Subgroups — C₃₁×#25274 #21482033

Order 7936.

986 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#486	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#476	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2541	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2369	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2065	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2041	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1982	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1982	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1939	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1926	C₂	1	2	yes	maximal, complemented
11	C₄×C₃₁×#102	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#127	C₂²	1	2	yes	—
15	C₂×C₃₁×#121	—	2	1	no	—
16	C₂×C₃₁×#121	—	2	1	no	—
17	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#486	—	2	4	no	—
19	C₃₁×#476	—	2	4	no	—
20	C₂×C₃₁×#106	C₂²	1	2	yes	—
21	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
23	C₂×C₃₁×#102	C₂²	1	2	yes	—
24	C₂×C₃₁×#102	C₂²	1	2	yes	—
25	C₂×C₃₁×#102	C₂²	1	2	yes	—
26	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#102	C₂²	1	2	yes	—
28	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#102	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
31	C₃₁×#392	—	2	2	no	—
32	C₃₁×#391	—	2	2	no	—
33	C₃₁×#390	C₂²	1	2	yes	—
34	C₃₁×#389	—	2	1	no	—
35	C₃₁×#389	C₂²	1	2	yes	—
36	C₄×C₄⋊C₄×C₃₁	—	2	1	no	—
37	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
38	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
39	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
40	C₃₁×#340	C₂²	1	1	yes	characteristic
41	C₃₁×#340	C₂²	1	1	yes	characteristic
42	C₃₁×#336	C₂²	1	2	yes	—
43	C₂²×Q₈×C₃₁	—	2	1	no	—
44	C₂²×D₄×C₃₁	—	2	1	no	—
45	C₂²×D₄×C₃₁	—	2	2	no	—
46	C₂²×D₄×C₃₁	—	2	2	no	—
47	C₂²×D₄×C₃₁	—	2	1	no	—
48	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
49	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₃₁	—	2	1	no	—
51	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
54	C₃₁×C₄⋊Q₈	—	2	2	no	—
55	C₃₁×C₄⋊Q₈	D₄	1	4	yes	complemented
56	C₃₁×#127	—	2	4	no	—
57	C₃₁×#127	—	2	2	no	—
58	C₃₁×#127	—	2	2	no	—
59	C₃₁×#127	—	4	2	no	—
60	C₃₁×#121	—	4	4	no	—
61	C₃₁×#121	—	4	4	no	—
62	C₄×D₄×C₃₁	—	2	2	no	—
63	C₄×D₄×C₃₁	—	2	2	no	—
64	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
65	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
66	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
68	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
71	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
72	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
73	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
74	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
75	C₂×C₄²×C₃₁	—	2	1	no	—
76	C₂×C₄²×C₃₁	—	2	2	no	—
77	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
78	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
79	C₃₁×#106	—	2	4	no	—
80	C₃₁×C₄⋊C₈	—	2	2	no	—
81	C₃₁×C₄⋊C₈	—	4	2	no	—
82	C₃₁×#102	—	2	2	no	—
83	C₃₁×#102	—	2	4	no	—
84	C₃₁×#102	—	2	4	no	—
85	C₃₁×#102	—	2	4	no	—
86	C₃₁×#102	—	2	2	no	—
87	C₃₁×#102	—	2	4	no	—
88	C₃₁×#102	—	4	2	no	—
89	C₄×C₈×C₃₁	—	2	2	no	—
90	C₃₁×#95	—	2	2	no	—
91	C₃₁×#95	—	2	2	no	—
92	C₃₁×#95	—	2	1	no	—
93	C₃₁×#95	D₄	1	2	yes	—
94	C₂⁴×C₃₁	—	2	1	no	—
95	C₂⁴×C₃₁	—	2	2	no	—
96	C₂×Q₈×C₃₁	—	2	4	no	—
97	C₂×Q₈×C₃₁	—	8	1	no	—
98	C₂×Q₈×C₃₁	—	8	1	no	—
99	C₂×D₄×C₃₁	—	2	1	no	—
100	C₂×D₄×C₃₁	—	2	2	no	—
101	C₂×D₄×C₃₁	—	2	1	no	—
102	C₂×D₄×C₃₁	—	2	1	no	—
103	C₂×D₄×C₃₁	—	2	2	no	—
104	C₂×D₄×C₃₁	—	2	2	no	—
105	C₂×D₄×C₃₁	—	2	2	no	—
106	C₂×D₄×C₃₁	—	2	1	no	—
107	C₂×D₄×C₃₁	—	2	2	no	—
108	C₂×D₄×C₃₁	—	2	1	no	—
109	C₂×D₄×C₃₁	—	2	2	no	—
110	C₂×D₄×C₃₁	—	4	4	no	—
111	C₂×D₄×C₃₁	—	4	4	no	—
112	C₂×D₄×C₃₁	—	4	4	no	—
113	C₂×D₄×C₃₁	—	4	2	no	—
114	C₂×D₄×C₃₁	—	4	2	no	—
115	C₂×D₄×C₃₁	—	4	4	no	—
116	C₂×D₄×C₃₁	—	4	2	no	—
117	C₂×D₄×C₃₁	—	4	2	no	—
118	C₂×D₄×C₃₁	—	8	2	no	—
119	C₂×D₄×C₃₁	—	8	1	no	—
120	C₂×D₄×C₃₁	—	8	1	no	—
121	C₂×D₄×C₃₁	—	8	2	no	—
122	C₂×D₄×C₃₁	#41	1	2	yes	—
123	C₂²×C₄×C₃₁	—	2	1	no	—
124	C₂²×C₄×C₃₁	—	2	2	no	—
125	C₂²×C₄×C₃₁	—	2	1	no	—
126	C₂²×C₄×C₃₁	—	2	1	no	—
127	C₂²×C₄×C₃₁	—	4	2	no	—
128	C₂²×C₄×C₃₁	—	4	2	no	—
129	C₂²×C₄×C₃₁	—	4	2	no	—
130	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
131	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
133	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
134	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
135	C₂×C₈×C₃₁	—	2	1	no	—
136	C₂×C₈×C₃₁	—	2	1	no	—
137	C₂×C₈×C₃₁	—	2	2	no	—
138	C₂×C₈×C₃₁	—	2	1	no	—
139	C₂×C₈×C₃₁	—	2	1	no	—
140	C₂×C₈×C₃₁	—	2	2	no	—
141	C₂×C₈×C₃₁	—	2	2	no	—
142	C₂×C₈×C₃₁	—	2	1	no	—
143	C₂×C₈×C₃₁	—	2	1	no	—
144	C₄⋊C₄×C₃₁	—	2	4	no	—
145	C₄⋊C₄×C₃₁	—	2	2	no	—
146	C₄⋊C₄×C₃₁	—	2	2	no	—
147	C₄⋊C₄×C₃₁	—	2	1	no	—
148	C₄⋊C₄×C₃₁	—	2	1	no	—
149	C₄⋊C₄×C₃₁	—	2	1	no	—
150	C₄⋊C₄×C₃₁	—	2	1	no	—
151	C₄⋊C₄×C₃₁	—	2	1	no	—
152	C₄⋊C₄×C₃₁	—	8	2	no	—
153	C₄⋊C₄×C₃₁	SD₁₆	1	4	yes	—
154	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
155	C₄⋊C₄×C₃₁	#41	1	2	yes	—
156	C₂²⋊C₄×C₃₁	—	4	4	no	—
157	C₂²⋊C₄×C₃₁	—	4	2	no	—
158	C₂²⋊C₄×C₃₁	—	4	4	no	—
159	C₄²×C₃₁	—	2	2	no	—
160	C₄²×C₃₁	—	2	2	no	—
161	C₄²×C₃₁	—	4	2	no	—
162	C₄²×C₃₁	—	8	2	no	—
163	C₄²×C₃₁	C₂×D₄	1	2	yes	—
164	C₄²×C₃₁	#41	1	2	yes	—
165	#25274	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
166	C₂³×C₃₁	—	2	2	no	—
167	C₂³×C₃₁	—	4	1	no	—
168	C₂³×C₃₁	—	4	1	no	—
169	C₂³×C₃₁	—	4	1	no	—
170	C₂³×C₃₁	—	4	1	no	—
171	C₂³×C₃₁	—	4	1	no	—
172	C₂³×C₃₁	—	4	1	no	—
173	C₂³×C₃₁	—	4	2	no	—
174	C₂³×C₃₁	—	4	2	no	—
175	C₂³×C₃₁	—	4	2	no	—
176	C₂³×C₃₁	—	4	2	no	—
177	C₂³×C₃₁	—	4	2	no	—
178	C₂³×C₃₁	—	4	2	no	—
179	C₂³×C₃₁	—	4	2	no	—
180	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
181	Q₈×C₃₁	—	8	4	no	—
182	D₄×C₃₁	—	2	2	no	—
183	D₄×C₃₁	—	2	2	no	—
184	D₄×C₃₁	—	4	2	no	—
185	D₄×C₃₁	—	4	1	no	—
186	D₄×C₃₁	—	4	2	no	—
187	D₄×C₃₁	—	4	2	no	—
188	D₄×C₃₁	—	4	1	no	—
189	D₄×C₃₁	—	4	2	no	—
190	D₄×C₃₁	—	8	8	no	—
191	D₄×C₃₁	—	8	4	no	—
192	D₄×C₃₁	—	8	8	no	—
193	D₄×C₃₁	—	8	4	no	—
194	C₂×C₄×C₃₁	—	2	2	no	—
195	C₂×C₄×C₃₁	—	2	2	no	—
196	C₂×C₄×C₃₁	—	2	1	no	—
197	C₂×C₄×C₃₁	—	2	1	no	—
198	C₂×C₄×C₃₁	—	2	4	no	—
199	C₂×C₄×C₃₁	—	2	2	no	—
200	C₂×C₄×C₃₁	—	2	2	no	—
201	C₂×C₄×C₃₁	—	2	2	no	—
202	C₂×C₄×C₃₁	—	2	2	no	—
203	C₂×C₄×C₃₁	—	4	1	no	—
204	C₂×C₄×C₃₁	—	4	1	no	—
205	C₂×C₄×C₃₁	—	4	2	no	—
206	C₂×C₄×C₃₁	—	4	2	no	—
207	C₂×C₄×C₃₁	—	4	1	no	—
208	C₂×C₄×C₃₁	—	4	1	no	—
209	C₂×C₄×C₃₁	—	4	1	no	—
210	C₂×C₄×C₃₁	—	4	1	no	—
211	C₂×C₄×C₃₁	—	4	1	no	—
212	C₂×C₄×C₃₁	—	4	1	no	—
213	C₂×C₄×C₃₁	—	8	4	no	—
214	C₂×C₄×C₃₁	#121	1	2	yes	—
215	C₂×C₄×C₃₁	#121	1	2	yes	—
216	C₂×C₄×C₃₁	C₂×SD₁₆	1	4	yes	—
217	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
218	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
219	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
220	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
221	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
222	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
223	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
224	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
225	C₈×C₃₁	—	4	1	no	—
226	C₈×C₃₁	—	4	1	no	—
227	C₈×C₃₁	—	4	2	no	—
228	C₂×#486	C₂×C₃₁	1	1	yes	characteristic, complemented
229	C₂×#476	C₂×C₃₁	1	1	yes	characteristic, complemented
230	#2541	C₂×C₃₁	1	1	yes	characteristic
231	#2369	C₂×C₃₁	1	1	yes	characteristic, complemented
232	#2065	C₂×C₃₁	1	2	yes	complemented
233	#2041	C₂×C₃₁	1	2	yes	complemented
234	#1982	C₂×C₃₁	1	1	yes	characteristic, complemented
235	#1982	C₂×C₃₁	1	2	yes	complemented
236	#1939	C₂×C₃₁	1	1	yes	characteristic, complemented
237	#1926	C₂×C₃₁	1	2	yes	complemented
238	C₄×#102	C₂×C₃₁	1	1	yes	characteristic, complemented
239	C₂²×C₃₁	—	4	2	no	—
240	C₂²×C₃₁	—	4	2	no	—
241	C₂²×C₃₁	—	4	2	no	—
242	C₂²×C₃₁	—	8	1	no	—
243	C₂²×C₃₁	—	8	1	no	—
244	C₂²×C₃₁	—	8	1	no	—
245	C₂²×C₃₁	—	8	1	no	—
246	C₂²×C₃₁	—	8	2	no	—
247	C₂²×C₃₁	—	8	2	no	—
248	C₂²×C₃₁	—	8	2	no	—
249	C₂²×C₃₁	—	8	2	no	—
250	C₂²×C₃₁	—	8	2	no	—
251	C₂²×C₃₁	—	8	2	no	—
252	C₂²×C₃₁	—	8	2	no	—
253	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
254	C₂²×C₃₁	#546	1	1	yes	characteristic
255	C₂²×C₃₁	#548	1	1	yes	characteristic
256	C₂²×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
257	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
258	C₂²×C₃₁	#576	1	1	yes	characteristic
259	C₂²×C₃₁	#576	1	1	yes	characteristic
260	C₄×C₃₁	—	2	4	no	—
261	C₄×C₃₁	—	2	2	no	—
262	C₄×C₃₁	—	4	2	no	—
263	C₄×C₃₁	—	4	4	no	—
264	C₄×C₃₁	—	4	2	no	—
265	C₄×C₃₁	—	8	1	no	—
266	C₄×C₃₁	—	8	1	no	—
267	C₄×C₃₁	#460	1	4	yes	—
268	C₄×C₃₁	#529	1	1	yes	characteristic
269	C₄×C₃₁	#542	1	1	yes	characteristic
270	C₄×C₃₁	#548	1	1	yes	characteristic
271	C₄×C₃₁	#550	1	1	yes	characteristic
272	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
273	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
274	C₂×#127	C₂²×C₃₁	1	2	yes	—
275	C₂×#121	—	2	1	no	—
276	C₂×#121	—	2	1	no	—
277	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
278	#486	—	2	4	no	—
279	#476	—	2	4	no	—
280	C₂×#106	C₂²×C₃₁	1	2	yes	—
281	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
282	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
283	C₂×#102	C₂²×C₃₁	1	2	yes	—
284	C₂×#102	C₂²×C₃₁	1	2	yes	—
285	C₂×#102	C₂²×C₃₁	1	2	yes	—
286	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
287	C₂×#102	C₂²×C₃₁	1	2	yes	—
288	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
289	C₂×#102	C₂²×C₃₁	1	2	yes	—
290	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
291	#392	—	2	2	no	—
292	#391	—	2	2	no	—
293	#390	C₂²×C₃₁	1	2	yes	—
294	#389	—	2	1	no	—
295	#389	C₂²×C₃₁	1	2	yes	—
296	C₄×C₄⋊C₄	—	2	1	no	—
297	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
298	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
299	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
300	#340	C₂²×C₃₁	1	1	yes	characteristic
301	#340	C₂²×C₃₁	1	1	yes	characteristic
302	#336	C₂²×C₃₁	1	2	yes	—
303	C₂×C₃₁	—	8	2	no	—
304	C₂×C₃₁	—	16	2	no	—
305	C₂×C₃₁	#2426	1	1	yes	characteristic
306	C₂×C₃₁	C₂×#460	1	1	yes	characteristic
307	C₂×C₃₁	#3041	1	1	yes	characteristic
308	C₂×C₃₁	#3295	1	1	yes	characteristic
309	C₂×C₃₁	#3325	1	1	yes	characteristic
310	C₂×C₃₁	#3362	1	1	yes	characteristic
311	C₂×C₃₁	#3366	1	1	yes	characteristic
312	C₂²×Q₈	—	2	1	no	—
313	C₂²×D₄	—	2	1	no	—
314	C₂²×D₄	—	2	2	no	—
315	C₂²×D₄	—	2	2	no	—
316	C₂²×D₄	—	2	1	no	—
317	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
318	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
319	C₂³×C₄	—	2	1	no	—
320	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
321	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
322	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
323	C₄⋊Q₈	—	2	2	no	—
324	C₄⋊Q₈	D₄×C₃₁	1	4	yes	complemented
325	#127	—	2	4	no	—
326	#127	—	2	2	no	—
327	#127	—	2	2	no	—
328	#127	—	4	2	no	—
329	#121	—	4	4	no	—
330	#121	—	4	4	no	—
331	C₄×D₄	—	2	2	no	—
332	C₄×D₄	—	2	2	no	—
333	C₂×C₄⋊C₄	—	2	1	no	—
334	C₂×C₄⋊C₄	—	2	2	no	—
335	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
336	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
337	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
338	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
339	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
340	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
341	C₂×C₂²⋊C₄	—	2	1	no	—
342	C₂×C₂²⋊C₄	—	2	2	no	—
343	C₂×C₂²⋊C₄	—	2	2	no	—
344	C₂×C₄²	—	2	1	no	—
345	C₂×C₄²	—	2	2	no	—
346	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
347	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
348	#106	—	2	4	no	—
349	C₄⋊C₈	—	2	2	no	—
350	C₄⋊C₈	—	4	2	no	—
351	#102	—	2	2	no	—
352	#102	—	2	4	no	—
353	#102	—	2	4	no	—
354	#102	—	2	4	no	—
355	#102	—	2	2	no	—
356	#102	—	2	4	no	—
357	#102	—	4	2	no	—
358	C₄×C₈	—	2	2	no	—
359	#95	—	2	2	no	—
360	#95	—	2	2	no	—
361	#95	—	2	1	no	—
362	#95	D₄×C₃₁	1	2	yes	—
363	C₃₁	#25274	1	1	yes	characteristic, direct product factor, Sylow
364	C₂⁴	—	2	1	no	—
365	C₂⁴	—	2	2	no	—
366	C₂×Q₈	—	2	4	no	—
367	C₂×Q₈	—	8	1	no	—
368	C₂×Q₈	—	8	1	no	—
369	C₂×D₄	—	2	1	no	—
370	C₂×D₄	—	2	2	no	—
371	C₂×D₄	—	2	1	no	—
372	C₂×D₄	—	2	1	no	—
373	C₂×D₄	—	2	2	no	—
374	C₂×D₄	—	2	2	no	—
375	C₂×D₄	—	2	2	no	—
376	C₂×D₄	—	2	1	no	—
377	C₂×D₄	—	2	2	no	—
378	C₂×D₄	—	2	1	no	—
379	C₂×D₄	—	2	2	no	—
380	C₂×D₄	—	4	4	no	—
381	C₂×D₄	—	4	4	no	—
382	C₂×D₄	—	4	4	no	—
383	C₂×D₄	—	4	2	no	—
384	C₂×D₄	—	4	2	no	—
385	C₂×D₄	—	4	4	no	—
386	C₂×D₄	—	4	2	no	—
387	C₂×D₄	—	4	2	no	—
388	C₂×D₄	—	8	2	no	—
389	C₂×D₄	—	8	1	no	—
390	C₂×D₄	—	8	1	no	—
391	C₂×D₄	—	8	2	no	—
392	C₂×D₄	#41×C₃₁	1	2	yes	—
393	C₂²×C₄	—	2	1	no	—
394	C₂²×C₄	—	2	2	no	—
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	4	2	no	—
398	C₂²×C₄	—	4	2	no	—
399	C₂²×C₄	—	4	2	no	—
400	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
401	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
402	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
403	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
404	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
405	C₂×C₈	—	2	1	no	—
406	C₂×C₈	—	2	1	no	—
407	C₂×C₈	—	2	2	no	—
408	C₂×C₈	—	2	1	no	—
409	C₂×C₈	—	2	2	no	—
410	C₂×C₈	—	2	1	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	2	1	no	—
413	C₂×C₈	—	2	1	no	—
414	C₄⋊C₄	—	2	4	no	—
415	C₄⋊C₄	—	2	2	no	—
416	C₄⋊C₄	—	2	2	no	—
417	C₄⋊C₄	—	2	1	no	—
418	C₄⋊C₄	—	2	1	no	—
419	C₄⋊C₄	—	2	1	no	—
420	C₄⋊C₄	—	2	1	no	—
421	C₄⋊C₄	—	2	1	no	—
422	C₄⋊C₄	—	8	2	no	—
423	C₄⋊C₄	SD₁₆×C₃₁	1	4	yes	—
424	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
425	C₄⋊C₄	#41×C₃₁	1	2	yes	—
426	C₂²⋊C₄	—	4	4	no	—
427	C₂²⋊C₄	—	4	2	no	—
428	C₂²⋊C₄	—	4	4	no	—
429	C₄²	—	2	2	no	—
430	C₄²	—	2	2	no	—
431	C₄²	—	4	2	no	—
432	C₄²	—	8	2	no	—
433	C₄²	C₂×D₄×C₃₁	1	2	yes	—
434	C₄²	#41×C₃₁	1	2	yes	—
435	C₂³	—	2	2	no	—
436	C₂³	—	4	1	no	—
437	C₂³	—	4	1	no	—
438	C₂³	—	4	1	no	—
439	C₂³	—	4	1	no	—
440	C₂³	—	4	1	no	—
441	C₂³	—	4	1	no	—
442	C₂³	—	4	2	no	—
443	C₂³	—	4	2	no	—
444	C₂³	—	4	2	no	—
445	C₂³	—	4	2	no	—
446	C₂³	—	4	2	no	—
447	C₂³	—	4	2	no	—
448	C₂³	—	4	2	no	—
449	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
450	Q₈	—	8	4	no	—
451	D₄	—	2	2	no	—
452	D₄	—	2	2	no	—
453	D₄	—	4	2	no	—
454	D₄	—	4	1	no	—
455	D₄	—	4	2	no	—
456	D₄	—	4	2	no	—
457	D₄	—	4	1	no	—
458	D₄	—	4	2	no	—
459	D₄	—	8	8	no	—
460	D₄	—	8	4	no	—
461	D₄	—	8	8	no	—
462	D₄	—	8	4	no	—
463	C₂×C₄	—	2	2	no	—
464	C₂×C₄	—	2	2	no	—
465	C₂×C₄	—	2	1	no	—
466	C₂×C₄	—	2	1	no	—
467	C₂×C₄	—	2	4	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	2	no	—
471	C₂×C₄	—	2	2	no	—
472	C₂×C₄	—	4	1	no	—
473	C₂×C₄	—	4	1	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	4	2	no	—
476	C₂×C₄	—	4	1	no	—
477	C₂×C₄	—	4	1	no	—
478	C₂×C₄	—	4	1	no	—
479	C₂×C₄	—	4	1	no	—
480	C₂×C₄	—	4	1	no	—
481	C₂×C₄	—	4	1	no	—
482	C₂×C₄	—	8	4	no	—
483	C₂×C₄	C₃₁×#121	1	2	yes	—
484	C₂×C₄	C₃₁×#121	1	2	yes	—
485	C₂×C₄	C₂×SD₁₆×C₃₁	1	4	yes	—
486	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
487	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
488	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
489	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
490	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
491	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
492	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
493	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
494	C₈	—	4	1	no	—
495	C₈	—	4	1	no	—
496	C₈	—	4	2	no	—
497	C₂²	—	4	2	no	—
498	C₂²	—	4	2	no	—
499	C₂²	—	4	2	no	—
500	C₂²	—	8	1	no	—
501	C₂²	—	8	1	no	—
502	C₂²	—	8	1	no	—
503	C₂²	—	8	1	no	—
504	C₂²	—	8	2	no	—
505	C₂²	—	8	2	no	—
506	C₂²	—	8	2	no	—
507	C₂²	—	8	2	no	—
508	C₂²	—	8	2	no	—
509	C₂²	—	8	2	no	—
510	C₂²	—	8	2	no	—
511	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
512	C₂²	C₃₁×#546	1	1	yes	characteristic
513	C₂²	C₃₁×#548	1	1	yes	characteristic
514	C₂²	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
515	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
516	C₂²	C₃₁×#576	1	1	yes	characteristic
517	C₂²	C₃₁×#576	1	1	yes	characteristic
518	C₄	—	2	4	no	—
519	C₄	—	2	2	no	—
520	C₄	—	4	2	no	—
521	C₄	—	4	4	no	—
522	C₄	—	4	2	no	—
523	C₄	—	8	1	no	—
524	C₄	—	8	1	no	—
525	C₄	C₃₁×#460	1	4	yes	—
526	C₄	C₃₁×#529	1	1	yes	characteristic
527	C₄	C₃₁×#542	1	1	yes	characteristic
528	C₄	C₃₁×#548	1	1	yes	characteristic
529	C₄	C₃₁×#550	1	1	yes	characteristic
530	C₂	—	8	2	no	—
531	C₂	—	16	2	no	—
532	C₂	C₃₁×#2426	1	1	yes	characteristic
533	C₂	C₂×C₃₁×#460	1	1	yes	characteristic
534	C₂	C₃₁×#3041	1	1	yes	characteristic
535	C₂	C₃₁×#3295	1	1	yes	characteristic
536	C₂	C₃₁×#3325	1	1	yes	characteristic
537	C₂	C₃₁×#3362	1	1	yes	characteristic
538	C₂	C₃₁×#3366	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#2541	C₂
LCS	G ▷ C₂²×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
1. C₂×C₃₁×#486	C₂	semidirect
2. C₂×C₃₁×#476	C₂	semidirect
4. C₃₁×#2369	C₂	semidirect
5. C₃₁×#2065	C₂	semidirect
6. C₃₁×#2041	C₂	semidirect
7. C₃₁×#1982	C₂	semidirect
8. C₃₁×#1982	C₂	semidirect
9. C₃₁×#1939	C₂	semidirect
10. C₃₁×#1926	C₂	semidirect
11. C₄×C₃₁×#102	C₂	semidirect
55. C₃₁×C₄⋊Q₈	D₄	semidirect
165. #25274	C₃₁	direct (N × H)
228. C₂×#486	C₂×C₃₁	semidirect
229. C₂×#476	C₂×C₃₁	semidirect
231. #2369	C₂×C₃₁	semidirect
232. #2065	C₂×C₃₁	semidirect
233. #2041	C₂×C₃₁	semidirect
234. #1982	C₂×C₃₁	semidirect
235. #1982	C₂×C₃₁	semidirect
236. #1939	C₂×C₃₁	semidirect
237. #1926	C₂×C₃₁	semidirect
238. C₄×#102	C₂×C₃₁	semidirect
324. C₄⋊Q₈	D₄×C₃₁	semidirect
363. C₃₁	#25274	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×#486	2
C₂×C₃₁×#476	2
C₃₁×#2541	2
C₃₁×#2369	2
C₃₁×#2065	2
C₃₁×#2041	2
C₃₁×#1982	2
C₃₁×#1939	2
C₃₁×#1926	2
C₄×C₃₁×#102	2
#25274	31

Subgroups — C31×#25274 #21482033

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#25274 #21482033