Subgroups — C₃₁×#60459 #21517218

Order 7936.

2688 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#578	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃₁×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3489	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3063	C₂	1	6	yes	maximal, complemented
5	C₃₁×#3061	C₂	1	3	yes	maximal, complemented
6	C₃₁×#3059	C₂	1	3	yes	maximal, complemented
7	C₃₁×#3056	C₂	1	6	yes	maximal, complemented
8	C₃₁×#3050	C₂	1	3	yes	maximal, complemented
9	C₂×C₃₁×#467	C₂	1	3	yes	maximal, complemented
10	C₃₁×#2913	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#2898	C₂	1	2	yes	maximal, complemented
12	C₂×C₃₁×2₋¹⁺⁴	C₂²	1	2	yes	—
13	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#578	—	2	8	no	—
15	C₂×C₃₁×#137	C₂²	1	6	yes	complemented
16	C₂×C₃₁×#135	C₂²	1	3	yes	complemented
17	C₂²×Q₁₆×C₃₁	C₂²	1	3	yes	complemented
18	C₂×C₃₁×#131	—	2	8	no	—
19	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#131	C₂²	1	3	yes	—
21	C₂×C₃₁×#131	C₂²	1	3	yes	—
22	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
23	C₂²×M₁₆×C₃₁	C₂²	1	3	yes	—
24	C₂³×C₈×C₃₁	C₂²	1	3	yes	—
25	D₄×Q₈×C₃₁	—	2	2	no	—
26	C₃₁×#548	—	2	2	no	—
27	C₃₁×#547	—	2	6	no	—
28	C₃₁×#546	—	2	6	no	—
29	C₃₁×#537	C₂²	1	6	yes	—
30	C₃₁×#536	C₂²	1	6	yes	—
31	C₃₁×#525	C₂²	1	6	yes	—
32	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	6	yes	—
33	C₃₁×#520	C₂²	1	2	yes	—
34	C₃₁×#470	C₂²	1	12	yes	complemented
35	C₃₁×#468	C₂²	1	12	yes	complemented
36	C₃₁×#467	C₂²	1	3	yes	—
37	C₃₁×#467	C₂²	1	6	yes	—
38	C₃₁×#467	C₂²	1	6	yes	complemented
39	C₃₁×#467	C₂²	1	6	yes	complemented
40	C₃₁×#467	C₂²	1	3	yes	complemented
41	C₃₁×#466	C₂²	1	1	yes	characteristic
42	C₃₁×#466	C₂²	1	3	yes	complemented
43	C₃₁×#465	C₂²	1	6	yes	—
44	C₃₁×#465	C₂²	1	6	yes	complemented
45	C₃₁×#428	C₂²	1	6	yes	complemented
46	C₃₁×#427	C₂²	1	3	yes	complemented
47	C₂×C₃₁×#107	C₂²	1	3	yes	complemented
48	C₃₁×#419	C₂²	1	6	yes	complemented
49	C₃₁×#417	C₂²	1	6	yes	complemented
50	C₃₁×#416	C₂²	1	6	yes	complemented
51	C₂×C₃₁×#103	C₂²	1	6	yes	complemented
52	C₃₁×2₋¹⁺⁴	—	2	8	no	—
53	C₃₁×2₋¹⁺⁴	—	2	8	no	—
54	C₂×#41×C₃₁	—	2	4	no	—
55	C₂×#41×C₃₁	—	2	4	no	—
56	C₂×#41×C₃₁	—	2	6	no	—
57	C₂×#41×C₃₁	D₄	1	3	yes	—
58	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
59	C₂×#41×C₃₁	C₂³	1	6	yes	—
60	C₂×#41×C₃₁	C₂³	1	3	yes	—
61	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
62	C₂²×Q₈×C₃₁	—	2	2	no	—
63	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
64	C₂²×Q₈×C₃₁	C₂³	1	6	yes	—
65	C₂²×D₄×C₃₁	D₄	1	3	yes	—
66	C₂³×C₄×C₃₁	C₂³	1	3	yes	—
67	C₃₁×#137	—	2	24	no	—
68	C₃₁×#135	—	2	12	no	—
69	C₂×Q₁₆×C₃₁	—	2	12	no	—
70	C₂×Q₁₆×C₃₁	C₂³	1	3	yes	—
71	C₂×Q₁₆×C₃₁	C₂³	1	6	yes	—
72	C₂×SD₁₆×C₃₁	C₂³	1	6	yes	—
73	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
74	C₃₁×#131	—	2	4	no	—
75	C₃₁×#131	—	2	12	no	—
76	C₃₁×#131	—	2	12	no	—
77	C₃₁×#131	D₄	1	8	yes	complemented
78	C₂×M₁₆×C₃₁	—	2	12	no	—
79	C₂×M₁₆×C₃₁	C₂³	1	3	yes	—
80	C₂×M₁₆×C₃₁	C₂³	1	6	yes	—
81	C₂×M₁₆×C₃₁	C₂³	1	3	yes	—
82	C₂²×C₈×C₃₁	—	2	12	no	—
83	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
84	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
85	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
86	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
87	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
88	C₂²×C₈×C₃₁	C₂³	1	3	yes	—
89	C₃₁×C₄⋊Q₈	—	2	6	no	—
90	C₃₁×#124	—	2	6	no	—
91	C₃₁×#123	—	2	12	no	—
92	C₃₁×#123	—	2	6	no	—
93	C₃₁×C₂²⋊Q₈	—	2	6	no	—
94	C₃₁×C₂²⋊Q₈	—	2	6	no	—
95	C₃₁×C₂²⋊Q₈	—	2	6	no	—
96	C₃₁×C₂²⋊Q₈	C₂³	1	6	yes	—
97	C₃₁×C₂²⋊Q₈	C₂³	1	6	yes	—
98	C₃₁×C₂²⋊Q₈	C₂³	1	12	yes	—
99	C₃₁×#121	—	2	12	no	—
100	C₃₁×#121	—	2	6	no	—
101	C₃₁×#121	C₂³	1	2	yes	—
102	C₃₁×#121	C₂³	1	6	yes	—
103	C₃₁×#120	—	2	6	no	—
104	C₄×Q₈×C₃₁	—	2	2	no	—
105	C₄×D₄×C₃₁	—	2	6	no	—
106	C₄×D₄×C₃₁	—	2	6	no	—
107	C₄×D₄×C₃₁	—	2	6	no	—
108	C₃₁×#117	C₂³	1	6	yes	—
109	C₂×C₄⋊C₄×C₃₁	—	2	6	no	—
110	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
111	C₂×C₂²⋊C₄×C₃₁	—	2	6	no	—
112	C₃₁×#107	C₂³	1	3	yes	—
113	C₃₁×#107	C₂³	1	6	yes	—
114	C₃₁×#107	C₂³	1	1	yes	characteristic
115	C₃₁×#106	C₂³	1	6	yes	—
116	C₃₁×#103	C₂³	1	6	yes	—
117	C₃₁×#103	C₂³	1	12	yes	—
118	C₃₁×#103	C₂³	1	6	yes	—
119	C₃₁×#102	C₂³	1	2	yes	—
120	C₃₁×#102	C₂³	1	6	yes	—
121	C₂⁴×C₃₁	C₂×D₄	1	3	yes	—
122	#41×C₃₁	—	2	4	no	—
123	#41×C₃₁	—	2	12	no	—
124	#41×C₃₁	—	2	6	no	—
125	#41×C₃₁	—	2	6	no	—
126	#41×C₃₁	—	2	12	no	—
127	#41×C₃₁	—	2	12	no	—
128	#41×C₃₁	—	4	24	no	—
129	#41×C₃₁	C₂×D₄	1	4	yes	—
130	#41×C₃₁	#41	1	4	yes	—
131	C₂×Q₈×C₃₁	—	2	4	no	—
132	C₂×Q₈×C₃₁	—	2	6	no	—
133	C₂×Q₈×C₃₁	—	2	12	no	—
134	C₂×Q₈×C₃₁	—	2	12	no	—
135	C₂×Q₈×C₃₁	—	4	8	no	—
136	C₂×Q₈×C₃₁	C₂×D₄	1	3	yes	—
137	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
138	C₂×Q₈×C₃₁	C₂⁴	1	6	yes	—
139	C₂×D₄×C₃₁	—	2	12	no	—
140	C₂×D₄×C₃₁	—	2	2	no	—
141	C₂×D₄×C₃₁	—	2	6	no	—
142	C₂×D₄×C₃₁	—	2	6	no	—
143	C₂×D₄×C₃₁	C₂×D₄	1	6	yes	—
144	C₂×D₄×C₃₁	C₂×D₄	1	3	yes	—
145	C₂×D₄×C₃₁	C₂×D₄	1	3	yes	—
146	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
147	C₂²×C₄×C₃₁	—	2	6	no	—
148	C₂²×C₄×C₃₁	—	2	6	no	—
149	C₂²×C₄×C₃₁	—	2	6	no	—
150	C₂²×C₄×C₃₁	—	2	6	no	—
151	C₂²×C₄×C₃₁	—	2	6	no	—
152	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
153	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
154	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
155	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
156	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
157	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
158	Q₁₆×C₃₁	—	2	6	no	—
159	Q₁₆×C₃₁	—	2	12	no	—
160	SD₁₆×C₃₁	—	2	12	no	—
161	D₈×C₃₁	—	2	2	no	—
162	M₁₆×C₃₁	—	2	6	no	—
163	M₁₆×C₃₁	—	2	12	no	—
164	M₁₆×C₃₁	C₂×D₄	1	12	yes	—
165	C₂×C₈×C₃₁	—	2	6	no	—
166	C₂×C₈×C₃₁	—	2	2	no	—
167	C₂×C₈×C₃₁	—	2	6	no	—
168	C₂×C₈×C₃₁	—	2	6	no	—
169	C₂×C₈×C₃₁	—	2	6	no	—
170	C₂×C₈×C₃₁	C₂×D₄	1	12	yes	—
171	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
172	C₂×C₈×C₃₁	C₂⁴	1	3	yes	—
173	C₂×C₈×C₃₁	C₂⁴	1	3	yes	—
174	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
175	C₄⋊C₄×C₃₁	—	2	6	no	—
176	C₄⋊C₄×C₃₁	—	2	12	no	—
177	C₄⋊C₄×C₃₁	—	2	6	no	—
178	C₄⋊C₄×C₃₁	—	2	6	no	—
179	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
180	C₄⋊C₄×C₃₁	C₂⁴	1	6	yes	—
181	C₂²⋊C₄×C₃₁	—	2	6	no	—
182	C₂²⋊C₄×C₃₁	—	2	12	no	—
183	C₂²⋊C₄×C₃₁	—	2	6	no	—
184	C₂²⋊C₄×C₃₁	—	2	6	no	—
185	C₂²⋊C₄×C₃₁	—	2	2	no	—
186	C₂²⋊C₄×C₃₁	—	2	6	no	—
187	C₄²×C₃₁	—	2	6	no	—
188	#60459	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
189	C₂³×C₃₁	—	2	6	no	—
190	C₂³×C₃₁	—	2	6	no	—
191	C₂³×C₃₁	—	2	2	no	—
192	C₂³×C₃₁	—	4	3	no	—
193	C₂³×C₃₁	C₂²×D₄	1	3	yes	—
194	C₂³×C₃₁	C₂²×D₄	1	3	yes	—
195	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
196	Q₈×C₃₁	—	2	6	no	—
197	Q₈×C₃₁	—	2	6	no	—
198	Q₈×C₃₁	—	2	6	no	—
199	Q₈×C₃₁	—	4	12	no	—
200	Q₈×C₃₁	#121	1	4	yes	—
201	D₄×C₃₁	—	2	12	no	—
202	D₄×C₃₁	—	2	6	no	—
203	D₄×C₃₁	—	2	2	no	—
204	D₄×C₃₁	—	2	2	no	—
205	D₄×C₃₁	—	4	12	no	—
206	D₄×C₃₁	—	4	4	no	—
207	D₄×C₃₁	—	8	6	no	—
208	D₄×C₃₁	#121	1	12	yes	—
209	C₂×C₄×C₃₁	—	2	6	no	—
210	C₂×C₄×C₃₁	—	2	6	no	—
211	C₂×C₄×C₃₁	—	2	6	no	—
212	C₂×C₄×C₃₁	—	2	1	no	—
213	C₂×C₄×C₃₁	—	2	3	no	—
214	C₂×C₄×C₃₁	—	2	1	no	—
215	C₂×C₄×C₃₁	—	2	3	no	—
216	C₂×C₄×C₃₁	—	2	2	no	—
217	C₂×C₄×C₃₁	—	2	6	no	—
218	C₂×C₄×C₃₁	—	2	6	no	—
219	C₂×C₄×C₃₁	—	4	12	no	—
220	C₂×C₄×C₃₁	—	4	12	no	—
221	C₂×C₄×C₃₁	—	8	6	no	—
222	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
223	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
224	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
225	C₂×C₄×C₃₁	C₂²×D₄	1	6	yes	—
226	C₂×C₄×C₃₁	C₂×#41	1	6	yes	—
227	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
228	C₈×C₃₁	—	2	1	no	—
229	C₈×C₃₁	—	2	3	no	—
230	C₈×C₃₁	C₂²×D₄	1	2	yes	—
231	C₈×C₃₁	C₂²×D₄	1	6	yes	—
232	C₂×#578	C₂×C₃₁	1	1	yes	characteristic, complemented
233	C₂²×#131	C₂×C₃₁	1	1	yes	characteristic, complemented
234	#3489	C₂×C₃₁	1	2	yes	complemented
235	#3063	C₂×C₃₁	1	6	yes	complemented
236	#3061	C₂×C₃₁	1	3	yes	complemented
237	#3059	C₂×C₃₁	1	3	yes	complemented
238	#3056	C₂×C₃₁	1	6	yes	complemented
239	#3050	C₂×C₃₁	1	3	yes	complemented
240	C₂×#467	C₂×C₃₁	1	3	yes	complemented
241	#2913	C₂×C₃₁	1	1	yes	characteristic, complemented
242	#2898	C₂×C₃₁	1	2	yes	complemented
243	C₂²×C₃₁	—	2	3	no	—
244	C₂²×C₃₁	—	2	1	no	—
245	C₂²×C₃₁	—	2	1	no	—
246	C₂²×C₃₁	—	2	3	no	—
247	C₂²×C₃₁	—	2	3	no	—
248	C₂²×C₃₁	—	2	3	no	—
249	C₂²×C₃₁	—	2	3	no	—
250	C₂²×C₃₁	—	4	12	no	—
251	C₂²×C₃₁	—	4	2	no	—
252	C₂²×C₃₁	—	4	2	no	—
253	C₂²×C₃₁	—	4	2	no	—
254	C₂²×C₃₁	C₂×#121	1	6	yes	—
255	C₂²×C₃₁	#578	1	2	yes	complemented
256	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
257	C₄×C₃₁	—	2	1	no	—
258	C₄×C₃₁	—	2	3	no	—
259	C₄×C₃₁	—	4	6	no	—
260	C₄×C₃₁	—	4	6	no	—
261	C₄×C₃₁	—	4	2	no	—
262	C₄×C₃₁	C₂×#121	1	6	yes	—
263	C₄×C₃₁	C₂³×D₄	1	1	yes	characteristic
264	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
265	C₂×2₋¹⁺⁴	C₂²×C₃₁	1	2	yes	—
266	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
267	#578	—	2	8	no	—
268	C₂×#137	C₂²×C₃₁	1	6	yes	complemented
269	C₂×#135	C₂²×C₃₁	1	3	yes	complemented
270	C₂²×Q₁₆	C₂²×C₃₁	1	3	yes	complemented
271	C₂×#131	—	2	8	no	—
272	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
273	C₂×#131	C₂²×C₃₁	1	3	yes	—
274	C₂×#131	C₂²×C₃₁	1	3	yes	—
275	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
276	C₂²×M₁₆	C₂²×C₃₁	1	3	yes	—
277	C₂³×C₈	C₂²×C₃₁	1	3	yes	—
278	D₄×Q₈	—	2	2	no	—
279	#548	—	2	2	no	—
280	#547	—	2	6	no	—
281	#546	—	2	6	no	—
282	#537	C₂²×C₃₁	1	6	yes	—
283	#536	C₂²×C₃₁	1	6	yes	—
284	#525	C₂²×C₃₁	1	6	yes	—
285	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	6	yes	—
286	#520	C₂²×C₃₁	1	2	yes	—
287	#470	C₂²×C₃₁	1	12	yes	complemented
288	#468	C₂²×C₃₁	1	12	yes	complemented
289	#467	C₂²×C₃₁	1	3	yes	—
290	#467	C₂²×C₃₁	1	6	yes	—
291	#467	C₂²×C₃₁	1	6	yes	complemented
292	#467	C₂²×C₃₁	1	6	yes	complemented
293	#467	C₂²×C₃₁	1	3	yes	complemented
294	#466	C₂²×C₃₁	1	1	yes	characteristic
295	#466	C₂²×C₃₁	1	3	yes	complemented
296	#465	C₂²×C₃₁	1	6	yes	—
297	#465	C₂²×C₃₁	1	6	yes	complemented
298	#428	C₂²×C₃₁	1	6	yes	complemented
299	#427	C₂²×C₃₁	1	3	yes	complemented
300	C₂×#107	C₂²×C₃₁	1	3	yes	complemented
301	#419	C₂²×C₃₁	1	6	yes	complemented
302	#417	C₂²×C₃₁	1	6	yes	complemented
303	#416	C₂²×C₃₁	1	6	yes	complemented
304	C₂×#103	C₂²×C₃₁	1	6	yes	complemented
305	C₂×C₃₁	—	2	2	no	—
306	C₂×C₃₁	—	2	6	no	—
307	C₂×C₃₁	—	4	3	no	—
308	C₂×C₃₁	—	8	2	no	—
309	C₂×C₃₁	C₂²×#121	1	1	yes	characteristic
310	C₂×C₃₁	C₂×#578	1	1	yes	characteristic
311	C₂×C₃₁	#3584	1	1	yes	characteristic
312	2₋¹⁺⁴	—	2	8	no	—
313	2₋¹⁺⁴	—	2	8	no	—
314	C₂×#41	—	2	4	no	—
315	C₂×#41	—	2	4	no	—
316	C₂×#41	—	2	6	no	—
317	C₂×#41	D₄×C₃₁	1	3	yes	—
318	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
319	C₂×#41	C₂³×C₃₁	1	6	yes	—
320	C₂×#41	C₂³×C₃₁	1	3	yes	—
321	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
322	C₂²×Q₈	—	2	2	no	—
323	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
324	C₂²×Q₈	C₂³×C₃₁	1	6	yes	—
325	C₂²×D₄	D₄×C₃₁	1	3	yes	—
326	C₂³×C₄	C₂³×C₃₁	1	3	yes	—
327	#137	—	2	24	no	—
328	#135	—	2	12	no	—
329	C₂×Q₁₆	—	2	12	no	—
330	C₂×Q₁₆	C₂³×C₃₁	1	3	yes	—
331	C₂×Q₁₆	C₂³×C₃₁	1	6	yes	—
332	C₂×SD₁₆	C₂³×C₃₁	1	6	yes	—
333	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
334	#131	—	2	4	no	—
335	#131	—	2	12	no	—
336	#131	—	2	12	no	—
337	#131	D₄×C₃₁	1	8	yes	complemented
338	C₂×M₁₆	—	2	12	no	—
339	C₂×M₁₆	C₂³×C₃₁	1	3	yes	—
340	C₂×M₁₆	C₂³×C₃₁	1	6	yes	—
341	C₂×M₁₆	C₂³×C₃₁	1	3	yes	—
342	C₂²×C₈	—	2	12	no	—
343	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
344	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
345	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
346	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
347	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
348	C₂²×C₈	C₂³×C₃₁	1	3	yes	—
349	C₄⋊Q₈	—	2	6	no	—
350	#124	—	2	6	no	—
351	#123	—	2	12	no	—
352	#123	—	2	6	no	—
353	C₂²⋊Q₈	—	2	6	no	—
354	C₂²⋊Q₈	—	2	6	no	—
355	C₂²⋊Q₈	—	2	6	no	—
356	C₂²⋊Q₈	C₂³×C₃₁	1	6	yes	—
357	C₂²⋊Q₈	C₂³×C₃₁	1	12	yes	—
358	C₂²⋊Q₈	C₂³×C₃₁	1	6	yes	—
359	#121	—	2	12	no	—
360	#121	—	2	6	no	—
361	#121	C₂³×C₃₁	1	2	yes	—
362	#121	C₂³×C₃₁	1	6	yes	—
363	#120	—	2	6	no	—
364	C₄×Q₈	—	2	2	no	—
365	C₄×D₄	—	2	6	no	—
366	C₄×D₄	—	2	6	no	—
367	C₄×D₄	—	2	6	no	—
368	#117	C₂³×C₃₁	1	6	yes	—
369	C₂×C₄⋊C₄	—	2	6	no	—
370	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
371	C₂×C₂²⋊C₄	—	2	6	no	—
372	#107	C₂³×C₃₁	1	3	yes	—
373	#107	C₂³×C₃₁	1	6	yes	—
374	#107	C₂³×C₃₁	1	1	yes	characteristic
375	#106	C₂³×C₃₁	1	6	yes	—
376	#103	C₂³×C₃₁	1	6	yes	—
377	#103	C₂³×C₃₁	1	12	yes	—
378	#103	C₂³×C₃₁	1	6	yes	—
379	#102	C₂³×C₃₁	1	2	yes	—
380	#102	C₂³×C₃₁	1	6	yes	—
381	C₃₁	#60459	1	1	yes	characteristic, direct product factor, Sylow
382	C₂⁴	C₂×D₄×C₃₁	1	3	yes	—
383	#41	—	2	4	no	—
384	#41	—	2	12	no	—
385	#41	—	2	6	no	—
386	#41	—	2	6	no	—
387	#41	—	2	12	no	—
388	#41	—	2	12	no	—
389	#41	—	4	24	no	—
390	#41	C₂×D₄×C₃₁	1	4	yes	—
391	#41	#41×C₃₁	1	4	yes	—
392	C₂×Q₈	—	2	4	no	—
393	C₂×Q₈	—	2	6	no	—
394	C₂×Q₈	—	2	12	no	—
395	C₂×Q₈	—	2	12	no	—
396	C₂×Q₈	—	4	8	no	—
397	C₂×Q₈	C₂×D₄×C₃₁	1	3	yes	—
398	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
399	C₂×Q₈	C₂⁴×C₃₁	1	6	yes	—
400	C₂×D₄	—	2	12	no	—
401	C₂×D₄	—	2	2	no	—
402	C₂×D₄	—	2	6	no	—
403	C₂×D₄	—	2	6	no	—
404	C₂×D₄	C₂×D₄×C₃₁	1	6	yes	—
405	C₂×D₄	C₂×D₄×C₃₁	1	3	yes	—
406	C₂×D₄	C₂×D₄×C₃₁	1	3	yes	—
407	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
408	C₂²×C₄	—	2	6	no	—
409	C₂²×C₄	—	2	6	no	—
410	C₂²×C₄	—	2	6	no	—
411	C₂²×C₄	—	2	6	no	—
412	C₂²×C₄	—	2	6	no	—
413	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
414	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
415	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
416	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
417	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
418	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
419	Q₁₆	—	2	6	no	—
420	Q₁₆	—	2	12	no	—
421	SD₁₆	—	2	12	no	—
422	D₈	—	2	2	no	—
423	M₁₆	—	2	6	no	—
424	M₁₆	—	2	12	no	—
425	M₁₆	C₂×D₄×C₃₁	1	12	yes	—
426	C₂×C₈	—	2	6	no	—
427	C₂×C₈	—	2	2	no	—
428	C₂×C₈	—	2	6	no	—
429	C₂×C₈	—	2	6	no	—
430	C₂×C₈	—	2	6	no	—
431	C₂×C₈	C₂×D₄×C₃₁	1	12	yes	—
432	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
433	C₂×C₈	C₂⁴×C₃₁	1	3	yes	—
434	C₂×C₈	C₂⁴×C₃₁	1	3	yes	—
435	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
436	C₄⋊C₄	—	2	6	no	—
437	C₄⋊C₄	—	2	12	no	—
438	C₄⋊C₄	—	2	6	no	—
439	C₄⋊C₄	—	2	6	no	—
440	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
441	C₄⋊C₄	C₂⁴×C₃₁	1	6	yes	—
442	C₂²⋊C₄	—	2	6	no	—
443	C₂²⋊C₄	—	2	12	no	—
444	C₂²⋊C₄	—	2	6	no	—
445	C₂²⋊C₄	—	2	6	no	—
446	C₂²⋊C₄	—	2	2	no	—
447	C₂²⋊C₄	—	2	6	no	—
448	C₄²	—	2	6	no	—
449	C₂³	—	2	6	no	—
450	C₂³	—	2	6	no	—
451	C₂³	—	2	2	no	—
452	C₂³	—	4	3	no	—
453	C₂³	C₂²×D₄×C₃₁	1	3	yes	—
454	C₂³	C₂²×D₄×C₃₁	1	3	yes	—
455	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
456	Q₈	—	2	6	no	—
457	Q₈	—	2	6	no	—
458	Q₈	—	2	6	no	—
459	Q₈	—	4	12	no	—
460	Q₈	C₃₁×#121	1	4	yes	—
461	D₄	—	2	12	no	—
462	D₄	—	2	6	no	—
463	D₄	—	2	2	no	—
464	D₄	—	2	2	no	—
465	D₄	—	4	12	no	—
466	D₄	—	4	4	no	—
467	D₄	—	8	6	no	—
468	D₄	C₃₁×#121	1	12	yes	—
469	C₂×C₄	—	2	6	no	—
470	C₂×C₄	—	2	6	no	—
471	C₂×C₄	—	2	6	no	—
472	C₂×C₄	—	2	1	no	—
473	C₂×C₄	—	2	3	no	—
474	C₂×C₄	—	2	1	no	—
475	C₂×C₄	—	2	3	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	2	6	no	—
478	C₂×C₄	—	2	6	no	—
479	C₂×C₄	—	4	12	no	—
480	C₂×C₄	—	4	12	no	—
481	C₂×C₄	—	8	6	no	—
482	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
483	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
484	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
485	C₂×C₄	C₂²×D₄×C₃₁	1	6	yes	—
486	C₂×C₄	C₂×#41×C₃₁	1	6	yes	—
487	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
488	C₈	—	2	1	no	—
489	C₈	—	2	3	no	—
490	C₈	C₂²×D₄×C₃₁	1	2	yes	—
491	C₈	C₂²×D₄×C₃₁	1	6	yes	—
492	C₂²	—	2	1	no	—
493	C₂²	—	2	3	no	—
494	C₂²	—	2	3	no	—
495	C₂²	—	2	1	no	—
496	C₂²	—	2	3	no	—
497	C₂²	—	2	3	no	—
498	C₂²	—	2	3	no	—
499	C₂²	—	4	12	no	—
500	C₂²	—	4	2	no	—
501	C₂²	—	4	2	no	—
502	C₂²	—	4	2	no	—
503	C₂²	C₂×C₃₁×#121	1	6	yes	—
504	C₂²	C₃₁×#578	1	2	yes	complemented
505	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
506	C₄	—	2	1	no	—
507	C₄	—	2	3	no	—
508	C₄	—	4	6	no	—
509	C₄	—	4	2	no	—
510	C₄	—	4	6	no	—
511	C₄	C₂×C₃₁×#121	1	6	yes	—
512	C₄	C₂³×D₄×C₃₁	1	1	yes	characteristic
513	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
514	C₂	—	2	2	no	—
515	C₂	—	2	6	no	—
516	C₂	—	4	3	no	—
517	C₂	—	8	2	no	—
518	C₂	C₂²×C₃₁×#121	1	1	yes	characteristic
519	C₂	C₂×C₃₁×#578	1	1	yes	characteristic
520	C₂	C₃₁×#3584	1	1	yes	characteristic

	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 ◁ C₂²×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#578	C₂	semidirect
2. C₂²×C₃₁×#131	C₂	semidirect
3. C₃₁×#3489	C₂	semidirect
4. C₃₁×#3063	C₂	semidirect
5. C₃₁×#3061	C₂	semidirect
6. C₃₁×#3059	C₂	semidirect
7. C₃₁×#3056	C₂	semidirect
8. C₃₁×#3050	C₂	semidirect
9. C₂×C₃₁×#467	C₂	semidirect
10. C₃₁×#2913	C₂	semidirect
11. C₃₁×#2898	C₂	semidirect
15. C₂×C₃₁×#137	C₂²	semidirect
16. C₂×C₃₁×#135	C₂²	semidirect
17. C₂²×Q₁₆×C₃₁	C₂²	semidirect
34. C₃₁×#470	C₂²	semidirect
35. C₃₁×#468	C₂²	semidirect
38. C₃₁×#467	C₂²	semidirect
39. C₃₁×#467	C₂²	semidirect
40. C₃₁×#467	C₂²	semidirect
42. C₃₁×#466	C₂²	semidirect
44. C₃₁×#465	C₂²	semidirect
45. C₃₁×#428	C₂²	semidirect
46. C₃₁×#427	C₂²	semidirect
47. C₂×C₃₁×#107	C₂²	semidirect
48. C₃₁×#419	C₂²	semidirect
49. C₃₁×#417	C₂²	semidirect
50. C₃₁×#416	C₂²	semidirect
51. C₂×C₃₁×#103	C₂²	semidirect
77. C₃₁×#131	D₄	semidirect
188. #60459	C₃₁	direct (N × H)
232. C₂×#578	C₂×C₃₁	semidirect
233. C₂²×#131	C₂×C₃₁	semidirect
234. #3489	C₂×C₃₁	semidirect
235. #3063	C₂×C₃₁	semidirect
236. #3061	C₂×C₃₁	semidirect
237. #3059	C₂×C₃₁	semidirect
238. #3056	C₂×C₃₁	semidirect
239. #3050	C₂×C₃₁	semidirect
240. C₂×#467	C₂×C₃₁	semidirect
241. #2913	C₂×C₃₁	semidirect
242. #2898	C₂×C₃₁	semidirect
255. C₂²×C₃₁	#578	semidirect
268. C₂×#137	C₂²×C₃₁	semidirect
269. C₂×#135	C₂²×C₃₁	semidirect
270. C₂²×Q₁₆	C₂²×C₃₁	semidirect
287. #470	C₂²×C₃₁	semidirect
288. #468	C₂²×C₃₁	semidirect
291. #467	C₂²×C₃₁	semidirect
292. #467	C₂²×C₃₁	semidirect
293. #467	C₂²×C₃₁	semidirect
295. #466	C₂²×C₃₁	semidirect
297. #465	C₂²×C₃₁	semidirect
298. #428	C₂²×C₃₁	semidirect
299. #427	C₂²×C₃₁	semidirect
300. C₂×#107	C₂²×C₃₁	semidirect
301. #419	C₂²×C₃₁	semidirect
302. #417	C₂²×C₃₁	semidirect
303. #416	C₂²×C₃₁	semidirect
304. C₂×#103	C₂²×C₃₁	semidirect
337. #131	D₄×C₃₁	semidirect
381. C₃₁	#60459	direct (N × H)
504. C₂²	C₃₁×#578	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#578	2
C₂²×C₃₁×#131	2
C₃₁×#3489	2
C₃₁×#3063	2
C₃₁×#3061	2
C₃₁×#3059	2
C₃₁×#3056	2
C₃₁×#3050	2
C₂×C₃₁×#467	2
C₃₁×#2913	2
C₃₁×#2898	2
#60459	31

Subgroups — C31×#60459 #21517218

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60459 #21517218