Subgroups — C₃₁×#21974 #21478733

Order 7936.

898 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#423	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1939	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1851	C₂	1	4	yes	maximal
6	C₃₁×#1842	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1831	C₂	1	2	yes	maximal, complemented
8	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
9	C₂×C₃₁×#117	C₄	1	2	yes	—
10	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#431	—	2	4	no	—
12	C₃₁×#423	—	2	4	no	—
13	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₃₁×C₄⋊C₈	C₂²	1	8	yes	—
18	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
20	C₄×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
22	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
23	C₄×C₂²⋊C₄×C₃₁	C₄	1	2	yes	—
24	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
25	C₃₁×#376	C₄	1	2	yes	—
26	C₄³×C₃₁	C₂²	1	2	yes	—
27	C₃₁×#336	C₂²	1	8	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	C₂×M₁₆×C₃₁	—	2	4	no	—
32	C₂×M₁₆×C₃₁	—	4	4	no	—
33	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
34	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
35	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
36	C₃₁×#117	—	2	8	no	—
37	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
38	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
39	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
41	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₃₁	D₄	1	4	yes	—
46	C₂×C₄²×C₃₁	Q₈	1	4	yes	—
47	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
48	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
49	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
50	C₃₁×C₄⋊C₈	—	2	4	no	—
51	C₃₁×C₄⋊C₈	—	2	4	no	—
52	C₃₁×C₄⋊C₈	—	2	2	no	—
53	C₃₁×C₄⋊C₈	—	2	2	no	—
54	C₃₁×C₄⋊C₈	—	4	8	no	—
55	C₃₁×C₂²⋊C₈	—	2	4	no	—
56	C₄×C₈×C₃₁	—	2	4	no	—
57	C₃₁×#95	C₂×C₄	1	4	yes	—
58	C₃₁×#95	C₂×C₄	1	2	yes	—
59	C₃₁×#95	C₂×C₄	1	2	yes	—
60	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	—	2	4	no	—
62	C₂²×C₄×C₃₁	—	2	2	no	—
63	C₂²×C₄×C₃₁	—	2	2	no	—
64	C₂²×C₄×C₃₁	—	2	2	no	—
65	C₂²×C₄×C₃₁	—	2	2	no	—
66	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
67	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
68	C₂²×C₄×C₃₁	C₂²×C₄	1	4	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	1	yes	characteristic
72	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
77	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
78	M₁₆×C₃₁	—	4	8	no	—
79	C₂×C₈×C₃₁	—	2	2	no	—
80	C₂×C₈×C₃₁	—	2	2	no	—
81	C₂×C₈×C₃₁	—	2	4	no	—
82	C₂×C₈×C₃₁	—	2	4	no	—
83	C₂×C₈×C₃₁	—	2	8	no	—
84	C₂×C₈×C₃₁	—	2	4	no	—
85	C₄⋊C₄×C₃₁	—	2	4	no	—
86	C₄⋊C₄×C₃₁	M₁₆	1	8	yes	complemented
87	C₂²⋊C₄×C₃₁	—	2	8	no	—
88	C₄²×C₃₁	—	2	8	no	—
89	C₄²×C₃₁	—	2	4	no	—
90	C₄²×C₃₁	—	2	4	no	—
91	C₄²×C₃₁	—	2	4	no	—
92	C₄²×C₃₁	—	2	4	no	—
93	C₄²×C₃₁	C₄⋊C₄	1	4	yes	—
94	C₄²×C₃₁	C₄⋊C₄	1	4	yes	—
95	C₄²×C₃₁	C₂×D₄	1	2	yes	—
96	C₄²×C₃₁	C₂×D₄	1	2	yes	—
97	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
98	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
99	#21974	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂³×C₃₁	—	2	2	no	—
101	C₂³×C₃₁	—	2	2	no	—
102	C₂³×C₃₁	—	2	1	no	—
103	C₂³×C₃₁	—	2	1	no	—
104	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
105	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₃₁	—	2	4	no	—
107	C₂×C₄×C₃₁	—	2	2	no	—
108	C₂×C₄×C₃₁	—	2	2	no	—
109	C₂×C₄×C₃₁	—	2	4	no	—
110	C₂×C₄×C₃₁	—	2	1	no	—
111	C₂×C₄×C₃₁	—	2	1	no	—
112	C₂×C₄×C₃₁	—	2	2	no	—
113	C₂×C₄×C₃₁	—	2	2	no	—
114	C₂×C₄×C₃₁	—	2	4	no	—
115	C₂×C₄×C₃₁	—	2	8	no	—
116	C₂×C₄×C₃₁	—	2	8	no	—
117	C₂×C₄×C₃₁	—	2	4	no	—
118	C₂×C₄×C₃₁	—	2	4	no	—
119	C₂×C₄×C₃₁	—	2	4	no	—
120	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
121	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
122	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
123	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×C₄×C₃₁	C₂×M₁₆	1	8	yes	—
126	C₂×C₄×C₃₁	C₂×M₁₆	1	4	yes	—
127	C₂×C₄×C₃₁	#131	1	2	yes	—
128	C₂×C₄×C₃₁	#131	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	H(5,𝔽₂)	1	2	yes	—
132	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
133	C₈×C₃₁	—	4	2	no	—
134	C₈×C₃₁	—	4	4	no	—
135	C₈×C₃₁	—	4	2	no	—
136	C₂×#431	C₂×C₃₁	1	1	yes	characteristic
137	C₂×#423	C₂×C₃₁	1	1	yes	characteristic
138	C₄×#117	C₂×C₃₁	1	1	yes	characteristic
139	#1939	C₂×C₃₁	1	4	yes	complemented
140	#1851	C₂×C₃₁	1	4	yes	—
141	#1842	C₂×C₃₁	1	2	yes	complemented
142	#1831	C₂×C₃₁	1	2	yes	complemented
143	C₂²×C₃₁	—	2	2	no	—
144	C₂²×C₃₁	—	2	2	no	—
145	C₂²×C₃₁	—	2	2	no	—
146	C₂²×C₃₁	—	4	2	no	—
147	C₂²×C₃₁	—	4	2	no	—
148	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
149	C₂²×C₃₁	#520	1	2	yes	—
150	C₂²×C₃₁	C₂²×M₁₆	1	2	yes	—
151	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
152	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
153	C₄×C₃₁	—	2	2	no	—
154	C₄×C₃₁	—	2	2	no	—
155	C₄×C₃₁	—	2	2	no	—
156	C₄×C₃₁	—	2	8	no	—
157	C₄×C₃₁	—	4	4	no	—
158	C₄×C₃₁	#423	1	4	yes	—
159	C₄×C₃₁	#424	1	4	yes	—
160	C₄×C₃₁	#535	1	1	yes	characteristic
161	C₄×C₃₁	#536	1	1	yes	characteristic
162	C₄×C₃₁	#544	1	2	yes	—
163	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
164	C₂×#117	C₄×C₃₁	1	2	yes	—
165	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
166	#431	—	2	4	no	—
167	#423	—	2	4	no	—
168	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
169	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
170	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
171	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
172	C₂×C₄⋊C₈	C₂²×C₃₁	1	8	yes	—
173	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
174	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
175	C₄×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
176	C₄×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
177	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
178	C₄×C₂²⋊C₄	C₄×C₃₁	1	2	yes	—
179	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
180	#376	C₄×C₃₁	1	2	yes	—
181	C₄³	C₂²×C₃₁	1	2	yes	—
182	#336	C₂²×C₃₁	1	8	yes	—
183	C₂×C₃₁	—	4	2	no	—
184	C₂×C₃₁	#2317	1	1	yes	characteristic
185	C₂×C₃₁	C₂×#423	1	1	yes	characteristic
186	C₂×C₃₁	C₂×#424	1	1	yes	characteristic
187	C₂×C₃₁	#2990	1	2	yes	—
188	C₂×C₃₁	#2991	1	2	yes	—
189	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
190	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
191	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
192	C₂×M₁₆	—	2	4	no	—
193	C₂×M₁₆	—	4	4	no	—
194	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
195	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
196	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
197	#117	—	2	8	no	—
198	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
199	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
200	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
201	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
202	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
203	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
204	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
205	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
206	C₂×C₄²	D₄×C₃₁	1	4	yes	—
207	C₂×C₄²	Q₈×C₃₁	1	4	yes	—
208	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
209	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
210	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
211	C₄⋊C₈	—	2	4	no	—
212	C₄⋊C₈	—	2	4	no	—
213	C₄⋊C₈	—	2	2	no	—
214	C₄⋊C₈	—	2	2	no	—
215	C₄⋊C₈	—	4	8	no	—
216	C₂²⋊C₈	—	2	4	no	—
217	C₄×C₈	—	2	4	no	—
218	#95	C₂×C₄×C₃₁	1	4	yes	—
219	#95	C₂×C₄×C₃₁	1	2	yes	—
220	#95	C₂×C₄×C₃₁	1	2	yes	—
221	C₃₁	#21974	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	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
229	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
230	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
231	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
232	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
233	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
234	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
235	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
236	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
238	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
239	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
240	M₁₆	—	4	8	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	2	4	no	—
245	C₂×C₈	—	2	8	no	—
246	C₂×C₈	—	2	4	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	M₁₆×C₃₁	1	8	yes	complemented
249	C₂²⋊C₄	—	2	8	no	—
250	C₄²	—	2	8	no	—
251	C₄²	—	2	4	no	—
252	C₄²	—	2	4	no	—
253	C₄²	—	2	4	no	—
254	C₄²	—	2	4	no	—
255	C₄²	C₄⋊C₄×C₃₁	1	4	yes	—
256	C₄²	C₄⋊C₄×C₃₁	1	4	yes	—
257	C₄²	C₂×D₄×C₃₁	1	2	yes	—
258	C₄²	C₂×D₄×C₃₁	1	2	yes	—
259	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
260	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
261	C₂³	—	2	2	no	—
262	C₂³	—	2	2	no	—
263	C₂³	—	2	1	no	—
264	C₂³	—	2	1	no	—
265	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
266	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	8	no	—
274	C₂×C₄	—	2	8	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	4	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
282	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
283	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
284	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
285	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
286	C₂×C₄	C₂×M₁₆×C₃₁	1	8	yes	—
287	C₂×C₄	C₂×M₁₆×C₃₁	1	4	yes	—
288	C₂×C₄	C₃₁×#131	1	2	yes	—
289	C₂×C₄	C₃₁×#131	1	2	yes	—
290	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
291	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
292	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
293	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
294	C₈	—	4	2	no	—
295	C₈	—	4	4	no	—
296	C₈	—	4	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	4	2	no	—
301	C₂²	—	4	2	no	—
302	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
303	C₂²	C₃₁×#520	1	2	yes	—
304	C₂²	C₂²×M₁₆×C₃₁	1	2	yes	—
305	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
306	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
307	C₄	—	2	2	no	—
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	2	8	no	—
311	C₄	—	4	4	no	—
312	C₄	C₃₁×#423	1	4	yes	—
313	C₄	C₃₁×#424	1	4	yes	—
314	C₄	C₃₁×#535	1	1	yes	characteristic
315	C₄	C₃₁×#536	1	1	yes	characteristic
316	C₄	C₃₁×#544	1	2	yes	—
317	C₂	—	4	2	no	—
318	C₂	C₃₁×#2317	1	1	yes	characteristic
319	C₂	C₂×C₃₁×#423	1	1	yes	characteristic
320	C₂	C₂×C₃₁×#424	1	1	yes	characteristic
321	C₂	C₃₁×#2990	1	2	yes	—
322	C₂	C₃₁×#2991	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	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×#1939	C₂	semidirect
6. C₃₁×#1842	C₂	semidirect
7. C₃₁×#1831	C₂	semidirect
86. C₄⋊C₄×C₃₁	M₁₆	semidirect
99. #21974	C₃₁	direct (N × H)
139. #1939	C₂×C₃₁	semidirect
141. #1842	C₂×C₃₁	semidirect
142. #1831	C₂×C₃₁	semidirect
221. C₃₁	#21974	direct (N × H)
248. C₄⋊C₄	M₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#431	2
C₂×C₃₁×#423	2
C₄×C₃₁×#117	2
C₃₁×#1939	2
C₃₁×#1851	2
C₃₁×#1842	2
C₃₁×#1831	2
#21974	31

Subgroups — C31×#21974 #21478733

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21974 #21478733