Subgroups — C₃₁×#24772 #21481531

Order 7936.

840 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#473	C₂	1	2	yes	maximal
2	C₃₁×#3039	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#462	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₃₁×#137	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2906	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#2064	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2047	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1966	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1950	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1908	C₂	1	2	yes	maximal
11	C₃₁×#1801	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#137	C₂²	1	2	yes	—
13	C₂×C₃₁×#137	C₂²	1	1	yes	characteristic
14	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
15	C₂×C₃₁×#131	C₂²	1	2	yes	—
16	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#473	—	2	8	no	—
20	C₃₁×#462	—	2	1	no	—
21	C₃₁×#462	—	2	1	no	—
22	C₃₁×#462	—	2	1	no	—
23	C₃₁×#462	—	2	1	no	—
24	C₃₁×#462	—	2	2	no	—
25	C₃₁×#461	—	2	1	no	—
26	C₃₁×#461	—	2	1	no	—
27	C₃₁×#441	—	2	1	no	—
28	C₃₁×#440	—	2	1	no	—
29	C₄×Q₁₆×C₃₁	—	2	1	no	—
30	C₄×SD₁₆×C₃₁	—	2	1	no	—
31	C₂×C₃₁×#108	C₂²	1	2	yes	—
32	C₃₁×#428	C₂²	1	1	yes	characteristic
33	C₃₁×#424	—	2	2	no	—
34	C₃₁×#424	C₂²	1	2	yes	—
35	C₃₁×#423	—	2	1	no	—
36	C₃₁×#423	C₂²	1	1	yes	characteristic
37	C₃₁×#423	C₂²	1	1	yes	characteristic
38	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
39	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
40	C₃₁×#419	C₂²	1	2	yes	—
41	C₃₁×#419	C₂²	1	1	yes	characteristic
42	C₃₁×#417	C₂²	1	1	yes	characteristic
43	C₂×C₃₁×#103	C₂²	1	2	yes	—
44	C₂×C₃₁×#101	C₂²	1	2	yes	—
45	C₂×C₃₁×#101	C₂²	1	2	yes	—
46	C₂×C₃₁×#101	C₂²	1	2	yes	—
47	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
48	C₃₁×#343	C₂²	1	2	yes	—
49	C₃₁×#341	C₂²	1	2	yes	—
50	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
51	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
52	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₃₁×#137	—	2	2	no	—
54	C₃₁×#137	—	2	4	no	—
55	C₃₁×#137	—	2	4	no	—
56	C₃₁×#137	D₄	1	4	yes	—
57	C₂×Q₁₆×C₃₁	—	2	1	no	—
58	C₂×Q₁₆×C₃₁	—	2	2	no	—
59	C₂×Q₁₆×C₃₁	—	2	4	no	—
60	C₂×Q₁₆×C₃₁	—	2	2	no	—
61	C₂×Q₁₆×C₃₁	—	2	2	no	—
62	C₂×Q₁₆×C₃₁	—	4	2	no	—
63	C₂×SD₁₆×C₃₁	—	2	1	no	—
64	C₂×SD₁₆×C₃₁	—	2	2	no	—
65	C₃₁×#131	—	2	4	no	—
66	C₃₁×#131	—	4	2	no	—
67	C₂×M₁₆×C₃₁	—	2	2	no	—
68	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
69	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
70	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
71	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₃₁	—	2	2	no	—
73	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
74	C₃₁×C₄⋊Q₈	—	2	1	no	—
75	C₃₁×C₄⋊Q₈	—	2	1	no	—
76	C₃₁×C₄⋊Q₈	—	2	1	no	—
77	C₃₁×C₄⋊Q₈	—	2	1	no	—
78	C₄×Q₈×C₃₁	—	2	1	no	—
79	C₄×Q₈×C₃₁	—	2	1	no	—
80	C₄×Q₈×C₃₁	—	2	1	no	—
81	C₄×Q₈×C₃₁	—	2	1	no	—
82	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
83	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
84	C₄×D₄×C₃₁	—	2	1	no	—
85	C₄×D₄×C₃₁	—	2	1	no	—
86	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
87	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
88	C₃₁×#117	—	2	1	no	—
89	C₃₁×#117	C₂³	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
91	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
92	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
93	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
94	C₂×C₄²×C₃₁	—	2	1	no	—
95	C₂×C₄²×C₃₁	—	2	1	no	—
96	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
97	C₃₁×#108	—	2	4	no	—
98	C₃₁×#107	—	2	1	no	—
99	C₃₁×#106	—	2	1	no	—
100	C₃₁×C₄⋊C₈	—	2	2	no	—
101	C₃₁×C₄⋊C₈	—	2	2	no	—
102	C₃₁×C₄⋊C₈	—	2	1	no	—
103	C₃₁×C₄⋊C₈	—	2	1	no	—
104	C₃₁×C₄⋊C₈	—	2	1	no	—
105	C₃₁×C₄⋊C₈	—	2	1	no	—
106	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
107	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
108	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
109	C₃₁×#103	—	2	1	no	—
110	C₃₁×#103	—	2	2	no	—
111	C₃₁×#103	—	2	2	no	—
112	C₃₁×#103	—	2	1	no	—
113	C₃₁×#103	—	2	1	no	—
114	C₃₁×#103	—	2	2	no	—
115	C₃₁×#103	—	2	2	no	—
116	C₃₁×#102	—	2	1	no	—
117	C₃₁×#101	—	2	4	no	—
118	C₃₁×#101	—	2	4	no	—
119	C₃₁×#101	—	4	2	no	—
120	C₃₁×#97	—	2	1	no	—
121	C₄×C₈×C₃₁	—	2	1	no	—
122	#41×C₃₁	—	2	1	no	—
123	#41×C₃₁	—	2	1	no	—
124	#41×C₃₁	C₂×D₄	1	2	yes	—
125	#41×C₃₁	#41	1	2	yes	—
126	C₂×Q₈×C₃₁	—	2	1	no	—
127	C₂×Q₈×C₃₁	—	2	1	no	—
128	C₂×Q₈×C₃₁	—	2	1	no	—
129	C₂×Q₈×C₃₁	—	2	2	no	—
130	C₂×Q₈×C₃₁	—	2	2	no	—
131	C₂×Q₈×C₃₁	—	2	2	no	—
132	C₂×Q₈×C₃₁	—	2	1	no	—
133	C₂×Q₈×C₃₁	—	2	2	no	—
134	C₂×Q₈×C₃₁	—	2	1	no	—
135	C₂×Q₈×C₃₁	—	2	2	no	—
136	C₂×Q₈×C₃₁	—	2	2	no	—
137	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
138	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
139	C₂×D₄×C₃₁	—	2	1	no	—
140	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₃₁	—	2	1	no	—
142	C₂²×C₄×C₃₁	—	2	1	no	—
143	C₂²×C₄×C₃₁	—	2	1	no	—
144	C₂²×C₄×C₃₁	—	2	1	no	—
145	C₂²×C₄×C₃₁	—	2	2	no	—
146	C₂²×C₄×C₃₁	—	2	1	no	—
147	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
148	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
149	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
150	Q₁₆×C₃₁	—	2	4	no	—
151	Q₁₆×C₃₁	—	4	4	no	—
152	Q₁₆×C₃₁	—	4	4	no	—
153	Q₁₆×C₃₁	—	4	4	no	—
154	SD₁₆×C₃₁	—	2	4	no	—
155	SD₁₆×C₃₁	—	4	4	no	—
156	M₁₆×C₃₁	—	2	1	no	—
157	M₁₆×C₃₁	—	2	2	no	—
158	M₁₆×C₃₁	—	2	2	no	—
159	M₁₆×C₃₁	—	2	2	no	—
160	M₁₆×C₃₁	—	2	1	no	—
161	M₁₆×C₃₁	—	2	2	no	—
162	M₁₆×C₃₁	—	2	1	no	—
163	M₁₆×C₃₁	—	4	4	no	—
164	M₁₆×C₃₁	C₂×D₄	1	2	yes	—
165	C₂×C₈×C₃₁	—	2	1	no	—
166	C₂×C₈×C₃₁	—	2	2	no	—
167	C₂×C₈×C₃₁	—	2	2	no	—
168	C₂×C₈×C₃₁	—	2	1	no	—
169	C₂×C₈×C₃₁	—	2	2	no	—
170	C₂×C₈×C₃₁	—	2	2	no	—
171	C₂×C₈×C₃₁	—	4	4	no	—
172	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
173	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
174	C₄⋊C₄×C₃₁	—	2	1	no	—
175	C₄⋊C₄×C₃₁	—	2	1	no	—
176	C₄⋊C₄×C₃₁	—	2	1	no	—
177	C₄⋊C₄×C₃₁	—	2	1	no	—
178	C₄⋊C₄×C₃₁	—	4	1	no	—
179	C₄⋊C₄×C₃₁	—	4	1	no	—
180	C₄⋊C₄×C₃₁	—	4	1	no	—
181	C₄⋊C₄×C₃₁	—	4	1	no	—
182	C₄⋊C₄×C₃₁	—	4	1	no	—
183	C₄⋊C₄×C₃₁	—	4	1	no	—
184	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
185	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
186	C₂²⋊C₄×C₃₁	—	2	1	no	—
187	C₂²⋊C₄×C₃₁	—	2	1	no	—
188	C₂²⋊C₄×C₃₁	—	2	1	no	—
189	C₄²×C₃₁	—	2	1	no	—
190	C₄²×C₃₁	—	2	1	no	—
191	C₄²×C₃₁	—	2	1	no	—
192	C₄²×C₃₁	—	4	1	no	—
193	C₄²×C₃₁	—	4	1	no	—
194	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
195	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
196	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
197	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
198	#24772	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
199	C₂³×C₃₁	—	2	1	no	—
200	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
201	Q₈×C₃₁	—	2	1	no	—
202	Q₈×C₃₁	—	2	2	no	—
203	Q₈×C₃₁	—	2	2	no	—
204	Q₈×C₃₁	—	4	1	no	—
205	Q₈×C₃₁	—	4	1	no	—
206	Q₈×C₃₁	—	4	2	no	—
207	Q₈×C₃₁	—	4	2	no	—
208	Q₈×C₃₁	—	4	2	no	—
209	Q₈×C₃₁	—	4	2	no	—
210	Q₈×C₃₁	#121	1	2	yes	—
211	D₄×C₃₁	—	2	2	no	—
212	D₄×C₃₁	—	2	2	no	—
213	D₄×C₃₁	—	2	1	no	—
214	D₄×C₃₁	#121	1	2	yes	—
215	C₂×C₄×C₃₁	—	2	1	no	—
216	C₂×C₄×C₃₁	—	2	2	no	—
217	C₂×C₄×C₃₁	—	2	2	no	—
218	C₂×C₄×C₃₁	—	2	2	no	—
219	C₂×C₄×C₃₁	—	2	1	no	—
220	C₂×C₄×C₃₁	—	2	1	no	—
221	C₂×C₄×C₃₁	—	2	2	no	—
222	C₂×C₄×C₃₁	—	4	1	no	—
223	C₂×C₄×C₃₁	—	4	1	no	—
224	C₂×C₄×C₃₁	—	4	1	no	—
225	C₂×C₄×C₃₁	—	4	2	no	—
226	C₂×C₄×C₃₁	—	4	1	no	—
227	C₂×C₄×C₃₁	—	4	2	no	—
228	C₂×C₄×C₃₁	—	4	2	no	—
229	C₂×C₄×C₃₁	—	4	2	no	—
230	C₂×C₄×C₃₁	—	8	1	no	—
231	C₂×C₄×C₃₁	—	8	1	no	—
232	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
233	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
234	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
235	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
236	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
238	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
239	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
240	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
241	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
242	C₈×C₃₁	—	2	2	no	—
243	C₈×C₃₁	—	2	2	no	—
244	C₈×C₃₁	—	2	2	no	—
245	C₈×C₃₁	—	4	2	no	—
246	C₈×C₃₁	—	4	1	no	—
247	C₈×C₃₁	—	4	2	no	—
248	C₂×#473	C₂×C₃₁	1	2	yes	—
249	#3039	C₂×C₃₁	1	1	yes	characteristic
250	C₂×#462	C₂×C₃₁	1	1	yes	characteristic, complemented
251	C₄×#137	C₂×C₃₁	1	1	yes	characteristic
252	#2906	C₂×C₃₁	1	1	yes	characteristic
253	#2064	C₂×C₃₁	1	2	yes	complemented
254	#2047	C₂×C₃₁	1	2	yes	complemented
255	#1966	C₂×C₃₁	1	1	yes	characteristic, complemented
256	#1950	C₂×C₃₁	1	1	yes	characteristic, complemented
257	#1908	C₂×C₃₁	1	2	yes	—
258	#1801	C₂×C₃₁	1	1	yes	characteristic, complemented
259	C₂²×C₃₁	—	2	1	no	—
260	C₂²×C₃₁	—	2	1	no	—
261	C₂²×C₃₁	—	2	2	no	—
262	C₂²×C₃₁	—	4	1	no	—
263	C₂²×C₃₁	—	4	1	no	—
264	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
265	C₂²×C₃₁	#547	1	1	yes	characteristic
266	C₂²×C₃₁	#553	1	1	yes	characteristic
267	C₄×C₃₁	—	2	2	no	—
268	C₄×C₃₁	—	2	1	no	—
269	C₄×C₃₁	—	2	1	no	—
270	C₄×C₃₁	—	2	1	no	—
271	C₄×C₃₁	—	4	2	no	—
272	C₄×C₃₁	—	4	1	no	—
273	C₄×C₃₁	—	4	1	no	—
274	C₄×C₃₁	—	8	2	no	—
275	C₄×C₃₁	—	8	1	no	—
276	C₄×C₃₁	#473	1	2	yes	complemented
277	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
278	C₄×C₃₁	#537	1	1	yes	characteristic
279	C₄×C₃₁	D₄²	1	1	yes	characteristic
280	C₄×C₃₁	#548	1	1	yes	characteristic
281	C₂×#137	C₂²×C₃₁	1	2	yes	—
282	C₂×#137	C₂²×C₃₁	1	1	yes	characteristic
283	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
284	C₂×#131	C₂²×C₃₁	1	2	yes	—
285	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
286	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
287	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
288	#473	—	2	8	no	—
289	#462	—	2	1	no	—
290	#462	—	2	1	no	—
291	#462	—	2	1	no	—
292	#462	—	2	1	no	—
293	#462	—	2	2	no	—
294	#461	—	2	1	no	—
295	#461	—	2	1	no	—
296	#441	—	2	1	no	—
297	#440	—	2	1	no	—
298	C₄×Q₁₆	—	2	1	no	—
299	C₄×SD₁₆	—	2	1	no	—
300	C₂×#108	C₂²×C₃₁	1	2	yes	—
301	#428	C₂²×C₃₁	1	1	yes	characteristic
302	#424	—	2	2	no	—
303	#424	C₂²×C₃₁	1	2	yes	—
304	#423	—	2	1	no	—
305	#423	C₂²×C₃₁	1	1	yes	characteristic
306	#423	C₂²×C₃₁	1	1	yes	characteristic
307	C₂×C₄⋊C₈	—	2	1	no	—
308	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
309	#419	C₂²×C₃₁	1	2	yes	—
310	#419	C₂²×C₃₁	1	1	yes	characteristic
311	#417	C₂²×C₃₁	1	1	yes	characteristic
312	C₂×#103	C₂²×C₃₁	1	2	yes	—
313	C₂×#101	C₂²×C₃₁	1	2	yes	—
314	C₂×#101	C₂²×C₃₁	1	2	yes	—
315	C₂×#101	C₂²×C₃₁	1	2	yes	—
316	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
317	#343	C₂²×C₃₁	1	2	yes	—
318	#341	C₂²×C₃₁	1	2	yes	—
319	C₂×C₃₁	—	2	1	no	—
320	C₂×C₃₁	—	2	1	no	—
321	C₂×C₃₁	—	4	2	no	—
322	C₂×C₃₁	#2416	1	1	yes	characteristic
323	C₂×C₃₁	C₂×#473	1	1	yes	characteristic
324	C₂×C₃₁	#3069	1	1	yes	characteristic
325	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
326	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
327	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
328	#137	—	2	2	no	—
329	#137	—	2	4	no	—
330	#137	—	2	4	no	—
331	#137	D₄×C₃₁	1	4	yes	—
332	C₂×Q₁₆	—	2	1	no	—
333	C₂×Q₁₆	—	2	4	no	—
334	C₂×Q₁₆	—	2	2	no	—
335	C₂×Q₁₆	—	2	2	no	—
336	C₂×Q₁₆	—	2	2	no	—
337	C₂×Q₁₆	—	4	2	no	—
338	C₂×SD₁₆	—	2	1	no	—
339	C₂×SD₁₆	—	2	2	no	—
340	#131	—	2	4	no	—
341	#131	—	4	2	no	—
342	C₂×M₁₆	—	2	2	no	—
343	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
344	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
345	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
346	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
347	C₂²×C₈	—	2	2	no	—
348	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
349	C₄⋊Q₈	—	2	1	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	C₄×Q₈	—	2	1	no	—
355	C₄×Q₈	—	2	1	no	—
356	C₄×Q₈	—	2	1	no	—
357	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
358	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
359	C₄×D₄	—	2	1	no	—
360	C₄×D₄	—	2	1	no	—
361	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
362	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
363	#117	—	2	1	no	—
364	#117	C₂³×C₃₁	1	1	yes	characteristic
365	C₂×C₄⋊C₄	—	2	1	no	—
366	C₂×C₄⋊C₄	—	2	1	no	—
367	C₂×C₄⋊C₄	—	2	1	no	—
368	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
369	C₂×C₄²	—	2	1	no	—
370	C₂×C₄²	—	2	1	no	—
371	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
372	#108	—	2	4	no	—
373	#107	—	2	1	no	—
374	#106	—	2	1	no	—
375	C₄⋊C₈	—	2	2	no	—
376	C₄⋊C₈	—	2	2	no	—
377	C₄⋊C₈	—	2	1	no	—
378	C₄⋊C₈	—	2	1	no	—
379	C₄⋊C₈	—	2	1	no	—
380	C₄⋊C₈	—	2	1	no	—
381	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
382	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
383	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
384	#103	—	2	1	no	—
385	#103	—	2	2	no	—
386	#103	—	2	2	no	—
387	#103	—	2	1	no	—
388	#103	—	2	1	no	—
389	#103	—	2	2	no	—
390	#103	—	2	2	no	—
391	#102	—	2	1	no	—
392	#101	—	2	4	no	—
393	#101	—	2	4	no	—
394	#101	—	4	2	no	—
395	#97	—	2	1	no	—
396	C₄×C₈	—	2	1	no	—
397	C₃₁	#24772	1	1	yes	characteristic, direct product factor, Sylow
398	#41	—	2	1	no	—
399	#41	—	2	1	no	—
400	#41	C₂×D₄×C₃₁	1	2	yes	—
401	#41	#41×C₃₁	1	2	yes	—
402	C₂×Q₈	—	2	1	no	—
403	C₂×Q₈	—	2	1	no	—
404	C₂×Q₈	—	2	1	no	—
405	C₂×Q₈	—	2	2	no	—
406	C₂×Q₈	—	2	2	no	—
407	C₂×Q₈	—	2	2	no	—
408	C₂×Q₈	—	2	1	no	—
409	C₂×Q₈	—	2	2	no	—
410	C₂×Q₈	—	2	1	no	—
411	C₂×Q₈	—	2	2	no	—
412	C₂×Q₈	—	2	2	no	—
413	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
414	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
415	C₂×D₄	—	2	1	no	—
416	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
417	C₂²×C₄	—	2	1	no	—
418	C₂²×C₄	—	2	1	no	—
419	C₂²×C₄	—	2	1	no	—
420	C₂²×C₄	—	2	1	no	—
421	C₂²×C₄	—	2	2	no	—
422	C₂²×C₄	—	2	1	no	—
423	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
424	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
425	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
426	Q₁₆	—	2	4	no	—
427	Q₁₆	—	4	4	no	—
428	Q₁₆	—	4	4	no	—
429	Q₁₆	—	4	4	no	—
430	SD₁₆	—	2	4	no	—
431	SD₁₆	—	4	4	no	—
432	M₁₆	—	2	1	no	—
433	M₁₆	—	2	2	no	—
434	M₁₆	—	2	2	no	—
435	M₁₆	—	2	2	no	—
436	M₁₆	—	2	1	no	—
437	M₁₆	—	2	2	no	—
438	M₁₆	—	2	1	no	—
439	M₁₆	—	4	4	no	—
440	M₁₆	C₂×D₄×C₃₁	1	2	yes	—
441	C₂×C₈	—	2	1	no	—
442	C₂×C₈	—	2	2	no	—
443	C₂×C₈	—	2	2	no	—
444	C₂×C₈	—	2	1	no	—
445	C₂×C₈	—	2	2	no	—
446	C₂×C₈	—	2	2	no	—
447	C₂×C₈	—	4	4	no	—
448	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
449	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
450	C₄⋊C₄	—	2	1	no	—
451	C₄⋊C₄	—	2	1	no	—
452	C₄⋊C₄	—	2	1	no	—
453	C₄⋊C₄	—	2	1	no	—
454	C₄⋊C₄	—	4	1	no	—
455	C₄⋊C₄	—	4	1	no	—
456	C₄⋊C₄	—	4	1	no	—
457	C₄⋊C₄	—	4	1	no	—
458	C₄⋊C₄	—	4	1	no	—
459	C₄⋊C₄	—	4	1	no	—
460	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
461	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
462	C₂²⋊C₄	—	2	1	no	—
463	C₂²⋊C₄	—	2	1	no	—
464	C₂²⋊C₄	—	2	1	no	—
465	C₄²	—	2	1	no	—
466	C₄²	—	2	1	no	—
467	C₄²	—	2	1	no	—
468	C₄²	—	4	1	no	—
469	C₄²	—	4	1	no	—
470	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
471	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
472	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
473	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
474	C₂³	—	2	1	no	—
475	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
476	Q₈	—	2	1	no	—
477	Q₈	—	2	2	no	—
478	Q₈	—	2	2	no	—
479	Q₈	—	4	1	no	—
480	Q₈	—	4	1	no	—
481	Q₈	—	4	2	no	—
482	Q₈	—	4	2	no	—
483	Q₈	—	4	2	no	—
484	Q₈	—	4	2	no	—
485	Q₈	C₃₁×#121	1	2	yes	—
486	D₄	—	2	2	no	—
487	D₄	—	2	2	no	—
488	D₄	—	2	1	no	—
489	D₄	C₃₁×#121	1	2	yes	—
490	C₂×C₄	—	2	1	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	2	1	no	—
495	C₂×C₄	—	2	1	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	4	1	no	—
498	C₂×C₄	—	4	1	no	—
499	C₂×C₄	—	4	1	no	—
500	C₂×C₄	—	4	2	no	—
501	C₂×C₄	—	4	1	no	—
502	C₂×C₄	—	4	2	no	—
503	C₂×C₄	—	4	2	no	—
504	C₂×C₄	—	4	2	no	—
505	C₂×C₄	—	8	1	no	—
506	C₂×C₄	—	8	1	no	—
507	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
508	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
509	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
510	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
511	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
512	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
513	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
514	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
515	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
516	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
517	C₈	—	2	2	no	—
518	C₈	—	2	2	no	—
519	C₈	—	2	2	no	—
520	C₈	—	4	2	no	—
521	C₈	—	4	1	no	—
522	C₈	—	4	2	no	—
523	C₂²	—	2	1	no	—
524	C₂²	—	2	1	no	—
525	C₂²	—	2	2	no	—
526	C₂²	—	4	1	no	—
527	C₂²	—	4	1	no	—
528	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
529	C₂²	C₃₁×#547	1	1	yes	characteristic
530	C₂²	C₃₁×#553	1	1	yes	characteristic
531	C₄	—	2	2	no	—
532	C₄	—	2	1	no	—
533	C₄	—	2	1	no	—
534	C₄	—	2	1	no	—
535	C₄	—	4	2	no	—
536	C₄	—	4	1	no	—
537	C₄	—	4	1	no	—
538	C₄	—	8	2	no	—
539	C₄	—	8	1	no	—
540	C₄	C₃₁×#473	1	2	yes	complemented
541	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
542	C₄	C₃₁×#537	1	1	yes	characteristic
543	C₄	D₄²×C₃₁	1	1	yes	characteristic
544	C₄	C₃₁×#548	1	1	yes	characteristic
545	C₂	—	2	1	no	—
546	C₂	—	2	1	no	—
547	C₂	—	4	2	no	—
548	C₂	C₃₁×#2416	1	1	yes	characteristic
549	C₂	C₂×C₃₁×#473	1	1	yes	characteristic
550	C₂	C₃₁×#3069	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×#41×C₃₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₃₁×#462	C₂	semidirect
6. C₃₁×#2064	C₂	semidirect
7. C₃₁×#2047	C₂	semidirect
8. C₃₁×#1966	C₂	semidirect
9. C₃₁×#1950	C₂	semidirect
11. C₃₁×#1801	C₂	semidirect
198. #24772	C₃₁	direct (N × H)
250. C₂×#462	C₂×C₃₁	semidirect
253. #2064	C₂×C₃₁	semidirect
254. #2047	C₂×C₃₁	semidirect
255. #1966	C₂×C₃₁	semidirect
256. #1950	C₂×C₃₁	semidirect
258. #1801	C₂×C₃₁	semidirect
276. C₄×C₃₁	#473	semidirect
397. C₃₁	#24772	direct (N × H)
540. C₄	C₃₁×#473	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#473	2
C₃₁×#3039	2
C₂×C₃₁×#462	2
C₄×C₃₁×#137	2
C₃₁×#2906	2
C₃₁×#2064	2
C₃₁×#2047	2
C₃₁×#1966	2
C₃₁×#1950	2
C₃₁×#1908	2
C₃₁×#1801	2
#24772	31

Subgroups — C31×#24772 #21481531

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24772 #21481531