Subgroups — C₃₁×#26274 #21483033

Order 7936.

796 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3155	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3096	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2952	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2051	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2037	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1978	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1975	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1972	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1951	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1883	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1773	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	2	yes	—
13	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
14	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#535	C₂²	1	2	yes	—
16	C₃₁×#520	C₂²	1	2	yes	—
17	C₃₁×#496	—	2	2	no	—
18	C₃₁×#495	—	2	1	no	—
19	C₃₁×#493	—	2	1	no	—
20	C₃₁×#482	—	2	2	no	—
21	C₃₁×#482	—	2	2	no	—
22	C₃₁×#481	—	2	2	no	—
23	C₃₁×#480	—	2	2	no	—
24	C₃₁×#442	—	2	1	no	—
25	C₃₁×#441	—	2	1	no	—
26	C₄×SD₁₆×C₃₁	—	2	2	no	—
27	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
29	C₃₁×#428	C₂²	1	2	yes	—
30	C₃₁×#427	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
32	C₂×C₃₁×#104	C₂²	1	2	yes	—
33	C₃₁×#418	C₂²	1	2	yes	—
34	C₃₁×#417	C₂²	1	2	yes	—
35	C₂×C₃₁×#102	C₂²	1	2	yes	—
36	C₂×C₃₁×#100	C₂²	1	2	yes	—
37	C₃₁×#408	C₂²	1	2	yes	—
38	C₃₁×#405	C₂²	1	1	yes	characteristic
39	C₃₁×#405	C₂²	1	1	yes	characteristic
40	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
41	C₃₁×#390	C₂²	1	2	yes	—
42	C₃₁×#343	C₂²	1	2	yes	—
43	C₃₁×#339	C₂²	1	1	yes	characteristic
44	C₃₁×#339	C₂²	1	1	yes	characteristic
45	C₂×#41×C₃₁	C₂³	1	2	yes	—
46	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
47	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
48	C₃₁×AGL(1,ℤ/8)	—	4	2	no	—
49	C₃₁×#135	—	2	2	no	—
50	C₃₁×#135	—	2	2	no	—
51	C₂×Q₁₆×C₃₁	D₄	1	1	yes	characteristic
52	C₂×SD₁₆×C₃₁	—	2	2	no	—
53	C₂×SD₁₆×C₃₁	D₄	1	2	yes	—
54	C₂×D₈×C₃₁	—	2	2	no	—
55	C₂×D₈×C₃₁	—	2	1	no	—
56	C₂×D₈×C₃₁	—	2	1	no	—
57	C₂×D₈×C₃₁	—	2	2	no	—
58	C₂×D₈×C₃₁	—	2	2	no	—
59	C₂×D₈×C₃₁	D₄	1	1	yes	characteristic
60	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
61	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
62	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
63	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
65	C₃₁×#127	—	2	2	no	—
66	C₃₁×#124	—	2	2	no	—
67	C₃₁×#121	—	2	2	no	—
68	C₃₁×#121	—	2	2	no	—
69	C₃₁×#120	—	4	2	no	—
70	C₄×Q₈×C₃₁	—	2	2	no	—
71	C₄×D₄×C₃₁	—	2	2	no	—
72	C₄×D₄×C₃₁	—	4	2	no	—
73	C₃₁×#117	—	2	2	no	—
74	C₃₁×#117	C₂³	1	1	yes	characteristic
75	C₃₁×#117	C₂³	1	1	yes	characteristic
76	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
77	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
78	C₂×C₂²⋊C₄×C₃₁	—	4	2	no	—
79	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
80	C₃₁×#108	—	2	2	no	—
81	C₃₁×#108	—	2	2	no	—
82	C₃₁×#107	—	2	2	no	—
83	C₃₁×#107	D₄	1	2	yes	—
84	C₃₁×#106	—	2	1	no	—
85	C₃₁×#106	—	2	1	no	—
86	C₃₁×#106	—	2	2	no	—
87	C₃₁×#106	D₄	1	2	yes	—
88	C₃₁×#104	—	4	2	no	—
89	C₃₁×#103	—	2	2	no	—
90	C₃₁×#102	—	2	2	no	—
91	C₃₁×#102	—	2	2	no	—
92	C₃₁×#102	—	2	2	no	—
93	C₃₁×#102	—	2	2	no	—
94	C₃₁×#102	—	2	2	no	—
95	C₃₁×#100	—	4	2	no	—
96	C₃₁×C₂²⋊C₈	—	2	2	no	—
97	C₃₁×C₂²⋊C₈	—	2	2	no	—
98	C₃₁×#97	—	2	1	no	—
99	C₃₁×#97	—	2	1	no	—
100	C₃₁×#97	—	2	2	no	—
101	C₄×C₈×C₃₁	—	2	1	no	—
102	C₄×C₈×C₃₁	—	2	1	no	—
103	C₂⁴×C₃₁	—	2	2	no	—
104	#41×C₃₁	—	2	2	no	—
105	#41×C₃₁	—	2	2	no	—
106	#41×C₃₁	—	2	2	no	—
107	#41×C₃₁	—	2	2	no	—
108	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
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	—	2	2	no	—
114	C₂×D₄×C₃₁	—	2	2	no	—
115	C₂×D₄×C₃₁	—	2	2	no	—
116	C₂×D₄×C₃₁	—	4	2	no	—
117	C₂×D₄×C₃₁	—	4	2	no	—
118	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
119	C₂²×C₄×C₃₁	—	2	2	no	—
120	C₂²×C₄×C₃₁	—	2	1	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	SD₁₆×C₃₁	—	4	4	no	—
127	D₈×C₃₁	—	2	2	no	—
128	D₈×C₃₁	—	4	4	no	—
129	D₈×C₃₁	—	4	2	no	—
130	D₈×C₃₁	—	4	2	no	—
131	M₁₆×C₃₁	—	2	1	no	—
132	M₁₆×C₃₁	—	2	1	no	—
133	M₁₆×C₃₁	—	2	2	no	—
134	M₁₆×C₃₁	—	2	1	no	—
135	M₁₆×C₃₁	—	2	2	no	—
136	M₁₆×C₃₁	#41	1	2	yes	—
137	C₂×C₈×C₃₁	—	2	1	no	—
138	C₂×C₈×C₃₁	—	2	1	no	—
139	C₂×C₈×C₃₁	—	2	1	no	—
140	C₂×C₈×C₃₁	—	2	1	no	—
141	C₂×C₈×C₃₁	—	2	2	no	—
142	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
143	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
144	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
145	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
146	C₂×C₈×C₃₁	#41	1	2	yes	—
147	C₂×C₈×C₃₁	#41	1	2	yes	—
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	C₂×D₄	1	2	yes	—
152	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
153	C₂²⋊C₄×C₃₁	—	2	1	no	—
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	C₂²⋊C₄×C₃₁	—	8	4	no	—
159	C₄²×C₃₁	—	2	1	no	—
160	C₄²×C₃₁	—	2	1	no	—
161	C₄²×C₃₁	—	2	1	no	—
162	C₄²×C₃₁	—	2	1	no	—
163	C₄²×C₃₁	—	4	2	no	—
164	#26274	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
165	C₂³×C₃₁	—	2	2	no	—
166	C₂³×C₃₁	—	4	2	no	—
167	C₂³×C₃₁	—	4	2	no	—
168	C₂³×C₃₁	—	4	2	no	—
169	C₂³×C₃₁	—	8	2	no	—
170	C₂³×C₃₁	—	8	2	no	—
171	C₂³×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
172	Q₈×C₃₁	—	2	2	no	—
173	Q₈×C₃₁	—	2	2	no	—
174	D₄×C₃₁	—	2	2	no	—
175	D₄×C₃₁	—	2	2	no	—
176	D₄×C₃₁	—	4	2	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	—	16	2	no	—
183	D₄×C₃₁	—	16	2	no	—
184	C₂×C₄×C₃₁	—	2	2	no	—
185	C₂×C₄×C₃₁	—	2	2	no	—
186	C₂×C₄×C₃₁	—	2	2	no	—
187	C₂×C₄×C₃₁	—	4	2	no	—
188	C₂×C₄×C₃₁	—	4	2	no	—
189	C₂×C₄×C₃₁	—	4	2	no	—
190	C₂×C₄×C₃₁	—	4	1	no	—
191	C₂×C₄×C₃₁	—	4	2	no	—
192	C₂×C₄×C₃₁	—	8	2	no	—
193	C₂×C₄×C₃₁	—	16	2	no	—
194	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
195	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
196	C₂×C₄×C₃₁	C₂×#41	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	H(5,𝔽₂)	1	1	yes	characteristic
200	C₈×C₃₁	—	2	1	no	—
201	C₈×C₃₁	—	2	1	no	—
202	C₈×C₃₁	—	2	2	no	—
203	C₈×C₃₁	—	4	1	no	—
204	C₈×C₃₁	—	4	2	no	—
205	C₈×C₃₁	#123	1	2	yes	—
206	C₈×C₃₁	#123	1	2	yes	—
207	#3155	C₂×C₃₁	1	1	yes	characteristic
208	#3096	C₂×C₃₁	1	2	yes	complemented
209	#2952	C₂×C₃₁	1	1	yes	characteristic, complemented
210	#2051	C₂×C₃₁	1	2	yes	complemented
211	#2037	C₂×C₃₁	1	2	yes	complemented
212	#1978	C₂×C₃₁	1	1	yes	characteristic, complemented
213	#1975	C₂×C₃₁	1	1	yes	characteristic, complemented
214	#1972	C₂×C₃₁	1	1	yes	characteristic, complemented
215	#1951	C₂×C₃₁	1	1	yes	characteristic, complemented
216	#1883	C₂×C₃₁	1	2	yes	complemented
217	#1773	C₂×C₃₁	1	1	yes	characteristic, complemented
218	C₂²×C₃₁	—	2	2	no	—
219	C₂²×C₃₁	—	4	2	no	—
220	C₂²×C₃₁	—	4	2	no	—
221	C₂²×C₃₁	—	4	2	no	—
222	C₂²×C₃₁	—	8	4	no	—
223	C₂²×C₃₁	—	8	2	no	—
224	C₂²×C₃₁	—	8	2	no	—
225	C₂²×C₃₁	—	8	2	no	—
226	C₂²×C₃₁	—	16	2	no	—
227	C₂²×C₃₁	#540	1	1	yes	characteristic
228	C₂²×C₃₁	D₄²	1	1	yes	characteristic
229	C₂²×C₃₁	#559	1	1	yes	characteristic
230	C₄×C₃₁	—	4	2	no	—
231	C₄×C₃₁	—	4	2	no	—
232	C₄×C₃₁	—	4	2	no	—
233	C₄×C₃₁	—	8	1	no	—
234	C₄×C₃₁	—	8	2	no	—
235	C₄×C₃₁	C₂×#123	1	1	yes	characteristic
236	C₄×C₃₁	#532	1	1	yes	characteristic
237	C₄×C₃₁	#546	1	1	yes	characteristic
238	C₄×C₃₁	#550	1	1	yes	characteristic
239	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	2	yes	—
240	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
241	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
242	#535	C₂²×C₃₁	1	2	yes	—
243	#520	C₂²×C₃₁	1	2	yes	—
244	#496	—	2	2	no	—
245	#495	—	2	1	no	—
246	#493	—	2	1	no	—
247	#482	—	2	2	no	—
248	#482	—	2	2	no	—
249	#481	—	2	2	no	—
250	#480	—	2	2	no	—
251	#442	—	2	1	no	—
252	#441	—	2	1	no	—
253	C₄×SD₁₆	—	2	2	no	—
254	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
255	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
256	#428	C₂²×C₃₁	1	2	yes	—
257	#427	C₂²×C₃₁	1	1	yes	characteristic
258	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
259	C₂×#104	C₂²×C₃₁	1	2	yes	—
260	#418	C₂²×C₃₁	1	2	yes	—
261	#417	C₂²×C₃₁	1	2	yes	—
262	C₂×#102	C₂²×C₃₁	1	2	yes	—
263	C₂×#100	C₂²×C₃₁	1	2	yes	—
264	#408	C₂²×C₃₁	1	2	yes	—
265	#405	C₂²×C₃₁	1	1	yes	characteristic
266	#405	C₂²×C₃₁	1	1	yes	characteristic
267	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
268	#390	C₂²×C₃₁	1	2	yes	—
269	#343	C₂²×C₃₁	1	2	yes	—
270	#339	C₂²×C₃₁	1	1	yes	characteristic
271	#339	C₂²×C₃₁	1	1	yes	characteristic
272	C₂×C₃₁	—	2	1	no	—
273	C₂×C₃₁	—	2	1	no	—
274	C₂×C₃₁	—	8	2	no	—
275	C₂×C₃₁	—	16	2	no	—
276	C₂×C₃₁	#2445	1	1	yes	characteristic
277	C₂×C₃₁	#3288	1	2	yes	—
278	C₂×#41	C₂³×C₃₁	1	2	yes	—
279	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
280	AGL(1,ℤ/8)	—	2	4	no	—
281	AGL(1,ℤ/8)	—	4	2	no	—
282	#135	—	2	2	no	—
283	#135	—	2	2	no	—
284	C₂×Q₁₆	D₄×C₃₁	1	1	yes	characteristic
285	C₂×SD₁₆	—	2	2	no	—
286	C₂×SD₁₆	D₄×C₃₁	1	2	yes	—
287	C₂×D₈	—	2	2	no	—
288	C₂×D₈	—	2	1	no	—
289	C₂×D₈	—	2	1	no	—
290	C₂×D₈	—	2	2	no	—
291	C₂×D₈	—	2	2	no	—
292	C₂×D₈	D₄×C₃₁	1	1	yes	characteristic
293	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
294	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
295	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
296	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
297	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
298	#127	—	2	2	no	—
299	#124	—	2	2	no	—
300	#121	—	2	2	no	—
301	#121	—	2	2	no	—
302	#120	—	4	2	no	—
303	C₄×Q₈	—	2	2	no	—
304	C₄×D₄	—	2	2	no	—
305	C₄×D₄	—	4	2	no	—
306	#117	—	2	2	no	—
307	#117	C₂³×C₃₁	1	1	yes	characteristic
308	#117	C₂³×C₃₁	1	1	yes	characteristic
309	C₂×C₄⋊C₄	—	2	2	no	—
310	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
311	C₂×C₂²⋊C₄	—	4	2	no	—
312	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
313	#108	—	2	2	no	—
314	#108	—	2	2	no	—
315	#107	—	2	2	no	—
316	#107	D₄×C₃₁	1	2	yes	—
317	#106	—	2	1	no	—
318	#106	—	2	1	no	—
319	#106	—	2	2	no	—
320	#106	D₄×C₃₁	1	2	yes	—
321	#104	—	4	2	no	—
322	#103	—	2	2	no	—
323	#102	—	2	2	no	—
324	#102	—	2	2	no	—
325	#102	—	2	2	no	—
326	#102	—	2	2	no	—
327	#102	—	2	2	no	—
328	#100	—	4	2	no	—
329	C₂²⋊C₈	—	2	2	no	—
330	C₂²⋊C₈	—	2	2	no	—
331	#97	—	2	1	no	—
332	#97	—	2	1	no	—
333	#97	—	2	2	no	—
334	C₄×C₈	—	2	1	no	—
335	C₄×C₈	—	2	1	no	—
336	C₃₁	#26274	1	1	yes	characteristic, direct product factor, Sylow
337	C₂⁴	—	2	2	no	—
338	#41	—	2	2	no	—
339	#41	—	2	2	no	—
340	#41	—	2	2	no	—
341	#41	—	2	2	no	—
342	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
343	C₂×D₄	—	2	2	no	—
344	C₂×D₄	—	2	2	no	—
345	C₂×D₄	—	2	2	no	—
346	C₂×D₄	—	2	2	no	—
347	C₂×D₄	—	2	2	no	—
348	C₂×D₄	—	2	2	no	—
349	C₂×D₄	—	2	2	no	—
350	C₂×D₄	—	4	2	no	—
351	C₂×D₄	—	4	2	no	—
352	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
353	C₂²×C₄	—	2	2	no	—
354	C₂²×C₄	—	2	1	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	4	2	no	—
357	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
358	Q₁₆	—	2	2	no	—
359	SD₁₆	—	2	4	no	—
360	SD₁₆	—	4	4	no	—
361	D₈	—	2	2	no	—
362	D₈	—	4	4	no	—
363	D₈	—	4	2	no	—
364	D₈	—	4	2	no	—
365	M₁₆	—	2	1	no	—
366	M₁₆	—	2	1	no	—
367	M₁₆	—	2	2	no	—
368	M₁₆	—	2	1	no	—
369	M₁₆	—	2	2	no	—
370	M₁₆	#41×C₃₁	1	2	yes	—
371	C₂×C₈	—	2	1	no	—
372	C₂×C₈	—	2	1	no	—
373	C₂×C₈	—	2	1	no	—
374	C₂×C₈	—	2	1	no	—
375	C₂×C₈	—	2	2	no	—
376	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
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	#41×C₃₁	1	2	yes	—
381	C₂×C₈	#41×C₃₁	1	2	yes	—
382	C₄⋊C₄	—	2	2	no	—
383	C₄⋊C₄	—	2	2	no	—
384	C₄⋊C₄	—	4	2	no	—
385	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
386	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
387	C₂²⋊C₄	—	2	1	no	—
388	C₂²⋊C₄	—	2	1	no	—
389	C₂²⋊C₄	—	4	2	no	—
390	C₂²⋊C₄	—	4	2	no	—
391	C₂²⋊C₄	—	4	2	no	—
392	C₂²⋊C₄	—	8	4	no	—
393	C₄²	—	2	1	no	—
394	C₄²	—	2	1	no	—
395	C₄²	—	2	1	no	—
396	C₄²	—	2	1	no	—
397	C₄²	—	4	2	no	—
398	C₂³	—	2	2	no	—
399	C₂³	—	4	2	no	—
400	C₂³	—	4	2	no	—
401	C₂³	—	4	2	no	—
402	C₂³	—	8	2	no	—
403	C₂³	—	8	2	no	—
404	C₂³	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
405	Q₈	—	2	2	no	—
406	Q₈	—	2	2	no	—
407	D₄	—	2	2	no	—
408	D₄	—	2	2	no	—
409	D₄	—	4	2	no	—
410	D₄	—	4	2	no	—
411	D₄	—	4	2	no	—
412	D₄	—	4	2	no	—
413	D₄	—	4	2	no	—
414	D₄	—	4	2	no	—
415	D₄	—	16	2	no	—
416	D₄	—	16	2	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	4	2	no	—
421	C₂×C₄	—	4	2	no	—
422	C₂×C₄	—	4	2	no	—
423	C₂×C₄	—	4	1	no	—
424	C₂×C₄	—	4	2	no	—
425	C₂×C₄	—	8	2	no	—
426	C₂×C₄	—	16	2	no	—
427	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
428	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
429	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
430	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
431	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
432	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
433	C₈	—	2	1	no	—
434	C₈	—	2	1	no	—
435	C₈	—	2	2	no	—
436	C₈	—	4	1	no	—
437	C₈	—	4	2	no	—
438	C₈	C₃₁×#123	1	2	yes	—
439	C₈	C₃₁×#123	1	2	yes	—
440	C₂²	—	2	2	no	—
441	C₂²	—	4	2	no	—
442	C₂²	—	4	2	no	—
443	C₂²	—	4	2	no	—
444	C₂²	—	8	4	no	—
445	C₂²	—	8	2	no	—
446	C₂²	—	8	2	no	—
447	C₂²	—	8	2	no	—
448	C₂²	—	16	2	no	—
449	C₂²	C₃₁×#540	1	1	yes	characteristic
450	C₂²	D₄²×C₃₁	1	1	yes	characteristic
451	C₂²	C₃₁×#559	1	1	yes	characteristic
452	C₄	—	4	2	no	—
453	C₄	—	4	2	no	—
454	C₄	—	4	2	no	—
455	C₄	—	8	1	no	—
456	C₄	—	8	2	no	—
457	C₄	C₂×C₃₁×#123	1	1	yes	characteristic
458	C₄	C₃₁×#532	1	1	yes	characteristic
459	C₄	C₃₁×#546	1	1	yes	characteristic
460	C₄	C₃₁×#550	1	1	yes	characteristic
461	C₂	—	2	1	no	—
462	C₂	—	2	1	no	—
463	C₂	—	8	2	no	—
464	C₂	—	16	2	no	—
465	C₂	C₃₁×#2445	1	1	yes	characteristic
466	C₂	C₃₁×#3288	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#3155	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
2. C₃₁×#3096	C₂	semidirect
3. C₃₁×#2952	C₂	semidirect
4. C₃₁×#2051	C₂	semidirect
5. C₃₁×#2037	C₂	semidirect
6. C₃₁×#1978	C₂	semidirect
7. C₃₁×#1975	C₂	semidirect
8. C₃₁×#1972	C₂	semidirect
9. C₃₁×#1951	C₂	semidirect
10. C₃₁×#1883	C₂	semidirect
11. C₃₁×#1773	C₂	semidirect
164. #26274	C₃₁	direct (N × H)
208. #3096	C₂×C₃₁	semidirect
209. #2952	C₂×C₃₁	semidirect
210. #2051	C₂×C₃₁	semidirect
211. #2037	C₂×C₃₁	semidirect
212. #1978	C₂×C₃₁	semidirect
213. #1975	C₂×C₃₁	semidirect
214. #1972	C₂×C₃₁	semidirect
215. #1951	C₂×C₃₁	semidirect
216. #1883	C₂×C₃₁	semidirect
217. #1773	C₂×C₃₁	semidirect
336. C₃₁	#26274	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×#3155	2
C₃₁×#3096	2
C₃₁×#2952	2
C₃₁×#2051	2
C₃₁×#2037	2
C₃₁×#1978	2
C₃₁×#1975	2
C₃₁×#1972	2
C₃₁×#1951	2
C₃₁×#1883	2
C₃₁×#1773	2
#26274	31

Subgroups — C31×#26274 #21483033

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#26274 #21483033