Subgroups — C₃₁×#32807 #21489566

Order 7936.

856 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×#2862	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#2858	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2055	C₂	1	4	yes	maximal, complemented
4	C₃₁×#2024	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2023	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2014	C₂	1	2	yes	maximal
7	C₃₁×#2013	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1831	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×SD₁₆×C₃₁	C₂²	1	4	yes	—
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#124	—	2	1	no	—
15	C₂×C₃₁×#124	—	2	1	no	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₃₁×#103	C₂²	1	4	yes	—
18	C₂×C₃₁×#102	C₂²	1	4	yes	—
19	C₃₁×#391	—	2	1	no	—
20	C₃₁×#391	—	2	1	no	—
21	C₃₁×#391	C₂²	1	2	yes	—
22	C₃₁×#391	C₂²	1	2	yes	—
23	C₃₁×#390	—	2	1	no	—
24	C₃₁×#390	—	2	1	no	—
25	C₃₁×#390	C₂²	1	2	yes	—
26	C₃₁×#390	C₂²	1	2	yes	—
27	C₃₁×#382	—	2	1	no	—
28	C₃₁×#382	—	2	1	no	—
29	C₄³×C₃₁	C₂²	1	1	yes	characteristic
30	C₃₁×#340	C₂²	1	4	yes	—
31	C₃₁×#336	C₂²	1	2	yes	—
32	C₃₁×#336	C₂²	1	2	yes	—
33	C₂²×Q₈×C₃₁	—	2	1	no	—
34	C₂²×Q₈×C₃₁	—	2	1	no	—
35	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
36	C₂²×D₄×C₃₁	—	2	1	no	—
37	C₂²×D₄×C₃₁	—	2	1	no	—
38	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
39	C₂×SD₁₆×C₃₁	—	4	4	no	—
40	C₂×SD₁₆×C₃₁	—	4	4	no	—
41	C₂×SD₁₆×C₃₁	—	4	4	no	—
42	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
43	C₃₁×C₄⋊Q₈	—	4	2	no	—
44	C₃₁×C₄⋊Q₈	—	4	2	no	—
45	C₃₁×C₄⋊Q₈	—	4	2	no	—
46	C₃₁×#127	—	4	2	no	—
47	C₃₁×#124	—	4	4	no	—
48	C₃₁×#124	—	4	4	no	—
49	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
50	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
51	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
52	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
53	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
55	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
56	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
57	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
58	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
59	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
60	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
61	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
62	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
63	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
66	C₃₁×C₄⋊C₈	—	4	2	no	—
67	C₃₁×#103	—	4	4	no	—
68	C₃₁×#102	—	4	4	no	—
69	C₃₁×#95	—	2	2	no	—
70	C₃₁×#95	—	2	2	no	—
71	C₃₁×#95	—	2	2	no	—
72	C₃₁×#95	—	2	2	no	—
73	C₂⁴×C₃₁	—	2	2	no	—
74	C₂×Q₈×C₃₁	—	2	4	no	—
75	C₂×Q₈×C₃₁	—	2	4	no	—
76	C₂×Q₈×C₃₁	—	2	2	no	—
77	C₂×Q₈×C₃₁	—	2	2	no	—
78	C₂×Q₈×C₃₁	—	4	2	no	—
79	C₂×Q₈×C₃₁	—	4	2	no	—
80	C₂×Q₈×C₃₁	—	4	2	no	—
81	C₂×Q₈×C₃₁	—	4	2	no	—
82	C₂×Q₈×C₃₁	—	4	2	no	—
83	C₂×Q₈×C₃₁	—	4	2	no	—
84	C₂×D₄×C₃₁	—	2	4	no	—
85	C₂×D₄×C₃₁	—	2	4	no	—
86	C₂×D₄×C₃₁	—	2	2	no	—
87	C₂×D₄×C₃₁	—	2	2	no	—
88	C₂×D₄×C₃₁	—	4	2	no	—
89	C₂×D₄×C₃₁	—	4	2	no	—
90	C₂×D₄×C₃₁	—	4	2	no	—
91	C₂×D₄×C₃₁	—	4	2	no	—
92	C₂×D₄×C₃₁	—	4	2	no	—
93	C₂×D₄×C₃₁	—	4	2	no	—
94	C₂²×C₄×C₃₁	—	2	2	no	—
95	C₂²×C₄×C₃₁	—	2	2	no	—
96	C₂²×C₄×C₃₁	—	2	2	no	—
97	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
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	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
104	SD₁₆×C₃₁	—	4	16	no	—
105	C₂×C₈×C₃₁	—	2	4	no	—
106	C₂×C₈×C₃₁	—	2	4	no	—
107	C₂×C₈×C₃₁	—	2	4	no	—
108	C₄⋊C₄×C₃₁	—	2	4	no	—
109	C₄⋊C₄×C₃₁	—	2	4	no	—
110	C₄⋊C₄×C₃₁	—	4	2	no	—
111	C₄⋊C₄×C₃₁	—	4	2	no	—
112	C₄⋊C₄×C₃₁	—	4	2	no	—
113	C₄⋊C₄×C₃₁	—	4	2	no	—
114	C₂²⋊C₄×C₃₁	—	4	4	no	—
115	C₂²⋊C₄×C₃₁	—	4	4	no	—
116	C₂²⋊C₄×C₃₁	—	4	4	no	—
117	C₂²⋊C₄×C₃₁	—	4	4	no	—
118	C₄²×C₃₁	—	2	2	no	—
119	C₄²×C₃₁	—	2	2	no	—
120	C₄²×C₃₁	—	2	1	no	—
121	C₄²×C₃₁	—	2	2	no	—
122	C₄²×C₃₁	—	2	2	no	—
123	C₄²×C₃₁	—	2	1	no	—
124	C₄²×C₃₁	—	4	1	no	—
125	C₄²×C₃₁	SD₁₆	1	4	yes	complemented
126	#32807	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
127	C₂³×C₃₁	—	4	2	no	—
128	C₂³×C₃₁	—	4	2	no	—
129	C₂³×C₃₁	—	4	2	no	—
130	C₂³×C₃₁	—	4	2	no	—
131	C₂³×C₃₁	—	4	2	no	—
132	C₂³×C₃₁	—	4	2	no	—
133	C₂³×C₃₁	—	4	2	no	—
134	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
135	Q₈×C₃₁	—	4	4	no	—
136	Q₈×C₃₁	—	4	4	no	—
137	Q₈×C₃₁	—	8	4	no	—
138	Q₈×C₃₁	—	8	4	no	—
139	D₄×C₃₁	—	4	4	no	—
140	D₄×C₃₁	—	4	4	no	—
141	D₄×C₃₁	—	8	4	no	—
142	D₄×C₃₁	—	8	4	no	—
143	C₂×C₄×C₃₁	—	2	1	no	—
144	C₂×C₄×C₃₁	—	2	1	no	—
145	C₂×C₄×C₃₁	—	2	1	no	—
146	C₂×C₄×C₃₁	—	2	1	no	—
147	C₂×C₄×C₃₁	—	2	1	no	—
148	C₂×C₄×C₃₁	—	2	1	no	—
149	C₂×C₄×C₃₁	—	2	1	no	—
150	C₂×C₄×C₃₁	—	2	1	no	—
151	C₂×C₄×C₃₁	—	2	1	no	—
152	C₂×C₄×C₃₁	—	2	1	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	—	4	2	no	—
157	C₂×C₄×C₃₁	—	4	2	no	—
158	C₂×C₄×C₃₁	—	4	2	no	—
159	C₂×C₄×C₃₁	—	4	2	no	—
160	C₂×C₄×C₃₁	—	4	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	—	4	2	no	—
165	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
166	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
167	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
168	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
169	C₂×C₄×C₃₁	#137	1	2	yes	—
170	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
171	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
172	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
173	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
174	C₈×C₃₁	—	4	4	no	—
175	#2862	C₂×C₃₁	1	1	yes	characteristic, complemented
176	#2858	C₂×C₃₁	1	1	yes	characteristic
177	#2055	C₂×C₃₁	1	4	yes	complemented
178	#2024	C₂×C₃₁	1	2	yes	complemented
179	#2023	C₂×C₃₁	1	2	yes	complemented
180	#2014	C₂×C₃₁	1	2	yes	—
181	#2013	C₂×C₃₁	1	2	yes	complemented
182	#1831	C₂×C₃₁	1	1	yes	characteristic, complemented
183	C₂²×C₃₁	—	8	2	no	—
184	C₂²×C₃₁	—	8	2	no	—
185	C₂²×C₃₁	—	8	2	no	—
186	C₂²×C₃₁	—	8	2	no	—
187	C₂²×C₃₁	—	8	2	no	—
188	C₂²×C₃₁	—	8	2	no	—
189	C₂²×C₃₁	—	8	2	no	—
190	C₂²×C₃₁	#535	1	1	yes	characteristic
191	C₂²×C₃₁	#535	1	1	yes	characteristic
192	C₂²×C₃₁	#537	1	1	yes	characteristic
193	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
194	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
195	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
196	C₂²×C₃₁	#575	1	1	yes	characteristic
197	C₄×C₃₁	—	2	2	no	—
198	C₄×C₃₁	—	2	2	no	—
199	C₄×C₃₁	—	2	2	no	—
200	C₄×C₃₁	—	2	2	no	—
201	C₄×C₃₁	—	2	2	no	—
202	C₄×C₃₁	—	4	1	no	—
203	C₄×C₃₁	—	8	2	no	—
204	C₄×C₃₁	—	8	2	no	—
205	C₄×C₃₁	—	8	2	no	—
206	C₄×C₃₁	H(3,𝔽₄)	1	2	yes	—
207	C₄×C₃₁	#562	1	2	yes	—
208	C₂²×SD₁₆	C₂²×C₃₁	1	4	yes	—
209	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
210	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
211	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
212	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
213	C₂×#124	—	2	1	no	—
214	C₂×#124	—	2	1	no	—
215	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
216	C₂×#103	C₂²×C₃₁	1	4	yes	—
217	C₂×#102	C₂²×C₃₁	1	4	yes	—
218	#391	—	2	1	no	—
219	#391	—	2	1	no	—
220	#391	C₂²×C₃₁	1	2	yes	—
221	#391	C₂²×C₃₁	1	2	yes	—
222	#390	—	2	1	no	—
223	#390	—	2	1	no	—
224	#390	C₂²×C₃₁	1	2	yes	—
225	#390	C₂²×C₃₁	1	2	yes	—
226	#382	—	2	1	no	—
227	#382	—	2	1	no	—
228	C₄³	C₂²×C₃₁	1	1	yes	characteristic
229	#340	C₂²×C₃₁	1	4	yes	—
230	#336	C₂²×C₃₁	1	2	yes	—
231	#336	C₂²×C₃₁	1	2	yes	—
232	C₂×C₃₁	—	16	2	no	—
233	C₂×C₃₁	#2819	1	1	yes	characteristic
234	C₂×C₃₁	#3206	1	1	yes	characteristic
235	C₂×C₃₁	#3208	1	1	yes	characteristic
236	C₂×C₃₁	#3213	1	1	yes	characteristic
237	C₂×C₃₁	#3215	1	1	yes	characteristic
238	C₂×C₃₁	#3246	1	1	yes	characteristic
239	C₂×C₃₁	#3254	1	1	yes	characteristic
240	C₂²×Q₈	—	2	1	no	—
241	C₂²×Q₈	—	2	1	no	—
242	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
243	C₂²×D₄	—	2	1	no	—
244	C₂²×D₄	—	2	1	no	—
245	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
246	C₂×SD₁₆	—	4	4	no	—
247	C₂×SD₁₆	—	4	4	no	—
248	C₂×SD₁₆	—	4	4	no	—
249	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
250	C₄⋊Q₈	—	4	2	no	—
251	C₄⋊Q₈	—	4	2	no	—
252	C₄⋊Q₈	—	4	2	no	—
253	#127	—	4	2	no	—
254	#124	—	4	4	no	—
255	#124	—	4	4	no	—
256	C₂×C₄⋊C₄	—	2	1	no	—
257	C₂×C₄⋊C₄	—	2	1	no	—
258	C₂×C₄⋊C₄	—	2	1	no	—
259	C₂×C₄⋊C₄	—	2	1	no	—
260	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
261	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
262	C₂×C₂²⋊C₄	—	2	2	no	—
263	C₂×C₂²⋊C₄	—	2	2	no	—
264	C₂×C₂²⋊C₄	—	2	2	no	—
265	C₂×C₂²⋊C₄	—	2	2	no	—
266	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
267	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
268	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
269	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
270	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
271	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
272	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
273	C₄⋊C₈	—	4	2	no	—
274	#103	—	4	4	no	—
275	#102	—	4	4	no	—
276	#95	—	2	2	no	—
277	#95	—	2	2	no	—
278	#95	—	2	2	no	—
279	#95	—	2	2	no	—
280	C₃₁	#32807	1	1	yes	characteristic, direct product factor, Sylow
281	C₂⁴	—	2	2	no	—
282	C₂×Q₈	—	2	4	no	—
283	C₂×Q₈	—	2	4	no	—
284	C₂×Q₈	—	2	2	no	—
285	C₂×Q₈	—	2	2	no	—
286	C₂×Q₈	—	4	2	no	—
287	C₂×Q₈	—	4	2	no	—
288	C₂×Q₈	—	4	2	no	—
289	C₂×Q₈	—	4	2	no	—
290	C₂×Q₈	—	4	2	no	—
291	C₂×Q₈	—	4	2	no	—
292	C₂×D₄	—	2	4	no	—
293	C₂×D₄	—	2	4	no	—
294	C₂×D₄	—	2	2	no	—
295	C₂×D₄	—	2	2	no	—
296	C₂×D₄	—	4	2	no	—
297	C₂×D₄	—	4	2	no	—
298	C₂×D₄	—	4	2	no	—
299	C₂×D₄	—	4	2	no	—
300	C₂×D₄	—	4	2	no	—
301	C₂×D₄	—	4	2	no	—
302	C₂²×C₄	—	2	2	no	—
303	C₂²×C₄	—	2	2	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	C₂×D₄×C₃₁	1	1	yes	characteristic
310	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
311	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
312	SD₁₆	—	4	16	no	—
313	C₂×C₈	—	2	4	no	—
314	C₂×C₈	—	2	4	no	—
315	C₂×C₈	—	2	4	no	—
316	C₄⋊C₄	—	2	4	no	—
317	C₄⋊C₄	—	2	4	no	—
318	C₄⋊C₄	—	4	2	no	—
319	C₄⋊C₄	—	4	2	no	—
320	C₄⋊C₄	—	4	2	no	—
321	C₄⋊C₄	—	4	2	no	—
322	C₂²⋊C₄	—	4	4	no	—
323	C₂²⋊C₄	—	4	4	no	—
324	C₂²⋊C₄	—	4	4	no	—
325	C₂²⋊C₄	—	4	4	no	—
326	C₄²	—	2	2	no	—
327	C₄²	—	2	2	no	—
328	C₄²	—	2	1	no	—
329	C₄²	—	2	2	no	—
330	C₄²	—	2	2	no	—
331	C₄²	—	2	1	no	—
332	C₄²	—	4	1	no	—
333	C₄²	SD₁₆×C₃₁	1	4	yes	complemented
334	C₂³	—	4	2	no	—
335	C₂³	—	4	2	no	—
336	C₂³	—	4	2	no	—
337	C₂³	—	4	2	no	—
338	C₂³	—	4	2	no	—
339	C₂³	—	4	2	no	—
340	C₂³	—	4	2	no	—
341	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
342	Q₈	—	4	4	no	—
343	Q₈	—	4	4	no	—
344	Q₈	—	8	4	no	—
345	Q₈	—	8	4	no	—
346	D₄	—	4	4	no	—
347	D₄	—	4	4	no	—
348	D₄	—	8	4	no	—
349	D₄	—	8	4	no	—
350	C₂×C₄	—	2	1	no	—
351	C₂×C₄	—	2	1	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	2	1	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	—	4	2	no	—
364	C₂×C₄	—	4	2	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	4	2	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	—	4	2	no	—
371	C₂×C₄	—	4	2	no	—
372	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
373	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
374	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
375	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
376	C₂×C₄	C₃₁×#137	1	2	yes	—
377	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
378	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
379	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
380	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
381	C₈	—	4	4	no	—
382	C₂²	—	8	2	no	—
383	C₂²	—	8	2	no	—
384	C₂²	—	8	2	no	—
385	C₂²	—	8	2	no	—
386	C₂²	—	8	2	no	—
387	C₂²	—	8	2	no	—
388	C₂²	—	8	2	no	—
389	C₂²	C₃₁×#535	1	1	yes	characteristic
390	C₂²	C₃₁×#535	1	1	yes	characteristic
391	C₂²	C₃₁×#537	1	1	yes	characteristic
392	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
393	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
394	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
395	C₂²	C₃₁×#575	1	1	yes	characteristic
396	C₄	—	2	2	no	—
397	C₄	—	2	2	no	—
398	C₄	—	2	2	no	—
399	C₄	—	2	2	no	—
400	C₄	—	2	2	no	—
401	C₄	—	4	1	no	—
402	C₄	—	8	2	no	—
403	C₄	—	8	2	no	—
404	C₄	—	8	2	no	—
405	C₄	C₃₁×H(3,𝔽₄)	1	2	yes	—
406	C₄	C₃₁×#562	1	2	yes	—
407	C₂	—	16	2	no	—
408	C₂	C₃₁×#2819	1	1	yes	characteristic
409	C₂	C₃₁×#3206	1	1	yes	characteristic
410	C₂	C₃₁×#3208	1	1	yes	characteristic
411	C₂	C₃₁×#3213	1	1	yes	characteristic
412	C₂	C₃₁×#3215	1	1	yes	characteristic
413	C₂	C₃₁×#3246	1	1	yes	characteristic
414	C₂	C₃₁×#3254	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#2862	C₂	semidirect
3. C₃₁×#2055	C₂	semidirect
4. C₃₁×#2024	C₂	semidirect
5. C₃₁×#2023	C₂	semidirect
7. C₃₁×#2013	C₂	semidirect
8. C₃₁×#1831	C₂	semidirect
125. C₄²×C₃₁	SD₁₆	semidirect
126. #32807	C₃₁	direct (N × H)
175. #2862	C₂×C₃₁	semidirect
177. #2055	C₂×C₃₁	semidirect
178. #2024	C₂×C₃₁	semidirect
179. #2023	C₂×C₃₁	semidirect
181. #2013	C₂×C₃₁	semidirect
182. #1831	C₂×C₃₁	semidirect
280. C₃₁	#32807	direct (N × H)
333. C₄²	SD₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#2862	2
C₃₁×#2858	2
C₃₁×#2055	2
C₃₁×#2024	2
C₃₁×#2023	2
C₃₁×#2014	2
C₃₁×#2013	2
C₃₁×#1831	2
#32807	31

Subgroups — C31×#32807 #21489566

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#32807 #21489566