Subgroups — C₃₁×#33247 #21490006

Order 7936.

922 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	C₂×C₃₁×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#495	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#489	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2640	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1973	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1970	C₂	1	2	yes	maximal
7	C₃₁×#1969	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1726	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	—
11	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₂×C₃₁×#124	C₂²	1	2	yes	—
14	C₃₁×#500	—	2	4	no	—
15	C₃₁×#495	—	2	4	no	—
16	C₃₁×#489	—	2	8	no	—
17	C₂×C₃₁×#107	C₂²	1	2	yes	—
18	C₂×C₃₁×#107	C₂²	1	2	yes	—
19	C₂×C₃₁×#103	C₂²	1	4	yes	—
20	C₂×C₃₁×#102	C₂²	1	4	yes	—
21	C₂×C₃₁×#97	C₂²	1	2	yes	—
22	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₃₁×#397	—	2	4	no	—
26	C₃₁×#393	—	2	4	no	—
27	C₃₁×#391	C₂²	1	4	yes	—
28	C₃₁×#390	C₂²	1	4	yes	—
29	C₃₁×#376	C₂²	1	1	yes	characteristic
30	C₂²×Q₈×C₃₁	—	2	2	no	—
31	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
32	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
33	C₂×Q₁₆×C₃₁	—	2	2	no	—
34	C₂×Q₁₆×C₃₁	—	2	4	no	—
35	C₂×SD₁₆×C₃₁	—	2	4	no	—
36	C₂×SD₁₆×C₃₁	—	2	8	no	—
37	C₂×D₈×C₃₁	—	2	2	no	—
38	C₂×D₈×C₃₁	—	2	4	no	—
39	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
40	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
42	C₃₁×C₄⋊Q₈	—	2	4	no	—
43	C₃₁×C₄⋊Q₈	—	2	4	no	—
44	C₃₁×#124	—	2	4	no	—
45	C₃₁×#124	—	2	4	no	—
46	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
47	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
48	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
49	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
50	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
51	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
52	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
53	C₃₁×#107	—	2	4	no	—
54	C₃₁×#107	—	2	4	no	—
55	C₃₁×#103	—	2	8	no	—
56	C₃₁×#102	—	2	8	no	—
57	C₃₁×#97	—	2	4	no	—
58	C₃₁×#97	D₄	1	4	yes	—
59	C₄×C₈×C₃₁	—	2	4	no	—
60	C₄×C₈×C₃₁	D₄	1	4	yes	—
61	C₃₁×#95	—	2	8	no	—
62	C₃₁×#95	D₄	1	2	yes	—
63	C₃₁×#95	D₄	1	2	yes	—
64	C₂⁴×C₃₁	—	2	2	no	—
65	C₂×Q₈×C₃₁	—	2	2	no	—
66	C₂×Q₈×C₃₁	—	2	2	no	—
67	C₂×Q₈×C₃₁	—	2	4	no	—
68	C₂×Q₈×C₃₁	—	2	4	no	—
69	C₂×Q₈×C₃₁	—	4	4	no	—
70	C₂×Q₈×C₃₁	—	8	4	no	—
71	C₂×D₄×C₃₁	—	2	2	no	—
72	C₂×D₄×C₃₁	—	2	2	no	—
73	C₂×D₄×C₃₁	—	2	4	no	—
74	C₂×D₄×C₃₁	—	2	4	no	—
75	C₂²×C₄×C₃₁	—	2	2	no	—
76	C₂²×C₄×C₃₁	—	2	4	no	—
77	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
78	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
81	Q₁₆×C₃₁	—	4	4	no	—
82	SD₁₆×C₃₁	—	4	8	no	—
83	D₈×C₃₁	—	4	4	no	—
84	C₂×C₈×C₃₁	—	2	2	no	—
85	C₂×C₈×C₃₁	—	2	2	no	—
86	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
87	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₃₁	#41	1	4	yes	—
90	C₂×C₈×C₃₁	#41	1	4	yes	—
91	C₄⋊C₄×C₃₁	—	2	4	no	—
92	C₄⋊C₄×C₃₁	—	2	4	no	—
93	C₄⋊C₄×C₃₁	—	4	4	no	—
94	C₂²⋊C₄×C₃₁	—	4	4	no	—
95	C₂²⋊C₄×C₃₁	—	4	4	no	—
96	C₂²⋊C₄×C₃₁	—	8	8	no	—
97	C₄²×C₃₁	—	2	2	no	—
98	C₄²×C₃₁	—	2	2	no	—
99	C₄²×C₃₁	C₂×D₄	1	2	yes	—
100	C₄²×C₃₁	C₂×D₄	1	2	yes	—
101	#33247	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
102	C₂³×C₃₁	—	4	2	no	—
103	C₂³×C₃₁	—	4	4	no	—
104	C₂³×C₃₁	—	4	4	no	—
105	C₂³×C₃₁	—	4	4	no	—
106	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
107	Q₈×C₃₁	—	4	2	no	—
108	Q₈×C₃₁	—	4	2	no	—
109	Q₈×C₃₁	—	4	4	no	—
110	Q₈×C₃₁	—	8	8	no	—
111	D₄×C₃₁	—	4	2	no	—
112	D₄×C₃₁	—	4	2	no	—
113	D₄×C₃₁	—	4	4	no	—
114	C₂×C₄×C₃₁	—	2	2	no	—
115	C₂×C₄×C₃₁	—	2	2	no	—
116	C₂×C₄×C₃₁	—	2	4	no	—
117	C₂×C₄×C₃₁	—	2	4	no	—
118	C₂×C₄×C₃₁	—	2	4	no	—
119	C₂×C₄×C₃₁	—	4	2	no	—
120	C₂×C₄×C₃₁	—	4	4	no	—
121	C₂×C₄×C₃₁	—	4	4	no	—
122	C₂×C₄×C₃₁	—	4	8	no	—
123	C₂×C₄×C₃₁	#127	1	2	yes	—
124	C₂×C₄×C₃₁	#127	1	2	yes	—
125	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
128	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
129	C₈×C₃₁	—	2	4	no	—
130	C₈×C₃₁	#124	1	4	yes	—
131	C₈×C₃₁	#124	1	4	yes	—
132	C₂×#500	C₂×C₃₁	1	1	yes	characteristic, complemented
133	C₂×#495	C₂×C₃₁	1	1	yes	characteristic
134	C₂×#489	C₂×C₃₁	1	2	yes	complemented
135	#2640	C₂×C₃₁	1	2	yes	complemented
136	#1973	C₂×C₃₁	1	4	yes	complemented
137	#1970	C₂×C₃₁	1	2	yes	—
138	#1969	C₂×C₃₁	1	2	yes	complemented
139	#1726	C₂×C₃₁	1	1	yes	characteristic, complemented
140	C₂²×C₃₁	—	8	2	no	—
141	C₂²×C₃₁	—	8	2	no	—
142	C₂²×C₃₁	—	8	2	no	—
143	C₂²×C₃₁	—	8	8	no	—
144	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
145	C₂²×C₃₁	#532	1	2	yes	—
146	C₂²×C₃₁	#576	1	2	yes	—
147	C₂²×C₃₁	#578	1	2	yes	—
148	C₄×C₃₁	—	2	2	no	—
149	C₄×C₃₁	—	2	2	no	—
150	C₄×C₃₁	—	4	4	no	—
151	C₄×C₃₁	—	8	2	no	—
152	C₄×C₃₁	—	8	4	no	—
153	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
154	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
155	C₄×C₃₁	#532	1	2	yes	—
156	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
157	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
158	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
159	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
160	C₂×#124	C₂²×C₃₁	1	2	yes	—
161	#500	—	2	4	no	—
162	#495	—	2	4	no	—
163	#489	—	2	8	no	—
164	C₂×#107	C₂²×C₃₁	1	2	yes	—
165	C₂×#107	C₂²×C₃₁	1	2	yes	—
166	C₂×#103	C₂²×C₃₁	1	4	yes	—
167	C₂×#102	C₂²×C₃₁	1	4	yes	—
168	C₂×#97	C₂²×C₃₁	1	2	yes	—
169	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
170	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
171	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
172	#397	—	2	4	no	—
173	#393	—	2	4	no	—
174	#391	C₂²×C₃₁	1	4	yes	—
175	#390	C₂²×C₃₁	1	4	yes	—
176	#376	C₂²×C₃₁	1	1	yes	characteristic
177	C₂×C₃₁	—	16	2	no	—
178	C₂×C₃₁	#2858	1	1	yes	characteristic
179	C₂×C₃₁	#3155	1	1	yes	characteristic
180	C₂×C₃₁	#3156	1	1	yes	characteristic
181	C₂×C₃₁	#3175	1	4	yes	—
182	C₂²×Q₈	—	2	2	no	—
183	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
184	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
185	C₂×Q₁₆	—	2	2	no	—
186	C₂×Q₁₆	—	2	4	no	—
187	C₂×SD₁₆	—	2	4	no	—
188	C₂×SD₁₆	—	2	8	no	—
189	C₂×D₈	—	2	2	no	—
190	C₂×D₈	—	2	4	no	—
191	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
192	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
193	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
194	C₄⋊Q₈	—	2	4	no	—
195	C₄⋊Q₈	—	2	4	no	—
196	#124	—	2	4	no	—
197	#124	—	2	4	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
200	C₂×C₂²⋊C₄	—	2	2	no	—
201	C₂×C₂²⋊C₄	—	2	2	no	—
202	C₂×C₂²⋊C₄	—	2	8	no	—
203	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
204	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
205	#107	—	2	4	no	—
206	#107	—	2	4	no	—
207	#103	—	2	8	no	—
208	#102	—	2	8	no	—
209	#97	—	2	4	no	—
210	#97	D₄×C₃₁	1	4	yes	—
211	C₄×C₈	—	2	4	no	—
212	C₄×C₈	D₄×C₃₁	1	4	yes	—
213	#95	—	2	8	no	—
214	#95	D₄×C₃₁	1	2	yes	—
215	#95	D₄×C₃₁	1	2	yes	—
216	C₃₁	#33247	1	1	yes	characteristic, direct product factor, Sylow
217	C₂⁴	—	2	2	no	—
218	C₂×Q₈	—	2	2	no	—
219	C₂×Q₈	—	2	2	no	—
220	C₂×Q₈	—	2	4	no	—
221	C₂×Q₈	—	2	4	no	—
222	C₂×Q₈	—	4	4	no	—
223	C₂×Q₈	—	8	4	no	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	2	4	no	—
227	C₂×D₄	—	2	4	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
231	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
233	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
234	Q₁₆	—	4	4	no	—
235	SD₁₆	—	4	8	no	—
236	D₈	—	4	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
240	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
241	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
242	C₂×C₈	#41×C₃₁	1	4	yes	—
243	C₂×C₈	#41×C₃₁	1	4	yes	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	2	4	no	—
246	C₄⋊C₄	—	4	4	no	—
247	C₂²⋊C₄	—	4	4	no	—
248	C₂²⋊C₄	—	4	4	no	—
249	C₂²⋊C₄	—	8	8	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	C₂×D₄×C₃₁	1	2	yes	—
253	C₄²	C₂×D₄×C₃₁	1	2	yes	—
254	C₂³	—	4	2	no	—
255	C₂³	—	4	4	no	—
256	C₂³	—	4	4	no	—
257	C₂³	—	4	4	no	—
258	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
259	Q₈	—	4	2	no	—
260	Q₈	—	4	2	no	—
261	Q₈	—	4	4	no	—
262	Q₈	—	8	8	no	—
263	D₄	—	4	2	no	—
264	D₄	—	4	2	no	—
265	D₄	—	4	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	8	no	—
275	C₂×C₄	C₃₁×#127	1	2	yes	—
276	C₂×C₄	C₃₁×#127	1	2	yes	—
277	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
278	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
279	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
280	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
281	C₈	—	2	4	no	—
282	C₈	C₃₁×#124	1	4	yes	—
283	C₈	C₃₁×#124	1	4	yes	—
284	C₂²	—	8	2	no	—
285	C₂²	—	8	2	no	—
286	C₂²	—	8	2	no	—
287	C₂²	—	8	8	no	—
288	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
289	C₂²	C₃₁×#532	1	2	yes	—
290	C₂²	C₃₁×#576	1	2	yes	—
291	C₂²	C₃₁×#578	1	2	yes	—
292	C₄	—	2	2	no	—
293	C₄	—	2	2	no	—
294	C₄	—	4	4	no	—
295	C₄	—	8	2	no	—
296	C₄	—	8	4	no	—
297	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
298	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
299	C₄	C₃₁×#532	1	2	yes	—
300	C₂	—	16	2	no	—
301	C₂	C₃₁×#2858	1	1	yes	characteristic
302	C₂	C₃₁×#3155	1	1	yes	characteristic
303	C₂	C₃₁×#3156	1	1	yes	characteristic
304	C₂	C₃₁×#3175	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×D₈×C₃₁	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₃₁×#376 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#500	C₂	semidirect
3. C₂×C₃₁×#489	C₂	semidirect
4. C₃₁×#2640	C₂	semidirect
5. C₃₁×#1973	C₂	semidirect
7. C₃₁×#1969	C₂	semidirect
8. C₃₁×#1726	C₂	semidirect
101. #33247	C₃₁	direct (N × H)
132. C₂×#500	C₂×C₃₁	semidirect
134. C₂×#489	C₂×C₃₁	semidirect
135. #2640	C₂×C₃₁	semidirect
136. #1973	C₂×C₃₁	semidirect
138. #1969	C₂×C₃₁	semidirect
139. #1726	C₂×C₃₁	semidirect
216. C₃₁	#33247	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×#500	2
C₂×C₃₁×#495	2
C₂×C₃₁×#489	2
C₃₁×#2640	2
C₃₁×#1973	2
C₃₁×#1970	2
C₃₁×#1969	2
C₃₁×#1726	2
#33247	31

Subgroups — C31×#33247 #21490006

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33247 #21490006