Subgroups — C₃₁×#22196 #21478955

Order 7936.

934 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#466	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2324	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1925	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1921	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1895	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1879	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1862	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1808	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1729	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#121	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
15	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
16	C₃₁×#466	C₄	1	8	yes	complemented
17	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#102	C₂²	1	2	yes	—
21	C₂×C₃₁×#102	C₂²	1	2	yes	—
22	C₂×C₃₁×#102	C₂²	1	2	yes	—
23	C₃₁×#390	—	2	2	no	—
24	C₃₁×#390	C₂²	1	2	yes	—
25	C₃₁×#389	C₂²	1	2	yes	—
26	C₃₁×#388	—	2	2	no	—
27	C₃₁×#386	—	2	2	no	—
28	C₃₁×#385	—	2	2	no	—
29	C₃₁×#380	C₂²	1	2	yes	—
30	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
31	C₃₁×#340	C₂²	1	2	yes	—
32	C₃₁×#340	C₂²	1	2	yes	—
33	C₃₁×#340	C₂²	1	2	yes	—
34	C₃₁×#336	C₂²	1	2	yes	—
35	C₃₁×#336	C₂²	1	2	yes	—
36	C₂²×D₄×C₃₁	—	2	2	no	—
37	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
38	C₂³×C₄×C₃₁	—	2	2	no	—
39	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
40	C₂×D₈×C₃₁	—	4	1	no	—
41	C₂×D₈×C₃₁	—	4	1	no	—
42	C₂×D₈×C₃₁	C₂×C₄	1	4	yes	—
43	C₂×M₁₆×C₃₁	—	2	2	no	—
44	C₂×M₁₆×C₃₁	—	4	1	no	—
45	C₂×M₁₆×C₃₁	—	4	1	no	—
46	C₂²×C₈×C₃₁	—	4	1	no	—
47	C₂²×C₈×C₃₁	—	4	1	no	—
48	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
49	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
50	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₃₁×#121	—	2	4	no	—
53	C₃₁×#121	C₂×C₄	1	8	yes	complemented
54	C₄×D₄×C₃₁	—	4	2	no	—
55	C₄×D₄×C₃₁	—	4	2	no	—
56	C₃₁×#117	—	4	1	no	—
57	C₃₁×#117	—	4	1	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
59	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
64	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
65	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
66	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
67	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
69	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
70	C₂×C₄²×C₃₁	—	2	2	no	—
71	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
72	C₃₁×#107	—	4	1	no	—
73	C₃₁×#107	—	4	1	no	—
74	C₃₁×#107	C₂×C₄	1	4	yes	complemented
75	C₃₁×#102	—	4	2	no	—
76	C₃₁×#102	—	4	2	no	—
77	C₃₁×#102	C₂×C₄	1	8	yes	complemented
78	C₃₁×#95	—	2	2	no	—
79	C₃₁×#95	—	2	2	no	—
80	C₃₁×#95	—	2	2	no	—
81	C₂⁴×C₃₁	—	2	2	no	—
82	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
83	C₂×D₄×C₃₁	—	2	8	no	—
84	C₂×D₄×C₃₁	—	2	2	no	—
85	C₂×D₄×C₃₁	—	2	2	no	—
86	C₂×D₄×C₃₁	—	2	2	no	—
87	C₂×D₄×C₃₁	—	2	2	no	—
88	C₂×D₄×C₃₁	—	2	2	no	—
89	C₂×D₄×C₃₁	—	8	2	no	—
90	C₂×D₄×C₃₁	—	8	2	no	—
91	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
92	C₂²×C₄×C₃₁	—	2	1	no	—
93	C₂²×C₄×C₃₁	—	2	1	no	—
94	C₂²×C₄×C₃₁	—	2	1	no	—
95	C₂²×C₄×C₃₁	—	2	1	no	—
96	C₂²×C₄×C₃₁	—	2	2	no	—
97	C₂²×C₄×C₃₁	—	2	2	no	—
98	C₂²×C₄×C₃₁	—	2	2	no	—
99	C₂²×C₄×C₃₁	—	4	4	no	—
100	C₂²×C₄×C₃₁	—	8	2	no	—
101	C₂²×C₄×C₃₁	—	8	2	no	—
102	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
103	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
105	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₃₁	#41	1	2	yes	—
107	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
108	D₈×C₃₁	—	4	4	no	—
109	M₁₆×C₃₁	—	4	4	no	—
110	C₂×C₈×C₃₁	—	2	1	no	—
111	C₂×C₈×C₃₁	—	2	1	no	—
112	C₂×C₈×C₃₁	—	2	1	no	—
113	C₂×C₈×C₃₁	—	2	1	no	—
114	C₂×C₈×C₃₁	—	2	2	no	—
115	C₂×C₈×C₃₁	—	2	2	no	—
116	C₂×C₈×C₃₁	—	2	2	no	—
117	C₂×C₈×C₃₁	—	4	4	no	—
118	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
119	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
120	C₄⋊C₄×C₃₁	—	2	2	no	—
121	C₄⋊C₄×C₃₁	—	2	2	no	—
122	C₄⋊C₄×C₃₁	—	2	2	no	—
123	C₄⋊C₄×C₃₁	—	2	2	no	—
124	C₄⋊C₄×C₃₁	—	2	2	no	—
125	C₄⋊C₄×C₃₁	—	4	1	no	—
126	C₄⋊C₄×C₃₁	—	4	1	no	—
127	C₄⋊C₄×C₃₁	—	8	2	no	—
128	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
129	C₂²⋊C₄×C₃₁	—	2	8	no	—
130	C₂²⋊C₄×C₃₁	—	4	2	no	—
131	C₂²⋊C₄×C₃₁	—	4	4	no	—
132	C₂²⋊C₄×C₃₁	—	8	2	no	—
133	C₂²⋊C₄×C₃₁	—	8	2	no	—
134	C₂²⋊C₄×C₃₁	—	8	2	no	—
135	C₄²×C₃₁	—	2	2	no	—
136	C₄²×C₃₁	—	2	2	no	—
137	C₄²×C₃₁	—	8	2	no	—
138	#22196	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂³×C₃₁	—	2	1	no	—
140	C₂³×C₃₁	—	2	1	no	—
141	C₂³×C₃₁	—	2	1	no	—
142	C₂³×C₃₁	—	2	1	no	—
143	C₂³×C₃₁	—	2	1	no	—
144	C₂³×C₃₁	—	2	1	no	—
145	C₂³×C₃₁	—	2	4	no	—
146	C₂³×C₃₁	—	4	2	no	—
147	C₂³×C₃₁	—	4	2	no	—
148	C₂³×C₃₁	—	4	2	no	—
149	C₂³×C₃₁	—	4	2	no	—
150	C₂³×C₃₁	—	4	2	no	—
151	C₂³×C₃₁	—	4	2	no	—
152	C₂³×C₃₁	C₄×D₄	1	2	yes	—
153	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
154	D₄×C₃₁	—	2	4	no	—
155	D₄×C₃₁	—	2	4	no	—
156	D₄×C₃₁	—	4	2	no	—
157	D₄×C₃₁	—	4	2	no	—
158	D₄×C₃₁	—	8	8	no	—
159	C₂×C₄×C₃₁	—	2	1	no	—
160	C₂×C₄×C₃₁	—	2	1	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	2	2	no	—
163	C₂×C₄×C₃₁	—	2	2	no	—
164	C₂×C₄×C₃₁	—	2	2	no	—
165	C₂×C₄×C₃₁	—	2	4	no	—
166	C₂×C₄×C₃₁	—	2	2	no	—
167	C₂×C₄×C₃₁	—	2	2	no	—
168	C₂×C₄×C₃₁	—	2	2	no	—
169	C₂×C₄×C₃₁	—	2	2	no	—
170	C₂×C₄×C₃₁	—	2	2	no	—
171	C₂×C₄×C₃₁	—	2	2	no	—
172	C₂×C₄×C₃₁	—	4	2	no	—
173	C₂×C₄×C₃₁	—	4	2	no	—
174	C₂×C₄×C₃₁	—	4	2	no	—
175	C₂×C₄×C₃₁	—	4	2	no	—
176	C₂×C₄×C₃₁	—	4	2	no	—
177	C₂×C₄×C₃₁	—	4	2	no	—
178	C₂×C₄×C₃₁	—	4	2	no	—
179	C₂×C₄×C₃₁	—	4	2	no	—
180	C₂×C₄×C₃₁	—	8	8	no	—
181	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
182	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
183	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
184	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
185	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
186	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
187	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
188	C₈×C₃₁	—	2	2	no	—
189	C₈×C₃₁	—	2	2	no	—
190	C₈×C₃₁	—	4	2	no	—
191	C₂×#466	C₂×C₃₁	1	1	yes	characteristic
192	#2324	C₂×C₃₁	1	2	yes	complemented
193	#1925	C₂×C₃₁	1	2	yes	complemented
194	#1921	C₂×C₃₁	1	2	yes	complemented
195	#1895	C₂×C₃₁	1	2	yes	complemented
196	#1879	C₂×C₃₁	1	2	yes	complemented
197	#1862	C₂×C₃₁	1	2	yes	complemented
198	#1808	C₂×C₃₁	1	1	yes	characteristic, complemented
199	#1729	C₂×C₃₁	1	1	yes	characteristic, complemented
200	C₂²×C₃₁	—	2	2	no	—
201	C₂²×C₃₁	—	2	2	no	—
202	C₂²×C₃₁	—	2	2	no	—
203	C₂²×C₃₁	—	4	1	no	—
204	C₂²×C₃₁	—	4	1	no	—
205	C₂²×C₃₁	—	4	1	no	—
206	C₂²×C₃₁	—	4	1	no	—
207	C₂²×C₃₁	—	4	4	no	—
208	C₂²×C₃₁	—	4	4	no	—
209	C₂²×C₃₁	—	4	4	no	—
210	C₂²×C₃₁	—	8	2	no	—
211	C₂²×C₃₁	—	8	2	no	—
212	C₂²×C₃₁	—	8	2	no	—
213	C₂²×C₃₁	—	8	2	no	—
214	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
215	C₂²×C₃₁	#534	1	1	yes	characteristic
216	C₂²×C₃₁	#537	1	1	yes	characteristic
217	C₂²×C₃₁	#576	1	1	yes	characteristic
218	C₂²×C₃₁	#576	1	1	yes	characteristic
219	C₂²×C₃₁	#576	1	1	yes	characteristic
220	C₂²×C₃₁	#578	1	1	yes	characteristic
221	C₄×C₃₁	—	2	2	no	—
222	C₄×C₃₁	—	4	2	no	—
223	C₄×C₃₁	—	4	2	no	—
224	C₄×C₃₁	—	4	4	no	—
225	C₄×C₃₁	—	8	2	no	—
226	C₄×C₃₁	—	8	2	no	—
227	C₄×C₃₁	#518	1	1	yes	characteristic
228	C₄×C₃₁	#520	1	1	yes	characteristic
229	C₄×C₃₁	#538	1	1	yes	characteristic
230	C₄×C₃₁	#539	1	1	yes	characteristic
231	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
232	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
233	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
234	C₂×#121	C₂²×C₃₁	1	2	yes	—
235	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
236	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
237	#466	C₄×C₃₁	1	8	yes	complemented
238	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
239	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
240	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
241	C₂×#102	C₂²×C₃₁	1	2	yes	—
242	C₂×#102	C₂²×C₃₁	1	2	yes	—
243	C₂×#102	C₂²×C₃₁	1	2	yes	—
244	#390	—	2	2	no	—
245	#390	C₂²×C₃₁	1	2	yes	—
246	#389	C₂²×C₃₁	1	2	yes	—
247	#388	—	2	2	no	—
248	#386	—	2	2	no	—
249	#385	—	2	2	no	—
250	#380	C₂²×C₃₁	1	2	yes	—
251	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
252	#340	C₂²×C₃₁	1	2	yes	—
253	#340	C₂²×C₃₁	1	2	yes	—
254	#340	C₂²×C₃₁	1	2	yes	—
255	#336	C₂²×C₃₁	1	2	yes	—
256	#336	C₂²×C₃₁	1	2	yes	—
257	C₂×C₃₁	—	4	2	no	—
258	C₂×C₃₁	—	8	4	no	—
259	C₂×C₃₁	#2325	1	1	yes	characteristic
260	C₂×C₃₁	#2948	1	1	yes	characteristic
261	C₂×C₃₁	#2951	1	1	yes	characteristic
262	C₂×C₃₁	#3199	1	1	yes	characteristic
263	C₂×C₃₁	#3203	1	1	yes	characteristic
264	C₂×C₃₁	#3259	1	1	yes	characteristic
265	C₂×C₃₁	#3262	1	1	yes	characteristic
266	C₂²×D₄	—	2	2	no	—
267	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
268	C₂³×C₄	—	2	2	no	—
269	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
270	C₂×D₈	—	4	1	no	—
271	C₂×D₈	—	4	1	no	—
272	C₂×D₈	C₂×C₄×C₃₁	1	4	yes	—
273	C₂×M₁₆	—	2	2	no	—
274	C₂×M₁₆	—	4	1	no	—
275	C₂×M₁₆	—	4	1	no	—
276	C₂²×C₈	—	4	1	no	—
277	C₂²×C₈	—	4	1	no	—
278	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	complemented
279	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
280	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
281	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
282	#121	—	2	4	no	—
283	#121	C₂×C₄×C₃₁	1	8	yes	complemented
284	C₄×D₄	—	4	2	no	—
285	C₄×D₄	—	4	2	no	—
286	#117	—	4	1	no	—
287	#117	—	4	1	no	—
288	C₂×C₄⋊C₄	—	2	2	no	—
289	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
290	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
291	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
292	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
293	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
294	C₂×C₂²⋊C₄	—	2	2	no	—
295	C₂×C₂²⋊C₄	—	2	2	no	—
296	C₂×C₂²⋊C₄	—	2	2	no	—
297	C₂×C₂²⋊C₄	—	2	2	no	—
298	C₂×C₂²⋊C₄	—	2	2	no	—
299	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
300	C₂×C₄²	—	2	2	no	—
301	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
302	#107	—	4	1	no	—
303	#107	—	4	1	no	—
304	#107	C₂×C₄×C₃₁	1	4	yes	complemented
305	#102	—	4	2	no	—
306	#102	—	4	2	no	—
307	#102	C₂×C₄×C₃₁	1	8	yes	complemented
308	#95	—	2	2	no	—
309	#95	—	2	2	no	—
310	#95	—	2	2	no	—
311	C₃₁	#22196	1	1	yes	characteristic, direct product factor, Sylow
312	C₂⁴	—	2	2	no	—
313	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
314	C₂×D₄	—	2	8	no	—
315	C₂×D₄	—	2	2	no	—
316	C₂×D₄	—	2	2	no	—
317	C₂×D₄	—	2	2	no	—
318	C₂×D₄	—	2	2	no	—
319	C₂×D₄	—	2	2	no	—
320	C₂×D₄	—	8	2	no	—
321	C₂×D₄	—	8	2	no	—
322	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
323	C₂²×C₄	—	2	1	no	—
324	C₂²×C₄	—	2	1	no	—
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	—	2	1	no	—
327	C₂²×C₄	—	2	2	no	—
328	C₂²×C₄	—	2	2	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	4	4	no	—
331	C₂²×C₄	—	8	2	no	—
332	C₂²×C₄	—	8	2	no	—
333	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
334	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
335	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
336	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
337	C₂²×C₄	#41×C₃₁	1	2	yes	—
338	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
339	D₈	—	4	4	no	—
340	M₁₆	—	4	4	no	—
341	C₂×C₈	—	2	1	no	—
342	C₂×C₈	—	2	1	no	—
343	C₂×C₈	—	2	1	no	—
344	C₂×C₈	—	2	1	no	—
345	C₂×C₈	—	2	2	no	—
346	C₂×C₈	—	2	2	no	—
347	C₂×C₈	—	2	2	no	—
348	C₂×C₈	—	4	4	no	—
349	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
350	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
351	C₄⋊C₄	—	2	2	no	—
352	C₄⋊C₄	—	2	2	no	—
353	C₄⋊C₄	—	2	2	no	—
354	C₄⋊C₄	—	2	2	no	—
355	C₄⋊C₄	—	2	2	no	—
356	C₄⋊C₄	—	4	1	no	—
357	C₄⋊C₄	—	4	1	no	—
358	C₄⋊C₄	—	8	2	no	—
359	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
360	C₂²⋊C₄	—	2	8	no	—
361	C₂²⋊C₄	—	4	2	no	—
362	C₂²⋊C₄	—	4	4	no	—
363	C₂²⋊C₄	—	8	2	no	—
364	C₂²⋊C₄	—	8	2	no	—
365	C₂²⋊C₄	—	8	2	no	—
366	C₄²	—	2	2	no	—
367	C₄²	—	2	2	no	—
368	C₄²	—	8	2	no	—
369	C₂³	—	2	1	no	—
370	C₂³	—	2	1	no	—
371	C₂³	—	2	1	no	—
372	C₂³	—	2	1	no	—
373	C₂³	—	2	1	no	—
374	C₂³	—	2	1	no	—
375	C₂³	—	2	4	no	—
376	C₂³	—	4	2	no	—
377	C₂³	—	4	2	no	—
378	C₂³	—	4	2	no	—
379	C₂³	—	4	2	no	—
380	C₂³	—	4	2	no	—
381	C₂³	—	4	2	no	—
382	C₂³	C₄×D₄×C₃₁	1	2	yes	—
383	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
384	D₄	—	2	4	no	—
385	D₄	—	2	4	no	—
386	D₄	—	4	2	no	—
387	D₄	—	4	2	no	—
388	D₄	—	8	8	no	—
389	C₂×C₄	—	2	1	no	—
390	C₂×C₄	—	2	1	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	2	4	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	—	4	2	no	—
405	C₂×C₄	—	4	2	no	—
406	C₂×C₄	—	4	2	no	—
407	C₂×C₄	—	4	2	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	4	2	no	—
410	C₂×C₄	—	8	8	no	—
411	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
412	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
413	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
414	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
415	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
416	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
417	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
418	C₈	—	2	2	no	—
419	C₈	—	2	2	no	—
420	C₈	—	4	2	no	—
421	C₂²	—	2	2	no	—
422	C₂²	—	2	2	no	—
423	C₂²	—	2	2	no	—
424	C₂²	—	4	1	no	—
425	C₂²	—	4	1	no	—
426	C₂²	—	4	1	no	—
427	C₂²	—	4	1	no	—
428	C₂²	—	4	4	no	—
429	C₂²	—	4	4	no	—
430	C₂²	—	4	4	no	—
431	C₂²	—	8	2	no	—
432	C₂²	—	8	2	no	—
433	C₂²	—	8	2	no	—
434	C₂²	—	8	2	no	—
435	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
436	C₂²	C₃₁×#534	1	1	yes	characteristic
437	C₂²	C₃₁×#537	1	1	yes	characteristic
438	C₂²	C₃₁×#576	1	1	yes	characteristic
439	C₂²	C₃₁×#576	1	1	yes	characteristic
440	C₂²	C₃₁×#576	1	1	yes	characteristic
441	C₂²	C₃₁×#578	1	1	yes	characteristic
442	C₄	—	2	2	no	—
443	C₄	—	4	2	no	—
444	C₄	—	4	2	no	—
445	C₄	—	4	4	no	—
446	C₄	—	8	2	no	—
447	C₄	—	8	2	no	—
448	C₄	C₃₁×#518	1	1	yes	characteristic
449	C₄	C₃₁×#520	1	1	yes	characteristic
450	C₄	C₃₁×#538	1	1	yes	characteristic
451	C₄	C₃₁×#539	1	1	yes	characteristic
452	C₂	—	4	2	no	—
453	C₂	—	8	4	no	—
454	C₂	C₃₁×#2325	1	1	yes	characteristic
455	C₂	C₃₁×#2948	1	1	yes	characteristic
456	C₂	C₃₁×#2951	1	1	yes	characteristic
457	C₂	C₃₁×#3199	1	1	yes	characteristic
458	C₂	C₃₁×#3203	1	1	yes	characteristic
459	C₂	C₃₁×#3259	1	1	yes	characteristic
460	C₂	C₃₁×#3262	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#466	C₂
LCS	G ▷ C₂×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
2. C₃₁×#2324	C₂	semidirect
3. C₃₁×#1925	C₂	semidirect
4. C₃₁×#1921	C₂	semidirect
5. C₃₁×#1895	C₂	semidirect
6. C₃₁×#1879	C₂	semidirect
7. C₃₁×#1862	C₂	semidirect
8. C₃₁×#1808	C₂	semidirect
9. C₃₁×#1729	C₂	semidirect
16. C₃₁×#466	C₄	semidirect
48. C₂²×C₈×C₃₁	C₂×C₄	semidirect
53. C₃₁×#121	C₂×C₄	semidirect
74. C₃₁×#107	C₂×C₄	semidirect
77. C₃₁×#102	C₂×C₄	semidirect
138. #22196	C₃₁	direct (N × H)
192. #2324	C₂×C₃₁	semidirect
193. #1925	C₂×C₃₁	semidirect
194. #1921	C₂×C₃₁	semidirect
195. #1895	C₂×C₃₁	semidirect
196. #1879	C₂×C₃₁	semidirect
197. #1862	C₂×C₃₁	semidirect
198. #1808	C₂×C₃₁	semidirect
199. #1729	C₂×C₃₁	semidirect
237. #466	C₄×C₃₁	semidirect
278. C₂²×C₈	C₂×C₄×C₃₁	semidirect
283. #121	C₂×C₄×C₃₁	semidirect
304. #107	C₂×C₄×C₃₁	semidirect
307. #102	C₂×C₄×C₃₁	semidirect
311. C₃₁	#22196	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×#466	2
C₃₁×#2324	2
C₃₁×#1925	2
C₃₁×#1921	2
C₃₁×#1895	2
C₃₁×#1879	2
C₃₁×#1862	2
C₃₁×#1808	2
C₃₁×#1729	2
#22196	31

Subgroups — C31×#22196 #21478955

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22196 #21478955