Subgroups — C₃₁×#17915 #21474674

Order 7936.

624 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×#3108	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2966	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1584	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1583	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1579	C₂	1	2	yes	maximal, complemented
7	C₃₁×SD₁₆⋊C₈	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1573	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1491	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1483	C₂	1	1	yes	maximal, characteristic
11	C₃₁×#1451	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#529	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#483	C₄	1	1	yes	characteristic
15	C₃₁×#480	C₄	1	1	yes	characteristic
16	C₃₁×#479	C₄	1	2	yes	—
17	C₃₁×#464	C₄	1	2	yes	—
18	C₃₁×#451	C₄	1	1	yes	characteristic
19	C₃₁×#448	C₄	1	1	yes	characteristic
20	C₃₁×#446	C₂²	1	1	yes	characteristic
21	C₈×Q₈×C₃₁	—	2	1	no	—
22	C₈×Q₈×C₃₁	—	2	1	no	—
23	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
24	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
26	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
27	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
28	C₃₁×#436	C₂²	1	1	yes	characteristic
29	C₃₁×#435	—	2	1	no	—
30	C₈×D₄×C₃₁	—	2	1	no	—
31	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
32	C₃₁×#433	—	2	1	no	—
33	C₃₁×#432	C₂²	1	1	yes	characteristic
34	C₃₁×#431	C₂²	1	1	yes	characteristic
35	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
36	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₃₁×#405	—	2	1	no	—
38	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈×C₃₁	—	2	1	no	—
40	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
41	C₃₁×#335	C₂²	1	2	yes	—
42	C₃₁×#334	C₂²	1	2	yes	—
43	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
44	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
45	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
46	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
47	C₃₁×#322	C₂²	1	2	yes	—
48	C₃₁×#322	C₂²	1	2	yes	—
49	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
50	C₂×Q₁₆×C₃₁	C₂×C₄	1	2	yes	—
51	C₂×SD₁₆×C₃₁	C₂×C₄	1	2	yes	—
52	C₂×M₁₆×C₃₁	—	2	1	no	—
53	C₂×M₁₆×C₃₁	—	2	1	no	—
54	C₂²×C₈×C₃₁	—	2	1	no	—
55	C₂²×C₈×C₃₁	—	2	1	no	—
56	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
58	C₃₁×#126	—	2	1	no	—
59	C₃₁×#125	C₂×C₄	1	1	yes	characteristic
60	C₃₁×#124	C₂×C₄	1	1	yes	characteristic
61	C₃₁×#123	—	2	1	no	—
62	C₃₁×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
63	C₃₁×#121	C₂×C₄	1	1	yes	characteristic
64	C₄×Q₈×C₃₁	—	2	1	no	—
65	C₄×Q₈×C₃₁	—	2	1	no	—
66	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
67	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
68	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
69	C₄×D₄×C₃₁	—	2	1	no	—
70	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₃₁×#117	—	2	1	no	—
72	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
73	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄²×C₃₁	—	2	1	no	—
75	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
76	C₃₁×#107	C₂×C₄	1	2	yes	—
77	C₃₁×#106	C₂×C₄	1	2	yes	—
78	C₃₁×C₄⋊C₈	—	2	2	no	—
79	C₃₁×C₄⋊C₈	—	2	2	no	—
80	C₃₁×C₄⋊C₈	—	2	1	no	—
81	C₃₁×C₄⋊C₈	—	2	1	no	—
82	C₃₁×C₄⋊C₈	—	2	1	no	—
83	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
84	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
85	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
86	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
87	C₃₁×#103	C₂×C₄	1	2	yes	—
88	C₃₁×#103	C₂×C₄	1	2	yes	—
89	C₃₁×#103	C₂×C₄	1	2	yes	—
90	C₃₁×#102	C₂×C₄	1	2	yes	—
91	C₃₁×C₂²⋊C₈	—	2	1	no	—
92	C₃₁×C₂²⋊C₈	—	2	1	no	—
93	C₃₁×C₂²⋊C₈	—	2	1	no	—
94	C₃₁×C₂²⋊C₈	C₂×C₄	1	1	yes	characteristic
95	C₃₁×C₂²⋊C₈	C₂×C₄	1	1	yes	characteristic
96	C₃₁×#97	—	2	1	no	—
97	C₃₁×#97	—	2	1	no	—
98	C₃₁×#97	—	2	1	no	—
99	C₄×C₈×C₃₁	—	2	2	no	—
100	C₄×C₈×C₃₁	—	2	2	no	—
101	C₄×C₈×C₃₁	—	2	1	no	—
102	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
103	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
104	C₄×C₈×C₃₁	C₂³	1	2	yes	—
105	C₄×C₈×C₃₁	C₂³	1	2	yes	—
106	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
107	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
108	C₂×Q₈×C₃₁	—	2	4	no	—
109	C₂×Q₈×C₃₁	—	2	1	no	—
110	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
111	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
112	C₂×D₄×C₃₁	—	2	1	no	—
113	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₄×C₃₁	—	2	1	no	—
115	C₂²×C₄×C₃₁	—	2	1	no	—
116	C₂²×C₄×C₃₁	—	2	1	no	—
117	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
120	Q₁₆×C₃₁	—	2	4	no	—
121	SD₁₆×C₃₁	—	2	4	no	—
122	M₁₆×C₃₁	—	4	2	no	—
123	M₁₆×C₃₁	—	4	2	no	—
124	C₂×C₈×C₃₁	—	2	2	no	—
125	C₂×C₈×C₃₁	—	2	2	no	—
126	C₂×C₈×C₃₁	—	2	1	no	—
127	C₂×C₈×C₃₁	—	2	1	no	—
128	C₂×C₈×C₃₁	—	2	2	no	—
129	C₂×C₈×C₃₁	—	4	1	no	—
130	C₂×C₈×C₃₁	—	4	2	no	—
131	C₂×C₈×C₃₁	—	4	2	no	—
132	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
133	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
134	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
138	C₄⋊C₄×C₃₁	—	2	2	no	—
139	C₄⋊C₄×C₃₁	—	2	1	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	2	no	—
144	C₄⋊C₄×C₃₁	—	2	1	no	—
145	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
146	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
147	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
148	C₂²⋊C₄×C₃₁	—	2	1	no	—
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	—	2	1	no	—
153	C₄²×C₃₁	—	2	2	no	—
154	C₄²×C₃₁	—	2	2	no	—
155	C₄²×C₃₁	—	2	1	no	—
156	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
157	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
158	C₄²×C₃₁	#41	1	1	yes	characteristic
159	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
160	#17915	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₂³×C₃₁	—	2	1	no	—
162	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
163	Q₈×C₃₁	—	2	4	no	—
164	Q₈×C₃₁	—	2	2	no	—
165	Q₈×C₃₁	—	2	2	no	—
166	Q₈×C₃₁	#131	1	4	yes	—
167	D₄×C₃₁	—	2	2	no	—
168	D₄×C₃₁	—	4	2	no	—
169	C₂×C₄×C₃₁	—	2	4	no	—
170	C₂×C₄×C₃₁	—	2	2	no	—
171	C₂×C₄×C₃₁	—	2	2	no	—
172	C₂×C₄×C₃₁	—	2	2	no	—
173	C₂×C₄×C₃₁	—	2	1	no	—
174	C₂×C₄×C₃₁	—	2	1	no	—
175	C₂×C₄×C₃₁	—	2	2	no	—
176	C₂×C₄×C₃₁	—	2	1	no	—
177	C₂×C₄×C₃₁	—	2	1	no	—
178	C₂×C₄×C₃₁	—	2	1	no	—
179	C₂×C₄×C₃₁	—	4	2	no	—
180	C₂×C₄×C₃₁	—	4	1	no	—
181	C₂×C₄×C₃₁	—	4	1	no	—
182	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
183	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
184	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
185	C₂×C₄×C₃₁	C₂³×C₄	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	—	2	1	no	—
191	C₈×C₃₁	—	2	1	no	—
192	C₈×C₃₁	—	2	1	no	—
193	C₈×C₃₁	—	4	2	no	—
194	C₈×C₃₁	—	4	1	no	—
195	C₈×C₃₁	—	4	2	no	—
196	C₈×C₃₁	—	4	1	no	—
197	C₈×C₃₁	#137	1	2	yes	—
198	#3108	C₂×C₃₁	1	1	yes	characteristic
199	#2966	C₂×C₃₁	1	1	yes	characteristic
200	C₂×C₈×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
201	#1584	C₂×C₃₁	1	2	yes	complemented
202	#1583	C₂×C₃₁	1	2	yes	complemented
203	#1579	C₂×C₃₁	1	2	yes	complemented
204	SD₁₆⋊C₈	C₂×C₃₁	1	2	yes	complemented
205	#1573	C₂×C₃₁	1	1	yes	characteristic, complemented
206	#1491	C₂×C₃₁	1	1	yes	characteristic, complemented
207	#1483	C₂×C₃₁	1	1	yes	characteristic
208	#1451	C₂×C₃₁	1	1	yes	characteristic, complemented
209	C₂²×C₃₁	—	2	1	no	—
210	C₂²×C₃₁	—	2	1	no	—
211	C₂²×C₃₁	—	4	1	no	—
212	C₂²×C₃₁	—	4	1	no	—
213	C₂²×C₃₁	—	4	1	no	—
214	C₂²×C₃₁	#444	1	2	yes	—
215	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
216	C₄×C₃₁	—	2	4	no	—
217	C₄×C₃₁	—	2	1	no	—
218	C₄×C₃₁	—	2	1	no	—
219	C₄×C₃₁	—	2	1	no	—
220	C₄×C₃₁	—	2	1	no	—
221	C₄×C₃₁	—	4	1	no	—
222	C₄×C₃₁	—	4	1	no	—
223	C₄×C₃₁	—	4	2	no	—
224	C₄×C₃₁	—	4	1	no	—
225	C₄×C₃₁	C₂×#131	1	2	yes	—
226	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
227	C₄×C₃₁	#575	1	1	yes	characteristic
228	#529	C₂²×C₃₁	1	1	yes	characteristic
229	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
230	#483	C₄×C₃₁	1	1	yes	characteristic
231	#480	C₄×C₃₁	1	1	yes	characteristic
232	#479	C₄×C₃₁	1	2	yes	—
233	#464	C₄×C₃₁	1	2	yes	—
234	#451	C₄×C₃₁	1	1	yes	characteristic
235	#448	C₄×C₃₁	1	1	yes	characteristic
236	#446	C₂²×C₃₁	1	1	yes	characteristic
237	C₈×Q₈	—	2	1	no	—
238	C₈×Q₈	—	2	1	no	—
239	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
240	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
241	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
242	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
243	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
244	#436	C₂²×C₃₁	1	1	yes	characteristic
245	#435	—	2	1	no	—
246	C₈×D₄	—	2	1	no	—
247	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
248	#433	—	2	1	no	—
249	#432	C₂²×C₃₁	1	1	yes	characteristic
250	#431	C₂²×C₃₁	1	1	yes	characteristic
251	C₂×C₄⋊C₈	—	2	1	no	—
252	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
253	#405	—	2	1	no	—
254	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
255	C₂×C₄×C₈	—	2	1	no	—
256	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
257	#335	C₂²×C₃₁	1	2	yes	—
258	#334	C₂²×C₃₁	1	2	yes	—
259	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
260	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
261	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
262	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
263	#322	C₂²×C₃₁	1	2	yes	—
264	#322	C₂²×C₃₁	1	2	yes	—
265	C₂×C₃₁	—	2	2	no	—
266	C₂×C₃₁	—	8	1	no	—
267	C₂×C₃₁	#2929	1	1	yes	characteristic
268	C₂×C₃₁	C₄×#137	1	1	yes	characteristic
269	C₂×C₃₁	C₂×#444	1	1	yes	characteristic
270	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
271	C₂×Q₁₆	C₂×C₄×C₃₁	1	2	yes	—
272	C₂×SD₁₆	C₂×C₄×C₃₁	1	2	yes	—
273	C₂×M₁₆	—	2	1	no	—
274	C₂×M₁₆	—	2	1	no	—
275	C₂²×C₈	—	2	1	no	—
276	C₂²×C₈	—	2	1	no	—
277	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
278	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
279	#126	—	2	1	no	—
280	#125	C₂×C₄×C₃₁	1	1	yes	characteristic
281	#124	C₂×C₄×C₃₁	1	1	yes	characteristic
282	#123	—	2	1	no	—
283	C₂²⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
284	#121	C₂×C₄×C₃₁	1	1	yes	characteristic
285	C₄×Q₈	—	2	1	no	—
286	C₄×Q₈	—	2	1	no	—
287	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
288	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
289	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
290	C₄×D₄	—	2	1	no	—
291	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
292	#117	—	2	1	no	—
293	C₂×C₄⋊C₄	—	2	1	no	—
294	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
295	C₂×C₄²	—	2	1	no	—
296	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
297	#107	C₂×C₄×C₃₁	1	2	yes	—
298	#106	C₂×C₄×C₃₁	1	2	yes	—
299	C₄⋊C₈	—	2	2	no	—
300	C₄⋊C₈	—	2	2	no	—
301	C₄⋊C₈	—	2	1	no	—
302	C₄⋊C₈	—	2	1	no	—
303	C₄⋊C₈	—	2	1	no	—
304	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
305	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
306	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
307	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
308	#103	C₂×C₄×C₃₁	1	2	yes	—
309	#103	C₂×C₄×C₃₁	1	2	yes	—
310	#103	C₂×C₄×C₃₁	1	2	yes	—
311	#102	C₂×C₄×C₃₁	1	2	yes	—
312	C₂²⋊C₈	—	2	1	no	—
313	C₂²⋊C₈	—	2	1	no	—
314	C₂²⋊C₈	—	2	1	no	—
315	C₂²⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
316	C₂²⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
317	#97	—	2	1	no	—
318	#97	—	2	1	no	—
319	#97	—	2	1	no	—
320	C₄×C₈	—	2	2	no	—
321	C₄×C₈	—	2	2	no	—
322	C₄×C₈	—	2	1	no	—
323	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
324	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
325	C₄×C₈	C₂³×C₃₁	1	2	yes	—
326	C₄×C₈	C₂³×C₃₁	1	2	yes	—
327	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
328	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
329	C₃₁	#17915	1	1	yes	characteristic, direct product factor, Sylow
330	C₂×Q₈	—	2	4	no	—
331	C₂×Q₈	—	2	1	no	—
332	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
333	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
334	C₂×D₄	—	2	1	no	—
335	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
336	C₂²×C₄	—	2	1	no	—
337	C₂²×C₄	—	2	1	no	—
338	C₂²×C₄	—	2	1	no	—
339	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
340	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
341	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
342	Q₁₆	—	2	4	no	—
343	SD₁₆	—	2	4	no	—
344	M₁₆	—	4	2	no	—
345	M₁₆	—	4	2	no	—
346	C₂×C₈	—	2	2	no	—
347	C₂×C₈	—	2	2	no	—
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	2	1	no	—
350	C₂×C₈	—	2	2	no	—
351	C₂×C₈	—	4	1	no	—
352	C₂×C₈	—	4	2	no	—
353	C₂×C₈	—	4	2	no	—
354	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
355	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
356	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
357	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
358	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
359	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	2	1	no	—
362	C₄⋊C₄	—	2	1	no	—
363	C₄⋊C₄	—	2	1	no	—
364	C₄⋊C₄	—	2	1	no	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	2	1	no	—
367	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
368	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
369	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
370	C₂²⋊C₄	—	2	1	no	—
371	C₂²⋊C₄	—	2	1	no	—
372	C₂²⋊C₄	—	2	1	no	—
373	C₂²⋊C₄	—	2	1	no	—
374	C₂²⋊C₄	—	2	1	no	—
375	C₄²	—	2	2	no	—
376	C₄²	—	2	2	no	—
377	C₄²	—	2	1	no	—
378	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
379	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
380	C₄²	#41×C₃₁	1	1	yes	characteristic
381	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
382	C₂³	—	2	1	no	—
383	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
384	Q₈	—	2	4	no	—
385	Q₈	—	2	2	no	—
386	Q₈	—	2	2	no	—
387	Q₈	C₃₁×#131	1	4	yes	—
388	D₄	—	2	2	no	—
389	D₄	—	4	2	no	—
390	C₂×C₄	—	2	4	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	1	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	2	1	no	—
398	C₂×C₄	—	2	1	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	4	2	no	—
401	C₂×C₄	—	4	1	no	—
402	C₂×C₄	—	4	1	no	—
403	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
404	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
405	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
406	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
407	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
408	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
409	C₈	—	2	2	no	—
410	C₈	—	2	2	no	—
411	C₈	—	2	1	no	—
412	C₈	—	2	1	no	—
413	C₈	—	2	1	no	—
414	C₈	—	4	2	no	—
415	C₈	—	4	1	no	—
416	C₈	—	4	2	no	—
417	C₈	—	4	1	no	—
418	C₈	C₃₁×#137	1	2	yes	—
419	C₂²	—	2	1	no	—
420	C₂²	—	2	1	no	—
421	C₂²	—	4	1	no	—
422	C₂²	—	4	1	no	—
423	C₂²	—	4	1	no	—
424	C₂²	C₃₁×#444	1	2	yes	—
425	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
426	C₄	—	2	4	no	—
427	C₄	—	2	1	no	—
428	C₄	—	2	1	no	—
429	C₄	—	2	1	no	—
430	C₄	—	2	1	no	—
431	C₄	—	4	1	no	—
432	C₄	—	4	1	no	—
433	C₄	—	4	2	no	—
434	C₄	—	4	1	no	—
435	C₄	C₂×C₃₁×#131	1	2	yes	—
436	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
437	C₄	C₃₁×#575	1	1	yes	characteristic
438	C₂	—	2	2	no	—
439	C₂	—	8	1	no	—
440	C₂	C₃₁×#2929	1	1	yes	characteristic
441	C₂	C₄×C₃₁×#137	1	1	yes	characteristic
442	C₂	C₂×C₃₁×#444	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×#121	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₈×Q₈×C₃₁	C₂	semidirect
4. C₃₁×#1584	C₂	semidirect
5. C₃₁×#1583	C₂	semidirect
6. C₃₁×#1579	C₂	semidirect
7. C₃₁×SD₁₆⋊C₈	C₂	semidirect
8. C₃₁×#1573	C₂	semidirect
9. C₃₁×#1491	C₂	semidirect
11. C₃₁×#1451	C₂	semidirect
160. #17915	C₃₁	direct (N × H)
200. C₂×C₈×Q₈	C₂×C₃₁	semidirect
201. #1584	C₂×C₃₁	semidirect
202. #1583	C₂×C₃₁	semidirect
203. #1579	C₂×C₃₁	semidirect
204. SD₁₆⋊C₈	C₂×C₃₁	semidirect
205. #1573	C₂×C₃₁	semidirect
206. #1491	C₂×C₃₁	semidirect
208. #1451	C₂×C₃₁	semidirect
329. C₃₁	#17915	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×#3108	2
C₃₁×#2966	2
C₂×C₈×Q₈×C₃₁	2
C₃₁×#1584	2
C₃₁×#1583	2
C₃₁×#1579	2
C₃₁×SD₁₆⋊C₈	2
C₃₁×#1573	2
C₃₁×#1491	2
C₃₁×#1483	2
C₃₁×#1451	2
#17915	31

Subgroups — C31×#17915 #21474674

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17915 #21474674