Subgroups — C₃₁×#25035 #21481794

Order 7936.

810 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#467	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2423	C₂	1	2	yes	maximal
4	C₃₁×#2072	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2036	C₂	1	2	yes	maximal
6	C₃₁×#1969	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1942	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1863	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1845	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1808	C₂	1	1	yes	maximal, characteristic
11	C₈×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
15	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
16	C₃₁×#467	—	2	4	no	—
17	C₄×Q₁₆×C₃₁	—	2	4	no	—
18	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#107	C₂²	1	2	yes	—
22	C₂×C₃₁×#103	C₂²	1	2	yes	—
23	C₂×C₃₁×#103	C₂²	1	2	yes	—
24	C₂×C₃₁×#103	C₂²	1	2	yes	—
25	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#399	—	2	2	no	—
29	C₃₁×#398	—	2	2	no	—
30	C₃₁×#391	C₂²	1	2	yes	—
31	C₃₁×#389	C₂²	1	2	yes	—
32	C₃₁×#388	—	2	2	no	—
33	C₃₁×#387	—	2	2	no	—
34	C₃₁×#384	C₂²	1	2	yes	—
35	C₃₁×#380	C₂²	1	2	yes	—
36	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#340	C₂²	1	2	yes	—
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#336	C₂²	1	1	yes	characteristic
40	C₃₁×#336	C₂²	1	1	yes	characteristic
41	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
42	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
43	C₂×Q₁₆×C₃₁	—	2	2	no	—
44	C₂×Q₁₆×C₃₁	—	2	2	no	—
45	C₂×Q₁₆×C₃₁	D₄	1	4	yes	—
46	C₂²×C₈×C₃₁	—	2	2	no	—
47	C₂²×C₈×C₃₁	—	2	2	no	—
48	C₂²×C₈×C₃₁	—	2	2	no	—
49	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₃₁×C₂²⋊Q₈	—	2	4	no	—
54	C₃₁×C₂²⋊Q₈	—	4	2	no	—
55	C₄×Q₈×C₃₁	—	2	4	no	—
56	C₄×Q₈×C₃₁	—	4	2	no	—
57	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
59	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
60	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
61	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
62	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
65	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
66	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
67	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
68	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
69	C₂×C₄²×C₃₁	—	2	2	no	—
70	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
71	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
72	C₃₁×#107	—	2	2	no	—
73	C₃₁×#107	—	2	2	no	—
74	C₃₁×#107	—	2	2	no	—
75	C₃₁×#107	—	4	2	no	—
76	C₃₁×#103	—	2	4	no	—
77	C₃₁×#103	—	2	4	no	—
78	C₃₁×#103	—	2	4	no	—
79	C₃₁×C₂²⋊C₈	—	2	2	no	—
80	C₄×C₈×C₃₁	—	2	2	no	—
81	C₄×C₈×C₃₁	—	2	2	no	—
82	C₃₁×#95	—	2	2	no	—
83	C₃₁×#95	—	2	2	no	—
84	C₃₁×#95	—	2	2	no	—
85	C₃₁×#95	—	2	2	no	—
86	C₃₁×#95	D₄	1	1	yes	characteristic
87	C₃₁×#95	D₄	1	1	yes	characteristic
88	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
89	C₂×Q₈×C₃₁	—	2	2	no	—
90	C₂×Q₈×C₃₁	—	2	2	no	—
91	C₂×Q₈×C₃₁	—	2	2	no	—
92	C₂×Q₈×C₃₁	—	2	2	no	—
93	C₂×Q₈×C₃₁	—	2	2	no	—
94	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
95	C₂²×C₄×C₃₁	—	2	2	no	—
96	C₂²×C₄×C₃₁	—	2	1	no	—
97	C₂²×C₄×C₃₁	—	2	1	no	—
98	C₂²×C₄×C₃₁	—	2	1	no	—
99	C₂²×C₄×C₃₁	—	2	1	no	—
100	C₂²×C₄×C₃₁	—	2	1	no	—
101	C₂²×C₄×C₃₁	—	2	1	no	—
102	C₂²×C₄×C₃₁	—	2	2	no	—
103	C₂²×C₄×C₃₁	—	2	2	no	—
104	C₂²×C₄×C₃₁	—	2	2	no	—
105	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
110	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
111	Q₁₆×C₃₁	—	2	8	no	—
112	C₂×C₈×C₃₁	—	2	1	no	—
113	C₂×C₈×C₃₁	—	2	1	no	—
114	C₂×C₈×C₃₁	—	2	1	no	—
115	C₂×C₈×C₃₁	—	2	1	no	—
116	C₂×C₈×C₃₁	—	2	1	no	—
117	C₂×C₈×C₃₁	—	2	1	no	—
118	C₂×C₈×C₃₁	—	2	1	no	—
119	C₂×C₈×C₃₁	—	2	1	no	—
120	C₂×C₈×C₃₁	—	4	4	no	—
121	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
122	C₂×C₈×C₃₁	#41	1	2	yes	—
123	C₂×C₈×C₃₁	#41	1	2	yes	—
124	C₂×C₈×C₃₁	#41	1	2	yes	—
125	C₄⋊C₄×C₃₁	—	2	2	no	—
126	C₄⋊C₄×C₃₁	—	2	2	no	—
127	C₄⋊C₄×C₃₁	—	2	2	no	—
128	C₄⋊C₄×C₃₁	—	2	2	no	—
129	C₄⋊C₄×C₃₁	—	2	2	no	—
130	C₄⋊C₄×C₃₁	—	2	2	no	—
131	C₄⋊C₄×C₃₁	—	4	4	no	—
132	C₄⋊C₄×C₃₁	—	4	4	no	—
133	C₄⋊C₄×C₃₁	—	4	4	no	—
134	C₄⋊C₄×C₃₁	—	8	2	no	—
135	C₄⋊C₄×C₃₁	—	8	2	no	—
136	C₂²⋊C₄×C₃₁	—	2	2	no	—
137	C₂²⋊C₄×C₃₁	—	4	4	no	—
138	C₂²⋊C₄×C₃₁	—	8	2	no	—
139	C₂²⋊C₄×C₃₁	Q₁₆	1	4	yes	complemented
140	C₄²×C₃₁	—	2	1	no	—
141	C₄²×C₃₁	—	2	1	no	—
142	C₄²×C₃₁	—	2	1	no	—
143	C₄²×C₃₁	—	2	1	no	—
144	C₄²×C₃₁	—	4	4	no	—
145	#25035	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
146	C₂³×C₃₁	—	2	1	no	—
147	C₂³×C₃₁	—	2	1	no	—
148	C₂³×C₃₁	—	2	1	no	—
149	C₂³×C₃₁	—	2	1	no	—
150	C₂³×C₃₁	—	2	1	no	—
151	C₂³×C₃₁	—	2	1	no	—
152	C₂³×C₃₁	C₂×Q₁₆	1	2	yes	—
153	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
154	Q₈×C₃₁	—	2	4	no	—
155	Q₈×C₃₁	—	2	4	no	—
156	Q₈×C₃₁	—	4	2	no	—
157	Q₈×C₃₁	—	4	2	no	—
158	C₂×C₄×C₃₁	—	2	1	no	—
159	C₂×C₄×C₃₁	—	2	1	no	—
160	C₂×C₄×C₃₁	—	2	1	no	—
161	C₂×C₄×C₃₁	—	2	1	no	—
162	C₂×C₄×C₃₁	—	2	1	no	—
163	C₂×C₄×C₃₁	—	2	1	no	—
164	C₂×C₄×C₃₁	—	2	1	no	—
165	C₂×C₄×C₃₁	—	2	1	no	—
166	C₂×C₄×C₃₁	—	2	4	no	—
167	C₂×C₄×C₃₁	—	2	1	no	—
168	C₂×C₄×C₃₁	—	2	1	no	—
169	C₂×C₄×C₃₁	—	2	1	no	—
170	C₂×C₄×C₃₁	—	2	1	no	—
171	C₂×C₄×C₃₁	—	4	2	no	—
172	C₂×C₄×C₃₁	—	4	2	no	—
173	C₂×C₄×C₃₁	—	4	1	no	—
174	C₂×C₄×C₃₁	—	4	1	no	—
175	C₂×C₄×C₃₁	—	4	1	no	—
176	C₂×C₄×C₃₁	—	4	1	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	—	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	2	no	—
186	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
187	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
188	C₂×C₄×C₃₁	#135	1	2	yes	—
189	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
191	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
192	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
193	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
194	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
195	C₈×C₃₁	—	2	2	no	—
196	C₈×C₃₁	—	4	1	no	—
197	C₈×C₃₁	—	4	1	no	—
198	C₈×C₃₁	#123	1	4	yes	—
199	C₂×#467	C₂×C₃₁	1	1	yes	characteristic
200	C₂×C₄×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
201	#2423	C₂×C₃₁	1	2	yes	—
202	#2072	C₂×C₃₁	1	2	yes	complemented
203	#2036	C₂×C₃₁	1	2	yes	—
204	#1969	C₂×C₃₁	1	1	yes	characteristic, complemented
205	#1942	C₂×C₃₁	1	1	yes	characteristic, complemented
206	#1863	C₂×C₃₁	1	2	yes	complemented
207	#1845	C₂×C₃₁	1	1	yes	characteristic, complemented
208	#1808	C₂×C₃₁	1	1	yes	characteristic
209	C₈×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic
210	C₂²×C₃₁	—	2	2	no	—
211	C₂²×C₃₁	—	2	2	no	—
212	C₂²×C₃₁	—	2	2	no	—
213	C₂²×C₃₁	—	4	1	no	—
214	C₂²×C₃₁	—	4	1	no	—
215	C₂²×C₃₁	—	4	1	no	—
216	C₂²×C₃₁	—	4	1	no	—
217	C₂²×C₃₁	#532	1	1	yes	characteristic
218	C₂²×C₃₁	D₄²	1	1	yes	characteristic
219	C₂²×C₃₁	#547	1	1	yes	characteristic
220	C₂²×C₃₁	C₂²×Q₁₆	1	1	yes	characteristic
221	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
222	C₂²×C₃₁	#576	1	1	yes	characteristic
223	C₂²×C₃₁	#578	1	1	yes	characteristic
224	C₄×C₃₁	—	2	2	no	—
225	C₄×C₃₁	—	2	2	no	—
226	C₄×C₃₁	—	2	2	no	—
227	C₄×C₃₁	—	4	1	no	—
228	C₄×C₃₁	—	4	4	no	—
229	C₄×C₃₁	—	4	1	no	—
230	C₄×C₃₁	—	8	2	no	—
231	C₄×C₃₁	—	8	2	no	—
232	C₄×C₃₁	—	8	2	no	—
233	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
234	C₄×C₃₁	#542	1	1	yes	characteristic
235	C₄×C₃₁	#547	1	1	yes	characteristic
236	C₄×C₃₁	#553	1	1	yes	characteristic
237	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
238	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
239	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
240	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
241	#467	—	2	4	no	—
242	C₄×Q₁₆	—	2	4	no	—
243	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
244	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
245	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
246	C₂×#107	C₂²×C₃₁	1	2	yes	—
247	C₂×#103	C₂²×C₃₁	1	2	yes	—
248	C₂×#103	C₂²×C₃₁	1	2	yes	—
249	C₂×#103	C₂²×C₃₁	1	2	yes	—
250	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
251	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
252	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
253	#399	—	2	2	no	—
254	#398	—	2	2	no	—
255	#391	C₂²×C₃₁	1	2	yes	—
256	#389	C₂²×C₃₁	1	2	yes	—
257	#388	—	2	2	no	—
258	#387	—	2	2	no	—
259	#384	C₂²×C₃₁	1	2	yes	—
260	#380	C₂²×C₃₁	1	2	yes	—
261	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
262	#340	C₂²×C₃₁	1	2	yes	—
263	#340	C₂²×C₃₁	1	2	yes	—
264	#336	C₂²×C₃₁	1	1	yes	characteristic
265	#336	C₂²×C₃₁	1	1	yes	characteristic
266	C₂×C₃₁	—	4	2	no	—
267	C₂×C₃₁	#2422	1	1	yes	characteristic
268	C₂×C₃₁	#3170	1	1	yes	characteristic
269	C₂×C₃₁	#3175	1	1	yes	characteristic
270	C₂×C₃₁	D₄×Q₁₆	1	1	yes	characteristic
271	C₂×C₃₁	#3287	1	1	yes	characteristic
272	C₂×C₃₁	#3336	1	1	yes	characteristic
273	C₂×C₃₁	#3338	1	1	yes	characteristic
274	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
275	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
276	C₂×Q₁₆	—	2	2	no	—
277	C₂×Q₁₆	—	2	2	no	—
278	C₂×Q₁₆	D₄×C₃₁	1	4	yes	—
279	C₂²×C₈	—	2	2	no	—
280	C₂²×C₈	—	2	2	no	—
281	C₂²×C₈	—	2	2	no	—
282	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
283	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
284	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
285	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
286	C₂²⋊Q₈	—	2	4	no	—
287	C₂²⋊Q₈	—	4	2	no	—
288	C₄×Q₈	—	2	4	no	—
289	C₄×Q₈	—	4	2	no	—
290	C₂×C₄⋊C₄	—	2	2	no	—
291	C₂×C₄⋊C₄	—	2	2	no	—
292	C₂×C₄⋊C₄	—	2	2	no	—
293	C₂×C₄⋊C₄	—	2	2	no	—
294	C₂×C₄⋊C₄	—	2	2	no	—
295	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
296	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
297	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
298	C₂×C₂²⋊C₄	—	2	2	no	—
299	C₂×C₂²⋊C₄	—	2	2	no	—
300	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
301	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
302	C₂×C₄²	—	2	2	no	—
303	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
304	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
305	#107	—	2	2	no	—
306	#107	—	2	2	no	—
307	#107	—	2	2	no	—
308	#107	—	4	2	no	—
309	#103	—	2	4	no	—
310	#103	—	2	4	no	—
311	#103	—	2	4	no	—
312	C₂²⋊C₈	—	2	2	no	—
313	C₄×C₈	—	2	2	no	—
314	C₄×C₈	—	2	2	no	—
315	#95	—	2	2	no	—
316	#95	—	2	2	no	—
317	#95	—	2	2	no	—
318	#95	—	2	2	no	—
319	#95	D₄×C₃₁	1	1	yes	characteristic
320	#95	D₄×C₃₁	1	1	yes	characteristic
321	C₃₁	#25035	1	1	yes	characteristic, direct product factor, Sylow
322	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
323	C₂×Q₈	—	2	2	no	—
324	C₂×Q₈	—	2	2	no	—
325	C₂×Q₈	—	2	2	no	—
326	C₂×Q₈	—	2	2	no	—
327	C₂×Q₈	—	2	2	no	—
328	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	2	1	no	—
331	C₂²×C₄	—	2	1	no	—
332	C₂²×C₄	—	2	1	no	—
333	C₂²×C₄	—	2	1	no	—
334	C₂²×C₄	—	2	1	no	—
335	C₂²×C₄	—	2	1	no	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	2	2	no	—
338	C₂²×C₄	—	2	2	no	—
339	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
340	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
341	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
342	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
343	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
344	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
345	Q₁₆	—	2	8	no	—
346	C₂×C₈	—	2	1	no	—
347	C₂×C₈	—	2	1	no	—
348	C₂×C₈	—	2	1	no	—
349	C₂×C₈	—	2	1	no	—
350	C₂×C₈	—	2	1	no	—
351	C₂×C₈	—	2	1	no	—
352	C₂×C₈	—	2	1	no	—
353	C₂×C₈	—	2	1	no	—
354	C₂×C₈	—	4	4	no	—
355	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
356	C₂×C₈	#41×C₃₁	1	2	yes	—
357	C₂×C₈	#41×C₃₁	1	2	yes	—
358	C₂×C₈	#41×C₃₁	1	2	yes	—
359	C₄⋊C₄	—	2	2	no	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	2	2	no	—
362	C₄⋊C₄	—	2	2	no	—
363	C₄⋊C₄	—	2	2	no	—
364	C₄⋊C₄	—	2	2	no	—
365	C₄⋊C₄	—	4	4	no	—
366	C₄⋊C₄	—	4	4	no	—
367	C₄⋊C₄	—	4	4	no	—
368	C₄⋊C₄	—	8	2	no	—
369	C₄⋊C₄	—	8	2	no	—
370	C₂²⋊C₄	—	2	2	no	—
371	C₂²⋊C₄	—	4	4	no	—
372	C₂²⋊C₄	—	8	2	no	—
373	C₂²⋊C₄	Q₁₆×C₃₁	1	4	yes	complemented
374	C₄²	—	2	1	no	—
375	C₄²	—	2	1	no	—
376	C₄²	—	2	1	no	—
377	C₄²	—	2	1	no	—
378	C₄²	—	4	4	no	—
379	C₂³	—	2	1	no	—
380	C₂³	—	2	1	no	—
381	C₂³	—	2	1	no	—
382	C₂³	—	2	1	no	—
383	C₂³	—	2	1	no	—
384	C₂³	—	2	1	no	—
385	C₂³	C₂×Q₁₆×C₃₁	1	2	yes	—
386	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
387	Q₈	—	2	4	no	—
388	Q₈	—	2	4	no	—
389	Q₈	—	4	2	no	—
390	Q₈	—	4	2	no	—
391	C₂×C₄	—	2	1	no	—
392	C₂×C₄	—	2	1	no	—
393	C₂×C₄	—	2	1	no	—
394	C₂×C₄	—	2	1	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	2	1	no	—
398	C₂×C₄	—	2	1	no	—
399	C₂×C₄	—	2	4	no	—
400	C₂×C₄	—	2	1	no	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	2	1	no	—
403	C₂×C₄	—	2	1	no	—
404	C₂×C₄	—	4	2	no	—
405	C₂×C₄	—	4	2	no	—
406	C₂×C₄	—	4	1	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	4	1	no	—
409	C₂×C₄	—	4	1	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	4	2	no	—
412	C₂×C₄	—	4	2	no	—
413	C₂×C₄	—	4	2	no	—
414	C₂×C₄	—	4	2	no	—
415	C₂×C₄	—	4	2	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
420	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
421	C₂×C₄	C₃₁×#135	1	2	yes	—
422	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
423	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
424	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
425	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
426	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
427	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
428	C₈	—	2	2	no	—
429	C₈	—	4	1	no	—
430	C₈	—	4	1	no	—
431	C₈	C₃₁×#123	1	4	yes	—
432	C₂²	—	2	2	no	—
433	C₂²	—	2	2	no	—
434	C₂²	—	2	2	no	—
435	C₂²	—	4	1	no	—
436	C₂²	—	4	1	no	—
437	C₂²	—	4	1	no	—
438	C₂²	—	4	1	no	—
439	C₂²	C₃₁×#532	1	1	yes	characteristic
440	C₂²	D₄²×C₃₁	1	1	yes	characteristic
441	C₂²	C₃₁×#547	1	1	yes	characteristic
442	C₂²	C₂²×Q₁₆×C₃₁	1	1	yes	characteristic
443	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
444	C₂²	C₃₁×#576	1	1	yes	characteristic
445	C₂²	C₃₁×#578	1	1	yes	characteristic
446	C₄	—	2	2	no	—
447	C₄	—	2	2	no	—
448	C₄	—	2	2	no	—
449	C₄	—	4	1	no	—
450	C₄	—	4	4	no	—
451	C₄	—	4	1	no	—
452	C₄	—	8	2	no	—
453	C₄	—	8	2	no	—
454	C₄	—	8	2	no	—
455	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
456	C₄	C₃₁×#542	1	1	yes	characteristic
457	C₄	C₃₁×#547	1	1	yes	characteristic
458	C₄	C₃₁×#553	1	1	yes	characteristic
459	C₂	—	4	2	no	—
460	C₂	C₃₁×#2422	1	1	yes	characteristic
461	C₂	C₃₁×#3170	1	1	yes	characteristic
462	C₂	C₃₁×#3175	1	1	yes	characteristic
463	C₂	D₄×Q₁₆×C₃₁	1	1	yes	characteristic
464	C₂	C₃₁×#3287	1	1	yes	characteristic
465	C₂	C₃₁×#3336	1	1	yes	characteristic
466	C₂	C₃₁×#3338	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 ◁ C₄×C₂²⋊C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×Q₁₆×C₃₁	C₂	semidirect
4. C₃₁×#2072	C₂	semidirect
6. C₃₁×#1969	C₂	semidirect
7. C₃₁×#1942	C₂	semidirect
8. C₃₁×#1863	C₂	semidirect
9. C₃₁×#1845	C₂	semidirect
139. C₂²⋊C₄×C₃₁	Q₁₆	semidirect
145. #25035	C₃₁	direct (N × H)
200. C₂×C₄×Q₁₆	C₂×C₃₁	semidirect
202. #2072	C₂×C₃₁	semidirect
204. #1969	C₂×C₃₁	semidirect
205. #1942	C₂×C₃₁	semidirect
206. #1863	C₂×C₃₁	semidirect
207. #1845	C₂×C₃₁	semidirect
321. C₃₁	#25035	direct (N × H)
373. 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×#467	2
C₂×C₄×Q₁₆×C₃₁	2
C₃₁×#2423	2
C₃₁×#2072	2
C₃₁×#2036	2
C₃₁×#1969	2
C₃₁×#1942	2
C₃₁×#1863	2
C₃₁×#1845	2
C₃₁×#1808	2
C₈×C₂²⋊C₄×C₃₁	2
#25035	31

Subgroups — C31×#25035 #21481794

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#25035 #21481794