Subgroups — C₃₁×#21878 #21478637

Order 7936.

1324 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#532	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2920	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#423	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1997	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1965	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1900	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1800	C₂	1	4	yes	maximal, complemented
8	C₂²×#41×C₃₁	C₄	1	2	yes	—
9	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
10	C₃₁×#532	—	2	1	no	—
11	C₃₁×#532	—	2	1	no	—
12	C₃₁×#532	—	2	1	no	—
13	C₃₁×#532	—	2	1	no	—
14	C₃₁×#532	—	2	1	no	—
15	C₃₁×#532	—	2	1	no	—
16	C₃₁×#532	—	2	1	no	—
17	C₃₁×#532	—	2	1	no	—
18	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#124	C₄	1	2	yes	—
21	C₂×C₃₁×#121	C₄	1	2	yes	—
22	C₂×C₃₁×#121	C₂²	1	2	yes	—
23	C₂×C₄×D₄×C₃₁	C₄	1	2	yes	—
24	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
25	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#432	—	2	1	no	—
27	C₃₁×#432	—	2	1	no	—
28	C₃₁×#431	—	2	1	no	—
29	C₃₁×#431	—	2	1	no	—
30	C₃₁×#423	—	2	1	no	—
31	C₃₁×#423	—	2	1	no	—
32	C₃₁×#423	C₂²	1	2	yes	—
33	C₃₁×#423	C₂²	1	2	yes	—
34	C₃₁×#423	C₂²	1	2	yes	—
35	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
36	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
37	C₂×C₃₁×#100	C₂²	1	8	yes	—
38	C₃₁×#407	C₂²	1	2	yes	—
39	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
40	C₃₁×#343	C₂²	1	8	yes	—
41	C₂×#41×C₃₁	—	2	2	no	—
42	C₂×#41×C₃₁	—	2	2	no	—
43	C₂×#41×C₃₁	—	2	2	no	—
44	C₂×#41×C₃₁	—	2	2	no	—
45	C₂×#41×C₃₁	—	4	8	no	—
46	C₂²×Q₈×C₃₁	C₂×C₄	1	2	yes	—
47	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
48	C₂²×D₄×C₃₁	C₂×C₄	1	4	yes	—
49	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
50	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
51	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
52	C₂³×C₄×C₃₁	C₂×C₄	1	4	yes	—
53	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
54	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
55	C₂×M₁₆×C₃₁	—	2	2	no	—
56	C₂×M₁₆×C₃₁	—	4	4	no	—
57	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
58	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
59	C₂²×C₈×C₃₁	—	2	2	no	—
60	C₃₁×C₄⋊Q₈	—	2	1	no	—
61	C₃₁×C₄⋊Q₈	—	2	1	no	—
62	C₃₁×C₄⋊Q₈	—	2	1	no	—
63	C₃₁×C₄⋊Q₈	—	2	1	no	—
64	C₃₁×#127	—	2	1	no	—
65	C₃₁×#127	—	2	1	no	—
66	C₃₁×#127	—	2	1	no	—
67	C₃₁×#127	—	2	1	no	—
68	C₃₁×#124	—	2	2	no	—
69	C₃₁×#124	—	2	2	no	—
70	C₃₁×#124	—	2	2	no	—
71	C₃₁×#124	—	2	2	no	—
72	C₃₁×#121	—	2	2	no	—
73	C₃₁×#121	—	2	2	no	—
74	C₃₁×#121	—	2	2	no	—
75	C₃₁×#121	—	2	2	no	—
76	C₃₁×#121	—	2	2	no	—
77	C₃₁×#121	—	2	2	no	—
78	C₃₁×#121	—	2	2	no	—
79	C₃₁×#121	—	2	2	no	—
80	C₄×D₄×C₃₁	—	2	2	no	—
81	C₄×D₄×C₃₁	—	2	2	no	—
82	C₄×D₄×C₃₁	—	2	2	no	—
83	C₄×D₄×C₃₁	—	2	2	no	—
84	C₄×D₄×C₃₁	—	2	2	no	—
85	C₄×D₄×C₃₁	—	2	2	no	—
86	C₄×D₄×C₃₁	—	2	2	no	—
87	C₄×D₄×C₃₁	—	2	2	no	—
88	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
89	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
90	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
91	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
92	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
93	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
94	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
95	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
96	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
97	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
98	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
99	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
100	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
101	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
102	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
104	C₃₁×C₄⋊C₈	—	2	2	no	—
105	C₃₁×C₄⋊C₈	—	2	2	no	—
106	C₃₁×C₄⋊C₈	—	2	2	no	—
107	C₃₁×C₄⋊C₈	—	2	2	no	—
108	C₃₁×C₄⋊C₈	—	2	2	no	—
109	C₃₁×C₄⋊C₈	—	2	2	no	—
110	C₃₁×#100	—	4	8	no	—
111	C₃₁×C₂²⋊C₈	—	2	2	no	—
112	C₃₁×C₂²⋊C₈	—	2	2	no	—
113	C₃₁×#97	—	2	2	no	—
114	C₄×C₈×C₃₁	—	2	2	no	—
115	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
116	C₂⁴×C₃₁	C₂²×C₄	1	4	yes	—
117	#41×C₃₁	—	4	8	no	—
118	#41×C₃₁	—	4	8	no	—
119	#41×C₃₁	—	4	8	no	—
120	#41×C₃₁	—	4	8	no	—
121	C₂×Q₈×C₃₁	—	2	2	no	—
122	C₂×Q₈×C₃₁	—	2	2	no	—
123	C₂×Q₈×C₃₁	—	4	4	no	—
124	C₂×D₄×C₃₁	—	2	4	no	—
125	C₂×D₄×C₃₁	—	2	4	no	—
126	C₂×D₄×C₃₁	—	2	2	no	—
127	C₂×D₄×C₃₁	—	2	2	no	—
128	C₂×D₄×C₃₁	—	2	2	no	—
129	C₂×D₄×C₃₁	—	2	2	no	—
130	C₂×D₄×C₃₁	—	2	2	no	—
131	C₂×D₄×C₃₁	—	2	2	no	—
132	C₂×D₄×C₃₁	—	2	4	no	—
133	C₂×D₄×C₃₁	—	2	4	no	—
134	C₂×D₄×C₃₁	—	4	4	no	—
135	C₂×D₄×C₃₁	—	4	4	no	—
136	C₂×D₄×C₃₁	—	4	8	no	—
137	C₂²×C₄×C₃₁	—	2	4	no	—
138	C₂²×C₄×C₃₁	—	2	2	no	—
139	C₂²×C₄×C₃₁	—	2	2	no	—
140	C₂²×C₄×C₃₁	—	2	4	no	—
141	C₂²×C₄×C₃₁	—	2	4	no	—
142	C₂²×C₄×C₃₁	—	4	2	no	—
143	C₂²×C₄×C₃₁	—	4	8	no	—
144	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
145	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
146	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
147	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
148	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
149	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
150	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
151	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
153	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
154	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
155	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
156	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
157	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
158	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
163	M₁₆×C₃₁	—	2	4	no	—
164	M₁₆×C₃₁	—	2	8	no	—
165	M₁₆×C₃₁	—	2	4	no	—
166	M₁₆×C₃₁	—	4	4	no	—
167	C₂×C₈×C₃₁	—	2	4	no	—
168	C₂×C₈×C₃₁	—	2	4	no	—
169	C₂×C₈×C₃₁	—	4	4	no	—
170	C₄⋊C₄×C₃₁	—	2	2	no	—
171	C₄⋊C₄×C₃₁	—	2	2	no	—
172	C₄⋊C₄×C₃₁	—	2	2	no	—
173	C₄⋊C₄×C₃₁	—	2	2	no	—
174	C₂²⋊C₄×C₃₁	—	2	4	no	—
175	C₂²⋊C₄×C₃₁	—	2	4	no	—
176	C₂²⋊C₄×C₃₁	—	2	4	no	—
177	C₂²⋊C₄×C₃₁	—	2	4	no	—
178	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
179	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
180	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
181	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
182	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
183	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
184	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
185	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
186	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
187	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
188	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
189	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
190	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
191	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
192	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
193	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
194	#21878	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
195	C₂³×C₃₁	—	2	2	no	—
196	C₂³×C₃₁	—	2	2	no	—
197	C₂³×C₃₁	—	2	4	no	—
198	C₂³×C₃₁	—	2	8	no	—
199	C₂³×C₃₁	—	4	1	no	—
200	C₂³×C₃₁	—	4	4	no	—
201	C₂³×C₃₁	—	4	4	no	—
202	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
203	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
204	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
205	Q₈×C₃₁	—	4	4	no	—
206	Q₈×C₃₁	—	4	4	no	—
207	D₄×C₃₁	—	4	4	no	—
208	D₄×C₃₁	—	4	4	no	—
209	D₄×C₃₁	—	4	8	no	—
210	D₄×C₃₁	—	4	8	no	—
211	D₄×C₃₁	—	4	2	no	—
212	D₄×C₃₁	—	4	2	no	—
213	D₄×C₃₁	—	4	4	no	—
214	D₄×C₃₁	—	4	4	no	—
215	D₄×C₃₁	—	4	4	no	—
216	C₂×C₄×C₃₁	—	2	2	no	—
217	C₂×C₄×C₃₁	—	2	2	no	—
218	C₂×C₄×C₃₁	—	2	1	no	—
219	C₂×C₄×C₃₁	—	2	1	no	—
220	C₂×C₄×C₃₁	—	2	1	no	—
221	C₂×C₄×C₃₁	—	2	1	no	—
222	C₂×C₄×C₃₁	—	2	2	no	—
223	C₂×C₄×C₃₁	—	2	2	no	—
224	C₂×C₄×C₃₁	—	2	2	no	—
225	C₂×C₄×C₃₁	—	2	2	no	—
226	C₂×C₄×C₃₁	—	2	2	no	—
227	C₂×C₄×C₃₁	—	2	2	no	—
228	C₂×C₄×C₃₁	—	2	2	no	—
229	C₂×C₄×C₃₁	—	2	2	no	—
230	C₂×C₄×C₃₁	—	2	2	no	—
231	C₂×C₄×C₃₁	—	2	4	no	—
232	C₂×C₄×C₃₁	—	4	1	no	—
233	C₂×C₄×C₃₁	—	4	1	no	—
234	C₂×C₄×C₃₁	—	4	1	no	—
235	C₂×C₄×C₃₁	—	4	8	no	—
236	C₂×C₄×C₃₁	—	4	8	no	—
237	C₂×C₄×C₃₁	—	4	4	no	—
238	C₂×C₄×C₃₁	#100	1	4	yes	complemented
239	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
240	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
241	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
242	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
243	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
244	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
245	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
246	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
247	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
248	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
249	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
250	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
251	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
252	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
253	C₈×C₃₁	—	4	4	no	—
254	C₈×C₃₁	—	4	4	no	—
255	C₂×#532	C₂×C₃₁	1	1	yes	characteristic
256	#2920	C₂×C₃₁	1	1	yes	characteristic, complemented
257	C₂×#423	C₂×C₃₁	1	1	yes	characteristic, complemented
258	#1997	C₂×C₃₁	1	2	yes	complemented
259	#1965	C₂×C₃₁	1	2	yes	complemented
260	#1900	C₂×C₃₁	1	4	yes	complemented
261	#1800	C₂×C₃₁	1	4	yes	complemented
262	C₂²×C₃₁	—	2	1	no	—
263	C₂²×C₃₁	—	2	1	no	—
264	C₂²×C₃₁	—	2	1	no	—
265	C₂²×C₃₁	—	2	1	no	—
266	C₂²×C₃₁	—	2	1	no	—
267	C₂²×C₃₁	—	2	1	no	—
268	C₂²×C₃₁	—	2	1	no	—
269	C₂²×C₃₁	—	4	4	no	—
270	C₂²×C₃₁	—	4	4	no	—
271	C₂²×C₃₁	—	4	4	no	—
272	C₂²×C₃₁	—	4	4	no	—
273	C₂²×C₃₁	—	8	8	no	—
274	C₂²×C₃₁	C₂×#100	1	2	yes	—
275	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
276	C₂²×C₃₁	#520	1	2	yes	—
277	C₄×C₃₁	—	2	1	no	—
278	C₄×C₃₁	—	2	1	no	—
279	C₄×C₃₁	—	2	1	no	—
280	C₄×C₃₁	—	2	1	no	—
281	C₄×C₃₁	—	2	1	no	—
282	C₄×C₃₁	—	2	1	no	—
283	C₄×C₃₁	—	2	1	no	—
284	C₄×C₃₁	—	4	4	no	—
285	C₄×C₃₁	C₂×#100	1	2	yes	—
286	C₄×C₃₁	C₂×#100	1	2	yes	—
287	C₄×C₃₁	#413	1	2	yes	—
288	C₄×C₃₁	#535	1	1	yes	characteristic
289	C₄×C₃₁	#536	1	1	yes	characteristic
290	C₄×C₃₁	#537	1	2	yes	—
291	C₂²×#41	C₄×C₃₁	1	2	yes	—
292	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
293	#532	—	2	1	no	—
294	#532	—	2	1	no	—
295	#532	—	2	1	no	—
296	#532	—	2	1	no	—
297	#532	—	2	1	no	—
298	#532	—	2	1	no	—
299	#532	—	2	1	no	—
300	#532	—	2	1	no	—
301	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
302	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
303	C₂×#124	C₄×C₃₁	1	2	yes	—
304	C₂×#121	C₄×C₃₁	1	2	yes	—
305	C₂×#121	C₂²×C₃₁	1	2	yes	—
306	C₂×C₄×D₄	C₄×C₃₁	1	2	yes	—
307	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
308	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
309	#432	—	2	1	no	—
310	#432	—	2	1	no	—
311	#431	—	2	1	no	—
312	#431	—	2	1	no	—
313	#423	—	2	1	no	—
314	#423	—	2	1	no	—
315	#423	C₂²×C₃₁	1	2	yes	—
316	#423	C₂²×C₃₁	1	2	yes	—
317	#423	C₂²×C₃₁	1	2	yes	—
318	C₂×C₄⋊C₈	—	2	1	no	—
319	C₂×C₄⋊C₈	—	2	1	no	—
320	C₂×#100	C₂²×C₃₁	1	8	yes	—
321	#407	C₂²×C₃₁	1	2	yes	—
322	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
323	#343	C₂²×C₃₁	1	8	yes	—
324	C₂×C₃₁	—	2	2	no	—
325	C₂×C₃₁	—	2	2	no	—
326	C₂×C₃₁	—	4	1	no	—
327	C₂×C₃₁	—	8	4	no	—
328	C₂×C₃₁	#2311	1	1	yes	characteristic
329	C₂×C₃₁	C₂²×#100	1	1	yes	characteristic
330	C₂×C₃₁	C₂×#413	1	1	yes	characteristic
331	C₂×#41	—	2	2	no	—
332	C₂×#41	—	2	2	no	—
333	C₂×#41	—	2	2	no	—
334	C₂×#41	—	2	2	no	—
335	C₂×#41	—	4	8	no	—
336	C₂²×Q₈	C₂×C₄×C₃₁	1	2	yes	—
337	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
338	C₂²×D₄	C₂×C₄×C₃₁	1	4	yes	—
339	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
340	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
341	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
342	C₂³×C₄	C₂×C₄×C₃₁	1	4	yes	—
343	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
344	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
345	C₂×M₁₆	—	2	2	no	—
346	C₂×M₁₆	—	4	4	no	—
347	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
348	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
349	C₂²×C₈	—	2	2	no	—
350	C₄⋊Q₈	—	2	1	no	—
351	C₄⋊Q₈	—	2	1	no	—
352	C₄⋊Q₈	—	2	1	no	—
353	C₄⋊Q₈	—	2	1	no	—
354	#127	—	2	1	no	—
355	#127	—	2	1	no	—
356	#127	—	2	1	no	—
357	#127	—	2	1	no	—
358	#124	—	2	2	no	—
359	#124	—	2	2	no	—
360	#124	—	2	2	no	—
361	#124	—	2	2	no	—
362	#121	—	2	2	no	—
363	#121	—	2	2	no	—
364	#121	—	2	2	no	—
365	#121	—	2	2	no	—
366	#121	—	2	2	no	—
367	#121	—	2	2	no	—
368	#121	—	2	2	no	—
369	#121	—	2	2	no	—
370	C₄×D₄	—	2	2	no	—
371	C₄×D₄	—	2	2	no	—
372	C₄×D₄	—	2	2	no	—
373	C₄×D₄	—	2	2	no	—
374	C₄×D₄	—	2	2	no	—
375	C₄×D₄	—	2	2	no	—
376	C₄×D₄	—	2	2	no	—
377	C₄×D₄	—	2	2	no	—
378	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
379	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
380	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
381	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
382	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
383	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
384	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
385	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
386	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
387	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
388	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
389	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
390	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
391	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
392	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
393	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
394	C₄⋊C₈	—	2	2	no	—
395	C₄⋊C₈	—	2	2	no	—
396	C₄⋊C₈	—	2	2	no	—
397	C₄⋊C₈	—	2	2	no	—
398	C₄⋊C₈	—	2	2	no	—
399	C₄⋊C₈	—	2	2	no	—
400	#100	—	4	8	no	—
401	C₂²⋊C₈	—	2	2	no	—
402	C₂²⋊C₈	—	2	2	no	—
403	#97	—	2	2	no	—
404	C₄×C₈	—	2	2	no	—
405	C₃₁	#21878	1	1	yes	characteristic, direct product factor, Sylow
406	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
407	C₂⁴	C₂²×C₄×C₃₁	1	4	yes	—
408	#41	—	4	8	no	—
409	#41	—	4	8	no	—
410	#41	—	4	8	no	—
411	#41	—	4	8	no	—
412	C₂×Q₈	—	2	2	no	—
413	C₂×Q₈	—	2	2	no	—
414	C₂×Q₈	—	4	4	no	—
415	C₂×D₄	—	2	4	no	—
416	C₂×D₄	—	2	4	no	—
417	C₂×D₄	—	2	2	no	—
418	C₂×D₄	—	2	2	no	—
419	C₂×D₄	—	2	2	no	—
420	C₂×D₄	—	2	2	no	—
421	C₂×D₄	—	2	2	no	—
422	C₂×D₄	—	2	2	no	—
423	C₂×D₄	—	2	4	no	—
424	C₂×D₄	—	2	4	no	—
425	C₂×D₄	—	4	4	no	—
426	C₂×D₄	—	4	4	no	—
427	C₂×D₄	—	4	8	no	—
428	C₂²×C₄	—	2	4	no	—
429	C₂²×C₄	—	2	2	no	—
430	C₂²×C₄	—	2	2	no	—
431	C₂²×C₄	—	2	4	no	—
432	C₂²×C₄	—	2	4	no	—
433	C₂²×C₄	—	4	2	no	—
434	C₂²×C₄	—	4	8	no	—
435	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
436	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
437	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
438	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
439	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
440	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
441	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
442	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
443	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
444	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
445	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
446	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
447	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
448	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
449	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
450	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
451	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
452	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
453	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
454	M₁₆	—	2	4	no	—
455	M₁₆	—	2	8	no	—
456	M₁₆	—	2	4	no	—
457	M₁₆	—	4	4	no	—
458	C₂×C₈	—	2	4	no	—
459	C₂×C₈	—	2	4	no	—
460	C₂×C₈	—	4	4	no	—
461	C₄⋊C₄	—	2	2	no	—
462	C₄⋊C₄	—	2	2	no	—
463	C₄⋊C₄	—	2	2	no	—
464	C₄⋊C₄	—	2	2	no	—
465	C₂²⋊C₄	—	2	4	no	—
466	C₂²⋊C₄	—	2	4	no	—
467	C₂²⋊C₄	—	2	4	no	—
468	C₂²⋊C₄	—	2	4	no	—
469	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
470	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
471	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
472	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
473	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
474	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
475	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
476	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
477	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
478	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
479	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
480	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
481	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
482	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
483	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
484	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
485	C₂³	—	2	2	no	—
486	C₂³	—	2	2	no	—
487	C₂³	—	2	4	no	—
488	C₂³	—	2	8	no	—
489	C₂³	—	4	1	no	—
490	C₂³	—	4	4	no	—
491	C₂³	—	4	4	no	—
492	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
493	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
494	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
495	Q₈	—	4	4	no	—
496	Q₈	—	4	4	no	—
497	D₄	—	4	4	no	—
498	D₄	—	4	4	no	—
499	D₄	—	4	8	no	—
500	D₄	—	4	8	no	—
501	D₄	—	4	4	no	—
502	D₄	—	4	4	no	—
503	D₄	—	4	2	no	—
504	D₄	—	4	2	no	—
505	D₄	—	4	4	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	2	no	—
508	C₂×C₄	—	2	2	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	2	no	—
515	C₂×C₄	—	2	1	no	—
516	C₂×C₄	—	2	1	no	—
517	C₂×C₄	—	2	1	no	—
518	C₂×C₄	—	2	1	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	2	4	no	—
522	C₂×C₄	—	4	1	no	—
523	C₂×C₄	—	4	1	no	—
524	C₂×C₄	—	4	1	no	—
525	C₂×C₄	—	4	8	no	—
526	C₂×C₄	—	4	8	no	—
527	C₂×C₄	—	4	4	no	—
528	C₂×C₄	C₃₁×#100	1	4	yes	complemented
529	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
530	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
531	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
532	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
533	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
534	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
535	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
536	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
537	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
538	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
539	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
540	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
541	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
542	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
543	C₈	—	4	4	no	—
544	C₈	—	4	4	no	—
545	C₂²	—	2	1	no	—
546	C₂²	—	2	1	no	—
547	C₂²	—	2	1	no	—
548	C₂²	—	2	1	no	—
549	C₂²	—	2	1	no	—
550	C₂²	—	2	1	no	—
551	C₂²	—	2	1	no	—
552	C₂²	—	4	4	no	—
553	C₂²	—	4	4	no	—
554	C₂²	—	4	4	no	—
555	C₂²	—	4	4	no	—
556	C₂²	—	8	8	no	—
557	C₂²	C₂×C₃₁×#100	1	2	yes	—
558	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
559	C₂²	C₃₁×#520	1	2	yes	—
560	C₄	—	2	1	no	—
561	C₄	—	2	1	no	—
562	C₄	—	2	1	no	—
563	C₄	—	2	1	no	—
564	C₄	—	2	1	no	—
565	C₄	—	2	1	no	—
566	C₄	—	2	1	no	—
567	C₄	—	4	4	no	—
568	C₄	C₂×C₃₁×#100	1	2	yes	—
569	C₄	C₂×C₃₁×#100	1	2	yes	—
570	C₄	C₃₁×#413	1	2	yes	—
571	C₄	C₃₁×#535	1	1	yes	characteristic
572	C₄	C₃₁×#536	1	1	yes	characteristic
573	C₄	C₃₁×#537	1	2	yes	—
574	C₂	—	2	2	no	—
575	C₂	—	2	2	no	—
576	C₂	—	4	1	no	—
577	C₂	—	8	4	no	—
578	C₂	C₃₁×#2311	1	1	yes	characteristic
579	C₂	C₂²×C₃₁×#100	1	1	yes	characteristic
580	C₂	C₂×C₃₁×#413	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#532	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2920	C₂	semidirect
3. C₂×C₃₁×#423	C₂	semidirect
4. C₃₁×#1997	C₂	semidirect
5. C₃₁×#1965	C₂	semidirect
6. C₃₁×#1900	C₂	semidirect
7. C₃₁×#1800	C₂	semidirect
194. #21878	C₃₁	direct (N × H)
238. C₂×C₄×C₃₁	#100	semidirect
256. #2920	C₂×C₃₁	semidirect
257. C₂×#423	C₂×C₃₁	semidirect
258. #1997	C₂×C₃₁	semidirect
259. #1965	C₂×C₃₁	semidirect
260. #1900	C₂×C₃₁	semidirect
261. #1800	C₂×C₃₁	semidirect
405. C₃₁	#21878	direct (N × H)
528. C₂×C₄	C₃₁×#100	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#532	2
C₃₁×#2920	2
C₂×C₃₁×#423	2
C₃₁×#1997	2
C₃₁×#1965	2
C₃₁×#1900	2
C₃₁×#1800	2
#21878	31

Subgroups — C31×#21878 #21478637

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21878 #21478637