Subgroups — C₃₁×#19554 #21476313

Order 7936.

688 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3253	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3205	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3205	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2979	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1697	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1687	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1677	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1667	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1574	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1545	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1543	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#532	C₂²	1	1	yes	characteristic
13	C₃₁×#532	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
15	C₃₁×#493	—	2	1	no	—
16	C₃₁×#493	C₂²	1	2	yes	—
17	C₃₁×#486	—	2	1	no	—
18	C₃₁×#486	C₂²	1	2	yes	—
19	C₃₁×#476	—	2	1	no	—
20	C₃₁×#476	C₂²	1	2	yes	complemented
21	C₃₁×#469	—	2	2	no	—
22	C₃₁×#459	—	2	1	no	—
23	C₃₁×#459	—	2	1	no	—
24	C₃₁×#459	—	2	1	no	—
25	C₃₁×#459	C₂²	1	2	yes	—
26	C₃₁×#459	C₂²	1	2	yes	—
27	C₃₁×#459	C₂²	1	2	yes	complemented
28	C₃₁×#447	—	2	2	no	—
29	C₃₁×#447	—	2	2	no	—
30	C₃₁×#436	—	2	1	no	—
31	C₃₁×#436	C₂²	1	2	yes	complemented
32	C₈×D₄×C₃₁	—	2	1	no	—
33	C₈×D₄×C₃₁	C₂²	1	2	yes	complemented
34	C₃₁×#431	C₂²	1	1	yes	characteristic
35	C₃₁×#431	C₂²	1	1	yes	characteristic
36	C₃₁×#431	C₂²	1	1	yes	characteristic
37	C₃₁×#423	C₂²	1	1	yes	characteristic
38	C₃₁×#408	—	2	2	no	—
39	C₃₁×#335	C₂²	1	2	yes	complemented
40	C₃₁×#334	C₂²	1	2	yes	complemented
41	C₃₁×#333	C₂²	1	1	yes	characteristic, complemented
42	C₃₁×#333	C₂²	1	1	yes	characteristic, complemented
43	C₃₁×#331	C₂²	1	2	yes	—
44	C₃₁×#331	C₂²	1	2	yes	—
45	C₃₁×#331	C₂²	1	1	yes	characteristic, complemented
46	C₃₁×#331	C₂²	1	1	yes	characteristic, complemented
47	C₂×#41×C₃₁	—	2	1	no	—
48	C₂×#41×C₃₁	—	2	1	no	—
49	C₂²×D₄×C₃₁	—	2	1	no	—
50	C₂²×D₄×C₃₁	—	2	1	no	—
51	C₂²×D₄×C₃₁	—	2	1	no	—
52	C₂×D₈×C₃₁	—	2	2	no	—
53	C₂×D₈×C₃₁	—	2	2	no	—
54	C₂×D₈×C₃₁	—	2	2	no	—
55	C₂×D₈×C₃₁	—	2	2	no	—
56	C₂×D₈×C₃₁	—	2	2	no	—
57	C₂×M₁₆×C₃₁	—	2	1	no	—
58	C₂×M₁₆×C₃₁	—	2	2	no	—
59	C₂²×C₈×C₃₁	—	2	2	no	—
60	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
61	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
62	C₃₁×#127	—	2	1	no	—
63	C₃₁×#127	—	2	1	no	—
64	C₃₁×#127	—	2	1	no	—
65	C₃₁×#127	D₄	1	1	yes	characteristic
66	C₃₁×#127	C₂³	1	2	yes	—
67	C₃₁×#127	C₂³	1	1	yes	characteristic
68	C₃₁×#124	—	2	1	no	—
69	C₃₁×#124	—	2	1	no	—
70	C₃₁×#121	—	2	1	no	—
71	C₃₁×#121	—	2	1	no	—
72	C₃₁×#121	—	2	1	no	—
73	C₃₁×#121	D₄	1	2	yes	—
74	C₄×D₄×C₃₁	—	2	1	no	—
75	C₄×D₄×C₃₁	—	2	1	no	—
76	C₄×D₄×C₃₁	—	2	1	no	—
77	C₄×D₄×C₃₁	C₂³	1	2	yes	—
78	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
79	C₃₁×#106	—	2	2	no	—
80	C₃₁×C₄⋊C₈	—	2	1	no	—
81	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
82	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
83	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
84	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
85	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
86	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
87	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
88	C₃₁×#102	—	2	2	no	—
89	C₃₁×#102	—	2	2	no	—
90	C₃₁×#102	—	2	2	no	—
91	C₃₁×#102	—	2	2	no	—
92	C₃₁×#102	—	2	2	no	—
93	C₃₁×#102	—	2	2	no	—
94	C₃₁×C₂²⋊C₈	—	2	1	no	—
95	C₃₁×C₂²⋊C₈	—	2	1	no	—
96	C₃₁×C₂²⋊C₈	—	2	2	no	—
97	C₃₁×C₂²⋊C₈	—	2	2	no	—
98	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
99	C₄×C₈×C₃₁	—	2	1	no	—
100	C₄×C₈×C₃₁	—	2	1	no	—
101	C₄×C₈×C₃₁	C₂³	1	2	yes	—
102	C₂⁴×C₃₁	—	4	1	no	—
103	#41×C₃₁	—	8	1	no	—
104	#41×C₃₁	—	8	1	no	—
105	#41×C₃₁	—	8	2	no	—
106	C₂×Q₈×C₃₁	—	2	1	no	—
107	C₂×Q₈×C₃₁	—	2	1	no	—
108	C₂×D₄×C₃₁	—	2	2	no	—
109	C₂×D₄×C₃₁	—	2	1	no	—
110	C₂×D₄×C₃₁	—	2	1	no	—
111	C₂×D₄×C₃₁	—	2	1	no	—
112	C₂×D₄×C₃₁	—	2	2	no	—
113	C₂×D₄×C₃₁	—	2	2	no	—
114	C₂×D₄×C₃₁	—	2	1	no	—
115	C₂×D₄×C₃₁	—	2	1	no	—
116	C₂×D₄×C₃₁	—	2	2	no	—
117	C₂×D₄×C₃₁	—	4	1	no	—
118	C₂×D₄×C₃₁	—	4	1	no	—
119	C₂×D₄×C₃₁	—	4	1	no	—
120	C₂×D₄×C₃₁	—	4	1	no	—
121	C₂×D₄×C₃₁	—	8	1	no	—
122	C₂×D₄×C₃₁	—	8	1	no	—
123	C₂×D₄×C₃₁	—	8	1	no	—
124	C₂×D₄×C₃₁	—	8	1	no	—
125	C₂×D₄×C₃₁	—	8	2	no	—
126	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
127	C₂²×C₄×C₃₁	—	2	2	no	—
128	C₂²×C₄×C₃₁	—	4	1	no	—
129	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
132	D₈×C₃₁	—	8	2	no	—
133	D₈×C₃₁	—	8	2	no	—
134	D₈×C₃₁	—	8	2	no	—
135	D₈×C₃₁	—	8	2	no	—
136	D₈×C₃₁	—	8	2	no	—
137	M₁₆×C₃₁	—	8	2	no	—
138	M₁₆×C₃₁	—	8	1	no	—
139	C₂×C₈×C₃₁	—	2	1	no	—
140	C₂×C₈×C₃₁	—	2	1	no	—
141	C₂×C₈×C₃₁	—	2	1	no	—
142	C₂×C₈×C₃₁	—	2	1	no	—
143	C₂×C₈×C₃₁	—	2	2	no	—
144	C₂×C₈×C₃₁	—	8	2	no	—
145	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
146	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
147	C₄⋊C₄×C₃₁	—	2	1	no	—
148	C₄⋊C₄×C₃₁	—	2	1	no	—
149	C₄⋊C₄×C₃₁	—	2	1	no	—
150	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
151	C₂²⋊C₄×C₃₁	—	2	2	no	—
152	C₂²⋊C₄×C₃₁	—	2	2	no	—
153	C₂²⋊C₄×C₃₁	—	4	1	no	—
154	C₂²⋊C₄×C₃₁	—	4	1	no	—
155	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
156	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₄²×C₃₁	#41	1	1	yes	characteristic
158	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
159	#19554	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂³×C₃₁	—	2	2	no	—
161	C₂³×C₃₁	—	4	1	no	—
162	C₂³×C₃₁	—	4	2	no	—
163	C₂³×C₃₁	—	8	2	no	—
164	C₂³×C₃₁	—	8	2	no	—
165	C₂³×C₃₁	—	8	2	no	—
166	C₂³×C₃₁	#121	1	1	yes	characteristic
167	Q₈×C₃₁	—	8	1	no	—
168	Q₈×C₃₁	—	8	1	no	—
169	D₄×C₃₁	—	2	2	no	—
170	D₄×C₃₁	—	2	2	no	—
171	D₄×C₃₁	—	4	1	no	—
172	D₄×C₃₁	—	4	1	no	—
173	D₄×C₃₁	—	4	1	no	—
174	D₄×C₃₁	—	4	2	no	—
175	D₄×C₃₁	—	4	2	no	—
176	D₄×C₃₁	—	4	2	no	—
177	D₄×C₃₁	—	4	2	no	—
178	D₄×C₃₁	—	4	1	no	—
179	D₄×C₃₁	—	4	2	no	—
180	D₄×C₃₁	—	8	1	no	—
181	D₄×C₃₁	—	8	2	no	—
182	D₄×C₃₁	—	8	1	no	—
183	D₄×C₃₁	—	8	2	no	—
184	D₄×C₃₁	—	8	2	no	—
185	D₄×C₃₁	—	8	2	no	—
186	D₄×C₃₁	—	8	2	no	—
187	D₄×C₃₁	—	8	2	no	—
188	C₂×C₄×C₃₁	—	2	1	no	—
189	C₂×C₄×C₃₁	—	2	1	no	—
190	C₂×C₄×C₃₁	—	2	1	no	—
191	C₂×C₄×C₃₁	—	2	1	no	—
192	C₂×C₄×C₃₁	—	2	2	no	—
193	C₂×C₄×C₃₁	—	2	2	no	—
194	C₂×C₄×C₃₁	—	4	2	no	—
195	C₂×C₄×C₃₁	—	4	2	no	—
196	C₂×C₄×C₃₁	—	4	1	no	—
197	C₂×C₄×C₃₁	—	8	2	no	—
198	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
199	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
200	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
201	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
202	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
203	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
204	C₈×C₃₁	—	2	2	no	—
205	C₈×C₃₁	—	2	2	no	—
206	C₈×C₃₁	—	4	1	no	—
207	C₈×C₃₁	—	4	1	no	—
208	C₈×C₃₁	—	4	2	no	—
209	C₈×C₃₁	—	4	1	no	—
210	C₈×C₃₁	—	4	1	no	—
211	#3253	C₂×C₃₁	1	1	yes	characteristic, complemented
212	#3205	C₂×C₃₁	1	1	yes	characteristic, complemented
213	#3205	C₂×C₃₁	1	1	yes	characteristic, complemented
214	#2979	C₂×C₃₁	1	1	yes	characteristic, complemented
215	#1697	C₂×C₃₁	1	2	yes	complemented
216	#1687	C₂×C₃₁	1	2	yes	complemented
217	#1677	C₂×C₃₁	1	2	yes	complemented
218	#1667	C₂×C₃₁	1	2	yes	complemented
219	#1574	C₂×C₃₁	1	1	yes	characteristic, complemented
220	#1545	C₂×C₃₁	1	1	yes	characteristic, complemented
221	#1543	C₂×C₃₁	1	1	yes	characteristic, complemented
222	C₂²×C₃₁	—	2	1	no	—
223	C₂²×C₃₁	—	2	1	no	—
224	C₂²×C₃₁	—	2	1	no	—
225	C₂²×C₃₁	—	4	2	no	—
226	C₂²×C₃₁	—	4	2	no	—
227	C₂²×C₃₁	—	4	2	no	—
228	C₂²×C₃₁	—	8	2	no	—
229	C₂²×C₃₁	—	8	1	no	—
230	C₂²×C₃₁	—	8	2	no	—
231	C₂²×C₃₁	—	8	1	no	—
232	C₂²×C₃₁	—	8	1	no	—
233	C₂²×C₃₁	—	8	2	no	—
234	C₂²×C₃₁	—	16	4	no	—
235	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
236	C₄×C₃₁	—	2	1	no	—
237	C₄×C₃₁	—	2	1	no	—
238	C₄×C₃₁	—	2	1	no	—
239	C₄×C₃₁	—	4	2	no	—
240	C₄×C₃₁	—	8	1	no	—
241	C₄×C₃₁	#472	1	2	yes	—
242	C₄×C₃₁	#576	1	1	yes	characteristic
243	C₄×C₃₁	#576	1	1	yes	characteristic
244	C₄×C₃₁	#576	1	1	yes	characteristic
245	C₄×C₃₁	#577	1	1	yes	characteristic
246	#532	C₂²×C₃₁	1	1	yes	characteristic
247	#532	C₂²×C₃₁	1	1	yes	characteristic
248	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
249	#493	—	2	1	no	—
250	#493	C₂²×C₃₁	1	2	yes	—
251	#486	—	2	1	no	—
252	#486	C₂²×C₃₁	1	2	yes	—
253	#476	—	2	1	no	—
254	#476	C₂²×C₃₁	1	2	yes	complemented
255	#469	—	2	2	no	—
256	#459	—	2	1	no	—
257	#459	—	2	1	no	—
258	#459	—	2	1	no	—
259	#459	C₂²×C₃₁	1	2	yes	—
260	#459	C₂²×C₃₁	1	2	yes	—
261	#459	C₂²×C₃₁	1	2	yes	complemented
262	#447	—	2	2	no	—
263	#447	—	2	2	no	—
264	#436	—	2	1	no	—
265	#436	C₂²×C₃₁	1	2	yes	complemented
266	C₈×D₄	—	2	1	no	—
267	C₈×D₄	C₂²×C₃₁	1	2	yes	complemented
268	#431	C₂²×C₃₁	1	1	yes	characteristic
269	#431	C₂²×C₃₁	1	1	yes	characteristic
270	#431	C₂²×C₃₁	1	1	yes	characteristic
271	#423	C₂²×C₃₁	1	1	yes	characteristic
272	#408	—	2	2	no	—
273	#335	C₂²×C₃₁	1	2	yes	complemented
274	#334	C₂²×C₃₁	1	2	yes	complemented
275	#333	C₂²×C₃₁	1	1	yes	characteristic, complemented
276	#333	C₂²×C₃₁	1	1	yes	characteristic, complemented
277	#331	C₂²×C₃₁	1	2	yes	—
278	#331	C₂²×C₃₁	1	2	yes	—
279	#331	C₂²×C₃₁	1	1	yes	characteristic, complemented
280	#331	C₂²×C₃₁	1	1	yes	characteristic, complemented
281	C₂×C₃₁	—	4	1	no	—
282	C₂×C₃₁	—	8	2	no	—
283	C₂×C₃₁	—	16	1	no	—
284	C₂×C₃₁	—	16	2	no	—
285	C₂×C₃₁	#3041	1	1	yes	characteristic
286	C₂×C₃₁	#3062	1	1	yes	characteristic
287	C₂×C₃₁	C₂×#472	1	1	yes	characteristic
288	C₂×#41	—	2	1	no	—
289	C₂×#41	—	2	1	no	—
290	C₂²×D₄	—	2	1	no	—
291	C₂²×D₄	—	2	1	no	—
292	C₂²×D₄	—	2	1	no	—
293	C₂×D₈	—	2	2	no	—
294	C₂×D₈	—	2	2	no	—
295	C₂×D₈	—	2	2	no	—
296	C₂×D₈	—	2	2	no	—
297	C₂×D₈	—	2	2	no	—
298	C₂×M₁₆	—	2	1	no	—
299	C₂×M₁₆	—	2	2	no	—
300	C₂²×C₈	—	2	2	no	—
301	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
302	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
303	#127	—	2	1	no	—
304	#127	—	2	1	no	—
305	#127	—	2	1	no	—
306	#127	D₄×C₃₁	1	1	yes	characteristic
307	#127	C₂³×C₃₁	1	2	yes	—
308	#127	C₂³×C₃₁	1	1	yes	characteristic
309	#124	—	2	1	no	—
310	#124	—	2	1	no	—
311	#121	—	2	1	no	—
312	#121	—	2	1	no	—
313	#121	—	2	1	no	—
314	#121	D₄×C₃₁	1	2	yes	—
315	C₄×D₄	—	2	1	no	—
316	C₄×D₄	—	2	1	no	—
317	C₄×D₄	—	2	1	no	—
318	C₄×D₄	C₂³×C₃₁	1	2	yes	—
319	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
320	#106	—	2	2	no	—
321	C₄⋊C₈	—	2	1	no	—
322	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
323	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
324	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
325	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
326	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
327	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
328	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
329	#102	—	2	2	no	—
330	#102	—	2	2	no	—
331	#102	—	2	2	no	—
332	#102	—	2	2	no	—
333	#102	—	2	2	no	—
334	#102	—	2	2	no	—
335	C₂²⋊C₈	—	2	1	no	—
336	C₂²⋊C₈	—	2	1	no	—
337	C₂²⋊C₈	—	2	2	no	—
338	C₂²⋊C₈	—	2	2	no	—
339	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
340	C₄×C₈	—	2	1	no	—
341	C₄×C₈	—	2	1	no	—
342	C₄×C₈	C₂³×C₃₁	1	2	yes	—
343	C₃₁	#19554	1	1	yes	characteristic, direct product factor, Sylow
344	C₂⁴	—	4	1	no	—
345	#41	—	8	1	no	—
346	#41	—	8	1	no	—
347	#41	—	8	2	no	—
348	C₂×Q₈	—	2	1	no	—
349	C₂×Q₈	—	2	1	no	—
350	C₂×D₄	—	2	2	no	—
351	C₂×D₄	—	2	1	no	—
352	C₂×D₄	—	2	1	no	—
353	C₂×D₄	—	2	1	no	—
354	C₂×D₄	—	2	2	no	—
355	C₂×D₄	—	2	2	no	—
356	C₂×D₄	—	2	1	no	—
357	C₂×D₄	—	2	1	no	—
358	C₂×D₄	—	2	2	no	—
359	C₂×D₄	—	4	1	no	—
360	C₂×D₄	—	4	1	no	—
361	C₂×D₄	—	4	1	no	—
362	C₂×D₄	—	4	1	no	—
363	C₂×D₄	—	8	1	no	—
364	C₂×D₄	—	8	1	no	—
365	C₂×D₄	—	8	1	no	—
366	C₂×D₄	—	8	1	no	—
367	C₂×D₄	—	8	2	no	—
368	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
369	C₂²×C₄	—	2	2	no	—
370	C₂²×C₄	—	4	1	no	—
371	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
372	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
373	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
374	D₈	—	8	2	no	—
375	D₈	—	8	2	no	—
376	D₈	—	8	2	no	—
377	D₈	—	8	2	no	—
378	D₈	—	8	2	no	—
379	M₁₆	—	8	2	no	—
380	M₁₆	—	8	1	no	—
381	C₂×C₈	—	2	1	no	—
382	C₂×C₈	—	2	1	no	—
383	C₂×C₈	—	2	1	no	—
384	C₂×C₈	—	2	1	no	—
385	C₂×C₈	—	2	2	no	—
386	C₂×C₈	—	8	2	no	—
387	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
388	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
389	C₄⋊C₄	—	2	1	no	—
390	C₄⋊C₄	—	2	1	no	—
391	C₄⋊C₄	—	2	1	no	—
392	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
393	C₂²⋊C₄	—	2	2	no	—
394	C₂²⋊C₄	—	2	2	no	—
395	C₂²⋊C₄	—	4	1	no	—
396	C₂²⋊C₄	—	4	1	no	—
397	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
398	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
399	C₄²	#41×C₃₁	1	1	yes	characteristic
400	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
401	C₂³	—	2	2	no	—
402	C₂³	—	4	1	no	—
403	C₂³	—	4	2	no	—
404	C₂³	—	8	2	no	—
405	C₂³	—	8	2	no	—
406	C₂³	—	8	2	no	—
407	C₂³	C₃₁×#121	1	1	yes	characteristic
408	Q₈	—	8	1	no	—
409	Q₈	—	8	1	no	—
410	D₄	—	2	2	no	—
411	D₄	—	2	2	no	—
412	D₄	—	4	2	no	—
413	D₄	—	4	1	no	—
414	D₄	—	4	1	no	—
415	D₄	—	4	1	no	—
416	D₄	—	4	2	no	—
417	D₄	—	4	2	no	—
418	D₄	—	4	2	no	—
419	D₄	—	4	2	no	—
420	D₄	—	4	1	no	—
421	D₄	—	8	1	no	—
422	D₄	—	8	2	no	—
423	D₄	—	8	1	no	—
424	D₄	—	8	2	no	—
425	D₄	—	8	2	no	—
426	D₄	—	8	2	no	—
427	D₄	—	8	2	no	—
428	D₄	—	8	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	4	2	no	—
436	C₂×C₄	—	4	2	no	—
437	C₂×C₄	—	4	1	no	—
438	C₂×C₄	—	8	2	no	—
439	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
440	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
441	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
442	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
443	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
444	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
445	C₈	—	2	2	no	—
446	C₈	—	2	2	no	—
447	C₈	—	4	1	no	—
448	C₈	—	4	1	no	—
449	C₈	—	4	2	no	—
450	C₈	—	4	1	no	—
451	C₈	—	4	1	no	—
452	C₂²	—	2	1	no	—
453	C₂²	—	2	1	no	—
454	C₂²	—	2	1	no	—
455	C₂²	—	4	2	no	—
456	C₂²	—	4	2	no	—
457	C₂²	—	4	2	no	—
458	C₂²	—	8	2	no	—
459	C₂²	—	8	1	no	—
460	C₂²	—	8	2	no	—
461	C₂²	—	8	1	no	—
462	C₂²	—	8	1	no	—
463	C₂²	—	8	2	no	—
464	C₂²	—	16	4	no	—
465	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
466	C₄	—	2	1	no	—
467	C₄	—	2	1	no	—
468	C₄	—	2	1	no	—
469	C₄	—	4	2	no	—
470	C₄	—	8	1	no	—
471	C₄	C₃₁×#472	1	2	yes	—
472	C₄	C₃₁×#576	1	1	yes	characteristic
473	C₄	C₃₁×#576	1	1	yes	characteristic
474	C₄	C₃₁×#576	1	1	yes	characteristic
475	C₄	C₃₁×#577	1	1	yes	characteristic
476	C₂	—	4	1	no	—
477	C₂	—	8	2	no	—
478	C₂	—	16	2	no	—
479	C₂	—	16	1	no	—
480	C₂	C₃₁×#3041	1	1	yes	characteristic
481	C₂	C₃₁×#3062	1	1	yes	characteristic
482	C₂	C₂×C₃₁×#472	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
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×#3253	C₂	semidirect
2. C₃₁×#3205	C₂	semidirect
3. C₃₁×#3205	C₂	semidirect
4. C₃₁×#2979	C₂	semidirect
5. C₃₁×#1697	C₂	semidirect
6. C₃₁×#1687	C₂	semidirect
7. C₃₁×#1677	C₂	semidirect
8. C₃₁×#1667	C₂	semidirect
9. C₃₁×#1574	C₂	semidirect
10. C₃₁×#1545	C₂	semidirect
11. C₃₁×#1543	C₂	semidirect
20. C₃₁×#476	C₂²	semidirect
27. C₃₁×#459	C₂²	semidirect
31. C₃₁×#436	C₂²	semidirect
33. C₈×D₄×C₃₁	C₂²	semidirect
39. C₃₁×#335	C₂²	semidirect
40. C₃₁×#334	C₂²	semidirect
41. C₃₁×#333	C₂²	semidirect
42. C₃₁×#333	C₂²	semidirect
45. C₃₁×#331	C₂²	semidirect
46. C₃₁×#331	C₂²	semidirect
159. #19554	C₃₁	direct (N × H)
211. #3253	C₂×C₃₁	semidirect
212. #3205	C₂×C₃₁	semidirect
213. #3205	C₂×C₃₁	semidirect
214. #2979	C₂×C₃₁	semidirect
215. #1697	C₂×C₃₁	semidirect
216. #1687	C₂×C₃₁	semidirect
217. #1677	C₂×C₃₁	semidirect
218. #1667	C₂×C₃₁	semidirect
219. #1574	C₂×C₃₁	semidirect
220. #1545	C₂×C₃₁	semidirect
221. #1543	C₂×C₃₁	semidirect
254. #476	C₂²×C₃₁	semidirect
261. #459	C₂²×C₃₁	semidirect
265. #436	C₂²×C₃₁	semidirect
267. C₈×D₄	C₂²×C₃₁	semidirect
273. #335	C₂²×C₃₁	semidirect
274. #334	C₂²×C₃₁	semidirect
275. #333	C₂²×C₃₁	semidirect
276. #333	C₂²×C₃₁	semidirect
279. #331	C₂²×C₃₁	semidirect
280. #331	C₂²×C₃₁	semidirect
343. C₃₁	#19554	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×#3253	2
C₃₁×#3205	2
C₃₁×#2979	2
C₃₁×#1697	2
C₃₁×#1687	2
C₃₁×#1677	2
C₃₁×#1667	2
C₃₁×#1574	2
C₃₁×#1545	2
C₃₁×#1543	2
#19554	31

Subgroups — C31×#19554 #21476313

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19554 #21476313