Subgroups — C₃₁×#25600 #21482359

Order 7936.

1158 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#487	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2857	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2001	C₂	1	4	yes	maximal
6	C₃₁×#1990	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1956	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1878	C₂	1	4	yes	maximal, complemented
9	C₄×C₃₁×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₁₆×C₃₁	C₂²	1	4	yes	—
11	C₃₁×#533	—	2	4	no	—
12	C₃₁×#533	—	2	4	no	—
13	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#125	—	2	1	no	—
17	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
18	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
19	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
20	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
21	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
22	C₂×C₃₁×#117	—	2	1	no	—
23	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#494	—	2	4	no	—
25	C₃₁×#487	—	2	4	no	—
26	C₂×C₃₁×#103	C₂²	1	4	yes	—
27	C₂×C₃₁×#103	C₂²	1	4	yes	—
28	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
31	C₃₁×#391	C₂²	1	2	yes	—
32	C₃₁×#389	—	2	1	no	—
33	C₃₁×#389	—	2	1	no	—
34	C₃₁×#384	C₂²	1	1	yes	characteristic
35	C₃₁×#380	—	2	1	no	—
36	C₃₁×#380	—	2	1	no	—
37	C₃₁×#340	C₂²	1	4	yes	—
38	C₃₁×#336	C₂²	1	2	yes	—
39	C₂²×Q₈×C₃₁	—	2	1	no	—
40	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
41	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
42	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
43	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
45	C₂×Q₁₆×C₃₁	—	2	8	no	—
46	C₂×Q₁₆×C₃₁	—	2	8	no	—
47	C₂×Q₁₆×C₃₁	—	4	4	no	—
48	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
49	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
50	C₃₁×C₄⋊Q₈	—	2	2	no	—
51	C₃₁×C₄⋊Q₈	—	2	2	no	—
52	C₃₁×C₄⋊Q₈	—	2	2	no	—
53	C₃₁×C₄⋊Q₈	—	2	2	no	—
54	C₃₁×C₄⋊Q₈	D₄	1	4	yes	—
55	C₃₁×C₄⋊Q₈	D₄	1	4	yes	—
56	C₃₁×#125	—	2	8	no	—
57	C₃₁×C₂²⋊Q₈	—	2	8	no	—
58	C₃₁×C₂²⋊Q₈	—	2	4	no	—
59	C₃₁×C₂²⋊Q₈	—	2	4	no	—
60	C₃₁×C₂²⋊Q₈	—	4	4	no	—
61	C₄×Q₈×C₃₁	—	2	8	no	—
62	C₄×Q₈×C₃₁	—	2	4	no	—
63	C₄×Q₈×C₃₁	—	2	4	no	—
64	C₃₁×#117	—	2	8	no	—
65	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
66	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
67	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
68	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
69	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
70	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
71	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
72	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
73	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
74	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
75	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
76	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
77	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
78	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
79	C₂×C₄²×C₃₁	—	2	2	no	—
80	C₂×C₄²×C₃₁	—	2	2	no	—
81	C₂×C₄²×C₃₁	—	2	2	no	—
82	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
83	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
84	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
85	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
86	C₃₁×#103	—	2	8	no	—
87	C₃₁×#103	—	4	4	no	—
88	C₃₁×C₂²⋊C₈	—	2	4	no	—
89	C₄×C₈×C₃₁	—	2	2	no	—
90	C₄×C₈×C₃₁	—	2	2	no	—
91	C₃₁×#95	—	2	2	no	—
92	C₃₁×#95	—	2	2	no	—
93	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
94	C₂×Q₈×C₃₁	—	2	4	no	—
95	C₂×Q₈×C₃₁	—	2	2	no	—
96	C₂×Q₈×C₃₁	—	2	2	no	—
97	C₂×Q₈×C₃₁	—	2	2	no	—
98	C₂×Q₈×C₃₁	—	2	4	no	—
99	C₂×Q₈×C₃₁	—	2	4	no	—
100	C₂×Q₈×C₃₁	—	2	4	no	—
101	C₂×Q₈×C₃₁	—	2	4	no	—
102	C₂×Q₈×C₃₁	—	4	2	no	—
103	C₂×Q₈×C₃₁	—	4	2	no	—
104	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
105	C₂²×C₄×C₃₁	—	2	2	no	—
106	C₂²×C₄×C₃₁	—	2	2	no	—
107	C₂²×C₄×C₃₁	—	2	2	no	—
108	C₂²×C₄×C₃₁	—	2	2	no	—
109	C₂²×C₄×C₃₁	—	2	2	no	—
110	C₂²×C₄×C₃₁	—	2	2	no	—
111	C₂²×C₄×C₃₁	—	2	2	no	—
112	C₂²×C₄×C₃₁	—	2	1	no	—
113	C₂²×C₄×C₃₁	—	2	1	no	—
114	C₂²×C₄×C₃₁	—	2	2	no	—
115	C₂²×C₄×C₃₁	—	2	2	no	—
116	C₂²×C₄×C₃₁	—	2	2	no	—
117	C₂²×C₄×C₃₁	—	2	2	no	—
118	C₂²×C₄×C₃₁	Q₁₆	1	4	yes	complemented
119	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
125	Q₁₆×C₃₁	—	4	16	no	—
126	C₂×C₈×C₃₁	—	2	2	no	—
127	C₂×C₈×C₃₁	—	2	2	no	—
128	C₂×C₈×C₃₁	—	2	2	no	—
129	C₂×C₈×C₃₁	—	2	2	no	—
130	C₂×C₈×C₃₁	—	2	2	no	—
131	C₂×C₈×C₃₁	#41	1	4	yes	—
132	C₄⋊C₄×C₃₁	—	2	8	no	—
133	C₄⋊C₄×C₃₁	—	2	8	no	—
134	C₄⋊C₄×C₃₁	—	2	4	no	—
135	C₄⋊C₄×C₃₁	—	2	4	no	—
136	C₄⋊C₄×C₃₁	—	2	4	no	—
137	C₄⋊C₄×C₃₁	—	2	2	no	—
138	C₄⋊C₄×C₃₁	—	2	4	no	—
139	C₄⋊C₄×C₃₁	—	4	4	no	—
140	C₄⋊C₄×C₃₁	—	4	2	no	—
141	C₄⋊C₄×C₃₁	—	4	2	no	—
142	C₄⋊C₄×C₃₁	—	4	2	no	—
143	C₄⋊C₄×C₃₁	—	4	2	no	—
144	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
145	C₂²⋊C₄×C₃₁	—	2	8	no	—
146	C₂²⋊C₄×C₃₁	—	4	4	no	—
147	C₄²×C₃₁	—	2	8	no	—
148	C₄²×C₃₁	—	2	2	no	—
149	C₄²×C₃₁	—	2	2	no	—
150	C₄²×C₃₁	—	2	1	no	—
151	C₄²×C₃₁	—	2	1	no	—
152	C₄²×C₃₁	C₂×D₄	1	2	yes	—
153	C₄²×C₃₁	C₂×D₄	1	2	yes	—
154	#25600	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
155	C₂³×C₃₁	—	2	1	no	—
156	C₂³×C₃₁	—	2	1	no	—
157	C₂³×C₃₁	—	2	1	no	—
158	C₂³×C₃₁	—	2	1	no	—
159	C₂³×C₃₁	—	2	1	no	—
160	C₂³×C₃₁	—	2	1	no	—
161	C₂³×C₃₁	C₂×Q₁₆	1	2	yes	—
162	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
163	Q₈×C₃₁	—	2	8	no	—
164	Q₈×C₃₁	—	4	4	no	—
165	Q₈×C₃₁	—	4	8	no	—
166	Q₈×C₃₁	—	4	4	no	—
167	Q₈×C₃₁	—	4	4	no	—
168	C₂×C₄×C₃₁	—	2	1	no	—
169	C₂×C₄×C₃₁	—	2	1	no	—
170	C₂×C₄×C₃₁	—	2	8	no	—
171	C₂×C₄×C₃₁	—	2	1	no	—
172	C₂×C₄×C₃₁	—	2	1	no	—
173	C₂×C₄×C₃₁	—	2	4	no	—
174	C₂×C₄×C₃₁	—	2	4	no	—
175	C₂×C₄×C₃₁	—	2	1	no	—
176	C₂×C₄×C₃₁	—	2	1	no	—
177	C₂×C₄×C₃₁	—	2	1	no	—
178	C₂×C₄×C₃₁	—	2	1	no	—
179	C₂×C₄×C₃₁	—	4	2	no	—
180	C₂×C₄×C₃₁	—	4	2	no	—
181	C₂×C₄×C₃₁	—	4	2	no	—
182	C₂×C₄×C₃₁	—	4	2	no	—
183	C₂×C₄×C₃₁	—	4	2	no	—
184	C₂×C₄×C₃₁	—	4	2	no	—
185	C₂×C₄×C₃₁	—	4	4	no	—
186	C₂×C₄×C₃₁	—	4	2	no	—
187	C₂×C₄×C₃₁	—	4	2	no	—
188	C₂×C₄×C₃₁	—	4	2	no	—
189	C₂×C₄×C₃₁	—	4	2	no	—
190	C₂×C₄×C₃₁	—	4	2	no	—
191	C₂×C₄×C₃₁	—	4	2	no	—
192	C₂×C₄×C₃₁	#120	1	2	yes	—
193	C₂×C₄×C₃₁	#120	1	2	yes	—
194	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
195	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
196	C₂×C₄×C₃₁	#135	1	2	yes	—
197	C₂×C₄×C₃₁	#137	1	2	yes	—
198	C₂×C₄×C₃₁	#137	1	2	yes	—
199	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
200	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
201	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
202	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
203	C₈×C₃₁	—	2	4	no	—
204	C₈×C₃₁	—	4	2	no	—
205	C₂×#533	C₂×C₃₁	1	1	yes	characteristic
206	C₂×#494	C₂×C₃₁	1	1	yes	characteristic, complemented
207	C₂×#487	C₂×C₃₁	1	1	yes	characteristic, complemented
208	#2857	C₂×C₃₁	1	1	yes	characteristic
209	#2001	C₂×C₃₁	1	4	yes	—
210	#1990	C₂×C₃₁	1	1	yes	characteristic, complemented
211	#1956	C₂×C₃₁	1	1	yes	characteristic, complemented
212	#1878	C₂×C₃₁	1	4	yes	complemented
213	C₄×C₂²⋊C₈	C₂×C₃₁	1	1	yes	characteristic
214	C₂²×C₃₁	—	2	2	no	—
215	C₂²×C₃₁	—	2	2	no	—
216	C₂²×C₃₁	—	2	2	no	—
217	C₂²×C₃₁	—	4	1	no	—
218	C₂²×C₃₁	—	4	1	no	—
219	C₂²×C₃₁	—	4	1	no	—
220	C₂²×C₃₁	—	4	1	no	—
221	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
222	C₂²×C₃₁	#532	1	1	yes	characteristic
223	C₂²×C₃₁	#535	1	1	yes	characteristic
224	C₂²×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
225	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
226	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
227	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
228	C₄×C₃₁	—	2	2	no	—
229	C₄×C₃₁	—	4	1	no	—
230	C₄×C₃₁	—	4	4	no	—
231	C₄×C₃₁	—	4	1	no	—
232	C₄×C₃₁	—	4	4	no	—
233	C₄×C₃₁	—	8	2	no	—
234	C₄×C₃₁	—	8	2	no	—
235	C₄×C₃₁	#451	1	4	yes	—
236	C₄×C₃₁	#452	1	4	yes	—
237	C₄×C₃₁	#546	1	2	yes	—
238	C₄×C₃₁	#550	1	2	yes	—
239	C₂²×Q₁₆	C₂²×C₃₁	1	4	yes	—
240	#533	—	2	4	no	—
241	#533	—	2	4	no	—
242	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
243	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
244	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
245	C₂×#125	—	2	1	no	—
246	C₂×C₂²⋊Q₈	—	2	1	no	—
247	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
248	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
249	C₂×C₄×Q₈	—	2	1	no	—
250	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
251	C₂×#117	—	2	1	no	—
252	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
253	#494	—	2	4	no	—
254	#487	—	2	4	no	—
255	C₂×#103	C₂²×C₃₁	1	4	yes	—
256	C₂×#103	C₂²×C₃₁	1	4	yes	—
257	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
258	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
259	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
260	#391	C₂²×C₃₁	1	2	yes	—
261	#389	—	2	1	no	—
262	#389	—	2	1	no	—
263	#384	C₂²×C₃₁	1	1	yes	characteristic
264	#380	—	2	1	no	—
265	#380	—	2	1	no	—
266	#340	C₂²×C₃₁	1	4	yes	—
267	#336	C₂²×C₃₁	1	2	yes	—
268	C₂×C₃₁	—	4	2	no	—
269	C₂×C₃₁	#2431	1	1	yes	characteristic
270	C₂×C₃₁	C₂×#451	1	1	yes	characteristic
271	C₂×C₃₁	C₂×#452	1	1	yes	characteristic
272	C₂×C₃₁	#3170	1	1	yes	characteristic
273	C₂×C₃₁	#3185	1	1	yes	characteristic
274	C₂×C₃₁	#3209	1	1	yes	characteristic
275	C₂×C₃₁	#3210	1	1	yes	characteristic
276	C₂²×Q₈	—	2	1	no	—
277	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
278	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
279	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
280	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
281	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
282	C₂×Q₁₆	—	2	8	no	—
283	C₂×Q₁₆	—	2	8	no	—
284	C₂×Q₁₆	—	4	4	no	—
285	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
286	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
287	C₄⋊Q₈	—	2	2	no	—
288	C₄⋊Q₈	—	2	2	no	—
289	C₄⋊Q₈	—	2	2	no	—
290	C₄⋊Q₈	—	2	2	no	—
291	C₄⋊Q₈	D₄×C₃₁	1	4	yes	—
292	C₄⋊Q₈	D₄×C₃₁	1	4	yes	—
293	#125	—	2	8	no	—
294	C₂²⋊Q₈	—	2	8	no	—
295	C₂²⋊Q₈	—	2	4	no	—
296	C₂²⋊Q₈	—	2	4	no	—
297	C₂²⋊Q₈	—	4	4	no	—
298	C₄×Q₈	—	2	8	no	—
299	C₄×Q₈	—	2	4	no	—
300	C₄×Q₈	—	2	4	no	—
301	#117	—	2	8	no	—
302	C₂×C₄⋊C₄	—	2	1	no	—
303	C₂×C₄⋊C₄	—	2	1	no	—
304	C₂×C₄⋊C₄	—	2	1	no	—
305	C₂×C₄⋊C₄	—	2	2	no	—
306	C₂×C₄⋊C₄	—	2	2	no	—
307	C₂×C₄⋊C₄	—	2	2	no	—
308	C₂×C₄⋊C₄	—	2	1	no	—
309	C₂×C₄⋊C₄	—	2	1	no	—
310	C₂×C₄⋊C₄	—	2	1	no	—
311	C₂×C₄⋊C₄	—	2	1	no	—
312	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
313	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
314	C₂×C₂²⋊C₄	—	2	2	no	—
315	C₂×C₂²⋊C₄	—	2	2	no	—
316	C₂×C₄²	—	2	2	no	—
317	C₂×C₄²	—	2	2	no	—
318	C₂×C₄²	—	2	2	no	—
319	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
320	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
321	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
322	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
323	#103	—	2	8	no	—
324	#103	—	4	4	no	—
325	C₂²⋊C₈	—	2	4	no	—
326	C₄×C₈	—	2	2	no	—
327	C₄×C₈	—	2	2	no	—
328	#95	—	2	2	no	—
329	#95	—	2	2	no	—
330	C₃₁	#25600	1	1	yes	characteristic, direct product factor, Sylow
331	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
332	C₂×Q₈	—	2	4	no	—
333	C₂×Q₈	—	2	2	no	—
334	C₂×Q₈	—	2	2	no	—
335	C₂×Q₈	—	2	2	no	—
336	C₂×Q₈	—	2	4	no	—
337	C₂×Q₈	—	2	4	no	—
338	C₂×Q₈	—	2	4	no	—
339	C₂×Q₈	—	2	4	no	—
340	C₂×Q₈	—	4	2	no	—
341	C₂×Q₈	—	4	2	no	—
342	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	—	2	2	no	—
345	C₂²×C₄	—	2	2	no	—
346	C₂²×C₄	—	2	2	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	—	2	2	no	—
349	C₂²×C₄	—	2	2	no	—
350	C₂²×C₄	—	2	1	no	—
351	C₂²×C₄	—	2	1	no	—
352	C₂²×C₄	—	2	2	no	—
353	C₂²×C₄	—	2	2	no	—
354	C₂²×C₄	—	2	2	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	Q₁₆×C₃₁	1	4	yes	complemented
357	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
358	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
359	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
360	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
361	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
362	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
363	Q₁₆	—	4	16	no	—
364	C₂×C₈	—	2	2	no	—
365	C₂×C₈	—	2	2	no	—
366	C₂×C₈	—	2	2	no	—
367	C₂×C₈	—	2	2	no	—
368	C₂×C₈	—	2	2	no	—
369	C₂×C₈	#41×C₃₁	1	4	yes	—
370	C₄⋊C₄	—	2	8	no	—
371	C₄⋊C₄	—	2	8	no	—
372	C₄⋊C₄	—	2	4	no	—
373	C₄⋊C₄	—	2	4	no	—
374	C₄⋊C₄	—	2	4	no	—
375	C₄⋊C₄	—	2	2	no	—
376	C₄⋊C₄	—	2	4	no	—
377	C₄⋊C₄	—	4	4	no	—
378	C₄⋊C₄	—	4	2	no	—
379	C₄⋊C₄	—	4	2	no	—
380	C₄⋊C₄	—	4	2	no	—
381	C₄⋊C₄	—	4	2	no	—
382	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
383	C₂²⋊C₄	—	2	8	no	—
384	C₂²⋊C₄	—	4	4	no	—
385	C₄²	—	2	8	no	—
386	C₄²	—	2	2	no	—
387	C₄²	—	2	2	no	—
388	C₄²	—	2	1	no	—
389	C₄²	—	2	1	no	—
390	C₄²	C₂×D₄×C₃₁	1	2	yes	—
391	C₄²	C₂×D₄×C₃₁	1	2	yes	—
392	C₂³	—	2	1	no	—
393	C₂³	—	2	1	no	—
394	C₂³	—	2	1	no	—
395	C₂³	—	2	1	no	—
396	C₂³	—	2	1	no	—
397	C₂³	—	2	1	no	—
398	C₂³	C₂×Q₁₆×C₃₁	1	2	yes	—
399	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
400	Q₈	—	2	8	no	—
401	Q₈	—	4	8	no	—
402	Q₈	—	4	4	no	—
403	Q₈	—	4	4	no	—
404	Q₈	—	4	4	no	—
405	C₂×C₄	—	2	1	no	—
406	C₂×C₄	—	2	1	no	—
407	C₂×C₄	—	2	8	no	—
408	C₂×C₄	—	2	1	no	—
409	C₂×C₄	—	2	1	no	—
410	C₂×C₄	—	2	4	no	—
411	C₂×C₄	—	2	4	no	—
412	C₂×C₄	—	2	1	no	—
413	C₂×C₄	—	2	1	no	—
414	C₂×C₄	—	2	1	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	4	4	no	—
421	C₂×C₄	—	4	2	no	—
422	C₂×C₄	—	4	2	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	—	4	2	no	—
425	C₂×C₄	—	4	2	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	—	4	2	no	—
429	C₂×C₄	C₃₁×#120	1	2	yes	—
430	C₂×C₄	C₃₁×#120	1	2	yes	—
431	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
432	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
433	C₂×C₄	C₃₁×#135	1	2	yes	—
434	C₂×C₄	C₃₁×#137	1	2	yes	—
435	C₂×C₄	C₃₁×#137	1	2	yes	—
436	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
437	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
438	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
439	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
440	C₈	—	2	4	no	—
441	C₈	—	4	2	no	—
442	C₂²	—	2	2	no	—
443	C₂²	—	2	2	no	—
444	C₂²	—	2	2	no	—
445	C₂²	—	4	1	no	—
446	C₂²	—	4	1	no	—
447	C₂²	—	4	1	no	—
448	C₂²	—	4	1	no	—
449	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
450	C₂²	C₃₁×#532	1	1	yes	characteristic
451	C₂²	C₃₁×#535	1	1	yes	characteristic
452	C₂²	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
453	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
454	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
455	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
456	C₄	—	2	2	no	—
457	C₄	—	4	1	no	—
458	C₄	—	4	4	no	—
459	C₄	—	4	1	no	—
460	C₄	—	4	4	no	—
461	C₄	—	8	2	no	—
462	C₄	—	8	2	no	—
463	C₄	C₃₁×#451	1	4	yes	—
464	C₄	C₃₁×#452	1	4	yes	—
465	C₄	C₃₁×#546	1	2	yes	—
466	C₄	C₃₁×#550	1	2	yes	—
467	C₂	—	4	2	no	—
468	C₂	C₃₁×#2431	1	1	yes	characteristic
469	C₂	C₂×C₃₁×#451	1	1	yes	characteristic
470	C₂	C₂×C₃₁×#452	1	1	yes	characteristic
471	C₂	C₃₁×#3170	1	1	yes	characteristic
472	C₂	C₃₁×#3185	1	1	yes	characteristic
473	C₂	C₃₁×#3209	1	1	yes	characteristic
474	C₂	C₃₁×#3210	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#494	C₂	semidirect
3. C₂×C₃₁×#487	C₂	semidirect
6. C₃₁×#1990	C₂	semidirect
7. C₃₁×#1956	C₂	semidirect
8. C₃₁×#1878	C₂	semidirect
118. C₂²×C₄×C₃₁	Q₁₆	semidirect
154. #25600	C₃₁	direct (N × H)
206. C₂×#494	C₂×C₃₁	semidirect
207. C₂×#487	C₂×C₃₁	semidirect
210. #1990	C₂×C₃₁	semidirect
211. #1956	C₂×C₃₁	semidirect
212. #1878	C₂×C₃₁	semidirect
330. C₃₁	#25600	direct (N × H)
356. C₂²×C₄	Q₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#533	2
C₂×C₃₁×#494	2
C₂×C₃₁×#487	2
C₃₁×#2857	2
C₃₁×#2001	2
C₃₁×#1990	2
C₃₁×#1956	2
C₃₁×#1878	2
C₄×C₃₁×C₂²⋊C₈	2
#25600	31

Subgroups — C31×#25600 #21482359

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#25600 #21482359