Subgroups — C₃₁×#22764 #21479523

Order 7936.

906 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2357	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2349	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1936	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1918	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1879	C₂	1	2	yes	maximal
7	C₃₁×#1835	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1804	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1804	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1770	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1762	C₂	1	2	yes	maximal, complemented
12	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₄×D₈×C₃₁	C₄	1	8	yes	complemented
17	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#102	C₂²	1	2	yes	—
20	C₂×C₃₁×#102	C₂²	1	2	yes	—
21	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#97	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₃₁×#390	C₂²	1	2	yes	—
26	C₃₁×#389	C₂²	1	2	yes	—
27	C₃₁×#388	—	2	2	no	—
28	C₃₁×#385	—	2	2	no	—
29	C₃₁×#382	C₂²	1	1	yes	characteristic
30	C₃₁×#382	C₂²	1	1	yes	characteristic
31	C₃₁×#381	—	2	1	no	—
32	C₃₁×#380	—	2	1	no	—
33	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
34	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
36	C₂×C₃₁×#95	—	2	1	no	—
37	C₃₁×#340	C₂²	1	2	yes	—
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#340	C₂²	1	2	yes	—
40	C₃₁×#340	C₂²	1	2	yes	—
41	C₃₁×#336	C₂²	1	2	yes	—
42	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
43	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₃₁	—	2	1	no	—
45	C₂³×C₄×C₃₁	—	2	1	no	—
46	C₂³×C₄×C₃₁	—	2	2	no	—
47	C₂×D₈×C₃₁	—	2	2	no	—
48	C₂×D₈×C₃₁	—	2	2	no	—
49	C₂×D₈×C₃₁	C₂×C₄	1	4	yes	—
50	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
51	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₄×D₄×C₃₁	—	2	4	no	—
54	C₄×D₄×C₃₁	—	2	2	no	—
55	C₄×D₄×C₃₁	—	2	2	no	—
56	C₄×D₄×C₃₁	—	4	2	no	—
57	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
58	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	complemented
59	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
60	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
61	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
66	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
67	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
69	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
70	C₂×C₄²×C₃₁	—	2	2	no	—
71	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
72	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
73	C₃₁×#107	C₂×C₄	1	4	yes	complemented
74	C₃₁×C₄⋊C₈	—	2	2	no	—
75	C₃₁×#102	—	2	4	no	—
76	C₃₁×#102	—	4	2	no	—
77	C₃₁×#102	C₂×C₄	1	4	yes	—
78	C₃₁×#102	C₂×C₄	1	4	yes	complemented
79	C₃₁×#97	—	4	2	no	—
80	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
81	C₃₁×#95	—	2	1	no	—
82	C₃₁×#95	—	2	2	no	—
83	C₃₁×#95	—	2	1	no	—
84	C₃₁×#95	—	2	2	no	—
85	C₃₁×#95	—	2	1	no	—
86	C₃₁×#95	—	2	1	no	—
87	C₃₁×#95	D₄	1	1	yes	characteristic
88	C₃₁×#95	D₄	1	1	yes	characteristic
89	C₂⁴×C₃₁	—	2	1	no	—
90	C₂⁴×C₃₁	—	2	1	no	—
91	C₂×D₄×C₃₁	—	2	1	no	—
92	C₂×D₄×C₃₁	—	2	1	no	—
93	C₂×D₄×C₃₁	—	2	1	no	—
94	C₂×D₄×C₃₁	—	2	1	no	—
95	C₂×D₄×C₃₁	—	2	1	no	—
96	C₂×D₄×C₃₁	—	2	1	no	—
97	C₂×D₄×C₃₁	—	2	1	no	—
98	C₂×D₄×C₃₁	—	2	1	no	—
99	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
100	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
101	C₂×D₄×C₃₁	#41	1	2	yes	—
102	C₂×D₄×C₃₁	#41	1	2	yes	—
103	C₂²×C₄×C₃₁	—	2	1	no	—
104	C₂²×C₄×C₃₁	—	2	1	no	—
105	C₂²×C₄×C₃₁	—	2	4	no	—
106	C₂²×C₄×C₃₁	—	2	4	no	—
107	C₂²×C₄×C₃₁	—	2	2	no	—
108	C₂²×C₄×C₃₁	—	2	2	no	—
109	C₂²×C₄×C₃₁	—	4	4	no	—
110	C₂²×C₄×C₃₁	—	4	4	no	—
111	C₂²×C₄×C₃₁	—	8	2	no	—
112	C₂²×C₄×C₃₁	—	8	2	no	—
113	C₂²×C₄×C₃₁	—	8	2	no	—
114	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
115	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
116	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
117	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
119	D₈×C₃₁	—	2	8	no	—
120	C₂×C₈×C₃₁	—	2	1	no	—
121	C₂×C₈×C₃₁	—	2	1	no	—
122	C₂×C₈×C₃₁	—	2	1	no	—
123	C₂×C₈×C₃₁	—	2	1	no	—
124	C₂×C₈×C₃₁	—	2	2	no	—
125	C₂×C₈×C₃₁	—	2	2	no	—
126	C₂×C₈×C₃₁	—	2	2	no	—
127	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
128	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
129	C₄⋊C₄×C₃₁	—	2	1	no	—
130	C₄⋊C₄×C₃₁	—	2	1	no	—
131	C₄⋊C₄×C₃₁	—	2	2	no	—
132	C₄⋊C₄×C₃₁	—	2	2	no	—
133	C₄⋊C₄×C₃₁	—	2	2	no	—
134	C₄⋊C₄×C₃₁	—	2	2	no	—
135	C₄⋊C₄×C₃₁	—	4	2	no	—
136	C₄⋊C₄×C₃₁	—	8	2	no	—
137	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
138	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
139	C₂²⋊C₄×C₃₁	—	2	4	no	—
140	C₂²⋊C₄×C₃₁	—	2	4	no	—
141	C₂²⋊C₄×C₃₁	—	4	4	no	—
142	C₂²⋊C₄×C₃₁	—	4	4	no	—
143	C₂²⋊C₄×C₃₁	—	8	2	no	—
144	C₄²×C₃₁	—	2	2	no	—
145	C₄²×C₃₁	—	2	2	no	—
146	C₄²×C₃₁	—	8	2	no	—
147	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
148	C₄²×C₃₁	#41	1	2	yes	—
149	#22764	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₂³×C₃₁	—	2	2	no	—
151	C₂³×C₃₁	—	2	2	no	—
152	C₂³×C₃₁	—	2	2	no	—
153	C₂³×C₃₁	—	2	2	no	—
154	C₂³×C₃₁	—	4	2	no	—
155	C₂³×C₃₁	—	4	2	no	—
156	C₂³×C₃₁	—	4	1	no	—
157	C₂³×C₃₁	—	4	1	no	—
158	C₂³×C₃₁	—	4	2	no	—
159	C₂³×C₃₁	—	4	2	no	—
160	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
161	D₄×C₃₁	—	2	2	no	—
162	D₄×C₃₁	—	2	2	no	—
163	D₄×C₃₁	—	2	2	no	—
164	D₄×C₃₁	—	2	2	no	—
165	D₄×C₃₁	—	2	2	no	—
166	D₄×C₃₁	—	4	1	no	—
167	D₄×C₃₁	—	4	1	no	—
168	D₄×C₃₁	#117	1	4	yes	—
169	C₂×C₄×C₃₁	—	2	2	no	—
170	C₂×C₄×C₃₁	—	2	2	no	—
171	C₂×C₄×C₃₁	—	2	2	no	—
172	C₂×C₄×C₃₁	—	2	2	no	—
173	C₂×C₄×C₃₁	—	2	2	no	—
174	C₂×C₄×C₃₁	—	2	2	no	—
175	C₂×C₄×C₃₁	—	2	2	no	—
176	C₂×C₄×C₃₁	—	2	2	no	—
177	C₂×C₄×C₃₁	—	2	2	no	—
178	C₂×C₄×C₃₁	—	2	2	no	—
179	C₂×C₄×C₃₁	—	4	2	no	—
180	C₂×C₄×C₃₁	—	4	2	no	—
181	C₂×C₄×C₃₁	—	4	2	no	—
182	C₂×C₄×C₃₁	—	4	2	no	—
183	C₂×C₄×C₃₁	—	4	2	no	—
184	C₂×C₄×C₃₁	—	4	2	no	—
185	C₂×C₄×C₃₁	—	4	2	no	—
186	C₂×C₄×C₃₁	—	4	2	no	—
187	C₂×C₄×C₃₁	—	8	4	no	—
188	C₂×C₄×C₃₁	—	8	4	no	—
189	C₂×C₄×C₃₁	—	8	8	no	—
190	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
191	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
192	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
193	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
194	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
195	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
196	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
197	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
198	C₈×C₃₁	—	2	2	no	—
199	C₈×C₃₁	—	2	2	no	—
200	C₈×C₃₁	—	4	2	no	—
201	C₂×C₄×D₈	C₂×C₃₁	1	1	yes	characteristic
202	#2357	C₂×C₃₁	1	1	yes	characteristic, complemented
203	#2349	C₂×C₃₁	1	1	yes	characteristic, complemented
204	#1936	C₂×C₃₁	1	2	yes	complemented
205	#1918	C₂×C₃₁	1	2	yes	complemented
206	#1879	C₂×C₃₁	1	2	yes	—
207	#1835	C₂×C₃₁	1	1	yes	characteristic, complemented
208	#1804	C₂×C₃₁	1	1	yes	characteristic, complemented
209	#1804	C₂×C₃₁	1	1	yes	characteristic, complemented
210	#1770	C₂×C₃₁	1	1	yes	characteristic, complemented
211	#1762	C₂×C₃₁	1	2	yes	complemented
212	C₂²×C₃₁	—	2	4	no	—
213	C₂²×C₃₁	—	4	4	no	—
214	C₂²×C₃₁	—	4	2	no	—
215	C₂²×C₃₁	—	4	2	no	—
216	C₂²×C₃₁	—	4	4	no	—
217	C₂²×C₃₁	—	4	2	no	—
218	C₂²×C₃₁	—	4	2	no	—
219	C₂²×C₃₁	—	4	2	no	—
220	C₂²×C₃₁	—	8	2	no	—
221	C₂²×C₃₁	—	8	2	no	—
222	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
223	C₂²×C₃₁	#546	1	1	yes	characteristic
224	C₂²×C₃₁	#548	1	1	yes	characteristic
225	C₂²×C₃₁	#575	1	2	yes	—
226	C₂²×C₃₁	#576	1	2	yes	—
227	C₄×C₃₁	—	2	2	no	—
228	C₄×C₃₁	—	2	2	no	—
229	C₄×C₃₁	—	4	2	no	—
230	C₄×C₃₁	—	4	2	no	—
231	C₄×C₃₁	—	4	2	no	—
232	C₄×C₃₁	—	4	2	no	—
233	C₄×C₃₁	—	8	2	no	—
234	C₄×C₃₁	—	8	2	no	—
235	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
236	C₄×C₃₁	#520	1	1	yes	characteristic
237	C₄×C₃₁	#552	1	1	yes	characteristic
238	C₄×C₃₁	#553	1	1	yes	characteristic
239	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
240	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
241	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
242	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
243	C₄×D₈	C₄×C₃₁	1	8	yes	complemented
244	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
245	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
246	C₂×#102	C₂²×C₃₁	1	2	yes	—
247	C₂×#102	C₂²×C₃₁	1	2	yes	—
248	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
249	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
250	C₂×#97	C₂²×C₃₁	1	2	yes	—
251	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
252	#390	C₂²×C₃₁	1	2	yes	—
253	#389	C₂²×C₃₁	1	2	yes	—
254	#388	—	2	2	no	—
255	#385	—	2	2	no	—
256	#382	C₂²×C₃₁	1	1	yes	characteristic
257	#382	C₂²×C₃₁	1	1	yes	characteristic
258	#381	—	2	1	no	—
259	#380	—	2	1	no	—
260	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
261	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
262	C₄×C₂²⋊C₄	—	2	1	no	—
263	C₂×#95	—	2	1	no	—
264	#340	C₂²×C₃₁	1	2	yes	—
265	#340	C₂²×C₃₁	1	2	yes	—
266	#340	C₂²×C₃₁	1	2	yes	—
267	#340	C₂²×C₃₁	1	2	yes	—
268	#336	C₂²×C₃₁	1	2	yes	—
269	C₂×C₃₁	—	4	4	no	—
270	C₂×C₃₁	—	8	2	no	—
271	C₂×C₃₁	#2352	1	1	yes	characteristic
272	C₂×C₃₁	#2943	1	1	yes	characteristic
273	C₂×C₃₁	#2948	1	1	yes	characteristic
274	C₂×C₃₁	#3322	1	2	yes	—
275	C₂×C₃₁	#3370	1	2	yes	—
276	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
277	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
278	C₂³×C₄	—	2	1	no	—
279	C₂³×C₄	—	2	1	no	—
280	C₂³×C₄	—	2	2	no	—
281	C₂×D₈	—	2	2	no	—
282	C₂×D₈	—	2	2	no	—
283	C₂×D₈	C₂×C₄×C₃₁	1	4	yes	—
284	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
285	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
286	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
287	C₄×D₄	—	2	4	no	—
288	C₄×D₄	—	2	2	no	—
289	C₄×D₄	—	2	2	no	—
290	C₄×D₄	—	4	2	no	—
291	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
292	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	complemented
293	C₂×C₄⋊C₄	—	2	2	no	—
294	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
295	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
296	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
297	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
298	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
299	C₂×C₂²⋊C₄	—	2	1	no	—
300	C₂×C₂²⋊C₄	—	2	1	no	—
301	C₂×C₂²⋊C₄	—	2	2	no	—
302	C₂×C₂²⋊C₄	—	2	2	no	—
303	C₂×C₂²⋊C₄	—	2	2	no	—
304	C₂×C₄²	—	2	2	no	—
305	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
306	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
307	#107	C₂×C₄×C₃₁	1	4	yes	complemented
308	C₄⋊C₈	—	2	2	no	—
309	#102	—	2	4	no	—
310	#102	—	4	2	no	—
311	#102	C₂×C₄×C₃₁	1	4	yes	—
312	#102	C₂×C₄×C₃₁	1	4	yes	complemented
313	#97	—	4	2	no	—
314	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	complemented
315	#95	—	2	1	no	—
316	#95	—	2	2	no	—
317	#95	—	2	1	no	—
318	#95	—	2	2	no	—
319	#95	—	2	1	no	—
320	#95	—	2	1	no	—
321	#95	D₄×C₃₁	1	1	yes	characteristic
322	#95	D₄×C₃₁	1	1	yes	characteristic
323	C₃₁	#22764	1	1	yes	characteristic, direct product factor, Sylow
324	C₂⁴	—	2	1	no	—
325	C₂⁴	—	2	1	no	—
326	C₂×D₄	—	2	1	no	—
327	C₂×D₄	—	2	1	no	—
328	C₂×D₄	—	2	1	no	—
329	C₂×D₄	—	2	1	no	—
330	C₂×D₄	—	2	1	no	—
331	C₂×D₄	—	2	1	no	—
332	C₂×D₄	—	2	1	no	—
333	C₂×D₄	—	2	1	no	—
334	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
335	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
336	C₂×D₄	#41×C₃₁	1	2	yes	—
337	C₂×D₄	#41×C₃₁	1	2	yes	—
338	C₂²×C₄	—	2	1	no	—
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	2	4	no	—
341	C₂²×C₄	—	2	4	no	—
342	C₂²×C₄	—	2	2	no	—
343	C₂²×C₄	—	2	2	no	—
344	C₂²×C₄	—	4	4	no	—
345	C₂²×C₄	—	4	4	no	—
346	C₂²×C₄	—	8	2	no	—
347	C₂²×C₄	—	8	2	no	—
348	C₂²×C₄	—	8	2	no	—
349	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
350	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
351	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
352	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
353	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
354	D₈	—	2	8	no	—
355	C₂×C₈	—	2	1	no	—
356	C₂×C₈	—	2	1	no	—
357	C₂×C₈	—	2	1	no	—
358	C₂×C₈	—	2	1	no	—
359	C₂×C₈	—	2	2	no	—
360	C₂×C₈	—	2	2	no	—
361	C₂×C₈	—	2	2	no	—
362	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
363	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
364	C₄⋊C₄	—	2	1	no	—
365	C₄⋊C₄	—	2	1	no	—
366	C₄⋊C₄	—	2	2	no	—
367	C₄⋊C₄	—	2	2	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	—	2	2	no	—
370	C₄⋊C₄	—	4	2	no	—
371	C₄⋊C₄	—	8	2	no	—
372	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
373	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
374	C₂²⋊C₄	—	2	4	no	—
375	C₂²⋊C₄	—	2	4	no	—
376	C₂²⋊C₄	—	4	4	no	—
377	C₂²⋊C₄	—	4	4	no	—
378	C₂²⋊C₄	—	8	2	no	—
379	C₄²	—	2	2	no	—
380	C₄²	—	2	2	no	—
381	C₄²	—	8	2	no	—
382	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
383	C₄²	#41×C₃₁	1	2	yes	—
384	C₂³	—	2	2	no	—
385	C₂³	—	2	2	no	—
386	C₂³	—	2	2	no	—
387	C₂³	—	2	2	no	—
388	C₂³	—	4	2	no	—
389	C₂³	—	4	2	no	—
390	C₂³	—	4	1	no	—
391	C₂³	—	4	1	no	—
392	C₂³	—	4	2	no	—
393	C₂³	—	4	2	no	—
394	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
395	D₄	—	2	2	no	—
396	D₄	—	2	2	no	—
397	D₄	—	2	2	no	—
398	D₄	—	2	2	no	—
399	D₄	—	2	2	no	—
400	D₄	—	4	1	no	—
401	D₄	—	4	1	no	—
402	D₄	C₃₁×#117	1	4	yes	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	2	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	4	2	no	—
414	C₂×C₄	—	4	2	no	—
415	C₂×C₄	—	4	2	no	—
416	C₂×C₄	—	4	2	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	4	2	no	—
421	C₂×C₄	—	8	4	no	—
422	C₂×C₄	—	8	4	no	—
423	C₂×C₄	—	8	8	no	—
424	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
425	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
426	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
427	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
428	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
429	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
430	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
431	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
432	C₈	—	2	2	no	—
433	C₈	—	2	2	no	—
434	C₈	—	4	2	no	—
435	C₂²	—	2	4	no	—
436	C₂²	—	4	4	no	—
437	C₂²	—	4	2	no	—
438	C₂²	—	4	2	no	—
439	C₂²	—	4	4	no	—
440	C₂²	—	4	2	no	—
441	C₂²	—	4	2	no	—
442	C₂²	—	4	2	no	—
443	C₂²	—	8	2	no	—
444	C₂²	—	8	2	no	—
445	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
446	C₂²	C₃₁×#546	1	1	yes	characteristic
447	C₂²	C₃₁×#548	1	1	yes	characteristic
448	C₂²	C₃₁×#575	1	2	yes	—
449	C₂²	C₃₁×#576	1	2	yes	—
450	C₄	—	2	2	no	—
451	C₄	—	2	2	no	—
452	C₄	—	4	2	no	—
453	C₄	—	4	2	no	—
454	C₄	—	4	2	no	—
455	C₄	—	4	2	no	—
456	C₄	—	8	2	no	—
457	C₄	—	8	2	no	—
458	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
459	C₄	C₃₁×#520	1	1	yes	characteristic
460	C₄	C₃₁×#552	1	1	yes	characteristic
461	C₄	C₃₁×#553	1	1	yes	characteristic
462	C₂	—	4	4	no	—
463	C₂	—	8	2	no	—
464	C₂	C₃₁×#2352	1	1	yes	characteristic
465	C₂	C₃₁×#2943	1	1	yes	characteristic
466	C₂	C₃₁×#2948	1	1	yes	characteristic
467	C₂	C₃₁×#3322	1	2	yes	—
468	C₂	C₃₁×#3370	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₈×C₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄×C₄⋊C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2357	C₂	semidirect
3. C₃₁×#2349	C₂	semidirect
4. C₃₁×#1936	C₂	semidirect
5. C₃₁×#1918	C₂	semidirect
7. C₃₁×#1835	C₂	semidirect
8. C₃₁×#1804	C₂	semidirect
9. C₃₁×#1804	C₂	semidirect
10. C₃₁×#1770	C₂	semidirect
11. C₃₁×#1762	C₂	semidirect
16. C₄×D₈×C₃₁	C₄	semidirect
58. C₄×D₄×C₃₁	C₂×C₄	semidirect
73. C₃₁×#107	C₂×C₄	semidirect
78. C₃₁×#102	C₂×C₄	semidirect
80. C₄×C₈×C₃₁	C₂×C₄	semidirect
149. #22764	C₃₁	direct (N × H)
202. #2357	C₂×C₃₁	semidirect
203. #2349	C₂×C₃₁	semidirect
204. #1936	C₂×C₃₁	semidirect
205. #1918	C₂×C₃₁	semidirect
207. #1835	C₂×C₃₁	semidirect
208. #1804	C₂×C₃₁	semidirect
209. #1804	C₂×C₃₁	semidirect
210. #1770	C₂×C₃₁	semidirect
211. #1762	C₂×C₃₁	semidirect
243. C₄×D₈	C₄×C₃₁	semidirect
292. C₄×D₄	C₂×C₄×C₃₁	semidirect
307. #107	C₂×C₄×C₃₁	semidirect
312. #102	C₂×C₄×C₃₁	semidirect
314. C₄×C₈	C₂×C₄×C₃₁	semidirect
323. C₃₁	#22764	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×C₄×D₈×C₃₁	2
C₃₁×#2357	2
C₃₁×#2349	2
C₃₁×#1936	2
C₃₁×#1918	2
C₃₁×#1879	2
C₃₁×#1835	2
C₃₁×#1804	2
C₃₁×#1770	2
C₃₁×#1762	2
#22764	31

Subgroups — C31×#22764 #21479523

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22764 #21479523