Subgroups — C₃₁×#60817 #21517576

Order 7936.

2258 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3508	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3498	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3220	C₂	1	8	yes	maximal, complemented
4	C₃₁×#3207	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3205	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3151	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₃₁×#493	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#3138	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#3131	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#3041	C₂	1	4	yes	maximal, complemented
11	C₃₁×#3008	C₂	1	4	yes	maximal, complemented
12	C₃₁×#2922	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₃₁×H(5,𝔽₂)	C₂²	1	2	yes	—
14	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	4	yes	complemented
15	C₂²×D₈×C₃₁	C₂²	1	4	yes	complemented
16	C₃₁×#559	—	2	2	no	—
17	C₃₁×#550	—	2	2	no	—
18	C₃₁×#550	—	2	2	no	—
19	C₃₁×#546	—	2	4	no	—
20	D₄²×C₃₁	—	2	2	no	—
21	D₄²×C₃₁	—	2	4	no	—
22	C₃₁×#540	C₂²	1	4	yes	complemented
23	C₃₁×#535	C₂²	1	2	yes	—
24	C₃₁×#535	C₂²	1	2	yes	—
25	C₃₁×#535	C₂²	1	4	yes	complemented
26	C₃₁×#533	C₂²	1	1	yes	characteristic, complemented
27	C₃₁×#532	C₂²	1	1	yes	characteristic
28	C₃₁×#532	C₂²	1	2	yes	complemented
29	C₃₁×#532	C₂²	1	4	yes	complemented
30	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic, complemented
32	C₂×C₃₁×#127	C₂²	1	2	yes	complemented
33	C₃₁×#518	C₂²	1	2	yes	complemented
34	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
35	C₃₁×#496	C₂²	1	4	yes	complemented
36	C₃₁×#493	C₂²	1	4	yes	complemented
37	C₃₁×#493	C₂²	1	4	yes	complemented
38	C₃₁×#493	C₂²	1	2	yes	complemented
39	C₃₁×#492	C₂²	1	2	yes	complemented
40	C₃₁×#490	C₂²	1	4	yes	complemented
41	C₃₁×#488	C₂²	1	4	yes	complemented
42	C₃₁×#487	C₂²	1	2	yes	complemented
43	C₃₁×#486	C₂²	1	2	yes	complemented
44	C₃₁×#486	C₂²	1	4	yes	complemented
45	C₃₁×#463	C₂²	1	16	yes	complemented
46	C₃₁×#459	C₂²	1	16	yes	complemented
47	C₃₁×#449	C₂²	1	16	yes	complemented
48	C₃₁×#447	C₂²	1	16	yes	complemented
49	C₃₁×#433	C₂²	1	4	yes	complemented
50	C₃₁×#431	C₂²	1	4	yes	complemented
51	C₃₁×#424	C₂²	1	2	yes	complemented
52	C₃₁×#418	C₂²	1	4	yes	complemented
53	C₃₁×#416	C₂²	1	4	yes	complemented
54	C₃₁×#408	C₂²	1	2	yes	—
55	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
56	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
57	C₃₁×H(5,𝔽₂)	—	4	8	no	—
58	C₂×#41×C₃₁	—	2	2	no	—
59	C₂×#41×C₃₁	—	2	4	no	—
60	C₂×#41×C₃₁	C₂³	1	2	yes	—
61	C₂×#41×C₃₁	C₂³	1	4	yes	—
62	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
63	C₂²×D₄×C₃₁	—	2	4	no	—
64	C₂²×D₄×C₃₁	—	2	2	no	—
65	C₂²×D₄×C₃₁	—	2	2	no	—
66	C₂²×D₄×C₃₁	—	2	4	no	—
67	C₂²×D₄×C₃₁	—	2	2	no	—
68	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
69	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
70	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
71	C₃₁×AGL(1,ℤ/8)	—	4	8	no	—
72	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
73	C₂×D₈×C₃₁	—	4	8	no	—
74	C₂×D₈×C₃₁	C₂³	1	8	yes	—
75	C₂×D₈×C₃₁	C₂³	1	16	yes	—
76	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
77	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
78	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
79	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
80	C₃₁×#127	—	2	2	no	—
81	C₃₁×#127	—	2	2	no	—
82	C₃₁×#127	—	2	4	no	—
83	C₃₁×#127	—	2	2	no	—
84	C₃₁×#127	C₂³	1	4	yes	—
85	C₃₁×#127	C₂³	1	2	yes	—
86	C₃₁×#127	C₂³	1	1	yes	characteristic
87	C₃₁×#127	C₂³	1	2	yes	—
88	C₃₁×#127	C₂³	1	2	yes	—
89	C₃₁×#127	C₂³	1	2	yes	—
90	C₃₁×#125	C₂³	1	2	yes	—
91	C₃₁×#124	—	2	2	no	—
92	C₃₁×#124	—	2	4	no	—
93	C₃₁×#124	C₂³	1	4	yes	—
94	C₃₁×#124	C₂³	1	2	yes	—
95	C₃₁×#123	—	2	8	no	—
96	C₃₁×C₂²⋊Q₈	—	2	2	no	—
97	C₃₁×#121	—	2	2	no	—
98	C₃₁×#121	—	2	4	no	—
99	C₃₁×#121	—	2	4	no	—
100	C₃₁×#121	—	2	4	no	—
101	C₃₁×#121	—	2	8	no	—
102	C₃₁×#121	—	2	8	no	—
103	C₃₁×#121	C₂³	1	8	yes	—
104	C₃₁×#120	—	2	4	no	—
105	C₃₁×#120	—	2	4	no	—
106	C₃₁×#120	—	2	8	no	—
107	C₄×Q₈×C₃₁	—	2	2	no	—
108	C₄×Q₈×C₃₁	D₄	1	2	yes	—
109	C₄×D₄×C₃₁	—	2	4	no	—
110	C₄×D₄×C₃₁	—	2	2	no	—
111	C₄×D₄×C₃₁	—	2	4	no	—
112	C₄×D₄×C₃₁	—	2	4	no	—
113	C₄×D₄×C₃₁	D₄	1	4	yes	—
114	C₄×D₄×C₃₁	D₄	1	2	yes	—
115	C₄×D₄×C₃₁	C₂³	1	8	yes	—
116	C₃₁×#117	C₂³	1	2	yes	—
117	C₃₁×#117	C₂³	1	1	yes	characteristic
118	C₃₁×#117	C₂³	1	2	yes	—
119	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
120	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
122	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
123	C₃₁×#103	C₂³	1	8	yes	—
124	C₃₁×#102	C₂³	1	8	yes	—
125	C₃₁×#102	C₂³	1	16	yes	—
126	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
127	C₃₁×#97	C₂³	1	2	yes	—
128	C₄×C₈×C₃₁	C₂³	1	2	yes	—
129	C₄×C₈×C₃₁	C₂³	1	2	yes	—
130	C₄×C₈×C₃₁	C₂³	1	2	yes	—
131	C₂⁴×C₃₁	—	2	2	no	—
132	C₂⁴×C₃₁	—	2	4	no	—
133	C₂⁴×C₃₁	—	2	2	no	—
134	#41×C₃₁	—	2	4	no	—
135	#41×C₃₁	—	2	8	no	—
136	#41×C₃₁	—	4	8	no	—
137	#41×C₃₁	—	4	8	no	—
138	#41×C₃₁	—	8	8	no	—
139	C₂×Q₈×C₃₁	—	2	2	no	—
140	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
141	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
142	C₂×D₄×C₃₁	—	2	4	no	—
143	C₂×D₄×C₃₁	—	2	2	no	—
144	C₂×D₄×C₃₁	—	2	8	no	—
145	C₂×D₄×C₃₁	—	2	2	no	—
146	C₂×D₄×C₃₁	—	2	4	no	—
147	C₂×D₄×C₃₁	—	2	2	no	—
148	C₂×D₄×C₃₁	—	2	4	no	—
149	C₂×D₄×C₃₁	—	2	4	no	—
150	C₂×D₄×C₃₁	—	2	2	no	—
151	C₂×D₄×C₃₁	—	2	8	no	—
152	C₂×D₄×C₃₁	—	2	4	no	—
153	C₂×D₄×C₃₁	—	2	4	no	—
154	C₂×D₄×C₃₁	—	2	8	no	—
155	C₂×D₄×C₃₁	—	2	8	no	—
156	C₂×D₄×C₃₁	—	4	8	no	—
157	C₂×D₄×C₃₁	—	4	16	no	—
158	C₂×D₄×C₃₁	—	4	8	no	—
159	C₂×D₄×C₃₁	—	8	4	no	—
160	C₂×D₄×C₃₁	—	8	4	no	—
161	C₂×D₄×C₃₁	—	8	8	no	—
162	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
163	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
164	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
165	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
166	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
167	C₂²×C₄×C₃₁	—	2	4	no	—
168	C₂²×C₄×C₃₁	—	2	2	no	—
169	C₂²×C₄×C₃₁	—	2	8	no	—
170	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
171	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
172	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
173	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
174	SD₁₆×C₃₁	—	4	8	no	—
175	D₈×C₃₁	—	4	16	no	—
176	D₈×C₃₁	—	4	8	no	—
177	M₁₆×C₃₁	—	2	4	no	—
178	C₂×C₈×C₃₁	—	2	4	no	—
179	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
180	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
181	C₄⋊C₄×C₃₁	—	2	2	no	—
182	C₄⋊C₄×C₃₁	—	2	2	no	—
183	C₄⋊C₄×C₃₁	—	2	4	no	—
184	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
185	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
186	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
187	C₂²⋊C₄×C₃₁	—	2	2	no	—
188	C₂²⋊C₄×C₃₁	—	2	8	no	—
189	C₂²⋊C₄×C₃₁	—	2	8	no	—
190	C₂²⋊C₄×C₃₁	—	2	4	no	—
191	C₂²⋊C₄×C₃₁	—	2	8	no	—
192	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
193	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
194	C₄²×C₃₁	—	2	2	no	—
195	C₄²×C₃₁	C₂×D₄	1	4	yes	—
196	C₄²×C₃₁	C₂×D₄	1	2	yes	—
197	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
198	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
199	C₄²×C₃₁	C₂⁴	1	2	yes	—
200	#60817	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
201	C₂³×C₃₁	—	2	4	no	—
202	C₂³×C₃₁	—	2	4	no	—
203	C₂³×C₃₁	—	2	2	no	—
204	C₂³×C₃₁	—	4	8	no	—
205	C₂³×C₃₁	—	4	4	no	—
206	C₂³×C₃₁	—	4	4	no	—
207	C₂³×C₃₁	—	8	4	no	—
208	C₂³×C₃₁	—	8	4	no	—
209	C₂³×C₃₁	—	8	8	no	—
210	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
211	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
212	Q₈×C₃₁	—	2	2	no	—
213	Q₈×C₃₁	—	2	4	no	—
214	Q₈×C₃₁	—	4	4	no	—
215	D₄×C₃₁	—	2	4	no	—
216	D₄×C₃₁	—	2	2	no	—
217	D₄×C₃₁	—	2	8	no	—
218	D₄×C₃₁	—	2	8	no	—
219	D₄×C₃₁	—	2	4	no	—
220	D₄×C₃₁	—	4	8	no	—
221	D₄×C₃₁	—	4	8	no	—
222	D₄×C₃₁	—	4	4	no	—
223	D₄×C₃₁	—	4	4	no	—
224	D₄×C₃₁	—	4	4	no	—
225	D₄×C₃₁	—	4	4	no	—
226	D₄×C₃₁	—	4	8	no	—
227	D₄×C₃₁	—	4	8	no	—
228	D₄×C₃₁	—	8	8	no	—
229	D₄×C₃₁	—	8	4	no	—
230	D₄×C₃₁	—	8	4	no	—
231	D₄×C₃₁	—	8	8	no	—
232	C₂×C₄×C₃₁	—	2	4	no	—
233	C₂×C₄×C₃₁	—	2	8	no	—
234	C₂×C₄×C₃₁	—	2	4	no	—
235	C₂×C₄×C₃₁	—	2	4	no	—
236	C₂×C₄×C₃₁	—	2	2	no	—
237	C₂×C₄×C₃₁	—	4	8	no	—
238	C₂×C₄×C₃₁	—	4	4	no	—
239	C₂×C₄×C₃₁	—	8	8	no	—
240	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
241	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
242	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
243	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
244	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
245	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
246	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
247	C₈×C₃₁	—	2	4	no	—
248	C₈×C₃₁	—	2	4	no	—
249	#3508	C₂×C₃₁	1	1	yes	characteristic, complemented
250	#3498	C₂×C₃₁	1	1	yes	characteristic, complemented
251	#3220	C₂×C₃₁	1	8	yes	complemented
252	#3207	C₂×C₃₁	1	4	yes	complemented
253	#3205	C₂×C₃₁	1	4	yes	complemented
254	#3151	C₂×C₃₁	1	1	yes	characteristic, complemented
255	C₂×#493	C₂×C₃₁	1	1	yes	characteristic, complemented
256	#3138	C₂×C₃₁	1	1	yes	characteristic, complemented
257	#3131	C₂×C₃₁	1	1	yes	characteristic, complemented
258	#3041	C₂×C₃₁	1	4	yes	complemented
259	#3008	C₂×C₃₁	1	4	yes	complemented
260	#2922	C₂×C₃₁	1	1	yes	characteristic, complemented
261	C₂²×C₃₁	—	2	2	no	—
262	C₂²×C₃₁	—	2	2	no	—
263	C₂²×C₃₁	—	2	4	no	—
264	C₂²×C₃₁	—	4	4	no	—
265	C₂²×C₃₁	—	4	2	no	—
266	C₂²×C₃₁	—	4	4	no	—
267	C₂²×C₃₁	—	4	8	no	—
268	C₂²×C₃₁	—	4	8	no	—
269	C₂²×C₃₁	—	4	4	no	—
270	C₂²×C₃₁	—	16	4	no	—
271	C₂²×C₃₁	—	16	4	no	—
272	C₂²×C₃₁	—	16	8	no	—
273	C₂²×C₃₁	#535	1	2	yes	—
274	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
275	C₄×C₃₁	—	2	2	no	—
276	C₄×C₃₁	—	2	4	no	—
277	C₄×C₃₁	—	2	2	no	—
278	C₄×C₃₁	—	4	2	no	—
279	C₄×C₃₁	—	4	4	no	—
280	C₄×C₃₁	#535	1	2	yes	—
281	C₄×C₃₁	#576	1	4	yes	—
282	C₄×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
283	C₂×H(5,𝔽₂)	C₂²×C₃₁	1	2	yes	—
284	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	4	yes	complemented
285	C₂²×D₈	C₂²×C₃₁	1	4	yes	complemented
286	#559	—	2	2	no	—
287	#550	—	2	2	no	—
288	#550	—	2	2	no	—
289	#546	—	2	4	no	—
290	D₄²	—	2	2	no	—
291	D₄²	—	2	4	no	—
292	#540	C₂²×C₃₁	1	4	yes	complemented
293	#535	C₂²×C₃₁	1	2	yes	—
294	#535	C₂²×C₃₁	1	2	yes	—
295	#535	C₂²×C₃₁	1	4	yes	complemented
296	#533	C₂²×C₃₁	1	1	yes	characteristic, complemented
297	#532	C₂²×C₃₁	1	1	yes	characteristic
298	#532	C₂²×C₃₁	1	2	yes	complemented
299	#532	C₂²×C₃₁	1	4	yes	complemented
300	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
301	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic, complemented
302	C₂×#127	C₂²×C₃₁	1	2	yes	complemented
303	#518	C₂²×C₃₁	1	2	yes	complemented
304	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
305	#496	C₂²×C₃₁	1	4	yes	complemented
306	#493	C₂²×C₃₁	1	4	yes	complemented
307	#493	C₂²×C₃₁	1	4	yes	complemented
308	#493	C₂²×C₃₁	1	2	yes	complemented
309	#492	C₂²×C₃₁	1	2	yes	complemented
310	#490	C₂²×C₃₁	1	4	yes	complemented
311	#488	C₂²×C₃₁	1	4	yes	complemented
312	#487	C₂²×C₃₁	1	2	yes	complemented
313	#486	C₂²×C₃₁	1	2	yes	complemented
314	#486	C₂²×C₃₁	1	4	yes	complemented
315	#463	C₂²×C₃₁	1	16	yes	complemented
316	#459	C₂²×C₃₁	1	16	yes	complemented
317	#449	C₂²×C₃₁	1	16	yes	complemented
318	#447	C₂²×C₃₁	1	16	yes	complemented
319	#433	C₂²×C₃₁	1	4	yes	complemented
320	#431	C₂²×C₃₁	1	4	yes	complemented
321	#424	C₂²×C₃₁	1	2	yes	complemented
322	#418	C₂²×C₃₁	1	4	yes	complemented
323	#416	C₂²×C₃₁	1	4	yes	complemented
324	#408	C₂²×C₃₁	1	2	yes	—
325	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
326	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
327	C₂×C₃₁	—	2	2	no	—
328	C₂×C₃₁	—	4	2	no	—
329	C₂×C₃₁	—	8	4	no	—
330	C₂×C₃₁	—	8	4	no	—
331	C₂×C₃₁	—	8	2	no	—
332	C₂×C₃₁	C₂×#535	1	1	yes	characteristic
333	C₂×C₃₁	C₂×#576	1	2	yes	—
334	H(5,𝔽₂)	—	4	8	no	—
335	C₂×#41	—	2	2	no	—
336	C₂×#41	—	2	4	no	—
337	C₂×#41	C₂³×C₃₁	1	2	yes	—
338	C₂×#41	C₂³×C₃₁	1	4	yes	—
339	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
340	C₂²×D₄	—	2	4	no	—
341	C₂²×D₄	—	2	2	no	—
342	C₂²×D₄	—	2	2	no	—
343	C₂²×D₄	—	2	4	no	—
344	C₂²×D₄	—	2	2	no	—
345	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
346	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
347	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
348	AGL(1,ℤ/8)	—	4	8	no	—
349	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
350	C₂×D₈	—	4	8	no	—
351	C₂×D₈	C₂³×C₃₁	1	16	yes	—
352	C₂×D₈	C₂³×C₃₁	1	8	yes	—
353	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
354	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
355	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
356	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
357	#127	—	2	2	no	—
358	#127	—	2	2	no	—
359	#127	—	2	4	no	—
360	#127	—	2	2	no	—
361	#127	C₂³×C₃₁	1	2	yes	—
362	#127	C₂³×C₃₁	1	2	yes	—
363	#127	C₂³×C₃₁	1	1	yes	characteristic
364	#127	C₂³×C₃₁	1	2	yes	—
365	#127	C₂³×C₃₁	1	4	yes	—
366	#127	C₂³×C₃₁	1	2	yes	—
367	#125	C₂³×C₃₁	1	2	yes	—
368	#124	—	2	2	no	—
369	#124	—	2	4	no	—
370	#124	C₂³×C₃₁	1	2	yes	—
371	#124	C₂³×C₃₁	1	4	yes	—
372	#123	—	2	8	no	—
373	C₂²⋊Q₈	—	2	2	no	—
374	#121	—	2	2	no	—
375	#121	—	2	4	no	—
376	#121	—	2	4	no	—
377	#121	—	2	4	no	—
378	#121	—	2	8	no	—
379	#121	—	2	8	no	—
380	#121	C₂³×C₃₁	1	8	yes	—
381	#120	—	2	4	no	—
382	#120	—	2	4	no	—
383	#120	—	2	8	no	—
384	C₄×Q₈	—	2	2	no	—
385	C₄×Q₈	D₄×C₃₁	1	2	yes	—
386	C₄×D₄	—	2	4	no	—
387	C₄×D₄	—	2	2	no	—
388	C₄×D₄	—	2	4	no	—
389	C₄×D₄	—	2	4	no	—
390	C₄×D₄	D₄×C₃₁	1	4	yes	—
391	C₄×D₄	D₄×C₃₁	1	2	yes	—
392	C₄×D₄	C₂³×C₃₁	1	8	yes	—
393	#117	C₂³×C₃₁	1	2	yes	—
394	#117	C₂³×C₃₁	1	1	yes	characteristic
395	#117	C₂³×C₃₁	1	2	yes	—
396	C₂×C₂²⋊C₄	—	2	4	no	—
397	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
398	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
399	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
400	#103	C₂³×C₃₁	1	8	yes	—
401	#102	C₂³×C₃₁	1	8	yes	—
402	#102	C₂³×C₃₁	1	16	yes	—
403	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
404	#97	C₂³×C₃₁	1	2	yes	—
405	C₄×C₈	C₂³×C₃₁	1	2	yes	—
406	C₄×C₈	C₂³×C₃₁	1	2	yes	—
407	C₄×C₈	C₂³×C₃₁	1	2	yes	—
408	C₃₁	#60817	1	1	yes	characteristic, direct product factor, Sylow
409	C₂⁴	—	2	2	no	—
410	C₂⁴	—	2	4	no	—
411	C₂⁴	—	2	2	no	—
412	#41	—	2	4	no	—
413	#41	—	2	8	no	—
414	#41	—	4	8	no	—
415	#41	—	4	8	no	—
416	#41	—	8	8	no	—
417	C₂×Q₈	—	2	2	no	—
418	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
419	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
420	C₂×D₄	—	2	4	no	—
421	C₂×D₄	—	2	2	no	—
422	C₂×D₄	—	2	8	no	—
423	C₂×D₄	—	2	2	no	—
424	C₂×D₄	—	2	4	no	—
425	C₂×D₄	—	2	2	no	—
426	C₂×D₄	—	2	4	no	—
427	C₂×D₄	—	2	4	no	—
428	C₂×D₄	—	2	2	no	—
429	C₂×D₄	—	2	8	no	—
430	C₂×D₄	—	2	4	no	—
431	C₂×D₄	—	2	4	no	—
432	C₂×D₄	—	2	8	no	—
433	C₂×D₄	—	2	8	no	—
434	C₂×D₄	—	4	8	no	—
435	C₂×D₄	—	4	16	no	—
436	C₂×D₄	—	4	8	no	—
437	C₂×D₄	—	8	4	no	—
438	C₂×D₄	—	8	4	no	—
439	C₂×D₄	—	8	8	no	—
440	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
441	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
442	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
443	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
444	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
445	C₂²×C₄	—	2	4	no	—
446	C₂²×C₄	—	2	2	no	—
447	C₂²×C₄	—	2	8	no	—
448	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
449	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
450	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
451	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
452	SD₁₆	—	4	8	no	—
453	D₈	—	4	16	no	—
454	D₈	—	4	8	no	—
455	M₁₆	—	2	4	no	—
456	C₂×C₈	—	2	4	no	—
457	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
458	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
459	C₄⋊C₄	—	2	2	no	—
460	C₄⋊C₄	—	2	2	no	—
461	C₄⋊C₄	—	2	4	no	—
462	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
463	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
464	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
465	C₂²⋊C₄	—	2	2	no	—
466	C₂²⋊C₄	—	2	8	no	—
467	C₂²⋊C₄	—	2	8	no	—
468	C₂²⋊C₄	—	2	4	no	—
469	C₂²⋊C₄	—	2	8	no	—
470	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
471	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
472	C₄²	—	2	2	no	—
473	C₄²	C₂×D₄×C₃₁	1	4	yes	—
474	C₄²	C₂×D₄×C₃₁	1	2	yes	—
475	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
476	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
477	C₄²	C₂⁴×C₃₁	1	2	yes	—
478	C₂³	—	2	4	no	—
479	C₂³	—	2	4	no	—
480	C₂³	—	2	2	no	—
481	C₂³	—	4	8	no	—
482	C₂³	—	4	4	no	—
483	C₂³	—	4	4	no	—
484	C₂³	—	8	4	no	—
485	C₂³	—	8	4	no	—
486	C₂³	—	8	8	no	—
487	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
488	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
489	Q₈	—	2	2	no	—
490	Q₈	—	2	4	no	—
491	Q₈	—	4	4	no	—
492	D₄	—	2	4	no	—
493	D₄	—	2	2	no	—
494	D₄	—	2	8	no	—
495	D₄	—	2	8	no	—
496	D₄	—	2	4	no	—
497	D₄	—	4	8	no	—
498	D₄	—	4	8	no	—
499	D₄	—	4	4	no	—
500	D₄	—	4	4	no	—
501	D₄	—	4	4	no	—
502	D₄	—	4	4	no	—
503	D₄	—	4	8	no	—
504	D₄	—	4	8	no	—
505	D₄	—	8	8	no	—
506	D₄	—	8	4	no	—
507	D₄	—	8	4	no	—
508	D₄	—	8	8	no	—
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	2	8	no	—
511	C₂×C₄	—	2	4	no	—
512	C₂×C₄	—	2	4	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	4	8	no	—
515	C₂×C₄	—	4	4	no	—
516	C₂×C₄	—	8	8	no	—
517	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
518	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
519	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
520	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
521	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
522	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
523	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
524	C₈	—	2	4	no	—
525	C₈	—	2	4	no	—
526	C₂²	—	2	2	no	—
527	C₂²	—	2	2	no	—
528	C₂²	—	2	4	no	—
529	C₂²	—	4	4	no	—
530	C₂²	—	4	2	no	—
531	C₂²	—	4	4	no	—
532	C₂²	—	4	8	no	—
533	C₂²	—	4	8	no	—
534	C₂²	—	4	4	no	—
535	C₂²	—	16	4	no	—
536	C₂²	—	16	4	no	—
537	C₂²	—	16	8	no	—
538	C₂²	C₃₁×#535	1	2	yes	—
539	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
540	C₄	—	2	2	no	—
541	C₄	—	2	4	no	—
542	C₄	—	2	2	no	—
543	C₄	—	4	2	no	—
544	C₄	—	4	4	no	—
545	C₄	C₃₁×#535	1	2	yes	—
546	C₄	C₃₁×#576	1	4	yes	—
547	C₄	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
548	C₂	—	2	2	no	—
549	C₂	—	4	2	no	—
550	C₂	—	8	4	no	—
551	C₂	—	8	2	no	—
552	C₂	—	8	4	no	—
553	C₂	C₂×C₃₁×#535	1	1	yes	characteristic
554	C₂	C₂×C₃₁×#576	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ 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
1. C₃₁×#3508	C₂	semidirect
2. C₃₁×#3498	C₂	semidirect
3. C₃₁×#3220	C₂	semidirect
4. C₃₁×#3207	C₂	semidirect
5. C₃₁×#3205	C₂	semidirect
6. C₃₁×#3151	C₂	semidirect
7. C₂×C₃₁×#493	C₂	semidirect
8. C₃₁×#3138	C₂	semidirect
9. C₃₁×#3131	C₂	semidirect
10. C₃₁×#3041	C₂	semidirect
11. C₃₁×#3008	C₂	semidirect
12. C₃₁×#2922	C₂	semidirect
14. C₂×C₃₁×AGL(1,ℤ/8)	C₂²	semidirect
15. C₂²×D₈×C₃₁	C₂²	semidirect
22. C₃₁×#540	C₂²	semidirect
25. C₃₁×#535	C₂²	semidirect
26. C₃₁×#533	C₂²	semidirect
28. C₃₁×#532	C₂²	semidirect
29. C₃₁×#532	C₂²	semidirect
31. C₂×C₃₁×#127	C₂²	semidirect
32. C₂×C₃₁×#127	C₂²	semidirect
33. C₃₁×#518	C₂²	semidirect
35. C₃₁×#496	C₂²	semidirect
36. C₃₁×#493	C₂²	semidirect
37. C₃₁×#493	C₂²	semidirect
38. C₃₁×#493	C₂²	semidirect
39. C₃₁×#492	C₂²	semidirect
40. C₃₁×#490	C₂²	semidirect
41. C₃₁×#488	C₂²	semidirect
42. C₃₁×#487	C₂²	semidirect
43. C₃₁×#486	C₂²	semidirect
44. C₃₁×#486	C₂²	semidirect
45. C₃₁×#463	C₂²	semidirect
46. C₃₁×#459	C₂²	semidirect
47. C₃₁×#449	C₂²	semidirect
48. C₃₁×#447	C₂²	semidirect
49. C₃₁×#433	C₂²	semidirect
50. C₃₁×#431	C₂²	semidirect
51. C₃₁×#424	C₂²	semidirect
52. C₃₁×#418	C₂²	semidirect
53. C₃₁×#416	C₂²	semidirect
55. C₄×M₁₆×C₃₁	C₂²	semidirect
56. C₂×C₄×C₈×C₃₁	C₂²	semidirect
200. #60817	C₃₁	direct (N × H)
249. #3508	C₂×C₃₁	semidirect
250. #3498	C₂×C₃₁	semidirect
251. #3220	C₂×C₃₁	semidirect
252. #3207	C₂×C₃₁	semidirect
253. #3205	C₂×C₃₁	semidirect
254. #3151	C₂×C₃₁	semidirect
255. C₂×#493	C₂×C₃₁	semidirect
256. #3138	C₂×C₃₁	semidirect
257. #3131	C₂×C₃₁	semidirect
258. #3041	C₂×C₃₁	semidirect
259. #3008	C₂×C₃₁	semidirect
260. #2922	C₂×C₃₁	semidirect
284. C₂×AGL(1,ℤ/8)	C₂²×C₃₁	semidirect
285. C₂²×D₈	C₂²×C₃₁	semidirect
292. #540	C₂²×C₃₁	semidirect
295. #535	C₂²×C₃₁	semidirect
296. #533	C₂²×C₃₁	semidirect
298. #532	C₂²×C₃₁	semidirect
299. #532	C₂²×C₃₁	semidirect
301. C₂×#127	C₂²×C₃₁	semidirect
302. C₂×#127	C₂²×C₃₁	semidirect
303. #518	C₂²×C₃₁	semidirect
305. #496	C₂²×C₃₁	semidirect
306. #493	C₂²×C₃₁	semidirect
307. #493	C₂²×C₃₁	semidirect
308. #493	C₂²×C₃₁	semidirect
309. #492	C₂²×C₃₁	semidirect
310. #490	C₂²×C₃₁	semidirect
311. #488	C₂²×C₃₁	semidirect
312. #487	C₂²×C₃₁	semidirect
313. #486	C₂²×C₃₁	semidirect
314. #486	C₂²×C₃₁	semidirect
315. #463	C₂²×C₃₁	semidirect
316. #459	C₂²×C₃₁	semidirect
317. #449	C₂²×C₃₁	semidirect
318. #447	C₂²×C₃₁	semidirect
319. #433	C₂²×C₃₁	semidirect
320. #431	C₂²×C₃₁	semidirect
321. #424	C₂²×C₃₁	semidirect
322. #418	C₂²×C₃₁	semidirect
323. #416	C₂²×C₃₁	semidirect
325. C₄×M₁₆	C₂²×C₃₁	semidirect
326. C₂×C₄×C₈	C₂²×C₃₁	semidirect
408. C₃₁	#60817	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×#3508	2
C₃₁×#3498	2
C₃₁×#3220	2
C₃₁×#3207	2
C₃₁×#3205	2
C₃₁×#3151	2
C₂×C₃₁×#493	2
C₃₁×#3138	2
C₃₁×#3131	2
C₃₁×#3041	2
C₃₁×#3008	2
C₃₁×#2922	2
#60817	31

Subgroups — C31×#60817 #21517576

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60817 #21517576