Subgroups — C₃₁×#24281 #21481040

Order 7936.

1564 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#472	C₂	1	2	yes	maximal, complemented
3	C₃₁×#3003	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2929	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2045	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2018	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1978	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1810	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1783	C₂	1	2	yes	maximal, complemented
10	C₂³×D₄×C₃₁	C₂²	1	2	yes	—
11	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	4	yes	complemented
12	C₂²×D₈×C₃₁	—	2	8	no	—
13	C₂²×D₈×C₃₁	—	2	2	no	—
14	C₂²×D₈×C₃₁	—	2	2	no	—
15	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
16	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
17	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic, complemented
18	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#525	C₂²	1	1	yes	characteristic
20	C₃₁×#472	—	2	8	no	—
21	C₃₁×#450	—	2	4	no	—
22	C₃₁×#447	—	2	4	no	—
23	C₃₁×#435	—	2	2	no	—
24	C₈×D₄×C₃₁	—	2	2	no	—
25	C₂×C₃₁×#108	C₂²	1	2	yes	complemented
26	C₃₁×#424	C₂²	1	1	yes	characteristic, complemented
27	C₃₁×#418	C₂²	1	4	yes	complemented
28	C₂×C₃₁×#100	C₂²	1	4	yes	complemented
29	C₂×C₃₁×#100	C₂²	1	4	yes	complemented
30	C₃₁×#408	C₂²	1	1	yes	characteristic, complemented
31	C₃₁×#407	C₂²	1	2	yes	—
32	C₃₁×#407	C₂²	1	1	yes	characteristic
33	C₃₁×#405	C₂²	1	1	yes	characteristic, complemented
34	C₃₁×#341	C₂²	1	2	yes	complemented
35	C₂⁵×C₃₁	—	2	2	no	—
36	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
37	C₂²×D₄×C₃₁	—	2	8	no	—
38	C₂²×D₄×C₃₁	—	2	4	no	—
39	C₂²×D₄×C₃₁	—	2	4	no	—
40	C₂²×D₄×C₃₁	—	2	2	no	—
41	C₂²×D₄×C₃₁	—	4	4	no	—
42	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
43	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
44	C₃₁×AGL(1,ℤ/8)	—	2	8	no	—
45	C₃₁×AGL(1,ℤ/8)	—	4	4	no	—
46	C₂×SD₁₆×C₃₁	—	2	4	no	—
47	C₂×D₈×C₃₁	—	2	16	no	—
48	C₂×D₈×C₃₁	—	2	16	no	—
49	C₂×D₈×C₃₁	—	2	8	no	—
50	C₂×D₈×C₃₁	—	2	4	no	—
51	C₂×D₈×C₃₁	—	2	4	no	—
52	C₂×D₈×C₃₁	—	2	2	no	—
53	C₂×D₈×C₃₁	—	2	2	no	—
54	C₂×D₈×C₃₁	—	2	4	no	—
55	C₂×D₈×C₃₁	D₄	1	8	yes	complemented
56	C₃₁×#131	—	2	4	no	—
57	C₂×M₁₆×C₃₁	—	2	1	no	—
58	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
59	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
60	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
61	C₂²×C₈×C₃₁	—	2	4	no	—
62	C₂²×C₈×C₃₁	—	2	1	no	—
63	C₂²×C₈×C₃₁	D₄	1	2	yes	—
64	C₂²×C₈×C₃₁	D₄	1	2	yes	—
65	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
66	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
67	C₃₁×#123	—	2	1	no	—
68	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
69	C₃₁×#121	D₄	1	2	yes	—
70	C₃₁×#120	—	2	1	no	—
71	C₄×D₄×C₃₁	—	2	2	no	—
72	C₃₁×#117	C₂³	1	1	yes	characteristic
73	C₃₁×#108	—	2	4	no	—
74	C₃₁×C₄⋊C₈	—	2	2	no	—
75	C₃₁×#102	—	2	8	no	—
76	C₃₁×#100	—	2	8	no	—
77	C₃₁×#100	—	4	4	no	—
78	C₃₁×C₂²⋊C₈	—	2	2	no	—
79	C₃₁×C₂²⋊C₈	—	2	4	no	—
80	C₃₁×C₂²⋊C₈	—	2	2	no	—
81	C₃₁×#97	—	2	1	no	—
82	C₄×C₈×C₃₁	—	2	1	no	—
83	C₂⁴×C₃₁	—	2	4	no	—
84	C₂⁴×C₃₁	—	4	8	no	—
85	C₂⁴×C₃₁	—	4	4	no	—
86	C₂⁴×C₃₁	—	8	8	no	—
87	C₂⁴×C₃₁	#41	1	1	yes	characteristic
88	#41×C₃₁	—	2	2	no	—
89	#41×C₃₁	—	2	2	no	—
90	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
91	C₂×D₄×C₃₁	—	2	16	no	—
92	C₂×D₄×C₃₁	—	2	2	no	—
93	C₂×D₄×C₃₁	—	2	1	no	—
94	C₂×D₄×C₃₁	—	2	4	no	—
95	C₂×D₄×C₃₁	—	2	4	no	—
96	C₂×D₄×C₃₁	—	2	4	no	—
97	C₂×D₄×C₃₁	—	2	4	no	—
98	C₂×D₄×C₃₁	—	2	4	no	—
99	C₂×D₄×C₃₁	—	4	16	no	—
100	C₂×D₄×C₃₁	—	4	8	no	—
101	C₂×D₄×C₃₁	—	4	8	no	—
102	C₂×D₄×C₃₁	—	4	4	no	—
103	C₂×D₄×C₃₁	C₂×D₄	1	8	yes	—
104	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₃₁	—	2	2	no	—
106	C₂²×C₄×C₃₁	—	2	2	no	—
107	C₂²×C₄×C₃₁	—	2	1	no	—
108	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
109	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
110	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
111	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
112	SD₁₆×C₃₁	—	4	8	no	—
113	D₈×C₃₁	—	2	16	no	—
114	D₈×C₃₁	—	2	8	no	—
115	D₈×C₃₁	—	2	8	no	—
116	D₈×C₃₁	—	4	8	no	—
117	M₁₆×C₃₁	—	2	2	no	—
118	M₁₆×C₃₁	—	2	4	no	—
119	M₁₆×C₃₁	—	2	2	no	—
120	M₁₆×C₃₁	—	2	4	no	—
121	M₁₆×C₃₁	—	4	2	no	—
122	C₂×C₈×C₃₁	—	2	2	no	—
123	C₂×C₈×C₃₁	—	2	4	no	—
124	C₂×C₈×C₃₁	—	2	4	no	—
125	C₂×C₈×C₃₁	—	2	1	no	—
126	C₂×C₈×C₃₁	—	2	4	no	—
127	C₂×C₈×C₃₁	—	2	1	no	—
128	C₂×C₈×C₃₁	—	4	2	no	—
129	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
130	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
132	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
133	C₄⋊C₄×C₃₁	—	2	2	no	—
134	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
135	C₂²⋊C₄×C₃₁	—	2	2	no	—
136	C₂²⋊C₄×C₃₁	—	2	1	no	—
137	C₂²⋊C₄×C₃₁	—	2	2	no	—
138	C₄²×C₃₁	—	2	1	no	—
139	#24281	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
140	C₂³×C₃₁	—	2	2	no	—
141	C₂³×C₃₁	—	2	4	no	—
142	C₂³×C₃₁	—	2	1	no	—
143	C₂³×C₃₁	—	2	8	no	—
144	C₂³×C₃₁	—	4	16	no	—
145	C₂³×C₃₁	—	4	4	no	—
146	C₂³×C₃₁	—	4	4	no	—
147	C₂³×C₃₁	—	8	8	no	—
148	C₂³×C₃₁	—	8	32	no	—
149	C₂³×C₃₁	—	8	8	no	—
150	C₂³×C₃₁	—	8	8	no	—
151	C₂³×C₃₁	#121	1	2	yes	—
152	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
153	Q₈×C₃₁	—	2	2	no	—
154	D₄×C₃₁	—	2	8	no	—
155	D₄×C₃₁	—	2	2	no	—
156	D₄×C₃₁	—	2	8	no	—
157	D₄×C₃₁	—	4	16	no	—
158	D₄×C₃₁	—	4	2	no	—
159	D₄×C₃₁	—	4	4	no	—
160	D₄×C₃₁	—	4	4	no	—
161	D₄×C₃₁	—	4	4	no	—
162	C₂×C₄×C₃₁	—	2	2	no	—
163	C₂×C₄×C₃₁	—	2	2	no	—
164	C₂×C₄×C₃₁	—	2	2	no	—
165	C₂×C₄×C₃₁	—	2	4	no	—
166	C₂×C₄×C₃₁	—	2	2	no	—
167	C₂×C₄×C₃₁	—	2	1	no	—
168	C₂×C₄×C₃₁	—	4	1	no	—
169	C₂×C₄×C₃₁	—	4	1	no	—
170	C₂×C₄×C₃₁	#121	1	2	yes	—
171	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
173	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
174	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
175	C₈×C₃₁	—	2	2	no	—
176	C₈×C₃₁	—	4	1	no	—
177	C₈×C₃₁	—	4	1	no	—
178	C₈×C₃₁	—	4	4	no	—
179	C₈×C₃₁	#120	1	2	yes	—
180	C₈×C₃₁	#120	1	2	yes	—
181	C₂³×D₈	C₂×C₃₁	1	1	yes	characteristic, complemented
182	C₂×#472	C₂×C₃₁	1	2	yes	complemented
183	#3003	C₂×C₃₁	1	2	yes	complemented
184	#2929	C₂×C₃₁	1	1	yes	characteristic, complemented
185	#2045	C₂×C₃₁	1	2	yes	complemented
186	#2018	C₂×C₃₁	1	2	yes	complemented
187	#1978	C₂×C₃₁	1	2	yes	complemented
188	#1810	C₂×C₃₁	1	1	yes	characteristic, complemented
189	#1783	C₂×C₃₁	1	2	yes	complemented
190	C₂²×C₃₁	—	2	8	no	—
191	C₂²×C₃₁	—	2	2	no	—
192	C₂²×C₃₁	—	2	2	no	—
193	C₂²×C₃₁	—	2	2	no	—
194	C₂²×C₃₁	—	4	1	no	—
195	C₂²×C₃₁	—	4	8	no	—
196	C₂²×C₃₁	—	4	2	no	—
197	C₂²×C₃₁	—	8	8	no	—
198	C₂²×C₃₁	—	8	8	no	—
199	C₂²×C₃₁	—	8	32	no	—
200	C₂²×C₃₁	—	8	8	no	—
201	C₂²×C₃₁	#472	1	4	yes	complemented
202	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
203	C₂²×C₃₁	#546	1	2	yes	—
204	C₄×C₃₁	—	2	2	no	—
205	C₄×C₃₁	—	4	1	no	—
206	C₄×C₃₁	—	4	2	no	—
207	C₄×C₃₁	C₂×#120	1	1	yes	characteristic
208	C₄×C₃₁	#535	1	1	yes	characteristic
209	C₄×C₃₁	D₄²	1	2	yes	—
210	C₂³×D₄	C₂²×C₃₁	1	2	yes	—
211	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	4	yes	complemented
212	C₂²×D₈	—	2	8	no	—
213	C₂²×D₈	—	2	2	no	—
214	C₂²×D₈	—	2	2	no	—
215	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
216	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
217	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic, complemented
218	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
219	#525	C₂²×C₃₁	1	1	yes	characteristic
220	#472	—	2	8	no	—
221	#450	—	2	4	no	—
222	#447	—	2	4	no	—
223	#435	—	2	2	no	—
224	C₈×D₄	—	2	2	no	—
225	C₂×#108	C₂²×C₃₁	1	2	yes	complemented
226	#424	C₂²×C₃₁	1	1	yes	characteristic, complemented
227	#418	C₂²×C₃₁	1	4	yes	complemented
228	C₂×#100	C₂²×C₃₁	1	4	yes	complemented
229	C₂×#100	C₂²×C₃₁	1	4	yes	complemented
230	#408	C₂²×C₃₁	1	1	yes	characteristic, complemented
231	#407	C₂²×C₃₁	1	2	yes	—
232	#407	C₂²×C₃₁	1	1	yes	characteristic
233	#405	C₂²×C₃₁	1	1	yes	characteristic, complemented
234	#341	C₂²×C₃₁	1	2	yes	complemented
235	C₂×C₃₁	—	2	2	no	—
236	C₂×C₃₁	—	2	4	no	—
237	C₂×C₃₁	—	8	1	no	—
238	C₂×C₃₁	—	8	8	no	—
239	C₂×C₃₁	#2408	1	1	yes	characteristic
240	C₂×C₃₁	C₂×#472	1	2	yes	—
241	C₂⁵	—	2	2	no	—
242	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
243	C₂²×D₄	—	2	8	no	—
244	C₂²×D₄	—	2	4	no	—
245	C₂²×D₄	—	2	4	no	—
246	C₂²×D₄	—	2	2	no	—
247	C₂²×D₄	—	4	4	no	—
248	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
249	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
250	AGL(1,ℤ/8)	—	2	8	no	—
251	AGL(1,ℤ/8)	—	4	4	no	—
252	C₂×SD₁₆	—	2	4	no	—
253	C₂×D₈	—	2	16	no	—
254	C₂×D₈	—	2	16	no	—
255	C₂×D₈	—	2	8	no	—
256	C₂×D₈	—	2	4	no	—
257	C₂×D₈	—	2	4	no	—
258	C₂×D₈	—	2	2	no	—
259	C₂×D₈	—	2	2	no	—
260	C₂×D₈	—	2	4	no	—
261	C₂×D₈	D₄×C₃₁	1	8	yes	complemented
262	#131	—	2	4	no	—
263	C₂×M₁₆	—	2	1	no	—
264	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
265	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
266	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
267	C₂²×C₈	—	2	4	no	—
268	C₂²×C₈	—	2	1	no	—
269	C₂²×C₈	D₄×C₃₁	1	2	yes	—
270	C₂²×C₈	D₄×C₃₁	1	2	yes	—
271	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
272	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
273	#123	—	2	1	no	—
274	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
275	#121	D₄×C₃₁	1	2	yes	—
276	#120	—	2	1	no	—
277	C₄×D₄	—	2	2	no	—
278	#117	C₂³×C₃₁	1	1	yes	characteristic
279	#108	—	2	4	no	—
280	C₄⋊C₈	—	2	2	no	—
281	#102	—	2	8	no	—
282	#100	—	2	8	no	—
283	#100	—	4	4	no	—
284	C₂²⋊C₈	—	2	2	no	—
285	C₂²⋊C₈	—	2	4	no	—
286	C₂²⋊C₈	—	2	2	no	—
287	#97	—	2	1	no	—
288	C₄×C₈	—	2	1	no	—
289	C₃₁	#24281	1	1	yes	characteristic, direct product factor, Sylow
290	C₂⁴	—	2	4	no	—
291	C₂⁴	—	4	8	no	—
292	C₂⁴	—	4	4	no	—
293	C₂⁴	—	8	8	no	—
294	C₂⁴	#41×C₃₁	1	1	yes	characteristic
295	#41	—	2	2	no	—
296	#41	—	2	2	no	—
297	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
298	C₂×D₄	—	2	16	no	—
299	C₂×D₄	—	2	2	no	—
300	C₂×D₄	—	2	1	no	—
301	C₂×D₄	—	2	4	no	—
302	C₂×D₄	—	2	4	no	—
303	C₂×D₄	—	2	4	no	—
304	C₂×D₄	—	2	4	no	—
305	C₂×D₄	—	2	4	no	—
306	C₂×D₄	—	4	16	no	—
307	C₂×D₄	—	4	8	no	—
308	C₂×D₄	—	4	8	no	—
309	C₂×D₄	—	4	4	no	—
310	C₂×D₄	C₂×D₄×C₃₁	1	8	yes	—
311	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	2	2	no	—
314	C₂²×C₄	—	2	1	no	—
315	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
316	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
317	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
318	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
319	SD₁₆	—	4	8	no	—
320	D₈	—	2	16	no	—
321	D₈	—	2	8	no	—
322	D₈	—	2	8	no	—
323	D₈	—	4	8	no	—
324	M₁₆	—	2	2	no	—
325	M₁₆	—	2	4	no	—
326	M₁₆	—	2	2	no	—
327	M₁₆	—	2	4	no	—
328	M₁₆	—	4	2	no	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	—	2	4	no	—
331	C₂×C₈	—	2	4	no	—
332	C₂×C₈	—	2	1	no	—
333	C₂×C₈	—	2	4	no	—
334	C₂×C₈	—	2	1	no	—
335	C₂×C₈	—	4	2	no	—
336	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
337	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
338	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
339	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
342	C₂²⋊C₄	—	2	2	no	—
343	C₂²⋊C₄	—	2	1	no	—
344	C₂²⋊C₄	—	2	2	no	—
345	C₄²	—	2	1	no	—
346	C₂³	—	2	4	no	—
347	C₂³	—	2	2	no	—
348	C₂³	—	2	1	no	—
349	C₂³	—	2	8	no	—
350	C₂³	—	4	16	no	—
351	C₂³	—	4	4	no	—
352	C₂³	—	4	4	no	—
353	C₂³	—	8	8	no	—
354	C₂³	—	8	32	no	—
355	C₂³	—	8	8	no	—
356	C₂³	—	8	8	no	—
357	C₂³	C₃₁×#121	1	2	yes	—
358	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
359	Q₈	—	2	2	no	—
360	D₄	—	2	8	no	—
361	D₄	—	2	2	no	—
362	D₄	—	2	8	no	—
363	D₄	—	4	16	no	—
364	D₄	—	4	4	no	—
365	D₄	—	4	2	no	—
366	D₄	—	4	4	no	—
367	D₄	—	4	4	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	4	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	1	no	—
374	C₂×C₄	—	4	1	no	—
375	C₂×C₄	—	4	1	no	—
376	C₂×C₄	C₃₁×#121	1	2	yes	—
377	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
378	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
379	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
380	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
381	C₈	—	2	2	no	—
382	C₈	—	4	1	no	—
383	C₈	—	4	1	no	—
384	C₈	—	4	4	no	—
385	C₈	C₃₁×#120	1	2	yes	—
386	C₈	C₃₁×#120	1	2	yes	—
387	C₂²	—	2	8	no	—
388	C₂²	—	2	2	no	—
389	C₂²	—	2	2	no	—
390	C₂²	—	2	2	no	—
391	C₂²	—	4	1	no	—
392	C₂²	—	4	8	no	—
393	C₂²	—	4	2	no	—
394	C₂²	—	8	8	no	—
395	C₂²	—	8	32	no	—
396	C₂²	—	8	8	no	—
397	C₂²	—	8	8	no	—
398	C₂²	C₃₁×#472	1	4	yes	complemented
399	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
400	C₂²	C₃₁×#546	1	2	yes	—
401	C₄	—	2	2	no	—
402	C₄	—	4	1	no	—
403	C₄	—	4	2	no	—
404	C₄	C₂×C₃₁×#120	1	1	yes	characteristic
405	C₄	C₃₁×#535	1	1	yes	characteristic
406	C₄	D₄²×C₃₁	1	2	yes	—
407	C₂	—	2	4	no	—
408	C₂	—	2	2	no	—
409	C₂	—	8	1	no	—
410	C₂	—	8	8	no	—
411	C₂	C₃₁×#2408	1	1	yes	characteristic
412	C₂	C₂×C₃₁×#472	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×D₈×C₃₁	C₂	semidirect
2. C₂×C₃₁×#472	C₂	semidirect
3. C₃₁×#3003	C₂	semidirect
4. C₃₁×#2929	C₂	semidirect
5. C₃₁×#2045	C₂	semidirect
6. C₃₁×#2018	C₂	semidirect
7. C₃₁×#1978	C₂	semidirect
8. C₃₁×#1810	C₂	semidirect
9. C₃₁×#1783	C₂	semidirect
11. C₂×C₃₁×AGL(1,ℤ/8)	C₂²	semidirect
17. C₂×C₃₁×#131	C₂²	semidirect
25. C₂×C₃₁×#108	C₂²	semidirect
26. C₃₁×#424	C₂²	semidirect
27. C₃₁×#418	C₂²	semidirect
28. C₂×C₃₁×#100	C₂²	semidirect
29. C₂×C₃₁×#100	C₂²	semidirect
30. C₃₁×#408	C₂²	semidirect
33. C₃₁×#405	C₂²	semidirect
34. C₃₁×#341	C₂²	semidirect
55. C₂×D₈×C₃₁	D₄	semidirect
139. #24281	C₃₁	direct (N × H)
181. C₂³×D₈	C₂×C₃₁	semidirect
182. C₂×#472	C₂×C₃₁	semidirect
183. #3003	C₂×C₃₁	semidirect
184. #2929	C₂×C₃₁	semidirect
185. #2045	C₂×C₃₁	semidirect
186. #2018	C₂×C₃₁	semidirect
187. #1978	C₂×C₃₁	semidirect
188. #1810	C₂×C₃₁	semidirect
189. #1783	C₂×C₃₁	semidirect
201. C₂²×C₃₁	#472	semidirect
211. C₂×AGL(1,ℤ/8)	C₂²×C₃₁	semidirect
217. C₂×#131	C₂²×C₃₁	semidirect
225. C₂×#108	C₂²×C₃₁	semidirect
226. #424	C₂²×C₃₁	semidirect
227. #418	C₂²×C₃₁	semidirect
228. C₂×#100	C₂²×C₃₁	semidirect
229. C₂×#100	C₂²×C₃₁	semidirect
230. #408	C₂²×C₃₁	semidirect
233. #405	C₂²×C₃₁	semidirect
234. #341	C₂²×C₃₁	semidirect
261. C₂×D₈	D₄×C₃₁	semidirect
289. C₃₁	#24281	direct (N × H)
398. C₂²	C₃₁×#472	semidirect

Maximal subgroups

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

subgroup	index
C₂³×D₈×C₃₁	2
C₂×C₃₁×#472	2
C₃₁×#3003	2
C₃₁×#2929	2
C₃₁×#2045	2
C₃₁×#2018	2
C₃₁×#1978	2
C₃₁×#1810	2
C₃₁×#1783	2
#24281	31

Subgroups — C31×#24281 #21481040

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#24281 #21481040