Subgroups — C₃₁×#26254 #21483013

Order 7936.

1050 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#493	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2445	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2054	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2041	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1970	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1970	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1970	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1942	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1879	C₂	1	2	yes	maximal, complemented
11	C₈×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
14	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#127	C₂²	1	2	yes	—
16	C₂×C₃₁×#123	—	2	2	no	—
17	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
18	C₃₁×#493	—	2	4	no	—
19	C₄×D₈×C₃₁	—	2	4	no	—
20	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
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	2	yes	—
27	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
30	C₃₁×#394	—	2	2	no	—
31	C₃₁×#394	—	2	2	no	—
32	C₃₁×#390	C₂²	1	2	yes	—
33	C₃₁×#390	C₂²	1	2	yes	—
34	C₃₁×#390	C₂²	1	2	yes	—
35	C₃₁×#389	C₂²	1	2	yes	—
36	C₃₁×#386	—	2	2	no	—
37	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#336	C₂²	1	1	yes	characteristic
40	C₃₁×#336	C₂²	1	1	yes	characteristic
41	C₂²×D₄×C₃₁	—	2	2	no	—
42	C₂²×D₄×C₃₁	—	2	2	no	—
43	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
44	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
45	C₂³×C₄×C₃₁	—	2	2	no	—
46	C₂×D₈×C₃₁	—	2	4	no	—
47	C₂×D₈×C₃₁	—	2	2	no	—
48	C₂×D₈×C₃₁	—	2	2	no	—
49	C₂×D₈×C₃₁	—	2	2	no	—
50	C₂×D₈×C₃₁	—	2	4	no	—
51	C₂×D₈×C₃₁	—	2	2	no	—
52	C₂×D₈×C₃₁	—	2	2	no	—
53	C₂×D₈×C₃₁	—	4	2	no	—
54	C₂×D₈×C₃₁	D₄	1	4	yes	—
55	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
59	C₃₁×#127	—	2	4	no	—
60	C₃₁×#127	—	4	2	no	—
61	C₃₁×#123	—	4	8	no	—
62	C₄×D₄×C₃₁	—	2	4	no	—
63	C₄×D₄×C₃₁	—	4	2	no	—
64	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
65	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
68	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
69	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
70	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
71	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
72	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
73	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
74	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
75	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
76	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
79	C₃₁×#107	—	2	2	no	—
80	C₃₁×#107	—	2	2	no	—
81	C₃₁×C₄⋊C₈	—	2	2	no	—
82	C₃₁×#102	—	2	4	no	—
83	C₃₁×#102	—	4	2	no	—
84	C₃₁×#102	—	4	2	no	—
85	C₃₁×#102	—	4	2	no	—
86	C₄×C₈×C₃₁	—	2	2	no	—
87	C₄×C₈×C₃₁	—	2	2	no	—
88	C₄×C₈×C₃₁	—	2	2	no	—
89	C₃₁×#95	—	2	2	no	—
90	C₃₁×#95	D₄	1	1	yes	characteristic
91	C₃₁×#95	D₄	1	1	yes	characteristic
92	C₂⁴×C₃₁	—	2	2	no	—
93	C₂⁴×C₃₁	—	2	2	no	—
94	C₂×D₄×C₃₁	—	2	2	no	—
95	C₂×D₄×C₃₁	—	2	2	no	—
96	C₂×D₄×C₃₁	—	2	2	no	—
97	C₂×D₄×C₃₁	—	2	2	no	—
98	C₂×D₄×C₃₁	—	2	2	no	—
99	C₂×D₄×C₃₁	—	2	2	no	—
100	C₂×D₄×C₃₁	—	2	2	no	—
101	C₂×D₄×C₃₁	—	2	2	no	—
102	C₂×D₄×C₃₁	—	2	2	no	—
103	C₂×D₄×C₃₁	—	2	2	no	—
104	C₂×D₄×C₃₁	—	2	2	no	—
105	C₂×D₄×C₃₁	—	4	4	no	—
106	C₂×D₄×C₃₁	—	4	4	no	—
107	C₂×D₄×C₃₁	—	4	4	no	—
108	C₂×D₄×C₃₁	—	4	4	no	—
109	C₂×D₄×C₃₁	—	8	2	no	—
110	C₂×D₄×C₃₁	—	8	2	no	—
111	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
112	C₂²×C₄×C₃₁	—	2	2	no	—
113	C₂²×C₄×C₃₁	—	2	2	no	—
114	C₂²×C₄×C₃₁	—	4	4	no	—
115	C₂²×C₄×C₃₁	—	4	4	no	—
116	C₂²×C₄×C₃₁	—	8	2	no	—
117	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
124	D₈×C₃₁	—	2	8	no	—
125	D₈×C₃₁	—	4	8	no	—
126	D₈×C₃₁	—	4	4	no	—
127	C₂×C₈×C₃₁	—	2	1	no	—
128	C₂×C₈×C₃₁	—	2	1	no	—
129	C₂×C₈×C₃₁	—	2	1	no	—
130	C₂×C₈×C₃₁	—	2	1	no	—
131	C₂×C₈×C₃₁	—	2	1	no	—
132	C₂×C₈×C₃₁	—	2	1	no	—
133	C₂×C₈×C₃₁	—	2	1	no	—
134	C₂×C₈×C₃₁	—	2	1	no	—
135	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
136	C₂×C₈×C₃₁	#41	1	2	yes	—
137	C₂×C₈×C₃₁	#41	1	2	yes	—
138	C₂×C₈×C₃₁	#41	1	2	yes	—
139	C₄⋊C₄×C₃₁	—	2	2	no	—
140	C₄⋊C₄×C₃₁	—	2	2	no	—
141	C₄⋊C₄×C₃₁	—	2	2	no	—
142	C₄⋊C₄×C₃₁	—	2	2	no	—
143	C₄⋊C₄×C₃₁	—	2	2	no	—
144	C₄⋊C₄×C₃₁	—	4	4	no	—
145	C₄⋊C₄×C₃₁	D₈	1	4	yes	complemented
146	C₂²⋊C₄×C₃₁	—	4	4	no	—
147	C₂²⋊C₄×C₃₁	—	4	4	no	—
148	C₂²⋊C₄×C₃₁	—	4	4	no	—
149	C₂²⋊C₄×C₃₁	—	4	4	no	—
150	C₂²⋊C₄×C₃₁	—	8	2	no	—
151	C₂²⋊C₄×C₃₁	—	8	2	no	—
152	C₂²⋊C₄×C₃₁	—	8	2	no	—
153	C₄²×C₃₁	—	2	1	no	—
154	C₄²×C₃₁	—	2	1	no	—
155	C₄²×C₃₁	—	2	1	no	—
156	C₄²×C₃₁	—	2	1	no	—
157	C₄²×C₃₁	—	2	1	no	—
158	C₄²×C₃₁	—	2	1	no	—
159	#26254	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂³×C₃₁	—	2	4	no	—
161	C₂³×C₃₁	—	4	2	no	—
162	C₂³×C₃₁	—	4	2	no	—
163	C₂³×C₃₁	—	4	2	no	—
164	C₂³×C₃₁	—	4	2	no	—
165	C₂³×C₃₁	—	4	2	no	—
166	C₂³×C₃₁	—	4	2	no	—
167	C₂³×C₃₁	—	4	2	no	—
168	C₂³×C₃₁	—	4	2	no	—
169	C₂³×C₃₁	—	4	2	no	—
170	C₂³×C₃₁	—	4	2	no	—
171	C₂³×C₃₁	—	4	2	no	—
172	C₂³×C₃₁	—	4	2	no	—
173	C₂³×C₃₁	—	4	2	no	—
174	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
175	D₄×C₃₁	—	2	4	no	—
176	D₄×C₃₁	—	2	4	no	—
177	D₄×C₃₁	—	4	2	no	—
178	D₄×C₃₁	—	4	2	no	—
179	D₄×C₃₁	—	4	2	no	—
180	D₄×C₃₁	—	4	2	no	—
181	D₄×C₃₁	—	4	2	no	—
182	D₄×C₃₁	—	4	2	no	—
183	D₄×C₃₁	—	8	8	no	—
184	D₄×C₃₁	—	8	8	no	—
185	C₂×C₄×C₃₁	—	2	1	no	—
186	C₂×C₄×C₃₁	—	2	1	no	—
187	C₂×C₄×C₃₁	—	2	1	no	—
188	C₂×C₄×C₃₁	—	2	1	no	—
189	C₂×C₄×C₃₁	—	2	1	no	—
190	C₂×C₄×C₃₁	—	2	1	no	—
191	C₂×C₄×C₃₁	—	2	1	no	—
192	C₂×C₄×C₃₁	—	2	1	no	—
193	C₂×C₄×C₃₁	—	2	1	no	—
194	C₂×C₄×C₃₁	—	2	1	no	—
195	C₂×C₄×C₃₁	—	2	1	no	—
196	C₂×C₄×C₃₁	—	2	1	no	—
197	C₂×C₄×C₃₁	—	2	1	no	—
198	C₂×C₄×C₃₁	—	2	1	no	—
199	C₂×C₄×C₃₁	—	4	2	no	—
200	C₂×C₄×C₃₁	—	4	2	no	—
201	C₂×C₄×C₃₁	—	4	2	no	—
202	C₂×C₄×C₃₁	—	4	2	no	—
203	C₂×C₄×C₃₁	—	4	2	no	—
204	C₂×C₄×C₃₁	—	4	2	no	—
205	C₂×C₄×C₃₁	—	8	8	no	—
206	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
207	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
208	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
209	C₂×C₄×C₃₁	#135	1	2	yes	—
210	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
211	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
212	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
213	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
214	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
215	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
216	C₈×C₃₁	—	2	2	no	—
217	C₈×C₃₁	—	4	1	no	—
218	C₈×C₃₁	—	4	1	no	—
219	C₈×C₃₁	#123	1	4	yes	complemented
220	C₂×#493	C₂×C₃₁	1	1	yes	characteristic
221	C₂×C₄×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
222	#2445	C₂×C₃₁	1	2	yes	complemented
223	#2054	C₂×C₃₁	1	2	yes	complemented
224	#2041	C₂×C₃₁	1	2	yes	complemented
225	#1970	C₂×C₃₁	1	1	yes	characteristic, complemented
226	#1970	C₂×C₃₁	1	1	yes	characteristic, complemented
227	#1970	C₂×C₃₁	1	1	yes	characteristic, complemented
228	#1942	C₂×C₃₁	1	1	yes	characteristic, complemented
229	#1879	C₂×C₃₁	1	2	yes	complemented
230	C₈×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
231	C₂²×C₃₁	—	4	4	no	—
232	C₂²×C₃₁	—	4	4	no	—
233	C₂²×C₃₁	—	4	4	no	—
234	C₂²×C₃₁	—	8	2	no	—
235	C₂²×C₃₁	—	8	2	no	—
236	C₂²×C₃₁	—	8	2	no	—
237	C₂²×C₃₁	—	8	2	no	—
238	C₂²×C₃₁	—	8	2	no	—
239	C₂²×C₃₁	—	8	2	no	—
240	C₂²×C₃₁	—	8	2	no	—
241	C₂²×C₃₁	—	8	2	no	—
242	C₂²×C₃₁	—	8	2	no	—
243	C₂²×C₃₁	—	8	2	no	—
244	C₂²×C₃₁	—	8	2	no	—
245	C₂²×C₃₁	#532	1	1	yes	characteristic
246	C₂²×C₃₁	D₄²	1	1	yes	characteristic
247	C₂²×C₃₁	#550	1	1	yes	characteristic
248	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
249	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
250	C₂²×C₃₁	#576	1	1	yes	characteristic
251	C₂²×C₃₁	#576	1	1	yes	characteristic
252	C₄×C₃₁	—	2	2	no	—
253	C₄×C₃₁	—	2	2	no	—
254	C₄×C₃₁	—	2	2	no	—
255	C₄×C₃₁	—	2	2	no	—
256	C₄×C₃₁	—	4	1	no	—
257	C₄×C₃₁	—	4	1	no	—
258	C₄×C₃₁	—	8	2	no	—
259	C₄×C₃₁	—	8	2	no	—
260	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
261	C₄×C₃₁	#540	1	1	yes	characteristic
262	C₄×C₃₁	#546	1	1	yes	characteristic
263	C₄×C₃₁	#559	1	1	yes	characteristic
264	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
265	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
266	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
267	C₂×#127	C₂²×C₃₁	1	2	yes	—
268	C₂×#123	—	2	2	no	—
269	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
270	#493	—	2	4	no	—
271	C₄×D₈	—	2	4	no	—
272	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
273	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
274	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
275	C₂×#102	C₂²×C₃₁	1	2	yes	—
276	C₂×#102	C₂²×C₃₁	1	2	yes	—
277	C₂×#102	C₂²×C₃₁	1	2	yes	—
278	C₂×#102	C₂²×C₃₁	1	2	yes	—
279	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
280	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
281	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
282	#394	—	2	2	no	—
283	#394	—	2	2	no	—
284	#390	C₂²×C₃₁	1	2	yes	—
285	#390	C₂²×C₃₁	1	2	yes	—
286	#390	C₂²×C₃₁	1	2	yes	—
287	#389	C₂²×C₃₁	1	2	yes	—
288	#386	—	2	2	no	—
289	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
290	#340	C₂²×C₃₁	1	2	yes	—
291	#336	C₂²×C₃₁	1	1	yes	characteristic
292	#336	C₂²×C₃₁	1	1	yes	characteristic
293	C₂×C₃₁	—	8	4	no	—
294	C₂×C₃₁	—	16	2	no	—
295	C₂×C₃₁	#2445	1	1	yes	characteristic
296	C₂×C₃₁	#3169	1	1	yes	characteristic
297	C₂×C₃₁	#3176	1	1	yes	characteristic
298	C₂×C₃₁	D₄×D₈	1	1	yes	characteristic
299	C₂×C₃₁	#3280	1	1	yes	characteristic
300	C₂×C₃₁	#3391	1	1	yes	characteristic
301	C₂×C₃₁	#3396	1	1	yes	characteristic
302	C₂²×D₄	—	2	2	no	—
303	C₂²×D₄	—	2	2	no	—
304	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
305	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
306	C₂³×C₄	—	2	2	no	—
307	C₂×D₈	—	2	4	no	—
308	C₂×D₈	—	2	2	no	—
309	C₂×D₈	—	2	2	no	—
310	C₂×D₈	—	2	2	no	—
311	C₂×D₈	—	2	4	no	—
312	C₂×D₈	—	2	2	no	—
313	C₂×D₈	—	2	2	no	—
314	C₂×D₈	—	4	2	no	—
315	C₂×D₈	D₄×C₃₁	1	4	yes	—
316	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
317	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
318	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
319	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
320	#127	—	2	4	no	—
321	#127	—	4	2	no	—
322	#123	—	4	8	no	—
323	C₄×D₄	—	2	4	no	—
324	C₄×D₄	—	4	2	no	—
325	C₂×C₄⋊C₄	—	2	2	no	—
326	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
327	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
328	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
329	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
330	C₂×C₂²⋊C₄	—	2	2	no	—
331	C₂×C₂²⋊C₄	—	2	2	no	—
332	C₂×C₂²⋊C₄	—	2	2	no	—
333	C₂×C₂²⋊C₄	—	2	2	no	—
334	C₂×C₂²⋊C₄	—	2	2	no	—
335	C₂×C₂²⋊C₄	—	2	2	no	—
336	C₂×C₂²⋊C₄	—	2	2	no	—
337	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
338	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
339	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
340	#107	—	2	2	no	—
341	#107	—	2	2	no	—
342	C₄⋊C₈	—	2	2	no	—
343	#102	—	2	4	no	—
344	#102	—	4	2	no	—
345	#102	—	4	2	no	—
346	#102	—	4	2	no	—
347	C₄×C₈	—	2	2	no	—
348	C₄×C₈	—	2	2	no	—
349	C₄×C₈	—	2	2	no	—
350	#95	—	2	2	no	—
351	#95	D₄×C₃₁	1	1	yes	characteristic
352	#95	D₄×C₃₁	1	1	yes	characteristic
353	C₃₁	#26254	1	1	yes	characteristic, direct product factor, Sylow
354	C₂⁴	—	2	2	no	—
355	C₂⁴	—	2	2	no	—
356	C₂×D₄	—	2	2	no	—
357	C₂×D₄	—	2	2	no	—
358	C₂×D₄	—	2	2	no	—
359	C₂×D₄	—	2	2	no	—
360	C₂×D₄	—	2	2	no	—
361	C₂×D₄	—	2	2	no	—
362	C₂×D₄	—	2	2	no	—
363	C₂×D₄	—	2	2	no	—
364	C₂×D₄	—	2	2	no	—
365	C₂×D₄	—	2	2	no	—
366	C₂×D₄	—	2	2	no	—
367	C₂×D₄	—	4	4	no	—
368	C₂×D₄	—	4	4	no	—
369	C₂×D₄	—	4	4	no	—
370	C₂×D₄	—	4	4	no	—
371	C₂×D₄	—	8	2	no	—
372	C₂×D₄	—	8	2	no	—
373	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
374	C₂²×C₄	—	2	2	no	—
375	C₂²×C₄	—	2	2	no	—
376	C₂²×C₄	—	4	4	no	—
377	C₂²×C₄	—	4	4	no	—
378	C₂²×C₄	—	8	2	no	—
379	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
380	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
381	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
382	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
383	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
384	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
385	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
386	D₈	—	2	8	no	—
387	D₈	—	4	8	no	—
388	D₈	—	4	4	no	—
389	C₂×C₈	—	2	1	no	—
390	C₂×C₈	—	2	1	no	—
391	C₂×C₈	—	2	1	no	—
392	C₂×C₈	—	2	1	no	—
393	C₂×C₈	—	2	1	no	—
394	C₂×C₈	—	2	1	no	—
395	C₂×C₈	—	2	1	no	—
396	C₂×C₈	—	2	1	no	—
397	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
398	C₂×C₈	#41×C₃₁	1	2	yes	—
399	C₂×C₈	#41×C₃₁	1	2	yes	—
400	C₂×C₈	#41×C₃₁	1	2	yes	—
401	C₄⋊C₄	—	2	2	no	—
402	C₄⋊C₄	—	2	2	no	—
403	C₄⋊C₄	—	2	2	no	—
404	C₄⋊C₄	—	2	2	no	—
405	C₄⋊C₄	—	2	2	no	—
406	C₄⋊C₄	—	4	4	no	—
407	C₄⋊C₄	D₈×C₃₁	1	4	yes	complemented
408	C₂²⋊C₄	—	4	4	no	—
409	C₂²⋊C₄	—	4	4	no	—
410	C₂²⋊C₄	—	4	4	no	—
411	C₂²⋊C₄	—	4	4	no	—
412	C₂²⋊C₄	—	8	2	no	—
413	C₂²⋊C₄	—	8	2	no	—
414	C₂²⋊C₄	—	8	2	no	—
415	C₄²	—	2	1	no	—
416	C₄²	—	2	1	no	—
417	C₄²	—	2	1	no	—
418	C₄²	—	2	1	no	—
419	C₄²	—	2	1	no	—
420	C₄²	—	2	1	no	—
421	C₂³	—	2	4	no	—
422	C₂³	—	4	2	no	—
423	C₂³	—	4	2	no	—
424	C₂³	—	4	2	no	—
425	C₂³	—	4	2	no	—
426	C₂³	—	4	2	no	—
427	C₂³	—	4	2	no	—
428	C₂³	—	4	2	no	—
429	C₂³	—	4	2	no	—
430	C₂³	—	4	2	no	—
431	C₂³	—	4	2	no	—
432	C₂³	—	4	2	no	—
433	C₂³	—	4	2	no	—
434	C₂³	—	4	2	no	—
435	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
436	D₄	—	2	4	no	—
437	D₄	—	2	4	no	—
438	D₄	—	4	2	no	—
439	D₄	—	4	2	no	—
440	D₄	—	4	2	no	—
441	D₄	—	4	2	no	—
442	D₄	—	4	2	no	—
443	D₄	—	4	2	no	—
444	D₄	—	8	8	no	—
445	D₄	—	8	8	no	—
446	C₂×C₄	—	2	1	no	—
447	C₂×C₄	—	2	1	no	—
448	C₂×C₄	—	2	1	no	—
449	C₂×C₄	—	2	1	no	—
450	C₂×C₄	—	2	1	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	1	no	—
453	C₂×C₄	—	2	1	no	—
454	C₂×C₄	—	2	1	no	—
455	C₂×C₄	—	2	1	no	—
456	C₂×C₄	—	2	1	no	—
457	C₂×C₄	—	2	1	no	—
458	C₂×C₄	—	2	1	no	—
459	C₂×C₄	—	2	1	no	—
460	C₂×C₄	—	4	2	no	—
461	C₂×C₄	—	4	2	no	—
462	C₂×C₄	—	4	2	no	—
463	C₂×C₄	—	4	2	no	—
464	C₂×C₄	—	4	2	no	—
465	C₂×C₄	—	4	2	no	—
466	C₂×C₄	—	8	8	no	—
467	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
468	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
469	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
470	C₂×C₄	C₃₁×#135	1	2	yes	—
471	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
472	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
473	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
474	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
475	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
476	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
477	C₈	—	2	2	no	—
478	C₈	—	4	1	no	—
479	C₈	—	4	1	no	—
480	C₈	C₃₁×#123	1	4	yes	complemented
481	C₂²	—	4	4	no	—
482	C₂²	—	4	4	no	—
483	C₂²	—	4	4	no	—
484	C₂²	—	8	2	no	—
485	C₂²	—	8	2	no	—
486	C₂²	—	8	2	no	—
487	C₂²	—	8	2	no	—
488	C₂²	—	8	2	no	—
489	C₂²	—	8	2	no	—
490	C₂²	—	8	2	no	—
491	C₂²	—	8	2	no	—
492	C₂²	—	8	2	no	—
493	C₂²	—	8	2	no	—
494	C₂²	—	8	2	no	—
495	C₂²	C₃₁×#532	1	1	yes	characteristic
496	C₂²	D₄²×C₃₁	1	1	yes	characteristic
497	C₂²	C₃₁×#550	1	1	yes	characteristic
498	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
499	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
500	C₂²	C₃₁×#576	1	1	yes	characteristic
501	C₂²	C₃₁×#576	1	1	yes	characteristic
502	C₄	—	2	2	no	—
503	C₄	—	2	2	no	—
504	C₄	—	2	2	no	—
505	C₄	—	2	2	no	—
506	C₄	—	4	1	no	—
507	C₄	—	4	1	no	—
508	C₄	—	8	2	no	—
509	C₄	—	8	2	no	—
510	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
511	C₄	C₃₁×#540	1	1	yes	characteristic
512	C₄	C₃₁×#546	1	1	yes	characteristic
513	C₄	C₃₁×#559	1	1	yes	characteristic
514	C₂	—	8	4	no	—
515	C₂	—	16	2	no	—
516	C₂	C₃₁×#2445	1	1	yes	characteristic
517	C₂	C₃₁×#3169	1	1	yes	characteristic
518	C₂	C₃₁×#3176	1	1	yes	characteristic
519	C₂	D₄×D₈×C₃₁	1	1	yes	characteristic
520	C₂	C₃₁×#3280	1	1	yes	characteristic
521	C₂	C₃₁×#3391	1	1	yes	characteristic
522	C₂	C₃₁×#3396	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#493	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
2. C₂×C₄×D₈×C₃₁	C₂	semidirect
3. C₃₁×#2445	C₂	semidirect
4. C₃₁×#2054	C₂	semidirect
5. C₃₁×#2041	C₂	semidirect
6. C₃₁×#1970	C₂	semidirect
7. C₃₁×#1970	C₂	semidirect
8. C₃₁×#1970	C₂	semidirect
9. C₃₁×#1942	C₂	semidirect
10. C₃₁×#1879	C₂	semidirect
11. C₈×C₄⋊C₄×C₃₁	C₂	semidirect
145. C₄⋊C₄×C₃₁	D₈	semidirect
159. #26254	C₃₁	direct (N × H)
219. C₈×C₃₁	#123	semidirect
221. C₂×C₄×D₈	C₂×C₃₁	semidirect
222. #2445	C₂×C₃₁	semidirect
223. #2054	C₂×C₃₁	semidirect
224. #2041	C₂×C₃₁	semidirect
225. #1970	C₂×C₃₁	semidirect
226. #1970	C₂×C₃₁	semidirect
227. #1970	C₂×C₃₁	semidirect
228. #1942	C₂×C₃₁	semidirect
229. #1879	C₂×C₃₁	semidirect
230. C₈×C₄⋊C₄	C₂×C₃₁	semidirect
353. C₃₁	#26254	direct (N × H)
407. C₄⋊C₄	D₈×C₃₁	semidirect
480. C₈	C₃₁×#123	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#493	2
C₂×C₄×D₈×C₃₁	2
C₃₁×#2445	2
C₃₁×#2054	2
C₃₁×#2041	2
C₃₁×#1970	2
C₃₁×#1942	2
C₃₁×#1879	2
C₈×C₄⋊C₄×C₃₁	2
#26254	31

Subgroups — C31×#26254 #21483013

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#26254 #21483013