Subgroups — C₃₁×#13007 #21469766

Order 7936.

492 conjugacy classes of subgroups.

The subgroup lattice has 338 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	C₃₁×#2916	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#361	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×#361	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#2117	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1388	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1385	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	C₃₁×#568	—	2	1	no	—
11	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
14	C₃₁×#472	—	2	2	no	—
15	C₃₁×#472	C₄	1	4	yes	—
16	C₃₁×#430	—	2	1	no	—
17	C₃₁×#430	—	2	2	no	—
18	C₂×C₃₁×#108	—	2	2	no	—
19	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
20	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#100	—	2	1	no	—
23	C₃₁×#361	—	2	2	no	—
24	C₃₁×#361	—	2	2	no	—
25	C₃₁×#350	—	2	2	no	—
26	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
27	C₂²×D₄×C₃₁	—	2	1	no	—
28	C₃₁×AGL(1,ℤ/8)	—	2	4	no	—
29	C₃₁×AGL(1,ℤ/8)	—	4	2	no	—
30	C₂×SD₁₆×C₃₁	—	4	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	—	2	1	no	—
36	C₂×D₈×C₃₁	—	4	1	no	—
37	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
38	C₃₁×#131	—	2	1	no	—
39	C₃₁×#131	—	2	1	no	—
40	C₃₁×#131	—	4	2	no	—
41	C₃₁×#131	—	4	2	no	—
42	C₃₁×#131	C₂×C₄	1	2	yes	—
43	C₃₁×#131	D₄	1	2	yes	—
44	C₂×M₁₆×C₃₁	—	2	1	no	—
45	C₂×M₁₆×C₃₁	—	2	4	no	—
46	C₂×M₁₆×C₃₁	—	2	1	no	—
47	C₂×M₁₆×C₃₁	—	2	1	no	—
48	C₂×M₁₆×C₃₁	—	2	1	no	—
49	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₃₁	—	2	1	no	—
51	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₃₁×M₃₂	—	2	1	no	—
53	C₃₁×M₃₂	—	2	1	no	—
54	C₃₁×M₃₂	—	2	1	no	—
55	C₃₁×M₃₂	—	2	1	no	—
56	C₂×C₁₆×C₃₁	—	2	1	no	—
57	C₂×C₁₆×C₃₁	—	2	1	no	—
58	C₃₁×#108	—	2	4	no	—
59	C₃₁×#108	—	2	1	no	—
60	C₃₁×#108	—	2	1	no	—
61	C₃₁×#108	—	2	1	no	—
62	C₃₁×#108	C₂×C₄	1	2	yes	—
63	C₃₁×#108	D₄	1	2	yes	—
64	C₃₁×#108	D₄	1	2	yes	—
65	C₃₁×#100	—	2	4	no	—
66	C₂⁴×C₃₁	—	4	1	no	—
67	#41×C₃₁	—	2	1	no	—
68	#41×C₃₁	—	2	1	no	—
69	#41×C₃₁	C₂²⋊C₄	1	2	yes	—
70	#41×C₃₁	D₈	1	2	yes	—
71	C₂×Q₈×C₃₁	SD₁₆	1	1	yes	characteristic
72	C₂×D₄×C₃₁	—	2	1	no	—
73	C₂×D₄×C₃₁	—	2	2	no	—
74	C₂×D₄×C₃₁	—	4	1	no	—
75	C₂×D₄×C₃₁	—	4	1	no	—
76	C₂×D₄×C₃₁	—	4	1	no	—
77	C₂×D₄×C₃₁	—	4	1	no	—
78	C₂×D₄×C₃₁	—	4	1	no	—
79	C₂×D₄×C₃₁	SD₁₆	1	1	yes	characteristic
80	C₂²×C₄×C₃₁	—	2	1	no	—
81	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
82	SD₁₆×C₃₁	—	4	4	no	—
83	D₈×C₃₁	—	2	2	no	—
84	D₈×C₃₁	—	2	2	no	—
85	D₈×C₃₁	—	4	4	no	—
86	D₈×C₃₁	—	4	1	no	—
87	D₈×C₃₁	—	4	1	no	—
88	M₁₆×C₃₁	—	2	1	no	—
89	M₁₆×C₃₁	—	2	2	no	—
90	M₁₆×C₃₁	—	2	2	no	—
91	M₁₆×C₃₁	—	2	2	no	—
92	M₁₆×C₃₁	—	2	2	no	—
93	M₁₆×C₃₁	—	2	2	no	—
94	M₁₆×C₃₁	—	4	4	no	—
95	M₁₆×C₃₁	—	4	1	no	—
96	M₁₆×C₃₁	—	4	1	no	—
97	M₁₆×C₃₁	—	4	1	no	—
98	M₁₆×C₃₁	C₂²⋊C₄	1	2	yes	—
99	C₂×C₈×C₃₁	—	2	1	no	—
100	C₂×C₈×C₃₁	—	2	2	no	—
101	C₂×C₈×C₃₁	—	2	2	no	—
102	C₂×C₈×C₃₁	—	4	4	no	—
103	C₂×C₈×C₃₁	—	4	1	no	—
104	C₂×C₈×C₃₁	—	4	1	no	—
105	C₂×C₈×C₃₁	—	4	1	no	—
106	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
111	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
112	C₁₆×C₃₁	—	4	1	no	—
113	C₁₆×C₃₁	—	4	1	no	—
114	#13007	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
115	C₂³×C₃₁	—	2	1	no	—
116	C₂³×C₃₁	—	4	2	no	—
117	C₂³×C₃₁	—	8	1	no	—
118	C₂³×C₃₁	—	8	1	no	—
119	C₂³×C₃₁	—	16	2	no	—
120	C₂³×C₃₁	#137	1	1	yes	characteristic
121	Q₈×C₃₁	—	2	1	no	—
122	Q₈×C₃₁	#102	1	2	yes	—
123	D₄×C₃₁	—	2	2	no	—
124	D₄×C₃₁	—	2	2	no	—
125	D₄×C₃₁	—	2	1	no	—
126	D₄×C₃₁	—	4	2	no	—
127	D₄×C₃₁	—	4	2	no	—
128	D₄×C₃₁	—	8	1	no	—
129	D₄×C₃₁	—	8	1	no	—
130	D₄×C₃₁	#102	1	2	yes	—
131	C₂×C₄×C₃₁	—	2	1	no	—
132	C₂×C₄×C₃₁	—	2	2	no	—
133	C₂×C₄×C₃₁	—	2	2	no	—
134	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
136	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
137	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
139	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
140	C₈×C₃₁	—	2	2	no	—
141	C₈×C₃₁	—	4	2	no	—
142	C₈×C₃₁	—	4	2	no	—
143	C₈×C₃₁	—	4	2	no	—
144	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
145	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
146	C₈×C₃₁	#121	1	1	yes	characteristic
147	C₈×C₃₁	#121	1	1	yes	characteristic
148	C₂×#472	C₂×C₃₁	1	1	yes	characteristic
149	#2916	C₂×C₃₁	1	1	yes	characteristic, complemented
150	C₂×#361	C₂×C₃₁	1	1	yes	characteristic, complemented
151	C₂×#361	C₂×C₃₁	1	1	yes	characteristic, complemented
152	#2117	C₂×C₃₁	1	1	yes	characteristic, complemented
153	#1388	C₂×C₃₁	1	1	yes	characteristic, complemented
154	#1385	C₂×C₃₁	1	1	yes	characteristic, complemented
155	C₂²×C₃₁	—	2	2	no	—
156	C₂²×C₃₁	—	4	1	no	—
157	C₂²×C₃₁	—	8	1	no	—
158	C₂²×C₃₁	—	8	2	no	—
159	C₂²×C₃₁	—	16	2	no	—
160	C₂²×C₃₁	—	16	2	no	—
161	C₂²×C₃₁	—	16	2	no	—
162	C₂²×C₃₁	#417	1	1	yes	characteristic
163	C₂²×C₃₁	#448	1	1	yes	characteristic
164	C₂²×C₃₁	#480	1	1	yes	characteristic
165	C₄×C₃₁	—	2	2	no	—
166	C₄×C₃₁	#386	1	1	yes	characteristic
167	C₄×C₃₁	C₂×#102	1	1	yes	characteristic
168	C₄×C₃₁	#447	1	1	yes	characteristic
169	C₄×C₃₁	#481	1	1	yes	characteristic
170	C₂×AGL(1,ℤ/8)	—	2	1	no	—
171	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
172	#568	—	2	1	no	—
173	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
174	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
175	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
176	#472	—	2	2	no	—
177	#472	C₄×C₃₁	1	4	yes	—
178	#430	—	2	1	no	—
179	#430	—	2	2	no	—
180	C₂×#108	—	2	2	no	—
181	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
182	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
183	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
184	C₂×#100	—	2	1	no	—
185	#361	—	2	2	no	—
186	#361	—	2	2	no	—
187	#350	—	2	2	no	—
188	C₂×C₃₁	—	2	1	no	—
189	C₂×C₃₁	—	2	1	no	—
190	C₂×C₃₁	—	2	1	no	—
191	C₂×C₃₁	—	4	2	no	—
192	C₂×C₃₁	—	16	2	no	—
193	C₂×C₃₁	#1874	1	1	yes	characteristic
194	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
195	C₂²×D₄	—	2	1	no	—
196	AGL(1,ℤ/8)	—	2	4	no	—
197	AGL(1,ℤ/8)	—	4	2	no	—
198	C₂×SD₁₆	—	4	1	no	—
199	C₂×D₈	—	2	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	—	4	1	no	—
205	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
206	#131	—	2	1	no	—
207	#131	—	2	1	no	—
208	#131	—	4	2	no	—
209	#131	—	4	2	no	—
210	#131	C₂×C₄×C₃₁	1	2	yes	—
211	#131	D₄×C₃₁	1	2	yes	—
212	C₂×M₁₆	—	2	1	no	—
213	C₂×M₁₆	—	2	4	no	—
214	C₂×M₁₆	—	2	1	no	—
215	C₂×M₁₆	—	2	1	no	—
216	C₂×M₁₆	—	2	1	no	—
217	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
218	C₂²×C₈	—	2	1	no	—
219	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
220	M₃₂	—	2	1	no	—
221	M₃₂	—	2	1	no	—
222	M₃₂	—	2	1	no	—
223	M₃₂	—	2	1	no	—
224	C₂×C₁₆	—	2	1	no	—
225	C₂×C₁₆	—	2	1	no	—
226	#108	—	2	4	no	—
227	#108	—	2	1	no	—
228	#108	—	2	1	no	—
229	#108	—	2	1	no	—
230	#108	C₂×C₄×C₃₁	1	2	yes	—
231	#108	D₄×C₃₁	1	2	yes	—
232	#108	D₄×C₃₁	1	2	yes	—
233	#100	—	2	4	no	—
234	C₃₁	#13007	1	1	yes	characteristic, direct product factor, Sylow
235	C₂⁴	—	4	1	no	—
236	#41	—	2	1	no	—
237	#41	—	2	1	no	—
238	#41	C₂²⋊C₄×C₃₁	1	2	yes	—
239	#41	D₈×C₃₁	1	2	yes	—
240	C₂×Q₈	SD₁₆×C₃₁	1	1	yes	characteristic
241	C₂×D₄	—	2	1	no	—
242	C₂×D₄	—	2	2	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	—	4	1	no	—
248	C₂×D₄	SD₁₆×C₃₁	1	1	yes	characteristic
249	C₂²×C₄	—	2	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	2	no	—
260	M₁₆	—	2	2	no	—
261	M₁₆	—	2	2	no	—
262	M₁₆	—	2	2	no	—
263	M₁₆	—	4	4	no	—
264	M₁₆	—	4	1	no	—
265	M₁₆	—	4	1	no	—
266	M₁₆	—	4	1	no	—
267	M₁₆	C₂²⋊C₄×C₃₁	1	2	yes	—
268	C₂×C₈	—	2	1	no	—
269	C₂×C₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	4	4	no	—
272	C₂×C₈	—	4	1	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	—	4	1	no	—
275	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
276	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
277	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
278	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
279	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
280	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
281	C₁₆	—	4	1	no	—
282	C₁₆	—	4	1	no	—
283	C₂³	—	2	1	no	—
284	C₂³	—	4	2	no	—
285	C₂³	—	8	1	no	—
286	C₂³	—	8	1	no	—
287	C₂³	—	16	2	no	—
288	C₂³	C₃₁×#137	1	1	yes	characteristic
289	Q₈	—	2	1	no	—
290	Q₈	C₃₁×#102	1	2	yes	—
291	D₄	—	2	2	no	—
292	D₄	—	2	2	no	—
293	D₄	—	2	1	no	—
294	D₄	—	4	2	no	—
295	D₄	—	4	2	no	—
296	D₄	—	8	1	no	—
297	D₄	—	8	1	no	—
298	D₄	C₃₁×#102	1	2	yes	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
303	C₂×C₄	C₃₁×#120	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	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
307	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
308	C₈	—	2	2	no	—
309	C₈	—	4	2	no	—
310	C₈	—	4	2	no	—
311	C₈	—	4	2	no	—
312	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
313	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
314	C₈	C₃₁×#121	1	1	yes	characteristic
315	C₈	C₃₁×#121	1	1	yes	characteristic
316	C₂²	—	2	2	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	8	1	no	—
319	C₂²	—	8	2	no	—
320	C₂²	—	16	2	no	—
321	C₂²	—	16	2	no	—
322	C₂²	—	16	2	no	—
323	C₂²	C₃₁×#417	1	1	yes	characteristic
324	C₂²	C₃₁×#448	1	1	yes	characteristic
325	C₂²	C₃₁×#480	1	1	yes	characteristic
326	C₄	—	2	2	no	—
327	C₄	C₃₁×#386	1	1	yes	characteristic
328	C₄	C₂×C₃₁×#102	1	1	yes	characteristic
329	C₄	C₃₁×#447	1	1	yes	characteristic
330	C₄	C₃₁×#481	1	1	yes	characteristic
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	2	1	no	—
334	C₂	—	4	2	no	—
335	C₂	—	16	2	no	—
336	C₂	C₃₁×#1874	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₃₁	#1874
commutator	C₂×C₈	C₂²×C₄×C₃₁
Frattini, cosocle	C₂²×C₈	C₂³×C₃₁
prime radical	C₂×C₃₁×#472	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×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. C₃₁×#2916	C₂	semidirect
3. C₂×C₃₁×#361	C₂	semidirect
4. C₂×C₃₁×#361	C₂	semidirect
5. C₃₁×#2117	C₂	semidirect
6. C₃₁×#1388	C₂	semidirect
7. C₃₁×#1385	C₂	semidirect
114. #13007	C₃₁	direct (N × H)
149. #2916	C₂×C₃₁	semidirect
150. C₂×#361	C₂×C₃₁	semidirect
151. C₂×#361	C₂×C₃₁	semidirect
152. #2117	C₂×C₃₁	semidirect
153. #1388	C₂×C₃₁	semidirect
154. #1385	C₂×C₃₁	semidirect
234. C₃₁	#13007	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
C₃₁×#2916	2
C₂×C₃₁×#361	2
C₃₁×#2117	2
C₃₁×#1388	2
C₃₁×#1385	2
#13007	31

Subgroups — C31×#13007 #21469766

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13007 #21469766