Subgroups — C₃₁×#21269 #21478028

Order 7936.

838 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#433	C₂	1	2	yes	maximal, complemented
2	C₃₁×#2293	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1955	C₂	1	2	yes	maximal
4	C₃₁×#1940	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1923	C₂	1	4	yes	maximal, complemented
6	C₄×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1750	C₂	1	2	yes	maximal, complemented
8	C₄×C₃₁×#97	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₃₁×#117	C₂²	1	2	yes	—
11	C₃₁×#433	—	2	8	no	—
12	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
13	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₃₁×C₂²⋊C₈	C₂²	1	4	yes	—
15	C₂×C₃₁×C₂²⋊C₈	C₂²	1	4	yes	—
16	C₂×C₃₁×#97	C₂²	1	2	yes	—
17	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
21	C₃₁×#390	C₄	1	2	yes	—
22	C₃₁×#389	C₄	1	2	yes	—
23	C₃₁×#388	C₄	1	4	yes	—
24	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
25	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
26	C₄³×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#336	C₂²	1	2	yes	—
28	C₃₁×#336	C₂²	1	4	yes	—
29	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
31	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
32	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
33	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
34	C₄×D₄×C₃₁	—	2	4	no	—
35	C₃₁×#117	—	2	4	no	—
36	C₃₁×#117	—	2	4	no	—
37	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
38	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
40	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
41	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
42	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
45	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
47	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
49	C₃₁×C₄⋊C₈	—	2	8	no	—
50	C₃₁×C₄⋊C₈	—	2	4	no	—
51	C₃₁×C₂²⋊C₈	—	2	8	no	—
52	C₃₁×C₂²⋊C₈	—	2	8	no	—
53	C₃₁×#97	—	2	4	no	—
54	C₃₁×#97	—	2	2	no	—
55	C₃₁×#97	D₄	1	4	yes	complemented
56	C₄×C₈×C₃₁	—	2	4	no	—
57	C₄×C₈×C₃₁	—	2	4	no	—
58	C₃₁×#95	C₂×C₄	1	4	yes	—
59	C₃₁×#95	C₂×C₄	1	2	yes	—
60	C₂⁴×C₃₁	C₂²×C₄	1	2	yes	—
61	C₂×D₄×C₃₁	—	2	2	no	—
62	C₂×D₄×C₃₁	—	4	2	no	—
63	C₂²×C₄×C₃₁	—	2	2	no	—
64	C₂²×C₄×C₃₁	—	2	4	no	—
65	C₂²×C₄×C₃₁	—	2	2	no	—
66	C₂²×C₄×C₃₁	—	2	4	no	—
67	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
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	2	yes	—
72	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
74	C₂×C₈×C₃₁	—	2	2	no	—
75	C₂×C₈×C₃₁	—	2	2	no	—
76	C₂×C₈×C₃₁	—	2	2	no	—
77	C₂×C₈×C₃₁	—	2	2	no	—
78	C₂×C₈×C₃₁	—	2	4	no	—
79	C₂×C₈×C₃₁	—	2	4	no	—
80	C₂×C₈×C₃₁	—	2	4	no	—
81	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
82	C₂×C₈×C₃₁	#41	1	4	yes	—
83	C₄⋊C₄×C₃₁	—	2	2	no	—
84	C₄⋊C₄×C₃₁	—	2	8	no	—
85	C₄⋊C₄×C₃₁	—	4	2	no	—
86	C₂²⋊C₄×C₃₁	—	2	4	no	—
87	C₂²⋊C₄×C₃₁	—	2	8	no	—
88	C₂²⋊C₄×C₃₁	—	2	8	no	—
89	C₄²×C₃₁	—	2	4	no	—
90	C₄²×C₃₁	—	2	2	no	—
91	C₄²×C₃₁	—	2	2	no	—
92	C₄²×C₃₁	—	2	2	no	—
93	C₄²×C₃₁	—	4	2	no	—
94	C₄²×C₃₁	C₂×D₄	1	2	yes	—
95	C₄²×C₃₁	#41	1	4	yes	—
96	C₄²×C₃₁	#41	1	2	yes	—
97	#21269	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂³×C₃₁	—	2	2	no	—
99	C₂³×C₃₁	—	2	2	no	—
100	C₂³×C₃₁	—	2	2	no	—
101	C₂³×C₃₁	—	2	4	no	—
102	C₂³×C₃₁	—	2	4	no	—
103	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
104	D₄×C₃₁	—	4	4	no	—
105	C₂×C₄×C₃₁	—	2	2	no	—
106	C₂×C₄×C₃₁	—	2	2	no	—
107	C₂×C₄×C₃₁	—	2	2	no	—
108	C₂×C₄×C₃₁	—	2	2	no	—
109	C₂×C₄×C₃₁	—	2	2	no	—
110	C₂×C₄×C₃₁	—	2	2	no	—
111	C₂×C₄×C₃₁	—	2	2	no	—
112	C₂×C₄×C₃₁	—	2	4	no	—
113	C₂×C₄×C₃₁	—	2	2	no	—
114	C₂×C₄×C₃₁	—	2	2	no	—
115	C₂×C₄×C₃₁	—	2	4	no	—
116	C₂×C₄×C₃₁	—	2	4	no	—
117	C₂×C₄×C₃₁	—	4	4	no	—
118	C₂×C₄×C₃₁	—	4	4	no	—
119	C₂×C₄×C₃₁	#117	1	2	yes	—
120	C₂×C₄×C₃₁	#117	1	2	yes	—
121	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
122	C₂×C₄×C₃₁	#131	1	4	yes	—
123	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
125	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
126	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
127	C₈×C₃₁	—	2	4	no	—
128	C₈×C₃₁	—	2	4	no	—
129	C₈×C₃₁	—	4	4	no	—
130	C₂×#433	C₂×C₃₁	1	2	yes	complemented
131	#2293	C₂×C₃₁	1	1	yes	characteristic
132	#1955	C₂×C₃₁	1	2	yes	—
133	#1940	C₂×C₃₁	1	2	yes	complemented
134	#1923	C₂×C₃₁	1	4	yes	complemented
135	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
136	#1750	C₂×C₃₁	1	2	yes	complemented
137	C₄×#97	C₂×C₃₁	1	1	yes	characteristic, complemented
138	C₂²×C₃₁	—	4	2	no	—
139	C₂²×C₃₁	—	4	2	no	—
140	C₂²×C₃₁	—	4	4	no	—
141	C₂²×C₃₁	—	4	4	no	—
142	C₂²×C₃₁	—	4	2	no	—
143	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
144	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
145	C₂²×C₃₁	C₄×#41	1	1	yes	characteristic
146	C₂²×C₃₁	C₂×#131	1	2	yes	—
147	C₂²×C₃₁	#568	1	2	yes	—
148	C₄×C₃₁	—	2	2	no	—
149	C₄×C₃₁	—	2	2	no	—
150	C₄×C₃₁	—	2	4	no	—
151	C₄×C₃₁	—	2	4	no	—
152	C₄×C₃₁	—	4	4	no	—
153	C₄×C₃₁	—	4	2	no	—
154	C₄×C₃₁	#529	1	2	yes	—
155	C₄×C₃₁	#532	1	2	yes	—
156	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
157	C₂×#117	C₂²×C₃₁	1	2	yes	—
158	#433	—	2	8	no	—
159	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
160	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
161	C₂×C₂²⋊C₈	C₂²×C₃₁	1	4	yes	—
162	C₂×C₂²⋊C₈	C₂²×C₃₁	1	4	yes	—
163	C₂×#97	C₂²×C₃₁	1	2	yes	—
164	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
165	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
166	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
167	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
168	#390	C₄×C₃₁	1	2	yes	—
169	#389	C₄×C₃₁	1	2	yes	—
170	#388	C₄×C₃₁	1	4	yes	—
171	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
172	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
173	C₄³	C₂²×C₃₁	1	1	yes	characteristic
174	#336	C₂²×C₃₁	1	2	yes	—
175	#336	C₂²×C₃₁	1	4	yes	—
176	C₂×C₃₁	—	8	2	no	—
177	C₂×C₃₁	#2293	1	1	yes	characteristic
178	C₂×C₃₁	#2921	1	1	yes	characteristic
179	C₂×C₃₁	#2922	1	1	yes	characteristic
180	C₂×C₃₁	#2935	1	2	yes	—
181	C₂×C₃₁	#2968	1	2	yes	—
182	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
183	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
184	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
185	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
186	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
187	C₄×D₄	—	2	4	no	—
188	#117	—	2	4	no	—
189	#117	—	2	4	no	—
190	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
191	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
192	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
193	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
194	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
195	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
196	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
197	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
198	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
199	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
200	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
201	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
202	C₄⋊C₈	—	2	8	no	—
203	C₄⋊C₈	—	2	4	no	—
204	C₂²⋊C₈	—	2	8	no	—
205	C₂²⋊C₈	—	2	8	no	—
206	#97	—	2	4	no	—
207	#97	—	2	2	no	—
208	#97	D₄×C₃₁	1	4	yes	complemented
209	C₄×C₈	—	2	4	no	—
210	C₄×C₈	—	2	4	no	—
211	#95	C₂×C₄×C₃₁	1	4	yes	—
212	#95	C₂×C₄×C₃₁	1	2	yes	—
213	C₃₁	#21269	1	1	yes	characteristic, direct product factor, Sylow
214	C₂⁴	C₂²×C₄×C₃₁	1	2	yes	—
215	C₂×D₄	—	2	2	no	—
216	C₂×D₄	—	4	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
222	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
223	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
224	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
225	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
226	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
227	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
236	C₂×C₈	#41×C₃₁	1	4	yes	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	8	no	—
239	C₄⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	2	4	no	—
241	C₂²⋊C₄	—	2	8	no	—
242	C₂²⋊C₄	—	2	8	no	—
243	C₄²	—	2	4	no	—
244	C₄²	—	2	2	no	—
245	C₄²	—	2	2	no	—
246	C₄²	—	2	2	no	—
247	C₄²	—	4	2	no	—
248	C₄²	C₂×D₄×C₃₁	1	2	yes	—
249	C₄²	#41×C₃₁	1	4	yes	—
250	C₄²	#41×C₃₁	1	2	yes	—
251	C₂³	—	2	2	no	—
252	C₂³	—	2	2	no	—
253	C₂³	—	2	2	no	—
254	C₂³	—	2	4	no	—
255	C₂³	—	2	4	no	—
256	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
257	D₄	—	4	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	C₃₁×#117	1	2	yes	—
273	C₂×C₄	C₃₁×#117	1	2	yes	—
274	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
275	C₂×C₄	C₃₁×#131	1	4	yes	—
276	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
277	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
278	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
279	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
280	C₈	—	2	4	no	—
281	C₈	—	2	4	no	—
282	C₈	—	4	4	no	—
283	C₂²	—	4	2	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	4	no	—
286	C₂²	—	4	4	no	—
287	C₂²	—	4	2	no	—
288	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
289	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
290	C₂²	C₄×#41×C₃₁	1	1	yes	characteristic
291	C₂²	C₂×C₃₁×#131	1	2	yes	—
292	C₂²	C₃₁×#568	1	2	yes	—
293	C₄	—	2	2	no	—
294	C₄	—	2	2	no	—
295	C₄	—	2	4	no	—
296	C₄	—	2	4	no	—
297	C₄	—	4	4	no	—
298	C₄	—	4	2	no	—
299	C₄	C₃₁×#529	1	2	yes	—
300	C₄	C₃₁×#532	1	2	yes	—
301	C₂	—	8	2	no	—
302	C₂	C₃₁×#2293	1	1	yes	characteristic
303	C₂	C₃₁×#2921	1	1	yes	characteristic
304	C₂	C₃₁×#2922	1	1	yes	characteristic
305	C₂	C₃₁×#2935	1	2	yes	—
306	C₂	C₃₁×#2968	1	2	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	C₂²×D₄×C₃₁	C₂×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
1. C₂×C₃₁×#433	C₂	semidirect
4. C₃₁×#1940	C₂	semidirect
5. C₃₁×#1923	C₂	semidirect
6. C₄×C₃₁×C₄⋊C₈	C₂	semidirect
7. C₃₁×#1750	C₂	semidirect
8. C₄×C₃₁×#97	C₂	semidirect
55. C₃₁×#97	D₄	semidirect
97. #21269	C₃₁	direct (N × H)
130. C₂×#433	C₂×C₃₁	semidirect
133. #1940	C₂×C₃₁	semidirect
134. #1923	C₂×C₃₁	semidirect
135. C₄×C₄⋊C₈	C₂×C₃₁	semidirect
136. #1750	C₂×C₃₁	semidirect
137. C₄×#97	C₂×C₃₁	semidirect
208. #97	D₄×C₃₁	semidirect
213. C₃₁	#21269	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×#433	2
C₃₁×#2293	2
C₃₁×#1955	2
C₃₁×#1940	2
C₃₁×#1923	2
C₄×C₃₁×C₄⋊C₈	2
C₃₁×#1750	2
C₄×C₃₁×#97	2
#21269	31

Subgroups — C31×#21269 #21478028

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21269 #21478028