Subgroups — C₃₁×#23067 #21479826

Order 7936.

814 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#476	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2369	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2354	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1990	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1947	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1936	C₂	1	2	yes	maximal
7	C₃₁×#1925	C₂	1	2	yes	maximal
8	C₃₁×#1919	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1776	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1770	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₃₁×#102	C₂	1	1	yes	maximal, characteristic
12	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#117	—	2	1	no	—
15	C₃₁×#476	C₄	1	8	yes	complemented
16	C₂×C₃₁×#107	C₂²	1	2	yes	—
17	C₂×C₃₁×#106	C₂²	1	2	yes	—
18	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#102	C₂²	1	2	yes	—
22	C₂×C₃₁×#97	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
24	C₃₁×#391	—	2	2	no	—
25	C₃₁×#390	C₂²	1	2	yes	—
26	C₃₁×#389	—	2	1	no	—
27	C₃₁×#389	C₂²	1	2	yes	—
28	C₃₁×#388	—	2	2	no	—
29	C₃₁×#384	C₂²	1	1	yes	characteristic
30	C₃₁×#380	—	2	1	no	—
31	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
32	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
33	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
34	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
36	C₃₁×#340	C₂²	1	1	yes	characteristic
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	1	yes	characteristic
41	C₃₁×#336	C₂²	1	2	yes	—
42	C₂²×Q₈×C₃₁	—	2	1	no	—
43	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₃₁	—	2	1	no	—
45	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
47	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
48	C₃₁×C₄⋊Q₈	—	2	2	no	—
49	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	—
50	C₄×D₄×C₃₁	—	2	2	no	—
51	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
52	C₃₁×#117	—	4	4	no	—
53	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
54	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
55	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
56	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
57	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
59	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
65	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
66	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
67	C₂×C₄²×C₃₁	—	2	2	no	—
68	C₂×C₄²×C₃₁	—	2	1	no	—
69	C₂×C₄²×C₃₁	—	2	2	no	—
70	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
71	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
72	C₃₁×#107	—	2	4	no	—
73	C₃₁×#106	C₂×C₄	1	8	yes	—
74	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
75	C₃₁×#102	—	2	2	no	—
76	C₃₁×#102	—	2	2	no	—
77	C₃₁×#102	C₂×C₄	1	8	yes	—
78	C₃₁×#97	—	2	4	no	—
79	C₄×C₈×C₃₁	—	2	2	no	—
80	C₃₁×#95	—	2	2	no	—
81	C₃₁×#95	—	2	2	no	—
82	C₃₁×#95	—	2	2	no	—
83	C₃₁×#95	—	2	1	no	—
84	C₃₁×#95	D₄	1	2	yes	—
85	C₂⁴×C₃₁	—	2	1	no	—
86	C₂×Q₈×C₃₁	—	2	4	no	—
87	C₂×Q₈×C₃₁	—	8	1	no	—
88	C₂×Q₈×C₃₁	—	8	1	no	—
89	C₂×D₄×C₃₁	—	2	1	no	—
90	C₂×D₄×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	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₃₁	—	2	1	no	—
96	C₂²×C₄×C₃₁	—	2	1	no	—
97	C₂²×C₄×C₃₁	—	2	2	no	—
98	C₂²×C₄×C₃₁	—	2	4	no	—
99	C₂²×C₄×C₃₁	—	2	1	no	—
100	C₂²×C₄×C₃₁	—	2	2	no	—
101	C₂²×C₄×C₃₁	—	4	2	no	—
102	C₂²×C₄×C₃₁	—	4	2	no	—
103	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
105	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
108	C₂×C₈×C₃₁	—	2	1	no	—
109	C₂×C₈×C₃₁	—	2	2	no	—
110	C₂×C₈×C₃₁	—	2	2	no	—
111	C₂×C₈×C₃₁	—	2	1	no	—
112	C₂×C₈×C₃₁	—	2	1	no	—
113	C₂×C₈×C₃₁	—	2	1	no	—
114	C₂×C₈×C₃₁	—	2	1	no	—
115	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
116	C₂×C₈×C₃₁	#41	1	2	yes	—
117	C₄⋊C₄×C₃₁	—	2	4	no	—
118	C₄⋊C₄×C₃₁	—	2	2	no	—
119	C₄⋊C₄×C₃₁	—	2	2	no	—
120	C₄⋊C₄×C₃₁	—	2	1	no	—
121	C₄⋊C₄×C₃₁	—	2	2	no	—
122	C₄⋊C₄×C₃₁	—	2	1	no	—
123	C₄⋊C₄×C₃₁	—	2	1	no	—
124	C₄⋊C₄×C₃₁	—	2	2	no	—
125	C₄⋊C₄×C₃₁	—	2	1	no	—
126	C₄⋊C₄×C₃₁	—	2	2	no	—
127	C₄⋊C₄×C₃₁	—	4	2	no	—
128	C₄⋊C₄×C₃₁	—	8	1	no	—
129	C₄⋊C₄×C₃₁	—	8	2	no	—
130	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
131	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
132	C₄⋊C₄×C₃₁	#41	1	2	yes	—
133	C₂²⋊C₄×C₃₁	—	2	4	no	—
134	C₂²⋊C₄×C₃₁	—	4	4	no	—
135	C₂²⋊C₄×C₃₁	—	8	2	no	—
136	C₄²×C₃₁	—	2	2	no	—
137	C₄²×C₃₁	—	2	2	no	—
138	C₄²×C₃₁	—	4	4	no	—
139	C₄²×C₃₁	—	4	2	no	—
140	C₄²×C₃₁	—	8	2	no	—
141	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
142	C₄²×C₃₁	#41	1	2	yes	—
143	#23067	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
144	C₂³×C₃₁	—	2	2	no	—
145	C₂³×C₃₁	—	4	1	no	—
146	C₂³×C₃₁	—	4	1	no	—
147	C₂³×C₃₁	—	4	1	no	—
148	C₂³×C₃₁	—	4	1	no	—
149	C₂³×C₃₁	—	4	1	no	—
150	C₂³×C₃₁	—	4	1	no	—
151	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
152	Q₈×C₃₁	—	8	4	no	—
153	D₄×C₃₁	—	2	2	no	—
154	D₄×C₃₁	—	2	2	no	—
155	D₄×C₃₁	—	4	1	no	—
156	D₄×C₃₁	—	4	1	no	—
157	C₂×C₄×C₃₁	—	2	1	no	—
158	C₂×C₄×C₃₁	—	2	1	no	—
159	C₂×C₄×C₃₁	—	2	2	no	—
160	C₂×C₄×C₃₁	—	2	4	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	2	2	no	—
163	C₂×C₄×C₃₁	—	2	2	no	—
164	C₂×C₄×C₃₁	—	2	2	no	—
165	C₂×C₄×C₃₁	—	2	2	no	—
166	C₂×C₄×C₃₁	—	2	2	no	—
167	C₂×C₄×C₃₁	—	2	2	no	—
168	C₂×C₄×C₃₁	—	4	1	no	—
169	C₂×C₄×C₃₁	—	4	1	no	—
170	C₂×C₄×C₃₁	—	4	1	no	—
171	C₂×C₄×C₃₁	—	4	1	no	—
172	C₂×C₄×C₃₁	—	4	2	no	—
173	C₂×C₄×C₃₁	—	4	2	no	—
174	C₂×C₄×C₃₁	—	4	8	no	—
175	C₂×C₄×C₃₁	—	4	1	no	—
176	C₂×C₄×C₃₁	—	4	2	no	—
177	C₂×C₄×C₃₁	—	4	2	no	—
178	C₂×C₄×C₃₁	—	4	1	no	—
179	C₂×C₄×C₃₁	—	4	2	no	—
180	C₂×C₄×C₃₁	—	4	1	no	—
181	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
182	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
183	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
184	C₂×C₄×C₃₁	#137	1	2	yes	—
185	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
186	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
187	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
188	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
189	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
190	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
191	C₈×C₃₁	—	2	4	no	—
192	C₈×C₃₁	—	2	2	no	—
193	C₈×C₃₁	—	4	1	no	—
194	C₂×#476	C₂×C₃₁	1	1	yes	characteristic
195	#2369	C₂×C₃₁	1	1	yes	characteristic, complemented
196	#2354	C₂×C₃₁	1	1	yes	characteristic
197	#1990	C₂×C₃₁	1	1	yes	characteristic, complemented
198	#1947	C₂×C₃₁	1	1	yes	characteristic, complemented
199	#1936	C₂×C₃₁	1	2	yes	—
200	#1925	C₂×C₃₁	1	2	yes	—
201	#1919	C₂×C₃₁	1	2	yes	complemented
202	#1776	C₂×C₃₁	1	2	yes	complemented
203	#1770	C₂×C₃₁	1	1	yes	characteristic, complemented
204	C₄×#102	C₂×C₃₁	1	1	yes	characteristic
205	C₂²×C₃₁	—	4	2	no	—
206	C₂²×C₃₁	—	4	2	no	—
207	C₂²×C₃₁	—	4	2	no	—
208	C₂²×C₃₁	—	8	1	no	—
209	C₂²×C₃₁	—	8	1	no	—
210	C₂²×C₃₁	—	8	1	no	—
211	C₂²×C₃₁	—	8	1	no	—
212	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
213	C₂²×C₃₁	#547	1	1	yes	characteristic
214	C₂²×C₃₁	#548	1	1	yes	characteristic
215	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
216	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
217	C₂²×C₃₁	#576	1	1	yes	characteristic
218	C₂²×C₃₁	#577	1	1	yes	characteristic
219	C₄×C₃₁	—	2	2	no	—
220	C₄×C₃₁	—	4	2	no	—
221	C₄×C₃₁	—	4	2	no	—
222	C₄×C₃₁	—	4	2	no	—
223	C₄×C₃₁	—	4	2	no	—
224	C₄×C₃₁	—	4	4	no	—
225	C₄×C₃₁	—	8	1	no	—
226	C₄×C₃₁	—	8	2	no	—
227	C₄×C₃₁	#440	1	4	yes	—
228	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
229	C₄×C₃₁	#519	1	1	yes	characteristic
230	C₄×C₃₁	#552	1	1	yes	characteristic
231	C₄×C₃₁	#556	1	1	yes	characteristic
232	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
233	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
234	C₂×#117	—	2	1	no	—
235	#476	C₄×C₃₁	1	8	yes	complemented
236	C₂×#107	C₂²×C₃₁	1	2	yes	—
237	C₂×#106	C₂²×C₃₁	1	2	yes	—
238	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
239	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
240	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
241	C₂×#102	C₂²×C₃₁	1	2	yes	—
242	C₂×#97	C₂²×C₃₁	1	2	yes	—
243	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
244	#391	—	2	2	no	—
245	#390	C₂²×C₃₁	1	2	yes	—
246	#389	—	2	1	no	—
247	#389	C₂²×C₃₁	1	2	yes	—
248	#388	—	2	2	no	—
249	#384	C₂²×C₃₁	1	1	yes	characteristic
250	#380	—	2	1	no	—
251	C₄×C₄⋊C₄	—	2	1	no	—
252	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
253	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
254	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
255	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
256	#340	C₂²×C₃₁	1	1	yes	characteristic
257	#340	C₂²×C₃₁	1	2	yes	—
258	#340	C₂²×C₃₁	1	2	yes	—
259	#340	C₂²×C₃₁	1	2	yes	—
260	#340	C₂²×C₃₁	1	1	yes	characteristic
261	#336	C₂²×C₃₁	1	2	yes	—
262	C₂×C₃₁	—	8	2	no	—
263	C₂×C₃₁	#2362	1	1	yes	characteristic
264	C₂×C₃₁	C₂×#440	1	1	yes	characteristic
265	C₂×C₃₁	#2946	1	1	yes	characteristic
266	C₂×C₃₁	#3354	1	1	yes	characteristic
267	C₂×C₃₁	#3357	1	1	yes	characteristic
268	C₂×C₃₁	#3366	1	1	yes	characteristic
269	C₂×C₃₁	#3369	1	1	yes	characteristic
270	C₂²×Q₈	—	2	1	no	—
271	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
272	C₂³×C₄	—	2	1	no	—
273	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
274	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
275	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
276	C₄⋊Q₈	—	2	2	no	—
277	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	—
278	C₄×D₄	—	2	2	no	—
279	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
280	#117	—	4	4	no	—
281	C₂×C₄⋊C₄	—	2	1	no	—
282	C₂×C₄⋊C₄	—	2	1	no	—
283	C₂×C₄⋊C₄	—	2	1	no	—
284	C₂×C₄⋊C₄	—	2	2	no	—
285	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
286	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
287	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
288	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
289	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
290	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
291	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
292	C₂×C₂²⋊C₄	—	2	1	no	—
293	C₂×C₂²⋊C₄	—	2	2	no	—
294	C₂×C₂²⋊C₄	—	2	2	no	—
295	C₂×C₄²	—	2	2	no	—
296	C₂×C₄²	—	2	1	no	—
297	C₂×C₄²	—	2	2	no	—
298	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
299	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
300	#107	—	2	4	no	—
301	#106	C₂×C₄×C₃₁	1	8	yes	—
302	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
303	#102	—	2	2	no	—
304	#102	—	2	2	no	—
305	#102	C₂×C₄×C₃₁	1	8	yes	—
306	#97	—	2	4	no	—
307	C₄×C₈	—	2	2	no	—
308	#95	—	2	2	no	—
309	#95	—	2	2	no	—
310	#95	—	2	2	no	—
311	#95	—	2	1	no	—
312	#95	D₄×C₃₁	1	2	yes	—
313	C₃₁	#23067	1	1	yes	characteristic, direct product factor, Sylow
314	C₂⁴	—	2	1	no	—
315	C₂×Q₈	—	2	4	no	—
316	C₂×Q₈	—	8	1	no	—
317	C₂×Q₈	—	8	1	no	—
318	C₂×D₄	—	2	1	no	—
319	C₂×D₄	—	2	1	no	—
320	C₂×D₄	—	2	1	no	—
321	C₂×D₄	—	2	1	no	—
322	C₂×D₄	—	2	1	no	—
323	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
324	C₂²×C₄	—	2	1	no	—
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	—	2	2	no	—
327	C₂²×C₄	—	2	4	no	—
328	C₂²×C₄	—	2	1	no	—
329	C₂²×C₄	—	2	2	no	—
330	C₂²×C₄	—	4	2	no	—
331	C₂²×C₄	—	4	2	no	—
332	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
333	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
334	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
335	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
336	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
337	C₂×C₈	—	2	1	no	—
338	C₂×C₈	—	2	2	no	—
339	C₂×C₈	—	2	2	no	—
340	C₂×C₈	—	2	1	no	—
341	C₂×C₈	—	2	1	no	—
342	C₂×C₈	—	2	1	no	—
343	C₂×C₈	—	2	1	no	—
344	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
345	C₂×C₈	#41×C₃₁	1	2	yes	—
346	C₄⋊C₄	—	2	4	no	—
347	C₄⋊C₄	—	2	2	no	—
348	C₄⋊C₄	—	2	2	no	—
349	C₄⋊C₄	—	2	1	no	—
350	C₄⋊C₄	—	2	2	no	—
351	C₄⋊C₄	—	2	1	no	—
352	C₄⋊C₄	—	2	1	no	—
353	C₄⋊C₄	—	2	2	no	—
354	C₄⋊C₄	—	2	1	no	—
355	C₄⋊C₄	—	2	2	no	—
356	C₄⋊C₄	—	4	2	no	—
357	C₄⋊C₄	—	8	1	no	—
358	C₄⋊C₄	—	8	2	no	—
359	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
360	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
361	C₄⋊C₄	#41×C₃₁	1	2	yes	—
362	C₂²⋊C₄	—	2	4	no	—
363	C₂²⋊C₄	—	4	4	no	—
364	C₂²⋊C₄	—	8	2	no	—
365	C₄²	—	2	2	no	—
366	C₄²	—	2	2	no	—
367	C₄²	—	4	4	no	—
368	C₄²	—	4	2	no	—
369	C₄²	—	8	2	no	—
370	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
371	C₄²	#41×C₃₁	1	2	yes	—
372	C₂³	—	2	2	no	—
373	C₂³	—	4	1	no	—
374	C₂³	—	4	1	no	—
375	C₂³	—	4	1	no	—
376	C₂³	—	4	1	no	—
377	C₂³	—	4	1	no	—
378	C₂³	—	4	1	no	—
379	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
380	Q₈	—	8	4	no	—
381	D₄	—	2	2	no	—
382	D₄	—	2	2	no	—
383	D₄	—	4	1	no	—
384	D₄	—	4	1	no	—
385	C₂×C₄	—	2	1	no	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	2	4	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	4	1	no	—
397	C₂×C₄	—	4	1	no	—
398	C₂×C₄	—	4	1	no	—
399	C₂×C₄	—	4	1	no	—
400	C₂×C₄	—	4	2	no	—
401	C₂×C₄	—	4	2	no	—
402	C₂×C₄	—	4	8	no	—
403	C₂×C₄	—	4	1	no	—
404	C₂×C₄	—	4	2	no	—
405	C₂×C₄	—	4	2	no	—
406	C₂×C₄	—	4	1	no	—
407	C₂×C₄	—	4	2	no	—
408	C₂×C₄	—	4	1	no	—
409	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
410	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
411	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
412	C₂×C₄	C₃₁×#137	1	2	yes	—
413	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
414	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
415	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
416	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
417	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
418	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
419	C₈	—	2	4	no	—
420	C₈	—	2	2	no	—
421	C₈	—	4	1	no	—
422	C₂²	—	4	2	no	—
423	C₂²	—	4	2	no	—
424	C₂²	—	4	2	no	—
425	C₂²	—	8	1	no	—
426	C₂²	—	8	1	no	—
427	C₂²	—	8	1	no	—
428	C₂²	—	8	1	no	—
429	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
430	C₂²	C₃₁×#547	1	1	yes	characteristic
431	C₂²	C₃₁×#548	1	1	yes	characteristic
432	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
433	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
434	C₂²	C₃₁×#576	1	1	yes	characteristic
435	C₂²	C₃₁×#577	1	1	yes	characteristic
436	C₄	—	2	2	no	—
437	C₄	—	4	2	no	—
438	C₄	—	4	2	no	—
439	C₄	—	4	2	no	—
440	C₄	—	4	2	no	—
441	C₄	—	4	4	no	—
442	C₄	—	8	1	no	—
443	C₄	—	8	2	no	—
444	C₄	C₃₁×#440	1	4	yes	—
445	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
446	C₄	C₃₁×#519	1	1	yes	characteristic
447	C₄	C₃₁×#552	1	1	yes	characteristic
448	C₄	C₃₁×#556	1	1	yes	characteristic
449	C₂	—	8	2	no	—
450	C₂	C₃₁×#2362	1	1	yes	characteristic
451	C₂	C₂×C₃₁×#440	1	1	yes	characteristic
452	C₂	C₃₁×#2946	1	1	yes	characteristic
453	C₂	C₃₁×#3354	1	1	yes	characteristic
454	C₂	C₃₁×#3357	1	1	yes	characteristic
455	C₂	C₃₁×#3366	1	1	yes	characteristic
456	C₂	C₃₁×#3369	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₄×C₃₁	C₂³
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×#2369	C₂	semidirect
4. C₃₁×#1990	C₂	semidirect
5. C₃₁×#1947	C₂	semidirect
8. C₃₁×#1919	C₂	semidirect
9. C₃₁×#1776	C₂	semidirect
10. C₃₁×#1770	C₂	semidirect
15. C₃₁×#476	C₄	semidirect
143. #23067	C₃₁	direct (N × H)
195. #2369	C₂×C₃₁	semidirect
197. #1990	C₂×C₃₁	semidirect
198. #1947	C₂×C₃₁	semidirect
201. #1919	C₂×C₃₁	semidirect
202. #1776	C₂×C₃₁	semidirect
203. #1770	C₂×C₃₁	semidirect
235. #476	C₄×C₃₁	semidirect
313. C₃₁	#23067	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×#476	2
C₃₁×#2369	2
C₃₁×#2354	2
C₃₁×#1990	2
C₃₁×#1947	2
C₃₁×#1936	2
C₃₁×#1925	2
C₃₁×#1919	2
C₃₁×#1776	2
C₃₁×#1770	2
C₄×C₃₁×#102	2
#23067	31

Subgroups — C31×#23067 #21479826

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23067 #21479826