Subgroups — C₃₁×#18692 #21475451

Order 7936.

924 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3548	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3255	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3254	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3131	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1644	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1643	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1642	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1641	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1560	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1493	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1492	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#562	—	2	2	no	—
13	C₃₁×#554	—	2	1	no	—
14	C₃₁×#554	C₂²	1	2	yes	—
15	C₃₁×#550	—	2	1	no	—
16	C₃₁×#550	C₂²	1	2	yes	—
17	C₃₁×#547	—	2	1	no	—
18	C₃₁×#547	—	2	1	no	—
19	C₃₁×#540	—	2	2	no	—
20	C₃₁×#537	—	2	2	no	—
21	C₃₁×#537	—	2	2	no	—
22	C₃₁×#533	C₂²	1	1	yes	characteristic
23	C₃₁×#533	C₂²	1	1	yes	characteristic
24	C₃₁×#532	C₂²	1	1	yes	characteristic
25	C₃₁×#532	C₂²	1	1	yes	characteristic
26	C₃₁×#488	—	2	2	no	—
27	C₃₁×#487	C₂²	1	1	yes	characteristic
28	C₃₁×#487	C₂²	1	1	yes	characteristic
29	C₃₁×#486	C₂²	1	1	yes	characteristic
30	C₃₁×#486	C₂²	1	1	yes	characteristic
31	C₃₁×#477	—	2	1	no	—
32	C₃₁×#475	—	2	1	no	—
33	C₃₁×#475	C₂²	1	2	yes	—
34	C₃₁×#474	C₂²	1	2	yes	—
35	C₃₁×#470	—	2	2	no	—
36	C₃₁×#468	—	2	2	no	—
37	C₃₁×#462	—	2	1	no	—
38	C₃₁×#462	C₂²	1	2	yes	—
39	C₃₁×#461	C₂²	1	2	yes	—
40	C₃₁×#460	—	2	1	no	—
41	C₃₁×#423	C₂²	1	1	yes	characteristic
42	C₃₁×#423	C₂²	1	1	yes	characteristic
43	C₃₁×#416	—	2	2	no	—
44	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
45	C₃₁×#333	C₂²	1	2	yes	—
46	C₃₁×#331	C₂²	1	2	yes	—
47	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
48	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
49	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
50	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
51	C₂×#41×C₃₁	—	2	2	no	—
52	C₂×#41×C₃₁	—	2	2	no	—
53	C₂×#41×C₃₁	—	2	1	no	—
54	C₂×#41×C₃₁	D₄	1	2	yes	—
55	C₂×Q₁₆×C₃₁	—	2	2	no	—
56	C₂×SD₁₆×C₃₁	—	2	2	no	—
57	C₂×M₁₆×C₃₁	—	2	1	no	—
58	C₂×M₁₆×C₃₁	—	2	1	no	—
59	C₂²×C₈×C₃₁	—	2	1	no	—
60	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
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	2	no	—
67	C₃₁×#127	D₄	1	1	yes	characteristic
68	C₃₁×#127	C₂³	1	1	yes	characteristic
69	C₃₁×#125	—	2	2	no	—
70	C₃₁×#125	—	2	1	no	—
71	C₃₁×#125	—	2	1	no	—
72	C₃₁×#124	—	2	1	no	—
73	C₃₁×#124	D₄	1	2	yes	—
74	C₃₁×#123	—	2	4	no	—
75	C₃₁×#123	—	2	4	no	—
76	C₃₁×C₂²⋊Q₈	—	2	2	no	—
77	C₃₁×C₂²⋊Q₈	—	2	4	no	—
78	C₃₁×C₂²⋊Q₈	—	2	2	no	—
79	C₃₁×C₂²⋊Q₈	—	2	4	no	—
80	C₃₁×C₂²⋊Q₈	—	2	1	no	—
81	C₃₁×C₂²⋊Q₈	—	2	1	no	—
82	C₃₁×C₂²⋊Q₈	—	2	1	no	—
83	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
84	C₃₁×#121	—	2	4	no	—
85	C₃₁×#121	—	2	4	no	—
86	C₃₁×#121	—	2	2	no	—
87	C₃₁×#121	—	2	4	no	—
88	C₃₁×#121	—	2	4	no	—
89	C₃₁×#121	—	2	1	no	—
90	C₃₁×#121	—	2	1	no	—
91	C₃₁×#121	—	2	1	no	—
92	C₃₁×#121	—	2	2	no	—
93	C₃₁×#121	D₄	1	2	yes	—
94	C₄×Q₈×C₃₁	—	2	1	no	—
95	C₄×Q₈×C₃₁	—	2	1	no	—
96	C₄×Q₈×C₃₁	—	2	1	no	—
97	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
98	C₄×D₄×C₃₁	—	2	2	no	—
99	C₄×D₄×C₃₁	—	2	2	no	—
100	C₄×D₄×C₃₁	—	2	1	no	—
101	C₄×D₄×C₃₁	—	2	1	no	—
102	C₄×D₄×C₃₁	—	2	1	no	—
103	C₄×D₄×C₃₁	C₂³	1	2	yes	—
104	C₃₁×#117	—	2	1	no	—
105	C₃₁×#117	—	2	1	no	—
106	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
107	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
108	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
109	C₃₁×#107	—	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×#103	—	2	2	no	—
118	C₃₁×#103	—	2	2	no	—
119	C₃₁×#103	—	2	2	no	—
120	C₃₁×#102	—	2	2	no	—
121	C₃₁×#102	—	2	2	no	—
122	C₃₁×#102	—	2	2	no	—
123	C₄×C₈×C₃₁	—	2	1	no	—
124	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
125	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
126	#41×C₃₁	—	4	4	no	—
127	#41×C₃₁	—	8	1	no	—
128	#41×C₃₁	—	8	1	no	—
129	#41×C₃₁	—	8	2	no	—
130	#41×C₃₁	—	8	2	no	—
131	#41×C₃₁	—	8	4	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	—	2	1	no	—
136	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
137	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
138	C₂×D₄×C₃₁	—	2	4	no	—
139	C₂×D₄×C₃₁	—	2	4	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	C₂×D₄	1	2	yes	—
147	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
148	C₂²×C₄×C₃₁	—	2	4	no	—
149	C₂²×C₄×C₃₁	—	2	4	no	—
150	C₂²×C₄×C₃₁	—	2	2	no	—
151	C₂²×C₄×C₃₁	—	2	2	no	—
152	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
155	Q₁₆×C₃₁	—	8	2	no	—
156	SD₁₆×C₃₁	—	8	2	no	—
157	M₁₆×C₃₁	—	8	1	no	—
158	M₁₆×C₃₁	—	8	1	no	—
159	C₂×C₈×C₃₁	—	2	1	no	—
160	C₂×C₈×C₃₁	—	2	1	no	—
161	C₂×C₈×C₃₁	—	2	2	no	—
162	C₂×C₈×C₃₁	—	2	2	no	—
163	C₂×C₈×C₃₁	—	4	2	no	—
164	C₄⋊C₄×C₃₁	—	2	4	no	—
165	C₄⋊C₄×C₃₁	—	2	4	no	—
166	C₄⋊C₄×C₃₁	—	2	1	no	—
167	C₄⋊C₄×C₃₁	—	2	1	no	—
168	C₄⋊C₄×C₃₁	—	2	1	no	—
169	C₄⋊C₄×C₃₁	—	2	1	no	—
170	C₄⋊C₄×C₃₁	—	2	2	no	—
171	C₄⋊C₄×C₃₁	—	2	1	no	—
172	C₄⋊C₄×C₃₁	—	2	1	no	—
173	C₄⋊C₄×C₃₁	—	2	2	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	—	2	1	no	—
178	C₄⋊C₄×C₃₁	—	4	1	no	—
179	C₄⋊C₄×C₃₁	—	4	1	no	—
180	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
181	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
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	—	4	1	no	—
190	C₄²×C₃₁	—	2	2	no	—
191	C₄²×C₃₁	—	4	1	no	—
192	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
193	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
194	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
195	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
196	#18692	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
197	C₂³×C₃₁	—	2	2	no	—
198	C₂³×C₃₁	—	2	2	no	—
199	C₂³×C₃₁	#120	1	1	yes	characteristic
200	Q₈×C₃₁	—	2	2	no	—
201	Q₈×C₃₁	—	2	2	no	—
202	Q₈×C₃₁	—	4	2	no	—
203	Q₈×C₃₁	—	4	2	no	—
204	Q₈×C₃₁	—	4	1	no	—
205	Q₈×C₃₁	—	4	1	no	—
206	Q₈×C₃₁	—	4	1	no	—
207	Q₈×C₃₁	—	4	1	no	—
208	Q₈×C₃₁	—	8	1	no	—
209	D₄×C₃₁	—	2	2	no	—
210	D₄×C₃₁	—	2	2	no	—
211	D₄×C₃₁	—	4	2	no	—
212	D₄×C₃₁	—	4	1	no	—
213	D₄×C₃₁	—	4	1	no	—
214	D₄×C₃₁	—	4	1	no	—
215	D₄×C₃₁	—	4	1	no	—
216	D₄×C₃₁	—	4	2	no	—
217	D₄×C₃₁	—	8	1	no	—
218	D₄×C₃₁	—	8	2	no	—
219	D₄×C₃₁	—	8	2	no	—
220	D₄×C₃₁	—	8	4	no	—
221	D₄×C₃₁	—	8	4	no	—
222	C₂×C₄×C₃₁	—	2	1	no	—
223	C₂×C₄×C₃₁	—	2	1	no	—
224	C₂×C₄×C₃₁	—	2	1	no	—
225	C₂×C₄×C₃₁	—	2	1	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	2	no	—
230	C₂×C₄×C₃₁	—	2	2	no	—
231	C₂×C₄×C₃₁	—	4	2	no	—
232	C₂×C₄×C₃₁	—	4	2	no	—
233	C₂×C₄×C₃₁	—	4	4	no	—
234	C₂×C₄×C₃₁	—	4	1	no	—
235	C₂×C₄×C₃₁	—	4	1	no	—
236	C₂×C₄×C₃₁	—	8	4	no	—
237	C₂×C₄×C₃₁	—	8	4	no	—
238	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
239	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
240	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
241	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
242	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
243	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
244	C₈×C₃₁	—	4	1	no	—
245	C₈×C₃₁	—	4	1	no	—
246	C₈×C₃₁	—	4	2	no	—
247	C₈×C₃₁	—	4	2	no	—
248	#3548	C₂×C₃₁	1	1	yes	characteristic
249	#3255	C₂×C₃₁	1	1	yes	characteristic, complemented
250	#3254	C₂×C₃₁	1	1	yes	characteristic, complemented
251	#3131	C₂×C₃₁	1	1	yes	characteristic, complemented
252	#1644	C₂×C₃₁	1	2	yes	complemented
253	#1643	C₂×C₃₁	1	2	yes	complemented
254	#1642	C₂×C₃₁	1	2	yes	complemented
255	#1641	C₂×C₃₁	1	2	yes	complemented
256	#1560	C₂×C₃₁	1	1	yes	characteristic, complemented
257	#1493	C₂×C₃₁	1	1	yes	characteristic, complemented
258	#1492	C₂×C₃₁	1	1	yes	characteristic, complemented
259	C₂²×C₃₁	—	2	1	no	—
260	C₂²×C₃₁	—	2	1	no	—
261	C₂²×C₃₁	—	2	1	no	—
262	C₂²×C₃₁	—	4	2	no	—
263	C₂²×C₃₁	—	4	2	no	—
264	C₂²×C₃₁	—	4	2	no	—
265	C₂²×C₃₁	—	4	2	no	—
266	C₂²×C₃₁	—	4	2	no	—
267	C₂²×C₃₁	—	4	2	no	—
268	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
269	C₄×C₃₁	—	2	1	no	—
270	C₄×C₃₁	—	2	1	no	—
271	C₄×C₃₁	—	2	1	no	—
272	C₄×C₃₁	—	4	2	no	—
273	C₄×C₃₁	—	4	2	no	—
274	C₄×C₃₁	—	4	2	no	—
275	C₄×C₃₁	—	4	2	no	—
276	C₄×C₃₁	—	8	1	no	—
277	C₄×C₃₁	—	8	1	no	—
278	C₄×C₃₁	#455	1	2	yes	—
279	C₄×C₃₁	#576	1	1	yes	characteristic
280	C₄×C₃₁	#577	1	1	yes	characteristic
281	C₄×C₃₁	#577	1	1	yes	characteristic
282	C₄×C₃₁	#578	1	1	yes	characteristic
283	#562	—	2	2	no	—
284	#554	—	2	1	no	—
285	#554	C₂²×C₃₁	1	2	yes	—
286	#550	—	2	1	no	—
287	#550	C₂²×C₃₁	1	2	yes	—
288	#547	—	2	1	no	—
289	#547	—	2	1	no	—
290	#540	—	2	2	no	—
291	#537	—	2	2	no	—
292	#537	—	2	2	no	—
293	#533	C₂²×C₃₁	1	1	yes	characteristic
294	#533	C₂²×C₃₁	1	1	yes	characteristic
295	#532	C₂²×C₃₁	1	1	yes	characteristic
296	#532	C₂²×C₃₁	1	1	yes	characteristic
297	#488	—	2	2	no	—
298	#487	C₂²×C₃₁	1	1	yes	characteristic
299	#487	C₂²×C₃₁	1	1	yes	characteristic
300	#486	C₂²×C₃₁	1	1	yes	characteristic
301	#486	C₂²×C₃₁	1	1	yes	characteristic
302	#477	—	2	1	no	—
303	#475	—	2	1	no	—
304	#475	C₂²×C₃₁	1	2	yes	—
305	#474	C₂²×C₃₁	1	2	yes	—
306	#470	—	2	2	no	—
307	#468	—	2	2	no	—
308	#462	—	2	1	no	—
309	#462	C₂²×C₃₁	1	2	yes	—
310	#461	C₂²×C₃₁	1	2	yes	—
311	#460	—	2	1	no	—
312	#423	C₂²×C₃₁	1	1	yes	characteristic
313	#423	C₂²×C₃₁	1	1	yes	characteristic
314	#416	—	2	2	no	—
315	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
316	#333	C₂²×C₃₁	1	2	yes	—
317	#331	C₂²×C₃₁	1	2	yes	—
318	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
319	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
320	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
321	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
322	C₂×C₃₁	—	4	1	no	—
323	C₂×C₃₁	—	8	2	no	—
324	C₂×C₃₁	—	8	2	no	—
325	C₂×C₃₁	#3012	1	1	yes	characteristic
326	C₂×C₃₁	#3013	1	1	yes	characteristic
327	C₂×C₃₁	C₂×#455	1	1	yes	characteristic
328	C₂×#41	—	2	2	no	—
329	C₂×#41	—	2	2	no	—
330	C₂×#41	—	2	1	no	—
331	C₂×#41	D₄×C₃₁	1	2	yes	—
332	C₂×Q₁₆	—	2	2	no	—
333	C₂×SD₁₆	—	2	2	no	—
334	C₂×M₁₆	—	2	1	no	—
335	C₂×M₁₆	—	2	1	no	—
336	C₂²×C₈	—	2	1	no	—
337	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
338	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
339	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
340	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
341	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
342	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
343	#127	—	2	2	no	—
344	#127	D₄×C₃₁	1	1	yes	characteristic
345	#127	C₂³×C₃₁	1	1	yes	characteristic
346	#125	—	2	2	no	—
347	#125	—	2	1	no	—
348	#125	—	2	1	no	—
349	#124	—	2	1	no	—
350	#124	D₄×C₃₁	1	2	yes	—
351	#123	—	2	4	no	—
352	#123	—	2	4	no	—
353	C₂²⋊Q₈	—	2	2	no	—
354	C₂²⋊Q₈	—	2	4	no	—
355	C₂²⋊Q₈	—	2	2	no	—
356	C₂²⋊Q₈	—	2	4	no	—
357	C₂²⋊Q₈	—	2	1	no	—
358	C₂²⋊Q₈	—	2	1	no	—
359	C₂²⋊Q₈	—	2	1	no	—
360	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
361	#121	—	2	4	no	—
362	#121	—	2	4	no	—
363	#121	—	2	2	no	—
364	#121	—	2	4	no	—
365	#121	—	2	4	no	—
366	#121	—	2	1	no	—
367	#121	—	2	1	no	—
368	#121	—	2	1	no	—
369	#121	—	2	2	no	—
370	#121	D₄×C₃₁	1	2	yes	—
371	C₄×Q₈	—	2	1	no	—
372	C₄×Q₈	—	2	1	no	—
373	C₄×Q₈	—	2	1	no	—
374	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
375	C₄×D₄	—	2	2	no	—
376	C₄×D₄	—	2	2	no	—
377	C₄×D₄	—	2	1	no	—
378	C₄×D₄	—	2	1	no	—
379	C₄×D₄	—	2	1	no	—
380	C₄×D₄	C₂³×C₃₁	1	2	yes	—
381	#117	—	2	1	no	—
382	#117	—	2	1	no	—
383	C₂×C₄⋊C₄	—	2	2	no	—
384	C₂×C₄⋊C₄	—	2	2	no	—
385	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
386	#107	—	2	2	no	—
387	#106	—	2	2	no	—
388	C₄⋊C₈	—	2	1	no	—
389	C₄⋊C₈	—	2	1	no	—
390	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
391	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
392	#103	—	2	2	no	—
393	#103	—	2	2	no	—
394	#103	—	2	2	no	—
395	#103	—	2	2	no	—
396	#103	—	2	2	no	—
397	#102	—	2	2	no	—
398	#102	—	2	2	no	—
399	#102	—	2	2	no	—
400	C₄×C₈	—	2	1	no	—
401	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
402	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
403	C₃₁	#18692	1	1	yes	characteristic, direct product factor, Sylow
404	#41	—	4	4	no	—
405	#41	—	8	1	no	—
406	#41	—	8	1	no	—
407	#41	—	8	2	no	—
408	#41	—	8	2	no	—
409	#41	—	8	4	no	—
410	C₂×Q₈	—	2	1	no	—
411	C₂×Q₈	—	2	1	no	—
412	C₂×Q₈	—	2	1	no	—
413	C₂×Q₈	—	2	1	no	—
414	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
415	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
416	C₂×D₄	—	2	4	no	—
417	C₂×D₄	—	2	4	no	—
418	C₂×D₄	—	2	2	no	—
419	C₂×D₄	—	2	2	no	—
420	C₂×D₄	—	2	1	no	—
421	C₂×D₄	—	2	1	no	—
422	C₂×D₄	—	2	1	no	—
423	C₂×D₄	—	2	1	no	—
424	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
425	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
426	C₂²×C₄	—	2	4	no	—
427	C₂²×C₄	—	2	4	no	—
428	C₂²×C₄	—	2	2	no	—
429	C₂²×C₄	—	2	2	no	—
430	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
431	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
432	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
433	Q₁₆	—	8	2	no	—
434	SD₁₆	—	8	2	no	—
435	M₁₆	—	8	1	no	—
436	M₁₆	—	8	1	no	—
437	C₂×C₈	—	2	1	no	—
438	C₂×C₈	—	2	1	no	—
439	C₂×C₈	—	2	2	no	—
440	C₂×C₈	—	2	2	no	—
441	C₂×C₈	—	4	2	no	—
442	C₄⋊C₄	—	2	4	no	—
443	C₄⋊C₄	—	2	4	no	—
444	C₄⋊C₄	—	2	1	no	—
445	C₄⋊C₄	—	2	1	no	—
446	C₄⋊C₄	—	2	1	no	—
447	C₄⋊C₄	—	2	1	no	—
448	C₄⋊C₄	—	2	2	no	—
449	C₄⋊C₄	—	2	1	no	—
450	C₄⋊C₄	—	2	1	no	—
451	C₄⋊C₄	—	2	2	no	—
452	C₄⋊C₄	—	2	1	no	—
453	C₄⋊C₄	—	2	1	no	—
454	C₄⋊C₄	—	2	1	no	—
455	C₄⋊C₄	—	2	1	no	—
456	C₄⋊C₄	—	4	1	no	—
457	C₄⋊C₄	—	4	1	no	—
458	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
459	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
460	C₂²⋊C₄	—	2	4	no	—
461	C₂²⋊C₄	—	2	4	no	—
462	C₂²⋊C₄	—	2	2	no	—
463	C₂²⋊C₄	—	2	2	no	—
464	C₂²⋊C₄	—	2	2	no	—
465	C₂²⋊C₄	—	2	2	no	—
466	C₂²⋊C₄	—	2	2	no	—
467	C₂²⋊C₄	—	4	1	no	—
468	C₄²	—	2	2	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	C₂⁴×C₃₁	1	1	yes	characteristic
474	C₂³	—	2	2	no	—
475	C₂³	—	2	2	no	—
476	C₂³	C₃₁×#120	1	1	yes	characteristic
477	Q₈	—	2	2	no	—
478	Q₈	—	2	2	no	—
479	Q₈	—	4	2	no	—
480	Q₈	—	4	2	no	—
481	Q₈	—	4	1	no	—
482	Q₈	—	4	1	no	—
483	Q₈	—	4	1	no	—
484	Q₈	—	4	1	no	—
485	Q₈	—	8	1	no	—
486	D₄	—	2	2	no	—
487	D₄	—	2	2	no	—
488	D₄	—	4	2	no	—
489	D₄	—	4	2	no	—
490	D₄	—	4	1	no	—
491	D₄	—	4	1	no	—
492	D₄	—	4	1	no	—
493	D₄	—	4	1	no	—
494	D₄	—	8	1	no	—
495	D₄	—	8	2	no	—
496	D₄	—	8	2	no	—
497	D₄	—	8	4	no	—
498	D₄	—	8	4	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	1	no	—
501	C₂×C₄	—	2	1	no	—
502	C₂×C₄	—	2	1	no	—
503	C₂×C₄	—	2	1	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	—	4	4	no	—
509	C₂×C₄	—	4	2	no	—
510	C₂×C₄	—	4	2	no	—
511	C₂×C₄	—	4	1	no	—
512	C₂×C₄	—	4	1	no	—
513	C₂×C₄	—	8	4	no	—
514	C₂×C₄	—	8	4	no	—
515	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
516	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
517	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
518	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
519	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
520	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
521	C₈	—	4	1	no	—
522	C₈	—	4	1	no	—
523	C₈	—	4	2	no	—
524	C₈	—	4	2	no	—
525	C₂²	—	2	1	no	—
526	C₂²	—	2	1	no	—
527	C₂²	—	2	1	no	—
528	C₂²	—	4	2	no	—
529	C₂²	—	4	2	no	—
530	C₂²	—	4	2	no	—
531	C₂²	—	4	2	no	—
532	C₂²	—	4	2	no	—
533	C₂²	—	4	2	no	—
534	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
535	C₄	—	2	1	no	—
536	C₄	—	2	1	no	—
537	C₄	—	2	1	no	—
538	C₄	—	4	2	no	—
539	C₄	—	4	2	no	—
540	C₄	—	4	2	no	—
541	C₄	—	4	2	no	—
542	C₄	—	8	1	no	—
543	C₄	—	8	1	no	—
544	C₄	C₃₁×#455	1	2	yes	—
545	C₄	C₃₁×#576	1	1	yes	characteristic
546	C₄	C₃₁×#577	1	1	yes	characteristic
547	C₄	C₃₁×#577	1	1	yes	characteristic
548	C₄	C₃₁×#578	1	1	yes	characteristic
549	C₂	—	4	1	no	—
550	C₂	—	8	2	no	—
551	C₂	—	8	2	no	—
552	C₂	C₃₁×#3012	1	1	yes	characteristic
553	C₂	C₃₁×#3013	1	1	yes	characteristic
554	C₂	C₂×C₃₁×#455	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3548	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×#3255	C₂	semidirect
3. C₃₁×#3254	C₂	semidirect
4. C₃₁×#3131	C₂	semidirect
5. C₃₁×#1644	C₂	semidirect
6. C₃₁×#1643	C₂	semidirect
7. C₃₁×#1642	C₂	semidirect
8. C₃₁×#1641	C₂	semidirect
9. C₃₁×#1560	C₂	semidirect
10. C₃₁×#1493	C₂	semidirect
11. C₃₁×#1492	C₂	semidirect
196. #18692	C₃₁	direct (N × H)
249. #3255	C₂×C₃₁	semidirect
250. #3254	C₂×C₃₁	semidirect
251. #3131	C₂×C₃₁	semidirect
252. #1644	C₂×C₃₁	semidirect
253. #1643	C₂×C₃₁	semidirect
254. #1642	C₂×C₃₁	semidirect
255. #1641	C₂×C₃₁	semidirect
256. #1560	C₂×C₃₁	semidirect
257. #1493	C₂×C₃₁	semidirect
258. #1492	C₂×C₃₁	semidirect
403. C₃₁	#18692	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×#3548	2
C₃₁×#3255	2
C₃₁×#3254	2
C₃₁×#3131	2
C₃₁×#1644	2
C₃₁×#1643	2
C₃₁×#1642	2
C₃₁×#1641	2
C₃₁×#1560	2
C₃₁×#1493	2
C₃₁×#1492	2
#18692	31

Subgroups — C31×#18692 #21475451

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#18692 #21475451