Subgroups — C₃₁×#22236 #21478995

Order 7936.

678 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3131	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2961	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#2932	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1995	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1963	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1929	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1866	C₂	1	4	yes	maximal, complemented
8	C₃₁×#1848	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1746	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#131	C₂²	1	2	yes	—
11	C₃₁×#533	C₂²	1	1	yes	characteristic
12	C₃₁×#532	C₂²	1	1	yes	characteristic
13	C₃₁×#488	C₄	1	4	yes	complemented
14	C₃₁×#487	C₄	1	2	yes	complemented
15	C₃₁×#486	C₄	1	2	yes	complemented
16	C₃₁×#446	—	2	4	no	—
17	C₃₁×#435	—	2	4	no	—
18	C₂×C₃₁×#108	C₂²	1	4	yes	—
19	C₃₁×#424	C₂²	1	2	yes	—
20	C₃₁×#423	C₂²	1	1	yes	characteristic
21	C₃₁×#423	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#104	C₂²	1	4	yes	—
23	C₃₁×#416	C₂²	1	4	yes	—
24	C₃₁×#408	C₂²	1	2	yes	—
25	C₃₁×#405	C₂²	1	2	yes	—
26	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#376	C₂²	1	1	yes	characteristic
29	C₃₁×#339	C₂²	1	4	yes	—
30	C₃₁×#336	C₂²	1	2	yes	—
31	C₃₁×#336	C₂²	1	2	yes	—
32	C₂×#41×C₃₁	C₂³	1	2	yes	—
33	C₃₁×#131	—	4	4	no	—
34	C₂×M₁₆×C₃₁	—	2	2	no	—
35	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
36	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
37	C₂²×C₈×C₃₁	—	2	2	no	—
38	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
39	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
40	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
41	C₃₁×C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
42	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
43	C₃₁×#127	C₂×C₄	1	1	yes	characteristic
44	C₃₁×#125	C₂×C₄	1	2	yes	—
45	C₃₁×#124	C₂×C₄	1	2	yes	—
46	C₃₁×C₂²⋊Q₈	—	2	2	no	—
47	C₃₁×#121	—	2	2	no	—
48	C₄×Q₈×C₃₁	—	2	2	no	—
49	C₄×D₄×C₃₁	—	2	2	no	—
50	C₃₁×#117	C₂³	1	2	yes	—
51	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
52	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
53	C₃₁×#108	—	4	4	no	—
54	C₃₁×C₄⋊C₈	—	2	4	no	—
55	C₃₁×C₄⋊C₈	—	2	2	no	—
56	C₃₁×C₄⋊C₈	—	2	2	no	—
57	C₃₁×#104	—	4	4	no	—
58	C₃₁×#103	C₂×C₄	1	8	yes	—
59	C₃₁×#102	C₂×C₄	1	8	yes	—
60	C₃₁×C₂²⋊C₈	—	2	4	no	—
61	C₃₁×#97	—	2	2	no	—
62	C₃₁×#97	D₄	1	4	yes	—
63	C₄×C₈×C₃₁	—	2	2	no	—
64	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
65	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
66	C₃₁×#95	—	2	2	no	—
67	#41×C₃₁	—	2	4	no	—
68	#41×C₃₁	—	2	4	no	—
69	C₂×Q₈×C₃₁	—	2	2	no	—
70	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
71	C₂×D₄×C₃₁	—	2	2	no	—
72	C₂×D₄×C₃₁	—	2	2	no	—
73	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₃₁	—	2	2	no	—
75	C₂²×C₄×C₃₁	—	2	2	no	—
76	C₂²×C₄×C₃₁	#41	1	2	yes	—
77	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
78	M₁₆×C₃₁	—	2	4	no	—
79	M₁₆×C₃₁	—	2	4	no	—
80	M₁₆×C₃₁	—	4	4	no	—
81	C₂×C₈×C₃₁	—	2	2	no	—
82	C₂×C₈×C₃₁	—	2	2	no	—
83	C₂×C₈×C₃₁	—	2	2	no	—
84	C₂×C₈×C₃₁	—	2	2	no	—
85	C₂×C₈×C₃₁	—	2	2	no	—
86	C₂×C₈×C₃₁	—	2	2	no	—
87	C₂×C₈×C₃₁	—	4	4	no	—
88	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
89	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
91	C₄⋊C₄×C₃₁	—	2	2	no	—
92	C₄⋊C₄×C₃₁	—	2	4	no	—
93	C₄⋊C₄×C₃₁	—	2	2	no	—
94	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
95	C₂²⋊C₄×C₃₁	—	2	2	no	—
96	C₂²⋊C₄×C₃₁	—	2	4	no	—
97	C₄²×C₃₁	—	2	2	no	—
98	C₄²×C₃₁	—	2	2	no	—
99	C₄²×C₃₁	—	2	2	no	—
100	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₃₁	C₂×D₄	1	2	yes	—
103	#22236	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂³×C₃₁	—	2	2	no	—
105	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
106	Q₈×C₃₁	—	2	4	no	—
107	Q₈×C₃₁	—	8	2	no	—
108	D₄×C₃₁	—	2	4	no	—
109	D₄×C₃₁	—	4	4	no	—
110	D₄×C₃₁	—	8	2	no	—
111	C₂×C₄×C₃₁	—	2	4	no	—
112	C₂×C₄×C₃₁	—	2	2	no	—
113	C₂×C₄×C₃₁	—	4	2	no	—
114	C₂×C₄×C₃₁	—	4	2	no	—
115	C₂×C₄×C₃₁	—	4	2	no	—
116	C₂×C₄×C₃₁	—	4	4	no	—
117	C₂×C₄×C₃₁	—	4	2	no	—
118	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
123	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
124	C₈×C₃₁	—	2	4	no	—
125	C₈×C₃₁	—	2	2	no	—
126	C₈×C₃₁	—	2	2	no	—
127	C₈×C₃₁	—	4	2	no	—
128	C₈×C₃₁	—	4	2	no	—
129	#3131	C₂×C₃₁	1	1	yes	characteristic
130	#2961	C₂×C₃₁	1	1	yes	characteristic, complemented
131	#2932	C₂×C₃₁	1	1	yes	characteristic
132	#1995	C₂×C₃₁	1	1	yes	characteristic, complemented
133	#1963	C₂×C₃₁	1	1	yes	characteristic
134	#1929	C₂×C₃₁	1	4	yes	complemented
135	#1866	C₂×C₃₁	1	4	yes	complemented
136	#1848	C₂×C₃₁	1	1	yes	characteristic, complemented
137	#1746	C₂×C₃₁	1	1	yes	characteristic, complemented
138	C₂²×C₃₁	—	2	2	no	—
139	C₂²×C₃₁	—	4	2	no	—
140	C₂²×C₃₁	—	4	4	no	—
141	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
142	C₂²×C₃₁	#542	1	2	yes	—
143	C₄×C₃₁	—	2	2	no	—
144	C₄×C₃₁	—	2	2	no	—
145	C₄×C₃₁	—	4	2	no	—
146	C₄×C₃₁	—	4	4	no	—
147	C₄×C₃₁	—	8	2	no	—
148	C₄×C₃₁	#520	1	2	yes	—
149	C₄×C₃₁	#535	1	1	yes	characteristic
150	C₄×C₃₁	#536	1	1	yes	characteristic
151	C₂×#131	C₂²×C₃₁	1	2	yes	—
152	#533	C₂²×C₃₁	1	1	yes	characteristic
153	#532	C₂²×C₃₁	1	1	yes	characteristic
154	#488	C₄×C₃₁	1	4	yes	complemented
155	#487	C₄×C₃₁	1	2	yes	complemented
156	#486	C₄×C₃₁	1	2	yes	complemented
157	#446	—	2	4	no	—
158	#435	—	2	4	no	—
159	C₂×#108	C₂²×C₃₁	1	4	yes	—
160	#424	C₂²×C₃₁	1	2	yes	—
161	#423	C₂²×C₃₁	1	1	yes	characteristic
162	#423	C₂²×C₃₁	1	1	yes	characteristic
163	C₂×#104	C₂²×C₃₁	1	4	yes	—
164	#416	C₂²×C₃₁	1	4	yes	—
165	#408	C₂²×C₃₁	1	2	yes	—
166	#405	C₂²×C₃₁	1	2	yes	—
167	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
168	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
169	#376	C₂²×C₃₁	1	1	yes	characteristic
170	#339	C₂²×C₃₁	1	4	yes	—
171	#336	C₂²×C₃₁	1	2	yes	—
172	#336	C₂²×C₃₁	1	2	yes	—
173	C₂×C₃₁	—	2	2	no	—
174	C₂×C₃₁	—	8	2	no	—
175	C₂×C₃₁	#2326	1	1	yes	characteristic
176	C₂×C₃₁	#2957	1	2	yes	—
177	C₂×#41	C₂³×C₃₁	1	2	yes	—
178	#131	—	4	4	no	—
179	C₂×M₁₆	—	2	2	no	—
180	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
181	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
184	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
185	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
186	C₄⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
187	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
188	#127	C₂×C₄×C₃₁	1	1	yes	characteristic
189	#125	C₂×C₄×C₃₁	1	2	yes	—
190	#124	C₂×C₄×C₃₁	1	2	yes	—
191	C₂²⋊Q₈	—	2	2	no	—
192	#121	—	2	2	no	—
193	C₄×Q₈	—	2	2	no	—
194	C₄×D₄	—	2	2	no	—
195	#117	C₂³×C₃₁	1	2	yes	—
196	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
197	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
198	#108	—	4	4	no	—
199	C₄⋊C₈	—	2	4	no	—
200	C₄⋊C₈	—	2	2	no	—
201	C₄⋊C₈	—	2	2	no	—
202	#104	—	4	4	no	—
203	#103	C₂×C₄×C₃₁	1	8	yes	—
204	#102	C₂×C₄×C₃₁	1	8	yes	—
205	C₂²⋊C₈	—	2	4	no	—
206	#97	—	2	2	no	—
207	#97	D₄×C₃₁	1	4	yes	—
208	C₄×C₈	—	2	2	no	—
209	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
210	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
211	#95	—	2	2	no	—
212	C₃₁	#22236	1	1	yes	characteristic, direct product factor, Sylow
213	#41	—	2	4	no	—
214	#41	—	2	4	no	—
215	C₂×Q₈	—	2	2	no	—
216	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
217	C₂×D₄	—	2	2	no	—
218	C₂×D₄	—	2	2	no	—
219	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	#41×C₃₁	1	2	yes	—
223	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
224	M₁₆	—	2	4	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	4	4	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	4	4	no	—
234	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
235	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
236	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
241	C₂²⋊C₄	—	2	2	no	—
242	C₂²⋊C₄	—	2	4	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	2	2	no	—
246	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
247	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
248	C₄²	C₂×D₄×C₃₁	1	2	yes	—
249	C₂³	—	2	2	no	—
250	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
251	Q₈	—	2	4	no	—
252	Q₈	—	8	2	no	—
253	D₄	—	2	4	no	—
254	D₄	—	4	4	no	—
255	D₄	—	8	2	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
264	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
265	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
266	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
267	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
268	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
269	C₈	—	2	4	no	—
270	C₈	—	2	2	no	—
271	C₈	—	2	2	no	—
272	C₈	—	4	2	no	—
273	C₈	—	4	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	4	2	no	—
276	C₂²	—	4	4	no	—
277	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
278	C₂²	C₃₁×#542	1	2	yes	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	4	2	no	—
282	C₄	—	4	4	no	—
283	C₄	—	8	2	no	—
284	C₄	C₃₁×#520	1	2	yes	—
285	C₄	C₃₁×#535	1	1	yes	characteristic
286	C₄	C₃₁×#536	1	1	yes	characteristic
287	C₂	—	2	2	no	—
288	C₂	—	8	2	no	—
289	C₂	C₃₁×#2326	1	1	yes	characteristic
290	C₂	C₃₁×#2957	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₃₁×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₃₁×#97 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2961	C₂	semidirect
4. C₃₁×#1995	C₂	semidirect
6. C₃₁×#1929	C₂	semidirect
7. C₃₁×#1866	C₂	semidirect
8. C₃₁×#1848	C₂	semidirect
9. C₃₁×#1746	C₂	semidirect
13. C₃₁×#488	C₄	semidirect
14. C₃₁×#487	C₄	semidirect
15. C₃₁×#486	C₄	semidirect
103. #22236	C₃₁	direct (N × H)
130. #2961	C₂×C₃₁	semidirect
132. #1995	C₂×C₃₁	semidirect
134. #1929	C₂×C₃₁	semidirect
135. #1866	C₂×C₃₁	semidirect
136. #1848	C₂×C₃₁	semidirect
137. #1746	C₂×C₃₁	semidirect
154. #488	C₄×C₃₁	semidirect
155. #487	C₄×C₃₁	semidirect
156. #486	C₄×C₃₁	semidirect
212. C₃₁	#22236	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×#3131	2
C₃₁×#2961	2
C₃₁×#2932	2
C₃₁×#1995	2
C₃₁×#1963	2
C₃₁×#1929	2
C₃₁×#1866	2
C₃₁×#1848	2
C₃₁×#1746	2
#22236	31

Subgroups — C31×#22236 #21478995

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22236 #21478995