Subgroups — C₃₁×#19235 #21475994

Order 7936.

640 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3246	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3226	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3139	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2935	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1703	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1699	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1683	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1679	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×#332	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#131	—	2	1	no	—
13	C₃₁×#533	C₂²	1	1	yes	characteristic
14	C₃₁×#532	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₃₁×#500	—	2	1	no	—
17	C₃₁×#491	—	2	2	no	—
18	C₃₁×#487	C₂²	1	2	yes	—
19	C₃₁×#487	C₂²	1	2	yes	—
20	C₃₁×#486	—	2	1	no	—
21	C₃₁×#480	—	2	2	no	—
22	C₃₁×#476	C₂²	1	2	yes	—
23	C₃₁×#475	—	2	1	no	—
24	C₃₁×#474	C₂²	1	2	yes	—
25	C₃₁×#474	C₂²	1	2	yes	—
26	C₃₁×#465	—	2	2	no	—
27	C₃₁×#461	C₂²	1	2	yes	—
28	C₃₁×#460	—	2	1	no	—
29	C₃₁×#436	C₂²	1	2	yes	—
30	C₃₁×#435	—	2	2	no	—
31	C₈×D₄×C₃₁	C₂²	1	2	yes	—
32	C₃₁×#431	C₂²	1	1	yes	characteristic
33	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
34	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₃₁×#419	—	2	1	no	—
36	C₃₁×#419	—	2	1	no	—
37	C₃₁×#408	—	2	1	no	—
38	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
39	C₃₁×#335	C₂²	1	2	yes	—
40	C₃₁×#334	C₂²	1	2	yes	—
41	C₃₁×#333	C₂²	1	1	yes	characteristic
42	C₃₁×#333	C₂²	1	1	yes	characteristic
43	C₃₁×#332	C₂²	1	2	yes	—
44	C₃₁×#332	C₂²	1	2	yes	—
45	C₃₁×#332	C₂²	1	1	yes	characteristic
46	C₃₁×#332	C₂²	1	1	yes	characteristic
47	C₂×#41×C₃₁	—	2	1	no	—
48	C₂²×Q₈×C₃₁	—	2	1	no	—
49	C₂×SD₁₆×C₃₁	—	2	2	no	—
50	C₃₁×#131	—	4	4	no	—
51	C₂×M₁₆×C₃₁	—	2	2	no	—
52	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
53	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₃₁	—	2	1	no	—
55	C₂²×C₈×C₃₁	—	2	1	no	—
56	C₂²×C₈×C₃₁	—	2	2	no	—
57	C₃₁×C₄⋊Q₈	—	2	1	no	—
58	C₃₁×C₄⋊Q₈	—	2	1	no	—
59	C₃₁×C₄⋊Q₈	—	2	1	no	—
60	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
61	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
62	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
63	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
64	C₃₁×#127	D₄	1	1	yes	characteristic
65	C₃₁×#125	—	2	1	no	—
66	C₃₁×#124	—	2	1	no	—
67	C₃₁×C₂²⋊Q₈	—	2	1	no	—
68	C₃₁×C₂²⋊Q₈	—	2	1	no	—
69	C₃₁×#121	—	2	1	no	—
70	C₃₁×#121	D₄	1	2	yes	—
71	C₄×Q₈×C₃₁	—	2	1	no	—
72	C₄×Q₈×C₃₁	—	2	1	no	—
73	C₄×D₄×C₃₁	—	2	1	no	—
74	C₄×D₄×C₃₁	C₂³	1	2	yes	—
75	C₃₁×#117	—	2	1	no	—
76	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
77	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
78	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
79	C₃₁×#107	—	2	2	no	—
80	C₃₁×#107	—	2	2	no	—
81	C₃₁×#106	—	2	2	no	—
82	C₃₁×C₄⋊C₈	—	2	1	no	—
83	C₃₁×C₄⋊C₈	—	2	1	no	—
84	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
85	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
86	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
87	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
88	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
89	C₃₁×#103	—	2	2	no	—
90	C₃₁×#103	—	2	2	no	—
91	C₃₁×#103	—	2	2	no	—
92	C₃₁×#103	—	2	2	no	—
93	C₃₁×#103	—	2	2	no	—
94	C₃₁×#102	—	2	2	no	—
95	C₃₁×#102	—	2	2	no	—
96	C₃₁×#102	—	2	2	no	—
97	C₃₁×C₂²⋊C₈	—	2	1	no	—
98	C₃₁×C₂²⋊C₈	—	2	2	no	—
99	C₃₁×C₂²⋊C₈	—	2	2	no	—
100	C₃₁×#97	D₄	1	2	yes	—
101	C₄×C₈×C₃₁	—	2	1	no	—
102	C₄×C₈×C₃₁	C₂³	1	2	yes	—
103	#41×C₃₁	—	4	2	no	—
104	#41×C₃₁	—	4	2	no	—
105	C₂×Q₈×C₃₁	—	2	1	no	—
106	C₂×Q₈×C₃₁	—	2	2	no	—
107	C₂×Q₈×C₃₁	—	2	1	no	—
108	C₂×Q₈×C₃₁	—	2	1	no	—
109	C₂×Q₈×C₃₁	—	4	1	no	—
110	C₂×Q₈×C₃₁	—	4	2	no	—
111	C₂×Q₈×C₃₁	—	4	2	no	—
112	C₂×D₄×C₃₁	—	2	2	no	—
113	C₂×D₄×C₃₁	—	2	1	no	—
114	C₂×D₄×C₃₁	—	2	1	no	—
115	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
116	C₂²×C₄×C₃₁	—	2	2	no	—
117	C₂²×C₄×C₃₁	—	4	1	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	SD₁₆×C₃₁	—	8	2	no	—
122	M₁₆×C₃₁	—	2	2	no	—
123	M₁₆×C₃₁	—	2	2	no	—
124	M₁₆×C₃₁	—	4	4	no	—
125	C₂×C₈×C₃₁	—	2	1	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	2	no	—
130	C₂×C₈×C₃₁	—	4	4	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	2	no	—
138	C₄⋊C₄×C₃₁	—	2	1	no	—
139	C₄⋊C₄×C₃₁	—	2	1	no	—
140	C₄⋊C₄×C₃₁	—	2	2	no	—
141	C₄⋊C₄×C₃₁	—	2	1	no	—
142	C₄⋊C₄×C₃₁	—	4	1	no	—
143	C₄⋊C₄×C₃₁	—	4	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	—	2	2	no	—
149	C₂²⋊C₄×C₃₁	—	4	1	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	#19235	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂³×C₃₁	—	2	2	no	—
157	C₂³×C₃₁	#121	1	1	yes	characteristic
158	Q₈×C₃₁	—	4	2	no	—
159	Q₈×C₃₁	—	4	2	no	—
160	Q₈×C₃₁	—	4	2	no	—
161	Q₈×C₃₁	—	4	1	no	—
162	Q₈×C₃₁	—	4	1	no	—
163	Q₈×C₃₁	—	8	1	no	—
164	Q₈×C₃₁	—	8	2	no	—
165	D₄×C₃₁	—	2	2	no	—
166	D₄×C₃₁	—	2	2	no	—
167	D₄×C₃₁	—	4	4	no	—
168	D₄×C₃₁	—	4	2	no	—
169	D₄×C₃₁	—	8	1	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	—	4	1	no	—
174	C₂×C₄×C₃₁	—	4	2	no	—
175	C₂×C₄×C₃₁	—	4	2	no	—
176	C₂×C₄×C₃₁	—	4	1	no	—
177	C₂×C₄×C₃₁	—	4	1	no	—
178	C₂×C₄×C₃₁	—	4	2	no	—
179	C₂×C₄×C₃₁	—	8	2	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	#135	1	2	yes	—
184	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
185	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
186	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
187	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
188	C₈×C₃₁	—	2	2	no	—
189	C₈×C₃₁	—	2	2	no	—
190	C₈×C₃₁	—	4	1	no	—
191	C₈×C₃₁	—	4	1	no	—
192	C₈×C₃₁	—	4	1	no	—
193	C₈×C₃₁	—	4	2	no	—
194	C₈×C₃₁	—	4	1	no	—
195	#3246	C₂×C₃₁	1	1	yes	characteristic
196	#3226	C₂×C₃₁	1	1	yes	characteristic
197	#3139	C₂×C₃₁	1	1	yes	characteristic, complemented
198	#2935	C₂×C₃₁	1	1	yes	characteristic
199	#1703	C₂×C₃₁	1	2	yes	complemented
200	#1699	C₂×C₃₁	1	2	yes	complemented
201	#1683	C₂×C₃₁	1	2	yes	complemented
202	#1679	C₂×C₃₁	1	2	yes	complemented
203	#1565	C₂×C₃₁	1	1	yes	characteristic, complemented
204	#1542	C₂×C₃₁	1	1	yes	characteristic, complemented
205	C₂×#332	C₂×C₃₁	1	1	yes	characteristic, complemented
206	C₂²×C₃₁	—	2	1	no	—
207	C₂²×C₃₁	—	2	1	no	—
208	C₂²×C₃₁	—	4	2	no	—
209	C₂²×C₃₁	—	4	2	no	—
210	C₂²×C₃₁	—	4	2	no	—
211	C₂²×C₃₁	#463	1	2	yes	—
212	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
213	C₄×C₃₁	—	2	1	no	—
214	C₄×C₃₁	—	2	1	no	—
215	C₄×C₃₁	—	2	1	no	—
216	C₄×C₃₁	—	4	2	no	—
217	C₄×C₃₁	—	8	2	no	—
218	C₄×C₃₁	—	8	1	no	—
219	C₄×C₃₁	—	8	1	no	—
220	C₄×C₃₁	#463	1	2	yes	—
221	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
222	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
223	C₄×C₃₁	#577	1	1	yes	characteristic
224	C₄×C₃₁	#578	1	1	yes	characteristic
225	C₂×#131	—	2	1	no	—
226	#533	C₂²×C₃₁	1	1	yes	characteristic
227	#532	C₂²×C₃₁	1	1	yes	characteristic
228	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
229	#500	—	2	1	no	—
230	#491	—	2	2	no	—
231	#487	C₂²×C₃₁	1	2	yes	—
232	#487	C₂²×C₃₁	1	2	yes	—
233	#486	—	2	1	no	—
234	#480	—	2	2	no	—
235	#476	C₂²×C₃₁	1	2	yes	—
236	#475	—	2	1	no	—
237	#474	C₂²×C₃₁	1	2	yes	—
238	#474	C₂²×C₃₁	1	2	yes	—
239	#465	—	2	2	no	—
240	#461	C₂²×C₃₁	1	2	yes	—
241	#460	—	2	1	no	—
242	#436	C₂²×C₃₁	1	2	yes	—
243	#435	—	2	2	no	—
244	C₈×D₄	C₂²×C₃₁	1	2	yes	—
245	#431	C₂²×C₃₁	1	1	yes	characteristic
246	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
247	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
248	#419	—	2	1	no	—
249	#419	—	2	1	no	—
250	#408	—	2	1	no	—
251	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
252	#335	C₂²×C₃₁	1	2	yes	—
253	#334	C₂²×C₃₁	1	2	yes	—
254	#333	C₂²×C₃₁	1	1	yes	characteristic
255	#333	C₂²×C₃₁	1	1	yes	characteristic
256	#332	C₂²×C₃₁	1	2	yes	—
257	#332	C₂²×C₃₁	1	2	yes	—
258	#332	C₂²×C₃₁	1	1	yes	characteristic
259	#332	C₂²×C₃₁	1	1	yes	characteristic
260	C₂×C₃₁	—	2	2	no	—
261	C₂×C₃₁	—	8	2	no	—
262	C₂×C₃₁	C₂×#463	1	1	yes	characteristic
263	C₂×C₃₁	#3063	1	1	yes	characteristic
264	C₂×C₃₁	#3069	1	1	yes	characteristic
265	C₂×#41	—	2	1	no	—
266	C₂²×Q₈	—	2	1	no	—
267	C₂×SD₁₆	—	2	2	no	—
268	#131	—	4	4	no	—
269	C₂×M₁₆	—	2	2	no	—
270	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
271	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
272	C₂²×C₈	—	2	1	no	—
273	C₂²×C₈	—	2	1	no	—
274	C₂²×C₈	—	2	2	no	—
275	C₄⋊Q₈	—	2	1	no	—
276	C₄⋊Q₈	—	2	1	no	—
277	C₄⋊Q₈	—	2	1	no	—
278	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
279	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
280	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
281	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
282	#127	D₄×C₃₁	1	1	yes	characteristic
283	#125	—	2	1	no	—
284	#124	—	2	1	no	—
285	C₂²⋊Q₈	—	2	1	no	—
286	C₂²⋊Q₈	—	2	1	no	—
287	#121	—	2	1	no	—
288	#121	D₄×C₃₁	1	2	yes	—
289	C₄×Q₈	—	2	1	no	—
290	C₄×Q₈	—	2	1	no	—
291	C₄×D₄	—	2	1	no	—
292	C₄×D₄	C₂³×C₃₁	1	2	yes	—
293	#117	—	2	1	no	—
294	C₂×C₄⋊C₄	—	2	1	no	—
295	C₂×C₄⋊C₄	—	2	1	no	—
296	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
297	#107	—	2	2	no	—
298	#107	—	2	2	no	—
299	#106	—	2	2	no	—
300	C₄⋊C₈	—	2	1	no	—
301	C₄⋊C₈	—	2	1	no	—
302	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
303	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
304	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
305	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
306	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
307	#103	—	2	2	no	—
308	#103	—	2	2	no	—
309	#103	—	2	2	no	—
310	#103	—	2	2	no	—
311	#103	—	2	2	no	—
312	#102	—	2	2	no	—
313	#102	—	2	2	no	—
314	#102	—	2	2	no	—
315	C₂²⋊C₈	—	2	1	no	—
316	C₂²⋊C₈	—	2	2	no	—
317	C₂²⋊C₈	—	2	2	no	—
318	#97	D₄×C₃₁	1	2	yes	—
319	C₄×C₈	—	2	1	no	—
320	C₄×C₈	C₂³×C₃₁	1	2	yes	—
321	C₃₁	#19235	1	1	yes	characteristic, direct product factor, Sylow
322	#41	—	4	2	no	—
323	#41	—	4	2	no	—
324	C₂×Q₈	—	2	1	no	—
325	C₂×Q₈	—	2	2	no	—
326	C₂×Q₈	—	2	1	no	—
327	C₂×Q₈	—	2	1	no	—
328	C₂×Q₈	—	4	1	no	—
329	C₂×Q₈	—	4	2	no	—
330	C₂×Q₈	—	4	2	no	—
331	C₂×D₄	—	2	2	no	—
332	C₂×D₄	—	2	1	no	—
333	C₂×D₄	—	2	1	no	—
334	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
335	C₂²×C₄	—	2	2	no	—
336	C₂²×C₄	—	4	1	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	—	8	2	no	—
341	M₁₆	—	2	2	no	—
342	M₁₆	—	2	2	no	—
343	M₁₆	—	4	4	no	—
344	C₂×C₈	—	2	1	no	—
345	C₂×C₈	—	2	1	no	—
346	C₂×C₈	—	2	1	no	—
347	C₂×C₈	—	2	1	no	—
348	C₂×C₈	—	2	2	no	—
349	C₂×C₈	—	4	4	no	—
350	C₂×C₈	—	4	2	no	—
351	C₂×C₈	—	8	1	no	—
352	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
353	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
354	C₄⋊C₄	—	2	1	no	—
355	C₄⋊C₄	—	2	1	no	—
356	C₄⋊C₄	—	2	2	no	—
357	C₄⋊C₄	—	2	1	no	—
358	C₄⋊C₄	—	2	1	no	—
359	C₄⋊C₄	—	2	2	no	—
360	C₄⋊C₄	—	2	1	no	—
361	C₄⋊C₄	—	4	1	no	—
362	C₄⋊C₄	—	4	1	no	—
363	C₄⋊C₄	—	4	1	no	—
364	C₄⋊C₄	—	4	1	no	—
365	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
366	C₂²⋊C₄	—	2	2	no	—
367	C₂²⋊C₄	—	2	2	no	—
368	C₂²⋊C₄	—	4	1	no	—
369	C₄²	—	4	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	2	no	—
375	C₂³	C₃₁×#121	1	1	yes	characteristic
376	Q₈	—	4	2	no	—
377	Q₈	—	4	2	no	—
378	Q₈	—	4	2	no	—
379	Q₈	—	4	1	no	—
380	Q₈	—	4	1	no	—
381	Q₈	—	8	1	no	—
382	Q₈	—	8	2	no	—
383	D₄	—	2	2	no	—
384	D₄	—	2	2	no	—
385	D₄	—	4	4	no	—
386	D₄	—	4	2	no	—
387	D₄	—	8	1	no	—
388	C₂×C₄	—	2	1	no	—
389	C₂×C₄	—	2	1	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	4	1	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	—	4	2	no	—
394	C₂×C₄	—	4	1	no	—
395	C₂×C₄	—	4	1	no	—
396	C₂×C₄	—	4	2	no	—
397	C₂×C₄	—	8	2	no	—
398	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
399	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
400	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
401	C₂×C₄	C₃₁×#135	1	2	yes	—
402	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
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	2	no	—
407	C₈	—	2	2	no	—
408	C₈	—	4	1	no	—
409	C₈	—	4	1	no	—
410	C₈	—	4	1	no	—
411	C₈	—	4	2	no	—
412	C₈	—	4	1	no	—
413	C₂²	—	2	1	no	—
414	C₂²	—	2	1	no	—
415	C₂²	—	4	2	no	—
416	C₂²	—	4	2	no	—
417	C₂²	—	4	2	no	—
418	C₂²	C₃₁×#463	1	2	yes	—
419	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
420	C₄	—	2	1	no	—
421	C₄	—	2	1	no	—
422	C₄	—	2	1	no	—
423	C₄	—	4	2	no	—
424	C₄	—	8	2	no	—
425	C₄	—	8	1	no	—
426	C₄	—	8	1	no	—
427	C₄	C₃₁×#463	1	2	yes	—
428	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
429	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
430	C₄	C₃₁×#577	1	1	yes	characteristic
431	C₄	C₃₁×#578	1	1	yes	characteristic
432	C₂	—	2	2	no	—
433	C₂	—	8	2	no	—
434	C₂	C₂×C₃₁×#463	1	1	yes	characteristic
435	C₂	C₃₁×#3063	1	1	yes	characteristic
436	C₂	C₃₁×#3069	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#532	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
3. C₃₁×#3139	C₂	semidirect
5. C₃₁×#1703	C₂	semidirect
6. C₃₁×#1699	C₂	semidirect
7. C₃₁×#1683	C₂	semidirect
8. C₃₁×#1679	C₂	semidirect
9. C₃₁×#1565	C₂	semidirect
10. C₃₁×#1542	C₂	semidirect
11. C₂×C₃₁×#332	C₂	semidirect
155. #19235	C₃₁	direct (N × H)
197. #3139	C₂×C₃₁	semidirect
199. #1703	C₂×C₃₁	semidirect
200. #1699	C₂×C₃₁	semidirect
201. #1683	C₂×C₃₁	semidirect
202. #1679	C₂×C₃₁	semidirect
203. #1565	C₂×C₃₁	semidirect
204. #1542	C₂×C₃₁	semidirect
205. C₂×#332	C₂×C₃₁	semidirect
321. C₃₁	#19235	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×#3246	2
C₃₁×#3226	2
C₃₁×#3139	2
C₃₁×#2935	2
C₃₁×#1703	2
C₃₁×#1699	2
C₃₁×#1683	2
C₃₁×#1679	2
C₃₁×#1565	2
C₃₁×#1542	2
C₂×C₃₁×#332	2
#19235	31

Subgroups — C31×#19235 #21475994

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19235 #21475994