Subgroups — C₃₁×#18069 #21474828

Order 7936.

600 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3168	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2969	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1591	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1589	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1583	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1582	C₂	1	2	yes	maximal, complemented
7	C₈×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1564	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1482	C₂	1	2	yes	maximal, complemented
11	C₈×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#529	C₂²	1	2	yes	—
13	C₃₁×#499	C₄	1	1	yes	characteristic
14	C₃₁×#495	C₄	1	1	yes	characteristic
15	C₃₁×#488	C₄	1	2	yes	—
16	C₃₁×#467	C₄	1	1	yes	characteristic
17	C₃₁×#466	C₄	1	1	yes	characteristic
18	C₃₁×#465	C₄	1	2	yes	—
19	C₃₁×#446	C₂²	1	2	yes	—
20	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
21	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
22	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
23	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#436	—	2	2	no	—
25	C₃₁×#436	C₂²	1	2	yes	—
26	C₃₁×#435	—	2	2	no	—
27	C₈×D₄×C₃₁	C₂²	1	2	yes	—
28	C₃₁×#433	—	2	2	no	—
29	C₃₁×#431	C₂²	1	2	yes	—
30	C₃₁×#431	C₂²	1	1	yes	characteristic
31	C₃₁×#405	—	2	2	no	—
32	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
34	C₃₁×#335	C₂²	1	1	yes	characteristic
35	C₃₁×#335	C₂²	1	1	yes	characteristic
36	C₃₁×#334	C₂²	1	2	yes	—
37	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
38	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
39	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
40	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
41	C₃₁×#322	C₂²	1	1	yes	characteristic
42	C₃₁×#322	C₂²	1	1	yes	characteristic
43	C₈²×C₃₁	C₂²	1	1	yes	characteristic
44	C₈²×C₃₁	C₂²	1	1	yes	characteristic
45	C₂×Q₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
46	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
47	C₂×D₈×C₃₁	C₂×C₄	1	1	yes	characteristic
48	C₂×M₁₆×C₃₁	—	2	2	no	—
49	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
50	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₃₁	—	2	2	no	—
52	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
54	C₃₁×#126	—	2	2	no	—
55	C₃₁×#125	C₂×C₄	1	2	yes	—
56	C₃₁×#124	C₂×C₄	1	2	yes	—
57	C₃₁×#123	—	2	2	no	—
58	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
59	C₃₁×#121	C₂×C₄	1	2	yes	—
60	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
61	C₄×D₄×C₃₁	—	2	2	no	—
62	C₄×D₄×C₃₁	C₂³	1	2	yes	—
63	C₃₁×#117	—	2	2	no	—
64	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
65	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
66	C₃₁×#107	C₂×C₄	1	1	yes	characteristic
67	C₃₁×#106	C₂×C₄	1	2	yes	—
68	C₃₁×C₄⋊C₈	—	2	2	no	—
69	C₃₁×C₄⋊C₈	—	2	2	no	—
70	C₃₁×C₄⋊C₈	—	2	1	no	—
71	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
72	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
73	C₃₁×#103	C₂×C₄	1	2	yes	—
74	C₃₁×#103	C₂×C₄	1	2	yes	—
75	C₃₁×#102	C₂×C₄	1	2	yes	—
76	C₃₁×#102	C₂×C₄	1	2	yes	—
77	C₃₁×C₂²⋊C₈	—	2	2	no	—
78	C₃₁×C₂²⋊C₈	—	2	1	no	—
79	C₃₁×C₂²⋊C₈	—	2	2	no	—
80	C₃₁×C₂²⋊C₈	—	2	2	no	—
81	C₃₁×#97	—	2	2	no	—
82	C₃₁×#97	D₄	1	1	yes	characteristic
83	C₃₁×#97	D₄	1	1	yes	characteristic
84	C₄×C₈×C₃₁	—	2	2	no	—
85	C₄×C₈×C₃₁	—	2	2	no	—
86	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
87	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
88	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
89	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
95	C₂×D₄×C₃₁	—	2	2	no	—
96	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₃₁	—	2	2	no	—
98	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
101	Q₁₆×C₃₁	—	2	2	no	—
102	SD₁₆×C₃₁	—	4	2	no	—
103	D₈×C₃₁	—	2	2	no	—
104	M₁₆×C₃₁	—	2	2	no	—
105	M₁₆×C₃₁	—	2	2	no	—
106	M₁₆×C₃₁	—	4	4	no	—
107	C₂×C₈×C₃₁	—	2	2	no	—
108	C₂×C₈×C₃₁	—	2	2	no	—
109	C₂×C₈×C₃₁	—	2	2	no	—
110	C₂×C₈×C₃₁	—	2	1	no	—
111	C₂×C₈×C₃₁	—	2	1	no	—
112	C₂×C₈×C₃₁	—	4	1	no	—
113	C₂×C₈×C₃₁	—	4	4	no	—
114	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
122	C₄⋊C₄×C₃₁	—	2	2	no	—
123	C₄⋊C₄×C₃₁	—	2	2	no	—
124	C₄⋊C₄×C₃₁	—	2	2	no	—
125	C₄⋊C₄×C₃₁	—	2	2	no	—
126	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
127	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
128	C₂²⋊C₄×C₃₁	—	2	2	no	—
129	C₂²⋊C₄×C₃₁	—	2	2	no	—
130	C₂²⋊C₄×C₃₁	—	2	2	no	—
131	C₂²⋊C₄×C₃₁	—	2	2	no	—
132	C₂²⋊C₄×C₃₁	—	2	2	no	—
133	C₄²×C₃₁	—	2	2	no	—
134	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
135	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₄²×C₃₁	#41	1	1	yes	characteristic
137	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
138	#18069	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂³×C₃₁	—	2	2	no	—
140	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
141	Q₈×C₃₁	—	2	2	no	—
142	Q₈×C₃₁	—	2	2	no	—
143	D₄×C₃₁	—	2	2	no	—
144	D₄×C₃₁	—	2	2	no	—
145	D₄×C₃₁	—	8	2	no	—
146	C₂×C₄×C₃₁	—	2	1	no	—
147	C₂×C₄×C₃₁	—	2	1	no	—
148	C₂×C₄×C₃₁	—	2	1	no	—
149	C₂×C₄×C₃₁	—	2	1	no	—
150	C₂×C₄×C₃₁	—	2	2	no	—
151	C₂×C₄×C₃₁	—	2	2	no	—
152	C₂×C₄×C₃₁	—	2	2	no	—
153	C₂×C₄×C₃₁	—	4	1	no	—
154	C₂×C₄×C₃₁	—	4	2	no	—
155	C₂×C₄×C₃₁	—	4	2	no	—
156	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
157	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
158	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
159	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
160	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
162	C₈×C₃₁	—	2	1	no	—
163	C₈×C₃₁	—	2	1	no	—
164	C₈×C₃₁	—	2	1	no	—
165	C₈×C₃₁	—	2	1	no	—
166	C₈×C₃₁	—	2	1	no	—
167	C₈×C₃₁	—	4	1	no	—
168	C₈×C₃₁	—	4	1	no	—
169	C₈×C₃₁	—	4	2	no	—
170	C₈×C₃₁	—	4	2	no	—
171	C₈×C₃₁	#131	1	2	yes	—
172	C₈×C₃₁	#135	1	2	yes	—
173	C₈×C₃₁	#135	1	2	yes	—
174	#3168	C₂×C₃₁	1	1	yes	characteristic
175	#2969	C₂×C₃₁	1	2	yes	complemented
176	#1591	C₂×C₃₁	1	1	yes	characteristic, complemented
177	#1589	C₂×C₃₁	1	1	yes	characteristic, complemented
178	#1583	C₂×C₃₁	1	2	yes	complemented
179	#1582	C₂×C₃₁	1	2	yes	complemented
180	C₈×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
181	C₈×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
182	#1564	C₂×C₃₁	1	1	yes	characteristic, complemented
183	#1482	C₂×C₃₁	1	2	yes	complemented
184	C₈×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
185	C₂²×C₃₁	—	2	1	no	—
186	C₂²×C₃₁	—	2	1	no	—
187	C₂²×C₃₁	—	2	1	no	—
188	C₂²×C₃₁	—	4	2	no	—
189	C₂²×C₃₁	—	4	2	no	—
190	C₂²×C₃₁	—	4	2	no	—
191	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
192	C₄×C₃₁	—	2	1	no	—
193	C₄×C₃₁	—	2	1	no	—
194	C₄×C₃₁	—	2	1	no	—
195	C₄×C₃₁	—	2	1	no	—
196	C₄×C₃₁	—	4	2	no	—
197	C₄×C₃₁	—	4	2	no	—
198	C₄×C₃₁	—	4	2	no	—
199	C₄×C₃₁	C₂×#131	1	1	yes	characteristic
200	C₄×C₃₁	#568	1	1	yes	characteristic
201	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
202	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
203	#529	C₂²×C₃₁	1	2	yes	—
204	#499	C₄×C₃₁	1	1	yes	characteristic
205	#495	C₄×C₃₁	1	1	yes	characteristic
206	#488	C₄×C₃₁	1	2	yes	—
207	#467	C₄×C₃₁	1	1	yes	characteristic
208	#466	C₄×C₃₁	1	1	yes	characteristic
209	#465	C₄×C₃₁	1	2	yes	—
210	#446	C₂²×C₃₁	1	2	yes	—
211	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
212	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
213	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
214	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
215	#436	—	2	2	no	—
216	#436	C₂²×C₃₁	1	2	yes	—
217	#435	—	2	2	no	—
218	C₈×D₄	C₂²×C₃₁	1	2	yes	—
219	#433	—	2	2	no	—
220	#431	C₂²×C₃₁	1	2	yes	—
221	#431	C₂²×C₃₁	1	1	yes	characteristic
222	#405	—	2	2	no	—
223	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
224	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
225	#335	C₂²×C₃₁	1	1	yes	characteristic
226	#335	C₂²×C₃₁	1	1	yes	characteristic
227	#334	C₂²×C₃₁	1	2	yes	—
228	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
229	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
230	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
231	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
232	#322	C₂²×C₃₁	1	1	yes	characteristic
233	#322	C₂²×C₃₁	1	1	yes	characteristic
234	C₈²	C₂²×C₃₁	1	1	yes	characteristic
235	C₈²	C₂²×C₃₁	1	1	yes	characteristic
236	C₂×C₃₁	—	4	1	no	—
237	C₂×C₃₁	—	8	2	no	—
238	C₂×C₃₁	#2935	1	1	yes	characteristic
239	C₂×C₃₁	C₄×#135	1	1	yes	characteristic
240	C₂×C₃₁	#2957	1	1	yes	characteristic
241	C₂×Q₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
242	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	—
243	C₂×D₈	C₂×C₄×C₃₁	1	1	yes	characteristic
244	C₂×M₁₆	—	2	2	no	—
245	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
246	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
247	C₂²×C₈	—	2	2	no	—
248	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
249	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
250	#126	—	2	2	no	—
251	#125	C₂×C₄×C₃₁	1	2	yes	—
252	#124	C₂×C₄×C₃₁	1	2	yes	—
253	#123	—	2	2	no	—
254	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
255	#121	C₂×C₄×C₃₁	1	2	yes	—
256	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
257	C₄×D₄	—	2	2	no	—
258	C₄×D₄	C₂³×C₃₁	1	2	yes	—
259	#117	—	2	2	no	—
260	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
261	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
262	#107	C₂×C₄×C₃₁	1	1	yes	characteristic
263	#106	C₂×C₄×C₃₁	1	2	yes	—
264	C₄⋊C₈	—	2	2	no	—
265	C₄⋊C₈	—	2	2	no	—
266	C₄⋊C₈	—	2	1	no	—
267	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
268	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
269	#103	C₂×C₄×C₃₁	1	2	yes	—
270	#103	C₂×C₄×C₃₁	1	2	yes	—
271	#102	C₂×C₄×C₃₁	1	2	yes	—
272	#102	C₂×C₄×C₃₁	1	2	yes	—
273	C₂²⋊C₈	—	2	2	no	—
274	C₂²⋊C₈	—	2	1	no	—
275	C₂²⋊C₈	—	2	2	no	—
276	C₂²⋊C₈	—	2	2	no	—
277	#97	—	2	2	no	—
278	#97	D₄×C₃₁	1	1	yes	characteristic
279	#97	D₄×C₃₁	1	1	yes	characteristic
280	C₄×C₈	—	2	2	no	—
281	C₄×C₈	—	2	2	no	—
282	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
283	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
284	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
285	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
286	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
287	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
288	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
289	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
290	C₃₁	#18069	1	1	yes	characteristic, direct product factor, Sylow
291	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
292	C₂×D₄	—	2	2	no	—
293	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
294	C₂²×C₄	—	2	2	no	—
295	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
296	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
297	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
298	Q₁₆	—	2	2	no	—
299	SD₁₆	—	4	2	no	—
300	D₈	—	2	2	no	—
301	M₁₆	—	2	2	no	—
302	M₁₆	—	2	2	no	—
303	M₁₆	—	4	4	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	2	1	no	—
308	C₂×C₈	—	2	1	no	—
309	C₂×C₈	—	4	1	no	—
310	C₂×C₈	—	4	4	no	—
311	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
312	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
313	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
314	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
315	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
316	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
317	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
318	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
319	C₄⋊C₄	—	2	2	no	—
320	C₄⋊C₄	—	2	2	no	—
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	2	2	no	—
323	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
324	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
325	C₂²⋊C₄	—	2	2	no	—
326	C₂²⋊C₄	—	2	2	no	—
327	C₂²⋊C₄	—	2	2	no	—
328	C₂²⋊C₄	—	2	2	no	—
329	C₂²⋊C₄	—	2	2	no	—
330	C₄²	—	2	2	no	—
331	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
332	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
333	C₄²	#41×C₃₁	1	1	yes	characteristic
334	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
335	C₂³	—	2	2	no	—
336	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
337	Q₈	—	2	2	no	—
338	Q₈	—	2	2	no	—
339	D₄	—	2	2	no	—
340	D₄	—	2	2	no	—
341	D₄	—	8	2	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	—	2	2	no	—
349	C₂×C₄	—	4	1	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	—	4	2	no	—
352	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
353	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
354	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
355	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
356	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
357	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	2	1	no	—
362	C₈	—	2	1	no	—
363	C₈	—	4	1	no	—
364	C₈	—	4	1	no	—
365	C₈	—	4	2	no	—
366	C₈	—	4	2	no	—
367	C₈	C₃₁×#131	1	2	yes	—
368	C₈	C₃₁×#135	1	2	yes	—
369	C₈	C₃₁×#135	1	2	yes	—
370	C₂²	—	2	1	no	—
371	C₂²	—	2	1	no	—
372	C₂²	—	2	1	no	—
373	C₂²	—	4	2	no	—
374	C₂²	—	4	2	no	—
375	C₂²	—	4	2	no	—
376	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
377	C₄	—	2	1	no	—
378	C₄	—	2	1	no	—
379	C₄	—	2	1	no	—
380	C₄	—	2	1	no	—
381	C₄	—	4	2	no	—
382	C₄	—	4	2	no	—
383	C₄	—	4	2	no	—
384	C₄	C₂×C₃₁×#131	1	1	yes	characteristic
385	C₄	C₃₁×#568	1	1	yes	characteristic
386	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
387	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
388	C₂	—	4	1	no	—
389	C₂	—	8	2	no	—
390	C₂	C₃₁×#2935	1	1	yes	characteristic
391	C₂	C₄×C₃₁×#135	1	1	yes	characteristic
392	C₂	C₃₁×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₃₁×#466	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₄×M₁₆×C₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₃₁×#2969	C₂	semidirect
3. C₃₁×#1591	C₂	semidirect
4. C₃₁×#1589	C₂	semidirect
5. C₃₁×#1583	C₂	semidirect
6. C₃₁×#1582	C₂	semidirect
7. C₈×Q₁₆×C₃₁	C₂	semidirect
8. C₈×D₈×C₃₁	C₂	semidirect
9. C₃₁×#1564	C₂	semidirect
10. C₃₁×#1482	C₂	semidirect
11. C₈×M₁₆×C₃₁	C₂	semidirect
138. #18069	C₃₁	direct (N × H)
175. #2969	C₂×C₃₁	semidirect
176. #1591	C₂×C₃₁	semidirect
177. #1589	C₂×C₃₁	semidirect
178. #1583	C₂×C₃₁	semidirect
179. #1582	C₂×C₃₁	semidirect
180. C₈×Q₁₆	C₂×C₃₁	semidirect
181. C₈×D₈	C₂×C₃₁	semidirect
182. #1564	C₂×C₃₁	semidirect
183. #1482	C₂×C₃₁	semidirect
184. C₈×M₁₆	C₂×C₃₁	semidirect
290. C₃₁	#18069	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×#3168	2
C₃₁×#2969	2
C₃₁×#1591	2
C₃₁×#1589	2
C₃₁×#1583	2
C₃₁×#1582	2
C₈×Q₁₆×C₃₁	2
C₈×D₈×C₃₁	2
C₃₁×#1564	2
C₃₁×#1482	2
C₈×M₁₆×C₃₁	2
#18069	31

Subgroups — C31×#18069 #21474828

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#18069 #21474828