Subgroups — C₃₁×#19719 #21476478

Order 7936.

840 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3228	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3208	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3207	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3205	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#3182	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#3151	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1721	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1721	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1719	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1717	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1715	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#1714	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#1706	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#1706	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#1453	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₃₁×AGL(1,ℤ/8)	—	2	2	no	—
17	C₃₁×#533	C₂²	1	1	yes	characteristic
18	C₃₁×#532	C₂²	1	1	yes	characteristic
19	C₃₁×#532	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
21	C₃₁×#500	C₂²	1	1	yes	characteristic, complemented
22	C₃₁×#500	C₂²	1	1	yes	characteristic, complemented
23	C₃₁×#498	C₂²	1	1	yes	characteristic, complemented
24	C₃₁×#496	—	2	2	no	—
25	C₃₁×#493	C₂²	1	1	yes	characteristic
26	C₃₁×#493	C₂²	1	1	yes	characteristic
27	C₃₁×#493	C₂²	1	1	yes	characteristic
28	C₃₁×#493	C₂²	1	1	yes	characteristic
29	C₃₁×#493	C₂²	1	1	yes	characteristic, complemented
30	C₃₁×#492	C₂²	1	1	yes	characteristic
31	C₃₁×#492	C₂²	1	1	yes	characteristic
32	C₃₁×#492	C₂²	1	1	yes	characteristic
33	C₃₁×#492	C₂²	1	1	yes	characteristic
34	C₃₁×#492	C₂²	1	1	yes	characteristic, complemented
35	C₃₁×#492	C₂²	1	1	yes	characteristic, complemented
36	C₃₁×#487	C₂²	1	1	yes	characteristic, complemented
37	C₃₁×#487	C₂²	1	1	yes	characteristic, complemented
38	C₃₁×#486	C₂²	1	1	yes	characteristic
39	C₃₁×#486	C₂²	1	1	yes	characteristic
40	C₃₁×#486	C₂²	1	1	yes	characteristic
41	C₃₁×#486	C₂²	1	1	yes	characteristic
42	C₃₁×#486	C₂²	1	1	yes	characteristic, complemented
43	C₃₁×#486	C₂²	1	1	yes	characteristic, complemented
44	C₃₁×#486	C₂²	1	1	yes	characteristic, complemented
45	C₃₁×#486	C₂²	1	1	yes	characteristic, complemented
46	C₃₁×#482	—	2	2	no	—
47	C₃₁×#478	—	2	2	no	—
48	C₃₁×#468	—	2	2	no	—
49	C₃₁×#463	—	2	2	no	—
50	C₃₁×#460	—	2	2	no	—
51	C₃₁×#459	—	2	2	no	—
52	C₃₁×#450	—	2	2	no	—
53	C₃₁×#449	—	2	2	no	—
54	C₃₁×#447	—	2	2	no	—
55	C₃₁×#440	—	2	2	no	—
56	C₃₁×#431	C₂²	1	1	yes	characteristic
57	C₃₁×#431	C₂²	1	1	yes	characteristic
58	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
59	C₃₁×#322	C₂²	1	1	yes	characteristic, complemented
60	C₃₁×#322	C₂²	1	1	yes	characteristic, complemented
61	C₃₁×#322	C₂²	1	1	yes	characteristic, complemented
62	C₈²×C₃₁	C₂²	1	1	yes	characteristic, complemented
63	C₂×#41×C₃₁	—	2	1	no	—
64	C₂×#41×C₃₁	—	2	1	no	—
65	C₂²×D₄×C₃₁	—	2	1	no	—
66	C₂²×D₄×C₃₁	—	2	1	no	—
67	C₂²×D₄×C₃₁	—	2	1	no	—
68	C₃₁×AGL(1,ℤ/8)	—	4	8	no	—
69	C₂×SD₁₆×C₃₁	—	2	2	no	—
70	C₂×SD₁₆×C₃₁	—	2	2	no	—
71	C₂×SD₁₆×C₃₁	—	2	2	no	—
72	C₂×SD₁₆×C₃₁	—	2	2	no	—
73	C₂×SD₁₆×C₃₁	—	2	2	no	—
74	C₂×SD₁₆×C₃₁	—	2	2	no	—
75	C₂×D₈×C₃₁	—	2	2	no	—
76	C₂×D₈×C₃₁	—	2	2	no	—
77	C₂×D₈×C₃₁	—	2	2	no	—
78	C₂×D₈×C₃₁	—	2	2	no	—
79	C₂×D₈×C₃₁	—	2	2	no	—
80	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
81	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
82	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
83	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
84	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
85	C₃₁×#127	—	2	1	no	—
86	C₃₁×#127	—	2	1	no	—
87	C₃₁×#127	—	2	1	no	—
88	C₃₁×#127	C₂³	1	1	yes	characteristic
89	C₃₁×#127	C₂³	1	1	yes	characteristic
90	C₃₁×#127	C₂³	1	1	yes	characteristic
91	C₃₁×#127	C₂³	1	1	yes	characteristic
92	C₃₁×#125	—	2	1	no	—
93	C₃₁×#124	—	2	1	no	—
94	C₃₁×#124	—	2	1	no	—
95	C₃₁×C₂²⋊Q₈	—	2	1	no	—
96	C₃₁×C₂²⋊Q₈	—	2	1	no	—
97	C₃₁×#121	—	2	1	no	—
98	C₃₁×#121	—	2	1	no	—
99	C₃₁×#121	—	2	1	no	—
100	C₃₁×#121	—	2	1	no	—
101	C₄×Q₈×C₃₁	—	2	1	no	—
102	C₄×Q₈×C₃₁	—	2	1	no	—
103	C₄×D₄×C₃₁	—	2	1	no	—
104	C₄×D₄×C₃₁	—	2	1	no	—
105	C₄×D₄×C₃₁	—	2	1	no	—
106	C₄×D₄×C₃₁	—	2	1	no	—
107	C₃₁×#117	—	2	1	no	—
108	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
109	C₃₁×#107	—	2	2	no	—
110	C₃₁×#107	—	2	2	no	—
111	C₃₁×#106	—	2	2	no	—
112	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
113	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
114	C₃₁×#103	—	2	2	no	—
115	C₃₁×#103	—	2	2	no	—
116	C₃₁×#102	—	2	2	no	—
117	C₃₁×#102	—	2	2	no	—
118	C₃₁×#102	—	2	2	no	—
119	C₃₁×#102	—	2	2	no	—
120	C₃₁×#102	—	2	2	no	—
121	C₃₁×#102	—	2	2	no	—
122	C₃₁×#102	—	2	2	no	—
123	C₃₁×#102	—	2	2	no	—
124	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
125	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
126	C₃₁×#97	D₄	1	2	yes	—
127	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
128	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
129	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
130	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
131	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
132	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
133	C₂⁴×C₃₁	—	4	1	no	—
134	#41×C₃₁	—	4	4	no	—
135	#41×C₃₁	—	8	2	no	—
136	C₂×Q₈×C₃₁	—	2	1	no	—
137	C₂×Q₈×C₃₁	—	2	1	no	—
138	C₂×Q₈×C₃₁	—	2	1	no	—
139	C₂×Q₈×C₃₁	—	2	1	no	—
140	C₂×D₄×C₃₁	—	2	1	no	—
141	C₂×D₄×C₃₁	—	2	1	no	—
142	C₂×D₄×C₃₁	—	2	1	no	—
143	C₂×D₄×C₃₁	—	2	1	no	—
144	C₂×D₄×C₃₁	—	2	1	no	—
145	C₂×D₄×C₃₁	—	2	1	no	—
146	C₂×D₄×C₃₁	—	2	1	no	—
147	C₂×D₄×C₃₁	—	2	1	no	—
148	C₂×D₄×C₃₁	—	2	1	no	—
149	C₂×D₄×C₃₁	—	2	1	no	—
150	C₂×D₄×C₃₁	—	2	1	no	—
151	C₂×D₄×C₃₁	—	2	1	no	—
152	C₂×D₄×C₃₁	—	4	4	no	—
153	C₂×D₄×C₃₁	—	4	1	no	—
154	C₂×D₄×C₃₁	—	4	1	no	—
155	C₂×D₄×C₃₁	—	4	1	no	—
156	C₂×D₄×C₃₁	—	4	1	no	—
157	C₂×D₄×C₃₁	—	4	1	no	—
158	C₂×D₄×C₃₁	—	8	2	no	—
159	C₂×D₄×C₃₁	—	8	2	no	—
160	C₂²×C₄×C₃₁	—	4	1	no	—
161	C₂²×C₄×C₃₁	—	4	1	no	—
162	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
165	SD₁₆×C₃₁	—	4	4	no	—
166	SD₁₆×C₃₁	—	4	4	no	—
167	SD₁₆×C₃₁	—	4	4	no	—
168	SD₁₆×C₃₁	—	8	2	no	—
169	SD₁₆×C₃₁	—	8	2	no	—
170	SD₁₆×C₃₁	—	8	2	no	—
171	D₈×C₃₁	—	4	4	no	—
172	D₈×C₃₁	—	4	4	no	—
173	D₈×C₃₁	—	8	2	no	—
174	D₈×C₃₁	—	8	2	no	—
175	D₈×C₃₁	—	8	2	no	—
176	M₁₆×C₃₁	—	2	4	no	—
177	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
178	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
179	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
180	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
181	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
182	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
183	C₄⋊C₄×C₃₁	—	2	1	no	—
184	C₄⋊C₄×C₃₁	—	2	1	no	—
185	C₄⋊C₄×C₃₁	—	2	1	no	—
186	C₄⋊C₄×C₃₁	—	2	1	no	—
187	C₄⋊C₄×C₃₁	—	2	1	no	—
188	C₄⋊C₄×C₃₁	—	2	1	no	—
189	C₄⋊C₄×C₃₁	—	2	1	no	—
190	C₄⋊C₄×C₃₁	—	2	1	no	—
191	C₄⋊C₄×C₃₁	—	4	1	no	—
192	C₄⋊C₄×C₃₁	—	4	1	no	—
193	C₂²⋊C₄×C₃₁	—	4	1	no	—
194	C₂²⋊C₄×C₃₁	—	4	1	no	—
195	C₂²⋊C₄×C₃₁	—	4	1	no	—
196	C₂²⋊C₄×C₃₁	—	4	1	no	—
197	C₂²⋊C₄×C₃₁	—	4	1	no	—
198	C₄²×C₃₁	—	4	1	no	—
199	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
200	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
201	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
202	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
203	#19719	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
204	C₂³×C₃₁	—	4	1	no	—
205	C₂³×C₃₁	—	4	1	no	—
206	C₂³×C₃₁	—	4	1	no	—
207	C₂³×C₃₁	—	4	1	no	—
208	C₂³×C₃₁	—	8	2	no	—
209	C₂³×C₃₁	—	8	2	no	—
210	C₂³×C₃₁	—	8	2	no	—
211	C₂³×C₃₁	#127	1	1	yes	characteristic
212	Q₈×C₃₁	—	4	2	no	—
213	Q₈×C₃₁	—	4	2	no	—
214	Q₈×C₃₁	—	4	2	no	—
215	Q₈×C₃₁	—	4	2	no	—
216	D₄×C₃₁	—	4	2	no	—
217	D₄×C₃₁	—	4	2	no	—
218	D₄×C₃₁	—	4	2	no	—
219	D₄×C₃₁	—	4	2	no	—
220	D₄×C₃₁	—	4	2	no	—
221	D₄×C₃₁	—	4	2	no	—
222	D₄×C₃₁	—	4	2	no	—
223	D₄×C₃₁	—	4	2	no	—
224	D₄×C₃₁	—	4	2	no	—
225	D₄×C₃₁	—	4	2	no	—
226	D₄×C₃₁	—	4	2	no	—
227	D₄×C₃₁	—	4	2	no	—
228	D₄×C₃₁	—	8	2	no	—
229	D₄×C₃₁	—	8	2	no	—
230	D₄×C₃₁	—	8	2	no	—
231	D₄×C₃₁	—	8	2	no	—
232	D₄×C₃₁	—	8	2	no	—
233	C₂×C₄×C₃₁	—	2	2	no	—
234	C₂×C₄×C₃₁	—	2	2	no	—
235	C₂×C₄×C₃₁	—	2	2	no	—
236	C₂×C₄×C₃₁	—	4	1	no	—
237	C₂×C₄×C₃₁	—	4	1	no	—
238	C₂×C₄×C₃₁	—	4	1	no	—
239	C₂×C₄×C₃₁	—	4	1	no	—
240	C₂×C₄×C₃₁	—	8	2	no	—
241	C₂×C₄×C₃₁	—	8	2	no	—
242	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
243	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
244	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
245	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
246	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
247	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
248	C₈×C₃₁	—	2	2	no	—
249	C₈×C₃₁	—	2	2	no	—
250	C₈×C₃₁	—	2	2	no	—
251	C₈×C₃₁	—	2	2	no	—
252	C₈×C₃₁	—	2	2	no	—
253	C₈×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
254	#3228	C₂×C₃₁	1	1	yes	characteristic, complemented
255	#3208	C₂×C₃₁	1	1	yes	characteristic, complemented
256	#3207	C₂×C₃₁	1	1	yes	characteristic, complemented
257	#3205	C₂×C₃₁	1	1	yes	characteristic, complemented
258	#3182	C₂×C₃₁	1	1	yes	characteristic, complemented
259	#3151	C₂×C₃₁	1	1	yes	characteristic, complemented
260	#1721	C₂×C₃₁	1	1	yes	characteristic, complemented
261	#1721	C₂×C₃₁	1	1	yes	characteristic, complemented
262	#1719	C₂×C₃₁	1	1	yes	characteristic, complemented
263	#1717	C₂×C₃₁	1	1	yes	characteristic, complemented
264	#1715	C₂×C₃₁	1	1	yes	characteristic, complemented
265	#1714	C₂×C₃₁	1	1	yes	characteristic, complemented
266	#1706	C₂×C₃₁	1	1	yes	characteristic, complemented
267	#1706	C₂×C₃₁	1	1	yes	characteristic, complemented
268	#1453	C₂×C₃₁	1	1	yes	characteristic, complemented
269	C₂²×C₃₁	—	2	1	no	—
270	C₂²×C₃₁	—	2	1	no	—
271	C₂²×C₃₁	—	2	1	no	—
272	C₂²×C₃₁	—	8	1	no	—
273	C₂²×C₃₁	—	8	1	no	—
274	C₂²×C₃₁	—	8	1	no	—
275	C₂²×C₃₁	—	8	1	no	—
276	C₂²×C₃₁	—	8	1	no	—
277	C₂²×C₃₁	—	8	1	no	—
278	C₂²×C₃₁	—	8	1	no	—
279	C₂²×C₃₁	—	8	1	no	—
280	C₂²×C₃₁	—	8	1	no	—
281	C₂²×C₃₁	—	8	1	no	—
282	C₂²×C₃₁	—	8	1	no	—
283	C₂²×C₃₁	—	8	1	no	—
284	C₂²×C₃₁	—	16	4	no	—
285	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
286	C₄×C₃₁	—	2	1	no	—
287	C₄×C₃₁	—	2	1	no	—
288	C₄×C₃₁	—	2	1	no	—
289	C₄×C₃₁	—	8	1	no	—
290	C₄×C₃₁	—	8	1	no	—
291	C₄×C₃₁	—	8	1	no	—
292	C₄×C₃₁	—	8	1	no	—
293	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
294	C₄×C₃₁	#576	1	2	yes	—
295	C₄×C₃₁	#577	1	2	yes	—
296	C₂×AGL(1,ℤ/8)	—	2	2	no	—
297	#533	C₂²×C₃₁	1	1	yes	characteristic
298	#532	C₂²×C₃₁	1	1	yes	characteristic
299	#532	C₂²×C₃₁	1	1	yes	characteristic
300	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
301	#500	C₂²×C₃₁	1	1	yes	characteristic, complemented
302	#500	C₂²×C₃₁	1	1	yes	characteristic, complemented
303	#498	C₂²×C₃₁	1	1	yes	characteristic, complemented
304	#496	—	2	2	no	—
305	#493	C₂²×C₃₁	1	1	yes	characteristic
306	#493	C₂²×C₃₁	1	1	yes	characteristic
307	#493	C₂²×C₃₁	1	1	yes	characteristic
308	#493	C₂²×C₃₁	1	1	yes	characteristic
309	#493	C₂²×C₃₁	1	1	yes	characteristic, complemented
310	#492	C₂²×C₃₁	1	1	yes	characteristic
311	#492	C₂²×C₃₁	1	1	yes	characteristic
312	#492	C₂²×C₃₁	1	1	yes	characteristic
313	#492	C₂²×C₃₁	1	1	yes	characteristic
314	#492	C₂²×C₃₁	1	1	yes	characteristic, complemented
315	#492	C₂²×C₃₁	1	1	yes	characteristic, complemented
316	#487	C₂²×C₃₁	1	1	yes	characteristic, complemented
317	#487	C₂²×C₃₁	1	1	yes	characteristic, complemented
318	#486	C₂²×C₃₁	1	1	yes	characteristic
319	#486	C₂²×C₃₁	1	1	yes	characteristic
320	#486	C₂²×C₃₁	1	1	yes	characteristic
321	#486	C₂²×C₃₁	1	1	yes	characteristic
322	#486	C₂²×C₃₁	1	1	yes	characteristic, complemented
323	#486	C₂²×C₃₁	1	1	yes	characteristic, complemented
324	#486	C₂²×C₃₁	1	1	yes	characteristic, complemented
325	#486	C₂²×C₃₁	1	1	yes	characteristic, complemented
326	#482	—	2	2	no	—
327	#478	—	2	2	no	—
328	#468	—	2	2	no	—
329	#463	—	2	2	no	—
330	#460	—	2	2	no	—
331	#459	—	2	2	no	—
332	#450	—	2	2	no	—
333	#449	—	2	2	no	—
334	#447	—	2	2	no	—
335	#440	—	2	2	no	—
336	#431	C₂²×C₃₁	1	1	yes	characteristic
337	#431	C₂²×C₃₁	1	1	yes	characteristic
338	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
339	#322	C₂²×C₃₁	1	1	yes	characteristic, complemented
340	#322	C₂²×C₃₁	1	1	yes	characteristic, complemented
341	#322	C₂²×C₃₁	1	1	yes	characteristic, complemented
342	C₈²	C₂²×C₃₁	1	1	yes	characteristic, complemented
343	C₂×C₃₁	—	4	1	no	—
344	C₂×C₃₁	—	16	1	no	—
345	C₂×C₃₁	—	16	1	no	—
346	C₂×C₃₁	—	16	1	no	—
347	C₂×C₃₁	—	16	1	no	—
348	C₂×C₃₁	#3151	1	1	yes	characteristic
349	C₂×C₃₁	#3154	1	1	yes	characteristic
350	C₂×C₃₁	#3155	1	1	yes	characteristic
351	C₂×#41	—	2	1	no	—
352	C₂×#41	—	2	1	no	—
353	C₂²×D₄	—	2	1	no	—
354	C₂²×D₄	—	2	1	no	—
355	C₂²×D₄	—	2	1	no	—
356	AGL(1,ℤ/8)	—	4	8	no	—
357	C₂×SD₁₆	—	2	2	no	—
358	C₂×SD₁₆	—	2	2	no	—
359	C₂×SD₁₆	—	2	2	no	—
360	C₂×SD₁₆	—	2	2	no	—
361	C₂×SD₁₆	—	2	2	no	—
362	C₂×SD₁₆	—	2	2	no	—
363	C₂×D₈	—	2	2	no	—
364	C₂×D₈	—	2	2	no	—
365	C₂×D₈	—	2	2	no	—
366	C₂×D₈	—	2	2	no	—
367	C₂×D₈	—	2	2	no	—
368	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
369	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
370	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
371	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
372	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
373	#127	—	2	1	no	—
374	#127	—	2	1	no	—
375	#127	—	2	1	no	—
376	#127	C₂³×C₃₁	1	1	yes	characteristic
377	#127	C₂³×C₃₁	1	1	yes	characteristic
378	#127	C₂³×C₃₁	1	1	yes	characteristic
379	#127	C₂³×C₃₁	1	1	yes	characteristic
380	#125	—	2	1	no	—
381	#124	—	2	1	no	—
382	#124	—	2	1	no	—
383	C₂²⋊Q₈	—	2	1	no	—
384	C₂²⋊Q₈	—	2	1	no	—
385	#121	—	2	1	no	—
386	#121	—	2	1	no	—
387	#121	—	2	1	no	—
388	#121	—	2	1	no	—
389	C₄×Q₈	—	2	1	no	—
390	C₄×Q₈	—	2	1	no	—
391	C₄×D₄	—	2	1	no	—
392	C₄×D₄	—	2	1	no	—
393	C₄×D₄	—	2	1	no	—
394	C₄×D₄	—	2	1	no	—
395	#117	—	2	1	no	—
396	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
397	#107	—	2	2	no	—
398	#107	—	2	2	no	—
399	#106	—	2	2	no	—
400	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
401	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
402	#103	—	2	2	no	—
403	#103	—	2	2	no	—
404	#102	—	2	2	no	—
405	#102	—	2	2	no	—
406	#102	—	2	2	no	—
407	#102	—	2	2	no	—
408	#102	—	2	2	no	—
409	#102	—	2	2	no	—
410	#102	—	2	2	no	—
411	#102	—	2	2	no	—
412	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
413	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
414	#97	D₄×C₃₁	1	2	yes	—
415	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
416	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
417	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
418	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
419	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
420	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
421	C₃₁	#19719	1	1	yes	characteristic, direct product factor, Sylow
422	C₂⁴	—	4	1	no	—
423	#41	—	4	4	no	—
424	#41	—	8	2	no	—
425	C₂×Q₈	—	2	1	no	—
426	C₂×Q₈	—	2	1	no	—
427	C₂×Q₈	—	2	1	no	—
428	C₂×Q₈	—	2	1	no	—
429	C₂×D₄	—	2	1	no	—
430	C₂×D₄	—	2	1	no	—
431	C₂×D₄	—	2	1	no	—
432	C₂×D₄	—	2	1	no	—
433	C₂×D₄	—	2	1	no	—
434	C₂×D₄	—	2	1	no	—
435	C₂×D₄	—	2	1	no	—
436	C₂×D₄	—	2	1	no	—
437	C₂×D₄	—	2	1	no	—
438	C₂×D₄	—	2	1	no	—
439	C₂×D₄	—	2	1	no	—
440	C₂×D₄	—	2	1	no	—
441	C₂×D₄	—	4	4	no	—
442	C₂×D₄	—	4	1	no	—
443	C₂×D₄	—	4	1	no	—
444	C₂×D₄	—	4	1	no	—
445	C₂×D₄	—	4	1	no	—
446	C₂×D₄	—	4	1	no	—
447	C₂×D₄	—	8	2	no	—
448	C₂×D₄	—	8	2	no	—
449	C₂²×C₄	—	4	1	no	—
450	C₂²×C₄	—	4	1	no	—
451	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
452	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
453	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
454	SD₁₆	—	4	4	no	—
455	SD₁₆	—	4	4	no	—
456	SD₁₆	—	4	4	no	—
457	SD₁₆	—	8	2	no	—
458	SD₁₆	—	8	2	no	—
459	SD₁₆	—	8	2	no	—
460	D₈	—	4	4	no	—
461	D₈	—	4	4	no	—
462	D₈	—	8	2	no	—
463	D₈	—	8	2	no	—
464	D₈	—	8	2	no	—
465	M₁₆	—	2	4	no	—
466	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
467	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
468	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
469	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
470	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
471	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
472	C₄⋊C₄	—	2	1	no	—
473	C₄⋊C₄	—	2	1	no	—
474	C₄⋊C₄	—	2	1	no	—
475	C₄⋊C₄	—	2	1	no	—
476	C₄⋊C₄	—	2	1	no	—
477	C₄⋊C₄	—	2	1	no	—
478	C₄⋊C₄	—	2	1	no	—
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	—	4	1	no	—
481	C₄⋊C₄	—	4	1	no	—
482	C₂²⋊C₄	—	4	1	no	—
483	C₂²⋊C₄	—	4	1	no	—
484	C₂²⋊C₄	—	4	1	no	—
485	C₂²⋊C₄	—	4	1	no	—
486	C₂²⋊C₄	—	4	1	no	—
487	C₄²	—	4	1	no	—
488	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
489	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
490	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
491	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
492	C₂³	—	4	1	no	—
493	C₂³	—	4	1	no	—
494	C₂³	—	4	1	no	—
495	C₂³	—	4	1	no	—
496	C₂³	—	8	2	no	—
497	C₂³	—	8	2	no	—
498	C₂³	—	8	2	no	—
499	C₂³	C₃₁×#127	1	1	yes	characteristic
500	Q₈	—	4	2	no	—
501	Q₈	—	4	2	no	—
502	Q₈	—	4	2	no	—
503	Q₈	—	4	2	no	—
504	D₄	—	4	2	no	—
505	D₄	—	4	2	no	—
506	D₄	—	4	2	no	—
507	D₄	—	4	2	no	—
508	D₄	—	4	2	no	—
509	D₄	—	4	2	no	—
510	D₄	—	4	2	no	—
511	D₄	—	4	2	no	—
512	D₄	—	4	2	no	—
513	D₄	—	4	2	no	—
514	D₄	—	4	2	no	—
515	D₄	—	4	2	no	—
516	D₄	—	8	2	no	—
517	D₄	—	8	2	no	—
518	D₄	—	8	2	no	—
519	D₄	—	8	2	no	—
520	D₄	—	8	2	no	—
521	C₂×C₄	—	2	2	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	2	2	no	—
524	C₂×C₄	—	4	1	no	—
525	C₂×C₄	—	4	1	no	—
526	C₂×C₄	—	4	1	no	—
527	C₂×C₄	—	4	1	no	—
528	C₂×C₄	—	8	2	no	—
529	C₂×C₄	—	8	2	no	—
530	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
531	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
532	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
533	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
534	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
535	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
536	C₈	—	2	2	no	—
537	C₈	—	2	2	no	—
538	C₈	—	2	2	no	—
539	C₈	—	2	2	no	—
540	C₈	—	2	2	no	—
541	C₈	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
542	C₂²	—	2	1	no	—
543	C₂²	—	2	1	no	—
544	C₂²	—	2	1	no	—
545	C₂²	—	8	1	no	—
546	C₂²	—	8	1	no	—
547	C₂²	—	8	1	no	—
548	C₂²	—	8	1	no	—
549	C₂²	—	8	1	no	—
550	C₂²	—	8	1	no	—
551	C₂²	—	8	1	no	—
552	C₂²	—	8	1	no	—
553	C₂²	—	8	1	no	—
554	C₂²	—	8	1	no	—
555	C₂²	—	8	1	no	—
556	C₂²	—	8	1	no	—
557	C₂²	—	16	4	no	—
558	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
559	C₄	—	2	1	no	—
560	C₄	—	2	1	no	—
561	C₄	—	2	1	no	—
562	C₄	—	8	1	no	—
563	C₄	—	8	1	no	—
564	C₄	—	8	1	no	—
565	C₄	—	8	1	no	—
566	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
567	C₄	C₃₁×#576	1	2	yes	—
568	C₄	C₃₁×#577	1	2	yes	—
569	C₂	—	4	1	no	—
570	C₂	—	16	1	no	—
571	C₂	—	16	1	no	—
572	C₂	—	16	1	no	—
573	C₂	—	16	1	no	—
574	C₂	C₃₁×#3151	1	1	yes	characteristic
575	C₂	C₃₁×#3154	1	1	yes	characteristic
576	C₂	C₃₁×#3155	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
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
1. C₃₁×#3228	C₂	semidirect
2. C₃₁×#3208	C₂	semidirect
3. C₃₁×#3207	C₂	semidirect
4. C₃₁×#3205	C₂	semidirect
5. C₃₁×#3182	C₂	semidirect
6. C₃₁×#3151	C₂	semidirect
7. C₃₁×#1721	C₂	semidirect
8. C₃₁×#1721	C₂	semidirect
9. C₃₁×#1719	C₂	semidirect
10. C₃₁×#1717	C₂	semidirect
11. C₃₁×#1715	C₂	semidirect
12. C₃₁×#1714	C₂	semidirect
13. C₃₁×#1706	C₂	semidirect
14. C₃₁×#1706	C₂	semidirect
15. C₃₁×#1453	C₂	semidirect
21. C₃₁×#500	C₂²	semidirect
22. C₃₁×#500	C₂²	semidirect
23. C₃₁×#498	C₂²	semidirect
29. C₃₁×#493	C₂²	semidirect
34. C₃₁×#492	C₂²	semidirect
35. C₃₁×#492	C₂²	semidirect
36. C₃₁×#487	C₂²	semidirect
37. C₃₁×#487	C₂²	semidirect
42. C₃₁×#486	C₂²	semidirect
43. C₃₁×#486	C₂²	semidirect
44. C₃₁×#486	C₂²	semidirect
45. C₃₁×#486	C₂²	semidirect
59. C₃₁×#322	C₂²	semidirect
60. C₃₁×#322	C₂²	semidirect
61. C₃₁×#322	C₂²	semidirect
62. C₈²×C₃₁	C₂²	semidirect
203. #19719	C₃₁	direct (N × H)
254. #3228	C₂×C₃₁	semidirect
255. #3208	C₂×C₃₁	semidirect
256. #3207	C₂×C₃₁	semidirect
257. #3205	C₂×C₃₁	semidirect
258. #3182	C₂×C₃₁	semidirect
259. #3151	C₂×C₃₁	semidirect
260. #1721	C₂×C₃₁	semidirect
261. #1721	C₂×C₃₁	semidirect
262. #1719	C₂×C₃₁	semidirect
263. #1717	C₂×C₃₁	semidirect
264. #1715	C₂×C₃₁	semidirect
265. #1714	C₂×C₃₁	semidirect
266. #1706	C₂×C₃₁	semidirect
267. #1706	C₂×C₃₁	semidirect
268. #1453	C₂×C₃₁	semidirect
301. #500	C₂²×C₃₁	semidirect
302. #500	C₂²×C₃₁	semidirect
303. #498	C₂²×C₃₁	semidirect
309. #493	C₂²×C₃₁	semidirect
314. #492	C₂²×C₃₁	semidirect
315. #492	C₂²×C₃₁	semidirect
316. #487	C₂²×C₃₁	semidirect
317. #487	C₂²×C₃₁	semidirect
322. #486	C₂²×C₃₁	semidirect
323. #486	C₂²×C₃₁	semidirect
324. #486	C₂²×C₃₁	semidirect
325. #486	C₂²×C₃₁	semidirect
339. #322	C₂²×C₃₁	semidirect
340. #322	C₂²×C₃₁	semidirect
341. #322	C₂²×C₃₁	semidirect
342. C₈²	C₂²×C₃₁	semidirect
421. C₃₁	#19719	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×#3228	2
C₃₁×#3208	2
C₃₁×#3207	2
C₃₁×#3205	2
C₃₁×#3182	2
C₃₁×#3151	2
C₃₁×#1721	2
C₃₁×#1719	2
C₃₁×#1717	2
C₃₁×#1715	2
C₃₁×#1714	2
C₃₁×#1706	2
C₃₁×#1453	2
#19719	31

Subgroups — C31×#19719 #21476478

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19719 #21476478