Subgroups — C₃₁×#17750 #21474509

Order 7936.

1276 conjugacy classes of subgroups.

The subgroup lattice has 578 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×#1560	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1551	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1548	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1545	C₂	1	2	yes	maximal, complemented
8	C₂²×#41×C₃₁	C₄	1	2	yes	—
9	C₂²×M₁₆×C₃₁	—	2	1	no	—
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	C₄	1	2	yes	—
15	C₃₁×#532	C₄	1	2	yes	—
16	C₃₁×#532	C₂²	1	2	yes	—
17	C₃₁×#532	C₂²	1	2	yes	—
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	—	2	1	no	—
22	C₂×C₃₁×#121	—	2	1	no	—
23	C₂×C₄×D₄×C₃₁	—	2	1	no	—
24	C₂×C₄×D₄×C₃₁	—	2	1	no	—
25	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#432	—	2	1	no	—
27	C₃₁×#432	C₂²	1	2	yes	complemented
28	C₃₁×#431	—	2	1	no	—
29	C₃₁×#431	C₂²	1	2	yes	—
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	complemented
34	C₃₁×#423	C₂²	1	2	yes	complemented
35	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
36	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	complemented
37	C₃₁×#407	—	2	1	no	—
38	C₄×M₁₆×C₃₁	—	2	1	no	—
39	C₃₁×#333	C₂²	1	8	yes	complemented
40	C₃₁×#331	C₂²	1	8	yes	complemented
41	C₂×#41×C₃₁	—	2	2	no	—
42	C₂×#41×C₃₁	—	2	2	no	—
43	C₂×#41×C₃₁	—	4	8	no	—
44	C₂×#41×C₃₁	C₂×C₄	1	4	yes	—
45	C₂×#41×C₃₁	C₂×C₄	1	4	yes	—
46	C₂²×Q₈×C₃₁	C₂×C₄	1	2	yes	—
47	C₂²×D₄×C₃₁	—	2	2	no	—
48	C₂²×D₄×C₃₁	—	2	1	no	—
49	C₂²×D₄×C₃₁	—	2	1	no	—
50	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
51	C₂³×C₄×C₃₁	—	2	2	no	—
52	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
53	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
54	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
55	C₂×M₁₆×C₃₁	—	2	2	no	—
56	C₂×M₁₆×C₃₁	—	2	2	no	—
57	C₂×M₁₆×C₃₁	—	2	2	no	—
58	C₂×M₁₆×C₃₁	—	4	4	no	—
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	C₂×C₄	1	2	yes	—
63	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
64	C₃₁×#127	—	2	1	no	—
65	C₃₁×#127	—	2	1	no	—
66	C₃₁×#127	C₂×C₄	1	2	yes	—
67	C₃₁×#127	C₂³	1	2	yes	—
68	C₃₁×#124	—	2	2	no	—
69	C₃₁×#124	—	2	2	no	—
70	C₃₁×#124	C₂×C₄	1	4	yes	—
71	C₃₁×#124	C₂×C₄	1	4	yes	—
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	—	2	1	no	—
89	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
90	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
91	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
92	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
93	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
94	C₂×C₄²×C₃₁	C₂×C₄	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	C₂³	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	C₂³	1	4	yes	—
109	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
110	C₃₁×C₂²⋊C₈	—	2	2	no	—
111	C₃₁×C₂²⋊C₈	—	2	2	no	—
112	C₃₁×#97	—	2	2	no	—
113	C₄×C₈×C₃₁	—	2	2	no	—
114	C₂⁴×C₃₁	—	2	2	no	—
115	C₂⁴×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
116	#41×C₃₁	—	4	8	no	—
117	#41×C₃₁	—	4	8	no	—
118	#41×C₃₁	—	4	8	no	—
119	#41×C₃₁	—	4	8	no	—
120	C₂×Q₈×C₃₁	—	2	2	no	—
121	C₂×Q₈×C₃₁	—	4	4	no	—
122	C₂×Q₈×C₃₁	C₂²×C₄	1	4	yes	—
123	C₂×D₄×C₃₁	—	2	4	no	—
124	C₂×D₄×C₃₁	—	2	4	no	—
125	C₂×D₄×C₃₁	—	2	2	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	—	4	4	no	—
131	C₂×D₄×C₃₁	—	4	4	no	—
132	C₂×D₄×C₃₁	—	4	8	no	—
133	C₂×D₄×C₃₁	—	4	2	no	—
134	C₂×D₄×C₃₁	—	4	2	no	—
135	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
136	C₂²×C₄×C₃₁	—	2	4	no	—
137	C₂²×C₄×C₃₁	—	2	2	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	C₂×D₄	1	1	yes	characteristic
157	C₂²×C₄×C₃₁	C₂×D₄	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	M₁₆×C₃₁	—	4	4	no	—
163	M₁₆×C₃₁	—	4	4	no	—
164	M₁₆×C₃₁	—	4	4	no	—
165	C₂×C₈×C₃₁	—	2	4	no	—
166	C₂×C₈×C₃₁	—	2	4	no	—
167	C₂×C₈×C₃₁	—	4	4	no	—
168	C₄⋊C₄×C₃₁	—	2	2	no	—
169	C₄⋊C₄×C₃₁	—	2	2	no	—
170	C₄⋊C₄×C₃₁	—	2	2	no	—
171	C₄⋊C₄×C₃₁	—	2	2	no	—
172	C₂²⋊C₄×C₃₁	—	2	4	no	—
173	C₂²⋊C₄×C₃₁	—	2	4	no	—
174	C₂²⋊C₄×C₃₁	—	2	4	no	—
175	C₂²⋊C₄×C₃₁	—	2	4	no	—
176	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
177	C₄²×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
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	C₂×D₄	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	C₂⁴	1	1	yes	characteristic
192	#17750	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
193	C₂³×C₃₁	—	2	2	no	—
194	C₂³×C₃₁	—	2	2	no	—
195	C₂³×C₃₁	—	2	4	no	—
196	C₂³×C₃₁	—	4	1	no	—
197	C₂³×C₃₁	—	4	4	no	—
198	C₂³×C₃₁	—	4	4	no	—
199	C₂³×C₃₁	—	4	4	no	—
200	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
201	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
202	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
203	Q₈×C₃₁	—	4	4	no	—
204	Q₈×C₃₁	—	4	4	no	—
205	D₄×C₃₁	—	4	4	no	—
206	D₄×C₃₁	—	4	4	no	—
207	D₄×C₃₁	—	4	8	no	—
208	D₄×C₃₁	—	4	8	no	—
209	D₄×C₃₁	—	4	2	no	—
210	D₄×C₃₁	—	4	2	no	—
211	D₄×C₃₁	—	4	4	no	—
212	D₄×C₃₁	—	4	4	no	—
213	D₄×C₃₁	—	4	4	no	—
214	C₂×C₄×C₃₁	—	2	2	no	—
215	C₂×C₄×C₃₁	—	2	2	no	—
216	C₂×C₄×C₃₁	—	2	1	no	—
217	C₂×C₄×C₃₁	—	2	1	no	—
218	C₂×C₄×C₃₁	—	2	1	no	—
219	C₂×C₄×C₃₁	—	2	1	no	—
220	C₂×C₄×C₃₁	—	2	2	no	—
221	C₂×C₄×C₃₁	—	2	2	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	4	no	—
230	C₂×C₄×C₃₁	—	4	1	no	—
231	C₂×C₄×C₃₁	—	4	1	no	—
232	C₂×C₄×C₃₁	—	4	1	no	—
233	C₂×C₄×C₃₁	—	4	8	no	—
234	C₂×C₄×C₃₁	—	4	4	no	—
235	C₂×C₄×C₃₁	—	4	8	no	—
236	C₂×C₄×C₃₁	#100	1	4	yes	—
237	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
238	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
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	AGL(1,ℤ/8)	1	2	yes	—
247	C₂×C₄×C₃₁	#137	1	2	yes	—
248	C₂×C₄×C₃₁	C₂³×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	C₈×C₃₁	—	4	4	no	—
252	C₈×C₃₁	—	4	4	no	—
253	C₂×#532	C₂×C₃₁	1	1	yes	characteristic
254	#2920	C₂×C₃₁	1	1	yes	characteristic, complemented
255	C₂×#423	C₂×C₃₁	1	1	yes	characteristic, complemented
256	#1560	C₂×C₃₁	1	4	yes	complemented
257	#1551	C₂×C₃₁	1	4	yes	complemented
258	#1548	C₂×C₃₁	1	2	yes	complemented
259	#1545	C₂×C₃₁	1	2	yes	complemented
260	C₂²×C₃₁	—	2	1	no	—
261	C₂²×C₃₁	—	2	1	no	—
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	—	4	4	no	—
268	C₂²×C₃₁	—	4	4	no	—
269	C₂²×C₃₁	—	4	4	no	—
270	C₂²×C₃₁	—	4	4	no	—
271	C₂²×C₃₁	—	8	8	no	—
272	C₂²×C₃₁	C₂×#100	1	2	yes	—
273	C₂²×C₃₁	#417	1	2	yes	—
274	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
275	C₄×C₃₁	—	2	1	no	—
276	C₄×C₃₁	—	2	1	no	—
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	—	4	4	no	—
283	C₄×C₃₁	C₂×#100	1	2	yes	—
284	C₄×C₃₁	C₂×#100	1	2	yes	—
285	C₄×C₃₁	#417	1	2	yes	—
286	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
287	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
288	C₄×C₃₁	#575	1	2	yes	—
289	C₂²×#41	C₄×C₃₁	1	2	yes	—
290	C₂²×M₁₆	—	2	1	no	—
291	#532	—	2	1	no	—
292	#532	—	2	1	no	—
293	#532	—	2	1	no	—
294	#532	—	2	1	no	—
295	#532	C₄×C₃₁	1	2	yes	—
296	#532	C₄×C₃₁	1	2	yes	—
297	#532	C₂²×C₃₁	1	2	yes	—
298	#532	C₂²×C₃₁	1	2	yes	—
299	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
300	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
301	C₂×#124	C₄×C₃₁	1	2	yes	—
302	C₂×#121	—	2	1	no	—
303	C₂×#121	—	2	1	no	—
304	C₂×C₄×D₄	—	2	1	no	—
305	C₂×C₄×D₄	—	2	1	no	—
306	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
307	#432	—	2	1	no	—
308	#432	C₂²×C₃₁	1	2	yes	complemented
309	#431	—	2	1	no	—
310	#431	C₂²×C₃₁	1	2	yes	—
311	#423	—	2	1	no	—
312	#423	—	2	1	no	—
313	#423	C₂²×C₃₁	1	2	yes	—
314	#423	C₂²×C₃₁	1	2	yes	complemented
315	#423	C₂²×C₃₁	1	2	yes	complemented
316	C₂×C₄⋊C₈	—	2	1	no	—
317	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
318	#407	—	2	1	no	—
319	C₄×M₁₆	—	2	1	no	—
320	#333	C₂²×C₃₁	1	8	yes	complemented
321	#331	C₂²×C₃₁	1	8	yes	complemented
322	C₂×C₃₁	—	2	2	no	—
323	C₂×C₃₁	—	2	2	no	—
324	C₂×C₃₁	—	4	1	no	—
325	C₂×C₃₁	—	8	4	no	—
326	C₂×C₃₁	C₂²×#100	1	1	yes	characteristic
327	C₂×C₃₁	C₂×#417	1	1	yes	characteristic
328	C₂×C₃₁	#2896	1	1	yes	characteristic
329	C₂×#41	—	2	2	no	—
330	C₂×#41	—	2	2	no	—
331	C₂×#41	—	4	8	no	—
332	C₂×#41	C₂×C₄×C₃₁	1	4	yes	—
333	C₂×#41	C₂×C₄×C₃₁	1	4	yes	—
334	C₂²×Q₈	C₂×C₄×C₃₁	1	2	yes	—
335	C₂²×D₄	—	2	2	no	—
336	C₂²×D₄	—	2	1	no	—
337	C₂²×D₄	—	2	1	no	—
338	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
339	C₂³×C₄	—	2	2	no	—
340	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
341	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
342	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
343	C₂×M₁₆	—	2	2	no	—
344	C₂×M₁₆	—	2	2	no	—
345	C₂×M₁₆	—	2	2	no	—
346	C₂×M₁₆	—	4	4	no	—
347	C₂²×C₈	—	2	2	no	—
348	C₄⋊Q₈	—	2	1	no	—
349	C₄⋊Q₈	—	2	1	no	—
350	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
351	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
352	#127	—	2	1	no	—
353	#127	—	2	1	no	—
354	#127	C₂×C₄×C₃₁	1	2	yes	—
355	#127	C₂³×C₃₁	1	2	yes	—
356	#124	—	2	2	no	—
357	#124	—	2	2	no	—
358	#124	C₂×C₄×C₃₁	1	4	yes	—
359	#124	C₂×C₄×C₃₁	1	4	yes	—
360	#121	—	2	2	no	—
361	#121	—	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	C₄×D₄	—	2	2	no	—
369	C₄×D₄	—	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	C₂×C₄⋊C₄	—	2	1	no	—
377	C₂×C₄⋊C₄	—	2	1	no	—
378	C₂×C₂²⋊C₄	—	2	2	no	—
379	C₂×C₂²⋊C₄	—	2	2	no	—
380	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
381	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
382	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
383	C₂×C₄²	D₄×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	C₂³×C₃₁	1	1	yes	characteristic
390	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
391	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
392	C₄⋊C₈	—	2	2	no	—
393	C₄⋊C₈	—	2	2	no	—
394	C₄⋊C₈	—	2	2	no	—
395	C₄⋊C₈	—	2	2	no	—
396	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
397	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
398	C₂²⋊C₈	—	2	2	no	—
399	C₂²⋊C₈	—	2	2	no	—
400	#97	—	2	2	no	—
401	C₄×C₈	—	2	2	no	—
402	C₃₁	#17750	1	1	yes	characteristic, direct product factor, Sylow
403	C₂⁴	—	2	2	no	—
404	C₂⁴	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
405	#41	—	4	8	no	—
406	#41	—	4	8	no	—
407	#41	—	4	8	no	—
408	#41	—	4	8	no	—
409	C₂×Q₈	—	2	2	no	—
410	C₂×Q₈	—	4	4	no	—
411	C₂×Q₈	C₂²×C₄×C₃₁	1	4	yes	—
412	C₂×D₄	—	2	4	no	—
413	C₂×D₄	—	2	4	no	—
414	C₂×D₄	—	2	2	no	—
415	C₂×D₄	—	2	2	no	—
416	C₂×D₄	—	2	2	no	—
417	C₂×D₄	—	2	2	no	—
418	C₂×D₄	—	2	2	no	—
419	C₂×D₄	—	4	4	no	—
420	C₂×D₄	—	4	4	no	—
421	C₂×D₄	—	4	8	no	—
422	C₂×D₄	—	4	2	no	—
423	C₂×D₄	—	4	2	no	—
424	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
425	C₂²×C₄	—	2	4	no	—
426	C₂²×C₄	—	2	2	no	—
427	C₂²×C₄	—	2	2	no	—
428	C₂²×C₄	—	2	2	no	—
429	C₂²×C₄	—	2	4	no	—
430	C₂²×C₄	—	2	4	no	—
431	C₂²×C₄	—	4	2	no	—
432	C₂²×C₄	—	4	8	no	—
433	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
434	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
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	C₂×D₄×C₃₁	1	1	yes	characteristic
446	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
447	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
448	C₂²×C₄	C₂×D₄×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	M₁₆	—	4	4	no	—
452	M₁₆	—	4	4	no	—
453	M₁₆	—	4	4	no	—
454	C₂×C₈	—	2	4	no	—
455	C₂×C₈	—	2	4	no	—
456	C₂×C₈	—	4	4	no	—
457	C₄⋊C₄	—	2	2	no	—
458	C₄⋊C₄	—	2	2	no	—
459	C₄⋊C₄	—	2	2	no	—
460	C₄⋊C₄	—	2	2	no	—
461	C₂²⋊C₄	—	2	4	no	—
462	C₂²⋊C₄	—	2	4	no	—
463	C₂²⋊C₄	—	2	4	no	—
464	C₂²⋊C₄	—	2	4	no	—
465	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
466	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
467	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
468	C₄²	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
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	C₂×D₄×C₃₁	1	1	yes	characteristic
475	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
476	C₄²	C₂×D₄×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	C₂⁴×C₃₁	1	1	yes	characteristic
481	C₂³	—	2	2	no	—
482	C₂³	—	2	2	no	—
483	C₂³	—	2	4	no	—
484	C₂³	—	4	1	no	—
485	C₂³	—	4	4	no	—
486	C₂³	—	4	4	no	—
487	C₂³	—	4	4	no	—
488	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
489	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
490	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
491	Q₈	—	4	4	no	—
492	Q₈	—	4	4	no	—
493	D₄	—	4	4	no	—
494	D₄	—	4	4	no	—
495	D₄	—	4	8	no	—
496	D₄	—	4	8	no	—
497	D₄	—	4	4	no	—
498	D₄	—	4	4	no	—
499	D₄	—	4	2	no	—
500	D₄	—	4	2	no	—
501	D₄	—	4	4	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	2	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	1	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	1	no	—
513	C₂×C₄	—	2	1	no	—
514	C₂×C₄	—	2	2	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	4	no	—
518	C₂×C₄	—	4	1	no	—
519	C₂×C₄	—	4	1	no	—
520	C₂×C₄	—	4	1	no	—
521	C₂×C₄	—	4	8	no	—
522	C₂×C₄	—	4	4	no	—
523	C₂×C₄	—	4	8	no	—
524	C₂×C₄	C₃₁×#100	1	4	yes	—
525	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
526	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
527	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
528	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
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	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
535	C₂×C₄	C₃₁×#137	1	2	yes	—
536	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
537	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
538	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
539	C₈	—	4	4	no	—
540	C₈	—	4	4	no	—
541	C₂²	—	2	1	no	—
542	C₂²	—	2	1	no	—
543	C₂²	—	2	1	no	—
544	C₂²	—	2	1	no	—
545	C₂²	—	2	1	no	—
546	C₂²	—	2	1	no	—
547	C₂²	—	2	1	no	—
548	C₂²	—	4	4	no	—
549	C₂²	—	4	4	no	—
550	C₂²	—	4	4	no	—
551	C₂²	—	4	4	no	—
552	C₂²	—	8	8	no	—
553	C₂²	C₂×C₃₁×#100	1	2	yes	—
554	C₂²	C₃₁×#417	1	2	yes	—
555	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
556	C₄	—	2	1	no	—
557	C₄	—	2	1	no	—
558	C₄	—	2	1	no	—
559	C₄	—	2	1	no	—
560	C₄	—	2	1	no	—
561	C₄	—	2	1	no	—
562	C₄	—	2	1	no	—
563	C₄	—	4	4	no	—
564	C₄	C₂×C₃₁×#100	1	2	yes	—
565	C₄	C₂×C₃₁×#100	1	2	yes	—
566	C₄	C₃₁×#417	1	2	yes	—
567	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
568	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
569	C₄	C₃₁×#575	1	2	yes	—
570	C₂	—	2	2	no	—
571	C₂	—	2	2	no	—
572	C₂	—	4	1	no	—
573	C₂	—	8	4	no	—
574	C₂	C₂²×C₃₁×#100	1	1	yes	characteristic
575	C₂	C₂×C₃₁×#417	1	1	yes	characteristic
576	C₂	C₃₁×#2896	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ 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×#1560	C₂	semidirect
5. C₃₁×#1551	C₂	semidirect
6. C₃₁×#1548	C₂	semidirect
7. C₃₁×#1545	C₂	semidirect
27. C₃₁×#432	C₂²	semidirect
33. C₃₁×#423	C₂²	semidirect
34. C₃₁×#423	C₂²	semidirect
36. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
39. C₃₁×#333	C₂²	semidirect
40. C₃₁×#331	C₂²	semidirect
192. #17750	C₃₁	direct (N × H)
254. #2920	C₂×C₃₁	semidirect
255. C₂×#423	C₂×C₃₁	semidirect
256. #1560	C₂×C₃₁	semidirect
257. #1551	C₂×C₃₁	semidirect
258. #1548	C₂×C₃₁	semidirect
259. #1545	C₂×C₃₁	semidirect
308. #432	C₂²×C₃₁	semidirect
314. #423	C₂²×C₃₁	semidirect
315. #423	C₂²×C₃₁	semidirect
317. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
320. #333	C₂²×C₃₁	semidirect
321. #331	C₂²×C₃₁	semidirect
402. C₃₁	#17750	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×#532	2
C₃₁×#2920	2
C₂×C₃₁×#423	2
C₃₁×#1560	2
C₃₁×#1551	2
C₃₁×#1548	2
C₃₁×#1545	2
#17750	31

Subgroups — C31×#17750 #21474509

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17750 #21474509