Subgroups — C₃₁×#18768 #21475527

Order 7936.

976 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈²×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3206	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3103	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#3037	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1645	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1623	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1548	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1487	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1479	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#137	—	2	1	no	—
11	Q₈²×C₃₁	—	2	2	no	—
12	Q₈²×C₃₁	—	2	2	no	—
13	Q₈²×C₃₁	—	2	2	no	—
14	Q₈²×C₃₁	C₂²	1	4	yes	—
15	C₃₁×#532	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×C₄⋊Q₈	—	2	2	no	—
17	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
18	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
19	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
20	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
21	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
22	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#492	C₂²	1	2	yes	—
25	C₃₁×#487	C₂²	1	2	yes	—
26	C₃₁×#481	—	2	1	no	—
27	C₃₁×#475	C₂²	1	2	yes	—
28	C₃₁×#464	—	2	2	no	—
29	C₃₁×#462	C₂²	1	2	yes	—
30	C₃₁×#461	—	2	2	no	—
31	C₃₁×#460	C₂²	1	2	yes	—
32	C₃₁×#460	C₂²	1	2	yes	—
33	C₃₁×#448	—	2	1	no	—
34	C₃₁×#448	—	2	2	no	—
35	C₄×SD₁₆×C₃₁	—	2	2	no	—
36	C₃₁×#431	C₂²	1	1	yes	characteristic
37	C₃₁×#431	C₂²	1	1	yes	characteristic
38	C₃₁×#423	C₂²	1	1	yes	characteristic
39	C₃₁×#417	—	2	1	no	—
40	C₃₁×#333	C₂²	1	4	yes	—
41	C₃₁×Q₈⋊C₈	C₂²	1	4	yes	—
42	C₃₁×Q₈⋊C₈	C₂²	1	4	yes	—
43	C₂×#41×C₃₁	—	2	1	no	—
44	C₂²×Q₈×C₃₁	—	2	2	no	—
45	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
46	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
47	C₂²×Q₈×C₃₁	D₄	1	2	yes	—
48	C₃₁×#137	—	4	4	no	—
49	C₂×Q₁₆×C₃₁	—	2	2	no	—
50	C₂×SD₁₆×C₃₁	—	2	2	no	—
51	C₂×SD₁₆×C₃₁	—	2	2	no	—
52	C₂×SD₁₆×C₃₁	—	2	4	no	—
53	C₂×M₁₆×C₃₁	—	2	1	no	—
54	C₃₁×C₄⋊Q₈	—	2	8	no	—
55	C₃₁×C₄⋊Q₈	—	2	4	no	—
56	C₃₁×C₄⋊Q₈	—	2	4	no	—
57	C₃₁×C₄⋊Q₈	—	2	8	no	—
58	C₃₁×C₄⋊Q₈	—	2	4	no	—
59	C₃₁×C₄⋊Q₈	—	2	4	no	—
60	C₃₁×C₄⋊Q₈	—	2	1	no	—
61	C₃₁×C₄⋊Q₈	—	2	1	no	—
62	C₃₁×C₄⋊Q₈	D₄	1	2	yes	—
63	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
64	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
65	C₃₁×#127	C₂³	1	1	yes	characteristic
66	C₃₁×#124	—	2	1	no	—
67	C₃₁×#121	—	2	1	no	—
68	C₃₁×#121	—	2	1	no	—
69	C₄×Q₈×C₃₁	—	2	4	no	—
70	C₄×Q₈×C₃₁	—	2	4	no	—
71	C₄×Q₈×C₃₁	—	2	4	no	—
72	C₄×Q₈×C₃₁	—	2	4	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	2	yes	—
78	C₄×Q₈×C₃₁	D₄	1	2	yes	—
79	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
80	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
81	C₄×D₄×C₃₁	—	2	1	no	—
82	C₄×D₄×C₃₁	—	2	1	no	—
83	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
84	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
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	—	2	1	no	—
89	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
90	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
91	C₂×C₄²×C₃₁	—	2	2	no	—
92	C₂×C₄²×C₃₁	—	2	1	no	—
93	C₂×C₄²×C₃₁	—	2	1	no	—
94	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
95	C₃₁×#106	—	2	2	no	—
96	C₃₁×C₄⋊C₈	—	2	1	no	—
97	C₃₁×C₄⋊C₈	—	2	1	no	—
98	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
99	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
100	C₃₁×#103	—	2	2	no	—
101	C₃₁×#103	—	2	4	no	—
102	C₃₁×#103	—	2	2	no	—
103	C₃₁×#102	—	2	2	no	—
104	C₃₁×#102	—	2	2	no	—
105	C₃₁×C₂²⋊C₈	—	2	1	no	—
106	C₃₁×C₂²⋊C₈	—	2	1	no	—
107	C₄×C₈×C₃₁	—	2	1	no	—
108	C₄×C₈×C₃₁	C₂³	1	2	yes	—
109	#41×C₃₁	—	4	2	no	—
110	#41×C₃₁	—	4	2	no	—
111	C₂×Q₈×C₃₁	—	2	4	no	—
112	C₂×Q₈×C₃₁	—	2	8	no	—
113	C₂×Q₈×C₃₁	—	2	1	no	—
114	C₂×Q₈×C₃₁	—	2	1	no	—
115	C₂×Q₈×C₃₁	—	2	2	no	—
116	C₂×Q₈×C₃₁	—	2	2	no	—
117	C₂×Q₈×C₃₁	—	2	1	no	—
118	C₂×Q₈×C₃₁	—	8	2	no	—
119	C₂×Q₈×C₃₁	—	8	4	no	—
120	C₂×Q₈×C₃₁	SD₁₆	1	4	yes	complemented
121	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
122	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
123	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
124	C₂×D₄×C₃₁	—	2	1	no	—
125	C₂×D₄×C₃₁	—	2	1	no	—
126	C₂×D₄×C₃₁	—	2	1	no	—
127	C₂×D₄×C₃₁	—	4	1	no	—
128	C₂²×C₄×C₃₁	—	2	1	no	—
129	C₂²×C₄×C₃₁	—	2	1	no	—
130	C₂²×C₄×C₃₁	—	2	2	no	—
131	C₂²×C₄×C₃₁	—	2	1	no	—
132	C₂²×C₄×C₃₁	—	2	1	no	—
133	C₂²×C₄×C₃₁	—	4	1	no	—
134	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
137	Q₁₆×C₃₁	—	4	4	no	—
138	SD₁₆×C₃₁	—	4	4	no	—
139	SD₁₆×C₃₁	—	4	4	no	—
140	SD₁₆×C₃₁	—	8	4	no	—
141	M₁₆×C₃₁	—	4	2	no	—
142	C₂×C₈×C₃₁	—	2	2	no	—
143	C₂×C₈×C₃₁	—	2	2	no	—
144	C₂×C₈×C₃₁	—	2	2	no	—
145	C₄⋊C₄×C₃₁	—	2	8	no	—
146	C₄⋊C₄×C₃₁	—	2	8	no	—
147	C₄⋊C₄×C₃₁	—	2	2	no	—
148	C₄⋊C₄×C₃₁	—	2	2	no	—
149	C₄⋊C₄×C₃₁	—	2	2	no	—
150	C₄⋊C₄×C₃₁	—	2	2	no	—
151	C₄⋊C₄×C₃₁	—	2	1	no	—
152	C₄⋊C₄×C₃₁	—	2	1	no	—
153	C₄⋊C₄×C₃₁	—	2	2	no	—
154	C₄⋊C₄×C₃₁	—	2	2	no	—
155	C₄⋊C₄×C₃₁	—	2	2	no	—
156	C₄⋊C₄×C₃₁	—	2	2	no	—
157	C₄⋊C₄×C₃₁	—	2	1	no	—
158	C₄⋊C₄×C₃₁	—	2	1	no	—
159	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
160	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
161	C₂²⋊C₄×C₃₁	—	4	1	no	—
162	C₂²⋊C₄×C₃₁	—	4	1	no	—
163	C₄²×C₃₁	—	2	8	no	—
164	C₄²×C₃₁	—	2	2	no	—
165	C₄²×C₃₁	—	2	2	no	—
166	C₄²×C₃₁	—	2	2	no	—
167	C₄²×C₃₁	—	2	2	no	—
168	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
169	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
170	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
171	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
172	#18768	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
173	C₂³×C₃₁	—	4	1	no	—
174	C₂³×C₃₁	#120	1	1	yes	characteristic
175	Q₈×C₃₁	—	2	4	no	—
176	Q₈×C₃₁	—	2	2	no	—
177	Q₈×C₃₁	—	2	2	no	—
178	Q₈×C₃₁	—	2	4	no	—
179	Q₈×C₃₁	—	4	4	no	—
180	Q₈×C₃₁	—	4	1	no	—
181	Q₈×C₃₁	—	4	1	no	—
182	Q₈×C₃₁	—	8	2	no	—
183	Q₈×C₃₁	—	8	8	no	—
184	Q₈×C₃₁	C₂×SD₁₆	1	4	yes	—
185	Q₈×C₃₁	#137	1	4	yes	complemented
186	D₄×C₃₁	—	4	2	no	—
187	D₄×C₃₁	—	4	1	no	—
188	D₄×C₃₁	—	4	2	no	—
189	D₄×C₃₁	—	4	1	no	—
190	D₄×C₃₁	—	8	2	no	—
191	C₂×C₄×C₃₁	—	2	4	no	—
192	C₂×C₄×C₃₁	—	2	2	no	—
193	C₂×C₄×C₃₁	—	2	2	no	—
194	C₂×C₄×C₃₁	—	2	4	no	—
195	C₂×C₄×C₃₁	—	2	2	no	—
196	C₂×C₄×C₃₁	—	2	4	no	—
197	C₂×C₄×C₃₁	—	2	2	no	—
198	C₂×C₄×C₃₁	—	2	2	no	—
199	C₂×C₄×C₃₁	—	4	1	no	—
200	C₂×C₄×C₃₁	—	4	4	no	—
201	C₂×C₄×C₃₁	—	4	2	no	—
202	C₂×C₄×C₃₁	—	4	2	no	—
203	C₂×C₄×C₃₁	—	4	1	no	—
204	C₂×C₄×C₃₁	—	8	2	no	—
205	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
206	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
207	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
208	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
209	C₂×C₄×C₃₁	#137	1	2	yes	—
210	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
211	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
212	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
213	C₈×C₃₁	—	4	2	no	—
214	C₈×C₃₁	—	4	2	no	—
215	C₈×C₃₁	—	4	2	no	—
216	C₂×Q₈²	C₂×C₃₁	1	1	yes	characteristic, complemented
217	#3206	C₂×C₃₁	1	1	yes	characteristic
218	#3103	C₂×C₃₁	1	1	yes	characteristic
219	#3037	C₂×C₃₁	1	1	yes	characteristic
220	#1645	C₂×C₃₁	1	4	yes	complemented
221	#1623	C₂×C₃₁	1	4	yes	complemented
222	#1548	C₂×C₃₁	1	1	yes	characteristic, complemented
223	#1487	C₂×C₃₁	1	1	yes	characteristic, complemented
224	#1479	C₂×C₃₁	1	1	yes	characteristic, complemented
225	C₂²×C₃₁	—	2	1	no	—
226	C₂²×C₃₁	—	2	1	no	—
227	C₂²×C₃₁	—	8	1	no	—
228	C₂²×C₃₁	—	8	1	no	—
229	C₂²×C₃₁	—	8	1	no	—
230	C₂²×C₃₁	#448	1	2	yes	—
231	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
232	C₄×C₃₁	—	2	4	no	—
233	C₄×C₃₁	—	2	4	no	—
234	C₄×C₃₁	—	2	1	no	—
235	C₄×C₃₁	—	2	1	no	—
236	C₄×C₃₁	—	2	1	no	—
237	C₄×C₃₁	—	4	4	no	—
238	C₄×C₃₁	—	4	2	no	—
239	C₄×C₃₁	—	4	2	no	—
240	C₄×C₃₁	—	8	1	no	—
241	C₄×C₃₁	#448	1	2	yes	—
242	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
243	C₄×C₃₁	C₂×#137	1	1	yes	characteristic
244	C₄×C₃₁	C₂×#137	1	2	yes	—
245	C₂×#137	—	2	1	no	—
246	Q₈²	—	2	2	no	—
247	Q₈²	—	2	2	no	—
248	Q₈²	—	2	2	no	—
249	Q₈²	C₂²×C₃₁	1	4	yes	—
250	#532	C₂²×C₃₁	1	1	yes	characteristic
251	C₂×C₄⋊Q₈	—	2	2	no	—
252	C₂×C₄⋊Q₈	—	2	1	no	—
253	C₂×C₄⋊Q₈	—	2	1	no	—
254	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
255	C₂×C₄×Q₈	—	2	1	no	—
256	C₂×C₄×Q₈	—	2	1	no	—
257	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
258	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
259	#492	C₂²×C₃₁	1	2	yes	—
260	#487	C₂²×C₃₁	1	2	yes	—
261	#481	—	2	1	no	—
262	#475	C₂²×C₃₁	1	2	yes	—
263	#464	—	2	2	no	—
264	#462	C₂²×C₃₁	1	2	yes	—
265	#461	—	2	2	no	—
266	#460	C₂²×C₃₁	1	2	yes	—
267	#460	C₂²×C₃₁	1	2	yes	—
268	#448	—	2	1	no	—
269	#448	—	2	2	no	—
270	C₄×SD₁₆	—	2	2	no	—
271	#431	C₂²×C₃₁	1	1	yes	characteristic
272	#431	C₂²×C₃₁	1	1	yes	characteristic
273	#423	C₂²×C₃₁	1	1	yes	characteristic
274	#417	—	2	1	no	—
275	#333	C₂²×C₃₁	1	4	yes	—
276	Q₈⋊C₈	C₂²×C₃₁	1	4	yes	—
277	Q₈⋊C₈	C₂²×C₃₁	1	4	yes	—
278	C₂×C₃₁	—	2	2	no	—
279	C₂×C₃₁	—	16	1	no	—
280	C₂×C₃₁	C₂×#448	1	1	yes	characteristic
281	C₂×C₃₁	#3004	1	1	yes	characteristic
282	C₂×C₃₁	#3019	1	1	yes	characteristic
283	C₂×#41	—	2	1	no	—
284	C₂²×Q₈	—	2	2	no	—
285	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
286	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
287	C₂²×Q₈	D₄×C₃₁	1	2	yes	—
288	#137	—	4	4	no	—
289	C₂×Q₁₆	—	2	2	no	—
290	C₂×SD₁₆	—	2	2	no	—
291	C₂×SD₁₆	—	2	2	no	—
292	C₂×SD₁₆	—	2	4	no	—
293	C₂×M₁₆	—	2	1	no	—
294	C₄⋊Q₈	—	2	8	no	—
295	C₄⋊Q₈	—	2	4	no	—
296	C₄⋊Q₈	—	2	4	no	—
297	C₄⋊Q₈	—	2	8	no	—
298	C₄⋊Q₈	—	2	4	no	—
299	C₄⋊Q₈	—	2	4	no	—
300	C₄⋊Q₈	—	2	1	no	—
301	C₄⋊Q₈	—	2	1	no	—
302	C₄⋊Q₈	D₄×C₃₁	1	2	yes	—
303	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
304	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
305	#127	C₂³×C₃₁	1	1	yes	characteristic
306	#124	—	2	1	no	—
307	#121	—	2	1	no	—
308	#121	—	2	1	no	—
309	C₄×Q₈	—	2	4	no	—
310	C₄×Q₈	—	2	4	no	—
311	C₄×Q₈	—	2	4	no	—
312	C₄×Q₈	—	2	4	no	—
313	C₄×Q₈	—	2	1	no	—
314	C₄×Q₈	—	2	1	no	—
315	C₄×Q₈	—	2	1	no	—
316	C₄×Q₈	—	2	1	no	—
317	C₄×Q₈	D₄×C₃₁	1	2	yes	—
318	C₄×Q₈	D₄×C₃₁	1	2	yes	—
319	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
320	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
321	C₄×D₄	—	2	1	no	—
322	C₄×D₄	—	2	1	no	—
323	C₂×C₄⋊C₄	—	2	2	no	—
324	C₂×C₄⋊C₄	—	2	2	no	—
325	C₂×C₄⋊C₄	—	2	1	no	—
326	C₂×C₄⋊C₄	—	2	1	no	—
327	C₂×C₄⋊C₄	—	2	1	no	—
328	C₂×C₄⋊C₄	—	2	1	no	—
329	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
330	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
331	C₂×C₄²	—	2	2	no	—
332	C₂×C₄²	—	2	1	no	—
333	C₂×C₄²	—	2	1	no	—
334	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
335	#106	—	2	2	no	—
336	C₄⋊C₈	—	2	1	no	—
337	C₄⋊C₈	—	2	1	no	—
338	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
339	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
340	#103	—	2	2	no	—
341	#103	—	2	4	no	—
342	#103	—	2	2	no	—
343	#102	—	2	2	no	—
344	#102	—	2	2	no	—
345	C₂²⋊C₈	—	2	1	no	—
346	C₂²⋊C₈	—	2	1	no	—
347	C₄×C₈	—	2	1	no	—
348	C₄×C₈	C₂³×C₃₁	1	2	yes	—
349	C₃₁	#18768	1	1	yes	characteristic, direct product factor, Sylow
350	#41	—	4	2	no	—
351	#41	—	4	2	no	—
352	C₂×Q₈	—	2	4	no	—
353	C₂×Q₈	—	2	8	no	—
354	C₂×Q₈	—	2	1	no	—
355	C₂×Q₈	—	2	1	no	—
356	C₂×Q₈	—	2	2	no	—
357	C₂×Q₈	—	2	2	no	—
358	C₂×Q₈	—	2	1	no	—
359	C₂×Q₈	—	8	2	no	—
360	C₂×Q₈	—	8	4	no	—
361	C₂×Q₈	SD₁₆×C₃₁	1	4	yes	complemented
362	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
363	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
364	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
365	C₂×D₄	—	2	1	no	—
366	C₂×D₄	—	2	1	no	—
367	C₂×D₄	—	2	1	no	—
368	C₂×D₄	—	4	1	no	—
369	C₂²×C₄	—	2	1	no	—
370	C₂²×C₄	—	2	1	no	—
371	C₂²×C₄	—	2	2	no	—
372	C₂²×C₄	—	2	1	no	—
373	C₂²×C₄	—	2	1	no	—
374	C₂²×C₄	—	4	1	no	—
375	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
376	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
377	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
378	Q₁₆	—	4	4	no	—
379	SD₁₆	—	4	4	no	—
380	SD₁₆	—	4	4	no	—
381	SD₁₆	—	8	4	no	—
382	M₁₆	—	4	2	no	—
383	C₂×C₈	—	2	2	no	—
384	C₂×C₈	—	2	2	no	—
385	C₂×C₈	—	2	2	no	—
386	C₄⋊C₄	—	2	8	no	—
387	C₄⋊C₄	—	2	8	no	—
388	C₄⋊C₄	—	2	2	no	—
389	C₄⋊C₄	—	2	2	no	—
390	C₄⋊C₄	—	2	2	no	—
391	C₄⋊C₄	—	2	2	no	—
392	C₄⋊C₄	—	2	1	no	—
393	C₄⋊C₄	—	2	1	no	—
394	C₄⋊C₄	—	2	2	no	—
395	C₄⋊C₄	—	2	2	no	—
396	C₄⋊C₄	—	2	2	no	—
397	C₄⋊C₄	—	2	2	no	—
398	C₄⋊C₄	—	2	1	no	—
399	C₄⋊C₄	—	2	1	no	—
400	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
401	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
402	C₂²⋊C₄	—	4	1	no	—
403	C₂²⋊C₄	—	4	1	no	—
404	C₄²	—	2	8	no	—
405	C₄²	—	2	2	no	—
406	C₄²	—	2	2	no	—
407	C₄²	—	2	2	no	—
408	C₄²	—	2	2	no	—
409	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
410	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
411	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
412	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
413	C₂³	—	4	1	no	—
414	C₂³	C₃₁×#120	1	1	yes	characteristic
415	Q₈	—	2	4	no	—
416	Q₈	—	2	4	no	—
417	Q₈	—	2	2	no	—
418	Q₈	—	2	2	no	—
419	Q₈	—	4	4	no	—
420	Q₈	—	4	1	no	—
421	Q₈	—	4	1	no	—
422	Q₈	—	8	2	no	—
423	Q₈	—	8	8	no	—
424	Q₈	C₂×SD₁₆×C₃₁	1	4	yes	—
425	Q₈	C₃₁×#137	1	4	yes	complemented
426	D₄	—	4	2	no	—
427	D₄	—	4	1	no	—
428	D₄	—	4	2	no	—
429	D₄	—	4	1	no	—
430	D₄	—	8	2	no	—
431	C₂×C₄	—	2	4	no	—
432	C₂×C₄	—	2	4	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	4	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	4	1	no	—
440	C₂×C₄	—	4	4	no	—
441	C₂×C₄	—	4	2	no	—
442	C₂×C₄	—	4	2	no	—
443	C₂×C₄	—	4	1	no	—
444	C₂×C₄	—	8	2	no	—
445	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
446	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
447	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
448	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
449	C₂×C₄	C₃₁×#137	1	2	yes	—
450	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
451	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
452	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
453	C₈	—	4	2	no	—
454	C₈	—	4	2	no	—
455	C₈	—	4	2	no	—
456	C₂²	—	2	1	no	—
457	C₂²	—	2	1	no	—
458	C₂²	—	8	1	no	—
459	C₂²	—	8	1	no	—
460	C₂²	—	8	1	no	—
461	C₂²	C₃₁×#448	1	2	yes	—
462	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
463	C₄	—	2	4	no	—
464	C₄	—	2	4	no	—
465	C₄	—	2	1	no	—
466	C₄	—	2	1	no	—
467	C₄	—	2	1	no	—
468	C₄	—	4	4	no	—
469	C₄	—	4	2	no	—
470	C₄	—	4	2	no	—
471	C₄	—	8	1	no	—
472	C₄	C₃₁×#448	1	2	yes	—
473	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
474	C₄	C₂×C₃₁×#137	1	1	yes	characteristic
475	C₄	C₂×C₃₁×#137	1	2	yes	—
476	C₂	—	2	2	no	—
477	C₂	—	16	1	no	—
478	C₂	C₂×C₃₁×#448	1	1	yes	characteristic
479	C₂	C₃₁×#3004	1	1	yes	characteristic
480	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×#532	C₂²
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
1. C₂×Q₈²×C₃₁	C₂	semidirect
5. C₃₁×#1645	C₂	semidirect
6. C₃₁×#1623	C₂	semidirect
7. C₃₁×#1548	C₂	semidirect
8. C₃₁×#1487	C₂	semidirect
9. C₃₁×#1479	C₂	semidirect
120. C₂×Q₈×C₃₁	SD₁₆	semidirect
172. #18768	C₃₁	direct (N × H)
185. Q₈×C₃₁	#137	semidirect
216. C₂×Q₈²	C₂×C₃₁	semidirect
220. #1645	C₂×C₃₁	semidirect
221. #1623	C₂×C₃₁	semidirect
222. #1548	C₂×C₃₁	semidirect
223. #1487	C₂×C₃₁	semidirect
224. #1479	C₂×C₃₁	semidirect
349. C₃₁	#18768	direct (N × H)
361. C₂×Q₈	SD₁₆×C₃₁	semidirect
425. Q₈	C₃₁×#137	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈²×C₃₁	2
C₃₁×#3206	2
C₃₁×#3103	2
C₃₁×#3037	2
C₃₁×#1645	2
C₃₁×#1623	2
C₃₁×#1548	2
C₃₁×#1487	2
C₃₁×#1479	2
#18768	31

Subgroups — C31×#18768 #21475527

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#18768 #21475527