Subgroups — C₃₁×#33516 #21490275

Order 7936.

1150 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#135	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#508	C₂	1	2	yes	maximal, complemented
3	C₃₁×#3050	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2189	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2188	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2187	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2186	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2185	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#2184	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2138	C₂	1	2	yes	maximal, complemented
11	C₂×C₃₁×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₃₁	—	2	1	no	—
13	C₂×C₃₁×#135	—	2	8	no	—
14	C₂×C₃₁×#135	—	2	1	no	—
15	C₂×C₃₁×#135	—	2	1	no	—
16	C₂×C₃₁×#135	C₂²	1	2	yes	—
17	C₂×C₃₁×#135	C₂²	1	2	yes	—
18	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic, complemented
19	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
20	C₂²×SD₁₆×C₃₁	—	2	1	no	—
21	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₃₁×#525	—	2	1	no	—
24	C₃₁×#508	—	2	8	no	—
25	C₂×C₃₁×Q₃₂	C₂²	1	2	yes	complemented
26	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	complemented
27	C₂×C₃₁×SD₃₂	C₂²	1	2	yes	complemented
28	C₂×C₃₁×D₁₆	C₂²	1	2	yes	complemented
29	C₂²×C₁₆×C₃₁	C₂²	1	2	yes	complemented
30	C₃₁×#467	C₂²	1	1	yes	characteristic
31	C₃₁×#467	C₂²	1	1	yes	characteristic, complemented
32	C₃₁×#466	C₂²	1	1	yes	characteristic
33	C₃₁×#466	C₂²	1	1	yes	characteristic, complemented
34	C₃₁×#465	—	2	1	no	—
35	C₃₁×#465	—	2	1	no	—
36	C₃₁×#427	C₂²	1	1	yes	characteristic, complemented
37	C₃₁×#416	—	2	1	no	—
38	C₃₁×#358	C₂²	1	2	yes	complemented
39	C₃₁×#358	C₂²	1	2	yes	complemented
40	C₃₁×#357	C₂²	1	2	yes	complemented
41	C₃₁×#357	C₂²	1	2	yes	complemented
42	C₃₁×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
43	C₃₁×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
44	C₃₁×C₂²⋊C₁₆	C₂²	1	2	yes	complemented
45	C₂×#41×C₃₁	—	2	8	no	—
46	C₂×#41×C₃₁	—	2	2	no	—
47	C₂×#41×C₃₁	—	2	2	no	—
48	C₂×#41×C₃₁	—	2	1	no	—
49	C₂×#41×C₃₁	—	4	4	no	—
50	C₂×#41×C₃₁	—	4	1	no	—
51	C₂×#41×C₃₁	—	4	1	no	—
52	C₂²×Q₈×C₃₁	—	2	1	no	—
53	C₂²×D₄×C₃₁	—	2	1	no	—
54	C₂²×D₄×C₃₁	—	4	1	no	—
55	C₂³×C₄×C₃₁	—	4	1	no	—
56	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
57	C₃₁×#135	—	2	8	no	—
58	C₃₁×#135	—	2	8	no	—
59	C₃₁×#135	—	2	4	no	—
60	C₃₁×#135	—	2	4	no	—
61	C₃₁×#135	—	2	4	no	—
62	C₃₁×#135	—	2	4	no	—
63	C₃₁×#135	D₄	1	8	yes	complemented
64	C₂×Q₁₆×C₃₁	—	2	4	no	—
65	C₂×Q₁₆×C₃₁	—	2	1	no	—
66	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
67	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
68	C₂×SD₁₆×C₃₁	—	2	8	no	—
69	C₂×SD₁₆×C₃₁	—	2	2	no	—
70	C₂×SD₁₆×C₃₁	—	2	2	no	—
71	C₂×SD₁₆×C₃₁	—	2	1	no	—
72	C₂×D₈×C₃₁	—	2	4	no	—
73	C₂×D₈×C₃₁	—	2	1	no	—
74	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
75	C₂×D₈×C₃₁	C₂³	1	2	yes	—
76	C₂²×C₈×C₃₁	—	2	4	no	—
77	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
78	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
79	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
80	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
81	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
82	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
83	C₃₁×#123	—	4	1	no	—
84	C₃₁×C₂²⋊Q₈	—	2	1	no	—
85	C₃₁×C₂²⋊Q₈	—	2	1	no	—
86	C₃₁×#121	—	2	1	no	—
87	C₃₁×#121	—	2	1	no	—
88	C₃₁×#120	—	4	1	no	—
89	C₄×D₄×C₃₁	—	4	1	no	—
90	C₄×D₄×C₃₁	—	4	1	no	—
91	C₃₁×#117	—	2	1	no	—
92	C₃₁×Q₃₂	—	2	4	no	—
93	C₃₁×SD₃₂	—	2	4	no	—
94	C₃₁×SD₃₂	—	2	4	no	—
95	C₃₁×D₁₆	—	2	4	no	—
96	C₂×C₁₆×C₃₁	—	2	4	no	—
97	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
98	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
99	C₃₁×#107	C₂³	1	1	yes	characteristic
100	C₃₁×#107	C₂³	1	1	yes	characteristic
101	C₃₁×#106	—	2	1	no	—
102	C₃₁×#103	—	2	1	no	—
103	C₃₁×#103	—	2	1	no	—
104	C₃₁×#102	—	2	1	no	—
105	C₃₁×#102	—	2	1	no	—
106	C₂⁴×C₃₁	—	4	1	no	—
107	C₂⁴×C₃₁	D₈	1	1	yes	characteristic
108	#41×C₃₁	—	2	8	no	—
109	#41×C₃₁	—	2	8	no	—
110	#41×C₃₁	—	4	8	no	—
111	#41×C₃₁	—	4	4	no	—
112	#41×C₃₁	—	4	8	no	—
113	#41×C₃₁	—	4	4	no	—
114	#41×C₃₁	—	4	2	no	—
115	#41×C₃₁	—	4	2	no	—
116	C₂×Q₈×C₃₁	—	2	4	no	—
117	C₂×Q₈×C₃₁	—	2	2	no	—
118	C₂×Q₈×C₃₁	—	2	1	no	—
119	C₂×Q₈×C₃₁	—	4	2	no	—
120	C₂×Q₈×C₃₁	—	4	1	no	—
121	C₂×D₄×C₃₁	—	2	4	no	—
122	C₂×D₄×C₃₁	—	2	2	no	—
123	C₂×D₄×C₃₁	—	2	1	no	—
124	C₂×D₄×C₃₁	—	4	2	no	—
125	C₂×D₄×C₃₁	—	4	8	no	—
126	C₂×D₄×C₃₁	—	4	2	no	—
127	C₂×D₄×C₃₁	—	4	2	no	—
128	C₂×D₄×C₃₁	—	4	1	no	—
129	C₂×D₄×C₃₁	—	4	1	no	—
130	C₂×D₄×C₃₁	—	4	1	no	—
131	C₂×D₄×C₃₁	—	4	1	no	—
132	C₂²×C₄×C₃₁	—	2	2	no	—
133	C₂²×C₄×C₃₁	—	2	2	no	—
134	C₂²×C₄×C₃₁	—	4	8	no	—
135	C₂²×C₄×C₃₁	—	4	1	no	—
136	C₂²×C₄×C₃₁	—	4	2	no	—
137	C₂²×C₄×C₃₁	—	4	2	no	—
138	C₂²×C₄×C₃₁	—	4	1	no	—
139	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
140	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
141	C₂²×C₄×C₃₁	D₈	1	1	yes	characteristic
142	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
144	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
145	Q₁₆×C₃₁	—	2	4	no	—
146	Q₁₆×C₃₁	—	2	2	no	—
147	Q₁₆×C₃₁	—	2	2	no	—
148	Q₁₆×C₃₁	C₂×D₄	1	4	yes	—
149	SD₁₆×C₃₁	—	2	8	no	—
150	SD₁₆×C₃₁	—	2	8	no	—
151	SD₁₆×C₃₁	—	4	2	no	—
152	D₈×C₃₁	—	2	4	no	—
153	D₈×C₃₁	—	2	2	no	—
154	D₈×C₃₁	—	2	2	no	—
155	D₈×C₃₁	C₂×D₄	1	4	yes	—
156	C₂×C₈×C₃₁	—	2	2	no	—
157	C₂×C₈×C₃₁	—	2	2	no	—
158	C₂×C₈×C₃₁	—	2	2	no	—
159	C₂×C₈×C₃₁	—	2	2	no	—
160	C₂×C₈×C₃₁	—	2	2	no	—
161	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
162	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
163	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
164	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
165	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
166	C₄⋊C₄×C₃₁	—	2	1	no	—
167	C₄⋊C₄×C₃₁	—	2	1	no	—
168	C₄⋊C₄×C₃₁	—	4	1	no	—
169	C₄⋊C₄×C₃₁	—	4	1	no	—
170	C₂²⋊C₄×C₃₁	—	4	1	no	—
171	C₂²⋊C₄×C₃₁	—	4	1	no	—
172	C₂²⋊C₄×C₃₁	—	4	1	no	—
173	C₂²⋊C₄×C₃₁	—	4	1	no	—
174	C₂²⋊C₄×C₃₁	—	4	1	no	—
175	C₄²×C₃₁	—	4	1	no	—
176	C₁₆×C₃₁	—	2	2	no	—
177	C₁₆×C₃₁	—	2	2	no	—
178	#33516	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
179	C₂³×C₃₁	—	2	2	no	—
180	C₂³×C₃₁	—	2	2	no	—
181	C₂³×C₃₁	—	4	1	no	—
182	C₂³×C₃₁	—	4	4	no	—
183	C₂³×C₃₁	—	4	2	no	—
184	C₂³×C₃₁	—	4	1	no	—
185	C₂³×C₃₁	—	8	2	no	—
186	C₂³×C₃₁	—	8	2	no	—
187	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
188	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
189	C₂³×C₃₁	C₂×D₈	1	1	yes	characteristic
190	Q₈×C₃₁	—	2	4	no	—
191	Q₈×C₃₁	—	4	4	no	—
192	Q₈×C₃₁	—	4	2	no	—
193	Q₈×C₃₁	—	4	1	no	—
194	Q₈×C₃₁	—	4	1	no	—
195	D₄×C₃₁	—	2	4	no	—
196	D₄×C₃₁	—	4	4	no	—
197	D₄×C₃₁	—	4	8	no	—
198	D₄×C₃₁	—	4	8	no	—
199	D₄×C₃₁	—	4	1	no	—
200	D₄×C₃₁	—	4	2	no	—
201	D₄×C₃₁	—	4	1	no	—
202	D₄×C₃₁	—	8	2	no	—
203	D₄×C₃₁	—	8	2	no	—
204	D₄×C₃₁	—	8	2	no	—
205	C₂×C₄×C₃₁	—	2	2	no	—
206	C₂×C₄×C₃₁	—	2	1	no	—
207	C₂×C₄×C₃₁	—	2	1	no	—
208	C₂×C₄×C₃₁	—	2	1	no	—
209	C₂×C₄×C₃₁	—	2	1	no	—
210	C₂×C₄×C₃₁	—	2	2	no	—
211	C₂×C₄×C₃₁	—	2	2	no	—
212	C₂×C₄×C₃₁	—	4	8	no	—
213	C₂×C₄×C₃₁	—	4	4	no	—
214	C₂×C₄×C₃₁	—	4	2	no	—
215	C₂×C₄×C₃₁	—	4	8	no	—
216	C₂×C₄×C₃₁	—	4	1	no	—
217	C₂×C₄×C₃₁	—	4	1	no	—
218	C₂×C₄×C₃₁	—	4	1	no	—
219	C₂×C₄×C₃₁	—	8	2	no	—
220	C₂×C₄×C₃₁	#120	1	2	yes	—
221	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
222	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
223	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
224	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
225	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
226	C₈×C₃₁	—	2	1	no	—
227	C₈×C₃₁	—	2	1	no	—
228	C₈×C₃₁	#120	1	2	yes	—
229	C₈×C₃₁	C₂²×D₄	1	2	yes	—
230	C₂²×#135	C₂×C₃₁	1	1	yes	characteristic, complemented
231	C₂×#508	C₂×C₃₁	1	2	yes	complemented
232	#3050	C₂×C₃₁	1	1	yes	characteristic, complemented
233	#2189	C₂×C₃₁	1	2	yes	complemented
234	#2188	C₂×C₃₁	1	1	yes	characteristic, complemented
235	#2187	C₂×C₃₁	1	1	yes	characteristic, complemented
236	#2186	C₂×C₃₁	1	2	yes	complemented
237	#2185	C₂×C₃₁	1	1	yes	characteristic, complemented
238	#2184	C₂×C₃₁	1	1	yes	characteristic, complemented
239	#2138	C₂×C₃₁	1	2	yes	complemented
240	C₂×C₂²⋊C₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
241	C₂²×C₃₁	—	2	1	no	—
242	C₂²×C₃₁	—	2	1	no	—
243	C₂²×C₃₁	—	2	1	no	—
244	C₂²×C₃₁	—	2	1	no	—
245	C₂²×C₃₁	—	2	1	no	—
246	C₂²×C₃₁	—	2	1	no	—
247	C₂²×C₃₁	—	2	1	no	—
248	C₂²×C₃₁	—	4	4	no	—
249	C₂²×C₃₁	—	4	4	no	—
250	C₂²×C₃₁	—	8	8	no	—
251	C₂²×C₃₁	—	8	2	no	—
252	C₂²×C₃₁	—	8	2	no	—
253	C₂²×C₃₁	—	8	1	no	—
254	C₂²×C₃₁	—	8	1	no	—
255	C₂²×C₃₁	—	8	1	no	—
256	C₂²×C₃₁	#447	1	2	yes	—
257	C₂²×C₃₁	#508	1	2	yes	complemented
258	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
259	C₄×C₃₁	—	2	1	no	—
260	C₄×C₃₁	—	2	1	no	—
261	C₄×C₃₁	—	4	4	no	—
262	C₄×C₃₁	—	8	1	no	—
263	C₄×C₃₁	—	8	1	no	—
264	C₄×C₃₁	—	8	1	no	—
265	C₄×C₃₁	#447	1	2	yes	—
266	C₄×C₃₁	C₂×#120	1	1	yes	characteristic
267	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
268	C₂²×#41	—	2	1	no	—
269	C₂×#135	—	2	8	no	—
270	C₂×#135	—	2	1	no	—
271	C₂×#135	—	2	1	no	—
272	C₂×#135	C₂²×C₃₁	1	2	yes	—
273	C₂×#135	C₂²×C₃₁	1	2	yes	—
274	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic, complemented
275	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
276	C₂²×SD₁₆	—	2	1	no	—
277	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
278	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
279	#525	—	2	1	no	—
280	#508	—	2	8	no	—
281	C₂×Q₃₂	C₂²×C₃₁	1	2	yes	complemented
282	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	complemented
283	C₂×SD₃₂	C₂²×C₃₁	1	2	yes	complemented
284	C₂×D₁₆	C₂²×C₃₁	1	2	yes	complemented
285	C₂²×C₁₆	C₂²×C₃₁	1	2	yes	complemented
286	#467	C₂²×C₃₁	1	1	yes	characteristic
287	#467	C₂²×C₃₁	1	1	yes	characteristic, complemented
288	#466	C₂²×C₃₁	1	1	yes	characteristic
289	#466	C₂²×C₃₁	1	1	yes	characteristic, complemented
290	#465	—	2	1	no	—
291	#465	—	2	1	no	—
292	#427	C₂²×C₃₁	1	1	yes	characteristic, complemented
293	#416	—	2	1	no	—
294	#358	C₂²×C₃₁	1	2	yes	complemented
295	#358	C₂²×C₃₁	1	2	yes	complemented
296	#357	C₂²×C₃₁	1	2	yes	complemented
297	#357	C₂²×C₃₁	1	2	yes	complemented
298	C₂²⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
299	C₂²⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
300	C₂²⋊C₁₆	C₂²×C₃₁	1	2	yes	complemented
301	C₂×C₃₁	—	2	2	no	—
302	C₂×C₃₁	—	2	2	no	—
303	C₂×C₃₁	—	4	1	no	—
304	C₂×C₃₁	—	8	4	no	—
305	C₂×C₃₁	—	16	1	no	—
306	C₂×C₃₁	C₂×#447	1	1	yes	characteristic
307	C₂×C₃₁	C₂×#508	1	1	yes	characteristic
308	C₂×C₃₁	#3414	1	1	yes	characteristic
309	C₂×#41	—	2	8	no	—
310	C₂×#41	—	2	2	no	—
311	C₂×#41	—	2	2	no	—
312	C₂×#41	—	2	1	no	—
313	C₂×#41	—	4	4	no	—
314	C₂×#41	—	4	1	no	—
315	C₂×#41	—	4	1	no	—
316	C₂²×Q₈	—	2	1	no	—
317	C₂²×D₄	—	2	1	no	—
318	C₂²×D₄	—	4	1	no	—
319	C₂³×C₄	—	4	1	no	—
320	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
321	#135	—	2	8	no	—
322	#135	—	2	8	no	—
323	#135	—	2	4	no	—
324	#135	—	2	4	no	—
325	#135	—	2	4	no	—
326	#135	—	2	4	no	—
327	#135	D₄×C₃₁	1	8	yes	complemented
328	C₂×Q₁₆	—	2	4	no	—
329	C₂×Q₁₆	—	2	1	no	—
330	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
331	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
332	C₂×SD₁₆	—	2	8	no	—
333	C₂×SD₁₆	—	2	2	no	—
334	C₂×SD₁₆	—	2	2	no	—
335	C₂×SD₁₆	—	2	1	no	—
336	C₂×D₈	—	2	4	no	—
337	C₂×D₈	—	2	1	no	—
338	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
339	C₂×D₈	C₂³×C₃₁	1	2	yes	—
340	C₂²×C₈	—	2	4	no	—
341	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
342	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
343	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
344	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
345	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
346	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
347	#123	—	4	1	no	—
348	C₂²⋊Q₈	—	2	1	no	—
349	C₂²⋊Q₈	—	2	1	no	—
350	#121	—	2	1	no	—
351	#121	—	2	1	no	—
352	#120	—	4	1	no	—
353	C₄×D₄	—	4	1	no	—
354	C₄×D₄	—	4	1	no	—
355	#117	—	2	1	no	—
356	Q₃₂	—	2	4	no	—
357	SD₃₂	—	2	4	no	—
358	SD₃₂	—	2	4	no	—
359	D₁₆	—	2	4	no	—
360	C₂×C₁₆	—	2	4	no	—
361	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
362	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
363	#107	C₂³×C₃₁	1	1	yes	characteristic
364	#107	C₂³×C₃₁	1	1	yes	characteristic
365	#106	—	2	1	no	—
366	#103	—	2	1	no	—
367	#103	—	2	1	no	—
368	#102	—	2	1	no	—
369	#102	—	2	1	no	—
370	C₃₁	#33516	1	1	yes	characteristic, direct product factor, Sylow
371	C₂⁴	—	4	1	no	—
372	C₂⁴	D₈×C₃₁	1	1	yes	characteristic
373	#41	—	2	8	no	—
374	#41	—	2	8	no	—
375	#41	—	4	8	no	—
376	#41	—	4	8	no	—
377	#41	—	4	4	no	—
378	#41	—	4	4	no	—
379	#41	—	4	2	no	—
380	#41	—	4	2	no	—
381	C₂×Q₈	—	2	4	no	—
382	C₂×Q₈	—	2	2	no	—
383	C₂×Q₈	—	2	1	no	—
384	C₂×Q₈	—	4	2	no	—
385	C₂×Q₈	—	4	1	no	—
386	C₂×D₄	—	2	4	no	—
387	C₂×D₄	—	2	2	no	—
388	C₂×D₄	—	2	1	no	—
389	C₂×D₄	—	4	2	no	—
390	C₂×D₄	—	4	8	no	—
391	C₂×D₄	—	4	2	no	—
392	C₂×D₄	—	4	2	no	—
393	C₂×D₄	—	4	1	no	—
394	C₂×D₄	—	4	1	no	—
395	C₂×D₄	—	4	1	no	—
396	C₂×D₄	—	4	1	no	—
397	C₂²×C₄	—	2	2	no	—
398	C₂²×C₄	—	2	2	no	—
399	C₂²×C₄	—	4	8	no	—
400	C₂²×C₄	—	4	1	no	—
401	C₂²×C₄	—	4	2	no	—
402	C₂²×C₄	—	4	2	no	—
403	C₂²×C₄	—	4	1	no	—
404	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
405	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
406	C₂²×C₄	D₈×C₃₁	1	1	yes	characteristic
407	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
408	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
409	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
410	Q₁₆	—	2	4	no	—
411	Q₁₆	—	2	2	no	—
412	Q₁₆	—	2	2	no	—
413	Q₁₆	C₂×D₄×C₃₁	1	4	yes	—
414	SD₁₆	—	2	8	no	—
415	SD₁₆	—	2	8	no	—
416	SD₁₆	—	4	2	no	—
417	D₈	—	2	4	no	—
418	D₈	—	2	2	no	—
419	D₈	—	2	2	no	—
420	D₈	C₂×D₄×C₃₁	1	4	yes	—
421	C₂×C₈	—	2	2	no	—
422	C₂×C₈	—	2	2	no	—
423	C₂×C₈	—	2	2	no	—
424	C₂×C₈	—	2	2	no	—
425	C₂×C₈	—	2	2	no	—
426	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
427	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
428	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
429	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
430	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
431	C₄⋊C₄	—	2	1	no	—
432	C₄⋊C₄	—	2	1	no	—
433	C₄⋊C₄	—	4	1	no	—
434	C₄⋊C₄	—	4	1	no	—
435	C₂²⋊C₄	—	4	1	no	—
436	C₂²⋊C₄	—	4	1	no	—
437	C₂²⋊C₄	—	4	1	no	—
438	C₂²⋊C₄	—	4	1	no	—
439	C₂²⋊C₄	—	4	1	no	—
440	C₄²	—	4	1	no	—
441	C₁₆	—	2	2	no	—
442	C₁₆	—	2	2	no	—
443	C₂³	—	2	2	no	—
444	C₂³	—	2	2	no	—
445	C₂³	—	4	1	no	—
446	C₂³	—	4	4	no	—
447	C₂³	—	4	2	no	—
448	C₂³	—	4	1	no	—
449	C₂³	—	8	2	no	—
450	C₂³	—	8	2	no	—
451	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
452	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
453	C₂³	C₂×D₈×C₃₁	1	1	yes	characteristic
454	Q₈	—	2	4	no	—
455	Q₈	—	4	4	no	—
456	Q₈	—	4	2	no	—
457	Q₈	—	4	1	no	—
458	Q₈	—	4	1	no	—
459	D₄	—	2	4	no	—
460	D₄	—	4	4	no	—
461	D₄	—	4	8	no	—
462	D₄	—	4	8	no	—
463	D₄	—	4	1	no	—
464	D₄	—	4	2	no	—
465	D₄	—	4	1	no	—
466	D₄	—	8	2	no	—
467	D₄	—	8	2	no	—
468	D₄	—	8	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	1	no	—
474	C₂×C₄	—	2	1	no	—
475	C₂×C₄	—	2	1	no	—
476	C₂×C₄	—	4	8	no	—
477	C₂×C₄	—	4	4	no	—
478	C₂×C₄	—	4	2	no	—
479	C₂×C₄	—	4	8	no	—
480	C₂×C₄	—	4	1	no	—
481	C₂×C₄	—	4	1	no	—
482	C₂×C₄	—	4	1	no	—
483	C₂×C₄	—	8	2	no	—
484	C₂×C₄	C₃₁×#120	1	2	yes	—
485	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
486	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
487	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
488	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
489	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
490	C₈	—	2	1	no	—
491	C₈	—	2	1	no	—
492	C₈	C₃₁×#120	1	2	yes	—
493	C₈	C₂²×D₄×C₃₁	1	2	yes	—
494	C₂²	—	2	1	no	—
495	C₂²	—	2	1	no	—
496	C₂²	—	2	1	no	—
497	C₂²	—	2	1	no	—
498	C₂²	—	2	1	no	—
499	C₂²	—	2	1	no	—
500	C₂²	—	2	1	no	—
501	C₂²	—	4	4	no	—
502	C₂²	—	4	4	no	—
503	C₂²	—	8	8	no	—
504	C₂²	—	8	2	no	—
505	C₂²	—	8	2	no	—
506	C₂²	—	8	1	no	—
507	C₂²	—	8	1	no	—
508	C₂²	—	8	1	no	—
509	C₂²	C₃₁×#447	1	2	yes	—
510	C₂²	C₃₁×#508	1	2	yes	complemented
511	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
512	C₄	—	2	1	no	—
513	C₄	—	2	1	no	—
514	C₄	—	4	4	no	—
515	C₄	—	8	1	no	—
516	C₄	—	8	1	no	—
517	C₄	—	8	1	no	—
518	C₄	C₃₁×#447	1	2	yes	—
519	C₄	C₂×C₃₁×#120	1	1	yes	characteristic
520	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
521	C₂	—	2	2	no	—
522	C₂	—	2	2	no	—
523	C₂	—	4	1	no	—
524	C₂	—	8	4	no	—
525	C₂	—	16	1	no	—
526	C₂	C₂×C₃₁×#447	1	1	yes	characteristic
527	C₂	C₂×C₃₁×#508	1	1	yes	characteristic
528	C₂	C₃₁×#3414	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃₁×#135	C₂	semidirect
2. C₂×C₃₁×#508	C₂	semidirect
3. C₃₁×#3050	C₂	semidirect
4. C₃₁×#2189	C₂	semidirect
5. C₃₁×#2188	C₂	semidirect
6. C₃₁×#2187	C₂	semidirect
7. C₃₁×#2186	C₂	semidirect
8. C₃₁×#2185	C₂	semidirect
9. C₃₁×#2184	C₂	semidirect
10. C₃₁×#2138	C₂	semidirect
11. C₂×C₃₁×C₂²⋊C₁₆	C₂	semidirect
18. C₂×C₃₁×#135	C₂²	semidirect
25. C₂×C₃₁×Q₃₂	C₂²	semidirect
26. C₂×C₃₁×SD₃₂	C₂²	semidirect
27. C₂×C₃₁×SD₃₂	C₂²	semidirect
28. C₂×C₃₁×D₁₆	C₂²	semidirect
29. C₂²×C₁₆×C₃₁	C₂²	semidirect
31. C₃₁×#467	C₂²	semidirect
33. C₃₁×#466	C₂²	semidirect
36. C₃₁×#427	C₂²	semidirect
38. C₃₁×#358	C₂²	semidirect
39. C₃₁×#358	C₂²	semidirect
40. C₃₁×#357	C₂²	semidirect
41. C₃₁×#357	C₂²	semidirect
44. C₃₁×C₂²⋊C₁₆	C₂²	semidirect
63. C₃₁×#135	D₄	semidirect
178. #33516	C₃₁	direct (N × H)
230. C₂²×#135	C₂×C₃₁	semidirect
231. C₂×#508	C₂×C₃₁	semidirect
232. #3050	C₂×C₃₁	semidirect
233. #2189	C₂×C₃₁	semidirect
234. #2188	C₂×C₃₁	semidirect
235. #2187	C₂×C₃₁	semidirect
236. #2186	C₂×C₃₁	semidirect
237. #2185	C₂×C₃₁	semidirect
238. #2184	C₂×C₃₁	semidirect
239. #2138	C₂×C₃₁	semidirect
240. C₂×C₂²⋊C₁₆	C₂×C₃₁	semidirect
257. C₂²×C₃₁	#508	semidirect
274. C₂×#135	C₂²×C₃₁	semidirect
281. C₂×Q₃₂	C₂²×C₃₁	semidirect
282. C₂×SD₃₂	C₂²×C₃₁	semidirect
283. C₂×SD₃₂	C₂²×C₃₁	semidirect
284. C₂×D₁₆	C₂²×C₃₁	semidirect
285. C₂²×C₁₆	C₂²×C₃₁	semidirect
287. #467	C₂²×C₃₁	semidirect
289. #466	C₂²×C₃₁	semidirect
292. #427	C₂²×C₃₁	semidirect
294. #358	C₂²×C₃₁	semidirect
295. #358	C₂²×C₃₁	semidirect
296. #357	C₂²×C₃₁	semidirect
297. #357	C₂²×C₃₁	semidirect
300. C₂²⋊C₁₆	C₂²×C₃₁	semidirect
327. #135	D₄×C₃₁	semidirect
370. C₃₁	#33516	direct (N × H)
510. C₂²	C₃₁×#508	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×#135	2
C₂×C₃₁×#508	2
C₃₁×#3050	2
C₃₁×#2189	2
C₃₁×#2188	2
C₃₁×#2187	2
C₃₁×#2186	2
C₃₁×#2185	2
C₃₁×#2184	2
C₃₁×#2138	2
C₂×C₃₁×C₂²⋊C₁₆	2
#33516	31

Subgroups — C31×#33516 #21490275

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33516 #21490275