Subgroups — C₃₁×#21325 #21478084

Order 7936.

1166 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#125	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#2285	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1992	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1991	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1736	C₂	1	4	yes	maximal
7	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
8	C₂×C₃₁×#125	—	2	2	no	—
9	C₂×C₃₁×#125	—	2	4	no	—
10	C₂×C₃₁×#125	C₄	1	8	yes	complemented
11	C₂×C₃₁×#125	C₂²	1	2	yes	—
12	C₂×C₃₁×#125	C₂²	1	2	yes	—
13	C₂×C₃₁×#117	C₂²	1	2	yes	—
14	C₂²×C₄⋊C₄×C₃₁	—	2	2	no	—
15	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
16	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
17	C₄×M₁₆×C₃₁	—	2	4	no	—
18	C₂×C₃₁×#97	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
20	C₃₁×#384	C₂²	1	2	yes	—
21	C₃₁×#382	C₂²	1	2	yes	—
22	C₃₁×#380	—	2	2	no	—
23	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
24	C₃₁×#340	C₂²	1	16	yes	—
25	C₂³×C₄×C₃₁	—	2	2	no	—
26	C₂³×C₄×C₃₁	D₄	1	2	yes	—
27	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
28	C₂×M₁₆×C₃₁	—	2	4	no	—
29	C₂×M₁₆×C₃₁	—	2	4	no	—
30	C₂×M₁₆×C₃₁	Q₈	1	8	yes	—
31	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
32	C₃₁×#125	—	2	16	no	—
33	C₃₁×#125	—	2	4	no	—
34	C₃₁×#125	—	2	4	no	—
35	C₃₁×#125	C₂×C₄	1	8	yes	—
36	C₃₁×#125	C₂×C₄	1	8	yes	—
37	C₃₁×#117	—	2	8	no	—
38	C₂×C₄⋊C₄×C₃₁	—	2	16	no	—
39	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
40	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
41	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
44	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	8	yes	—
45	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
46	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
47	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
48	C₂×C₄²×C₃₁	—	2	2	no	—
49	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
50	C₂×C₄²×C₃₁	D₄	1	2	yes	—
51	C₂×C₄²×C₃₁	D₄	1	4	yes	—
52	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
54	C₃₁×#97	—	2	4	no	—
55	C₄×C₈×C₃₁	—	2	4	no	—
56	C₃₁×#95	—	2	4	no	—
57	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	—	2	8	no	—
59	C₂²×C₄×C₃₁	—	2	4	no	—
60	C₂²×C₄×C₃₁	—	2	4	no	—
61	C₂²×C₄×C₃₁	—	4	4	no	—
62	C₂²×C₄×C₃₁	—	4	8	no	—
63	C₂²×C₄×C₃₁	C₂²⋊C₄	1	8	yes	—
64	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
70	C₂²×C₄×C₃₁	#41	1	4	yes	—
71	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
72	M₁₆×C₃₁	—	2	16	no	—
73	C₂×C₈×C₃₁	—	2	4	no	—
74	C₂×C₈×C₃₁	—	2	4	no	—
75	C₂×C₈×C₃₁	C₂×Q₈	1	8	yes	—
76	C₄⋊C₄×C₃₁	—	2	16	no	—
77	C₄⋊C₄×C₃₁	—	2	16	no	—
78	C₄⋊C₄×C₃₁	—	2	8	no	—
79	C₄⋊C₄×C₃₁	—	2	4	no	—
80	C₄⋊C₄×C₃₁	—	2	8	no	—
81	C₄⋊C₄×C₃₁	—	4	8	no	—
82	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
83	C₂²⋊C₄×C₃₁	—	4	4	no	—
84	C₄²×C₃₁	—	4	4	no	—
85	C₄²×C₃₁	C₂²⋊C₄	1	8	yes	—
86	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
87	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
88	C₄²×C₃₁	C₂×D₄	1	2	yes	—
89	C₄²×C₃₁	C₂×D₄	1	2	yes	—
90	#21325	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₃₁	—	2	4	no	—
92	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
93	C₂³×C₃₁	#124	1	2	yes	—
94	C₂³×C₃₁	C₄⋊Q₈	1	2	yes	—
95	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
96	C₂×C₄×C₃₁	—	2	4	no	—
97	C₂×C₄×C₃₁	—	2	4	no	—
98	C₂×C₄×C₃₁	—	2	8	no	—
99	C₂×C₄×C₃₁	—	4	8	no	—
100	C₂×C₄×C₃₁	—	4	16	no	—
101	C₂×C₄×C₃₁	—	4	8	no	—
102	C₂×C₄×C₃₁	—	4	4	no	—
103	C₂×C₄×C₃₁	—	4	8	no	—
104	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
105	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
106	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
107	C₂×C₄×C₃₁	C₄×Q₈	1	8	yes	—
108	C₂×C₄×C₃₁	C₂²⋊Q₈	1	8	yes	—
109	C₂×C₄×C₃₁	#124	1	2	yes	—
110	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
111	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
114	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
115	C₈×C₃₁	—	2	8	no	—
116	C₂²×#125	C₂×C₃₁	1	1	yes	characteristic
117	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic
118	#2285	C₂×C₃₁	1	1	yes	characteristic
119	#1992	C₂×C₃₁	1	4	yes	complemented
120	#1991	C₂×C₃₁	1	4	yes	complemented
121	#1736	C₂×C₃₁	1	4	yes	—
122	C₂²×C₃₁	—	2	4	no	—
123	C₂²×C₃₁	—	2	4	no	—
124	C₂²×C₃₁	—	2	4	no	—
125	C₂²×C₃₁	#391	1	4	yes	—
126	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
127	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
128	C₂²×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
129	C₂²×C₃₁	#575	1	4	yes	—
130	C₄×C₃₁	—	2	4	no	—
131	C₄×C₃₁	—	2	4	no	—
132	C₄×C₃₁	—	4	8	no	—
133	C₄×C₃₁	—	8	4	no	—
134	C₄×C₃₁	#391	1	4	yes	—
135	C₄×C₃₁	C₂×C₄×Q₈	1	2	yes	—
136	C₄×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
137	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
138	C₂×#125	—	2	2	no	—
139	C₂×#125	—	2	4	no	—
140	C₂×#125	C₄×C₃₁	1	8	yes	complemented
141	C₂×#125	C₂²×C₃₁	1	2	yes	—
142	C₂×#125	C₂²×C₃₁	1	2	yes	—
143	C₂×#117	C₂²×C₃₁	1	2	yes	—
144	C₂²×C₄⋊C₄	—	2	2	no	—
145	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
146	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
147	C₄×M₁₆	—	2	4	no	—
148	C₂×#97	C₂²×C₃₁	1	2	yes	—
149	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
150	#384	C₂²×C₃₁	1	2	yes	—
151	#382	C₂²×C₃₁	1	2	yes	—
152	#380	—	2	2	no	—
153	C₄×C₄⋊C₄	—	2	2	no	—
154	#340	C₂²×C₃₁	1	16	yes	—
155	C₂×C₃₁	—	2	4	no	—
156	C₂×C₃₁	C₂×#391	1	1	yes	characteristic
157	C₂×C₃₁	#2896	1	2	yes	—
158	C₂×C₃₁	#3142	1	2	yes	—
159	C₂×C₃₁	#3166	1	2	yes	—
160	C₂³×C₄	—	2	2	no	—
161	C₂³×C₄	D₄×C₃₁	1	2	yes	—
162	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
163	C₂×M₁₆	—	2	4	no	—
164	C₂×M₁₆	—	2	4	no	—
165	C₂×M₁₆	Q₈×C₃₁	1	8	yes	—
166	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
167	#125	—	2	16	no	—
168	#125	—	2	4	no	—
169	#125	—	2	4	no	—
170	#125	C₂×C₄×C₃₁	1	8	yes	—
171	#125	C₂×C₄×C₃₁	1	8	yes	—
172	#117	—	2	8	no	—
173	C₂×C₄⋊C₄	—	2	16	no	—
174	C₂×C₄⋊C₄	—	2	4	no	—
175	C₂×C₄⋊C₄	—	2	4	no	—
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₄⋊C₄	—	2	4	no	—
178	C₂×C₄⋊C₄	—	2	4	no	—
179	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	8	yes	—
180	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
181	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
182	C₂×C₂²⋊C₄	—	2	2	no	—
183	C₂×C₄²	—	2	2	no	—
184	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
185	C₂×C₄²	D₄×C₃₁	1	2	yes	—
186	C₂×C₄²	D₄×C₃₁	1	4	yes	—
187	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
188	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
189	#97	—	2	4	no	—
190	C₄×C₈	—	2	4	no	—
191	#95	—	2	4	no	—
192	C₃₁	#21325	1	1	yes	characteristic, direct product factor, Sylow
193	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
194	C₂²×C₄	—	2	8	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	4	4	no	—
198	C₂²×C₄	—	4	8	no	—
199	C₂²×C₄	C₂²⋊C₄×C₃₁	1	8	yes	—
200	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
201	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
202	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
203	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
204	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
205	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
206	C₂²×C₄	#41×C₃₁	1	4	yes	—
207	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
208	M₁₆	—	2	16	no	—
209	C₂×C₈	—	2	4	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	C₂×Q₈×C₃₁	1	8	yes	—
212	C₄⋊C₄	—	2	16	no	—
213	C₄⋊C₄	—	2	16	no	—
214	C₄⋊C₄	—	2	8	no	—
215	C₄⋊C₄	—	2	4	no	—
216	C₄⋊C₄	—	2	8	no	—
217	C₄⋊C₄	—	4	8	no	—
218	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
219	C₂²⋊C₄	—	4	4	no	—
220	C₄²	—	4	4	no	—
221	C₄²	C₂²⋊C₄×C₃₁	1	8	yes	—
222	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
223	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
224	C₄²	C₂×D₄×C₃₁	1	2	yes	—
225	C₄²	C₂×D₄×C₃₁	1	2	yes	—
226	C₂³	—	2	4	no	—
227	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
228	C₂³	C₃₁×#124	1	2	yes	—
229	C₂³	C₃₁×C₄⋊Q₈	1	2	yes	—
230	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	4	8	no	—
235	C₂×C₄	—	4	16	no	—
236	C₂×C₄	—	4	8	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	8	no	—
239	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
240	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
241	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
242	C₂×C₄	C₄×Q₈×C₃₁	1	8	yes	—
243	C₂×C₄	C₃₁×C₂²⋊Q₈	1	8	yes	—
244	C₂×C₄	C₃₁×#124	1	2	yes	—
245	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
246	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
247	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
248	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
249	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
250	C₈	—	2	8	no	—
251	C₂²	—	2	4	no	—
252	C₂²	—	2	4	no	—
253	C₂²	—	2	4	no	—
254	C₂²	C₃₁×#391	1	4	yes	—
255	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
256	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
257	C₂²	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
258	C₂²	C₃₁×#575	1	4	yes	—
259	C₄	—	2	4	no	—
260	C₄	—	2	4	no	—
261	C₄	—	4	8	no	—
262	C₄	—	8	4	no	—
263	C₄	C₃₁×#391	1	4	yes	—
264	C₄	C₂×C₄×Q₈×C₃₁	1	2	yes	—
265	C₄	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
266	C₂	—	2	4	no	—
267	C₂	C₂×C₃₁×#391	1	1	yes	characteristic
268	C₂	C₃₁×#2896	1	2	yes	—
269	C₂	C₃₁×#3142	1	2	yes	—
270	C₂	C₃₁×#3166	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1992	C₂	semidirect
5. C₃₁×#1991	C₂	semidirect
10. C₂×C₃₁×#125	C₄	semidirect
90. #21325	C₃₁	direct (N × H)
119. #1992	C₂×C₃₁	semidirect
120. #1991	C₂×C₃₁	semidirect
140. C₂×#125	C₄×C₃₁	semidirect
192. C₃₁	#21325	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₃₁×#125	2
C₂×C₄×M₁₆×C₃₁	2
C₃₁×#2285	2
C₃₁×#1992	2
C₃₁×#1991	2
C₃₁×#1736	2
#21325	31

Subgroups — C31×#21325 #21478084

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21325 #21478084