Subgroups — C₃₁×#33465 #21490224

Order 7936.

642 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#508	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2957	C₂	1	2	yes	maximal, complemented
3	C₃₁×#2179	C₂	1	4	yes	maximal, complemented
4	C₃₁×#2178	C₂	1	4	yes	maximal, complemented
5	C₂×C₃₁×#368	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#2143	C₂	1	2	yes	maximal, complemented
7	C₃₁×#2108	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₃₁×#135	C₂²	1	2	yes	—
9	C₃₁×#568	—	2	2	no	—
10	C₃₁×#508	C₄	1	2	yes	complemented
11	C₃₁×#508	C₄	1	2	yes	complemented
12	C₃₁×#508	C₂²	1	2	yes	—
13	C₃₁×#508	C₂²	1	2	yes	—
14	C₂×C₃₁×Q₃₂	C₄	1	1	yes	characteristic, complemented
15	C₂×C₃₁×SD₃₂	C₄	1	2	yes	complemented
16	C₂×C₃₁×D₁₆	C₄	1	1	yes	characteristic, complemented
17	C₂×C₃₁×M₃₂	C₂²	1	1	yes	characteristic
18	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#444	—	2	4	no	—
20	C₃₁×#443	C₂²	1	8	yes	complemented
21	C₂×C₃₁×#108	C₂²	1	2	yes	—
22	C₃₁×#421	—	2	2	no	—
23	C₃₁×#405	C₂²	1	1	yes	characteristic
24	C₃₁×#368	C₂²	1	2	yes	complemented
25	C₃₁×#368	C₂²	1	2	yes	complemented
26	C₃₁×#359	C₂²	1	8	yes	complemented
27	C₃₁×#346	C₂²	1	2	yes	complemented
28	C₄×C₁₆×C₃₁	C₂²	1	2	yes	complemented
29	C₂×#41×C₃₁	—	2	2	no	—
30	C₃₁×#135	—	2	2	no	—
31	C₃₁×#135	—	2	2	no	—
32	C₃₁×#135	C₂×C₄	1	4	yes	—
33	C₃₁×#135	C₂³	1	4	yes	—
34	C₂×Q₁₆×C₃₁	C₂×C₄	1	2	yes	—
35	C₂×SD₁₆×C₃₁	—	2	2	no	—
36	C₂×D₈×C₃₁	C₂×C₄	1	2	yes	—
37	C₃₁×#131	—	2	8	no	—
38	C₃₁×#131	—	4	4	no	—
39	C₂×M₁₆×C₃₁	—	2	2	no	—
40	C₂×M₁₆×C₃₁	—	4	4	no	—
41	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
43	C₃₁×#117	D₄	1	1	yes	characteristic
44	C₃₁×Q₃₂	C₂×C₄	1	2	yes	—
45	C₃₁×Q₃₂	C₂×C₄	1	2	yes	—
46	C₃₁×SD₃₂	C₂×C₄	1	4	yes	—
47	C₃₁×SD₃₂	C₂×C₄	1	4	yes	—
48	C₃₁×D₁₆	C₂×C₄	1	2	yes	—
49	C₃₁×D₁₆	C₂×C₄	1	2	yes	—
50	C₃₁×M₃₂	—	2	1	no	—
51	C₃₁×M₃₂	—	2	1	no	—
52	C₂×C₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
53	C₂×C₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
54	C₂×C₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
55	C₂×C₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
56	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
57	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
58	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
59	C₃₁×#108	—	2	2	no	—
60	C₃₁×#108	C₂³	1	4	yes	—
61	C₃₁×#104	—	2	8	no	—
62	C₃₁×#97	D₄	1	2	yes	—
63	C₄×C₈×C₃₁	C₂³	1	2	yes	—
64	#41×C₃₁	—	2	4	no	—
65	#41×C₃₁	—	2	4	no	—
66	#41×C₃₁	—	4	2	no	—
67	#41×C₃₁	—	4	2	no	—
68	C₂×Q₈×C₃₁	—	2	2	no	—
69	C₂×D₄×C₃₁	—	2	2	no	—
70	C₂×D₄×C₃₁	—	4	2	no	—
71	C₂²×C₄×C₃₁	—	4	2	no	—
72	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
73	Q₁₆×C₃₁	—	2	2	no	—
74	Q₁₆×C₃₁	C₂²×C₄	1	4	yes	—
75	SD₁₆×C₃₁	—	2	4	no	—
76	SD₁₆×C₃₁	—	2	4	no	—
77	D₈×C₃₁	—	2	2	no	—
78	D₈×C₃₁	C₂²×C₄	1	4	yes	—
79	M₁₆×C₃₁	—	2	4	no	—
80	M₁₆×C₃₁	—	4	2	no	—
81	M₁₆×C₃₁	—	4	8	no	—
82	M₁₆×C₃₁	D₈	1	2	yes	—
83	M₁₆×C₃₁	D₈	1	2	yes	—
84	C₂×C₈×C₃₁	—	4	2	no	—
85	C₂×C₈×C₃₁	—	4	8	no	—
86	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
91	C₂×C₈×C₃₁	#41	1	1	yes	characteristic
92	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₃₁	—	2	1	no	—
94	C₂²⋊C₄×C₃₁	—	2	1	no	—
95	C₄²×C₃₁	C₂×D₄	1	2	yes	—
96	C₁₆×C₃₁	—	2	2	no	—
97	C₁₆×C₃₁	C₂²×C₄	1	1	yes	characteristic
98	C₁₆×C₃₁	C₂²×C₄	1	1	yes	characteristic
99	C₁₆×C₃₁	C₂²×C₄	1	1	yes	characteristic
100	C₁₆×C₃₁	C₂²×C₄	1	1	yes	characteristic
101	#33465	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂³×C₃₁	—	4	2	no	—
103	C₂³×C₃₁	#135	1	1	yes	characteristic
104	Q₈×C₃₁	—	2	4	no	—
105	Q₈×C₃₁	—	4	2	no	—
106	D₄×C₃₁	—	2	4	no	—
107	D₄×C₃₁	—	4	4	no	—
108	D₄×C₃₁	—	4	4	no	—
109	D₄×C₃₁	—	4	2	no	—
110	C₂×C₄×C₃₁	—	2	2	no	—
111	C₂×C₄×C₃₁	—	4	2	no	—
112	C₂×C₄×C₃₁	—	4	4	no	—
113	C₂×C₄×C₃₁	—	4	4	no	—
114	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₃₁	#135	1	1	yes	characteristic
119	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
120	C₈×C₃₁	—	4	4	no	—
121	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
122	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
123	C₈×C₃₁	C₂×D₈	1	4	yes	—
124	C₈×C₃₁	C₂³×C₄	1	1	yes	characteristic
125	C₈×C₃₁	C₂×#41	1	1	yes	characteristic
126	C₂×#508	C₂×C₃₁	1	1	yes	characteristic
127	#2957	C₂×C₃₁	1	2	yes	complemented
128	#2179	C₂×C₃₁	1	4	yes	complemented
129	#2178	C₂×C₃₁	1	4	yes	complemented
130	C₂×#368	C₂×C₃₁	1	1	yes	characteristic, complemented
131	#2143	C₂×C₃₁	1	2	yes	complemented
132	#2108	C₂×C₃₁	1	1	yes	characteristic, complemented
133	C₂²×C₃₁	—	4	1	no	—
134	C₂²×C₃₁	—	4	4	no	—
135	C₂²×C₃₁	—	16	2	no	—
136	C₂²×C₃₁	C₄×D₈	1	1	yes	characteristic
137	C₂²×C₃₁	C₄×D₈	1	1	yes	characteristic
138	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
139	C₄×C₃₁	—	4	2	no	—
140	C₄×C₃₁	—	4	4	no	—
141	C₄×C₃₁	C₄×D₈	1	1	yes	characteristic
142	C₄×C₃₁	C₄×D₈	1	1	yes	characteristic
143	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
144	C₄×C₃₁	C₂²×D₈	1	1	yes	characteristic
145	C₂×#135	C₂²×C₃₁	1	2	yes	—
146	#568	—	2	2	no	—
147	#508	C₄×C₃₁	1	2	yes	complemented
148	#508	C₄×C₃₁	1	2	yes	complemented
149	#508	C₂²×C₃₁	1	2	yes	—
150	#508	C₂²×C₃₁	1	2	yes	—
151	C₂×Q₃₂	C₄×C₃₁	1	1	yes	characteristic, complemented
152	C₂×SD₃₂	C₄×C₃₁	1	2	yes	complemented
153	C₂×D₁₆	C₄×C₃₁	1	1	yes	characteristic, complemented
154	C₂×M₃₂	C₂²×C₃₁	1	1	yes	characteristic
155	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
156	#444	—	2	4	no	—
157	#443	C₂²×C₃₁	1	8	yes	complemented
158	C₂×#108	C₂²×C₃₁	1	2	yes	—
159	#421	—	2	2	no	—
160	#405	C₂²×C₃₁	1	1	yes	characteristic
161	#368	C₂²×C₃₁	1	2	yes	complemented
162	#368	C₂²×C₃₁	1	2	yes	complemented
163	#359	C₂²×C₃₁	1	8	yes	complemented
164	#346	C₂²×C₃₁	1	2	yes	complemented
165	C₄×C₁₆	C₂²×C₃₁	1	2	yes	complemented
166	C₂×C₃₁	—	2	1	no	—
167	C₂×C₃₁	—	2	1	no	—
168	C₂×C₃₁	—	2	1	no	—
169	C₂×C₃₁	—	8	4	no	—
170	C₂×C₃₁	C₂×C₄×D₈	1	1	yes	characteristic
171	C₂×#41	—	2	2	no	—
172	#135	—	2	2	no	—
173	#135	—	2	2	no	—
174	#135	C₂×C₄×C₃₁	1	4	yes	—
175	#135	C₂³×C₃₁	1	4	yes	—
176	C₂×Q₁₆	C₂×C₄×C₃₁	1	2	yes	—
177	C₂×SD₁₆	—	2	2	no	—
178	C₂×D₈	C₂×C₄×C₃₁	1	2	yes	—
179	#131	—	2	8	no	—
180	#131	—	4	4	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	—	4	4	no	—
183	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
184	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
185	#117	D₄×C₃₁	1	1	yes	characteristic
186	Q₃₂	C₂×C₄×C₃₁	1	2	yes	—
187	Q₃₂	C₂×C₄×C₃₁	1	2	yes	—
188	SD₃₂	C₂×C₄×C₃₁	1	4	yes	—
189	SD₃₂	C₂×C₄×C₃₁	1	4	yes	—
190	D₁₆	C₂×C₄×C₃₁	1	2	yes	—
191	D₁₆	C₂×C₄×C₃₁	1	2	yes	—
192	M₃₂	—	2	1	no	—
193	M₃₂	—	2	1	no	—
194	C₂×C₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
195	C₂×C₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
196	C₂×C₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
197	C₂×C₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
198	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
199	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
200	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
201	#108	—	2	2	no	—
202	#108	C₂³×C₃₁	1	4	yes	—
203	#104	—	2	8	no	—
204	#97	D₄×C₃₁	1	2	yes	—
205	C₄×C₈	C₂³×C₃₁	1	2	yes	—
206	C₃₁	#33465	1	1	yes	characteristic, direct product factor, Sylow
207	#41	—	2	4	no	—
208	#41	—	2	4	no	—
209	#41	—	4	2	no	—
210	#41	—	4	2	no	—
211	C₂×Q₈	—	2	2	no	—
212	C₂×D₄	—	2	2	no	—
213	C₂×D₄	—	4	2	no	—
214	C₂²×C₄	—	4	2	no	—
215	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
216	Q₁₆	—	2	2	no	—
217	Q₁₆	C₂²×C₄×C₃₁	1	4	yes	—
218	SD₁₆	—	2	4	no	—
219	SD₁₆	—	2	4	no	—
220	D₈	—	2	2	no	—
221	D₈	C₂²×C₄×C₃₁	1	4	yes	—
222	M₁₆	—	2	4	no	—
223	M₁₆	—	4	2	no	—
224	M₁₆	—	4	8	no	—
225	M₁₆	D₈×C₃₁	1	2	yes	—
226	M₁₆	D₈×C₃₁	1	2	yes	—
227	C₂×C₈	—	4	2	no	—
228	C₂×C₈	—	4	8	no	—
229	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
230	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
231	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
232	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
233	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
234	C₂×C₈	#41×C₃₁	1	1	yes	characteristic
235	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
236	C₄⋊C₄	—	2	1	no	—
237	C₂²⋊C₄	—	2	1	no	—
238	C₄²	C₂×D₄×C₃₁	1	2	yes	—
239	C₁₆	—	2	2	no	—
240	C₁₆	C₂²×C₄×C₃₁	1	1	yes	characteristic
241	C₁₆	C₂²×C₄×C₃₁	1	1	yes	characteristic
242	C₁₆	C₂²×C₄×C₃₁	1	1	yes	characteristic
243	C₁₆	C₂²×C₄×C₃₁	1	1	yes	characteristic
244	C₂³	—	4	2	no	—
245	C₂³	C₃₁×#135	1	1	yes	characteristic
246	Q₈	—	2	4	no	—
247	Q₈	—	4	2	no	—
248	D₄	—	2	4	no	—
249	D₄	—	4	4	no	—
250	D₄	—	4	4	no	—
251	D₄	—	4	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
257	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
258	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
259	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
260	C₂×C₄	C₃₁×#135	1	1	yes	characteristic
261	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
262	C₈	—	4	4	no	—
263	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
264	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
265	C₈	C₂×D₈×C₃₁	1	4	yes	—
266	C₈	C₂³×C₄×C₃₁	1	1	yes	characteristic
267	C₈	C₂×#41×C₃₁	1	1	yes	characteristic
268	C₂²	—	4	1	no	—
269	C₂²	—	4	4	no	—
270	C₂²	—	16	2	no	—
271	C₂²	C₄×D₈×C₃₁	1	1	yes	characteristic
272	C₂²	C₄×D₈×C₃₁	1	1	yes	characteristic
273	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
274	C₄	—	4	2	no	—
275	C₄	—	4	4	no	—
276	C₄	C₄×D₈×C₃₁	1	1	yes	characteristic
277	C₄	C₄×D₈×C₃₁	1	1	yes	characteristic
278	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
279	C₄	C₂²×D₈×C₃₁	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	2	1	no	—
283	C₂	—	8	4	no	—
284	C₂	C₂×C₄×D₈×C₃₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₃₁	C₂²×D₈
commutator	C₈	C₂³×C₄×C₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
socle	C₂×C₃₁	C₂×C₄×D₈
prime radical	C₂×C₃₁×#508	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₁ ◁ C₂×M₁₆×C₃₁ ◁ C₃₁×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2957	C₂	semidirect
3. C₃₁×#2179	C₂	semidirect
4. C₃₁×#2178	C₂	semidirect
5. C₂×C₃₁×#368	C₂	semidirect
6. C₃₁×#2143	C₂	semidirect
7. C₃₁×#2108	C₂	semidirect
10. C₃₁×#508	C₄	semidirect
11. C₃₁×#508	C₄	semidirect
14. C₂×C₃₁×Q₃₂	C₄	semidirect
15. C₂×C₃₁×SD₃₂	C₄	semidirect
16. C₂×C₃₁×D₁₆	C₄	semidirect
20. C₃₁×#443	C₂²	semidirect
24. C₃₁×#368	C₂²	semidirect
25. C₃₁×#368	C₂²	semidirect
26. C₃₁×#359	C₂²	semidirect
27. C₃₁×#346	C₂²	semidirect
28. C₄×C₁₆×C₃₁	C₂²	semidirect
101. #33465	C₃₁	direct (N × H)
127. #2957	C₂×C₃₁	semidirect
128. #2179	C₂×C₃₁	semidirect
129. #2178	C₂×C₃₁	semidirect
130. C₂×#368	C₂×C₃₁	semidirect
131. #2143	C₂×C₃₁	semidirect
132. #2108	C₂×C₃₁	semidirect
147. #508	C₄×C₃₁	semidirect
148. #508	C₄×C₃₁	semidirect
151. C₂×Q₃₂	C₄×C₃₁	semidirect
152. C₂×SD₃₂	C₄×C₃₁	semidirect
153. C₂×D₁₆	C₄×C₃₁	semidirect
157. #443	C₂²×C₃₁	semidirect
161. #368	C₂²×C₃₁	semidirect
162. #368	C₂²×C₃₁	semidirect
163. #359	C₂²×C₃₁	semidirect
164. #346	C₂²×C₃₁	semidirect
165. C₄×C₁₆	C₂²×C₃₁	semidirect
206. C₃₁	#33465	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×#508	2
C₃₁×#2957	2
C₃₁×#2179	2
C₃₁×#2178	2
C₂×C₃₁×#368	2
C₃₁×#2143	2
C₃₁×#2108	2
#33465	31

Subgroups — C31×#33465 #21490224

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33465 #21490224