Subgroups — C₃₁×#60450 #21517209

Order 7936.

2722 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#135	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃₁×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3449	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3061	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3060	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3059	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3058	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3057	C₂	1	4	yes	maximal, complemented
9	C₃₁×#3054	C₂	1	4	yes	maximal, complemented
10	C₃₁×#3050	C₂	1	4	yes	maximal, complemented
11	C₂×C₃₁×#427	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#2896	C₂	1	2	yes	maximal, complemented
13	C₂²×#41×C₃₁	C₂²	1	2	yes	—
14	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#135	—	2	8	no	—
16	C₂×C₃₁×#135	C₂²	1	2	yes	—
17	C₂×C₃₁×#135	C₂²	1	2	yes	—
18	C₂×C₃₁×#135	C₂²	1	2	yes	complemented
19	C₂×C₃₁×#135	C₂²	1	2	yes	complemented
20	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
21	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	complemented
22	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
23	C₂×C₃₁×#131	—	2	8	no	—
24	C₂×C₃₁×#131	C₂²	1	2	yes	—
25	C₂×C₃₁×#131	C₂²	1	2	yes	—
26	C₂×C₃₁×#131	C₂²	1	2	yes	complemented
27	C₂×C₃₁×#131	C₂²	1	2	yes	complemented
28	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
29	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	complemented
30	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
31	C₂³×C₈×C₃₁	C₂²	1	2	yes	complemented
32	C₃₁×#537	C₂²	1	8	yes	complemented
33	C₃₁×#536	—	2	2	no	—
34	C₃₁×#536	C₂²	1	4	yes	—
35	C₃₁×#535	—	2	2	no	—
36	C₃₁×#535	C₂²	1	4	yes	—
37	C₃₁×#534	—	2	4	no	—
38	C₃₁×#532	—	2	4	no	—
39	C₃₁×#525	—	2	4	no	—
40	C₃₁×#525	C₂²	1	8	yes	complemented
41	C₂×C₃₁×#117	C₂²	1	2	yes	—
42	C₃₁×#470	C₂²	1	8	yes	complemented
43	C₃₁×#469	C₂²	1	8	yes	complemented
44	C₃₁×#468	C₂²	1	16	yes	complemented
45	C₃₁×#467	C₂²	1	8	yes	complemented
46	C₃₁×#466	C₂²	1	8	yes	complemented
47	C₃₁×#465	C₂²	1	16	yes	complemented
48	C₃₁×#427	C₂²	1	2	yes	complemented
49	C₃₁×#427	C₂²	1	2	yes	complemented
50	C₃₁×#427	C₂²	1	2	yes	complemented
51	C₃₁×#427	C₂²	1	2	yes	complemented
52	C₂×C₃₁×#107	C₂²	1	2	yes	complemented
53	C₂×C₃₁×#106	C₂²	1	2	yes	complemented
54	C₃₁×#419	C₂²	1	4	yes	complemented
55	C₃₁×#418	C₂²	1	4	yes	complemented
56	C₃₁×#417	C₂²	1	8	yes	complemented
57	C₃₁×#416	C₂²	1	8	yes	complemented
58	C₂×#41×C₃₁	—	2	4	no	—
59	C₂×#41×C₃₁	—	2	2	no	—
60	C₂×#41×C₃₁	—	2	2	no	—
61	C₂×#41×C₃₁	—	2	4	no	—
62	C₂×#41×C₃₁	—	2	8	no	—
63	C₂×#41×C₃₁	—	2	8	no	—
64	C₂×#41×C₃₁	D₄	1	2	yes	—
65	C₂×#41×C₃₁	D₄	1	2	yes	—
66	C₂×#41×C₃₁	C₂³	1	4	yes	—
67	C₂×#41×C₃₁	C₂³	1	4	yes	—
68	C₂×#41×C₃₁	C₂³	1	2	yes	—
69	C₂×#41×C₃₁	C₂³	1	2	yes	—
70	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
71	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
72	C₂²×D₄×C₃₁	—	2	2	no	—
73	C₂²×D₄×C₃₁	D₄	1	2	yes	—
74	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
75	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
76	C₂³×C₄×C₃₁	—	2	2	no	—
77	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
78	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
79	C₃₁×#135	—	2	8	no	—
80	C₃₁×#135	—	2	8	no	—
81	C₃₁×#135	—	2	8	no	—
82	C₃₁×#135	—	2	8	no	—
83	C₂×Q₁₆×C₃₁	—	2	4	no	—
84	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
85	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
86	C₂×SD₁₆×C₃₁	—	2	8	no	—
87	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
88	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
89	C₂×D₈×C₃₁	—	2	4	no	—
90	C₂×D₈×C₃₁	C₂³	1	2	yes	—
91	C₂×D₈×C₃₁	C₂³	1	2	yes	—
92	C₃₁×#131	—	2	8	no	—
93	C₃₁×#131	—	2	8	no	—
94	C₃₁×#131	—	2	8	no	—
95	C₃₁×#131	—	2	8	no	—
96	C₂×M₁₆×C₃₁	—	2	4	no	—
97	C₂×M₁₆×C₃₁	—	2	8	no	—
98	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
99	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
100	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
101	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
102	C₂²×C₈×C₃₁	—	2	8	no	—
103	C₂²×C₈×C₃₁	D₄	1	8	yes	complemented
104	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
105	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
106	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
107	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
108	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
109	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
110	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
111	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
112	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
113	C₃₁×C₄⋊Q₈	—	2	4	no	—
114	C₃₁×#127	—	2	4	no	—
115	C₃₁×#124	—	2	4	no	—
116	C₃₁×#124	—	2	4	no	—
117	C₃₁×#123	—	2	8	no	—
118	C₃₁×#123	—	2	8	no	—
119	C₃₁×C₂²⋊Q₈	—	2	8	no	—
120	C₃₁×C₂²⋊Q₈	C₂³	1	16	yes	—
121	C₃₁×#121	—	2	8	no	—
122	C₃₁×#121	—	2	16	no	—
123	C₃₁×#121	C₂³	1	16	yes	—
124	C₃₁×#120	—	2	8	no	—
125	C₃₁×#120	—	2	8	no	—
126	C₄×D₄×C₃₁	—	2	16	no	—
127	C₃₁×#117	—	2	2	no	—
128	C₃₁×#117	—	2	2	no	—
129	C₃₁×#117	C₂³	1	4	yes	—
130	C₃₁×#117	C₂³	1	4	yes	—
131	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
132	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
133	C₂×C₄²×C₃₁	—	2	2	no	—
134	C₃₁×#107	C₂³	1	4	yes	—
135	C₃₁×#107	C₂³	1	4	yes	—
136	C₃₁×#106	C₂³	1	4	yes	—
137	C₃₁×#106	C₂³	1	4	yes	—
138	C₃₁×#103	C₂³	1	16	yes	—
139	C₃₁×#102	C₂³	1	16	yes	—
140	C₂⁴×C₃₁	—	2	2	no	—
141	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
142	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
143	#41×C₃₁	—	2	8	no	—
144	#41×C₃₁	—	2	8	no	—
145	#41×C₃₁	—	2	4	no	—
146	#41×C₃₁	—	2	4	no	—
147	#41×C₃₁	—	2	4	no	—
148	#41×C₃₁	—	2	4	no	—
149	#41×C₃₁	—	2	8	no	—
150	#41×C₃₁	—	2	8	no	—
151	#41×C₃₁	—	2	8	no	—
152	#41×C₃₁	—	2	8	no	—
153	#41×C₃₁	—	4	8	no	—
154	#41×C₃₁	—	4	8	no	—
155	C₂×Q₈×C₃₁	—	2	4	no	—
156	C₂×Q₈×C₃₁	—	2	2	no	—
157	C₂×Q₈×C₃₁	—	2	4	no	—
158	C₂×Q₈×C₃₁	—	2	4	no	—
159	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
160	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
161	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
162	C₂×D₄×C₃₁	—	2	8	no	—
163	C₂×D₄×C₃₁	—	2	4	no	—
164	C₂×D₄×C₃₁	—	2	4	no	—
165	C₂×D₄×C₃₁	—	2	16	no	—
166	C₂×D₄×C₃₁	—	2	4	no	—
167	C₂×D₄×C₃₁	—	2	4	no	—
168	C₂×D₄×C₃₁	—	2	2	no	—
169	C₂×D₄×C₃₁	—	2	4	no	—
170	C₂×D₄×C₃₁	—	4	8	no	—
171	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
172	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
173	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
174	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
175	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
176	C₂²×C₄×C₃₁	—	2	4	no	—
177	C₂²×C₄×C₃₁	—	2	4	no	—
178	C₂²×C₄×C₃₁	—	2	2	no	—
179	C₂²×C₄×C₃₁	—	2	4	no	—
180	C₂²×C₄×C₃₁	—	2	4	no	—
181	C₂²×C₄×C₃₁	—	2	4	no	—
182	C₂²×C₄×C₃₁	—	4	8	no	—
183	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
184	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
185	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
186	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
187	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
188	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
189	C₂²×C₄×C₃₁	#41	1	4	yes	—
190	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
191	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
192	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
193	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
194	Q₁₆×C₃₁	—	2	4	no	—
195	Q₁₆×C₃₁	—	2	4	no	—
196	SD₁₆×C₃₁	—	2	8	no	—
197	SD₁₆×C₃₁	—	2	8	no	—
198	D₈×C₃₁	—	2	4	no	—
199	D₈×C₃₁	—	2	4	no	—
200	M₁₆×C₃₁	—	2	4	no	—
201	M₁₆×C₃₁	—	2	4	no	—
202	M₁₆×C₃₁	—	2	8	no	—
203	M₁₆×C₃₁	—	2	8	no	—
204	C₂×C₈×C₃₁	—	2	4	no	—
205	C₂×C₈×C₃₁	—	2	8	no	—
206	C₂×C₈×C₃₁	—	2	8	no	—
207	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
208	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
209	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
210	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
211	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
212	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
213	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
214	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
215	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
216	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
217	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
218	C₄⋊C₄×C₃₁	—	2	4	no	—
219	C₄⋊C₄×C₃₁	—	2	16	no	—
220	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
221	C₂²⋊C₄×C₃₁	—	2	4	no	—
222	C₂²⋊C₄×C₃₁	—	2	4	no	—
223	C₂²⋊C₄×C₃₁	—	2	16	no	—
224	C₂²⋊C₄×C₃₁	—	2	16	no	—
225	C₄²×C₃₁	—	2	4	no	—
226	C₄²×C₃₁	—	2	4	no	—
227	#60450	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
228	C₂³×C₃₁	—	2	4	no	—
229	C₂³×C₃₁	—	2	4	no	—
230	C₂³×C₃₁	—	4	2	no	—
231	C₂³×C₃₁	—	4	4	no	—
232	C₂³×C₃₁	—	4	4	no	—
233	C₂³×C₃₁	—	8	4	no	—
234	C₂³×C₃₁	#121	1	4	yes	—
235	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
236	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
237	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
238	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
239	Q₈×C₃₁	—	2	4	no	—
240	Q₈×C₃₁	—	2	4	no	—
241	Q₈×C₃₁	—	2	4	no	—
242	Q₈×C₃₁	—	2	4	no	—
243	Q₈×C₃₁	—	4	4	no	—
244	D₄×C₃₁	—	2	8	no	—
245	D₄×C₃₁	—	2	8	no	—
246	D₄×C₃₁	—	2	4	no	—
247	D₄×C₃₁	—	2	4	no	—
248	D₄×C₃₁	—	2	4	no	—
249	D₄×C₃₁	—	2	4	no	—
250	D₄×C₃₁	—	4	8	no	—
251	D₄×C₃₁	—	4	8	no	—
252	D₄×C₃₁	—	4	4	no	—
253	D₄×C₃₁	—	8	16	no	—
254	C₂×C₄×C₃₁	—	2	4	no	—
255	C₂×C₄×C₃₁	—	2	8	no	—
256	C₂×C₄×C₃₁	—	2	4	no	—
257	C₂×C₄×C₃₁	—	2	4	no	—
258	C₂×C₄×C₃₁	—	2	8	no	—
259	C₂×C₄×C₃₁	—	2	4	no	—
260	C₂×C₄×C₃₁	—	4	4	no	—
261	C₂×C₄×C₃₁	—	4	8	no	—
262	C₂×C₄×C₃₁	—	4	8	no	—
263	C₂×C₄×C₃₁	—	8	4	no	—
264	C₂×C₄×C₃₁	#121	1	4	yes	—
265	C₂×C₄×C₃₁	#121	1	4	yes	—
266	C₂×C₄×C₃₁	#121	1	4	yes	—
267	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
268	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
269	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
270	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
271	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
272	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
273	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
274	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
275	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
276	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
277	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
278	C₈×C₃₁	—	2	4	no	—
279	C₈×C₃₁	C₂²×D₄	1	2	yes	—
280	C₈×C₃₁	C₂²×D₄	1	2	yes	—
281	C₈×C₃₁	C₂²×D₄	1	2	yes	—
282	C₈×C₃₁	C₂²×D₄	1	2	yes	—
283	C₂²×#135	C₂×C₃₁	1	1	yes	characteristic, complemented
284	C₂²×#131	C₂×C₃₁	1	1	yes	characteristic, complemented
285	#3449	C₂×C₃₁	1	2	yes	complemented
286	#3061	C₂×C₃₁	1	4	yes	complemented
287	#3060	C₂×C₃₁	1	4	yes	complemented
288	#3059	C₂×C₃₁	1	2	yes	complemented
289	#3058	C₂×C₃₁	1	2	yes	complemented
290	#3057	C₂×C₃₁	1	4	yes	complemented
291	#3054	C₂×C₃₁	1	4	yes	complemented
292	#3050	C₂×C₃₁	1	4	yes	complemented
293	C₂×#427	C₂×C₃₁	1	1	yes	characteristic, complemented
294	#2896	C₂×C₃₁	1	2	yes	complemented
295	C₂²×C₃₁	—	2	2	no	—
296	C₂²×C₃₁	—	2	2	no	—
297	C₂²×C₃₁	—	2	2	no	—
298	C₂²×C₃₁	—	2	8	no	—
299	C₂²×C₃₁	—	2	4	no	—
300	C₂²×C₃₁	—	4	4	no	—
301	C₂²×C₃₁	—	4	8	no	—
302	C₂²×C₃₁	—	4	8	no	—
303	C₂²×C₃₁	—	4	4	no	—
304	C₂²×C₃₁	—	8	8	no	—
305	C₂²×C₃₁	C₂×#121	1	2	yes	—
306	C₂²×C₃₁	C₂×#121	1	2	yes	—
307	C₂²×C₃₁	C₂×#121	1	2	yes	—
308	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
309	C₄×C₃₁	—	2	4	no	—
310	C₄×C₃₁	—	4	8	no	—
311	C₄×C₃₁	—	4	4	no	—
312	C₄×C₃₁	C₂×#121	1	2	yes	—
313	C₄×C₃₁	C₂×#121	1	2	yes	—
314	C₄×C₃₁	C₂×#121	1	2	yes	—
315	C₄×C₃₁	C₂³×D₄	1	1	yes	characteristic
316	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
317	C₂²×#41	C₂²×C₃₁	1	2	yes	—
318	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
319	C₂×#135	—	2	8	no	—
320	C₂×#135	C₂²×C₃₁	1	2	yes	—
321	C₂×#135	C₂²×C₃₁	1	2	yes	—
322	C₂×#135	C₂²×C₃₁	1	2	yes	complemented
323	C₂×#135	C₂²×C₃₁	1	2	yes	complemented
324	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
325	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	complemented
326	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
327	C₂×#131	—	2	8	no	—
328	C₂×#131	C₂²×C₃₁	1	2	yes	—
329	C₂×#131	C₂²×C₃₁	1	2	yes	—
330	C₂×#131	C₂²×C₃₁	1	2	yes	complemented
331	C₂×#131	C₂²×C₃₁	1	2	yes	complemented
332	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
333	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	complemented
334	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
335	C₂³×C₈	C₂²×C₃₁	1	2	yes	complemented
336	#537	C₂²×C₃₁	1	8	yes	complemented
337	#536	—	2	2	no	—
338	#536	C₂²×C₃₁	1	4	yes	—
339	#535	—	2	2	no	—
340	#535	C₂²×C₃₁	1	4	yes	—
341	#534	—	2	4	no	—
342	#532	—	2	4	no	—
343	#525	—	2	4	no	—
344	#525	C₂²×C₃₁	1	8	yes	complemented
345	C₂×#117	C₂²×C₃₁	1	2	yes	—
346	#470	C₂²×C₃₁	1	8	yes	complemented
347	#469	C₂²×C₃₁	1	8	yes	complemented
348	#468	C₂²×C₃₁	1	16	yes	complemented
349	#467	C₂²×C₃₁	1	8	yes	complemented
350	#466	C₂²×C₃₁	1	8	yes	complemented
351	#465	C₂²×C₃₁	1	16	yes	complemented
352	#427	C₂²×C₃₁	1	2	yes	complemented
353	#427	C₂²×C₃₁	1	2	yes	complemented
354	#427	C₂²×C₃₁	1	2	yes	complemented
355	#427	C₂²×C₃₁	1	2	yes	complemented
356	C₂×#107	C₂²×C₃₁	1	2	yes	complemented
357	C₂×#106	C₂²×C₃₁	1	2	yes	complemented
358	#419	C₂²×C₃₁	1	4	yes	complemented
359	#418	C₂²×C₃₁	1	4	yes	complemented
360	#417	C₂²×C₃₁	1	8	yes	complemented
361	#416	C₂²×C₃₁	1	8	yes	complemented
362	C₂×C₃₁	—	2	2	no	—
363	C₂×C₃₁	—	2	2	no	—
364	C₂×C₃₁	—	2	2	no	—
365	C₂×C₃₁	—	4	4	no	—
366	C₂×C₃₁	—	8	4	no	—
367	C₂×C₃₁	C₂²×#121	1	1	yes	characteristic
368	C₂×C₃₁	#3584	1	2	yes	—
369	C₂×#41	—	2	4	no	—
370	C₂×#41	—	2	2	no	—
371	C₂×#41	—	2	2	no	—
372	C₂×#41	—	2	4	no	—
373	C₂×#41	—	2	8	no	—
374	C₂×#41	—	2	8	no	—
375	C₂×#41	D₄×C₃₁	1	2	yes	—
376	C₂×#41	D₄×C₃₁	1	2	yes	—
377	C₂×#41	C₂³×C₃₁	1	4	yes	—
378	C₂×#41	C₂³×C₃₁	1	4	yes	—
379	C₂×#41	C₂³×C₃₁	1	2	yes	—
380	C₂×#41	C₂³×C₃₁	1	2	yes	—
381	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
382	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
383	C₂²×D₄	—	2	2	no	—
384	C₂²×D₄	D₄×C₃₁	1	2	yes	—
385	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
386	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
387	C₂³×C₄	—	2	2	no	—
388	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
389	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
390	#135	—	2	8	no	—
391	#135	—	2	8	no	—
392	#135	—	2	8	no	—
393	#135	—	2	8	no	—
394	C₂×Q₁₆	—	2	4	no	—
395	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
396	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
397	C₂×SD₁₆	—	2	8	no	—
398	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
399	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
400	C₂×D₈	—	2	4	no	—
401	C₂×D₈	C₂³×C₃₁	1	2	yes	—
402	C₂×D₈	C₂³×C₃₁	1	2	yes	—
403	#131	—	2	8	no	—
404	#131	—	2	8	no	—
405	#131	—	2	8	no	—
406	#131	—	2	8	no	—
407	C₂×M₁₆	—	2	4	no	—
408	C₂×M₁₆	—	2	8	no	—
409	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
410	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
411	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
412	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
413	C₂²×C₈	—	2	8	no	—
414	C₂²×C₈	D₄×C₃₁	1	8	yes	complemented
415	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
416	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
417	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
418	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
419	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
420	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
421	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
422	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
423	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
424	C₄⋊Q₈	—	2	4	no	—
425	#127	—	2	4	no	—
426	#124	—	2	4	no	—
427	#124	—	2	4	no	—
428	#123	—	2	8	no	—
429	#123	—	2	8	no	—
430	C₂²⋊Q₈	—	2	8	no	—
431	C₂²⋊Q₈	C₂³×C₃₁	1	16	yes	—
432	#121	—	2	8	no	—
433	#121	—	2	16	no	—
434	#121	C₂³×C₃₁	1	16	yes	—
435	#120	—	2	8	no	—
436	#120	—	2	8	no	—
437	C₄×D₄	—	2	16	no	—
438	#117	—	2	2	no	—
439	#117	—	2	2	no	—
440	#117	C₂³×C₃₁	1	4	yes	—
441	#117	C₂³×C₃₁	1	4	yes	—
442	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
443	C₂×C₂²⋊C₄	—	2	2	no	—
444	C₂×C₄²	—	2	2	no	—
445	#107	C₂³×C₃₁	1	4	yes	—
446	#107	C₂³×C₃₁	1	4	yes	—
447	#106	C₂³×C₃₁	1	4	yes	—
448	#106	C₂³×C₃₁	1	4	yes	—
449	#103	C₂³×C₃₁	1	16	yes	—
450	#102	C₂³×C₃₁	1	16	yes	—
451	C₃₁	#60450	1	1	yes	characteristic, direct product factor, Sylow
452	C₂⁴	—	2	2	no	—
453	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
454	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
455	#41	—	2	8	no	—
456	#41	—	2	8	no	—
457	#41	—	2	4	no	—
458	#41	—	2	4	no	—
459	#41	—	2	4	no	—
460	#41	—	2	4	no	—
461	#41	—	2	8	no	—
462	#41	—	2	8	no	—
463	#41	—	2	8	no	—
464	#41	—	2	8	no	—
465	#41	—	4	8	no	—
466	#41	—	4	8	no	—
467	C₂×Q₈	—	2	4	no	—
468	C₂×Q₈	—	2	2	no	—
469	C₂×Q₈	—	2	4	no	—
470	C₂×Q₈	—	2	4	no	—
471	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
472	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
473	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
474	C₂×D₄	—	2	8	no	—
475	C₂×D₄	—	2	4	no	—
476	C₂×D₄	—	2	4	no	—
477	C₂×D₄	—	2	16	no	—
478	C₂×D₄	—	2	4	no	—
479	C₂×D₄	—	2	4	no	—
480	C₂×D₄	—	2	2	no	—
481	C₂×D₄	—	2	4	no	—
482	C₂×D₄	—	4	8	no	—
483	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
484	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
485	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
486	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
487	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
488	C₂²×C₄	—	2	4	no	—
489	C₂²×C₄	—	2	4	no	—
490	C₂²×C₄	—	2	2	no	—
491	C₂²×C₄	—	2	4	no	—
492	C₂²×C₄	—	2	4	no	—
493	C₂²×C₄	—	2	4	no	—
494	C₂²×C₄	—	4	8	no	—
495	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
496	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
497	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
498	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
499	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
500	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
501	C₂²×C₄	#41×C₃₁	1	4	yes	—
502	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
503	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
504	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
505	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
506	Q₁₆	—	2	4	no	—
507	Q₁₆	—	2	4	no	—
508	SD₁₆	—	2	8	no	—
509	SD₁₆	—	2	8	no	—
510	D₈	—	2	4	no	—
511	D₈	—	2	4	no	—
512	M₁₆	—	2	4	no	—
513	M₁₆	—	2	4	no	—
514	M₁₆	—	2	8	no	—
515	M₁₆	—	2	8	no	—
516	C₂×C₈	—	2	4	no	—
517	C₂×C₈	—	2	8	no	—
518	C₂×C₈	—	2	8	no	—
519	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
520	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
521	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
522	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
523	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
524	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
525	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
526	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
527	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
528	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
529	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
530	C₄⋊C₄	—	2	4	no	—
531	C₄⋊C₄	—	2	16	no	—
532	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
533	C₂²⋊C₄	—	2	4	no	—
534	C₂²⋊C₄	—	2	4	no	—
535	C₂²⋊C₄	—	2	16	no	—
536	C₂²⋊C₄	—	2	16	no	—
537	C₄²	—	2	4	no	—
538	C₄²	—	2	4	no	—
539	C₂³	—	2	4	no	—
540	C₂³	—	2	4	no	—
541	C₂³	—	4	2	no	—
542	C₂³	—	4	4	no	—
543	C₂³	—	4	4	no	—
544	C₂³	—	8	4	no	—
545	C₂³	C₃₁×#121	1	4	yes	—
546	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
547	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
548	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
549	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
550	Q₈	—	2	4	no	—
551	Q₈	—	2	4	no	—
552	Q₈	—	2	4	no	—
553	Q₈	—	2	4	no	—
554	Q₈	—	4	4	no	—
555	D₄	—	2	8	no	—
556	D₄	—	2	8	no	—
557	D₄	—	2	4	no	—
558	D₄	—	2	4	no	—
559	D₄	—	2	4	no	—
560	D₄	—	2	4	no	—
561	D₄	—	4	8	no	—
562	D₄	—	4	8	no	—
563	D₄	—	4	4	no	—
564	D₄	—	8	16	no	—
565	C₂×C₄	—	2	4	no	—
566	C₂×C₄	—	2	8	no	—
567	C₂×C₄	—	2	4	no	—
568	C₂×C₄	—	2	4	no	—
569	C₂×C₄	—	2	8	no	—
570	C₂×C₄	—	2	4	no	—
571	C₂×C₄	—	4	4	no	—
572	C₂×C₄	—	4	8	no	—
573	C₂×C₄	—	4	8	no	—
574	C₂×C₄	—	8	4	no	—
575	C₂×C₄	C₃₁×#121	1	4	yes	—
576	C₂×C₄	C₃₁×#121	1	4	yes	—
577	C₂×C₄	C₃₁×#121	1	4	yes	—
578	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
579	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
580	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
581	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
582	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
583	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
584	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
585	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
586	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
587	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
588	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
589	C₈	—	2	4	no	—
590	C₈	C₂²×D₄×C₃₁	1	2	yes	—
591	C₈	C₂²×D₄×C₃₁	1	2	yes	—
592	C₈	C₂²×D₄×C₃₁	1	2	yes	—
593	C₈	C₂²×D₄×C₃₁	1	2	yes	—
594	C₂²	—	2	2	no	—
595	C₂²	—	2	2	no	—
596	C₂²	—	2	2	no	—
597	C₂²	—	2	4	no	—
598	C₂²	—	2	8	no	—
599	C₂²	—	4	4	no	—
600	C₂²	—	4	8	no	—
601	C₂²	—	4	4	no	—
602	C₂²	—	4	8	no	—
603	C₂²	—	8	8	no	—
604	C₂²	C₂×C₃₁×#121	1	2	yes	—
605	C₂²	C₂×C₃₁×#121	1	2	yes	—
606	C₂²	C₂×C₃₁×#121	1	2	yes	—
607	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
608	C₄	—	2	4	no	—
609	C₄	—	4	4	no	—
610	C₄	—	4	8	no	—
611	C₄	C₂×C₃₁×#121	1	2	yes	—
612	C₄	C₂×C₃₁×#121	1	2	yes	—
613	C₄	C₂×C₃₁×#121	1	2	yes	—
614	C₄	C₂³×D₄×C₃₁	1	1	yes	characteristic
615	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
616	C₂	—	2	2	no	—
617	C₂	—	2	2	no	—
618	C₂	—	2	2	no	—
619	C₂	—	4	4	no	—
620	C₂	—	8	4	no	—
621	C₂	C₂²×C₃₁×#121	1	1	yes	characteristic
622	C₂	C₃₁×#3584	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
socle	C₂²×C₃₁	C₂³×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂²×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃₁×#135	C₂	semidirect
2. C₂²×C₃₁×#131	C₂	semidirect
3. C₃₁×#3449	C₂	semidirect
4. C₃₁×#3061	C₂	semidirect
5. C₃₁×#3060	C₂	semidirect
6. C₃₁×#3059	C₂	semidirect
7. C₃₁×#3058	C₂	semidirect
8. C₃₁×#3057	C₂	semidirect
9. C₃₁×#3054	C₂	semidirect
10. C₃₁×#3050	C₂	semidirect
11. C₂×C₃₁×#427	C₂	semidirect
12. C₃₁×#2896	C₂	semidirect
18. C₂×C₃₁×#135	C₂²	semidirect
19. C₂×C₃₁×#135	C₂²	semidirect
20. C₂²×Q₁₆×C₃₁	C₂²	semidirect
21. C₂²×SD₁₆×C₃₁	C₂²	semidirect
22. C₂²×D₈×C₃₁	C₂²	semidirect
26. C₂×C₃₁×#131	C₂²	semidirect
27. C₂×C₃₁×#131	C₂²	semidirect
29. C₂²×M₁₆×C₃₁	C₂²	semidirect
31. C₂³×C₈×C₃₁	C₂²	semidirect
32. C₃₁×#537	C₂²	semidirect
40. C₃₁×#525	C₂²	semidirect
42. C₃₁×#470	C₂²	semidirect
43. C₃₁×#469	C₂²	semidirect
44. C₃₁×#468	C₂²	semidirect
45. C₃₁×#467	C₂²	semidirect
46. C₃₁×#466	C₂²	semidirect
47. C₃₁×#465	C₂²	semidirect
48. C₃₁×#427	C₂²	semidirect
49. C₃₁×#427	C₂²	semidirect
50. C₃₁×#427	C₂²	semidirect
51. C₃₁×#427	C₂²	semidirect
52. C₂×C₃₁×#107	C₂²	semidirect
53. C₂×C₃₁×#106	C₂²	semidirect
54. C₃₁×#419	C₂²	semidirect
55. C₃₁×#418	C₂²	semidirect
56. C₃₁×#417	C₂²	semidirect
57. C₃₁×#416	C₂²	semidirect
103. C₂²×C₈×C₃₁	D₄	semidirect
227. #60450	C₃₁	direct (N × H)
283. C₂²×#135	C₂×C₃₁	semidirect
284. C₂²×#131	C₂×C₃₁	semidirect
285. #3449	C₂×C₃₁	semidirect
286. #3061	C₂×C₃₁	semidirect
287. #3060	C₂×C₃₁	semidirect
288. #3059	C₂×C₃₁	semidirect
289. #3058	C₂×C₃₁	semidirect
290. #3057	C₂×C₃₁	semidirect
291. #3054	C₂×C₃₁	semidirect
292. #3050	C₂×C₃₁	semidirect
293. C₂×#427	C₂×C₃₁	semidirect
294. #2896	C₂×C₃₁	semidirect
322. C₂×#135	C₂²×C₃₁	semidirect
323. C₂×#135	C₂²×C₃₁	semidirect
324. C₂²×Q₁₆	C₂²×C₃₁	semidirect
325. C₂²×SD₁₆	C₂²×C₃₁	semidirect
326. C₂²×D₈	C₂²×C₃₁	semidirect
330. C₂×#131	C₂²×C₃₁	semidirect
331. C₂×#131	C₂²×C₃₁	semidirect
333. C₂²×M₁₆	C₂²×C₃₁	semidirect
335. C₂³×C₈	C₂²×C₃₁	semidirect
336. #537	C₂²×C₃₁	semidirect
344. #525	C₂²×C₃₁	semidirect
346. #470	C₂²×C₃₁	semidirect
347. #469	C₂²×C₃₁	semidirect
348. #468	C₂²×C₃₁	semidirect
349. #467	C₂²×C₃₁	semidirect
350. #466	C₂²×C₃₁	semidirect
351. #465	C₂²×C₃₁	semidirect
352. #427	C₂²×C₃₁	semidirect
353. #427	C₂²×C₃₁	semidirect
354. #427	C₂²×C₃₁	semidirect
355. #427	C₂²×C₃₁	semidirect
356. C₂×#107	C₂²×C₃₁	semidirect
357. C₂×#106	C₂²×C₃₁	semidirect
358. #419	C₂²×C₃₁	semidirect
359. #418	C₂²×C₃₁	semidirect
360. #417	C₂²×C₃₁	semidirect
361. #416	C₂²×C₃₁	semidirect
414. C₂²×C₈	D₄×C₃₁	semidirect
451. C₃₁	#60450	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×#135	2
C₂²×C₃₁×#131	2
C₃₁×#3449	2
C₃₁×#3061	2
C₃₁×#3060	2
C₃₁×#3059	2
C₃₁×#3058	2
C₃₁×#3057	2
C₃₁×#3054	2
C₃₁×#3050	2
C₂×C₃₁×#427	2
C₃₁×#2896	2
#60450	31

Subgroups — C31×#60450 #21517209

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60450 #21517209