Subgroups — C₃₁×#33062 #21489821

Order 7936.

764 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#488	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3080	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3040	C₂	1	1	yes	maximal, characteristic
4	C₄×C₃₁×#131	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1998	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1967	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1931	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1930	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1908	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1907	C₂	1	2	yes	maximal
11	C₃₁×#1837	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#137	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#131	C₂²	1	2	yes	—
15	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#124	C₂²	1	1	yes	characteristic
17	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#488	—	2	1	no	—
19	C₃₁×#488	—	2	1	no	—
20	C₃₁×#488	—	2	2	no	—
21	C₃₁×#479	—	2	2	no	—
22	C₃₁×#478	—	2	2	no	—
23	C₃₁×#464	—	2	2	no	—
24	C₃₁×#463	—	2	2	no	—
25	C₃₁×#428	C₂²	1	2	yes	—
26	C₃₁×#423	C₂²	1	1	yes	characteristic
27	C₃₁×#423	C₂²	1	1	yes	characteristic
28	C₃₁×#419	C₂²	1	1	yes	characteristic
29	C₃₁×#418	C₂²	1	1	yes	characteristic
30	C₃₁×#417	C₂²	1	2	yes	—
31	C₂×C₃₁×#103	C₂²	1	2	yes	—
32	C₂×C₃₁×#102	C₂²	1	2	yes	—
33	C₂×C₃₁×#101	C₂²	1	2	yes	—
34	C₂×C₃₁×#100	C₂²	1	2	yes	—
35	C₃₁×#405	—	2	2	no	—
36	C₃₁×#405	C₂²	1	2	yes	—
37	C₄×M₁₆×C₃₁	—	2	1	no	—
38	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈×C₃₁	—	2	1	no	—
40	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
41	C₃₁×#343	C₂²	1	2	yes	—
42	C₃₁×#343	C₂²	1	2	yes	—
43	C₃₁×#341	C₂²	1	2	yes	—
44	C₃₁×#341	C₂²	1	2	yes	—
45	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
46	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₃₁×#137	—	2	4	no	—
49	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
50	C₂×Q₁₆×C₃₁	—	2	1	no	—
51	C₂×SD₁₆×C₃₁	—	2	1	no	—
52	C₂×SD₁₆×C₃₁	—	2	1	no	—
53	C₂×D₈×C₃₁	—	2	1	no	—
54	C₃₁×#131	—	2	4	no	—
55	C₃₁×#131	—	2	4	no	—
56	C₂×M₁₆×C₃₁	—	2	2	no	—
57	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
58	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
59	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
60	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
61	C₂²×C₈×C₃₁	—	2	2	no	—
62	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
63	C₃₁×#125	—	2	2	no	—
64	C₃₁×#125	—	2	1	no	—
65	C₃₁×#125	—	2	1	no	—
66	C₃₁×#124	—	2	2	no	—
67	C₃₁×#124	—	2	1	no	—
68	C₃₁×#124	—	2	1	no	—
69	C₄×Q₈×C₃₁	D₄	1	2	yes	—
70	C₄×D₄×C₃₁	—	2	2	no	—
71	C₄×D₄×C₃₁	D₄	1	2	yes	—
72	C₃₁×#117	—	2	1	no	—
73	C₃₁×#117	C₂³	1	1	yes	characteristic
74	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
75	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
76	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
77	C₂×C₄²×C₃₁	—	2	1	no	—
78	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
79	C₃₁×#107	—	2	2	no	—
80	C₃₁×#106	—	2	2	no	—
81	C₃₁×C₄⋊C₈	—	2	2	no	—
82	C₃₁×C₄⋊C₈	—	2	2	no	—
83	C₃₁×#103	—	2	2	no	—
84	C₃₁×#103	—	2	2	no	—
85	C₃₁×#103	—	2	2	no	—
86	C₃₁×#103	—	2	2	no	—
87	C₃₁×#102	—	2	2	no	—
88	C₃₁×#102	—	2	2	no	—
89	C₃₁×#102	—	2	2	no	—
90	C₃₁×#102	—	2	2	no	—
91	C₃₁×#101	—	2	4	no	—
92	C₃₁×#100	—	2	4	no	—
93	C₃₁×#97	—	2	2	no	—
94	C₃₁×#97	—	2	1	no	—
95	C₄×C₈×C₃₁	—	2	2	no	—
96	C₄×C₈×C₃₁	—	2	1	no	—
97	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
98	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
99	C₄×C₈×C₃₁	D₄	1	2	yes	—
100	C₂⁴×C₃₁	—	2	1	no	—
101	#41×C₃₁	—	2	2	no	—
102	#41×C₃₁	#41	1	4	yes	—
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×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
108	C₂×D₄×C₃₁	—	2	1	no	—
109	C₂×D₄×C₃₁	—	2	1	no	—
110	C₂×D₄×C₃₁	—	2	1	no	—
111	C₂×D₄×C₃₁	—	2	2	no	—
112	C₂×D₄×C₃₁	—	2	2	no	—
113	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₃₁	—	2	1	no	—
115	C₂²×C₄×C₃₁	—	2	2	no	—
116	C₂²×C₄×C₃₁	—	2	1	no	—
117	C₂²×C₄×C₃₁	—	2	2	no	—
118	C₂²×C₄×C₃₁	#41	1	2	yes	—
119	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
120	Q₁₆×C₃₁	—	4	2	no	—
121	SD₁₆×C₃₁	—	4	2	no	—
122	SD₁₆×C₃₁	—	4	2	no	—
123	D₈×C₃₁	—	4	2	no	—
124	M₁₆×C₃₁	—	2	8	no	—
125	M₁₆×C₃₁	—	2	2	no	—
126	M₁₆×C₃₁	—	2	2	no	—
127	M₁₆×C₃₁	—	2	2	no	—
128	M₁₆×C₃₁	—	2	2	no	—
129	M₁₆×C₃₁	—	2	2	no	—
130	M₁₆×C₃₁	#41	1	4	yes	—
131	C₂×C₈×C₃₁	—	2	8	no	—
132	C₂×C₈×C₃₁	—	2	2	no	—
133	C₂×C₈×C₃₁	—	2	2	no	—
134	C₂×C₈×C₃₁	—	2	2	no	—
135	C₂×C₈×C₃₁	—	2	1	no	—
136	C₂×C₈×C₃₁	—	2	2	no	—
137	C₂×C₈×C₃₁	—	2	1	no	—
138	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
139	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
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	—	4	2	no	—
144	C₄⋊C₄×C₃₁	—	4	2	no	—
145	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
146	C₂²⋊C₄×C₃₁	—	2	1	no	—
147	C₂²⋊C₄×C₃₁	—	2	2	no	—
148	C₂²⋊C₄×C₃₁	—	4	2	no	—
149	C₂²⋊C₄×C₃₁	—	4	2	no	—
150	C₄²×C₃₁	—	2	2	no	—
151	C₄²×C₃₁	—	2	1	no	—
152	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
153	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
154	C₄²×C₃₁	C₂×D₄	1	2	yes	—
155	#33062	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
156	C₂³×C₃₁	—	2	1	no	—
157	C₂³×C₃₁	—	4	1	no	—
158	C₂³×C₃₁	—	4	2	no	—
159	C₂³×C₃₁	—	8	2	no	—
160	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
161	Q₈×C₃₁	—	2	1	no	—
162	Q₈×C₃₁	—	4	2	no	—
163	Q₈×C₃₁	—	4	1	no	—
164	Q₈×C₃₁	—	4	1	no	—
165	Q₈×C₃₁	#124	1	2	yes	—
166	D₄×C₃₁	—	2	4	no	—
167	D₄×C₃₁	—	2	1	no	—
168	D₄×C₃₁	—	4	2	no	—
169	D₄×C₃₁	—	4	1	no	—
170	D₄×C₃₁	—	4	1	no	—
171	D₄×C₃₁	#124	1	2	yes	—
172	C₂×C₄×C₃₁	—	2	1	no	—
173	C₂×C₄×C₃₁	—	2	4	no	—
174	C₂×C₄×C₃₁	—	2	2	no	—
175	C₂×C₄×C₃₁	—	4	2	no	—
176	C₂×C₄×C₃₁	—	4	4	no	—
177	C₂×C₄×C₃₁	—	4	1	no	—
178	C₂×C₄×C₃₁	—	4	2	no	—
179	C₂×C₄×C₃₁	—	4	2	no	—
180	C₂×C₄×C₃₁	—	8	2	no	—
181	C₂×C₄×C₃₁	#121	1	2	yes	—
182	C₂×C₄×C₃₁	#121	1	2	yes	—
183	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
184	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
186	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
187	C₈×C₃₁	—	2	2	no	—
188	C₈×C₃₁	—	2	4	no	—
189	C₈×C₃₁	—	2	2	no	—
190	C₈×C₃₁	—	4	1	no	—
191	C₈×C₃₁	—	4	1	no	—
192	C₈×C₃₁	—	4	2	no	—
193	C₂×#488	C₂×C₃₁	1	1	yes	characteristic, complemented
194	#3080	C₂×C₃₁	1	1	yes	characteristic, complemented
195	#3040	C₂×C₃₁	1	1	yes	characteristic
196	C₄×#131	C₂×C₃₁	1	1	yes	characteristic, complemented
197	#1998	C₂×C₃₁	1	1	yes	characteristic, complemented
198	#1967	C₂×C₃₁	1	1	yes	characteristic, complemented
199	#1931	C₂×C₃₁	1	2	yes	complemented
200	#1930	C₂×C₃₁	1	2	yes	complemented
201	#1908	C₂×C₃₁	1	2	yes	complemented
202	#1907	C₂×C₃₁	1	2	yes	—
203	#1837	C₂×C₃₁	1	1	yes	characteristic, complemented
204	C₂²×C₃₁	—	2	2	no	—
205	C₂²×C₃₁	—	2	2	no	—
206	C₂²×C₃₁	—	4	2	no	—
207	C₂²×C₃₁	—	8	1	no	—
208	C₂²×C₃₁	—	8	2	no	—
209	C₂²×C₃₁	—	16	2	no	—
210	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
211	C₂²×C₃₁	#529	1	2	yes	—
212	C₄×C₃₁	—	2	2	no	—
213	C₄×C₃₁	—	2	2	no	—
214	C₄×C₃₁	—	2	2	no	—
215	C₄×C₃₁	—	4	2	no	—
216	C₄×C₃₁	—	8	1	no	—
217	C₄×C₃₁	—	8	2	no	—
218	C₄×C₃₁	#525	1	2	yes	—
219	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
220	C₄×C₃₁	#532	1	1	yes	characteristic
221	C₂×#137	C₂²×C₃₁	1	1	yes	characteristic
222	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	1	yes	characteristic
223	C₂×#131	C₂²×C₃₁	1	2	yes	—
224	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic
225	C₂×#124	C₂²×C₃₁	1	1	yes	characteristic
226	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
227	#488	—	2	1	no	—
228	#488	—	2	1	no	—
229	#488	—	2	2	no	—
230	#479	—	2	2	no	—
231	#478	—	2	2	no	—
232	#464	—	2	2	no	—
233	#463	—	2	2	no	—
234	#428	C₂²×C₃₁	1	2	yes	—
235	#423	C₂²×C₃₁	1	1	yes	characteristic
236	#423	C₂²×C₃₁	1	1	yes	characteristic
237	#419	C₂²×C₃₁	1	1	yes	characteristic
238	#418	C₂²×C₃₁	1	1	yes	characteristic
239	#417	C₂²×C₃₁	1	2	yes	—
240	C₂×#103	C₂²×C₃₁	1	2	yes	—
241	C₂×#102	C₂²×C₃₁	1	2	yes	—
242	C₂×#101	C₂²×C₃₁	1	2	yes	—
243	C₂×#100	C₂²×C₃₁	1	2	yes	—
244	#405	—	2	2	no	—
245	#405	C₂²×C₃₁	1	2	yes	—
246	C₄×M₁₆	—	2	1	no	—
247	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
248	C₂×C₄×C₈	—	2	1	no	—
249	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
250	#343	C₂²×C₃₁	1	2	yes	—
251	#343	C₂²×C₃₁	1	2	yes	—
252	#341	C₂²×C₃₁	1	2	yes	—
253	#341	C₂²×C₃₁	1	2	yes	—
254	C₂×C₃₁	—	2	2	no	—
255	C₂×C₃₁	—	4	2	no	—
256	C₂×C₃₁	—	16	1	no	—
257	C₂×C₃₁	#2850	1	1	yes	characteristic
258	C₂×C₃₁	#3067	1	2	yes	—
259	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
260	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
261	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
262	#137	—	2	4	no	—
263	AGL(1,ℤ/8)	—	2	4	no	—
264	C₂×Q₁₆	—	2	1	no	—
265	C₂×SD₁₆	—	2	1	no	—
266	C₂×SD₁₆	—	2	1	no	—
267	C₂×D₈	—	2	1	no	—
268	#131	—	2	4	no	—
269	#131	—	2	4	no	—
270	C₂×M₁₆	—	2	2	no	—
271	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
272	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
273	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
274	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
275	C₂²×C₈	—	2	2	no	—
276	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
277	#125	—	2	2	no	—
278	#125	—	2	1	no	—
279	#125	—	2	1	no	—
280	#124	—	2	2	no	—
281	#124	—	2	1	no	—
282	#124	—	2	1	no	—
283	C₄×Q₈	D₄×C₃₁	1	2	yes	—
284	C₄×D₄	—	2	2	no	—
285	C₄×D₄	D₄×C₃₁	1	2	yes	—
286	#117	—	2	1	no	—
287	#117	C₂³×C₃₁	1	1	yes	characteristic
288	C₂×C₄⋊C₄	—	2	2	no	—
289	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
290	C₂×C₂²⋊C₄	—	2	2	no	—
291	C₂×C₄²	—	2	1	no	—
292	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
293	#107	—	2	2	no	—
294	#106	—	2	2	no	—
295	C₄⋊C₈	—	2	2	no	—
296	C₄⋊C₈	—	2	2	no	—
297	#103	—	2	2	no	—
298	#103	—	2	2	no	—
299	#103	—	2	2	no	—
300	#103	—	2	2	no	—
301	#102	—	2	2	no	—
302	#102	—	2	2	no	—
303	#102	—	2	2	no	—
304	#102	—	2	2	no	—
305	#101	—	2	4	no	—
306	#100	—	2	4	no	—
307	#97	—	2	2	no	—
308	#97	—	2	1	no	—
309	C₄×C₈	—	2	2	no	—
310	C₄×C₈	—	2	1	no	—
311	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
312	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
313	C₄×C₈	D₄×C₃₁	1	2	yes	—
314	C₃₁	#33062	1	1	yes	characteristic, direct product factor, Sylow
315	C₂⁴	—	2	1	no	—
316	#41	—	2	2	no	—
317	#41	#41×C₃₁	1	4	yes	—
318	C₂×Q₈	—	2	1	no	—
319	C₂×Q₈	—	2	1	no	—
320	C₂×Q₈	—	2	2	no	—
321	C₂×Q₈	—	2	2	no	—
322	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
323	C₂×D₄	—	2	1	no	—
324	C₂×D₄	—	2	1	no	—
325	C₂×D₄	—	2	1	no	—
326	C₂×D₄	—	2	2	no	—
327	C₂×D₄	—	2	2	no	—
328	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
329	C₂²×C₄	—	2	1	no	—
330	C₂²×C₄	—	2	2	no	—
331	C₂²×C₄	—	2	1	no	—
332	C₂²×C₄	—	2	2	no	—
333	C₂²×C₄	#41×C₃₁	1	2	yes	—
334	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
335	Q₁₆	—	4	2	no	—
336	SD₁₆	—	4	2	no	—
337	SD₁₆	—	4	2	no	—
338	D₈	—	4	2	no	—
339	M₁₆	—	2	8	no	—
340	M₁₆	—	2	2	no	—
341	M₁₆	—	2	2	no	—
342	M₁₆	—	2	2	no	—
343	M₁₆	—	2	2	no	—
344	M₁₆	—	2	2	no	—
345	M₁₆	#41×C₃₁	1	4	yes	—
346	C₂×C₈	—	2	8	no	—
347	C₂×C₈	—	2	2	no	—
348	C₂×C₈	—	2	2	no	—
349	C₂×C₈	—	2	2	no	—
350	C₂×C₈	—	2	1	no	—
351	C₂×C₈	—	2	2	no	—
352	C₂×C₈	—	2	1	no	—
353	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
354	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
355	C₄⋊C₄	—	2	2	no	—
356	C₄⋊C₄	—	2	2	no	—
357	C₄⋊C₄	—	2	2	no	—
358	C₄⋊C₄	—	4	2	no	—
359	C₄⋊C₄	—	4	2	no	—
360	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
361	C₂²⋊C₄	—	2	1	no	—
362	C₂²⋊C₄	—	2	2	no	—
363	C₂²⋊C₄	—	4	2	no	—
364	C₂²⋊C₄	—	4	2	no	—
365	C₄²	—	2	2	no	—
366	C₄²	—	2	1	no	—
367	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
368	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
369	C₄²	C₂×D₄×C₃₁	1	2	yes	—
370	C₂³	—	2	1	no	—
371	C₂³	—	4	1	no	—
372	C₂³	—	4	2	no	—
373	C₂³	—	8	2	no	—
374	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
375	Q₈	—	2	1	no	—
376	Q₈	—	4	2	no	—
377	Q₈	—	4	1	no	—
378	Q₈	—	4	1	no	—
379	Q₈	C₃₁×#124	1	2	yes	—
380	D₄	—	2	4	no	—
381	D₄	—	2	1	no	—
382	D₄	—	4	2	no	—
383	D₄	—	4	1	no	—
384	D₄	—	4	1	no	—
385	D₄	C₃₁×#124	1	2	yes	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	2	4	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	4	4	no	—
391	C₂×C₄	—	4	1	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	—	4	2	no	—
394	C₂×C₄	—	8	2	no	—
395	C₂×C₄	C₃₁×#121	1	2	yes	—
396	C₂×C₄	C₃₁×#121	1	2	yes	—
397	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
398	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
399	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
400	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
401	C₈	—	2	2	no	—
402	C₈	—	2	4	no	—
403	C₈	—	2	2	no	—
404	C₈	—	4	1	no	—
405	C₈	—	4	1	no	—
406	C₈	—	4	2	no	—
407	C₂²	—	2	2	no	—
408	C₂²	—	2	2	no	—
409	C₂²	—	4	2	no	—
410	C₂²	—	8	1	no	—
411	C₂²	—	8	2	no	—
412	C₂²	—	16	2	no	—
413	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
414	C₂²	C₃₁×#529	1	2	yes	—
415	C₄	—	2	2	no	—
416	C₄	—	2	2	no	—
417	C₄	—	2	2	no	—
418	C₄	—	4	2	no	—
419	C₄	—	8	1	no	—
420	C₄	—	8	2	no	—
421	C₄	C₃₁×#525	1	2	yes	—
422	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
423	C₄	C₃₁×#532	1	1	yes	characteristic
424	C₂	—	2	2	no	—
425	C₂	—	4	2	no	—
426	C₂	—	16	1	no	—
427	C₂	C₃₁×#2850	1	1	yes	characteristic
428	C₂	C₃₁×#3067	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#488	C₂	semidirect
2. C₃₁×#3080	C₂	semidirect
4. C₄×C₃₁×#131	C₂	semidirect
5. C₃₁×#1998	C₂	semidirect
6. C₃₁×#1967	C₂	semidirect
7. C₃₁×#1931	C₂	semidirect
8. C₃₁×#1930	C₂	semidirect
9. C₃₁×#1908	C₂	semidirect
11. C₃₁×#1837	C₂	semidirect
155. #33062	C₃₁	direct (N × H)
193. C₂×#488	C₂×C₃₁	semidirect
194. #3080	C₂×C₃₁	semidirect
196. C₄×#131	C₂×C₃₁	semidirect
197. #1998	C₂×C₃₁	semidirect
198. #1967	C₂×C₃₁	semidirect
199. #1931	C₂×C₃₁	semidirect
200. #1930	C₂×C₃₁	semidirect
201. #1908	C₂×C₃₁	semidirect
203. #1837	C₂×C₃₁	semidirect
314. C₃₁	#33062	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×#488	2
C₃₁×#3080	2
C₃₁×#3040	2
C₄×C₃₁×#131	2
C₃₁×#1998	2
C₃₁×#1967	2
C₃₁×#1931	2
C₃₁×#1930	2
C₃₁×#1908	2
C₃₁×#1907	2
C₃₁×#1837	2
#33062	31

Subgroups — C31×#33062 #21489821

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33062 #21489821