Subgroups — C₃₁×#17686 #21474445

Order 7936.

902 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3510	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2970	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1535	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1533	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1532	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1531	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1529	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1524	C₂	1	2	yes	maximal, complemented
9	C₂×C₃₁×#329	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#559	—	2	2	no	—
11	C₃₁×#556	—	2	2	no	—
12	C₃₁×#551	—	2	2	no	—
13	C₃₁×#551	—	2	2	no	—
14	C₃₁×#542	C₄	1	2	yes	—
15	C₃₁×#542	C₄	1	2	yes	—
16	C₃₁×#536	C₄	1	1	yes	characteristic
17	C₃₁×#535	C₄	1	1	yes	characteristic
18	C₃₁×#533	C₂²	1	1	yes	characteristic
19	C₃₁×#529	C₂²	1	2	yes	—
20	C₃₁×#529	C₂²	1	2	yes	—
21	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
22	C₃₁×#519	C₄	1	1	yes	characteristic
23	C₃₁×#518	C₄	1	1	yes	characteristic
24	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
25	C₃₁×#446	—	2	2	no	—
26	C₃₁×#435	—	2	2	no	—
27	C₃₁×#435	—	2	2	no	—
28	C₈×D₄×C₃₁	—	2	2	no	—
29	C₃₁×#432	C₂²	1	2	yes	—
30	C₃₁×#432	C₂²	1	2	yes	complemented
31	C₃₁×#432	C₂²	1	2	yes	complemented
32	C₄×M₁₆×C₃₁	C₂²	1	2	yes	complemented
33	C₂×C₃₁×#97	C₂²	1	2	yes	—
34	C₂×C₃₁×#97	C₂²	1	2	yes	complemented
35	C₃₁×#330	C₂²	1	2	yes	complemented
36	C₃₁×#330	C₂²	1	2	yes	complemented
37	C₃₁×#330	C₂²	1	2	yes	complemented
38	C₃₁×#330	C₂²	1	2	yes	complemented
39	C₃₁×#329	C₂²	1	1	yes	characteristic
40	C₃₁×#329	C₂²	1	2	yes	—
41	C₃₁×#329	C₂²	1	1	yes	characteristic
42	C₃₁×#329	C₂²	1	2	yes	—
43	C₃₁×#329	C₂²	1	1	yes	characteristic
44	C₃₁×#329	C₂²	1	1	yes	characteristic
45	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
46	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
47	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
48	C₂×M₁₆×C₃₁	—	2	2	no	—
49	C₂²×C₈×C₃₁	—	2	2	no	—
50	C₂²×C₈×C₃₁	—	2	2	no	—
51	C₃₁×C₄⋊Q₈	—	2	1	no	—
52	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
53	C₃₁×#127	—	2	1	no	—
54	C₃₁×#126	—	2	2	no	—
55	C₃₁×#126	—	2	2	no	—
56	C₃₁×#126	—	2	2	no	—
57	C₃₁×#126	—	2	2	no	—
58	C₃₁×#125	C₂³	1	2	yes	—
59	C₃₁×#124	—	2	1	no	—
60	C₃₁×#124	—	2	1	no	—
61	C₃₁×#124	—	2	2	no	—
62	C₃₁×#124	—	2	2	no	—
63	C₃₁×#124	—	2	2	no	—
64	C₃₁×#124	C₂×C₄	1	1	yes	characteristic
65	C₃₁×#124	C₂×C₄	1	1	yes	characteristic
66	C₃₁×#124	C₂³	1	1	yes	characteristic
67	C₃₁×#124	C₂³	1	1	yes	characteristic
68	C₃₁×#123	—	2	2	no	—
69	C₃₁×#123	—	2	2	no	—
70	C₃₁×#123	—	2	2	no	—
71	C₃₁×#123	—	2	2	no	—
72	C₃₁×#123	—	2	2	no	—
73	C₃₁×C₂²⋊Q₈	—	2	2	no	—
74	C₃₁×C₂²⋊Q₈	—	2	2	no	—
75	C₃₁×C₂²⋊Q₈	—	2	2	no	—
76	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
77	C₃₁×C₂²⋊Q₈	C₂×C₄	1	2	yes	—
78	C₃₁×#121	C₂×C₄	1	2	yes	—
79	C₃₁×#121	C₂×C₄	1	2	yes	—
80	C₃₁×#120	—	2	2	no	—
81	C₄×Q₈×C₃₁	—	2	2	no	—
82	C₄×Q₈×C₃₁	—	2	2	no	—
83	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
84	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
85	C₄×Q₈×C₃₁	D₄	1	2	yes	—
86	C₄×D₄×C₃₁	—	2	2	no	—
87	C₄×D₄×C₃₁	—	2	2	no	—
88	C₄×D₄×C₃₁	—	2	2	no	—
89	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
90	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
91	C₄×D₄×C₃₁	D₄	1	2	yes	—
92	C₄×D₄×C₃₁	D₄	1	2	yes	—
93	C₄×D₄×C₃₁	D₄	1	2	yes	—
94	C₃₁×#117	—	2	1	no	—
95	C₃₁×#117	—	2	1	no	—
96	C₃₁×#117	—	2	2	no	—
97	C₃₁×#117	—	2	2	no	—
98	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
99	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
100	C₃₁×#117	C₂×C₄	1	1	yes	characteristic
101	C₃₁×#117	C₂×C₄	1	2	yes	—
102	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
103	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
104	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
105	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
106	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
107	C₃₁×C₄⋊C₈	—	2	2	no	—
108	C₃₁×C₄⋊C₈	—	2	2	no	—
109	C₃₁×C₄⋊C₈	—	2	2	no	—
110	C₃₁×C₂²⋊C₈	—	2	2	no	—
111	C₃₁×C₂²⋊C₈	—	2	2	no	—
112	C₃₁×C₂²⋊C₈	—	2	2	no	—
113	C₃₁×#97	—	2	2	no	—
114	C₃₁×#97	—	2	2	no	—
115	C₃₁×#97	C₂³	1	2	yes	—
116	C₃₁×#97	C₂³	1	2	yes	—
117	C₃₁×#97	C₂³	1	2	yes	—
118	C₃₁×#97	C₂³	1	2	yes	—
119	C₄×C₈×C₃₁	—	2	2	no	—
120	C₂⁴×C₃₁	—	2	1	no	—
121	#41×C₃₁	—	4	2	no	—
122	C₂×Q₈×C₃₁	—	2	1	no	—
123	C₂×Q₈×C₃₁	—	2	2	no	—
124	C₂×Q₈×C₃₁	—	4	2	no	—
125	C₂×Q₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
126	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
127	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
128	C₂×D₄×C₃₁	—	2	2	no	—
129	C₂×D₄×C₃₁	—	2	2	no	—
130	C₂×D₄×C₃₁	—	2	1	no	—
131	C₂×D₄×C₃₁	—	4	2	no	—
132	C₂×D₄×C₃₁	C₂²⋊C₄	1	2	yes	—
133	C₂×D₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
134	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
135	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
136	C₂²×C₄×C₃₁	—	2	1	no	—
137	C₂²×C₄×C₃₁	—	2	2	no	—
138	C₂²×C₄×C₃₁	—	2	2	no	—
139	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
140	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
141	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
142	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
143	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
144	M₁₆×C₃₁	—	4	4	no	—
145	C₂×C₈×C₃₁	—	2	2	no	—
146	C₂×C₈×C₃₁	—	2	2	no	—
147	C₂×C₈×C₃₁	—	2	2	no	—
148	C₂×C₈×C₃₁	—	2	2	no	—
149	C₂×C₈×C₃₁	—	4	4	no	—
150	C₂×C₈×C₃₁	—	4	4	no	—
151	C₄⋊C₄×C₃₁	—	2	2	no	—
152	C₄⋊C₄×C₃₁	—	2	2	no	—
153	C₄⋊C₄×C₃₁	—	2	2	no	—
154	C₄⋊C₄×C₃₁	—	2	2	no	—
155	C₄⋊C₄×C₃₁	—	2	2	no	—
156	C₄⋊C₄×C₃₁	—	2	2	no	—
157	C₄⋊C₄×C₃₁	—	2	2	no	—
158	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
159	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
160	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
161	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
162	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
163	C₂²⋊C₄×C₃₁	—	2	1	no	—
164	C₂²⋊C₄×C₃₁	—	2	1	no	—
165	C₂²⋊C₄×C₃₁	—	2	2	no	—
166	C₂²⋊C₄×C₃₁	—	2	2	no	—
167	C₂²⋊C₄×C₃₁	—	2	2	no	—
168	C₂²⋊C₄×C₃₁	—	2	2	no	—
169	C₂²⋊C₄×C₃₁	—	2	2	no	—
170	C₂²⋊C₄×C₃₁	—	2	2	no	—
171	C₂²⋊C₄×C₃₁	—	2	2	no	—
172	C₂²⋊C₄×C₃₁	—	2	2	no	—
173	C₂²⋊C₄×C₃₁	—	2	2	no	—
174	C₂²⋊C₄×C₃₁	—	2	2	no	—
175	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
176	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
177	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
178	C₂²⋊C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
179	C₄²×C₃₁	—	2	2	no	—
180	C₄²×C₃₁	—	2	2	no	—
181	C₄²×C₃₁	—	2	1	no	—
182	C₄²×C₃₁	—	2	1	no	—
183	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
184	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
185	C₄²×C₃₁	C₂×D₄	1	2	yes	—
186	C₄²×C₃₁	C₂×D₄	1	2	yes	—
187	C₄²×C₃₁	C₂×D₄	1	2	yes	—
188	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
189	#17686	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
190	C₂³×C₃₁	—	2	1	no	—
191	C₂³×C₃₁	—	2	1	no	—
192	C₂³×C₃₁	—	4	2	no	—
193	C₂³×C₃₁	—	8	2	no	—
194	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
195	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
196	Q₈×C₃₁	—	4	1	no	—
197	Q₈×C₃₁	—	4	1	no	—
198	Q₈×C₃₁	—	4	1	no	—
199	Q₈×C₃₁	—	4	2	no	—
200	Q₈×C₃₁	—	8	2	no	—
201	D₄×C₃₁	—	2	4	no	—
202	D₄×C₃₁	—	4	1	no	—
203	D₄×C₃₁	—	4	1	no	—
204	D₄×C₃₁	—	4	1	no	—
205	D₄×C₃₁	—	4	2	no	—
206	D₄×C₃₁	—	8	2	no	—
207	D₄×C₃₁	—	8	2	no	—
208	C₂×C₄×C₃₁	—	2	2	no	—
209	C₂×C₄×C₃₁	—	2	2	no	—
210	C₂×C₄×C₃₁	—	2	1	no	—
211	C₂×C₄×C₃₁	—	2	1	no	—
212	C₂×C₄×C₃₁	—	2	2	no	—
213	C₂×C₄×C₃₁	—	4	2	no	—
214	C₂×C₄×C₃₁	—	4	2	no	—
215	C₂×C₄×C₃₁	—	4	2	no	—
216	C₂×C₄×C₃₁	—	4	2	no	—
217	C₂×C₄×C₃₁	—	4	2	no	—
218	C₂×C₄×C₃₁	—	8	2	no	—
219	C₂×C₄×C₃₁	—	8	2	no	—
220	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
221	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
222	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
223	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
224	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
225	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
226	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
227	C₈×C₃₁	—	2	4	no	—
228	C₈×C₃₁	—	4	2	no	—
229	C₈×C₃₁	—	4	2	no	—
230	C₈×C₃₁	—	4	2	no	—
231	#3510	C₂×C₃₁	1	1	yes	characteristic
232	#2970	C₂×C₃₁	1	2	yes	complemented
233	#1535	C₂×C₃₁	1	2	yes	complemented
234	#1533	C₂×C₃₁	1	1	yes	characteristic, complemented
235	#1532	C₂×C₃₁	1	2	yes	complemented
236	#1531	C₂×C₃₁	1	2	yes	complemented
237	#1529	C₂×C₃₁	1	2	yes	complemented
238	#1524	C₂×C₃₁	1	2	yes	complemented
239	C₂×#329	C₂×C₃₁	1	1	yes	characteristic, complemented
240	C₂²×C₃₁	—	2	2	no	—
241	C₂²×C₃₁	—	2	2	no	—
242	C₂²×C₃₁	—	2	2	no	—
243	C₂²×C₃₁	—	2	2	no	—
244	C₂²×C₃₁	—	4	1	no	—
245	C₂²×C₃₁	—	4	1	no	—
246	C₂²×C₃₁	—	4	2	no	—
247	C₂²×C₃₁	—	4	2	no	—
248	C₂²×C₃₁	—	16	2	no	—
249	C₂²×C₃₁	#413	1	2	yes	—
250	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
251	C₄×C₃₁	—	2	1	no	—
252	C₄×C₃₁	—	2	1	no	—
253	C₄×C₃₁	—	2	2	no	—
254	C₄×C₃₁	—	2	2	no	—
255	C₄×C₃₁	—	2	2	no	—
256	C₄×C₃₁	—	2	2	no	—
257	C₄×C₃₁	—	2	2	no	—
258	C₄×C₃₁	—	4	1	no	—
259	C₄×C₃₁	—	4	1	no	—
260	C₄×C₃₁	—	4	2	no	—
261	C₄×C₃₁	—	4	2	no	—
262	#559	—	2	2	no	—
263	#556	—	2	2	no	—
264	#551	—	2	2	no	—
265	#551	—	2	2	no	—
266	#542	C₄×C₃₁	1	2	yes	—
267	#542	C₄×C₃₁	1	2	yes	—
268	#536	C₄×C₃₁	1	1	yes	characteristic
269	#535	C₄×C₃₁	1	1	yes	characteristic
270	#533	C₂²×C₃₁	1	1	yes	characteristic
271	#529	C₂²×C₃₁	1	2	yes	—
272	#529	C₂²×C₃₁	1	2	yes	—
273	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
274	#519	C₄×C₃₁	1	1	yes	characteristic
275	#518	C₄×C₃₁	1	1	yes	characteristic
276	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
277	#446	—	2	2	no	—
278	#435	—	2	2	no	—
279	#435	—	2	2	no	—
280	C₈×D₄	—	2	2	no	—
281	#432	C₂²×C₃₁	1	2	yes	—
282	#432	C₂²×C₃₁	1	2	yes	complemented
283	#432	C₂²×C₃₁	1	2	yes	complemented
284	C₄×M₁₆	C₂²×C₃₁	1	2	yes	complemented
285	C₂×#97	C₂²×C₃₁	1	2	yes	—
286	C₂×#97	C₂²×C₃₁	1	2	yes	complemented
287	#330	C₂²×C₃₁	1	2	yes	complemented
288	#330	C₂²×C₃₁	1	2	yes	complemented
289	#330	C₂²×C₃₁	1	2	yes	complemented
290	#330	C₂²×C₃₁	1	2	yes	complemented
291	#329	C₂²×C₃₁	1	1	yes	characteristic
292	#329	C₂²×C₃₁	1	2	yes	—
293	#329	C₂²×C₃₁	1	1	yes	characteristic
294	#329	C₂²×C₃₁	1	2	yes	—
295	#329	C₂²×C₃₁	1	1	yes	characteristic
296	#329	C₂²×C₃₁	1	1	yes	characteristic
297	C₂×C₃₁	—	2	2	no	—
298	C₂×C₃₁	—	4	2	no	—
299	C₂×C₃₁	—	8	1	no	—
300	C₂×C₃₁	—	8	1	no	—
301	C₂×C₃₁	C₂×#413	1	1	yes	characteristic
302	C₂×C₃₁	#2901	1	2	yes	—
303	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
304	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
305	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
306	C₂×M₁₆	—	2	2	no	—
307	C₂²×C₈	—	2	2	no	—
308	C₂²×C₈	—	2	2	no	—
309	C₄⋊Q₈	—	2	1	no	—
310	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
311	#127	—	2	1	no	—
312	#126	—	2	2	no	—
313	#126	—	2	2	no	—
314	#126	—	2	2	no	—
315	#126	—	2	2	no	—
316	#125	C₂³×C₃₁	1	2	yes	—
317	#124	—	2	1	no	—
318	#124	—	2	1	no	—
319	#124	—	2	2	no	—
320	#124	—	2	2	no	—
321	#124	—	2	2	no	—
322	#124	C₂×C₄×C₃₁	1	1	yes	characteristic
323	#124	C₂×C₄×C₃₁	1	1	yes	characteristic
324	#124	C₂³×C₃₁	1	1	yes	characteristic
325	#124	C₂³×C₃₁	1	1	yes	characteristic
326	#123	—	2	2	no	—
327	#123	—	2	2	no	—
328	#123	—	2	2	no	—
329	#123	—	2	2	no	—
330	#123	—	2	2	no	—
331	C₂²⋊Q₈	—	2	2	no	—
332	C₂²⋊Q₈	—	2	2	no	—
333	C₂²⋊Q₈	—	2	2	no	—
334	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
335	C₂²⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
336	#121	C₂×C₄×C₃₁	1	2	yes	—
337	#121	C₂×C₄×C₃₁	1	2	yes	—
338	#120	—	2	2	no	—
339	C₄×Q₈	—	2	2	no	—
340	C₄×Q₈	—	2	2	no	—
341	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
342	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
343	C₄×Q₈	D₄×C₃₁	1	2	yes	—
344	C₄×D₄	—	2	2	no	—
345	C₄×D₄	—	2	2	no	—
346	C₄×D₄	—	2	2	no	—
347	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
348	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
349	C₄×D₄	D₄×C₃₁	1	2	yes	—
350	C₄×D₄	D₄×C₃₁	1	2	yes	—
351	C₄×D₄	D₄×C₃₁	1	2	yes	—
352	#117	—	2	1	no	—
353	#117	—	2	1	no	—
354	#117	—	2	2	no	—
355	#117	—	2	2	no	—
356	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
357	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
358	#117	C₂×C₄×C₃₁	1	1	yes	characteristic
359	#117	C₂×C₄×C₃₁	1	2	yes	—
360	C₂×C₂²⋊C₄	—	2	2	no	—
361	C₂×C₂²⋊C₄	—	2	1	no	—
362	C₂×C₂²⋊C₄	—	2	1	no	—
363	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
364	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
365	C₄⋊C₈	—	2	2	no	—
366	C₄⋊C₈	—	2	2	no	—
367	C₄⋊C₈	—	2	2	no	—
368	C₂²⋊C₈	—	2	2	no	—
369	C₂²⋊C₈	—	2	2	no	—
370	C₂²⋊C₈	—	2	2	no	—
371	#97	—	2	2	no	—
372	#97	—	2	2	no	—
373	#97	C₂³×C₃₁	1	2	yes	—
374	#97	C₂³×C₃₁	1	2	yes	—
375	#97	C₂³×C₃₁	1	2	yes	—
376	#97	C₂³×C₃₁	1	2	yes	—
377	C₄×C₈	—	2	2	no	—
378	C₃₁	#17686	1	1	yes	characteristic, direct product factor, Sylow
379	C₂⁴	—	2	1	no	—
380	#41	—	4	2	no	—
381	C₂×Q₈	—	2	1	no	—
382	C₂×Q₈	—	2	2	no	—
383	C₂×Q₈	—	4	2	no	—
384	C₂×Q₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
385	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
386	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
387	C₂×D₄	—	2	2	no	—
388	C₂×D₄	—	2	2	no	—
389	C₂×D₄	—	2	1	no	—
390	C₂×D₄	—	4	2	no	—
391	C₂×D₄	C₂²⋊C₄×C₃₁	1	2	yes	—
392	C₂×D₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
393	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
394	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	2	2	no	—
397	C₂²×C₄	—	2	2	no	—
398	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
399	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
400	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
401	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
402	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
403	M₁₆	—	4	4	no	—
404	C₂×C₈	—	2	2	no	—
405	C₂×C₈	—	2	2	no	—
406	C₂×C₈	—	2	2	no	—
407	C₂×C₈	—	2	2	no	—
408	C₂×C₈	—	4	4	no	—
409	C₂×C₈	—	4	4	no	—
410	C₄⋊C₄	—	2	2	no	—
411	C₄⋊C₄	—	2	2	no	—
412	C₄⋊C₄	—	2	2	no	—
413	C₄⋊C₄	—	2	2	no	—
414	C₄⋊C₄	—	2	2	no	—
415	C₄⋊C₄	—	2	2	no	—
416	C₄⋊C₄	—	2	2	no	—
417	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
418	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
419	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
420	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
421	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
422	C₂²⋊C₄	—	2	1	no	—
423	C₂²⋊C₄	—	2	1	no	—
424	C₂²⋊C₄	—	2	2	no	—
425	C₂²⋊C₄	—	2	2	no	—
426	C₂²⋊C₄	—	2	2	no	—
427	C₂²⋊C₄	—	2	2	no	—
428	C₂²⋊C₄	—	2	2	no	—
429	C₂²⋊C₄	—	2	2	no	—
430	C₂²⋊C₄	—	2	2	no	—
431	C₂²⋊C₄	—	2	2	no	—
432	C₂²⋊C₄	—	2	2	no	—
433	C₂²⋊C₄	—	2	2	no	—
434	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
435	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
436	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
437	C₂²⋊C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
438	C₄²	—	2	2	no	—
439	C₄²	—	2	2	no	—
440	C₄²	—	2	1	no	—
441	C₄²	—	2	1	no	—
442	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
443	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
444	C₄²	C₂×D₄×C₃₁	1	2	yes	—
445	C₄²	C₂×D₄×C₃₁	1	2	yes	—
446	C₄²	C₂×D₄×C₃₁	1	2	yes	—
447	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
448	C₂³	—	2	1	no	—
449	C₂³	—	2	1	no	—
450	C₂³	—	4	2	no	—
451	C₂³	—	8	2	no	—
452	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
453	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
454	Q₈	—	4	1	no	—
455	Q₈	—	4	1	no	—
456	Q₈	—	4	1	no	—
457	Q₈	—	4	2	no	—
458	Q₈	—	8	2	no	—
459	D₄	—	2	4	no	—
460	D₄	—	4	1	no	—
461	D₄	—	4	1	no	—
462	D₄	—	4	1	no	—
463	D₄	—	4	2	no	—
464	D₄	—	8	2	no	—
465	D₄	—	8	2	no	—
466	C₂×C₄	—	2	2	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	1	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	4	2	no	—
472	C₂×C₄	—	4	2	no	—
473	C₂×C₄	—	4	2	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	4	2	no	—
476	C₂×C₄	—	8	2	no	—
477	C₂×C₄	—	8	2	no	—
478	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
479	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
480	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
481	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
482	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
483	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
484	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
485	C₈	—	2	4	no	—
486	C₈	—	4	2	no	—
487	C₈	—	4	2	no	—
488	C₈	—	4	2	no	—
489	C₂²	—	2	2	no	—
490	C₂²	—	2	2	no	—
491	C₂²	—	2	2	no	—
492	C₂²	—	2	2	no	—
493	C₂²	—	4	1	no	—
494	C₂²	—	4	1	no	—
495	C₂²	—	4	2	no	—
496	C₂²	—	4	2	no	—
497	C₂²	—	16	2	no	—
498	C₂²	C₃₁×#413	1	2	yes	—
499	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
500	C₄	—	2	1	no	—
501	C₄	—	2	1	no	—
502	C₄	—	2	2	no	—
503	C₄	—	2	2	no	—
504	C₄	—	2	2	no	—
505	C₄	—	2	2	no	—
506	C₄	—	2	2	no	—
507	C₄	—	4	1	no	—
508	C₄	—	4	1	no	—
509	C₄	—	4	2	no	—
510	C₄	—	4	2	no	—
511	C₂	—	2	2	no	—
512	C₂	—	4	2	no	—
513	C₂	—	8	1	no	—
514	C₂	—	8	1	no	—
515	C₂	C₂×C₃₁×#413	1	1	yes	characteristic
516	C₂	C₃₁×#2901	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#535	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2970	C₂	semidirect
3. C₃₁×#1535	C₂	semidirect
4. C₃₁×#1533	C₂	semidirect
5. C₃₁×#1532	C₂	semidirect
6. C₃₁×#1531	C₂	semidirect
7. C₃₁×#1529	C₂	semidirect
8. C₃₁×#1524	C₂	semidirect
9. C₂×C₃₁×#329	C₂	semidirect
30. C₃₁×#432	C₂²	semidirect
31. C₃₁×#432	C₂²	semidirect
32. C₄×M₁₆×C₃₁	C₂²	semidirect
34. C₂×C₃₁×#97	C₂²	semidirect
35. C₃₁×#330	C₂²	semidirect
36. C₃₁×#330	C₂²	semidirect
37. C₃₁×#330	C₂²	semidirect
38. C₃₁×#330	C₂²	semidirect
189. #17686	C₃₁	direct (N × H)
232. #2970	C₂×C₃₁	semidirect
233. #1535	C₂×C₃₁	semidirect
234. #1533	C₂×C₃₁	semidirect
235. #1532	C₂×C₃₁	semidirect
236. #1531	C₂×C₃₁	semidirect
237. #1529	C₂×C₃₁	semidirect
238. #1524	C₂×C₃₁	semidirect
239. C₂×#329	C₂×C₃₁	semidirect
282. #432	C₂²×C₃₁	semidirect
283. #432	C₂²×C₃₁	semidirect
284. C₄×M₁₆	C₂²×C₃₁	semidirect
286. C₂×#97	C₂²×C₃₁	semidirect
287. #330	C₂²×C₃₁	semidirect
288. #330	C₂²×C₃₁	semidirect
289. #330	C₂²×C₃₁	semidirect
290. #330	C₂²×C₃₁	semidirect
378. C₃₁	#17686	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×#3510	2
C₃₁×#2970	2
C₃₁×#1535	2
C₃₁×#1533	2
C₃₁×#1532	2
C₃₁×#1531	2
C₃₁×#1529	2
C₃₁×#1524	2
C₂×C₃₁×#329	2
#17686	31

Subgroups — C31×#17686 #21474445

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17686 #21474445