Subgroups — C₃₁×#60699 #21517458

Order 7936.

1842 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3461	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3185	C₂	1	2	yes	maximal, complemented
3	C₃₁×#3183	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3181	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3178	C₂	1	4	yes	maximal, complemented
6	C₃₁×#3177	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3173	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3163	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#3156	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#3137	C₂	1	2	yes	maximal, complemented
11	C₃₁×#3063	C₂	1	2	yes	maximal, complemented
12	C₃₁×#3060	C₂	1	1	yes	maximal, characteristic
13	C₃₁×#3059	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×C₃₁×#137	C₂	1	2	yes	maximal, complemented
15	C₃₁×#2943	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₃₁×#441	C₂	1	1	yes	maximal, characteristic, complemented
17	C₃₁×#2875	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×C₃₁×#137	C₂²	1	2	yes	—
19	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
20	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
23	C₃₁×#544	C₂²	1	2	yes	—
24	C₃₁×#542	—	2	2	no	—
25	C₃₁×#542	C₂²	1	4	yes	—
26	C₃₁×#541	—	2	2	no	—
27	C₃₁×#541	C₂²	1	4	yes	—
28	C₃₁×#539	—	2	4	no	—
29	C₃₁×#537	C₂²	1	2	yes	—
30	C₃₁×#536	C₂²	1	2	yes	—
31	C₃₁×#536	C₂²	1	2	yes	—
32	C₃₁×#533	—	2	2	no	—
33	C₃₁×#533	C₂²	1	4	yes	—
34	C₃₁×#529	—	2	4	no	—
35	C₃₁×#529	—	2	2	no	—
36	C₃₁×#529	C₂²	1	4	yes	—
37	C₄×#41×C₃₁	C₂²	1	2	yes	—
38	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
39	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
40	C₃₁×#501	C₂²	1	4	yes	—
41	C₃₁×#499	C₂²	1	2	yes	—
42	C₃₁×#498	C₂²	1	2	yes	—
43	C₃₁×#497	C₂²	1	4	yes	—
44	C₃₁×#495	C₂²	1	2	yes	—
45	C₃₁×#494	C₂²	1	2	yes	—
46	C₃₁×#491	C₂²	1	4	yes	—
47	C₃₁×#489	C₂²	1	4	yes	—
48	C₃₁×#488	C₂²	1	4	yes	—
49	C₃₁×#487	C₂²	1	4	yes	—
50	C₃₁×#470	C₂²	1	2	yes	—
51	C₃₁×#470	C₂²	1	2	yes	—
52	C₃₁×#470	C₂²	1	2	yes	—
53	C₃₁×#470	C₂²	1	4	yes	—
54	C₃₁×#469	C₂²	1	2	yes	—
55	C₃₁×#468	C₂²	1	4	yes	—
56	C₃₁×#467	C₂²	1	4	yes	—
57	C₃₁×#467	C₂²	1	4	yes	—
58	C₃₁×#465	C₂²	1	4	yes	—
59	C₃₁×#465	C₂²	1	4	yes	—
60	C₃₁×#442	C₂²	1	1	yes	characteristic
61	C₃₁×#442	C₂²	1	1	yes	characteristic
62	C₃₁×#441	C₂²	1	2	yes	—
63	C₃₁×#441	C₂²	1	1	yes	characteristic
64	C₃₁×#441	C₂²	1	2	yes	—
65	C₃₁×#441	C₂²	1	2	yes	—
66	C₃₁×#441	C₂²	1	4	yes	—
67	C₃₁×#441	C₂²	1	2	yes	—
68	C₃₁×#441	C₂²	1	1	yes	characteristic
69	C₃₁×#440	C₂²	1	2	yes	—
70	C₃₁×#440	C₂²	1	2	yes	—
71	C₃₁×#440	C₂²	1	4	yes	—
72	C₄×Q₁₆×C₃₁	C₂²	1	4	yes	—
73	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	—
74	C₃₁×#428	C₂²	1	2	yes	—
75	C₃₁×#427	C₂²	1	1	yes	characteristic
76	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
77	C₃₁×#419	C₂²	1	2	yes	—
78	C₃₁×#417	C₂²	1	2	yes	—
79	C₃₁×#416	C₂²	1	2	yes	—
80	C₂×C₃₁×#103	C₂²	1	2	yes	—
81	C₃₁×#405	C₂²	1	1	yes	characteristic
82	C₃₁×#405	C₂²	1	1	yes	characteristic
83	C₃₁×#405	C₂²	1	2	yes	—
84	C₃₁×#405	C₂²	1	2	yes	—
85	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
86	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
87	C₂×C₃₁×#97	C₂²	1	2	yes	—
88	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
89	C₂×#41×C₃₁	C₂³	1	2	yes	—
90	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
91	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
92	C₃₁×#137	—	2	8	no	—
93	C₃₁×#135	—	2	4	no	—
94	C₂×Q₁₆×C₃₁	—	2	4	no	—
95	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
96	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
97	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
98	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
99	C₂×SD₁₆×C₃₁	C₂³	1	4	yes	—
100	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
101	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
102	C₃₁×#131	—	2	4	no	—
103	C₃₁×#131	—	2	4	no	—
104	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
105	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
106	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
107	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
108	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
109	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
110	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
111	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
112	C₃₁×C₄⋊Q₈	—	2	2	no	—
113	C₃₁×C₄⋊Q₈	—	2	2	no	—
114	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
115	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
116	C₃₁×#126	—	2	4	no	—
117	C₃₁×#126	—	2	4	no	—
118	C₃₁×#126	—	2	4	no	—
119	C₃₁×#126	—	2	4	no	—
120	C₃₁×#125	—	2	4	no	—
121	C₃₁×#125	—	2	2	no	—
122	C₃₁×#125	—	2	4	no	—
123	C₃₁×#125	—	2	4	no	—
124	C₃₁×#125	C₂³	1	4	yes	—
125	C₃₁×#124	—	2	2	no	—
126	C₃₁×#124	—	2	4	no	—
127	C₃₁×#124	C₂³	1	4	yes	—
128	C₃₁×#123	—	2	4	no	—
129	C₃₁×#123	—	2	4	no	—
130	C₃₁×#123	—	2	4	no	—
131	C₃₁×#123	—	2	4	no	—
132	C₃₁×C₂²⋊Q₈	—	2	2	no	—
133	C₃₁×C₂²⋊Q₈	—	2	4	no	—
134	C₃₁×C₂²⋊Q₈	—	2	2	no	—
135	C₃₁×C₂²⋊Q₈	—	2	2	no	—
136	C₃₁×C₂²⋊Q₈	—	2	4	no	—
137	C₃₁×C₂²⋊Q₈	—	2	4	no	—
138	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
139	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
140	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
141	C₃₁×#121	—	2	4	no	—
142	C₃₁×#121	—	2	2	no	—
143	C₃₁×#121	C₂³	1	4	yes	—
144	C₄×Q₈×C₃₁	—	2	2	no	—
145	C₄×Q₈×C₃₁	—	2	2	no	—
146	C₄×Q₈×C₃₁	—	2	4	no	—
147	C₄×Q₈×C₃₁	D₄	1	2	yes	—
148	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
149	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
150	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
151	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
152	C₄×D₄×C₃₁	—	2	2	no	—
153	C₄×D₄×C₃₁	—	2	2	no	—
154	C₄×D₄×C₃₁	—	2	4	no	—
155	C₄×D₄×C₃₁	D₄	1	4	yes	—
156	C₄×D₄×C₃₁	D₄	1	2	yes	—
157	C₄×D₄×C₃₁	C₂³	1	2	yes	—
158	C₄×D₄×C₃₁	C₂³	1	2	yes	—
159	C₃₁×#117	—	2	4	no	—
160	C₃₁×#117	—	2	2	no	—
161	C₃₁×#117	—	2	4	no	—
162	C₃₁×#117	C₂³	1	2	yes	—
163	C₃₁×#117	C₂³	1	1	yes	characteristic
164	C₃₁×#117	C₂³	1	2	yes	—
165	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
166	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
167	C₂×C₄²×C₃₁	—	2	2	no	—
168	C₂×C₄²×C₃₁	—	2	2	no	—
169	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
170	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
171	C₃₁×#107	C₂³	1	2	yes	—
172	C₃₁×#107	C₂³	1	4	yes	—
173	C₃₁×#106	C₂³	1	2	yes	—
174	C₃₁×#106	C₂³	1	2	yes	—
175	C₃₁×#106	C₂³	1	1	yes	characteristic
176	C₃₁×#106	C₂³	1	1	yes	characteristic
177	C₃₁×#106	C₂³	1	4	yes	—
178	C₃₁×#103	C₂³	1	4	yes	—
179	C₃₁×#103	C₂³	1	4	yes	—
180	C₃₁×#103	C₂³	1	2	yes	—
181	C₃₁×#103	C₂³	1	2	yes	—
182	C₃₁×#103	C₂³	1	4	yes	—
183	C₃₁×#103	C₂³	1	4	yes	—
184	C₃₁×#103	C₂³	1	4	yes	—
185	C₃₁×#102	C₂³	1	2	yes	—
186	C₃₁×#102	C₂³	1	2	yes	—
187	C₃₁×#102	C₂³	1	4	yes	—
188	C₃₁×#97	C₂³	1	1	yes	characteristic
189	C₃₁×#97	C₂³	1	2	yes	—
190	C₃₁×#97	C₂³	1	1	yes	characteristic
191	C₃₁×#97	C₂³	1	1	yes	characteristic
192	C₃₁×#97	C₂³	1	1	yes	characteristic
193	C₃₁×#97	C₂³	1	2	yes	—
194	C₃₁×#97	C₂³	1	2	yes	—
195	C₄×C₈×C₃₁	C₂³	1	2	yes	—
196	C₄×C₈×C₃₁	C₂³	1	2	yes	—
197	C₄×C₈×C₃₁	C₂³	1	2	yes	—
198	#41×C₃₁	—	2	2	no	—
199	#41×C₃₁	—	2	2	no	—
200	#41×C₃₁	—	2	4	no	—
201	#41×C₃₁	—	2	4	no	—
202	C₂×Q₈×C₃₁	—	2	2	no	—
203	C₂×Q₈×C₃₁	—	2	4	no	—
204	C₂×Q₈×C₃₁	—	2	4	no	—
205	C₂×Q₈×C₃₁	—	2	4	no	—
206	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
207	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
208	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
209	C₂×D₄×C₃₁	—	2	2	no	—
210	C₂×D₄×C₃₁	—	2	4	no	—
211	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
212	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
213	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
214	C₂²×C₄×C₃₁	—	2	2	no	—
215	C₂²×C₄×C₃₁	—	2	2	no	—
216	C₂²×C₄×C₃₁	—	2	2	no	—
217	C₂²×C₄×C₃₁	—	2	2	no	—
218	C₂²×C₄×C₃₁	—	2	2	no	—
219	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
220	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
221	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
222	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
223	Q₁₆×C₃₁	—	2	4	no	—
224	Q₁₆×C₃₁	—	2	4	no	—
225	Q₁₆×C₃₁	—	2	2	no	—
226	Q₁₆×C₃₁	—	2	8	no	—
227	SD₁₆×C₃₁	—	2	4	no	—
228	SD₁₆×C₃₁	—	2	8	no	—
229	D₈×C₃₁	—	2	2	no	—
230	M₁₆×C₃₁	—	2	2	no	—
231	M₁₆×C₃₁	—	2	4	no	—
232	M₁₆×C₃₁	—	2	4	no	—
233	M₁₆×C₃₁	#41	1	4	yes	—
234	C₂×C₈×C₃₁	—	2	2	no	—
235	C₂×C₈×C₃₁	—	2	4	no	—
236	C₂×C₈×C₃₁	—	2	4	no	—
237	C₂×C₈×C₃₁	#41	1	4	yes	—
238	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
239	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
240	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
241	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
242	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
243	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
244	C₄⋊C₄×C₃₁	—	2	4	no	—
245	C₄⋊C₄×C₃₁	—	2	4	no	—
246	C₄⋊C₄×C₃₁	—	2	2	no	—
247	C₄⋊C₄×C₃₁	—	2	2	no	—
248	C₄⋊C₄×C₃₁	—	2	2	no	—
249	C₄⋊C₄×C₃₁	—	2	4	no	—
250	C₄⋊C₄×C₃₁	—	2	4	no	—
251	C₄⋊C₄×C₃₁	—	2	4	no	—
252	C₄⋊C₄×C₃₁	—	2	4	no	—
253	C₄⋊C₄×C₃₁	—	2	4	no	—
254	C₄⋊C₄×C₃₁	—	2	4	no	—
255	C₄⋊C₄×C₃₁	—	2	4	no	—
256	C₄⋊C₄×C₃₁	—	2	4	no	—
257	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
258	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
259	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
260	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
261	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
262	C₂²⋊C₄×C₃₁	—	2	4	no	—
263	C₂²⋊C₄×C₃₁	—	2	4	no	—
264	C₂²⋊C₄×C₃₁	—	2	2	no	—
265	C₂²⋊C₄×C₃₁	—	2	4	no	—
266	C₂²⋊C₄×C₃₁	—	2	2	no	—
267	C₂²⋊C₄×C₃₁	—	2	2	no	—
268	C₂²⋊C₄×C₃₁	—	2	4	no	—
269	C₂²⋊C₄×C₃₁	—	2	4	no	—
270	C₂²⋊C₄×C₃₁	—	2	4	no	—
271	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
272	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
273	C₄²×C₃₁	—	2	4	no	—
274	C₄²×C₃₁	—	2	2	no	—
275	C₄²×C₃₁	—	2	2	no	—
276	C₄²×C₃₁	—	2	2	no	—
277	C₄²×C₃₁	—	2	4	no	—
278	C₄²×C₃₁	C₂×D₄	1	2	yes	—
279	C₄²×C₃₁	C₂×D₄	1	4	yes	—
280	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
281	C₄²×C₃₁	C₂⁴	1	2	yes	—
282	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
283	#60699	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
284	C₂³×C₃₁	—	2	2	no	—
285	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
286	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
287	Q₈×C₃₁	—	2	2	no	—
288	Q₈×C₃₁	—	2	4	no	—
289	Q₈×C₃₁	—	2	2	no	—
290	Q₈×C₃₁	—	2	4	no	—
291	Q₈×C₃₁	—	2	2	no	—
292	Q₈×C₃₁	—	4	4	no	—
293	Q₈×C₃₁	—	8	4	no	—
294	D₄×C₃₁	—	2	4	no	—
295	D₄×C₃₁	—	2	2	no	—
296	D₄×C₃₁	—	2	2	no	—
297	D₄×C₃₁	—	2	2	no	—
298	D₄×C₃₁	—	4	4	no	—
299	D₄×C₃₁	—	8	4	no	—
300	C₂×C₄×C₃₁	—	2	2	no	—
301	C₂×C₄×C₃₁	—	2	2	no	—
302	C₂×C₄×C₃₁	—	2	2	no	—
303	C₂×C₄×C₃₁	—	2	4	no	—
304	C₂×C₄×C₃₁	—	2	2	no	—
305	C₂×C₄×C₃₁	—	2	2	no	—
306	C₂×C₄×C₃₁	—	2	4	no	—
307	C₂×C₄×C₃₁	—	4	2	no	—
308	C₂×C₄×C₃₁	—	4	4	no	—
309	C₂×C₄×C₃₁	—	4	4	no	—
310	C₂×C₄×C₃₁	—	4	4	no	—
311	C₂×C₄×C₃₁	—	4	4	no	—
312	C₂×C₄×C₃₁	—	8	2	no	—
313	C₂×C₄×C₃₁	—	8	2	no	—
314	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
315	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
316	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
317	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
318	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
319	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
320	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
321	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
322	C₈×C₃₁	—	2	1	no	—
323	C₈×C₃₁	—	2	1	no	—
324	C₈×C₃₁	—	2	2	no	—
325	C₈×C₃₁	C₂×#41	1	2	yes	—
326	C₈×C₃₁	C₂×#41	1	4	yes	—
327	C₈×C₃₁	C₂×#41	1	2	yes	—
328	#3461	C₂×C₃₁	1	2	yes	complemented
329	#3185	C₂×C₃₁	1	2	yes	complemented
330	#3183	C₂×C₃₁	1	2	yes	complemented
331	#3181	C₂×C₃₁	1	4	yes	complemented
332	#3178	C₂×C₃₁	1	4	yes	complemented
333	#3177	C₂×C₃₁	1	2	yes	complemented
334	#3173	C₂×C₃₁	1	2	yes	complemented
335	#3163	C₂×C₃₁	1	1	yes	characteristic, complemented
336	#3156	C₂×C₃₁	1	1	yes	characteristic, complemented
337	#3137	C₂×C₃₁	1	2	yes	complemented
338	#3063	C₂×C₃₁	1	2	yes	complemented
339	#3060	C₂×C₃₁	1	1	yes	characteristic
340	#3059	C₂×C₃₁	1	1	yes	characteristic, complemented
341	C₄×#137	C₂×C₃₁	1	2	yes	complemented
342	#2943	C₂×C₃₁	1	1	yes	characteristic, complemented
343	C₂×#441	C₂×C₃₁	1	1	yes	characteristic, complemented
344	#2875	C₂×C₃₁	1	1	yes	characteristic, complemented
345	C₂²×C₃₁	—	2	2	no	—
346	C₂²×C₃₁	—	2	4	no	—
347	C₂²×C₃₁	—	2	2	no	—
348	C₂²×C₃₁	—	4	2	no	—
349	C₂²×C₃₁	—	4	2	no	—
350	C₂²×C₃₁	—	4	2	no	—
351	C₂²×C₃₁	#532	1	2	yes	—
352	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
353	C₄×C₃₁	—	2	2	no	—
354	C₄×C₃₁	—	2	2	no	—
355	C₄×C₃₁	—	2	4	no	—
356	C₄×C₃₁	—	2	2	no	—
357	C₄×C₃₁	—	4	2	no	—
358	C₄×C₃₁	—	4	4	no	—
359	C₄×C₃₁	—	4	2	no	—
360	C₄×C₃₁	—	4	2	no	—
361	C₄×C₃₁	—	4	4	no	—
362	C₄×C₃₁	#532	1	2	yes	—
363	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
364	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
365	C₂×#137	C₂²×C₃₁	1	2	yes	—
366	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
367	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
368	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
369	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
370	#544	C₂²×C₃₁	1	2	yes	—
371	#542	—	2	2	no	—
372	#542	C₂²×C₃₁	1	4	yes	—
373	#541	—	2	2	no	—
374	#541	C₂²×C₃₁	1	4	yes	—
375	#539	—	2	4	no	—
376	#537	C₂²×C₃₁	1	2	yes	—
377	#536	C₂²×C₃₁	1	2	yes	—
378	#536	C₂²×C₃₁	1	2	yes	—
379	#533	—	2	2	no	—
380	#533	C₂²×C₃₁	1	4	yes	—
381	#529	—	2	4	no	—
382	#529	—	2	2	no	—
383	#529	C₂²×C₃₁	1	4	yes	—
384	C₄×#41	C₂²×C₃₁	1	2	yes	—
385	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
386	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
387	#501	C₂²×C₃₁	1	4	yes	—
388	#499	C₂²×C₃₁	1	2	yes	—
389	#498	C₂²×C₃₁	1	2	yes	—
390	#497	C₂²×C₃₁	1	4	yes	—
391	#495	C₂²×C₃₁	1	2	yes	—
392	#494	C₂²×C₃₁	1	2	yes	—
393	#491	C₂²×C₃₁	1	4	yes	—
394	#489	C₂²×C₃₁	1	4	yes	—
395	#488	C₂²×C₃₁	1	4	yes	—
396	#487	C₂²×C₃₁	1	4	yes	—
397	#470	C₂²×C₃₁	1	2	yes	—
398	#470	C₂²×C₃₁	1	2	yes	—
399	#470	C₂²×C₃₁	1	2	yes	—
400	#470	C₂²×C₃₁	1	4	yes	—
401	#469	C₂²×C₃₁	1	2	yes	—
402	#468	C₂²×C₃₁	1	4	yes	—
403	#467	C₂²×C₃₁	1	4	yes	—
404	#467	C₂²×C₃₁	1	4	yes	—
405	#465	C₂²×C₃₁	1	4	yes	—
406	#465	C₂²×C₃₁	1	4	yes	—
407	#442	C₂²×C₃₁	1	1	yes	characteristic
408	#442	C₂²×C₃₁	1	1	yes	characteristic
409	#441	C₂²×C₃₁	1	2	yes	—
410	#441	C₂²×C₃₁	1	1	yes	characteristic
411	#441	C₂²×C₃₁	1	2	yes	—
412	#441	C₂²×C₃₁	1	2	yes	—
413	#441	C₂²×C₃₁	1	4	yes	—
414	#441	C₂²×C₃₁	1	2	yes	—
415	#441	C₂²×C₃₁	1	1	yes	characteristic
416	#440	C₂²×C₃₁	1	2	yes	—
417	#440	C₂²×C₃₁	1	2	yes	—
418	#440	C₂²×C₃₁	1	4	yes	—
419	C₄×Q₁₆	C₂²×C₃₁	1	4	yes	—
420	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	—
421	#428	C₂²×C₃₁	1	2	yes	—
422	#427	C₂²×C₃₁	1	1	yes	characteristic
423	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
424	#419	C₂²×C₃₁	1	2	yes	—
425	#417	C₂²×C₃₁	1	2	yes	—
426	#416	C₂²×C₃₁	1	2	yes	—
427	C₂×#103	C₂²×C₃₁	1	2	yes	—
428	#405	C₂²×C₃₁	1	1	yes	characteristic
429	#405	C₂²×C₃₁	1	1	yes	characteristic
430	#405	C₂²×C₃₁	1	2	yes	—
431	#405	C₂²×C₃₁	1	2	yes	—
432	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
433	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
434	C₂×#97	C₂²×C₃₁	1	2	yes	—
435	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
436	C₂×C₃₁	—	2	2	no	—
437	C₂×C₃₁	—	4	2	no	—
438	C₂×C₃₁	—	8	2	no	—
439	C₂×C₃₁	C₂×#532	1	1	yes	characteristic
440	C₂×C₃₁	#3586	1	2	yes	—
441	C₂×#41	C₂³×C₃₁	1	2	yes	—
442	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
443	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
444	#137	—	2	8	no	—
445	#135	—	2	4	no	—
446	C₂×Q₁₆	—	2	4	no	—
447	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
448	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
449	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
450	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
451	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
452	C₂×SD₁₆	C₂³×C₃₁	1	4	yes	—
453	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
454	#131	—	2	4	no	—
455	#131	—	2	4	no	—
456	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
457	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
458	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
459	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
460	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
461	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
462	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
463	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
464	C₄⋊Q₈	—	2	2	no	—
465	C₄⋊Q₈	—	2	2	no	—
466	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
467	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
468	#126	—	2	4	no	—
469	#126	—	2	4	no	—
470	#126	—	2	4	no	—
471	#126	—	2	4	no	—
472	#125	—	2	4	no	—
473	#125	—	2	2	no	—
474	#125	—	2	4	no	—
475	#125	—	2	4	no	—
476	#125	C₂³×C₃₁	1	4	yes	—
477	#124	—	2	2	no	—
478	#124	—	2	4	no	—
479	#124	C₂³×C₃₁	1	4	yes	—
480	#123	—	2	4	no	—
481	#123	—	2	4	no	—
482	#123	—	2	4	no	—
483	#123	—	2	4	no	—
484	C₂²⋊Q₈	—	2	2	no	—
485	C₂²⋊Q₈	—	2	4	no	—
486	C₂²⋊Q₈	—	2	2	no	—
487	C₂²⋊Q₈	—	2	2	no	—
488	C₂²⋊Q₈	—	2	4	no	—
489	C₂²⋊Q₈	—	2	4	no	—
490	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
491	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
492	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
493	#121	—	2	4	no	—
494	#121	—	2	2	no	—
495	#121	C₂³×C₃₁	1	4	yes	—
496	C₄×Q₈	—	2	2	no	—
497	C₄×Q₈	—	2	2	no	—
498	C₄×Q₈	—	2	4	no	—
499	C₄×Q₈	D₄×C₃₁	1	2	yes	—
500	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
501	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
502	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
503	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
504	C₄×D₄	—	2	2	no	—
505	C₄×D₄	—	2	2	no	—
506	C₄×D₄	—	2	4	no	—
507	C₄×D₄	D₄×C₃₁	1	4	yes	—
508	C₄×D₄	D₄×C₃₁	1	2	yes	—
509	C₄×D₄	C₂³×C₃₁	1	2	yes	—
510	C₄×D₄	C₂³×C₃₁	1	2	yes	—
511	#117	—	2	4	no	—
512	#117	—	2	2	no	—
513	#117	—	2	4	no	—
514	#117	C₂³×C₃₁	1	2	yes	—
515	#117	C₂³×C₃₁	1	1	yes	characteristic
516	#117	C₂³×C₃₁	1	2	yes	—
517	C₂×C₄⋊C₄	—	2	2	no	—
518	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
519	C₂×C₄²	—	2	2	no	—
520	C₂×C₄²	—	2	2	no	—
521	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
522	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
523	#107	C₂³×C₃₁	1	2	yes	—
524	#107	C₂³×C₃₁	1	4	yes	—
525	#106	C₂³×C₃₁	1	2	yes	—
526	#106	C₂³×C₃₁	1	2	yes	—
527	#106	C₂³×C₃₁	1	1	yes	characteristic
528	#106	C₂³×C₃₁	1	1	yes	characteristic
529	#106	C₂³×C₃₁	1	4	yes	—
530	#103	C₂³×C₃₁	1	4	yes	—
531	#103	C₂³×C₃₁	1	4	yes	—
532	#103	C₂³×C₃₁	1	2	yes	—
533	#103	C₂³×C₃₁	1	2	yes	—
534	#103	C₂³×C₃₁	1	4	yes	—
535	#103	C₂³×C₃₁	1	4	yes	—
536	#103	C₂³×C₃₁	1	4	yes	—
537	#102	C₂³×C₃₁	1	2	yes	—
538	#102	C₂³×C₃₁	1	2	yes	—
539	#102	C₂³×C₃₁	1	4	yes	—
540	#97	C₂³×C₃₁	1	1	yes	characteristic
541	#97	C₂³×C₃₁	1	2	yes	—
542	#97	C₂³×C₃₁	1	1	yes	characteristic
543	#97	C₂³×C₃₁	1	1	yes	characteristic
544	#97	C₂³×C₃₁	1	1	yes	characteristic
545	#97	C₂³×C₃₁	1	2	yes	—
546	#97	C₂³×C₃₁	1	2	yes	—
547	C₄×C₈	C₂³×C₃₁	1	2	yes	—
548	C₄×C₈	C₂³×C₃₁	1	2	yes	—
549	C₄×C₈	C₂³×C₃₁	1	2	yes	—
550	C₃₁	#60699	1	1	yes	characteristic, direct product factor, Sylow
551	#41	—	2	2	no	—
552	#41	—	2	2	no	—
553	#41	—	2	4	no	—
554	#41	—	2	4	no	—
555	C₂×Q₈	—	2	2	no	—
556	C₂×Q₈	—	2	4	no	—
557	C₂×Q₈	—	2	4	no	—
558	C₂×Q₈	—	2	4	no	—
559	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
560	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
561	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
562	C₂×D₄	—	2	2	no	—
563	C₂×D₄	—	2	4	no	—
564	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
565	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
566	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
567	C₂²×C₄	—	2	2	no	—
568	C₂²×C₄	—	2	2	no	—
569	C₂²×C₄	—	2	2	no	—
570	C₂²×C₄	—	2	2	no	—
571	C₂²×C₄	—	2	2	no	—
572	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
573	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
574	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
575	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
576	Q₁₆	—	2	4	no	—
577	Q₁₆	—	2	4	no	—
578	Q₁₆	—	2	2	no	—
579	Q₁₆	—	2	8	no	—
580	SD₁₆	—	2	4	no	—
581	SD₁₆	—	2	8	no	—
582	D₈	—	2	2	no	—
583	M₁₆	—	2	2	no	—
584	M₁₆	—	2	4	no	—
585	M₁₆	—	2	4	no	—
586	M₁₆	#41×C₃₁	1	4	yes	—
587	C₂×C₈	—	2	2	no	—
588	C₂×C₈	—	2	4	no	—
589	C₂×C₈	—	2	4	no	—
590	C₂×C₈	#41×C₃₁	1	4	yes	—
591	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
592	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
593	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
594	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
595	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
596	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
597	C₄⋊C₄	—	2	4	no	—
598	C₄⋊C₄	—	2	4	no	—
599	C₄⋊C₄	—	2	2	no	—
600	C₄⋊C₄	—	2	2	no	—
601	C₄⋊C₄	—	2	2	no	—
602	C₄⋊C₄	—	2	4	no	—
603	C₄⋊C₄	—	2	4	no	—
604	C₄⋊C₄	—	2	4	no	—
605	C₄⋊C₄	—	2	4	no	—
606	C₄⋊C₄	—	2	4	no	—
607	C₄⋊C₄	—	2	4	no	—
608	C₄⋊C₄	—	2	4	no	—
609	C₄⋊C₄	—	2	4	no	—
610	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
611	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
612	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
613	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
614	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
615	C₂²⋊C₄	—	2	4	no	—
616	C₂²⋊C₄	—	2	4	no	—
617	C₂²⋊C₄	—	2	2	no	—
618	C₂²⋊C₄	—	2	4	no	—
619	C₂²⋊C₄	—	2	2	no	—
620	C₂²⋊C₄	—	2	2	no	—
621	C₂²⋊C₄	—	2	4	no	—
622	C₂²⋊C₄	—	2	4	no	—
623	C₂²⋊C₄	—	2	4	no	—
624	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
625	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
626	C₄²	—	2	4	no	—
627	C₄²	—	2	2	no	—
628	C₄²	—	2	2	no	—
629	C₄²	—	2	2	no	—
630	C₄²	—	2	4	no	—
631	C₄²	C₂×D₄×C₃₁	1	2	yes	—
632	C₄²	C₂×D₄×C₃₁	1	4	yes	—
633	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
634	C₄²	C₂⁴×C₃₁	1	2	yes	—
635	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
636	C₂³	—	2	2	no	—
637	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
638	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
639	Q₈	—	2	2	no	—
640	Q₈	—	2	4	no	—
641	Q₈	—	2	2	no	—
642	Q₈	—	2	4	no	—
643	Q₈	—	2	2	no	—
644	Q₈	—	4	4	no	—
645	Q₈	—	8	4	no	—
646	D₄	—	2	4	no	—
647	D₄	—	2	2	no	—
648	D₄	—	2	2	no	—
649	D₄	—	2	2	no	—
650	D₄	—	4	4	no	—
651	D₄	—	8	4	no	—
652	C₂×C₄	—	2	2	no	—
653	C₂×C₄	—	2	2	no	—
654	C₂×C₄	—	2	2	no	—
655	C₂×C₄	—	2	4	no	—
656	C₂×C₄	—	2	2	no	—
657	C₂×C₄	—	2	2	no	—
658	C₂×C₄	—	2	4	no	—
659	C₂×C₄	—	4	2	no	—
660	C₂×C₄	—	4	4	no	—
661	C₂×C₄	—	4	4	no	—
662	C₂×C₄	—	4	4	no	—
663	C₂×C₄	—	4	4	no	—
664	C₂×C₄	—	8	2	no	—
665	C₂×C₄	—	8	2	no	—
666	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
667	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
668	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
669	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
670	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
671	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
672	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
673	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
674	C₈	—	2	1	no	—
675	C₈	—	2	1	no	—
676	C₈	—	2	2	no	—
677	C₈	C₂×#41×C₃₁	1	2	yes	—
678	C₈	C₂×#41×C₃₁	1	4	yes	—
679	C₈	C₂×#41×C₃₁	1	2	yes	—
680	C₂²	—	2	2	no	—
681	C₂²	—	2	2	no	—
682	C₂²	—	2	4	no	—
683	C₂²	—	4	2	no	—
684	C₂²	—	4	2	no	—
685	C₂²	—	4	2	no	—
686	C₂²	C₃₁×#532	1	2	yes	—
687	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
688	C₄	—	2	2	no	—
689	C₄	—	2	2	no	—
690	C₄	—	2	2	no	—
691	C₄	—	2	4	no	—
692	C₄	—	4	2	no	—
693	C₄	—	4	4	no	—
694	C₄	—	4	2	no	—
695	C₄	—	4	4	no	—
696	C₄	—	4	2	no	—
697	C₄	C₃₁×#532	1	2	yes	—
698	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
699	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
700	C₂	—	2	2	no	—
701	C₂	—	4	2	no	—
702	C₂	—	8	2	no	—
703	C₂	C₂×C₃₁×#532	1	1	yes	characteristic
704	C₂	C₃₁×#3586	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₃₁×#3060	C₂
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×#3461	C₂	semidirect
2. C₃₁×#3185	C₂	semidirect
3. C₃₁×#3183	C₂	semidirect
4. C₃₁×#3181	C₂	semidirect
5. C₃₁×#3178	C₂	semidirect
6. C₃₁×#3177	C₂	semidirect
7. C₃₁×#3173	C₂	semidirect
8. C₃₁×#3163	C₂	semidirect
9. C₃₁×#3156	C₂	semidirect
10. C₃₁×#3137	C₂	semidirect
11. C₃₁×#3063	C₂	semidirect
13. C₃₁×#3059	C₂	semidirect
14. C₄×C₃₁×#137	C₂	semidirect
15. C₃₁×#2943	C₂	semidirect
16. C₂×C₃₁×#441	C₂	semidirect
17. C₃₁×#2875	C₂	semidirect
283. #60699	C₃₁	direct (N × H)
328. #3461	C₂×C₃₁	semidirect
329. #3185	C₂×C₃₁	semidirect
330. #3183	C₂×C₃₁	semidirect
331. #3181	C₂×C₃₁	semidirect
332. #3178	C₂×C₃₁	semidirect
333. #3177	C₂×C₃₁	semidirect
334. #3173	C₂×C₃₁	semidirect
335. #3163	C₂×C₃₁	semidirect
336. #3156	C₂×C₃₁	semidirect
337. #3137	C₂×C₃₁	semidirect
338. #3063	C₂×C₃₁	semidirect
340. #3059	C₂×C₃₁	semidirect
341. C₄×#137	C₂×C₃₁	semidirect
342. #2943	C₂×C₃₁	semidirect
343. C₂×#441	C₂×C₃₁	semidirect
344. #2875	C₂×C₃₁	semidirect
550. C₃₁	#60699	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×#3461	2
C₃₁×#3185	2
C₃₁×#3183	2
C₃₁×#3181	2
C₃₁×#3178	2
C₃₁×#3177	2
C₃₁×#3173	2
C₃₁×#3163	2
C₃₁×#3156	2
C₃₁×#3137	2
C₃₁×#3063	2
C₃₁×#3060	2
C₃₁×#3059	2
C₄×C₃₁×#137	2
C₃₁×#2943	2
C₂×C₃₁×#441	2
C₃₁×#2875	2
#60699	31

Subgroups — C31×#60699 #21517458

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60699 #21517458