Subgroups — C₃₁×#34377 #21491136

Order 7936.

2586 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#124	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2379	C₂	1	4	yes	maximal, complemented
3	C₃₁×#2367	C₂	1	4	yes	maximal, complemented
4	C₃₁×#2342	C₂	1	4	yes	maximal, complemented
5	Q₈×C₂²⋊C₄×C₃₁	C₂	1	2	yes	maximal, complemented
6	D₄×C₂²⋊C₄×C₃₁	C₂	1	2	yes	maximal
7	C₃₁×#2337	C₂	1	4	yes	maximal, complemented
8	C₃₁×#2326	C₂	1	4	yes	maximal, complemented
9	C₃₁×#2315	C₂	1	2	yes	maximal, complemented
10	C₃₁×#2312	C₂	1	2	yes	maximal, complemented
11	C₃₁×#2285	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#2283	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#124	C₄	1	16	yes	complemented
16	C₂×C₃₁×#124	C₂²	1	2	yes	—
17	C₂×C₃₁×#124	C₂²	1	2	yes	—
18	C₂×C₃₁×#124	C₂²	1	2	yes	—
19	C₂×C₃₁×#124	C₂²	1	2	yes	—
20	C₂×C₄×Q₈×C₃₁	C₂²	1	4	yes	complemented
21	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
22	C₂×C₃₁×#117	C₂²	1	2	yes	—
23	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
24	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
25	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
26	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#391	C₂²	1	4	yes	complemented
28	C₃₁×#391	C₂²	1	8	yes	complemented
29	C₃₁×#390	C₂²	1	4	yes	—
30	C₃₁×#388	C₂²	1	8	yes	complemented
31	C₃₁×#388	C₂²	1	8	yes	complemented
32	C₃₁×#388	C₂²	1	8	yes	complemented
33	C₃₁×#388	C₂²	1	8	yes	complemented
34	C₃₁×#386	C₂²	1	8	yes	—
35	C₃₁×#385	C₂²	1	8	yes	complemented
36	C₃₁×#385	C₂²	1	8	yes	complemented
37	C₃₁×#385	C₂²	1	8	yes	complemented
38	C₃₁×#384	C₂²	1	2	yes	complemented
39	C₃₁×#383	C₂²	1	2	yes	complemented
40	C₃₁×#382	C₂²	1	2	yes	complemented
41	C₃₁×#382	C₂²	1	2	yes	complemented
42	C₃₁×#381	C₂²	1	2	yes	—
43	C₃₁×#380	C₂²	1	1	yes	characteristic
44	C₃₁×#380	C₂²	1	2	yes	—
45	C₃₁×#380	C₂²	1	1	yes	characteristic
46	C₃₁×#380	C₂²	1	4	yes	—
47	C₃₁×#380	C₂²	1	2	yes	—
48	C₃₁×#379	C₂²	1	4	yes	—
49	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
50	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
51	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
52	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
53	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
54	C₄×C₂²⋊C₄×C₃₁	C₂²	1	8	yes	complemented
55	C₃₁×#376	C₂²	1	2	yes	complemented
56	C₂⁵×C₃₁	C₂³	1	2	yes	—
57	C₂²×Q₈×C₃₁	—	4	4	no	—
58	C₂²×Q₈×C₃₁	C₂×C₄	1	8	yes	complemented
59	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
60	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
61	C₂²×D₄×C₃₁	—	4	4	no	—
62	C₂²×D₄×C₃₁	C₂×C₄	1	8	yes	—
63	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
64	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
65	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₃₁	C₂³	1	8	yes	—
68	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
70	C₃₁×#124	C₂×C₄	1	16	yes	complemented
71	C₃₁×#124	C₂×C₄	1	16	yes	complemented
72	C₃₁×#124	C₂×C₄	1	16	yes	complemented
73	C₃₁×#124	C₂×C₄	1	16	yes	complemented
74	C₄×Q₈×C₃₁	—	4	8	no	—
75	C₄×D₄×C₃₁	—	4	8	no	—
76	C₃₁×#117	—	4	4	no	—
77	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
78	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
79	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
80	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
81	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
82	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
83	C₂×C₂²⋊C₄×C₃₁	—	4	8	no	—
84	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	16	yes	complemented
85	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	16	yes	complemented
86	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
87	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
88	C₂×C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
89	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
90	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
91	C₂×C₂²⋊C₄×C₃₁	C₂³	1	8	yes	—
92	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
93	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
94	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
95	C₂×C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
96	C₂×C₄²×C₃₁	C₂×C₄	1	8	yes	complemented
97	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
98	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
99	C₂×C₄²×C₃₁	C₂³	1	8	yes	—
100	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
102	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
103	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
104	C₃₁×#95	C₂³	1	2	yes	—
105	C₃₁×#95	C₂³	1	2	yes	—
106	C₃₁×#95	C₂³	1	4	yes	—
107	C₃₁×#95	C₂³	1	4	yes	—
108	C₃₁×#95	C₂³	1	8	yes	—
109	C₃₁×#95	C₂³	1	4	yes	—
110	C₃₁×#95	C₂³	1	8	yes	—
111	C₃₁×#95	C₂³	1	4	yes	—
112	C₃₁×#95	C₂³	1	4	yes	—
113	C₂⁴×C₃₁	—	2	4	no	—
114	C₂⁴×C₃₁	—	2	4	no	—
115	C₂⁴×C₃₁	—	4	4	no	—
116	C₂⁴×C₃₁	—	4	4	no	—
117	C₂⁴×C₃₁	C₂²×C₄	1	8	yes	—
118	C₂⁴×C₃₁	C₂⁴	1	4	yes	—
119	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
120	C₂×Q₈×C₃₁	—	2	8	no	—
121	C₂×Q₈×C₃₁	—	4	4	no	—
122	C₂×Q₈×C₃₁	—	4	16	no	—
123	C₂×Q₈×C₃₁	C₂²×C₄	1	8	yes	—
124	C₂×Q₈×C₃₁	C₂²×C₄	1	8	yes	—
125	C₂×D₄×C₃₁	—	2	8	no	—
126	C₂×D₄×C₃₁	—	4	16	no	—
127	C₂×D₄×C₃₁	—	4	4	no	—
128	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
129	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
130	C₂²×C₄×C₃₁	—	2	2	no	—
131	C₂²×C₄×C₃₁	—	2	2	no	—
132	C₂²×C₄×C₃₁	—	2	4	no	—
133	C₂²×C₄×C₃₁	—	4	2	no	—
134	C₂²×C₄×C₃₁	—	4	4	no	—
135	C₂²×C₄×C₃₁	—	4	8	no	—
136	C₂²×C₄×C₃₁	—	4	8	no	—
137	C₂²×C₄×C₃₁	—	4	8	no	—
138	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
139	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
140	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
141	C₂²×C₄×C₃₁	C₂²×C₄	1	8	yes	—
142	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
143	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
144	C₂²×C₄×C₃₁	C₂⁴	1	8	yes	—
145	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
146	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
147	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
148	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
149	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
150	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
151	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
152	C₄⋊C₄×C₃₁	—	2	8	no	—
153	C₄⋊C₄×C₃₁	—	4	2	no	—
154	C₄⋊C₄×C₃₁	—	4	4	no	—
155	C₄⋊C₄×C₃₁	—	4	2	no	—
156	C₄⋊C₄×C₃₁	—	4	4	no	—
157	C₄⋊C₄×C₃₁	—	4	8	no	—
158	C₂²⋊C₄×C₃₁	—	2	4	no	—
159	C₂²⋊C₄×C₃₁	—	2	8	no	—
160	C₂²⋊C₄×C₃₁	—	2	4	no	—
161	C₂²⋊C₄×C₃₁	—	4	8	no	—
162	C₂²⋊C₄×C₃₁	—	4	8	no	—
163	C₂²⋊C₄×C₃₁	—	4	8	no	—
164	C₂²⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
165	C₂²⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
166	C₂²⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
167	C₂²⋊C₄×C₃₁	C₂²×C₄	1	16	yes	—
168	C₄²×C₃₁	—	2	8	no	—
169	C₄²×C₃₁	—	4	4	no	—
170	C₄²×C₃₁	—	4	8	no	—
171	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
172	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
173	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
174	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
175	#34377	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
176	C₂³×C₃₁	—	2	2	no	—
177	C₂³×C₃₁	—	2	2	no	—
178	C₂³×C₃₁	—	2	4	no	—
179	C₂³×C₃₁	—	2	8	no	—
180	C₂³×C₃₁	—	2	4	no	—
181	C₂³×C₃₁	—	2	8	no	—
182	C₂³×C₃₁	—	2	16	no	—
183	C₂³×C₃₁	—	4	8	no	—
184	C₂³×C₃₁	—	4	8	no	—
185	C₂³×C₃₁	—	8	8	no	—
186	C₂³×C₃₁	—	8	8	no	—
187	C₂³×C₃₁	C₂³×C₄	1	8	yes	—
188	C₂³×C₃₁	C₂³×C₄	1	2	yes	—
189	C₂³×C₃₁	H(5,𝔽₂)	1	2	yes	—
190	C₂³×C₃₁	2₋¹⁺⁴	1	2	yes	—
191	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
192	Q₈×C₃₁	—	2	16	no	—
193	Q₈×C₃₁	—	4	8	no	—
194	D₄×C₃₁	—	2	16	no	—
195	D₄×C₃₁	—	4	8	no	—
196	C₂×C₄×C₃₁	—	2	8	no	—
197	C₂×C₄×C₃₁	—	2	1	no	—
198	C₂×C₄×C₃₁	—	2	8	no	—
199	C₂×C₄×C₃₁	—	2	1	no	—
200	C₂×C₄×C₃₁	—	2	2	no	—
201	C₂×C₄×C₃₁	—	2	1	no	—
202	C₂×C₄×C₃₁	—	2	1	no	—
203	C₂×C₄×C₃₁	—	2	2	no	—
204	C₂×C₄×C₃₁	—	2	2	no	—
205	C₂×C₄×C₃₁	—	2	8	no	—
206	C₂×C₄×C₃₁	—	2	4	no	—
207	C₂×C₄×C₃₁	—	2	4	no	—
208	C₂×C₄×C₃₁	—	2	8	no	—
209	C₂×C₄×C₃₁	—	2	8	no	—
210	C₂×C₄×C₃₁	—	2	4	no	—
211	C₂×C₄×C₃₁	—	2	4	no	—
212	C₂×C₄×C₃₁	—	2	4	no	—
213	C₂×C₄×C₃₁	—	2	8	no	—
214	C₂×C₄×C₃₁	—	2	4	no	—
215	C₂×C₄×C₃₁	—	4	4	no	—
216	C₂×C₄×C₃₁	—	4	8	no	—
217	C₂×C₄×C₃₁	—	8	16	no	—
218	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
219	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
220	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
221	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
222	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
223	C₂×C₄×C₃₁	C₂³×C₄	1	8	yes	—
224	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
225	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
226	C₂²×#124	C₂×C₃₁	1	1	yes	characteristic
227	#2379	C₂×C₃₁	1	4	yes	complemented
228	#2367	C₂×C₃₁	1	4	yes	complemented
229	#2342	C₂×C₃₁	1	4	yes	complemented
230	Q₈×C₂²⋊C₄	C₂×C₃₁	1	2	yes	complemented
231	D₄×C₂²⋊C₄	C₂×C₃₁	1	2	yes	—
232	#2337	C₂×C₃₁	1	4	yes	complemented
233	#2326	C₂×C₃₁	1	4	yes	complemented
234	#2315	C₂×C₃₁	1	2	yes	complemented
235	#2312	C₂×C₃₁	1	2	yes	complemented
236	#2285	C₂×C₃₁	1	1	yes	characteristic, complemented
237	#2283	C₂×C₃₁	1	1	yes	characteristic, complemented
238	C₂²×C₃₁	—	2	2	no	—
239	C₂²×C₃₁	—	2	2	no	—
240	C₂²×C₃₁	—	2	4	no	—
241	C₂²×C₃₁	—	2	4	no	—
242	C₂²×C₃₁	—	2	8	no	—
243	C₂²×C₃₁	—	2	8	no	—
244	C₂²×C₃₁	—	2	8	no	—
245	C₂²×C₃₁	—	4	8	no	—
246	C₂²×C₃₁	—	4	8	no	—
247	C₂²×C₃₁	—	8	16	no	—
248	C₂²×C₃₁	#520	1	4	yes	—
249	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
250	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
251	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
252	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
253	C₂²×C₃₁	#585	1	2	yes	—
254	C₄×C₃₁	—	2	2	no	—
255	C₄×C₃₁	—	2	8	no	—
256	C₄×C₃₁	—	2	2	no	—
257	C₄×C₃₁	—	2	2	no	—
258	C₄×C₃₁	—	2	2	no	—
259	C₄×C₃₁	—	2	2	no	—
260	C₄×C₃₁	—	4	8	no	—
261	C₄×C₃₁	—	4	4	no	—
262	C₄×C₃₁	—	4	4	no	—
263	C₄×C₃₁	#518	1	4	yes	—
264	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
265	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
266	C₂×#124	C₄×C₃₁	1	16	yes	complemented
267	C₂×#124	C₂²×C₃₁	1	2	yes	—
268	C₂×#124	C₂²×C₃₁	1	2	yes	—
269	C₂×#124	C₂²×C₃₁	1	2	yes	—
270	C₂×#124	C₂²×C₃₁	1	2	yes	—
271	C₂×C₄×Q₈	C₂²×C₃₁	1	4	yes	complemented
272	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
273	C₂×#117	C₂²×C₃₁	1	2	yes	—
274	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
275	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
276	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
277	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
278	#391	C₂²×C₃₁	1	4	yes	complemented
279	#391	C₂²×C₃₁	1	8	yes	complemented
280	#390	C₂²×C₃₁	1	4	yes	—
281	#388	C₂²×C₃₁	1	8	yes	complemented
282	#388	C₂²×C₃₁	1	8	yes	complemented
283	#388	C₂²×C₃₁	1	8	yes	complemented
284	#388	C₂²×C₃₁	1	8	yes	complemented
285	#386	C₂²×C₃₁	1	8	yes	—
286	#385	C₂²×C₃₁	1	8	yes	complemented
287	#385	C₂²×C₃₁	1	8	yes	complemented
288	#385	C₂²×C₃₁	1	8	yes	complemented
289	#384	C₂²×C₃₁	1	2	yes	complemented
290	#383	C₂²×C₃₁	1	2	yes	complemented
291	#382	C₂²×C₃₁	1	2	yes	complemented
292	#382	C₂²×C₃₁	1	2	yes	complemented
293	#381	C₂²×C₃₁	1	2	yes	—
294	#380	C₂²×C₃₁	1	1	yes	characteristic
295	#380	C₂²×C₃₁	1	2	yes	—
296	#380	C₂²×C₃₁	1	1	yes	characteristic
297	#380	C₂²×C₃₁	1	4	yes	—
298	#380	C₂²×C₃₁	1	2	yes	—
299	#379	C₂²×C₃₁	1	4	yes	—
300	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
301	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
302	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
303	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
304	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
305	C₄×C₂²⋊C₄	C₂²×C₃₁	1	8	yes	complemented
306	#376	C₂²×C₃₁	1	2	yes	complemented
307	C₂×C₃₁	—	2	4	no	—
308	C₂×C₃₁	—	4	8	no	—
309	C₂×C₃₁	C₂×#518	1	1	yes	characteristic
310	C₂×C₃₁	C₂×#520	1	1	yes	characteristic
311	C₂×C₃₁	#3428	1	1	yes	characteristic
312	C₂×C₃₁	#3558	1	2	yes	—
313	C₂×C₃₁	#3561	1	2	yes	—
314	C₂⁵	C₂³×C₃₁	1	2	yes	—
315	C₂²×Q₈	—	4	4	no	—
316	C₂²×Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
317	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
318	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
319	C₂²×D₄	—	4	4	no	—
320	C₂²×D₄	C₂×C₄×C₃₁	1	8	yes	—
321	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
322	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
323	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
324	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
325	C₂³×C₄	C₂³×C₃₁	1	8	yes	—
326	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
327	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
328	#124	C₂×C₄×C₃₁	1	16	yes	complemented
329	#124	C₂×C₄×C₃₁	1	16	yes	complemented
330	#124	C₂×C₄×C₃₁	1	16	yes	complemented
331	#124	C₂×C₄×C₃₁	1	16	yes	complemented
332	C₄×Q₈	—	4	8	no	—
333	C₄×D₄	—	4	8	no	—
334	#117	—	4	4	no	—
335	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
336	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
337	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
338	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
339	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
340	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
341	C₂×C₂²⋊C₄	—	4	8	no	—
342	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	16	yes	complemented
343	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	16	yes	complemented
344	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
345	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
346	C₂×C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
347	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
348	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
349	C₂×C₂²⋊C₄	C₂³×C₃₁	1	8	yes	—
350	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
351	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
352	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
353	C₂×C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
354	C₂×C₄²	C₂×C₄×C₃₁	1	8	yes	complemented
355	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
356	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
357	C₂×C₄²	C₂³×C₃₁	1	8	yes	—
358	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
359	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
360	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
361	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
362	#95	C₂³×C₃₁	1	2	yes	—
363	#95	C₂³×C₃₁	1	2	yes	—
364	#95	C₂³×C₃₁	1	4	yes	—
365	#95	C₂³×C₃₁	1	4	yes	—
366	#95	C₂³×C₃₁	1	8	yes	—
367	#95	C₂³×C₃₁	1	4	yes	—
368	#95	C₂³×C₃₁	1	8	yes	—
369	#95	C₂³×C₃₁	1	4	yes	—
370	#95	C₂³×C₃₁	1	4	yes	—
371	C₃₁	#34377	1	1	yes	characteristic, direct product factor, Sylow
372	C₂⁴	—	2	4	no	—
373	C₂⁴	—	2	4	no	—
374	C₂⁴	—	4	4	no	—
375	C₂⁴	—	4	4	no	—
376	C₂⁴	C₂²×C₄×C₃₁	1	8	yes	—
377	C₂⁴	C₂⁴×C₃₁	1	4	yes	—
378	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
379	C₂×Q₈	—	2	8	no	—
380	C₂×Q₈	—	4	4	no	—
381	C₂×Q₈	—	4	16	no	—
382	C₂×Q₈	C₂²×C₄×C₃₁	1	8	yes	—
383	C₂×Q₈	C₂²×C₄×C₃₁	1	8	yes	—
384	C₂×D₄	—	2	8	no	—
385	C₂×D₄	—	4	16	no	—
386	C₂×D₄	—	4	4	no	—
387	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
388	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	2	2	no	—
391	C₂²×C₄	—	2	4	no	—
392	C₂²×C₄	—	4	2	no	—
393	C₂²×C₄	—	4	4	no	—
394	C₂²×C₄	—	4	8	no	—
395	C₂²×C₄	—	4	8	no	—
396	C₂²×C₄	—	4	8	no	—
397	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
398	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
399	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
400	C₂²×C₄	C₂²×C₄×C₃₁	1	8	yes	—
401	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
402	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
403	C₂²×C₄	C₂⁴×C₃₁	1	8	yes	—
404	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
405	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
406	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
407	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
408	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
409	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
410	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
411	C₄⋊C₄	—	2	8	no	—
412	C₄⋊C₄	—	4	2	no	—
413	C₄⋊C₄	—	4	4	no	—
414	C₄⋊C₄	—	4	2	no	—
415	C₄⋊C₄	—	4	4	no	—
416	C₄⋊C₄	—	4	8	no	—
417	C₂²⋊C₄	—	2	4	no	—
418	C₂²⋊C₄	—	2	8	no	—
419	C₂²⋊C₄	—	2	4	no	—
420	C₂²⋊C₄	—	4	8	no	—
421	C₂²⋊C₄	—	4	8	no	—
422	C₂²⋊C₄	—	4	8	no	—
423	C₂²⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
424	C₂²⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
425	C₂²⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
426	C₂²⋊C₄	C₂²×C₄×C₃₁	1	16	yes	—
427	C₄²	—	2	8	no	—
428	C₄²	—	4	4	no	—
429	C₄²	—	4	8	no	—
430	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
431	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
432	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
433	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
434	C₂³	—	2	2	no	—
435	C₂³	—	2	2	no	—
436	C₂³	—	2	4	no	—
437	C₂³	—	2	8	no	—
438	C₂³	—	2	4	no	—
439	C₂³	—	2	8	no	—
440	C₂³	—	2	16	no	—
441	C₂³	—	4	8	no	—
442	C₂³	—	4	8	no	—
443	C₂³	—	8	8	no	—
444	C₂³	—	8	8	no	—
445	C₂³	C₂³×C₄×C₃₁	1	8	yes	—
446	C₂³	C₂³×C₄×C₃₁	1	2	yes	—
447	C₂³	C₃₁×H(5,𝔽₂)	1	2	yes	—
448	C₂³	C₃₁×2₋¹⁺⁴	1	2	yes	—
449	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
450	Q₈	—	2	16	no	—
451	Q₈	—	4	8	no	—
452	D₄	—	2	16	no	—
453	D₄	—	4	8	no	—
454	C₂×C₄	—	2	8	no	—
455	C₂×C₄	—	2	1	no	—
456	C₂×C₄	—	2	8	no	—
457	C₂×C₄	—	2	1	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	1	no	—
460	C₂×C₄	—	2	1	no	—
461	C₂×C₄	—	2	2	no	—
462	C₂×C₄	—	2	2	no	—
463	C₂×C₄	—	2	8	no	—
464	C₂×C₄	—	2	4	no	—
465	C₂×C₄	—	2	4	no	—
466	C₂×C₄	—	2	8	no	—
467	C₂×C₄	—	2	8	no	—
468	C₂×C₄	—	2	4	no	—
469	C₂×C₄	—	2	4	no	—
470	C₂×C₄	—	2	4	no	—
471	C₂×C₄	—	2	8	no	—
472	C₂×C₄	—	2	4	no	—
473	C₂×C₄	—	4	4	no	—
474	C₂×C₄	—	4	8	no	—
475	C₂×C₄	—	8	16	no	—
476	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
477	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
478	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
479	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
480	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
481	C₂×C₄	C₂³×C₄×C₃₁	1	8	yes	—
482	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
483	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
484	C₂²	—	2	2	no	—
485	C₂²	—	2	2	no	—
486	C₂²	—	2	4	no	—
487	C₂²	—	2	4	no	—
488	C₂²	—	2	8	no	—
489	C₂²	—	2	8	no	—
490	C₂²	—	2	8	no	—
491	C₂²	—	4	8	no	—
492	C₂²	—	4	8	no	—
493	C₂²	—	8	16	no	—
494	C₂²	C₃₁×#520	1	4	yes	—
495	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
496	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
497	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
498	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
499	C₂²	C₃₁×#585	1	2	yes	—
500	C₄	—	2	2	no	—
501	C₄	—	2	8	no	—
502	C₄	—	2	2	no	—
503	C₄	—	2	2	no	—
504	C₄	—	2	2	no	—
505	C₄	—	2	2	no	—
506	C₄	—	4	4	no	—
507	C₄	—	4	8	no	—
508	C₄	—	4	4	no	—
509	C₄	C₃₁×#518	1	4	yes	—
510	C₂	—	2	4	no	—
511	C₂	—	4	8	no	—
512	C₂	C₂×C₃₁×#518	1	1	yes	characteristic
513	C₂	C₂×C₃₁×#520	1	1	yes	characteristic
514	C₂	C₃₁×#3428	1	1	yes	characteristic
515	C₂	C₃₁×#3558	1	2	yes	—
516	C₂	C₃₁×#3561	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁵×C₃₁
prime radical	C₂³×D₄×C₃₁	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2379	C₂	semidirect
3. C₃₁×#2367	C₂	semidirect
4. C₃₁×#2342	C₂	semidirect
5. Q₈×C₂²⋊C₄×C₃₁	C₂	semidirect
7. C₃₁×#2337	C₂	semidirect
8. C₃₁×#2326	C₂	semidirect
9. C₃₁×#2315	C₂	semidirect
10. C₃₁×#2312	C₂	semidirect
11. C₃₁×#2285	C₂	semidirect
12. C₃₁×#2283	C₂	semidirect
15. C₂×C₃₁×#124	C₄	semidirect
20. C₂×C₄×Q₈×C₃₁	C₂²	semidirect
27. C₃₁×#391	C₂²	semidirect
28. C₃₁×#391	C₂²	semidirect
30. C₃₁×#388	C₂²	semidirect
31. C₃₁×#388	C₂²	semidirect
32. C₃₁×#388	C₂²	semidirect
33. C₃₁×#388	C₂²	semidirect
35. C₃₁×#385	C₂²	semidirect
36. C₃₁×#385	C₂²	semidirect
37. C₃₁×#385	C₂²	semidirect
38. C₃₁×#384	C₂²	semidirect
39. C₃₁×#383	C₂²	semidirect
40. C₃₁×#382	C₂²	semidirect
41. C₃₁×#382	C₂²	semidirect
49. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
50. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
51. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
54. C₄×C₂²⋊C₄×C₃₁	C₂²	semidirect
55. C₃₁×#376	C₂²	semidirect
58. C₂²×Q₈×C₃₁	C₂×C₄	semidirect
70. C₃₁×#124	C₂×C₄	semidirect
71. C₃₁×#124	C₂×C₄	semidirect
72. C₃₁×#124	C₂×C₄	semidirect
73. C₃₁×#124	C₂×C₄	semidirect
84. C₂×C₂²⋊C₄×C₃₁	C₂×C₄	semidirect
85. C₂×C₂²⋊C₄×C₃₁	C₂×C₄	semidirect
96. C₂×C₄²×C₃₁	C₂×C₄	semidirect
175. #34377	C₃₁	direct (N × H)
227. #2379	C₂×C₃₁	semidirect
228. #2367	C₂×C₃₁	semidirect
229. #2342	C₂×C₃₁	semidirect
230. Q₈×C₂²⋊C₄	C₂×C₃₁	semidirect
232. #2337	C₂×C₃₁	semidirect
233. #2326	C₂×C₃₁	semidirect
234. #2315	C₂×C₃₁	semidirect
235. #2312	C₂×C₃₁	semidirect
236. #2285	C₂×C₃₁	semidirect
237. #2283	C₂×C₃₁	semidirect
266. C₂×#124	C₄×C₃₁	semidirect
271. C₂×C₄×Q₈	C₂²×C₃₁	semidirect
278. #391	C₂²×C₃₁	semidirect
279. #391	C₂²×C₃₁	semidirect
281. #388	C₂²×C₃₁	semidirect
282. #388	C₂²×C₃₁	semidirect
283. #388	C₂²×C₃₁	semidirect
284. #388	C₂²×C₃₁	semidirect
286. #385	C₂²×C₃₁	semidirect
287. #385	C₂²×C₃₁	semidirect
288. #385	C₂²×C₃₁	semidirect
289. #384	C₂²×C₃₁	semidirect
290. #383	C₂²×C₃₁	semidirect
291. #382	C₂²×C₃₁	semidirect
292. #382	C₂²×C₃₁	semidirect
300. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
301. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
302. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
305. C₄×C₂²⋊C₄	C₂²×C₃₁	semidirect
306. #376	C₂²×C₃₁	semidirect
316. C₂²×Q₈	C₂×C₄×C₃₁	semidirect
328. #124	C₂×C₄×C₃₁	semidirect
329. #124	C₂×C₄×C₃₁	semidirect
330. #124	C₂×C₄×C₃₁	semidirect
331. #124	C₂×C₄×C₃₁	semidirect
342. C₂×C₂²⋊C₄	C₂×C₄×C₃₁	semidirect
343. C₂×C₂²⋊C₄	C₂×C₄×C₃₁	semidirect
354. C₂×C₄²	C₂×C₄×C₃₁	semidirect
371. C₃₁	#34377	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×#124	2
C₃₁×#2379	2
C₃₁×#2367	2
C₃₁×#2342	2
Q₈×C₂²⋊C₄×C₃₁	2
D₄×C₂²⋊C₄×C₃₁	2
C₃₁×#2337	2
C₃₁×#2326	2
C₃₁×#2315	2
C₃₁×#2312	2
C₃₁×#2285	2
C₃₁×#2283	2
#34377	31

Subgroups — C31×#34377 #21491136

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#34377 #21491136