Subgroups — C₃₁×#18599 #21475358

Order 7936.

1190 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3496	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3235	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3182	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3101	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1647	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1639	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1635	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1629	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1553	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1490	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1484	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×H(5,𝔽₂)	—	2	1	no	—
13	C₃₁×#547	—	2	1	no	—
14	C₃₁×#547	C₂²	1	2	yes	—
15	C₃₁×#546	—	2	2	no	—
16	C₃₁×#546	—	2	2	no	—
17	D₄²×C₃₁	—	2	1	no	—
18	D₄²×C₃₁	C₂²	1	2	yes	—
19	C₃₁×#537	—	2	1	no	—
20	C₃₁×#534	—	2	2	no	—
21	C₃₁×#533	C₂²	1	1	yes	characteristic, complemented
22	C₃₁×#532	C₂²	1	1	yes	characteristic
23	C₃₁×#532	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#121	—	2	2	no	—
25	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
26	C₃₁×#501	—	2	1	no	—
27	C₃₁×#500	C₂²	1	1	yes	characteristic, complemented
28	C₃₁×#492	C₂²	1	1	yes	characteristic, complemented
29	C₃₁×#490	—	2	1	no	—
30	C₃₁×#487	C₂²	1	1	yes	characteristic, complemented
31	C₃₁×#486	C₂²	1	1	yes	characteristic, complemented
32	C₃₁×#483	—	2	1	no	—
33	C₃₁×#482	—	2	2	no	—
34	C₃₁×#476	C₂²	1	2	yes	complemented
35	C₃₁×#476	C₂²	1	2	yes	complemented
36	C₃₁×#474	C₂²	1	2	yes	complemented
37	C₃₁×#468	—	2	2	no	—
38	C₃₁×#461	C₂²	1	2	yes	complemented
39	C₃₁×#450	—	2	1	no	—
40	C₃₁×#440	—	2	2	no	—
41	C₄×SD₁₆×C₃₁	—	2	2	no	—
42	C₃₁×#432	C₂²	1	1	yes	characteristic, complemented
43	C₃₁×#431	C₂²	1	1	yes	characteristic, complemented
44	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
45	C₃₁×#332	C₂²	1	2	yes	complemented
46	C₃₁×#331	C₂²	1	2	yes	complemented
47	C₃₁×D₄⋊C₈	C₂²	1	2	yes	complemented
48	C₃₁×D₄⋊C₈	C₂²	1	2	yes	complemented
49	C₃₁×D₄⋊C₈	C₂²	1	2	yes	complemented
50	C₃₁×D₄⋊C₈	C₂²	1	2	yes	complemented
51	C₃₁×H(5,𝔽₂)	—	4	8	no	—
52	C₂×#41×C₃₁	—	2	2	no	—
53	C₂×#41×C₃₁	—	2	2	no	—
54	C₂×#41×C₃₁	—	2	1	no	—
55	C₂×#41×C₃₁	—	2	1	no	—
56	C₂²×D₄×C₃₁	—	2	4	no	—
57	C₂²×D₄×C₃₁	—	2	2	no	—
58	C₂²×D₄×C₃₁	—	2	2	no	—
59	C₂²×D₄×C₃₁	—	2	2	no	—
60	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
61	C₂³×C₄×C₃₁	—	2	1	no	—
62	C₂×SD₁₆×C₃₁	—	2	2	no	—
63	C₂×SD₁₆×C₃₁	—	2	2	no	—
64	C₂×M₁₆×C₃₁	—	2	1	no	—
65	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
66	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
67	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
68	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
69	C₃₁×#127	D₄	1	1	yes	characteristic
70	C₃₁×#127	C₂³	1	1	yes	characteristic
71	C₃₁×#125	—	2	1	no	—
72	C₃₁×#124	—	2	1	no	—
73	C₃₁×#124	—	2	1	no	—
74	C₃₁×#123	—	2	4	no	—
75	C₃₁×#123	—	2	4	no	—
76	C₃₁×C₂²⋊Q₈	—	2	2	no	—
77	C₃₁×C₂²⋊Q₈	—	2	2	no	—
78	C₃₁×C₂²⋊Q₈	—	2	1	no	—
79	C₃₁×C₂²⋊Q₈	—	2	1	no	—
80	C₃₁×#121	—	2	4	no	—
81	C₃₁×#121	—	2	4	no	—
82	C₃₁×#121	—	2	2	no	—
83	C₃₁×#121	—	2	2	no	—
84	C₃₁×#121	—	2	1	no	—
85	C₃₁×#121	—	2	1	no	—
86	C₃₁×#121	—	2	2	no	—
87	C₃₁×#121	—	2	2	no	—
88	C₃₁×#121	D₄	1	2	yes	—
89	C₃₁×#121	D₄	1	2	yes	—
90	C₃₁×#120	—	2	4	no	—
91	C₃₁×#120	—	2	4	no	—
92	C₄×Q₈×C₃₁	—	2	1	no	—
93	C₄×Q₈×C₃₁	—	2	1	no	—
94	C₄×D₄×C₃₁	—	2	1	no	—
95	C₄×D₄×C₃₁	—	2	1	no	—
96	C₄×D₄×C₃₁	—	2	1	no	—
97	C₄×D₄×C₃₁	—	2	1	no	—
98	C₄×D₄×C₃₁	D₄	1	2	yes	—
99	C₄×D₄×C₃₁	D₄	1	2	yes	—
100	C₄×D₄×C₃₁	C₂³	1	2	yes	—
101	C₄×D₄×C₃₁	C₂³	1	2	yes	—
102	C₃₁×#117	—	2	1	no	—
103	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
104	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
105	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
106	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
107	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
108	C₃₁×#107	—	2	2	no	—
109	C₃₁×#107	—	2	2	no	—
110	C₃₁×#106	—	2	2	no	—
111	C₃₁×#106	—	2	2	no	—
112	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
113	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
114	C₃₁×#103	—	2	2	no	—
115	C₃₁×#103	—	2	2	no	—
116	C₃₁×#102	—	2	2	no	—
117	C₃₁×#102	—	2	2	no	—
118	C₃₁×#102	—	2	2	no	—
119	C₃₁×#102	—	2	2	no	—
120	C₃₁×C₂²⋊C₈	—	2	1	no	—
121	C₃₁×C₂²⋊C₈	—	2	1	no	—
122	C₃₁×#97	—	2	1	no	—
123	C₃₁×#97	—	2	1	no	—
124	C₄×C₈×C₃₁	—	2	1	no	—
125	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
126	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
127	C₂⁴×C₃₁	—	2	1	no	—
128	C₂⁴×C₃₁	—	2	4	no	—
129	#41×C₃₁	—	4	4	no	—
130	#41×C₃₁	—	4	4	no	—
131	#41×C₃₁	—	4	8	no	—
132	#41×C₃₁	—	4	4	no	—
133	#41×C₃₁	—	4	4	no	—
134	C₂×Q₈×C₃₁	—	2	1	no	—
135	C₂×Q₈×C₃₁	—	2	1	no	—
136	C₂×Q₈×C₃₁	—	2	1	no	—
137	C₂×Q₈×C₃₁	—	2	1	no	—
138	C₂×D₄×C₃₁	—	2	4	no	—
139	C₂×D₄×C₃₁	—	2	4	no	—
140	C₂×D₄×C₃₁	—	2	4	no	—
141	C₂×D₄×C₃₁	—	2	2	no	—
142	C₂×D₄×C₃₁	—	2	2	no	—
143	C₂×D₄×C₃₁	—	2	4	no	—
144	C₂×D₄×C₃₁	—	2	1	no	—
145	C₂×D₄×C₃₁	—	2	1	no	—
146	C₂×D₄×C₃₁	—	2	1	no	—
147	C₂×D₄×C₃₁	—	2	1	no	—
148	C₂×D₄×C₃₁	—	2	1	no	—
149	C₂×D₄×C₃₁	—	4	8	no	—
150	C₂×D₄×C₃₁	—	4	4	no	—
151	C₂×D₄×C₃₁	—	4	16	no	—
152	C₂×D₄×C₃₁	—	4	4	no	—
153	C₂×D₄×C₃₁	—	8	4	no	—
154	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
155	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
156	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
157	C₂²×C₄×C₃₁	—	2	4	no	—
158	C₂²×C₄×C₃₁	—	2	1	no	—
159	C₂²×C₄×C₃₁	—	2	2	no	—
160	C₂²×C₄×C₃₁	—	2	2	no	—
161	C₂²×C₄×C₃₁	—	2	2	no	—
162	C₂²×C₄×C₃₁	—	2	2	no	—
163	C₂²×C₄×C₃₁	—	8	2	no	—
164	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
167	SD₁₆×C₃₁	—	4	4	no	—
168	SD₁₆×C₃₁	—	4	4	no	—
169	M₁₆×C₃₁	—	4	2	no	—
170	C₂×C₈×C₃₁	—	2	1	no	—
171	C₂×C₈×C₃₁	—	2	2	no	—
172	C₂×C₈×C₃₁	—	2	1	no	—
173	C₂×C₈×C₃₁	—	2	2	no	—
174	C₄⋊C₄×C₃₁	—	2	4	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	—	2	1	no	—
179	C₄⋊C₄×C₃₁	—	2	1	no	—
180	C₄⋊C₄×C₃₁	—	2	1	no	—
181	C₄⋊C₄×C₃₁	—	2	1	no	—
182	C₄⋊C₄×C₃₁	—	4	1	no	—
183	C₄⋊C₄×C₃₁	—	4	1	no	—
184	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
185	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
186	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
187	C₂²⋊C₄×C₃₁	—	2	4	no	—
188	C₂²⋊C₄×C₃₁	—	2	4	no	—
189	C₂²⋊C₄×C₃₁	—	2	2	no	—
190	C₂²⋊C₄×C₃₁	—	2	2	no	—
191	C₂²⋊C₄×C₃₁	—	2	2	no	—
192	C₂²⋊C₄×C₃₁	—	2	2	no	—
193	C₂²⋊C₄×C₃₁	—	2	2	no	—
194	C₂²⋊C₄×C₃₁	—	2	2	no	—
195	C₂²⋊C₄×C₃₁	—	4	1	no	—
196	C₄²×C₃₁	—	4	1	no	—
197	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
198	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
199	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
200	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
201	#18599	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
202	C₂³×C₃₁	—	2	4	no	—
203	C₂³×C₃₁	—	2	2	no	—
204	C₂³×C₃₁	—	2	2	no	—
205	C₂³×C₃₁	—	2	2	no	—
206	C₂³×C₃₁	—	4	8	no	—
207	C₂³×C₃₁	—	8	1	no	—
208	C₂³×C₃₁	—	8	1	no	—
209	C₂³×C₃₁	—	8	4	no	—
210	C₂³×C₃₁	—	8	4	no	—
211	C₂³×C₃₁	#120	1	1	yes	characteristic
212	Q₈×C₃₁	—	4	2	no	—
213	Q₈×C₃₁	—	4	2	no	—
214	Q₈×C₃₁	—	4	2	no	—
215	Q₈×C₃₁	—	4	2	no	—
216	D₄×C₃₁	—	2	4	no	—
217	D₄×C₃₁	—	2	2	no	—
218	D₄×C₃₁	—	2	4	no	—
219	D₄×C₃₁	—	4	2	no	—
220	D₄×C₃₁	—	4	2	no	—
221	D₄×C₃₁	—	4	4	no	—
222	D₄×C₃₁	—	4	4	no	—
223	D₄×C₃₁	—	4	8	no	—
224	D₄×C₃₁	—	4	8	no	—
225	D₄×C₃₁	—	4	2	no	—
226	D₄×C₃₁	—	4	2	no	—
227	D₄×C₃₁	—	8	1	no	—
228	D₄×C₃₁	—	8	4	no	—
229	D₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
230	C₂×C₄×C₃₁	—	2	2	no	—
231	C₂×C₄×C₃₁	—	2	2	no	—
232	C₂×C₄×C₃₁	—	2	2	no	—
233	C₂×C₄×C₃₁	—	2	2	no	—
234	C₂×C₄×C₃₁	—	2	2	no	—
235	C₂×C₄×C₃₁	—	4	1	no	—
236	C₂×C₄×C₃₁	—	4	4	no	—
237	C₂×C₄×C₃₁	—	4	8	no	—
238	C₂×C₄×C₃₁	—	4	2	no	—
239	C₂×C₄×C₃₁	—	4	2	no	—
240	C₂×C₄×C₃₁	—	4	4	no	—
241	C₂×C₄×C₃₁	—	4	1	no	—
242	C₂×C₄×C₃₁	—	4	1	no	—
243	C₂×C₄×C₃₁	—	8	2	no	—
244	C₂×C₄×C₃₁	—	8	1	no	—
245	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
246	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
247	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
248	C₂×C₄×C₃₁	C₂²×D₄	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	—	2	2	no	—
252	C₈×C₃₁	—	4	1	no	—
253	C₈×C₃₁	—	4	2	no	—
254	C₈×C₃₁	—	4	2	no	—
255	#3496	C₂×C₃₁	1	1	yes	characteristic
256	#3235	C₂×C₃₁	1	1	yes	characteristic, complemented
257	#3182	C₂×C₃₁	1	1	yes	characteristic, complemented
258	#3101	C₂×C₃₁	1	1	yes	characteristic, complemented
259	#1647	C₂×C₃₁	1	2	yes	complemented
260	#1639	C₂×C₃₁	1	2	yes	complemented
261	#1635	C₂×C₃₁	1	2	yes	complemented
262	#1629	C₂×C₃₁	1	2	yes	complemented
263	#1553	C₂×C₃₁	1	1	yes	characteristic, complemented
264	#1490	C₂×C₃₁	1	1	yes	characteristic, complemented
265	#1484	C₂×C₃₁	1	1	yes	characteristic, complemented
266	C₂²×C₃₁	—	2	1	no	—
267	C₂²×C₃₁	—	2	4	no	—
268	C₂²×C₃₁	—	2	1	no	—
269	C₂²×C₃₁	—	2	1	no	—
270	C₂²×C₃₁	—	4	2	no	—
271	C₂²×C₃₁	—	4	2	no	—
272	C₂²×C₃₁	—	4	2	no	—
273	C₂²×C₃₁	—	4	2	no	—
274	C₂²×C₃₁	—	4	2	no	—
275	C₂²×C₃₁	—	4	2	no	—
276	C₂²×C₃₁	—	4	2	no	—
277	C₂²×C₃₁	—	4	2	no	—
278	C₂²×C₃₁	—	8	4	no	—
279	C₂²×C₃₁	—	16	8	no	—
280	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
281	C₄×C₃₁	—	2	1	no	—
282	C₄×C₃₁	—	2	1	no	—
283	C₄×C₃₁	—	2	1	no	—
284	C₄×C₃₁	—	2	1	no	—
285	C₄×C₃₁	—	4	2	no	—
286	C₄×C₃₁	—	4	2	no	—
287	C₄×C₃₁	—	4	2	no	—
288	C₄×C₃₁	—	8	1	no	—
289	C₄×C₃₁	—	8	1	no	—
290	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
291	C₄×C₃₁	#577	1	2	yes	—
292	C₂×H(5,𝔽₂)	—	2	1	no	—
293	#547	—	2	1	no	—
294	#547	C₂²×C₃₁	1	2	yes	—
295	#546	—	2	2	no	—
296	#546	—	2	2	no	—
297	D₄²	—	2	1	no	—
298	D₄²	C₂²×C₃₁	1	2	yes	—
299	#537	—	2	1	no	—
300	#534	—	2	2	no	—
301	#533	C₂²×C₃₁	1	1	yes	characteristic, complemented
302	#532	C₂²×C₃₁	1	1	yes	characteristic
303	#532	C₂²×C₃₁	1	1	yes	characteristic
304	C₂×#121	—	2	2	no	—
305	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
306	#501	—	2	1	no	—
307	#500	C₂²×C₃₁	1	1	yes	characteristic, complemented
308	#492	C₂²×C₃₁	1	1	yes	characteristic, complemented
309	#490	—	2	1	no	—
310	#487	C₂²×C₃₁	1	1	yes	characteristic, complemented
311	#486	C₂²×C₃₁	1	1	yes	characteristic, complemented
312	#483	—	2	1	no	—
313	#482	—	2	2	no	—
314	#476	C₂²×C₃₁	1	2	yes	complemented
315	#476	C₂²×C₃₁	1	2	yes	complemented
316	#474	C₂²×C₃₁	1	2	yes	complemented
317	#468	—	2	2	no	—
318	#461	C₂²×C₃₁	1	2	yes	complemented
319	#450	—	2	1	no	—
320	#440	—	2	2	no	—
321	C₄×SD₁₆	—	2	2	no	—
322	#432	C₂²×C₃₁	1	1	yes	characteristic, complemented
323	#431	C₂²×C₃₁	1	1	yes	characteristic, complemented
324	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
325	#332	C₂²×C₃₁	1	2	yes	complemented
326	#331	C₂²×C₃₁	1	2	yes	complemented
327	D₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
328	D₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
329	D₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
330	D₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
331	C₂×C₃₁	—	4	1	no	—
332	C₂×C₃₁	—	4	4	no	—
333	C₂×C₃₁	—	8	2	no	—
334	C₂×C₃₁	—	8	2	no	—
335	C₂×C₃₁	#3003	1	1	yes	characteristic
336	C₂×C₃₁	#3009	1	1	yes	characteristic
337	C₂×C₃₁	#3018	1	1	yes	characteristic
338	H(5,𝔽₂)	—	4	8	no	—
339	C₂×#41	—	2	2	no	—
340	C₂×#41	—	2	2	no	—
341	C₂×#41	—	2	1	no	—
342	C₂×#41	—	2	1	no	—
343	C₂²×D₄	—	2	4	no	—
344	C₂²×D₄	—	2	2	no	—
345	C₂²×D₄	—	2	2	no	—
346	C₂²×D₄	—	2	2	no	—
347	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
348	C₂³×C₄	—	2	1	no	—
349	C₂×SD₁₆	—	2	2	no	—
350	C₂×SD₁₆	—	2	2	no	—
351	C₂×M₁₆	—	2	1	no	—
352	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
353	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
354	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
355	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
356	#127	D₄×C₃₁	1	1	yes	characteristic
357	#127	C₂³×C₃₁	1	1	yes	characteristic
358	#125	—	2	1	no	—
359	#124	—	2	1	no	—
360	#124	—	2	1	no	—
361	#123	—	2	4	no	—
362	#123	—	2	4	no	—
363	C₂²⋊Q₈	—	2	2	no	—
364	C₂²⋊Q₈	—	2	2	no	—
365	C₂²⋊Q₈	—	2	1	no	—
366	C₂²⋊Q₈	—	2	1	no	—
367	#121	—	2	4	no	—
368	#121	—	2	4	no	—
369	#121	—	2	2	no	—
370	#121	—	2	2	no	—
371	#121	—	2	1	no	—
372	#121	—	2	1	no	—
373	#121	—	2	2	no	—
374	#121	—	2	2	no	—
375	#121	D₄×C₃₁	1	2	yes	—
376	#121	D₄×C₃₁	1	2	yes	—
377	#120	—	2	4	no	—
378	#120	—	2	4	no	—
379	C₄×Q₈	—	2	1	no	—
380	C₄×Q₈	—	2	1	no	—
381	C₄×D₄	—	2	1	no	—
382	C₄×D₄	—	2	1	no	—
383	C₄×D₄	—	2	1	no	—
384	C₄×D₄	—	2	1	no	—
385	C₄×D₄	D₄×C₃₁	1	2	yes	—
386	C₄×D₄	D₄×C₃₁	1	2	yes	—
387	C₄×D₄	C₂³×C₃₁	1	2	yes	—
388	C₄×D₄	C₂³×C₃₁	1	2	yes	—
389	#117	—	2	1	no	—
390	C₂×C₄⋊C₄	—	2	2	no	—
391	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
392	C₂×C₂²⋊C₄	—	2	4	no	—
393	C₂×C₂²⋊C₄	—	2	1	no	—
394	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
395	#107	—	2	2	no	—
396	#107	—	2	2	no	—
397	#106	—	2	2	no	—
398	#106	—	2	2	no	—
399	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
400	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
401	#103	—	2	2	no	—
402	#103	—	2	2	no	—
403	#102	—	2	2	no	—
404	#102	—	2	2	no	—
405	#102	—	2	2	no	—
406	#102	—	2	2	no	—
407	C₂²⋊C₈	—	2	1	no	—
408	C₂²⋊C₈	—	2	1	no	—
409	#97	—	2	1	no	—
410	#97	—	2	1	no	—
411	C₄×C₈	—	2	1	no	—
412	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
413	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
414	C₃₁	#18599	1	1	yes	characteristic, direct product factor, Sylow
415	C₂⁴	—	2	1	no	—
416	C₂⁴	—	2	4	no	—
417	#41	—	4	4	no	—
418	#41	—	4	4	no	—
419	#41	—	4	8	no	—
420	#41	—	4	4	no	—
421	#41	—	4	4	no	—
422	C₂×Q₈	—	2	1	no	—
423	C₂×Q₈	—	2	1	no	—
424	C₂×Q₈	—	2	1	no	—
425	C₂×Q₈	—	2	1	no	—
426	C₂×D₄	—	2	4	no	—
427	C₂×D₄	—	2	4	no	—
428	C₂×D₄	—	2	4	no	—
429	C₂×D₄	—	2	2	no	—
430	C₂×D₄	—	2	2	no	—
431	C₂×D₄	—	2	4	no	—
432	C₂×D₄	—	2	1	no	—
433	C₂×D₄	—	2	1	no	—
434	C₂×D₄	—	2	1	no	—
435	C₂×D₄	—	2	1	no	—
436	C₂×D₄	—	2	1	no	—
437	C₂×D₄	—	4	8	no	—
438	C₂×D₄	—	4	4	no	—
439	C₂×D₄	—	4	16	no	—
440	C₂×D₄	—	4	4	no	—
441	C₂×D₄	—	8	4	no	—
442	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
443	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
444	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
445	C₂²×C₄	—	2	4	no	—
446	C₂²×C₄	—	2	1	no	—
447	C₂²×C₄	—	2	2	no	—
448	C₂²×C₄	—	2	2	no	—
449	C₂²×C₄	—	2	2	no	—
450	C₂²×C₄	—	2	2	no	—
451	C₂²×C₄	—	8	2	no	—
452	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
453	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
454	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
455	SD₁₆	—	4	4	no	—
456	SD₁₆	—	4	4	no	—
457	M₁₆	—	4	2	no	—
458	C₂×C₈	—	2	1	no	—
459	C₂×C₈	—	2	2	no	—
460	C₂×C₈	—	2	1	no	—
461	C₂×C₈	—	2	2	no	—
462	C₄⋊C₄	—	2	4	no	—
463	C₄⋊C₄	—	2	1	no	—
464	C₄⋊C₄	—	2	1	no	—
465	C₄⋊C₄	—	2	1	no	—
466	C₄⋊C₄	—	2	1	no	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	—	2	1	no	—
470	C₄⋊C₄	—	4	1	no	—
471	C₄⋊C₄	—	4	1	no	—
472	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
473	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
474	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
475	C₂²⋊C₄	—	2	4	no	—
476	C₂²⋊C₄	—	2	4	no	—
477	C₂²⋊C₄	—	2	2	no	—
478	C₂²⋊C₄	—	2	2	no	—
479	C₂²⋊C₄	—	2	2	no	—
480	C₂²⋊C₄	—	2	2	no	—
481	C₂²⋊C₄	—	2	2	no	—
482	C₂²⋊C₄	—	2	2	no	—
483	C₂²⋊C₄	—	4	1	no	—
484	C₄²	—	4	1	no	—
485	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
486	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
487	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
488	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
489	C₂³	—	2	4	no	—
490	C₂³	—	2	2	no	—
491	C₂³	—	2	2	no	—
492	C₂³	—	2	2	no	—
493	C₂³	—	4	8	no	—
494	C₂³	—	8	1	no	—
495	C₂³	—	8	1	no	—
496	C₂³	—	8	4	no	—
497	C₂³	—	8	4	no	—
498	C₂³	C₃₁×#120	1	1	yes	characteristic
499	Q₈	—	4	2	no	—
500	Q₈	—	4	2	no	—
501	Q₈	—	4	2	no	—
502	Q₈	—	4	2	no	—
503	D₄	—	2	4	no	—
504	D₄	—	2	2	no	—
505	D₄	—	2	4	no	—
506	D₄	—	4	2	no	—
507	D₄	—	4	2	no	—
508	D₄	—	4	4	no	—
509	D₄	—	4	4	no	—
510	D₄	—	4	8	no	—
511	D₄	—	4	8	no	—
512	D₄	—	4	2	no	—
513	D₄	—	4	2	no	—
514	D₄	—	8	1	no	—
515	D₄	—	8	4	no	—
516	D₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	2	2	no	—
522	C₂×C₄	—	4	1	no	—
523	C₂×C₄	—	4	4	no	—
524	C₂×C₄	—	4	8	no	—
525	C₂×C₄	—	4	2	no	—
526	C₂×C₄	—	4	2	no	—
527	C₂×C₄	—	4	4	no	—
528	C₂×C₄	—	4	1	no	—
529	C₂×C₄	—	4	1	no	—
530	C₂×C₄	—	8	2	no	—
531	C₂×C₄	—	8	1	no	—
532	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
533	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
534	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
535	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
536	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
537	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
538	C₈	—	2	2	no	—
539	C₈	—	4	1	no	—
540	C₈	—	4	2	no	—
541	C₈	—	4	2	no	—
542	C₂²	—	2	1	no	—
543	C₂²	—	2	4	no	—
544	C₂²	—	2	1	no	—
545	C₂²	—	2	1	no	—
546	C₂²	—	4	2	no	—
547	C₂²	—	4	2	no	—
548	C₂²	—	4	2	no	—
549	C₂²	—	4	2	no	—
550	C₂²	—	4	2	no	—
551	C₂²	—	4	2	no	—
552	C₂²	—	4	2	no	—
553	C₂²	—	4	2	no	—
554	C₂²	—	8	4	no	—
555	C₂²	—	16	8	no	—
556	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
557	C₄	—	2	1	no	—
558	C₄	—	2	1	no	—
559	C₄	—	2	1	no	—
560	C₄	—	2	1	no	—
561	C₄	—	4	2	no	—
562	C₄	—	4	2	no	—
563	C₄	—	4	2	no	—
564	C₄	—	8	1	no	—
565	C₄	—	8	1	no	—
566	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
567	C₄	C₃₁×#577	1	2	yes	—
568	C₂	—	4	1	no	—
569	C₂	—	4	4	no	—
570	C₂	—	8	2	no	—
571	C₂	—	8	2	no	—
572	C₂	C₃₁×#3003	1	1	yes	characteristic
573	C₂	C₃₁×#3009	1	1	yes	characteristic
574	C₂	C₃₁×#3018	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3496	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ 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. C₃₁×#3235	C₂	semidirect
3. C₃₁×#3182	C₂	semidirect
4. C₃₁×#3101	C₂	semidirect
5. C₃₁×#1647	C₂	semidirect
6. C₃₁×#1639	C₂	semidirect
7. C₃₁×#1635	C₂	semidirect
8. C₃₁×#1629	C₂	semidirect
9. C₃₁×#1553	C₂	semidirect
10. C₃₁×#1490	C₂	semidirect
11. C₃₁×#1484	C₂	semidirect
21. C₃₁×#533	C₂²	semidirect
27. C₃₁×#500	C₂²	semidirect
28. C₃₁×#492	C₂²	semidirect
30. C₃₁×#487	C₂²	semidirect
31. C₃₁×#486	C₂²	semidirect
34. C₃₁×#476	C₂²	semidirect
35. C₃₁×#476	C₂²	semidirect
36. C₃₁×#474	C₂²	semidirect
38. C₃₁×#461	C₂²	semidirect
42. C₃₁×#432	C₂²	semidirect
43. C₃₁×#431	C₂²	semidirect
44. C₄×M₁₆×C₃₁	C₂²	semidirect
45. C₃₁×#332	C₂²	semidirect
46. C₃₁×#331	C₂²	semidirect
47. C₃₁×D₄⋊C₈	C₂²	semidirect
48. C₃₁×D₄⋊C₈	C₂²	semidirect
49. C₃₁×D₄⋊C₈	C₂²	semidirect
50. C₃₁×D₄⋊C₈	C₂²	semidirect
201. #18599	C₃₁	direct (N × H)
256. #3235	C₂×C₃₁	semidirect
257. #3182	C₂×C₃₁	semidirect
258. #3101	C₂×C₃₁	semidirect
259. #1647	C₂×C₃₁	semidirect
260. #1639	C₂×C₃₁	semidirect
261. #1635	C₂×C₃₁	semidirect
262. #1629	C₂×C₃₁	semidirect
263. #1553	C₂×C₃₁	semidirect
264. #1490	C₂×C₃₁	semidirect
265. #1484	C₂×C₃₁	semidirect
301. #533	C₂²×C₃₁	semidirect
307. #500	C₂²×C₃₁	semidirect
308. #492	C₂²×C₃₁	semidirect
310. #487	C₂²×C₃₁	semidirect
311. #486	C₂²×C₃₁	semidirect
314. #476	C₂²×C₃₁	semidirect
315. #476	C₂²×C₃₁	semidirect
316. #474	C₂²×C₃₁	semidirect
318. #461	C₂²×C₃₁	semidirect
322. #432	C₂²×C₃₁	semidirect
323. #431	C₂²×C₃₁	semidirect
324. C₄×M₁₆	C₂²×C₃₁	semidirect
325. #332	C₂²×C₃₁	semidirect
326. #331	C₂²×C₃₁	semidirect
327. D₄⋊C₈	C₂²×C₃₁	semidirect
328. D₄⋊C₈	C₂²×C₃₁	semidirect
329. D₄⋊C₈	C₂²×C₃₁	semidirect
330. D₄⋊C₈	C₂²×C₃₁	semidirect
414. C₃₁	#18599	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×#3496	2
C₃₁×#3235	2
C₃₁×#3182	2
C₃₁×#3101	2
C₃₁×#1647	2
C₃₁×#1639	2
C₃₁×#1635	2
C₃₁×#1629	2
C₃₁×#1553	2
C₃₁×#1490	2
C₃₁×#1484	2
#18599	31

Subgroups — C31×#18599 #21475358

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#18599 #21475358