Subgroups — C₃₁×#25612 #21482371

Order 7936.

1094 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#541	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#497	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#488	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2620	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2001	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2000	C₂	1	2	yes	maximal
7	C₃₁×#1992	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1956	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1880	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1878	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1750	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
14	C₃₁×#541	—	2	4	no	—
15	C₃₁×#541	—	2	4	no	—
16	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#126	—	2	2	no	—
18	C₂×C₃₁×#125	—	2	2	no	—
19	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
22	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
25	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
26	C₃₁×#497	—	2	4	no	—
27	C₃₁×#488	—	2	4	no	—
28	C₂×C₃₁×#103	C₂²	1	2	yes	—
29	C₂×C₃₁×#103	C₂²	1	2	yes	—
30	C₂×C₃₁×#103	C₂²	1	2	yes	—
31	C₂×C₃₁×#102	C₂²	1	2	yes	—
32	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
33	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
35	C₃₁×#396	—	2	2	no	—
36	C₃₁×#394	—	2	2	no	—
37	C₃₁×#391	C₂²	1	1	yes	characteristic
38	C₃₁×#390	C₂²	1	1	yes	characteristic
39	C₃₁×#384	C₂²	1	1	yes	characteristic
40	C₃₁×#340	C₂²	1	2	yes	—
41	C₃₁×#340	C₂²	1	2	yes	—
42	C₃₁×#336	C₂²	1	2	yes	—
43	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
44	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
45	C₂²×D₄×C₃₁	—	2	1	no	—
46	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₂×Q₁₆×C₃₁	—	2	4	no	—
49	C₂×Q₁₆×C₃₁	—	2	4	no	—
50	C₂×Q₁₆×C₃₁	—	4	2	no	—
51	C₂×SD₁₆×C₃₁	—	2	4	no	—
52	C₂×SD₁₆×C₃₁	—	2	4	no	—
53	C₂×SD₁₆×C₃₁	—	4	2	no	—
54	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
55	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
56	C₃₁×C₄⋊Q₈	—	2	2	no	—
57	C₃₁×C₄⋊Q₈	D₄	1	4	yes	complemented
58	C₃₁×#126	—	2	16	no	—
59	C₃₁×#125	—	2	16	no	—
60	C₃₁×#125	—	2	2	no	—
61	C₃₁×#125	D₄	1	4	yes	complemented
62	C₃₁×#124	—	2	2	no	—
63	C₃₁×#124	—	2	2	no	—
64	C₃₁×C₂²⋊Q₈	—	2	4	no	—
65	C₃₁×C₂²⋊Q₈	—	2	4	no	—
66	C₃₁×C₂²⋊Q₈	—	4	2	no	—
67	C₃₁×#121	—	4	2	no	—
68	C₄×Q₈×C₃₁	—	2	4	no	—
69	C₄×Q₈×C₃₁	—	2	4	no	—
70	C₃₁×#117	—	2	2	no	—
71	C₃₁×#117	—	2	2	no	—
72	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
73	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
74	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
75	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
76	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
77	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
78	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
79	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
81	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
82	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
83	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
84	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
85	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
86	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
87	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
88	C₂×C₄²×C₃₁	—	2	2	no	—
89	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
90	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
91	C₃₁×#103	—	2	4	no	—
92	C₃₁×#103	—	4	2	no	—
93	C₃₁×#103	—	4	2	no	—
94	C₃₁×#102	—	2	4	no	—
95	C₃₁×C₂²⋊C₈	—	2	4	no	—
96	C₃₁×#97	—	2	2	no	—
97	C₄×C₈×C₃₁	—	2	2	no	—
98	C₃₁×#95	—	2	2	no	—
99	C₂⁴×C₃₁	—	2	1	no	—
100	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₂×Q₈×C₃₁	—	2	2	no	—
102	C₂×Q₈×C₃₁	—	2	1	no	—
103	C₂×Q₈×C₃₁	—	2	1	no	—
104	C₂×Q₈×C₃₁	—	2	2	no	—
105	C₂×Q₈×C₃₁	—	2	2	no	—
106	C₂×Q₈×C₃₁	—	2	2	no	—
107	C₂×Q₈×C₃₁	—	2	2	no	—
108	C₂×Q₈×C₃₁	—	2	2	no	—
109	C₂×Q₈×C₃₁	—	2	2	no	—
110	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
111	C₂×D₄×C₃₁	—	2	1	no	—
112	C₂×D₄×C₃₁	—	2	1	no	—
113	C₂×D₄×C₃₁	—	2	2	no	—
114	C₂×D₄×C₃₁	—	2	2	no	—
115	C₂×D₄×C₃₁	—	4	2	no	—
116	C₂×D₄×C₃₁	—	8	2	no	—
117	C₂²×C₄×C₃₁	—	2	2	no	—
118	C₂²×C₄×C₃₁	—	2	2	no	—
119	C₂²×C₄×C₃₁	—	2	1	no	—
120	C₂²×C₄×C₃₁	—	2	1	no	—
121	C₂²×C₄×C₃₁	—	2	2	no	—
122	C₂²×C₄×C₃₁	—	2	2	no	—
123	C₂²×C₄×C₃₁	—	2	1	no	—
124	C₂²×C₄×C₃₁	—	2	1	no	—
125	C₂²×C₄×C₃₁	—	2	1	no	—
126	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
127	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
129	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
130	Q₁₆×C₃₁	—	4	8	no	—
131	SD₁₆×C₃₁	—	4	8	no	—
132	C₂×C₈×C₃₁	—	2	2	no	—
133	C₂×C₈×C₃₁	—	2	2	no	—
134	C₂×C₈×C₃₁	—	2	2	no	—
135	C₂×C₈×C₃₁	—	2	2	no	—
136	C₂×C₈×C₃₁	—	2	2	no	—
137	C₂×C₈×C₃₁	#41	1	4	yes	—
138	C₄⋊C₄×C₃₁	—	2	8	no	—
139	C₄⋊C₄×C₃₁	—	2	8	no	—
140	C₄⋊C₄×C₃₁	—	2	8	no	—
141	C₄⋊C₄×C₃₁	—	2	8	no	—
142	C₄⋊C₄×C₃₁	—	2	4	no	—
143	C₄⋊C₄×C₃₁	—	2	2	no	—
144	C₄⋊C₄×C₃₁	—	2	2	no	—
145	C₄⋊C₄×C₃₁	—	2	2	no	—
146	C₄⋊C₄×C₃₁	—	4	2	no	—
147	C₄⋊C₄×C₃₁	—	8	2	no	—
148	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
149	C₂²⋊C₄×C₃₁	—	2	8	no	—
150	C₂²⋊C₄×C₃₁	—	2	4	no	—
151	C₂²⋊C₄×C₃₁	—	4	2	no	—
152	C₂²⋊C₄×C₃₁	—	4	2	no	—
153	C₂²⋊C₄×C₃₁	—	4	4	no	—
154	C₂²⋊C₄×C₃₁	—	8	2	no	—
155	C₄²×C₃₁	—	2	8	no	—
156	C₄²×C₃₁	—	2	1	no	—
157	C₄²×C₃₁	—	2	1	no	—
158	C₄²×C₃₁	C₂×D₄	1	2	yes	—
159	C₄²×C₃₁	C₂×D₄	1	2	yes	—
160	#25612	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
161	C₂³×C₃₁	—	2	1	no	—
162	C₂³×C₃₁	—	2	1	no	—
163	C₂³×C₃₁	—	2	1	no	—
164	C₂³×C₃₁	—	2	2	no	—
165	C₂³×C₃₁	—	2	2	no	—
166	C₂³×C₃₁	—	4	1	no	—
167	C₂³×C₃₁	—	4	1	no	—
168	C₂³×C₃₁	—	4	1	no	—
169	C₂³×C₃₁	—	4	2	no	—
170	C₂³×C₃₁	—	4	2	no	—
171	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
172	Q₈×C₃₁	—	2	4	no	—
173	Q₈×C₃₁	—	2	4	no	—
174	Q₈×C₃₁	—	4	2	no	—
175	Q₈×C₃₁	—	4	2	no	—
176	Q₈×C₃₁	—	4	2	no	—
177	Q₈×C₃₁	—	4	2	no	—
178	D₄×C₃₁	—	4	2	no	—
179	D₄×C₃₁	—	4	2	no	—
180	D₄×C₃₁	—	8	4	no	—
181	C₂×C₄×C₃₁	—	2	2	no	—
182	C₂×C₄×C₃₁	—	2	2	no	—
183	C₂×C₄×C₃₁	—	2	2	no	—
184	C₂×C₄×C₃₁	—	2	4	no	—
185	C₂×C₄×C₃₁	—	2	4	no	—
186	C₂×C₄×C₃₁	—	2	1	no	—
187	C₂×C₄×C₃₁	—	2	1	no	—
188	C₂×C₄×C₃₁	—	2	1	no	—
189	C₂×C₄×C₃₁	—	2	2	no	—
190	C₂×C₄×C₃₁	—	2	2	no	—
191	C₂×C₄×C₃₁	—	4	2	no	—
192	C₂×C₄×C₃₁	—	4	2	no	—
193	C₂×C₄×C₃₁	—	4	2	no	—
194	C₂×C₄×C₃₁	—	4	2	no	—
195	C₂×C₄×C₃₁	—	4	2	no	—
196	C₂×C₄×C₃₁	—	4	2	no	—
197	C₂×C₄×C₃₁	—	4	1	no	—
198	C₂×C₄×C₃₁	—	4	1	no	—
199	C₂×C₄×C₃₁	—	4	1	no	—
200	C₂×C₄×C₃₁	—	4	1	no	—
201	C₂×C₄×C₃₁	—	4	1	no	—
202	C₂×C₄×C₃₁	—	4	2	no	—
203	C₂×C₄×C₃₁	—	4	2	no	—
204	C₂×C₄×C₃₁	#120	1	4	yes	—
205	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
206	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
207	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
208	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
209	C₈×C₃₁	—	2	4	no	—
210	C₈×C₃₁	—	4	2	no	—
211	C₂×#541	C₂×C₃₁	1	1	yes	characteristic, complemented
212	C₂×#497	C₂×C₃₁	1	1	yes	characteristic, complemented
213	C₂×#488	C₂×C₃₁	1	1	yes	characteristic, complemented
214	#2620	C₂×C₃₁	1	1	yes	characteristic
215	#2001	C₂×C₃₁	1	2	yes	complemented
216	#2000	C₂×C₃₁	1	2	yes	—
217	#1992	C₂×C₃₁	1	1	yes	characteristic, complemented
218	#1956	C₂×C₃₁	1	1	yes	characteristic, complemented
219	#1880	C₂×C₃₁	1	2	yes	complemented
220	#1878	C₂×C₃₁	1	2	yes	complemented
221	#1750	C₂×C₃₁	1	1	yes	characteristic, complemented
222	C₂²×C₃₁	—	4	2	no	—
223	C₂²×C₃₁	—	4	1	no	—
224	C₂²×C₃₁	—	4	1	no	—
225	C₂²×C₃₁	—	4	1	no	—
226	C₂²×C₃₁	—	4	2	no	—
227	C₂²×C₃₁	—	8	2	no	—
228	C₂²×C₃₁	—	8	1	no	—
229	C₂²×C₃₁	—	8	1	no	—
230	C₂²×C₃₁	—	8	1	no	—
231	C₂²×C₃₁	—	8	2	no	—
232	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
233	C₂²×C₃₁	#532	1	1	yes	characteristic
234	C₂²×C₃₁	#535	1	1	yes	characteristic
235	C₂²×C₃₁	#577	1	2	yes	—
236	C₂²×C₃₁	#578	1	2	yes	—
237	C₄×C₃₁	—	2	4	no	—
238	C₄×C₃₁	—	4	1	no	—
239	C₄×C₃₁	—	4	4	no	—
240	C₄×C₃₁	—	4	2	no	—
241	C₄×C₃₁	—	4	4	no	—
242	C₄×C₃₁	—	8	1	no	—
243	C₄×C₃₁	—	8	1	no	—
244	C₄×C₃₁	—	8	1	no	—
245	C₄×C₃₁	#546	1	2	yes	—
246	C₄×C₃₁	#550	1	2	yes	—
247	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
248	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
249	#541	—	2	4	no	—
250	#541	—	2	4	no	—
251	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
252	C₂×#126	—	2	2	no	—
253	C₂×#125	—	2	2	no	—
254	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
255	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
256	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
257	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
258	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
259	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
260	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
261	#497	—	2	4	no	—
262	#488	—	2	4	no	—
263	C₂×#103	C₂²×C₃₁	1	2	yes	—
264	C₂×#103	C₂²×C₃₁	1	2	yes	—
265	C₂×#103	C₂²×C₃₁	1	2	yes	—
266	C₂×#102	C₂²×C₃₁	1	2	yes	—
267	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
268	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
269	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
270	#396	—	2	2	no	—
271	#394	—	2	2	no	—
272	#391	C₂²×C₃₁	1	1	yes	characteristic
273	#390	C₂²×C₃₁	1	1	yes	characteristic
274	#384	C₂²×C₃₁	1	1	yes	characteristic
275	#340	C₂²×C₃₁	1	2	yes	—
276	#340	C₂²×C₃₁	1	2	yes	—
277	#336	C₂²×C₃₁	1	2	yes	—
278	C₂×C₃₁	—	8	1	no	—
279	C₂×C₃₁	—	16	1	no	—
280	C₂×C₃₁	#2431	1	1	yes	characteristic
281	C₂×C₃₁	#3013	1	2	yes	—
282	C₂×C₃₁	#3178	1	2	yes	—
283	C₂×C₃₁	#3222	1	1	yes	characteristic
284	C₂×C₃₁	#3223	1	1	yes	characteristic
285	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
286	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
287	C₂²×D₄	—	2	1	no	—
288	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
289	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
290	C₂×Q₁₆	—	2	4	no	—
291	C₂×Q₁₆	—	2	4	no	—
292	C₂×Q₁₆	—	4	2	no	—
293	C₂×SD₁₆	—	2	4	no	—
294	C₂×SD₁₆	—	2	4	no	—
295	C₂×SD₁₆	—	4	2	no	—
296	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
297	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
298	C₄⋊Q₈	—	2	2	no	—
299	C₄⋊Q₈	D₄×C₃₁	1	4	yes	complemented
300	#126	—	2	16	no	—
301	#125	—	2	16	no	—
302	#125	—	2	2	no	—
303	#125	D₄×C₃₁	1	4	yes	complemented
304	#124	—	2	2	no	—
305	#124	—	2	2	no	—
306	C₂²⋊Q₈	—	2	4	no	—
307	C₂²⋊Q₈	—	2	4	no	—
308	C₂²⋊Q₈	—	4	2	no	—
309	#121	—	4	2	no	—
310	C₄×Q₈	—	2	4	no	—
311	C₄×Q₈	—	2	4	no	—
312	#117	—	2	2	no	—
313	#117	—	2	2	no	—
314	C₂×C₄⋊C₄	—	2	2	no	—
315	C₂×C₄⋊C₄	—	2	2	no	—
316	C₂×C₄⋊C₄	—	2	2	no	—
317	C₂×C₄⋊C₄	—	2	2	no	—
318	C₂×C₄⋊C₄	—	2	1	no	—
319	C₂×C₄⋊C₄	—	2	2	no	—
320	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
321	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
322	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
323	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
324	C₂×C₂²⋊C₄	—	2	2	no	—
325	C₂×C₂²⋊C₄	—	2	1	no	—
326	C₂×C₂²⋊C₄	—	2	1	no	—
327	C₂×C₂²⋊C₄	—	2	2	no	—
328	C₂×C₂²⋊C₄	—	2	2	no	—
329	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
330	C₂×C₄²	—	2	2	no	—
331	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
332	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
333	#103	—	2	4	no	—
334	#103	—	4	2	no	—
335	#103	—	4	2	no	—
336	#102	—	2	4	no	—
337	C₂²⋊C₈	—	2	4	no	—
338	#97	—	2	2	no	—
339	C₄×C₈	—	2	2	no	—
340	#95	—	2	2	no	—
341	C₃₁	#25612	1	1	yes	characteristic, direct product factor, Sylow
342	C₂⁴	—	2	1	no	—
343	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
344	C₂×Q₈	—	2	2	no	—
345	C₂×Q₈	—	2	1	no	—
346	C₂×Q₈	—	2	1	no	—
347	C₂×Q₈	—	2	2	no	—
348	C₂×Q₈	—	2	2	no	—
349	C₂×Q₈	—	2	2	no	—
350	C₂×Q₈	—	2	2	no	—
351	C₂×Q₈	—	2	2	no	—
352	C₂×Q₈	—	2	2	no	—
353	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
354	C₂×D₄	—	2	1	no	—
355	C₂×D₄	—	2	1	no	—
356	C₂×D₄	—	2	2	no	—
357	C₂×D₄	—	2	2	no	—
358	C₂×D₄	—	4	2	no	—
359	C₂×D₄	—	8	2	no	—
360	C₂²×C₄	—	2	2	no	—
361	C₂²×C₄	—	2	2	no	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	2	1	no	—
364	C₂²×C₄	—	2	2	no	—
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	—	2	1	no	—
367	C₂²×C₄	—	2	1	no	—
368	C₂²×C₄	—	2	1	no	—
369	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
370	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
371	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
372	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
373	Q₁₆	—	4	8	no	—
374	SD₁₆	—	4	8	no	—
375	C₂×C₈	—	2	2	no	—
376	C₂×C₈	—	2	2	no	—
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	—	2	2	no	—
379	C₂×C₈	—	2	2	no	—
380	C₂×C₈	#41×C₃₁	1	4	yes	—
381	C₄⋊C₄	—	2	8	no	—
382	C₄⋊C₄	—	2	8	no	—
383	C₄⋊C₄	—	2	8	no	—
384	C₄⋊C₄	—	2	8	no	—
385	C₄⋊C₄	—	2	4	no	—
386	C₄⋊C₄	—	2	2	no	—
387	C₄⋊C₄	—	2	2	no	—
388	C₄⋊C₄	—	2	2	no	—
389	C₄⋊C₄	—	4	2	no	—
390	C₄⋊C₄	—	8	2	no	—
391	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
392	C₂²⋊C₄	—	2	8	no	—
393	C₂²⋊C₄	—	2	4	no	—
394	C₂²⋊C₄	—	4	2	no	—
395	C₂²⋊C₄	—	4	2	no	—
396	C₂²⋊C₄	—	4	4	no	—
397	C₂²⋊C₄	—	8	2	no	—
398	C₄²	—	2	8	no	—
399	C₄²	—	2	1	no	—
400	C₄²	—	2	1	no	—
401	C₄²	C₂×D₄×C₃₁	1	2	yes	—
402	C₄²	C₂×D₄×C₃₁	1	2	yes	—
403	C₂³	—	2	1	no	—
404	C₂³	—	2	1	no	—
405	C₂³	—	2	1	no	—
406	C₂³	—	2	2	no	—
407	C₂³	—	2	2	no	—
408	C₂³	—	4	1	no	—
409	C₂³	—	4	1	no	—
410	C₂³	—	4	1	no	—
411	C₂³	—	4	2	no	—
412	C₂³	—	4	2	no	—
413	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
414	Q₈	—	2	4	no	—
415	Q₈	—	2	4	no	—
416	Q₈	—	4	2	no	—
417	Q₈	—	4	2	no	—
418	Q₈	—	4	2	no	—
419	Q₈	—	4	2	no	—
420	D₄	—	4	2	no	—
421	D₄	—	4	2	no	—
422	D₄	—	8	4	no	—
423	C₂×C₄	—	2	2	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	4	no	—
427	C₂×C₄	—	2	4	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	1	no	—
430	C₂×C₄	—	2	1	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	2	no	—
435	C₂×C₄	—	4	2	no	—
436	C₂×C₄	—	4	2	no	—
437	C₂×C₄	—	4	2	no	—
438	C₂×C₄	—	4	2	no	—
439	C₂×C₄	—	4	1	no	—
440	C₂×C₄	—	4	1	no	—
441	C₂×C₄	—	4	1	no	—
442	C₂×C₄	—	4	1	no	—
443	C₂×C₄	—	4	1	no	—
444	C₂×C₄	—	4	2	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	C₃₁×#120	1	4	yes	—
447	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
448	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
449	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
450	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
451	C₈	—	2	4	no	—
452	C₈	—	4	2	no	—
453	C₂²	—	4	2	no	—
454	C₂²	—	4	1	no	—
455	C₂²	—	4	1	no	—
456	C₂²	—	4	1	no	—
457	C₂²	—	4	2	no	—
458	C₂²	—	8	2	no	—
459	C₂²	—	8	1	no	—
460	C₂²	—	8	1	no	—
461	C₂²	—	8	1	no	—
462	C₂²	—	8	2	no	—
463	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
464	C₂²	C₃₁×#532	1	1	yes	characteristic
465	C₂²	C₃₁×#535	1	1	yes	characteristic
466	C₂²	C₃₁×#577	1	2	yes	—
467	C₂²	C₃₁×#578	1	2	yes	—
468	C₄	—	2	4	no	—
469	C₄	—	4	1	no	—
470	C₄	—	4	4	no	—
471	C₄	—	4	2	no	—
472	C₄	—	4	4	no	—
473	C₄	—	8	1	no	—
474	C₄	—	8	1	no	—
475	C₄	—	8	1	no	—
476	C₄	C₃₁×#546	1	2	yes	—
477	C₄	C₃₁×#550	1	2	yes	—
478	C₂	—	8	1	no	—
479	C₂	—	16	1	no	—
480	C₂	C₃₁×#2431	1	1	yes	characteristic
481	C₂	C₃₁×#3013	1	2	yes	—
482	C₂	C₃₁×#3178	1	2	yes	—
483	C₂	C₃₁×#3222	1	1	yes	characteristic
484	C₂	C₃₁×#3223	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#541	C₂	semidirect
2. C₂×C₃₁×#497	C₂	semidirect
3. C₂×C₃₁×#488	C₂	semidirect
5. C₃₁×#2001	C₂	semidirect
7. C₃₁×#1992	C₂	semidirect
8. C₃₁×#1956	C₂	semidirect
9. C₃₁×#1880	C₂	semidirect
10. C₃₁×#1878	C₂	semidirect
11. C₃₁×#1750	C₂	semidirect
57. C₃₁×C₄⋊Q₈	D₄	semidirect
61. C₃₁×#125	D₄	semidirect
160. #25612	C₃₁	direct (N × H)
211. C₂×#541	C₂×C₃₁	semidirect
212. C₂×#497	C₂×C₃₁	semidirect
213. C₂×#488	C₂×C₃₁	semidirect
215. #2001	C₂×C₃₁	semidirect
217. #1992	C₂×C₃₁	semidirect
218. #1956	C₂×C₃₁	semidirect
219. #1880	C₂×C₃₁	semidirect
220. #1878	C₂×C₃₁	semidirect
221. #1750	C₂×C₃₁	semidirect
299. C₄⋊Q₈	D₄×C₃₁	semidirect
303. #125	D₄×C₃₁	semidirect
341. C₃₁	#25612	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃₁×#541	2
C₂×C₃₁×#497	2
C₂×C₃₁×#488	2
C₃₁×#2620	2
C₃₁×#2001	2
C₃₁×#2000	2
C₃₁×#1992	2
C₃₁×#1956	2
C₃₁×#1880	2
C₃₁×#1878	2
C₃₁×#1750	2
#25612	31

Subgroups — C31×#25612 #21482371

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#25612 #21482371