Subgroups — C₃₁×#24828 #21481587

Order 7936.

824 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#475	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#440	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2563	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2351	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2085	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2065	C₂	1	2	yes	maximal
7	C₃₁×#1993	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1940	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1880	C₂	1	2	yes	maximal
10	C₃₁×#1805	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1804	C₂	1	1	yes	maximal, characteristic
12	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
15	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#475	—	2	4	no	—
18	C₃₁×#440	—	2	4	no	—
19	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#106	C₂²	1	2	yes	—
21	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
23	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#103	C₂²	1	2	yes	—
25	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×#102	C₂²	1	2	yes	—
27	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
28	C₃₁×#393	—	2	1	no	—
29	C₃₁×#393	—	2	1	no	—
30	C₃₁×#390	C₂²	1	2	yes	—
31	C₃₁×#389	—	2	2	no	—
32	C₃₁×#389	C₂²	1	2	yes	—
33	C₃₁×#382	—	2	1	no	—
34	C₃₁×#382	C₂²	1	2	yes	—
35	C₃₁×#382	C₂²	1	1	yes	characteristic
36	C₃₁×#380	—	2	2	no	—
37	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
38	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
39	C₃₁×#340	C₂²	1	2	yes	—
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	—	2	1	no	—
44	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₃₁	—	2	1	no	—
47	C₂×SD₁₆×C₃₁	—	2	2	no	—
48	C₂×SD₁₆×C₃₁	—	2	2	no	—
49	C₂×SD₁₆×C₃₁	Q₈	1	4	yes	complemented
50	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
52	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₃₁×C₄⋊Q₈	—	2	2	no	—
54	C₃₁×C₄⋊Q₈	—	2	2	no	—
55	C₄×Q₈×C₃₁	—	2	2	no	—
56	C₄×Q₈×C₃₁	—	2	4	no	—
57	C₄×Q₈×C₃₁	—	4	2	no	—
58	C₄×Q₈×C₃₁	D₄	1	4	yes	—
59	C₄×D₄×C₃₁	—	2	2	no	—
60	C₄×D₄×C₃₁	—	2	2	no	—
61	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
62	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
63	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
64	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
65	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
66	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
68	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
71	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
72	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
73	C₂×C₄²×C₃₁	—	2	1	no	—
74	C₂×C₄²×C₃₁	—	2	2	no	—
75	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
76	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
77	C₃₁×#107	—	2	2	no	—
78	C₃₁×#106	—	2	4	no	—
79	C₃₁×C₄⋊C₈	—	2	2	no	—
80	C₃₁×C₄⋊C₈	—	4	2	no	—
81	C₃₁×#103	—	2	2	no	—
82	C₃₁×#103	—	2	4	no	—
83	C₃₁×#102	—	2	2	no	—
84	C₃₁×#102	—	4	2	no	—
85	C₃₁×#97	—	2	2	no	—
86	C₃₁×#95	—	2	1	no	—
87	C₃₁×#95	—	2	2	no	—
88	C₃₁×#95	—	2	2	no	—
89	C₃₁×#95	—	2	2	no	—
90	C₃₁×#95	—	2	1	no	—
91	C₃₁×#95	D₄	1	1	yes	characteristic
92	C₃₁×#95	D₄	1	1	yes	characteristic
93	C₂⁴×C₃₁	—	2	1	no	—
94	C₂×Q₈×C₃₁	—	2	1	no	—
95	C₂×Q₈×C₃₁	—	2	1	no	—
96	C₂×Q₈×C₃₁	—	2	1	no	—
97	C₂×Q₈×C₃₁	—	4	2	no	—
98	C₂×Q₈×C₃₁	—	8	2	no	—
99	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
100	C₂×Q₈×C₃₁	C₂×Q₈	1	2	yes	—
101	C₂×Q₈×C₃₁	#41	1	2	yes	—
102	C₂×D₄×C₃₁	—	2	1	no	—
103	C₂×D₄×C₃₁	—	2	1	no	—
104	C₂×D₄×C₃₁	—	2	1	no	—
105	C₂×D₄×C₃₁	—	2	1	no	—
106	C₂×D₄×C₃₁	—	2	1	no	—
107	C₂×D₄×C₃₁	C₂×Q₈	1	2	yes	—
108	C₂²×C₄×C₃₁	—	2	1	no	—
109	C₂²×C₄×C₃₁	—	2	1	no	—
110	C₂²×C₄×C₃₁	—	2	1	no	—
111	C₂²×C₄×C₃₁	—	2	2	no	—
112	C₂²×C₄×C₃₁	—	2	2	no	—
113	C₂²×C₄×C₃₁	—	4	4	no	—
114	C₂²×C₄×C₃₁	—	4	2	no	—
115	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
116	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
117	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
120	SD₁₆×C₃₁	—	2	8	no	—
121	C₂×C₈×C₃₁	—	2	1	no	—
122	C₂×C₈×C₃₁	—	2	1	no	—
123	C₂×C₈×C₃₁	—	2	1	no	—
124	C₂×C₈×C₃₁	—	2	2	no	—
125	C₂×C₈×C₃₁	—	2	2	no	—
126	C₂×C₈×C₃₁	—	2	1	no	—
127	C₂×C₈×C₃₁	—	2	1	no	—
128	C₂×C₈×C₃₁	—	2	2	no	—
129	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
130	C₄⋊C₄×C₃₁	—	2	4	no	—
131	C₄⋊C₄×C₃₁	—	2	1	no	—
132	C₄⋊C₄×C₃₁	—	2	2	no	—
133	C₄⋊C₄×C₃₁	—	2	1	no	—
134	C₄⋊C₄×C₃₁	—	2	1	no	—
135	C₄⋊C₄×C₃₁	—	2	2	no	—
136	C₄⋊C₄×C₃₁	—	2	2	no	—
137	C₄⋊C₄×C₃₁	—	2	2	no	—
138	C₄⋊C₄×C₃₁	—	4	2	no	—
139	C₄⋊C₄×C₃₁	—	4	4	no	—
140	C₄⋊C₄×C₃₁	—	4	4	no	—
141	C₄⋊C₄×C₃₁	—	4	2	no	—
142	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
143	C₂²⋊C₄×C₃₁	—	4	2	no	—
144	C₂²⋊C₄×C₃₁	—	8	2	no	—
145	C₂²⋊C₄×C₃₁	—	8	2	no	—
146	C₄²×C₃₁	—	2	4	no	—
147	C₄²×C₃₁	—	2	2	no	—
148	C₄²×C₃₁	—	2	2	no	—
149	C₄²×C₃₁	—	4	4	no	—
150	C₄²×C₃₁	C₂×D₄	1	2	yes	—
151	C₄²×C₃₁	#41	1	2	yes	—
152	#24828	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₂³×C₃₁	—	2	2	no	—
154	C₂³×C₃₁	—	4	1	no	—
155	C₂³×C₃₁	—	4	1	no	—
156	C₂³×C₃₁	—	4	2	no	—
157	C₂³×C₃₁	—	4	2	no	—
158	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
159	Q₈×C₃₁	—	2	2	no	—
160	Q₈×C₃₁	—	2	2	no	—
161	Q₈×C₃₁	—	2	2	no	—
162	Q₈×C₃₁	—	8	4	no	—
163	Q₈×C₃₁	C₂²⋊Q₈	1	4	yes	—
164	D₄×C₃₁	—	2	2	no	—
165	D₄×C₃₁	—	2	2	no	—
166	D₄×C₃₁	—	4	1	no	—
167	D₄×C₃₁	—	4	1	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	—	2	2	no	—
173	C₂×C₄×C₃₁	—	2	2	no	—
174	C₂×C₄×C₃₁	—	2	2	no	—
175	C₂×C₄×C₃₁	—	2	2	no	—
176	C₂×C₄×C₃₁	—	2	2	no	—
177	C₂×C₄×C₃₁	—	2	2	no	—
178	C₂×C₄×C₃₁	—	2	2	no	—
179	C₂×C₄×C₃₁	—	2	2	no	—
180	C₂×C₄×C₃₁	—	4	2	no	—
181	C₂×C₄×C₃₁	—	4	1	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	—	4	2	no	—
187	C₂×C₄×C₃₁	—	4	2	no	—
188	C₂×C₄×C₃₁	—	4	2	no	—
189	C₂×C₄×C₃₁	—	8	4	no	—
190	C₂×C₄×C₃₁	#121	1	2	yes	—
191	C₂×C₄×C₃₁	#121	1	2	yes	—
192	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
193	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
194	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
195	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
196	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
197	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
198	C₈×C₃₁	—	2	2	no	—
199	C₈×C₃₁	—	4	1	no	—
200	C₈×C₃₁	—	4	2	no	—
201	C₂×#475	C₂×C₃₁	1	1	yes	characteristic, complemented
202	C₂×#440	C₂×C₃₁	1	1	yes	characteristic
203	#2563	C₂×C₃₁	1	1	yes	characteristic
204	#2351	C₂×C₃₁	1	1	yes	characteristic, complemented
205	#2085	C₂×C₃₁	1	2	yes	complemented
206	#2065	C₂×C₃₁	1	2	yes	—
207	#1993	C₂×C₃₁	1	1	yes	characteristic, complemented
208	#1940	C₂×C₃₁	1	1	yes	characteristic, complemented
209	#1880	C₂×C₃₁	1	2	yes	—
210	#1805	C₂×C₃₁	1	2	yes	complemented
211	#1804	C₂×C₃₁	1	1	yes	characteristic
212	C₂²×C₃₁	—	4	4	no	—
213	C₂²×C₃₁	—	4	2	no	—
214	C₂²×C₃₁	—	8	1	no	—
215	C₂²×C₃₁	—	8	1	no	—
216	C₂²×C₃₁	—	8	2	no	—
217	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
218	C₂²×C₃₁	#546	1	1	yes	characteristic
219	C₂²×C₃₁	D₄×Q₈	1	1	yes	characteristic
220	C₂²×C₃₁	#575	1	2	yes	—
221	C₂²×C₃₁	#577	1	2	yes	—
222	C₄×C₃₁	—	2	4	no	—
223	C₄×C₃₁	—	2	2	no	—
224	C₄×C₃₁	—	2	2	no	—
225	C₄×C₃₁	—	4	2	no	—
226	C₄×C₃₁	—	4	2	no	—
227	C₄×C₃₁	—	4	2	no	—
228	C₄×C₃₁	—	8	1	no	—
229	C₄×C₃₁	—	8	2	no	—
230	C₄×C₃₁	—	8	2	no	—
231	C₄×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
232	C₄×C₃₁	#537	1	1	yes	characteristic
233	C₄×C₃₁	#552	1	1	yes	characteristic
234	C₄×C₃₁	#554	1	1	yes	characteristic
235	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
236	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
237	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
238	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
239	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
240	#475	—	2	4	no	—
241	#440	—	2	4	no	—
242	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
243	C₂×#106	C₂²×C₃₁	1	2	yes	—
244	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
245	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
246	C₂×#103	C₂²×C₃₁	1	2	yes	—
247	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
248	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
249	C₂×#102	C₂²×C₃₁	1	2	yes	—
250	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
251	#393	—	2	1	no	—
252	#393	—	2	1	no	—
253	#390	C₂²×C₃₁	1	2	yes	—
254	#389	—	2	2	no	—
255	#389	C₂²×C₃₁	1	2	yes	—
256	#382	—	2	1	no	—
257	#382	C₂²×C₃₁	1	2	yes	—
258	#382	C₂²×C₃₁	1	1	yes	characteristic
259	#380	—	2	2	no	—
260	C₄×C₄⋊C₄	—	2	1	no	—
261	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
262	#340	C₂²×C₃₁	1	2	yes	—
263	#340	C₂²×C₃₁	1	2	yes	—
264	#340	C₂²×C₃₁	1	2	yes	—
265	#336	C₂²×C₃₁	1	2	yes	—
266	C₂×C₃₁	—	8	2	no	—
267	C₂×C₃₁	#2417	1	1	yes	characteristic
268	C₂×C₃₁	#3040	1	1	yes	characteristic
269	C₂×C₃₁	#3043	1	1	yes	characteristic
270	C₂×C₃₁	#3321	1	2	yes	—
271	C₂×C₃₁	#3388	1	2	yes	—
272	C₂²×Q₈	—	2	1	no	—
273	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
274	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
275	C₂³×C₄	—	2	1	no	—
276	C₂×SD₁₆	—	2	2	no	—
277	C₂×SD₁₆	—	2	2	no	—
278	C₂×SD₁₆	Q₈×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	C₄⋊Q₈	—	2	2	no	—
283	C₄⋊Q₈	—	2	2	no	—
284	C₄×Q₈	—	2	2	no	—
285	C₄×Q₈	—	2	4	no	—
286	C₄×Q₈	—	4	2	no	—
287	C₄×Q₈	D₄×C₃₁	1	4	yes	—
288	C₄×D₄	—	2	2	no	—
289	C₄×D₄	—	2	2	no	—
290	C₂×C₄⋊C₄	—	2	1	no	—
291	C₂×C₄⋊C₄	—	2	2	no	—
292	C₂×C₄⋊C₄	—	2	1	no	—
293	C₂×C₄⋊C₄	—	2	2	no	—
294	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
295	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
296	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
297	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
298	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
299	C₂×C₂²⋊C₄	—	2	1	no	—
300	C₂×C₂²⋊C₄	—	2	2	no	—
301	C₂×C₂²⋊C₄	—	2	2	no	—
302	C₂×C₄²	—	2	1	no	—
303	C₂×C₄²	—	2	2	no	—
304	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
305	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
306	#107	—	2	2	no	—
307	#106	—	2	4	no	—
308	C₄⋊C₈	—	2	2	no	—
309	C₄⋊C₈	—	4	2	no	—
310	#103	—	2	2	no	—
311	#103	—	2	4	no	—
312	#102	—	2	2	no	—
313	#102	—	4	2	no	—
314	#97	—	2	2	no	—
315	#95	—	2	1	no	—
316	#95	—	2	2	no	—
317	#95	—	2	2	no	—
318	#95	—	2	2	no	—
319	#95	—	2	1	no	—
320	#95	D₄×C₃₁	1	1	yes	characteristic
321	#95	D₄×C₃₁	1	1	yes	characteristic
322	C₃₁	#24828	1	1	yes	characteristic, direct product factor, Sylow
323	C₂⁴	—	2	1	no	—
324	C₂×Q₈	—	2	1	no	—
325	C₂×Q₈	—	2	1	no	—
326	C₂×Q₈	—	2	1	no	—
327	C₂×Q₈	—	4	2	no	—
328	C₂×Q₈	—	8	2	no	—
329	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
330	C₂×Q₈	C₂×Q₈×C₃₁	1	2	yes	—
331	C₂×Q₈	#41×C₃₁	1	2	yes	—
332	C₂×D₄	—	2	1	no	—
333	C₂×D₄	—	2	1	no	—
334	C₂×D₄	—	2	1	no	—
335	C₂×D₄	—	2	1	no	—
336	C₂×D₄	—	2	1	no	—
337	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	—
338	C₂²×C₄	—	2	1	no	—
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	2	1	no	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	—	2	2	no	—
343	C₂²×C₄	—	4	4	no	—
344	C₂²×C₄	—	4	2	no	—
345	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
346	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
347	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
348	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
349	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
350	SD₁₆	—	2	8	no	—
351	C₂×C₈	—	2	1	no	—
352	C₂×C₈	—	2	1	no	—
353	C₂×C₈	—	2	1	no	—
354	C₂×C₈	—	2	2	no	—
355	C₂×C₈	—	2	2	no	—
356	C₂×C₈	—	2	1	no	—
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	—	2	1	no	—
359	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
360	C₄⋊C₄	—	2	4	no	—
361	C₄⋊C₄	—	2	1	no	—
362	C₄⋊C₄	—	2	2	no	—
363	C₄⋊C₄	—	2	1	no	—
364	C₄⋊C₄	—	2	1	no	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	2	2	no	—
367	C₄⋊C₄	—	2	2	no	—
368	C₄⋊C₄	—	4	2	no	—
369	C₄⋊C₄	—	4	4	no	—
370	C₄⋊C₄	—	4	4	no	—
371	C₄⋊C₄	—	4	2	no	—
372	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
373	C₂²⋊C₄	—	4	2	no	—
374	C₂²⋊C₄	—	8	2	no	—
375	C₂²⋊C₄	—	8	2	no	—
376	C₄²	—	2	4	no	—
377	C₄²	—	2	2	no	—
378	C₄²	—	2	2	no	—
379	C₄²	—	4	4	no	—
380	C₄²	C₂×D₄×C₃₁	1	2	yes	—
381	C₄²	#41×C₃₁	1	2	yes	—
382	C₂³	—	2	2	no	—
383	C₂³	—	4	1	no	—
384	C₂³	—	4	1	no	—
385	C₂³	—	4	2	no	—
386	C₂³	—	4	2	no	—
387	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
388	Q₈	—	2	2	no	—
389	Q₈	—	2	2	no	—
390	Q₈	—	2	2	no	—
391	Q₈	—	8	4	no	—
392	Q₈	C₃₁×C₂²⋊Q₈	1	4	yes	—
393	D₄	—	2	2	no	—
394	D₄	—	2	2	no	—
395	D₄	—	4	1	no	—
396	D₄	—	4	1	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	—	2	2	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	4	2	no	—
410	C₂×C₄	—	4	1	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	—	8	4	no	—
419	C₂×C₄	C₃₁×#121	1	2	yes	—
420	C₂×C₄	C₃₁×#121	1	2	yes	—
421	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
422	C₂×C₄	C₂²×Q₈×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	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
426	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
427	C₈	—	2	2	no	—
428	C₈	—	4	1	no	—
429	C₈	—	4	2	no	—
430	C₂²	—	4	4	no	—
431	C₂²	—	4	2	no	—
432	C₂²	—	8	1	no	—
433	C₂²	—	8	1	no	—
434	C₂²	—	8	2	no	—
435	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
436	C₂²	C₃₁×#546	1	1	yes	characteristic
437	C₂²	D₄×Q₈×C₃₁	1	1	yes	characteristic
438	C₂²	C₃₁×#575	1	2	yes	—
439	C₂²	C₃₁×#577	1	2	yes	—
440	C₄	—	2	4	no	—
441	C₄	—	2	2	no	—
442	C₄	—	2	2	no	—
443	C₄	—	4	2	no	—
444	C₄	—	4	2	no	—
445	C₄	—	4	2	no	—
446	C₄	—	8	1	no	—
447	C₄	—	8	2	no	—
448	C₄	—	8	2	no	—
449	C₄	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
450	C₄	C₃₁×#537	1	1	yes	characteristic
451	C₄	C₃₁×#552	1	1	yes	characteristic
452	C₄	C₃₁×#554	1	1	yes	characteristic
453	C₂	—	8	2	no	—
454	C₂	C₃₁×#2417	1	1	yes	characteristic
455	C₂	C₃₁×#3040	1	1	yes	characteristic
456	C₂	C₃₁×#3043	1	1	yes	characteristic
457	C₂	C₃₁×#3321	1	2	yes	—
458	C₂	C₃₁×#3388	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#475	C₂	semidirect
4. C₃₁×#2351	C₂	semidirect
5. C₃₁×#2085	C₂	semidirect
7. C₃₁×#1993	C₂	semidirect
8. C₃₁×#1940	C₂	semidirect
10. C₃₁×#1805	C₂	semidirect
49. C₂×SD₁₆×C₃₁	Q₈	semidirect
152. #24828	C₃₁	direct (N × H)
201. C₂×#475	C₂×C₃₁	semidirect
204. #2351	C₂×C₃₁	semidirect
205. #2085	C₂×C₃₁	semidirect
207. #1993	C₂×C₃₁	semidirect
208. #1940	C₂×C₃₁	semidirect
210. #1805	C₂×C₃₁	semidirect
278. C₂×SD₁₆	Q₈×C₃₁	semidirect
322. C₃₁	#24828	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×#475	2
C₂×C₃₁×#440	2
C₃₁×#2563	2
C₃₁×#2351	2
C₃₁×#2085	2
C₃₁×#2065	2
C₃₁×#1993	2
C₃₁×#1940	2
C₃₁×#1880	2
C₃₁×#1805	2
C₃₁×#1804	2
#24828	31

Subgroups — C31×#24828 #21481587

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24828 #21481587