Subgroups — C₃₁×#25434 #21482193

Order 7936.

1012 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	C₂×C₃₁×#493	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#472	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3058	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3041	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2948	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2045	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2002	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1978	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1945	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1887	C₂	1	2	yes	maximal, complemented
11	C₃₁×#1764	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	2	yes	—
13	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
15	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
17	C₃₁×#535	C₂²	1	2	yes	—
18	C₂×C₃₁×#127	C₂²	1	2	yes	—
19	C₃₁×#518	C₂²	1	2	yes	—
20	C₃₁×#493	—	2	4	no	—
21	C₃₁×#472	—	2	4	no	—
22	C₃₁×#469	—	2	2	no	—
23	C₃₁×#466	—	2	2	no	—
24	C₃₁×#463	—	2	4	no	—
25	C₃₁×#459	—	2	4	no	—
26	C₃₁×#442	—	2	2	no	—
27	C₄×D₈×C₃₁	—	2	2	no	—
28	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
29	C₃₁×#427	C₂²	1	1	yes	characteristic
30	C₃₁×#424	C₂²	1	2	yes	—
31	C₂×C₃₁×#104	C₂²	1	1	yes	characteristic
32	C₃₁×#418	C₂²	1	2	yes	—
33	C₃₁×#418	C₂²	1	2	yes	—
34	C₃₁×#418	C₂²	1	2	yes	—
35	C₃₁×#416	C₂²	1	1	yes	characteristic
36	C₂×C₃₁×#100	C₂²	1	2	yes	—
37	C₂×C₃₁×#100	C₂²	1	2	yes	—
38	C₃₁×#405	C₂²	1	1	yes	characteristic
39	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
40	C₃₁×#341	C₂²	1	1	yes	characteristic
41	C₃₁×#339	C₂²	1	1	yes	characteristic
42	C₃₁×#339	C₂²	1	2	yes	—
43	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
44	C₂²×D₄×C₃₁	—	4	2	no	—
45	C₂²×D₄×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	C₂×SD₁₆×C₃₁	—	2	2	no	—
50	C₂×D₈×C₃₁	—	2	2	no	—
51	C₂×D₈×C₃₁	—	2	4	no	—
52	C₂×D₈×C₃₁	—	2	1	no	—
53	C₂×D₈×C₃₁	—	2	1	no	—
54	C₂×D₈×C₃₁	—	2	4	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	—	2	2	no	—
60	C₂×D₈×C₃₁	—	4	2	no	—
61	C₂×D₈×C₃₁	D₄	1	4	yes	complemented
62	C₃₁×#131	—	2	2	no	—
63	C₃₁×#131	—	2	2	no	—
64	C₂×M₁₆×C₃₁	—	2	1	no	—
65	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
66	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
67	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
68	C₂²×C₈×C₃₁	—	2	1	no	—
69	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
71	C₃₁×#127	—	2	2	no	—
72	C₃₁×#127	—	2	4	no	—
73	C₃₁×#127	—	2	4	no	—
74	C₃₁×#124	—	2	2	no	—
75	C₃₁×#121	—	2	4	no	—
76	C₃₁×#120	—	4	2	no	—
77	C₄×D₄×C₃₁	—	2	4	no	—
78	C₄×D₄×C₃₁	—	2	4	no	—
79	C₃₁×#117	C₂³	1	1	yes	characteristic
80	C₃₁×#117	C₂³	1	2	yes	—
81	C₂×C₂²⋊C₄×C₃₁	—	4	2	no	—
82	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
83	C₃₁×#108	—	2	2	no	—
84	C₃₁×#107	D₄	1	2	yes	complemented
85	C₃₁×#106	D₄	1	2	yes	complemented
86	C₃₁×C₄⋊C₈	—	2	4	no	—
87	C₃₁×#104	—	2	2	no	—
88	C₃₁×#103	D₄	1	2	yes	complemented
89	C₃₁×#102	—	2	4	no	—
90	C₃₁×#102	—	2	4	no	—
91	C₃₁×#102	—	2	4	no	—
92	C₃₁×#102	D₄	1	2	yes	complemented
93	C₃₁×#100	—	2	4	no	—
94	C₃₁×#100	—	4	2	no	—
95	C₃₁×#97	—	2	1	no	—
96	C₄×C₈×C₃₁	—	2	2	no	—
97	C₄×C₈×C₃₁	—	2	1	no	—
98	C₂⁴×C₃₁	—	2	2	no	—
99	C₂⁴×C₃₁	—	2	2	no	—
100	#41×C₃₁	—	2	1	no	—
101	#41×C₃₁	—	2	1	no	—
102	#41×C₃₁	—	2	1	no	—
103	#41×C₃₁	—	2	1	no	—
104	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂×D₄×C₃₁	—	2	1	no	—
106	C₂×D₄×C₃₁	—	2	2	no	—
107	C₂×D₄×C₃₁	—	2	2	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	—	2	2	no	—
114	C₂×D₄×C₃₁	—	2	2	no	—
115	C₂×D₄×C₃₁	—	2	2	no	—
116	C₂×D₄×C₃₁	—	2	2	no	—
117	C₂×D₄×C₃₁	—	4	8	no	—
118	C₂×D₄×C₃₁	—	4	4	no	—
119	C₂×D₄×C₃₁	—	8	8	no	—
120	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
121	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₃₁	—	2	1	no	—
123	C₂²×C₄×C₃₁	—	2	1	no	—
124	C₂²×C₄×C₃₁	—	4	4	no	—
125	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
126	SD₁₆×C₃₁	—	4	4	no	—
127	D₈×C₃₁	—	2	4	no	—
128	D₈×C₃₁	—	2	4	no	—
129	D₈×C₃₁	—	4	8	no	—
130	D₈×C₃₁	—	4	4	no	—
131	D₈×C₃₁	—	4	2	no	—
132	D₈×C₃₁	—	4	2	no	—
133	M₁₆×C₃₁	—	2	1	no	—
134	M₁₆×C₃₁	—	2	2	no	—
135	M₁₆×C₃₁	—	2	1	no	—
136	M₁₆×C₃₁	—	2	1	no	—
137	M₁₆×C₃₁	—	2	1	no	—
138	M₁₆×C₃₁	—	2	2	no	—
139	M₁₆×C₃₁	—	4	2	no	—
140	C₂×C₈×C₃₁	—	2	1	no	—
141	C₂×C₈×C₃₁	—	2	1	no	—
142	C₂×C₈×C₃₁	—	2	1	no	—
143	C₂×C₈×C₃₁	—	2	1	no	—
144	C₂×C₈×C₃₁	—	2	2	no	—
145	C₂×C₈×C₃₁	—	4	2	no	—
146	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
147	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
148	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
149	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
150	C₂×C₈×C₃₁	#41	1	2	yes	—
151	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
152	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
153	C₂²⋊C₄×C₃₁	—	2	1	no	—
154	C₂²⋊C₄×C₃₁	—	2	2	no	—
155	C₂²⋊C₄×C₃₁	—	4	4	no	—
156	C₂²⋊C₄×C₃₁	—	4	4	no	—
157	C₄²×C₃₁	—	2	1	no	—
158	C₄²×C₃₁	—	2	1	no	—
159	C₄²×C₃₁	—	2	2	no	—
160	C₄²×C₃₁	#41	1	2	yes	—
161	#25434	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
162	C₂³×C₃₁	—	2	1	no	—
163	C₂³×C₃₁	—	2	4	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	—	8	4	no	—
170	C₂³×C₃₁	—	8	4	no	—
171	C₂³×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
172	Q₈×C₃₁	—	2	1	no	—
173	Q₈×C₃₁	—	2	1	no	—
174	D₄×C₃₁	—	2	4	no	—
175	D₄×C₃₁	—	2	1	no	—
176	D₄×C₃₁	—	2	1	no	—
177	D₄×C₃₁	—	2	4	no	—
178	D₄×C₃₁	—	4	1	no	—
179	D₄×C₃₁	—	4	1	no	—
180	D₄×C₃₁	—	4	2	no	—
181	D₄×C₃₁	—	4	2	no	—
182	D₄×C₃₁	—	4	2	no	—
183	D₄×C₃₁	—	4	2	no	—
184	D₄×C₃₁	—	4	2	no	—
185	D₄×C₃₁	—	4	2	no	—
186	D₄×C₃₁	—	8	16	no	—
187	D₄×C₃₁	—	16	4	no	—
188	C₂×C₄×C₃₁	—	2	4	no	—
189	C₂×C₄×C₃₁	—	2	2	no	—
190	C₂×C₄×C₃₁	—	2	2	no	—
191	C₂×C₄×C₃₁	—	2	4	no	—
192	C₂×C₄×C₃₁	—	2	1	no	—
193	C₂×C₄×C₃₁	—	4	1	no	—
194	C₂×C₄×C₃₁	—	4	1	no	—
195	C₂×C₄×C₃₁	—	16	4	no	—
196	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
197	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
198	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
199	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
200	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
201	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
202	C₈×C₃₁	—	2	2	no	—
203	C₈×C₃₁	—	4	1	no	—
204	C₈×C₃₁	—	4	2	no	—
205	C₈×C₃₁	—	4	1	no	—
206	C₈×C₃₁	#121	1	2	yes	—
207	C₈×C₃₁	#121	1	2	yes	—
208	C₂×#493	C₂×C₃₁	1	1	yes	characteristic, complemented
209	C₂×#472	C₂×C₃₁	1	1	yes	characteristic, complemented
210	#3058	C₂×C₃₁	1	1	yes	characteristic, complemented
211	#3041	C₂×C₃₁	1	2	yes	complemented
212	#2948	C₂×C₃₁	1	1	yes	characteristic, complemented
213	#2045	C₂×C₃₁	1	2	yes	complemented
214	#2002	C₂×C₃₁	1	2	yes	complemented
215	#1978	C₂×C₃₁	1	1	yes	characteristic, complemented
216	#1945	C₂×C₃₁	1	1	yes	characteristic, complemented
217	#1887	C₂×C₃₁	1	2	yes	complemented
218	#1764	C₂×C₃₁	1	1	yes	characteristic, complemented
219	C₂²×C₃₁	—	2	2	no	—
220	C₂²×C₃₁	—	4	1	no	—
221	C₂²×C₃₁	—	4	4	no	—
222	C₂²×C₃₁	—	4	2	no	—
223	C₂²×C₃₁	—	8	4	no	—
224	C₂²×C₃₁	—	8	4	no	—
225	C₂²×C₃₁	—	8	4	no	—
226	C₂²×C₃₁	—	8	2	no	—
227	C₂²×C₃₁	—	8	4	no	—
228	C₂²×C₃₁	—	16	2	no	—
229	C₂²×C₃₁	—	16	2	no	—
230	C₂²×C₃₁	#540	1	1	yes	characteristic
231	C₂²×C₃₁	D₄²	1	1	yes	characteristic
232	C₂²×C₃₁	#546	1	1	yes	characteristic
233	C₄×C₃₁	—	2	4	no	—
234	C₄×C₃₁	—	4	1	no	—
235	C₄×C₃₁	—	4	2	no	—
236	C₄×C₃₁	—	4	4	no	—
237	C₄×C₃₁	C₂×#121	1	1	yes	characteristic
238	C₄×C₃₁	#532	1	1	yes	characteristic
239	C₄×C₃₁	D₄²	1	1	yes	characteristic
240	C₄×C₃₁	#550	1	1	yes	characteristic
241	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	2	yes	—
242	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
243	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
244	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
245	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
246	#535	C₂²×C₃₁	1	2	yes	—
247	C₂×#127	C₂²×C₃₁	1	2	yes	—
248	#518	C₂²×C₃₁	1	2	yes	—
249	#493	—	2	4	no	—
250	#472	—	2	4	no	—
251	#469	—	2	2	no	—
252	#466	—	2	2	no	—
253	#463	—	2	4	no	—
254	#459	—	2	4	no	—
255	#442	—	2	2	no	—
256	C₄×D₈	—	2	2	no	—
257	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
258	#427	C₂²×C₃₁	1	1	yes	characteristic
259	#424	C₂²×C₃₁	1	2	yes	—
260	C₂×#104	C₂²×C₃₁	1	1	yes	characteristic
261	#418	C₂²×C₃₁	1	2	yes	—
262	#418	C₂²×C₃₁	1	2	yes	—
263	#418	C₂²×C₃₁	1	2	yes	—
264	#416	C₂²×C₃₁	1	1	yes	characteristic
265	C₂×#100	C₂²×C₃₁	1	2	yes	—
266	C₂×#100	C₂²×C₃₁	1	2	yes	—
267	#405	C₂²×C₃₁	1	1	yes	characteristic
268	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
269	#341	C₂²×C₃₁	1	1	yes	characteristic
270	#339	C₂²×C₃₁	1	1	yes	characteristic
271	#339	C₂²×C₃₁	1	2	yes	—
272	C₂×C₃₁	—	2	1	no	—
273	C₂×C₃₁	—	2	1	no	—
274	C₂×C₃₁	—	8	1	no	—
275	C₂×C₃₁	—	8	4	no	—
276	C₂×C₃₁	—	16	2	no	—
277	C₂×C₃₁	#2428	1	1	yes	characteristic
278	C₂×C₃₁	#3292	1	2	yes	—
279	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
280	C₂²×D₄	—	4	2	no	—
281	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
282	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
283	AGL(1,ℤ/8)	—	2	4	no	—
284	AGL(1,ℤ/8)	—	4	2	no	—
285	C₂×SD₁₆	—	2	2	no	—
286	C₂×D₈	—	2	4	no	—
287	C₂×D₈	—	2	2	no	—
288	C₂×D₈	—	2	1	no	—
289	C₂×D₈	—	2	1	no	—
290	C₂×D₈	—	2	4	no	—
291	C₂×D₈	—	2	1	no	—
292	C₂×D₈	—	2	1	no	—
293	C₂×D₈	—	2	2	no	—
294	C₂×D₈	—	2	2	no	—
295	C₂×D₈	—	2	2	no	—
296	C₂×D₈	—	4	2	no	—
297	C₂×D₈	D₄×C₃₁	1	4	yes	complemented
298	#131	—	2	2	no	—
299	#131	—	2	2	no	—
300	C₂×M₁₆	—	2	1	no	—
301	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
302	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
303	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
304	C₂²×C₈	—	2	1	no	—
305	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
306	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
307	#127	—	2	4	no	—
308	#127	—	2	2	no	—
309	#127	—	2	4	no	—
310	#124	—	2	2	no	—
311	#121	—	2	4	no	—
312	#120	—	4	2	no	—
313	C₄×D₄	—	2	4	no	—
314	C₄×D₄	—	2	4	no	—
315	#117	C₂³×C₃₁	1	1	yes	characteristic
316	#117	C₂³×C₃₁	1	2	yes	—
317	C₂×C₂²⋊C₄	—	4	2	no	—
318	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
319	#108	—	2	2	no	—
320	#107	D₄×C₃₁	1	2	yes	complemented
321	#106	D₄×C₃₁	1	2	yes	complemented
322	C₄⋊C₈	—	2	4	no	—
323	#104	—	2	2	no	—
324	#103	D₄×C₃₁	1	2	yes	complemented
325	#102	—	2	4	no	—
326	#102	—	2	4	no	—
327	#102	—	2	4	no	—
328	#102	D₄×C₃₁	1	2	yes	complemented
329	#100	—	2	4	no	—
330	#100	—	4	2	no	—
331	#97	—	2	1	no	—
332	C₄×C₈	—	2	2	no	—
333	C₄×C₈	—	2	1	no	—
334	C₃₁	#25434	1	1	yes	characteristic, direct product factor, Sylow
335	C₂⁴	—	2	2	no	—
336	C₂⁴	—	2	2	no	—
337	#41	—	2	1	no	—
338	#41	—	2	1	no	—
339	#41	—	2	1	no	—
340	#41	—	2	1	no	—
341	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
342	C₂×D₄	—	2	1	no	—
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	—	2	2	no	—
351	C₂×D₄	—	2	2	no	—
352	C₂×D₄	—	2	2	no	—
353	C₂×D₄	—	2	2	no	—
354	C₂×D₄	—	4	8	no	—
355	C₂×D₄	—	4	4	no	—
356	C₂×D₄	—	8	8	no	—
357	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
358	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
359	C₂²×C₄	—	2	1	no	—
360	C₂²×C₄	—	2	1	no	—
361	C₂²×C₄	—	4	4	no	—
362	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
363	SD₁₆	—	4	4	no	—
364	D₈	—	2	4	no	—
365	D₈	—	2	4	no	—
366	D₈	—	4	8	no	—
367	D₈	—	4	4	no	—
368	D₈	—	4	2	no	—
369	D₈	—	4	2	no	—
370	M₁₆	—	2	1	no	—
371	M₁₆	—	2	2	no	—
372	M₁₆	—	2	1	no	—
373	M₁₆	—	2	1	no	—
374	M₁₆	—	2	1	no	—
375	M₁₆	—	2	2	no	—
376	M₁₆	—	4	2	no	—
377	C₂×C₈	—	2	1	no	—
378	C₂×C₈	—	2	1	no	—
379	C₂×C₈	—	2	1	no	—
380	C₂×C₈	—	2	1	no	—
381	C₂×C₈	—	2	2	no	—
382	C₂×C₈	—	4	2	no	—
383	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
384	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
385	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
386	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
387	C₂×C₈	#41×C₃₁	1	2	yes	—
388	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
389	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
390	C₂²⋊C₄	—	2	1	no	—
391	C₂²⋊C₄	—	2	2	no	—
392	C₂²⋊C₄	—	4	4	no	—
393	C₂²⋊C₄	—	4	4	no	—
394	C₄²	—	2	1	no	—
395	C₄²	—	2	1	no	—
396	C₄²	—	2	2	no	—
397	C₄²	#41×C₃₁	1	2	yes	—
398	C₂³	—	2	1	no	—
399	C₂³	—	2	4	no	—
400	C₂³	—	4	2	no	—
401	C₂³	—	4	2	no	—
402	C₂³	—	4	2	no	—
403	C₂³	—	4	2	no	—
404	C₂³	—	4	2	no	—
405	C₂³	—	8	4	no	—
406	C₂³	—	8	4	no	—
407	C₂³	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
408	Q₈	—	2	1	no	—
409	Q₈	—	2	1	no	—
410	D₄	—	2	4	no	—
411	D₄	—	2	1	no	—
412	D₄	—	2	1	no	—
413	D₄	—	2	4	no	—
414	D₄	—	4	1	no	—
415	D₄	—	4	1	no	—
416	D₄	—	4	2	no	—
417	D₄	—	4	2	no	—
418	D₄	—	4	2	no	—
419	D₄	—	4	2	no	—
420	D₄	—	4	2	no	—
421	D₄	—	4	2	no	—
422	D₄	—	8	16	no	—
423	D₄	—	16	4	no	—
424	C₂×C₄	—	2	4	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	4	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	4	1	no	—
430	C₂×C₄	—	4	1	no	—
431	C₂×C₄	—	16	4	no	—
432	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
433	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
434	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
435	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
436	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
437	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
438	C₈	—	2	2	no	—
439	C₈	—	4	1	no	—
440	C₈	—	4	2	no	—
441	C₈	—	4	1	no	—
442	C₈	C₃₁×#121	1	2	yes	—
443	C₈	C₃₁×#121	1	2	yes	—
444	C₂²	—	2	2	no	—
445	C₂²	—	4	1	no	—
446	C₂²	—	4	4	no	—
447	C₂²	—	4	2	no	—
448	C₂²	—	8	4	no	—
449	C₂²	—	8	4	no	—
450	C₂²	—	8	4	no	—
451	C₂²	—	8	2	no	—
452	C₂²	—	8	4	no	—
453	C₂²	—	16	2	no	—
454	C₂²	—	16	2	no	—
455	C₂²	C₃₁×#540	1	1	yes	characteristic
456	C₂²	D₄²×C₃₁	1	1	yes	characteristic
457	C₂²	C₃₁×#546	1	1	yes	characteristic
458	C₄	—	2	4	no	—
459	C₄	—	4	1	no	—
460	C₄	—	4	2	no	—
461	C₄	—	4	4	no	—
462	C₄	C₂×C₃₁×#121	1	1	yes	characteristic
463	C₄	C₃₁×#532	1	1	yes	characteristic
464	C₄	D₄²×C₃₁	1	1	yes	characteristic
465	C₄	C₃₁×#550	1	1	yes	characteristic
466	C₂	—	2	1	no	—
467	C₂	—	2	1	no	—
468	C₂	—	8	1	no	—
469	C₂	—	8	4	no	—
470	C₂	—	16	2	no	—
471	C₂	C₃₁×#2428	1	1	yes	characteristic
472	C₂	C₃₁×#3292	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
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. C₂×C₃₁×#493	C₂	semidirect
2. C₂×C₃₁×#472	C₂	semidirect
3. C₃₁×#3058	C₂	semidirect
4. C₃₁×#3041	C₂	semidirect
5. C₃₁×#2948	C₂	semidirect
6. C₃₁×#2045	C₂	semidirect
7. C₃₁×#2002	C₂	semidirect
8. C₃₁×#1978	C₂	semidirect
9. C₃₁×#1945	C₂	semidirect
10. C₃₁×#1887	C₂	semidirect
11. C₃₁×#1764	C₂	semidirect
61. C₂×D₈×C₃₁	D₄	semidirect
84. C₃₁×#107	D₄	semidirect
85. C₃₁×#106	D₄	semidirect
88. C₃₁×#103	D₄	semidirect
92. C₃₁×#102	D₄	semidirect
161. #25434	C₃₁	direct (N × H)
208. C₂×#493	C₂×C₃₁	semidirect
209. C₂×#472	C₂×C₃₁	semidirect
210. #3058	C₂×C₃₁	semidirect
211. #3041	C₂×C₃₁	semidirect
212. #2948	C₂×C₃₁	semidirect
213. #2045	C₂×C₃₁	semidirect
214. #2002	C₂×C₃₁	semidirect
215. #1978	C₂×C₃₁	semidirect
216. #1945	C₂×C₃₁	semidirect
217. #1887	C₂×C₃₁	semidirect
218. #1764	C₂×C₃₁	semidirect
297. C₂×D₈	D₄×C₃₁	semidirect
320. #107	D₄×C₃₁	semidirect
321. #106	D₄×C₃₁	semidirect
324. #103	D₄×C₃₁	semidirect
328. #102	D₄×C₃₁	semidirect
334. C₃₁	#25434	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×#493	2
C₂×C₃₁×#472	2
C₃₁×#3058	2
C₃₁×#3041	2
C₃₁×#2948	2
C₃₁×#2045	2
C₃₁×#2002	2
C₃₁×#1978	2
C₃₁×#1945	2
C₃₁×#1887	2
C₃₁×#1764	2
#25434	31

Subgroups — C31×#25434 #21482193

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#25434 #21482193