Subgroups — C₃₁×#18829 #21475588

Order 7936.

872 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×#3503	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3237	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3229	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3116	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1653	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1651	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1637	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1636	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1556	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1497	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1488	C₂	1	1	yes	maximal, characteristic
12	Q₈²×C₃₁	—	2	1	no	—
13	C₃₁×#557	C₂²	1	2	yes	—
14	C₃₁×#554	—	2	1	no	—
15	C₃₁×#554	—	2	2	no	—
16	D₄×Q₈×C₃₁	—	2	2	no	—
17	D₄×Q₈×C₃₁	C₂²	1	2	yes	—
18	C₃₁×#544	—	2	2	no	—
19	C₃₁×#544	—	2	1	no	—
20	C₃₁×#533	—	2	1	no	—
21	C₃₁×#533	C₂²	1	1	yes	characteristic
22	C₃₁×#533	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
24	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
25	C₃₁×#520	—	2	1	no	—
26	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#501	—	2	1	no	—
28	C₃₁×#498	C₂²	1	2	yes	—
29	C₃₁×#491	—	2	1	no	—
30	C₃₁×#487	C₂²	1	2	yes	—
31	C₃₁×#485	—	2	2	no	—
32	C₃₁×#485	—	2	1	no	—
33	C₃₁×#483	—	2	2	no	—
34	C₃₁×#477	C₂²	1	1	yes	characteristic
35	C₃₁×#477	C₂²	1	2	yes	—
36	C₃₁×#476	C₂²	1	1	yes	characteristic
37	C₃₁×#475	—	2	2	no	—
38	C₃₁×#475	C₂²	1	2	yes	—
39	C₃₁×#462	C₂²	1	1	yes	characteristic
40	C₃₁×#461	C₂²	1	1	yes	characteristic
41	C₃₁×#452	—	2	1	no	—
42	C₃₁×#441	—	2	1	no	—
43	C₃₁×#440	—	2	1	no	—
44	C₃₁×#432	C₂²	1	1	yes	characteristic
45	C₃₁×#432	C₂²	1	1	yes	characteristic
46	C₃₁×#431	C₂²	1	1	yes	characteristic
47	C₃₁×#332	C₂²	1	2	yes	—
48	C₃₁×#332	C₂²	1	2	yes	—
49	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
50	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
51	C₃₁×Q₈⋊C₈	C₂²	1	2	yes	—
52	C₃₁×D₄⋊C₈	C₂²	1	2	yes	—
53	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
54	C₂²×Q₈×C₃₁	—	2	2	no	—
55	C₂²×Q₈×C₃₁	—	2	1	no	—
56	C₂×Q₁₆×C₃₁	—	2	1	no	—
57	C₂×SD₁₆×C₃₁	—	2	1	no	—
58	C₃₁×C₄⋊Q₈	—	2	2	no	—
59	C₃₁×C₄⋊Q₈	—	2	2	no	—
60	C₃₁×C₄⋊Q₈	—	2	4	no	—
61	C₃₁×C₄⋊Q₈	—	2	2	no	—
62	C₃₁×C₄⋊Q₈	—	2	2	no	—
63	C₃₁×C₄⋊Q₈	—	2	1	no	—
64	C₃₁×C₄⋊Q₈	—	2	1	no	—
65	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
66	C₃₁×C₄⋊Q₈	D₄	1	2	yes	—
67	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
68	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
69	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
70	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
71	C₃₁×#125	—	2	2	no	—
72	C₃₁×#125	—	2	4	no	—
73	C₃₁×#125	—	2	2	no	—
74	C₃₁×#125	—	2	1	no	—
75	C₃₁×#125	—	2	1	no	—
76	C₃₁×C₂²⋊Q₈	—	2	2	no	—
77	C₃₁×C₂²⋊Q₈	—	2	4	no	—
78	C₃₁×C₂²⋊Q₈	—	2	4	no	—
79	C₃₁×C₂²⋊Q₈	—	2	1	no	—
80	C₃₁×C₂²⋊Q₈	—	2	2	no	—
81	C₃₁×C₂²⋊Q₈	—	2	4	no	—
82	C₃₁×C₂²⋊Q₈	—	2	1	no	—
83	C₃₁×C₂²⋊Q₈	—	2	1	no	—
84	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
85	C₄×Q₈×C₃₁	—	2	2	no	—
86	C₄×Q₈×C₃₁	—	2	2	no	—
87	C₄×Q₈×C₃₁	—	2	2	no	—
88	C₄×Q₈×C₃₁	—	2	1	no	—
89	C₄×Q₈×C₃₁	—	2	1	no	—
90	C₄×Q₈×C₃₁	—	2	1	no	—
91	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
92	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
93	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
94	C₄×D₄×C₃₁	—	2	4	no	—
95	C₄×D₄×C₃₁	—	2	2	no	—
96	C₄×D₄×C₃₁	—	2	1	no	—
97	C₄×D₄×C₃₁	—	2	1	no	—
98	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
99	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
100	C₃₁×#117	—	2	2	no	—
101	C₃₁×#117	—	2	1	no	—
102	C₃₁×#117	—	2	1	no	—
103	C₃₁×#117	—	2	1	no	—
104	C₃₁×#117	D₄	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
106	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
107	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
108	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
109	C₂×C₄²×C₃₁	—	2	1	no	—
110	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
111	C₃₁×#107	—	2	1	no	—
112	C₃₁×#107	—	2	2	no	—
113	C₃₁×#106	—	2	1	no	—
114	C₃₁×#106	—	2	4	no	—
115	C₃₁×#106	—	2	2	no	—
116	C₃₁×C₄⋊C₈	—	2	1	no	—
117	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
118	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
119	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
120	C₃₁×#103	—	2	1	no	—
121	C₃₁×#103	—	2	2	no	—
122	C₃₁×#103	—	2	2	no	—
123	C₃₁×#103	—	2	1	no	—
124	C₃₁×#103	—	2	1	no	—
125	C₃₁×#103	—	2	4	no	—
126	C₃₁×#102	—	2	1	no	—
127	C₃₁×C₂²⋊C₈	—	2	1	no	—
128	C₃₁×C₂²⋊C₈	—	2	1	no	—
129	C₃₁×C₂²⋊C₈	—	2	1	no	—
130	C₃₁×#97	—	2	1	no	—
131	C₃₁×#97	—	2	1	no	—
132	C₄×C₈×C₃₁	C₂³	1	2	yes	—
133	#41×C₃₁	—	2	2	no	—
134	#41×C₃₁	—	2	2	no	—
135	C₂×Q₈×C₃₁	—	2	4	no	—
136	C₂×Q₈×C₃₁	—	2	1	no	—
137	C₂×Q₈×C₃₁	—	2	2	no	—
138	C₂×Q₈×C₃₁	—	2	2	no	—
139	C₂×Q₈×C₃₁	—	2	1	no	—
140	C₂×Q₈×C₃₁	—	2	1	no	—
141	C₂×Q₈×C₃₁	—	8	2	no	—
142	C₂×Q₈×C₃₁	—	8	4	no	—
143	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
144	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
145	C₂×D₄×C₃₁	—	2	1	no	—
146	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
147	C₂²×C₄×C₃₁	—	2	1	no	—
148	C₂²×C₄×C₃₁	—	2	4	no	—
149	C₂²×C₄×C₃₁	—	2	2	no	—
150	C₂²×C₄×C₃₁	—	2	1	no	—
151	C₂²×C₄×C₃₁	—	2	1	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	—	4	2	no	—
156	SD₁₆×C₃₁	—	4	2	no	—
157	C₂×C₈×C₃₁	—	2	2	no	—
158	C₂×C₈×C₃₁	—	2	1	no	—
159	C₂×C₈×C₃₁	—	2	2	no	—
160	C₂×C₈×C₃₁	—	2	1	no	—
161	C₄⋊C₄×C₃₁	—	2	4	no	—
162	C₄⋊C₄×C₃₁	—	2	4	no	—
163	C₄⋊C₄×C₃₁	—	2	4	no	—
164	C₄⋊C₄×C₃₁	—	2	4	no	—
165	C₄⋊C₄×C₃₁	—	2	1	no	—
166	C₄⋊C₄×C₃₁	—	2	2	no	—
167	C₄⋊C₄×C₃₁	—	2	1	no	—
168	C₄⋊C₄×C₃₁	—	2	2	no	—
169	C₄⋊C₄×C₃₁	—	2	1	no	—
170	C₄⋊C₄×C₃₁	—	2	1	no	—
171	C₄⋊C₄×C₃₁	—	2	2	no	—
172	C₄⋊C₄×C₃₁	—	2	2	no	—
173	C₄⋊C₄×C₃₁	—	2	1	no	—
174	C₄⋊C₄×C₃₁	—	2	2	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	—	4	1	no	—
180	C₄⋊C₄×C₃₁	—	4	1	no	—
181	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
182	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
183	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
184	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
185	C₂²⋊C₄×C₃₁	—	2	4	no	—
186	C₂²⋊C₄×C₃₁	—	2	2	no	—
187	C₂²⋊C₄×C₃₁	—	2	1	no	—
188	C₂²⋊C₄×C₃₁	—	2	1	no	—
189	C₂²⋊C₄×C₃₁	—	2	1	no	—
190	C₂²⋊C₄×C₃₁	—	4	1	no	—
191	C₄²×C₃₁	—	2	4	no	—
192	C₄²×C₃₁	—	2	1	no	—
193	C₄²×C₃₁	—	2	1	no	—
194	C₄²×C₃₁	—	2	1	no	—
195	C₄²×C₃₁	—	2	2	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	#18829	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
202	C₂³×C₃₁	—	2	1	no	—
203	C₂³×C₃₁	#120	1	1	yes	characteristic
204	Q₈×C₃₁	—	2	1	no	—
205	Q₈×C₃₁	—	2	4	no	—
206	Q₈×C₃₁	—	4	2	no	—
207	Q₈×C₃₁	—	4	2	no	—
208	Q₈×C₃₁	—	4	4	no	—
209	Q₈×C₃₁	—	4	1	no	—
210	Q₈×C₃₁	—	4	1	no	—
211	Q₈×C₃₁	—	8	2	no	—
212	Q₈×C₃₁	—	8	4	no	—
213	Q₈×C₃₁	#137	1	2	yes	—
214	D₄×C₃₁	—	2	4	no	—
215	D₄×C₃₁	—	2	1	no	—
216	D₄×C₃₁	#137	1	2	yes	—
217	C₂×C₄×C₃₁	—	2	4	no	—
218	C₂×C₄×C₃₁	—	2	1	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	—	2	2	no	—
223	C₂×C₄×C₃₁	—	2	1	no	—
224	C₂×C₄×C₃₁	—	2	1	no	—
225	C₂×C₄×C₃₁	—	2	2	no	—
226	C₂×C₄×C₃₁	—	2	4	no	—
227	C₂×C₄×C₃₁	—	4	1	no	—
228	C₂×C₄×C₃₁	—	4	2	no	—
229	C₂×C₄×C₃₁	—	4	2	no	—
230	C₂×C₄×C₃₁	—	4	1	no	—
231	C₂×C₄×C₃₁	—	4	1	no	—
232	C₂×C₄×C₃₁	—	8	2	no	—
233	C₂×C₄×C₃₁	—	8	4	no	—
234	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
235	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
236	C₂×C₄×C₃₁	#120	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	C₂²×D₄	1	1	yes	characteristic
240	C₈×C₃₁	—	4	2	no	—
241	C₈×C₃₁	—	4	1	no	—
242	C₈×C₃₁	—	4	1	no	—
243	C₈×C₃₁	—	4	2	no	—
244	#3503	C₂×C₃₁	1	1	yes	characteristic
245	#3237	C₂×C₃₁	1	1	yes	characteristic, complemented
246	#3229	C₂×C₃₁	1	1	yes	characteristic, complemented
247	#3116	C₂×C₃₁	1	1	yes	characteristic
248	#1653	C₂×C₃₁	1	2	yes	complemented
249	#1651	C₂×C₃₁	1	2	yes	complemented
250	#1637	C₂×C₃₁	1	2	yes	complemented
251	#1636	C₂×C₃₁	1	2	yes	complemented
252	#1556	C₂×C₃₁	1	1	yes	characteristic, complemented
253	#1497	C₂×C₃₁	1	1	yes	characteristic, complemented
254	#1488	C₂×C₃₁	1	1	yes	characteristic
255	C₂²×C₃₁	—	2	1	no	—
256	C₂²×C₃₁	—	2	2	no	—
257	C₂²×C₃₁	—	2	1	no	—
258	C₂²×C₃₁	—	2	1	no	—
259	C₂²×C₃₁	—	4	1	no	—
260	C₂²×C₃₁	—	4	1	no	—
261	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
262	C₄×C₃₁	—	2	1	no	—
263	C₄×C₃₁	—	2	2	no	—
264	C₄×C₃₁	—	2	1	no	—
265	C₄×C₃₁	—	2	1	no	—
266	C₄×C₃₁	—	2	1	no	—
267	C₄×C₃₁	—	4	1	no	—
268	C₄×C₃₁	—	4	2	no	—
269	C₄×C₃₁	—	4	4	no	—
270	C₄×C₃₁	—	4	1	no	—
271	C₄×C₃₁	—	4	2	no	—
272	C₄×C₃₁	—	8	1	no	—
273	C₄×C₃₁	—	8	1	no	—
274	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
275	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
276	C₄×C₃₁	#577	1	2	yes	—
277	Q₈²	—	2	1	no	—
278	#557	C₂²×C₃₁	1	2	yes	—
279	#554	—	2	1	no	—
280	#554	—	2	2	no	—
281	D₄×Q₈	—	2	2	no	—
282	D₄×Q₈	C₂²×C₃₁	1	2	yes	—
283	#544	—	2	2	no	—
284	#544	—	2	1	no	—
285	#533	—	2	1	no	—
286	#533	C₂²×C₃₁	1	1	yes	characteristic
287	#533	C₂²×C₃₁	1	1	yes	characteristic
288	C₂×C₄⋊Q₈	—	2	1	no	—
289	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
290	#520	—	2	1	no	—
291	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
292	#501	—	2	1	no	—
293	#498	C₂²×C₃₁	1	2	yes	—
294	#491	—	2	1	no	—
295	#487	C₂²×C₃₁	1	2	yes	—
296	#485	—	2	2	no	—
297	#485	—	2	1	no	—
298	#483	—	2	2	no	—
299	#477	C₂²×C₃₁	1	1	yes	characteristic
300	#477	C₂²×C₃₁	1	2	yes	—
301	#476	C₂²×C₃₁	1	1	yes	characteristic
302	#475	—	2	2	no	—
303	#475	C₂²×C₃₁	1	2	yes	—
304	#462	C₂²×C₃₁	1	1	yes	characteristic
305	#461	C₂²×C₃₁	1	1	yes	characteristic
306	#452	—	2	1	no	—
307	#441	—	2	1	no	—
308	#440	—	2	1	no	—
309	#432	C₂²×C₃₁	1	1	yes	characteristic
310	#432	C₂²×C₃₁	1	1	yes	characteristic
311	#431	C₂²×C₃₁	1	1	yes	characteristic
312	#332	C₂²×C₃₁	1	2	yes	—
313	#332	C₂²×C₃₁	1	2	yes	—
314	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
315	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
316	Q₈⋊C₈	C₂²×C₃₁	1	2	yes	—
317	D₄⋊C₈	C₂²×C₃₁	1	2	yes	—
318	C₂×C₃₁	—	4	1	no	—
319	C₂×C₃₁	—	4	2	no	—
320	C₂×C₃₁	#3004	1	1	yes	characteristic
321	C₂×C₃₁	#3009	1	1	yes	characteristic
322	C₂×C₃₁	#3020	1	1	yes	characteristic
323	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
324	C₂²×Q₈	—	2	2	no	—
325	C₂²×Q₈	—	2	1	no	—
326	C₂×Q₁₆	—	2	1	no	—
327	C₂×SD₁₆	—	2	1	no	—
328	C₄⋊Q₈	—	2	2	no	—
329	C₄⋊Q₈	—	2	2	no	—
330	C₄⋊Q₈	—	2	4	no	—
331	C₄⋊Q₈	—	2	2	no	—
332	C₄⋊Q₈	—	2	2	no	—
333	C₄⋊Q₈	—	2	1	no	—
334	C₄⋊Q₈	—	2	1	no	—
335	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
336	C₄⋊Q₈	D₄×C₃₁	1	2	yes	—
337	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
338	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
339	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
340	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
341	#125	—	2	2	no	—
342	#125	—	2	4	no	—
343	#125	—	2	2	no	—
344	#125	—	2	1	no	—
345	#125	—	2	1	no	—
346	C₂²⋊Q₈	—	2	2	no	—
347	C₂²⋊Q₈	—	2	4	no	—
348	C₂²⋊Q₈	—	2	4	no	—
349	C₂²⋊Q₈	—	2	1	no	—
350	C₂²⋊Q₈	—	2	2	no	—
351	C₂²⋊Q₈	—	2	4	no	—
352	C₂²⋊Q₈	—	2	1	no	—
353	C₂²⋊Q₈	—	2	1	no	—
354	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
355	C₄×Q₈	—	2	2	no	—
356	C₄×Q₈	—	2	2	no	—
357	C₄×Q₈	—	2	2	no	—
358	C₄×Q₈	—	2	1	no	—
359	C₄×Q₈	—	2	1	no	—
360	C₄×Q₈	—	2	1	no	—
361	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
362	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
363	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
364	C₄×D₄	—	2	4	no	—
365	C₄×D₄	—	2	2	no	—
366	C₄×D₄	—	2	1	no	—
367	C₄×D₄	—	2	1	no	—
368	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
369	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
370	#117	—	2	2	no	—
371	#117	—	2	1	no	—
372	#117	—	2	1	no	—
373	#117	—	2	1	no	—
374	#117	D₄×C₃₁	1	1	yes	characteristic
375	C₂×C₄⋊C₄	—	2	2	no	—
376	C₂×C₄⋊C₄	—	2	2	no	—
377	C₂×C₄⋊C₄	—	2	1	no	—
378	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
379	C₂×C₄²	—	2	1	no	—
380	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
381	#107	—	2	1	no	—
382	#107	—	2	2	no	—
383	#106	—	2	1	no	—
384	#106	—	2	4	no	—
385	#106	—	2	2	no	—
386	C₄⋊C₈	—	2	1	no	—
387	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
388	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
389	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
390	#103	—	2	1	no	—
391	#103	—	2	2	no	—
392	#103	—	2	2	no	—
393	#103	—	2	1	no	—
394	#103	—	2	1	no	—
395	#103	—	2	4	no	—
396	#102	—	2	1	no	—
397	C₂²⋊C₈	—	2	1	no	—
398	C₂²⋊C₈	—	2	1	no	—
399	C₂²⋊C₈	—	2	1	no	—
400	#97	—	2	1	no	—
401	#97	—	2	1	no	—
402	C₄×C₈	C₂³×C₃₁	1	2	yes	—
403	C₃₁	#18829	1	1	yes	characteristic, direct product factor, Sylow
404	#41	—	2	2	no	—
405	#41	—	2	2	no	—
406	C₂×Q₈	—	2	4	no	—
407	C₂×Q₈	—	2	1	no	—
408	C₂×Q₈	—	2	2	no	—
409	C₂×Q₈	—	2	2	no	—
410	C₂×Q₈	—	2	1	no	—
411	C₂×Q₈	—	2	1	no	—
412	C₂×Q₈	—	8	2	no	—
413	C₂×Q₈	—	8	4	no	—
414	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
415	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
416	C₂×D₄	—	2	1	no	—
417	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
418	C₂²×C₄	—	2	1	no	—
419	C₂²×C₄	—	2	4	no	—
420	C₂²×C₄	—	2	2	no	—
421	C₂²×C₄	—	2	1	no	—
422	C₂²×C₄	—	2	1	no	—
423	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
424	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
425	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
426	Q₁₆	—	4	2	no	—
427	SD₁₆	—	4	2	no	—
428	C₂×C₈	—	2	2	no	—
429	C₂×C₈	—	2	1	no	—
430	C₂×C₈	—	2	2	no	—
431	C₂×C₈	—	2	1	no	—
432	C₄⋊C₄	—	2	4	no	—
433	C₄⋊C₄	—	2	4	no	—
434	C₄⋊C₄	—	2	4	no	—
435	C₄⋊C₄	—	2	4	no	—
436	C₄⋊C₄	—	2	1	no	—
437	C₄⋊C₄	—	2	2	no	—
438	C₄⋊C₄	—	2	1	no	—
439	C₄⋊C₄	—	2	2	no	—
440	C₄⋊C₄	—	2	1	no	—
441	C₄⋊C₄	—	2	1	no	—
442	C₄⋊C₄	—	2	2	no	—
443	C₄⋊C₄	—	2	2	no	—
444	C₄⋊C₄	—	2	1	no	—
445	C₄⋊C₄	—	2	2	no	—
446	C₄⋊C₄	—	2	1	no	—
447	C₄⋊C₄	—	2	1	no	—
448	C₄⋊C₄	—	2	1	no	—
449	C₄⋊C₄	—	2	1	no	—
450	C₄⋊C₄	—	4	1	no	—
451	C₄⋊C₄	—	4	1	no	—
452	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
453	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
454	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
455	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
456	C₂²⋊C₄	—	2	4	no	—
457	C₂²⋊C₄	—	2	2	no	—
458	C₂²⋊C₄	—	2	1	no	—
459	C₂²⋊C₄	—	2	1	no	—
460	C₂²⋊C₄	—	2	1	no	—
461	C₂²⋊C₄	—	4	1	no	—
462	C₄²	—	2	4	no	—
463	C₄²	—	2	1	no	—
464	C₄²	—	2	1	no	—
465	C₄²	—	2	1	no	—
466	C₄²	—	2	2	no	—
467	C₄²	—	4	1	no	—
468	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
469	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
470	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
471	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
472	C₂³	—	2	1	no	—
473	C₂³	C₃₁×#120	1	1	yes	characteristic
474	Q₈	—	2	1	no	—
475	Q₈	—	2	4	no	—
476	Q₈	—	4	2	no	—
477	Q₈	—	4	2	no	—
478	Q₈	—	4	4	no	—
479	Q₈	—	4	1	no	—
480	Q₈	—	4	1	no	—
481	Q₈	—	8	2	no	—
482	Q₈	—	8	4	no	—
483	Q₈	C₃₁×#137	1	2	yes	—
484	D₄	—	2	4	no	—
485	D₄	—	2	1	no	—
486	D₄	C₃₁×#137	1	2	yes	—
487	C₂×C₄	—	2	4	no	—
488	C₂×C₄	—	2	1	no	—
489	C₂×C₄	—	2	1	no	—
490	C₂×C₄	—	2	1	no	—
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	1	no	—
494	C₂×C₄	—	2	1	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	4	no	—
497	C₂×C₄	—	4	1	no	—
498	C₂×C₄	—	4	2	no	—
499	C₂×C₄	—	4	2	no	—
500	C₂×C₄	—	4	1	no	—
501	C₂×C₄	—	4	1	no	—
502	C₂×C₄	—	8	2	no	—
503	C₂×C₄	—	8	4	no	—
504	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
505	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
506	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
507	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
508	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
509	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
510	C₈	—	4	2	no	—
511	C₈	—	4	1	no	—
512	C₈	—	4	1	no	—
513	C₈	—	4	2	no	—
514	C₂²	—	2	1	no	—
515	C₂²	—	2	2	no	—
516	C₂²	—	2	1	no	—
517	C₂²	—	2	1	no	—
518	C₂²	—	4	1	no	—
519	C₂²	—	4	1	no	—
520	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
521	C₄	—	2	1	no	—
522	C₄	—	2	2	no	—
523	C₄	—	2	1	no	—
524	C₄	—	2	1	no	—
525	C₄	—	2	1	no	—
526	C₄	—	4	1	no	—
527	C₄	—	4	2	no	—
528	C₄	—	4	4	no	—
529	C₄	—	4	1	no	—
530	C₄	—	4	2	no	—
531	C₄	—	8	1	no	—
532	C₄	—	8	1	no	—
533	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
534	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
535	C₄	C₃₁×#577	1	2	yes	—
536	C₂	—	4	1	no	—
537	C₂	—	4	2	no	—
538	C₂	C₃₁×#3004	1	1	yes	characteristic
539	C₂	C₃₁×#3009	1	1	yes	characteristic
540	C₂	C₃₁×#3020	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₂×#41×C₃₁	D₄
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×#3237	C₂	semidirect
3. C₃₁×#3229	C₂	semidirect
5. C₃₁×#1653	C₂	semidirect
6. C₃₁×#1651	C₂	semidirect
7. C₃₁×#1637	C₂	semidirect
8. C₃₁×#1636	C₂	semidirect
9. C₃₁×#1556	C₂	semidirect
10. C₃₁×#1497	C₂	semidirect
201. #18829	C₃₁	direct (N × H)
245. #3237	C₂×C₃₁	semidirect
246. #3229	C₂×C₃₁	semidirect
248. #1653	C₂×C₃₁	semidirect
249. #1651	C₂×C₃₁	semidirect
250. #1637	C₂×C₃₁	semidirect
251. #1636	C₂×C₃₁	semidirect
252. #1556	C₂×C₃₁	semidirect
253. #1497	C₂×C₃₁	semidirect
403. C₃₁	#18829	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×#3503	2
C₃₁×#3237	2
C₃₁×#3229	2
C₃₁×#3116	2
C₃₁×#1653	2
C₃₁×#1651	2
C₃₁×#1637	2
C₃₁×#1636	2
C₃₁×#1556	2
C₃₁×#1497	2
C₃₁×#1488	2
#18829	31

Subgroups — C31×#18829 #21475588

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#18829 #21475588