Subgroups — C₃₁×#13393 #21470152

Order 7936.

412 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#472	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#368	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#361	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×#357	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2113	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1816	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1387	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×AGL(1,ℤ/8)	—	2	1	no	—
9	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#472	—	2	2	no	—
14	C₃₁×#472	—	2	2	no	—
15	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
18	C₃₁×#417	—	2	1	no	—
19	C₂×C₃₁×#100	—	2	1	no	—
20	C₃₁×#368	—	2	2	no	—
21	C₃₁×#361	—	2	2	no	—
22	C₃₁×#357	—	2	2	no	—
23	C₃₁×C₂²⋊C₁₆	—	2	2	no	—
24	C₃₁×#343	—	2	1	no	—
25	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
26	C₂²×D₄×C₃₁	—	2	1	no	—
27	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
28	C₃₁×AGL(1,ℤ/8)	—	4	2	no	—
29	C₂×SD₁₆×C₃₁	—	4	1	no	—
30	C₂×D₈×C₃₁	—	2	1	no	—
31	C₂×D₈×C₃₁	—	2	1	no	—
32	C₂×D₈×C₃₁	—	2	1	no	—
33	C₂×D₈×C₃₁	—	2	1	no	—
34	C₂×D₈×C₃₁	—	2	1	no	—
35	C₂×D₈×C₃₁	—	4	1	no	—
36	C₂×D₈×C₃₁	D₄	1	2	yes	—
37	C₃₁×#131	—	2	1	no	—
38	C₃₁×#131	—	2	1	no	—
39	C₃₁×#131	—	2	1	no	—
40	C₃₁×#131	—	2	1	no	—
41	C₂×M₁₆×C₃₁	—	2	1	no	—
42	C₂×M₁₆×C₃₁	—	2	1	no	—
43	C₂×M₁₆×C₃₁	—	2	1	no	—
44	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
45	C₂²×C₈×C₃₁	—	2	1	no	—
46	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
48	C₃₁×M₃₂	—	2	1	no	—
49	C₃₁×M₃₂	—	2	1	no	—
50	C₂×C₁₆×C₃₁	—	2	1	no	—
51	C₂×C₁₆×C₃₁	—	2	1	no	—
52	C₂×C₁₆×C₃₁	—	2	1	no	—
53	C₂×C₁₆×C₃₁	D₄	1	2	yes	—
54	C₃₁×#108	—	2	1	no	—
55	C₃₁×#108	—	2	1	no	—
56	C₃₁×#108	—	2	1	no	—
57	C₃₁×#108	D₄	1	2	yes	—
58	C₃₁×#107	—	2	1	no	—
59	C₃₁×#107	—	2	1	no	—
60	C₃₁×#103	—	4	1	no	—
61	C₃₁×#102	—	4	1	no	—
62	C₃₁×#100	—	2	4	no	—
63	C₂⁴×C₃₁	—	4	1	no	—
64	#41×C₃₁	—	2	1	no	—
65	#41×C₃₁	—	2	1	no	—
66	#41×C₃₁	—	2	1	no	—
67	#41×C₃₁	—	2	1	no	—
68	C₂×Q₈×C₃₁	D₈	1	1	yes	characteristic
69	C₂×D₄×C₃₁	—	2	1	no	—
70	C₂×D₄×C₃₁	—	2	2	no	—
71	C₂×D₄×C₃₁	—	4	1	no	—
72	C₂×D₄×C₃₁	—	4	1	no	—
73	C₂×D₄×C₃₁	—	4	1	no	—
74	C₂×D₄×C₃₁	—	4	1	no	—
75	C₂×D₄×C₃₁	—	4	1	no	—
76	C₂×D₄×C₃₁	D₈	1	1	yes	characteristic
77	C₂²×C₄×C₃₁	—	2	1	no	—
78	C₂²×C₄×C₃₁	—	4	1	no	—
79	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
80	SD₁₆×C₃₁	—	4	4	no	—
81	D₈×C₃₁	—	2	2	no	—
82	D₈×C₃₁	—	2	2	no	—
83	D₈×C₃₁	—	4	4	no	—
84	D₈×C₃₁	—	4	1	no	—
85	D₈×C₃₁	—	4	1	no	—
86	M₁₆×C₃₁	—	2	1	no	—
87	M₁₆×C₃₁	—	2	2	no	—
88	M₁₆×C₃₁	—	2	1	no	—
89	M₁₆×C₃₁	—	4	1	no	—
90	M₁₆×C₃₁	—	4	1	no	—
91	M₁₆×C₃₁	—	4	1	no	—
92	M₁₆×C₃₁	—	4	1	no	—
93	M₁₆×C₃₁	—	4	1	no	—
94	C₂×C₈×C₃₁	—	2	1	no	—
95	C₂×C₈×C₃₁	—	4	1	no	—
96	C₂×C₈×C₃₁	—	4	1	no	—
97	C₂×C₈×C₃₁	—	4	1	no	—
98	C₂×C₈×C₃₁	—	4	1	no	—
99	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
103	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
104	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
105	C₄⋊C₄×C₃₁	—	4	1	no	—
106	C₄⋊C₄×C₃₁	—	4	1	no	—
107	C₁₆×C₃₁	—	2	2	no	—
108	C₁₆×C₃₁	—	4	1	no	—
109	#13393	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂³×C₃₁	—	2	1	no	—
111	C₂³×C₃₁	—	4	2	no	—
112	C₂³×C₃₁	—	8	1	no	—
113	C₂³×C₃₁	—	8	1	no	—
114	C₂³×C₃₁	—	16	1	no	—
115	C₂³×C₃₁	—	16	1	no	—
116	C₂³×C₃₁	#135	1	1	yes	characteristic
117	Q₈×C₃₁	—	2	1	no	—
118	Q₈×C₃₁	—	2	1	no	—
119	D₄×C₃₁	—	2	1	no	—
120	D₄×C₃₁	—	2	1	no	—
121	D₄×C₃₁	—	4	1	no	—
122	D₄×C₃₁	—	4	1	no	—
123	D₄×C₃₁	—	4	2	no	—
124	D₄×C₃₁	—	4	2	no	—
125	D₄×C₃₁	—	8	1	no	—
126	D₄×C₃₁	—	8	1	no	—
127	C₂×C₄×C₃₁	—	2	1	no	—
128	C₂×C₄×C₃₁	—	4	1	no	—
129	C₂×C₄×C₃₁	—	4	1	no	—
130	C₂×C₄×C₃₁	—	8	1	no	—
131	C₂×C₄×C₃₁	—	16	1	no	—
132	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
133	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
134	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
135	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
137	C₂×C₄×C₃₁	#137	1	1	yes	characteristic
138	C₈×C₃₁	—	4	1	no	—
139	C₈×C₃₁	—	4	2	no	—
140	C₈×C₃₁	—	8	1	no	—
141	C₈×C₃₁	#121	1	1	yes	characteristic
142	C₈×C₃₁	#121	1	1	yes	characteristic
143	C₈×C₃₁	#123	1	1	yes	characteristic
144	C₈×C₃₁	#124	1	1	yes	characteristic
145	C₂×#472	C₂×C₃₁	1	1	yes	characteristic
146	C₂×#368	C₂×C₃₁	1	1	yes	characteristic, complemented
147	C₂×#361	C₂×C₃₁	1	1	yes	characteristic, complemented
148	C₂×#357	C₂×C₃₁	1	1	yes	characteristic, complemented
149	#2113	C₂×C₃₁	1	1	yes	characteristic, complemented
150	#1816	C₂×C₃₁	1	1	yes	characteristic, complemented
151	#1387	C₂×C₃₁	1	1	yes	characteristic, complemented
152	C₂²×C₃₁	—	2	1	no	—
153	C₂²×C₃₁	—	2	1	no	—
154	C₂²×C₃₁	—	4	1	no	—
155	C₂²×C₃₁	—	4	1	no	—
156	C₂²×C₃₁	—	4	1	no	—
157	C₂²×C₃₁	—	8	2	no	—
158	C₂²×C₃₁	—	16	2	no	—
159	C₂²×C₃₁	—	16	2	no	—
160	C₂²×C₃₁	—	16	1	no	—
161	C₂²×C₃₁	—	16	1	no	—
162	C₂²×C₃₁	#452	1	1	yes	characteristic
163	C₂²×C₃₁	#466	1	1	yes	characteristic
164	C₂²×C₃₁	#482	1	1	yes	characteristic
165	C₄×C₃₁	—	4	1	no	—
166	C₄×C₃₁	—	16	1	no	—
167	C₄×C₃₁	#394	1	1	yes	characteristic
168	C₄×C₃₁	#447	1	1	yes	characteristic
169	C₄×C₃₁	#468	1	1	yes	characteristic
170	C₄×C₃₁	#480	1	1	yes	characteristic
171	C₂×AGL(1,ℤ/8)	—	2	1	no	—
172	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
173	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
174	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
175	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
176	#472	—	2	2	no	—
177	#472	—	2	2	no	—
178	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
179	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
180	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
181	#417	—	2	1	no	—
182	C₂×#100	—	2	1	no	—
183	#368	—	2	2	no	—
184	#361	—	2	2	no	—
185	#357	—	2	2	no	—
186	C₂²⋊C₁₆	—	2	2	no	—
187	#343	—	2	1	no	—
188	C₂×C₃₁	—	2	1	no	—
189	C₂×C₃₁	—	2	1	no	—
190	C₂×C₃₁	—	8	1	no	—
191	C₂×C₃₁	—	16	2	no	—
192	C₂×C₃₁	#2033	1	1	yes	characteristic
193	C₂×C₃₁	#2221	1	1	yes	characteristic
194	C₂×C₃₁	#2223	1	1	yes	characteristic
195	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
196	C₂²×D₄	—	2	1	no	—
197	AGL(1,ℤ/8)	—	2	4	no	—
198	AGL(1,ℤ/8)	—	4	2	no	—
199	C₂×SD₁₆	—	4	1	no	—
200	C₂×D₈	—	2	1	no	—
201	C₂×D₈	—	2	1	no	—
202	C₂×D₈	—	2	1	no	—
203	C₂×D₈	—	2	1	no	—
204	C₂×D₈	—	2	1	no	—
205	C₂×D₈	—	4	1	no	—
206	C₂×D₈	D₄×C₃₁	1	2	yes	—
207	#131	—	2	1	no	—
208	#131	—	2	1	no	—
209	#131	—	2	1	no	—
210	#131	—	2	1	no	—
211	C₂×M₁₆	—	2	1	no	—
212	C₂×M₁₆	—	2	1	no	—
213	C₂×M₁₆	—	2	1	no	—
214	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
215	C₂²×C₈	—	2	1	no	—
216	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
217	C₂×C₄⋊C₄	—	2	1	no	—
218	M₃₂	—	2	1	no	—
219	M₃₂	—	2	1	no	—
220	C₂×C₁₆	—	2	1	no	—
221	C₂×C₁₆	—	2	1	no	—
222	C₂×C₁₆	—	2	1	no	—
223	C₂×C₁₆	D₄×C₃₁	1	2	yes	—
224	#108	—	2	1	no	—
225	#108	—	2	1	no	—
226	#108	—	2	1	no	—
227	#108	D₄×C₃₁	1	2	yes	—
228	#107	—	2	1	no	—
229	#107	—	2	1	no	—
230	#103	—	4	1	no	—
231	#102	—	4	1	no	—
232	#100	—	2	4	no	—
233	C₃₁	#13393	1	1	yes	characteristic, direct product factor, Sylow
234	C₂⁴	—	4	1	no	—
235	#41	—	2	1	no	—
236	#41	—	2	1	no	—
237	#41	—	2	1	no	—
238	#41	—	2	1	no	—
239	C₂×Q₈	D₈×C₃₁	1	1	yes	characteristic
240	C₂×D₄	—	2	1	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	4	1	no	—
243	C₂×D₄	—	4	1	no	—
244	C₂×D₄	—	4	1	no	—
245	C₂×D₄	—	4	1	no	—
246	C₂×D₄	—	4	1	no	—
247	C₂×D₄	D₈×C₃₁	1	1	yes	characteristic
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
251	SD₁₆	—	4	4	no	—
252	D₈	—	2	2	no	—
253	D₈	—	2	2	no	—
254	D₈	—	4	4	no	—
255	D₈	—	4	1	no	—
256	D₈	—	4	1	no	—
257	M₁₆	—	2	1	no	—
258	M₁₆	—	2	2	no	—
259	M₁₆	—	2	1	no	—
260	M₁₆	—	4	1	no	—
261	M₁₆	—	4	1	no	—
262	M₁₆	—	4	1	no	—
263	M₁₆	—	4	1	no	—
264	M₁₆	—	4	1	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	4	1	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	—	4	1	no	—
270	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
271	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
272	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
273	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
274	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
275	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₁₆	—	2	2	no	—
279	C₁₆	—	4	1	no	—
280	C₂³	—	2	1	no	—
281	C₂³	—	4	2	no	—
282	C₂³	—	8	1	no	—
283	C₂³	—	8	1	no	—
284	C₂³	—	16	1	no	—
285	C₂³	—	16	1	no	—
286	C₂³	C₃₁×#135	1	1	yes	characteristic
287	Q₈	—	2	1	no	—
288	Q₈	—	2	1	no	—
289	D₄	—	2	1	no	—
290	D₄	—	2	1	no	—
291	D₄	—	4	1	no	—
292	D₄	—	4	1	no	—
293	D₄	—	4	2	no	—
294	D₄	—	4	2	no	—
295	D₄	—	8	1	no	—
296	D₄	—	8	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	8	1	no	—
301	C₂×C₄	—	16	1	no	—
302	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
303	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
304	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
305	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
306	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
307	C₂×C₄	C₃₁×#137	1	1	yes	characteristic
308	C₈	—	4	2	no	—
309	C₈	—	4	1	no	—
310	C₈	—	8	1	no	—
311	C₈	C₃₁×#121	1	1	yes	characteristic
312	C₈	C₃₁×#121	1	1	yes	characteristic
313	C₈	C₃₁×#123	1	1	yes	characteristic
314	C₈	C₃₁×#124	1	1	yes	characteristic
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	16	2	no	—
322	C₂²	—	16	2	no	—
323	C₂²	—	16	1	no	—
324	C₂²	—	16	1	no	—
325	C₂²	C₃₁×#452	1	1	yes	characteristic
326	C₂²	C₃₁×#466	1	1	yes	characteristic
327	C₂²	C₃₁×#482	1	1	yes	characteristic
328	C₄	—	4	1	no	—
329	C₄	—	16	1	no	—
330	C₄	C₃₁×#394	1	1	yes	characteristic
331	C₄	C₃₁×#447	1	1	yes	characteristic
332	C₄	C₃₁×#468	1	1	yes	characteristic
333	C₄	C₃₁×#480	1	1	yes	characteristic
334	C₂	—	2	1	no	—
335	C₂	—	2	1	no	—
336	C₂	—	8	1	no	—
337	C₂	—	16	2	no	—
338	C₂	C₃₁×#2033	1	1	yes	characteristic
339	C₂	C₃₁×#2221	1	1	yes	characteristic
340	C₂	C₃₁×#2223	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
prime radical	C₂×C₃₁×#472	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ C₂×C₃₁×#131 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#368	C₂	semidirect
3. C₂×C₃₁×#361	C₂	semidirect
4. C₂×C₃₁×#357	C₂	semidirect
5. C₃₁×#2113	C₂	semidirect
6. C₃₁×#1816	C₂	semidirect
7. C₃₁×#1387	C₂	semidirect
109. #13393	C₃₁	direct (N × H)
146. C₂×#368	C₂×C₃₁	semidirect
147. C₂×#361	C₂×C₃₁	semidirect
148. C₂×#357	C₂×C₃₁	semidirect
149. #2113	C₂×C₃₁	semidirect
150. #1816	C₂×C₃₁	semidirect
151. #1387	C₂×C₃₁	semidirect
233. C₃₁	#13393	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×#472	2
C₂×C₃₁×#368	2
C₂×C₃₁×#361	2
C₂×C₃₁×#357	2
C₃₁×#2113	2
C₃₁×#1816	2
C₃₁×#1387	2
#13393	31

Subgroups — C31×#13393 #21470152

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13393 #21470152