Subgroups — C₃₁×#23551 #21480310

Order 7936.

1154 conjugacy classes of subgroups.

The subgroup lattice has 342 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×C₃₁×#427	C₂	1	2	yes	maximal
3	C₃₁×#2296	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2059	C₂	1	4	yes	maximal, complemented
5	C₃₁×#2058	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2057	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1735	C₂	1	2	yes	maximal
8	C₂×C₃₁×#336	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
10	C₃₁×#533	—	2	8	no	—
11	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₂×C₃₁×#125	C₂²	1	2	yes	—
13	C₂×C₃₁×C₂²⋊Q₈	—	2	2	no	—
14	C₂×C₄×Q₈×C₃₁	—	2	2	no	—
15	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#117	C₂²	1	2	yes	—
17	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#427	—	2	8	no	—
20	C₂×C₃₁×#107	C₂²	1	4	yes	—
21	C₂×C₃₁×#106	C₂²	1	4	yes	—
22	C₃₁×#389	C₂²	1	2	yes	—
23	C₃₁×#389	C₂²	1	2	yes	—
24	C₃₁×#385	—	2	2	no	—
25	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
26	C₃₁×#340	C₂²	1	4	yes	—
27	C₃₁×#340	C₂²	1	4	yes	—
28	C₃₁×#336	C₂²	1	1	yes	characteristic
29	C₃₁×#336	C₂²	1	1	yes	characteristic
30	C₃₁×#336	C₂²	1	2	yes	—
31	C₂²×Q₈×C₃₁	—	2	2	no	—
32	C₂³×C₄×C₃₁	D₄	1	2	yes	—
33	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₃₁	—	2	4	no	—
35	C₂²×C₈×C₃₁	—	2	4	no	—
36	C₂²×C₈×C₃₁	Q₈	1	8	yes	complemented
37	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
38	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
39	C₃₁×C₄⋊Q₈	—	2	8	no	—
40	C₃₁×#125	—	2	8	no	—
41	C₃₁×C₂²⋊Q₈	—	2	16	no	—
42	C₄×Q₈×C₃₁	—	2	16	no	—
43	C₃₁×#117	—	2	4	no	—
44	C₃₁×#117	—	2	2	no	—
45	C₃₁×#117	—	2	4	no	—
46	C₃₁×#117	D₄	1	8	yes	—
47	C₃₁×#117	Q₈	1	4	yes	complemented
48	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
49	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
50	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
51	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
52	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
53	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
54	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
55	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
56	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
57	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
58	C₂×C₄²×C₃₁	—	2	4	no	—
59	C₂×C₄²×C₃₁	—	2	2	no	—
60	C₂×C₄²×C₃₁	—	2	1	no	—
61	C₂×C₄²×C₃₁	—	2	1	no	—
62	C₂×C₄²×C₃₁	D₄	1	2	yes	—
63	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
65	C₃₁×#107	—	2	8	no	—
66	C₃₁×#106	—	2	8	no	—
67	C₃₁×#95	—	2	2	no	—
68	C₃₁×#95	—	2	2	no	—
69	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
70	C₂×Q₈×C₃₁	—	2	8	no	—
71	C₂×Q₈×C₃₁	—	4	4	no	—
72	C₂×Q₈×C₃₁	—	4	4	no	—
73	C₂²×C₄×C₃₁	—	2	4	no	—
74	C₂²×C₄×C₃₁	—	2	2	no	—
75	C₂²×C₄×C₃₁	—	2	4	no	—
76	C₂²×C₄×C₃₁	—	2	4	no	—
77	C₂²×C₄×C₃₁	—	2	2	no	—
78	C₂²×C₄×C₃₁	—	4	2	no	—
79	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
82	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₃₁	C₂×Q₈	1	4	yes	—
84	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₃₁	#41	1	2	yes	—
86	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
87	C₂×C₈×C₃₁	—	2	16	no	—
88	C₂×C₈×C₃₁	—	2	2	no	—
89	C₂×C₈×C₃₁	—	2	2	no	—
90	C₂×C₈×C₃₁	—	2	2	no	—
91	C₂×C₈×C₃₁	—	2	2	no	—
92	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
93	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
94	C₄⋊C₄×C₃₁	—	2	16	no	—
95	C₄⋊C₄×C₃₁	—	2	8	no	—
96	C₄⋊C₄×C₃₁	—	2	4	no	—
97	C₄⋊C₄×C₃₁	—	2	2	no	—
98	C₄⋊C₄×C₃₁	—	2	4	no	—
99	C₄⋊C₄×C₃₁	—	4	4	no	—
100	C₄⋊C₄×C₃₁	—	4	4	no	—
101	C₄⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
102	C₄⋊C₄×C₃₁	C₂×Q₈	1	4	yes	—
103	C₂²⋊C₄×C₃₁	—	2	8	no	—
104	C₂²⋊C₄×C₃₁	—	2	4	no	—
105	C₂²⋊C₄×C₃₁	—	4	2	no	—
106	C₄²×C₃₁	—	2	4	no	—
107	C₄²×C₃₁	—	2	8	no	—
108	C₄²×C₃₁	—	2	4	no	—
109	C₄²×C₃₁	—	2	2	no	—
110	C₄²×C₃₁	—	2	2	no	—
111	C₄²×C₃₁	—	4	2	no	—
112	#23551	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂³×C₃₁	—	2	2	no	—
114	C₂³×C₃₁	—	2	2	no	—
115	C₂³×C₃₁	#120	1	2	yes	—
116	C₂³×C₃₁	C₂²⋊Q₈	1	2	yes	—
117	C₂³×C₃₁	C₄⋊Q₈	1	2	yes	—
118	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
119	Q₈×C₃₁	—	4	16	no	—
120	C₂×C₄×C₃₁	—	2	2	no	—
121	C₂×C₄×C₃₁	—	2	2	no	—
122	C₂×C₄×C₃₁	—	2	2	no	—
123	C₂×C₄×C₃₁	—	2	8	no	—
124	C₂×C₄×C₃₁	—	2	4	no	—
125	C₂×C₄×C₃₁	—	2	2	no	—
126	C₂×C₄×C₃₁	—	2	2	no	—
127	C₂×C₄×C₃₁	—	4	8	no	—
128	C₂×C₄×C₃₁	—	4	2	no	—
129	C₂×C₄×C₃₁	—	4	4	no	—
130	C₂×C₄×C₃₁	—	4	2	no	—
131	C₂×C₄×C₃₁	—	4	4	no	—
132	C₂×C₄×C₃₁	—	4	2	no	—
133	C₂×C₄×C₃₁	—	4	2	no	—
134	C₂×C₄×C₃₁	—	4	2	no	—
135	C₂×C₄×C₃₁	#120	1	2	yes	—
136	C₂×C₄×C₃₁	C₂²⋊Q₈	1	8	yes	—
137	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
138	C₂×C₄×C₃₁	C₄⋊Q₈	1	8	yes	—
139	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
140	C₂×C₄×C₃₁	#135	1	4	yes	—
141	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
142	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
143	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
144	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
145	C₈×C₃₁	—	2	4	no	—
146	C₈×C₃₁	—	2	4	no	—
147	C₂×#533	C₂×C₃₁	1	1	yes	characteristic
148	C₂×#427	C₂×C₃₁	1	2	yes	—
149	#2296	C₂×C₃₁	1	1	yes	characteristic
150	#2059	C₂×C₃₁	1	4	yes	complemented
151	#2058	C₂×C₃₁	1	2	yes	complemented
152	#2057	C₂×C₃₁	1	2	yes	complemented
153	#1735	C₂×C₃₁	1	2	yes	—
154	C₂×#336	C₂×C₃₁	1	1	yes	characteristic
155	C₂²×C₃₁	—	2	4	no	—
156	C₂²×C₃₁	—	2	2	no	—
157	C₂²×C₃₁	—	2	2	no	—
158	C₂²×C₃₁	—	2	4	no	—
159	C₂²×C₃₁	#395	1	4	yes	—
160	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
161	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
162	C₂²×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
163	C₂²×C₃₁	C₂×#135	1	2	yes	—
164	C₂²×C₃₁	#575	1	2	yes	—
165	C₄×C₃₁	—	2	4	no	—
166	C₄×C₃₁	—	4	4	no	—
167	C₄×C₃₁	—	4	2	no	—
168	C₄×C₃₁	—	4	8	no	—
169	C₄×C₃₁	—	8	2	no	—
170	C₄×C₃₁	#395	1	4	yes	—
171	C₄×C₃₁	C₂×C₂²⋊Q₈	1	2	yes	—
172	C₄×C₃₁	C₂×C₄⋊Q₈	1	2	yes	—
173	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
174	#533	—	2	8	no	—
175	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
176	C₂×#125	C₂²×C₃₁	1	2	yes	—
177	C₂×C₂²⋊Q₈	—	2	2	no	—
178	C₂×C₄×Q₈	—	2	2	no	—
179	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
180	C₂×#117	C₂²×C₃₁	1	2	yes	—
181	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
182	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
183	#427	—	2	8	no	—
184	C₂×#107	C₂²×C₃₁	1	4	yes	—
185	C₂×#106	C₂²×C₃₁	1	4	yes	—
186	#389	C₂²×C₃₁	1	2	yes	—
187	#389	C₂²×C₃₁	1	2	yes	—
188	#385	—	2	2	no	—
189	C₄×C₄⋊C₄	—	2	2	no	—
190	#340	C₂²×C₃₁	1	4	yes	—
191	#340	C₂²×C₃₁	1	4	yes	—
192	#336	C₂²×C₃₁	1	1	yes	characteristic
193	#336	C₂²×C₃₁	1	1	yes	characteristic
194	#336	C₂²×C₃₁	1	2	yes	—
195	C₂×C₃₁	—	2	4	no	—
196	C₂×C₃₁	C₂×#395	1	1	yes	characteristic
197	C₂×C₃₁	#3002	1	2	yes	—
198	C₂×C₃₁	#3076	1	2	yes	—
199	C₂×C₃₁	#3160	1	1	yes	characteristic
200	C₂×C₃₁	#3162	1	1	yes	characteristic
201	C₂²×Q₈	—	2	2	no	—
202	C₂³×C₄	D₄×C₃₁	1	2	yes	—
203	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
204	C₂²×C₈	—	2	4	no	—
205	C₂²×C₈	—	2	4	no	—
206	C₂²×C₈	Q₈×C₃₁	1	8	yes	complemented
207	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
208	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
209	C₄⋊Q₈	—	2	8	no	—
210	#125	—	2	8	no	—
211	C₂²⋊Q₈	—	2	16	no	—
212	C₄×Q₈	—	2	16	no	—
213	#117	—	2	4	no	—
214	#117	—	2	2	no	—
215	#117	—	2	4	no	—
216	#117	D₄×C₃₁	1	8	yes	—
217	#117	Q₈×C₃₁	1	4	yes	complemented
218	C₂×C₄⋊C₄	—	2	4	no	—
219	C₂×C₄⋊C₄	—	2	2	no	—
220	C₂×C₄⋊C₄	—	2	2	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
223	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
224	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
225	C₂×C₂²⋊C₄	—	2	2	no	—
226	C₂×C₂²⋊C₄	—	2	1	no	—
227	C₂×C₂²⋊C₄	—	2	1	no	—
228	C₂×C₄²	—	2	2	no	—
229	C₂×C₄²	—	2	1	no	—
230	C₂×C₄²	—	2	4	no	—
231	C₂×C₄²	—	2	1	no	—
232	C₂×C₄²	D₄×C₃₁	1	2	yes	—
233	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
234	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
235	#107	—	2	8	no	—
236	#106	—	2	8	no	—
237	#95	—	2	2	no	—
238	#95	—	2	2	no	—
239	C₃₁	#23551	1	1	yes	characteristic, direct product factor, Sylow
240	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
241	C₂×Q₈	—	2	8	no	—
242	C₂×Q₈	—	4	4	no	—
243	C₂×Q₈	—	4	4	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	2	2	no	—
246	C₂²×C₄	—	2	4	no	—
247	C₂²×C₄	—	2	4	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	—	4	2	no	—
250	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
251	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
252	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
254	C₂²×C₄	C₂×Q₈×C₃₁	1	4	yes	—
255	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
256	C₂²×C₄	#41×C₃₁	1	2	yes	—
257	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
258	C₂×C₈	—	2	16	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
264	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
265	C₄⋊C₄	—	2	16	no	—
266	C₄⋊C₄	—	2	8	no	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	2	no	—
269	C₄⋊C₄	—	2	4	no	—
270	C₄⋊C₄	—	4	4	no	—
271	C₄⋊C₄	—	4	4	no	—
272	C₄⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
273	C₄⋊C₄	C₂×Q₈×C₃₁	1	4	yes	—
274	C₂²⋊C₄	—	2	8	no	—
275	C₂²⋊C₄	—	2	4	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₄²	—	2	4	no	—
278	C₄²	—	2	8	no	—
279	C₄²	—	2	4	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	4	2	no	—
283	C₂³	—	2	2	no	—
284	C₂³	—	2	2	no	—
285	C₂³	C₃₁×#120	1	2	yes	—
286	C₂³	C₃₁×C₂²⋊Q₈	1	2	yes	—
287	C₂³	C₃₁×C₄⋊Q₈	1	2	yes	—
288	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
289	Q₈	—	4	16	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	8	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	4	8	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	C₃₁×#120	1	2	yes	—
306	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
307	C₂×C₄	C₃₁×C₂²⋊Q₈	1	8	yes	—
308	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
309	C₂×C₄	C₃₁×C₄⋊Q₈	1	8	yes	—
310	C₂×C₄	C₃₁×#135	1	4	yes	—
311	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
312	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
313	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
314	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
315	C₈	—	2	4	no	—
316	C₈	—	2	4	no	—
317	C₂²	—	2	4	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	4	no	—
321	C₂²	C₃₁×#395	1	4	yes	—
322	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
323	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
324	C₂²	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
325	C₂²	C₂×C₃₁×#135	1	2	yes	—
326	C₂²	C₃₁×#575	1	2	yes	—
327	C₄	—	2	4	no	—
328	C₄	—	4	4	no	—
329	C₄	—	4	2	no	—
330	C₄	—	4	8	no	—
331	C₄	—	8	2	no	—
332	C₄	C₃₁×#395	1	4	yes	—
333	C₄	C₂×C₃₁×C₂²⋊Q₈	1	2	yes	—
334	C₄	C₂×C₃₁×C₄⋊Q₈	1	2	yes	—
335	C₂	—	2	4	no	—
336	C₂	C₂×C₃₁×#395	1	1	yes	characteristic
337	C₂	C₃₁×#3002	1	2	yes	—
338	C₂	C₃₁×#3076	1	2	yes	—
339	C₂	C₃₁×#3160	1	1	yes	characteristic
340	C₂	C₃₁×#3162	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂²×D₄
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂²×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×#2059	C₂	semidirect
5. C₃₁×#2058	C₂	semidirect
6. C₃₁×#2057	C₂	semidirect
36. C₂²×C₈×C₃₁	Q₈	semidirect
47. C₃₁×#117	Q₈	semidirect
112. #23551	C₃₁	direct (N × H)
150. #2059	C₂×C₃₁	semidirect
151. #2058	C₂×C₃₁	semidirect
152. #2057	C₂×C₃₁	semidirect
206. C₂²×C₈	Q₈×C₃₁	semidirect
217. #117	Q₈×C₃₁	semidirect
239. C₃₁	#23551	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×C₃₁×#427	2
C₃₁×#2296	2
C₃₁×#2059	2
C₃₁×#2058	2
C₃₁×#2057	2
C₃₁×#1735	2
C₂×C₃₁×#336	2
#23551	31

Subgroups — C31×#23551 #21480310

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23551 #21480310