Subgroups — C₃₁×#21322 #21478081

Order 7936.

1270 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#533	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#384	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1992	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1990	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1736	C₂	1	4	yes	maximal
7	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
8	C₃₁×#533	—	2	4	no	—
9	C₃₁×#533	C₄	1	8	yes	complemented
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂×C₃₁×#125	C₂²	1	2	yes	—
12	C₂×C₃₁×C₂²⋊Q₈	—	2	2	no	—
13	C₂×C₄×Q₈×C₃₁	—	2	2	no	—
14	C₂×C₃₁×#117	C₂²	1	2	yes	—
15	C₂²×C₄⋊C₄×C₃₁	—	2	2	no	—
16	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
17	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
18	C₄×M₁₆×C₃₁	—	2	4	no	—
19	C₂×C₃₁×#97	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
21	C₃₁×#384	—	2	2	no	—
22	C₃₁×#384	C₂²	1	2	yes	—
23	C₃₁×#384	C₂²	1	2	yes	—
24	C₃₁×#340	C₂²	1	16	yes	—
25	C₂²×Q₈×C₃₁	—	2	2	no	—
26	C₂³×C₄×C₃₁	—	2	2	no	—
27	C₂³×C₄×C₃₁	D₄	1	2	yes	—
28	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
29	C₂×M₁₆×C₃₁	—	2	4	no	—
30	C₂×M₁₆×C₃₁	—	2	4	no	—
31	C₂×M₁₆×C₃₁	Q₈	1	8	yes	complemented
32	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
33	C₃₁×C₄⋊Q₈	—	2	4	no	—
34	C₃₁×C₄⋊Q₈	C₂×C₄	1	8	yes	complemented
35	C₃₁×#125	—	2	4	no	—
36	C₃₁×#125	C₂×C₄	1	8	yes	complemented
37	C₃₁×C₂²⋊Q₈	—	2	16	no	—
38	C₄×Q₈×C₃₁	—	2	16	no	—
39	C₃₁×#117	—	2	4	no	—
40	C₃₁×#117	C₂×C₄	1	8	yes	—
41	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
42	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
44	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
45	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
46	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
47	C₂×C₄⋊C₄×C₃₁	—	4	8	no	—
48	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
49	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
50	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
51	C₂×C₄²×C₃₁	—	2	2	no	—
52	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
53	C₂×C₄²×C₃₁	D₄	1	2	yes	—
54	C₂×C₄²×C₃₁	D₄	1	4	yes	—
55	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
57	C₃₁×#97	—	2	4	no	—
58	C₄×C₈×C₃₁	—	2	4	no	—
59	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
60	C₂×Q₈×C₃₁	—	2	8	no	—
61	C₂×Q₈×C₃₁	—	4	4	no	—
62	C₂×Q₈×C₃₁	—	4	4	no	—
63	C₂²×C₄×C₃₁	—	2	4	no	—
64	C₂²×C₄×C₃₁	—	2	4	no	—
65	C₂²×C₄×C₃₁	—	2	4	no	—
66	C₂²×C₄×C₃₁	—	2	4	no	—
67	C₂²×C₄×C₃₁	—	4	8	no	—
68	C₂²×C₄×C₃₁	—	4	4	no	—
69	C₂²×C₄×C₃₁	C₂²⋊C₄	1	8	yes	—
70	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₃₁	C₂×Q₈	1	4	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	4	no	—
80	C₂×C₈×C₃₁	—	2	4	no	—
81	C₂×C₈×C₃₁	C₂×Q₈	1	8	yes	—
82	C₄⋊C₄×C₃₁	—	2	16	no	—
83	C₄⋊C₄×C₃₁	—	2	8	no	—
84	C₄⋊C₄×C₃₁	—	2	4	no	—
85	C₄⋊C₄×C₃₁	—	2	8	no	—
86	C₄⋊C₄×C₃₁	—	4	8	no	—
87	C₄⋊C₄×C₃₁	—	4	4	no	—
88	C₄⋊C₄×C₃₁	—	4	8	no	—
89	C₄⋊C₄×C₃₁	C₂²×C₄	1	8	yes	—
90	C₂²⋊C₄×C₃₁	—	2	8	no	—
91	C₄²×C₃₁	—	2	8	no	—
92	C₄²×C₃₁	C₂²⋊C₄	1	8	yes	—
93	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
94	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
95	C₄²×C₃₁	C₂×D₄	1	2	yes	—
96	C₄²×C₃₁	C₂×D₄	1	2	yes	—
97	#21322	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	C₂×C₂²⋊C₄	1	2	yes	—
101	C₂³×C₃₁	#124	1	2	yes	—
102	C₂³×C₃₁	C₄⋊Q₈	1	2	yes	—
103	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
104	Q₈×C₃₁	—	4	16	no	—
105	C₂×C₄×C₃₁	—	2	2	no	—
106	C₂×C₄×C₃₁	—	2	2	no	—
107	C₂×C₄×C₃₁	—	2	16	no	—
108	C₂×C₄×C₃₁	—	2	8	no	—
109	C₂×C₄×C₃₁	—	4	4	no	—
110	C₂×C₄×C₃₁	—	4	4	no	—
111	C₂×C₄×C₃₁	—	4	4	no	—
112	C₂×C₄×C₃₁	—	4	8	no	—
113	C₂×C₄×C₃₁	—	8	8	no	—
114	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
115	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
116	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
117	C₂×C₄×C₃₁	C₄×Q₈	1	8	yes	—
118	C₂×C₄×C₃₁	C₂²⋊Q₈	1	8	yes	—
119	C₂×C₄×C₃₁	#124	1	2	yes	—
120	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
121	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
122	C₂×C₄×C₃₁	#137	1	4	yes	—
123	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
126	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
127	C₈×C₃₁	—	2	8	no	—
128	C₂×#533	C₂×C₃₁	1	1	yes	characteristic
129	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic
130	C₂×#384	C₂×C₃₁	1	1	yes	characteristic
131	#1992	C₂×C₃₁	1	4	yes	complemented
132	#1990	C₂×C₃₁	1	4	yes	complemented
133	#1736	C₂×C₃₁	1	4	yes	—
134	C₂²×C₃₁	—	2	2	no	—
135	C₂²×C₃₁	—	2	2	no	—
136	C₂²×C₃₁	—	2	4	no	—
137	C₂²×C₃₁	—	2	4	no	—
138	C₂²×C₃₁	#391	1	4	yes	—
139	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
140	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
141	C₂²×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
142	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
143	C₂²×C₃₁	C₂×#137	1	2	yes	—
144	C₄×C₃₁	—	2	4	no	—
145	C₄×C₃₁	—	4	8	no	—
146	C₄×C₃₁	—	8	4	no	—
147	C₄×C₃₁	#391	1	4	yes	—
148	C₄×C₃₁	#417	1	8	yes	—
149	C₄×C₃₁	C₂×C₄×Q₈	1	2	yes	—
150	C₄×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
151	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
152	#533	—	2	4	no	—
153	#533	C₄×C₃₁	1	8	yes	complemented
154	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
155	C₂×#125	C₂²×C₃₁	1	2	yes	—
156	C₂×C₂²⋊Q₈	—	2	2	no	—
157	C₂×C₄×Q₈	—	2	2	no	—
158	C₂×#117	C₂²×C₃₁	1	2	yes	—
159	C₂²×C₄⋊C₄	—	2	2	no	—
160	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
161	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
162	C₄×M₁₆	—	2	4	no	—
163	C₂×#97	C₂²×C₃₁	1	2	yes	—
164	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
165	#384	—	2	2	no	—
166	#384	C₂²×C₃₁	1	2	yes	—
167	#384	C₂²×C₃₁	1	2	yes	—
168	#340	C₂²×C₃₁	1	16	yes	—
169	C₂×C₃₁	—	2	4	no	—
170	C₂×C₃₁	C₂×#391	1	1	yes	characteristic
171	C₂×C₃₁	C₂×#417	1	2	yes	—
172	C₂×C₃₁	#3138	1	1	yes	characteristic
173	C₂×C₃₁	#3139	1	1	yes	characteristic
174	C₂×C₃₁	#3165	1	2	yes	—
175	C₂²×Q₈	—	2	2	no	—
176	C₂³×C₄	—	2	2	no	—
177	C₂³×C₄	D₄×C₃₁	1	2	yes	—
178	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
179	C₂×M₁₆	—	2	4	no	—
180	C₂×M₁₆	—	2	4	no	—
181	C₂×M₁₆	Q₈×C₃₁	1	8	yes	complemented
182	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
183	C₄⋊Q₈	—	2	4	no	—
184	C₄⋊Q₈	C₂×C₄×C₃₁	1	8	yes	complemented
185	#125	—	2	4	no	—
186	#125	C₂×C₄×C₃₁	1	8	yes	complemented
187	C₂²⋊Q₈	—	2	16	no	—
188	C₄×Q₈	—	2	16	no	—
189	#117	—	2	4	no	—
190	#117	C₂×C₄×C₃₁	1	8	yes	—
191	C₂×C₄⋊C₄	—	2	4	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	8	no	—
194	C₂×C₄⋊C₄	—	2	4	no	—
195	C₂×C₄⋊C₄	—	2	4	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	4	8	no	—
198	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
199	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
200	C₂×C₂²⋊C₄	—	2	2	no	—
201	C₂×C₄²	—	2	2	no	—
202	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
203	C₂×C₄²	D₄×C₃₁	1	2	yes	—
204	C₂×C₄²	D₄×C₃₁	1	4	yes	—
205	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
206	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
207	#97	—	2	4	no	—
208	C₄×C₈	—	2	4	no	—
209	C₃₁	#21322	1	1	yes	characteristic, direct product factor, Sylow
210	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
211	C₂×Q₈	—	2	8	no	—
212	C₂×Q₈	—	4	4	no	—
213	C₂×Q₈	—	4	4	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	4	8	no	—
219	C₂²×C₄	—	4	4	no	—
220	C₂²×C₄	C₂²⋊C₄×C₃₁	1	8	yes	—
221	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
222	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
223	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
224	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
225	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
226	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
227	C₂²×C₄	#41×C₃₁	1	4	yes	—
228	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
229	M₁₆	—	2	16	no	—
230	C₂×C₈	—	2	4	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	C₂×Q₈×C₃₁	1	8	yes	—
233	C₄⋊C₄	—	2	16	no	—
234	C₄⋊C₄	—	2	8	no	—
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	2	8	no	—
237	C₄⋊C₄	—	4	8	no	—
238	C₄⋊C₄	—	4	4	no	—
239	C₄⋊C₄	—	4	8	no	—
240	C₄⋊C₄	C₂²×C₄×C₃₁	1	8	yes	—
241	C₂²⋊C₄	—	2	8	no	—
242	C₄²	—	2	8	no	—
243	C₄²	C₂²⋊C₄×C₃₁	1	8	yes	—
244	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
245	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
246	C₄²	C₂×D₄×C₃₁	1	2	yes	—
247	C₄²	C₂×D₄×C₃₁	1	2	yes	—
248	C₂³	—	2	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
251	C₂³	C₃₁×#124	1	2	yes	—
252	C₂³	C₃₁×C₄⋊Q₈	1	2	yes	—
253	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
254	Q₈	—	4	16	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	16	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	4	no	—
262	C₂×C₄	—	4	8	no	—
263	C₂×C₄	—	8	8	no	—
264	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
265	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
266	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
267	C₂×C₄	C₄×Q₈×C₃₁	1	8	yes	—
268	C₂×C₄	C₃₁×C₂²⋊Q₈	1	8	yes	—
269	C₂×C₄	C₃₁×#124	1	2	yes	—
270	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
271	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
272	C₂×C₄	C₃₁×#137	1	4	yes	—
273	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
274	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
275	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
276	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
277	C₈	—	2	8	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	4	no	—
282	C₂²	C₃₁×#391	1	4	yes	—
283	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
284	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
285	C₂²	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
286	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
287	C₂²	C₂×C₃₁×#137	1	2	yes	—
288	C₄	—	2	4	no	—
289	C₄	—	4	8	no	—
290	C₄	—	8	4	no	—
291	C₄	C₃₁×#391	1	4	yes	—
292	C₄	C₃₁×#417	1	8	yes	—
293	C₄	C₂×C₄×Q₈×C₃₁	1	2	yes	—
294	C₄	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
295	C₂	—	2	4	no	—
296	C₂	C₂×C₃₁×#391	1	1	yes	characteristic
297	C₂	C₂×C₃₁×#417	1	2	yes	—
298	C₂	C₃₁×#3138	1	1	yes	characteristic
299	C₂	C₃₁×#3139	1	1	yes	characteristic
300	C₂	C₃₁×#3165	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1992	C₂	semidirect
5. C₃₁×#1990	C₂	semidirect
9. C₃₁×#533	C₄	semidirect
31. C₂×M₁₆×C₃₁	Q₈	semidirect
34. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
36. C₃₁×#125	C₂×C₄	semidirect
97. #21322	C₃₁	direct (N × H)
131. #1992	C₂×C₃₁	semidirect
132. #1990	C₂×C₃₁	semidirect
153. #533	C₄×C₃₁	semidirect
181. C₂×M₁₆	Q₈×C₃₁	semidirect
184. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
186. #125	C₂×C₄×C₃₁	semidirect
209. C₃₁	#21322	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×#533	2
C₂×C₄×M₁₆×C₃₁	2
C₂×C₃₁×#384	2
C₃₁×#1992	2
C₃₁×#1990	2
C₃₁×#1736	2
#21322	31

Subgroups — C31×#21322 #21478081

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21322 #21478081