Subgroups — C₃₁×#61893 #21518652

Order 7936.

1698 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3507	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#557	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3400	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3398	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3374	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3372	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3218	C₂	1	4	yes	maximal, complemented
8	C₃₁×#3209	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#3206	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#3117	C₂	1	4	yes	maximal, complemented
11	C₃₁×#3104	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#3103	C₂	1	1	yes	maximal, characteristic
13	C₃₁×#2983	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#559	C₂²	1	2	yes	—
15	C₃₁×#557	—	2	2	no	—
16	C₃₁×#557	—	2	2	no	—
17	C₃₁×#557	C₂²	1	4	yes	—
18	C₃₁×#557	C₂²	1	4	yes	—
19	C₃₁×#557	C₂²	1	2	yes	—
20	C₃₁×#556	—	2	2	no	—
21	C₃₁×#556	C₂²	1	2	yes	—
22	C₃₁×#554	C₂²	1	2	yes	—
23	C₃₁×#552	—	2	2	no	—
24	C₃₁×#539	—	2	2	no	—
25	C₃₁×#536	—	2	1	no	—
26	C₃₁×#533	C₂²	1	1	yes	characteristic
27	C₃₁×#532	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
30	C₃₁×#529	C₂²	1	4	yes	—
31	C₂×C₃₁×#125	—	2	1	no	—
32	C₂×C₃₁×#125	—	2	2	no	—
33	C₃₁×#520	—	2	1	no	—
34	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
35	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
36	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
37	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
38	C₃₁×#497	C₂²	1	8	yes	—
39	C₃₁×#494	C₂²	1	2	yes	—
40	C₃₁×#492	C₂²	1	2	yes	—
41	C₃₁×#491	C₂²	1	8	yes	—
42	C₃₁×#487	C₂²	1	2	yes	—
43	C₃₁×#487	C₂²	1	2	yes	—
44	C₃₁×#484	C₂²	1	2	yes	—
45	C₃₁×#481	C₂²	1	2	yes	—
46	C₃₁×#479	C₂²	1	8	yes	—
47	C₃₁×#477	C₂²	1	2	yes	—
48	C₃₁×#475	C₂²	1	2	yes	—
49	C₃₁×#464	C₂²	1	8	yes	—
50	C₃₁×#462	C₂²	1	2	yes	—
51	C₃₁×#462	C₂²	1	8	yes	—
52	C₃₁×#462	C₂²	1	4	yes	—
53	C₃₁×#461	C₂²	1	4	yes	—
54	C₃₁×#461	C₂²	1	8	yes	—
55	C₃₁×#460	C₂²	1	2	yes	—
56	C₃₁×#451	C₂²	1	2	yes	—
57	C₃₁×#451	C₂²	1	4	yes	—
58	C₃₁×#448	C₂²	1	2	yes	—
59	C₃₁×#448	C₂²	1	4	yes	—
60	C₃₁×#446	C₂²	1	4	yes	—
61	C₃₁×#446	C₂²	1	4	yes	—
62	C₃₁×#441	C₂²	1	8	yes	—
63	C₃₁×#440	C₂²	1	8	yes	—
64	C₄×Q₁₆×C₃₁	C₂²	1	4	yes	—
65	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	—
66	C₃₁×#432	C₂²	1	4	yes	—
67	C₃₁×#431	C₂²	1	1	yes	characteristic
68	C₃₁×#431	C₂²	1	1	yes	characteristic
69	C₂×#41×C₃₁	—	2	1	no	—
70	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
71	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
72	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
73	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
74	C₂×Q₁₆×C₃₁	C₂³	1	8	yes	—
75	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
76	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
77	C₃₁×C₄⋊Q₈	—	2	2	no	—
78	C₃₁×C₄⋊Q₈	—	2	4	no	—
79	C₃₁×C₄⋊Q₈	—	2	1	no	—
80	C₃₁×C₄⋊Q₈	—	2	1	no	—
81	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
82	C₃₁×C₄⋊Q₈	C₂³	1	8	yes	—
83	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
84	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
85	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
86	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
87	C₃₁×#127	C₂³	1	1	yes	characteristic
88	C₃₁×#126	—	2	4	no	—
89	C₃₁×#126	—	2	4	no	—
90	C₃₁×#125	—	2	4	no	—
91	C₃₁×#125	—	2	4	no	—
92	C₃₁×#125	—	2	8	no	—
93	C₃₁×#125	—	2	8	no	—
94	C₃₁×#125	—	2	4	no	—
95	C₃₁×#125	—	2	1	no	—
96	C₃₁×#125	C₂³	1	4	yes	—
97	C₃₁×#124	—	2	2	no	—
98	C₃₁×#124	—	2	1	no	—
99	C₃₁×#124	C₂³	1	4	yes	—
100	C₃₁×#123	—	2	4	no	—
101	C₃₁×#123	—	2	4	no	—
102	C₃₁×C₂²⋊Q₈	—	2	2	no	—
103	C₃₁×C₂²⋊Q₈	—	2	2	no	—
104	C₃₁×C₂²⋊Q₈	—	2	1	no	—
105	C₃₁×C₂²⋊Q₈	—	2	4	no	—
106	C₃₁×C₂²⋊Q₈	C₂³	1	2	yes	—
107	C₃₁×#121	—	2	1	no	—
108	C₃₁×#121	C₂³	1	2	yes	—
109	C₄×Q₈×C₃₁	—	2	4	no	—
110	C₄×Q₈×C₃₁	—	2	4	no	—
111	C₄×Q₈×C₃₁	—	2	2	no	—
112	C₄×Q₈×C₃₁	—	2	4	no	—
113	C₄×Q₈×C₃₁	—	2	1	no	—
114	C₄×Q₈×C₃₁	—	2	1	no	—
115	C₄×Q₈×C₃₁	—	2	1	no	—
116	C₄×Q₈×C₃₁	D₄	1	2	yes	—
117	C₄×Q₈×C₃₁	D₄	1	2	yes	—
118	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
119	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
120	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
121	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
122	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
123	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
124	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
125	C₄×D₄×C₃₁	—	2	4	no	—
126	C₄×D₄×C₃₁	—	2	2	no	—
127	C₄×D₄×C₃₁	—	2	1	no	—
128	C₄×D₄×C₃₁	C₂³	1	2	yes	—
129	C₄×D₄×C₃₁	C₂³	1	4	yes	—
130	C₃₁×#117	—	2	1	no	—
131	C₃₁×#117	—	2	2	no	—
132	C₃₁×#117	—	2	4	no	—
133	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
134	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
135	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
136	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
137	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
138	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
139	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
140	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
141	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
142	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
143	C₂×C₄²×C₃₁	—	2	2	no	—
144	C₂×C₄²×C₃₁	—	2	1	no	—
145	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
146	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
147	C₃₁×#107	C₂³	1	4	yes	—
148	C₃₁×#106	C₂³	1	4	yes	—
149	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
150	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
151	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
152	C₃₁×#103	C₂³	1	4	yes	—
153	C₃₁×#103	C₂³	1	4	yes	—
154	C₃₁×#103	C₂³	1	4	yes	—
155	C₃₁×#103	C₂³	1	8	yes	—
156	C₃₁×#103	C₂³	1	8	yes	—
157	C₃₁×#102	C₂³	1	4	yes	—
158	C₃₁×C₂²⋊C₈	C₂³	1	2	yes	—
159	C₃₁×C₂²⋊C₈	C₂³	1	2	yes	—
160	C₃₁×#97	C₂³	1	8	yes	—
161	C₄×C₈×C₃₁	C₂³	1	2	yes	—
162	C₄×C₈×C₃₁	C₂³	1	2	yes	—
163	#41×C₃₁	—	4	4	no	—
164	C₂×Q₈×C₃₁	—	2	4	no	—
165	C₂×Q₈×C₃₁	—	2	1	no	—
166	C₂×Q₈×C₃₁	—	2	2	no	—
167	C₂×Q₈×C₃₁	—	2	1	no	—
168	C₂×Q₈×C₃₁	—	2	1	no	—
169	C₂×Q₈×C₃₁	—	4	2	no	—
170	C₂×Q₈×C₃₁	—	4	4	no	—
171	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
172	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
173	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
174	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
175	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
176	C₂×D₄×C₃₁	—	2	2	no	—
177	C₂×D₄×C₃₁	—	2	1	no	—
178	C₂×D₄×C₃₁	—	2	1	no	—
179	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
180	C₂²×C₄×C₃₁	—	2	1	no	—
181	C₂²×C₄×C₃₁	—	2	2	no	—
182	C₂²×C₄×C₃₁	—	2	1	no	—
183	C₂²×C₄×C₃₁	—	2	2	no	—
184	C₂²×C₄×C₃₁	—	2	4	no	—
185	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
186	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
187	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
188	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
189	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
190	Q₁₆×C₃₁	—	4	4	no	—
191	Q₁₆×C₃₁	—	4	8	no	—
192	SD₁₆×C₃₁	—	4	8	no	—
193	SD₁₆×C₃₁	—	4	4	no	—
194	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
195	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
196	C₄⋊C₄×C₃₁	—	2	1	no	—
197	C₄⋊C₄×C₃₁	—	2	4	no	—
198	C₄⋊C₄×C₃₁	—	2	8	no	—
199	C₄⋊C₄×C₃₁	—	2	2	no	—
200	C₄⋊C₄×C₃₁	—	2	2	no	—
201	C₄⋊C₄×C₃₁	—	2	8	no	—
202	C₄⋊C₄×C₃₁	—	2	2	no	—
203	C₄⋊C₄×C₃₁	—	2	2	no	—
204	C₄⋊C₄×C₃₁	—	2	8	no	—
205	C₄⋊C₄×C₃₁	—	2	2	no	—
206	C₄⋊C₄×C₃₁	—	2	2	no	—
207	C₄⋊C₄×C₃₁	—	2	1	no	—
208	C₄⋊C₄×C₃₁	—	2	1	no	—
209	C₄⋊C₄×C₃₁	—	2	2	no	—
210	C₄⋊C₄×C₃₁	—	2	4	no	—
211	C₄⋊C₄×C₃₁	—	2	1	no	—
212	C₄⋊C₄×C₃₁	—	2	1	no	—
213	C₄⋊C₄×C₃₁	—	2	2	no	—
214	C₄⋊C₄×C₃₁	—	2	4	no	—
215	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
216	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
217	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
218	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
219	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
220	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
221	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
222	C₂²⋊C₄×C₃₁	—	2	2	no	—
223	C₂²⋊C₄×C₃₁	—	2	2	no	—
224	C₂²⋊C₄×C₃₁	—	2	4	no	—
225	C₂²⋊C₄×C₃₁	—	2	4	no	—
226	C₂²⋊C₄×C₃₁	—	2	2	no	—
227	C₂²⋊C₄×C₃₁	—	2	4	no	—
228	C₄²×C₃₁	—	2	8	no	—
229	C₄²×C₃₁	—	2	2	no	—
230	C₄²×C₃₁	—	2	2	no	—
231	C₄²×C₃₁	—	2	4	no	—
232	C₄²×C₃₁	—	2	2	no	—
233	C₄²×C₃₁	C₂×D₄	1	4	yes	—
234	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
235	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
236	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
237	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
238	C₄²×C₃₁	C₂⁴	1	4	yes	—
239	#61893	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
240	C₂³×C₃₁	—	2	2	no	—
241	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
242	Q₈×C₃₁	—	2	4	no	—
243	Q₈×C₃₁	—	2	4	no	—
244	Q₈×C₃₁	—	2	2	no	—
245	Q₈×C₃₁	—	2	8	no	—
246	Q₈×C₃₁	—	2	4	no	—
247	Q₈×C₃₁	—	2	2	no	—
248	Q₈×C₃₁	—	4	2	no	—
249	Q₈×C₃₁	—	4	4	no	—
250	Q₈×C₃₁	—	4	2	no	—
251	Q₈×C₃₁	—	8	1	no	—
252	D₄×C₃₁	—	2	4	no	—
253	D₄×C₃₁	—	4	4	no	—
254	D₄×C₃₁	—	4	2	no	—
255	D₄×C₃₁	—	8	1	no	—
256	C₂×C₄×C₃₁	—	2	4	no	—
257	C₂×C₄×C₃₁	—	2	2	no	—
258	C₂×C₄×C₃₁	—	2	2	no	—
259	C₂×C₄×C₃₁	—	2	2	no	—
260	C₂×C₄×C₃₁	—	2	2	no	—
261	C₂×C₄×C₃₁	—	2	2	no	—
262	C₂×C₄×C₃₁	—	2	4	no	—
263	C₂×C₄×C₃₁	—	2	2	no	—
264	C₂×C₄×C₃₁	—	4	2	no	—
265	C₂×C₄×C₃₁	—	4	1	no	—
266	C₂×C₄×C₃₁	—	4	2	no	—
267	C₂×C₄×C₃₁	—	4	2	no	—
268	C₂×C₄×C₃₁	—	4	4	no	—
269	C₂×C₄×C₃₁	—	4	4	no	—
270	C₂×C₄×C₃₁	—	8	4	no	—
271	C₂×C₄×C₃₁	#137	1	4	yes	—
272	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
273	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
274	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
275	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
276	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
277	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
278	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
279	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
280	C₈×C₃₁	—	2	4	no	—
281	C₈×C₃₁	—	2	4	no	—
282	#3507	C₂×C₃₁	1	1	yes	characteristic
283	C₂×#557	C₂×C₃₁	1	1	yes	characteristic, complemented
284	#3400	C₂×C₃₁	1	4	yes	complemented
285	#3398	C₂×C₃₁	1	4	yes	complemented
286	#3374	C₂×C₃₁	1	4	yes	complemented
287	#3372	C₂×C₃₁	1	4	yes	complemented
288	#3218	C₂×C₃₁	1	4	yes	complemented
289	#3209	C₂×C₃₁	1	1	yes	characteristic, complemented
290	#3206	C₂×C₃₁	1	1	yes	characteristic
291	#3117	C₂×C₃₁	1	4	yes	complemented
292	#3104	C₂×C₃₁	1	1	yes	characteristic, complemented
293	#3103	C₂×C₃₁	1	1	yes	characteristic
294	#2983	C₂×C₃₁	1	1	yes	characteristic, complemented
295	C₂²×C₃₁	—	2	1	no	—
296	C₂²×C₃₁	—	2	1	no	—
297	C₂²×C₃₁	—	4	2	no	—
298	C₂²×C₃₁	—	4	2	no	—
299	C₂²×C₃₁	—	4	2	no	—
300	C₂²×C₃₁	C₂×#137	1	2	yes	—
301	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
302	C₄×C₃₁	—	2	4	no	—
303	C₄×C₃₁	—	2	1	no	—
304	C₄×C₃₁	—	2	1	no	—
305	C₄×C₃₁	—	2	1	no	—
306	C₄×C₃₁	—	2	4	no	—
307	C₄×C₃₁	—	4	2	no	—
308	C₄×C₃₁	—	4	2	no	—
309	C₄×C₃₁	—	4	2	no	—
310	C₄×C₃₁	—	4	4	no	—
311	C₄×C₃₁	—	4	2	no	—
312	C₄×C₃₁	—	4	2	no	—
313	C₄×C₃₁	C₂×#137	1	2	yes	—
314	C₄×C₃₁	C₂×#137	1	2	yes	—
315	C₄×C₃₁	#585	1	2	yes	—
316	#559	C₂²×C₃₁	1	2	yes	—
317	#557	—	2	2	no	—
318	#557	—	2	2	no	—
319	#557	C₂²×C₃₁	1	4	yes	—
320	#557	C₂²×C₃₁	1	4	yes	—
321	#557	C₂²×C₃₁	1	2	yes	—
322	#556	—	2	2	no	—
323	#556	C₂²×C₃₁	1	2	yes	—
324	#554	C₂²×C₃₁	1	2	yes	—
325	#552	—	2	2	no	—
326	#539	—	2	2	no	—
327	#536	—	2	1	no	—
328	#533	C₂²×C₃₁	1	1	yes	characteristic
329	#532	C₂²×C₃₁	1	1	yes	characteristic
330	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
331	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
332	#529	C₂²×C₃₁	1	4	yes	—
333	C₂×#125	—	2	1	no	—
334	C₂×#125	—	2	2	no	—
335	#520	—	2	1	no	—
336	C₂×C₄×Q₈	—	2	1	no	—
337	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
338	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
339	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
340	#497	C₂²×C₃₁	1	8	yes	—
341	#494	C₂²×C₃₁	1	2	yes	—
342	#492	C₂²×C₃₁	1	2	yes	—
343	#491	C₂²×C₃₁	1	8	yes	—
344	#487	C₂²×C₃₁	1	2	yes	—
345	#487	C₂²×C₃₁	1	2	yes	—
346	#484	C₂²×C₃₁	1	2	yes	—
347	#481	C₂²×C₃₁	1	2	yes	—
348	#479	C₂²×C₃₁	1	8	yes	—
349	#477	C₂²×C₃₁	1	2	yes	—
350	#475	C₂²×C₃₁	1	2	yes	—
351	#464	C₂²×C₃₁	1	8	yes	—
352	#462	C₂²×C₃₁	1	2	yes	—
353	#462	C₂²×C₃₁	1	8	yes	—
354	#462	C₂²×C₃₁	1	4	yes	—
355	#461	C₂²×C₃₁	1	4	yes	—
356	#461	C₂²×C₃₁	1	8	yes	—
357	#460	C₂²×C₃₁	1	2	yes	—
358	#451	C₂²×C₃₁	1	2	yes	—
359	#451	C₂²×C₃₁	1	4	yes	—
360	#448	C₂²×C₃₁	1	2	yes	—
361	#448	C₂²×C₃₁	1	4	yes	—
362	#446	C₂²×C₃₁	1	4	yes	—
363	#446	C₂²×C₃₁	1	4	yes	—
364	#441	C₂²×C₃₁	1	8	yes	—
365	#440	C₂²×C₃₁	1	8	yes	—
366	C₄×Q₁₆	C₂²×C₃₁	1	4	yes	—
367	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	—
368	#432	C₂²×C₃₁	1	4	yes	—
369	#431	C₂²×C₃₁	1	1	yes	characteristic
370	#431	C₂²×C₃₁	1	1	yes	characteristic
371	C₂×C₃₁	—	2	2	no	—
372	C₂×C₃₁	—	8	2	no	—
373	C₂×C₃₁	#3487	1	1	yes	characteristic
374	C₂×C₃₁	C₂²×#137	1	1	yes	characteristic
375	C₂×C₃₁	#3586	1	1	yes	characteristic
376	C₂×#41	—	2	1	no	—
377	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
378	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
379	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
380	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
381	C₂×Q₁₆	C₂³×C₃₁	1	8	yes	—
382	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
383	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
384	C₄⋊Q₈	—	2	2	no	—
385	C₄⋊Q₈	—	2	4	no	—
386	C₄⋊Q₈	—	2	1	no	—
387	C₄⋊Q₈	—	2	1	no	—
388	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
389	C₄⋊Q₈	C₂³×C₃₁	1	8	yes	—
390	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
391	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
392	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
393	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
394	#127	C₂³×C₃₁	1	1	yes	characteristic
395	#126	—	2	4	no	—
396	#126	—	2	4	no	—
397	#125	—	2	4	no	—
398	#125	—	2	4	no	—
399	#125	—	2	8	no	—
400	#125	—	2	8	no	—
401	#125	—	2	4	no	—
402	#125	—	2	1	no	—
403	#125	C₂³×C₃₁	1	4	yes	—
404	#124	—	2	2	no	—
405	#124	—	2	1	no	—
406	#124	C₂³×C₃₁	1	4	yes	—
407	#123	—	2	4	no	—
408	#123	—	2	4	no	—
409	C₂²⋊Q₈	—	2	2	no	—
410	C₂²⋊Q₈	—	2	2	no	—
411	C₂²⋊Q₈	—	2	1	no	—
412	C₂²⋊Q₈	—	2	4	no	—
413	C₂²⋊Q₈	C₂³×C₃₁	1	2	yes	—
414	#121	—	2	1	no	—
415	#121	C₂³×C₃₁	1	2	yes	—
416	C₄×Q₈	—	2	4	no	—
417	C₄×Q₈	—	2	4	no	—
418	C₄×Q₈	—	2	2	no	—
419	C₄×Q₈	—	2	4	no	—
420	C₄×Q₈	—	2	1	no	—
421	C₄×Q₈	—	2	1	no	—
422	C₄×Q₈	—	2	1	no	—
423	C₄×Q₈	D₄×C₃₁	1	2	yes	—
424	C₄×Q₈	D₄×C₃₁	1	2	yes	—
425	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
426	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
427	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
428	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
429	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
430	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
431	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
432	C₄×D₄	—	2	4	no	—
433	C₄×D₄	—	2	2	no	—
434	C₄×D₄	—	2	1	no	—
435	C₄×D₄	C₂³×C₃₁	1	4	yes	—
436	C₄×D₄	C₂³×C₃₁	1	2	yes	—
437	#117	—	2	1	no	—
438	#117	—	2	2	no	—
439	#117	—	2	4	no	—
440	C₂×C₄⋊C₄	—	2	2	no	—
441	C₂×C₄⋊C₄	—	2	1	no	—
442	C₂×C₄⋊C₄	—	2	2	no	—
443	C₂×C₄⋊C₄	—	2	1	no	—
444	C₂×C₄⋊C₄	—	2	2	no	—
445	C₂×C₄⋊C₄	—	2	1	no	—
446	C₂×C₄⋊C₄	—	2	2	no	—
447	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
448	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
449	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
450	C₂×C₄²	—	2	2	no	—
451	C₂×C₄²	—	2	1	no	—
452	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
453	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
454	#107	C₂³×C₃₁	1	4	yes	—
455	#106	C₂³×C₃₁	1	4	yes	—
456	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
457	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
458	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
459	#103	C₂³×C₃₁	1	4	yes	—
460	#103	C₂³×C₃₁	1	4	yes	—
461	#103	C₂³×C₃₁	1	4	yes	—
462	#103	C₂³×C₃₁	1	8	yes	—
463	#103	C₂³×C₃₁	1	8	yes	—
464	#102	C₂³×C₃₁	1	4	yes	—
465	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
466	C₂²⋊C₈	C₂³×C₃₁	1	2	yes	—
467	#97	C₂³×C₃₁	1	8	yes	—
468	C₄×C₈	C₂³×C₃₁	1	2	yes	—
469	C₄×C₈	C₂³×C₃₁	1	2	yes	—
470	C₃₁	#61893	1	1	yes	characteristic, direct product factor, Sylow
471	#41	—	4	4	no	—
472	C₂×Q₈	—	2	4	no	—
473	C₂×Q₈	—	2	1	no	—
474	C₂×Q₈	—	2	2	no	—
475	C₂×Q₈	—	2	1	no	—
476	C₂×Q₈	—	2	1	no	—
477	C₂×Q₈	—	4	2	no	—
478	C₂×Q₈	—	4	4	no	—
479	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
480	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
481	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
482	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
483	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
484	C₂×D₄	—	2	2	no	—
485	C₂×D₄	—	2	1	no	—
486	C₂×D₄	—	2	1	no	—
487	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
488	C₂²×C₄	—	2	1	no	—
489	C₂²×C₄	—	2	2	no	—
490	C₂²×C₄	—	2	1	no	—
491	C₂²×C₄	—	2	2	no	—
492	C₂²×C₄	—	2	4	no	—
493	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
494	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
495	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
496	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
497	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
498	Q₁₆	—	4	4	no	—
499	Q₁₆	—	4	8	no	—
500	SD₁₆	—	4	8	no	—
501	SD₁₆	—	4	4	no	—
502	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
503	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
504	C₄⋊C₄	—	2	1	no	—
505	C₄⋊C₄	—	2	4	no	—
506	C₄⋊C₄	—	2	8	no	—
507	C₄⋊C₄	—	2	2	no	—
508	C₄⋊C₄	—	2	2	no	—
509	C₄⋊C₄	—	2	8	no	—
510	C₄⋊C₄	—	2	2	no	—
511	C₄⋊C₄	—	2	2	no	—
512	C₄⋊C₄	—	2	8	no	—
513	C₄⋊C₄	—	2	2	no	—
514	C₄⋊C₄	—	2	2	no	—
515	C₄⋊C₄	—	2	1	no	—
516	C₄⋊C₄	—	2	1	no	—
517	C₄⋊C₄	—	2	2	no	—
518	C₄⋊C₄	—	2	4	no	—
519	C₄⋊C₄	—	2	1	no	—
520	C₄⋊C₄	—	2	1	no	—
521	C₄⋊C₄	—	2	2	no	—
522	C₄⋊C₄	—	2	4	no	—
523	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
524	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
525	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
526	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
527	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
528	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
529	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
530	C₂²⋊C₄	—	2	2	no	—
531	C₂²⋊C₄	—	2	2	no	—
532	C₂²⋊C₄	—	2	4	no	—
533	C₂²⋊C₄	—	2	4	no	—
534	C₂²⋊C₄	—	2	2	no	—
535	C₂²⋊C₄	—	2	4	no	—
536	C₄²	—	2	8	no	—
537	C₄²	—	2	2	no	—
538	C₄²	—	2	2	no	—
539	C₄²	—	2	4	no	—
540	C₄²	—	2	2	no	—
541	C₄²	C₂×D₄×C₃₁	1	4	yes	—
542	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
543	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
544	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
545	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
546	C₄²	C₂⁴×C₃₁	1	4	yes	—
547	C₂³	—	2	2	no	—
548	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
549	Q₈	—	2	4	no	—
550	Q₈	—	2	4	no	—
551	Q₈	—	2	2	no	—
552	Q₈	—	2	8	no	—
553	Q₈	—	2	4	no	—
554	Q₈	—	2	2	no	—
555	Q₈	—	4	2	no	—
556	Q₈	—	4	4	no	—
557	Q₈	—	4	2	no	—
558	Q₈	—	8	1	no	—
559	D₄	—	2	4	no	—
560	D₄	—	4	4	no	—
561	D₄	—	4	2	no	—
562	D₄	—	8	1	no	—
563	C₂×C₄	—	2	4	no	—
564	C₂×C₄	—	2	2	no	—
565	C₂×C₄	—	2	2	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	2	no	—
568	C₂×C₄	—	2	2	no	—
569	C₂×C₄	—	2	4	no	—
570	C₂×C₄	—	2	2	no	—
571	C₂×C₄	—	4	2	no	—
572	C₂×C₄	—	4	1	no	—
573	C₂×C₄	—	4	2	no	—
574	C₂×C₄	—	4	2	no	—
575	C₂×C₄	—	4	4	no	—
576	C₂×C₄	—	4	4	no	—
577	C₂×C₄	—	8	4	no	—
578	C₂×C₄	C₃₁×#137	1	4	yes	—
579	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
580	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
581	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
582	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
583	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
584	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
585	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
586	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
587	C₈	—	2	4	no	—
588	C₈	—	2	4	no	—
589	C₂²	—	2	1	no	—
590	C₂²	—	2	1	no	—
591	C₂²	—	4	2	no	—
592	C₂²	—	4	2	no	—
593	C₂²	—	4	2	no	—
594	C₂²	C₂×C₃₁×#137	1	2	yes	—
595	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
596	C₄	—	2	1	no	—
597	C₄	—	2	4	no	—
598	C₄	—	2	4	no	—
599	C₄	—	2	1	no	—
600	C₄	—	2	1	no	—
601	C₄	—	4	2	no	—
602	C₄	—	4	4	no	—
603	C₄	—	4	2	no	—
604	C₄	—	4	2	no	—
605	C₄	—	4	2	no	—
606	C₄	—	4	2	no	—
607	C₄	C₂×C₃₁×#137	1	2	yes	—
608	C₄	C₂×C₃₁×#137	1	2	yes	—
609	C₄	C₃₁×#585	1	2	yes	—
610	C₂	—	2	2	no	—
611	C₂	—	8	2	no	—
612	C₂	C₃₁×#3487	1	1	yes	characteristic
613	C₂	C₂²×C₃₁×#137	1	1	yes	characteristic
614	C₂	C₃₁×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₃₁×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#557	C₂	semidirect
3. C₃₁×#3400	C₂	semidirect
4. C₃₁×#3398	C₂	semidirect
5. C₃₁×#3374	C₂	semidirect
6. C₃₁×#3372	C₂	semidirect
7. C₃₁×#3218	C₂	semidirect
8. C₃₁×#3209	C₂	semidirect
10. C₃₁×#3117	C₂	semidirect
11. C₃₁×#3104	C₂	semidirect
13. C₃₁×#2983	C₂	semidirect
239. #61893	C₃₁	direct (N × H)
283. C₂×#557	C₂×C₃₁	semidirect
284. #3400	C₂×C₃₁	semidirect
285. #3398	C₂×C₃₁	semidirect
286. #3374	C₂×C₃₁	semidirect
287. #3372	C₂×C₃₁	semidirect
288. #3218	C₂×C₃₁	semidirect
289. #3209	C₂×C₃₁	semidirect
291. #3117	C₂×C₃₁	semidirect
292. #3104	C₂×C₃₁	semidirect
294. #2983	C₂×C₃₁	semidirect
470. C₃₁	#61893	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×#3507	2
C₂×C₃₁×#557	2
C₃₁×#3400	2
C₃₁×#3398	2
C₃₁×#3374	2
C₃₁×#3372	2
C₃₁×#3218	2
C₃₁×#3209	2
C₃₁×#3206	2
C₃₁×#3117	2
C₃₁×#3104	2
C₃₁×#3103	2
C₃₁×#2983	2
#61893	31

Subgroups — C31×#61893 #21518652

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61893 #21518652