Subgroups — C₃₁×#9854 #21466613

Order 7936.

460 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3167	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1994	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#1962	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1744	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1567	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1493	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1492	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#533	C₂²	1	1	yes	characteristic
9	C₃₁×#532	C₂²	1	1	yes	characteristic
10	C₃₁×#498	C₄	1	1	yes	characteristic, complemented
11	C₃₁×#492	C₄	1	1	yes	characteristic, complemented
12	C₃₁×#487	C₄	1	1	yes	characteristic, complemented
13	C₃₁×#486	C₄	1	1	yes	characteristic, complemented
14	C₃₁×#465	—	2	2	no	—
15	C₄×SD₁₆×C₃₁	—	2	2	no	—
16	C₃₁×#423	C₂²	1	1	yes	characteristic
17	C₃₁×#423	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#104	—	2	2	no	—
19	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₄³×C₃₁	C₂²	1	1	yes	characteristic
22	C₃₁×#339	—	2	2	no	—
23	C₃₁×#336	—	2	2	no	—
24	C₃₁×#335	—	2	2	no	—
25	C₃₁×Q₈⋊C₈	—	2	2	no	—
26	C₃₁×D₄⋊C₈	—	2	2	no	—
27	C₂×#41×C₃₁	—	2	1	no	—
28	C₂×SD₁₆×C₃₁	—	2	2	no	—
29	C₂×M₁₆×C₃₁	—	2	1	no	—
30	C₂×M₁₆×C₃₁	—	2	1	no	—
31	C₂²×C₈×C₃₁	—	2	1	no	—
32	C₂²×C₈×C₃₁	D₄	1	2	yes	—
33	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
34	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
35	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
36	C₃₁×#127	C₂×C₄	1	1	yes	characteristic
37	C₃₁×#125	—	2	1	no	—
38	C₃₁×#124	—	2	1	no	—
39	C₃₁×C₂²⋊Q₈	—	2	1	no	—
40	C₃₁×C₂²⋊Q₈	—	2	1	no	—
41	C₃₁×#121	—	2	1	no	—
42	C₃₁×#121	—	2	1	no	—
43	C₄×Q₈×C₃₁	—	2	1	no	—
44	C₄×Q₈×C₃₁	—	2	1	no	—
45	C₄×D₄×C₃₁	—	2	1	no	—
46	C₄×D₄×C₃₁	—	2	1	no	—
47	C₃₁×#117	—	2	1	no	—
48	C₂×C₄²×C₃₁	—	2	1	no	—
49	C₂×C₄²×C₃₁	—	2	1	no	—
50	C₂×C₄²×C₃₁	D₄	1	2	yes	—
51	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
52	C₃₁×#106	—	2	2	no	—
53	C₃₁×C₄⋊C₈	—	2	1	no	—
54	C₃₁×C₄⋊C₈	—	2	1	no	—
55	C₃₁×C₄⋊C₈	—	2	1	no	—
56	C₃₁×C₄⋊C₈	—	2	1	no	—
57	C₃₁×#104	—	4	4	no	—
58	C₃₁×#103	—	2	2	no	—
59	C₃₁×#102	—	2	2	no	—
60	C₄×C₈×C₃₁	—	2	1	no	—
61	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₃₁	D₄	1	2	yes	—
64	C₄×C₈×C₃₁	D₄	1	2	yes	—
65	#41×C₃₁	—	4	2	no	—
66	#41×C₃₁	—	4	2	no	—
67	C₂×Q₈×C₃₁	—	2	1	no	—
68	C₂×Q₈×C₃₁	—	2	1	no	—
69	C₂×Q₈×C₃₁	—	2	1	no	—
70	C₂×D₄×C₃₁	—	2	1	no	—
71	C₂×D₄×C₃₁	—	2	1	no	—
72	C₂×D₄×C₃₁	—	2	1	no	—
73	C₂×D₄×C₃₁	—	4	1	no	—
74	C₂²×C₄×C₃₁	—	2	2	no	—
75	C₂²×C₄×C₃₁	—	4	1	no	—
76	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
79	SD₁₆×C₃₁	—	4	4	no	—
80	M₁₆×C₃₁	—	4	2	no	—
81	M₁₆×C₃₁	—	4	2	no	—
82	C₂×C₈×C₃₁	—	2	2	no	—
83	C₂×C₈×C₃₁	—	4	2	no	—
84	C₂×C₈×C₃₁	—	4	2	no	—
85	C₂×C₈×C₃₁	—	4	1	no	—
86	C₂×C₈×C₃₁	—	4	1	no	—
87	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
88	C₂×C₈×C₃₁	C₂²⋊C₄	1	2	yes	—
89	C₄⋊C₄×C₃₁	—	2	1	no	—
90	C₄⋊C₄×C₃₁	—	2	1	no	—
91	C₄⋊C₄×C₃₁	—	2	1	no	—
92	C₄⋊C₄×C₃₁	—	2	1	no	—
93	C₄⋊C₄×C₃₁	—	2	1	no	—
94	C₄⋊C₄×C₃₁	—	2	1	no	—
95	C₄⋊C₄×C₃₁	—	4	1	no	—
96	C₄⋊C₄×C₃₁	—	4	1	no	—
97	C₂²⋊C₄×C₃₁	—	4	1	no	—
98	C₂²⋊C₄×C₃₁	—	4	1	no	—
99	C₂²⋊C₄×C₃₁	—	4	1	no	—
100	C₄²×C₃₁	—	2	4	no	—
101	C₄²×C₃₁	—	2	2	no	—
102	C₄²×C₃₁	—	2	2	no	—
103	C₄²×C₃₁	—	4	2	no	—
104	C₄²×C₃₁	—	4	1	no	—
105	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
106	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₃₁	#41	1	1	yes	characteristic
109	#9854	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂³×C₃₁	—	4	1	no	—
111	C₂³×C₃₁	#121	1	1	yes	characteristic
112	Q₈×C₃₁	—	4	2	no	—
113	Q₈×C₃₁	—	4	2	no	—
114	Q₈×C₃₁	—	4	2	no	—
115	D₄×C₃₁	—	4	2	no	—
116	D₄×C₃₁	—	4	2	no	—
117	D₄×C₃₁	—	4	2	no	—
118	D₄×C₃₁	—	8	2	no	—
119	C₂×C₄×C₃₁	—	2	2	no	—
120	C₂×C₄×C₃₁	—	2	2	no	—
121	C₂×C₄×C₃₁	—	2	4	no	—
122	C₂×C₄×C₃₁	—	4	2	no	—
123	C₂×C₄×C₃₁	—	4	2	no	—
124	C₂×C₄×C₃₁	—	4	1	no	—
125	C₂×C₄×C₃₁	—	4	1	no	—
126	C₂×C₄×C₃₁	—	4	1	no	—
127	C₂×C₄×C₃₁	—	8	1	no	—
128	C₂×C₄×C₃₁	—	8	1	no	—
129	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
133	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
134	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
135	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
136	C₂×C₄×C₃₁	#137	1	2	yes	—
137	C₈×C₃₁	—	2	2	no	—
138	C₈×C₃₁	—	2	2	no	—
139	C₈×C₃₁	—	4	2	no	—
140	C₈×C₃₁	—	8	1	no	—
141	C₈×C₃₁	—	8	1	no	—
142	#3167	C₂×C₃₁	1	1	yes	characteristic
143	#1994	C₂×C₃₁	1	1	yes	characteristic, complemented
144	#1962	C₂×C₃₁	1	1	yes	characteristic
145	#1744	C₂×C₃₁	1	1	yes	characteristic, complemented
146	#1567	C₂×C₃₁	1	1	yes	characteristic, complemented
147	#1493	C₂×C₃₁	1	1	yes	characteristic, complemented
148	#1492	C₂×C₃₁	1	1	yes	characteristic
149	C₂²×C₃₁	—	2	1	no	—
150	C₂²×C₃₁	—	2	1	no	—
151	C₂²×C₃₁	—	8	1	no	—
152	C₂²×C₃₁	—	8	1	no	—
153	C₂²×C₃₁	—	8	1	no	—
154	C₂²×C₃₁	#390	1	1	yes	characteristic
155	C₂²×C₃₁	#468	1	2	yes	—
156	C₄×C₃₁	—	2	1	no	—
157	C₄×C₃₁	—	2	1	no	—
158	C₄×C₃₁	—	4	4	no	—
159	C₄×C₃₁	—	8	1	no	—
160	C₄×C₃₁	—	8	1	no	—
161	C₄×C₃₁	—	8	1	no	—
162	C₄×C₃₁	#421	1	2	yes	—
163	C₄×C₃₁	#440	1	2	yes	—
164	C₄×C₃₁	#443	1	2	yes	—
165	C₄×C₃₁	#496	1	1	yes	characteristic
166	C₄×C₃₁	#497	1	1	yes	characteristic
167	#533	C₂²×C₃₁	1	1	yes	characteristic
168	#532	C₂²×C₃₁	1	1	yes	characteristic
169	#498	C₄×C₃₁	1	1	yes	characteristic, complemented
170	#492	C₄×C₃₁	1	1	yes	characteristic, complemented
171	#487	C₄×C₃₁	1	1	yes	characteristic, complemented
172	#486	C₄×C₃₁	1	1	yes	characteristic, complemented
173	#465	—	2	2	no	—
174	C₄×SD₁₆	—	2	2	no	—
175	#423	C₂²×C₃₁	1	1	yes	characteristic
176	#423	C₂²×C₃₁	1	1	yes	characteristic
177	C₂×#104	—	2	2	no	—
178	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
179	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
180	C₄³	C₂²×C₃₁	1	1	yes	characteristic
181	#339	—	2	2	no	—
182	#336	—	2	2	no	—
183	#335	—	2	2	no	—
184	Q₈⋊C₈	—	2	2	no	—
185	D₄⋊C₈	—	2	2	no	—
186	C₂×C₃₁	—	2	2	no	—
187	C₂×C₃₁	—	16	1	no	—
188	C₂×C₃₁	#1963	1	1	yes	characteristic
189	C₂×C₃₁	#1973	1	1	yes	characteristic
190	C₂×C₃₁	#1974	1	1	yes	characteristic
191	C₂×#41	—	2	1	no	—
192	C₂×SD₁₆	—	2	2	no	—
193	C₂×M₁₆	—	2	1	no	—
194	C₂×M₁₆	—	2	1	no	—
195	C₂²×C₈	—	2	1	no	—
196	C₂²×C₈	D₄×C₃₁	1	2	yes	—
197	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
198	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
199	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
200	#127	C₂×C₄×C₃₁	1	1	yes	characteristic
201	#125	—	2	1	no	—
202	#124	—	2	1	no	—
203	C₂²⋊Q₈	—	2	1	no	—
204	C₂²⋊Q₈	—	2	1	no	—
205	#121	—	2	1	no	—
206	#121	—	2	1	no	—
207	C₄×Q₈	—	2	1	no	—
208	C₄×Q₈	—	2	1	no	—
209	C₄×D₄	—	2	1	no	—
210	C₄×D₄	—	2	1	no	—
211	#117	—	2	1	no	—
212	C₂×C₄²	—	2	1	no	—
213	C₂×C₄²	—	2	1	no	—
214	C₂×C₄²	D₄×C₃₁	1	2	yes	—
215	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
216	#106	—	2	2	no	—
217	C₄⋊C₈	—	2	1	no	—
218	C₄⋊C₈	—	2	1	no	—
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	2	1	no	—
221	#104	—	4	4	no	—
222	#103	—	2	2	no	—
223	#102	—	2	2	no	—
224	C₄×C₈	—	2	1	no	—
225	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
226	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
227	C₄×C₈	D₄×C₃₁	1	2	yes	—
228	C₄×C₈	D₄×C₃₁	1	2	yes	—
229	C₃₁	#9854	1	1	yes	characteristic, direct product factor, Sylow
230	#41	—	4	2	no	—
231	#41	—	4	2	no	—
232	C₂×Q₈	—	2	1	no	—
233	C₂×Q₈	—	2	1	no	—
234	C₂×Q₈	—	2	1	no	—
235	C₂×D₄	—	2	1	no	—
236	C₂×D₄	—	2	1	no	—
237	C₂×D₄	—	2	1	no	—
238	C₂×D₄	—	4	1	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	4	1	no	—
241	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
243	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
244	SD₁₆	—	4	4	no	—
245	M₁₆	—	4	2	no	—
246	M₁₆	—	4	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	4	2	no	—
249	C₂×C₈	—	4	2	no	—
250	C₂×C₈	—	4	1	no	—
251	C₂×C₈	—	4	1	no	—
252	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
253	C₂×C₈	C₂²⋊C₄×C₃₁	1	2	yes	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	2	1	no	—
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	4	1	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₂²⋊C₄	—	4	1	no	—
263	C₂²⋊C₄	—	4	1	no	—
264	C₂²⋊C₄	—	4	1	no	—
265	C₄²	—	2	4	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	4	2	no	—
269	C₄²	—	4	1	no	—
270	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
271	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
272	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
273	C₄²	#41×C₃₁	1	1	yes	characteristic
274	C₂³	—	4	1	no	—
275	C₂³	C₃₁×#121	1	1	yes	characteristic
276	Q₈	—	4	2	no	—
277	Q₈	—	4	2	no	—
278	Q₈	—	4	2	no	—
279	D₄	—	4	2	no	—
280	D₄	—	4	2	no	—
281	D₄	—	4	2	no	—
282	D₄	—	8	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	4	1	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	8	1	no	—
292	C₂×C₄	—	8	1	no	—
293	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
294	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
295	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
296	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
297	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
298	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
299	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
300	C₂×C₄	C₃₁×#137	1	2	yes	—
301	C₈	—	2	2	no	—
302	C₈	—	2	2	no	—
303	C₈	—	4	2	no	—
304	C₈	—	8	1	no	—
305	C₈	—	8	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	8	1	no	—
309	C₂²	—	8	1	no	—
310	C₂²	—	8	1	no	—
311	C₂²	C₃₁×#390	1	1	yes	characteristic
312	C₂²	C₃₁×#468	1	2	yes	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	4	4	no	—
316	C₄	—	8	1	no	—
317	C₄	—	8	1	no	—
318	C₄	—	8	1	no	—
319	C₄	C₃₁×#421	1	2	yes	—
320	C₄	C₃₁×#440	1	2	yes	—
321	C₄	C₃₁×#443	1	2	yes	—
322	C₄	C₃₁×#496	1	1	yes	characteristic
323	C₄	C₃₁×#497	1	1	yes	characteristic
324	C₂	—	2	2	no	—
325	C₂	—	16	1	no	—
326	C₂	C₃₁×#1963	1	1	yes	characteristic
327	C₂	C₃₁×#1973	1	1	yes	characteristic
328	C₂	C₃₁×#1974	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	#127
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle	C₂²×C₃₁	#390
prime radical	C₃₁×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1994	C₂	semidirect
4. C₃₁×#1744	C₂	semidirect
5. C₃₁×#1567	C₂	semidirect
6. C₃₁×#1493	C₂	semidirect
10. C₃₁×#498	C₄	semidirect
11. C₃₁×#492	C₄	semidirect
12. C₃₁×#487	C₄	semidirect
13. C₃₁×#486	C₄	semidirect
109. #9854	C₃₁	direct (N × H)
143. #1994	C₂×C₃₁	semidirect
145. #1744	C₂×C₃₁	semidirect
146. #1567	C₂×C₃₁	semidirect
147. #1493	C₂×C₃₁	semidirect
169. #498	C₄×C₃₁	semidirect
170. #492	C₄×C₃₁	semidirect
171. #487	C₄×C₃₁	semidirect
172. #486	C₄×C₃₁	semidirect
229. C₃₁	#9854	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×#3167	2
C₃₁×#1994	2
C₃₁×#1962	2
C₃₁×#1744	2
C₃₁×#1567	2
C₃₁×#1493	2
C₃₁×#1492	2
#9854	31

Subgroups — C31×#9854 #21466613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9854 #21466613