Subgroups — C₃₁×#19242 #21476001

Order 7936.

636 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×#3249	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3208	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3138	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2962	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1702	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1698	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1682	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1678	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1565	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1542	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#331	C₂	1	1	yes	maximal, characteristic
12	C₃₁×#533	C₂²	1	1	yes	characteristic
13	C₃₁×#533	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
15	C₃₁×#492	—	2	1	no	—
16	C₃₁×#490	—	2	2	no	—
17	C₃₁×#486	—	2	1	no	—
18	C₃₁×#486	C₂²	1	2	yes	—
19	C₃₁×#486	C₂²	1	2	yes	—
20	C₃₁×#477	—	2	1	no	—
21	C₃₁×#477	C₂²	1	2	yes	complemented
22	C₃₁×#467	—	2	2	no	—
23	C₃₁×#462	—	2	1	no	—
24	C₃₁×#462	C₂²	1	2	yes	complemented
25	C₃₁×#460	C₂²	1	2	yes	—
26	C₃₁×#460	C₂²	1	2	yes	—
27	C₃₁×#450	—	2	2	no	—
28	C₃₁×#446	—	2	2	no	—
29	C₃₁×#446	C₂²	1	2	yes	complemented
30	C₈×Q₈×C₃₁	C₂²	1	2	yes	complemented
31	C₃₁×#431	C₂²	1	1	yes	characteristic
32	C₃₁×#424	—	2	1	no	—
33	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₃₁×#418	—	2	1	no	—
36	C₃₁×#418	—	2	1	no	—
37	C₃₁×#405	—	2	1	no	—
38	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
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×#332	C₂²	1	1	yes	characteristic, complemented
44	C₃₁×#332	C₂²	1	1	yes	characteristic, complemented
45	C₃₁×#331	C₂²	1	2	yes	—
46	C₃₁×#331	C₂²	1	2	yes	—
47	C₂²×D₄×C₃₁	—	2	1	no	—
48	C₂²×D₄×C₃₁	—	2	1	no	—
49	C₂²×D₄×C₃₁	—	2	1	no	—
50	C₂×Q₁₆×C₃₁	—	2	2	no	—
51	C₂×SD₁₆×C₃₁	—	2	2	no	—
52	C₂×SD₁₆×C₃₁	—	2	2	no	—
53	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
54	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₃₁	—	2	1	no	—
56	C₂²×C₈×C₃₁	—	2	1	no	—
57	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
58	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
59	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
60	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
61	C₃₁×#127	—	2	1	no	—
62	C₃₁×#127	—	2	1	no	—
63	C₃₁×#127	—	2	1	no	—
64	C₃₁×#127	C₂³	1	2	yes	—
65	C₃₁×#125	—	2	1	no	—
66	C₃₁×#125	—	2	1	no	—
67	C₃₁×C₂²⋊Q₈	—	2	1	no	—
68	C₃₁×C₂²⋊Q₈	—	2	1	no	—
69	C₃₁×C₂²⋊Q₈	—	2	1	no	—
70	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
71	C₄×Q₈×C₃₁	—	2	1	no	—
72	C₄×Q₈×C₃₁	—	2	1	no	—
73	C₄×Q₈×C₃₁	—	2	1	no	—
74	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
75	C₃₁×#117	—	2	1	no	—
76	C₃₁×#117	—	2	1	no	—
77	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
78	C₃₁×#107	—	2	2	no	—
79	C₃₁×C₄⋊C₈	—	2	2	no	—
80	C₃₁×C₄⋊C₈	—	2	2	no	—
81	C₃₁×C₄⋊C₈	—	2	1	no	—
82	C₃₁×C₄⋊C₈	—	2	1	no	—
83	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
84	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
85	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
86	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
87	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
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×#102	—	2	2	no	—
93	C₃₁×#102	—	2	2	no	—
94	C₃₁×#102	—	2	2	no	—
95	C₃₁×#102	—	2	2	no	—
96	C₃₁×C₂²⋊C₈	—	2	1	no	—
97	C₃₁×#97	—	2	2	no	—
98	C₃₁×#97	D₄	1	2	yes	—
99	C₄×C₈×C₃₁	—	2	2	no	—
100	C₄×C₈×C₃₁	—	2	1	no	—
101	C₄×C₈×C₃₁	C₂³	1	2	yes	—
102	C₂⁴×C₃₁	—	4	1	no	—
103	C₂×Q₈×C₃₁	—	2	1	no	—
104	C₂×Q₈×C₃₁	—	2	1	no	—
105	C₂×Q₈×C₃₁	—	2	1	no	—
106	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
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	1	no	—
111	C₂×D₄×C₃₁	—	4	1	no	—
112	C₂×D₄×C₃₁	—	4	1	no	—
113	C₂×D₄×C₃₁	—	4	2	no	—
114	C₂×D₄×C₃₁	—	4	2	no	—
115	C₂×D₄×C₃₁	—	8	1	no	—
116	C₂×D₄×C₃₁	—	8	1	no	—
117	C₂×D₄×C₃₁	—	8	2	no	—
118	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
121	Q₁₆×C₃₁	—	8	2	no	—
122	SD₁₆×C₃₁	—	8	2	no	—
123	SD₁₆×C₃₁	—	8	2	no	—
124	M₁₆×C₃₁	—	2	2	no	—
125	M₁₆×C₃₁	—	2	2	no	—
126	C₂×C₈×C₃₁	—	2	1	no	—
127	C₂×C₈×C₃₁	—	2	1	no	—
128	C₂×C₈×C₃₁	—	2	1	no	—
129	C₂×C₈×C₃₁	—	2	1	no	—
130	C₂×C₈×C₃₁	—	2	2	no	—
131	C₂×C₈×C₃₁	—	4	2	no	—
132	C₂×C₈×C₃₁	—	8	1	no	—
133	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
134	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
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	2	no	—
139	C₄⋊C₄×C₃₁	—	2	2	no	—
140	C₄⋊C₄×C₃₁	—	2	1	no	—
141	C₄⋊C₄×C₃₁	—	2	1	no	—
142	C₄⋊C₄×C₃₁	—	2	1	no	—
143	C₄⋊C₄×C₃₁	—	2	1	no	—
144	C₄⋊C₄×C₃₁	—	4	1	no	—
145	C₄⋊C₄×C₃₁	—	4	1	no	—
146	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
147	C₂²⋊C₄×C₃₁	—	2	2	no	—
148	C₂²⋊C₄×C₃₁	—	4	1	no	—
149	C₄²×C₃₁	—	2	2	no	—
150	C₄²×C₃₁	—	4	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	#19242	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂³×C₃₁	—	4	2	no	—
157	C₂³×C₃₁	—	8	2	no	—
158	C₂³×C₃₁	—	8	2	no	—
159	C₂³×C₃₁	—	8	2	no	—
160	C₂³×C₃₁	#121	1	1	yes	characteristic
161	Q₈×C₃₁	—	2	2	no	—
162	Q₈×C₃₁	—	2	2	no	—
163	Q₈×C₃₁	—	4	1	no	—
164	Q₈×C₃₁	—	4	1	no	—
165	Q₈×C₃₁	—	8	1	no	—
166	Q₈×C₃₁	—	8	1	no	—
167	D₄×C₃₁	—	4	2	no	—
168	D₄×C₃₁	—	4	2	no	—
169	D₄×C₃₁	—	4	2	no	—
170	D₄×C₃₁	—	4	2	no	—
171	D₄×C₃₁	—	8	2	no	—
172	D₄×C₃₁	—	8	2	no	—
173	D₄×C₃₁	—	8	2	no	—
174	D₄×C₃₁	—	8	2	no	—
175	C₂×C₄×C₃₁	—	2	1	no	—
176	C₂×C₄×C₃₁	—	2	1	no	—
177	C₂×C₄×C₃₁	—	2	2	no	—
178	C₂×C₄×C₃₁	—	2	2	no	—
179	C₂×C₄×C₃₁	—	4	1	no	—
180	C₂×C₄×C₃₁	—	4	1	no	—
181	C₂×C₄×C₃₁	—	4	1	no	—
182	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
183	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
184	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
185	C₂×C₄×C₃₁	#135	1	2	yes	—
186	C₂×C₄×C₃₁	#137	1	2	yes	—
187	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
188	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
189	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
190	C₈×C₃₁	—	2	2	no	—
191	C₈×C₃₁	—	2	2	no	—
192	C₈×C₃₁	—	4	1	no	—
193	C₈×C₃₁	—	4	1	no	—
194	C₈×C₃₁	—	4	2	no	—
195	C₈×C₃₁	—	4	1	no	—
196	C₈×C₃₁	—	4	1	no	—
197	#3249	C₂×C₃₁	1	1	yes	characteristic, complemented
198	#3208	C₂×C₃₁	1	1	yes	characteristic
199	#3138	C₂×C₃₁	1	1	yes	characteristic
200	#2962	C₂×C₃₁	1	1	yes	characteristic, complemented
201	#1702	C₂×C₃₁	1	2	yes	complemented
202	#1698	C₂×C₃₁	1	2	yes	complemented
203	#1682	C₂×C₃₁	1	2	yes	complemented
204	#1678	C₂×C₃₁	1	2	yes	complemented
205	#1565	C₂×C₃₁	1	1	yes	characteristic, complemented
206	#1542	C₂×C₃₁	1	1	yes	characteristic, complemented
207	C₂×#331	C₂×C₃₁	1	1	yes	characteristic
208	C₂²×C₃₁	—	2	1	no	—
209	C₂²×C₃₁	—	2	1	no	—
210	C₂²×C₃₁	—	8	2	no	—
211	C₂²×C₃₁	—	8	2	no	—
212	C₂²×C₃₁	—	8	2	no	—
213	C₂²×C₃₁	—	16	4	no	—
214	C₂²×C₃₁	#464	1	2	yes	—
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	—	4	2	no	—
220	C₄×C₃₁	—	4	2	no	—
221	C₄×C₃₁	—	8	1	no	—
222	C₄×C₃₁	—	8	1	no	—
223	C₄×C₃₁	#464	1	2	yes	—
224	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
225	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
226	C₄×C₃₁	#576	1	1	yes	characteristic
227	C₄×C₃₁	#577	1	1	yes	characteristic
228	#533	C₂²×C₃₁	1	1	yes	characteristic
229	#533	C₂²×C₃₁	1	1	yes	characteristic
230	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
231	#492	—	2	1	no	—
232	#490	—	2	2	no	—
233	#486	—	2	1	no	—
234	#486	C₂²×C₃₁	1	2	yes	—
235	#486	C₂²×C₃₁	1	2	yes	—
236	#477	—	2	1	no	—
237	#477	C₂²×C₃₁	1	2	yes	complemented
238	#467	—	2	2	no	—
239	#462	—	2	1	no	—
240	#462	C₂²×C₃₁	1	2	yes	complemented
241	#460	C₂²×C₃₁	1	2	yes	—
242	#460	C₂²×C₃₁	1	2	yes	—
243	#450	—	2	2	no	—
244	#446	—	2	2	no	—
245	#446	C₂²×C₃₁	1	2	yes	complemented
246	C₈×Q₈	C₂²×C₃₁	1	2	yes	complemented
247	#431	C₂²×C₃₁	1	1	yes	characteristic
248	#424	—	2	1	no	—
249	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
250	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
251	#418	—	2	1	no	—
252	#418	—	2	1	no	—
253	#405	—	2	1	no	—
254	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
255	#335	C₂²×C₃₁	1	2	yes	complemented
256	#334	C₂²×C₃₁	1	2	yes	complemented
257	#333	C₂²×C₃₁	1	1	yes	characteristic, complemented
258	#333	C₂²×C₃₁	1	1	yes	characteristic, complemented
259	#332	C₂²×C₃₁	1	1	yes	characteristic, complemented
260	#332	C₂²×C₃₁	1	1	yes	characteristic, complemented
261	#331	C₂²×C₃₁	1	2	yes	—
262	#331	C₂²×C₃₁	1	2	yes	—
263	C₂×C₃₁	—	2	2	no	—
264	C₂×C₃₁	—	16	2	no	—
265	C₂×C₃₁	C₂×#464	1	1	yes	characteristic
266	C₂×C₃₁	#3062	1	1	yes	characteristic
267	C₂×C₃₁	#3068	1	1	yes	characteristic
268	C₂²×D₄	—	2	1	no	—
269	C₂²×D₄	—	2	1	no	—
270	C₂²×D₄	—	2	1	no	—
271	C₂×Q₁₆	—	2	2	no	—
272	C₂×SD₁₆	—	2	2	no	—
273	C₂×SD₁₆	—	2	2	no	—
274	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
275	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
276	C₂²×C₈	—	2	1	no	—
277	C₂²×C₈	—	2	1	no	—
278	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
279	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
280	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
281	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
282	#127	—	2	1	no	—
283	#127	—	2	1	no	—
284	#127	—	2	1	no	—
285	#127	C₂³×C₃₁	1	2	yes	—
286	#125	—	2	1	no	—
287	#125	—	2	1	no	—
288	C₂²⋊Q₈	—	2	1	no	—
289	C₂²⋊Q₈	—	2	1	no	—
290	C₂²⋊Q₈	—	2	1	no	—
291	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
292	C₄×Q₈	—	2	1	no	—
293	C₄×Q₈	—	2	1	no	—
294	C₄×Q₈	—	2	1	no	—
295	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
296	#117	—	2	1	no	—
297	#117	—	2	1	no	—
298	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
299	#107	—	2	2	no	—
300	C₄⋊C₈	—	2	2	no	—
301	C₄⋊C₈	—	2	2	no	—
302	C₄⋊C₈	—	2	1	no	—
303	C₄⋊C₈	—	2	1	no	—
304	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
305	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
306	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
307	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
308	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
309	#103	—	2	2	no	—
310	#103	—	2	2	no	—
311	#102	—	2	2	no	—
312	#102	—	2	2	no	—
313	#102	—	2	2	no	—
314	#102	—	2	2	no	—
315	#102	—	2	2	no	—
316	#102	—	2	2	no	—
317	C₂²⋊C₈	—	2	1	no	—
318	#97	—	2	2	no	—
319	#97	D₄×C₃₁	1	2	yes	—
320	C₄×C₈	—	2	2	no	—
321	C₄×C₈	—	2	1	no	—
322	C₄×C₈	C₂³×C₃₁	1	2	yes	—
323	C₃₁	#19242	1	1	yes	characteristic, direct product factor, Sylow
324	C₂⁴	—	4	1	no	—
325	C₂×Q₈	—	2	1	no	—
326	C₂×Q₈	—	2	1	no	—
327	C₂×Q₈	—	2	1	no	—
328	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
329	C₂×D₄	—	2	2	no	—
330	C₂×D₄	—	2	2	no	—
331	C₂×D₄	—	2	2	no	—
332	C₂×D₄	—	4	1	no	—
333	C₂×D₄	—	4	1	no	—
334	C₂×D₄	—	4	1	no	—
335	C₂×D₄	—	4	2	no	—
336	C₂×D₄	—	4	2	no	—
337	C₂×D₄	—	8	1	no	—
338	C₂×D₄	—	8	1	no	—
339	C₂×D₄	—	8	2	no	—
340	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
341	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
342	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
343	Q₁₆	—	8	2	no	—
344	SD₁₆	—	8	2	no	—
345	SD₁₆	—	8	2	no	—
346	M₁₆	—	2	2	no	—
347	M₁₆	—	2	2	no	—
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	2	1	no	—
350	C₂×C₈	—	2	1	no	—
351	C₂×C₈	—	2	1	no	—
352	C₂×C₈	—	2	2	no	—
353	C₂×C₈	—	4	2	no	—
354	C₂×C₈	—	8	1	no	—
355	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
356	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
357	C₄⋊C₄	—	2	1	no	—
358	C₄⋊C₄	—	2	1	no	—
359	C₄⋊C₄	—	2	1	no	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	2	2	no	—
362	C₄⋊C₄	—	2	1	no	—
363	C₄⋊C₄	—	2	1	no	—
364	C₄⋊C₄	—	2	1	no	—
365	C₄⋊C₄	—	2	1	no	—
366	C₄⋊C₄	—	4	1	no	—
367	C₄⋊C₄	—	4	1	no	—
368	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
369	C₂²⋊C₄	—	2	2	no	—
370	C₂²⋊C₄	—	4	1	no	—
371	C₄²	—	2	2	no	—
372	C₄²	—	4	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	—	4	2	no	—
378	C₂³	—	8	2	no	—
379	C₂³	—	8	2	no	—
380	C₂³	—	8	2	no	—
381	C₂³	C₃₁×#121	1	1	yes	characteristic
382	Q₈	—	2	2	no	—
383	Q₈	—	2	2	no	—
384	Q₈	—	4	1	no	—
385	Q₈	—	4	1	no	—
386	Q₈	—	8	1	no	—
387	Q₈	—	8	1	no	—
388	D₄	—	4	2	no	—
389	D₄	—	4	2	no	—
390	D₄	—	4	2	no	—
391	D₄	—	4	2	no	—
392	D₄	—	8	2	no	—
393	D₄	—	8	2	no	—
394	D₄	—	8	2	no	—
395	D₄	—	8	2	no	—
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	2	1	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	4	1	no	—
401	C₂×C₄	—	4	1	no	—
402	C₂×C₄	—	4	1	no	—
403	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
404	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
405	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
406	C₂×C₄	C₃₁×#135	1	2	yes	—
407	C₂×C₄	C₃₁×#137	1	2	yes	—
408	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
409	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
410	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
411	C₈	—	2	2	no	—
412	C₈	—	2	2	no	—
413	C₈	—	4	1	no	—
414	C₈	—	4	1	no	—
415	C₈	—	4	2	no	—
416	C₈	—	4	1	no	—
417	C₈	—	4	1	no	—
418	C₂²	—	2	1	no	—
419	C₂²	—	2	1	no	—
420	C₂²	—	8	2	no	—
421	C₂²	—	8	2	no	—
422	C₂²	—	8	2	no	—
423	C₂²	—	16	4	no	—
424	C₂²	C₃₁×#464	1	2	yes	—
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	—	4	2	no	—
430	C₄	—	4	2	no	—
431	C₄	—	8	1	no	—
432	C₄	—	8	1	no	—
433	C₄	C₃₁×#464	1	2	yes	—
434	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
435	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
436	C₄	C₃₁×#576	1	1	yes	characteristic
437	C₄	C₃₁×#577	1	1	yes	characteristic
438	C₂	—	2	2	no	—
439	C₂	—	16	2	no	—
440	C₂	C₂×C₃₁×#464	1	1	yes	characteristic
441	C₂	C₃₁×#3062	1	1	yes	characteristic
442	C₂	C₃₁×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
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×#3249	C₂	semidirect
4. C₃₁×#2962	C₂	semidirect
5. C₃₁×#1702	C₂	semidirect
6. C₃₁×#1698	C₂	semidirect
7. C₃₁×#1682	C₂	semidirect
8. C₃₁×#1678	C₂	semidirect
9. C₃₁×#1565	C₂	semidirect
10. C₃₁×#1542	C₂	semidirect
21. C₃₁×#477	C₂²	semidirect
24. C₃₁×#462	C₂²	semidirect
29. C₃₁×#446	C₂²	semidirect
30. C₈×Q₈×C₃₁	C₂²	semidirect
39. C₃₁×#335	C₂²	semidirect
40. C₃₁×#334	C₂²	semidirect
41. C₃₁×#333	C₂²	semidirect
42. C₃₁×#333	C₂²	semidirect
43. C₃₁×#332	C₂²	semidirect
44. C₃₁×#332	C₂²	semidirect
155. #19242	C₃₁	direct (N × H)
197. #3249	C₂×C₃₁	semidirect
200. #2962	C₂×C₃₁	semidirect
201. #1702	C₂×C₃₁	semidirect
202. #1698	C₂×C₃₁	semidirect
203. #1682	C₂×C₃₁	semidirect
204. #1678	C₂×C₃₁	semidirect
205. #1565	C₂×C₃₁	semidirect
206. #1542	C₂×C₃₁	semidirect
237. #477	C₂²×C₃₁	semidirect
240. #462	C₂²×C₃₁	semidirect
245. #446	C₂²×C₃₁	semidirect
246. C₈×Q₈	C₂²×C₃₁	semidirect
255. #335	C₂²×C₃₁	semidirect
256. #334	C₂²×C₃₁	semidirect
257. #333	C₂²×C₃₁	semidirect
258. #333	C₂²×C₃₁	semidirect
259. #332	C₂²×C₃₁	semidirect
260. #332	C₂²×C₃₁	semidirect
323. C₃₁	#19242	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×#3249	2
C₃₁×#3208	2
C₃₁×#3138	2
C₃₁×#2962	2
C₃₁×#1702	2
C₃₁×#1698	2
C₃₁×#1682	2
C₃₁×#1678	2
C₃₁×#1565	2
C₃₁×#1542	2
C₂×C₃₁×#331	2
#19242	31

Subgroups — C31×#19242 #21476001

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19242 #21476001