Subgroups — C₃₁×#36591 #21493350

Order 7936.

1838 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2861	C₂	1	2	yes	maximal, complemented
2	C₃₁×#2858	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2595	C₂	1	4	yes	maximal, complemented
4	C₃₁×#2568	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2548	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#2516	C₂	1	4	yes	maximal, complemented
7	C₃₁×#2513	C₂	1	2	yes	maximal, complemented
8	C₃₁×#2448	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2448	C₂	1	2	yes	maximal, complemented
10	C₃₁×#2447	C₂	1	2	yes	maximal, complemented
11	C₃₁×#2367	C₂	1	2	yes	maximal, complemented
12	C₃₁×#2366	C₂	1	1	yes	maximal, characteristic
13	C₃₁×#2365	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#2357	C₂	1	2	yes	maximal, complemented
15	C₄×C₃₁×#125	C₂	1	1	yes	maximal, characteristic, complemented
16	C₄×C₃₁×#124	C₂	1	2	yes	maximal, complemented
17	C₄²×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#124	C₂²	1	2	yes	—
22	C₂×C₃₁×#124	C₂²	1	2	yes	—
23	C₂×C₃₁×#124	C₂²	1	2	yes	—
24	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
26	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
28	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
29	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
30	C₃₁×#400	C₂²	1	4	yes	—
31	C₃₁×#399	C₂²	1	4	yes	—
32	C₃₁×#397	C₂²	1	4	yes	—
33	C₃₁×#397	C₂²	1	4	yes	—
34	C₃₁×#397	C₂²	1	4	yes	—
35	C₃₁×#394	C₂²	1	4	yes	—
36	C₃₁×#391	C₂²	1	2	yes	—
37	C₃₁×#391	C₂²	1	2	yes	—
38	C₃₁×#391	C₂²	1	2	yes	—
39	C₃₁×#391	C₂²	1	2	yes	—
40	C₃₁×#390	C₂²	1	2	yes	—
41	C₃₁×#390	C₂²	1	2	yes	—
42	C₃₁×#390	C₂²	1	2	yes	—
43	C₃₁×#390	C₂²	1	2	yes	—
44	C₃₁×#390	C₂²	1	2	yes	—
45	C₃₁×#390	C₂²	1	2	yes	—
46	C₃₁×#390	C₂²	1	2	yes	—
47	C₃₁×#390	C₂²	1	2	yes	—
48	C₃₁×#390	C₂²	1	2	yes	—
49	C₃₁×#389	C₂²	1	2	yes	—
50	C₃₁×#389	C₂²	1	1	yes	characteristic
51	C₃₁×#389	C₂²	1	1	yes	characteristic
52	C₃₁×#388	C₂²	1	4	yes	—
53	C₃₁×#388	C₂²	1	4	yes	—
54	C₃₁×#388	C₂²	1	4	yes	—
55	C₃₁×#388	C₂²	1	4	yes	—
56	C₃₁×#388	C₂²	1	4	yes	—
57	C₃₁×#388	C₂²	1	4	yes	—
58	C₃₁×#388	C₂²	1	4	yes	—
59	C₃₁×#388	C₂²	1	4	yes	—
60	C₃₁×#387	C₂²	1	2	yes	—
61	C₃₁×#387	C₂²	1	4	yes	—
62	C₃₁×#387	C₂²	1	2	yes	—
63	C₃₁×#386	C₂²	1	4	yes	—
64	C₃₁×#385	C₂²	1	4	yes	—
65	C₃₁×#382	C₂²	1	2	yes	—
66	C₃₁×#382	C₂²	1	1	yes	characteristic
67	C₃₁×#381	C₂²	1	4	yes	—
68	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
69	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
70	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
71	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
72	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
73	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
74	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
75	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
76	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
77	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
78	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
79	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
80	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
81	C₄³×C₃₁	C₂²	1	1	yes	characteristic
82	C₄³×C₃₁	C₂²	1	2	yes	—
83	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
84	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
85	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
86	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
87	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
88	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
89	C₃₁×C₄⋊Q₈	—	2	4	no	—
90	C₃₁×#127	—	2	4	no	—
91	C₃₁×#125	—	2	4	no	—
92	C₃₁×#124	—	2	8	no	—
93	C₃₁×#124	—	2	8	no	—
94	C₃₁×#124	—	4	4	no	—
95	C₄×Q₈×C₃₁	—	2	4	no	—
96	C₄×Q₈×C₃₁	—	2	4	no	—
97	C₄×Q₈×C₃₁	—	2	4	no	—
98	C₄×Q₈×C₃₁	—	4	4	no	—
99	C₄×D₄×C₃₁	—	2	8	no	—
100	C₄×D₄×C₃₁	—	4	4	no	—
101	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
102	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
104	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
108	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
109	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
110	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
111	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
113	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
114	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
115	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
116	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
117	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
118	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
119	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
120	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
121	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
122	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
123	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
124	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
125	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
126	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
127	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
128	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
129	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
130	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
131	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
132	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
133	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
134	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
135	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
136	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
137	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
138	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
139	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
140	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
141	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
142	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
143	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
144	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
145	C₃₁×#95	C₂³	1	2	yes	—
146	C₃₁×#95	C₂³	1	2	yes	—
147	C₃₁×#95	C₂³	1	2	yes	—
148	C₃₁×#95	C₂³	1	4	yes	—
149	C₃₁×#95	C₂³	1	4	yes	—
150	C₃₁×#95	C₂³	1	4	yes	—
151	C₃₁×#95	C₂³	1	4	yes	—
152	C₃₁×#95	C₂³	1	4	yes	—
153	C₃₁×#95	C₂³	1	2	yes	—
154	C₃₁×#95	C₂³	1	2	yes	—
155	C₃₁×#95	C₂³	1	2	yes	—
156	C₃₁×#95	C₂³	1	2	yes	—
157	C₃₁×#95	C₂³	1	2	yes	—
158	C₃₁×#95	C₂³	1	4	yes	—
159	C₃₁×#95	C₂³	1	2	yes	—
160	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
161	C₂×Q₈×C₃₁	—	2	2	no	—
162	C₂×Q₈×C₃₁	—	2	2	no	—
163	C₂×Q₈×C₃₁	—	2	4	no	—
164	C₂×Q₈×C₃₁	—	2	4	no	—
165	C₂×Q₈×C₃₁	—	2	4	no	—
166	C₂×Q₈×C₃₁	#41	1	4	yes	—
167	C₂×D₄×C₃₁	—	2	4	no	—
168	C₂×D₄×C₃₁	—	2	4	no	—
169	C₂×D₄×C₃₁	—	2	4	no	—
170	C₂×D₄×C₃₁	—	2	4	no	—
171	C₂×D₄×C₃₁	—	2	4	no	—
172	C₂×D₄×C₃₁	—	2	4	no	—
173	C₂×D₄×C₃₁	—	2	4	no	—
174	C₂×D₄×C₃₁	—	4	2	no	—
175	C₂×D₄×C₃₁	—	4	2	no	—
176	C₂²×C₄×C₃₁	—	2	8	no	—
177	C₂²×C₄×C₃₁	—	4	4	no	—
178	C₂²×C₄×C₃₁	—	4	4	no	—
179	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
180	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
181	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
182	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
183	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
184	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
185	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
186	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
187	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
188	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
189	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
190	C₂²×C₄×C₃₁	C₂⁴	1	2	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	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
195	C₄⋊C₄×C₃₁	—	2	4	no	—
196	C₄⋊C₄×C₃₁	—	2	4	no	—
197	C₄⋊C₄×C₃₁	—	2	4	no	—
198	C₄⋊C₄×C₃₁	—	2	2	no	—
199	C₄⋊C₄×C₃₁	—	2	2	no	—
200	C₄⋊C₄×C₃₁	—	2	4	no	—
201	C₄⋊C₄×C₃₁	—	2	4	no	—
202	C₄⋊C₄×C₃₁	—	2	2	no	—
203	C₄⋊C₄×C₃₁	—	2	2	no	—
204	C₄⋊C₄×C₃₁	—	2	4	no	—
205	C₄⋊C₄×C₃₁	—	2	4	no	—
206	C₄⋊C₄×C₃₁	—	2	2	no	—
207	C₄⋊C₄×C₃₁	—	2	2	no	—
208	C₄⋊C₄×C₃₁	—	4	2	no	—
209	C₄⋊C₄×C₃₁	—	4	2	no	—
210	C₄⋊C₄×C₃₁	—	4	4	no	—
211	C₄⋊C₄×C₃₁	#41	1	4	yes	—
212	C₂²⋊C₄×C₃₁	—	2	8	no	—
213	C₂²⋊C₄×C₃₁	—	2	8	no	—
214	C₂²⋊C₄×C₃₁	—	2	8	no	—
215	C₂²⋊C₄×C₃₁	—	2	8	no	—
216	C₂²⋊C₄×C₃₁	—	2	8	no	—
217	C₂²⋊C₄×C₃₁	—	4	4	no	—
218	C₂²⋊C₄×C₃₁	—	4	4	no	—
219	C₂²⋊C₄×C₃₁	—	4	4	no	—
220	C₂²⋊C₄×C₃₁	—	4	4	no	—
221	C₂²⋊C₄×C₃₁	—	4	4	no	—
222	C₂²⋊C₄×C₃₁	—	4	4	no	—
223	C₄²×C₃₁	—	2	4	no	—
224	C₄²×C₃₁	—	2	4	no	—
225	C₄²×C₃₁	—	2	4	no	—
226	C₄²×C₃₁	—	2	2	no	—
227	C₄²×C₃₁	—	2	2	no	—
228	C₄²×C₃₁	—	2	4	no	—
229	C₄²×C₃₁	—	2	4	no	—
230	C₄²×C₃₁	—	2	4	no	—
231	C₄²×C₃₁	—	2	2	no	—
232	C₄²×C₃₁	—	4	2	no	—
233	C₄²×C₃₁	—	4	1	no	—
234	C₄²×C₃₁	—	4	1	no	—
235	C₄²×C₃₁	—	4	2	no	—
236	C₄²×C₃₁	—	4	4	no	—
237	C₄²×C₃₁	#41	1	4	yes	—
238	C₄²×C₃₁	#41	1	4	yes	—
239	#36591	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
240	C₂³×C₃₁	—	2	4	no	—
241	C₂³×C₃₁	—	2	4	no	—
242	C₂³×C₃₁	—	2	4	no	—
243	C₂³×C₃₁	—	2	4	no	—
244	C₂³×C₃₁	—	2	4	no	—
245	C₂³×C₃₁	—	2	4	no	—
246	C₂³×C₃₁	—	2	4	no	—
247	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
248	Q₈×C₃₁	—	2	8	no	—
249	Q₈×C₃₁	—	4	8	no	—
250	D₄×C₃₁	—	4	8	no	—
251	D₄×C₃₁	—	4	8	no	—
252	D₄×C₃₁	—	8	4	no	—
253	C₂×C₄×C₃₁	—	2	1	no	—
254	C₂×C₄×C₃₁	—	2	1	no	—
255	C₂×C₄×C₃₁	—	2	2	no	—
256	C₂×C₄×C₃₁	—	2	2	no	—
257	C₂×C₄×C₃₁	—	2	2	no	—
258	C₂×C₄×C₃₁	—	2	1	no	—
259	C₂×C₄×C₃₁	—	2	1	no	—
260	C₂×C₄×C₃₁	—	2	2	no	—
261	C₂×C₄×C₃₁	—	2	1	no	—
262	C₂×C₄×C₃₁	—	2	1	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	1	no	—
268	C₂×C₄×C₃₁	—	2	1	no	—
269	C₂×C₄×C₃₁	—	2	2	no	—
270	C₂×C₄×C₃₁	—	2	2	no	—
271	C₂×C₄×C₃₁	—	2	4	no	—
272	C₂×C₄×C₃₁	—	2	2	no	—
273	C₂×C₄×C₃₁	—	2	2	no	—
274	C₂×C₄×C₃₁	—	2	4	no	—
275	C₂×C₄×C₃₁	—	2	1	no	—
276	C₂×C₄×C₃₁	—	2	1	no	—
277	C₂×C₄×C₃₁	—	2	2	no	—
278	C₂×C₄×C₃₁	—	2	2	no	—
279	C₂×C₄×C₃₁	—	2	4	no	—
280	C₂×C₄×C₃₁	—	2	2	no	—
281	C₂×C₄×C₃₁	—	2	2	no	—
282	C₂×C₄×C₃₁	—	2	2	no	—
283	C₂×C₄×C₃₁	—	2	2	no	—
284	C₂×C₄×C₃₁	—	2	1	no	—
285	C₂×C₄×C₃₁	—	2	1	no	—
286	C₂×C₄×C₃₁	—	2	2	no	—
287	C₂×C₄×C₃₁	—	2	2	no	—
288	C₂×C₄×C₃₁	—	2	2	no	—
289	C₂×C₄×C₃₁	—	2	2	no	—
290	C₂×C₄×C₃₁	—	8	8	no	—
291	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
292	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
293	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
294	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
295	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
296	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
297	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
298	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
299	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
300	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
301	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
302	#2861	C₂×C₃₁	1	2	yes	complemented
303	#2858	C₂×C₃₁	1	1	yes	characteristic
304	#2595	C₂×C₃₁	1	4	yes	complemented
305	#2568	C₂×C₃₁	1	1	yes	characteristic, complemented
306	#2548	C₂×C₃₁	1	1	yes	characteristic
307	#2516	C₂×C₃₁	1	4	yes	complemented
308	#2513	C₂×C₃₁	1	2	yes	complemented
309	#2448	C₂×C₃₁	1	2	yes	complemented
310	#2448	C₂×C₃₁	1	2	yes	complemented
311	#2447	C₂×C₃₁	1	2	yes	complemented
312	#2367	C₂×C₃₁	1	2	yes	complemented
313	#2366	C₂×C₃₁	1	1	yes	characteristic
314	#2365	C₂×C₃₁	1	1	yes	characteristic, complemented
315	#2357	C₂×C₃₁	1	2	yes	complemented
316	C₄×#125	C₂×C₃₁	1	1	yes	characteristic, complemented
317	C₄×#124	C₂×C₃₁	1	2	yes	complemented
318	C₄²×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
319	C₂²×C₃₁	—	4	4	no	—
320	C₂²×C₃₁	—	4	4	no	—
321	C₂²×C₃₁	—	4	4	no	—
322	C₂²×C₃₁	—	4	4	no	—
323	C₂²×C₃₁	—	4	4	no	—
324	C₂²×C₃₁	—	4	4	no	—
325	C₂²×C₃₁	—	4	4	no	—
326	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
327	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
328	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
329	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
330	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
331	C₂²×C₃₁	#585	1	1	yes	characteristic
332	C₂²×C₃₁	#585	1	1	yes	characteristic
333	C₄×C₃₁	—	2	4	no	—
334	C₄×C₃₁	—	2	2	no	—
335	C₄×C₃₁	—	2	2	no	—
336	C₄×C₃₁	—	2	2	no	—
337	C₄×C₃₁	—	2	4	no	—
338	C₄×C₃₁	—	4	4	no	—
339	C₄×C₃₁	—	4	2	no	—
340	C₄×C₃₁	—	4	2	no	—
341	C₄×C₃₁	—	4	2	no	—
342	C₄×C₃₁	—	4	1	no	—
343	C₄×C₃₁	—	4	2	no	—
344	C₄×C₃₁	—	4	2	no	—
345	C₄×C₃₁	—	4	1	no	—
346	C₄×C₃₁	—	4	2	no	—
347	C₄×C₃₁	#529	1	4	yes	—
348	C₄×C₃₁	#559	1	4	yes	—
349	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
350	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
351	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
352	C₂×#124	C₂²×C₃₁	1	2	yes	—
353	C₂×#124	C₂²×C₃₁	1	2	yes	—
354	C₂×#124	C₂²×C₃₁	1	2	yes	—
355	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
356	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
357	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
358	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
359	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
360	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
361	#400	C₂²×C₃₁	1	4	yes	—
362	#399	C₂²×C₃₁	1	4	yes	—
363	#397	C₂²×C₃₁	1	4	yes	—
364	#397	C₂²×C₃₁	1	4	yes	—
365	#397	C₂²×C₃₁	1	4	yes	—
366	#394	C₂²×C₃₁	1	4	yes	—
367	#391	C₂²×C₃₁	1	2	yes	—
368	#391	C₂²×C₃₁	1	2	yes	—
369	#391	C₂²×C₃₁	1	2	yes	—
370	#391	C₂²×C₃₁	1	2	yes	—
371	#390	C₂²×C₃₁	1	2	yes	—
372	#390	C₂²×C₃₁	1	2	yes	—
373	#390	C₂²×C₃₁	1	2	yes	—
374	#390	C₂²×C₃₁	1	2	yes	—
375	#390	C₂²×C₃₁	1	2	yes	—
376	#390	C₂²×C₃₁	1	2	yes	—
377	#390	C₂²×C₃₁	1	2	yes	—
378	#390	C₂²×C₃₁	1	2	yes	—
379	#390	C₂²×C₃₁	1	2	yes	—
380	#389	C₂²×C₃₁	1	1	yes	characteristic
381	#389	C₂²×C₃₁	1	1	yes	characteristic
382	#389	C₂²×C₃₁	1	2	yes	—
383	#388	C₂²×C₃₁	1	4	yes	—
384	#388	C₂²×C₃₁	1	4	yes	—
385	#388	C₂²×C₃₁	1	4	yes	—
386	#388	C₂²×C₃₁	1	4	yes	—
387	#388	C₂²×C₃₁	1	4	yes	—
388	#388	C₂²×C₃₁	1	4	yes	—
389	#388	C₂²×C₃₁	1	4	yes	—
390	#388	C₂²×C₃₁	1	4	yes	—
391	#387	C₂²×C₃₁	1	2	yes	—
392	#387	C₂²×C₃₁	1	2	yes	—
393	#387	C₂²×C₃₁	1	4	yes	—
394	#386	C₂²×C₃₁	1	4	yes	—
395	#385	C₂²×C₃₁	1	4	yes	—
396	#382	C₂²×C₃₁	1	1	yes	characteristic
397	#382	C₂²×C₃₁	1	2	yes	—
398	#381	C₂²×C₃₁	1	4	yes	—
399	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
400	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
401	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
402	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
403	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
404	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
405	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
406	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
407	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
408	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
409	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
410	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
411	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
412	C₄³	C₂²×C₃₁	1	1	yes	characteristic
413	C₄³	C₂²×C₃₁	1	2	yes	—
414	C₂×C₃₁	—	8	4	no	—
415	C₂×C₃₁	C₂×#529	1	1	yes	characteristic
416	C₂×C₃₁	C₂×#559	1	1	yes	characteristic
417	C₂×C₃₁	#3480	1	1	yes	characteristic
418	C₂×C₃₁	#3480	1	1	yes	characteristic
419	C₂×C₃₁	#3510	1	1	yes	characteristic
420	C₂×C₃₁	#3514	1	1	yes	characteristic
421	C₂×C₃₁	#3517	1	1	yes	characteristic
422	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
423	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
424	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
425	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
426	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
427	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
428	C₄⋊Q₈	—	2	4	no	—
429	#127	—	2	4	no	—
430	#125	—	2	4	no	—
431	#124	—	2	8	no	—
432	#124	—	2	8	no	—
433	#124	—	4	4	no	—
434	C₄×Q₈	—	2	4	no	—
435	C₄×Q₈	—	2	4	no	—
436	C₄×Q₈	—	2	4	no	—
437	C₄×Q₈	—	4	4	no	—
438	C₄×D₄	—	2	8	no	—
439	C₄×D₄	—	4	4	no	—
440	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
441	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
442	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
443	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
444	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
445	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
446	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
447	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
448	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
449	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
450	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
451	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
452	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
453	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
454	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
455	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
456	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
457	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
458	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
459	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
460	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
461	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
462	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
463	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
464	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
465	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
466	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
467	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
468	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
469	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
470	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
471	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
472	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
473	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
474	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
475	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
476	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
477	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
478	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
479	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
480	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
481	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
482	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
483	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
484	#95	C₂³×C₃₁	1	2	yes	—
485	#95	C₂³×C₃₁	1	2	yes	—
486	#95	C₂³×C₃₁	1	4	yes	—
487	#95	C₂³×C₃₁	1	4	yes	—
488	#95	C₂³×C₃₁	1	2	yes	—
489	#95	C₂³×C₃₁	1	2	yes	—
490	#95	C₂³×C₃₁	1	2	yes	—
491	#95	C₂³×C₃₁	1	2	yes	—
492	#95	C₂³×C₃₁	1	4	yes	—
493	#95	C₂³×C₃₁	1	2	yes	—
494	#95	C₂³×C₃₁	1	2	yes	—
495	#95	C₂³×C₃₁	1	2	yes	—
496	#95	C₂³×C₃₁	1	4	yes	—
497	#95	C₂³×C₃₁	1	4	yes	—
498	#95	C₂³×C₃₁	1	4	yes	—
499	C₃₁	#36591	1	1	yes	characteristic, direct product factor, Sylow
500	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
501	C₂×Q₈	—	2	2	no	—
502	C₂×Q₈	—	2	2	no	—
503	C₂×Q₈	—	2	4	no	—
504	C₂×Q₈	—	2	4	no	—
505	C₂×Q₈	—	2	4	no	—
506	C₂×Q₈	#41×C₃₁	1	4	yes	—
507	C₂×D₄	—	2	4	no	—
508	C₂×D₄	—	2	4	no	—
509	C₂×D₄	—	2	4	no	—
510	C₂×D₄	—	2	4	no	—
511	C₂×D₄	—	2	4	no	—
512	C₂×D₄	—	2	4	no	—
513	C₂×D₄	—	2	4	no	—
514	C₂×D₄	—	4	2	no	—
515	C₂×D₄	—	4	2	no	—
516	C₂²×C₄	—	2	8	no	—
517	C₂²×C₄	—	4	4	no	—
518	C₂²×C₄	—	4	4	no	—
519	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
520	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
521	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
522	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
523	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
524	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
525	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
526	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
527	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
528	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
529	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
530	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
531	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
532	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
533	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
534	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
535	C₄⋊C₄	—	2	4	no	—
536	C₄⋊C₄	—	2	4	no	—
537	C₄⋊C₄	—	2	4	no	—
538	C₄⋊C₄	—	2	2	no	—
539	C₄⋊C₄	—	2	2	no	—
540	C₄⋊C₄	—	2	4	no	—
541	C₄⋊C₄	—	2	4	no	—
542	C₄⋊C₄	—	2	2	no	—
543	C₄⋊C₄	—	2	2	no	—
544	C₄⋊C₄	—	2	4	no	—
545	C₄⋊C₄	—	2	4	no	—
546	C₄⋊C₄	—	2	2	no	—
547	C₄⋊C₄	—	2	2	no	—
548	C₄⋊C₄	—	4	2	no	—
549	C₄⋊C₄	—	4	2	no	—
550	C₄⋊C₄	—	4	4	no	—
551	C₄⋊C₄	#41×C₃₁	1	4	yes	—
552	C₂²⋊C₄	—	2	8	no	—
553	C₂²⋊C₄	—	2	8	no	—
554	C₂²⋊C₄	—	2	8	no	—
555	C₂²⋊C₄	—	2	8	no	—
556	C₂²⋊C₄	—	2	8	no	—
557	C₂²⋊C₄	—	4	4	no	—
558	C₂²⋊C₄	—	4	4	no	—
559	C₂²⋊C₄	—	4	4	no	—
560	C₂²⋊C₄	—	4	4	no	—
561	C₂²⋊C₄	—	4	4	no	—
562	C₂²⋊C₄	—	4	4	no	—
563	C₄²	—	2	4	no	—
564	C₄²	—	2	4	no	—
565	C₄²	—	2	4	no	—
566	C₄²	—	2	2	no	—
567	C₄²	—	2	2	no	—
568	C₄²	—	2	4	no	—
569	C₄²	—	2	4	no	—
570	C₄²	—	2	4	no	—
571	C₄²	—	2	2	no	—
572	C₄²	—	4	2	no	—
573	C₄²	—	4	1	no	—
574	C₄²	—	4	1	no	—
575	C₄²	—	4	2	no	—
576	C₄²	—	4	4	no	—
577	C₄²	#41×C₃₁	1	4	yes	—
578	C₄²	#41×C₃₁	1	4	yes	—
579	C₂³	—	2	4	no	—
580	C₂³	—	2	4	no	—
581	C₂³	—	2	4	no	—
582	C₂³	—	2	4	no	—
583	C₂³	—	2	4	no	—
584	C₂³	—	2	4	no	—
585	C₂³	—	2	4	no	—
586	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
587	Q₈	—	2	8	no	—
588	Q₈	—	4	8	no	—
589	D₄	—	4	8	no	—
590	D₄	—	4	8	no	—
591	D₄	—	8	4	no	—
592	C₂×C₄	—	2	1	no	—
593	C₂×C₄	—	2	1	no	—
594	C₂×C₄	—	2	2	no	—
595	C₂×C₄	—	2	2	no	—
596	C₂×C₄	—	2	2	no	—
597	C₂×C₄	—	2	1	no	—
598	C₂×C₄	—	2	1	no	—
599	C₂×C₄	—	2	2	no	—
600	C₂×C₄	—	2	1	no	—
601	C₂×C₄	—	2	1	no	—
602	C₂×C₄	—	2	2	no	—
603	C₂×C₄	—	2	2	no	—
604	C₂×C₄	—	2	2	no	—
605	C₂×C₄	—	2	2	no	—
606	C₂×C₄	—	2	1	no	—
607	C₂×C₄	—	2	1	no	—
608	C₂×C₄	—	2	2	no	—
609	C₂×C₄	—	2	2	no	—
610	C₂×C₄	—	2	4	no	—
611	C₂×C₄	—	2	2	no	—
612	C₂×C₄	—	2	2	no	—
613	C₂×C₄	—	2	4	no	—
614	C₂×C₄	—	2	1	no	—
615	C₂×C₄	—	2	1	no	—
616	C₂×C₄	—	2	2	no	—
617	C₂×C₄	—	2	2	no	—
618	C₂×C₄	—	2	4	no	—
619	C₂×C₄	—	2	2	no	—
620	C₂×C₄	—	2	2	no	—
621	C₂×C₄	—	2	2	no	—
622	C₂×C₄	—	2	2	no	—
623	C₂×C₄	—	2	1	no	—
624	C₂×C₄	—	2	1	no	—
625	C₂×C₄	—	2	2	no	—
626	C₂×C₄	—	2	2	no	—
627	C₂×C₄	—	2	2	no	—
628	C₂×C₄	—	2	2	no	—
629	C₂×C₄	—	8	8	no	—
630	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
631	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
632	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
633	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
634	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
635	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
636	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
637	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
638	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
639	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
640	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
641	C₂²	—	4	4	no	—
642	C₂²	—	4	4	no	—
643	C₂²	—	4	4	no	—
644	C₂²	—	4	4	no	—
645	C₂²	—	4	4	no	—
646	C₂²	—	4	4	no	—
647	C₂²	—	4	4	no	—
648	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
649	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
650	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
651	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
652	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
653	C₂²	C₃₁×#585	1	1	yes	characteristic
654	C₂²	C₃₁×#585	1	1	yes	characteristic
655	C₄	—	2	4	no	—
656	C₄	—	2	2	no	—
657	C₄	—	2	2	no	—
658	C₄	—	2	2	no	—
659	C₄	—	2	4	no	—
660	C₄	—	4	4	no	—
661	C₄	—	4	2	no	—
662	C₄	—	4	2	no	—
663	C₄	—	4	2	no	—
664	C₄	—	4	2	no	—
665	C₄	—	4	1	no	—
666	C₄	—	4	2	no	—
667	C₄	—	4	1	no	—
668	C₄	—	4	2	no	—
669	C₄	C₃₁×#529	1	4	yes	—
670	C₄	C₃₁×#559	1	4	yes	—
671	C₂	—	8	4	no	—
672	C₂	C₂×C₃₁×#529	1	1	yes	characteristic
673	C₂	C₂×C₃₁×#559	1	1	yes	characteristic
674	C₂	C₃₁×#3480	1	1	yes	characteristic
675	C₂	C₃₁×#3480	1	1	yes	characteristic
676	C₂	C₃₁×#3510	1	1	yes	characteristic
677	C₂	C₃₁×#3514	1	1	yes	characteristic
678	C₂	C₃₁×#3517	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#2861	C₂	semidirect
3. C₃₁×#2595	C₂	semidirect
4. C₃₁×#2568	C₂	semidirect
6. C₃₁×#2516	C₂	semidirect
7. C₃₁×#2513	C₂	semidirect
8. C₃₁×#2448	C₂	semidirect
9. C₃₁×#2448	C₂	semidirect
10. C₃₁×#2447	C₂	semidirect
11. C₃₁×#2367	C₂	semidirect
13. C₃₁×#2365	C₂	semidirect
14. C₃₁×#2357	C₂	semidirect
15. C₄×C₃₁×#125	C₂	semidirect
16. C₄×C₃₁×#124	C₂	semidirect
17. C₄²×Q₈×C₃₁	C₂	semidirect
239. #36591	C₃₁	direct (N × H)
302. #2861	C₂×C₃₁	semidirect
304. #2595	C₂×C₃₁	semidirect
305. #2568	C₂×C₃₁	semidirect
307. #2516	C₂×C₃₁	semidirect
308. #2513	C₂×C₃₁	semidirect
309. #2448	C₂×C₃₁	semidirect
310. #2448	C₂×C₃₁	semidirect
311. #2447	C₂×C₃₁	semidirect
312. #2367	C₂×C₃₁	semidirect
314. #2365	C₂×C₃₁	semidirect
315. #2357	C₂×C₃₁	semidirect
316. C₄×#125	C₂×C₃₁	semidirect
317. C₄×#124	C₂×C₃₁	semidirect
318. C₄²×Q₈	C₂×C₃₁	semidirect
499. C₃₁	#36591	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×#2861	2
C₃₁×#2858	2
C₃₁×#2595	2
C₃₁×#2568	2
C₃₁×#2548	2
C₃₁×#2516	2
C₃₁×#2513	2
C₃₁×#2448	2
C₃₁×#2447	2
C₃₁×#2367	2
C₃₁×#2366	2
C₃₁×#2365	2
C₃₁×#2357	2
C₄×C₃₁×#125	2
C₄×C₃₁×#124	2
C₄²×Q₈×C₃₁	2
#36591	31

Subgroups — C31×#36591 #21493350

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#36591 #21493350