Subgroups — C₃₁×#61288 #21518047

Order 7936.

1846 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3511	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3356	C₂	1	4	yes	maximal, complemented
3	C₃₁×#3354	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3340	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3335	C₂	1	4	yes	maximal, complemented
6	C₂×C₃₁×#498	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#3099	C₂	1	4	yes	maximal, complemented
8	C₃₁×#3079	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3060	C₂	1	2	yes	maximal, complemented
10	C₄×C₃₁×#135	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2931	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#2912	C₂	1	2	yes	maximal, complemented
13	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#131	C₂²	1	2	yes	—
15	C₃₁×#556	C₂²	1	4	yes	—
16	C₃₁×#555	C₂²	1	4	yes	—
17	C₃₁×#554	C₂²	1	4	yes	—
18	C₃₁×#553	—	2	4	no	—
19	C₃₁×#548	—	2	4	no	—
20	C₃₁×#547	C₂²	1	4	yes	—
21	C₃₁×#542	—	2	4	no	—
22	C₃₁×#537	C₂²	1	4	yes	—
23	C₃₁×#533	—	2	2	no	—
24	C₃₁×#532	—	2	2	no	—
25	C₃₁×#532	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
27	C₃₁×#520	C₂²	1	4	yes	—
28	C₄×#41×C₃₁	C₂²	1	2	yes	—
29	C₃₁×#498	C₂²	1	4	yes	—
30	C₃₁×#498	C₂²	1	4	yes	—
31	C₃₁×#485	C₂²	1	8	yes	—
32	C₃₁×#483	C₂²	1	8	yes	—
33	C₃₁×#482	C₂²	1	8	yes	—
34	C₃₁×#481	C₂²	1	8	yes	—
35	C₃₁×#477	C₂²	1	4	yes	—
36	C₃₁×#476	C₂²	1	4	yes	—
37	C₃₁×#475	C₂²	1	4	yes	—
38	C₃₁×#474	C₂²	1	4	yes	—
39	C₃₁×#470	C₂²	1	4	yes	—
40	C₃₁×#469	C₂²	1	4	yes	—
41	C₃₁×#465	C₂²	1	8	yes	—
42	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
43	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
44	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
45	C₄×D₈×C₃₁	C₂²	1	2	yes	—
46	C₃₁×#436	C₂²	1	4	yes	—
47	C₈×D₄×C₃₁	C₂²	1	4	yes	—
48	C₃₁×#428	C₂²	1	4	yes	—
49	C₃₁×#428	C₂²	1	8	yes	—
50	C₃₁×#427	C₂²	1	4	yes	—
51	C₃₁×#427	C₂²	1	1	yes	characteristic
52	C₂×C₃₁×#106	C₂²	1	4	yes	—
53	C₂×C₃₁×#106	C₂²	1	2	yes	—
54	C₂×C₃₁×#106	C₂²	1	2	yes	—
55	C₃₁×#423	C₂²	1	1	yes	characteristic
56	C₃₁×#417	C₂²	1	4	yes	—
57	C₃₁×#416	C₂²	1	2	yes	—
58	C₃₁×#408	C₂²	1	2	yes	—
59	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
60	C₂×#41×C₃₁	—	2	4	no	—
61	C₂×#41×C₃₁	C₂³	1	2	yes	—
62	C₂×#41×C₃₁	C₂³	1	2	yes	—
63	C₂²×Q₈×C₃₁	—	2	2	no	—
64	C₃₁×#135	—	2	4	no	—
65	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
66	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
67	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
68	C₃₁×#131	—	2	8	no	—
69	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
70	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
71	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
72	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
73	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
74	C₃₁×C₄⋊Q₈	—	2	2	no	—
75	C₃₁×C₄⋊Q₈	—	2	2	no	—
76	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
77	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
78	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
79	C₃₁×#127	D₄	1	1	yes	characteristic
80	C₃₁×#126	—	2	8	no	—
81	C₃₁×#125	—	2	4	no	—
82	C₃₁×#124	—	2	4	no	—
83	C₃₁×#124	—	2	8	no	—
84	C₃₁×#124	D₄	1	2	yes	—
85	C₃₁×#123	—	2	8	no	—
86	C₃₁×C₂²⋊Q₈	—	2	4	no	—
87	C₃₁×C₂²⋊Q₈	—	2	4	no	—
88	C₃₁×C₂²⋊Q₈	—	2	8	no	—
89	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	—
90	C₃₁×#121	—	2	4	no	—
91	C₃₁×#121	—	2	4	no	—
92	C₃₁×#121	—	2	8	no	—
93	C₃₁×#121	D₄	1	4	yes	—
94	C₃₁×#121	C₂³	1	8	yes	—
95	C₄×Q₈×C₃₁	—	2	4	no	—
96	C₄×Q₈×C₃₁	—	2	4	no	—
97	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
98	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
99	C₄×D₄×C₃₁	—	2	4	no	—
100	C₄×D₄×C₃₁	—	2	8	no	—
101	C₄×D₄×C₃₁	—	2	4	no	—
102	C₄×D₄×C₃₁	—	2	2	no	—
103	C₄×D₄×C₃₁	—	2	2	no	—
104	C₄×D₄×C₃₁	C₂³	1	2	yes	—
105	C₄×D₄×C₃₁	C₂³	1	2	yes	—
106	C₄×D₄×C₃₁	C₂³	1	4	yes	—
107	C₃₁×#117	—	2	4	no	—
108	C₃₁×#117	—	2	2	no	—
109	C₃₁×#117	C₂³	1	2	yes	—
110	C₃₁×#117	C₂³	1	8	yes	—
111	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
112	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
113	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
114	C₂×C₄²×C₃₁	—	2	2	no	—
115	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
116	C₃₁×#107	C₂³	1	2	yes	—
117	C₃₁×#107	C₂³	1	8	yes	—
118	C₃₁×#106	C₂³	1	1	yes	characteristic
119	C₃₁×#106	C₂³	1	4	yes	—
120	C₃₁×#106	C₂³	1	4	yes	—
121	C₃₁×#106	C₂³	1	8	yes	—
122	C₃₁×#106	C₂³	1	8	yes	—
123	C₃₁×#106	C₂³	1	1	yes	characteristic
124	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
125	C₃₁×#103	C₂³	1	2	yes	—
126	C₃₁×#103	C₂³	1	2	yes	—
127	C₃₁×#103	C₂³	1	8	yes	—
128	C₃₁×#102	C₂³	1	2	yes	—
129	C₃₁×#102	C₂³	1	2	yes	—
130	C₃₁×#102	C₂³	1	8	yes	—
131	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
132	C₄×C₈×C₃₁	C₂³	1	2	yes	—
133	C₄×C₈×C₃₁	C₂³	1	2	yes	—
134	#41×C₃₁	—	2	4	no	—
135	#41×C₃₁	—	2	4	no	—
136	#41×C₃₁	—	2	4	no	—
137	#41×C₃₁	—	2	4	no	—
138	#41×C₃₁	—	8	8	no	—
139	C₂×Q₈×C₃₁	—	2	4	no	—
140	C₂×Q₈×C₃₁	—	2	4	no	—
141	C₂×Q₈×C₃₁	—	8	4	no	—
142	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
143	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
144	C₂×D₄×C₃₁	—	2	2	no	—
145	C₂×D₄×C₃₁	—	2	8	no	—
146	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
147	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
148	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
149	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
150	C₂²×C₄×C₃₁	—	2	4	no	—
151	C₂²×C₄×C₃₁	—	2	8	no	—
152	C₂²×C₄×C₃₁	—	2	2	no	—
153	C₂²×C₄×C₃₁	—	2	2	no	—
154	C₂²×C₄×C₃₁	—	2	2	no	—
155	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
156	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
157	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
158	Q₁₆×C₃₁	—	2	2	no	—
159	SD₁₆×C₃₁	—	2	4	no	—
160	D₈×C₃₁	—	2	2	no	—
161	M₁₆×C₃₁	—	2	4	no	—
162	M₁₆×C₃₁	—	2	8	no	—
163	C₂×C₈×C₃₁	—	2	2	no	—
164	C₂×C₈×C₃₁	—	2	8	no	—
165	C₂×C₈×C₃₁	#41	1	4	yes	complemented
166	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
167	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
168	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
169	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
170	C₄⋊C₄×C₃₁	—	2	4	no	—
171	C₄⋊C₄×C₃₁	—	2	4	no	—
172	C₄⋊C₄×C₃₁	—	2	4	no	—
173	C₄⋊C₄×C₃₁	—	2	8	no	—
174	C₄⋊C₄×C₃₁	—	2	2	no	—
175	C₄⋊C₄×C₃₁	—	2	2	no	—
176	C₄⋊C₄×C₃₁	—	2	8	no	—
177	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
178	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
179	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
180	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
181	C₂²⋊C₄×C₃₁	—	2	2	no	—
182	C₂²⋊C₄×C₃₁	—	2	8	no	—
183	C₂²⋊C₄×C₃₁	—	2	8	no	—
184	C₂²⋊C₄×C₃₁	—	2	2	no	—
185	C₂²⋊C₄×C₃₁	—	2	2	no	—
186	C₂²⋊C₄×C₃₁	—	2	8	no	—
187	C₂²⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
188	C₄²×C₃₁	—	2	2	no	—
189	C₄²×C₃₁	—	2	2	no	—
190	C₄²×C₃₁	—	2	2	no	—
191	C₄²×C₃₁	—	2	8	no	—
192	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
193	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
194	C₄²×C₃₁	C₂⁴	1	2	yes	—
195	#61288	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
196	C₂³×C₃₁	—	2	2	no	—
197	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
198	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
199	Q₈×C₃₁	—	2	4	no	—
200	Q₈×C₃₁	—	2	2	no	—
201	Q₈×C₃₁	—	2	2	no	—
202	Q₈×C₃₁	—	4	8	no	—
203	Q₈×C₃₁	—	8	4	no	—
204	D₄×C₃₁	—	2	8	no	—
205	D₄×C₃₁	—	2	4	no	—
206	D₄×C₃₁	—	2	8	no	—
207	D₄×C₃₁	—	2	2	no	—
208	D₄×C₃₁	—	2	2	no	—
209	D₄×C₃₁	—	4	4	no	—
210	D₄×C₃₁	—	8	8	no	—
211	C₂×C₄×C₃₁	—	2	4	no	—
212	C₂×C₄×C₃₁	—	2	4	no	—
213	C₂×C₄×C₃₁	—	2	4	no	—
214	C₂×C₄×C₃₁	—	2	2	no	—
215	C₂×C₄×C₃₁	—	2	2	no	—
216	C₂×C₄×C₃₁	—	2	2	no	—
217	C₂×C₄×C₃₁	—	2	8	no	—
218	C₂×C₄×C₃₁	—	4	4	no	—
219	C₂×C₄×C₃₁	—	4	4	no	—
220	C₂×C₄×C₃₁	—	4	4	no	—
221	C₂×C₄×C₃₁	—	8	4	no	—
222	C₂×C₄×C₃₁	—	8	8	no	—
223	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
224	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
225	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
226	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
227	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
228	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
229	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
230	C₈×C₃₁	—	2	4	no	—
231	C₈×C₃₁	—	2	2	no	—
232	C₈×C₃₁	C₂×#41	1	2	yes	—
233	C₈×C₃₁	C₂×#41	1	2	yes	—
234	#3511	C₂×C₃₁	1	2	yes	complemented
235	#3356	C₂×C₃₁	1	4	yes	complemented
236	#3354	C₂×C₃₁	1	4	yes	complemented
237	#3340	C₂×C₃₁	1	4	yes	complemented
238	#3335	C₂×C₃₁	1	4	yes	complemented
239	C₂×#498	C₂×C₃₁	1	1	yes	characteristic, complemented
240	#3099	C₂×C₃₁	1	4	yes	complemented
241	#3079	C₂×C₃₁	1	2	yes	complemented
242	#3060	C₂×C₃₁	1	2	yes	complemented
243	C₄×#135	C₂×C₃₁	1	1	yes	characteristic, complemented
244	#2931	C₂×C₃₁	1	1	yes	characteristic, complemented
245	#2912	C₂×C₃₁	1	2	yes	complemented
246	C₂²×C₃₁	—	2	2	no	—
247	C₂²×C₃₁	—	2	4	no	—
248	C₂²×C₃₁	—	2	8	no	—
249	C₂²×C₃₁	—	4	2	no	—
250	C₂²×C₃₁	—	4	2	no	—
251	C₂²×C₃₁	—	4	2	no	—
252	C₂²×C₃₁	#547	1	2	yes	—
253	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
254	C₄×C₃₁	—	2	4	no	—
255	C₄×C₃₁	—	2	2	no	—
256	C₄×C₃₁	—	4	2	no	—
257	C₄×C₃₁	—	4	2	no	—
258	C₄×C₃₁	—	4	2	no	—
259	C₄×C₃₁	—	4	8	no	—
260	C₄×C₃₁	#547	1	2	yes	—
261	C₄×C₃₁	#577	1	4	yes	—
262	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
263	C₄×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
264	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
265	C₂×#131	C₂²×C₃₁	1	2	yes	—
266	#556	C₂²×C₃₁	1	4	yes	—
267	#555	C₂²×C₃₁	1	4	yes	—
268	#554	C₂²×C₃₁	1	4	yes	—
269	#553	—	2	4	no	—
270	#548	—	2	4	no	—
271	#547	C₂²×C₃₁	1	4	yes	—
272	#542	—	2	4	no	—
273	#537	C₂²×C₃₁	1	4	yes	—
274	#533	—	2	2	no	—
275	#532	—	2	2	no	—
276	#532	C₂²×C₃₁	1	1	yes	characteristic
277	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
278	#520	C₂²×C₃₁	1	4	yes	—
279	C₄×#41	C₂²×C₃₁	1	2	yes	—
280	#498	C₂²×C₃₁	1	4	yes	—
281	#498	C₂²×C₃₁	1	4	yes	—
282	#485	C₂²×C₃₁	1	8	yes	—
283	#483	C₂²×C₃₁	1	8	yes	—
284	#482	C₂²×C₃₁	1	8	yes	—
285	#481	C₂²×C₃₁	1	8	yes	—
286	#477	C₂²×C₃₁	1	4	yes	—
287	#476	C₂²×C₃₁	1	4	yes	—
288	#475	C₂²×C₃₁	1	4	yes	—
289	#474	C₂²×C₃₁	1	4	yes	—
290	#470	C₂²×C₃₁	1	4	yes	—
291	#469	C₂²×C₃₁	1	4	yes	—
292	#465	C₂²×C₃₁	1	8	yes	—
293	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
294	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
295	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
296	C₄×D₈	C₂²×C₃₁	1	2	yes	—
297	#436	C₂²×C₃₁	1	4	yes	—
298	C₈×D₄	C₂²×C₃₁	1	4	yes	—
299	#428	C₂²×C₃₁	1	4	yes	—
300	#428	C₂²×C₃₁	1	8	yes	—
301	#427	C₂²×C₃₁	1	4	yes	—
302	#427	C₂²×C₃₁	1	1	yes	characteristic
303	C₂×#106	C₂²×C₃₁	1	4	yes	—
304	C₂×#106	C₂²×C₃₁	1	2	yes	—
305	C₂×#106	C₂²×C₃₁	1	2	yes	—
306	#423	C₂²×C₃₁	1	1	yes	characteristic
307	#417	C₂²×C₃₁	1	4	yes	—
308	#416	C₂²×C₃₁	1	2	yes	—
309	#408	C₂²×C₃₁	1	2	yes	—
310	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
311	C₂×C₃₁	—	2	2	no	—
312	C₂×C₃₁	—	4	4	no	—
313	C₂×C₃₁	—	8	2	no	—
314	C₂×C₃₁	C₂×#547	1	1	yes	characteristic
315	C₂×C₃₁	C₂×#577	1	2	yes	—
316	C₂×#41	—	2	4	no	—
317	C₂×#41	C₂³×C₃₁	1	2	yes	—
318	C₂×#41	C₂³×C₃₁	1	2	yes	—
319	C₂²×Q₈	—	2	2	no	—
320	#135	—	2	4	no	—
321	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
322	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
323	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
324	#131	—	2	8	no	—
325	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
326	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
327	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
328	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
329	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
330	C₄⋊Q₈	—	2	2	no	—
331	C₄⋊Q₈	—	2	2	no	—
332	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
333	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
334	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
335	#127	D₄×C₃₁	1	1	yes	characteristic
336	#126	—	2	8	no	—
337	#125	—	2	4	no	—
338	#124	—	2	4	no	—
339	#124	—	2	8	no	—
340	#124	D₄×C₃₁	1	2	yes	—
341	#123	—	2	8	no	—
342	C₂²⋊Q₈	—	2	4	no	—
343	C₂²⋊Q₈	—	2	4	no	—
344	C₂²⋊Q₈	—	2	8	no	—
345	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
346	#121	—	2	4	no	—
347	#121	—	2	4	no	—
348	#121	—	2	8	no	—
349	#121	D₄×C₃₁	1	4	yes	—
350	#121	C₂³×C₃₁	1	8	yes	—
351	C₄×Q₈	—	2	4	no	—
352	C₄×Q₈	—	2	4	no	—
353	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
354	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
355	C₄×D₄	—	2	4	no	—
356	C₄×D₄	—	2	8	no	—
357	C₄×D₄	—	2	4	no	—
358	C₄×D₄	—	2	2	no	—
359	C₄×D₄	—	2	2	no	—
360	C₄×D₄	C₂³×C₃₁	1	2	yes	—
361	C₄×D₄	C₂³×C₃₁	1	2	yes	—
362	C₄×D₄	C₂³×C₃₁	1	4	yes	—
363	#117	—	2	4	no	—
364	#117	—	2	2	no	—
365	#117	C₂³×C₃₁	1	2	yes	—
366	#117	C₂³×C₃₁	1	8	yes	—
367	C₂×C₄⋊C₄	—	2	2	no	—
368	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
369	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
370	C₂×C₄²	—	2	2	no	—
371	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
372	#107	C₂³×C₃₁	1	2	yes	—
373	#107	C₂³×C₃₁	1	8	yes	—
374	#106	C₂³×C₃₁	1	1	yes	characteristic
375	#106	C₂³×C₃₁	1	4	yes	—
376	#106	C₂³×C₃₁	1	4	yes	—
377	#106	C₂³×C₃₁	1	8	yes	—
378	#106	C₂³×C₃₁	1	8	yes	—
379	#106	C₂³×C₃₁	1	1	yes	characteristic
380	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
381	#103	C₂³×C₃₁	1	2	yes	—
382	#103	C₂³×C₃₁	1	2	yes	—
383	#103	C₂³×C₃₁	1	8	yes	—
384	#102	C₂³×C₃₁	1	2	yes	—
385	#102	C₂³×C₃₁	1	2	yes	—
386	#102	C₂³×C₃₁	1	8	yes	—
387	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
388	C₄×C₈	C₂³×C₃₁	1	2	yes	—
389	C₄×C₈	C₂³×C₃₁	1	2	yes	—
390	C₃₁	#61288	1	1	yes	characteristic, direct product factor, Sylow
391	#41	—	2	4	no	—
392	#41	—	2	4	no	—
393	#41	—	2	4	no	—
394	#41	—	2	4	no	—
395	#41	—	8	8	no	—
396	C₂×Q₈	—	2	4	no	—
397	C₂×Q₈	—	2	4	no	—
398	C₂×Q₈	—	8	4	no	—
399	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
400	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
401	C₂×D₄	—	2	2	no	—
402	C₂×D₄	—	2	8	no	—
403	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
404	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
405	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
406	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
407	C₂²×C₄	—	2	4	no	—
408	C₂²×C₄	—	2	8	no	—
409	C₂²×C₄	—	2	2	no	—
410	C₂²×C₄	—	2	2	no	—
411	C₂²×C₄	—	2	2	no	—
412	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
413	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
414	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
415	Q₁₆	—	2	2	no	—
416	SD₁₆	—	2	4	no	—
417	D₈	—	2	2	no	—
418	M₁₆	—	2	4	no	—
419	M₁₆	—	2	8	no	—
420	C₂×C₈	—	2	2	no	—
421	C₂×C₈	—	2	8	no	—
422	C₂×C₈	#41×C₃₁	1	4	yes	complemented
423	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
424	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
425	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
426	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
427	C₄⋊C₄	—	2	4	no	—
428	C₄⋊C₄	—	2	4	no	—
429	C₄⋊C₄	—	2	4	no	—
430	C₄⋊C₄	—	2	8	no	—
431	C₄⋊C₄	—	2	2	no	—
432	C₄⋊C₄	—	2	2	no	—
433	C₄⋊C₄	—	2	8	no	—
434	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
435	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
436	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
437	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
438	C₂²⋊C₄	—	2	2	no	—
439	C₂²⋊C₄	—	2	8	no	—
440	C₂²⋊C₄	—	2	8	no	—
441	C₂²⋊C₄	—	2	2	no	—
442	C₂²⋊C₄	—	2	2	no	—
443	C₂²⋊C₄	—	2	8	no	—
444	C₂²⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
445	C₄²	—	2	2	no	—
446	C₄²	—	2	2	no	—
447	C₄²	—	2	2	no	—
448	C₄²	—	2	8	no	—
449	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
450	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
451	C₄²	C₂⁴×C₃₁	1	2	yes	—
452	C₂³	—	2	2	no	—
453	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
454	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
455	Q₈	—	2	4	no	—
456	Q₈	—	2	2	no	—
457	Q₈	—	2	2	no	—
458	Q₈	—	4	8	no	—
459	Q₈	—	8	4	no	—
460	D₄	—	2	8	no	—
461	D₄	—	2	4	no	—
462	D₄	—	2	8	no	—
463	D₄	—	2	2	no	—
464	D₄	—	2	2	no	—
465	D₄	—	4	4	no	—
466	D₄	—	8	8	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	2	4	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	2	8	no	—
474	C₂×C₄	—	4	4	no	—
475	C₂×C₄	—	4	4	no	—
476	C₂×C₄	—	4	4	no	—
477	C₂×C₄	—	8	4	no	—
478	C₂×C₄	—	8	8	no	—
479	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
480	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
481	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
482	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
483	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
484	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
485	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
486	C₈	—	2	4	no	—
487	C₈	—	2	2	no	—
488	C₈	C₂×#41×C₃₁	1	2	yes	—
489	C₈	C₂×#41×C₃₁	1	2	yes	—
490	C₂²	—	2	2	no	—
491	C₂²	—	2	4	no	—
492	C₂²	—	2	8	no	—
493	C₂²	—	4	2	no	—
494	C₂²	—	4	2	no	—
495	C₂²	—	4	2	no	—
496	C₂²	C₃₁×#547	1	2	yes	—
497	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
498	C₄	—	2	4	no	—
499	C₄	—	2	2	no	—
500	C₄	—	4	2	no	—
501	C₄	—	4	2	no	—
502	C₄	—	4	2	no	—
503	C₄	—	4	8	no	—
504	C₄	C₃₁×#547	1	2	yes	—
505	C₄	C₃₁×#577	1	4	yes	—
506	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
507	C₄	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
508	C₂	—	2	2	no	—
509	C₂	—	4	4	no	—
510	C₂	—	8	2	no	—
511	C₂	C₂×C₃₁×#547	1	1	yes	characteristic
512	C₂	C₂×C₃₁×#577	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#532 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₃₁×#3511	C₂	semidirect
2. C₃₁×#3356	C₂	semidirect
3. C₃₁×#3354	C₂	semidirect
4. C₃₁×#3340	C₂	semidirect
5. C₃₁×#3335	C₂	semidirect
6. C₂×C₃₁×#498	C₂	semidirect
7. C₃₁×#3099	C₂	semidirect
8. C₃₁×#3079	C₂	semidirect
9. C₃₁×#3060	C₂	semidirect
10. C₄×C₃₁×#135	C₂	semidirect
11. C₃₁×#2931	C₂	semidirect
12. C₃₁×#2912	C₂	semidirect
165. C₂×C₈×C₃₁	#41	semidirect
195. #61288	C₃₁	direct (N × H)
234. #3511	C₂×C₃₁	semidirect
235. #3356	C₂×C₃₁	semidirect
236. #3354	C₂×C₃₁	semidirect
237. #3340	C₂×C₃₁	semidirect
238. #3335	C₂×C₃₁	semidirect
239. C₂×#498	C₂×C₃₁	semidirect
240. #3099	C₂×C₃₁	semidirect
241. #3079	C₂×C₃₁	semidirect
242. #3060	C₂×C₃₁	semidirect
243. C₄×#135	C₂×C₃₁	semidirect
244. #2931	C₂×C₃₁	semidirect
245. #2912	C₂×C₃₁	semidirect
390. C₃₁	#61288	direct (N × H)
422. C₂×C₈	#41×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3511	2
C₃₁×#3356	2
C₃₁×#3354	2
C₃₁×#3340	2
C₃₁×#3335	2
C₂×C₃₁×#498	2
C₃₁×#3099	2
C₃₁×#3079	2
C₃₁×#3060	2
C₄×C₃₁×#135	2
C₃₁×#2931	2
C₃₁×#2912	2
#61288	31

Subgroups — C31×#61288 #21518047

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61288 #21518047