Subgroups — C₃₁×#24672 #21481431

Order 7936.

754 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3039	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#462	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₃₁×#137	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2925	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#2052	C₂	1	2	yes	maximal
6	C₃₁×#2020	C₂	1	2	yes	maximal
7	C₃₁×#1979	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×#1930	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1801	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#137	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#137	C₂²	1	2	yes	—
14	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	—
15	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#462	—	2	1	no	—
19	C₃₁×#462	—	2	1	no	—
20	C₃₁×#462	—	2	1	no	—
21	C₃₁×#462	—	2	1	no	—
22	C₃₁×#462	—	2	2	no	—
23	C₃₁×#461	—	2	1	no	—
24	C₃₁×#461	—	2	1	no	—
25	C₃₁×#441	—	2	1	no	—
26	C₃₁×#440	—	2	1	no	—
27	C₄×Q₁₆×C₃₁	—	2	1	no	—
28	C₄×SD₁₆×C₃₁	—	2	1	no	—
29	C₃₁×#433	—	2	2	no	—
30	C₃₁×#432	—	2	1	no	—
31	C₃₁×#431	—	2	1	no	—
32	C₂×C₃₁×#108	C₂²	1	2	yes	—
33	C₃₁×#428	C₂²	1	1	yes	characteristic
34	C₃₁×#423	C₂²	1	1	yes	characteristic
35	C₃₁×#423	C₂²	1	1	yes	characteristic
36	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₃₁×#419	C₂²	1	1	yes	characteristic
38	C₃₁×#419	C₂²	1	2	yes	—
39	C₃₁×#417	C₂²	1	1	yes	characteristic
40	C₂×C₃₁×#103	C₂²	1	2	yes	—
41	C₂×C₃₁×#101	C₂²	1	2	yes	—
42	C₂×C₃₁×#101	C₂²	1	2	yes	—
43	C₂×C₃₁×#101	C₂²	1	2	yes	—
44	C₃₁×#408	C₂²	1	2	yes	—
45	C₃₁×#405	C₂²	1	2	yes	—
46	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
47	C₃₁×#343	C₂²	1	2	yes	—
48	C₃₁×#341	C₂²	1	2	yes	—
49	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
50	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
51	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₃₁×#137	—	2	2	no	—
53	C₃₁×#137	—	2	2	no	—
54	C₃₁×#137	—	4	2	no	—
55	C₃₁×#137	—	4	2	no	—
56	C₂×Q₁₆×C₃₁	—	2	1	no	—
57	C₂×Q₁₆×C₃₁	—	2	2	no	—
58	C₂×Q₁₆×C₃₁	—	2	4	no	—
59	C₂×Q₁₆×C₃₁	—	2	2	no	—
60	C₂×Q₁₆×C₃₁	—	2	2	no	—
61	C₂×Q₁₆×C₃₁	—	4	2	no	—
62	C₂×SD₁₆×C₃₁	—	2	1	no	—
63	C₂×SD₁₆×C₃₁	—	2	2	no	—
64	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
65	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
66	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
67	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
69	C₃₁×C₄⋊Q₈	—	2	1	no	—
70	C₃₁×C₄⋊Q₈	—	2	1	no	—
71	C₃₁×C₄⋊Q₈	—	2	1	no	—
72	C₃₁×C₄⋊Q₈	—	2	1	no	—
73	C₄×Q₈×C₃₁	—	2	1	no	—
74	C₄×Q₈×C₃₁	—	2	1	no	—
75	C₄×Q₈×C₃₁	—	2	1	no	—
76	C₄×Q₈×C₃₁	—	2	1	no	—
77	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
78	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
79	C₄×D₄×C₃₁	—	2	1	no	—
80	C₄×D₄×C₃₁	—	2	1	no	—
81	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
82	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
83	C₃₁×#117	—	2	1	no	—
84	C₃₁×#117	C₂³	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
86	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
87	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
88	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
89	C₂×C₄²×C₃₁	—	2	1	no	—
90	C₂×C₄²×C₃₁	—	2	1	no	—
91	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
92	C₃₁×#108	—	4	2	no	—
93	C₃₁×#107	—	2	1	no	—
94	C₃₁×#106	—	2	1	no	—
95	C₃₁×C₄⋊C₈	—	2	1	no	—
96	C₃₁×C₄⋊C₈	—	2	1	no	—
97	C₃₁×C₄⋊C₈	—	2	1	no	—
98	C₃₁×C₄⋊C₈	—	2	1	no	—
99	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
100	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
101	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
102	C₃₁×#103	—	2	1	no	—
103	C₃₁×#103	—	2	2	no	—
104	C₃₁×#103	—	2	2	no	—
105	C₃₁×#103	—	2	1	no	—
106	C₃₁×#103	—	2	1	no	—
107	C₃₁×#103	—	2	2	no	—
108	C₃₁×#103	—	2	2	no	—
109	C₃₁×#102	—	2	1	no	—
110	C₃₁×#101	—	2	4	no	—
111	C₃₁×#101	—	4	2	no	—
112	C₃₁×#101	—	4	2	no	—
113	C₃₁×C₂²⋊C₈	—	2	2	no	—
114	C₃₁×C₂²⋊C₈	—	2	2	no	—
115	C₃₁×#97	—	2	2	no	—
116	C₃₁×#97	—	2	1	no	—
117	C₄×C₈×C₃₁	—	2	2	no	—
118	C₄×C₈×C₃₁	—	2	1	no	—
119	#41×C₃₁	—	2	1	no	—
120	#41×C₃₁	—	2	1	no	—
121	#41×C₃₁	—	2	1	no	—
122	#41×C₃₁	—	2	1	no	—
123	C₂×Q₈×C₃₁	—	2	1	no	—
124	C₂×Q₈×C₃₁	—	2	1	no	—
125	C₂×Q₈×C₃₁	—	2	1	no	—
126	C₂×Q₈×C₃₁	—	2	2	no	—
127	C₂×Q₈×C₃₁	—	2	2	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	1	no	—
132	C₂×Q₈×C₃₁	—	2	2	no	—
133	C₂×Q₈×C₃₁	—	2	2	no	—
134	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂×Q₈×C₃₁	#41	1	2	yes	—
136	C₂×D₄×C₃₁	—	2	1	no	—
137	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₃₁	—	2	1	no	—
139	C₂²×C₄×C₃₁	—	2	1	no	—
140	C₂²×C₄×C₃₁	—	2	1	no	—
141	C₂²×C₄×C₃₁	—	2	1	no	—
142	C₂²×C₄×C₃₁	—	2	2	no	—
143	C₂²×C₄×C₃₁	—	2	1	no	—
144	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
145	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
146	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
147	Q₁₆×C₃₁	—	2	4	no	—
148	Q₁₆×C₃₁	—	4	4	no	—
149	Q₁₆×C₃₁	—	4	4	no	—
150	Q₁₆×C₃₁	—	4	4	no	—
151	SD₁₆×C₃₁	—	2	4	no	—
152	SD₁₆×C₃₁	—	4	4	no	—
153	M₁₆×C₃₁	—	2	1	no	—
154	M₁₆×C₃₁	—	2	2	no	—
155	M₁₆×C₃₁	—	2	1	no	—
156	M₁₆×C₃₁	—	2	2	no	—
157	M₁₆×C₃₁	—	2	2	no	—
158	M₁₆×C₃₁	—	2	1	no	—
159	M₁₆×C₃₁	—	2	2	no	—
160	M₁₆×C₃₁	#41	1	2	yes	—
161	C₂×C₈×C₃₁	—	2	1	no	—
162	C₂×C₈×C₃₁	—	2	2	no	—
163	C₂×C₈×C₃₁	—	2	2	no	—
164	C₂×C₈×C₃₁	—	2	2	no	—
165	C₂×C₈×C₃₁	—	2	1	no	—
166	C₂×C₈×C₃₁	—	2	2	no	—
167	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
168	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
169	C₄⋊C₄×C₃₁	—	2	1	no	—
170	C₄⋊C₄×C₃₁	—	2	1	no	—
171	C₄⋊C₄×C₃₁	—	2	1	no	—
172	C₄⋊C₄×C₃₁	—	2	1	no	—
173	C₄⋊C₄×C₃₁	—	4	1	no	—
174	C₄⋊C₄×C₃₁	—	4	1	no	—
175	C₄⋊C₄×C₃₁	—	4	1	no	—
176	C₄⋊C₄×C₃₁	—	4	1	no	—
177	C₄⋊C₄×C₃₁	—	4	1	no	—
178	C₄⋊C₄×C₃₁	—	4	1	no	—
179	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
180	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
181	C₂²⋊C₄×C₃₁	—	2	1	no	—
182	C₂²⋊C₄×C₃₁	—	2	1	no	—
183	C₂²⋊C₄×C₃₁	—	2	1	no	—
184	C₄²×C₃₁	—	2	1	no	—
185	C₄²×C₃₁	—	2	1	no	—
186	C₄²×C₃₁	—	2	1	no	—
187	C₄²×C₃₁	—	4	1	no	—
188	C₄²×C₃₁	—	4	1	no	—
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	#24672	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
194	C₂³×C₃₁	—	2	1	no	—
195	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
196	Q₈×C₃₁	—	2	2	no	—
197	Q₈×C₃₁	—	2	1	no	—
198	Q₈×C₃₁	—	2	1	no	—
199	Q₈×C₃₁	—	2	2	no	—
200	Q₈×C₃₁	—	4	1	no	—
201	Q₈×C₃₁	—	4	1	no	—
202	Q₈×C₃₁	—	4	2	no	—
203	Q₈×C₃₁	—	4	2	no	—
204	Q₈×C₃₁	—	4	2	no	—
205	Q₈×C₃₁	—	4	2	no	—
206	D₄×C₃₁	—	2	1	no	—
207	D₄×C₃₁	—	2	1	no	—
208	D₄×C₃₁	—	4	1	no	—
209	D₄×C₃₁	—	4	1	no	—
210	C₂×C₄×C₃₁	—	2	2	no	—
211	C₂×C₄×C₃₁	—	2	1	no	—
212	C₂×C₄×C₃₁	—	2	1	no	—
213	C₂×C₄×C₃₁	—	2	1	no	—
214	C₂×C₄×C₃₁	—	2	2	no	—
215	C₂×C₄×C₃₁	—	4	1	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	—	4	1	no	—
220	C₂×C₄×C₃₁	—	4	2	no	—
221	C₂×C₄×C₃₁	—	4	1	no	—
222	C₂×C₄×C₃₁	—	4	2	no	—
223	C₂×C₄×C₃₁	—	4	2	no	—
224	C₂×C₄×C₃₁	—	4	2	no	—
225	C₂×C₄×C₃₁	—	8	1	no	—
226	C₂×C₄×C₃₁	—	8	1	no	—
227	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
228	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
229	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
230	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
231	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
232	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
233	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
234	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
235	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
236	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
237	C₈×C₃₁	—	2	2	no	—
238	C₈×C₃₁	—	2	2	no	—
239	C₈×C₃₁	—	2	2	no	—
240	C₈×C₃₁	—	4	1	no	—
241	C₈×C₃₁	—	4	2	no	—
242	C₈×C₃₁	—	4	2	no	—
243	#3039	C₂×C₃₁	1	1	yes	characteristic
244	C₂×#462	C₂×C₃₁	1	1	yes	characteristic, complemented
245	C₄×#137	C₂×C₃₁	1	1	yes	characteristic
246	#2925	C₂×C₃₁	1	1	yes	characteristic
247	#2052	C₂×C₃₁	1	2	yes	—
248	#2020	C₂×C₃₁	1	2	yes	—
249	#1979	C₂×C₃₁	1	2	yes	complemented
250	#1966	C₂×C₃₁	1	1	yes	characteristic, complemented
251	#1950	C₂×C₃₁	1	1	yes	characteristic, complemented
252	#1930	C₂×C₃₁	1	2	yes	complemented
253	#1801	C₂×C₃₁	1	1	yes	characteristic, complemented
254	C₂²×C₃₁	—	2	1	no	—
255	C₂²×C₃₁	—	2	1	no	—
256	C₂²×C₃₁	—	4	1	no	—
257	C₂²×C₃₁	—	4	1	no	—
258	C₂²×C₃₁	—	4	1	no	—
259	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
260	C₂²×C₃₁	#553	1	1	yes	characteristic
261	C₂²×C₃₁	#555	1	1	yes	characteristic
262	C₄×C₃₁	—	2	1	no	—
263	C₄×C₃₁	—	2	1	no	—
264	C₄×C₃₁	—	2	1	no	—
265	C₄×C₃₁	—	4	1	no	—
266	C₄×C₃₁	—	4	2	no	—
267	C₄×C₃₁	—	4	1	no	—
268	C₄×C₃₁	—	4	1	no	—
269	C₄×C₃₁	—	8	2	no	—
270	C₄×C₃₁	—	8	1	no	—
271	C₄×C₃₁	#473	1	2	yes	—
272	C₄×C₃₁	#525	1	1	yes	characteristic
273	C₄×C₃₁	#535	1	1	yes	characteristic
274	C₄×C₃₁	#546	1	1	yes	characteristic
275	C₄×C₃₁	#550	1	1	yes	characteristic
276	C₂×#137	C₂²×C₃₁	1	1	yes	characteristic
277	C₂×#137	C₂²×C₃₁	1	2	yes	—
278	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	—
279	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
280	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
281	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
282	#462	—	2	1	no	—
283	#462	—	2	1	no	—
284	#462	—	2	1	no	—
285	#462	—	2	1	no	—
286	#462	—	2	2	no	—
287	#461	—	2	1	no	—
288	#461	—	2	1	no	—
289	#441	—	2	1	no	—
290	#440	—	2	1	no	—
291	C₄×Q₁₆	—	2	1	no	—
292	C₄×SD₁₆	—	2	1	no	—
293	#433	—	2	2	no	—
294	#432	—	2	1	no	—
295	#431	—	2	1	no	—
296	C₂×#108	C₂²×C₃₁	1	2	yes	—
297	#428	C₂²×C₃₁	1	1	yes	characteristic
298	#423	C₂²×C₃₁	1	1	yes	characteristic
299	#423	C₂²×C₃₁	1	1	yes	characteristic
300	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
301	#419	C₂²×C₃₁	1	1	yes	characteristic
302	#419	C₂²×C₃₁	1	2	yes	—
303	#417	C₂²×C₃₁	1	1	yes	characteristic
304	C₂×#103	C₂²×C₃₁	1	2	yes	—
305	C₂×#101	C₂²×C₃₁	1	2	yes	—
306	C₂×#101	C₂²×C₃₁	1	2	yes	—
307	C₂×#101	C₂²×C₃₁	1	2	yes	—
308	#408	C₂²×C₃₁	1	2	yes	—
309	#405	C₂²×C₃₁	1	2	yes	—
310	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
311	#343	C₂²×C₃₁	1	2	yes	—
312	#341	C₂²×C₃₁	1	2	yes	—
313	C₂×C₃₁	—	2	1	no	—
314	C₂×C₃₁	—	2	1	no	—
315	C₂×C₃₁	—	8	1	no	—
316	C₂×C₃₁	#2414	1	1	yes	characteristic
317	C₂×C₃₁	C₂×#473	1	1	yes	characteristic
318	C₂×C₃₁	#3069	1	1	yes	characteristic
319	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
320	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
321	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
322	#137	—	2	2	no	—
323	#137	—	2	2	no	—
324	#137	—	4	2	no	—
325	#137	—	4	2	no	—
326	C₂×Q₁₆	—	2	1	no	—
327	C₂×Q₁₆	—	2	2	no	—
328	C₂×Q₁₆	—	2	4	no	—
329	C₂×Q₁₆	—	2	2	no	—
330	C₂×Q₁₆	—	2	2	no	—
331	C₂×Q₁₆	—	4	2	no	—
332	C₂×SD₁₆	—	2	1	no	—
333	C₂×SD₁₆	—	2	2	no	—
334	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
335	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
336	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
337	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
338	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
339	C₄⋊Q₈	—	2	1	no	—
340	C₄⋊Q₈	—	2	1	no	—
341	C₄⋊Q₈	—	2	1	no	—
342	C₄⋊Q₈	—	2	1	no	—
343	C₄×Q₈	—	2	1	no	—
344	C₄×Q₈	—	2	1	no	—
345	C₄×Q₈	—	2	1	no	—
346	C₄×Q₈	—	2	1	no	—
347	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
348	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
349	C₄×D₄	—	2	1	no	—
350	C₄×D₄	—	2	1	no	—
351	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
352	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
353	#117	—	2	1	no	—
354	#117	C₂³×C₃₁	1	1	yes	characteristic
355	C₂×C₄⋊C₄	—	2	1	no	—
356	C₂×C₄⋊C₄	—	2	1	no	—
357	C₂×C₄⋊C₄	—	2	1	no	—
358	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
359	C₂×C₄²	—	2	1	no	—
360	C₂×C₄²	—	2	1	no	—
361	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
362	#108	—	4	2	no	—
363	#107	—	2	1	no	—
364	#106	—	2	1	no	—
365	C₄⋊C₈	—	2	1	no	—
366	C₄⋊C₈	—	2	1	no	—
367	C₄⋊C₈	—	2	1	no	—
368	C₄⋊C₈	—	2	1	no	—
369	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
370	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
371	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
372	#103	—	2	1	no	—
373	#103	—	2	2	no	—
374	#103	—	2	2	no	—
375	#103	—	2	1	no	—
376	#103	—	2	1	no	—
377	#103	—	2	2	no	—
378	#103	—	2	2	no	—
379	#102	—	2	1	no	—
380	#101	—	2	4	no	—
381	#101	—	4	2	no	—
382	#101	—	4	2	no	—
383	C₂²⋊C₈	—	2	2	no	—
384	C₂²⋊C₈	—	2	2	no	—
385	#97	—	2	2	no	—
386	#97	—	2	1	no	—
387	C₄×C₈	—	2	2	no	—
388	C₄×C₈	—	2	1	no	—
389	C₃₁	#24672	1	1	yes	characteristic, direct product factor, Sylow
390	#41	—	2	1	no	—
391	#41	—	2	1	no	—
392	#41	—	2	1	no	—
393	#41	—	2	1	no	—
394	C₂×Q₈	—	2	1	no	—
395	C₂×Q₈	—	2	1	no	—
396	C₂×Q₈	—	2	1	no	—
397	C₂×Q₈	—	2	2	no	—
398	C₂×Q₈	—	2	2	no	—
399	C₂×Q₈	—	2	1	no	—
400	C₂×Q₈	—	2	2	no	—
401	C₂×Q₈	—	2	2	no	—
402	C₂×Q₈	—	2	1	no	—
403	C₂×Q₈	—	2	2	no	—
404	C₂×Q₈	—	2	2	no	—
405	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
406	C₂×Q₈	#41×C₃₁	1	2	yes	—
407	C₂×D₄	—	2	1	no	—
408	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
409	C₂²×C₄	—	2	1	no	—
410	C₂²×C₄	—	2	1	no	—
411	C₂²×C₄	—	2	1	no	—
412	C₂²×C₄	—	2	1	no	—
413	C₂²×C₄	—	2	2	no	—
414	C₂²×C₄	—	2	1	no	—
415	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
416	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
417	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
418	Q₁₆	—	2	4	no	—
419	Q₁₆	—	4	4	no	—
420	Q₁₆	—	4	4	no	—
421	Q₁₆	—	4	4	no	—
422	SD₁₆	—	2	4	no	—
423	SD₁₆	—	4	4	no	—
424	M₁₆	—	2	1	no	—
425	M₁₆	—	2	2	no	—
426	M₁₆	—	2	1	no	—
427	M₁₆	—	2	2	no	—
428	M₁₆	—	2	2	no	—
429	M₁₆	—	2	1	no	—
430	M₁₆	—	2	2	no	—
431	M₁₆	#41×C₃₁	1	2	yes	—
432	C₂×C₈	—	2	1	no	—
433	C₂×C₈	—	2	2	no	—
434	C₂×C₈	—	2	2	no	—
435	C₂×C₈	—	2	2	no	—
436	C₂×C₈	—	2	1	no	—
437	C₂×C₈	—	2	2	no	—
438	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
439	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
440	C₄⋊C₄	—	2	1	no	—
441	C₄⋊C₄	—	2	1	no	—
442	C₄⋊C₄	—	2	1	no	—
443	C₄⋊C₄	—	2	1	no	—
444	C₄⋊C₄	—	4	1	no	—
445	C₄⋊C₄	—	4	1	no	—
446	C₄⋊C₄	—	4	1	no	—
447	C₄⋊C₄	—	4	1	no	—
448	C₄⋊C₄	—	4	1	no	—
449	C₄⋊C₄	—	4	1	no	—
450	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
451	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
452	C₂²⋊C₄	—	2	1	no	—
453	C₂²⋊C₄	—	2	1	no	—
454	C₂²⋊C₄	—	2	1	no	—
455	C₄²	—	2	1	no	—
456	C₄²	—	2	1	no	—
457	C₄²	—	2	1	no	—
458	C₄²	—	4	1	no	—
459	C₄²	—	4	1	no	—
460	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
461	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
462	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
463	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
464	C₂³	—	2	1	no	—
465	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
466	Q₈	—	2	2	no	—
467	Q₈	—	2	1	no	—
468	Q₈	—	2	1	no	—
469	Q₈	—	2	2	no	—
470	Q₈	—	4	1	no	—
471	Q₈	—	4	1	no	—
472	Q₈	—	4	2	no	—
473	Q₈	—	4	2	no	—
474	Q₈	—	4	2	no	—
475	Q₈	—	4	2	no	—
476	D₄	—	2	1	no	—
477	D₄	—	2	1	no	—
478	D₄	—	4	1	no	—
479	D₄	—	4	1	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	1	no	—
482	C₂×C₄	—	2	1	no	—
483	C₂×C₄	—	2	1	no	—
484	C₂×C₄	—	2	2	no	—
485	C₂×C₄	—	4	1	no	—
486	C₂×C₄	—	4	1	no	—
487	C₂×C₄	—	4	1	no	—
488	C₂×C₄	—	4	2	no	—
489	C₂×C₄	—	4	1	no	—
490	C₂×C₄	—	4	1	no	—
491	C₂×C₄	—	4	1	no	—
492	C₂×C₄	—	4	2	no	—
493	C₂×C₄	—	4	2	no	—
494	C₂×C₄	—	4	2	no	—
495	C₂×C₄	—	8	1	no	—
496	C₂×C₄	—	8	1	no	—
497	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
498	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
499	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
500	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
501	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
502	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
503	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
504	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
505	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
506	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
507	C₈	—	2	2	no	—
508	C₈	—	2	2	no	—
509	C₈	—	2	2	no	—
510	C₈	—	4	1	no	—
511	C₈	—	4	2	no	—
512	C₈	—	4	2	no	—
513	C₂²	—	2	1	no	—
514	C₂²	—	2	1	no	—
515	C₂²	—	4	1	no	—
516	C₂²	—	4	1	no	—
517	C₂²	—	4	1	no	—
518	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
519	C₂²	C₃₁×#553	1	1	yes	characteristic
520	C₂²	C₃₁×#555	1	1	yes	characteristic
521	C₄	—	2	1	no	—
522	C₄	—	2	1	no	—
523	C₄	—	2	1	no	—
524	C₄	—	4	1	no	—
525	C₄	—	4	2	no	—
526	C₄	—	4	1	no	—
527	C₄	—	4	1	no	—
528	C₄	—	8	2	no	—
529	C₄	—	8	1	no	—
530	C₄	C₃₁×#473	1	2	yes	—
531	C₄	C₃₁×#525	1	1	yes	characteristic
532	C₄	C₃₁×#535	1	1	yes	characteristic
533	C₄	C₃₁×#546	1	1	yes	characteristic
534	C₄	C₃₁×#550	1	1	yes	characteristic
535	C₂	—	2	1	no	—
536	C₂	—	2	1	no	—
537	C₂	—	8	1	no	—
538	C₂	C₃₁×#2414	1	1	yes	characteristic
539	C₂	C₂×C₃₁×#473	1	1	yes	characteristic
540	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
2. C₂×C₃₁×#462	C₂	semidirect
7. C₃₁×#1979	C₂	semidirect
8. C₃₁×#1966	C₂	semidirect
9. C₃₁×#1950	C₂	semidirect
10. C₃₁×#1930	C₂	semidirect
11. C₃₁×#1801	C₂	semidirect
193. #24672	C₃₁	direct (N × H)
244. C₂×#462	C₂×C₃₁	semidirect
249. #1979	C₂×C₃₁	semidirect
250. #1966	C₂×C₃₁	semidirect
251. #1950	C₂×C₃₁	semidirect
252. #1930	C₂×C₃₁	semidirect
253. #1801	C₂×C₃₁	semidirect
389. C₃₁	#24672	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3039	2
C₂×C₃₁×#462	2
C₄×C₃₁×#137	2
C₃₁×#2925	2
C₃₁×#2052	2
C₃₁×#2020	2
C₃₁×#1979	2
C₃₁×#1966	2
C₃₁×#1950	2
C₃₁×#1930	2
C₃₁×#1801	2
#24672	31

Subgroups — C31×#24672 #21481431

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24672 #21481431