Subgroups — C₃₁×#18601 #21475360

Order 7936.

910 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3501	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3237	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3185	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3101	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1651	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1643	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1637	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1632	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1554	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1490	C₂	1	1	yes	maximal, characteristic
11	C₃₁×#1485	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#554	—	2	1	no	—
13	C₃₁×#554	—	2	2	no	—
14	C₃₁×#554	—	2	2	no	—
15	C₃₁×#554	C₂²	1	2	yes	—
16	D₄×Q₈×C₃₁	—	2	1	no	—
17	D₄×Q₈×C₃₁	C₂²	1	2	yes	—
18	C₃₁×#544	—	2	1	no	—
19	C₃₁×#543	—	2	2	no	—
20	C₃₁×#533	C₂²	1	1	yes	characteristic
21	C₃₁×#533	C₂²	1	1	yes	characteristic
22	C₃₁×#533	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×C₂²⋊Q₈	—	2	2	no	—
24	C₃₁×#518	—	2	1	no	—
25	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#501	—	2	1	no	—
27	C₃₁×#498	C₂²	1	1	yes	characteristic, complemented
28	C₃₁×#494	C₂²	1	1	yes	characteristic, complemented
29	C₃₁×#491	—	2	1	no	—
30	C₃₁×#487	C₂²	1	1	yes	characteristic, complemented
31	C₃₁×#487	C₂²	1	1	yes	characteristic, complemented
32	C₃₁×#485	—	2	2	no	—
33	C₃₁×#483	—	2	1	no	—
34	C₃₁×#477	C₂²	1	2	yes	complemented
35	C₃₁×#476	C₂²	1	2	yes	—
36	C₃₁×#475	C₂²	1	2	yes	complemented
37	C₃₁×#470	—	2	2	no	—
38	C₃₁×#461	C₂²	1	2	yes	—
39	C₃₁×#450	—	2	1	no	—
40	C₃₁×#441	—	2	2	no	—
41	C₄×SD₁₆×C₃₁	—	2	2	no	—
42	C₃₁×#432	C₂²	1	1	yes	characteristic
43	C₃₁×#431	C₂²	1	1	yes	characteristic
44	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
45	C₃₁×#333	C₂²	1	2	yes	complemented
46	C₃₁×#332	C₂²	1	2	yes	complemented
47	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	complemented
48	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	complemented
49	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
50	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
51	C₂²×Q₈×C₃₁	—	2	2	no	—
52	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
53	C₂³×C₄×C₃₁	—	2	1	no	—
54	C₂×Q₁₆×C₃₁	—	2	2	no	—
55	C₂×SD₁₆×C₃₁	—	2	2	no	—
56	C₂×M₁₆×C₃₁	—	2	1	no	—
57	C₃₁×C₄⋊Q₈	—	2	2	no	—
58	C₃₁×C₄⋊Q₈	—	2	2	no	—
59	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
60	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
61	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
62	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
63	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
64	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
65	C₃₁×#125	—	2	2	no	—
66	C₃₁×#125	—	2	2	no	—
67	C₃₁×#125	—	2	1	no	—
68	C₃₁×#125	—	2	1	no	—
69	C₃₁×#125	—	2	1	no	—
70	C₃₁×C₂²⋊Q₈	—	2	4	no	—
71	C₃₁×C₂²⋊Q₈	—	2	4	no	—
72	C₃₁×C₂²⋊Q₈	—	2	4	no	—
73	C₃₁×C₂²⋊Q₈	—	2	4	no	—
74	C₃₁×C₂²⋊Q₈	—	2	2	no	—
75	C₃₁×C₂²⋊Q₈	—	2	4	no	—
76	C₃₁×C₂²⋊Q₈	—	2	1	no	—
77	C₃₁×C₂²⋊Q₈	—	2	1	no	—
78	C₃₁×C₂²⋊Q₈	—	2	2	no	—
79	C₃₁×C₂²⋊Q₈	—	2	4	no	—
80	C₃₁×C₂²⋊Q₈	—	2	1	no	—
81	C₃₁×C₂²⋊Q₈	—	2	1	no	—
82	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
83	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
84	C₄×Q₈×C₃₁	—	2	1	no	—
85	C₄×Q₈×C₃₁	—	2	1	no	—
86	C₄×Q₈×C₃₁	—	2	1	no	—
87	C₄×Q₈×C₃₁	—	2	1	no	—
88	C₄×Q₈×C₃₁	D₄	1	2	yes	—
89	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
90	C₄×D₄×C₃₁	—	2	2	no	—
91	C₄×D₄×C₃₁	—	2	2	no	—
92	C₄×D₄×C₃₁	—	2	2	no	—
93	C₄×D₄×C₃₁	—	2	2	no	—
94	C₄×D₄×C₃₁	—	2	1	no	—
95	C₄×D₄×C₃₁	—	2	1	no	—
96	C₄×D₄×C₃₁	D₄	1	2	yes	—
97	C₄×D₄×C₃₁	C₂³	1	2	yes	—
98	C₃₁×#117	—	2	1	no	—
99	C₃₁×#117	—	2	1	no	—
100	C₃₁×#117	—	2	1	no	—
101	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
102	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
103	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
104	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
105	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
106	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
107	C₃₁×#107	—	2	2	no	—
108	C₃₁×#106	—	2	2	no	—
109	C₃₁×#106	—	2	2	no	—
110	C₃₁×#106	—	2	2	no	—
111	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
112	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
113	C₃₁×#103	—	2	2	no	—
114	C₃₁×#103	—	2	2	no	—
115	C₃₁×#103	—	2	2	no	—
116	C₃₁×#103	—	2	2	no	—
117	C₃₁×#103	—	2	2	no	—
118	C₃₁×#102	—	2	2	no	—
119	C₃₁×C₂²⋊C₈	—	2	1	no	—
120	C₃₁×C₂²⋊C₈	—	2	1	no	—
121	C₃₁×#97	—	2	1	no	—
122	C₃₁×#97	—	2	1	no	—
123	C₄×C₈×C₃₁	—	2	1	no	—
124	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
125	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
126	C₂⁴×C₃₁	—	2	1	no	—
127	C₂×Q₈×C₃₁	—	2	4	no	—
128	C₂×Q₈×C₃₁	—	2	1	no	—
129	C₂×Q₈×C₃₁	—	2	1	no	—
130	C₂×Q₈×C₃₁	—	2	1	no	—
131	C₂×Q₈×C₃₁	—	2	1	no	—
132	C₂×Q₈×C₃₁	—	8	4	no	—
133	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
134	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
135	C₂×D₄×C₃₁	—	2	4	no	—
136	C₂×D₄×C₃₁	—	2	1	no	—
137	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
138	C₂²×C₄×C₃₁	—	2	4	no	—
139	C₂²×C₄×C₃₁	—	2	4	no	—
140	C₂²×C₄×C₃₁	—	2	1	no	—
141	C₂²×C₄×C₃₁	—	2	2	no	—
142	C₂²×C₄×C₃₁	—	2	2	no	—
143	C₂²×C₄×C₃₁	—	8	2	no	—
144	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
146	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
147	Q₁₆×C₃₁	—	4	4	no	—
148	SD₁₆×C₃₁	—	4	4	no	—
149	M₁₆×C₃₁	—	4	2	no	—
150	C₂×C₈×C₃₁	—	2	1	no	—
151	C₂×C₈×C₃₁	—	2	2	no	—
152	C₂×C₈×C₃₁	—	2	1	no	—
153	C₂×C₈×C₃₁	—	2	2	no	—
154	C₄⋊C₄×C₃₁	—	2	4	no	—
155	C₄⋊C₄×C₃₁	—	2	4	no	—
156	C₄⋊C₄×C₃₁	—	2	4	no	—
157	C₄⋊C₄×C₃₁	—	2	2	no	—
158	C₄⋊C₄×C₃₁	—	2	2	no	—
159	C₄⋊C₄×C₃₁	—	2	1	no	—
160	C₄⋊C₄×C₃₁	—	2	1	no	—
161	C₄⋊C₄×C₃₁	—	2	1	no	—
162	C₄⋊C₄×C₃₁	—	2	1	no	—
163	C₄⋊C₄×C₃₁	—	2	2	no	—
164	C₄⋊C₄×C₃₁	—	2	1	no	—
165	C₄⋊C₄×C₃₁	—	2	1	no	—
166	C₄⋊C₄×C₃₁	—	2	1	no	—
167	C₄⋊C₄×C₃₁	—	2	2	no	—
168	C₄⋊C₄×C₃₁	—	2	1	no	—
169	C₄⋊C₄×C₃₁	—	2	1	no	—
170	C₄⋊C₄×C₃₁	—	2	1	no	—
171	C₄⋊C₄×C₃₁	—	2	1	no	—
172	C₄⋊C₄×C₃₁	—	4	1	no	—
173	C₄⋊C₄×C₃₁	—	4	1	no	—
174	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
175	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
176	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
177	C₂²⋊C₄×C₃₁	—	2	4	no	—
178	C₂²⋊C₄×C₃₁	—	2	4	no	—
179	C₂²⋊C₄×C₃₁	—	2	2	no	—
180	C₂²⋊C₄×C₃₁	—	2	2	no	—
181	C₂²⋊C₄×C₃₁	—	2	2	no	—
182	C₂²⋊C₄×C₃₁	—	2	2	no	—
183	C₂²⋊C₄×C₃₁	—	4	1	no	—
184	C₄²×C₃₁	—	2	2	no	—
185	C₄²×C₃₁	—	2	2	no	—
186	C₄²×C₃₁	—	4	1	no	—
187	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
188	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
189	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
190	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
191	#18601	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
192	C₂³×C₃₁	—	2	4	no	—
193	C₂³×C₃₁	—	2	2	no	—
194	C₂³×C₃₁	—	8	1	no	—
195	C₂³×C₃₁	—	8	1	no	—
196	C₂³×C₃₁	#120	1	1	yes	characteristic
197	Q₈×C₃₁	—	2	4	no	—
198	Q₈×C₃₁	—	4	2	no	—
199	Q₈×C₃₁	—	4	2	no	—
200	Q₈×C₃₁	—	4	2	no	—
201	Q₈×C₃₁	—	4	2	no	—
202	Q₈×C₃₁	—	8	1	no	—
203	Q₈×C₃₁	—	8	4	no	—
204	D₄×C₃₁	—	2	4	no	—
205	D₄×C₃₁	—	2	2	no	—
206	D₄×C₃₁	#137	1	4	yes	—
207	C₂×C₄×C₃₁	—	2	2	no	—
208	C₂×C₄×C₃₁	—	2	2	no	—
209	C₂×C₄×C₃₁	—	2	2	no	—
210	C₂×C₄×C₃₁	—	2	2	no	—
211	C₂×C₄×C₃₁	—	2	2	no	—
212	C₂×C₄×C₃₁	—	2	2	no	—
213	C₂×C₄×C₃₁	—	2	2	no	—
214	C₂×C₄×C₃₁	—	4	1	no	—
215	C₂×C₄×C₃₁	—	4	4	no	—
216	C₂×C₄×C₃₁	—	4	1	no	—
217	C₂×C₄×C₃₁	—	4	1	no	—
218	C₂×C₄×C₃₁	—	8	2	no	—
219	C₂×C₄×C₃₁	—	8	1	no	—
220	C₂×C₄×C₃₁	—	8	4	no	—
221	C₂×C₄×C₃₁	—	8	4	no	—
222	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
223	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
224	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
225	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
226	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
227	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
228	C₈×C₃₁	—	2	2	no	—
229	C₈×C₃₁	—	4	1	no	—
230	C₈×C₃₁	—	4	2	no	—
231	C₈×C₃₁	—	4	2	no	—
232	#3501	C₂×C₃₁	1	1	yes	characteristic
233	#3237	C₂×C₃₁	1	1	yes	characteristic, complemented
234	#3185	C₂×C₃₁	1	1	yes	characteristic, complemented
235	#3101	C₂×C₃₁	1	1	yes	characteristic
236	#1651	C₂×C₃₁	1	2	yes	complemented
237	#1643	C₂×C₃₁	1	2	yes	complemented
238	#1637	C₂×C₃₁	1	2	yes	complemented
239	#1632	C₂×C₃₁	1	2	yes	complemented
240	#1554	C₂×C₃₁	1	1	yes	characteristic, complemented
241	#1490	C₂×C₃₁	1	1	yes	characteristic
242	#1485	C₂×C₃₁	1	1	yes	characteristic, complemented
243	C₂²×C₃₁	—	2	1	no	—
244	C₂²×C₃₁	—	2	4	no	—
245	C₂²×C₃₁	—	2	1	no	—
246	C₂²×C₃₁	—	2	1	no	—
247	C₂²×C₃₁	—	4	2	no	—
248	C₂²×C₃₁	—	4	2	no	—
249	C₂²×C₃₁	—	8	4	no	—
250	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
251	C₄×C₃₁	—	2	1	no	—
252	C₄×C₃₁	—	2	1	no	—
253	C₄×C₃₁	—	2	1	no	—
254	C₄×C₃₁	—	2	1	no	—
255	C₄×C₃₁	—	4	2	no	—
256	C₄×C₃₁	—	4	2	no	—
257	C₄×C₃₁	—	4	2	no	—
258	C₄×C₃₁	—	4	2	no	—
259	C₄×C₃₁	—	4	2	no	—
260	C₄×C₃₁	—	8	1	no	—
261	C₄×C₃₁	—	8	1	no	—
262	C₄×C₃₁	C₂×#137	1	2	yes	—
263	C₄×C₃₁	#577	1	2	yes	—
264	#554	—	2	1	no	—
265	#554	—	2	2	no	—
266	#554	—	2	2	no	—
267	#554	C₂²×C₃₁	1	2	yes	—
268	D₄×Q₈	—	2	1	no	—
269	D₄×Q₈	C₂²×C₃₁	1	2	yes	—
270	#544	—	2	1	no	—
271	#543	—	2	2	no	—
272	#533	C₂²×C₃₁	1	1	yes	characteristic
273	#533	C₂²×C₃₁	1	1	yes	characteristic
274	#533	C₂²×C₃₁	1	1	yes	characteristic
275	C₂×C₂²⋊Q₈	—	2	2	no	—
276	#518	—	2	1	no	—
277	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
278	#501	—	2	1	no	—
279	#498	C₂²×C₃₁	1	1	yes	characteristic, complemented
280	#494	C₂²×C₃₁	1	1	yes	characteristic, complemented
281	#491	—	2	1	no	—
282	#487	C₂²×C₃₁	1	1	yes	characteristic, complemented
283	#487	C₂²×C₃₁	1	1	yes	characteristic, complemented
284	#485	—	2	2	no	—
285	#483	—	2	1	no	—
286	#477	C₂²×C₃₁	1	2	yes	complemented
287	#476	C₂²×C₃₁	1	2	yes	—
288	#475	C₂²×C₃₁	1	2	yes	complemented
289	#470	—	2	2	no	—
290	#461	C₂²×C₃₁	1	2	yes	—
291	#450	—	2	1	no	—
292	#441	—	2	2	no	—
293	C₄×SD₁₆	—	2	2	no	—
294	#432	C₂²×C₃₁	1	1	yes	characteristic
295	#431	C₂²×C₃₁	1	1	yes	characteristic
296	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
297	#333	C₂²×C₃₁	1	2	yes	complemented
298	#332	C₂²×C₃₁	1	2	yes	complemented
299	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	complemented
300	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	complemented
301	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
302	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
303	C₂×C₃₁	—	4	1	no	—
304	C₂×C₃₁	—	4	4	no	—
305	C₂×C₃₁	#3004	1	1	yes	characteristic
306	C₂×C₃₁	#3009	1	1	yes	characteristic
307	C₂×C₃₁	#3018	1	1	yes	characteristic
308	C₂²×Q₈	—	2	2	no	—
309	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
310	C₂³×C₄	—	2	1	no	—
311	C₂×Q₁₆	—	2	2	no	—
312	C₂×SD₁₆	—	2	2	no	—
313	C₂×M₁₆	—	2	1	no	—
314	C₄⋊Q₈	—	2	2	no	—
315	C₄⋊Q₈	—	2	2	no	—
316	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
317	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
318	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
319	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
320	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
321	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
322	#125	—	2	2	no	—
323	#125	—	2	2	no	—
324	#125	—	2	1	no	—
325	#125	—	2	1	no	—
326	#125	—	2	1	no	—
327	C₂²⋊Q₈	—	2	4	no	—
328	C₂²⋊Q₈	—	2	4	no	—
329	C₂²⋊Q₈	—	2	4	no	—
330	C₂²⋊Q₈	—	2	4	no	—
331	C₂²⋊Q₈	—	2	2	no	—
332	C₂²⋊Q₈	—	2	4	no	—
333	C₂²⋊Q₈	—	2	1	no	—
334	C₂²⋊Q₈	—	2	1	no	—
335	C₂²⋊Q₈	—	2	2	no	—
336	C₂²⋊Q₈	—	2	4	no	—
337	C₂²⋊Q₈	—	2	1	no	—
338	C₂²⋊Q₈	—	2	1	no	—
339	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
340	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
341	C₄×Q₈	—	2	1	no	—
342	C₄×Q₈	—	2	1	no	—
343	C₄×Q₈	—	2	1	no	—
344	C₄×Q₈	—	2	1	no	—
345	C₄×Q₈	D₄×C₃₁	1	2	yes	—
346	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
347	C₄×D₄	—	2	2	no	—
348	C₄×D₄	—	2	2	no	—
349	C₄×D₄	—	2	2	no	—
350	C₄×D₄	—	2	2	no	—
351	C₄×D₄	—	2	1	no	—
352	C₄×D₄	—	2	1	no	—
353	C₄×D₄	D₄×C₃₁	1	2	yes	—
354	C₄×D₄	C₂³×C₃₁	1	2	yes	—
355	#117	—	2	1	no	—
356	#117	—	2	1	no	—
357	#117	—	2	1	no	—
358	C₂×C₄⋊C₄	—	2	4	no	—
359	C₂×C₄⋊C₄	—	2	2	no	—
360	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
361	C₂×C₂²⋊C₄	—	2	4	no	—
362	C₂×C₂²⋊C₄	—	2	1	no	—
363	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
364	#107	—	2	2	no	—
365	#106	—	2	2	no	—
366	#106	—	2	2	no	—
367	#106	—	2	2	no	—
368	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
369	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
370	#103	—	2	2	no	—
371	#103	—	2	2	no	—
372	#103	—	2	2	no	—
373	#103	—	2	2	no	—
374	#103	—	2	2	no	—
375	#102	—	2	2	no	—
376	C₂²⋊C₈	—	2	1	no	—
377	C₂²⋊C₈	—	2	1	no	—
378	#97	—	2	1	no	—
379	#97	—	2	1	no	—
380	C₄×C₈	—	2	1	no	—
381	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
382	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
383	C₃₁	#18601	1	1	yes	characteristic, direct product factor, Sylow
384	C₂⁴	—	2	1	no	—
385	C₂×Q₈	—	2	4	no	—
386	C₂×Q₈	—	2	1	no	—
387	C₂×Q₈	—	2	1	no	—
388	C₂×Q₈	—	2	1	no	—
389	C₂×Q₈	—	2	1	no	—
390	C₂×Q₈	—	8	4	no	—
391	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
392	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
393	C₂×D₄	—	2	4	no	—
394	C₂×D₄	—	2	1	no	—
395	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
396	C₂²×C₄	—	2	4	no	—
397	C₂²×C₄	—	2	4	no	—
398	C₂²×C₄	—	2	1	no	—
399	C₂²×C₄	—	2	2	no	—
400	C₂²×C₄	—	2	2	no	—
401	C₂²×C₄	—	8	2	no	—
402	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
403	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
404	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
405	Q₁₆	—	4	4	no	—
406	SD₁₆	—	4	4	no	—
407	M₁₆	—	4	2	no	—
408	C₂×C₈	—	2	1	no	—
409	C₂×C₈	—	2	2	no	—
410	C₂×C₈	—	2	1	no	—
411	C₂×C₈	—	2	2	no	—
412	C₄⋊C₄	—	2	4	no	—
413	C₄⋊C₄	—	2	4	no	—
414	C₄⋊C₄	—	2	4	no	—
415	C₄⋊C₄	—	2	2	no	—
416	C₄⋊C₄	—	2	2	no	—
417	C₄⋊C₄	—	2	1	no	—
418	C₄⋊C₄	—	2	1	no	—
419	C₄⋊C₄	—	2	1	no	—
420	C₄⋊C₄	—	2	1	no	—
421	C₄⋊C₄	—	2	2	no	—
422	C₄⋊C₄	—	2	1	no	—
423	C₄⋊C₄	—	2	1	no	—
424	C₄⋊C₄	—	2	1	no	—
425	C₄⋊C₄	—	2	2	no	—
426	C₄⋊C₄	—	2	1	no	—
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	—	2	1	no	—
429	C₄⋊C₄	—	2	1	no	—
430	C₄⋊C₄	—	4	1	no	—
431	C₄⋊C₄	—	4	1	no	—
432	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
433	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
434	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
435	C₂²⋊C₄	—	2	4	no	—
436	C₂²⋊C₄	—	2	4	no	—
437	C₂²⋊C₄	—	2	2	no	—
438	C₂²⋊C₄	—	2	2	no	—
439	C₂²⋊C₄	—	2	2	no	—
440	C₂²⋊C₄	—	2	2	no	—
441	C₂²⋊C₄	—	4	1	no	—
442	C₄²	—	2	2	no	—
443	C₄²	—	2	2	no	—
444	C₄²	—	4	1	no	—
445	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
446	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
447	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
448	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
449	C₂³	—	2	4	no	—
450	C₂³	—	2	2	no	—
451	C₂³	—	8	1	no	—
452	C₂³	—	8	1	no	—
453	C₂³	C₃₁×#120	1	1	yes	characteristic
454	Q₈	—	2	4	no	—
455	Q₈	—	4	2	no	—
456	Q₈	—	4	2	no	—
457	Q₈	—	4	2	no	—
458	Q₈	—	4	2	no	—
459	Q₈	—	8	1	no	—
460	Q₈	—	8	4	no	—
461	D₄	—	2	4	no	—
462	D₄	—	2	2	no	—
463	D₄	C₃₁×#137	1	4	yes	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	2	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	4	1	no	—
472	C₂×C₄	—	4	4	no	—
473	C₂×C₄	—	4	1	no	—
474	C₂×C₄	—	4	1	no	—
475	C₂×C₄	—	8	2	no	—
476	C₂×C₄	—	8	1	no	—
477	C₂×C₄	—	8	4	no	—
478	C₂×C₄	—	8	4	no	—
479	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
480	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
481	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
482	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
483	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
484	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
485	C₈	—	2	2	no	—
486	C₈	—	4	1	no	—
487	C₈	—	4	2	no	—
488	C₈	—	4	2	no	—
489	C₂²	—	2	1	no	—
490	C₂²	—	2	4	no	—
491	C₂²	—	2	1	no	—
492	C₂²	—	2	1	no	—
493	C₂²	—	4	2	no	—
494	C₂²	—	4	2	no	—
495	C₂²	—	8	4	no	—
496	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
497	C₄	—	2	1	no	—
498	C₄	—	2	1	no	—
499	C₄	—	2	1	no	—
500	C₄	—	2	1	no	—
501	C₄	—	4	2	no	—
502	C₄	—	4	2	no	—
503	C₄	—	4	2	no	—
504	C₄	—	4	2	no	—
505	C₄	—	4	2	no	—
506	C₄	—	8	1	no	—
507	C₄	—	8	1	no	—
508	C₄	C₂×C₃₁×#137	1	2	yes	—
509	C₄	C₃₁×#577	1	2	yes	—
510	C₂	—	4	1	no	—
511	C₂	—	4	4	no	—
512	C₂	C₃₁×#3004	1	1	yes	characteristic
513	C₂	C₃₁×#3009	1	1	yes	characteristic
514	C₂	C₃₁×#3018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₂²×D₄×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
2. C₃₁×#3237	C₂	semidirect
3. C₃₁×#3185	C₂	semidirect
5. C₃₁×#1651	C₂	semidirect
6. C₃₁×#1643	C₂	semidirect
7. C₃₁×#1637	C₂	semidirect
8. C₃₁×#1632	C₂	semidirect
9. C₃₁×#1554	C₂	semidirect
11. C₃₁×#1485	C₂	semidirect
27. C₃₁×#498	C₂²	semidirect
28. C₃₁×#494	C₂²	semidirect
30. C₃₁×#487	C₂²	semidirect
31. C₃₁×#487	C₂²	semidirect
34. C₃₁×#477	C₂²	semidirect
36. C₃₁×#475	C₂²	semidirect
45. C₃₁×#333	C₂²	semidirect
46. C₃₁×#332	C₂²	semidirect
47. C₃₁×Q₈⋊C₈	C₂²	semidirect
48. C₃₁×Q₈⋊C₈	C₂²	semidirect
191. #18601	C₃₁	direct (N × H)
233. #3237	C₂×C₃₁	semidirect
234. #3185	C₂×C₃₁	semidirect
236. #1651	C₂×C₃₁	semidirect
237. #1643	C₂×C₃₁	semidirect
238. #1637	C₂×C₃₁	semidirect
239. #1632	C₂×C₃₁	semidirect
240. #1554	C₂×C₃₁	semidirect
242. #1485	C₂×C₃₁	semidirect
279. #498	C₂²×C₃₁	semidirect
280. #494	C₂²×C₃₁	semidirect
282. #487	C₂²×C₃₁	semidirect
283. #487	C₂²×C₃₁	semidirect
286. #477	C₂²×C₃₁	semidirect
288. #475	C₂²×C₃₁	semidirect
297. #333	C₂²×C₃₁	semidirect
298. #332	C₂²×C₃₁	semidirect
299. Q₈⋊C₈	C₂²×C₃₁	semidirect
300. Q₈⋊C₈	C₂²×C₃₁	semidirect
383. C₃₁	#18601	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×#3501	2
C₃₁×#3237	2
C₃₁×#3185	2
C₃₁×#3101	2
C₃₁×#1651	2
C₃₁×#1643	2
C₃₁×#1637	2
C₃₁×#1632	2
C₃₁×#1554	2
C₃₁×#1490	2
C₃₁×#1485	2
#18601	31

Subgroups — C31×#18601 #21475360

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#18601 #21475360