(1,ℤ/8)	—	2	1	no	—
9	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
10	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#460	—	2	1	no	—
12	C₃₁×#460	C₄	1	2	yes	complemented
13	C₃₁×#459	—	2	1	no	—
14	C₃₁×#459	C₄	1	2	yes	complemented
15	C₈×Q₈×C₃₁	—	2	1	no	—
16	C₈×D₄×C₃₁	—	2	2	no	—
17	C₈×D₄×C₃₁	—	2	1	no	—
18	C₃₁×#431	—	2	1	no	—
19	C₃₁×#431	C₂²	1	1	yes	characteristic
20	C₃₁×#423	C₂²	1	1	yes	characteristic
21	C₃₁×#423	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
23	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#104	—	2	1	no	—
25	C₃₁×#418	—	2	1	no	—
26	C₂×C₃₁×#102	—	2	2	no	—
27	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#339	—	2	1	no	—
29	C₃₁×#331	—	2	1	no	—
30	C₃₁×#331	—	2	1	no	—
31	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
32	C₂²×D₄×C₃₁	—	2	1	no	—
33	C₂²×D₄×C₃₁	—	2	1	no	—
34	C₂²×D₄×C₃₁	—	2	1	no	—
35	C₃₁×AGL(1,ℤ/8)	—	4	2	no	—
36	C₃₁×AGL(1,ℤ/8)	—	4	2	no	—
37	C₂×SD₁₆×C₃₁	—	2	2	no	—
38	C₂×D₈×C₃₁	—	2	2	no	—
39	C₂×M₁₆×C₃₁	—	2	1	no	—
40	C₂×M₁₆×C₃₁	—	2	1	no	—
41	C₂²×C₈×C₃₁	—	2	2	no	—
42	C₂²×C₈×C₃₁	D₄	1	2	yes	—
43	C₃₁×#127	—	2	1	no	—
44	C₃₁×#127	—	2	1	no	—
45	C₃₁×#127	—	2	1	no	—
46	C₃₁×#127	C₂×C₄	1	2	yes	—
47	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
48	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
49	C₄×D₄×C₃₁	—	2	1	no	—
50	C₄×D₄×C₃₁	—	2	1	no	—
51	C₄×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
52	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
53	C₃₁×#117	—	2	1	no	—
54	C₃₁×#117	D₄	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
56	C₂×C₄²×C₃₁	—	2	1	no	—
57	C₂×C₄²×C₃₁	—	2	1	no	—
58	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
59	C₃₁×C₄⋊C₈	—	2	2	no	—
60	C₃₁×C₄⋊C₈	—	2	1	no	—
61	C₃₁×C₄⋊C₈	—	2	1	no	—
62	C₃₁×C₄⋊C₈	—	2	1	no	—
63	C₃₁×C₄⋊C₈	—	2	1	no	—
64	C₃₁×C₄⋊C₈	—	2	1	no	—
65	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
66	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
67	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
68	C₃₁×#104	—	4	2	no	—
69	C₃₁×#102	—	2	2	no	—
70	C₃₁×#102	—	2	2	no	—
71	C₃₁×#102	—	4	4	no	—
72	C₃₁×C₂²⋊C₈	—	2	2	no	—
73	C₃₁×C₂²⋊C₈	—	2	1	no	—
74	C₄×C₈×C₃₁	—	2	2	no	—
75	C₄×C₈×C₃₁	—	2	1	no	—
76	C₃₁×#95	—	2	1	no	—
77	C₂⁴×C₃₁	—	4	1	no	—
78	#41×C₃₁	—	2	1	no	—
79	#41×C₃₁	—	2	1	no	—
80	#41×C₃₁	D₈	1	2	yes	—
81	#41×C₃₁	SD₁₆	1	2	yes	—
82	C₂×Q₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×D₄×C₃₁	—	2	1	no	—
84	C₂×D₄×C₃₁	—	2	2	no	—
85	C₂×D₄×C₃₁	—	2	2	no	—
86	C₂×D₄×C₃₁	—	2	2	no	—
87	C₂×D₄×C₃₁	—	4	2	no	—
88	C₂×D₄×C₃₁	—	4	2	no	—
89	C₂×D₄×C₃₁	—	4	1	no	—
90	C₂×D₄×C₃₁	—	4	1	no	—
91	C₂×D₄×C₃₁	—	4	1	no	—
92	C₂×D₄×C₃₁	—	4	1	no	—
93	C₂×D₄×C₃₁	—	4	1	no	—
94	C₂×D₄×C₃₁	—	4	1	no	—
95	C₂×D₄×C₃₁	—	4	1	no	—
96	C₂×D₄×C₃₁	—	8	2	no	—
97	C₂×D₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
98	C₂²×C₄×C₃₁	—	2	1	no	—
99	C₂²×C₄×C₃₁	—	2	1	no	—
100	C₂²×C₄×C₃₁	—	2	1	no	—
101	C₂²×C₄×C₃₁	—	4	1	no	—
102	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
105	SD₁₆×C₃₁	—	4	4	no	—
106	D₈×C₃₁	—	4	4	no	—
107	M₁₆×C₃₁	—	4	1	no	—
108	M₁₆×C₃₁	—	4	1	no	—
109	M₁₆×C₃₁	—	4	1	no	—
110	M₁₆×C₃₁	—	4	1	no	—
111	C₂×C₈×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	—	4	2	no	—
116	C₂×C₈×C₃₁	—	4	1	no	—
117	C₂×C₈×C₃₁	—	8	2	no	—
118	C₄⋊C₄×C₃₁	—	2	1	no	—
119	C₄⋊C₄×C₃₁	—	2	1	no	—
120	C₄⋊C₄×C₃₁	—	2	2	no	—
121	C₄⋊C₄×C₃₁	—	4	1	no	—
122	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
123	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
124	C₂²⋊C₄×C₃₁	—	2	1	no	—
125	C₂²⋊C₄×C₃₁	—	2	1	no	—
126	C₂²⋊C₄×C₃₁	—	2	1	no	—
127	C₄²×C₃₁	—	2	1	no	—
128	C₄²×C₃₁	—	2	1	no	—
129	C₄²×C₃₁	—	2	1	no	—
130	C₄²×C₃₁	—	4	1	no	—
131	C₄²×C₃₁	—	4	1	no	—
132	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
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	#9318	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂³×C₃₁	—	2	1	no	—
138	C₂³×C₃₁	—	4	2	no	—
139	C₂³×C₃₁	—	8	2	no	—
140	C₂³×C₃₁	—	8	2	no	—
141	C₂³×C₃₁	—	8	1	no	—
142	C₂³×C₃₁	—	8	1	no	—
143	C₂³×C₃₁	#121	1	1	yes	characteristic
144	Q₈×C₃₁	—	2	1	no	—
145	Q₈×C₃₁	#102	1	2	yes	complemented
146	D₄×C₃₁	—	2	2	no	—
147	D₄×C₃₁	—	2	2	no	—
148	D₄×C₃₁	—	2	1	no	—
149	D₄×C₃₁	—	4	4	no	—
150	D₄×C₃₁	—	4	2	no	—
151	D₄×C₃₁	—	4	2	no	—
152	D₄×C₃₁	—	8	2	no	—
153	D₄×C₃₁	—	8	2	no	—
154	D₄×C₃₁	—	8	1	no	—
155	D₄×C₃₁	—	8	1	no	—
156	D₄×C₃₁	—	8	1	no	—
157	D₄×C₃₁	—	8	1	no	—
158	D₄×C₃₁	#102	1	2	yes	complemented
159	C₂×C₄×C₃₁	—	2	1	no	—
160	C₂×C₄×C₃₁	—	2	1	no	—
161	C₂×C₄×C₃₁	—	2	1	no	—
162	C₂×C₄×C₃₁	—	2	2	no	—
163	C₂×C₄×C₃₁	—	2	2	no	—
164	C₂×C₄×C₃₁	—	2	1	no	—
165	C₂×C₄×C₃₁	—	2	1	no	—
166	C₂×C₄×C₃₁	—	4	1	no	—
167	C₂×C₄×C₃₁	—	4	1	no	—
168	C₂×C₄×C₃₁	—	4	1	no	—
169	C₂×C₄×C₃₁	—	4	2	no	—
170	C₂×C₄×C₃₁	—	8	1	no	—
171	C₂×C₄×C₃₁	—	8	1	no	—
172	C₂×C₄×C₃₁	—	8	1	no	—
173	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
174	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
175	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
176	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
177	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
178	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
179	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
180	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
181	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
182	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
183	C₈×C₃₁	—	4	2	no	—
184	C₈×C₃₁	—	4	2	no	—
185	C₈×C₃₁	—	4	1	no	—
186	C₈×C₃₁	—	4	1	no	—
187	C₈×C₃₁	—	8	1	no	—
188	#3038	C₂×C₃₁	1	1	yes	characteristic
189	#2988	C₂×C₃₁	1	1	yes	characteristic, complemented
190	#1982	C₂×C₃₁	1	1	yes	characteristic, complemented
191	#1806	C₂×C₃₁	1	1	yes	characteristic, complemented
192	#1545	C₂×C₃₁	1	1	yes	characteristic, complemented
193	#1303	C₂×C₃₁	1	1	yes	characteristic, complemented
194	#1292	C₂×C₃₁	1	1	yes	characteristic, complemented
195	C₂²×C₃₁	—	2	1	no	—
196	C₂²×C₃₁	—	2	1	no	—
197	C₂²×C₃₁	—	2	2	no	—
198	C₂²×C₃₁	—	4	1	no	—
199	C₂²×C₃₁	—	4	1	no	—
200	C₂²×C₃₁	—	8	2	no	—
201	C₂²×C₃₁	—	8	2	no	—
202	C₂²×C₃₁	—	8	2	no	—
203	C₂²×C₃₁	—	16	2	no	—
204	C₂²×C₃₁	—	16	2	no	—
205	C₂²×C₃₁	#386	1	1	yes	characteristic
206	C₂²×C₃₁	#465	1	1	yes	characteristic
207	C₂²×C₃₁	#466	1	1	yes	characteristic
208	C₄×C₃₁	—	2	1	no	—
209	C₄×C₃₁	—	2	2	no	—
210	C₄×C₃₁	—	2	1	no	—
211	C₄×C₃₁	—	2	1	no	—
212	C₄×C₃₁	—	4	1	no	—
213	C₄×C₃₁	—	4	1	no	—
214	C₄×C₃₁	—	8	2	no	—
215	C₄×C₃₁	C₂×#102	1	1	yes	characteristic
216	C₄×C₃₁	#416	1	1	yes	characteristic
217	C₄×C₃₁	#453	1	1	yes	characteristic
218	C₄×C₃₁	#454	1	1	yes	characteristic
219	C₄×C₃₁	#465	1	1	yes	characteristic
220	C₄×C₃₁	#466	1	1	yes	characteristic
221	C₂×AGL(1,ℤ/8)	—	2	1	no	—
222	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
223	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
224	#460	—	2	1	no	—
225	#460	C₄×C₃₁	1	2	yes	complemented
226	#459	—	2	1	no	—
227	#459	C₄×C₃₁	1	2	yes	complemented
228	C₈×Q₈	—	2	1	no	—
229	C₈×D₄	—	2	2	no	—
230	C₈×D₄	—	2	1	no	—
231	#431	—	2	1	no	—
232	#431	C₂²×C₃₁	1	1	yes	characteristic
233	#423	C₂²×C₃₁	1	1	yes	characteristic
234	#423	C₂²×C₃₁	1	1	yes	characteristic
235	C₂×C₄⋊C₈	—	2	1	no	—
236	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
237	C₂×#104	—	2	1	no	—
238	#418	—	2	1	no	—
239	C₂×#102	—	2	2	no	—
240	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
241	#339	—	2	1	no	—
242	#331	—	2	1	no	—
243	#331	—	2	1	no	—
244	C₂×C₃₁	—	2	1	no	—
245	C₂×C₃₁	—	2	1	no	—
246	C₂×C₃₁	—	4	2	no	—
247	C₂×C₃₁	—	16	2	no	—
248	C₂×C₃₁	#1883	1	1	yes	characteristic
249	C₂×C₃₁	#1893	1	1	yes	characteristic
250	C₂×C₃₁	#1903	1	1	yes	characteristic
251	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
252	C₂²×D₄	—	2	1	no	—
253	C₂²×D₄	—	2	1	no	—
254	C₂²×D₄	—	2	1	no	—
255	AGL(1,ℤ/8)	—	4	2	no	—
256	AGL(1,ℤ/8)	—	4	2	no	—
257	C₂×SD₁₆	—	2	2	no	—
258	C₂×D₈	—	2	2	no	—
259	C₂×M₁₆	—	2	1	no	—
260	C₂×M₁₆	—	2	1	no	—
261	C₂²×C₈	—	2	2	no	—
262	C₂²×C₈	D₄×C₃₁	1	2	yes	—
263	#127	—	2	1	no	—
264	#127	—	2	1	no	—
265	#127	—	2	1	no	—
266	#127	C₂×C₄×C₃₁	1	2	yes	—
267	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
268	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
269	C₄×D₄	—	2	1	no	—
270	C₄×D₄	—	2	1	no	—
271	C₄×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
272	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
273	#117	—	2	1	no	—
274	#117	D₄×C₃₁	1	1	yes	characteristic
275	C₂×C₄⋊C₄	—	2	1	no	—
276	C₂×C₄²	—	2	1	no	—
277	C₂×C₄²	—	2	1	no	—
278	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
279	C₄⋊C₈	—	2	2	no	—
280	C₄⋊C₈	—	2	1	no	—
281	C₄⋊C₈	—	2	1	no	—
282	C₄⋊C₈	—	2	1	no	—
283	C₄⋊C₈	—	2	1	no	—
284	C₄⋊C₈	—	2	1	no	—
285	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
286	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
287	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
288	#104	—	4	2	no	—
289	#102	—	2	2	no	—
290	#102	—	2	2	no	—
291	#102	—	4	4	no	—
292	C₂²⋊C₈	—	2	2	no	—
293	C₂²⋊C₈	—	2	1	no	—
294	C₄×C₈	—	2	2	no	—
295	C₄×C₈	—	2	1	no	—
296	#95	—	2	1	no	—
297	C₃₁	#9318	1	1	yes	characteristic, direct product factor, Sylow
298	C₂⁴	—	4	1	no	—
299	#41	—	2	1	no	—
300	#41	—	2	1	no	—
301	#41	D₈×C₃₁	1	2	yes	—
302	#41	SD₁₆×C₃₁	1	2	yes	—
303	C₂×Q₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
304	C₂×D₄	—	2	1	no	—
305	C₂×D₄	—	2	2	no	—
306	C₂×D₄	—	2	2	no	—
307	C₂×D₄	—	2	2	no	—
308	C₂×D₄	—	4	2	no	—
309	C₂×D₄	—	4	2	no	—
310	C₂×D₄	—	4	1	no	—
311	C₂×D₄	—	4	1	no	—
312	C₂×D₄	—	4	1	no	—
313	C₂×D₄	—	4	1	no	—
314	C₂×D₄	—	4	1	no	—
315	C₂×D₄	—	4	1	no	—
316	C₂×D₄	—	4	1	no	—
317	C₂×D₄	—	8	2	no	—
318	C₂×D₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
319	C₂²×C₄	—	2	1	no	—
320	C₂²×C₄	—	2	1	no	—
321	C₂²×C₄	—	2	1	no	—
322	C₂²×C₄	—	4	1	no	—
323	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
324	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
325	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
326	SD₁₆	—	4	4	no	—
327	D₈	—	4	4	no	—
328	M₁₆	—	4	1	no	—
329	M₁₆	—	4	1	no	—
330	M₁₆	—	4	1	no	—
331	M₁₆	—	4	1	no	—
332	C₂×C₈	—	2	2	no	—
333	C₂×C₈	—	2	1	no	—
334	C₂×C₈	—	2	2	no	—
335	C₂×C₈	—	2	1	no	—
336	C₂×C₈	—	4	2	no	—
337	C₂×C₈	—	4	1	no	—
338	C₂×C₈	—	8	2	no	—
339	C₄⋊C₄	—	2	1	no	—
340	C₄⋊C₄	—	2	1	no	—
341	C₄⋊C₄	—	2	2	no	—
342	C₄⋊C₄	—	4	1	no	—
343	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
344	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
345	C₂²⋊C₄	—	2	1	no	—
346	C₂²⋊C₄	—	2	1	no	—
347	C₂²⋊C₄	—	2	1	no	—
348	C₄²	—	2	1	no	—
349	C₄²	—	2	1	no	—
350	C₄²	—	2	1	no	—
351	C₄²	—	4	1	no	—
352	C₄²	—	4	1	no	—
353	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
354	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
355	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
356	C₄²	#41×C₃₁	1	1	yes	characteristic
357	C₂³	—	2	1	no	—
358	C₂³	—	4	2	no	—
359	C₂³	—	8	2	no	—
360	C₂³	—	8	2	no	—
361	C₂³	—	8	1	no	—
362	C₂³	—	8	1	no	—
363	C₂³	C₃₁×#121	1	1	yes	characteristic
364	Q₈	—	2	1	no	—
365	Q₈	C₃₁×#102	1	2	yes	complemented
366	D₄	—	2	2	no	—
367	D₄	—	2	2	no	—
368	D₄	—	2	1	no	—
369	D₄	—	4	4	no	—
370	D₄	—	4	2	no	—
371	D₄	—	4	2	no	—
372	D₄	—	8	2	no	—
373	D₄	—	8	2	no	—
374	D₄	—	8	1	no	—
375	D₄	—	8	1	no	—
376	D₄	—	8	1	no	—
377	D₄	—	8	1	no	—
378	D₄	C₃₁×#102	1	2	yes	complemented
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	1	no	—
385	C₂×C₄	—	2	1	no	—
386	C₂×C₄	—	4	1	no	—
387	C₂×C₄	—	4	1	no	—
388	C₂×C₄	—	4	1	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	8	1	no	—
391	C₂×C₄	—	8	1	no	—
392	C₂×C₄	—	8	1	no	—
393	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
394	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
395	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
396	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
397	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
398	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
399	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
400	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
401	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
402	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
403	C₈	—	4	2	no	—
404	C₈	—	4	2	no	—
405	C₈	—	4	1	no	—
406	C₈	—	4	1	no	—
407	C₈	—	8	1	no	—
408	C₂²	—	2	1	no	—
409	C₂²	—	2	1	no	—
410	C₂²	—	2	2	no	—
411	C₂²	—	4	1	no	—
412	C₂²	—	4	1	no	—
413	C₂²	—	8	2	no	—
414	C₂²	—	8	2	no	—
415	C₂²	—	8	2	no	—
416	C₂²	—	16	2	no	—
417	C₂²	—	16	2	no	—
418	C₂²	C₃₁×#386	1	1	yes	characteristic
419	C₂²	C₃₁×#465	1	1	yes	characteristic
420	C₂²	C₃₁×#466	1	1	yes	characteristic
421	C₄	—	2	1	no	—
422	C₄	—	2	2	no	—
423	C₄	—	2	1	no	—
424	C₄	—	2	1	no	—
425	C₄	—	4	1	no	—
426	C₄	—	4	1	no	—
427	C₄	—	8	2	no	—
428	C₄	C₂×C₃₁×#102	1	1	yes	characteristic
429	C₄	C₃₁×#416	1	1	yes	characteristic
430	C₄	C₃₁×#453	1	1	yes	characteristic
431	C₄	C₃₁×#454	1	1	yes	characteristic
432	C₄	C₃₁×#465	1	1	yes	characteristic
433	C₄	C₃₁×#466	1	1	yes	characteristic
434	C₂	—	2	1	no	—
435	C₂	—	2	1	no	—
436	C₂	—	4	2	no	—
437	C₂	—	16	2	no	—
438	C₂	C₃₁×#1883	1	1	yes	characteristic
439	C₂	C₃₁×#1893	1	1	yes	characteristic
440	C₂	C₃₁×#1903	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	#386
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₃₁×#3038	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×#2988	C₂	semidirect
3. C₃₁×#1982	C₂	semidirect
4. C₃₁×#1806	C₂	semidirect
5. C₃₁×#1545	C₂	semidirect
6. C₃₁×#1303	C₂	semidirect
7. C₃₁×#1292	C₂	semidirect
12. C₃₁×#460	C₄	semidirect
14. C₃₁×#459	C₄	semidirect
136. #9318	C₃₁	direct (N × H)
145. Q₈×C₃₁	#102	semidirect
158. D₄×C₃₁	#102	semidirect
189. #2988	C₂×C₃₁	semidirect
190. #1982	C₂×C₃₁	semidirect
191. #1806	C₂×C₃₁	semidirect
192. #1545	C₂×C₃₁	semidirect
193. #1303	C₂×C₃₁	semidirect
194. #1292	C₂×C₃₁	semidirect
225. #460	C₄×C₃₁	semidirect
227. #459	C₄×C₃₁	semidirect
297. C₃₁	#9318	direct (N × H)
365. Q₈	C₃₁×#102	semidirect
378. D₄	C₃₁×#102	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#3038	2
C₃₁×#2988	2
C₃₁×#1982	2
C₃₁×#1806	2
C₃₁×#1545	2
C₃₁×#1303	2
C₃₁×#1292	2
#9318	31

Subgroups — C31×#9318 #21466077

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9318 #21466077