Subgroups — C₃₁×#19225 #21475984

Order 7936.

630 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3212	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3169	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2966	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1694	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1693	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1688	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1687	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1682	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1681	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1564	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1553	C₂	1	2	yes	maximal, complemented
12	C₃₁×#532	C₂²	1	2	yes	—
13	C₃₁×#529	C₂²	1	1	yes	characteristic
14	C₃₁×#500	C₂²	1	1	yes	characteristic
15	C₃₁×#495	—	2	2	no	—
16	C₃₁×#493	C₂²	1	1	yes	characteristic
17	C₃₁×#488	—	2	2	no	—
18	C₃₁×#486	C₂²	1	2	yes	—
19	C₃₁×#466	—	2	1	no	—
20	C₃₁×#466	—	2	1	no	—
21	C₃₁×#462	C₂²	1	2	yes	—
22	C₃₁×#461	C₂²	1	2	yes	—
23	C₃₁×#460	C₂²	1	2	yes	—
24	C₃₁×#459	C₂²	1	2	yes	—
25	C₃₁×#452	—	2	2	no	—
26	C₃₁×#449	—	2	2	no	—
27	C₃₁×#446	C₂²	1	1	yes	characteristic
28	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₄×D₈×C₃₁	—	2	1	no	—
30	C₄×D₈×C₃₁	—	2	1	no	—
31	C₃₁×#436	C₂²	1	1	yes	characteristic
32	C₃₁×#435	—	2	1	no	—
33	C₈×D₄×C₃₁	—	2	1	no	—
34	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₃₁×#433	—	2	1	no	—
36	C₃₁×#432	C₂²	1	1	yes	characteristic
37	C₃₁×#431	C₂²	1	2	yes	—
38	C₃₁×#405	—	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×#332	C₂²	1	2	yes	—
44	C₃₁×#332	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×Q₁₆×C₃₁	—	2	2	no	—
49	C₂×SD₁₆×C₃₁	—	2	2	no	—
50	C₂×SD₁₆×C₃₁	—	2	2	no	—
51	C₂×D₈×C₃₁	—	2	2	no	—
52	C₂×D₈×C₃₁	—	2	1	no	—
53	C₂×D₈×C₃₁	—	2	1	no	—
54	C₂×M₁₆×C₃₁	—	2	1	no	—
55	C₂²×C₈×C₃₁	—	2	1	no	—
56	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
58	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
59	C₃₁×#127	C₂³	1	2	yes	—
60	C₃₁×#126	—	2	1	no	—
61	C₃₁×#125	D₄	1	1	yes	characteristic
62	C₃₁×#124	—	2	2	no	—
63	C₃₁×#124	D₄	1	1	yes	characteristic
64	C₃₁×#123	—	2	1	no	—
65	C₃₁×C₂²⋊Q₈	D₄	1	1	yes	characteristic
66	C₃₁×#121	—	2	2	no	—
67	C₃₁×#121	—	2	2	no	—
68	C₃₁×#121	D₄	1	1	yes	characteristic
69	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
70	C₄×D₄×C₃₁	—	2	1	no	—
71	C₄×D₄×C₃₁	—	2	2	no	—
72	C₄×D₄×C₃₁	—	2	2	no	—
73	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
74	C₃₁×#117	—	2	1	no	—
75	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
76	C₃₁×#107	—	2	1	no	—
77	C₃₁×#107	—	2	1	no	—
78	C₃₁×C₄⋊C₈	—	2	1	no	—
79	C₃₁×C₄⋊C₈	—	2	1	no	—
80	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
81	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
82	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
83	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
84	C₃₁×#103	—	2	2	no	—
85	C₃₁×#103	—	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	2	no	—
91	C₃₁×C₂²⋊C₈	—	2	1	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	—	8	2	no	—
103	#41×C₃₁	—	8	2	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	—	4	2	no	—
114	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
117	Q₁₆×C₃₁	—	8	2	no	—
118	SD₁₆×C₃₁	—	8	2	no	—
119	SD₁₆×C₃₁	—	8	2	no	—
120	D₈×C₃₁	—	4	2	no	—
121	D₈×C₃₁	—	4	2	no	—
122	D₈×C₃₁	—	8	2	no	—
123	M₁₆×C₃₁	—	4	2	no	—
124	C₂×C₈×C₃₁	—	2	2	no	—
125	C₂×C₈×C₃₁	—	2	1	no	—
126	C₂×C₈×C₃₁	—	2	2	no	—
127	C₂×C₈×C₃₁	—	2	2	no	—
128	C₂×C₈×C₃₁	—	2	1	no	—
129	C₂×C₈×C₃₁	—	4	1	no	—
130	C₂×C₈×C₃₁	—	4	2	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	1	no	—
138	C₄⋊C₄×C₃₁	—	2	1	no	—
139	C₄⋊C₄×C₃₁	—	2	2	no	—
140	C₄⋊C₄×C₃₁	—	2	2	no	—
141	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
142	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
143	C₂²⋊C₄×C₃₁	—	2	1	no	—
144	C₂²⋊C₄×C₃₁	—	2	1	no	—
145	C₂²⋊C₄×C₃₁	—	2	1	no	—
146	C₂²⋊C₄×C₃₁	—	2	1	no	—
147	C₂²⋊C₄×C₃₁	—	2	1	no	—
148	C₂²⋊C₄×C₃₁	—	4	2	no	—
149	C₂²⋊C₄×C₃₁	—	4	2	no	—
150	C₄²×C₃₁	—	2	1	no	—
151	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
152	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
153	C₄²×C₃₁	#41	1	1	yes	characteristic
154	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
155	#19225	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂³×C₃₁	—	2	1	no	—
157	C₂³×C₃₁	—	4	2	no	—
158	C₂³×C₃₁	#121	1	1	yes	characteristic
159	Q₈×C₃₁	—	2	2	no	—
160	Q₈×C₃₁	—	4	2	no	—
161	Q₈×C₃₁	—	4	2	no	—
162	D₄×C₃₁	—	2	2	no	—
163	D₄×C₃₁	—	4	2	no	—
164	D₄×C₃₁	—	4	2	no	—
165	D₄×C₃₁	—	4	2	no	—
166	D₄×C₃₁	—	4	2	no	—
167	D₄×C₃₁	—	4	2	no	—
168	D₄×C₃₁	—	8	2	no	—
169	D₄×C₃₁	—	8	4	no	—
170	C₂×C₄×C₃₁	—	2	1	no	—
171	C₂×C₄×C₃₁	—	2	1	no	—
172	C₂×C₄×C₃₁	—	2	2	no	—
173	C₂×C₄×C₃₁	—	2	1	no	—
174	C₂×C₄×C₃₁	—	2	1	no	—
175	C₂×C₄×C₃₁	—	2	1	no	—
176	C₂×C₄×C₃₁	—	4	1	no	—
177	C₂×C₄×C₃₁	—	4	1	no	—
178	C₂×C₄×C₃₁	—	4	1	no	—
179	C₂×C₄×C₃₁	—	4	2	no	—
180	C₂×C₄×C₃₁	—	8	4	no	—
181	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
182	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
183	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
184	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
186	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
187	C₈×C₃₁	—	2	1	no	—
188	C₈×C₃₁	—	2	1	no	—
189	C₈×C₃₁	—	2	1	no	—
190	C₈×C₃₁	—	4	1	no	—
191	C₈×C₃₁	—	4	1	no	—
192	C₈×C₃₁	—	4	2	no	—
193	C₈×C₃₁	—	4	2	no	—
194	C₈×C₃₁	#135	1	2	yes	—
195	#3212	C₂×C₃₁	1	2	yes	complemented
196	#3169	C₂×C₃₁	1	1	yes	characteristic, complemented
197	#2966	C₂×C₃₁	1	1	yes	characteristic, complemented
198	#1694	C₂×C₃₁	1	1	yes	characteristic, complemented
199	#1693	C₂×C₃₁	1	1	yes	characteristic, complemented
200	#1688	C₂×C₃₁	1	1	yes	characteristic, complemented
201	#1687	C₂×C₃₁	1	1	yes	characteristic, complemented
202	#1682	C₂×C₃₁	1	2	yes	complemented
203	#1681	C₂×C₃₁	1	2	yes	complemented
204	#1564	C₂×C₃₁	1	1	yes	characteristic, complemented
205	#1553	C₂×C₃₁	1	2	yes	complemented
206	C₂²×C₃₁	—	2	1	no	—
207	C₂²×C₃₁	—	2	1	no	—
208	C₂²×C₃₁	—	2	1	no	—
209	C₂²×C₃₁	—	4	1	no	—
210	C₂²×C₃₁	—	4	1	no	—
211	C₂²×C₃₁	—	4	1	no	—
212	C₂²×C₃₁	—	8	2	no	—
213	C₂²×C₃₁	—	8	2	no	—
214	C₂²×C₃₁	—	8	2	no	—
215	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
216	C₄×C₃₁	—	2	1	no	—
217	C₄×C₃₁	—	2	1	no	—
218	C₄×C₃₁	—	2	1	no	—
219	C₄×C₃₁	—	2	1	no	—
220	C₄×C₃₁	—	4	1	no	—
221	C₄×C₃₁	—	4	1	no	—
222	C₄×C₃₁	—	4	1	no	—
223	C₄×C₃₁	—	8	2	no	—
224	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
225	C₄×C₃₁	#575	1	1	yes	characteristic
226	C₄×C₃₁	#576	1	2	yes	—
227	#532	C₂²×C₃₁	1	2	yes	—
228	#529	C₂²×C₃₁	1	1	yes	characteristic
229	#500	C₂²×C₃₁	1	1	yes	characteristic
230	#495	—	2	2	no	—
231	#493	C₂²×C₃₁	1	1	yes	characteristic
232	#488	—	2	2	no	—
233	#486	C₂²×C₃₁	1	2	yes	—
234	#466	—	2	1	no	—
235	#466	—	2	1	no	—
236	#462	C₂²×C₃₁	1	2	yes	—
237	#461	C₂²×C₃₁	1	2	yes	—
238	#460	C₂²×C₃₁	1	2	yes	—
239	#459	C₂²×C₃₁	1	2	yes	—
240	#452	—	2	2	no	—
241	#449	—	2	2	no	—
242	#446	C₂²×C₃₁	1	1	yes	characteristic
243	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
244	C₄×D₈	—	2	1	no	—
245	C₄×D₈	—	2	1	no	—
246	#436	C₂²×C₃₁	1	1	yes	characteristic
247	#435	—	2	1	no	—
248	C₈×D₄	—	2	1	no	—
249	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
250	#433	—	2	1	no	—
251	#432	C₂²×C₃₁	1	1	yes	characteristic
252	#431	C₂²×C₃₁	1	2	yes	—
253	#405	—	2	1	no	—
254	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
255	#335	C₂²×C₃₁	1	2	yes	—
256	#334	C₂²×C₃₁	1	1	yes	characteristic
257	#334	C₂²×C₃₁	1	1	yes	characteristic
258	#332	C₂²×C₃₁	1	2	yes	—
259	#332	C₂²×C₃₁	1	2	yes	—
260	#331	C₂²×C₃₁	1	2	yes	—
261	#331	C₂²×C₃₁	1	2	yes	—
262	C₂×C₃₁	—	4	1	no	—
263	C₂×C₃₁	—	8	1	no	—
264	C₂×C₃₁	—	16	2	no	—
265	C₂×C₃₁	#3035	1	1	yes	characteristic
266	C₂×C₃₁	#3058	1	1	yes	characteristic
267	C₂×C₃₁	#3067	1	1	yes	characteristic
268	C₂×#41	—	2	2	no	—
269	C₂×Q₁₆	—	2	2	no	—
270	C₂×SD₁₆	—	2	2	no	—
271	C₂×SD₁₆	—	2	2	no	—
272	C₂×D₈	—	2	2	no	—
273	C₂×D₈	—	2	1	no	—
274	C₂×D₈	—	2	1	no	—
275	C₂×M₁₆	—	2	1	no	—
276	C₂²×C₈	—	2	1	no	—
277	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
278	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
279	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
280	#127	C₂³×C₃₁	1	2	yes	—
281	#126	—	2	1	no	—
282	#125	D₄×C₃₁	1	1	yes	characteristic
283	#124	—	2	2	no	—
284	#124	D₄×C₃₁	1	1	yes	characteristic
285	#123	—	2	1	no	—
286	C₂²⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
287	#121	—	2	2	no	—
288	#121	—	2	2	no	—
289	#121	D₄×C₃₁	1	1	yes	characteristic
290	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
291	C₄×D₄	—	2	1	no	—
292	C₄×D₄	—	2	2	no	—
293	C₄×D₄	—	2	2	no	—
294	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
295	#117	—	2	1	no	—
296	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
297	#107	—	2	1	no	—
298	#107	—	2	1	no	—
299	C₄⋊C₈	—	2	1	no	—
300	C₄⋊C₈	—	2	1	no	—
301	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
302	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
303	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
304	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
305	#103	—	2	2	no	—
306	#103	—	2	2	no	—
307	#102	—	2	2	no	—
308	#102	—	2	2	no	—
309	#102	—	2	2	no	—
310	#102	—	2	2	no	—
311	C₂²⋊C₈	—	2	2	no	—
312	C₂²⋊C₈	—	2	1	no	—
313	C₂²⋊C₈	—	2	1	no	—
314	C₂²⋊C₈	—	2	1	no	—
315	#97	—	2	1	no	—
316	#97	—	2	1	no	—
317	C₄×C₈	—	2	1	no	—
318	C₄×C₈	—	2	2	no	—
319	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
320	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
321	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
322	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
323	C₃₁	#19225	1	1	yes	characteristic, direct product factor, Sylow
324	#41	—	8	2	no	—
325	#41	—	8	2	no	—
326	C₂×Q₈	—	2	2	no	—
327	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
328	C₂×D₄	—	2	1	no	—
329	C₂×D₄	—	2	2	no	—
330	C₂×D₄	—	2	2	no	—
331	C₂×D₄	—	2	2	no	—
332	C₂×D₄	—	4	2	no	—
333	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
334	C₂²×C₄	—	2	1	no	—
335	C₂²×C₄	—	4	2	no	—
336	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
337	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
338	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
339	Q₁₆	—	8	2	no	—
340	SD₁₆	—	8	2	no	—
341	SD₁₆	—	8	2	no	—
342	D₈	—	4	2	no	—
343	D₈	—	4	2	no	—
344	D₈	—	8	2	no	—
345	M₁₆	—	4	2	no	—
346	C₂×C₈	—	2	2	no	—
347	C₂×C₈	—	2	1	no	—
348	C₂×C₈	—	2	2	no	—
349	C₂×C₈	—	2	2	no	—
350	C₂×C₈	—	2	1	no	—
351	C₂×C₈	—	4	1	no	—
352	C₂×C₈	—	4	2	no	—
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	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
357	C₄⋊C₄	—	2	1	no	—
358	C₄⋊C₄	—	2	1	no	—
359	C₄⋊C₄	—	2	1	no	—
360	C₄⋊C₄	—	2	1	no	—
361	C₄⋊C₄	—	2	2	no	—
362	C₄⋊C₄	—	2	2	no	—
363	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
364	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
365	C₂²⋊C₄	—	2	1	no	—
366	C₂²⋊C₄	—	2	1	no	—
367	C₂²⋊C₄	—	2	1	no	—
368	C₂²⋊C₄	—	2	1	no	—
369	C₂²⋊C₄	—	2	1	no	—
370	C₂²⋊C₄	—	4	2	no	—
371	C₂²⋊C₄	—	4	2	no	—
372	C₄²	—	2	1	no	—
373	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
374	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
375	C₄²	#41×C₃₁	1	1	yes	characteristic
376	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
377	C₂³	—	2	1	no	—
378	C₂³	—	4	2	no	—
379	C₂³	C₃₁×#121	1	1	yes	characteristic
380	Q₈	—	2	2	no	—
381	Q₈	—	4	2	no	—
382	Q₈	—	4	2	no	—
383	D₄	—	2	2	no	—
384	D₄	—	4	2	no	—
385	D₄	—	4	2	no	—
386	D₄	—	4	2	no	—
387	D₄	—	4	2	no	—
388	D₄	—	4	2	no	—
389	D₄	—	8	2	no	—
390	D₄	—	8	4	no	—
391	C₂×C₄	—	2	1	no	—
392	C₂×C₄	—	2	1	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	1	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	4	1	no	—
398	C₂×C₄	—	4	1	no	—
399	C₂×C₄	—	4	1	no	—
400	C₂×C₄	—	4	2	no	—
401	C₂×C₄	—	8	4	no	—
402	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
403	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
404	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
405	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
406	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
407	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
408	C₈	—	2	1	no	—
409	C₈	—	2	1	no	—
410	C₈	—	2	1	no	—
411	C₈	—	4	1	no	—
412	C₈	—	4	1	no	—
413	C₈	—	4	2	no	—
414	C₈	—	4	2	no	—
415	C₈	C₃₁×#135	1	2	yes	—
416	C₂²	—	2	1	no	—
417	C₂²	—	2	1	no	—
418	C₂²	—	2	1	no	—
419	C₂²	—	4	1	no	—
420	C₂²	—	4	1	no	—
421	C₂²	—	4	1	no	—
422	C₂²	—	8	2	no	—
423	C₂²	—	8	2	no	—
424	C₂²	—	8	2	no	—
425	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
426	C₄	—	2	1	no	—
427	C₄	—	2	1	no	—
428	C₄	—	2	1	no	—
429	C₄	—	2	1	no	—
430	C₄	—	4	1	no	—
431	C₄	—	4	1	no	—
432	C₄	—	4	1	no	—
433	C₄	—	8	2	no	—
434	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
435	C₄	C₃₁×#575	1	1	yes	characteristic
436	C₄	C₃₁×#576	1	2	yes	—
437	C₂	—	4	1	no	—
438	C₂	—	8	1	no	—
439	C₂	—	16	2	no	—
440	C₂	C₃₁×#3035	1	1	yes	characteristic
441	C₂	C₃₁×#3058	1	1	yes	characteristic
442	C₂	C₃₁×#3067	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×#3212	C₂	semidirect
2. C₃₁×#3169	C₂	semidirect
3. C₃₁×#2966	C₂	semidirect
4. C₃₁×#1694	C₂	semidirect
5. C₃₁×#1693	C₂	semidirect
6. C₃₁×#1688	C₂	semidirect
7. C₃₁×#1687	C₂	semidirect
8. C₃₁×#1682	C₂	semidirect
9. C₃₁×#1681	C₂	semidirect
10. C₃₁×#1564	C₂	semidirect
11. C₃₁×#1553	C₂	semidirect
155. #19225	C₃₁	direct (N × H)
195. #3212	C₂×C₃₁	semidirect
196. #3169	C₂×C₃₁	semidirect
197. #2966	C₂×C₃₁	semidirect
198. #1694	C₂×C₃₁	semidirect
199. #1693	C₂×C₃₁	semidirect
200. #1688	C₂×C₃₁	semidirect
201. #1687	C₂×C₃₁	semidirect
202. #1682	C₂×C₃₁	semidirect
203. #1681	C₂×C₃₁	semidirect
204. #1564	C₂×C₃₁	semidirect
205. #1553	C₂×C₃₁	semidirect
323. C₃₁	#19225	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×#3212	2
C₃₁×#3169	2
C₃₁×#2966	2
C₃₁×#1694	2
C₃₁×#1693	2
C₃₁×#1688	2
C₃₁×#1687	2
C₃₁×#1682	2
C₃₁×#1681	2
C₃₁×#1564	2
C₃₁×#1553	2
#19225	31

Subgroups — C31×#19225 #21475984

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19225 #21475984