Subgroups — C₃₁×#18935 #21475694

Order 7936.

986 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3549	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3246	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3235	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3210	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1646	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1639	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1634	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1629	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1551	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1496	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1480	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×H(3,𝔽₄)	—	2	2	no	—
13	C₃₁×#555	—	2	1	no	—
14	C₃₁×#555	—	2	1	no	—
15	C₃₁×#555	C₂²	1	2	yes	—
16	D₄²×C₃₁	—	2	1	no	—
17	D₄²×C₃₁	—	2	1	no	—
18	D₄²×C₃₁	C₂²	1	2	yes	—
19	C₃₁×#535	—	2	2	no	—
20	C₃₁×#535	—	2	2	no	—
21	C₃₁×#535	—	2	2	no	—
22	C₃₁×#533	C₂²	1	1	yes	characteristic, complemented
23	C₃₁×#532	C₂²	1	1	yes	characteristic
24	C₃₁×#532	C₂²	1	1	yes	characteristic
25	C₃₁×#532	C₂²	1	1	yes	characteristic
26	C₃₁×#501	—	2	1	no	—
27	C₃₁×#494	C₂²	1	1	yes	characteristic, complemented
28	C₃₁×#492	C₂²	1	1	yes	characteristic
29	C₃₁×#490	—	2	1	no	—
30	C₃₁×#487	C₂²	1	1	yes	characteristic, complemented
31	C₃₁×#486	C₂²	1	1	yes	characteristic
32	C₃₁×#482	—	2	2	no	—
33	C₃₁×#476	C₂²	1	2	yes	complemented
34	C₃₁×#476	C₂²	1	2	yes	complemented
35	C₃₁×#475	—	2	1	no	—
36	C₃₁×#474	C₂²	1	2	yes	complemented
37	C₃₁×#465	—	2	2	no	—
38	C₃₁×#464	—	2	2	no	—
39	C₃₁×#461	C₂²	1	2	yes	complemented
40	C₃₁×#460	—	2	1	no	—
41	C₃₁×#452	—	2	2	no	—
42	C₃₁×#432	C₂²	1	1	yes	characteristic, complemented
43	C₃₁×#431	C₂²	1	1	yes	characteristic, complemented
44	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic, complemented
45	C₃₁×#333	C₂²	1	2	yes	complemented
46	C₃₁×#331	C₂²	1	2	yes	—
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	C₂×#41×C₃₁	—	2	1	no	—
52	C₂×#41×C₃₁	—	2	1	no	—
53	C₂×#41×C₃₁	D₄	1	2	yes	—
54	C₂²×D₄×C₃₁	—	2	2	no	—
55	C₂²×D₄×C₃₁	—	2	2	no	—
56	C₂²×D₄×C₃₁	—	2	2	no	—
57	C₂×Q₁₆×C₃₁	—	2	2	no	—
58	C₂×SD₁₆×C₃₁	—	2	2	no	—
59	C₂×SD₁₆×C₃₁	—	2	2	no	—
60	C₂²×C₈×C₃₁	—	2	1	no	—
61	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
62	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
63	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
64	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
65	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
66	C₃₁×#127	—	2	4	no	—
67	C₃₁×#127	D₄	1	1	yes	characteristic
68	C₃₁×#127	D₄	1	1	yes	characteristic
69	C₃₁×#127	C₂³	1	1	yes	characteristic
70	C₃₁×#125	—	2	1	no	—
71	C₃₁×#124	—	2	4	no	—
72	C₃₁×#124	—	2	4	no	—
73	C₃₁×#124	—	2	4	no	—
74	C₃₁×#124	—	2	1	no	—
75	C₃₁×#124	—	2	1	no	—
76	C₃₁×#124	D₄	1	2	yes	—
77	C₃₁×C₂²⋊Q₈	—	2	1	no	—
78	C₃₁×C₂²⋊Q₈	—	2	1	no	—
79	C₃₁×#121	—	2	2	no	—
80	C₃₁×#121	—	2	2	no	—
81	C₃₁×#121	—	2	2	no	—
82	C₃₁×#121	—	2	2	no	—
83	C₃₁×#121	—	2	1	no	—
84	C₃₁×#121	—	2	1	no	—
85	C₃₁×#121	—	2	1	no	—
86	C₃₁×#121	—	2	1	no	—
87	C₃₁×#121	—	2	2	no	—
88	C₃₁×#121	—	2	2	no	—
89	C₃₁×#121	D₄	1	2	yes	—
90	C₃₁×#121	D₄	1	2	yes	—
91	C₃₁×#120	—	2	4	no	—
92	C₃₁×#120	—	2	4	no	—
93	C₃₁×#120	—	2	4	no	—
94	C₃₁×#120	—	2	4	no	—
95	C₄×Q₈×C₃₁	—	2	1	no	—
96	C₄×Q₈×C₃₁	—	2	1	no	—
97	C₄×D₄×C₃₁	—	2	1	no	—
98	C₄×D₄×C₃₁	—	2	1	no	—
99	C₄×D₄×C₃₁	—	2	1	no	—
100	C₄×D₄×C₃₁	—	2	1	no	—
101	C₄×D₄×C₃₁	C₂³	1	2	yes	—
102	C₄×D₄×C₃₁	C₂³	1	2	yes	—
103	C₃₁×#117	—	2	2	no	—
104	C₃₁×#117	—	2	2	no	—
105	C₃₁×#117	—	2	2	no	—
106	C₃₁×#117	—	2	1	no	—
107	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
108	C₃₁×#107	—	2	2	no	—
109	C₃₁×#106	—	2	2	no	—
110	C₃₁×#106	—	2	2	no	—
111	C₃₁×C₄⋊C₈	—	2	1	no	—
112	C₃₁×C₄⋊C₈	—	2	1	no	—
113	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
114	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
115	C₃₁×#103	—	2	2	no	—
116	C₃₁×#103	—	2	2	no	—
117	C₃₁×#102	—	2	2	no	—
118	C₃₁×#102	—	2	2	no	—
119	C₃₁×#102	—	2	2	no	—
120	C₃₁×#102	—	2	2	no	—
121	C₃₁×C₂²⋊C₈	—	2	1	no	—
122	C₃₁×C₂²⋊C₈	—	2	1	no	—
123	C₃₁×#97	—	2	1	no	—
124	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
125	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
126	C₂⁴×C₃₁	—	2	4	no	—
127	#41×C₃₁	—	8	1	no	—
128	#41×C₃₁	—	8	2	no	—
129	#41×C₃₁	—	8	1	no	—
130	#41×C₃₁	—	8	2	no	—
131	C₂×Q₈×C₃₁	—	2	1	no	—
132	C₂×Q₈×C₃₁	—	2	1	no	—
133	C₂×Q₈×C₃₁	—	2	1	no	—
134	C₂×Q₈×C₃₁	—	2	1	no	—
135	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
136	C₂×D₄×C₃₁	—	2	4	no	—
137	C₂×D₄×C₃₁	—	2	4	no	—
138	C₂×D₄×C₃₁	—	2	4	no	—
139	C₂×D₄×C₃₁	—	2	2	no	—
140	C₂×D₄×C₃₁	—	2	2	no	—
141	C₂×D₄×C₃₁	—	2	2	no	—
142	C₂×D₄×C₃₁	—	2	1	no	—
143	C₂×D₄×C₃₁	—	2	1	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	—	8	2	no	—
149	C₂×D₄×C₃₁	—	8	4	no	—
150	C₂×D₄×C₃₁	—	8	2	no	—
151	C₂×D₄×C₃₁	—	8	4	no	—
152	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
153	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
154	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
155	C₂²×C₄×C₃₁	—	2	2	no	—
156	C₂²×C₄×C₃₁	—	2	2	no	—
157	C₂²×C₄×C₃₁	—	2	2	no	—
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	Q₁₆×C₃₁	—	8	2	no	—
162	SD₁₆×C₃₁	—	8	2	no	—
163	SD₁₆×C₃₁	—	8	2	no	—
164	C₂×C₈×C₃₁	—	2	1	no	—
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	—	8	1	no	—
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	—	2	1	no	—
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	—	4	1	no	—
178	C₄⋊C₄×C₃₁	—	4	1	no	—
179	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
180	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
181	C₂²⋊C₄×C₃₁	—	2	4	no	—
182	C₂²⋊C₄×C₃₁	—	2	4	no	—
183	C₂²⋊C₄×C₃₁	—	2	4	no	—
184	C₂²⋊C₄×C₃₁	—	2	2	no	—
185	C₂²⋊C₄×C₃₁	—	2	2	no	—
186	C₂²⋊C₄×C₃₁	—	2	2	no	—
187	C₂²⋊C₄×C₃₁	—	2	2	no	—
188	C₂²⋊C₄×C₃₁	—	2	2	no	—
189	C₂²⋊C₄×C₃₁	—	2	2	no	—
190	C₂²⋊C₄×C₃₁	—	4	1	no	—
191	C₄²×C₃₁	—	2	4	no	—
192	C₄²×C₃₁	—	4	1	no	—
193	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
194	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
195	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
196	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
197	#18935	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
198	C₂³×C₃₁	—	2	2	no	—
199	C₂³×C₃₁	—	2	2	no	—
200	C₂³×C₃₁	—	2	2	no	—
201	C₂³×C₃₁	—	8	4	no	—
202	C₂³×C₃₁	—	8	4	no	—
203	C₂³×C₃₁	—	8	4	no	—
204	C₂³×C₃₁	#120	1	1	yes	characteristic
205	Q₈×C₃₁	—	4	2	no	—
206	Q₈×C₃₁	—	4	2	no	—
207	Q₈×C₃₁	—	4	1	no	—
208	Q₈×C₃₁	—	4	1	no	—
209	Q₈×C₃₁	—	8	1	no	—
210	Q₈×C₃₁	—	8	1	no	—
211	D₄×C₃₁	—	2	2	no	—
212	D₄×C₃₁	—	2	4	no	—
213	D₄×C₃₁	—	2	2	no	—
214	D₄×C₃₁	—	4	2	no	—
215	D₄×C₃₁	—	4	2	no	—
216	D₄×C₃₁	—	4	2	no	—
217	D₄×C₃₁	—	4	1	no	—
218	D₄×C₃₁	—	4	2	no	—
219	D₄×C₃₁	—	4	1	no	—
220	D₄×C₃₁	—	8	1	no	—
221	D₄×C₃₁	—	8	1	no	—
222	D₄×C₃₁	—	8	2	no	—
223	D₄×C₃₁	—	8	2	no	—
224	D₄×C₃₁	—	8	2	no	—
225	D₄×C₃₁	—	8	4	no	—
226	D₄×C₃₁	—	8	4	no	—
227	D₄×C₃₁	—	8	4	no	—
228	C₂×C₄×C₃₁	—	2	1	no	—
229	C₂×C₄×C₃₁	—	2	2	no	—
230	C₂×C₄×C₃₁	—	2	1	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	—	4	1	no	—
235	C₂×C₄×C₃₁	—	4	4	no	—
236	C₂×C₄×C₃₁	—	4	2	no	—
237	C₂×C₄×C₃₁	—	4	4	no	—
238	C₂×C₄×C₃₁	—	4	2	no	—
239	C₂×C₄×C₃₁	—	4	1	no	—
240	C₂×C₄×C₃₁	—	4	1	no	—
241	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
242	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
243	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
244	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
245	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
246	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
247	C₈×C₃₁	—	4	1	no	—
248	C₈×C₃₁	—	4	1	no	—
249	C₈×C₃₁	—	4	2	no	—
250	C₈×C₃₁	—	4	2	no	—
251	#3549	C₂×C₃₁	1	1	yes	characteristic
252	#3246	C₂×C₃₁	1	1	yes	characteristic, complemented
253	#3235	C₂×C₃₁	1	1	yes	characteristic, complemented
254	#3210	C₂×C₃₁	1	1	yes	characteristic, complemented
255	#1646	C₂×C₃₁	1	2	yes	complemented
256	#1639	C₂×C₃₁	1	2	yes	complemented
257	#1634	C₂×C₃₁	1	2	yes	complemented
258	#1629	C₂×C₃₁	1	2	yes	complemented
259	#1551	C₂×C₃₁	1	1	yes	characteristic, complemented
260	#1496	C₂×C₃₁	1	1	yes	characteristic, complemented
261	#1480	C₂×C₃₁	1	1	yes	characteristic, complemented
262	C₂²×C₃₁	—	2	1	no	—
263	C₂²×C₃₁	—	2	1	no	—
264	C₂²×C₃₁	—	2	1	no	—
265	C₂²×C₃₁	—	4	2	no	—
266	C₂²×C₃₁	—	4	2	no	—
267	C₂²×C₃₁	—	4	2	no	—
268	C₂²×C₃₁	—	4	2	no	—
269	C₂²×C₃₁	—	4	2	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	—	16	8	no	—
275	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
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	—	4	2	no	—
281	C₄×C₃₁	—	4	2	no	—
282	C₄×C₃₁	—	4	2	no	—
283	C₄×C₃₁	—	8	1	no	—
284	C₄×C₃₁	—	8	1	no	—
285	C₄×C₃₁	#576	1	1	yes	characteristic
286	C₄×C₃₁	#577	1	2	yes	—
287	C₄×C₃₁	#577	1	1	yes	characteristic
288	H(3,𝔽₄)	—	2	2	no	—
289	#555	—	2	1	no	—
290	#555	—	2	1	no	—
291	#555	C₂²×C₃₁	1	2	yes	—
292	D₄²	—	2	1	no	—
293	D₄²	—	2	1	no	—
294	D₄²	C₂²×C₃₁	1	2	yes	—
295	#535	—	2	2	no	—
296	#535	—	2	2	no	—
297	#535	—	2	2	no	—
298	#533	C₂²×C₃₁	1	1	yes	characteristic, complemented
299	#532	C₂²×C₃₁	1	1	yes	characteristic
300	#532	C₂²×C₃₁	1	1	yes	characteristic
301	#532	C₂²×C₃₁	1	1	yes	characteristic
302	#501	—	2	1	no	—
303	#494	C₂²×C₃₁	1	1	yes	characteristic, complemented
304	#492	C₂²×C₃₁	1	1	yes	characteristic
305	#490	—	2	1	no	—
306	#487	C₂²×C₃₁	1	1	yes	characteristic, complemented
307	#486	C₂²×C₃₁	1	1	yes	characteristic
308	#482	—	2	2	no	—
309	#476	C₂²×C₃₁	1	2	yes	complemented
310	#476	C₂²×C₃₁	1	2	yes	complemented
311	#475	—	2	1	no	—
312	#474	C₂²×C₃₁	1	2	yes	complemented
313	#465	—	2	2	no	—
314	#464	—	2	2	no	—
315	#461	C₂²×C₃₁	1	2	yes	complemented
316	#460	—	2	1	no	—
317	#452	—	2	2	no	—
318	#432	C₂²×C₃₁	1	1	yes	characteristic, complemented
319	#431	C₂²×C₃₁	1	1	yes	characteristic, complemented
320	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
321	#333	C₂²×C₃₁	1	2	yes	complemented
322	#331	C₂²×C₃₁	1	2	yes	—
323	D₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
324	D₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
325	D₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
326	D₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
327	C₂×C₃₁	—	4	1	no	—
328	C₂×C₃₁	—	8	2	no	—
329	C₂×C₃₁	—	8	2	no	—
330	C₂×C₃₁	—	8	2	no	—
331	C₂×C₃₁	#3009	1	1	yes	characteristic
332	C₂×C₃₁	#3012	1	1	yes	characteristic
333	C₂×C₃₁	#3019	1	1	yes	characteristic
334	C₂×#41	—	2	1	no	—
335	C₂×#41	—	2	1	no	—
336	C₂×#41	D₄×C₃₁	1	2	yes	—
337	C₂²×D₄	—	2	2	no	—
338	C₂²×D₄	—	2	2	no	—
339	C₂²×D₄	—	2	2	no	—
340	C₂×Q₁₆	—	2	2	no	—
341	C₂×SD₁₆	—	2	2	no	—
342	C₂×SD₁₆	—	2	2	no	—
343	C₂²×C₈	—	2	1	no	—
344	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
345	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
346	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
347	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
348	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
349	#127	—	2	4	no	—
350	#127	D₄×C₃₁	1	1	yes	characteristic
351	#127	D₄×C₃₁	1	1	yes	characteristic
352	#127	C₂³×C₃₁	1	1	yes	characteristic
353	#125	—	2	1	no	—
354	#124	—	2	4	no	—
355	#124	—	2	4	no	—
356	#124	—	2	4	no	—
357	#124	—	2	1	no	—
358	#124	—	2	1	no	—
359	#124	D₄×C₃₁	1	2	yes	—
360	C₂²⋊Q₈	—	2	1	no	—
361	C₂²⋊Q₈	—	2	1	no	—
362	#121	—	2	2	no	—
363	#121	—	2	2	no	—
364	#121	—	2	2	no	—
365	#121	—	2	2	no	—
366	#121	—	2	1	no	—
367	#121	—	2	1	no	—
368	#121	—	2	1	no	—
369	#121	—	2	1	no	—
370	#121	—	2	2	no	—
371	#121	—	2	2	no	—
372	#121	D₄×C₃₁	1	2	yes	—
373	#121	D₄×C₃₁	1	2	yes	—
374	#120	—	2	4	no	—
375	#120	—	2	4	no	—
376	#120	—	2	4	no	—
377	#120	—	2	4	no	—
378	C₄×Q₈	—	2	1	no	—
379	C₄×Q₈	—	2	1	no	—
380	C₄×D₄	—	2	1	no	—
381	C₄×D₄	—	2	1	no	—
382	C₄×D₄	—	2	1	no	—
383	C₄×D₄	—	2	1	no	—
384	C₄×D₄	C₂³×C₃₁	1	2	yes	—
385	C₄×D₄	C₂³×C₃₁	1	2	yes	—
386	#117	—	2	2	no	—
387	#117	—	2	2	no	—
388	#117	—	2	2	no	—
389	#117	—	2	1	no	—
390	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
391	#107	—	2	2	no	—
392	#106	—	2	2	no	—
393	#106	—	2	2	no	—
394	C₄⋊C₈	—	2	1	no	—
395	C₄⋊C₈	—	2	1	no	—
396	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
397	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
398	#103	—	2	2	no	—
399	#103	—	2	2	no	—
400	#102	—	2	2	no	—
401	#102	—	2	2	no	—
402	#102	—	2	2	no	—
403	#102	—	2	2	no	—
404	C₂²⋊C₈	—	2	1	no	—
405	C₂²⋊C₈	—	2	1	no	—
406	#97	—	2	1	no	—
407	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
408	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
409	C₃₁	#18935	1	1	yes	characteristic, direct product factor, Sylow
410	C₂⁴	—	2	4	no	—
411	#41	—	8	1	no	—
412	#41	—	8	2	no	—
413	#41	—	8	1	no	—
414	#41	—	8	2	no	—
415	C₂×Q₈	—	2	1	no	—
416	C₂×Q₈	—	2	1	no	—
417	C₂×Q₈	—	2	1	no	—
418	C₂×Q₈	—	2	1	no	—
419	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
420	C₂×D₄	—	2	4	no	—
421	C₂×D₄	—	2	4	no	—
422	C₂×D₄	—	2	4	no	—
423	C₂×D₄	—	2	2	no	—
424	C₂×D₄	—	2	2	no	—
425	C₂×D₄	—	2	2	no	—
426	C₂×D₄	—	2	1	no	—
427	C₂×D₄	—	2	1	no	—
428	C₂×D₄	—	2	1	no	—
429	C₂×D₄	—	2	1	no	—
430	C₂×D₄	—	2	1	no	—
431	C₂×D₄	—	2	1	no	—
432	C₂×D₄	—	8	2	no	—
433	C₂×D₄	—	8	4	no	—
434	C₂×D₄	—	8	2	no	—
435	C₂×D₄	—	8	4	no	—
436	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
437	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
438	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
439	C₂²×C₄	—	2	2	no	—
440	C₂²×C₄	—	2	2	no	—
441	C₂²×C₄	—	2	2	no	—
442	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
443	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
444	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
445	Q₁₆	—	8	2	no	—
446	SD₁₆	—	8	2	no	—
447	SD₁₆	—	8	2	no	—
448	C₂×C₈	—	2	1	no	—
449	C₂×C₈	—	2	1	no	—
450	C₂×C₈	—	2	2	no	—
451	C₂×C₈	—	2	2	no	—
452	C₂×C₈	—	8	1	no	—
453	C₄⋊C₄	—	2	1	no	—
454	C₄⋊C₄	—	2	1	no	—
455	C₄⋊C₄	—	2	1	no	—
456	C₄⋊C₄	—	2	1	no	—
457	C₄⋊C₄	—	2	1	no	—
458	C₄⋊C₄	—	2	1	no	—
459	C₄⋊C₄	—	2	1	no	—
460	C₄⋊C₄	—	2	1	no	—
461	C₄⋊C₄	—	4	1	no	—
462	C₄⋊C₄	—	4	1	no	—
463	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
464	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
465	C₂²⋊C₄	—	2	4	no	—
466	C₂²⋊C₄	—	2	4	no	—
467	C₂²⋊C₄	—	2	4	no	—
468	C₂²⋊C₄	—	2	2	no	—
469	C₂²⋊C₄	—	2	2	no	—
470	C₂²⋊C₄	—	2	2	no	—
471	C₂²⋊C₄	—	2	2	no	—
472	C₂²⋊C₄	—	2	2	no	—
473	C₂²⋊C₄	—	2	2	no	—
474	C₂²⋊C₄	—	4	1	no	—
475	C₄²	—	2	4	no	—
476	C₄²	—	4	1	no	—
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	2	no	—
484	C₂³	—	8	4	no	—
485	C₂³	—	8	4	no	—
486	C₂³	—	8	4	no	—
487	C₂³	C₃₁×#120	1	1	yes	characteristic
488	Q₈	—	4	2	no	—
489	Q₈	—	4	2	no	—
490	Q₈	—	4	1	no	—
491	Q₈	—	4	1	no	—
492	Q₈	—	8	1	no	—
493	Q₈	—	8	1	no	—
494	D₄	—	2	2	no	—
495	D₄	—	2	4	no	—
496	D₄	—	2	2	no	—
497	D₄	—	4	2	no	—
498	D₄	—	4	2	no	—
499	D₄	—	4	2	no	—
500	D₄	—	4	1	no	—
501	D₄	—	4	2	no	—
502	D₄	—	4	1	no	—
503	D₄	—	8	1	no	—
504	D₄	—	8	1	no	—
505	D₄	—	8	2	no	—
506	D₄	—	8	2	no	—
507	D₄	—	8	2	no	—
508	D₄	—	8	4	no	—
509	D₄	—	8	4	no	—
510	D₄	—	8	4	no	—
511	C₂×C₄	—	2	1	no	—
512	C₂×C₄	—	2	2	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	—	4	1	no	—
518	C₂×C₄	—	4	4	no	—
519	C₂×C₄	—	4	2	no	—
520	C₂×C₄	—	4	4	no	—
521	C₂×C₄	—	4	2	no	—
522	C₂×C₄	—	4	1	no	—
523	C₂×C₄	—	4	1	no	—
524	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
525	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
526	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
527	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
528	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
529	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
530	C₈	—	4	1	no	—
531	C₈	—	4	1	no	—
532	C₈	—	4	2	no	—
533	C₈	—	4	2	no	—
534	C₂²	—	2	1	no	—
535	C₂²	—	2	1	no	—
536	C₂²	—	2	1	no	—
537	C₂²	—	4	2	no	—
538	C₂²	—	4	2	no	—
539	C₂²	—	4	2	no	—
540	C₂²	—	4	2	no	—
541	C₂²	—	4	2	no	—
542	C₂²	—	4	2	no	—
543	C₂²	—	4	2	no	—
544	C₂²	—	4	2	no	—
545	C₂²	—	4	2	no	—
546	C₂²	—	16	8	no	—
547	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
548	C₄	—	2	1	no	—
549	C₄	—	2	1	no	—
550	C₄	—	2	1	no	—
551	C₄	—	2	1	no	—
552	C₄	—	4	2	no	—
553	C₄	—	4	2	no	—
554	C₄	—	4	2	no	—
555	C₄	—	8	1	no	—
556	C₄	—	8	1	no	—
557	C₄	C₃₁×#576	1	1	yes	characteristic
558	C₄	C₃₁×#577	1	2	yes	—
559	C₄	C₃₁×#577	1	1	yes	characteristic
560	C₂	—	4	1	no	—
561	C₂	—	8	2	no	—
562	C₂	—	8	2	no	—
563	C₂	—	8	2	no	—
564	C₂	C₃₁×#3009	1	1	yes	characteristic
565	C₂	C₃₁×#3012	1	1	yes	characteristic
566	C₂	C₃₁×#3019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3549	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×#3246	C₂	semidirect
3. C₃₁×#3235	C₂	semidirect
4. C₃₁×#3210	C₂	semidirect
5. C₃₁×#1646	C₂	semidirect
6. C₃₁×#1639	C₂	semidirect
7. C₃₁×#1634	C₂	semidirect
8. C₃₁×#1629	C₂	semidirect
9. C₃₁×#1551	C₂	semidirect
10. C₃₁×#1496	C₂	semidirect
11. C₃₁×#1480	C₂	semidirect
22. C₃₁×#533	C₂²	semidirect
27. C₃₁×#494	C₂²	semidirect
30. C₃₁×#487	C₂²	semidirect
33. C₃₁×#476	C₂²	semidirect
34. C₃₁×#476	C₂²	semidirect
36. C₃₁×#474	C₂²	semidirect
39. C₃₁×#461	C₂²	semidirect
42. C₃₁×#432	C₂²	semidirect
43. C₃₁×#431	C₂²	semidirect
44. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
45. C₃₁×#333	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
197. #18935	C₃₁	direct (N × H)
252. #3246	C₂×C₃₁	semidirect
253. #3235	C₂×C₃₁	semidirect
254. #3210	C₂×C₃₁	semidirect
255. #1646	C₂×C₃₁	semidirect
256. #1639	C₂×C₃₁	semidirect
257. #1634	C₂×C₃₁	semidirect
258. #1629	C₂×C₃₁	semidirect
259. #1551	C₂×C₃₁	semidirect
260. #1496	C₂×C₃₁	semidirect
261. #1480	C₂×C₃₁	semidirect
298. #533	C₂²×C₃₁	semidirect
303. #494	C₂²×C₃₁	semidirect
306. #487	C₂²×C₃₁	semidirect
309. #476	C₂²×C₃₁	semidirect
310. #476	C₂²×C₃₁	semidirect
312. #474	C₂²×C₃₁	semidirect
315. #461	C₂²×C₃₁	semidirect
318. #432	C₂²×C₃₁	semidirect
319. #431	C₂²×C₃₁	semidirect
320. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
321. #333	C₂²×C₃₁	semidirect
323. D₄⋊C₈	C₂²×C₃₁	semidirect
324. D₄⋊C₈	C₂²×C₃₁	semidirect
325. D₄⋊C₈	C₂²×C₃₁	semidirect
326. D₄⋊C₈	C₂²×C₃₁	semidirect
409. C₃₁	#18935	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×#3549	2
C₃₁×#3246	2
C₃₁×#3235	2
C₃₁×#3210	2
C₃₁×#1646	2
C₃₁×#1639	2
C₃₁×#1634	2
C₃₁×#1629	2
C₃₁×#1551	2
C₃₁×#1496	2
C₃₁×#1480	2
#18935	31

Subgroups — C31×#18935 #21475694

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#18935 #21475694