Subgroups — C₃₁×#19273 #21476032

Order 7936.

608 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3160	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3116	C₂	1	2	yes	maximal
3	C₈×#41×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1676	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1675	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1674	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1673	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1672	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1671	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1568	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1554	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×#500	C₂²	1	1	yes	characteristic
16	C₃₁×#499	—	2	1	no	—
17	C₃₁×#499	—	2	1	no	—
18	C₃₁×#498	C₂²	1	2	yes	—
19	C₃₁×#485	—	2	2	no	—
20	C₃₁×#477	C₂²	1	2	yes	—
21	C₃₁×#476	C₂²	1	2	yes	—
22	C₃₁×#462	C₂²	1	2	yes	—
23	C₃₁×#461	C₂²	1	2	yes	—
24	C₃₁×#452	—	2	2	no	—
25	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
26	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#441	—	2	2	no	—
28	C₃₁×#440	—	2	2	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×#432	C₂²	1	2	yes	—
34	C₃₁×#431	—	2	1	no	—
35	C₃₁×#431	C₂²	1	1	yes	characteristic
36	C₃₁×#427	—	2	1	no	—
37	C₃₁×#427	—	2	1	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×#334	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	2	no	—
49	C₂×SD₁₆×C₃₁	—	2	2	no	—
50	C₂²×C₈×C₃₁	—	2	1	no	—
51	C₂²×C₈×C₃₁	—	2	1	no	—
52	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
54	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
55	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
56	C₃₁×#125	—	2	2	no	—
57	C₃₁×C₂²⋊Q₈	—	2	2	no	—
58	C₃₁×C₂²⋊Q₈	—	2	2	no	—
59	C₄×Q₈×C₃₁	—	2	2	no	—
60	C₄×Q₈×C₃₁	—	2	2	no	—
61	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
62	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
63	C₄×D₄×C₃₁	—	2	1	no	—
64	C₄×D₄×C₃₁	—	2	1	no	—
65	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
66	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₃₁×#117	—	2	1	no	—
68	C₃₁×#117	—	2	2	no	—
69	C₃₁×#117	D₄	1	1	yes	characteristic
70	C₂×C₄²×C₃₁	—	2	1	no	—
71	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
72	C₃₁×#107	—	2	1	no	—
73	C₃₁×#107	—	2	1	no	—
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×#106	—	2	2	no	—
79	C₃₁×#106	—	2	2	no	—
80	C₃₁×C₄⋊C₈	—	2	1	no	—
81	C₃₁×C₄⋊C₈	—	2	1	no	—
82	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
83	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
84	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
85	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
86	C₃₁×#103	—	2	2	no	—
87	C₃₁×#103	—	2	2	no	—
88	C₃₁×#103	—	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	2	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	—	2	2	no	—
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	4	no	—
116	SD₁₆×C₃₁	—	4	4	no	—
117	C₂×C₈×C₃₁	—	2	2	no	—
118	C₂×C₈×C₃₁	—	2	2	no	—
119	C₂×C₈×C₃₁	—	2	1	no	—
120	C₂×C₈×C₃₁	—	2	1	no	—
121	C₂×C₈×C₃₁	—	2	2	no	—
122	C₂×C₈×C₃₁	—	2	2	no	—
123	C₂×C₈×C₃₁	—	4	2	no	—
124	C₂×C₈×C₃₁	—	4	2	no	—
125	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
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	C₄⋊C₄×C₃₁	—	2	1	no	—
130	C₄⋊C₄×C₃₁	—	2	1	no	—
131	C₄⋊C₄×C₃₁	—	2	2	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	—	4	2	no	—
136	C₄⋊C₄×C₃₁	—	4	2	no	—
137	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
138	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
139	C₂²⋊C₄×C₃₁	—	2	1	no	—
140	C₂²⋊C₄×C₃₁	—	2	1	no	—
141	C₂²⋊C₄×C₃₁	—	2	1	no	—
142	C₂²⋊C₄×C₃₁	—	4	2	no	—
143	C₄²×C₃₁	—	2	1	no	—
144	C₄²×C₃₁	—	2	1	no	—
145	C₄²×C₃₁	—	2	1	no	—
146	C₄²×C₃₁	—	4	2	no	—
147	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
148	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
149	C₄²×C₃₁	#41	1	1	yes	characteristic
150	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
151	#19273	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
152	C₂³×C₃₁	—	2	1	no	—
153	C₂³×C₃₁	#121	1	1	yes	characteristic
154	Q₈×C₃₁	—	2	1	no	—
155	Q₈×C₃₁	—	4	2	no	—
156	Q₈×C₃₁	—	4	2	no	—
157	Q₈×C₃₁	—	8	2	no	—
158	Q₈×C₃₁	#135	1	2	yes	—
159	D₄×C₃₁	—	2	4	no	—
160	D₄×C₃₁	—	2	1	no	—
161	D₄×C₃₁	#135	1	2	yes	—
162	C₂×C₄×C₃₁	—	2	4	no	—
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	1	no	—
167	C₂×C₄×C₃₁	—	2	1	no	—
168	C₂×C₄×C₃₁	—	2	1	no	—
169	C₂×C₄×C₃₁	—	2	1	no	—
170	C₂×C₄×C₃₁	—	4	1	no	—
171	C₂×C₄×C₃₁	—	4	2	no	—
172	C₂×C₄×C₃₁	—	4	1	no	—
173	C₂×C₄×C₃₁	—	4	1	no	—
174	C₂×C₄×C₃₁	—	4	2	no	—
175	C₂×C₄×C₃₁	—	4	2	no	—
176	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
177	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
178	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
179	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
180	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
181	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
182	C₈×C₃₁	—	2	1	no	—
183	C₈×C₃₁	—	2	1	no	—
184	C₈×C₃₁	—	4	2	no	—
185	C₈×C₃₁	—	4	2	no	—
186	C₈×C₃₁	—	4	1	no	—
187	C₈×C₃₁	—	4	1	no	—
188	C₈×C₃₁	#137	1	2	yes	—
189	C₈×C₃₁	#137	1	2	yes	—
190	#3160	C₂×C₃₁	1	1	yes	characteristic, complemented
191	#3116	C₂×C₃₁	1	2	yes	—
192	C₈×#41	C₂×C₃₁	1	1	yes	characteristic
193	#1676	C₂×C₃₁	1	1	yes	characteristic, complemented
194	#1675	C₂×C₃₁	1	1	yes	characteristic, complemented
195	#1674	C₂×C₃₁	1	1	yes	characteristic, complemented
196	#1673	C₂×C₃₁	1	1	yes	characteristic, complemented
197	#1672	C₂×C₃₁	1	2	yes	complemented
198	#1671	C₂×C₃₁	1	2	yes	complemented
199	#1568	C₂×C₃₁	1	1	yes	characteristic, complemented
200	#1554	C₂×C₃₁	1	2	yes	complemented
201	C₂²×C₃₁	—	2	1	no	—
202	C₂²×C₃₁	—	2	1	no	—
203	C₂²×C₃₁	—	2	2	no	—
204	C₂²×C₃₁	—	2	1	no	—
205	C₂²×C₃₁	—	4	1	no	—
206	C₂²×C₃₁	—	4	1	no	—
207	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
208	C₄×C₃₁	—	2	1	no	—
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	C₂×#135	1	1	yes	characteristic
218	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
219	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
220	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
221	#533	C₂²×C₃₁	1	2	yes	—
222	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
223	#500	C₂²×C₃₁	1	1	yes	characteristic
224	#500	C₂²×C₃₁	1	1	yes	characteristic
225	#499	—	2	1	no	—
226	#499	—	2	1	no	—
227	#498	C₂²×C₃₁	1	2	yes	—
228	#485	—	2	2	no	—
229	#477	C₂²×C₃₁	1	2	yes	—
230	#476	C₂²×C₃₁	1	2	yes	—
231	#462	C₂²×C₃₁	1	2	yes	—
232	#461	C₂²×C₃₁	1	2	yes	—
233	#452	—	2	2	no	—
234	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
235	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
236	#441	—	2	2	no	—
237	#440	—	2	2	no	—
238	C₈×D₄	—	2	1	no	—
239	C₈×D₄	—	2	1	no	—
240	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
241	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
242	#432	C₂²×C₃₁	1	2	yes	—
243	#431	—	2	1	no	—
244	#431	C₂²×C₃₁	1	1	yes	characteristic
245	#427	—	2	1	no	—
246	#427	—	2	1	no	—
247	C₂×C₄×C₈	—	2	1	no	—
248	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
249	#335	C₂²×C₃₁	1	1	yes	characteristic
250	#335	C₂²×C₃₁	1	1	yes	characteristic
251	#334	C₂²×C₃₁	1	2	yes	—
252	#333	C₂²×C₃₁	1	2	yes	—
253	#333	C₂²×C₃₁	1	2	yes	—
254	#332	C₂²×C₃₁	1	2	yes	—
255	#332	C₂²×C₃₁	1	2	yes	—
256	C₂×C₃₁	—	4	1	no	—
257	C₂×C₃₁	—	4	2	no	—
258	C₂×C₃₁	#3039	1	1	yes	characteristic
259	C₂×C₃₁	#3050	1	1	yes	characteristic
260	C₂×C₃₁	#3067	1	1	yes	characteristic
261	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
262	C₂×Q₁₆	—	2	2	no	—
263	C₂×SD₁₆	—	2	2	no	—
264	C₂²×C₈	—	2	1	no	—
265	C₂²×C₈	—	2	1	no	—
266	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
267	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
268	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
269	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
270	#125	—	2	2	no	—
271	C₂²⋊Q₈	—	2	2	no	—
272	C₂²⋊Q₈	—	2	2	no	—
273	C₄×Q₈	—	2	2	no	—
274	C₄×Q₈	—	2	2	no	—
275	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
276	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
277	C₄×D₄	—	2	1	no	—
278	C₄×D₄	—	2	1	no	—
279	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
280	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
281	#117	—	2	1	no	—
282	#117	—	2	2	no	—
283	#117	D₄×C₃₁	1	1	yes	characteristic
284	C₂×C₄²	—	2	1	no	—
285	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
286	#107	—	2	1	no	—
287	#107	—	2	1	no	—
288	#107	—	2	1	no	—
289	#107	—	2	1	no	—
290	#107	—	2	2	no	—
291	#106	—	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	C₂³×C₃₁	1	1	yes	characteristic
297	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
298	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
299	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
300	#103	—	2	2	no	—
301	#103	—	2	2	no	—
302	#103	—	2	2	no	—
303	#102	—	2	2	no	—
304	C₂²⋊C₈	—	2	1	no	—
305	C₂²⋊C₈	—	2	2	no	—
306	C₂²⋊C₈	—	2	1	no	—
307	C₂²⋊C₈	—	2	1	no	—
308	#97	—	2	2	no	—
309	C₄×C₈	—	2	1	no	—
310	C₄×C₈	—	2	1	no	—
311	C₄×C₈	—	2	1	no	—
312	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
313	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
314	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
315	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
316	C₃₁	#19273	1	1	yes	characteristic, direct product factor, Sylow
317	#41	—	2	2	no	—
318	#41	—	2	2	no	—
319	C₂×Q₈	—	2	2	no	—
320	C₂×Q₈	—	2	2	no	—
321	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
322	C₂×D₄	—	2	1	no	—
323	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
324	C₂²×C₄	—	2	1	no	—
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	—	2	1	no	—
327	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
328	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
329	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
330	Q₁₆	—	4	4	no	—
331	SD₁₆	—	4	4	no	—
332	C₂×C₈	—	2	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	2	no	—
339	C₂×C₈	—	4	2	no	—
340	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
341	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
342	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
343	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
344	C₄⋊C₄	—	2	1	no	—
345	C₄⋊C₄	—	2	1	no	—
346	C₄⋊C₄	—	2	2	no	—
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	—	2	2	no	—
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	—	4	2	no	—
351	C₄⋊C₄	—	4	2	no	—
352	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
353	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
354	C₂²⋊C₄	—	2	1	no	—
355	C₂²⋊C₄	—	2	1	no	—
356	C₂²⋊C₄	—	2	1	no	—
357	C₂²⋊C₄	—	4	2	no	—
358	C₄²	—	2	1	no	—
359	C₄²	—	2	1	no	—
360	C₄²	—	2	1	no	—
361	C₄²	—	4	2	no	—
362	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
363	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
364	C₄²	#41×C₃₁	1	1	yes	characteristic
365	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
366	C₂³	—	2	1	no	—
367	C₂³	C₃₁×#121	1	1	yes	characteristic
368	Q₈	—	2	1	no	—
369	Q₈	—	4	2	no	—
370	Q₈	—	4	2	no	—
371	Q₈	—	8	2	no	—
372	Q₈	C₃₁×#135	1	2	yes	—
373	D₄	—	2	4	no	—
374	D₄	—	2	1	no	—
375	D₄	C₃₁×#135	1	2	yes	—
376	C₂×C₄	—	2	4	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	2	1	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	C₂²×D₄×C₃₁	1	1	yes	characteristic
394	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
395	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
396	C₈	—	2	1	no	—
397	C₈	—	2	1	no	—
398	C₈	—	4	2	no	—
399	C₈	—	4	2	no	—
400	C₈	—	4	1	no	—
401	C₈	—	4	1	no	—
402	C₈	C₃₁×#137	1	2	yes	—
403	C₈	C₃₁×#137	1	2	yes	—
404	C₂²	—	2	1	no	—
405	C₂²	—	2	1	no	—
406	C₂²	—	2	2	no	—
407	C₂²	—	2	1	no	—
408	C₂²	—	4	1	no	—
409	C₂²	—	4	1	no	—
410	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
411	C₄	—	2	1	no	—
412	C₄	—	2	1	no	—
413	C₄	—	2	2	no	—
414	C₄	—	2	1	no	—
415	C₄	—	2	1	no	—
416	C₄	—	4	1	no	—
417	C₄	—	4	1	no	—
418	C₄	—	8	2	no	—
419	C₄	—	8	2	no	—
420	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
421	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
422	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
423	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
424	C₂	—	4	1	no	—
425	C₂	—	4	2	no	—
426	C₂	C₃₁×#3039	1	1	yes	characteristic
427	C₂	C₃₁×#3050	1	1	yes	characteristic
428	C₂	C₃₁×#3067	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₂×#41×C₃₁	D₄
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×#3160	C₂	semidirect
4. C₃₁×#1676	C₂	semidirect
5. C₃₁×#1675	C₂	semidirect
6. C₃₁×#1674	C₂	semidirect
7. C₃₁×#1673	C₂	semidirect
8. C₃₁×#1672	C₂	semidirect
9. C₃₁×#1671	C₂	semidirect
10. C₃₁×#1568	C₂	semidirect
11. C₃₁×#1554	C₂	semidirect
151. #19273	C₃₁	direct (N × H)
190. #3160	C₂×C₃₁	semidirect
193. #1676	C₂×C₃₁	semidirect
194. #1675	C₂×C₃₁	semidirect
195. #1674	C₂×C₃₁	semidirect
196. #1673	C₂×C₃₁	semidirect
197. #1672	C₂×C₃₁	semidirect
198. #1671	C₂×C₃₁	semidirect
199. #1568	C₂×C₃₁	semidirect
200. #1554	C₂×C₃₁	semidirect
316. C₃₁	#19273	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×#3160	2
C₃₁×#3116	2
C₈×#41×C₃₁	2
C₃₁×#1676	2
C₃₁×#1675	2
C₃₁×#1674	2
C₃₁×#1673	2
C₃₁×#1672	2
C₃₁×#1671	2
C₃₁×#1568	2
C₃₁×#1554	2
#19273	31

Subgroups — C31×#19273 #21476032

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19273 #21476032