Subgroups — C₃₁×#60682 #21517441

Order 7936.

1878 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3455	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3177	C₂	1	2	yes	maximal, complemented
3	C₃₁×#3176	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3175	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3174	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3173	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3160	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#3147	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#3135	C₂	1	2	yes	maximal, complemented
10	C₃₁×#3054	C₂	1	2	yes	maximal, complemented
11	C₃₁×#3050	C₂	1	2	yes	maximal, complemented
12	C₃₁×#2943	C₂	1	2	yes	maximal, complemented
13	C₄×C₃₁×#135	C₂	1	2	yes	maximal, complemented
14	C₂×C₃₁×#405	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×C₃₁×#135	C₂²	1	2	yes	—
16	C₂×C₃₁×#135	C₂²	1	2	yes	—
17	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
18	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#542	C₂²	1	8	yes	—
20	C₃₁×#541	C₂²	1	4	yes	—
21	C₃₁×#540	C₂²	1	4	yes	—
22	C₃₁×#539	—	2	4	no	—
23	C₃₁×#538	—	2	4	no	—
24	C₃₁×#533	C₂²	1	2	yes	—
25	C₃₁×#532	C₂²	1	2	yes	—
26	C₃₁×#529	—	2	8	no	—
27	C₃₁×#525	C₂²	1	4	yes	—
28	C₄×#41×C₃₁	C₂²	1	4	yes	—
29	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
30	C₃₁×#500	C₂²	1	2	yes	complemented
31	C₃₁×#499	C₂²	1	2	yes	complemented
32	C₃₁×#499	C₂²	1	2	yes	complemented
33	C₃₁×#498	C₂²	1	2	yes	complemented
34	C₃₁×#495	C₂²	1	2	yes	complemented
35	C₃₁×#495	C₂²	1	2	yes	complemented
36	C₃₁×#494	C₂²	1	1	yes	characteristic, complemented
37	C₃₁×#493	C₂²	1	1	yes	characteristic, complemented
38	C₃₁×#492	C₂²	1	2	yes	complemented
39	C₃₁×#491	C₂²	1	4	yes	complemented
40	C₃₁×#490	C₂²	1	4	yes	complemented
41	C₃₁×#489	C₂²	1	8	yes	complemented
42	C₃₁×#470	C₂²	1	4	yes	—
43	C₃₁×#469	C₂²	1	4	yes	—
44	C₃₁×#468	C₂²	1	8	yes	—
45	C₃₁×#467	C₂²	1	4	yes	—
46	C₃₁×#466	C₂²	1	4	yes	—
47	C₃₁×#465	C₂²	1	8	yes	—
48	C₃₁×#442	C₂²	1	4	yes	complemented
49	C₃₁×#441	C₂²	1	4	yes	complemented
50	C₃₁×#440	C₂²	1	8	yes	complemented
51	C₄×Q₁₆×C₃₁	C₂²	1	4	yes	complemented
52	C₄×SD₁₆×C₃₁	C₂²	1	8	yes	complemented
53	C₄×D₈×C₃₁	C₂²	1	4	yes	complemented
54	C₃₁×#427	C₂²	1	2	yes	—
55	C₃₁×#427	C₂²	1	2	yes	—
56	C₃₁×#416	C₂²	1	4	yes	—
57	C₃₁×#416	C₂²	1	4	yes	—
58	C₃₁×#405	C₂²	1	2	yes	—
59	C₃₁×#405	C₂²	1	2	yes	—
60	C₃₁×#405	C₂²	1	2	yes	—
61	C₃₁×#405	C₂²	1	2	yes	—
62	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
63	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
64	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
65	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
66	C₂×#41×C₃₁	C₂³	1	4	yes	—
67	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
68	C₃₁×#135	—	2	8	no	—
69	C₃₁×#135	—	2	8	no	—
70	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
71	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
72	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
73	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
74	C₂×D₈×C₃₁	C₂³	1	2	yes	—
75	C₂×D₈×C₃₁	C₂³	1	2	yes	—
76	C₂×M₁₆×C₃₁	—	2	4	no	—
77	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
78	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
79	C₂²×C₈×C₃₁	—	2	4	no	—
80	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
81	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
82	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
83	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
84	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
85	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
86	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
87	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
88	C₃₁×#127	C₂³	1	2	yes	—
89	C₃₁×#126	—	2	8	no	—
90	C₃₁×#126	—	2	8	no	—
91	C₃₁×#125	—	2	8	no	—
92	C₃₁×#125	C₂³	1	4	yes	—
93	C₃₁×#124	—	2	8	no	—
94	C₃₁×#124	C₂³	1	4	yes	—
95	C₃₁×#123	—	2	4	no	—
96	C₃₁×#123	—	2	8	no	—
97	C₃₁×#123	—	2	8	no	—
98	C₃₁×C₂²⋊Q₈	—	2	4	no	—
99	C₃₁×C₂²⋊Q₈	—	2	8	no	—
100	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	—
101	C₃₁×#121	—	2	4	no	—
102	C₃₁×#121	—	2	8	no	—
103	C₃₁×#121	C₂³	1	8	yes	—
104	C₃₁×#120	—	2	4	no	—
105	C₄×Q₈×C₃₁	—	2	4	no	—
106	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
107	C₄×D₄×C₃₁	—	2	8	no	—
108	C₄×D₄×C₃₁	—	2	8	no	—
109	C₄×D₄×C₃₁	—	2	4	no	—
110	C₄×D₄×C₃₁	C₂³	1	8	yes	—
111	C₃₁×#117	—	2	4	no	—
112	C₃₁×#117	D₄	1	4	yes	—
113	C₃₁×#117	C₂³	1	4	yes	—
114	C₃₁×#117	C₂³	1	2	yes	—
115	C₃₁×#117	C₂³	1	2	yes	—
116	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
117	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
118	C₂×C₄²×C₃₁	—	2	4	no	—
119	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
120	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
121	C₃₁×#107	C₂³	1	4	yes	—
122	C₃₁×#107	C₂³	1	4	yes	—
123	C₃₁×#106	C₂³	1	4	yes	—
124	C₃₁×#106	C₂³	1	4	yes	—
125	C₃₁×#103	C₂³	1	8	yes	—
126	C₃₁×#103	C₂³	1	8	yes	—
127	C₃₁×#102	C₂³	1	8	yes	—
128	C₃₁×#102	C₂³	1	8	yes	—
129	C₃₁×#97	C₂³	1	2	yes	—
130	C₃₁×#97	C₂³	1	2	yes	—
131	C₃₁×#97	C₂³	1	2	yes	—
132	C₃₁×#97	C₂³	1	2	yes	—
133	C₄×C₈×C₃₁	C₂³	1	2	yes	—
134	C₄×C₈×C₃₁	C₂³	1	2	yes	—
135	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
136	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
137	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
138	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
139	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
140	#41×C₃₁	—	2	8	no	—
141	#41×C₃₁	—	2	8	no	—
142	C₂×Q₈×C₃₁	—	2	4	no	—
143	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
144	C₂×D₄×C₃₁	—	2	4	no	—
145	C₂×D₄×C₃₁	—	2	8	no	—
146	C₂×D₄×C₃₁	—	2	4	no	—
147	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
148	C₂²×C₄×C₃₁	—	2	2	no	—
149	C₂²×C₄×C₃₁	—	2	2	no	—
150	C₂²×C₄×C₃₁	—	2	4	no	—
151	C₂²×C₄×C₃₁	—	2	8	no	—
152	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
154	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
155	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
156	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
157	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
158	Q₁₆×C₃₁	—	2	4	no	—
159	Q₁₆×C₃₁	—	2	4	no	—
160	SD₁₆×C₃₁	—	2	8	no	—
161	SD₁₆×C₃₁	—	2	8	no	—
162	D₈×C₃₁	—	2	4	no	—
163	D₈×C₃₁	—	2	4	no	—
164	M₁₆×C₃₁	—	2	4	no	—
165	M₁₆×C₃₁	—	2	4	no	—
166	C₂×C₈×C₃₁	—	2	2	no	—
167	C₂×C₈×C₃₁	—	2	2	no	—
168	C₂×C₈×C₃₁	—	2	4	no	—
169	C₂×C₈×C₃₁	—	2	4	no	—
170	C₂×C₈×C₃₁	#41	1	4	yes	—
171	C₂×C₈×C₃₁	#41	1	4	yes	—
172	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
173	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
174	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
175	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
176	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
177	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
178	C₄⋊C₄×C₃₁	—	2	8	no	—
179	C₄⋊C₄×C₃₁	—	2	4	no	—
180	C₄⋊C₄×C₃₁	—	2	8	no	—
181	C₄⋊C₄×C₃₁	—	2	8	no	—
182	C₄⋊C₄×C₃₁	—	2	4	no	—
183	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
184	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
185	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
186	C₂²⋊C₄×C₃₁	—	2	4	no	—
187	C₂²⋊C₄×C₃₁	—	2	8	no	—
188	C₂²⋊C₄×C₃₁	—	2	8	no	—
189	C₂²⋊C₄×C₃₁	—	2	8	no	—
190	C₂²⋊C₄×C₃₁	—	2	8	no	—
191	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
192	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
193	C₄²×C₃₁	—	2	8	no	—
194	C₄²×C₃₁	—	2	4	no	—
195	C₄²×C₃₁	C₂×D₄	1	2	yes	—
196	C₄²×C₃₁	C₂×D₄	1	2	yes	—
197	C₄²×C₃₁	C₂⁴	1	2	yes	—
198	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
199	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
200	#60682	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
201	C₂³×C₃₁	—	2	2	no	—
202	C₂³×C₃₁	—	2	4	no	—
203	C₂³×C₃₁	—	4	2	no	—
204	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
205	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
206	Q₈×C₃₁	—	2	4	no	—
207	Q₈×C₃₁	—	2	4	no	—
208	Q₈×C₃₁	—	8	4	no	—
209	D₄×C₃₁	—	2	4	no	—
210	D₄×C₃₁	—	2	4	no	—
211	D₄×C₃₁	—	4	8	no	—
212	D₄×C₃₁	—	8	4	no	—
213	D₄×C₃₁	—	8	8	no	—
214	C₂×C₄×C₃₁	—	2	2	no	—
215	C₂×C₄×C₃₁	—	2	4	no	—
216	C₂×C₄×C₃₁	—	2	2	no	—
217	C₂×C₄×C₃₁	—	2	2	no	—
218	C₂×C₄×C₃₁	—	2	4	no	—
219	C₂×C₄×C₃₁	—	2	8	no	—
220	C₂×C₄×C₃₁	—	4	2	no	—
221	C₂×C₄×C₃₁	—	4	2	no	—
222	C₂×C₄×C₃₁	—	4	8	no	—
223	C₂×C₄×C₃₁	—	8	8	no	—
224	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
225	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
226	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
227	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
228	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
229	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
230	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
231	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
232	C₈×C₃₁	—	2	2	no	—
233	C₈×C₃₁	—	2	2	no	—
234	C₈×C₃₁	C₂×#41	1	2	yes	—
235	C₈×C₃₁	C₂×#41	1	2	yes	—
236	C₈×C₃₁	C₂×#41	1	2	yes	—
237	C₈×C₃₁	C₂×#41	1	2	yes	—
238	#3455	C₂×C₃₁	1	2	yes	complemented
239	#3177	C₂×C₃₁	1	2	yes	complemented
240	#3176	C₂×C₃₁	1	2	yes	complemented
241	#3175	C₂×C₃₁	1	4	yes	complemented
242	#3174	C₂×C₃₁	1	4	yes	complemented
243	#3173	C₂×C₃₁	1	4	yes	complemented
244	#3160	C₂×C₃₁	1	1	yes	characteristic, complemented
245	#3147	C₂×C₃₁	1	1	yes	characteristic, complemented
246	#3135	C₂×C₃₁	1	2	yes	complemented
247	#3054	C₂×C₃₁	1	2	yes	complemented
248	#3050	C₂×C₃₁	1	2	yes	complemented
249	#2943	C₂×C₃₁	1	2	yes	complemented
250	C₄×#135	C₂×C₃₁	1	2	yes	complemented
251	C₂×#405	C₂×C₃₁	1	1	yes	characteristic, complemented
252	C₂²×C₃₁	—	2	2	no	—
253	C₂²×C₃₁	—	2	4	no	—
254	C₂²×C₃₁	—	2	2	no	—
255	C₂²×C₃₁	—	4	4	no	—
256	C₂²×C₃₁	—	4	8	no	—
257	C₂²×C₃₁	—	4	4	no	—
258	C₂²×C₃₁	#532	1	2	yes	—
259	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
260	C₄×C₃₁	—	2	4	no	—
261	C₄×C₃₁	—	2	2	no	—
262	C₄×C₃₁	—	2	2	no	—
263	C₄×C₃₁	—	2	2	no	—
264	C₄×C₃₁	—	4	8	no	—
265	C₄×C₃₁	—	4	4	no	—
266	C₄×C₃₁	#532	1	2	yes	—
267	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
268	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
269	C₂×#135	C₂²×C₃₁	1	2	yes	—
270	C₂×#135	C₂²×C₃₁	1	2	yes	—
271	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
272	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
273	#542	C₂²×C₃₁	1	8	yes	—
274	#541	C₂²×C₃₁	1	4	yes	—
275	#540	C₂²×C₃₁	1	4	yes	—
276	#539	—	2	4	no	—
277	#538	—	2	4	no	—
278	#533	C₂²×C₃₁	1	2	yes	—
279	#532	C₂²×C₃₁	1	2	yes	—
280	#529	—	2	8	no	—
281	#525	C₂²×C₃₁	1	4	yes	—
282	C₄×#41	C₂²×C₃₁	1	4	yes	—
283	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
284	#500	C₂²×C₃₁	1	2	yes	complemented
285	#499	C₂²×C₃₁	1	2	yes	complemented
286	#499	C₂²×C₃₁	1	2	yes	complemented
287	#498	C₂²×C₃₁	1	2	yes	complemented
288	#495	C₂²×C₃₁	1	2	yes	complemented
289	#495	C₂²×C₃₁	1	2	yes	complemented
290	#494	C₂²×C₃₁	1	1	yes	characteristic, complemented
291	#493	C₂²×C₃₁	1	1	yes	characteristic, complemented
292	#492	C₂²×C₃₁	1	2	yes	complemented
293	#491	C₂²×C₃₁	1	4	yes	complemented
294	#490	C₂²×C₃₁	1	4	yes	complemented
295	#489	C₂²×C₃₁	1	8	yes	complemented
296	#470	C₂²×C₃₁	1	4	yes	—
297	#469	C₂²×C₃₁	1	4	yes	—
298	#468	C₂²×C₃₁	1	8	yes	—
299	#467	C₂²×C₃₁	1	4	yes	—
300	#466	C₂²×C₃₁	1	4	yes	—
301	#465	C₂²×C₃₁	1	8	yes	—
302	#442	C₂²×C₃₁	1	4	yes	complemented
303	#441	C₂²×C₃₁	1	4	yes	complemented
304	#440	C₂²×C₃₁	1	8	yes	complemented
305	C₄×Q₁₆	C₂²×C₃₁	1	4	yes	complemented
306	C₄×SD₁₆	C₂²×C₃₁	1	8	yes	complemented
307	C₄×D₈	C₂²×C₃₁	1	4	yes	complemented
308	#427	C₂²×C₃₁	1	2	yes	—
309	#427	C₂²×C₃₁	1	2	yes	—
310	#416	C₂²×C₃₁	1	4	yes	—
311	#416	C₂²×C₃₁	1	4	yes	—
312	#405	C₂²×C₃₁	1	2	yes	—
313	#405	C₂²×C₃₁	1	2	yes	—
314	#405	C₂²×C₃₁	1	2	yes	—
315	#405	C₂²×C₃₁	1	2	yes	—
316	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
317	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
318	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
319	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
320	C₂×C₃₁	—	2	2	no	—
321	C₂×C₃₁	—	4	2	no	—
322	C₂×C₃₁	—	8	4	no	—
323	C₂×C₃₁	C₂×#532	1	1	yes	characteristic
324	C₂×C₃₁	#3584	1	2	yes	—
325	C₂×#41	C₂³×C₃₁	1	4	yes	—
326	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
327	#135	—	2	8	no	—
328	#135	—	2	8	no	—
329	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
330	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
331	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
332	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
333	C₂×D₈	C₂³×C₃₁	1	2	yes	—
334	C₂×D₈	C₂³×C₃₁	1	2	yes	—
335	C₂×M₁₆	—	2	4	no	—
336	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
337	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
338	C₂²×C₈	—	2	4	no	—
339	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
340	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
341	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
342	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
343	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
344	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
345	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
346	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
347	#127	C₂³×C₃₁	1	2	yes	—
348	#126	—	2	8	no	—
349	#126	—	2	8	no	—
350	#125	—	2	8	no	—
351	#125	C₂³×C₃₁	1	4	yes	—
352	#124	—	2	8	no	—
353	#124	C₂³×C₃₁	1	4	yes	—
354	#123	—	2	4	no	—
355	#123	—	2	8	no	—
356	#123	—	2	8	no	—
357	C₂²⋊Q₈	—	2	4	no	—
358	C₂²⋊Q₈	—	2	8	no	—
359	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
360	#121	—	2	4	no	—
361	#121	—	2	8	no	—
362	#121	C₂³×C₃₁	1	8	yes	—
363	#120	—	2	4	no	—
364	C₄×Q₈	—	2	4	no	—
365	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
366	C₄×D₄	—	2	8	no	—
367	C₄×D₄	—	2	8	no	—
368	C₄×D₄	—	2	4	no	—
369	C₄×D₄	C₂³×C₃₁	1	8	yes	—
370	#117	—	2	4	no	—
371	#117	D₄×C₃₁	1	4	yes	—
372	#117	C₂³×C₃₁	1	4	yes	—
373	#117	C₂³×C₃₁	1	2	yes	—
374	#117	C₂³×C₃₁	1	2	yes	—
375	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
376	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
377	C₂×C₄²	—	2	4	no	—
378	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
379	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
380	#107	C₂³×C₃₁	1	4	yes	—
381	#107	C₂³×C₃₁	1	4	yes	—
382	#106	C₂³×C₃₁	1	4	yes	—
383	#106	C₂³×C₃₁	1	4	yes	—
384	#103	C₂³×C₃₁	1	8	yes	—
385	#103	C₂³×C₃₁	1	8	yes	—
386	#102	C₂³×C₃₁	1	8	yes	—
387	#102	C₂³×C₃₁	1	8	yes	—
388	#97	C₂³×C₃₁	1	2	yes	—
389	#97	C₂³×C₃₁	1	2	yes	—
390	#97	C₂³×C₃₁	1	2	yes	—
391	#97	C₂³×C₃₁	1	2	yes	—
392	C₄×C₈	C₂³×C₃₁	1	2	yes	—
393	C₄×C₈	C₂³×C₃₁	1	2	yes	—
394	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
395	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
396	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
397	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
398	C₃₁	#60682	1	1	yes	characteristic, direct product factor, Sylow
399	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
400	#41	—	2	8	no	—
401	#41	—	2	8	no	—
402	C₂×Q₈	—	2	4	no	—
403	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
404	C₂×D₄	—	2	4	no	—
405	C₂×D₄	—	2	8	no	—
406	C₂×D₄	—	2	4	no	—
407	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
408	C₂²×C₄	—	2	2	no	—
409	C₂²×C₄	—	2	2	no	—
410	C₂²×C₄	—	2	4	no	—
411	C₂²×C₄	—	2	8	no	—
412	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
413	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
414	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
415	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
416	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
417	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
418	Q₁₆	—	2	4	no	—
419	Q₁₆	—	2	4	no	—
420	SD₁₆	—	2	8	no	—
421	SD₁₆	—	2	8	no	—
422	D₈	—	2	4	no	—
423	D₈	—	2	4	no	—
424	M₁₆	—	2	4	no	—
425	M₁₆	—	2	4	no	—
426	C₂×C₈	—	2	2	no	—
427	C₂×C₈	—	2	2	no	—
428	C₂×C₈	—	2	4	no	—
429	C₂×C₈	—	2	4	no	—
430	C₂×C₈	#41×C₃₁	1	4	yes	—
431	C₂×C₈	#41×C₃₁	1	4	yes	—
432	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
433	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
434	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
435	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
436	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
437	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
438	C₄⋊C₄	—	2	8	no	—
439	C₄⋊C₄	—	2	4	no	—
440	C₄⋊C₄	—	2	8	no	—
441	C₄⋊C₄	—	2	8	no	—
442	C₄⋊C₄	—	2	4	no	—
443	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
444	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
445	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
446	C₂²⋊C₄	—	2	4	no	—
447	C₂²⋊C₄	—	2	8	no	—
448	C₂²⋊C₄	—	2	8	no	—
449	C₂²⋊C₄	—	2	8	no	—
450	C₂²⋊C₄	—	2	8	no	—
451	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
452	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
453	C₄²	—	2	4	no	—
454	C₄²	—	2	8	no	—
455	C₄²	C₂×D₄×C₃₁	1	2	yes	—
456	C₄²	C₂×D₄×C₃₁	1	2	yes	—
457	C₄²	C₂⁴×C₃₁	1	2	yes	—
458	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
459	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
460	C₂³	—	2	2	no	—
461	C₂³	—	2	4	no	—
462	C₂³	—	4	2	no	—
463	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
464	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
465	Q₈	—	2	4	no	—
466	Q₈	—	2	4	no	—
467	Q₈	—	8	4	no	—
468	D₄	—	2	4	no	—
469	D₄	—	2	4	no	—
470	D₄	—	4	8	no	—
471	D₄	—	8	4	no	—
472	D₄	—	8	8	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	2	4	no	—
475	C₂×C₄	—	2	2	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	2	4	no	—
478	C₂×C₄	—	2	8	no	—
479	C₂×C₄	—	4	2	no	—
480	C₂×C₄	—	4	2	no	—
481	C₂×C₄	—	4	8	no	—
482	C₂×C₄	—	8	8	no	—
483	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
484	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
485	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
486	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
487	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
488	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
489	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
490	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
491	C₈	—	2	2	no	—
492	C₈	—	2	2	no	—
493	C₈	C₂×#41×C₃₁	1	2	yes	—
494	C₈	C₂×#41×C₃₁	1	2	yes	—
495	C₈	C₂×#41×C₃₁	1	2	yes	—
496	C₈	C₂×#41×C₃₁	1	2	yes	—
497	C₂²	—	2	2	no	—
498	C₂²	—	2	2	no	—
499	C₂²	—	2	4	no	—
500	C₂²	—	4	4	no	—
501	C₂²	—	4	4	no	—
502	C₂²	—	4	8	no	—
503	C₂²	C₃₁×#532	1	2	yes	—
504	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
505	C₄	—	2	2	no	—
506	C₄	—	2	4	no	—
507	C₄	—	2	2	no	—
508	C₄	—	2	2	no	—
509	C₄	—	4	4	no	—
510	C₄	—	4	8	no	—
511	C₄	C₃₁×#532	1	2	yes	—
512	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
513	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
514	C₂	—	2	2	no	—
515	C₂	—	4	2	no	—
516	C₂	—	8	4	no	—
517	C₂	C₂×C₃₁×#532	1	1	yes	characteristic
518	C₂	C₃₁×#3584	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
socle	C₂²×C₃₁	C₂³×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3455	C₂	semidirect
2. C₃₁×#3177	C₂	semidirect
3. C₃₁×#3176	C₂	semidirect
4. C₃₁×#3175	C₂	semidirect
5. C₃₁×#3174	C₂	semidirect
6. C₃₁×#3173	C₂	semidirect
7. C₃₁×#3160	C₂	semidirect
8. C₃₁×#3147	C₂	semidirect
9. C₃₁×#3135	C₂	semidirect
10. C₃₁×#3054	C₂	semidirect
11. C₃₁×#3050	C₂	semidirect
12. C₃₁×#2943	C₂	semidirect
13. C₄×C₃₁×#135	C₂	semidirect
14. C₂×C₃₁×#405	C₂	semidirect
30. C₃₁×#500	C₂²	semidirect
31. C₃₁×#499	C₂²	semidirect
32. C₃₁×#499	C₂²	semidirect
33. C₃₁×#498	C₂²	semidirect
34. C₃₁×#495	C₂²	semidirect
35. C₃₁×#495	C₂²	semidirect
36. C₃₁×#494	C₂²	semidirect
37. C₃₁×#493	C₂²	semidirect
38. C₃₁×#492	C₂²	semidirect
39. C₃₁×#491	C₂²	semidirect
40. C₃₁×#490	C₂²	semidirect
41. C₃₁×#489	C₂²	semidirect
48. C₃₁×#442	C₂²	semidirect
49. C₃₁×#441	C₂²	semidirect
50. C₃₁×#440	C₂²	semidirect
51. C₄×Q₁₆×C₃₁	C₂²	semidirect
52. C₄×SD₁₆×C₃₁	C₂²	semidirect
53. C₄×D₈×C₃₁	C₂²	semidirect
200. #60682	C₃₁	direct (N × H)
238. #3455	C₂×C₃₁	semidirect
239. #3177	C₂×C₃₁	semidirect
240. #3176	C₂×C₃₁	semidirect
241. #3175	C₂×C₃₁	semidirect
242. #3174	C₂×C₃₁	semidirect
243. #3173	C₂×C₃₁	semidirect
244. #3160	C₂×C₃₁	semidirect
245. #3147	C₂×C₃₁	semidirect
246. #3135	C₂×C₃₁	semidirect
247. #3054	C₂×C₃₁	semidirect
248. #3050	C₂×C₃₁	semidirect
249. #2943	C₂×C₃₁	semidirect
250. C₄×#135	C₂×C₃₁	semidirect
251. C₂×#405	C₂×C₃₁	semidirect
284. #500	C₂²×C₃₁	semidirect
285. #499	C₂²×C₃₁	semidirect
286. #499	C₂²×C₃₁	semidirect
287. #498	C₂²×C₃₁	semidirect
288. #495	C₂²×C₃₁	semidirect
289. #495	C₂²×C₃₁	semidirect
290. #494	C₂²×C₃₁	semidirect
291. #493	C₂²×C₃₁	semidirect
292. #492	C₂²×C₃₁	semidirect
293. #491	C₂²×C₃₁	semidirect
294. #490	C₂²×C₃₁	semidirect
295. #489	C₂²×C₃₁	semidirect
302. #442	C₂²×C₃₁	semidirect
303. #441	C₂²×C₃₁	semidirect
304. #440	C₂²×C₃₁	semidirect
305. C₄×Q₁₆	C₂²×C₃₁	semidirect
306. C₄×SD₁₆	C₂²×C₃₁	semidirect
307. C₄×D₈	C₂²×C₃₁	semidirect
398. C₃₁	#60682	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×#3455	2
C₃₁×#3177	2
C₃₁×#3176	2
C₃₁×#3175	2
C₃₁×#3174	2
C₃₁×#3173	2
C₃₁×#3160	2
C₃₁×#3147	2
C₃₁×#3135	2
C₃₁×#3054	2
C₃₁×#3050	2
C₃₁×#2943	2
C₄×C₃₁×#135	2
C₂×C₃₁×#405	2
#60682	31

Subgroups — C31×#60682 #21517441

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60682 #21517441