Subgroups — C₃₁×#8684 #21465443

Order 7936.

430 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2931	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1848	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#335	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1547	C₂	1	2	yes	maximal
5	C₃₁×#1292	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#131	—	2	1	no	—
7	C₃₁×#532	C₂²	1	1	yes	characteristic
8	C₃₁×#436	—	2	1	no	—
9	C₃₁×#436	C₄	1	2	yes	—
10	C₈×D₄×C₃₁	—	2	1	no	—
11	C₈×D₄×C₃₁	C₄	1	2	yes	—
12	C₂×C₃₁×#108	—	2	1	no	—
13	C₂×C₃₁×#108	—	2	1	no	—
14	C₃₁×#423	C₂²	1	2	yes	—
15	C₃₁×#423	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
17	C₃₁×#408	—	2	1	no	—
18	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#335	—	2	1	no	—
20	C₃₁×#335	—	2	1	no	—
21	C₃₁×#332	—	2	2	no	—
22	C₃₁×#331	—	2	2	no	—
23	C₂×#41×C₃₁	—	2	1	no	—
24	C₃₁×#131	—	4	2	no	—
25	C₃₁×#131	—	4	2	no	—
26	C₂×M₁₆×C₃₁	—	2	1	no	—
27	C₂×M₁₆×C₃₁	—	2	2	no	—
28	C₂×M₁₆×C₃₁	—	2	2	no	—
29	C₂²×C₈×C₃₁	—	2	2	no	—
30	C₂²×C₈×C₃₁	—	2	2	no	—
31	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
33	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
34	C₃₁×#127	D₄	1	1	yes	characteristic
35	C₃₁×#124	—	2	1	no	—
36	C₃₁×#121	—	2	1	no	—
37	C₃₁×#121	Q₈	1	2	yes	—
38	C₄×D₄×C₃₁	—	2	1	no	—
39	C₄×D₄×C₃₁	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
41	C₃₁×#108	—	4	2	no	—
42	C₃₁×#108	—	4	2	no	—
43	C₃₁×C₄⋊C₈	—	2	1	no	—
44	C₃₁×C₄⋊C₈	—	2	2	no	—
45	C₃₁×C₄⋊C₈	—	2	2	no	—
46	C₃₁×C₄⋊C₈	—	2	2	no	—
47	C₃₁×C₄⋊C₈	—	2	2	no	—
48	C₃₁×C₄⋊C₈	C₂×C₄	1	2	yes	—
49	C₃₁×C₂²⋊C₈	—	2	2	no	—
50	C₃₁×C₂²⋊C₈	—	2	2	no	—
51	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
54	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
55	#41×C₃₁	—	4	1	no	—
56	#41×C₃₁	—	4	1	no	—
57	#41×C₃₁	—	4	1	no	—
58	#41×C₃₁	—	4	1	no	—
59	C₂×Q₈×C₃₁	—	2	1	no	—
60	C₂×D₄×C₃₁	—	2	2	no	—
61	C₂×D₄×C₃₁	—	2	1	no	—
62	C₂×D₄×C₃₁	—	2	1	no	—
63	C₂×D₄×C₃₁	C₄⋊C₄	1	2	yes	—
64	C₂²×C₄×C₃₁	—	2	2	no	—
65	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
68	M₁₆×C₃₁	—	4	1	no	—
69	M₁₆×C₃₁	—	4	1	no	—
70	M₁₆×C₃₁	—	4	4	no	—
71	M₁₆×C₃₁	—	4	2	no	—
72	M₁₆×C₃₁	—	4	2	no	—
73	C₂×C₈×C₃₁	—	2	1	no	—
74	C₂×C₈×C₃₁	—	2	1	no	—
75	C₂×C₈×C₃₁	—	2	2	no	—
76	C₂×C₈×C₃₁	—	4	4	no	—
77	C₂×C₈×C₃₁	—	4	2	no	—
78	C₂×C₈×C₃₁	—	4	2	no	—
79	C₂×C₈×C₃₁	—	8	2	no	—
80	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₃₁	D₈	1	2	yes	—
85	C₂×C₈×C₃₁	SD₁₆	1	2	yes	—
86	C₄⋊C₄×C₃₁	—	2	1	no	—
87	C₄⋊C₄×C₃₁	C₄⋊C₄	1	2	yes	—
88	C₂²⋊C₄×C₃₁	—	2	2	no	—
89	C₂²⋊C₄×C₃₁	—	2	2	no	—
90	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₃₁	#41	1	1	yes	characteristic
94	#8684	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
95	C₂³×C₃₁	—	2	2	no	—
96	C₂³×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
97	Q₈×C₃₁	—	4	1	no	—
98	Q₈×C₃₁	—	4	1	no	—
99	D₄×C₃₁	—	4	2	no	—
100	D₄×C₃₁	—	4	2	no	—
101	D₄×C₃₁	—	4	2	no	—
102	D₄×C₃₁	—	4	1	no	—
103	D₄×C₃₁	—	4	1	no	—
104	D₄×C₃₁	—	4	1	no	—
105	D₄×C₃₁	—	4	1	no	—
106	C₂×C₄×C₃₁	—	2	1	no	—
107	C₂×C₄×C₃₁	—	2	1	no	—
108	C₂×C₄×C₃₁	—	2	2	no	—
109	C₂×C₄×C₃₁	—	4	1	no	—
110	C₂×C₄×C₃₁	—	4	2	no	—
111	C₂×C₄×C₃₁	—	4	2	no	—
112	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₃₁	#117	1	1	yes	characteristic
115	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
116	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
117	C₂×C₄×C₃₁	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	1	yes	characteristic
121	C₂×C₄×C₃₁	#137	1	1	yes	characteristic
122	C₈×C₃₁	—	2	1	no	—
123	C₈×C₃₁	—	2	1	no	—
124	C₈×C₃₁	—	4	2	no	—
125	C₈×C₃₁	—	8	2	no	—
126	C₈×C₃₁	—	8	2	no	—
127	C₈×C₃₁	#102	1	2	yes	—
128	C₈×C₃₁	#102	1	2	yes	—
129	#2931	C₂×C₃₁	1	1	yes	characteristic
130	#1848	C₂×C₃₁	1	1	yes	characteristic, complemented
131	C₂×#335	C₂×C₃₁	1	1	yes	characteristic, complemented
132	#1547	C₂×C₃₁	1	2	yes	—
133	#1292	C₂×C₃₁	1	2	yes	complemented
134	C₂²×C₃₁	—	2	1	no	—
135	C₂²×C₃₁	—	2	1	no	—
136	C₂²×C₃₁	—	4	2	no	—
137	C₂²×C₃₁	—	4	2	no	—
138	C₂²×C₃₁	—	4	2	no	—
139	C₂²×C₃₁	#380	1	1	yes	characteristic
140	C₂²×C₃₁	#474	1	1	yes	characteristic
141	C₂²×C₃₁	#476	1	1	yes	characteristic
142	C₄×C₃₁	—	2	1	no	—
143	C₄×C₃₁	—	2	1	no	—
144	C₄×C₃₁	—	2	1	no	—
145	C₄×C₃₁	—	4	2	no	—
146	C₄×C₃₁	C₂×#102	1	1	yes	characteristic
147	C₄×C₃₁	#417	1	1	yes	characteristic
148	C₄×C₃₁	#459	1	1	yes	characteristic
149	C₄×C₃₁	#461	1	1	yes	characteristic
150	C₄×C₃₁	#472	1	1	yes	characteristic
151	C₄×C₃₁	#473	1	1	yes	characteristic
152	C₂×#131	—	2	1	no	—
153	#532	C₂²×C₃₁	1	1	yes	characteristic
154	#436	—	2	1	no	—
155	#436	C₄×C₃₁	1	2	yes	—
156	C₈×D₄	—	2	1	no	—
157	C₈×D₄	C₄×C₃₁	1	2	yes	—
158	C₂×#108	—	2	1	no	—
159	C₂×#108	—	2	1	no	—
160	#423	C₂²×C₃₁	1	2	yes	—
161	#423	C₂²×C₃₁	1	1	yes	characteristic
162	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
163	#408	—	2	1	no	—
164	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
165	#335	—	2	1	no	—
166	#335	—	2	1	no	—
167	#332	—	2	2	no	—
168	#331	—	2	2	no	—
169	C₂×C₃₁	—	2	1	no	—
170	C₂×C₃₁	—	2	1	no	—
171	C₂×C₃₁	—	8	2	no	—
172	C₂×C₃₁	#1802	1	1	yes	characteristic
173	C₂×C₃₁	#1813	1	1	yes	characteristic
174	C₂×C₃₁	#1816	1	1	yes	characteristic
175	C₂×#41	—	2	1	no	—
176	#131	—	4	2	no	—
177	#131	—	4	2	no	—
178	C₂×M₁₆	—	2	1	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
184	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
185	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
186	#127	D₄×C₃₁	1	1	yes	characteristic
187	#124	—	2	1	no	—
188	#121	—	2	1	no	—
189	#121	Q₈×C₃₁	1	2	yes	—
190	C₄×D₄	—	2	1	no	—
191	C₄×D₄	C₂×C₄×C₃₁	1	2	yes	—
192	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
193	#108	—	4	2	no	—
194	#108	—	4	2	no	—
195	C₄⋊C₈	—	2	1	no	—
196	C₄⋊C₈	—	2	2	no	—
197	C₄⋊C₈	—	2	2	no	—
198	C₄⋊C₈	—	2	2	no	—
199	C₄⋊C₈	—	2	2	no	—
200	C₄⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
201	C₂²⋊C₈	—	2	2	no	—
202	C₂²⋊C₈	—	2	2	no	—
203	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
204	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
205	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
206	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
207	C₃₁	#8684	1	1	yes	characteristic, direct product factor, Sylow
208	#41	—	4	1	no	—
209	#41	—	4	1	no	—
210	#41	—	4	1	no	—
211	#41	—	4	1	no	—
212	C₂×Q₈	—	2	1	no	—
213	C₂×D₄	—	2	2	no	—
214	C₂×D₄	—	2	1	no	—
215	C₂×D₄	—	2	1	no	—
216	C₂×D₄	C₄⋊C₄×C₃₁	1	2	yes	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
219	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
220	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
221	M₁₆	—	4	1	no	—
222	M₁₆	—	4	1	no	—
223	M₁₆	—	4	4	no	—
224	M₁₆	—	4	2	no	—
225	M₁₆	—	4	2	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	4	4	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	—	4	2	no	—
232	C₂×C₈	—	8	2	no	—
233	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
234	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
235	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
236	C₂×C₈	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
237	C₂×C₈	D₈×C₃₁	1	2	yes	—
238	C₂×C₈	SD₁₆×C₃₁	1	2	yes	—
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	C₄⋊C₄×C₃₁	1	2	yes	—
241	C₂²⋊C₄	—	2	2	no	—
242	C₂²⋊C₄	—	2	2	no	—
243	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
244	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
245	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
246	C₄²	#41×C₃₁	1	1	yes	characteristic
247	C₂³	—	2	2	no	—
248	C₂³	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
249	Q₈	—	4	1	no	—
250	Q₈	—	4	1	no	—
251	D₄	—	4	2	no	—
252	D₄	—	4	2	no	—
253	D₄	—	4	2	no	—
254	D₄	—	4	1	no	—
255	D₄	—	4	1	no	—
256	D₄	—	4	1	no	—
257	D₄	—	4	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
265	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
266	C₂×C₄	C₃₁×#117	1	1	yes	characteristic
267	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
268	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
269	C₂×C₄	C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
270	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
271	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
272	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
273	C₂×C₄	C₃₁×#137	1	1	yes	characteristic
274	C₈	—	2	1	no	—
275	C₈	—	2	1	no	—
276	C₈	—	4	2	no	—
277	C₈	—	8	2	no	—
278	C₈	—	8	2	no	—
279	C₈	C₃₁×#102	1	2	yes	—
280	C₈	C₃₁×#102	1	2	yes	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	C₃₁×#380	1	1	yes	characteristic
287	C₂²	C₃₁×#474	1	1	yes	characteristic
288	C₂²	C₃₁×#476	1	1	yes	characteristic
289	C₄	—	2	1	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	4	2	no	—
293	C₄	C₂×C₃₁×#102	1	1	yes	characteristic
294	C₄	C₃₁×#417	1	1	yes	characteristic
295	C₄	C₃₁×#459	1	1	yes	characteristic
296	C₄	C₃₁×#461	1	1	yes	characteristic
297	C₄	C₃₁×#472	1	1	yes	characteristic
298	C₄	C₃₁×#473	1	1	yes	characteristic
299	C₂	—	2	1	no	—
300	C₂	—	2	1	no	—
301	C₂	—	8	2	no	—
302	C₂	C₃₁×#1802	1	1	yes	characteristic
303	C₂	C₃₁×#1813	1	1	yes	characteristic
304	C₂	C₃₁×#1816	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	#380
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₃₁×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#1848	C₂	semidirect
3. C₂×C₃₁×#335	C₂	semidirect
5. C₃₁×#1292	C₂	semidirect
94. #8684	C₃₁	direct (N × H)
130. #1848	C₂×C₃₁	semidirect
131. C₂×#335	C₂×C₃₁	semidirect
133. #1292	C₂×C₃₁	semidirect
207. C₃₁	#8684	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×#2931	2
C₃₁×#1848	2
C₂×C₃₁×#335	2
C₃₁×#1547	2
C₃₁×#1292	2
#8684	31

Subgroups — C31×#8684 #21465443

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8684 #21465443