Subgroups — C₃₁×#19160 #21475919

Order 7936.

642 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×#3169	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3106	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2965	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1698	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1697	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1696	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1695	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1688	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1687	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1564	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1543	C₂	1	2	yes	maximal, complemented
12	C₃₁×#532	C₂²	1	2	yes	—
13	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#500	C₂²	1	1	yes	characteristic
15	C₃₁×#493	C₂²	1	1	yes	characteristic
16	C₃₁×#486	C₂²	1	2	yes	—
17	C₃₁×#482	—	2	2	no	—
18	C₃₁×#477	C₂²	1	2	yes	—
19	C₃₁×#476	C₂²	1	2	yes	—
20	C₃₁×#466	—	2	1	no	—
21	C₃₁×#466	—	2	1	no	—
22	C₃₁×#460	C₂²	1	2	yes	—
23	C₃₁×#459	C₂²	1	2	yes	—
24	C₃₁×#449	—	2	2	no	—
25	C₃₁×#446	C₂²	1	1	yes	characteristic
26	C₃₁×#446	C₂²	1	1	yes	characteristic
27	C₄×SD₁₆×C₃₁	—	2	2	no	—
28	C₄×D₈×C₃₁	—	2	2	no	—
29	C₄×D₈×C₃₁	—	2	1	no	—
30	C₄×D₈×C₃₁	—	2	1	no	—
31	C₃₁×#436	C₂²	1	1	yes	characteristic
32	C₃₁×#436	C₂²	1	1	yes	characteristic
33	C₃₁×#435	—	2	1	no	—
34	C₃₁×#435	—	2	1	no	—
35	C₃₁×#431	—	2	1	no	—
36	C₃₁×#431	C₂²	1	1	yes	characteristic
37	C₃₁×#431	C₂²	1	2	yes	—
38	C₄×M₁₆×C₃₁	—	2	1	no	—
39	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
40	C₃₁×#335	C₂²	1	2	yes	—
41	C₃₁×#334	C₂²	1	1	yes	characteristic
42	C₃₁×#334	C₂²	1	1	yes	characteristic
43	C₃₁×#333	C₂²	1	2	yes	—
44	C₃₁×#333	C₂²	1	2	yes	—
45	C₃₁×#331	C₂²	1	2	yes	—
46	C₃₁×#331	C₂²	1	2	yes	—
47	C₂×#41×C₃₁	—	2	2	no	—
48	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
49	C₂×SD₁₆×C₃₁	—	2	2	no	—
50	C₂×D₈×C₃₁	—	2	2	no	—
51	C₂×D₈×C₃₁	—	2	1	no	—
52	C₂×D₈×C₃₁	—	2	1	no	—
53	C₂×M₁₆×C₃₁	—	2	1	no	—
54	C₂×M₁₆×C₃₁	—	2	1	no	—
55	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
56	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
57	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
58	C₃₁×#127	C₂³	1	2	yes	—
59	C₃₁×#124	—	2	2	no	—
60	C₃₁×#121	—	2	2	no	—
61	C₃₁×#121	—	2	2	no	—
62	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
63	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
64	C₄×D₄×C₃₁	—	2	1	no	—
65	C₄×D₄×C₃₁	—	2	1	no	—
66	C₄×D₄×C₃₁	—	2	2	no	—
67	C₄×D₄×C₃₁	—	2	2	no	—
68	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
69	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₃₁×#117	—	2	1	no	—
71	C₃₁×#117	D₄	1	1	yes	characteristic
72	C₂×C₄²×C₃₁	—	2	1	no	—
73	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
74	C₃₁×#107	—	2	1	no	—
75	C₃₁×#107	—	2	1	no	—
76	C₃₁×#107	—	2	2	no	—
77	C₃₁×#106	—	2	2	no	—
78	C₃₁×C₄⋊C₈	—	2	1	no	—
79	C₃₁×C₄⋊C₈	—	2	1	no	—
80	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
81	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
82	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
83	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
84	C₃₁×#103	—	2	2	no	—
85	C₃₁×#102	—	2	2	no	—
86	C₃₁×#102	—	2	2	no	—
87	C₃₁×#102	—	2	2	no	—
88	C₃₁×#102	—	2	2	no	—
89	C₃₁×#102	—	2	2	no	—
90	C₃₁×C₂²⋊C₈	—	2	1	no	—
91	C₃₁×C₂²⋊C₈	—	2	2	no	—
92	C₃₁×C₂²⋊C₈	—	2	1	no	—
93	C₃₁×C₂²⋊C₈	—	2	1	no	—
94	C₃₁×#97	—	2	1	no	—
95	C₃₁×#97	—	2	1	no	—
96	C₄×C₈×C₃₁	—	2	1	no	—
97	C₄×C₈×C₃₁	—	2	2	no	—
98	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
99	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
100	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
101	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
102	#41×C₃₁	—	4	2	no	—
103	#41×C₃₁	—	8	4	no	—
104	C₂×Q₈×C₃₁	—	2	2	no	—
105	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂×D₄×C₃₁	—	2	1	no	—
107	C₂×D₄×C₃₁	—	2	2	no	—
108	C₂×D₄×C₃₁	—	2	2	no	—
109	C₂×D₄×C₃₁	—	2	2	no	—
110	C₂×D₄×C₃₁	—	4	2	no	—
111	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₃₁	—	2	1	no	—
113	C₂²×C₄×C₃₁	—	2	1	no	—
114	C₂²×C₄×C₃₁	—	2	1	no	—
115	C₂²×C₄×C₃₁	—	4	2	no	—
116	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
119	SD₁₆×C₃₁	—	4	4	no	—
120	D₈×C₃₁	—	4	2	no	—
121	D₈×C₃₁	—	4	4	no	—
122	D₈×C₃₁	—	4	2	no	—
123	M₁₆×C₃₁	—	4	2	no	—
124	M₁₆×C₃₁	—	4	2	no	—
125	C₂×C₈×C₃₁	—	2	2	no	—
126	C₂×C₈×C₃₁	—	2	1	no	—
127	C₂×C₈×C₃₁	—	2	1	no	—
128	C₂×C₈×C₃₁	—	2	2	no	—
129	C₂×C₈×C₃₁	—	2	2	no	—
130	C₂×C₈×C₃₁	—	4	1	no	—
131	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
132	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
133	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
134	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₄⋊C₄×C₃₁	—	2	1	no	—
136	C₄⋊C₄×C₃₁	—	2	1	no	—
137	C₄⋊C₄×C₃₁	—	2	2	no	—
138	C₄⋊C₄×C₃₁	—	2	2	no	—
139	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
140	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
141	C₂²⋊C₄×C₃₁	—	2	1	no	—
142	C₂²⋊C₄×C₃₁	—	2	1	no	—
143	C₂²⋊C₄×C₃₁	—	2	1	no	—
144	C₂²⋊C₄×C₃₁	—	4	2	no	—
145	C₂²⋊C₄×C₃₁	—	4	2	no	—
146	C₄²×C₃₁	—	2	1	no	—
147	C₄²×C₃₁	—	2	1	no	—
148	C₄²×C₃₁	—	2	1	no	—
149	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
150	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
151	C₄²×C₃₁	#41	1	1	yes	characteristic
152	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
153	#19160	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
154	C₂³×C₃₁	—	2	1	no	—
155	C₂³×C₃₁	—	4	2	no	—
156	C₂³×C₃₁	#121	1	1	yes	characteristic
157	Q₈×C₃₁	—	2	2	no	—
158	Q₈×C₃₁	—	8	2	no	—
159	D₄×C₃₁	—	2	2	no	—
160	D₄×C₃₁	—	4	2	no	—
161	D₄×C₃₁	—	4	2	no	—
162	D₄×C₃₁	—	4	2	no	—
163	D₄×C₃₁	—	4	2	no	—
164	D₄×C₃₁	—	4	2	no	—
165	D₄×C₃₁	—	8	2	no	—
166	D₄×C₃₁	—	8	4	no	—
167	C₂×C₄×C₃₁	—	2	1	no	—
168	C₂×C₄×C₃₁	—	2	1	no	—
169	C₂×C₄×C₃₁	—	2	2	no	—
170	C₂×C₄×C₃₁	—	2	2	no	—
171	C₂×C₄×C₃₁	—	2	1	no	—
172	C₂×C₄×C₃₁	—	2	1	no	—
173	C₂×C₄×C₃₁	—	2	1	no	—
174	C₂×C₄×C₃₁	—	4	2	no	—
175	C₂×C₄×C₃₁	—	4	1	no	—
176	C₂×C₄×C₃₁	—	4	1	no	—
177	C₂×C₄×C₃₁	—	4	2	no	—
178	C₂×C₄×C₃₁	—	4	2	no	—
179	C₂×C₄×C₃₁	—	8	4	no	—
180	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
181	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
182	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
183	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
186	C₈×C₃₁	—	2	1	no	—
187	C₈×C₃₁	—	2	1	no	—
188	C₈×C₃₁	—	2	1	no	—
189	C₈×C₃₁	—	4	1	no	—
190	C₈×C₃₁	—	4	1	no	—
191	C₈×C₃₁	—	4	2	no	—
192	C₈×C₃₁	—	4	2	no	—
193	C₈×C₃₁	#135	1	2	yes	—
194	#3169	C₂×C₃₁	1	1	yes	characteristic, complemented
195	#3106	C₂×C₃₁	1	2	yes	complemented
196	#2965	C₂×C₃₁	1	1	yes	characteristic, complemented
197	#1698	C₂×C₃₁	1	2	yes	complemented
198	#1697	C₂×C₃₁	1	2	yes	complemented
199	#1696	C₂×C₃₁	1	1	yes	characteristic, complemented
200	#1695	C₂×C₃₁	1	1	yes	characteristic, complemented
201	#1688	C₂×C₃₁	1	1	yes	characteristic, complemented
202	#1687	C₂×C₃₁	1	1	yes	characteristic, complemented
203	#1564	C₂×C₃₁	1	1	yes	characteristic, complemented
204	#1543	C₂×C₃₁	1	2	yes	complemented
205	C₂²×C₃₁	—	2	1	no	—
206	C₂²×C₃₁	—	2	1	no	—
207	C₂²×C₃₁	—	2	1	no	—
208	C₂²×C₃₁	—	4	1	no	—
209	C₂²×C₃₁	—	4	1	no	—
210	C₂²×C₃₁	—	4	1	no	—
211	C₂²×C₃₁	—	8	2	no	—
212	C₂²×C₃₁	—	8	2	no	—
213	C₂²×C₃₁	—	8	2	no	—
214	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
215	C₄×C₃₁	—	2	1	no	—
216	C₄×C₃₁	—	2	1	no	—
217	C₄×C₃₁	—	2	1	no	—
218	C₄×C₃₁	—	4	1	no	—
219	C₄×C₃₁	—	4	1	no	—
220	C₄×C₃₁	—	4	1	no	—
221	C₄×C₃₁	—	8	2	no	—
222	C₄×C₃₁	#472	1	2	yes	—
223	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
224	C₄×C₃₁	#575	1	1	yes	characteristic
225	C₄×C₃₁	#575	1	2	yes	—
226	#532	C₂²×C₃₁	1	2	yes	—
227	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
228	#500	C₂²×C₃₁	1	1	yes	characteristic
229	#493	C₂²×C₃₁	1	1	yes	characteristic
230	#486	C₂²×C₃₁	1	2	yes	—
231	#482	—	2	2	no	—
232	#477	C₂²×C₃₁	1	2	yes	—
233	#476	C₂²×C₃₁	1	2	yes	—
234	#466	—	2	1	no	—
235	#466	—	2	1	no	—
236	#460	C₂²×C₃₁	1	2	yes	—
237	#459	C₂²×C₃₁	1	2	yes	—
238	#449	—	2	2	no	—
239	#446	C₂²×C₃₁	1	1	yes	characteristic
240	#446	C₂²×C₃₁	1	1	yes	characteristic
241	C₄×SD₁₆	—	2	2	no	—
242	C₄×D₈	—	2	2	no	—
243	C₄×D₈	—	2	1	no	—
244	C₄×D₈	—	2	1	no	—
245	#436	C₂²×C₃₁	1	1	yes	characteristic
246	#436	C₂²×C₃₁	1	1	yes	characteristic
247	#435	—	2	1	no	—
248	#435	—	2	1	no	—
249	#431	—	2	1	no	—
250	#431	C₂²×C₃₁	1	1	yes	characteristic
251	#431	C₂²×C₃₁	1	2	yes	—
252	C₄×M₁₆	—	2	1	no	—
253	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
254	#335	C₂²×C₃₁	1	2	yes	—
255	#334	C₂²×C₃₁	1	1	yes	characteristic
256	#334	C₂²×C₃₁	1	1	yes	characteristic
257	#333	C₂²×C₃₁	1	2	yes	—
258	#333	C₂²×C₃₁	1	2	yes	—
259	#331	C₂²×C₃₁	1	2	yes	—
260	#331	C₂²×C₃₁	1	2	yes	—
261	C₂×C₃₁	—	4	1	no	—
262	C₂×C₃₁	—	8	1	no	—
263	C₂×C₃₁	—	16	2	no	—
264	C₂×C₃₁	#3035	1	1	yes	characteristic
265	C₂×C₃₁	#3054	1	1	yes	characteristic
266	C₂×C₃₁	C₂×#472	1	1	yes	characteristic
267	C₂×#41	—	2	2	no	—
268	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
269	C₂×SD₁₆	—	2	2	no	—
270	C₂×D₈	—	2	2	no	—
271	C₂×D₈	—	2	1	no	—
272	C₂×D₈	—	2	1	no	—
273	C₂×M₁₆	—	2	1	no	—
274	C₂×M₁₆	—	2	1	no	—
275	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
276	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
277	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
278	#127	C₂³×C₃₁	1	2	yes	—
279	#124	—	2	2	no	—
280	#121	—	2	2	no	—
281	#121	—	2	2	no	—
282	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
283	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
284	C₄×D₄	—	2	1	no	—
285	C₄×D₄	—	2	1	no	—
286	C₄×D₄	—	2	2	no	—
287	C₄×D₄	—	2	2	no	—
288	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
289	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
290	#117	—	2	1	no	—
291	#117	D₄×C₃₁	1	1	yes	characteristic
292	C₂×C₄²	—	2	1	no	—
293	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
294	#107	—	2	1	no	—
295	#107	—	2	1	no	—
296	#107	—	2	2	no	—
297	#106	—	2	2	no	—
298	C₄⋊C₈	—	2	1	no	—
299	C₄⋊C₈	—	2	1	no	—
300	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
301	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
302	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
303	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
304	#103	—	2	2	no	—
305	#102	—	2	2	no	—
306	#102	—	2	2	no	—
307	#102	—	2	2	no	—
308	#102	—	2	2	no	—
309	#102	—	2	2	no	—
310	C₂²⋊C₈	—	2	1	no	—
311	C₂²⋊C₈	—	2	2	no	—
312	C₂²⋊C₈	—	2	1	no	—
313	C₂²⋊C₈	—	2	1	no	—
314	#97	—	2	1	no	—
315	#97	—	2	1	no	—
316	C₄×C₈	—	2	1	no	—
317	C₄×C₈	—	2	2	no	—
318	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
319	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
320	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
321	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
322	C₃₁	#19160	1	1	yes	characteristic, direct product factor, Sylow
323	#41	—	4	2	no	—
324	#41	—	8	4	no	—
325	C₂×Q₈	—	2	2	no	—
326	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
327	C₂×D₄	—	2	1	no	—
328	C₂×D₄	—	2	2	no	—
329	C₂×D₄	—	2	2	no	—
330	C₂×D₄	—	2	2	no	—
331	C₂×D₄	—	4	2	no	—
332	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
333	C₂²×C₄	—	2	1	no	—
334	C₂²×C₄	—	2	1	no	—
335	C₂²×C₄	—	2	1	no	—
336	C₂²×C₄	—	4	2	no	—
337	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
338	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
339	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
340	SD₁₆	—	4	4	no	—
341	D₈	—	4	2	no	—
342	D₈	—	4	4	no	—
343	D₈	—	4	2	no	—
344	M₁₆	—	4	2	no	—
345	M₁₆	—	4	2	no	—
346	C₂×C₈	—	2	2	no	—
347	C₂×C₈	—	2	1	no	—
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	2	2	no	—
350	C₂×C₈	—	2	2	no	—
351	C₂×C₈	—	4	1	no	—
352	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
353	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
354	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
355	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
356	C₄⋊C₄	—	2	1	no	—
357	C₄⋊C₄	—	2	1	no	—
358	C₄⋊C₄	—	2	2	no	—
359	C₄⋊C₄	—	2	2	no	—
360	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
361	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
362	C₂²⋊C₄	—	2	1	no	—
363	C₂²⋊C₄	—	2	1	no	—
364	C₂²⋊C₄	—	2	1	no	—
365	C₂²⋊C₄	—	4	2	no	—
366	C₂²⋊C₄	—	4	2	no	—
367	C₄²	—	2	1	no	—
368	C₄²	—	2	1	no	—
369	C₄²	—	2	1	no	—
370	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
371	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
372	C₄²	#41×C₃₁	1	1	yes	characteristic
373	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
374	C₂³	—	2	1	no	—
375	C₂³	—	4	2	no	—
376	C₂³	C₃₁×#121	1	1	yes	characteristic
377	Q₈	—	2	2	no	—
378	Q₈	—	8	2	no	—
379	D₄	—	2	2	no	—
380	D₄	—	4	2	no	—
381	D₄	—	4	2	no	—
382	D₄	—	4	2	no	—
383	D₄	—	4	2	no	—
384	D₄	—	4	2	no	—
385	D₄	—	8	2	no	—
386	D₄	—	8	4	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	2	1	no	—
389	C₂×C₄	—	2	1	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	1	no	—
392	C₂×C₄	—	2	1	no	—
393	C₂×C₄	—	2	1	no	—
394	C₂×C₄	—	4	2	no	—
395	C₂×C₄	—	4	2	no	—
396	C₂×C₄	—	4	1	no	—
397	C₂×C₄	—	4	1	no	—
398	C₂×C₄	—	4	2	no	—
399	C₂×C₄	—	8	4	no	—
400	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
401	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
402	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
403	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
404	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
405	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
406	C₈	—	2	1	no	—
407	C₈	—	2	1	no	—
408	C₈	—	2	1	no	—
409	C₈	—	4	1	no	—
410	C₈	—	4	1	no	—
411	C₈	—	4	2	no	—
412	C₈	—	4	2	no	—
413	C₈	C₃₁×#135	1	2	yes	—
414	C₂²	—	2	1	no	—
415	C₂²	—	2	1	no	—
416	C₂²	—	2	1	no	—
417	C₂²	—	4	1	no	—
418	C₂²	—	4	1	no	—
419	C₂²	—	4	1	no	—
420	C₂²	—	8	2	no	—
421	C₂²	—	8	2	no	—
422	C₂²	—	8	2	no	—
423	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
424	C₄	—	2	1	no	—
425	C₄	—	2	1	no	—
426	C₄	—	2	1	no	—
427	C₄	—	4	1	no	—
428	C₄	—	4	1	no	—
429	C₄	—	4	1	no	—
430	C₄	—	8	2	no	—
431	C₄	C₃₁×#472	1	2	yes	—
432	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
433	C₄	C₃₁×#575	1	1	yes	characteristic
434	C₄	C₃₁×#575	1	2	yes	—
435	C₂	—	4	1	no	—
436	C₂	—	8	1	no	—
437	C₂	—	16	2	no	—
438	C₂	C₃₁×#3035	1	1	yes	characteristic
439	C₂	C₃₁×#3054	1	1	yes	characteristic
440	C₂	C₂×C₃₁×#472	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3169	C₂	semidirect
2. C₃₁×#3106	C₂	semidirect
3. C₃₁×#2965	C₂	semidirect
4. C₃₁×#1698	C₂	semidirect
5. C₃₁×#1697	C₂	semidirect
6. C₃₁×#1696	C₂	semidirect
7. C₃₁×#1695	C₂	semidirect
8. C₃₁×#1688	C₂	semidirect
9. C₃₁×#1687	C₂	semidirect
10. C₃₁×#1564	C₂	semidirect
11. C₃₁×#1543	C₂	semidirect
153. #19160	C₃₁	direct (N × H)
194. #3169	C₂×C₃₁	semidirect
195. #3106	C₂×C₃₁	semidirect
196. #2965	C₂×C₃₁	semidirect
197. #1698	C₂×C₃₁	semidirect
198. #1697	C₂×C₃₁	semidirect
199. #1696	C₂×C₃₁	semidirect
200. #1695	C₂×C₃₁	semidirect
201. #1688	C₂×C₃₁	semidirect
202. #1687	C₂×C₃₁	semidirect
203. #1564	C₂×C₃₁	semidirect
204. #1543	C₂×C₃₁	semidirect
322. C₃₁	#19160	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×#3169	2
C₃₁×#3106	2
C₃₁×#2965	2
C₃₁×#1698	2
C₃₁×#1697	2
C₃₁×#1696	2
C₃₁×#1695	2
C₃₁×#1688	2
C₃₁×#1687	2
C₃₁×#1564	2
C₃₁×#1543	2
#19160	31

Subgroups — C31×#19160 #21475919

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19160 #21475919