Subgroups — C₃₁×#22256 #21479015

Order 7936.

614 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3160	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2960	C₂	1	2	yes	maximal
3	C₃₁×#1994	C₂	1	2	yes	maximal
4	C₃₁×#1922	C₂	1	8	yes	maximal
5	C₃₁×#1837	C₂	1	1	yes	maximal, characteristic
6	C₃₁×#1744	C₂	1	1	yes	maximal, characteristic
7	C₃₁×#533	C₂²	1	2	yes	—
8	C₃₁×#500	C₄	1	1	yes	characteristic, complemented
9	C₃₁×#500	C₄	1	1	yes	characteristic, complemented
10	C₃₁×#499	C₄	1	4	yes	complemented
11	C₃₁×#498	C₄	1	2	yes	complemented
12	C₃₁×#446	—	2	2	no	—
13	C₃₁×#446	—	2	2	no	—
14	C₈×Q₈×C₃₁	—	2	2	no	—
15	C₈×Q₈×C₃₁	—	2	2	no	—
16	C₃₁×#427	C₂²	1	4	yes	—
17	C₃₁×#424	C₂²	1	4	yes	—
18	C₃₁×#423	C₂²	1	2	yes	—
19	C₃₁×#405	C₂²	1	4	yes	—
20	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
22	C₄³×C₃₁	C₂²	1	1	yes	characteristic
23	C₃₁×#341	C₂²	1	4	yes	—
24	C₃₁×#339	C₂²	1	8	yes	—
25	C₃₁×#336	C₂²	1	4	yes	—
26	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
27	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
28	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
29	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
30	C₃₁×C₄⋊Q₈	C₂×C₄	1	2	yes	—
31	C₃₁×#125	C₂×C₄	1	4	yes	—
32	C₃₁×C₂²⋊Q₈	—	2	2	no	—
33	C₃₁×C₂²⋊Q₈	—	2	2	no	—
34	C₄×Q₈×C₃₁	—	2	2	no	—
35	C₄×Q₈×C₃₁	—	2	2	no	—
36	C₃₁×#117	C₂³	1	4	yes	—
37	C₂×C₄²×C₃₁	—	2	1	no	—
38	C₂×C₄²×C₃₁	—	2	1	no	—
39	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
40	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
41	C₃₁×#107	C₂×C₄	1	4	yes	—
42	C₃₁×#107	C₂×C₄	1	4	yes	—
43	C₃₁×#106	C₂×C₄	1	8	yes	—
44	C₃₁×C₄⋊C₈	—	2	4	no	—
45	C₃₁×C₄⋊C₈	—	2	4	no	—
46	C₃₁×C₄⋊C₈	—	2	2	no	—
47	C₃₁×C₄⋊C₈	—	2	2	no	—
48	C₃₁×#97	—	2	4	no	—
49	C₄×C₈×C₃₁	—	2	4	no	—
50	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
51	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
54	C₄×C₈×C₃₁	D₄	1	2	yes	—
55	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
56	C₂×Q₈×C₃₁	—	2	2	no	—
57	C₂×Q₈×C₃₁	—	2	2	no	—
58	C₂²×C₄×C₃₁	—	2	2	no	—
59	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
60	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
62	M₁₆×C₃₁	—	2	8	no	—
63	C₂×C₈×C₃₁	—	2	4	no	—
64	C₂×C₈×C₃₁	—	2	4	no	—
65	C₂×C₈×C₃₁	—	2	4	no	—
66	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
70	C₄⋊C₄×C₃₁	—	2	4	no	—
71	C₄⋊C₄×C₃₁	—	2	4	no	—
72	C₄⋊C₄×C₃₁	—	2	2	no	—
73	C₄⋊C₄×C₃₁	—	2	2	no	—
74	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
75	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
76	C₂²⋊C₄×C₃₁	—	2	4	no	—
77	C₄²×C₃₁	—	2	4	no	—
78	C₄²×C₃₁	—	2	4	no	—
79	C₄²×C₃₁	—	2	4	no	—
80	C₄²×C₃₁	—	4	2	no	—
81	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
85	#22256	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
87	Q₈×C₃₁	—	4	4	no	—
88	Q₈×C₃₁	—	4	4	no	—
89	C₂×C₄×C₃₁	—	2	2	no	—
90	C₂×C₄×C₃₁	—	2	2	no	—
91	C₂×C₄×C₃₁	—	2	4	no	—
92	C₂×C₄×C₃₁	—	2	4	no	—
93	C₂×C₄×C₃₁	—	2	4	no	—
94	C₂×C₄×C₃₁	—	4	2	no	—
95	C₂×C₄×C₃₁	—	4	2	no	—
96	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
102	C₂×C₄×C₃₁	2₋¹⁺⁴	1	4	yes	—
103	C₈×C₃₁	—	2	2	no	—
104	C₈×C₃₁	—	2	2	no	—
105	C₈×C₃₁	—	2	2	no	—
106	C₈×C₃₁	—	2	2	no	—
107	C₈×C₃₁	—	4	4	no	—
108	#3160	C₂×C₃₁	1	1	yes	characteristic
109	#2960	C₂×C₃₁	1	2	yes	—
110	#1994	C₂×C₃₁	1	2	yes	—
111	#1922	C₂×C₃₁	1	8	yes	—
112	#1837	C₂×C₃₁	1	1	yes	characteristic
113	#1744	C₂×C₃₁	1	1	yes	characteristic
114	C₂²×C₃₁	—	2	1	no	—
115	C₂²×C₃₁	—	2	1	no	—
116	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
117	C₂²×C₃₁	#541	1	2	yes	—
118	C₄×C₃₁	—	2	1	no	—
119	C₄×C₃₁	—	2	1	no	—
120	C₄×C₃₁	—	4	4	no	—
121	C₄×C₃₁	—	4	4	no	—
122	C₄×C₃₁	—	4	4	no	—
123	C₄×C₃₁	#443	1	2	yes	—
124	C₄×C₃₁	#444	1	2	yes	—
125	C₄×C₃₁	#519	1	2	yes	—
126	C₄×C₃₁	#536	1	2	yes	—
127	#533	C₂²×C₃₁	1	2	yes	—
128	#500	C₄×C₃₁	1	1	yes	characteristic, complemented
129	#500	C₄×C₃₁	1	1	yes	characteristic, complemented
130	#499	C₄×C₃₁	1	4	yes	complemented
131	#498	C₄×C₃₁	1	2	yes	complemented
132	#446	—	2	2	no	—
133	#446	—	2	2	no	—
134	C₈×Q₈	—	2	2	no	—
135	C₈×Q₈	—	2	2	no	—
136	#427	C₂²×C₃₁	1	4	yes	—
137	#424	C₂²×C₃₁	1	4	yes	—
138	#423	C₂²×C₃₁	1	2	yes	—
139	#405	C₂²×C₃₁	1	4	yes	—
140	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
141	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
142	C₄³	C₂²×C₃₁	1	1	yes	characteristic
143	#341	C₂²×C₃₁	1	4	yes	—
144	#339	C₂²×C₃₁	1	8	yes	—
145	#336	C₂²×C₃₁	1	4	yes	—
146	C₂×C₃₁	—	2	2	no	—
147	C₂×C₃₁	#2327	1	1	yes	characteristic
148	C₂×C₃₁	C₂×#443	1	1	yes	characteristic
149	C₂×C₃₁	C₂×#444	1	1	yes	characteristic
150	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
151	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
152	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
153	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
154	C₄⋊Q₈	C₂×C₄×C₃₁	1	2	yes	—
155	#125	C₂×C₄×C₃₁	1	4	yes	—
156	C₂²⋊Q₈	—	2	2	no	—
157	C₂²⋊Q₈	—	2	2	no	—
158	C₄×Q₈	—	2	2	no	—
159	C₄×Q₈	—	2	2	no	—
160	#117	C₂³×C₃₁	1	4	yes	—
161	C₂×C₄²	—	2	1	no	—
162	C₂×C₄²	—	2	1	no	—
163	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
164	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
165	#107	C₂×C₄×C₃₁	1	4	yes	—
166	#107	C₂×C₄×C₃₁	1	4	yes	—
167	#106	C₂×C₄×C₃₁	1	8	yes	—
168	C₄⋊C₈	—	2	4	no	—
169	C₄⋊C₈	—	2	4	no	—
170	C₄⋊C₈	—	2	2	no	—
171	C₄⋊C₈	—	2	2	no	—
172	#97	—	2	4	no	—
173	C₄×C₈	—	2	4	no	—
174	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
175	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
176	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
177	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
178	C₄×C₈	D₄×C₃₁	1	2	yes	—
179	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
180	C₃₁	#22256	1	1	yes	characteristic, direct product factor, Sylow
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	—	2	2	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
185	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
186	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
187	M₁₆	—	2	8	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
192	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
193	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
194	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	2	4	no	—
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
200	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
201	C₂²⋊C₄	—	2	4	no	—
202	C₄²	—	2	4	no	—
203	C₄²	—	2	4	no	—
204	C₄²	—	2	4	no	—
205	C₄²	—	4	2	no	—
206	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
207	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
208	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
209	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
210	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
211	Q₈	—	4	4	no	—
212	Q₈	—	4	4	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	2	4	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
221	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
222	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
223	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
224	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
225	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
226	C₂×C₄	C₃₁×2₋¹⁺⁴	1	4	yes	—
227	C₈	—	2	2	no	—
228	C₈	—	2	2	no	—
229	C₈	—	2	2	no	—
230	C₈	—	2	2	no	—
231	C₈	—	4	4	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	1	no	—
234	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
235	C₂²	C₃₁×#541	1	2	yes	—
236	C₄	—	2	1	no	—
237	C₄	—	2	1	no	—
238	C₄	—	4	4	no	—
239	C₄	—	4	4	no	—
240	C₄	—	4	4	no	—
241	C₄	C₃₁×#443	1	2	yes	—
242	C₄	C₃₁×#444	1	2	yes	—
243	C₄	C₃₁×#519	1	2	yes	—
244	C₄	C₃₁×#536	1	2	yes	—
245	C₂	—	2	2	no	—
246	C₂	C₃₁×#2327	1	1	yes	characteristic
247	C₂	C₂×C₃₁×#443	1	1	yes	characteristic
248	C₂	C₂×C₃₁×#444	1	1	yes	characteristic

	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	C₂³×C₃₁	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₃₁×#500	C₄	semidirect
9. C₃₁×#500	C₄	semidirect
10. C₃₁×#499	C₄	semidirect
11. C₃₁×#498	C₄	semidirect
85. #22256	C₃₁	direct (N × H)
128. #500	C₄×C₃₁	semidirect
129. #500	C₄×C₃₁	semidirect
130. #499	C₄×C₃₁	semidirect
131. #498	C₄×C₃₁	semidirect
180. C₃₁	#22256	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×#3160	2
C₃₁×#2960	2
C₃₁×#1994	2
C₃₁×#1922	2
C₃₁×#1837	2
C₃₁×#1744	2
#22256	31

Subgroups — C31×#22256 #21479015

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22256 #21479015