Subgroups — C₃₁×#22146 #21478905

Order 7936.

896 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#461	C₂	1	1	yes	maximal, characteristic
2	C₄×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2293	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1981	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1968	C₂	1	2	yes	maximal
6	C₃₁×#1948	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1933	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1880	C₂	1	2	yes	maximal
9	C₃₁×#1865	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1802	C₂	1	1	yes	maximal, characteristic
11	C₄×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
13	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
15	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#117	—	2	1	no	—
17	C₃₁×#461	C₄	1	8	yes	complemented
18	C₂×C₃₁×#106	C₂²	1	2	yes	—
19	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#103	C₂²	1	2	yes	—
22	C₂×C₃₁×#103	C₂²	1	2	yes	—
23	C₂×C₃₁×#103	C₂²	1	2	yes	—
24	C₂×C₃₁×#102	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
26	C₃₁×#391	C₂²	1	2	yes	—
27	C₃₁×#390	C₂²	1	1	yes	characteristic
28	C₃₁×#390	C₂²	1	1	yes	characteristic
29	C₃₁×#389	—	2	1	no	—
30	C₃₁×#389	—	2	1	no	—
31	C₃₁×#389	C₂²	1	1	yes	characteristic
32	C₃₁×#389	C₂²	1	1	yes	characteristic
33	C₃₁×#388	—	2	1	no	—
34	C₃₁×#388	—	2	1	no	—
35	C₃₁×#384	C₂²	1	1	yes	characteristic
36	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
37	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
38	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
39	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
40	C₄³×C₃₁	C₂²	1	1	yes	characteristic
41	C₃₁×#340	C₂²	1	2	yes	—
42	C₃₁×#340	C₂²	1	2	yes	—
43	C₃₁×#336	C₂²	1	2	yes	—
44	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
45	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
46	C₂³×C₄×C₃₁	—	2	1	no	—
47	C₂×SD₁₆×C₃₁	—	4	2	no	—
48	C₂×SD₁₆×C₃₁	—	4	2	no	—
49	C₂×SD₁₆×C₃₁	C₂×C₄	1	8	yes	—
50	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
51	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
52	C₃₁×C₄⋊Q₈	—	2	2	no	—
53	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	—
54	C₄×Q₈×C₃₁	—	4	2	no	—
55	C₄×Q₈×C₃₁	—	4	2	no	—
56	C₄×D₄×C₃₁	—	2	2	no	—
57	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
58	C₃₁×#117	—	4	4	no	—
59	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
60	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
61	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
62	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
63	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
64	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
65	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
66	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
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	1	no	—
72	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
73	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
74	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
75	C₂×C₄²×C₃₁	—	2	1	no	—
76	C₂×C₄²×C₃₁	—	2	2	no	—
77	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
78	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
79	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
80	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
81	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
82	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
83	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
84	C₃₁×#106	—	4	2	no	—
85	C₃₁×C₄⋊C₈	—	2	2	no	—
86	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
87	C₃₁×#103	—	4	2	no	—
88	C₃₁×#103	—	4	2	no	—
89	C₃₁×#103	C₂×C₄	1	8	yes	—
90	C₃₁×#102	—	4	2	no	—
91	C₄×C₈×C₃₁	—	4	2	no	—
92	C₃₁×#95	—	2	1	no	—
93	C₃₁×#95	—	2	1	no	—
94	C₃₁×#95	—	2	2	no	—
95	C₃₁×#95	—	2	2	no	—
96	C₃₁×#95	—	2	1	no	—
97	C₂⁴×C₃₁	—	2	1	no	—
98	C₂×Q₈×C₃₁	—	2	2	no	—
99	C₂×Q₈×C₃₁	—	2	2	no	—
100	C₂×Q₈×C₃₁	—	2	2	no	—
101	C₂×Q₈×C₃₁	—	2	2	no	—
102	C₂×Q₈×C₃₁	—	2	2	no	—
103	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
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	—	2	1	no	—
108	C₂×D₄×C₃₁	—	2	1	no	—
109	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
110	C₂²×C₄×C₃₁	—	2	2	no	—
111	C₂²×C₄×C₃₁	—	2	1	no	—
112	C₂²×C₄×C₃₁	—	2	4	no	—
113	C₂²×C₄×C₃₁	—	2	1	no	—
114	C₂²×C₄×C₃₁	—	2	1	no	—
115	C₂²×C₄×C₃₁	—	2	2	no	—
116	C₂²×C₄×C₃₁	—	4	2	no	—
117	C₂²×C₄×C₃₁	—	4	2	no	—
118	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
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	SD₁₆×C₃₁	—	4	8	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	C₂²×C₄	1	4	yes	—
132	C₄⋊C₄×C₃₁	—	2	4	no	—
133	C₄⋊C₄×C₃₁	—	2	4	no	—
134	C₄⋊C₄×C₃₁	—	2	1	no	—
135	C₄⋊C₄×C₃₁	—	2	2	no	—
136	C₄⋊C₄×C₃₁	—	2	1	no	—
137	C₄⋊C₄×C₃₁	—	2	1	no	—
138	C₄⋊C₄×C₃₁	—	2	1	no	—
139	C₄⋊C₄×C₃₁	—	2	2	no	—
140	C₄⋊C₄×C₃₁	—	2	1	no	—
141	C₄⋊C₄×C₃₁	—	4	2	no	—
142	C₄⋊C₄×C₃₁	—	4	4	no	—
143	C₄⋊C₄×C₃₁	—	4	2	no	—
144	C₄⋊C₄×C₃₁	—	4	2	no	—
145	C₄⋊C₄×C₃₁	—	4	2	no	—
146	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
147	C₂²⋊C₄×C₃₁	—	2	4	no	—
148	C₂²⋊C₄×C₃₁	—	4	2	no	—
149	C₂²⋊C₄×C₃₁	—	4	2	no	—
150	C₂²⋊C₄×C₃₁	—	4	2	no	—
151	C₂²⋊C₄×C₃₁	—	4	2	no	—
152	C₄²×C₃₁	—	2	2	no	—
153	C₄²×C₃₁	—	2	2	no	—
154	C₄²×C₃₁	—	2	1	no	—
155	C₄²×C₃₁	—	2	1	no	—
156	C₄²×C₃₁	—	2	1	no	—
157	C₄²×C₃₁	—	2	1	no	—
158	C₄²×C₃₁	—	4	1	no	—
159	C₄²×C₃₁	—	4	4	no	—
160	C₄²×C₃₁	—	4	2	no	—
161	C₄²×C₃₁	SD₁₆	1	4	yes	complemented
162	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
163	C₄²×C₃₁	#41	1	2	yes	—
164	#22146	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
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	1	no	—
170	C₂³×C₃₁	—	4	1	no	—
171	C₂³×C₃₁	—	4	1	no	—
172	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
173	Q₈×C₃₁	—	2	4	no	—
174	Q₈×C₃₁	—	2	4	no	—
175	Q₈×C₃₁	—	4	2	no	—
176	Q₈×C₃₁	—	4	2	no	—
177	D₄×C₃₁	—	2	2	no	—
178	D₄×C₃₁	—	2	2	no	—
179	D₄×C₃₁	—	4	1	no	—
180	D₄×C₃₁	—	4	1	no	—
181	C₂×C₄×C₃₁	—	2	1	no	—
182	C₂×C₄×C₃₁	—	2	1	no	—
183	C₂×C₄×C₃₁	—	2	1	no	—
184	C₂×C₄×C₃₁	—	2	1	no	—
185	C₂×C₄×C₃₁	—	2	1	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	4	no	—
191	C₂×C₄×C₃₁	—	2	1	no	—
192	C₂×C₄×C₃₁	—	2	1	no	—
193	C₂×C₄×C₃₁	—	2	1	no	—
194	C₂×C₄×C₃₁	—	4	2	no	—
195	C₂×C₄×C₃₁	—	4	2	no	—
196	C₂×C₄×C₃₁	—	4	1	no	—
197	C₂×C₄×C₃₁	—	4	1	no	—
198	C₂×C₄×C₃₁	—	4	8	no	—
199	C₂×C₄×C₃₁	—	4	2	no	—
200	C₂×C₄×C₃₁	—	4	1	no	—
201	C₂×C₄×C₃₁	—	4	1	no	—
202	C₂×C₄×C₃₁	—	4	1	no	—
203	C₂×C₄×C₃₁	—	4	1	no	—
204	C₂×C₄×C₃₁	—	4	2	no	—
205	C₂×C₄×C₃₁	—	4	1	no	—
206	C₂×C₄×C₃₁	—	4	1	no	—
207	C₂×C₄×C₃₁	—	4	2	no	—
208	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
209	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
210	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
211	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
212	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
213	C₂×C₄×C₃₁	#135	1	2	yes	—
214	C₂×C₄×C₃₁	#137	1	2	yes	—
215	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
216	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
217	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
218	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
219	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
220	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
221	C₈×C₃₁	—	2	4	no	—
222	C₈×C₃₁	—	4	2	no	—
223	C₂×#461	C₂×C₃₁	1	1	yes	characteristic
224	C₄×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
225	#2293	C₂×C₃₁	1	1	yes	characteristic
226	#1981	C₂×C₃₁	1	1	yes	characteristic, complemented
227	#1968	C₂×C₃₁	1	2	yes	—
228	#1948	C₂×C₃₁	1	1	yes	characteristic, complemented
229	#1933	C₂×C₃₁	1	2	yes	complemented
230	#1880	C₂×C₃₁	1	2	yes	—
231	#1865	C₂×C₃₁	1	2	yes	complemented
232	#1802	C₂×C₃₁	1	1	yes	characteristic
233	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
234	C₂²×C₃₁	—	4	2	no	—
235	C₂²×C₃₁	—	4	2	no	—
236	C₂²×C₃₁	—	4	2	no	—
237	C₂²×C₃₁	—	8	1	no	—
238	C₂²×C₃₁	—	8	1	no	—
239	C₂²×C₃₁	—	8	1	no	—
240	C₂²×C₃₁	—	8	1	no	—
241	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
242	C₂²×C₃₁	#535	1	1	yes	characteristic
243	C₂²×C₃₁	#537	1	1	yes	characteristic
244	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
245	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
246	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
247	C₂²×C₃₁	#575	1	1	yes	characteristic
248	C₄×C₃₁	—	2	2	no	—
249	C₄×C₃₁	—	2	2	no	—
250	C₄×C₃₁	—	2	2	no	—
251	C₄×C₃₁	—	2	2	no	—
252	C₄×C₃₁	—	4	4	no	—
253	C₄×C₃₁	—	4	2	no	—
254	C₄×C₃₁	—	4	1	no	—
255	C₄×C₃₁	—	8	1	no	—
256	C₄×C₃₁	—	8	2	no	—
257	C₄×C₃₁	—	8	1	no	—
258	C₄×C₃₁	C₄×SD₁₆	1	4	yes	—
259	C₄×C₃₁	#518	1	1	yes	characteristic
260	C₄×C₃₁	#520	1	1	yes	characteristic
261	C₄×C₃₁	#540	1	1	yes	characteristic
262	C₄×C₃₁	#542	1	1	yes	characteristic
263	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
264	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
265	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
266	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
267	C₂×#117	—	2	1	no	—
268	#461	C₄×C₃₁	1	8	yes	complemented
269	C₂×#106	C₂²×C₃₁	1	2	yes	—
270	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
271	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
272	C₂×#103	C₂²×C₃₁	1	2	yes	—
273	C₂×#103	C₂²×C₃₁	1	2	yes	—
274	C₂×#103	C₂²×C₃₁	1	2	yes	—
275	C₂×#102	C₂²×C₃₁	1	2	yes	—
276	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
277	#391	C₂²×C₃₁	1	2	yes	—
278	#390	C₂²×C₃₁	1	1	yes	characteristic
279	#390	C₂²×C₃₁	1	1	yes	characteristic
280	#389	—	2	1	no	—
281	#389	—	2	1	no	—
282	#389	C₂²×C₃₁	1	1	yes	characteristic
283	#389	C₂²×C₃₁	1	1	yes	characteristic
284	#388	—	2	1	no	—
285	#388	—	2	1	no	—
286	#384	C₂²×C₃₁	1	1	yes	characteristic
287	C₄×C₄⋊C₄	—	2	1	no	—
288	C₄×C₄⋊C₄	—	2	1	no	—
289	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
290	C₄×C₂²⋊C₄	—	2	1	no	—
291	C₄³	C₂²×C₃₁	1	1	yes	characteristic
292	#340	C₂²×C₃₁	1	2	yes	—
293	#340	C₂²×C₃₁	1	2	yes	—
294	#336	C₂²×C₃₁	1	2	yes	—
295	C₂×C₃₁	—	8	2	no	—
296	C₂×C₃₁	#2324	1	1	yes	characteristic
297	C₂×C₃₁	C₂×C₄×SD₁₆	1	1	yes	characteristic
298	C₂×C₃₁	C₄×#137	1	1	yes	characteristic
299	C₂×C₃₁	#3206	1	1	yes	characteristic
300	C₂×C₃₁	#3212	1	1	yes	characteristic
301	C₂×C₃₁	#3246	1	1	yes	characteristic
302	C₂×C₃₁	#3248	1	1	yes	characteristic
303	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
304	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
305	C₂³×C₄	—	2	1	no	—
306	C₂×SD₁₆	—	4	2	no	—
307	C₂×SD₁₆	—	4	2	no	—
308	C₂×SD₁₆	C₂×C₄×C₃₁	1	8	yes	—
309	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
310	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
311	C₄⋊Q₈	—	2	2	no	—
312	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
313	C₄×Q₈	—	4	2	no	—
314	C₄×Q₈	—	4	2	no	—
315	C₄×D₄	—	2	2	no	—
316	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
317	#117	—	4	4	no	—
318	C₂×C₄⋊C₄	—	2	1	no	—
319	C₂×C₄⋊C₄	—	2	1	no	—
320	C₂×C₄⋊C₄	—	2	1	no	—
321	C₂×C₄⋊C₄	—	2	1	no	—
322	C₂×C₄⋊C₄	—	2	2	no	—
323	C₂×C₄⋊C₄	—	2	1	no	—
324	C₂×C₄⋊C₄	—	2	1	no	—
325	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
326	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
327	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
328	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
329	C₂×C₂²⋊C₄	—	2	1	no	—
330	C₂×C₂²⋊C₄	—	2	1	no	—
331	C₂×C₂²⋊C₄	—	2	1	no	—
332	C₂×C₂²⋊C₄	—	2	1	no	—
333	C₂×C₂²⋊C₄	—	2	1	no	—
334	C₂×C₄²	—	2	1	no	—
335	C₂×C₄²	—	2	2	no	—
336	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
337	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
338	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
339	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
340	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
341	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
342	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
343	#106	—	4	2	no	—
344	C₄⋊C₈	—	2	2	no	—
345	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
346	#103	—	4	2	no	—
347	#103	—	4	2	no	—
348	#103	C₂×C₄×C₃₁	1	8	yes	—
349	#102	—	4	2	no	—
350	C₄×C₈	—	4	2	no	—
351	#95	—	2	1	no	—
352	#95	—	2	1	no	—
353	#95	—	2	2	no	—
354	#95	—	2	2	no	—
355	#95	—	2	1	no	—
356	C₃₁	#22146	1	1	yes	characteristic, direct product factor, Sylow
357	C₂⁴	—	2	1	no	—
358	C₂×Q₈	—	2	2	no	—
359	C₂×Q₈	—	2	2	no	—
360	C₂×Q₈	—	2	2	no	—
361	C₂×Q₈	—	2	2	no	—
362	C₂×Q₈	—	2	2	no	—
363	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
364	C₂×D₄	—	2	1	no	—
365	C₂×D₄	—	2	1	no	—
366	C₂×D₄	—	2	1	no	—
367	C₂×D₄	—	2	1	no	—
368	C₂×D₄	—	2	1	no	—
369	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
370	C₂²×C₄	—	2	2	no	—
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	2	4	no	—
373	C₂²×C₄	—	2	1	no	—
374	C₂²×C₄	—	2	1	no	—
375	C₂²×C₄	—	2	2	no	—
376	C₂²×C₄	—	4	2	no	—
377	C₂²×C₄	—	4	2	no	—
378	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
379	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
380	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
381	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
382	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
383	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
384	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
385	SD₁₆	—	4	8	no	—
386	C₂×C₈	—	2	2	no	—
387	C₂×C₈	—	2	2	no	—
388	C₂×C₈	—	2	2	no	—
389	C₂×C₈	—	2	2	no	—
390	C₂×C₈	—	2	2	no	—
391	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
392	C₄⋊C₄	—	2	4	no	—
393	C₄⋊C₄	—	2	4	no	—
394	C₄⋊C₄	—	2	1	no	—
395	C₄⋊C₄	—	2	2	no	—
396	C₄⋊C₄	—	2	1	no	—
397	C₄⋊C₄	—	2	1	no	—
398	C₄⋊C₄	—	2	1	no	—
399	C₄⋊C₄	—	2	2	no	—
400	C₄⋊C₄	—	2	1	no	—
401	C₄⋊C₄	—	4	2	no	—
402	C₄⋊C₄	—	4	4	no	—
403	C₄⋊C₄	—	4	2	no	—
404	C₄⋊C₄	—	4	2	no	—
405	C₄⋊C₄	—	4	2	no	—
406	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
407	C₂²⋊C₄	—	2	4	no	—
408	C₂²⋊C₄	—	4	2	no	—
409	C₂²⋊C₄	—	4	2	no	—
410	C₂²⋊C₄	—	4	2	no	—
411	C₂²⋊C₄	—	4	2	no	—
412	C₄²	—	2	2	no	—
413	C₄²	—	2	2	no	—
414	C₄²	—	2	1	no	—
415	C₄²	—	2	1	no	—
416	C₄²	—	2	1	no	—
417	C₄²	—	2	1	no	—
418	C₄²	—	4	1	no	—
419	C₄²	—	4	4	no	—
420	C₄²	—	4	2	no	—
421	C₄²	SD₁₆×C₃₁	1	4	yes	complemented
422	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
423	C₄²	#41×C₃₁	1	2	yes	—
424	C₂³	—	2	2	no	—
425	C₂³	—	4	1	no	—
426	C₂³	—	4	1	no	—
427	C₂³	—	4	1	no	—
428	C₂³	—	4	1	no	—
429	C₂³	—	4	1	no	—
430	C₂³	—	4	1	no	—
431	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
432	Q₈	—	2	4	no	—
433	Q₈	—	2	4	no	—
434	Q₈	—	4	2	no	—
435	Q₈	—	4	2	no	—
436	D₄	—	2	2	no	—
437	D₄	—	2	2	no	—
438	D₄	—	4	1	no	—
439	D₄	—	4	1	no	—
440	C₂×C₄	—	2	1	no	—
441	C₂×C₄	—	2	1	no	—
442	C₂×C₄	—	2	1	no	—
443	C₂×C₄	—	2	1	no	—
444	C₂×C₄	—	2	1	no	—
445	C₂×C₄	—	2	1	no	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	2	2	no	—
449	C₂×C₄	—	2	4	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	4	2	no	—
454	C₂×C₄	—	4	2	no	—
455	C₂×C₄	—	4	1	no	—
456	C₂×C₄	—	4	1	no	—
457	C₂×C₄	—	4	8	no	—
458	C₂×C₄	—	4	2	no	—
459	C₂×C₄	—	4	1	no	—
460	C₂×C₄	—	4	1	no	—
461	C₂×C₄	—	4	1	no	—
462	C₂×C₄	—	4	1	no	—
463	C₂×C₄	—	4	2	no	—
464	C₂×C₄	—	4	1	no	—
465	C₂×C₄	—	4	1	no	—
466	C₂×C₄	—	4	2	no	—
467	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
468	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
469	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
470	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
471	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
472	C₂×C₄	C₃₁×#135	1	2	yes	—
473	C₂×C₄	C₃₁×#137	1	2	yes	—
474	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
475	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
476	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
477	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
478	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
479	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
480	C₈	—	2	4	no	—
481	C₈	—	4	2	no	—
482	C₂²	—	4	2	no	—
483	C₂²	—	4	2	no	—
484	C₂²	—	4	2	no	—
485	C₂²	—	8	1	no	—
486	C₂²	—	8	1	no	—
487	C₂²	—	8	1	no	—
488	C₂²	—	8	1	no	—
489	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
490	C₂²	C₃₁×#535	1	1	yes	characteristic
491	C₂²	C₃₁×#537	1	1	yes	characteristic
492	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
493	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
494	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
495	C₂²	C₃₁×#575	1	1	yes	characteristic
496	C₄	—	2	2	no	—
497	C₄	—	2	2	no	—
498	C₄	—	2	2	no	—
499	C₄	—	2	2	no	—
500	C₄	—	4	4	no	—
501	C₄	—	4	2	no	—
502	C₄	—	4	1	no	—
503	C₄	—	8	1	no	—
504	C₄	—	8	2	no	—
505	C₄	—	8	1	no	—
506	C₄	C₄×SD₁₆×C₃₁	1	4	yes	—
507	C₄	C₃₁×#518	1	1	yes	characteristic
508	C₄	C₃₁×#520	1	1	yes	characteristic
509	C₄	C₃₁×#540	1	1	yes	characteristic
510	C₄	C₃₁×#542	1	1	yes	characteristic
511	C₂	—	8	2	no	—
512	C₂	C₃₁×#2324	1	1	yes	characteristic
513	C₂	C₂×C₄×SD₁₆×C₃₁	1	1	yes	characteristic
514	C₂	C₄×C₃₁×#137	1	1	yes	characteristic
515	C₂	C₃₁×#3206	1	1	yes	characteristic
516	C₂	C₃₁×#3212	1	1	yes	characteristic
517	C₂	C₃₁×#3246	1	1	yes	characteristic
518	C₂	C₃₁×#3248	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	C₂²×D₄×C₃₁	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄×C₃₁×C₄⋊Q₈	C₂	semidirect
4. C₃₁×#1981	C₂	semidirect
6. C₃₁×#1948	C₂	semidirect
7. C₃₁×#1933	C₂	semidirect
9. C₃₁×#1865	C₂	semidirect
11. C₄×C₃₁×C₄⋊C₈	C₂	semidirect
17. C₃₁×#461	C₄	semidirect
161. C₄²×C₃₁	SD₁₆	semidirect
164. #22146	C₃₁	direct (N × H)
224. C₄×C₄⋊Q₈	C₂×C₃₁	semidirect
226. #1981	C₂×C₃₁	semidirect
228. #1948	C₂×C₃₁	semidirect
229. #1933	C₂×C₃₁	semidirect
231. #1865	C₂×C₃₁	semidirect
233. C₄×C₄⋊C₈	C₂×C₃₁	semidirect
268. #461	C₄×C₃₁	semidirect
356. C₃₁	#22146	direct (N × H)
421. C₄²	SD₁₆×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×#461	2
C₄×C₃₁×C₄⋊Q₈	2
C₃₁×#2293	2
C₃₁×#1981	2
C₃₁×#1968	2
C₃₁×#1948	2
C₃₁×#1933	2
C₃₁×#1880	2
C₃₁×#1865	2
C₃₁×#1802	2
C₄×C₃₁×C₄⋊C₈	2
#22146	31

Subgroups — C31×#22146 #21478905

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22146 #21478905