Subgroups — C₃₁×#33245 #21490004

Order 7936.

922 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#498	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×#1972	C₂	1	2	yes	maximal
6	C₃₁×#1971	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1968	C₂	1	4	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×#498	—	2	4	no	—
15	C₃₁×#495	—	2	4	no	—
16	C₃₁×#489	—	2	8	no	—
17	C₂×C₃₁×#106	C₂²	1	2	yes	—
18	C₂×C₃₁×#106	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×#106	—	2	4	no	—
54	C₃₁×#106	—	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	4	yes	—
63	C₂⁴×C₃₁	—	2	2	no	—
64	C₂×Q₈×C₃₁	—	2	2	no	—
65	C₂×Q₈×C₃₁	—	2	2	no	—
66	C₂×Q₈×C₃₁	—	2	4	no	—
67	C₂×Q₈×C₃₁	—	2	4	no	—
68	C₂×Q₈×C₃₁	—	4	4	no	—
69	C₂×Q₈×C₃₁	—	8	4	no	—
70	C₂×D₄×C₃₁	—	2	2	no	—
71	C₂×D₄×C₃₁	—	2	2	no	—
72	C₂×D₄×C₃₁	—	2	4	no	—
73	C₂×D₄×C₃₁	—	2	4	no	—
74	C₂²×C₄×C₃₁	—	2	2	no	—
75	C₂²×C₄×C₃₁	—	2	4	no	—
76	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
77	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
79	Q₁₆×C₃₁	—	4	4	no	—
80	SD₁₆×C₃₁	—	4	8	no	—
81	D₈×C₃₁	—	4	4	no	—
82	C₂×C₈×C₃₁	—	2	2	no	—
83	C₂×C₈×C₃₁	—	2	2	no	—
84	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
85	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₃₁	#41	1	4	yes	—
88	C₂×C₈×C₃₁	#41	1	4	yes	—
89	C₄⋊C₄×C₃₁	—	2	4	no	—
90	C₄⋊C₄×C₃₁	—	2	4	no	—
91	C₄⋊C₄×C₃₁	—	4	4	no	—
92	C₂²⋊C₄×C₃₁	—	4	4	no	—
93	C₂²⋊C₄×C₃₁	—	4	4	no	—
94	C₂²⋊C₄×C₃₁	—	8	8	no	—
95	C₄²×C₃₁	—	2	2	no	—
96	C₄²×C₃₁	—	2	2	no	—
97	C₄²×C₃₁	C₂×D₄	1	2	yes	—
98	C₄²×C₃₁	C₂×D₄	1	2	yes	—
99	#33245	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂³×C₃₁	—	4	2	no	—
101	C₂³×C₃₁	—	4	4	no	—
102	C₂³×C₃₁	—	4	4	no	—
103	C₂³×C₃₁	—	4	4	no	—
104	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
105	Q₈×C₃₁	—	4	2	no	—
106	Q₈×C₃₁	—	4	2	no	—
107	Q₈×C₃₁	—	4	4	no	—
108	Q₈×C₃₁	—	8	8	no	—
109	D₄×C₃₁	—	4	2	no	—
110	D₄×C₃₁	—	4	2	no	—
111	D₄×C₃₁	—	4	4	no	—
112	C₂×C₄×C₃₁	—	2	4	no	—
113	C₂×C₄×C₃₁	—	2	4	no	—
114	C₂×C₄×C₃₁	—	2	8	no	—
115	C₂×C₄×C₃₁	—	4	2	no	—
116	C₂×C₄×C₃₁	—	4	4	no	—
117	C₂×C₄×C₃₁	—	4	4	no	—
118	C₂×C₄×C₃₁	—	4	8	no	—
119	C₂×C₄×C₃₁	#127	1	4	yes	—
120	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
123	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
124	C₈×C₃₁	—	2	4	no	—
125	C₈×C₃₁	#124	1	4	yes	—
126	C₈×C₃₁	#124	1	4	yes	—
127	C₂×#498	C₂×C₃₁	1	1	yes	characteristic, complemented
128	C₂×#495	C₂×C₃₁	1	1	yes	characteristic
129	C₂×#489	C₂×C₃₁	1	2	yes	complemented
130	#2640	C₂×C₃₁	1	2	yes	complemented
131	#1972	C₂×C₃₁	1	2	yes	—
132	#1971	C₂×C₃₁	1	2	yes	complemented
133	#1968	C₂×C₃₁	1	4	yes	complemented
134	#1726	C₂×C₃₁	1	1	yes	characteristic, complemented
135	C₂²×C₃₁	—	8	2	no	—
136	C₂²×C₃₁	—	8	2	no	—
137	C₂²×C₃₁	—	8	2	no	—
138	C₂²×C₃₁	—	8	8	no	—
139	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
140	C₂²×C₃₁	#532	1	2	yes	—
141	C₂²×C₃₁	#577	1	4	yes	—
142	C₄×C₃₁	—	2	4	no	—
143	C₄×C₃₁	—	4	4	no	—
144	C₄×C₃₁	—	8	2	no	—
145	C₄×C₃₁	—	8	4	no	—
146	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
147	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
148	C₄×C₃₁	#532	1	2	yes	—
149	C₂²×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
150	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	—
151	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
152	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
153	C₂×#124	C₂²×C₃₁	1	2	yes	—
154	#498	—	2	4	no	—
155	#495	—	2	4	no	—
156	#489	—	2	8	no	—
157	C₂×#106	C₂²×C₃₁	1	2	yes	—
158	C₂×#106	C₂²×C₃₁	1	2	yes	—
159	C₂×#103	C₂²×C₃₁	1	4	yes	—
160	C₂×#102	C₂²×C₃₁	1	4	yes	—
161	C₂×#97	C₂²×C₃₁	1	2	yes	—
162	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
163	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
164	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
165	#397	—	2	4	no	—
166	#393	—	2	4	no	—
167	#391	C₂²×C₃₁	1	4	yes	—
168	#390	C₂²×C₃₁	1	4	yes	—
169	#376	C₂²×C₃₁	1	1	yes	characteristic
170	C₂×C₃₁	—	16	2	no	—
171	C₂×C₃₁	#2858	1	1	yes	characteristic
172	C₂×C₃₁	#3154	1	2	yes	—
173	C₂×C₃₁	#3173	1	4	yes	—
174	C₂²×Q₈	—	2	2	no	—
175	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
176	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
177	C₂×Q₁₆	—	2	2	no	—
178	C₂×Q₁₆	—	2	4	no	—
179	C₂×SD₁₆	—	2	4	no	—
180	C₂×SD₁₆	—	2	8	no	—
181	C₂×D₈	—	2	2	no	—
182	C₂×D₈	—	2	4	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	—	2	4	no	—
187	C₄⋊Q₈	—	2	4	no	—
188	#124	—	2	4	no	—
189	#124	—	2	4	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
192	C₂×C₂²⋊C₄	—	2	2	no	—
193	C₂×C₂²⋊C₄	—	2	2	no	—
194	C₂×C₂²⋊C₄	—	2	8	no	—
195	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
196	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
197	#106	—	2	4	no	—
198	#106	—	2	4	no	—
199	#103	—	2	8	no	—
200	#102	—	2	8	no	—
201	#97	—	2	4	no	—
202	#97	D₄×C₃₁	1	4	yes	—
203	C₄×C₈	—	2	4	no	—
204	C₄×C₈	D₄×C₃₁	1	4	yes	—
205	#95	—	2	8	no	—
206	#95	D₄×C₃₁	1	4	yes	—
207	C₃₁	#33245	1	1	yes	characteristic, direct product factor, Sylow
208	C₂⁴	—	2	2	no	—
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	—	2	2	no	—
211	C₂×Q₈	—	2	4	no	—
212	C₂×Q₈	—	2	4	no	—
213	C₂×Q₈	—	4	4	no	—
214	C₂×Q₈	—	8	4	no	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	—	2	2	no	—
217	C₂×D₄	—	2	4	no	—
218	C₂×D₄	—	2	4	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
222	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
223	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
224	Q₁₆	—	4	4	no	—
225	SD₁₆	—	4	8	no	—
226	D₈	—	4	4	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
230	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
231	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
232	C₂×C₈	#41×C₃₁	1	4	yes	—
233	C₂×C₈	#41×C₃₁	1	4	yes	—
234	C₄⋊C₄	—	2	4	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	4	4	no	—
237	C₂²⋊C₄	—	4	4	no	—
238	C₂²⋊C₄	—	4	4	no	—
239	C₂²⋊C₄	—	8	8	no	—
240	C₄²	—	2	2	no	—
241	C₄²	—	2	2	no	—
242	C₄²	C₂×D₄×C₃₁	1	2	yes	—
243	C₄²	C₂×D₄×C₃₁	1	2	yes	—
244	C₂³	—	4	2	no	—
245	C₂³	—	4	4	no	—
246	C₂³	—	4	4	no	—
247	C₂³	—	4	4	no	—
248	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
249	Q₈	—	4	2	no	—
250	Q₈	—	4	2	no	—
251	Q₈	—	4	4	no	—
252	Q₈	—	8	8	no	—
253	D₄	—	4	2	no	—
254	D₄	—	4	2	no	—
255	D₄	—	4	4	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	8	no	—
263	C₂×C₄	C₃₁×#127	1	4	yes	—
264	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
265	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
266	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
267	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
268	C₈	—	2	4	no	—
269	C₈	C₃₁×#124	1	4	yes	—
270	C₈	C₃₁×#124	1	4	yes	—
271	C₂²	—	8	2	no	—
272	C₂²	—	8	2	no	—
273	C₂²	—	8	2	no	—
274	C₂²	—	8	8	no	—
275	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
276	C₂²	C₃₁×#532	1	2	yes	—
277	C₂²	C₃₁×#577	1	4	yes	—
278	C₄	—	2	4	no	—
279	C₄	—	4	4	no	—
280	C₄	—	8	2	no	—
281	C₄	—	8	4	no	—
282	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
283	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
284	C₄	C₃₁×#532	1	2	yes	—
285	C₂	—	16	2	no	—
286	C₂	C₃₁×#2858	1	1	yes	characteristic
287	C₂	C₃₁×#3154	1	2	yes	—
288	C₂	C₃₁×#3173	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×#498	C₂	semidirect
3. C₂×C₃₁×#489	C₂	semidirect
4. C₃₁×#2640	C₂	semidirect
6. C₃₁×#1971	C₂	semidirect
7. C₃₁×#1968	C₂	semidirect
8. C₃₁×#1726	C₂	semidirect
99. #33245	C₃₁	direct (N × H)
127. C₂×#498	C₂×C₃₁	semidirect
129. C₂×#489	C₂×C₃₁	semidirect
130. #2640	C₂×C₃₁	semidirect
132. #1971	C₂×C₃₁	semidirect
133. #1968	C₂×C₃₁	semidirect
134. #1726	C₂×C₃₁	semidirect
207. C₃₁	#33245	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×#498	2
C₂×C₃₁×#495	2
C₂×C₃₁×#489	2
C₃₁×#2640	2
C₃₁×#1972	2
C₃₁×#1971	2
C₃₁×#1968	2
C₃₁×#1726	2
#33245	31

Subgroups — C31×#33245 #21490004

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33245 #21490004