Subgroups — C₃₁×#61086 #21517845

Order 7936.

2666 conjugacy classes of subgroups.

The subgroup lattice has 604 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	2	yes	maximal, complemented
2	C₃₁×#3497	C₂	1	2	yes	maximal, complemented
3	C₃₁×#3287	C₂	1	2	yes	maximal, complemented
4	D₄×Q₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3285	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3278	C₂	1	4	yes	maximal, complemented
7	C₃₁×#3277	C₂	1	4	yes	maximal, complemented
8	C₃₁×#3147	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#3061	C₂	1	2	yes	maximal, complemented
10	C₃₁×#3037	C₂	1	2	yes	maximal, complemented
11	C₃₁×#3013	C₂	1	4	yes	maximal, complemented
12	C₂×C₄×Q₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#2931	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₃₁×2₋¹⁺⁴	C₂²	1	4	yes	—
15	C₃₁×#578	—	2	16	no	—
16	C₂×C₃₁×#137	C₂²	1	4	yes	—
17	C₂×C₃₁×#137	C₂²	1	8	yes	—
18	C₂×C₃₁×#135	C₂²	1	2	yes	—
19	C₂×C₃₁×#135	C₂²	1	4	yes	—
20	C₂×C₃₁×#135	C₂²	1	2	yes	—
21	C₂²×Q₁₆×C₃₁	C₂²	1	4	yes	—
22	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#131	C₂²	1	2	yes	—
24	C₃₁×#550	—	2	2	no	—
25	C₃₁×#550	C₂²	1	4	yes	—
26	D₄×Q₈×C₃₁	—	2	2	no	—
27	D₄×Q₈×C₃₁	C₂²	1	4	yes	—
28	C₃₁×#548	—	2	4	no	—
29	C₃₁×#548	C₂²	1	8	yes	—
30	C₃₁×#537	—	2	4	no	—
31	C₃₁×#537	C₂²	1	4	yes	—
32	C₃₁×#532	—	2	4	no	—
33	C₃₁×#532	C₂²	1	2	yes	—
34	C₃₁×#532	C₂²	1	1	yes	characteristic
35	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
36	C₃₁×#495	C₂²	1	2	yes	—
37	C₃₁×#495	C₂²	1	2	yes	—
38	C₃₁×#494	C₂²	1	1	yes	characteristic
39	C₃₁×#493	C₂²	1	1	yes	characteristic
40	C₃₁×#492	C₂²	1	2	yes	—
41	C₃₁×#468	C₂²	1	8	yes	—
42	C₃₁×#467	C₂²	1	4	yes	—
43	C₃₁×#466	C₂²	1	4	yes	—
44	C₃₁×#464	C₂²	1	8	yes	—
45	C₃₁×#462	C₂²	1	4	yes	—
46	C₃₁×#460	C₂²	1	4	yes	—
47	C₃₁×#452	C₂²	1	8	yes	—
48	C₃₁×#451	C₂²	1	8	yes	—
49	C₃₁×#449	C₂²	1	8	yes	—
50	C₃₁×#448	C₂²	1	8	yes	—
51	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
52	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
53	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
54	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
55	C₃₁×#436	C₂²	1	2	yes	—
56	C₃₁×#436	C₂²	1	2	yes	—
57	C₈×D₄×C₃₁	C₂²	1	2	yes	—
58	C₈×D₄×C₃₁	C₂²	1	2	yes	—
59	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
60	C₃₁×#423	C₂²	1	1	yes	characteristic
61	C₃₁×#417	C₂²	1	4	yes	—
62	C₂×C₃₁×#103	C₂²	1	2	yes	—
63	C₃₁×#408	C₂²	1	2	yes	—
64	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
65	C₃₁×2₋¹⁺⁴	—	2	16	no	—
66	C₃₁×2₋¹⁺⁴	—	4	8	no	—
67	C₂×#41×C₃₁	—	2	8	no	—
68	C₂×#41×C₃₁	—	2	4	no	—
69	C₂×#41×C₃₁	C₂³	1	4	yes	—
70	C₂×#41×C₃₁	C₂³	1	8	yes	—
71	C₂×#41×C₃₁	C₂³	1	2	yes	—
72	C₂²×Q₈×C₃₁	—	2	4	no	—
73	C₂²×Q₈×C₃₁	C₂³	1	8	yes	—
74	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
75	C₃₁×#137	—	2	16	no	—
76	C₃₁×#137	—	2	32	no	—
77	C₃₁×#135	—	2	16	no	—
78	C₃₁×#135	—	2	8	no	—
79	C₃₁×#135	—	4	4	no	—
80	C₂×Q₁₆×C₃₁	—	2	16	no	—
81	C₂×Q₁₆×C₃₁	D₄	1	8	yes	complemented
82	C₂×Q₁₆×C₃₁	C₂³	1	8	yes	—
83	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
84	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
85	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
86	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
87	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
88	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
89	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
90	C₂×D₈×C₃₁	C₂³	1	2	yes	—
91	C₂×D₈×C₃₁	C₂³	1	2	yes	—
92	C₃₁×#131	—	2	8	no	—
93	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
94	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
95	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
96	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
97	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
98	C₃₁×C₄⋊Q₈	—	2	4	no	—
99	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
100	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
101	C₃₁×#127	—	2	4	no	—
102	C₃₁×#127	D₄	1	1	yes	characteristic
103	C₃₁×#127	C₂³	1	2	yes	—
104	C₃₁×#124	—	2	8	no	—
105	C₃₁×#124	D₄	1	2	yes	—
106	C₃₁×#124	C₂³	1	4	yes	—
107	C₃₁×C₂²⋊Q₈	—	2	4	no	—
108	C₃₁×C₂²⋊Q₈	—	2	8	no	—
109	C₃₁×C₂²⋊Q₈	—	2	8	no	—
110	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	—
111	C₃₁×#121	—	2	8	no	—
112	C₃₁×#121	—	2	8	no	—
113	C₃₁×#121	—	2	4	no	—
114	C₃₁×#121	—	2	2	no	—
115	C₃₁×#121	—	2	2	no	—
116	C₃₁×#121	—	2	8	no	—
117	C₃₁×#121	D₄	1	2	yes	—
118	C₃₁×#121	D₄	1	2	yes	—
119	C₃₁×#121	C₂³	1	8	yes	—
120	C₄×Q₈×C₃₁	—	2	2	no	—
121	C₄×Q₈×C₃₁	—	2	2	no	—
122	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
123	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
124	C₄×D₄×C₃₁	—	2	4	no	—
125	C₄×D₄×C₃₁	—	2	4	no	—
126	C₄×D₄×C₃₁	—	2	8	no	—
127	C₄×D₄×C₃₁	—	2	2	no	—
128	C₄×D₄×C₃₁	—	2	2	no	—
129	C₄×D₄×C₃₁	C₂³	1	2	yes	—
130	C₄×D₄×C₃₁	C₂³	1	2	yes	—
131	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
132	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
133	C₂×C₄²×C₃₁	—	2	2	no	—
134	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
135	C₃₁×#107	C₂³	1	2	yes	—
136	C₃₁×#107	C₂³	1	2	yes	—
137	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
138	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
139	C₃₁×#103	C₂³	1	4	yes	—
140	C₃₁×#103	C₂³	1	4	yes	—
141	C₃₁×#103	C₂³	1	8	yes	—
142	C₃₁×#102	C₂³	1	8	yes	—
143	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
144	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
145	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
146	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
147	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
148	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
149	#41×C₃₁	—	2	4	no	—
150	#41×C₃₁	—	2	4	no	—
151	#41×C₃₁	—	2	16	no	—
152	#41×C₃₁	—	2	8	no	—
153	#41×C₃₁	—	2	8	no	—
154	#41×C₃₁	—	4	8	no	—
155	#41×C₃₁	—	4	32	no	—
156	#41×C₃₁	—	4	16	no	—
157	C₂×Q₈×C₃₁	—	2	2	no	—
158	C₂×Q₈×C₃₁	—	2	8	no	—
159	C₂×Q₈×C₃₁	—	2	4	no	—
160	C₂×Q₈×C₃₁	—	2	16	no	—
161	C₂×Q₈×C₃₁	—	4	8	no	—
162	C₂×Q₈×C₃₁	—	4	16	no	—
163	C₂×Q₈×C₃₁	C₂×D₄	1	8	yes	—
164	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
165	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
166	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
167	C₂×D₄×C₃₁	—	2	4	no	—
168	C₂×D₄×C₃₁	—	2	8	no	—
169	C₂×D₄×C₃₁	—	2	8	no	—
170	C₂×D₄×C₃₁	—	2	2	no	—
171	C₂×D₄×C₃₁	—	2	2	no	—
172	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
173	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
174	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
175	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
176	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
177	C₂²×C₄×C₃₁	—	2	2	no	—
178	C₂²×C₄×C₃₁	—	2	8	no	—
179	C₂²×C₄×C₃₁	—	2	2	no	—
180	C₂²×C₄×C₃₁	—	2	8	no	—
181	C₂²×C₄×C₃₁	—	2	4	no	—
182	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
183	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
184	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
185	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
186	Q₁₆×C₃₁	—	2	16	no	—
187	Q₁₆×C₃₁	—	2	4	no	—
188	Q₁₆×C₃₁	—	4	2	no	—
189	Q₁₆×C₃₁	C₂×D₄	1	8	yes	—
190	Q₁₆×C₃₁	C₂×D₄	1	8	yes	—
191	SD₁₆×C₃₁	—	2	16	no	—
192	SD₁₆×C₃₁	—	2	8	no	—
193	SD₁₆×C₃₁	—	4	4	no	—
194	D₈×C₃₁	—	2	4	no	—
195	D₈×C₃₁	—	4	2	no	—
196	M₁₆×C₃₁	—	2	4	no	—
197	M₁₆×C₃₁	—	2	8	no	—
198	C₂×C₈×C₃₁	—	2	2	no	—
199	C₂×C₈×C₃₁	—	2	8	no	—
200	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
201	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
202	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
203	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
204	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
205	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
206	C₄⋊C₄×C₃₁	—	2	4	no	—
207	C₄⋊C₄×C₃₁	—	2	4	no	—
208	C₄⋊C₄×C₃₁	—	2	2	no	—
209	C₄⋊C₄×C₃₁	—	2	2	no	—
210	C₄⋊C₄×C₃₁	—	2	2	no	—
211	C₄⋊C₄×C₃₁	—	2	8	no	—
212	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
213	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
214	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
215	C₂²⋊C₄×C₃₁	—	2	8	no	—
216	C₂²⋊C₄×C₃₁	—	2	4	no	—
217	C₂²⋊C₄×C₃₁	—	2	4	no	—
218	C₂²⋊C₄×C₃₁	—	2	8	no	—
219	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
220	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
221	C₄²×C₃₁	—	2	4	no	—
222	C₄²×C₃₁	—	2	4	no	—
223	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
224	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
225	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
226	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
227	#61086	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
228	C₂³×C₃₁	—	2	4	no	—
229	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
230	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
231	Q₈×C₃₁	—	2	8	no	—
232	Q₈×C₃₁	—	2	4	no	—
233	Q₈×C₃₁	—	2	4	no	—
234	Q₈×C₃₁	—	2	4	no	—
235	Q₈×C₃₁	—	2	4	no	—
236	Q₈×C₃₁	—	4	16	no	—
237	Q₈×C₃₁	C₂²×D₄	1	8	yes	—
238	D₄×C₃₁	—	2	8	no	—
239	D₄×C₃₁	—	2	4	no	—
240	D₄×C₃₁	—	2	4	no	—
241	D₄×C₃₁	—	2	4	no	—
242	D₄×C₃₁	—	2	4	no	—
243	D₄×C₃₁	—	4	2	no	—
244	D₄×C₃₁	—	4	16	no	—
245	D₄×C₃₁	—	4	8	no	—
246	D₄×C₃₁	—	4	4	no	—
247	D₄×C₃₁	—	8	2	no	—
248	D₄×C₃₁	—	8	8	no	—
249	C₂×C₄×C₃₁	—	2	8	no	—
250	C₂×C₄×C₃₁	—	2	4	no	—
251	C₂×C₄×C₃₁	—	2	2	no	—
252	C₂×C₄×C₃₁	—	2	4	no	—
253	C₂×C₄×C₃₁	—	2	4	no	—
254	C₂×C₄×C₃₁	—	2	4	no	—
255	C₂×C₄×C₃₁	—	2	4	no	—
256	C₂×C₄×C₃₁	—	4	1	no	—
257	C₂×C₄×C₃₁	—	4	16	no	—
258	C₂×C₄×C₃₁	—	4	16	no	—
259	C₂×C₄×C₃₁	—	4	8	no	—
260	C₂×C₄×C₃₁	—	8	2	no	—
261	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
262	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
263	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
264	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
265	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
266	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
267	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
268	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
269	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
270	C₈×C₃₁	—	2	4	no	—
271	C₈×C₃₁	—	2	1	no	—
272	C₈×C₃₁	—	2	1	no	—
273	C₈×C₃₁	C₂²×D₄	1	2	yes	—
274	C₈×C₃₁	C₂²×D₄	1	2	yes	—
275	C₂×#578	C₂×C₃₁	1	2	yes	complemented
276	#3497	C₂×C₃₁	1	2	yes	complemented
277	#3287	C₂×C₃₁	1	2	yes	complemented
278	D₄×Q₁₆	C₂×C₃₁	1	2	yes	complemented
279	#3285	C₂×C₃₁	1	4	yes	complemented
280	#3278	C₂×C₃₁	1	4	yes	complemented
281	#3277	C₂×C₃₁	1	4	yes	complemented
282	#3147	C₂×C₃₁	1	1	yes	characteristic, complemented
283	#3061	C₂×C₃₁	1	2	yes	complemented
284	#3037	C₂×C₃₁	1	2	yes	complemented
285	#3013	C₂×C₃₁	1	4	yes	complemented
286	C₂×C₄×Q₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
287	#2931	C₂×C₃₁	1	1	yes	characteristic, complemented
288	C₂²×C₃₁	—	2	1	no	—
289	C₂²×C₃₁	—	2	1	no	—
290	C₂²×C₃₁	—	2	4	no	—
291	C₂²×C₃₁	—	2	4	no	—
292	C₂²×C₃₁	—	2	4	no	—
293	C₂²×C₃₁	—	4	4	no	—
294	C₂²×C₃₁	—	4	4	no	—
295	C₂²×C₃₁	—	4	4	no	—
296	C₂²×C₃₁	D₄²	1	2	yes	—
297	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
298	C₄×C₃₁	—	2	1	no	—
299	C₄×C₃₁	—	2	1	no	—
300	C₄×C₃₁	—	2	8	no	—
301	C₄×C₃₁	—	2	4	no	—
302	C₄×C₃₁	—	2	1	no	—
303	C₄×C₃₁	—	4	4	no	—
304	C₄×C₃₁	—	4	4	no	—
305	C₄×C₃₁	D₄²	1	2	yes	—
306	C₄×C₃₁	#578	1	2	yes	complemented
307	C₄×C₃₁	C₂³×D₄	1	1	yes	characteristic
308	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
309	C₂×2₋¹⁺⁴	C₂²×C₃₁	1	4	yes	—
310	#578	—	2	16	no	—
311	C₂×#137	C₂²×C₃₁	1	4	yes	—
312	C₂×#137	C₂²×C₃₁	1	8	yes	—
313	C₂×#135	C₂²×C₃₁	1	2	yes	—
314	C₂×#135	C₂²×C₃₁	1	4	yes	—
315	C₂×#135	C₂²×C₃₁	1	2	yes	—
316	C₂²×Q₁₆	C₂²×C₃₁	1	4	yes	—
317	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
318	C₂×#131	C₂²×C₃₁	1	2	yes	—
319	#550	—	2	2	no	—
320	#550	C₂²×C₃₁	1	4	yes	—
321	D₄×Q₈	—	2	2	no	—
322	D₄×Q₈	C₂²×C₃₁	1	4	yes	—
323	#548	—	2	4	no	—
324	#548	C₂²×C₃₁	1	8	yes	—
325	#537	—	2	4	no	—
326	#537	C₂²×C₃₁	1	4	yes	—
327	#532	—	2	4	no	—
328	#532	C₂²×C₃₁	1	2	yes	—
329	#532	C₂²×C₃₁	1	1	yes	characteristic
330	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
331	#495	C₂²×C₃₁	1	2	yes	—
332	#495	C₂²×C₃₁	1	2	yes	—
333	#494	C₂²×C₃₁	1	1	yes	characteristic
334	#493	C₂²×C₃₁	1	1	yes	characteristic
335	#492	C₂²×C₃₁	1	2	yes	—
336	#468	C₂²×C₃₁	1	8	yes	—
337	#467	C₂²×C₃₁	1	4	yes	—
338	#466	C₂²×C₃₁	1	4	yes	—
339	#464	C₂²×C₃₁	1	8	yes	—
340	#462	C₂²×C₃₁	1	4	yes	—
341	#460	C₂²×C₃₁	1	4	yes	—
342	#452	C₂²×C₃₁	1	8	yes	—
343	#451	C₂²×C₃₁	1	8	yes	—
344	#449	C₂²×C₃₁	1	8	yes	—
345	#448	C₂²×C₃₁	1	8	yes	—
346	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
347	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
348	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
349	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
350	#436	C₂²×C₃₁	1	2	yes	—
351	#436	C₂²×C₃₁	1	2	yes	—
352	C₈×D₄	C₂²×C₃₁	1	2	yes	—
353	C₈×D₄	C₂²×C₃₁	1	2	yes	—
354	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
355	#423	C₂²×C₃₁	1	1	yes	characteristic
356	#417	C₂²×C₃₁	1	4	yes	—
357	C₂×#103	C₂²×C₃₁	1	2	yes	—
358	#408	C₂²×C₃₁	1	2	yes	—
359	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
360	C₂×C₃₁	—	2	2	no	—
361	C₂×C₃₁	—	4	4	no	—
362	C₂×C₃₁	—	8	4	no	—
363	C₂×C₃₁	C₂×D₄²	1	1	yes	characteristic
364	C₂×C₃₁	C₂×#578	1	1	yes	characteristic
365	C₂×C₃₁	#3584	1	1	yes	characteristic
366	2₋¹⁺⁴	—	2	16	no	—
367	2₋¹⁺⁴	—	4	8	no	—
368	C₂×#41	—	2	8	no	—
369	C₂×#41	—	2	4	no	—
370	C₂×#41	C₂³×C₃₁	1	4	yes	—
371	C₂×#41	C₂³×C₃₁	1	8	yes	—
372	C₂×#41	C₂³×C₃₁	1	2	yes	—
373	C₂²×Q₈	—	2	4	no	—
374	C₂²×Q₈	C₂³×C₃₁	1	8	yes	—
375	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
376	#137	—	2	16	no	—
377	#137	—	2	32	no	—
378	#135	—	2	16	no	—
379	#135	—	2	8	no	—
380	#135	—	4	4	no	—
381	C₂×Q₁₆	—	2	16	no	—
382	C₂×Q₁₆	D₄×C₃₁	1	8	yes	complemented
383	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
384	C₂×Q₁₆	C₂³×C₃₁	1	8	yes	—
385	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
386	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
387	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
388	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
389	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
390	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
391	C₂×D₈	C₂³×C₃₁	1	2	yes	—
392	C₂×D₈	C₂³×C₃₁	1	2	yes	—
393	#131	—	2	8	no	—
394	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
395	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
396	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
397	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
398	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
399	C₄⋊Q₈	—	2	4	no	—
400	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
401	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
402	#127	—	2	4	no	—
403	#127	D₄×C₃₁	1	1	yes	characteristic
404	#127	C₂³×C₃₁	1	2	yes	—
405	#124	—	2	8	no	—
406	#124	D₄×C₃₁	1	2	yes	—
407	#124	C₂³×C₃₁	1	4	yes	—
408	C₂²⋊Q₈	—	2	4	no	—
409	C₂²⋊Q₈	—	2	8	no	—
410	C₂²⋊Q₈	—	2	8	no	—
411	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
412	#121	—	2	8	no	—
413	#121	—	2	8	no	—
414	#121	—	2	4	no	—
415	#121	—	2	2	no	—
416	#121	—	2	2	no	—
417	#121	—	2	8	no	—
418	#121	D₄×C₃₁	1	2	yes	—
419	#121	D₄×C₃₁	1	2	yes	—
420	#121	C₂³×C₃₁	1	8	yes	—
421	C₄×Q₈	—	2	2	no	—
422	C₄×Q₈	—	2	2	no	—
423	C₄×Q₈	C₂³×C₃₁	1	4	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	8	no	—
428	C₄×D₄	—	2	2	no	—
429	C₄×D₄	—	2	2	no	—
430	C₄×D₄	C₂³×C₃₁	1	2	yes	—
431	C₄×D₄	C₂³×C₃₁	1	2	yes	—
432	C₂×C₄⋊C₄	—	2	2	no	—
433	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
434	C₂×C₄²	—	2	2	no	—
435	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
436	#107	C₂³×C₃₁	1	2	yes	—
437	#107	C₂³×C₃₁	1	2	yes	—
438	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
439	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
440	#103	C₂³×C₃₁	1	4	yes	—
441	#103	C₂³×C₃₁	1	4	yes	—
442	#103	C₂³×C₃₁	1	8	yes	—
443	#102	C₂³×C₃₁	1	8	yes	—
444	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
445	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
446	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
447	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
448	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
449	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
450	C₃₁	#61086	1	1	yes	characteristic, direct product factor, Sylow
451	#41	—	2	4	no	—
452	#41	—	2	4	no	—
453	#41	—	2	16	no	—
454	#41	—	2	8	no	—
455	#41	—	2	8	no	—
456	#41	—	4	8	no	—
457	#41	—	4	32	no	—
458	#41	—	4	16	no	—
459	C₂×Q₈	—	2	2	no	—
460	C₂×Q₈	—	2	8	no	—
461	C₂×Q₈	—	2	4	no	—
462	C₂×Q₈	—	2	16	no	—
463	C₂×Q₈	—	4	8	no	—
464	C₂×Q₈	—	4	16	no	—
465	C₂×Q₈	C₂×D₄×C₃₁	1	8	yes	—
466	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
467	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
468	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
469	C₂×D₄	—	2	4	no	—
470	C₂×D₄	—	2	8	no	—
471	C₂×D₄	—	2	8	no	—
472	C₂×D₄	—	2	2	no	—
473	C₂×D₄	—	2	2	no	—
474	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
475	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
476	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
477	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
478	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
479	C₂²×C₄	—	2	2	no	—
480	C₂²×C₄	—	2	8	no	—
481	C₂²×C₄	—	2	2	no	—
482	C₂²×C₄	—	2	8	no	—
483	C₂²×C₄	—	2	4	no	—
484	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
485	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
486	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
487	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
488	Q₁₆	—	2	16	no	—
489	Q₁₆	—	2	4	no	—
490	Q₁₆	—	4	2	no	—
491	Q₁₆	C₂×D₄×C₃₁	1	8	yes	—
492	Q₁₆	C₂×D₄×C₃₁	1	8	yes	—
493	SD₁₆	—	2	16	no	—
494	SD₁₆	—	2	8	no	—
495	SD₁₆	—	4	4	no	—
496	D₈	—	2	4	no	—
497	D₈	—	4	2	no	—
498	M₁₆	—	2	4	no	—
499	M₁₆	—	2	8	no	—
500	C₂×C₈	—	2	2	no	—
501	C₂×C₈	—	2	8	no	—
502	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
503	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
504	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
505	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
506	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
507	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
508	C₄⋊C₄	—	2	4	no	—
509	C₄⋊C₄	—	2	4	no	—
510	C₄⋊C₄	—	2	2	no	—
511	C₄⋊C₄	—	2	2	no	—
512	C₄⋊C₄	—	2	2	no	—
513	C₄⋊C₄	—	2	8	no	—
514	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
515	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
516	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
517	C₂²⋊C₄	—	2	8	no	—
518	C₂²⋊C₄	—	2	4	no	—
519	C₂²⋊C₄	—	2	4	no	—
520	C₂²⋊C₄	—	2	8	no	—
521	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
522	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
523	C₄²	—	2	4	no	—
524	C₄²	—	2	4	no	—
525	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
526	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
527	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
528	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
529	C₂³	—	2	4	no	—
530	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
531	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
532	Q₈	—	2	8	no	—
533	Q₈	—	2	4	no	—
534	Q₈	—	2	4	no	—
535	Q₈	—	2	4	no	—
536	Q₈	—	2	4	no	—
537	Q₈	—	4	16	no	—
538	Q₈	C₂²×D₄×C₃₁	1	8	yes	—
539	D₄	—	2	8	no	—
540	D₄	—	2	4	no	—
541	D₄	—	2	4	no	—
542	D₄	—	2	4	no	—
543	D₄	—	2	4	no	—
544	D₄	—	4	2	no	—
545	D₄	—	4	16	no	—
546	D₄	—	4	8	no	—
547	D₄	—	4	4	no	—
548	D₄	—	8	2	no	—
549	D₄	—	8	8	no	—
550	C₂×C₄	—	2	8	no	—
551	C₂×C₄	—	2	4	no	—
552	C₂×C₄	—	2	2	no	—
553	C₂×C₄	—	2	4	no	—
554	C₂×C₄	—	2	4	no	—
555	C₂×C₄	—	2	4	no	—
556	C₂×C₄	—	2	4	no	—
557	C₂×C₄	—	4	1	no	—
558	C₂×C₄	—	4	16	no	—
559	C₂×C₄	—	4	16	no	—
560	C₂×C₄	—	4	8	no	—
561	C₂×C₄	—	8	2	no	—
562	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
563	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
564	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
565	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
566	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
567	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
568	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
569	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
570	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
571	C₈	—	2	4	no	—
572	C₈	—	2	1	no	—
573	C₈	—	2	1	no	—
574	C₈	C₂²×D₄×C₃₁	1	2	yes	—
575	C₈	C₂²×D₄×C₃₁	1	2	yes	—
576	C₂²	—	2	1	no	—
577	C₂²	—	2	1	no	—
578	C₂²	—	2	4	no	—
579	C₂²	—	2	4	no	—
580	C₂²	—	2	4	no	—
581	C₂²	—	4	4	no	—
582	C₂²	—	4	4	no	—
583	C₂²	—	4	4	no	—
584	C₂²	D₄²×C₃₁	1	2	yes	—
585	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
586	C₄	—	2	8	no	—
587	C₄	—	2	4	no	—
588	C₄	—	2	1	no	—
589	C₄	—	2	1	no	—
590	C₄	—	2	1	no	—
591	C₄	—	4	4	no	—
592	C₄	—	4	4	no	—
593	C₄	D₄²×C₃₁	1	2	yes	—
594	C₄	C₃₁×#578	1	2	yes	complemented
595	C₄	C₂³×D₄×C₃₁	1	1	yes	characteristic
596	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
597	C₂	—	2	2	no	—
598	C₂	—	4	4	no	—
599	C₂	—	8	4	no	—
600	C₂	C₂×D₄²×C₃₁	1	1	yes	characteristic
601	C₂	C₂×C₃₁×#578	1	1	yes	characteristic
602	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 ◁ C₃₁×#532 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#578	C₂	semidirect
2. C₃₁×#3497	C₂	semidirect
3. C₃₁×#3287	C₂	semidirect
4. D₄×Q₁₆×C₃₁	C₂	semidirect
5. C₃₁×#3285	C₂	semidirect
6. C₃₁×#3278	C₂	semidirect
7. C₃₁×#3277	C₂	semidirect
8. C₃₁×#3147	C₂	semidirect
9. C₃₁×#3061	C₂	semidirect
10. C₃₁×#3037	C₂	semidirect
11. C₃₁×#3013	C₂	semidirect
12. C₂×C₄×Q₁₆×C₃₁	C₂	semidirect
13. C₃₁×#2931	C₂	semidirect
81. C₂×Q₁₆×C₃₁	D₄	semidirect
227. #61086	C₃₁	direct (N × H)
275. C₂×#578	C₂×C₃₁	semidirect
276. #3497	C₂×C₃₁	semidirect
277. #3287	C₂×C₃₁	semidirect
278. D₄×Q₁₆	C₂×C₃₁	semidirect
279. #3285	C₂×C₃₁	semidirect
280. #3278	C₂×C₃₁	semidirect
281. #3277	C₂×C₃₁	semidirect
282. #3147	C₂×C₃₁	semidirect
283. #3061	C₂×C₃₁	semidirect
284. #3037	C₂×C₃₁	semidirect
285. #3013	C₂×C₃₁	semidirect
286. C₂×C₄×Q₁₆	C₂×C₃₁	semidirect
287. #2931	C₂×C₃₁	semidirect
306. C₄×C₃₁	#578	semidirect
382. C₂×Q₁₆	D₄×C₃₁	semidirect
450. C₃₁	#61086	direct (N × H)
594. C₄	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
C₃₁×#3497	2
C₃₁×#3287	2
D₄×Q₁₆×C₃₁	2
C₃₁×#3285	2
C₃₁×#3278	2
C₃₁×#3277	2
C₃₁×#3147	2
C₃₁×#3061	2
C₃₁×#3037	2
C₃₁×#3013	2
C₂×C₄×Q₁₆×C₃₁	2
C₃₁×#2931	2
#61086	31

Subgroups — C31×#61086 #21517845

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61086 #21517845