Subgroups — C₃₁×#7903 #21464662

Order 7936.

794 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#423	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1279	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₃₁	—	2	1	no	—
6	C₂²×M₁₆×C₃₁	—	2	1	no	—
7	C₂×C₃₁×#117	C₄	1	2	yes	—
8	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
9	C₃₁×#423	—	2	2	no	—
10	C₃₁×#423	C₄	1	4	yes	complemented
11	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
12	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
13	C₄×M₁₆×C₃₁	—	2	2	no	—
14	C₄×M₁₆×C₃₁	C₄	1	4	yes	complemented
15	C₂×C₃₁×#97	—	2	1	no	—
16	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
17	C₄×C₄⋊C₄×C₃₁	C₄	1	2	yes	—
18	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
19	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
20	C₄×C₂²⋊C₄×C₃₁	—	2	1	no	—
21	C₃₁×#376	—	2	1	no	—
22	C₄³×C₃₁	—	2	1	no	—
23	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
25	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
26	C₂×M₁₆×C₃₁	—	2	8	no	—
27	C₂×M₁₆×C₃₁	—	2	4	no	—
28	C₂×M₁₆×C₃₁	—	2	4	no	—
29	C₂×M₁₆×C₃₁	—	4	4	no	—
30	C₂²×C₈×C₃₁	—	2	2	no	—
31	C₂²×C₈×C₃₁	—	2	2	no	—
32	C₃₁×#117	—	2	8	no	—
33	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
34	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
35	C₂×C₄²×C₃₁	—	2	2	no	—
36	C₂×C₄²×C₃₁	—	2	2	no	—
37	C₂×C₄²×C₃₁	—	2	2	no	—
38	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
43	C₂×C₄²×C₃₁	D₄	1	2	yes	—
44	C₂×C₄²×C₃₁	Q₈	1	2	yes	—
45	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
46	C₃₁×C₄⋊C₈	—	2	4	no	—
47	C₃₁×C₄⋊C₈	—	2	2	no	—
48	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
49	C₃₁×#97	—	2	4	no	—
50	C₄×C₈×C₃₁	—	2	2	no	—
51	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
52	C₃₁×#95	—	2	2	no	—
53	C₃₁×#95	—	2	2	no	—
54	C₂⁴×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄×C₃₁	—	2	4	no	—
56	C₂²×C₄×C₃₁	—	2	4	no	—
57	C₂²×C₄×C₃₁	—	2	2	no	—
58	C₂²×C₄×C₃₁	—	2	2	no	—
59	C₂²×C₄×C₃₁	—	2	4	no	—
60	C₂²×C₄×C₃₁	C₄²	1	2	yes	—
61	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
62	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₃₁	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
72	M₁₆×C₃₁	—	2	16	no	—
73	M₁₆×C₃₁	—	4	8	no	—
74	C₂×C₈×C₃₁	—	2	8	no	—
75	C₂×C₈×C₃₁	—	2	4	no	—
76	C₂×C₈×C₃₁	—	2	4	no	—
77	C₂×C₈×C₃₁	—	4	4	no	—
78	C₄⋊C₄×C₃₁	—	2	4	no	—
79	C₄⋊C₄×C₃₁	M₁₆	1	8	yes	complemented
80	C₂²⋊C₄×C₃₁	—	2	8	no	—
81	C₄²×C₃₁	—	2	8	no	—
82	C₄²×C₃₁	—	4	4	no	—
83	C₄²×C₃₁	—	4	2	no	—
84	C₄²×C₃₁	—	4	2	no	—
85	C₄²×C₃₁	C₄²	1	2	yes	—
86	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
87	C₄²×C₃₁	C₂²⋊C₄	1	2	yes	—
88	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
89	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
90	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
91	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
95	#7903	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
96	C₂³×C₃₁	—	2	2	no	—
97	C₂³×C₃₁	—	2	2	no	—
98	C₂³×C₃₁	—	2	2	no	—
99	C₂³×C₃₁	#95	1	2	yes	—
100	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₃₁	—	2	2	no	—
102	C₂×C₄×C₃₁	—	2	2	no	—
103	C₂×C₄×C₃₁	—	2	2	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	—	2	2	no	—
108	C₂×C₄×C₃₁	—	2	2	no	—
109	C₂×C₄×C₃₁	—	2	4	no	—
110	C₂×C₄×C₃₁	—	2	4	no	—
111	C₂×C₄×C₃₁	—	2	8	no	—
112	C₂×C₄×C₃₁	—	4	4	no	—
113	C₂×C₄×C₃₁	—	4	4	no	—
114	C₂×C₄×C₃₁	#95	1	2	yes	—
115	C₂×C₄×C₃₁	#95	1	2	yes	—
116	C₂×C₄×C₃₁	#95	1	2	yes	—
117	C₂×C₄×C₃₁	#104	1	4	yes	—
118	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
119	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₃₁	C₂×M₁₆	1	4	yes	—
125	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
126	C₂×C₄×C₃₁	#137	1	2	yes	—
127	C₈×C₃₁	—	2	8	no	—
128	C₈×C₃₁	—	4	4	no	—
129	C₂×#423	C₂×C₃₁	1	1	yes	characteristic
130	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic
131	C₄×#117	C₂×C₃₁	1	1	yes	characteristic
132	#1279	C₂×C₃₁	1	4	yes	complemented
133	C₂²×C₃₁	—	2	2	no	—
134	C₂²×C₃₁	—	2	2	no	—
135	C₂²×C₃₁	—	2	2	no	—
136	C₂²×C₃₁	—	4	4	no	—
137	C₂²×C₃₁	C₂×#95	1	1	yes	characteristic
138	C₂²×C₃₁	#407	1	2	yes	—
139	C₂²×C₃₁	#417	1	2	yes	—
140	C₂²×C₃₁	C₂×#104	1	2	yes	—
141	C₄×C₃₁	—	2	2	no	—
142	C₄×C₃₁	—	2	1	no	—
143	C₄×C₃₁	—	2	1	no	—
144	C₄×C₃₁	—	2	2	no	—
145	C₄×C₃₁	—	4	8	no	—
146	C₄×C₃₁	#339	1	4	yes	—
147	C₄×C₃₁	C₄×M₁₆	1	2	yes	—
148	C₄×C₃₁	#418	1	1	yes	characteristic
149	C₄×C₃₁	#419	1	1	yes	characteristic
150	C₄×C₃₁	#423	1	2	yes	—
151	C₄×C₃₁	#428	1	2	yes	—
152	C₂²×M₁₆	—	2	1	no	—
153	C₂²×M₁₆	—	2	1	no	—
154	C₂×#117	C₄×C₃₁	1	2	yes	—
155	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
156	#423	—	2	2	no	—
157	#423	C₄×C₃₁	1	4	yes	complemented
158	C₂×C₄⋊C₈	—	2	1	no	—
159	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
160	C₄×M₁₆	—	2	2	no	—
161	C₄×M₁₆	C₄×C₃₁	1	4	yes	complemented
162	C₂×#97	—	2	1	no	—
163	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
164	C₄×C₄⋊C₄	C₄×C₃₁	1	2	yes	—
165	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
166	C₄×C₂²⋊C₄	—	2	1	no	—
167	C₄×C₂²⋊C₄	—	2	1	no	—
168	#376	—	2	1	no	—
169	C₄³	—	2	1	no	—
170	C₂×C₃₁	—	4	2	no	—
171	C₂×C₃₁	#1489	1	4	yes	—
172	C₂×C₃₁	#1728	1	1	yes	characteristic
173	C₂×C₃₁	C₂×#339	1	1	yes	characteristic
174	C₂×C₃₁	#1736	1	1	yes	characteristic
175	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
176	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
177	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
178	C₂×M₁₆	—	2	8	no	—
179	C₂×M₁₆	—	2	4	no	—
180	C₂×M₁₆	—	2	4	no	—
181	C₂×M₁₆	—	4	4	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	2	no	—
184	#117	—	2	8	no	—
185	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
186	C₂×C₂²⋊C₄	—	2	2	no	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	—	2	2	no	—
190	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
191	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
192	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
193	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
194	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
195	C₂×C₄²	D₄×C₃₁	1	2	yes	—
196	C₂×C₄²	Q₈×C₃₁	1	2	yes	—
197	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
198	C₄⋊C₈	—	2	4	no	—
199	C₄⋊C₈	—	2	2	no	—
200	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
201	#97	—	2	4	no	—
202	C₄×C₈	—	2	2	no	—
203	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
204	#95	—	2	2	no	—
205	#95	—	2	2	no	—
206	C₃₁	#7903	1	1	yes	characteristic, direct product factor, Sylow
207	C₂⁴	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	C₄²×C₃₁	1	2	yes	—
214	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
215	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
216	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
217	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
218	C₂²×C₄	C₂²⋊C₄×C₃₁	1	1	yes	characteristic
219	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
220	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
221	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
222	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
225	M₁₆	—	2	16	no	—
226	M₁₆	—	4	8	no	—
227	C₂×C₈	—	2	8	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	4	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	M₁₆×C₃₁	1	8	yes	complemented
233	C₂²⋊C₄	—	2	8	no	—
234	C₄²	—	2	8	no	—
235	C₄²	—	4	4	no	—
236	C₄²	—	4	2	no	—
237	C₄²	—	4	2	no	—
238	C₄²	C₄²×C₃₁	1	2	yes	—
239	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
240	C₄²	C₂²⋊C₄×C₃₁	1	2	yes	—
241	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
242	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
243	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
244	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
245	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
246	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
247	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
248	C₂³	—	2	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	—	2	2	no	—
251	C₂³	C₃₁×#95	1	2	yes	—
252	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	C₃₁×#95	1	2	yes	—
267	C₂×C₄	C₃₁×#95	1	2	yes	—
268	C₂×C₄	C₃₁×#95	1	2	yes	—
269	C₂×C₄	C₃₁×#104	1	4	yes	—
270	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
271	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
272	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
273	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
274	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
275	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
276	C₂×C₄	C₂×M₁₆×C₃₁	1	4	yes	—
277	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
278	C₂×C₄	C₃₁×#137	1	2	yes	—
279	C₈	—	2	8	no	—
280	C₈	—	4	4	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	4	4	no	—
285	C₂²	C₂×C₃₁×#95	1	1	yes	characteristic
286	C₂²	C₃₁×#407	1	2	yes	—
287	C₂²	C₃₁×#417	1	2	yes	—
288	C₂²	C₂×C₃₁×#104	1	2	yes	—
289	C₄	—	2	2	no	—
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	2	no	—
293	C₄	—	4	8	no	—
294	C₄	C₃₁×#339	1	4	yes	—
295	C₄	C₄×M₁₆×C₃₁	1	2	yes	—
296	C₄	C₃₁×#418	1	1	yes	characteristic
297	C₄	C₃₁×#419	1	1	yes	characteristic
298	C₄	C₃₁×#423	1	2	yes	—
299	C₄	C₃₁×#428	1	2	yes	—
300	C₂	—	4	2	no	—
301	C₂	C₃₁×#1489	1	4	yes	—
302	C₂	C₃₁×#1728	1	1	yes	characteristic
303	C₂	C₂×C₃₁×#339	1	1	yes	characteristic
304	C₂	C₃₁×#1736	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1279	C₂	semidirect
10. C₃₁×#423	C₄	semidirect
14. C₄×M₁₆×C₃₁	C₄	semidirect
79. C₄⋊C₄×C₃₁	M₁₆	semidirect
95. #7903	C₃₁	direct (N × H)
132. #1279	C₂×C₃₁	semidirect
157. #423	C₄×C₃₁	semidirect
161. C₄×M₁₆	C₄×C₃₁	semidirect
206. C₃₁	#7903	direct (N × H)
232. 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×#423	2
C₂×C₄×M₁₆×C₃₁	2
C₄×C₃₁×#117	2
C₃₁×#1279	2
#7903	31

Subgroups — C31×#7903 #21464662

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#7903 #21464662