Subgroups — C₃₁×#24288 #21481047

Order 7936.

1290 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3584	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3067	C₂	1	2	yes	maximal, complemented
3	C₃₁×#3018	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2932	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2050	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2017	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1975	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1813	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1791	C₂	1	2	yes	maximal, complemented
10	C₃₁×#585	C₂²	1	2	yes	—
11	C₃₁×#578	—	2	2	no	—
12	C₃₁×#577	—	2	4	no	—
13	C₃₁×#576	—	2	2	no	—
14	C₃₁×#575	—	2	2	no	—
15	C₃₁×#575	C₂²	1	2	yes	—
16	C₃₁×#575	C₂²	1	4	yes	complemented
17	C₂×C₃₁×#135	—	2	2	no	—
18	C₂×C₃₁×#135	C₂²	1	2	yes	—
19	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic, complemented
21	C₃₁×#532	C₂²	1	1	yes	characteristic
22	C₃₁×#473	—	2	2	no	—
23	C₃₁×#472	—	2	2	no	—
24	C₃₁×#471	—	2	4	no	—
25	C₃₁×#456	—	2	2	no	—
26	C₃₁×#455	—	2	2	no	—
27	C₃₁×#454	—	2	2	no	—
28	C₃₁×#453	—	2	2	no	—
29	C₃₁×#435	—	2	2	no	—
30	C₃₁×#435	—	2	2	no	—
31	C₂×C₃₁×#108	C₂²	1	2	yes	complemented
32	C₃₁×#423	C₂²	1	1	yes	characteristic, complemented
33	C₂×C₃₁×#104	C₂²	1	4	yes	complemented
34	C₃₁×#413	C₂²	1	2	yes	—
35	C₃₁×#413	C₂²	1	4	yes	complemented
36	C₃₁×#408	C₂²	1	1	yes	characteristic
37	C₃₁×#408	C₂²	1	1	yes	characteristic, complemented
38	C₃₁×#408	C₂²	1	4	yes	complemented
39	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic, complemented
40	C₃₁×#338	C₂²	1	2	yes	complemented
41	C₃₁×2₋¹⁺⁴	—	2	4	no	—
42	C₃₁×2₋¹⁺⁴	—	2	2	no	—
43	C₃₁×H(5,𝔽₂)	—	2	4	no	—
44	C₃₁×H(5,𝔽₂)	—	2	4	no	—
45	C₃₁×H(5,𝔽₂)	—	2	2	no	—
46	C₂×#41×C₃₁	—	2	4	no	—
47	C₂×#41×C₃₁	—	4	4	no	—
48	C₂×#41×C₃₁	C₂³	1	4	yes	—
49	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
50	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
51	C₃₁×#137	—	2	8	no	—
52	C₃₁×#137	—	2	4	no	—
53	C₃₁×#137	—	2	4	no	—
54	C₃₁×#137	—	2	4	no	—
55	C₃₁×AGL(1,ℤ/8)	—	2	8	no	—
56	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
57	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
58	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
59	C₃₁×#135	—	2	4	no	—
60	C₃₁×#135	—	2	4	no	—
61	C₃₁×#135	—	2	2	no	—
62	C₃₁×#135	—	2	4	no	—
63	C₃₁×#135	—	2	2	no	—
64	C₃₁×#135	—	2	2	no	—
65	C₃₁×#135	—	2	2	no	—
66	C₃₁×#135	—	4	4	no	—
67	C₂×Q₁₆×C₃₁	—	2	2	no	—
68	C₂×Q₁₆×C₃₁	D₄	1	2	yes	complemented
69	C₂×SD₁₆×C₃₁	—	2	4	no	—
70	C₂×SD₁₆×C₃₁	D₄	1	4	yes	complemented
71	C₂×D₈×C₃₁	—	2	2	no	—
72	C₂×D₈×C₃₁	D₄	1	2	yes	complemented
73	C₃₁×#131	—	2	2	no	—
74	C₃₁×#131	—	2	2	no	—
75	C₃₁×#131	—	2	2	no	—
76	C₃₁×#131	—	2	2	no	—
77	C₂×M₁₆×C₃₁	—	2	1	no	—
78	C₂×M₁₆×C₃₁	D₄	1	2	yes	—
79	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
80	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
81	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
82	C₂²×C₈×C₃₁	—	2	1	no	—
83	C₂²×C₈×C₃₁	D₄	1	2	yes	—
84	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
85	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
86	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
87	C₃₁×#127	D₄	1	1	yes	characteristic
88	C₃₁×#124	D₄	1	1	yes	characteristic
89	C₃₁×#124	D₄	1	1	yes	characteristic
90	C₃₁×#121	—	2	2	no	—
91	C₄×D₄×C₃₁	—	2	2	no	—
92	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
93	C₃₁×#108	—	2	2	no	—
94	C₃₁×#108	—	2	2	no	—
95	C₃₁×C₄⋊C₈	—	2	2	no	—
96	C₃₁×#104	—	2	4	no	—
97	C₃₁×#104	—	2	4	no	—
98	C₃₁×#101	—	2	2	no	—
99	C₃₁×#101	—	2	4	no	—
100	C₃₁×#100	—	2	2	no	—
101	C₃₁×#100	—	2	4	no	—
102	C₃₁×C₂²⋊C₈	—	2	2	no	—
103	C₃₁×C₂²⋊C₈	—	2	2	no	—
104	C₃₁×C₂²⋊C₈	—	4	4	no	—
105	C₃₁×#97	—	2	1	no	—
106	C₃₁×#97	—	2	1	no	—
107	#41×C₃₁	—	2	2	no	—
108	#41×C₃₁	—	2	2	no	—
109	#41×C₃₁	—	2	8	no	—
110	#41×C₃₁	—	2	4	no	—
111	#41×C₃₁	—	2	2	no	—
112	#41×C₃₁	—	2	4	no	—
113	#41×C₃₁	—	2	4	no	—
114	#41×C₃₁	—	2	2	no	—
115	#41×C₃₁	—	2	4	no	—
116	#41×C₃₁	—	4	8	no	—
117	#41×C₃₁	—	4	4	no	—
118	#41×C₃₁	—	4	4	no	—
119	#41×C₃₁	—	4	4	no	—
120	#41×C₃₁	—	4	4	no	—
121	C₂×Q₈×C₃₁	—	2	4	no	—
122	C₂×Q₈×C₃₁	—	4	4	no	—
123	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
124	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
125	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
126	C₂×D₄×C₃₁	—	2	4	no	—
127	C₂×D₄×C₃₁	—	2	1	no	—
128	C₂×D₄×C₃₁	—	2	4	no	—
129	C₂×D₄×C₃₁	—	2	2	no	—
130	C₂×D₄×C₃₁	—	4	4	no	—
131	C₂×D₄×C₃₁	—	4	8	no	—
132	C₂×D₄×C₃₁	—	4	4	no	—
133	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
134	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
136	C₂×D₄×C₃₁	#41	1	2	yes	—
137	C₂²×C₄×C₃₁	—	2	1	no	—
138	C₂²×C₄×C₃₁	—	2	1	no	—
139	C₂²×C₄×C₃₁	—	2	4	no	—
140	C₂²×C₄×C₃₁	—	4	4	no	—
141	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
142	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
143	Q₁₆×C₃₁	—	2	4	no	—
144	Q₁₆×C₃₁	—	2	2	no	—
145	Q₁₆×C₃₁	—	2	2	no	—
146	Q₁₆×C₃₁	—	4	4	no	—
147	SD₁₆×C₃₁	—	2	8	no	—
148	SD₁₆×C₃₁	—	2	4	no	—
149	SD₁₆×C₃₁	—	2	4	no	—
150	SD₁₆×C₃₁	—	4	4	no	—
151	SD₁₆×C₃₁	—	4	4	no	—
152	D₈×C₃₁	—	2	4	no	—
153	D₈×C₃₁	—	2	2	no	—
154	D₈×C₃₁	—	2	2	no	—
155	D₈×C₃₁	—	4	4	no	—
156	M₁₆×C₃₁	—	2	2	no	—
157	M₁₆×C₃₁	—	2	4	no	—
158	M₁₆×C₃₁	—	2	2	no	—
159	M₁₆×C₃₁	—	2	4	no	—
160	M₁₆×C₃₁	—	2	4	no	—
161	M₁₆×C₃₁	—	4	1	no	—
162	M₁₆×C₃₁	—	4	1	no	—
163	C₂×C₈×C₃₁	—	2	2	no	—
164	C₂×C₈×C₃₁	—	2	2	no	—
165	C₂×C₈×C₃₁	—	2	1	no	—
166	C₂×C₈×C₃₁	—	2	1	no	—
167	C₂×C₈×C₃₁	—	2	2	no	—
168	C₂×C₈×C₃₁	—	2	4	no	—
169	C₂×C₈×C₃₁	—	4	2	no	—
170	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
171	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
172	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
173	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
174	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
175	C₄⋊C₄×C₃₁	—	2	2	no	—
176	C₂²⋊C₄×C₃₁	—	2	2	no	—
177	C₂²⋊C₄×C₃₁	—	2	2	no	—
178	C₄²×C₃₁	—	2	1	no	—
179	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
180	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
181	#24288	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
182	C₂³×C₃₁	—	2	1	no	—
183	C₂³×C₃₁	—	2	4	no	—
184	C₂³×C₃₁	—	4	4	no	—
185	C₂³×C₃₁	#121	1	2	yes	—
186	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
187	Q₈×C₃₁	—	2	2	no	—
188	Q₈×C₃₁	—	2	4	no	—
189	Q₈×C₃₁	—	2	2	no	—
190	Q₈×C₃₁	—	2	4	no	—
191	Q₈×C₃₁	—	4	4	no	—
192	D₄×C₃₁	—	2	4	no	—
193	D₄×C₃₁	—	2	4	no	—
194	D₄×C₃₁	—	2	2	no	—
195	D₄×C₃₁	—	2	2	no	—
196	D₄×C₃₁	—	2	4	no	—
197	D₄×C₃₁	—	4	4	no	—
198	D₄×C₃₁	—	4	8	no	—
199	D₄×C₃₁	—	4	4	no	—
200	D₄×C₃₁	—	4	4	no	—
201	D₄×C₃₁	—	4	2	no	—
202	D₄×C₃₁	—	8	2	no	—
203	C₂×C₄×C₃₁	—	2	2	no	—
204	C₂×C₄×C₃₁	—	2	2	no	—
205	C₂×C₄×C₃₁	—	2	1	no	—
206	C₂×C₄×C₃₁	—	2	1	no	—
207	C₂×C₄×C₃₁	—	2	1	no	—
208	C₂×C₄×C₃₁	—	2	4	no	—
209	C₂×C₄×C₃₁	—	4	8	no	—
210	C₂×C₄×C₃₁	—	4	4	no	—
211	C₂×C₄×C₃₁	—	4	1	no	—
212	C₂×C₄×C₃₁	—	4	4	no	—
213	C₂×C₄×C₃₁	—	4	1	no	—
214	C₂×C₄×C₃₁	—	4	4	no	—
215	C₂×C₄×C₃₁	—	8	1	no	—
216	C₂×C₄×C₃₁	#121	1	2	yes	—
217	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
218	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
219	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
220	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
221	C₈×C₃₁	—	2	2	no	—
222	C₈×C₃₁	—	4	1	no	—
223	C₈×C₃₁	—	4	1	no	—
224	C₈×C₃₁	—	4	4	no	—
225	C₈×C₃₁	#120	1	1	yes	characteristic
226	C₈×C₃₁	#120	1	1	yes	characteristic
227	C₈×C₃₁	#120	1	2	yes	—
228	#3584	C₂×C₃₁	1	1	yes	characteristic, complemented
229	#3067	C₂×C₃₁	1	2	yes	complemented
230	#3018	C₂×C₃₁	1	2	yes	complemented
231	#2932	C₂×C₃₁	1	1	yes	characteristic, complemented
232	#2050	C₂×C₃₁	1	2	yes	complemented
233	#2017	C₂×C₃₁	1	2	yes	complemented
234	#1975	C₂×C₃₁	1	2	yes	complemented
235	#1813	C₂×C₃₁	1	1	yes	characteristic, complemented
236	#1791	C₂×C₃₁	1	2	yes	complemented
237	C₂²×C₃₁	—	2	2	no	—
238	C₂²×C₃₁	—	4	1	no	—
239	C₂²×C₃₁	—	4	2	no	—
240	C₂²×C₃₁	—	4	2	no	—
241	C₂²×C₃₁	—	4	4	no	—
242	C₂²×C₃₁	—	4	1	no	—
243	C₂²×C₃₁	—	8	4	no	—
244	C₂²×C₃₁	—	16	4	no	—
245	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
246	C₂²×C₃₁	#546	1	2	yes	—
247	C₄×C₃₁	—	2	2	no	—
248	C₄×C₃₁	—	4	1	no	—
249	C₄×C₃₁	—	4	1	no	—
250	C₄×C₃₁	—	4	4	no	—
251	C₄×C₃₁	—	4	1	no	—
252	C₄×C₃₁	C₂×#120	1	1	yes	characteristic
253	C₄×C₃₁	#535	1	1	yes	characteristic
254	C₄×C₃₁	D₄²	1	2	yes	—
255	#585	C₂²×C₃₁	1	2	yes	—
256	#578	—	2	2	no	—
257	#577	—	2	4	no	—
258	#576	—	2	2	no	—
259	#575	—	2	2	no	—
260	#575	C₂²×C₃₁	1	2	yes	—
261	#575	C₂²×C₃₁	1	4	yes	complemented
262	C₂×#135	—	2	2	no	—
263	C₂×#135	C₂²×C₃₁	1	2	yes	—
264	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
265	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic, complemented
266	#532	C₂²×C₃₁	1	1	yes	characteristic
267	#473	—	2	2	no	—
268	#472	—	2	2	no	—
269	#471	—	2	4	no	—
270	#456	—	2	2	no	—
271	#455	—	2	2	no	—
272	#454	—	2	2	no	—
273	#453	—	2	2	no	—
274	#435	—	2	2	no	—
275	#435	—	2	2	no	—
276	C₂×#108	C₂²×C₃₁	1	2	yes	complemented
277	#423	C₂²×C₃₁	1	1	yes	characteristic, complemented
278	C₂×#104	C₂²×C₃₁	1	4	yes	complemented
279	#413	C₂²×C₃₁	1	2	yes	—
280	#413	C₂²×C₃₁	1	4	yes	complemented
281	#408	C₂²×C₃₁	1	1	yes	characteristic
282	#408	C₂²×C₃₁	1	1	yes	characteristic, complemented
283	#408	C₂²×C₃₁	1	4	yes	complemented
284	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic, complemented
285	#338	C₂²×C₃₁	1	2	yes	complemented
286	C₂×C₃₁	—	2	1	no	—
287	C₂×C₃₁	—	2	2	no	—
288	C₂×C₃₁	—	4	2	no	—
289	C₂×C₃₁	—	8	1	no	—
290	C₂×C₃₁	—	8	4	no	—
291	C₂×C₃₁	#2408	1	1	yes	characteristic
292	2₋¹⁺⁴	—	2	4	no	—
293	2₋¹⁺⁴	—	2	2	no	—
294	H(5,𝔽₂)	—	2	4	no	—
295	H(5,𝔽₂)	—	2	4	no	—
296	H(5,𝔽₂)	—	2	2	no	—
297	C₂×#41	—	2	4	no	—
298	C₂×#41	—	4	4	no	—
299	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
300	C₂×#41	C₂³×C₃₁	1	4	yes	—
301	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
302	#137	—	2	8	no	—
303	#137	—	2	4	no	—
304	#137	—	2	4	no	—
305	#137	—	2	4	no	—
306	AGL(1,ℤ/8)	—	2	8	no	—
307	AGL(1,ℤ/8)	—	2	4	no	—
308	AGL(1,ℤ/8)	—	2	4	no	—
309	AGL(1,ℤ/8)	—	2	4	no	—
310	#135	—	2	4	no	—
311	#135	—	2	4	no	—
312	#135	—	2	2	no	—
313	#135	—	2	4	no	—
314	#135	—	2	2	no	—
315	#135	—	2	2	no	—
316	#135	—	2	2	no	—
317	#135	—	4	4	no	—
318	C₂×Q₁₆	—	2	2	no	—
319	C₂×Q₁₆	D₄×C₃₁	1	2	yes	complemented
320	C₂×SD₁₆	—	2	4	no	—
321	C₂×SD₁₆	D₄×C₃₁	1	4	yes	complemented
322	C₂×D₈	—	2	2	no	—
323	C₂×D₈	D₄×C₃₁	1	2	yes	complemented
324	#131	—	2	2	no	—
325	#131	—	2	2	no	—
326	#131	—	2	2	no	—
327	#131	—	2	2	no	—
328	C₂×M₁₆	—	2	1	no	—
329	C₂×M₁₆	D₄×C₃₁	1	2	yes	—
330	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
331	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
332	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
333	C₂²×C₈	—	2	1	no	—
334	C₂²×C₈	D₄×C₃₁	1	2	yes	—
335	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
336	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
337	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
338	#127	D₄×C₃₁	1	1	yes	characteristic
339	#124	D₄×C₃₁	1	1	yes	characteristic
340	#124	D₄×C₃₁	1	1	yes	characteristic
341	#121	—	2	2	no	—
342	C₄×D₄	—	2	2	no	—
343	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
344	#108	—	2	2	no	—
345	#108	—	2	2	no	—
346	C₄⋊C₈	—	2	2	no	—
347	#104	—	2	4	no	—
348	#104	—	2	4	no	—
349	#101	—	2	2	no	—
350	#101	—	2	4	no	—
351	#100	—	2	2	no	—
352	#100	—	2	4	no	—
353	C₂²⋊C₈	—	2	2	no	—
354	C₂²⋊C₈	—	2	2	no	—
355	C₂²⋊C₈	—	4	4	no	—
356	#97	—	2	1	no	—
357	#97	—	2	1	no	—
358	C₃₁	#24288	1	1	yes	characteristic, direct product factor, Sylow
359	#41	—	2	2	no	—
360	#41	—	2	2	no	—
361	#41	—	2	8	no	—
362	#41	—	2	4	no	—
363	#41	—	2	2	no	—
364	#41	—	2	4	no	—
365	#41	—	2	4	no	—
366	#41	—	2	2	no	—
367	#41	—	2	4	no	—
368	#41	—	4	8	no	—
369	#41	—	4	4	no	—
370	#41	—	4	4	no	—
371	#41	—	4	4	no	—
372	#41	—	4	4	no	—
373	C₂×Q₈	—	2	4	no	—
374	C₂×Q₈	—	4	4	no	—
375	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
376	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
377	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
378	C₂×D₄	—	2	4	no	—
379	C₂×D₄	—	2	1	no	—
380	C₂×D₄	—	2	4	no	—
381	C₂×D₄	—	2	2	no	—
382	C₂×D₄	—	4	4	no	—
383	C₂×D₄	—	4	8	no	—
384	C₂×D₄	—	4	4	no	—
385	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
386	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
387	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
388	C₂×D₄	#41×C₃₁	1	2	yes	—
389	C₂²×C₄	—	2	1	no	—
390	C₂²×C₄	—	2	1	no	—
391	C₂²×C₄	—	2	4	no	—
392	C₂²×C₄	—	4	4	no	—
393	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
394	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
395	Q₁₆	—	2	4	no	—
396	Q₁₆	—	2	2	no	—
397	Q₁₆	—	2	2	no	—
398	Q₁₆	—	4	4	no	—
399	SD₁₆	—	2	8	no	—
400	SD₁₆	—	2	4	no	—
401	SD₁₆	—	2	4	no	—
402	SD₁₆	—	4	4	no	—
403	SD₁₆	—	4	4	no	—
404	D₈	—	2	4	no	—
405	D₈	—	2	2	no	—
406	D₈	—	2	2	no	—
407	D₈	—	4	4	no	—
408	M₁₆	—	2	2	no	—
409	M₁₆	—	2	4	no	—
410	M₁₆	—	2	2	no	—
411	M₁₆	—	2	4	no	—
412	M₁₆	—	2	4	no	—
413	M₁₆	—	4	1	no	—
414	M₁₆	—	4	1	no	—
415	C₂×C₈	—	2	2	no	—
416	C₂×C₈	—	2	2	no	—
417	C₂×C₈	—	2	1	no	—
418	C₂×C₈	—	2	1	no	—
419	C₂×C₈	—	2	2	no	—
420	C₂×C₈	—	2	4	no	—
421	C₂×C₈	—	4	2	no	—
422	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
423	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
424	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
425	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
426	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
427	C₄⋊C₄	—	2	2	no	—
428	C₂²⋊C₄	—	2	2	no	—
429	C₂²⋊C₄	—	2	2	no	—
430	C₄²	—	2	1	no	—
431	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
432	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
433	C₂³	—	2	1	no	—
434	C₂³	—	2	4	no	—
435	C₂³	—	4	4	no	—
436	C₂³	C₃₁×#121	1	2	yes	—
437	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
438	Q₈	—	2	2	no	—
439	Q₈	—	2	4	no	—
440	Q₈	—	2	2	no	—
441	Q₈	—	2	4	no	—
442	Q₈	—	4	4	no	—
443	D₄	—	2	4	no	—
444	D₄	—	2	4	no	—
445	D₄	—	2	2	no	—
446	D₄	—	2	2	no	—
447	D₄	—	2	4	no	—
448	D₄	—	4	4	no	—
449	D₄	—	4	8	no	—
450	D₄	—	4	4	no	—
451	D₄	—	4	4	no	—
452	D₄	—	4	2	no	—
453	D₄	—	8	2	no	—
454	C₂×C₄	—	2	2	no	—
455	C₂×C₄	—	2	2	no	—
456	C₂×C₄	—	2	1	no	—
457	C₂×C₄	—	2	1	no	—
458	C₂×C₄	—	2	1	no	—
459	C₂×C₄	—	2	4	no	—
460	C₂×C₄	—	4	8	no	—
461	C₂×C₄	—	4	4	no	—
462	C₂×C₄	—	4	1	no	—
463	C₂×C₄	—	4	4	no	—
464	C₂×C₄	—	4	1	no	—
465	C₂×C₄	—	4	4	no	—
466	C₂×C₄	—	8	1	no	—
467	C₂×C₄	C₃₁×#121	1	2	yes	—
468	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
469	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
470	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
471	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
472	C₈	—	2	2	no	—
473	C₈	—	4	1	no	—
474	C₈	—	4	1	no	—
475	C₈	—	4	4	no	—
476	C₈	C₃₁×#120	1	1	yes	characteristic
477	C₈	C₃₁×#120	1	1	yes	characteristic
478	C₈	C₃₁×#120	1	2	yes	—
479	C₂²	—	2	2	no	—
480	C₂²	—	4	1	no	—
481	C₂²	—	4	2	no	—
482	C₂²	—	4	2	no	—
483	C₂²	—	4	4	no	—
484	C₂²	—	4	1	no	—
485	C₂²	—	8	4	no	—
486	C₂²	—	16	4	no	—
487	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
488	C₂²	C₃₁×#546	1	2	yes	—
489	C₄	—	2	2	no	—
490	C₄	—	4	1	no	—
491	C₄	—	4	1	no	—
492	C₄	—	4	4	no	—
493	C₄	—	4	1	no	—
494	C₄	C₂×C₃₁×#120	1	1	yes	characteristic
495	C₄	C₃₁×#535	1	1	yes	characteristic
496	C₄	D₄²×C₃₁	1	2	yes	—
497	C₂	—	2	1	no	—
498	C₂	—	2	2	no	—
499	C₂	—	4	2	no	—
500	C₂	—	8	1	no	—
501	C₂	—	8	4	no	—
502	C₂	C₃₁×#2408	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₃₁	#535
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂×C₃₁	#2408
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3584	C₂	semidirect
2. C₃₁×#3067	C₂	semidirect
3. C₃₁×#3018	C₂	semidirect
4. C₃₁×#2932	C₂	semidirect
5. C₃₁×#2050	C₂	semidirect
6. C₃₁×#2017	C₂	semidirect
7. C₃₁×#1975	C₂	semidirect
8. C₃₁×#1813	C₂	semidirect
9. C₃₁×#1791	C₂	semidirect
16. C₃₁×#575	C₂²	semidirect
20. C₂×C₃₁×#131	C₂²	semidirect
31. C₂×C₃₁×#108	C₂²	semidirect
32. C₃₁×#423	C₂²	semidirect
33. C₂×C₃₁×#104	C₂²	semidirect
35. C₃₁×#413	C₂²	semidirect
37. C₃₁×#408	C₂²	semidirect
38. C₃₁×#408	C₂²	semidirect
39. C₂×C₃₁×#97	C₂²	semidirect
40. C₃₁×#338	C₂²	semidirect
68. C₂×Q₁₆×C₃₁	D₄	semidirect
70. C₂×SD₁₆×C₃₁	D₄	semidirect
72. C₂×D₈×C₃₁	D₄	semidirect
181. #24288	C₃₁	direct (N × H)
228. #3584	C₂×C₃₁	semidirect
229. #3067	C₂×C₃₁	semidirect
230. #3018	C₂×C₃₁	semidirect
231. #2932	C₂×C₃₁	semidirect
232. #2050	C₂×C₃₁	semidirect
233. #2017	C₂×C₃₁	semidirect
234. #1975	C₂×C₃₁	semidirect
235. #1813	C₂×C₃₁	semidirect
236. #1791	C₂×C₃₁	semidirect
261. #575	C₂²×C₃₁	semidirect
265. C₂×#131	C₂²×C₃₁	semidirect
276. C₂×#108	C₂²×C₃₁	semidirect
277. #423	C₂²×C₃₁	semidirect
278. C₂×#104	C₂²×C₃₁	semidirect
280. #413	C₂²×C₃₁	semidirect
282. #408	C₂²×C₃₁	semidirect
283. #408	C₂²×C₃₁	semidirect
284. C₂×#97	C₂²×C₃₁	semidirect
285. #338	C₂²×C₃₁	semidirect
319. C₂×Q₁₆	D₄×C₃₁	semidirect
321. C₂×SD₁₆	D₄×C₃₁	semidirect
323. C₂×D₈	D₄×C₃₁	semidirect
358. C₃₁	#24288	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×#3584	2
C₃₁×#3067	2
C₃₁×#3018	2
C₃₁×#2932	2
C₃₁×#2050	2
C₃₁×#2017	2
C₃₁×#1975	2
C₃₁×#1813	2
C₃₁×#1791	2
#24288	31

Subgroups — C31×#24288 #21481047

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24288 #21481047