Subgroups — C₃₁×#22695 #21479454

Order 7936.

808 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2957	C₂	1	2	yes	maximal
2	C₂×C₃₁×#444	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2916	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1953	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1861	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1849	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1814	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1773	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
10	C₃₁×#568	C₂²	1	4	yes	—
11	C₂×C₃₁×#131	C₂²	1	2	yes	—
12	C₃₁×#444	—	2	4	no	—
13	C₃₁×#444	C₄	1	8	yes	complemented
14	C₃₁×#443	—	2	4	no	—
15	C₃₁×#430	—	2	4	no	—
16	C₃₁×#430	C₂²	1	4	yes	—
17	C₂×C₃₁×#108	—	2	4	no	—
18	C₂×C₃₁×#108	C₂²	1	2	yes	—
19	C₂×C₃₁×#108	C₂²	1	2	yes	—
20	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
21	C₂×C₃₁×#108	C₂²	1	1	yes	characteristic
22	C₃₁×#427	C₂²	1	1	yes	characteristic
23	C₃₁×#423	C₂²	1	2	yes	—
24	C₃₁×#421	C₂²	1	4	yes	—
25	C₂×C₃₁×#104	C₂²	1	2	yes	—
26	C₃₁×#405	C₂²	1	2	yes	—
27	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
28	C₃₁×#344	C₂²	1	4	yes	—
29	C₃₁×#339	C₂²	1	2	yes	—
30	C₂×#41×C₃₁	C₂³	1	2	yes	—
31	C₃₁×#135	C₂×C₄	1	4	yes	—
32	C₃₁×#135	D₄	1	4	yes	—
33	C₂×Q₁₆×C₃₁	Q₈	1	1	yes	characteristic
34	C₂×SD₁₆×C₃₁	Q₈	1	2	yes	—
35	C₂×D₈×C₃₁	Q₈	1	1	yes	characteristic
36	C₃₁×#131	—	2	2	no	—
37	C₃₁×#131	—	2	2	no	—
38	C₃₁×#131	—	2	8	no	—
39	C₃₁×#131	—	4	4	no	—
40	C₃₁×#131	C₂×C₄	1	4	yes	—
41	C₃₁×#131	C₂×C₄	1	4	yes	—
42	C₂×M₁₆×C₃₁	—	2	2	no	—
43	C₂×M₁₆×C₃₁	—	2	8	no	—
44	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
45	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
46	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
47	C₂²×C₈×C₃₁	—	2	2	no	—
48	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
50	C₃₁×#117	C₂³	1	2	yes	—
51	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
52	C₃₁×#108	—	2	8	no	—
53	C₃₁×#108	—	2	2	no	—
54	C₃₁×#108	—	2	2	no	—
55	C₃₁×#108	—	2	2	no	—
56	C₃₁×#108	—	2	2	no	—
57	C₃₁×#108	C₂×C₄	1	4	yes	—
58	C₃₁×#108	D₄	1	4	yes	—
59	C₃₁×#107	—	2	1	no	—
60	C₃₁×#106	—	2	1	no	—
61	C₃₁×C₄⋊C₈	—	4	2	no	—
62	C₃₁×#104	—	2	8	no	—
63	C₃₁×#104	C₂×C₄	1	8	yes	—
64	C₃₁×#97	—	2	2	no	—
65	C₃₁×#97	C₂×C₄	1	4	yes	—
66	C₄×C₈×C₃₁	—	2	2	no	—
67	#41×C₃₁	—	2	2	no	—
68	#41×C₃₁	—	2	2	no	—
69	#41×C₃₁	C₂²×C₄	1	4	yes	—
70	#41×C₃₁	C₂×D₄	1	4	yes	—
71	C₂×Q₈×C₃₁	C₂×Q₈	1	2	yes	—
72	C₂×D₄×C₃₁	—	2	2	no	—
73	C₂×D₄×C₃₁	C₂×Q₈	1	2	yes	—
74	C₂²×C₄×C₃₁	—	2	2	no	—
75	C₂²×C₄×C₃₁	—	2	1	no	—
76	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
77	Q₁₆×C₃₁	C₄⋊C₄	1	4	yes	—
78	SD₁₆×C₃₁	C₄⋊C₄	1	8	yes	—
79	D₈×C₃₁	C₄⋊C₄	1	4	yes	—
80	M₁₆×C₃₁	—	2	2	no	—
81	M₁₆×C₃₁	—	2	2	no	—
82	M₁₆×C₃₁	—	2	4	no	—
83	M₁₆×C₃₁	—	2	4	no	—
84	M₁₆×C₃₁	—	2	4	no	—
85	M₁₆×C₃₁	—	2	4	no	—
86	M₁₆×C₃₁	—	4	8	no	—
87	M₁₆×C₃₁	C₂²×C₄	1	4	yes	—
88	M₁₆×C₃₁	C₂×D₄	1	4	yes	—
89	C₂×C₈×C₃₁	—	2	2	no	—
90	C₂×C₈×C₃₁	—	2	2	no	—
91	C₂×C₈×C₃₁	—	2	4	no	—
92	C₂×C₈×C₃₁	—	2	4	no	—
93	C₂×C₈×C₃₁	—	2	4	no	—
94	C₂×C₈×C₃₁	—	4	8	no	—
95	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
100	C₂×C₈×C₃₁	C₂×Q₈	1	1	yes	characteristic
101	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
102	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
103	C₄⋊C₄×C₃₁	—	2	2	no	—
104	C₂²⋊C₄×C₃₁	—	2	2	no	—
105	C₄²×C₃₁	—	2	1	no	—
106	C₄²×C₃₁	—	2	2	no	—
107	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
108	#22695	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
109	C₂³×C₃₁	—	2	2	no	—
110	C₂³×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
111	Q₈×C₃₁	—	2	2	no	—
112	Q₈×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
113	D₄×C₃₁	—	2	4	no	—
114	D₄×C₃₁	—	2	4	no	—
115	D₄×C₃₁	—	2	2	no	—
116	D₄×C₃₁	C₂×C₄⋊C₄	1	4	yes	—
117	C₂×C₄×C₃₁	—	2	2	no	—
118	C₂×C₄×C₃₁	—	2	4	no	—
119	C₂×C₄×C₃₁	—	2	4	no	—
120	C₂×C₄×C₃₁	—	2	2	no	—
121	C₂×C₄×C₃₁	—	4	2	no	—
122	C₂×C₄×C₃₁	—	8	1	no	—
123	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
127	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
128	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
129	C₈×C₃₁	—	2	4	no	—
130	C₈×C₃₁	—	2	4	no	—
131	C₈×C₃₁	—	4	4	no	—
132	C₈×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
133	C₈×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
134	C₈×C₃₁	C₄×D₄	1	2	yes	—
135	C₈×C₃₁	C₄×D₄	1	2	yes	—
136	#2957	C₂×C₃₁	1	2	yes	—
137	C₂×#444	C₂×C₃₁	1	1	yes	characteristic
138	#2916	C₂×C₃₁	1	2	yes	complemented
139	#1953	C₂×C₃₁	1	2	yes	complemented
140	#1861	C₂×C₃₁	1	4	yes	complemented
141	#1849	C₂×C₃₁	1	1	yes	characteristic, complemented
142	#1814	C₂×C₃₁	1	2	yes	complemented
143	#1773	C₂×C₃₁	1	1	yes	characteristic, complemented
144	C₂²×C₃₁	—	2	4	no	—
145	C₂²×C₃₁	—	4	1	no	—
146	C₂²×C₃₁	—	8	2	no	—
147	C₂²×C₃₁	#520	1	1	yes	characteristic
148	C₂²×C₃₁	#547	1	1	yes	characteristic
149	C₂²×C₃₁	D₄×Q₈	1	1	yes	characteristic
150	C₄×C₃₁	—	2	4	no	—
151	C₄×C₃₁	—	4	2	no	—
152	C₄×C₃₁	—	8	2	no	—
153	C₄×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
154	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
155	C₄×C₃₁	D₄²	1	1	yes	characteristic
156	C₄×C₃₁	#554	1	1	yes	characteristic
157	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
158	#568	C₂²×C₃₁	1	4	yes	—
159	C₂×#131	C₂²×C₃₁	1	2	yes	—
160	#444	—	2	4	no	—
161	#444	C₄×C₃₁	1	8	yes	complemented
162	#443	—	2	4	no	—
163	#430	—	2	4	no	—
164	#430	C₂²×C₃₁	1	4	yes	—
165	C₂×#108	—	2	4	no	—
166	C₂×#108	C₂²×C₃₁	1	2	yes	—
167	C₂×#108	C₂²×C₃₁	1	2	yes	—
168	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
169	C₂×#108	C₂²×C₃₁	1	1	yes	characteristic
170	#427	C₂²×C₃₁	1	1	yes	characteristic
171	#423	C₂²×C₃₁	1	2	yes	—
172	#421	C₂²×C₃₁	1	4	yes	—
173	C₂×#104	C₂²×C₃₁	1	2	yes	—
174	#405	C₂²×C₃₁	1	2	yes	—
175	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
176	#344	C₂²×C₃₁	1	4	yes	—
177	#339	C₂²×C₃₁	1	2	yes	—
178	C₂×C₃₁	—	2	1	no	—
179	C₂×C₃₁	—	2	1	no	—
180	C₂×C₃₁	—	2	1	no	—
181	C₂×C₃₁	—	4	4	no	—
182	C₂×C₃₁	D₄×C₄⋊C₄	1	1	yes	characteristic
183	C₂×#41	C₂³×C₃₁	1	2	yes	—
184	#135	C₂×C₄×C₃₁	1	4	yes	—
185	#135	D₄×C₃₁	1	4	yes	—
186	C₂×Q₁₆	Q₈×C₃₁	1	1	yes	characteristic
187	C₂×SD₁₆	Q₈×C₃₁	1	2	yes	—
188	C₂×D₈	Q₈×C₃₁	1	1	yes	characteristic
189	#131	—	2	2	no	—
190	#131	—	2	2	no	—
191	#131	—	2	8	no	—
192	#131	—	4	4	no	—
193	#131	C₂×C₄×C₃₁	1	4	yes	—
194	#131	C₂×C₄×C₃₁	1	4	yes	—
195	C₂×M₁₆	—	2	2	no	—
196	C₂×M₁₆	—	2	8	no	—
197	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
198	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
199	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
200	C₂²×C₈	—	2	2	no	—
201	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
202	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
203	#117	C₂³×C₃₁	1	2	yes	—
204	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
205	#108	—	2	8	no	—
206	#108	—	2	2	no	—
207	#108	—	2	2	no	—
208	#108	—	2	2	no	—
209	#108	—	2	2	no	—
210	#108	C₂×C₄×C₃₁	1	4	yes	—
211	#108	D₄×C₃₁	1	4	yes	—
212	#107	—	2	1	no	—
213	#106	—	2	1	no	—
214	C₄⋊C₈	—	4	2	no	—
215	#104	—	2	8	no	—
216	#104	C₂×C₄×C₃₁	1	8	yes	—
217	#97	—	2	2	no	—
218	#97	C₂×C₄×C₃₁	1	4	yes	—
219	C₄×C₈	—	2	2	no	—
220	C₃₁	#22695	1	1	yes	characteristic, direct product factor, Sylow
221	#41	—	2	2	no	—
222	#41	—	2	2	no	—
223	#41	C₂²×C₄×C₃₁	1	4	yes	—
224	#41	C₂×D₄×C₃₁	1	4	yes	—
225	C₂×Q₈	C₂×Q₈×C₃₁	1	2	yes	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
231	Q₁₆	C₄⋊C₄×C₃₁	1	4	yes	—
232	SD₁₆	C₄⋊C₄×C₃₁	1	8	yes	—
233	D₈	C₄⋊C₄×C₃₁	1	4	yes	—
234	M₁₆	—	2	2	no	—
235	M₁₆	—	2	2	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	2	4	no	—
240	M₁₆	—	4	8	no	—
241	M₁₆	C₂²×C₄×C₃₁	1	4	yes	—
242	M₁₆	C₂×D₄×C₃₁	1	4	yes	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	2	4	no	—
248	C₂×C₈	—	4	8	no	—
249	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
250	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
251	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
252	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
253	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
254	C₂×C₈	C₂×Q₈×C₃₁	1	1	yes	characteristic
255	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
256	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
257	C₄⋊C₄	—	2	2	no	—
258	C₂²⋊C₄	—	2	2	no	—
259	C₄²	—	2	1	no	—
260	C₄²	—	2	2	no	—
261	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
262	C₂³	—	2	2	no	—
263	C₂³	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
264	Q₈	—	2	2	no	—
265	Q₈	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
266	D₄	—	2	4	no	—
267	D₄	—	2	4	no	—
268	D₄	—	2	2	no	—
269	D₄	C₂×C₄⋊C₄×C₃₁	1	4	yes	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	8	1	no	—
276	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
277	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
278	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
279	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
280	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
281	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
282	C₈	—	2	4	no	—
283	C₈	—	2	4	no	—
284	C₈	—	4	4	no	—
285	C₈	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
286	C₈	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
287	C₈	C₄×D₄×C₃₁	1	2	yes	—
288	C₈	C₄×D₄×C₃₁	1	2	yes	—
289	C₂²	—	2	4	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	8	2	no	—
292	C₂²	C₃₁×#520	1	1	yes	characteristic
293	C₂²	C₃₁×#547	1	1	yes	characteristic
294	C₂²	D₄×Q₈×C₃₁	1	1	yes	characteristic
295	C₄	—	2	4	no	—
296	C₄	—	4	2	no	—
297	C₄	—	8	2	no	—
298	C₄	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
299	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
300	C₄	D₄²×C₃₁	1	1	yes	characteristic
301	C₄	C₃₁×#554	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	2	1	no	—
305	C₂	—	4	4	no	—
306	C₂	D₄×C₄⋊C₄×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₃₁	D₄²
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂×C₃₁	D₄×C₄⋊C₄
prime radical	C₂×C₃₁×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#2916	C₂	semidirect
4. C₃₁×#1953	C₂	semidirect
5. C₃₁×#1861	C₂	semidirect
6. C₃₁×#1849	C₂	semidirect
7. C₃₁×#1814	C₂	semidirect
8. C₃₁×#1773	C₂	semidirect
13. C₃₁×#444	C₄	semidirect
108. #22695	C₃₁	direct (N × H)
138. #2916	C₂×C₃₁	semidirect
139. #1953	C₂×C₃₁	semidirect
140. #1861	C₂×C₃₁	semidirect
141. #1849	C₂×C₃₁	semidirect
142. #1814	C₂×C₃₁	semidirect
143. #1773	C₂×C₃₁	semidirect
161. #444	C₄×C₃₁	semidirect
220. C₃₁	#22695	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×#2957	2
C₂×C₃₁×#444	2
C₃₁×#2916	2
C₃₁×#1953	2
C₃₁×#1861	2
C₃₁×#1849	2
C₃₁×#1814	2
C₃₁×#1773	2
#22695	31

Subgroups — C31×#22695 #21479454

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22695 #21479454