Subgroups — C₃₁×#24479 #21481238

Order 7936.

1102 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#484	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃₁×#102	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2435	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2345	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2067	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2042	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1983	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1925	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1892	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1850	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1786	C₂	1	1	yes	maximal, characteristic
12	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#124	C₂²	1	2	yes	—
15	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#484	—	2	4	no	—
18	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic, complemented
19	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic, complemented
20	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	complemented
21	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic, complemented
22	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic, complemented
23	C₂×C₃₁×#103	C₂²	1	2	yes	complemented
24	C₂×C₃₁×#102	—	2	4	no	—
25	C₂×C₃₁×#102	C₂²	1	2	yes	—
26	C₂×C₃₁×#102	C₂²	1	2	yes	—
27	C₂×C₃₁×#102	C₂²	1	2	yes	—
28	C₂×C₃₁×#102	C₂²	1	2	yes	—
29	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
30	C₃₁×#394	—	2	2	no	—
31	C₃₁×#390	C₂²	1	2	yes	—
32	C₃₁×#389	C₂²	1	2	yes	complemented
33	C₃₁×#387	—	2	2	no	—
34	C₃₁×#380	—	2	1	no	—
35	C₃₁×#380	C₂²	1	1	yes	characteristic
36	C₃₁×#379	—	2	1	no	—
37	C₃₁×#379	—	2	1	no	—
38	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
39	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
40	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
41	C₃₁×#340	C₂²	1	2	yes	complemented
42	C₃₁×#336	C₂²	1	2	yes	complemented
43	C₂⁵×C₃₁	—	2	1	no	—
44	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
45	C₂²×D₄×C₃₁	—	2	4	no	—
46	C₂²×D₄×C₃₁	—	2	2	no	—
47	C₂²×D₄×C₃₁	—	2	1	no	—
48	C₂²×D₄×C₃₁	—	2	2	no	—
49	C₂²×D₄×C₃₁	—	4	2	no	—
50	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
51	C₂³×C₄×C₃₁	—	2	1	no	—
52	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₃₁	—	2	2	no	—
54	C₂²×C₈×C₃₁	—	4	2	no	—
55	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
57	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
58	C₃₁×#124	—	2	4	no	—
59	C₃₁×#124	—	4	2	no	—
60	C₃₁×C₂²⋊Q₈	—	2	2	no	—
61	C₃₁×C₂²⋊Q₈	—	2	2	no	—
62	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
63	C₂×C₄⋊C₄×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	4	yes	—
67	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
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	1	yes	characteristic
71	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
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×C₂²⋊C₄×C₃₁	—	2	2	no	—
76	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
77	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
78	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
79	C₂×C₄²×C₃₁	—	2	2	no	—
80	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
81	C₃₁×#107	—	2	2	no	—
82	C₃₁×#107	—	2	2	no	—
83	C₃₁×C₄⋊C₈	—	4	2	no	—
84	C₃₁×#103	—	2	2	no	—
85	C₃₁×#103	—	2	2	no	—
86	C₃₁×#103	—	4	2	no	—
87	C₃₁×#102	—	2	4	no	—
88	C₃₁×#102	—	2	4	no	—
89	C₃₁×#102	—	2	4	no	—
90	C₃₁×#102	—	4	2	no	—
91	C₃₁×C₂²⋊C₈	—	2	2	no	—
92	C₃₁×#95	—	2	2	no	—
93	C₃₁×#95	—	2	2	no	—
94	C₃₁×#95	—	2	1	no	—
95	C₃₁×#95	D₄	1	2	yes	—
96	C₂⁴×C₃₁	—	2	2	no	—
97	C₂⁴×C₃₁	—	4	4	no	—
98	C₂⁴×C₃₁	—	4	2	no	—
99	C₂⁴×C₃₁	—	4	2	no	—
100	C₂⁴×C₃₁	—	4	2	no	—
101	C₂⁴×C₃₁	—	8	2	no	—
102	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂×Q₈×C₃₁	—	2	1	no	—
104	C₂×Q₈×C₃₁	—	2	1	no	—
105	C₂×Q₈×C₃₁	—	2	2	no	—
106	C₂×Q₈×C₃₁	—	2	2	no	—
107	C₂×D₄×C₃₁	—	2	8	no	—
108	C₂×D₄×C₃₁	—	2	2	no	—
109	C₂×D₄×C₃₁	—	2	2	no	—
110	C₂×D₄×C₃₁	—	2	2	no	—
111	C₂×D₄×C₃₁	—	2	2	no	—
112	C₂×D₄×C₃₁	—	2	2	no	—
113	C₂×D₄×C₃₁	—	4	8	no	—
114	C₂×D₄×C₃₁	—	4	4	no	—
115	C₂×D₄×C₃₁	—	4	4	no	—
116	C₂×D₄×C₃₁	—	4	2	no	—
117	C₂×D₄×C₃₁	#41	1	4	yes	—
118	C₂²×C₄×C₃₁	—	2	2	no	—
119	C₂²×C₄×C₃₁	—	2	2	no	—
120	C₂²×C₄×C₃₁	—	2	1	no	—
121	C₂²×C₄×C₃₁	—	2	1	no	—
122	C₂²×C₄×C₃₁	—	2	4	no	—
123	C₂²×C₄×C₃₁	—	2	2	no	—
124	C₂²×C₄×C₃₁	—	2	1	no	—
125	C₂²×C₄×C₃₁	—	2	1	no	—
126	C₂²×C₄×C₃₁	—	8	2	no	—
127	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
128	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
129	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
131	C₂²×C₄×C₃₁	#41	1	2	yes	—
132	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
133	C₂×C₈×C₃₁	—	2	1	no	—
134	C₂×C₈×C₃₁	—	2	1	no	—
135	C₂×C₈×C₃₁	—	2	2	no	—
136	C₂×C₈×C₃₁	—	2	2	no	—
137	C₂×C₈×C₃₁	—	2	2	no	—
138	C₂×C₈×C₃₁	—	2	2	no	—
139	C₂×C₈×C₃₁	—	2	2	no	—
140	C₂×C₈×C₃₁	—	4	4	no	—
141	C₄⋊C₄×C₃₁	—	2	4	no	—
142	C₄⋊C₄×C₃₁	—	2	1	no	—
143	C₄⋊C₄×C₃₁	—	2	1	no	—
144	C₄⋊C₄×C₃₁	—	2	1	no	—
145	C₄⋊C₄×C₃₁	—	2	2	no	—
146	C₄⋊C₄×C₃₁	—	2	1	no	—
147	C₄⋊C₄×C₃₁	—	2	2	no	—
148	C₄⋊C₄×C₃₁	—	4	2	no	—
149	C₄⋊C₄×C₃₁	—	8	2	no	—
150	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
151	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
152	C₂²⋊C₄×C₃₁	—	2	2	no	—
153	C₂²⋊C₄×C₃₁	—	2	2	no	—
154	C₂²⋊C₄×C₃₁	—	4	4	no	—
155	C₂²⋊C₄×C₃₁	—	4	4	no	—
156	C₂²⋊C₄×C₃₁	—	4	2	no	—
157	C₂²⋊C₄×C₃₁	—	4	2	no	—
158	C₂²⋊C₄×C₃₁	—	8	2	no	—
159	C₂²⋊C₄×C₃₁	—	8	2	no	—
160	C₄²×C₃₁	—	2	2	no	—
161	C₄²×C₃₁	—	2	2	no	—
162	C₄²×C₃₁	—	8	2	no	—
163	#24479	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
164	C₂³×C₃₁	—	2	2	no	—
165	C₂³×C₃₁	—	2	2	no	—
166	C₂³×C₃₁	—	2	4	no	—
167	C₂³×C₃₁	—	4	8	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	—	8	8	no	—
175	C₂³×C₃₁	—	8	8	no	—
176	C₂³×C₃₁	—	8	4	no	—
177	C₂³×C₃₁	—	8	4	no	—
178	C₂³×C₃₁	#121	1	2	yes	—
179	C₂³×C₃₁	#135	1	2	yes	—
180	C₂³×C₃₁	#137	1	2	yes	—
181	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
182	Q₈×C₃₁	—	4	1	no	—
183	Q₈×C₃₁	—	4	1	no	—
184	Q₈×C₃₁	—	4	2	no	—
185	D₄×C₃₁	—	2	4	no	—
186	D₄×C₃₁	—	2	4	no	—
187	D₄×C₃₁	—	4	8	no	—
188	D₄×C₃₁	—	4	2	no	—
189	D₄×C₃₁	—	4	2	no	—
190	C₂×C₄×C₃₁	—	2	2	no	—
191	C₂×C₄×C₃₁	—	2	2	no	—
192	C₂×C₄×C₃₁	—	2	2	no	—
193	C₂×C₄×C₃₁	—	2	4	no	—
194	C₂×C₄×C₃₁	—	2	2	no	—
195	C₂×C₄×C₃₁	—	2	2	no	—
196	C₂×C₄×C₃₁	—	2	2	no	—
197	C₂×C₄×C₃₁	—	2	2	no	—
198	C₂×C₄×C₃₁	—	2	2	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	—	4	2	no	—
203	C₂×C₄×C₃₁	—	4	2	no	—
204	C₂×C₄×C₃₁	—	4	1	no	—
205	C₂×C₄×C₃₁	—	4	1	no	—
206	C₂×C₄×C₃₁	—	4	2	no	—
207	C₂×C₄×C₃₁	—	4	2	no	—
208	C₂×C₄×C₃₁	—	4	2	no	—
209	C₂×C₄×C₃₁	—	4	2	no	—
210	C₂×C₄×C₃₁	—	4	2	no	—
211	C₂×C₄×C₃₁	—	8	4	no	—
212	C₂×C₄×C₃₁	#121	1	2	yes	—
213	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
215	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
216	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
217	C₈×C₃₁	—	4	1	no	—
218	C₈×C₃₁	—	4	1	no	—
219	C₈×C₃₁	—	4	2	no	—
220	C₂×#484	C₂×C₃₁	1	1	yes	characteristic, complemented
221	C₂²×#102	C₂×C₃₁	1	1	yes	characteristic
222	#2435	C₂×C₃₁	1	1	yes	characteristic
223	#2345	C₂×C₃₁	1	1	yes	characteristic, complemented
224	#2067	C₂×C₃₁	1	2	yes	complemented
225	#2042	C₂×C₃₁	1	2	yes	complemented
226	#1983	C₂×C₃₁	1	2	yes	complemented
227	#1925	C₂×C₃₁	1	2	yes	complemented
228	#1892	C₂×C₃₁	1	1	yes	characteristic, complemented
229	#1850	C₂×C₃₁	1	1	yes	characteristic, complemented
230	#1786	C₂×C₃₁	1	1	yes	characteristic
231	C₂²×C₃₁	—	2	2	no	—
232	C₂²×C₃₁	—	2	2	no	—
233	C₂²×C₃₁	—	2	4	no	—
234	C₂²×C₃₁	—	2	4	no	—
235	C₂²×C₃₁	—	4	4	no	—
236	C₂²×C₃₁	—	4	4	no	—
237	C₂²×C₃₁	—	4	4	no	—
238	C₂²×C₃₁	—	8	16	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	#464	1	4	yes	—
244	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
245	C₂²×C₃₁	#546	1	2	yes	—
246	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
247	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
248	C₂²×C₃₁	#576	1	2	yes	—
249	C₄×C₃₁	—	2	2	no	—
250	C₄×C₃₁	—	4	2	no	—
251	C₄×C₃₁	—	4	2	no	—
252	C₄×C₃₁	—	4	2	no	—
253	C₄×C₃₁	—	4	2	no	—
254	C₄×C₃₁	—	8	1	no	—
255	C₄×C₃₁	—	8	1	no	—
256	C₄×C₃₁	—	8	2	no	—
257	C₄×C₃₁	#525	1	1	yes	characteristic
258	C₄×C₃₁	#535	1	1	yes	characteristic
259	C₄×C₃₁	#551	1	2	yes	—
260	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
261	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
262	C₂×#124	C₂²×C₃₁	1	2	yes	—
263	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
264	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
265	#484	—	2	4	no	—
266	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic, complemented
267	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic, complemented
268	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	complemented
269	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic, complemented
270	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic, complemented
271	C₂×#103	C₂²×C₃₁	1	2	yes	complemented
272	C₂×#102	—	2	4	no	—
273	C₂×#102	C₂²×C₃₁	1	2	yes	—
274	C₂×#102	C₂²×C₃₁	1	2	yes	—
275	C₂×#102	C₂²×C₃₁	1	2	yes	—
276	C₂×#102	C₂²×C₃₁	1	2	yes	—
277	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
278	#394	—	2	2	no	—
279	#390	C₂²×C₃₁	1	2	yes	—
280	#389	C₂²×C₃₁	1	2	yes	complemented
281	#387	—	2	2	no	—
282	#380	—	2	1	no	—
283	#380	C₂²×C₃₁	1	1	yes	characteristic
284	#379	—	2	1	no	—
285	#379	—	2	1	no	—
286	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
287	C₄×C₂²⋊C₄	—	2	1	no	—
288	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
289	#340	C₂²×C₃₁	1	2	yes	complemented
290	#336	C₂²×C₃₁	1	2	yes	complemented
291	C₂×C₃₁	—	2	4	no	—
292	C₂×C₃₁	—	8	4	no	—
293	C₂×C₃₁	#2411	1	1	yes	characteristic
294	C₂×C₃₁	C₂×#464	1	1	yes	characteristic
295	C₂×C₃₁	#3041	1	1	yes	characteristic
296	C₂×C₃₁	#3297	1	2	yes	—
297	C₂×C₃₁	#3314	1	2	yes	—
298	C₂⁵	—	2	1	no	—
299	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
300	C₂²×D₄	—	2	4	no	—
301	C₂²×D₄	—	2	2	no	—
302	C₂²×D₄	—	2	1	no	—
303	C₂²×D₄	—	2	2	no	—
304	C₂²×D₄	—	4	2	no	—
305	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
306	C₂³×C₄	—	2	1	no	—
307	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
308	C₂²×C₈	—	2	2	no	—
309	C₂²×C₈	—	4	2	no	—
310	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
311	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
312	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
313	#124	—	2	4	no	—
314	#124	—	4	2	no	—
315	C₂²⋊Q₈	—	2	2	no	—
316	C₂²⋊Q₈	—	2	2	no	—
317	C₂×C₄⋊C₄	—	2	1	no	—
318	C₂×C₄⋊C₄	—	2	2	no	—
319	C₂×C₄⋊C₄	—	2	1	no	—
320	C₂×C₄⋊C₄	—	2	2	no	—
321	C₂×C₄⋊C₄	D₄×C₃₁	1	4	yes	—
322	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
323	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
324	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
325	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
326	C₂×C₂²⋊C₄	—	2	1	no	—
327	C₂×C₂²⋊C₄	—	2	1	no	—
328	C₂×C₂²⋊C₄	—	2	2	no	—
329	C₂×C₂²⋊C₄	—	2	2	no	—
330	C₂×C₂²⋊C₄	—	2	2	no	—
331	C₂×C₂²⋊C₄	—	2	2	no	—
332	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
333	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
334	C₂×C₄²	—	2	2	no	—
335	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
336	#107	—	2	2	no	—
337	#107	—	2	2	no	—
338	C₄⋊C₈	—	4	2	no	—
339	#103	—	2	2	no	—
340	#103	—	2	2	no	—
341	#103	—	4	2	no	—
342	#102	—	2	4	no	—
343	#102	—	2	4	no	—
344	#102	—	2	4	no	—
345	#102	—	4	2	no	—
346	C₂²⋊C₈	—	2	2	no	—
347	#95	—	2	2	no	—
348	#95	—	2	2	no	—
349	#95	—	2	1	no	—
350	#95	D₄×C₃₁	1	2	yes	—
351	C₃₁	#24479	1	1	yes	characteristic, direct product factor, Sylow
352	C₂⁴	—	2	2	no	—
353	C₂⁴	—	4	4	no	—
354	C₂⁴	—	4	2	no	—
355	C₂⁴	—	4	2	no	—
356	C₂⁴	—	4	2	no	—
357	C₂⁴	—	8	2	no	—
358	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
359	C₂×Q₈	—	2	1	no	—
360	C₂×Q₈	—	2	1	no	—
361	C₂×Q₈	—	2	2	no	—
362	C₂×Q₈	—	2	2	no	—
363	C₂×D₄	—	2	8	no	—
364	C₂×D₄	—	2	2	no	—
365	C₂×D₄	—	2	2	no	—
366	C₂×D₄	—	2	2	no	—
367	C₂×D₄	—	2	2	no	—
368	C₂×D₄	—	2	2	no	—
369	C₂×D₄	—	4	8	no	—
370	C₂×D₄	—	4	4	no	—
371	C₂×D₄	—	4	4	no	—
372	C₂×D₄	—	4	2	no	—
373	C₂×D₄	#41×C₃₁	1	4	yes	—
374	C₂²×C₄	—	2	2	no	—
375	C₂²×C₄	—	2	2	no	—
376	C₂²×C₄	—	2	1	no	—
377	C₂²×C₄	—	2	1	no	—
378	C₂²×C₄	—	2	4	no	—
379	C₂²×C₄	—	2	2	no	—
380	C₂²×C₄	—	2	1	no	—
381	C₂²×C₄	—	2	1	no	—
382	C₂²×C₄	—	8	2	no	—
383	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
384	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
385	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
386	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
387	C₂²×C₄	#41×C₃₁	1	2	yes	—
388	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
389	C₂×C₈	—	2	1	no	—
390	C₂×C₈	—	2	1	no	—
391	C₂×C₈	—	2	2	no	—
392	C₂×C₈	—	2	2	no	—
393	C₂×C₈	—	2	2	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	2	2	no	—
396	C₂×C₈	—	4	4	no	—
397	C₄⋊C₄	—	2	4	no	—
398	C₄⋊C₄	—	2	1	no	—
399	C₄⋊C₄	—	2	1	no	—
400	C₄⋊C₄	—	2	1	no	—
401	C₄⋊C₄	—	2	2	no	—
402	C₄⋊C₄	—	2	1	no	—
403	C₄⋊C₄	—	2	2	no	—
404	C₄⋊C₄	—	4	2	no	—
405	C₄⋊C₄	—	8	2	no	—
406	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
407	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
408	C₂²⋊C₄	—	2	2	no	—
409	C₂²⋊C₄	—	2	2	no	—
410	C₂²⋊C₄	—	4	4	no	—
411	C₂²⋊C₄	—	4	4	no	—
412	C₂²⋊C₄	—	4	2	no	—
413	C₂²⋊C₄	—	4	2	no	—
414	C₂²⋊C₄	—	8	2	no	—
415	C₂²⋊C₄	—	8	2	no	—
416	C₄²	—	2	2	no	—
417	C₄²	—	2	2	no	—
418	C₄²	—	8	2	no	—
419	C₂³	—	2	2	no	—
420	C₂³	—	2	2	no	—
421	C₂³	—	2	4	no	—
422	C₂³	—	4	8	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	—	8	8	no	—
430	C₂³	—	8	8	no	—
431	C₂³	—	8	4	no	—
432	C₂³	—	8	4	no	—
433	C₂³	C₃₁×#121	1	2	yes	—
434	C₂³	C₃₁×#135	1	2	yes	—
435	C₂³	C₃₁×#137	1	2	yes	—
436	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
437	Q₈	—	4	1	no	—
438	Q₈	—	4	1	no	—
439	Q₈	—	4	2	no	—
440	D₄	—	2	4	no	—
441	D₄	—	2	4	no	—
442	D₄	—	4	8	no	—
443	D₄	—	4	2	no	—
444	D₄	—	4	2	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	2	no	—
457	C₂×C₄	—	4	2	no	—
458	C₂×C₄	—	4	2	no	—
459	C₂×C₄	—	4	1	no	—
460	C₂×C₄	—	4	1	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	4	no	—
467	C₂×C₄	C₃₁×#121	1	2	yes	—
468	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
469	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
470	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
471	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
472	C₈	—	4	1	no	—
473	C₈	—	4	1	no	—
474	C₈	—	4	2	no	—
475	C₂²	—	2	2	no	—
476	C₂²	—	2	2	no	—
477	C₂²	—	2	4	no	—
478	C₂²	—	2	4	no	—
479	C₂²	—	4	4	no	—
480	C₂²	—	4	4	no	—
481	C₂²	—	4	4	no	—
482	C₂²	—	8	16	no	—
483	C₂²	—	8	2	no	—
484	C₂²	—	8	2	no	—
485	C₂²	—	8	2	no	—
486	C₂²	—	8	2	no	—
487	C₂²	C₃₁×#464	1	4	yes	—
488	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
489	C₂²	C₃₁×#546	1	2	yes	—
490	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
491	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
492	C₂²	C₃₁×#576	1	2	yes	—
493	C₄	—	2	2	no	—
494	C₄	—	4	2	no	—
495	C₄	—	4	2	no	—
496	C₄	—	4	2	no	—
497	C₄	—	4	2	no	—
498	C₄	—	8	1	no	—
499	C₄	—	8	1	no	—
500	C₄	—	8	2	no	—
501	C₄	C₃₁×#525	1	1	yes	characteristic
502	C₄	C₃₁×#535	1	1	yes	characteristic
503	C₄	C₃₁×#551	1	2	yes	—
504	C₂	—	2	4	no	—
505	C₂	—	8	4	no	—
506	C₂	C₃₁×#2411	1	1	yes	characteristic
507	C₂	C₂×C₃₁×#464	1	1	yes	characteristic
508	C₂	C₃₁×#3041	1	1	yes	characteristic
509	C₂	C₃₁×#3297	1	2	yes	—
510	C₂	C₃₁×#3314	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#484	C₂	semidirect
4. C₃₁×#2345	C₂	semidirect
5. C₃₁×#2067	C₂	semidirect
6. C₃₁×#2042	C₂	semidirect
7. C₃₁×#1983	C₂	semidirect
8. C₃₁×#1925	C₂	semidirect
9. C₃₁×#1892	C₂	semidirect
10. C₃₁×#1850	C₂	semidirect
18. C₂×C₃₁×#107	C₂²	semidirect
19. C₂×C₃₁×#107	C₂²	semidirect
20. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
21. C₂×C₃₁×#103	C₂²	semidirect
22. C₂×C₃₁×#103	C₂²	semidirect
23. C₂×C₃₁×#103	C₂²	semidirect
32. C₃₁×#389	C₂²	semidirect
38. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
41. C₃₁×#340	C₂²	semidirect
42. C₃₁×#336	C₂²	semidirect
163. #24479	C₃₁	direct (N × H)
220. C₂×#484	C₂×C₃₁	semidirect
223. #2345	C₂×C₃₁	semidirect
224. #2067	C₂×C₃₁	semidirect
225. #2042	C₂×C₃₁	semidirect
226. #1983	C₂×C₃₁	semidirect
227. #1925	C₂×C₃₁	semidirect
228. #1892	C₂×C₃₁	semidirect
229. #1850	C₂×C₃₁	semidirect
266. C₂×#107	C₂²×C₃₁	semidirect
267. C₂×#107	C₂²×C₃₁	semidirect
268. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
269. C₂×#103	C₂²×C₃₁	semidirect
270. C₂×#103	C₂²×C₃₁	semidirect
271. C₂×#103	C₂²×C₃₁	semidirect
280. #389	C₂²×C₃₁	semidirect
286. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
289. #340	C₂²×C₃₁	semidirect
290. #336	C₂²×C₃₁	semidirect
351. C₃₁	#24479	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×#484	2
C₂²×C₃₁×#102	2
C₃₁×#2435	2
C₃₁×#2345	2
C₃₁×#2067	2
C₃₁×#2042	2
C₃₁×#1983	2
C₃₁×#1925	2
C₃₁×#1892	2
C₃₁×#1850	2
C₃₁×#1786	2
#24479	31

Subgroups — C31×#24479 #21481238

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24479 #21481238