Subgroups — C₃₁×#25052 #21481811

Order 7936.

720 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3099	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3061	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2951	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2079	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2040	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1974	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1952	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1860	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1846	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1816	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1764	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
14	C₃₁×#537	C₂²	1	2	yes	—
15	C₃₁×#520	C₂²	1	2	yes	—
16	C₃₁×#485	—	2	2	no	—
17	C₃₁×#483	—	2	2	no	—
18	C₃₁×#482	—	2	2	no	—
19	C₃₁×#481	—	2	2	no	—
20	C₃₁×#468	—	2	2	no	—
21	C₃₁×#467	—	2	1	no	—
22	C₃₁×#466	—	2	1	no	—
23	C₃₁×#440	—	2	2	no	—
24	C₄×Q₁₆×C₃₁	—	2	1	no	—
25	C₄×D₈×C₃₁	—	2	1	no	—
26	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
27	C₃₁×#428	C₂²	1	2	yes	—
28	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
30	C₂×C₃₁×#106	C₂²	1	2	yes	—
31	C₂×C₃₁×#104	C₂²	1	2	yes	—
32	C₂×C₃₁×#104	C₂²	1	1	yes	characteristic
33	C₃₁×#417	C₂²	1	2	yes	—
34	C₃₁×#417	C₂²	1	2	yes	—
35	C₃₁×#417	C₂²	1	2	yes	—
36	C₃₁×#416	C₂²	1	1	yes	characteristic
37	C₃₁×#408	C₂²	1	2	yes	—
38	C₃₁×#405	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
40	C₃₁×#382	C₂²	1	2	yes	—
41	C₃₁×#343	C₂²	1	2	yes	—
42	C₃₁×#343	C₂²	1	2	yes	—
43	C₃₁×#341	C₂²	1	1	yes	characteristic
44	C₃₁×#339	C₂²	1	1	yes	characteristic
45	C₂×#41×C₃₁	C₂³	1	2	yes	—
46	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
47	C₃₁×#135	—	2	2	no	—
48	C₃₁×#135	—	2	2	no	—
49	C₂×Q₁₆×C₃₁	D₄	1	1	yes	characteristic
50	C₂×SD₁₆×C₃₁	D₄	1	2	yes	—
51	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
52	C₃₁×#131	—	2	2	no	—
53	C₃₁×#131	—	2	2	no	—
54	C₂×M₁₆×C₃₁	—	2	1	no	—
55	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
56	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
57	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
58	C₂²×C₈×C₃₁	—	2	1	no	—
59	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
61	C₃₁×C₂²⋊Q₈	—	2	2	no	—
62	C₃₁×C₂²⋊Q₈	—	2	2	no	—
63	C₃₁×#121	—	2	2	no	—
64	C₃₁×#121	—	2	2	no	—
65	C₃₁×#121	—	4	2	no	—
66	C₄×Q₈×C₃₁	—	2	2	no	—
67	C₄×D₄×C₃₁	—	2	2	no	—
68	C₄×D₄×C₃₁	—	4	2	no	—
69	C₃₁×#117	—	2	2	no	—
70	C₃₁×#117	C₂³	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
72	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
73	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
74	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
75	C₃₁×#108	—	2	2	no	—
76	C₃₁×#107	—	2	1	no	—
77	C₃₁×#107	—	2	1	no	—
78	C₃₁×#107	—	2	1	no	—
79	C₃₁×#107	—	2	1	no	—
80	C₃₁×#107	—	2	2	no	—
81	C₃₁×#106	—	2	2	no	—
82	C₃₁×#106	—	2	2	no	—
83	C₃₁×#106	—	2	2	no	—
84	C₃₁×#104	—	2	4	no	—
85	C₃₁×#104	—	2	2	no	—
86	C₃₁×#103	—	2	2	no	—
87	C₃₁×#103	—	2	2	no	—
88	C₃₁×#103	—	2	2	no	—
89	C₃₁×#103	D₄	1	1	yes	characteristic
90	C₃₁×#103	D₄	1	1	yes	characteristic
91	C₃₁×#102	—	2	2	no	—
92	C₃₁×#102	—	2	2	no	—
93	C₃₁×#102	—	2	2	no	—
94	C₃₁×#102	D₄	1	1	yes	characteristic
95	C₃₁×#102	D₄	1	1	yes	characteristic
96	C₃₁×C₂²⋊C₈	—	2	2	no	—
97	C₃₁×C₂²⋊C₈	—	2	2	no	—
98	C₃₁×#97	—	2	1	no	—
99	C₄×C₈×C₃₁	—	2	1	no	—
100	C₄×C₈×C₃₁	—	2	2	no	—
101	C₃₁×#95	—	4	2	no	—
102	#41×C₃₁	—	2	2	no	—
103	#41×C₃₁	—	2	1	no	—
104	#41×C₃₁	—	2	1	no	—
105	#41×C₃₁	—	2	1	no	—
106	#41×C₃₁	—	2	2	no	—
107	#41×C₃₁	—	2	2	no	—
108	#41×C₃₁	—	2	2	no	—
109	#41×C₃₁	—	2	1	no	—
110	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
111	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
112	C₂×D₄×C₃₁	—	2	2	no	—
113	C₂×D₄×C₃₁	—	2	1	no	—
114	C₂×D₄×C₃₁	—	4	2	no	—
115	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
116	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
117	C₂²×C₄×C₃₁	—	2	1	no	—
118	C₂²×C₄×C₃₁	—	2	2	no	—
119	C₂²×C₄×C₃₁	—	2	1	no	—
120	C₂²×C₄×C₃₁	—	2	2	no	—
121	C₂²×C₄×C₃₁	—	2	2	no	—
122	C₂²×C₄×C₃₁	—	4	2	no	—
123	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
124	Q₁₆×C₃₁	—	2	2	no	—
125	SD₁₆×C₃₁	—	2	4	no	—
126	D₈×C₃₁	—	2	2	no	—
127	M₁₆×C₃₁	—	2	2	no	—
128	M₁₆×C₃₁	—	2	1	no	—
129	M₁₆×C₃₁	—	2	2	no	—
130	M₁₆×C₃₁	—	2	1	no	—
131	M₁₆×C₃₁	—	2	1	no	—
132	M₁₆×C₃₁	—	2	1	no	—
133	M₁₆×C₃₁	—	4	2	no	—
134	C₂×C₈×C₃₁	—	2	1	no	—
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	—	4	2	no	—
140	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
141	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
142	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
143	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
144	C₂×C₈×C₃₁	#41	1	2	yes	—
145	C₂×C₈×C₃₁	#41	1	2	yes	—
146	C₄⋊C₄×C₃₁	—	2	2	no	—
147	C₄⋊C₄×C₃₁	—	2	2	no	—
148	C₄⋊C₄×C₃₁	—	2	2	no	—
149	C₄⋊C₄×C₃₁	—	2	2	no	—
150	C₄⋊C₄×C₃₁	—	4	2	no	—
151	C₄⋊C₄×C₃₁	—	4	2	no	—
152	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
153	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
154	C₂²⋊C₄×C₃₁	—	2	1	no	—
155	C₂²⋊C₄×C₃₁	—	4	2	no	—
156	C₂²⋊C₄×C₃₁	—	4	2	no	—
157	C₂²⋊C₄×C₃₁	—	4	2	no	—
158	C₄²×C₃₁	—	2	1	no	—
159	C₄²×C₃₁	—	2	1	no	—
160	C₄²×C₃₁	—	4	2	no	—
161	C₄²×C₃₁	#41	1	2	yes	—
162	#25052	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
163	C₂³×C₃₁	—	2	2	no	—
164	C₂³×C₃₁	—	2	1	no	—
165	C₂³×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
166	Q₈×C₃₁	—	2	2	no	—
167	Q₈×C₃₁	—	2	1	no	—
168	Q₈×C₃₁	—	2	1	no	—
169	Q₈×C₃₁	—	2	2	no	—
170	D₄×C₃₁	—	2	2	no	—
171	D₄×C₃₁	—	2	1	no	—
172	D₄×C₃₁	—	2	1	no	—
173	D₄×C₃₁	—	2	2	no	—
174	D₄×C₃₁	—	4	2	no	—
175	D₄×C₃₁	—	4	1	no	—
176	D₄×C₃₁	—	4	2	no	—
177	D₄×C₃₁	—	4	1	no	—
178	D₄×C₃₁	—	16	2	no	—
179	C₂×C₄×C₃₁	—	2	1	no	—
180	C₂×C₄×C₃₁	—	2	1	no	—
181	C₂×C₄×C₃₁	—	2	2	no	—
182	C₂×C₄×C₃₁	—	2	1	no	—
183	C₂×C₄×C₃₁	—	2	2	no	—
184	C₂×C₄×C₃₁	—	4	1	no	—
185	C₂×C₄×C₃₁	—	4	1	no	—
186	C₂×C₄×C₃₁	—	4	1	no	—
187	C₂×C₄×C₃₁	—	4	2	no	—
188	C₂×C₄×C₃₁	—	4	2	no	—
189	C₂×C₄×C₃₁	—	4	1	no	—
190	C₂×C₄×C₃₁	—	4	2	no	—
191	C₂×C₄×C₃₁	—	4	2	no	—
192	C₂×C₄×C₃₁	—	8	2	no	—
193	C₂×C₄×C₃₁	—	8	2	no	—
194	C₂×C₄×C₃₁	—	16	2	no	—
195	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
196	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
197	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
198	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
199	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
200	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
201	C₈×C₃₁	—	2	1	no	—
202	C₈×C₃₁	—	2	1	no	—
203	C₈×C₃₁	—	4	1	no	—
204	C₈×C₃₁	—	4	2	no	—
205	C₈×C₃₁	—	4	1	no	—
206	C₈×C₃₁	#123	1	2	yes	—
207	C₈×C₃₁	#123	1	2	yes	—
208	#3099	C₂×C₃₁	1	2	yes	complemented
209	#3061	C₂×C₃₁	1	1	yes	characteristic
210	#2951	C₂×C₃₁	1	1	yes	characteristic, complemented
211	#2079	C₂×C₃₁	1	2	yes	complemented
212	#2040	C₂×C₃₁	1	2	yes	complemented
213	#1974	C₂×C₃₁	1	1	yes	characteristic, complemented
214	#1952	C₂×C₃₁	1	1	yes	characteristic, complemented
215	#1860	C₂×C₃₁	1	2	yes	complemented
216	#1846	C₂×C₃₁	1	1	yes	characteristic, complemented
217	#1816	C₂×C₃₁	1	1	yes	characteristic, complemented
218	#1764	C₂×C₃₁	1	1	yes	characteristic, complemented
219	C₂²×C₃₁	—	2	1	no	—
220	C₂²×C₃₁	—	2	1	no	—
221	C₂²×C₃₁	—	4	1	no	—
222	C₂²×C₃₁	—	4	2	no	—
223	C₂²×C₃₁	—	4	2	no	—
224	C₂²×C₃₁	—	4	1	no	—
225	C₂²×C₃₁	—	4	2	no	—
226	C₂²×C₃₁	—	4	1	no	—
227	C₂²×C₃₁	#542	1	1	yes	characteristic
228	C₂²×C₃₁	D₄²	1	1	yes	characteristic
229	C₂²×C₃₁	#553	1	1	yes	characteristic
230	C₄×C₃₁	—	4	2	no	—
231	C₄×C₃₁	—	4	2	no	—
232	C₄×C₃₁	—	4	1	no	—
233	C₄×C₃₁	—	4	1	no	—
234	C₄×C₃₁	—	4	1	no	—
235	C₄×C₃₁	—	8	2	no	—
236	C₄×C₃₁	—	8	2	no	—
237	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
238	C₄×C₃₁	#532	1	1	yes	characteristic
239	C₄×C₃₁	#547	1	1	yes	characteristic
240	C₄×C₃₁	#547	1	1	yes	characteristic
241	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
242	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
243	#537	C₂²×C₃₁	1	2	yes	—
244	#520	C₂²×C₃₁	1	2	yes	—
245	#485	—	2	2	no	—
246	#483	—	2	2	no	—
247	#482	—	2	2	no	—
248	#481	—	2	2	no	—
249	#468	—	2	2	no	—
250	#467	—	2	1	no	—
251	#466	—	2	1	no	—
252	#440	—	2	2	no	—
253	C₄×Q₁₆	—	2	1	no	—
254	C₄×D₈	—	2	1	no	—
255	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
256	#428	C₂²×C₃₁	1	2	yes	—
257	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
258	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
259	C₂×#106	C₂²×C₃₁	1	2	yes	—
260	C₂×#104	C₂²×C₃₁	1	2	yes	—
261	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic
262	#417	C₂²×C₃₁	1	2	yes	—
263	#417	C₂²×C₃₁	1	2	yes	—
264	#417	C₂²×C₃₁	1	2	yes	—
265	#416	C₂²×C₃₁	1	1	yes	characteristic
266	#408	C₂²×C₃₁	1	2	yes	—
267	#405	C₂²×C₃₁	1	1	yes	characteristic
268	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
269	#382	C₂²×C₃₁	1	2	yes	—
270	#343	C₂²×C₃₁	1	2	yes	—
271	#343	C₂²×C₃₁	1	2	yes	—
272	#341	C₂²×C₃₁	1	1	yes	characteristic
273	#339	C₂²×C₃₁	1	1	yes	characteristic
274	C₂×C₃₁	—	2	1	no	—
275	C₂×C₃₁	—	2	1	no	—
276	C₂×C₃₁	—	8	2	no	—
277	C₂×C₃₁	—	8	1	no	—
278	C₂×C₃₁	#2422	1	1	yes	characteristic
279	C₂×C₃₁	#3290	1	1	yes	characteristic
280	C₂×C₃₁	#3290	1	1	yes	characteristic
281	C₂×#41	C₂³×C₃₁	1	2	yes	—
282	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
283	#135	—	2	2	no	—
284	#135	—	2	2	no	—
285	C₂×Q₁₆	D₄×C₃₁	1	1	yes	characteristic
286	C₂×SD₁₆	D₄×C₃₁	1	2	yes	—
287	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
288	#131	—	2	2	no	—
289	#131	—	2	2	no	—
290	C₂×M₁₆	—	2	1	no	—
291	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
292	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
293	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
294	C₂²×C₈	—	2	1	no	—
295	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
296	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
297	C₂²⋊Q₈	—	2	2	no	—
298	C₂²⋊Q₈	—	2	2	no	—
299	#121	—	2	2	no	—
300	#121	—	2	2	no	—
301	#121	—	4	2	no	—
302	C₄×Q₈	—	2	2	no	—
303	C₄×D₄	—	2	2	no	—
304	C₄×D₄	—	4	2	no	—
305	#117	—	2	2	no	—
306	#117	C₂³×C₃₁	1	1	yes	characteristic
307	C₂×C₄⋊C₄	—	2	2	no	—
308	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
309	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
310	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
311	#108	—	2	2	no	—
312	#107	—	2	1	no	—
313	#107	—	2	1	no	—
314	#107	—	2	1	no	—
315	#107	—	2	1	no	—
316	#107	—	2	2	no	—
317	#106	—	2	2	no	—
318	#106	—	2	2	no	—
319	#106	—	2	2	no	—
320	#104	—	2	4	no	—
321	#104	—	2	2	no	—
322	#103	—	2	2	no	—
323	#103	—	2	2	no	—
324	#103	—	2	2	no	—
325	#103	D₄×C₃₁	1	1	yes	characteristic
326	#103	D₄×C₃₁	1	1	yes	characteristic
327	#102	—	2	2	no	—
328	#102	—	2	2	no	—
329	#102	—	2	2	no	—
330	#102	D₄×C₃₁	1	1	yes	characteristic
331	#102	D₄×C₃₁	1	1	yes	characteristic
332	C₂²⋊C₈	—	2	2	no	—
333	C₂²⋊C₈	—	2	2	no	—
334	#97	—	2	1	no	—
335	C₄×C₈	—	2	1	no	—
336	C₄×C₈	—	2	2	no	—
337	#95	—	4	2	no	—
338	C₃₁	#25052	1	1	yes	characteristic, direct product factor, Sylow
339	#41	—	2	2	no	—
340	#41	—	2	1	no	—
341	#41	—	2	1	no	—
342	#41	—	2	1	no	—
343	#41	—	2	2	no	—
344	#41	—	2	2	no	—
345	#41	—	2	2	no	—
346	#41	—	2	1	no	—
347	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
348	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
349	C₂×D₄	—	2	2	no	—
350	C₂×D₄	—	2	1	no	—
351	C₂×D₄	—	4	2	no	—
352	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
353	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
354	C₂²×C₄	—	2	1	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	2	1	no	—
357	C₂²×C₄	—	2	2	no	—
358	C₂²×C₄	—	2	2	no	—
359	C₂²×C₄	—	4	2	no	—
360	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
361	Q₁₆	—	2	2	no	—
362	SD₁₆	—	2	4	no	—
363	D₈	—	2	2	no	—
364	M₁₆	—	2	2	no	—
365	M₁₆	—	2	1	no	—
366	M₁₆	—	2	2	no	—
367	M₁₆	—	2	1	no	—
368	M₁₆	—	2	1	no	—
369	M₁₆	—	2	1	no	—
370	M₁₆	—	4	2	no	—
371	C₂×C₈	—	2	1	no	—
372	C₂×C₈	—	2	1	no	—
373	C₂×C₈	—	2	1	no	—
374	C₂×C₈	—	2	2	no	—
375	C₂×C₈	—	2	1	no	—
376	C₂×C₈	—	4	2	no	—
377	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
378	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
379	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
380	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
381	C₂×C₈	#41×C₃₁	1	2	yes	—
382	C₂×C₈	#41×C₃₁	1	2	yes	—
383	C₄⋊C₄	—	2	2	no	—
384	C₄⋊C₄	—	2	2	no	—
385	C₄⋊C₄	—	2	2	no	—
386	C₄⋊C₄	—	2	2	no	—
387	C₄⋊C₄	—	4	2	no	—
388	C₄⋊C₄	—	4	2	no	—
389	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
390	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
391	C₂²⋊C₄	—	2	1	no	—
392	C₂²⋊C₄	—	4	2	no	—
393	C₂²⋊C₄	—	4	2	no	—
394	C₂²⋊C₄	—	4	2	no	—
395	C₄²	—	2	1	no	—
396	C₄²	—	2	1	no	—
397	C₄²	—	4	2	no	—
398	C₄²	#41×C₃₁	1	2	yes	—
399	C₂³	—	2	2	no	—
400	C₂³	—	2	1	no	—
401	C₂³	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
402	Q₈	—	2	2	no	—
403	Q₈	—	2	1	no	—
404	Q₈	—	2	1	no	—
405	Q₈	—	2	2	no	—
406	D₄	—	2	2	no	—
407	D₄	—	2	1	no	—
408	D₄	—	2	1	no	—
409	D₄	—	2	2	no	—
410	D₄	—	4	2	no	—
411	D₄	—	4	1	no	—
412	D₄	—	4	2	no	—
413	D₄	—	4	1	no	—
414	D₄	—	16	2	no	—
415	C₂×C₄	—	2	1	no	—
416	C₂×C₄	—	2	1	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	4	1	no	—
421	C₂×C₄	—	4	1	no	—
422	C₂×C₄	—	4	1	no	—
423	C₂×C₄	—	4	2	no	—
424	C₂×C₄	—	4	2	no	—
425	C₂×C₄	—	4	1	no	—
426	C₂×C₄	—	4	2	no	—
427	C₂×C₄	—	4	2	no	—
428	C₂×C₄	—	8	2	no	—
429	C₂×C₄	—	8	2	no	—
430	C₂×C₄	—	16	2	no	—
431	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
432	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
433	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
434	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
435	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
436	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
437	C₈	—	2	1	no	—
438	C₈	—	2	1	no	—
439	C₈	—	4	1	no	—
440	C₈	—	4	2	no	—
441	C₈	—	4	1	no	—
442	C₈	C₃₁×#123	1	2	yes	—
443	C₈	C₃₁×#123	1	2	yes	—
444	C₂²	—	2	1	no	—
445	C₂²	—	2	1	no	—
446	C₂²	—	4	1	no	—
447	C₂²	—	4	2	no	—
448	C₂²	—	4	2	no	—
449	C₂²	—	4	1	no	—
450	C₂²	—	4	2	no	—
451	C₂²	—	4	1	no	—
452	C₂²	C₃₁×#542	1	1	yes	characteristic
453	C₂²	D₄²×C₃₁	1	1	yes	characteristic
454	C₂²	C₃₁×#553	1	1	yes	characteristic
455	C₄	—	4	2	no	—
456	C₄	—	4	2	no	—
457	C₄	—	4	1	no	—
458	C₄	—	4	1	no	—
459	C₄	—	4	1	no	—
460	C₄	—	8	2	no	—
461	C₄	—	8	2	no	—
462	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
463	C₄	C₃₁×#532	1	1	yes	characteristic
464	C₄	C₃₁×#547	1	1	yes	characteristic
465	C₄	C₃₁×#547	1	1	yes	characteristic
466	C₂	—	2	1	no	—
467	C₂	—	2	1	no	—
468	C₂	—	8	2	no	—
469	C₂	—	8	1	no	—
470	C₂	C₃₁×#2422	1	1	yes	characteristic
471	C₂	C₃₁×#3290	1	1	yes	characteristic
472	C₂	C₃₁×#3290	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#3061	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3099	C₂	semidirect
3. C₃₁×#2951	C₂	semidirect
4. C₃₁×#2079	C₂	semidirect
5. C₃₁×#2040	C₂	semidirect
6. C₃₁×#1974	C₂	semidirect
7. C₃₁×#1952	C₂	semidirect
8. C₃₁×#1860	C₂	semidirect
9. C₃₁×#1846	C₂	semidirect
10. C₃₁×#1816	C₂	semidirect
11. C₃₁×#1764	C₂	semidirect
162. #25052	C₃₁	direct (N × H)
208. #3099	C₂×C₃₁	semidirect
210. #2951	C₂×C₃₁	semidirect
211. #2079	C₂×C₃₁	semidirect
212. #2040	C₂×C₃₁	semidirect
213. #1974	C₂×C₃₁	semidirect
214. #1952	C₂×C₃₁	semidirect
215. #1860	C₂×C₃₁	semidirect
216. #1846	C₂×C₃₁	semidirect
217. #1816	C₂×C₃₁	semidirect
218. #1764	C₂×C₃₁	semidirect
338. C₃₁	#25052	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3099	2
C₃₁×#3061	2
C₃₁×#2951	2
C₃₁×#2079	2
C₃₁×#2040	2
C₃₁×#1974	2
C₃₁×#1952	2
C₃₁×#1860	2
C₃₁×#1846	2
C₃₁×#1816	2
C₃₁×#1764	2
#25052	31

Subgroups — C31×#25052 #21481811

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#25052 #21481811