Subgroups — C₃₁×#20814 #21477573

Order 7936.

906 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#432	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#431	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#376	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1843	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1832	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1797	C₂	1	4	yes	maximal
7	C₃₁×#1768	C₂	1	4	yes	maximal, complemented
8	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
9	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
10	C₃₁×#432	—	2	4	no	—
11	C₃₁×#431	—	2	4	no	—
12	C₂×C₃₁×C₄⋊C₈	C₂²	1	8	yes	—
13	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
17	C₂×C₃₁×#97	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
19	C₃₁×#376	C₄	1	2	yes	—
20	C₃₁×#376	C₄	1	2	yes	—
21	C₃₁×#376	C₂²	1	1	yes	characteristic
22	C₃₁×#376	C₂²	1	1	yes	characteristic
23	C₃₁×#376	C₂²	1	2	yes	—
24	C₂×C₃₁×#95	C₄	1	2	yes	—
25	C₂×C₃₁×#95	C₄	1	2	yes	—
26	C₃₁×#336	C₂²	1	8	yes	—
27	C₂³×C₄×C₃₁	C₂×C₄	1	4	yes	—
28	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
32	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
33	C₂×C₄²×C₃₁	—	2	8	no	—
34	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
35	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₃₁	D₄	1	4	yes	—
38	C₂×C₄²×C₃₁	Q₈	1	4	yes	—
39	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
40	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
41	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
43	C₃₁×C₄⋊C₈	—	2	4	no	—
44	C₃₁×C₄⋊C₈	—	2	4	no	—
45	C₃₁×C₄⋊C₈	—	2	16	no	—
46	C₃₁×C₂²⋊C₈	—	2	4	no	—
47	C₃₁×C₂²⋊C₈	—	2	4	no	—
48	C₃₁×#97	—	2	4	no	—
49	C₄×C₈×C₃₁	—	2	4	no	—
50	C₃₁×#95	C₂×C₄	1	2	yes	—
51	C₃₁×#95	C₂×C₄	1	2	yes	—
52	C₃₁×#95	C₂×C₄	1	4	yes	—
53	C₃₁×#95	C₂×C₄	1	4	yes	—
54	C₃₁×#95	C₂×C₄	1	2	yes	—
55	C₃₁×#95	C₂×C₄	1	2	yes	—
56	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	—	2	4	no	—
58	C₂²×C₄×C₃₁	—	2	4	no	—
59	C₂²×C₄×C₃₁	—	2	8	no	—
60	C₂²×C₄×C₃₁	—	2	8	no	—
61	C₂²×C₄×C₃₁	—	4	2	no	—
62	C₂²×C₄×C₃₁	—	4	4	no	—
63	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
64	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
65	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
74	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
75	C₂×C₈×C₃₁	—	2	4	no	—
76	C₂×C₈×C₃₁	—	2	4	no	—
77	C₂×C₈×C₃₁	—	2	8	no	—
78	C₂×C₈×C₃₁	—	2	8	no	—
79	C₄²×C₃₁	—	2	8	no	—
80	C₄²×C₃₁	—	2	4	no	—
81	C₄²×C₃₁	C₄⋊C₄	1	4	yes	—
82	C₄²×C₃₁	C₄⋊C₄	1	4	yes	—
83	C₄²×C₃₁	C₂×D₄	1	2	yes	—
84	C₄²×C₃₁	C₂×D₄	1	2	yes	—
85	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
86	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
87	C₄²×C₃₁	#41	1	8	yes	—
88	#20814	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂³×C₃₁	—	2	2	no	—
90	C₂³×C₃₁	—	2	2	no	—
91	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
92	C₂³×C₃₁	#131	1	4	yes	—
93	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₃₁	—	2	2	no	—
95	C₂×C₄×C₃₁	—	2	2	no	—
96	C₂×C₄×C₃₁	—	2	2	no	—
97	C₂×C₄×C₃₁	—	2	4	no	—
98	C₂×C₄×C₃₁	—	2	2	no	—
99	C₂×C₄×C₃₁	—	2	2	no	—
100	C₂×C₄×C₃₁	—	2	4	no	—
101	C₂×C₄×C₃₁	—	2	8	no	—
102	C₂×C₄×C₃₁	—	2	4	no	—
103	C₂×C₄×C₃₁	—	2	4	no	—
104	C₂×C₄×C₃₁	—	2	4	no	—
105	C₂×C₄×C₃₁	—	2	4	no	—
106	C₂×C₄×C₃₁	—	2	4	no	—
107	C₂×C₄×C₃₁	—	4	4	no	—
108	C₂×C₄×C₃₁	—	4	16	no	—
109	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₃₁	#117	1	8	yes	—
115	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
117	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
118	C₈×C₃₁	—	4	4	no	—
119	C₈×C₃₁	—	4	4	no	—
120	C₂×#432	C₂×C₃₁	1	1	yes	characteristic
121	C₂×#431	C₂×C₃₁	1	1	yes	characteristic
122	C₂×#376	C₂×C₃₁	1	1	yes	characteristic
123	#1843	C₂×C₃₁	1	2	yes	complemented
124	#1832	C₂×C₃₁	1	2	yes	complemented
125	#1797	C₂×C₃₁	1	4	yes	—
126	#1768	C₂×C₃₁	1	4	yes	complemented
127	C₂²×C₃₁	—	2	2	no	—
128	C₂²×C₃₁	—	2	8	no	—
129	C₂²×C₃₁	—	2	2	no	—
130	C₂²×C₃₁	#424	1	4	yes	—
131	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
132	C₂²×C₃₁	C₂×#117	1	2	yes	—
133	C₂²×C₃₁	C₂×#131	1	2	yes	—
134	C₂²×C₃₁	#568	1	2	yes	—
135	C₄×C₃₁	—	2	2	no	—
136	C₄×C₃₁	—	2	2	no	—
137	C₄×C₃₁	—	2	2	no	—
138	C₄×C₃₁	—	2	2	no	—
139	C₄×C₃₁	—	2	8	no	—
140	C₄×C₃₁	—	2	2	no	—
141	C₄×C₃₁	—	4	4	no	—
142	C₄×C₃₁	#532	1	2	yes	—
143	C₄×C₃₁	#533	1	2	yes	—
144	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
145	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
146	#432	—	2	4	no	—
147	#431	—	2	4	no	—
148	C₂×C₄⋊C₈	C₂²×C₃₁	1	8	yes	—
149	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
150	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
151	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
152	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
153	C₂×#97	C₂²×C₃₁	1	2	yes	—
154	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
155	#376	C₄×C₃₁	1	2	yes	—
156	#376	C₄×C₃₁	1	2	yes	—
157	#376	C₂²×C₃₁	1	1	yes	characteristic
158	#376	C₂²×C₃₁	1	1	yes	characteristic
159	#376	C₂²×C₃₁	1	2	yes	—
160	C₂×#95	C₄×C₃₁	1	2	yes	—
161	C₂×#95	C₄×C₃₁	1	2	yes	—
162	#336	C₂²×C₃₁	1	8	yes	—
163	C₂×C₃₁	—	2	4	no	—
164	C₂×C₃₁	#2285	1	1	yes	characteristic
165	C₂×C₃₁	C₂×#424	1	1	yes	characteristic
166	C₂×C₃₁	#2905	1	1	yes	characteristic
167	C₂×C₃₁	#2924	1	4	yes	—
168	C₂³×C₄	C₂×C₄×C₃₁	1	4	yes	—
169	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
170	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
171	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
172	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
173	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
174	C₂×C₄²	—	2	8	no	—
175	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
176	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
177	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
178	C₂×C₄²	D₄×C₃₁	1	4	yes	—
179	C₂×C₄²	Q₈×C₃₁	1	4	yes	—
180	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
181	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
182	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
183	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
184	C₄⋊C₈	—	2	4	no	—
185	C₄⋊C₈	—	2	4	no	—
186	C₄⋊C₈	—	2	16	no	—
187	C₂²⋊C₈	—	2	4	no	—
188	C₂²⋊C₈	—	2	4	no	—
189	#97	—	2	4	no	—
190	C₄×C₈	—	2	4	no	—
191	#95	C₂×C₄×C₃₁	1	2	yes	—
192	#95	C₂×C₄×C₃₁	1	2	yes	—
193	#95	C₂×C₄×C₃₁	1	4	yes	—
194	#95	C₂×C₄×C₃₁	1	4	yes	—
195	#95	C₂×C₄×C₃₁	1	2	yes	—
196	#95	C₂×C₄×C₃₁	1	2	yes	—
197	C₃₁	#20814	1	1	yes	characteristic, direct product factor, Sylow
198	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	8	no	—
202	C₂²×C₄	—	2	8	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	—	4	4	no	—
205	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
206	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
207	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
208	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
209	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
210	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
211	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
212	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
213	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
214	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
215	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
216	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	8	no	—
220	C₂×C₈	—	2	8	no	—
221	C₄²	—	2	8	no	—
222	C₄²	—	2	4	no	—
223	C₄²	C₄⋊C₄×C₃₁	1	4	yes	—
224	C₄²	C₄⋊C₄×C₃₁	1	4	yes	—
225	C₄²	C₂×D₄×C₃₁	1	2	yes	—
226	C₄²	C₂×D₄×C₃₁	1	2	yes	—
227	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
228	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
229	C₄²	#41×C₃₁	1	8	yes	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	2	no	—
232	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
233	C₂³	C₃₁×#131	1	4	yes	—
234	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	16	no	—
250	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
251	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
252	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
253	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
254	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
255	C₂×C₄	C₃₁×#117	1	8	yes	—
256	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
257	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
258	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
259	C₈	—	4	4	no	—
260	C₈	—	4	4	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	8	no	—
263	C₂²	—	2	2	no	—
264	C₂²	C₃₁×#424	1	4	yes	—
265	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
266	C₂²	C₂×C₃₁×#117	1	2	yes	—
267	C₂²	C₂×C₃₁×#131	1	2	yes	—
268	C₂²	C₃₁×#568	1	2	yes	—
269	C₄	—	2	2	no	—
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	8	no	—
274	C₄	—	2	2	no	—
275	C₄	—	4	4	no	—
276	C₄	C₃₁×#532	1	2	yes	—
277	C₄	C₃₁×#533	1	2	yes	—
278	C₂	—	2	4	no	—
279	C₂	C₃₁×#2285	1	1	yes	characteristic
280	C₂	C₂×C₃₁×#424	1	1	yes	characteristic
281	C₂	C₃₁×#2905	1	1	yes	characteristic
282	C₂	C₃₁×#2924	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1843	C₂	semidirect
5. C₃₁×#1832	C₂	semidirect
7. C₃₁×#1768	C₂	semidirect
88. #20814	C₃₁	direct (N × H)
123. #1843	C₂×C₃₁	semidirect
124. #1832	C₂×C₃₁	semidirect
126. #1768	C₂×C₃₁	semidirect
197. C₃₁	#20814	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×#432	2
C₂×C₃₁×#431	2
C₂×C₃₁×#376	2
C₃₁×#1843	2
C₃₁×#1832	2
C₃₁×#1797	2
C₃₁×#1768	2
#20814	31

Subgroups — C31×#20814 #21477573

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20814 #21477573