Subgroups — C₃₁×#33188 #21489947

Order 7936.

1068 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#497	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#470	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#468	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#467	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₁×#465	C₂	1	1	yes	maximal, characteristic
6	C₂×C₃₁×#405	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#2621	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#2620	C₂	1	1	yes	maximal, characteristic
9	C₃₁×#1973	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1971	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1969	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#1968	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#1958	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#1956	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#1847	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
17	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
18	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
19	C₂²×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
20	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
21	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
29	C₃₁×#497	—	2	4	no	—
30	C₃₁×#470	—	2	4	no	—
31	C₃₁×#468	—	2	4	no	—
32	C₃₁×#467	—	2	4	no	—
33	C₃₁×#465	—	2	4	no	—
34	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
35	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
36	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
37	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
38	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
39	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
40	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
41	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
42	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
43	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
44	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
45	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
46	C₃₁×#405	—	2	4	no	—
47	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
48	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
49	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
50	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
51	C₃₁×#398	—	2	2	no	—
52	C₃₁×#397	—	2	2	no	—
53	C₃₁×#396	—	2	2	no	—
54	C₃₁×#394	—	2	2	no	—
55	C₃₁×#391	C₂²	1	1	yes	characteristic
56	C₃₁×#391	C₂²	1	1	yes	characteristic
57	C₃₁×#391	C₂²	1	1	yes	characteristic
58	C₃₁×#390	C₂²	1	1	yes	characteristic
59	C₃₁×#384	C₂²	1	1	yes	characteristic
60	C₃₁×#382	C₂²	1	1	yes	characteristic
61	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
62	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
63	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
64	C₂²×D₄×C₃₁	—	2	1	no	—
65	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₂×Q₁₆×C₃₁	—	2	2	no	—
68	C₂×Q₁₆×C₃₁	—	2	2	no	—
69	C₂×Q₁₆×C₃₁	—	2	2	no	—
70	C₂×Q₁₆×C₃₁	—	2	2	no	—
71	C₂×Q₁₆×C₃₁	—	2	2	no	—
72	C₂×Q₁₆×C₃₁	—	2	2	no	—
73	C₂×SD₁₆×C₃₁	—	2	2	no	—
74	C₂×SD₁₆×C₃₁	—	2	2	no	—
75	C₂×SD₁₆×C₃₁	—	2	2	no	—
76	C₂×SD₁₆×C₃₁	—	2	2	no	—
77	C₂×SD₁₆×C₃₁	—	2	2	no	—
78	C₂×SD₁₆×C₃₁	—	2	2	no	—
79	C₂×M₁₆×C₃₁	—	2	2	no	—
80	C₂×M₁₆×C₃₁	—	2	2	no	—
81	C₂×M₁₆×C₃₁	D₄	1	4	yes	—
82	C₂²×C₈×C₃₁	—	2	2	no	—
83	C₂²×C₈×C₃₁	—	2	2	no	—
84	C₂²×C₈×C₃₁	D₄	1	4	yes	—
85	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
86	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
87	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
88	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
89	C₃₁×C₄⋊Q₈	—	2	2	no	—
90	C₃₁×C₄⋊Q₈	—	2	2	no	—
91	C₃₁×#124	—	2	2	no	—
92	C₃₁×#124	—	2	2	no	—
93	C₃₁×C₂²⋊Q₈	—	2	2	no	—
94	C₃₁×C₂²⋊Q₈	—	2	2	no	—
95	C₃₁×C₂²⋊Q₈	—	2	2	no	—
96	C₃₁×C₂²⋊Q₈	—	2	2	no	—
97	C₃₁×C₂²⋊Q₈	—	2	2	no	—
98	C₃₁×C₂²⋊Q₈	—	2	2	no	—
99	C₃₁×#121	—	2	2	no	—
100	C₃₁×#121	—	2	2	no	—
101	C₃₁×#117	—	2	2	no	—
102	C₃₁×#117	—	2	2	no	—
103	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
104	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
105	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
106	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
107	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
108	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
109	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
110	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
111	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
113	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
114	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
115	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
116	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
117	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
118	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
119	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
120	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
121	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
122	C₃₁×#107	—	2	2	no	—
123	C₃₁×#107	—	2	2	no	—
124	C₃₁×#106	—	2	2	no	—
125	C₃₁×#106	—	2	2	no	—
126	C₃₁×#103	—	2	2	no	—
127	C₃₁×#103	—	2	2	no	—
128	C₃₁×#103	—	2	2	no	—
129	C₃₁×#103	—	2	2	no	—
130	C₃₁×#103	—	2	2	no	—
131	C₃₁×#103	—	2	2	no	—
132	C₃₁×#102	—	2	2	no	—
133	C₃₁×#102	—	2	2	no	—
134	C₃₁×#97	—	2	2	no	—
135	C₃₁×#97	D₄	1	4	yes	complemented
136	C₄×C₈×C₃₁	—	2	2	no	—
137	C₄×C₈×C₃₁	—	2	2	no	—
138	C₃₁×#95	—	2	2	no	—
139	C₃₁×#95	—	2	2	no	—
140	C₃₁×#95	—	2	2	no	—
141	C₃₁×#95	—	2	2	no	—
142	C₂⁴×C₃₁	—	2	1	no	—
143	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
144	C₂×Q₈×C₃₁	—	2	1	no	—
145	C₂×Q₈×C₃₁	—	2	1	no	—
146	C₂×Q₈×C₃₁	—	2	1	no	—
147	C₂×Q₈×C₃₁	—	2	1	no	—
148	C₂×Q₈×C₃₁	—	2	1	no	—
149	C₂×Q₈×C₃₁	—	2	1	no	—
150	C₂×Q₈×C₃₁	—	2	1	no	—
151	C₂×Q₈×C₃₁	—	2	1	no	—
152	C₂×Q₈×C₃₁	—	2	1	no	—
153	C₂×Q₈×C₃₁	—	2	1	no	—
154	C₂×Q₈×C₃₁	—	2	1	no	—
155	C₂×Q₈×C₃₁	—	2	1	no	—
156	C₂×Q₈×C₃₁	—	2	1	no	—
157	C₂×Q₈×C₃₁	—	2	1	no	—
158	C₂×Q₈×C₃₁	—	2	1	no	—
159	C₂×Q₈×C₃₁	—	2	1	no	—
160	C₂×Q₈×C₃₁	—	2	1	no	—
161	C₂×Q₈×C₃₁	—	2	1	no	—
162	C₂×D₄×C₃₁	—	2	1	no	—
163	C₂×D₄×C₃₁	—	2	1	no	—
164	C₂×D₄×C₃₁	—	2	1	no	—
165	C₂×D₄×C₃₁	—	2	1	no	—
166	C₂×D₄×C₃₁	—	2	1	no	—
167	C₂×D₄×C₃₁	—	2	1	no	—
168	C₂×D₄×C₃₁	—	4	2	no	—
169	C₂×D₄×C₃₁	—	8	2	no	—
170	C₂²×C₄×C₃₁	—	2	1	no	—
171	C₂²×C₄×C₃₁	—	2	1	no	—
172	C₂²×C₄×C₃₁	—	2	1	no	—
173	C₂²×C₄×C₃₁	—	2	1	no	—
174	C₂²×C₄×C₃₁	—	2	1	no	—
175	C₂²×C₄×C₃₁	—	2	1	no	—
176	C₂²×C₄×C₃₁	—	2	1	no	—
177	C₂²×C₄×C₃₁	—	2	1	no	—
178	C₂²×C₄×C₃₁	—	2	1	no	—
179	C₂²×C₄×C₃₁	—	2	1	no	—
180	C₂²×C₄×C₃₁	—	2	1	no	—
181	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
182	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
183	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
184	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
185	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
186	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
187	Q₁₆×C₃₁	—	4	4	no	—
188	Q₁₆×C₃₁	—	4	4	no	—
189	SD₁₆×C₃₁	—	4	4	no	—
190	SD₁₆×C₃₁	—	4	4	no	—
191	M₁₆×C₃₁	—	2	8	no	—
192	C₂×C₈×C₃₁	—	2	1	no	—
193	C₂×C₈×C₃₁	—	2	1	no	—
194	C₂×C₈×C₃₁	—	2	1	no	—
195	C₂×C₈×C₃₁	—	2	1	no	—
196	C₂×C₈×C₃₁	—	2	8	no	—
197	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
198	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
199	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
200	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
201	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
202	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
203	C₂×C₈×C₃₁	#41	1	2	yes	—
204	C₂×C₈×C₃₁	#41	1	2	yes	—
205	C₄⋊C₄×C₃₁	—	2	4	no	—
206	C₄⋊C₄×C₃₁	—	2	1	no	—
207	C₄⋊C₄×C₃₁	—	2	1	no	—
208	C₄⋊C₄×C₃₁	—	2	1	no	—
209	C₄⋊C₄×C₃₁	—	2	1	no	—
210	C₄⋊C₄×C₃₁	—	2	1	no	—
211	C₄⋊C₄×C₃₁	—	2	1	no	—
212	C₄⋊C₄×C₃₁	—	2	1	no	—
213	C₄⋊C₄×C₃₁	—	2	1	no	—
214	C₄⋊C₄×C₃₁	—	4	4	no	—
215	C₄⋊C₄×C₃₁	—	4	2	no	—
216	C₄⋊C₄×C₃₁	—	4	2	no	—
217	C₄⋊C₄×C₃₁	—	4	2	no	—
218	C₄⋊C₄×C₃₁	—	8	2	no	—
219	C₂²⋊C₄×C₃₁	—	2	4	no	—
220	C₂²⋊C₄×C₃₁	—	4	2	no	—
221	C₂²⋊C₄×C₃₁	—	4	2	no	—
222	C₂²⋊C₄×C₃₁	—	4	2	no	—
223	C₂²⋊C₄×C₃₁	—	4	4	no	—
224	C₂²⋊C₄×C₃₁	—	4	2	no	—
225	C₂²⋊C₄×C₃₁	—	8	2	no	—
226	C₄²×C₃₁	—	2	1	no	—
227	C₄²×C₃₁	—	2	1	no	—
228	C₄²×C₃₁	C₂×D₄	1	2	yes	—
229	C₄²×C₃₁	#41	1	2	yes	—
230	#33188	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
231	C₂³×C₃₁	—	2	1	no	—
232	C₂³×C₃₁	—	2	1	no	—
233	C₂³×C₃₁	—	2	2	no	—
234	C₂³×C₃₁	—	2	2	no	—
235	C₂³×C₃₁	—	4	1	no	—
236	C₂³×C₃₁	—	4	1	no	—
237	C₂³×C₃₁	—	4	1	no	—
238	C₂³×C₃₁	—	4	2	no	—
239	C₂³×C₃₁	—	4	2	no	—
240	C₂³×C₃₁	#127	1	2	yes	—
241	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
242	Q₈×C₃₁	—	4	1	no	—
243	Q₈×C₃₁	—	4	1	no	—
244	Q₈×C₃₁	—	4	1	no	—
245	Q₈×C₃₁	—	4	1	no	—
246	Q₈×C₃₁	—	4	1	no	—
247	Q₈×C₃₁	—	4	1	no	—
248	Q₈×C₃₁	—	4	1	no	—
249	Q₈×C₃₁	—	4	1	no	—
250	Q₈×C₃₁	—	4	1	no	—
251	Q₈×C₃₁	—	4	1	no	—
252	Q₈×C₃₁	—	4	1	no	—
253	Q₈×C₃₁	—	4	1	no	—
254	D₄×C₃₁	—	4	1	no	—
255	D₄×C₃₁	—	4	1	no	—
256	D₄×C₃₁	—	4	1	no	—
257	D₄×C₃₁	—	4	1	no	—
258	D₄×C₃₁	—	8	4	no	—
259	C₂×C₄×C₃₁	—	2	1	no	—
260	C₂×C₄×C₃₁	—	2	1	no	—
261	C₂×C₄×C₃₁	—	2	2	no	—
262	C₂×C₄×C₃₁	—	2	2	no	—
263	C₂×C₄×C₃₁	—	2	2	no	—
264	C₂×C₄×C₃₁	—	2	2	no	—
265	C₂×C₄×C₃₁	—	2	2	no	—
266	C₂×C₄×C₃₁	—	2	2	no	—
267	C₂×C₄×C₃₁	—	2	2	no	—
268	C₂×C₄×C₃₁	—	2	2	no	—
269	C₂×C₄×C₃₁	—	2	2	no	—
270	C₂×C₄×C₃₁	—	4	1	no	—
271	C₂×C₄×C₃₁	—	4	1	no	—
272	C₂×C₄×C₃₁	—	4	1	no	—
273	C₂×C₄×C₃₁	—	4	1	no	—
274	C₂×C₄×C₃₁	—	4	1	no	—
275	C₂×C₄×C₃₁	—	4	1	no	—
276	C₂×C₄×C₃₁	—	4	1	no	—
277	C₂×C₄×C₃₁	—	4	1	no	—
278	C₂×C₄×C₃₁	—	4	1	no	—
279	C₂×C₄×C₃₁	—	4	1	no	—
280	C₂×C₄×C₃₁	—	4	1	no	—
281	C₂×C₄×C₃₁	—	4	1	no	—
282	C₂×C₄×C₃₁	—	4	1	no	—
283	C₂×C₄×C₃₁	—	4	2	no	—
284	C₂×C₄×C₃₁	—	4	2	no	—
285	C₂×C₄×C₃₁	—	4	2	no	—
286	C₂×C₄×C₃₁	—	4	2	no	—
287	C₂×C₄×C₃₁	#121	1	4	yes	—
288	C₂×C₄×C₃₁	#127	1	2	yes	—
289	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
290	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
291	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
292	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
293	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
294	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
295	C₈×C₃₁	—	2	2	no	—
296	C₈×C₃₁	—	2	2	no	—
297	C₈×C₃₁	#121	1	4	yes	—
298	C₈×C₃₁	#121	1	4	yes	—
299	C₂×#497	C₂×C₃₁	1	1	yes	characteristic, complemented
300	C₂×#470	C₂×C₃₁	1	1	yes	characteristic, complemented
301	C₂×#468	C₂×C₃₁	1	1	yes	characteristic
302	C₂×#467	C₂×C₃₁	1	1	yes	characteristic, complemented
303	C₂×#465	C₂×C₃₁	1	1	yes	characteristic
304	C₂×#405	C₂×C₃₁	1	1	yes	characteristic, complemented
305	#2621	C₂×C₃₁	1	1	yes	characteristic, complemented
306	#2620	C₂×C₃₁	1	1	yes	characteristic
307	#1973	C₂×C₃₁	1	1	yes	characteristic, complemented
308	#1971	C₂×C₃₁	1	1	yes	characteristic, complemented
309	#1969	C₂×C₃₁	1	1	yes	characteristic, complemented
310	#1968	C₂×C₃₁	1	1	yes	characteristic, complemented
311	#1958	C₂×C₃₁	1	1	yes	characteristic, complemented
312	#1956	C₂×C₃₁	1	1	yes	characteristic, complemented
313	#1847	C₂×C₃₁	1	1	yes	characteristic, complemented
314	C₂²×C₃₁	—	2	2	no	—
315	C₂²×C₃₁	—	2	2	no	—
316	C₂²×C₃₁	—	2	2	no	—
317	C₂²×C₃₁	—	4	2	no	—
318	C₂²×C₃₁	—	4	2	no	—
319	C₂²×C₃₁	—	8	1	no	—
320	C₂²×C₃₁	—	8	1	no	—
321	C₂²×C₃₁	—	8	1	no	—
322	C₂²×C₃₁	—	8	2	no	—
323	C₂²×C₃₁	—	8	2	no	—
324	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
325	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
326	C₂²×C₃₁	#532	1	1	yes	characteristic
327	C₂²×C₃₁	#577	1	2	yes	—
328	C₂²×C₃₁	#578	1	2	yes	—
329	C₄×C₃₁	—	2	2	no	—
330	C₄×C₃₁	—	2	4	no	—
331	C₄×C₃₁	—	4	2	no	—
332	C₄×C₃₁	—	8	1	no	—
333	C₄×C₃₁	—	8	1	no	—
334	C₄×C₃₁	—	8	1	no	—
335	C₄×C₃₁	—	8	1	no	—
336	C₄×C₃₁	—	8	1	no	—
337	C₄×C₃₁	—	8	1	no	—
338	C₄×C₃₁	—	8	1	no	—
339	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
340	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
341	C₄×C₃₁	#532	1	1	yes	characteristic
342	C₄×C₃₁	#532	1	1	yes	characteristic
343	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
344	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
345	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
346	C₂²×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
347	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
348	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
349	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
350	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
351	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
352	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
353	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
354	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
355	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
356	#497	—	2	4	no	—
357	#470	—	2	4	no	—
358	#468	—	2	4	no	—
359	#467	—	2	4	no	—
360	#465	—	2	4	no	—
361	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
362	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
363	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
364	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
365	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
366	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
367	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
368	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
369	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
370	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
371	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
372	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
373	#405	—	2	4	no	—
374	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
375	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
376	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
377	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
378	#398	—	2	2	no	—
379	#397	—	2	2	no	—
380	#396	—	2	2	no	—
381	#394	—	2	2	no	—
382	#391	C₂²×C₃₁	1	1	yes	characteristic
383	#391	C₂²×C₃₁	1	1	yes	characteristic
384	#391	C₂²×C₃₁	1	1	yes	characteristic
385	#390	C₂²×C₃₁	1	1	yes	characteristic
386	#384	C₂²×C₃₁	1	1	yes	characteristic
387	#382	C₂²×C₃₁	1	1	yes	characteristic
388	C₂×C₃₁	—	4	2	no	—
389	C₂×C₃₁	—	16	1	no	—
390	C₂×C₃₁	#2856	1	1	yes	characteristic
391	C₂×C₃₁	#3063	1	2	yes	—
392	C₂×C₃₁	#3154	1	1	yes	characteristic
393	C₂×C₃₁	#3156	1	1	yes	characteristic
394	C₂×C₃₁	#3178	1	2	yes	—
395	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
396	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
397	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
398	C₂²×D₄	—	2	1	no	—
399	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
400	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
401	C₂×Q₁₆	—	2	2	no	—
402	C₂×Q₁₆	—	2	2	no	—
403	C₂×Q₁₆	—	2	2	no	—
404	C₂×Q₁₆	—	2	2	no	—
405	C₂×Q₁₆	—	2	2	no	—
406	C₂×Q₁₆	—	2	2	no	—
407	C₂×SD₁₆	—	2	2	no	—
408	C₂×SD₁₆	—	2	2	no	—
409	C₂×SD₁₆	—	2	2	no	—
410	C₂×SD₁₆	—	2	2	no	—
411	C₂×SD₁₆	—	2	2	no	—
412	C₂×SD₁₆	—	2	2	no	—
413	C₂×M₁₆	—	2	2	no	—
414	C₂×M₁₆	—	2	2	no	—
415	C₂×M₁₆	D₄×C₃₁	1	4	yes	—
416	C₂²×C₈	—	2	2	no	—
417	C₂²×C₈	—	2	2	no	—
418	C₂²×C₈	D₄×C₃₁	1	4	yes	—
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	C₄⋊Q₈	—	2	2	no	—
424	C₄⋊Q₈	—	2	2	no	—
425	#124	—	2	2	no	—
426	#124	—	2	2	no	—
427	C₂²⋊Q₈	—	2	2	no	—
428	C₂²⋊Q₈	—	2	2	no	—
429	C₂²⋊Q₈	—	2	2	no	—
430	C₂²⋊Q₈	—	2	2	no	—
431	C₂²⋊Q₈	—	2	2	no	—
432	C₂²⋊Q₈	—	2	2	no	—
433	#121	—	2	2	no	—
434	#121	—	2	2	no	—
435	#117	—	2	2	no	—
436	#117	—	2	2	no	—
437	C₂×C₄⋊C₄	—	2	1	no	—
438	C₂×C₄⋊C₄	—	2	1	no	—
439	C₂×C₄⋊C₄	—	2	1	no	—
440	C₂×C₄⋊C₄	—	2	2	no	—
441	C₂×C₄⋊C₄	—	2	2	no	—
442	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
443	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
444	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
445	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
446	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
447	C₂×C₂²⋊C₄	—	2	1	no	—
448	C₂×C₂²⋊C₄	—	2	1	no	—
449	C₂×C₂²⋊C₄	—	2	1	no	—
450	C₂×C₂²⋊C₄	—	2	1	no	—
451	C₂×C₂²⋊C₄	—	2	2	no	—
452	C₂×C₂²⋊C₄	—	2	2	no	—
453	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
454	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
455	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
456	#107	—	2	2	no	—
457	#107	—	2	2	no	—
458	#106	—	2	2	no	—
459	#106	—	2	2	no	—
460	#103	—	2	2	no	—
461	#103	—	2	2	no	—
462	#103	—	2	2	no	—
463	#103	—	2	2	no	—
464	#103	—	2	2	no	—
465	#103	—	2	2	no	—
466	#102	—	2	2	no	—
467	#102	—	2	2	no	—
468	#97	—	2	2	no	—
469	#97	D₄×C₃₁	1	4	yes	complemented
470	C₄×C₈	—	2	2	no	—
471	C₄×C₈	—	2	2	no	—
472	#95	—	2	2	no	—
473	#95	—	2	2	no	—
474	#95	—	2	2	no	—
475	#95	—	2	2	no	—
476	C₃₁	#33188	1	1	yes	characteristic, direct product factor, Sylow
477	C₂⁴	—	2	1	no	—
478	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
479	C₂×Q₈	—	2	1	no	—
480	C₂×Q₈	—	2	1	no	—
481	C₂×Q₈	—	2	1	no	—
482	C₂×Q₈	—	2	1	no	—
483	C₂×Q₈	—	2	1	no	—
484	C₂×Q₈	—	2	1	no	—
485	C₂×Q₈	—	2	1	no	—
486	C₂×Q₈	—	2	1	no	—
487	C₂×Q₈	—	2	1	no	—
488	C₂×Q₈	—	2	1	no	—
489	C₂×Q₈	—	2	1	no	—
490	C₂×Q₈	—	2	1	no	—
491	C₂×Q₈	—	2	1	no	—
492	C₂×Q₈	—	2	1	no	—
493	C₂×Q₈	—	2	1	no	—
494	C₂×Q₈	—	2	1	no	—
495	C₂×Q₈	—	2	1	no	—
496	C₂×Q₈	—	2	1	no	—
497	C₂×D₄	—	2	1	no	—
498	C₂×D₄	—	2	1	no	—
499	C₂×D₄	—	2	1	no	—
500	C₂×D₄	—	2	1	no	—
501	C₂×D₄	—	2	1	no	—
502	C₂×D₄	—	2	1	no	—
503	C₂×D₄	—	4	2	no	—
504	C₂×D₄	—	8	2	no	—
505	C₂²×C₄	—	2	1	no	—
506	C₂²×C₄	—	2	1	no	—
507	C₂²×C₄	—	2	1	no	—
508	C₂²×C₄	—	2	1	no	—
509	C₂²×C₄	—	2	1	no	—
510	C₂²×C₄	—	2	1	no	—
511	C₂²×C₄	—	2	1	no	—
512	C₂²×C₄	—	2	1	no	—
513	C₂²×C₄	—	2	1	no	—
514	C₂²×C₄	—	2	1	no	—
515	C₂²×C₄	—	2	1	no	—
516	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
517	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
518	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
519	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
520	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
521	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
522	Q₁₆	—	4	4	no	—
523	Q₁₆	—	4	4	no	—
524	SD₁₆	—	4	4	no	—
525	SD₁₆	—	4	4	no	—
526	M₁₆	—	2	8	no	—
527	C₂×C₈	—	2	1	no	—
528	C₂×C₈	—	2	1	no	—
529	C₂×C₈	—	2	1	no	—
530	C₂×C₈	—	2	1	no	—
531	C₂×C₈	—	2	8	no	—
532	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
533	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
534	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
535	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
536	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
537	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
538	C₂×C₈	#41×C₃₁	1	2	yes	—
539	C₂×C₈	#41×C₃₁	1	2	yes	—
540	C₄⋊C₄	—	2	4	no	—
541	C₄⋊C₄	—	2	1	no	—
542	C₄⋊C₄	—	2	1	no	—
543	C₄⋊C₄	—	2	1	no	—
544	C₄⋊C₄	—	2	1	no	—
545	C₄⋊C₄	—	2	1	no	—
546	C₄⋊C₄	—	2	1	no	—
547	C₄⋊C₄	—	2	1	no	—
548	C₄⋊C₄	—	2	1	no	—
549	C₄⋊C₄	—	4	4	no	—
550	C₄⋊C₄	—	4	2	no	—
551	C₄⋊C₄	—	4	2	no	—
552	C₄⋊C₄	—	4	2	no	—
553	C₄⋊C₄	—	8	2	no	—
554	C₂²⋊C₄	—	2	4	no	—
555	C₂²⋊C₄	—	4	2	no	—
556	C₂²⋊C₄	—	4	2	no	—
557	C₂²⋊C₄	—	4	2	no	—
558	C₂²⋊C₄	—	4	4	no	—
559	C₂²⋊C₄	—	4	2	no	—
560	C₂²⋊C₄	—	8	2	no	—
561	C₄²	—	2	1	no	—
562	C₄²	—	2	1	no	—
563	C₄²	C₂×D₄×C₃₁	1	2	yes	—
564	C₄²	#41×C₃₁	1	2	yes	—
565	C₂³	—	2	1	no	—
566	C₂³	—	2	1	no	—
567	C₂³	—	2	2	no	—
568	C₂³	—	2	2	no	—
569	C₂³	—	4	1	no	—
570	C₂³	—	4	1	no	—
571	C₂³	—	4	1	no	—
572	C₂³	—	4	2	no	—
573	C₂³	—	4	2	no	—
574	C₂³	C₃₁×#127	1	2	yes	—
575	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
576	Q₈	—	4	1	no	—
577	Q₈	—	4	1	no	—
578	Q₈	—	4	1	no	—
579	Q₈	—	4	1	no	—
580	Q₈	—	4	1	no	—
581	Q₈	—	4	1	no	—
582	Q₈	—	4	1	no	—
583	Q₈	—	4	1	no	—
584	Q₈	—	4	1	no	—
585	Q₈	—	4	1	no	—
586	Q₈	—	4	1	no	—
587	Q₈	—	4	1	no	—
588	D₄	—	4	1	no	—
589	D₄	—	4	1	no	—
590	D₄	—	4	1	no	—
591	D₄	—	4	1	no	—
592	D₄	—	8	4	no	—
593	C₂×C₄	—	2	1	no	—
594	C₂×C₄	—	2	1	no	—
595	C₂×C₄	—	2	2	no	—
596	C₂×C₄	—	2	2	no	—
597	C₂×C₄	—	2	2	no	—
598	C₂×C₄	—	2	2	no	—
599	C₂×C₄	—	2	2	no	—
600	C₂×C₄	—	2	2	no	—
601	C₂×C₄	—	2	2	no	—
602	C₂×C₄	—	2	2	no	—
603	C₂×C₄	—	2	2	no	—
604	C₂×C₄	—	4	1	no	—
605	C₂×C₄	—	4	1	no	—
606	C₂×C₄	—	4	1	no	—
607	C₂×C₄	—	4	1	no	—
608	C₂×C₄	—	4	1	no	—
609	C₂×C₄	—	4	1	no	—
610	C₂×C₄	—	4	1	no	—
611	C₂×C₄	—	4	1	no	—
612	C₂×C₄	—	4	1	no	—
613	C₂×C₄	—	4	1	no	—
614	C₂×C₄	—	4	1	no	—
615	C₂×C₄	—	4	1	no	—
616	C₂×C₄	—	4	1	no	—
617	C₂×C₄	—	4	2	no	—
618	C₂×C₄	—	4	2	no	—
619	C₂×C₄	—	4	2	no	—
620	C₂×C₄	—	4	2	no	—
621	C₂×C₄	C₃₁×#121	1	4	yes	—
622	C₂×C₄	C₃₁×#127	1	2	yes	—
623	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
624	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
625	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
626	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
627	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
628	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
629	C₈	—	2	2	no	—
630	C₈	—	2	2	no	—
631	C₈	C₃₁×#121	1	4	yes	—
632	C₈	C₃₁×#121	1	4	yes	—
633	C₂²	—	2	2	no	—
634	C₂²	—	2	2	no	—
635	C₂²	—	2	2	no	—
636	C₂²	—	4	2	no	—
637	C₂²	—	4	2	no	—
638	C₂²	—	8	1	no	—
639	C₂²	—	8	1	no	—
640	C₂²	—	8	1	no	—
641	C₂²	—	8	2	no	—
642	C₂²	—	8	2	no	—
643	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
644	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
645	C₂²	C₃₁×#532	1	1	yes	characteristic
646	C₂²	C₃₁×#577	1	2	yes	—
647	C₂²	C₃₁×#578	1	2	yes	—
648	C₄	—	2	2	no	—
649	C₄	—	2	4	no	—
650	C₄	—	4	2	no	—
651	C₄	—	8	1	no	—
652	C₄	—	8	1	no	—
653	C₄	—	8	1	no	—
654	C₄	—	8	1	no	—
655	C₄	—	8	1	no	—
656	C₄	—	8	1	no	—
657	C₄	—	8	1	no	—
658	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
659	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
660	C₄	C₃₁×#532	1	1	yes	characteristic
661	C₄	C₃₁×#532	1	1	yes	characteristic
662	C₂	—	4	2	no	—
663	C₂	—	16	1	no	—
664	C₂	C₃₁×#2856	1	1	yes	characteristic
665	C₂	C₃₁×#3063	1	2	yes	—
666	C₂	C₃₁×#3154	1	1	yes	characteristic
667	C₂	C₃₁×#3156	1	1	yes	characteristic
668	C₂	C₃₁×#3178	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#497	C₂	semidirect
2. C₂×C₃₁×#470	C₂	semidirect
4. C₂×C₃₁×#467	C₂	semidirect
6. C₂×C₃₁×#405	C₂	semidirect
7. C₃₁×#2621	C₂	semidirect
9. C₃₁×#1973	C₂	semidirect
10. C₃₁×#1971	C₂	semidirect
11. C₃₁×#1969	C₂	semidirect
12. C₃₁×#1968	C₂	semidirect
13. C₃₁×#1958	C₂	semidirect
14. C₃₁×#1956	C₂	semidirect
15. C₃₁×#1847	C₂	semidirect
135. C₃₁×#97	D₄	semidirect
230. #33188	C₃₁	direct (N × H)
299. C₂×#497	C₂×C₃₁	semidirect
300. C₂×#470	C₂×C₃₁	semidirect
302. C₂×#467	C₂×C₃₁	semidirect
304. C₂×#405	C₂×C₃₁	semidirect
305. #2621	C₂×C₃₁	semidirect
307. #1973	C₂×C₃₁	semidirect
308. #1971	C₂×C₃₁	semidirect
309. #1969	C₂×C₃₁	semidirect
310. #1968	C₂×C₃₁	semidirect
311. #1958	C₂×C₃₁	semidirect
312. #1956	C₂×C₃₁	semidirect
313. #1847	C₂×C₃₁	semidirect
469. #97	D₄×C₃₁	semidirect
476. C₃₁	#33188	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#497	2
C₂×C₃₁×#470	2
C₂×C₃₁×#468	2
C₂×C₃₁×#467	2
C₂×C₃₁×#465	2
C₂×C₃₁×#405	2
C₃₁×#2621	2
C₃₁×#2620	2
C₃₁×#1973	2
C₃₁×#1971	2
C₃₁×#1969	2
C₃₁×#1968	2
C₃₁×#1958	2
C₃₁×#1956	2
C₃₁×#1847	2
#33188	31

Subgroups — C31×#33188 #21489947

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33188 #21489947