Subgroups — C₃₁×#19126 #21475885

Order 7936.

618 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3170	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3076	C₂	1	2	yes	maximal
3	C₈×#41×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1684	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1683	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1676	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1675	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1670	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1669	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#335	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1542	C₂	1	2	yes	maximal, complemented
12	C₃₁×#533	C₂²	1	2	yes	—
13	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#500	C₂²	1	1	yes	characteristic
15	C₃₁×#494	C₂²	1	1	yes	characteristic
16	C₃₁×#487	C₂²	1	2	yes	—
17	C₃₁×#479	—	2	2	no	—
18	C₃₁×#478	—	2	2	no	—
19	C₃₁×#477	C₂²	1	2	yes	—
20	C₃₁×#476	C₂²	1	2	yes	—
21	C₃₁×#475	C₂²	1	2	yes	—
22	C₃₁×#474	C₂²	1	2	yes	—
23	C₃₁×#467	—	2	1	no	—
24	C₃₁×#467	—	2	1	no	—
25	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
26	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₄×Q₁₆×C₃₁	—	2	1	no	—
28	C₄×Q₁₆×C₃₁	—	2	1	no	—
29	C₈×D₄×C₃₁	—	2	1	no	—
30	C₈×D₄×C₃₁	—	2	1	no	—
31	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
32	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₃₁×#431	—	2	1	no	—
34	C₃₁×#431	C₂²	1	1	yes	characteristic
35	C₃₁×#427	—	2	2	no	—
36	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
37	C₃₁×#416	—	2	2	no	—
38	C₂×C₄×C₈×C₃₁	—	2	1	no	—
39	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
40	C₃₁×#335	C₂²	1	1	yes	characteristic
41	C₃₁×#335	C₂²	1	1	yes	characteristic
42	C₃₁×#335	C₂²	1	2	yes	—
43	C₃₁×#333	C₂²	1	2	yes	—
44	C₃₁×#333	C₂²	1	2	yes	—
45	C₃₁×#332	C₂²	1	2	yes	—
46	C₃₁×#332	C₂²	1	2	yes	—
47	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
48	C₂×Q₁₆×C₃₁	—	2	1	no	—
49	C₂×Q₁₆×C₃₁	—	2	1	no	—
50	C₂²×C₈×C₃₁	—	2	2	no	—
51	C₂²×C₈×C₃₁	—	2	1	no	—
52	C₂²×C₈×C₃₁	—	2	1	no	—
53	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
55	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
56	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
57	C₃₁×#125	—	2	2	no	—
58	C₃₁×C₂²⋊Q₈	—	2	2	no	—
59	C₃₁×C₂²⋊Q₈	—	2	2	no	—
60	C₄×Q₈×C₃₁	—	2	2	no	—
61	C₄×Q₈×C₃₁	—	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	D₄	1	1	yes	characteristic
67	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₃₁×#117	—	2	1	no	—
69	C₃₁×#117	—	2	2	no	—
70	C₃₁×#117	D₄	1	1	yes	characteristic
71	C₂×C₄²×C₃₁	—	2	1	no	—
72	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
73	C₃₁×#107	—	2	1	no	—
74	C₃₁×#107	—	2	1	no	—
75	C₃₁×#107	—	2	2	no	—
76	C₃₁×#107	—	2	2	no	—
77	C₃₁×#106	—	2	2	no	—
78	C₃₁×#106	—	2	2	no	—
79	C₃₁×C₄⋊C₈	—	2	1	no	—
80	C₃₁×C₄⋊C₈	—	2	1	no	—
81	C₃₁×C₄⋊C₈	—	2	2	no	—
82	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
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×#103	—	2	2	no	—
87	C₃₁×#103	—	2	2	no	—
88	C₃₁×#103	—	2	2	no	—
89	C₃₁×#103	—	2	2	no	—
90	C₃₁×#102	—	2	2	no	—
91	C₃₁×#102	—	2	2	no	—
92	C₃₁×C₂²⋊C₈	—	2	1	no	—
93	C₃₁×C₂²⋊C₈	—	2	1	no	—
94	C₃₁×C₂²⋊C₈	—	2	1	no	—
95	C₄×C₈×C₃₁	—	2	1	no	—
96	C₄×C₈×C₃₁	—	2	1	no	—
97	C₄×C₈×C₃₁	—	2	1	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	—	2	2	no	—
103	#41×C₃₁	Q₁₆	1	4	yes	complemented
104	C₂×Q₈×C₃₁	—	2	2	no	—
105	C₂×Q₈×C₃₁	—	2	2	no	—
106	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂×D₄×C₃₁	—	2	1	no	—
108	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₃₁	—	2	1	no	—
110	C₂²×C₄×C₃₁	—	2	1	no	—
111	C₂²×C₄×C₃₁	—	2	1	no	—
112	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
115	Q₁₆×C₃₁	—	4	2	no	—
116	Q₁₆×C₃₁	—	4	2	no	—
117	C₂×C₈×C₃₁	—	2	2	no	—
118	C₂×C₈×C₃₁	—	2	1	no	—
119	C₂×C₈×C₃₁	—	2	1	no	—
120	C₂×C₈×C₃₁	—	2	2	no	—
121	C₂×C₈×C₃₁	—	2	2	no	—
122	C₂×C₈×C₃₁	—	4	1	no	—
123	C₂×C₈×C₃₁	—	4	2	no	—
124	C₂×C₈×C₃₁	—	4	4	no	—
125	C₂×C₈×C₃₁	—	4	2	no	—
126	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
130	C₄⋊C₄×C₃₁	—	2	1	no	—
131	C₄⋊C₄×C₃₁	—	2	1	no	—
132	C₄⋊C₄×C₃₁	—	2	2	no	—
133	C₄⋊C₄×C₃₁	—	2	2	no	—
134	C₄⋊C₄×C₃₁	—	2	2	no	—
135	C₄⋊C₄×C₃₁	—	2	2	no	—
136	C₄⋊C₄×C₃₁	—	4	2	no	—
137	C₄⋊C₄×C₃₁	—	4	2	no	—
138	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
139	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
140	C₂²⋊C₄×C₃₁	—	2	1	no	—
141	C₂²⋊C₄×C₃₁	—	2	1	no	—
142	C₂²⋊C₄×C₃₁	—	2	1	no	—
143	C₂²⋊C₄×C₃₁	—	4	2	no	—
144	C₄²×C₃₁	—	2	1	no	—
145	C₄²×C₃₁	—	2	1	no	—
146	C₄²×C₃₁	—	2	1	no	—
147	C₄²×C₃₁	—	4	2	no	—
148	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
149	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
150	C₄²×C₃₁	#41	1	1	yes	characteristic
151	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
152	#19126	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₂³×C₃₁	—	2	1	no	—
154	C₂³×C₃₁	#121	1	1	yes	characteristic
155	Q₈×C₃₁	—	2	1	no	—
156	Q₈×C₃₁	—	4	2	no	—
157	Q₈×C₃₁	—	4	2	no	—
158	Q₈×C₃₁	—	8	2	no	—
159	Q₈×C₃₁	C₂×Q₁₆	1	2	yes	—
160	D₄×C₃₁	—	2	4	no	—
161	D₄×C₃₁	—	2	1	no	—
162	D₄×C₃₁	C₂×Q₁₆	1	2	yes	—
163	C₂×C₄×C₃₁	—	2	1	no	—
164	C₂×C₄×C₃₁	—	2	1	no	—
165	C₂×C₄×C₃₁	—	2	1	no	—
166	C₂×C₄×C₃₁	—	2	4	no	—
167	C₂×C₄×C₃₁	—	2	1	no	—
168	C₂×C₄×C₃₁	—	2	1	no	—
169	C₂×C₄×C₃₁	—	4	1	no	—
170	C₂×C₄×C₃₁	—	4	2	no	—
171	C₂×C₄×C₃₁	—	4	1	no	—
172	C₂×C₄×C₃₁	—	4	1	no	—
173	C₂×C₄×C₃₁	—	4	2	no	—
174	C₂×C₄×C₃₁	—	4	2	no	—
175	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
176	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
177	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
178	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
179	C₂×C₄×C₃₁	#135	1	2	yes	—
180	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
181	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
182	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
183	C₈×C₃₁	—	2	1	no	—
184	C₈×C₃₁	—	2	1	no	—
185	C₈×C₃₁	—	2	1	no	—
186	C₈×C₃₁	—	4	1	no	—
187	C₈×C₃₁	—	4	2	no	—
188	C₈×C₃₁	—	4	2	no	—
189	C₈×C₃₁	—	4	1	no	—
190	C₈×C₃₁	#135	1	2	yes	—
191	#3170	C₂×C₃₁	1	1	yes	characteristic, complemented
192	#3076	C₂×C₃₁	1	2	yes	—
193	C₈×#41	C₂×C₃₁	1	1	yes	characteristic
194	#1684	C₂×C₃₁	1	2	yes	complemented
195	#1683	C₂×C₃₁	1	2	yes	complemented
196	#1676	C₂×C₃₁	1	1	yes	characteristic, complemented
197	#1675	C₂×C₃₁	1	1	yes	characteristic, complemented
198	#1670	C₂×C₃₁	1	1	yes	characteristic, complemented
199	#1669	C₂×C₃₁	1	1	yes	characteristic, complemented
200	C₂×#335	C₂×C₃₁	1	1	yes	characteristic, complemented
201	#1542	C₂×C₃₁	1	2	yes	complemented
202	C₂²×C₃₁	—	2	1	no	—
203	C₂²×C₃₁	—	2	1	no	—
204	C₂²×C₃₁	—	2	2	no	—
205	C₂²×C₃₁	—	4	1	no	—
206	C₂²×C₃₁	—	4	1	no	—
207	C₂²×C₃₁	#467	1	2	yes	—
208	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
209	C₄×C₃₁	—	2	1	no	—
210	C₄×C₃₁	—	2	2	no	—
211	C₄×C₃₁	—	2	1	no	—
212	C₄×C₃₁	—	2	1	no	—
213	C₄×C₃₁	—	4	1	no	—
214	C₄×C₃₁	—	4	1	no	—
215	C₄×C₃₁	—	8	2	no	—
216	C₄×C₃₁	—	8	2	no	—
217	C₄×C₃₁	#467	1	2	yes	—
218	C₄×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
219	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
220	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
221	C₄×C₃₁	#575	1	1	yes	characteristic
222	#533	C₂²×C₃₁	1	2	yes	—
223	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
224	#500	C₂²×C₃₁	1	1	yes	characteristic
225	#494	C₂²×C₃₁	1	1	yes	characteristic
226	#487	C₂²×C₃₁	1	2	yes	—
227	#479	—	2	2	no	—
228	#478	—	2	2	no	—
229	#477	C₂²×C₃₁	1	2	yes	—
230	#476	C₂²×C₃₁	1	2	yes	—
231	#475	C₂²×C₃₁	1	2	yes	—
232	#474	C₂²×C₃₁	1	2	yes	—
233	#467	—	2	1	no	—
234	#467	—	2	1	no	—
235	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
236	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
237	C₄×Q₁₆	—	2	1	no	—
238	C₄×Q₁₆	—	2	1	no	—
239	C₈×D₄	—	2	1	no	—
240	C₈×D₄	—	2	1	no	—
241	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
242	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
243	#431	—	2	1	no	—
244	#431	C₂²×C₃₁	1	1	yes	characteristic
245	#427	—	2	2	no	—
246	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
247	#416	—	2	2	no	—
248	C₂×C₄×C₈	—	2	1	no	—
249	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
250	#335	C₂²×C₃₁	1	1	yes	characteristic
251	#335	C₂²×C₃₁	1	1	yes	characteristic
252	#335	C₂²×C₃₁	1	2	yes	—
253	#333	C₂²×C₃₁	1	2	yes	—
254	#333	C₂²×C₃₁	1	2	yes	—
255	#332	C₂²×C₃₁	1	2	yes	—
256	#332	C₂²×C₃₁	1	2	yes	—
257	C₂×C₃₁	—	2	2	no	—
258	C₂×C₃₁	—	4	2	no	—
259	C₂×C₃₁	#3035	1	1	yes	characteristic
260	C₂×C₃₁	C₂×#467	1	1	yes	characteristic
261	C₂×C₃₁	#3067	1	1	yes	characteristic
262	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
263	C₂×Q₁₆	—	2	1	no	—
264	C₂×Q₁₆	—	2	1	no	—
265	C₂²×C₈	—	2	2	no	—
266	C₂²×C₈	—	2	1	no	—
267	C₂²×C₈	—	2	1	no	—
268	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
269	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
270	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
271	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
272	#125	—	2	2	no	—
273	C₂²⋊Q₈	—	2	2	no	—
274	C₂²⋊Q₈	—	2	2	no	—
275	C₄×Q₈	—	2	2	no	—
276	C₄×Q₈	—	2	2	no	—
277	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
278	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
279	C₄×D₄	—	2	1	no	—
280	C₄×D₄	—	2	1	no	—
281	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
282	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
283	#117	—	2	1	no	—
284	#117	—	2	2	no	—
285	#117	D₄×C₃₁	1	1	yes	characteristic
286	C₂×C₄²	—	2	1	no	—
287	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
288	#107	—	2	1	no	—
289	#107	—	2	1	no	—
290	#107	—	2	2	no	—
291	#107	—	2	2	no	—
292	#106	—	2	2	no	—
293	#106	—	2	2	no	—
294	C₄⋊C₈	—	2	1	no	—
295	C₄⋊C₈	—	2	1	no	—
296	C₄⋊C₈	—	2	2	no	—
297	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
298	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
299	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
300	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
301	#103	—	2	2	no	—
302	#103	—	2	2	no	—
303	#103	—	2	2	no	—
304	#103	—	2	2	no	—
305	#102	—	2	2	no	—
306	#102	—	2	2	no	—
307	C₂²⋊C₈	—	2	1	no	—
308	C₂²⋊C₈	—	2	1	no	—
309	C₂²⋊C₈	—	2	1	no	—
310	C₄×C₈	—	2	1	no	—
311	C₄×C₈	—	2	1	no	—
312	C₄×C₈	—	2	1	no	—
313	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
314	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
315	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
316	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
317	C₃₁	#19126	1	1	yes	characteristic, direct product factor, Sylow
318	#41	—	2	2	no	—
319	#41	Q₁₆×C₃₁	1	4	yes	complemented
320	C₂×Q₈	—	2	2	no	—
321	C₂×Q₈	—	2	2	no	—
322	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
323	C₂×D₄	—	2	1	no	—
324	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	—	2	1	no	—
327	C₂²×C₄	—	2	1	no	—
328	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
329	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
330	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
331	Q₁₆	—	4	2	no	—
332	Q₁₆	—	4	2	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	2	1	no	—
335	C₂×C₈	—	2	1	no	—
336	C₂×C₈	—	2	2	no	—
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	—	4	1	no	—
339	C₂×C₈	—	4	2	no	—
340	C₂×C₈	—	4	4	no	—
341	C₂×C₈	—	4	2	no	—
342	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
343	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
344	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
345	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
346	C₄⋊C₄	—	2	1	no	—
347	C₄⋊C₄	—	2	1	no	—
348	C₄⋊C₄	—	2	2	no	—
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	—	2	2	no	—
351	C₄⋊C₄	—	2	2	no	—
352	C₄⋊C₄	—	4	2	no	—
353	C₄⋊C₄	—	4	2	no	—
354	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
355	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
356	C₂²⋊C₄	—	2	1	no	—
357	C₂²⋊C₄	—	2	1	no	—
358	C₂²⋊C₄	—	2	1	no	—
359	C₂²⋊C₄	—	4	2	no	—
360	C₄²	—	2	1	no	—
361	C₄²	—	2	1	no	—
362	C₄²	—	2	1	no	—
363	C₄²	—	4	2	no	—
364	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
365	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
366	C₄²	#41×C₃₁	1	1	yes	characteristic
367	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
368	C₂³	—	2	1	no	—
369	C₂³	C₃₁×#121	1	1	yes	characteristic
370	Q₈	—	2	1	no	—
371	Q₈	—	4	2	no	—
372	Q₈	—	4	2	no	—
373	Q₈	—	8	2	no	—
374	Q₈	C₂×Q₁₆×C₃₁	1	2	yes	—
375	D₄	—	2	4	no	—
376	D₄	—	2	1	no	—
377	D₄	C₂×Q₁₆×C₃₁	1	2	yes	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	2	4	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	2	1	no	—
383	C₂×C₄	—	2	1	no	—
384	C₂×C₄	—	4	1	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	1	no	—
387	C₂×C₄	—	4	1	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
391	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
392	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
393	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
394	C₂×C₄	C₃₁×#135	1	2	yes	—
395	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
396	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
397	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
398	C₈	—	2	1	no	—
399	C₈	—	2	1	no	—
400	C₈	—	2	1	no	—
401	C₈	—	4	1	no	—
402	C₈	—	4	2	no	—
403	C₈	—	4	2	no	—
404	C₈	—	4	1	no	—
405	C₈	C₃₁×#135	1	2	yes	—
406	C₂²	—	2	1	no	—
407	C₂²	—	2	1	no	—
408	C₂²	—	2	2	no	—
409	C₂²	—	4	1	no	—
410	C₂²	—	4	1	no	—
411	C₂²	C₃₁×#467	1	2	yes	—
412	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
413	C₄	—	2	1	no	—
414	C₄	—	2	2	no	—
415	C₄	—	2	1	no	—
416	C₄	—	2	1	no	—
417	C₄	—	4	1	no	—
418	C₄	—	4	1	no	—
419	C₄	—	8	2	no	—
420	C₄	—	8	2	no	—
421	C₄	C₃₁×#467	1	2	yes	—
422	C₄	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
423	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
424	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
425	C₄	C₃₁×#575	1	1	yes	characteristic
426	C₂	—	2	2	no	—
427	C₂	—	4	2	no	—
428	C₂	C₃₁×#3035	1	1	yes	characteristic
429	C₂	C₂×C₃₁×#467	1	1	yes	characteristic
430	C₂	C₃₁×#3067	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×#121
prime radical	C₂×#41×C₃₁	D₄
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×#3170	C₂	semidirect
4. C₃₁×#1684	C₂	semidirect
5. C₃₁×#1683	C₂	semidirect
6. C₃₁×#1676	C₂	semidirect
7. C₃₁×#1675	C₂	semidirect
8. C₃₁×#1670	C₂	semidirect
9. C₃₁×#1669	C₂	semidirect
10. C₂×C₃₁×#335	C₂	semidirect
11. C₃₁×#1542	C₂	semidirect
103. #41×C₃₁	Q₁₆	semidirect
152. #19126	C₃₁	direct (N × H)
191. #3170	C₂×C₃₁	semidirect
194. #1684	C₂×C₃₁	semidirect
195. #1683	C₂×C₃₁	semidirect
196. #1676	C₂×C₃₁	semidirect
197. #1675	C₂×C₃₁	semidirect
198. #1670	C₂×C₃₁	semidirect
199. #1669	C₂×C₃₁	semidirect
200. C₂×#335	C₂×C₃₁	semidirect
201. #1542	C₂×C₃₁	semidirect
317. C₃₁	#19126	direct (N × H)
319. #41	Q₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3170	2
C₃₁×#3076	2
C₈×#41×C₃₁	2
C₃₁×#1684	2
C₃₁×#1683	2
C₃₁×#1676	2
C₃₁×#1675	2
C₃₁×#1670	2
C₃₁×#1669	2
C₂×C₃₁×#335	2
C₃₁×#1542	2
#19126	31

Subgroups — C31×#19126 #21475885

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19126 #21475885