Subgroups — C₃₁×#60830 #21517589

Order 7936.

2150 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3512	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3498	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3222	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3220	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3219	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3217	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3151	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#3150	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#3142	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#490	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#3045	C₂	1	4	yes	maximal, complemented
12	C₃₁×#3012	C₂	1	4	yes	maximal, complemented
13	C₃₁×#2923	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₃₁×H(5,𝔽₂)	C₂²	1	2	yes	—
15	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	4	yes	complemented
16	C₂×C₃₁×#135	C₂²	1	4	yes	complemented
17	C₃₁×#553	—	2	4	no	—
18	C₃₁×#553	—	2	2	no	—
19	C₃₁×#552	—	2	2	no	—
20	C₃₁×#550	—	2	2	no	—
21	D₄²×C₃₁	—	2	4	no	—
22	D₄²×C₃₁	—	2	2	no	—
23	C₃₁×#541	C₂²	1	2	yes	complemented
24	C₃₁×#540	C₂²	1	2	yes	complemented
25	C₃₁×#537	C₂²	1	2	yes	—
26	C₃₁×#535	C₂²	1	4	yes	—
27	C₃₁×#535	C₂²	1	2	yes	—
28	C₃₁×#532	C₂²	1	1	yes	characteristic
29	C₃₁×#532	C₂²	1	2	yes	—
30	C₃₁×#532	C₂²	1	1	yes	characteristic, complemented
31	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
32	C₂×C₃₁×#127	C₂²	1	2	yes	—
33	C₃₁×#529	C₂²	1	4	yes	complemented
34	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic, complemented
35	C₃₁×#520	C₂²	1	2	yes	complemented
36	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#497	C₂²	1	2	yes	complemented
38	C₃₁×#497	C₂²	1	2	yes	complemented
39	C₃₁×#496	C₂²	1	2	yes	complemented
40	C₃₁×#496	C₂²	1	2	yes	complemented
41	C₃₁×#496	C₂²	1	4	yes	complemented
42	C₃₁×#493	C₂²	1	2	yes	complemented
43	C₃₁×#492	C₂²	1	2	yes	complemented
44	C₃₁×#491	C₂²	1	2	yes	complemented
45	C₃₁×#490	C₂²	1	4	yes	complemented
46	C₃₁×#490	C₂²	1	4	yes	complemented
47	C₃₁×#490	C₂²	1	2	yes	complemented
48	C₃₁×#488	C₂²	1	4	yes	complemented
49	C₃₁×#464	C₂²	1	8	yes	complemented
50	C₃₁×#463	C₂²	1	8	yes	complemented
51	C₃₁×#460	C₂²	1	8	yes	complemented
52	C₃₁×#459	C₂²	1	8	yes	complemented
53	C₃₁×#452	C₂²	1	8	yes	complemented
54	C₃₁×#450	C₂²	1	8	yes	complemented
55	C₃₁×#449	C₂²	1	8	yes	complemented
56	C₃₁×#447	C₂²	1	8	yes	complemented
57	C₃₁×#433	C₂²	1	4	yes	complemented
58	C₃₁×#432	C₂²	1	4	yes	complemented
59	C₃₁×#424	C₂²	1	2	yes	complemented
60	C₃₁×#417	C₂²	1	4	yes	complemented
61	C₂×C₃₁×#102	C₂²	1	4	yes	complemented
62	C₃₁×#408	C₂²	1	2	yes	—
63	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic, complemented
64	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic, complemented
65	C₃₁×H(5,𝔽₂)	—	4	8	no	—
66	C₂×#41×C₃₁	—	2	2	no	—
67	C₂×#41×C₃₁	C₂³	1	2	yes	—
68	C₂×#41×C₃₁	C₂³	1	2	yes	—
69	C₂×#41×C₃₁	C₂³	1	4	yes	—
70	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
71	C₂²×D₄×C₃₁	—	2	4	no	—
72	C₂²×D₄×C₃₁	—	2	2	no	—
73	C₂²×D₄×C₃₁	—	2	2	no	—
74	C₂²×D₄×C₃₁	—	2	4	no	—
75	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
76	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
77	C₃₁×AGL(1,ℤ/8)	—	4	8	no	—
78	C₃₁×#135	—	4	8	no	—
79	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
80	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
81	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
82	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
83	C₂×D₈×C₃₁	C₂³	1	4	yes	—
84	C₂×D₈×C₃₁	C₂³	1	8	yes	—
85	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
86	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
87	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
88	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
89	C₃₁×#127	—	2	4	no	—
90	C₃₁×#127	—	2	2	no	—
91	C₃₁×#127	C₂³	1	4	yes	—
92	C₃₁×#127	C₂³	1	2	yes	—
93	C₃₁×#127	C₂³	1	2	yes	—
94	C₃₁×#127	C₂³	1	1	yes	characteristic
95	C₃₁×#127	C₂³	1	1	yes	characteristic
96	C₃₁×#126	—	2	4	no	—
97	C₃₁×#126	—	2	4	no	—
98	C₃₁×#125	—	2	4	no	—
99	C₃₁×#125	—	2	2	no	—
100	C₃₁×#125	C₂³	1	2	yes	—
101	C₃₁×#125	C₂³	1	2	yes	—
102	C₃₁×#124	C₂³	1	4	yes	—
103	C₃₁×#124	C₂³	1	2	yes	—
104	C₃₁×#124	C₂³	1	2	yes	—
105	C₃₁×#123	—	2	4	no	—
106	C₃₁×#123	—	2	4	no	—
107	C₃₁×C₂²⋊Q₈	—	2	2	no	—
108	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
109	C₃₁×#121	—	2	2	no	—
110	C₃₁×#121	—	2	4	no	—
111	C₃₁×#121	—	2	2	no	—
112	C₃₁×#121	—	2	4	no	—
113	C₃₁×#121	—	2	2	no	—
114	C₃₁×#121	—	2	4	no	—
115	C₃₁×#121	—	2	8	no	—
116	C₃₁×#121	—	2	4	no	—
117	C₃₁×#121	C₂³	1	4	yes	—
118	C₃₁×#120	—	2	4	no	—
119	C₃₁×#120	—	2	8	no	—
120	C₄×Q₈×C₃₁	D₄	1	2	yes	—
121	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
122	C₄×D₄×C₃₁	—	2	4	no	—
123	C₄×D₄×C₃₁	—	2	4	no	—
124	C₄×D₄×C₃₁	—	2	4	no	—
125	C₄×D₄×C₃₁	—	2	2	no	—
126	C₄×D₄×C₃₁	—	2	2	no	—
127	C₄×D₄×C₃₁	D₄	1	4	yes	—
128	C₄×D₄×C₃₁	D₄	1	2	yes	—
129	C₄×D₄×C₃₁	C₂³	1	4	yes	—
130	C₃₁×#117	—	2	2	no	—
131	C₃₁×#117	—	2	4	no	—
132	C₃₁×#117	C₂³	1	1	yes	characteristic
133	C₃₁×#117	C₂³	1	2	yes	—
134	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
135	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
136	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
137	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
138	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
139	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
140	C₃₁×#103	C₂³	1	8	yes	—
141	C₃₁×#102	C₂³	1	8	yes	—
142	C₃₁×#102	C₂³	1	8	yes	—
143	C₃₁×#102	C₂³	1	8	yes	—
144	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
145	C₃₁×#97	C₂³	1	2	yes	—
146	C₃₁×#97	C₂³	1	2	yes	—
147	C₃₁×#97	C₂³	1	2	yes	—
148	C₄×C₈×C₃₁	C₂³	1	2	yes	—
149	C₂⁴×C₃₁	—	2	4	no	—
150	C₂⁴×C₃₁	—	2	2	no	—
151	#41×C₃₁	—	2	4	no	—
152	#41×C₃₁	—	2	8	no	—
153	#41×C₃₁	—	2	8	no	—
154	#41×C₃₁	—	4	8	no	—
155	#41×C₃₁	—	4	8	no	—
156	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
158	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
159	C₂×D₄×C₃₁	—	2	2	no	—
160	C₂×D₄×C₃₁	—	2	4	no	—
161	C₂×D₄×C₃₁	—	2	2	no	—
162	C₂×D₄×C₃₁	—	2	4	no	—
163	C₂×D₄×C₃₁	—	2	4	no	—
164	C₂×D₄×C₃₁	—	2	2	no	—
165	C₂×D₄×C₃₁	—	2	4	no	—
166	C₂×D₄×C₃₁	—	2	2	no	—
167	C₂×D₄×C₃₁	—	2	2	no	—
168	C₂×D₄×C₃₁	—	2	8	no	—
169	C₂×D₄×C₃₁	—	2	4	no	—
170	C₂×D₄×C₃₁	—	2	8	no	—
171	C₂×D₄×C₃₁	—	4	8	no	—
172	C₂×D₄×C₃₁	—	4	16	no	—
173	C₂×D₄×C₃₁	—	4	8	no	—
174	C₂×D₄×C₃₁	—	8	8	no	—
175	C₂×D₄×C₃₁	—	8	4	no	—
176	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
177	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
178	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
179	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
180	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
181	C₂²×C₄×C₃₁	—	2	2	no	—
182	C₂²×C₄×C₃₁	—	2	2	no	—
183	C₂²×C₄×C₃₁	—	2	4	no	—
184	C₂²×C₄×C₃₁	—	2	2	no	—
185	C₂²×C₄×C₃₁	—	2	4	no	—
186	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
187	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
188	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
189	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
190	Q₁₆×C₃₁	—	4	4	no	—
191	SD₁₆×C₃₁	—	4	4	no	—
192	SD₁₆×C₃₁	—	4	4	no	—
193	SD₁₆×C₃₁	—	4	8	no	—
194	D₈×C₃₁	—	4	4	no	—
195	D₈×C₃₁	—	4	8	no	—
196	M₁₆×C₃₁	—	2	4	no	—
197	C₂×C₈×C₃₁	—	2	4	no	—
198	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
199	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
200	C₄⋊C₄×C₃₁	—	2	2	no	—
201	C₄⋊C₄×C₃₁	—	2	4	no	—
202	C₄⋊C₄×C₃₁	—	2	4	no	—
203	C₄⋊C₄×C₃₁	—	2	4	no	—
204	C₄⋊C₄×C₃₁	—	2	4	no	—
205	C₄⋊C₄×C₃₁	—	2	2	no	—
206	C₄⋊C₄×C₃₁	—	2	2	no	—
207	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
208	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
209	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
210	C₂²⋊C₄×C₃₁	—	2	4	no	—
211	C₂²⋊C₄×C₃₁	—	2	2	no	—
212	C₂²⋊C₄×C₃₁	—	2	4	no	—
213	C₂²⋊C₄×C₃₁	—	2	4	no	—
214	C₂²⋊C₄×C₃₁	—	2	2	no	—
215	C₂²⋊C₄×C₃₁	—	2	8	no	—
216	C₂²⋊C₄×C₃₁	—	2	4	no	—
217	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
218	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
219	C₄²×C₃₁	—	2	4	no	—
220	C₄²×C₃₁	C₂×D₄	1	4	yes	—
221	C₄²×C₃₁	C₂×D₄	1	2	yes	—
222	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
223	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
224	C₄²×C₃₁	C₂⁴	1	2	yes	—
225	#60830	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
226	C₂³×C₃₁	—	2	2	no	—
227	C₂³×C₃₁	—	2	4	no	—
228	C₂³×C₃₁	—	2	2	no	—
229	C₂³×C₃₁	—	4	8	no	—
230	C₂³×C₃₁	—	4	4	no	—
231	C₂³×C₃₁	—	8	4	no	—
232	C₂³×C₃₁	—	8	8	no	—
233	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
234	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
235	Q₈×C₃₁	—	2	2	no	—
236	Q₈×C₃₁	—	2	4	no	—
237	Q₈×C₃₁	—	2	4	no	—
238	D₄×C₃₁	—	2	4	no	—
239	D₄×C₃₁	—	2	2	no	—
240	D₄×C₃₁	—	2	4	no	—
241	D₄×C₃₁	—	2	8	no	—
242	D₄×C₃₁	—	2	4	no	—
243	D₄×C₃₁	—	4	8	no	—
244	D₄×C₃₁	—	4	8	no	—
245	D₄×C₃₁	—	4	4	no	—
246	D₄×C₃₁	—	4	8	no	—
247	D₄×C₃₁	—	4	4	no	—
248	D₄×C₃₁	—	4	4	no	—
249	D₄×C₃₁	—	4	4	no	—
250	D₄×C₃₁	—	8	2	no	—
251	D₄×C₃₁	—	8	8	no	—
252	D₄×C₃₁	—	8	4	no	—
253	D₄×C₃₁	—	8	4	no	—
254	C₂×C₄×C₃₁	—	2	4	no	—
255	C₂×C₄×C₃₁	—	2	2	no	—
256	C₂×C₄×C₃₁	—	2	4	no	—
257	C₂×C₄×C₃₁	—	2	2	no	—
258	C₂×C₄×C₃₁	—	4	4	no	—
259	C₂×C₄×C₃₁	—	4	2	no	—
260	C₂×C₄×C₃₁	—	4	8	no	—
261	C₂×C₄×C₃₁	—	4	4	no	—
262	C₂×C₄×C₃₁	—	4	4	no	—
263	C₂×C₄×C₃₁	—	8	4	no	—
264	C₂×C₄×C₃₁	—	8	2	no	—
265	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
266	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
267	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
268	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
269	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
270	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
271	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
272	C₈×C₃₁	—	2	4	no	—
273	C₈×C₃₁	—	2	4	no	—
274	#3512	C₂×C₃₁	1	1	yes	characteristic, complemented
275	#3498	C₂×C₃₁	1	1	yes	characteristic, complemented
276	#3222	C₂×C₃₁	1	4	yes	complemented
277	#3220	C₂×C₃₁	1	4	yes	complemented
278	#3219	C₂×C₃₁	1	4	yes	complemented
279	#3217	C₂×C₃₁	1	4	yes	complemented
280	#3151	C₂×C₃₁	1	1	yes	characteristic, complemented
281	#3150	C₂×C₃₁	1	1	yes	characteristic, complemented
282	#3142	C₂×C₃₁	1	1	yes	characteristic, complemented
283	C₂×#490	C₂×C₃₁	1	1	yes	characteristic, complemented
284	#3045	C₂×C₃₁	1	4	yes	complemented
285	#3012	C₂×C₃₁	1	4	yes	complemented
286	#2923	C₂×C₃₁	1	1	yes	characteristic, complemented
287	C₂²×C₃₁	—	2	2	no	—
288	C₂²×C₃₁	—	2	4	no	—
289	C₂²×C₃₁	—	2	2	no	—
290	C₂²×C₃₁	—	4	2	no	—
291	C₂²×C₃₁	—	4	8	no	—
292	C₂²×C₃₁	—	4	4	no	—
293	C₂²×C₃₁	—	4	2	no	—
294	C₂²×C₃₁	—	4	4	no	—
295	C₂²×C₃₁	—	4	2	no	—
296	C₂²×C₃₁	—	4	2	no	—
297	C₂²×C₃₁	—	16	4	no	—
298	C₂²×C₃₁	—	16	8	no	—
299	C₂²×C₃₁	#535	1	2	yes	—
300	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
301	C₄×C₃₁	—	2	2	no	—
302	C₄×C₃₁	—	2	4	no	—
303	C₄×C₃₁	—	2	2	no	—
304	C₄×C₃₁	—	2	2	no	—
305	C₄×C₃₁	—	4	4	no	—
306	C₄×C₃₁	—	4	2	no	—
307	C₄×C₃₁	—	4	2	no	—
308	C₄×C₃₁	#535	1	2	yes	—
309	C₄×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
310	C₂×H(5,𝔽₂)	C₂²×C₃₁	1	2	yes	—
311	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	4	yes	complemented
312	C₂×#135	C₂²×C₃₁	1	4	yes	complemented
313	#553	—	2	4	no	—
314	#553	—	2	2	no	—
315	#552	—	2	2	no	—
316	#550	—	2	2	no	—
317	D₄²	—	2	4	no	—
318	D₄²	—	2	2	no	—
319	#541	C₂²×C₃₁	1	2	yes	complemented
320	#540	C₂²×C₃₁	1	2	yes	complemented
321	#537	C₂²×C₃₁	1	2	yes	—
322	#535	C₂²×C₃₁	1	4	yes	—
323	#535	C₂²×C₃₁	1	2	yes	—
324	#532	C₂²×C₃₁	1	1	yes	characteristic
325	#532	C₂²×C₃₁	1	2	yes	—
326	#532	C₂²×C₃₁	1	1	yes	characteristic, complemented
327	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
328	C₂×#127	C₂²×C₃₁	1	2	yes	—
329	#529	C₂²×C₃₁	1	4	yes	complemented
330	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic, complemented
331	#520	C₂²×C₃₁	1	2	yes	complemented
332	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
333	#497	C₂²×C₃₁	1	2	yes	complemented
334	#497	C₂²×C₃₁	1	2	yes	complemented
335	#496	C₂²×C₃₁	1	2	yes	complemented
336	#496	C₂²×C₃₁	1	2	yes	complemented
337	#496	C₂²×C₃₁	1	4	yes	complemented
338	#493	C₂²×C₃₁	1	2	yes	complemented
339	#492	C₂²×C₃₁	1	2	yes	complemented
340	#491	C₂²×C₃₁	1	2	yes	complemented
341	#490	C₂²×C₃₁	1	4	yes	complemented
342	#490	C₂²×C₃₁	1	4	yes	complemented
343	#490	C₂²×C₃₁	1	2	yes	complemented
344	#488	C₂²×C₃₁	1	4	yes	complemented
345	#464	C₂²×C₃₁	1	8	yes	complemented
346	#463	C₂²×C₃₁	1	8	yes	complemented
347	#460	C₂²×C₃₁	1	8	yes	complemented
348	#459	C₂²×C₃₁	1	8	yes	complemented
349	#452	C₂²×C₃₁	1	8	yes	complemented
350	#450	C₂²×C₃₁	1	8	yes	complemented
351	#449	C₂²×C₃₁	1	8	yes	complemented
352	#447	C₂²×C₃₁	1	8	yes	complemented
353	#433	C₂²×C₃₁	1	4	yes	complemented
354	#432	C₂²×C₃₁	1	4	yes	complemented
355	#424	C₂²×C₃₁	1	2	yes	complemented
356	#417	C₂²×C₃₁	1	4	yes	complemented
357	C₂×#102	C₂²×C₃₁	1	4	yes	complemented
358	#408	C₂²×C₃₁	1	2	yes	—
359	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic, complemented
360	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic, complemented
361	C₂×C₃₁	—	2	2	no	—
362	C₂×C₃₁	—	4	2	no	—
363	C₂×C₃₁	—	8	2	no	—
364	C₂×C₃₁	—	8	4	no	—
365	C₂×C₃₁	—	8	2	no	—
366	C₂×C₃₁	C₂×#535	1	1	yes	characteristic
367	C₂×C₃₁	#3585	1	2	yes	—
368	H(5,𝔽₂)	—	4	8	no	—
369	C₂×#41	—	2	2	no	—
370	C₂×#41	C₂³×C₃₁	1	2	yes	—
371	C₂×#41	C₂³×C₃₁	1	2	yes	—
372	C₂×#41	C₂³×C₃₁	1	4	yes	—
373	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
374	C₂²×D₄	—	2	4	no	—
375	C₂²×D₄	—	2	2	no	—
376	C₂²×D₄	—	2	2	no	—
377	C₂²×D₄	—	2	4	no	—
378	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
379	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
380	AGL(1,ℤ/8)	—	4	8	no	—
381	#135	—	4	8	no	—
382	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
383	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
384	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
385	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
386	C₂×D₈	C₂³×C₃₁	1	4	yes	—
387	C₂×D₈	C₂³×C₃₁	1	8	yes	—
388	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
389	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
390	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
391	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
392	#127	—	2	4	no	—
393	#127	—	2	2	no	—
394	#127	C₂³×C₃₁	1	2	yes	—
395	#127	C₂³×C₃₁	1	1	yes	characteristic
396	#127	C₂³×C₃₁	1	4	yes	—
397	#127	C₂³×C₃₁	1	2	yes	—
398	#127	C₂³×C₃₁	1	1	yes	characteristic
399	#126	—	2	4	no	—
400	#126	—	2	4	no	—
401	#125	—	2	4	no	—
402	#125	—	2	2	no	—
403	#125	C₂³×C₃₁	1	2	yes	—
404	#125	C₂³×C₃₁	1	2	yes	—
405	#124	C₂³×C₃₁	1	2	yes	—
406	#124	C₂³×C₃₁	1	4	yes	—
407	#124	C₂³×C₃₁	1	2	yes	—
408	#123	—	2	4	no	—
409	#123	—	2	4	no	—
410	C₂²⋊Q₈	—	2	2	no	—
411	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
412	#121	—	2	2	no	—
413	#121	—	2	4	no	—
414	#121	—	2	2	no	—
415	#121	—	2	4	no	—
416	#121	—	2	2	no	—
417	#121	—	2	4	no	—
418	#121	—	2	8	no	—
419	#121	—	2	4	no	—
420	#121	C₂³×C₃₁	1	4	yes	—
421	#120	—	2	4	no	—
422	#120	—	2	8	no	—
423	C₄×Q₈	D₄×C₃₁	1	2	yes	—
424	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
425	C₄×D₄	—	2	4	no	—
426	C₄×D₄	—	2	4	no	—
427	C₄×D₄	—	2	4	no	—
428	C₄×D₄	—	2	2	no	—
429	C₄×D₄	—	2	2	no	—
430	C₄×D₄	D₄×C₃₁	1	4	yes	—
431	C₄×D₄	D₄×C₃₁	1	2	yes	—
432	C₄×D₄	C₂³×C₃₁	1	4	yes	—
433	#117	—	2	2	no	—
434	#117	—	2	4	no	—
435	#117	C₂³×C₃₁	1	1	yes	characteristic
436	#117	C₂³×C₃₁	1	2	yes	—
437	C₂×C₄⋊C₄	—	2	2	no	—
438	C₂×C₄⋊C₄	—	2	2	no	—
439	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
440	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
441	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
442	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
443	#103	C₂³×C₃₁	1	8	yes	—
444	#102	C₂³×C₃₁	1	8	yes	—
445	#102	C₂³×C₃₁	1	8	yes	—
446	#102	C₂³×C₃₁	1	8	yes	—
447	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
448	#97	C₂³×C₃₁	1	2	yes	—
449	#97	C₂³×C₃₁	1	2	yes	—
450	#97	C₂³×C₃₁	1	2	yes	—
451	C₄×C₈	C₂³×C₃₁	1	2	yes	—
452	C₃₁	#60830	1	1	yes	characteristic, direct product factor, Sylow
453	C₂⁴	—	2	4	no	—
454	C₂⁴	—	2	2	no	—
455	#41	—	2	4	no	—
456	#41	—	2	8	no	—
457	#41	—	2	8	no	—
458	#41	—	4	8	no	—
459	#41	—	4	8	no	—
460	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
461	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
462	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
463	C₂×D₄	—	2	2	no	—
464	C₂×D₄	—	2	4	no	—
465	C₂×D₄	—	2	2	no	—
466	C₂×D₄	—	2	4	no	—
467	C₂×D₄	—	2	4	no	—
468	C₂×D₄	—	2	2	no	—
469	C₂×D₄	—	2	4	no	—
470	C₂×D₄	—	2	2	no	—
471	C₂×D₄	—	2	2	no	—
472	C₂×D₄	—	2	8	no	—
473	C₂×D₄	—	2	4	no	—
474	C₂×D₄	—	2	8	no	—
475	C₂×D₄	—	4	8	no	—
476	C₂×D₄	—	4	16	no	—
477	C₂×D₄	—	4	8	no	—
478	C₂×D₄	—	8	8	no	—
479	C₂×D₄	—	8	4	no	—
480	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
481	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
482	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
483	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
484	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
485	C₂²×C₄	—	2	2	no	—
486	C₂²×C₄	—	2	2	no	—
487	C₂²×C₄	—	2	4	no	—
488	C₂²×C₄	—	2	2	no	—
489	C₂²×C₄	—	2	4	no	—
490	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
491	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
492	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
493	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
494	Q₁₆	—	4	4	no	—
495	SD₁₆	—	4	4	no	—
496	SD₁₆	—	4	4	no	—
497	SD₁₆	—	4	8	no	—
498	D₈	—	4	4	no	—
499	D₈	—	4	8	no	—
500	M₁₆	—	2	4	no	—
501	C₂×C₈	—	2	4	no	—
502	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
503	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
504	C₄⋊C₄	—	2	2	no	—
505	C₄⋊C₄	—	2	4	no	—
506	C₄⋊C₄	—	2	4	no	—
507	C₄⋊C₄	—	2	4	no	—
508	C₄⋊C₄	—	2	4	no	—
509	C₄⋊C₄	—	2	2	no	—
510	C₄⋊C₄	—	2	2	no	—
511	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
512	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
513	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
514	C₂²⋊C₄	—	2	4	no	—
515	C₂²⋊C₄	—	2	2	no	—
516	C₂²⋊C₄	—	2	4	no	—
517	C₂²⋊C₄	—	2	4	no	—
518	C₂²⋊C₄	—	2	2	no	—
519	C₂²⋊C₄	—	2	8	no	—
520	C₂²⋊C₄	—	2	4	no	—
521	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
522	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
523	C₄²	—	2	4	no	—
524	C₄²	C₂×D₄×C₃₁	1	4	yes	—
525	C₄²	C₂×D₄×C₃₁	1	2	yes	—
526	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
527	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
528	C₄²	C₂⁴×C₃₁	1	2	yes	—
529	C₂³	—	2	2	no	—
530	C₂³	—	2	4	no	—
531	C₂³	—	2	2	no	—
532	C₂³	—	4	8	no	—
533	C₂³	—	4	4	no	—
534	C₂³	—	8	4	no	—
535	C₂³	—	8	8	no	—
536	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
537	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
538	Q₈	—	2	2	no	—
539	Q₈	—	2	4	no	—
540	Q₈	—	2	4	no	—
541	D₄	—	2	4	no	—
542	D₄	—	2	2	no	—
543	D₄	—	2	4	no	—
544	D₄	—	2	8	no	—
545	D₄	—	2	4	no	—
546	D₄	—	4	8	no	—
547	D₄	—	4	8	no	—
548	D₄	—	4	4	no	—
549	D₄	—	4	8	no	—
550	D₄	—	4	4	no	—
551	D₄	—	4	4	no	—
552	D₄	—	4	4	no	—
553	D₄	—	8	2	no	—
554	D₄	—	8	8	no	—
555	D₄	—	8	4	no	—
556	D₄	—	8	4	no	—
557	C₂×C₄	—	2	4	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	4	no	—
560	C₂×C₄	—	2	2	no	—
561	C₂×C₄	—	4	4	no	—
562	C₂×C₄	—	4	2	no	—
563	C₂×C₄	—	4	8	no	—
564	C₂×C₄	—	4	4	no	—
565	C₂×C₄	—	4	4	no	—
566	C₂×C₄	—	8	4	no	—
567	C₂×C₄	—	8	2	no	—
568	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
569	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
570	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
571	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
572	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
573	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
574	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
575	C₈	—	2	4	no	—
576	C₈	—	2	4	no	—
577	C₂²	—	2	2	no	—
578	C₂²	—	2	2	no	—
579	C₂²	—	2	4	no	—
580	C₂²	—	4	4	no	—
581	C₂²	—	4	2	no	—
582	C₂²	—	4	2	no	—
583	C₂²	—	4	2	no	—
584	C₂²	—	4	8	no	—
585	C₂²	—	4	4	no	—
586	C₂²	—	4	2	no	—
587	C₂²	—	16	4	no	—
588	C₂²	—	16	8	no	—
589	C₂²	C₃₁×#535	1	2	yes	—
590	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
591	C₄	—	2	2	no	—
592	C₄	—	2	2	no	—
593	C₄	—	2	4	no	—
594	C₄	—	2	2	no	—
595	C₄	—	4	2	no	—
596	C₄	—	4	2	no	—
597	C₄	—	4	4	no	—
598	C₄	C₃₁×#535	1	2	yes	—
599	C₄	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
600	C₂	—	2	2	no	—
601	C₂	—	4	2	no	—
602	C₂	—	8	4	no	—
603	C₂	—	8	2	no	—
604	C₂	—	8	2	no	—
605	C₂	C₂×C₃₁×#535	1	1	yes	characteristic
606	C₂	C₃₁×#3585	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 ◁ C₄×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3512	C₂	semidirect
2. C₃₁×#3498	C₂	semidirect
3. C₃₁×#3222	C₂	semidirect
4. C₃₁×#3220	C₂	semidirect
5. C₃₁×#3219	C₂	semidirect
6. C₃₁×#3217	C₂	semidirect
7. C₃₁×#3151	C₂	semidirect
8. C₃₁×#3150	C₂	semidirect
9. C₃₁×#3142	C₂	semidirect
10. C₂×C₃₁×#490	C₂	semidirect
11. C₃₁×#3045	C₂	semidirect
12. C₃₁×#3012	C₂	semidirect
13. C₃₁×#2923	C₂	semidirect
15. C₂×C₃₁×AGL(1,ℤ/8)	C₂²	semidirect
16. C₂×C₃₁×#135	C₂²	semidirect
23. C₃₁×#541	C₂²	semidirect
24. C₃₁×#540	C₂²	semidirect
30. C₃₁×#532	C₂²	semidirect
33. C₃₁×#529	C₂²	semidirect
34. C₂×C₃₁×#125	C₂²	semidirect
35. C₃₁×#520	C₂²	semidirect
37. C₃₁×#497	C₂²	semidirect
38. C₃₁×#497	C₂²	semidirect
39. C₃₁×#496	C₂²	semidirect
40. C₃₁×#496	C₂²	semidirect
41. C₃₁×#496	C₂²	semidirect
42. C₃₁×#493	C₂²	semidirect
43. C₃₁×#492	C₂²	semidirect
44. C₃₁×#491	C₂²	semidirect
45. C₃₁×#490	C₂²	semidirect
46. C₃₁×#490	C₂²	semidirect
47. C₃₁×#490	C₂²	semidirect
48. C₃₁×#488	C₂²	semidirect
49. C₃₁×#464	C₂²	semidirect
50. C₃₁×#463	C₂²	semidirect
51. C₃₁×#460	C₂²	semidirect
52. C₃₁×#459	C₂²	semidirect
53. C₃₁×#452	C₂²	semidirect
54. C₃₁×#450	C₂²	semidirect
55. C₃₁×#449	C₂²	semidirect
56. C₃₁×#447	C₂²	semidirect
57. C₃₁×#433	C₂²	semidirect
58. C₃₁×#432	C₂²	semidirect
59. C₃₁×#424	C₂²	semidirect
60. C₃₁×#417	C₂²	semidirect
61. C₂×C₃₁×#102	C₂²	semidirect
63. C₄×M₁₆×C₃₁	C₂²	semidirect
64. C₂×C₃₁×#97	C₂²	semidirect
225. #60830	C₃₁	direct (N × H)
274. #3512	C₂×C₃₁	semidirect
275. #3498	C₂×C₃₁	semidirect
276. #3222	C₂×C₃₁	semidirect
277. #3220	C₂×C₃₁	semidirect
278. #3219	C₂×C₃₁	semidirect
279. #3217	C₂×C₃₁	semidirect
280. #3151	C₂×C₃₁	semidirect
281. #3150	C₂×C₃₁	semidirect
282. #3142	C₂×C₃₁	semidirect
283. C₂×#490	C₂×C₃₁	semidirect
284. #3045	C₂×C₃₁	semidirect
285. #3012	C₂×C₃₁	semidirect
286. #2923	C₂×C₃₁	semidirect
311. C₂×AGL(1,ℤ/8)	C₂²×C₃₁	semidirect
312. C₂×#135	C₂²×C₃₁	semidirect
319. #541	C₂²×C₃₁	semidirect
320. #540	C₂²×C₃₁	semidirect
326. #532	C₂²×C₃₁	semidirect
329. #529	C₂²×C₃₁	semidirect
330. C₂×#125	C₂²×C₃₁	semidirect
331. #520	C₂²×C₃₁	semidirect
333. #497	C₂²×C₃₁	semidirect
334. #497	C₂²×C₃₁	semidirect
335. #496	C₂²×C₃₁	semidirect
336. #496	C₂²×C₃₁	semidirect
337. #496	C₂²×C₃₁	semidirect
338. #493	C₂²×C₃₁	semidirect
339. #492	C₂²×C₃₁	semidirect
340. #491	C₂²×C₃₁	semidirect
341. #490	C₂²×C₃₁	semidirect
342. #490	C₂²×C₃₁	semidirect
343. #490	C₂²×C₃₁	semidirect
344. #488	C₂²×C₃₁	semidirect
345. #464	C₂²×C₃₁	semidirect
346. #463	C₂²×C₃₁	semidirect
347. #460	C₂²×C₃₁	semidirect
348. #459	C₂²×C₃₁	semidirect
349. #452	C₂²×C₃₁	semidirect
350. #450	C₂²×C₃₁	semidirect
351. #449	C₂²×C₃₁	semidirect
352. #447	C₂²×C₃₁	semidirect
353. #433	C₂²×C₃₁	semidirect
354. #432	C₂²×C₃₁	semidirect
355. #424	C₂²×C₃₁	semidirect
356. #417	C₂²×C₃₁	semidirect
357. C₂×#102	C₂²×C₃₁	semidirect
359. C₄×M₁₆	C₂²×C₃₁	semidirect
360. C₂×#97	C₂²×C₃₁	semidirect
452. C₃₁	#60830	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×#3512	2
C₃₁×#3498	2
C₃₁×#3222	2
C₃₁×#3220	2
C₃₁×#3219	2
C₃₁×#3217	2
C₃₁×#3151	2
C₃₁×#3150	2
C₃₁×#3142	2
C₂×C₃₁×#490	2
C₃₁×#3045	2
C₃₁×#3012	2
C₃₁×#2923	2
#60830	31

Subgroups — C31×#60830 #21517589

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60830 #21517589