Subgroups — C₃₁×#20430 #21477189

Order 7936.

1138 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#424	C₂	1	2	yes	maximal
2	C₂×C₃₁×#405	C₂	1	2	yes	maximal
3	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1768	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1767	C₂	1	2	yes	maximal, complemented
6	C₄×C₃₁×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1728	C₂	1	1	yes	maximal, characteristic
8	C₂×C₃₁×#336	C₂	1	1	yes	maximal, characteristic
9	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
10	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
11	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
12	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
14	C₃₁×#424	C₄	1	16	yes	complemented
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
17	C₃₁×#405	—	2	8	no	—
18	C₂×C₃₁×#97	C₂²	1	4	yes	—
19	C₂×C₄×C₈×C₃₁	C₂²	1	4	yes	—
20	C₄×C₄⋊C₄×C₃₁	C₄	1	2	yes	—
21	C₄×C₄⋊C₄×C₃₁	C₄	1	2	yes	—
22	C₄×C₂²⋊C₄×C₃₁	C₄	1	4	yes	—
23	C₃₁×#376	C₂²	1	2	yes	—
24	C₄³×C₃₁	C₂²	1	2	yes	—
25	C₃₁×#336	C₂²	1	1	yes	characteristic
26	C₃₁×#336	C₂²	1	1	yes	characteristic
27	C₃₁×#336	C₂²	1	2	yes	—
28	C₃₁×#336	C₂²	1	2	yes	—
29	C₃₁×#336	C₂²	1	2	yes	—
30	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
31	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
32	C₂×M₁₆×C₃₁	—	2	4	no	—
33	C₂×M₁₆×C₃₁	—	2	4	no	—
34	C₂×M₁₆×C₃₁	C₂×C₄	1	8	yes	—
35	C₂²×C₈×C₃₁	—	2	4	no	—
36	C₂²×C₈×C₃₁	—	2	4	no	—
37	C₂²×C₈×C₃₁	C₂×C₄	1	8	yes	—
38	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
39	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
40	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
41	C₃₁×#117	C₂×C₄	1	8	yes	—
42	C₃₁×#117	D₄	1	4	yes	—
43	C₃₁×#117	Q₈	1	4	yes	—
44	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
48	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₃₁	—	2	4	no	—
50	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
51	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
52	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
53	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
56	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
57	C₃₁×C₄⋊C₈	C₂×C₄	1	16	yes	complemented
58	C₃₁×C₄⋊C₈	C₂×C₄	1	16	yes	complemented
59	C₃₁×#97	—	2	8	no	—
60	C₄×C₈×C₃₁	—	2	8	no	—
61	C₃₁×#95	C₂×C₄	1	4	yes	—
62	C₃₁×#95	C₂×C₄	1	4	yes	—
63	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	—	2	4	no	—
65	C₂²×C₄×C₃₁	—	2	4	no	—
66	C₂²×C₄×C₃₁	—	4	2	no	—
67	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₃₁	#41	1	4	yes	—
77	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
78	M₁₆×C₃₁	—	2	16	no	—
79	C₂×C₈×C₃₁	—	2	16	no	—
80	C₂×C₈×C₃₁	—	2	2	no	—
81	C₂×C₈×C₃₁	—	2	4	no	—
82	C₂×C₈×C₃₁	—	2	2	no	—
83	C₂×C₈×C₃₁	—	2	2	no	—
84	C₂×C₈×C₃₁	—	2	2	no	—
85	C₂×C₈×C₃₁	—	2	4	no	—
86	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₃₁	C₂²×C₄	1	8	yes	—
89	C₄⋊C₄×C₃₁	C₄²	1	4	yes	—
90	C₄⋊C₄×C₃₁	C₄²	1	4	yes	—
91	C₄⋊C₄×C₃₁	C₄⋊C₄	1	8	yes	—
92	C₂²⋊C₄×C₃₁	C₄²	1	8	yes	—
93	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	8	yes	—
94	C₄²×C₃₁	—	2	4	no	—
95	C₄²×C₃₁	—	2	8	no	—
96	C₄²×C₃₁	—	2	2	no	—
97	C₄²×C₃₁	—	2	2	no	—
98	C₄²×C₃₁	—	4	4	no	—
99	C₄²×C₃₁	C₂²×C₄	1	8	yes	—
100	C₄²×C₃₁	C₂×D₄	1	4	yes	—
101	C₄²×C₃₁	C₂×Q₈	1	4	yes	—
102	#20430	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂³×C₃₁	—	2	2	no	—
104	C₂³×C₃₁	—	2	2	no	—
105	C₂³×C₃₁	C₂×C₄²	1	2	yes	—
106	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
107	C₂³×C₃₁	#117	1	2	yes	—
108	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₃₁	—	2	2	no	—
110	C₂×C₄×C₃₁	—	2	4	no	—
111	C₂×C₄×C₃₁	—	2	4	no	—
112	C₂×C₄×C₃₁	—	2	2	no	—
113	C₂×C₄×C₃₁	—	2	4	no	—
114	C₂×C₄×C₃₁	—	2	4	no	—
115	C₂×C₄×C₃₁	—	2	2	no	—
116	C₂×C₄×C₃₁	—	2	2	no	—
117	C₂×C₄×C₃₁	—	2	2	no	—
118	C₂×C₄×C₃₁	—	4	8	no	—
119	C₂×C₄×C₃₁	C₂×C₄²	1	4	yes	—
120	C₂×C₄×C₃₁	C₂×C₄²	1	4	yes	—
121	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
122	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	8	yes	—
123	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×C₄×C₃₁	#117	1	2	yes	—
125	C₂×C₄×C₃₁	C₄×D₄	1	8	yes	—
126	C₂×C₄×C₃₁	C₄×Q₈	1	8	yes	—
127	C₂×C₄×C₃₁	#131	1	4	yes	—
128	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
129	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
130	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
131	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
132	C₈×C₃₁	—	2	4	no	—
133	C₈×C₃₁	—	2	4	no	—
134	C₈×C₃₁	—	2	8	no	—
135	C₂×#424	C₂×C₃₁	1	2	yes	—
136	C₂×#405	C₂×C₃₁	1	2	yes	—
137	C₄×#117	C₂×C₃₁	1	1	yes	characteristic
138	#1768	C₂×C₃₁	1	4	yes	complemented
139	#1767	C₂×C₃₁	1	2	yes	complemented
140	C₄×C₄⋊C₈	C₂×C₃₁	1	2	yes	complemented
141	#1728	C₂×C₃₁	1	1	yes	characteristic
142	C₂×#336	C₂×C₃₁	1	1	yes	characteristic
143	C₂²×C₃₁	—	2	2	no	—
144	C₂²×C₃₁	—	2	2	no	—
145	C₂²×C₃₁	—	2	4	no	—
146	C₂²×C₃₁	—	2	4	no	—
147	C₂²×C₃₁	C₄×C₄⋊C₄	1	4	yes	—
148	C₂²×C₃₁	C₂²×C₄²	1	1	yes	characteristic
149	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
150	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
151	C₂²×C₃₁	C₂×#131	1	2	yes	—
152	C₂²×C₃₁	#568	1	2	yes	—
153	C₄×C₃₁	—	2	4	no	—
154	C₄×C₃₁	—	2	4	no	—
155	C₄×C₃₁	—	2	8	no	—
156	C₄×C₃₁	—	2	4	no	—
157	C₄×C₃₁	—	4	2	no	—
158	C₄×C₃₁	C₄×C₄⋊C₄	1	4	yes	—
159	C₄×C₃₁	C₂×C₄×D₄	1	2	yes	—
160	C₄×C₃₁	C₂×C₄×Q₈	1	2	yes	—
161	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
162	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
163	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
164	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
165	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
166	#424	C₄×C₃₁	1	16	yes	complemented
167	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
168	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
169	#405	—	2	8	no	—
170	C₂×#97	C₂²×C₃₁	1	4	yes	—
171	C₂×C₄×C₈	C₂²×C₃₁	1	4	yes	—
172	C₄×C₄⋊C₄	C₄×C₃₁	1	2	yes	—
173	C₄×C₄⋊C₄	C₄×C₃₁	1	2	yes	—
174	C₄×C₂²⋊C₄	C₄×C₃₁	1	4	yes	—
175	#376	C₂²×C₃₁	1	2	yes	—
176	C₄³	C₂²×C₃₁	1	2	yes	—
177	#336	C₂²×C₃₁	1	1	yes	characteristic
178	#336	C₂²×C₃₁	1	1	yes	characteristic
179	#336	C₂²×C₃₁	1	2	yes	—
180	#336	C₂²×C₃₁	1	2	yes	—
181	#336	C₂²×C₃₁	1	2	yes	—
182	C₂×C₃₁	—	2	4	no	—
183	C₂×C₃₁	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
184	C₂×C₃₁	C₄×#131	1	1	yes	characteristic
185	C₂×C₃₁	#2875	1	1	yes	characteristic
186	C₂×C₃₁	#2906	1	2	yes	—
187	C₂×C₃₁	#2924	1	2	yes	—
188	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
189	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
190	C₂×M₁₆	—	2	4	no	—
191	C₂×M₁₆	—	2	4	no	—
192	C₂×M₁₆	C₂×C₄×C₃₁	1	8	yes	—
193	C₂²×C₈	—	2	4	no	—
194	C₂²×C₈	—	2	4	no	—
195	C₂²×C₈	C₂×C₄×C₃₁	1	8	yes	—
196	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
197	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
198	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
199	#117	C₂×C₄×C₃₁	1	8	yes	—
200	#117	D₄×C₃₁	1	4	yes	—
201	#117	Q₈×C₃₁	1	4	yes	—
202	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
203	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
204	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
205	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
206	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
207	C₂×C₄²	—	2	4	no	—
208	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
209	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
210	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
211	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
212	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
213	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
214	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
215	C₄⋊C₈	C₂×C₄×C₃₁	1	16	yes	complemented
216	C₄⋊C₈	C₂×C₄×C₃₁	1	16	yes	complemented
217	#97	—	2	8	no	—
218	C₄×C₈	—	2	8	no	—
219	#95	C₂×C₄×C₃₁	1	4	yes	—
220	#95	C₂×C₄×C₃₁	1	4	yes	—
221	C₃₁	#20430	1	1	yes	characteristic, direct product factor, Sylow
222	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	4	2	no	—
226	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
227	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
228	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
229	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
230	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
231	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
232	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
233	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
234	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
235	C₂²×C₄	#41×C₃₁	1	4	yes	—
236	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
237	M₁₆	—	2	16	no	—
238	C₂×C₈	—	2	16	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
246	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
247	C₂×C₈	C₂²×C₄×C₃₁	1	8	yes	—
248	C₄⋊C₄	C₄²×C₃₁	1	4	yes	—
249	C₄⋊C₄	C₄²×C₃₁	1	4	yes	—
250	C₄⋊C₄	C₄⋊C₄×C₃₁	1	8	yes	—
251	C₂²⋊C₄	C₄²×C₃₁	1	8	yes	—
252	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	8	yes	—
253	C₄²	—	2	4	no	—
254	C₄²	—	2	8	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	4	4	no	—
258	C₄²	C₂²×C₄×C₃₁	1	8	yes	—
259	C₄²	C₂×D₄×C₃₁	1	4	yes	—
260	C₄²	C₂×Q₈×C₃₁	1	4	yes	—
261	C₂³	—	2	2	no	—
262	C₂³	—	2	2	no	—
263	C₂³	C₂×C₄²×C₃₁	1	2	yes	—
264	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
265	C₂³	C₃₁×#117	1	2	yes	—
266	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	4	no	—
272	C₂×C₄	—	2	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	4	8	no	—
277	C₂×C₄	C₂×C₄²×C₃₁	1	4	yes	—
278	C₂×C₄	C₂×C₄²×C₃₁	1	4	yes	—
279	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
280	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	8	yes	—
281	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
282	C₂×C₄	C₃₁×#117	1	2	yes	—
283	C₂×C₄	C₄×D₄×C₃₁	1	8	yes	—
284	C₂×C₄	C₄×Q₈×C₃₁	1	8	yes	—
285	C₂×C₄	C₃₁×#131	1	4	yes	—
286	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
287	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
288	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
289	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
290	C₈	—	2	4	no	—
291	C₈	—	2	4	no	—
292	C₈	—	2	8	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	4	no	—
296	C₂²	—	2	4	no	—
297	C₂²	C₄×C₄⋊C₄×C₃₁	1	4	yes	—
298	C₂²	C₂²×C₄²×C₃₁	1	1	yes	characteristic
299	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
300	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
301	C₂²	C₂×C₃₁×#131	1	2	yes	—
302	C₂²	C₃₁×#568	1	2	yes	—
303	C₄	—	2	4	no	—
304	C₄	—	2	4	no	—
305	C₄	—	2	8	no	—
306	C₄	—	2	4	no	—
307	C₄	—	4	2	no	—
308	C₄	C₄×C₄⋊C₄×C₃₁	1	4	yes	—
309	C₄	C₂×C₄×D₄×C₃₁	1	2	yes	—
310	C₄	C₂×C₄×Q₈×C₃₁	1	2	yes	—
311	C₂	—	2	4	no	—
312	C₂	C₂×C₄×C₄⋊C₄×C₃₁	1	1	yes	characteristic
313	C₂	C₄×C₃₁×#131	1	1	yes	characteristic
314	C₂	C₃₁×#2875	1	1	yes	characteristic
315	C₂	C₃₁×#2906	1	2	yes	—
316	C₂	C₃₁×#2924	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator	C₂²	C₂²×C₄²×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂³×C₄
prime radical	C₂⁴×C₃₁	C₂²×C₄
LCS	G ▷ C₂² ▷ 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×#1768	C₂	semidirect
5. C₃₁×#1767	C₂	semidirect
6. C₄×C₃₁×C₄⋊C₈	C₂	semidirect
14. C₃₁×#424	C₄	semidirect
57. C₃₁×C₄⋊C₈	C₂×C₄	semidirect
58. C₃₁×C₄⋊C₈	C₂×C₄	semidirect
102. #20430	C₃₁	direct (N × H)
138. #1768	C₂×C₃₁	semidirect
139. #1767	C₂×C₃₁	semidirect
140. C₄×C₄⋊C₈	C₂×C₃₁	semidirect
166. #424	C₄×C₃₁	semidirect
215. C₄⋊C₈	C₂×C₄×C₃₁	semidirect
216. C₄⋊C₈	C₂×C₄×C₃₁	semidirect
221. C₃₁	#20430	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×#424	2
C₂×C₃₁×#405	2
C₄×C₃₁×#117	2
C₃₁×#1768	2
C₃₁×#1767	2
C₄×C₃₁×C₄⋊C₈	2
C₃₁×#1728	2
C₂×C₃₁×#336	2
#20430	31

Subgroups — C31×#20430 #21477189

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#20430 #21477189