Subgroups — C₃₁×#19269 #21476028

Order 7936.

716 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3147	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3115	C₂	1	2	yes	maximal, complemented
3	C₈×#41×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1670	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1669	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1668	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1667	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1666	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1665	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1568	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1553	C₂	1	2	yes	maximal, complemented
12	C₂×C₃₁×#135	—	2	1	no	—
13	C₂×C₃₁×#135	—	2	1	no	—
14	C₃₁×#532	C₂²	1	2	yes	—
15	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#495	—	2	1	no	—
17	C₃₁×#495	—	2	1	no	—
18	C₃₁×#494	C₂²	1	1	yes	characteristic
19	C₃₁×#493	C₂²	1	1	yes	characteristic
20	C₃₁×#492	C₂²	1	2	yes	—
21	C₃₁×#482	—	2	2	no	—
22	C₃₁×#475	C₂²	1	2	yes	—
23	C₃₁×#474	C₂²	1	2	yes	—
24	C₃₁×#460	C₂²	1	2	yes	—
25	C₃₁×#459	C₂²	1	2	yes	—
26	C₃₁×#449	—	2	2	no	—
27	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₃₁×#442	—	2	2	no	—
30	C₃₁×#440	—	2	2	no	—
31	C₈×D₄×C₃₁	—	2	1	no	—
32	C₈×D₄×C₃₁	—	2	1	no	—
33	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
34	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₃₁×#432	C₂²	1	2	yes	—
36	C₃₁×#431	—	2	1	no	—
37	C₃₁×#431	C₂²	1	1	yes	characteristic
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×#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×#41×C₃₁	D₄	1	1	yes	characteristic
49	C₃₁×#135	—	4	4	no	—
50	C₃₁×#135	—	4	4	no	—
51	C₂×Q₁₆×C₃₁	—	2	1	no	—
52	C₂×Q₁₆×C₃₁	—	2	1	no	—
53	C₂×SD₁₆×C₃₁	—	2	2	no	—
54	C₂×SD₁₆×C₃₁	—	2	2	no	—
55	C₂×SD₁₆×C₃₁	—	2	2	no	—
56	C₂×D₈×C₃₁	—	2	2	no	—
57	C₂×D₈×C₃₁	—	2	1	no	—
58	C₂×D₈×C₃₁	—	2	1	no	—
59	C₂²×C₈×C₃₁	—	2	1	no	—
60	C₂²×C₈×C₃₁	—	2	1	no	—
61	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
62	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
63	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
64	C₃₁×#127	C₂³	1	2	yes	—
65	C₃₁×#124	—	2	2	no	—
66	C₃₁×#121	—	2	2	no	—
67	C₃₁×#121	—	2	2	no	—
68	C₄×Q₈×C₃₁	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	1	no	—
72	C₄×D₄×C₃₁	—	2	2	no	—
73	C₄×D₄×C₃₁	—	2	2	no	—
74	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
75	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
76	C₃₁×#117	—	2	1	no	—
77	C₃₁×#117	D₄	1	1	yes	characteristic
78	C₂×C₄²×C₃₁	—	2	1	no	—
79	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
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	2	yes	—
88	C₃₁×#103	—	2	2	no	—
89	C₃₁×#102	—	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	2	no	—
94	C₃₁×C₂²⋊C₈	—	2	1	no	—
95	C₃₁×C₂²⋊C₈	—	2	1	no	—
96	C₃₁×#97	—	2	2	no	—
97	C₄×C₈×C₃₁	—	2	1	no	—
98	C₄×C₈×C₃₁	—	2	1	no	—
99	C₄×C₈×C₃₁	—	2	1	no	—
100	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
101	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
102	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
103	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
104	#41×C₃₁	—	2	2	no	—
105	#41×C₃₁	—	2	2	no	—
106	#41×C₃₁	—	4	4	no	—
107	#41×C₃₁	—	4	4	no	—
108	C₂×Q₈×C₃₁	—	2	2	no	—
109	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
110	C₂×D₄×C₃₁	—	2	1	no	—
111	C₂×D₄×C₃₁	—	2	2	no	—
112	C₂×D₄×C₃₁	—	2	2	no	—
113	C₂×D₄×C₃₁	—	2	2	no	—
114	C₂×D₄×C₃₁	—	4	2	no	—
115	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₃₁	—	2	1	no	—
117	C₂²×C₄×C₃₁	—	2	1	no	—
118	C₂²×C₄×C₃₁	—	2	1	no	—
119	C₂²×C₄×C₃₁	—	4	2	no	—
120	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
123	Q₁₆×C₃₁	—	4	2	no	—
124	Q₁₆×C₃₁	—	4	2	no	—
125	SD₁₆×C₃₁	—	4	4	no	—
126	SD₁₆×C₃₁	—	4	4	no	—
127	SD₁₆×C₃₁	—	4	4	no	—
128	D₈×C₃₁	—	4	2	no	—
129	D₈×C₃₁	—	4	4	no	—
130	D₈×C₃₁	—	4	2	no	—
131	C₂×C₈×C₃₁	—	2	2	no	—
132	C₂×C₈×C₃₁	—	2	2	no	—
133	C₂×C₈×C₃₁	—	2	1	no	—
134	C₂×C₈×C₃₁	—	2	1	no	—
135	C₂×C₈×C₃₁	—	2	2	no	—
136	C₂×C₈×C₃₁	—	2	2	no	—
137	C₂×C₈×C₃₁	—	4	2	no	—
138	C₂×C₈×C₃₁	—	4	2	no	—
139	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
140	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
141	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
142	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
143	C₄⋊C₄×C₃₁	—	2	1	no	—
144	C₄⋊C₄×C₃₁	—	2	1	no	—
145	C₄⋊C₄×C₃₁	—	2	2	no	—
146	C₄⋊C₄×C₃₁	—	2	2	no	—
147	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
148	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
149	C₂²⋊C₄×C₃₁	—	2	1	no	—
150	C₂²⋊C₄×C₃₁	—	2	1	no	—
151	C₂²⋊C₄×C₃₁	—	2	1	no	—
152	C₂²⋊C₄×C₃₁	—	4	2	no	—
153	C₂²⋊C₄×C₃₁	—	4	2	no	—
154	C₄²×C₃₁	—	2	1	no	—
155	C₄²×C₃₁	—	2	1	no	—
156	C₄²×C₃₁	—	2	1	no	—
157	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
158	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
159	C₄²×C₃₁	#41	1	1	yes	characteristic
160	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
161	#19269	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
162	C₂³×C₃₁	—	2	1	no	—
163	C₂³×C₃₁	—	4	2	no	—
164	C₂³×C₃₁	#121	1	1	yes	characteristic
165	Q₈×C₃₁	—	2	1	no	—
166	Q₈×C₃₁	—	4	2	no	—
167	Q₈×C₃₁	—	4	2	no	—
168	Q₈×C₃₁	#135	1	2	yes	complemented
169	D₄×C₃₁	—	2	4	no	—
170	D₄×C₃₁	—	2	1	no	—
171	D₄×C₃₁	—	4	2	no	—
172	D₄×C₃₁	—	4	2	no	—
173	D₄×C₃₁	—	4	2	no	—
174	D₄×C₃₁	—	4	2	no	—
175	D₄×C₃₁	—	8	2	no	—
176	D₄×C₃₁	—	8	4	no	—
177	D₄×C₃₁	#135	1	2	yes	complemented
178	C₂×C₄×C₃₁	—	2	4	no	—
179	C₂×C₄×C₃₁	—	2	1	no	—
180	C₂×C₄×C₃₁	—	2	1	no	—
181	C₂×C₄×C₃₁	—	2	1	no	—
182	C₂×C₄×C₃₁	—	2	1	no	—
183	C₂×C₄×C₃₁	—	2	1	no	—
184	C₂×C₄×C₃₁	—	2	1	no	—
185	C₂×C₄×C₃₁	—	2	1	no	—
186	C₂×C₄×C₃₁	—	4	1	no	—
187	C₂×C₄×C₃₁	—	4	2	no	—
188	C₂×C₄×C₃₁	—	4	1	no	—
189	C₂×C₄×C₃₁	—	4	1	no	—
190	C₂×C₄×C₃₁	—	4	2	no	—
191	C₂×C₄×C₃₁	—	8	4	no	—
192	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
193	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
194	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
195	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
196	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
197	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
198	C₈×C₃₁	—	2	1	no	—
199	C₈×C₃₁	—	2	1	no	—
200	C₈×C₃₁	—	4	2	no	—
201	C₈×C₃₁	—	4	2	no	—
202	C₈×C₃₁	—	4	1	no	—
203	C₈×C₃₁	—	4	1	no	—
204	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
205	C₈×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
206	#3147	C₂×C₃₁	1	1	yes	characteristic, complemented
207	#3115	C₂×C₃₁	1	2	yes	complemented
208	C₈×#41	C₂×C₃₁	1	1	yes	characteristic, complemented
209	#1670	C₂×C₃₁	1	1	yes	characteristic, complemented
210	#1669	C₂×C₃₁	1	1	yes	characteristic, complemented
211	#1668	C₂×C₃₁	1	1	yes	characteristic, complemented
212	#1667	C₂×C₃₁	1	1	yes	characteristic, complemented
213	#1666	C₂×C₃₁	1	2	yes	complemented
214	#1665	C₂×C₃₁	1	2	yes	complemented
215	#1568	C₂×C₃₁	1	1	yes	characteristic, complemented
216	#1553	C₂×C₃₁	1	2	yes	complemented
217	C₂²×C₃₁	—	2	1	no	—
218	C₂²×C₃₁	—	2	1	no	—
219	C₂²×C₃₁	—	2	2	no	—
220	C₂²×C₃₁	—	2	1	no	—
221	C₂²×C₃₁	—	4	1	no	—
222	C₂²×C₃₁	—	4	1	no	—
223	C₂²×C₃₁	—	8	2	no	—
224	C₂²×C₃₁	—	8	2	no	—
225	C₂²×C₃₁	—	8	2	no	—
226	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
227	C₄×C₃₁	—	2	1	no	—
228	C₄×C₃₁	—	2	1	no	—
229	C₄×C₃₁	—	2	2	no	—
230	C₄×C₃₁	—	2	1	no	—
231	C₄×C₃₁	—	2	1	no	—
232	C₄×C₃₁	—	4	1	no	—
233	C₄×C₃₁	—	4	1	no	—
234	C₄×C₃₁	—	8	2	no	—
235	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
236	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
237	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
238	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
239	C₂×#135	—	2	1	no	—
240	C₂×#135	—	2	1	no	—
241	#532	C₂²×C₃₁	1	2	yes	—
242	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
243	#495	—	2	1	no	—
244	#495	—	2	1	no	—
245	#494	C₂²×C₃₁	1	1	yes	characteristic
246	#493	C₂²×C₃₁	1	1	yes	characteristic
247	#492	C₂²×C₃₁	1	2	yes	—
248	#482	—	2	2	no	—
249	#475	C₂²×C₃₁	1	2	yes	—
250	#474	C₂²×C₃₁	1	2	yes	—
251	#460	C₂²×C₃₁	1	2	yes	—
252	#459	C₂²×C₃₁	1	2	yes	—
253	#449	—	2	2	no	—
254	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
255	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
256	#442	—	2	2	no	—
257	#440	—	2	2	no	—
258	C₈×D₄	—	2	1	no	—
259	C₈×D₄	—	2	1	no	—
260	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
261	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
262	#432	C₂²×C₃₁	1	2	yes	—
263	#431	—	2	1	no	—
264	#431	C₂²×C₃₁	1	1	yes	characteristic
265	C₂×C₄×C₈	—	2	1	no	—
266	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
267	#335	C₂²×C₃₁	1	1	yes	characteristic
268	#335	C₂²×C₃₁	1	1	yes	characteristic
269	#334	C₂²×C₃₁	1	2	yes	—
270	#332	C₂²×C₃₁	1	2	yes	—
271	#332	C₂²×C₃₁	1	2	yes	—
272	#331	C₂²×C₃₁	1	2	yes	—
273	#331	C₂²×C₃₁	1	2	yes	—
274	C₂×C₃₁	—	4	1	no	—
275	C₂×C₃₁	—	4	2	no	—
276	C₂×C₃₁	—	16	2	no	—
277	C₂×C₃₁	#3038	1	1	yes	characteristic
278	C₂×C₃₁	#3050	1	1	yes	characteristic
279	C₂×C₃₁	#3067	1	1	yes	characteristic
280	C₂×#41	—	2	2	no	—
281	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
282	#135	—	4	4	no	—
283	#135	—	4	4	no	—
284	C₂×Q₁₆	—	2	1	no	—
285	C₂×Q₁₆	—	2	1	no	—
286	C₂×SD₁₆	—	2	2	no	—
287	C₂×SD₁₆	—	2	2	no	—
288	C₂×SD₁₆	—	2	2	no	—
289	C₂×D₈	—	2	2	no	—
290	C₂×D₈	—	2	1	no	—
291	C₂×D₈	—	2	1	no	—
292	C₂²×C₈	—	2	1	no	—
293	C₂²×C₈	—	2	1	no	—
294	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
295	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
296	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
297	#127	C₂³×C₃₁	1	2	yes	—
298	#124	—	2	2	no	—
299	#121	—	2	2	no	—
300	#121	—	2	2	no	—
301	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
302	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
303	C₄×D₄	—	2	1	no	—
304	C₄×D₄	—	2	1	no	—
305	C₄×D₄	—	2	2	no	—
306	C₄×D₄	—	2	2	no	—
307	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
308	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
309	#117	—	2	1	no	—
310	#117	D₄×C₃₁	1	1	yes	characteristic
311	C₂×C₄²	—	2	1	no	—
312	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
313	#107	—	2	2	no	—
314	#106	—	2	2	no	—
315	C₄⋊C₈	—	2	1	no	—
316	C₄⋊C₈	—	2	1	no	—
317	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
318	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
319	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
320	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
321	#103	—	2	2	no	—
322	#102	—	2	2	no	—
323	#102	—	2	2	no	—
324	#102	—	2	2	no	—
325	C₂²⋊C₈	—	2	1	no	—
326	C₂²⋊C₈	—	2	2	no	—
327	C₂²⋊C₈	—	2	1	no	—
328	C₂²⋊C₈	—	2	1	no	—
329	#97	—	2	2	no	—
330	C₄×C₈	—	2	1	no	—
331	C₄×C₈	—	2	1	no	—
332	C₄×C₈	—	2	1	no	—
333	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
334	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
335	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
336	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
337	C₃₁	#19269	1	1	yes	characteristic, direct product factor, Sylow
338	#41	—	2	2	no	—
339	#41	—	2	2	no	—
340	#41	—	4	4	no	—
341	#41	—	4	4	no	—
342	C₂×Q₈	—	2	2	no	—
343	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
344	C₂×D₄	—	2	1	no	—
345	C₂×D₄	—	2	2	no	—
346	C₂×D₄	—	2	2	no	—
347	C₂×D₄	—	2	2	no	—
348	C₂×D₄	—	4	2	no	—
349	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
350	C₂²×C₄	—	2	1	no	—
351	C₂²×C₄	—	2	1	no	—
352	C₂²×C₄	—	2	1	no	—
353	C₂²×C₄	—	4	2	no	—
354	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
355	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
356	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
357	Q₁₆	—	4	2	no	—
358	Q₁₆	—	4	2	no	—
359	SD₁₆	—	4	4	no	—
360	SD₁₆	—	4	4	no	—
361	SD₁₆	—	4	4	no	—
362	D₈	—	4	2	no	—
363	D₈	—	4	4	no	—
364	D₈	—	4	2	no	—
365	C₂×C₈	—	2	2	no	—
366	C₂×C₈	—	2	2	no	—
367	C₂×C₈	—	2	1	no	—
368	C₂×C₈	—	2	1	no	—
369	C₂×C₈	—	2	2	no	—
370	C₂×C₈	—	2	2	no	—
371	C₂×C₈	—	4	2	no	—
372	C₂×C₈	—	4	2	no	—
373	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
374	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
375	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
376	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
377	C₄⋊C₄	—	2	1	no	—
378	C₄⋊C₄	—	2	1	no	—
379	C₄⋊C₄	—	2	2	no	—
380	C₄⋊C₄	—	2	2	no	—
381	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
382	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
383	C₂²⋊C₄	—	2	1	no	—
384	C₂²⋊C₄	—	2	1	no	—
385	C₂²⋊C₄	—	2	1	no	—
386	C₂²⋊C₄	—	4	2	no	—
387	C₂²⋊C₄	—	4	2	no	—
388	C₄²	—	2	1	no	—
389	C₄²	—	2	1	no	—
390	C₄²	—	2	1	no	—
391	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
392	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
393	C₄²	#41×C₃₁	1	1	yes	characteristic
394	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
395	C₂³	—	2	1	no	—
396	C₂³	—	4	2	no	—
397	C₂³	C₃₁×#121	1	1	yes	characteristic
398	Q₈	—	2	1	no	—
399	Q₈	—	4	2	no	—
400	Q₈	—	4	2	no	—
401	Q₈	C₃₁×#135	1	2	yes	complemented
402	D₄	—	2	4	no	—
403	D₄	—	2	1	no	—
404	D₄	—	4	2	no	—
405	D₄	—	4	2	no	—
406	D₄	—	4	2	no	—
407	D₄	—	4	2	no	—
408	D₄	—	8	2	no	—
409	D₄	—	8	4	no	—
410	D₄	C₃₁×#135	1	2	yes	complemented
411	C₂×C₄	—	2	4	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	1	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	4	1	no	—
420	C₂×C₄	—	4	2	no	—
421	C₂×C₄	—	4	1	no	—
422	C₂×C₄	—	4	1	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	—	8	4	no	—
425	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
426	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
427	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
428	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
429	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
430	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
431	C₈	—	2	1	no	—
432	C₈	—	2	1	no	—
433	C₈	—	4	2	no	—
434	C₈	—	4	2	no	—
435	C₈	—	4	1	no	—
436	C₈	—	4	1	no	—
437	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
438	C₈	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
439	C₂²	—	2	1	no	—
440	C₂²	—	2	1	no	—
441	C₂²	—	2	2	no	—
442	C₂²	—	2	1	no	—
443	C₂²	—	4	1	no	—
444	C₂²	—	4	1	no	—
445	C₂²	—	8	2	no	—
446	C₂²	—	8	2	no	—
447	C₂²	—	8	2	no	—
448	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
449	C₄	—	2	1	no	—
450	C₄	—	2	1	no	—
451	C₄	—	2	2	no	—
452	C₄	—	2	1	no	—
453	C₄	—	2	1	no	—
454	C₄	—	4	1	no	—
455	C₄	—	4	1	no	—
456	C₄	—	8	2	no	—
457	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
458	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
459	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
460	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
461	C₂	—	4	1	no	—
462	C₂	—	4	2	no	—
463	C₂	—	16	2	no	—
464	C₂	C₃₁×#3038	1	1	yes	characteristic
465	C₂	C₃₁×#3050	1	1	yes	characteristic
466	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×#3147	C₂	semidirect
2. C₃₁×#3115	C₂	semidirect
3. C₈×#41×C₃₁	C₂	semidirect
4. C₃₁×#1670	C₂	semidirect
5. C₃₁×#1669	C₂	semidirect
6. C₃₁×#1668	C₂	semidirect
7. C₃₁×#1667	C₂	semidirect
8. C₃₁×#1666	C₂	semidirect
9. C₃₁×#1665	C₂	semidirect
10. C₃₁×#1568	C₂	semidirect
11. C₃₁×#1553	C₂	semidirect
161. #19269	C₃₁	direct (N × H)
168. Q₈×C₃₁	#135	semidirect
177. D₄×C₃₁	#135	semidirect
206. #3147	C₂×C₃₁	semidirect
207. #3115	C₂×C₃₁	semidirect
208. C₈×#41	C₂×C₃₁	semidirect
209. #1670	C₂×C₃₁	semidirect
210. #1669	C₂×C₃₁	semidirect
211. #1668	C₂×C₃₁	semidirect
212. #1667	C₂×C₃₁	semidirect
213. #1666	C₂×C₃₁	semidirect
214. #1665	C₂×C₃₁	semidirect
215. #1568	C₂×C₃₁	semidirect
216. #1553	C₂×C₃₁	semidirect
337. C₃₁	#19269	direct (N × H)
401. Q₈	C₃₁×#135	semidirect
410. D₄	C₃₁×#135	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3147	2
C₃₁×#3115	2
C₈×#41×C₃₁	2
C₃₁×#1670	2
C₃₁×#1669	2
C₃₁×#1668	2
C₃₁×#1667	2
C₃₁×#1666	2
C₃₁×#1665	2
C₃₁×#1568	2
C₃₁×#1553	2
#19269	31

Subgroups — C31×#19269 #21476028

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19269 #21476028