Subgroups — C₃₁×#23799 #21480558

Order 7936.

1154 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×SD₁₆×C₃₁	C₂	1	4	yes	maximal, complemented
2	C₄×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×C₃₁×#127	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2096	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2094	C₂	1	2	yes	maximal, complemented
6	C₃₁×#2014	C₂	1	2	yes	maximal
7	C₃₁×#2013	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1744	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×SD₁₆×C₃₁	C₂²	1	4	yes	—
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
13	C₂×C₄×D₄×C₃₁	—	2	2	no	—
14	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
15	C₄×SD₁₆×C₃₁	—	2	16	no	—
16	C₂×C₃₁×#106	C₂²	1	4	yes	—
17	C₂×C₃₁×#103	C₂²	1	4	yes	—
18	C₂×C₃₁×#102	C₂²	1	4	yes	—
19	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
20	C₃₁×#391	C₂²	1	2	yes	—
21	C₃₁×#390	—	2	2	no	—
22	C₃₁×#390	C₂²	1	2	yes	—
23	C₃₁×#389	—	2	2	no	—
24	C₃₁×#384	C₂²	1	1	yes	characteristic
25	C₃₁×#384	C₂²	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₃₁	—	2	2	no	—
27	C₄³×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#336	C₂²	1	2	yes	—
29	C₃₁×#336	C₂²	1	2	yes	—
30	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
31	C₂²×D₄×C₃₁	—	2	2	no	—
32	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
33	C₂³×C₄×C₃₁	—	2	2	no	—
34	C₂×SD₁₆×C₃₁	—	2	8	no	—
35	C₂×SD₁₆×C₃₁	—	2	8	no	—
36	C₂×SD₁₆×C₃₁	—	2	8	no	—
37	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
38	C₃₁×C₄⋊Q₈	—	2	4	no	—
39	C₃₁×#127	—	2	4	no	—
40	C₄×Q₈×C₃₁	—	2	8	no	—
41	C₄×D₄×C₃₁	—	2	8	no	—
42	C₄×D₄×C₃₁	—	4	8	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
45	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
46	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
47	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
49	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
50	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
51	C₂×C₄²×C₃₁	—	2	2	no	—
52	C₂×C₄²×C₃₁	D₄	1	2	yes	—
53	C₂×C₄²×C₃₁	D₄	1	2	yes	—
54	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
57	C₃₁×#106	—	2	8	no	—
58	C₃₁×#103	—	2	8	no	—
59	C₃₁×#102	—	2	8	no	—
60	C₄×C₈×C₃₁	D₄	1	8	yes	complemented
61	C₃₁×#95	—	2	4	no	—
62	C₂⁴×C₃₁	—	2	2	no	—
63	C₂×Q₈×C₃₁	—	2	2	no	—
64	C₂×Q₈×C₃₁	—	2	4	no	—
65	C₂×Q₈×C₃₁	—	2	4	no	—
66	C₂×Q₈×C₃₁	#41	1	4	yes	—
67	C₂×D₄×C₃₁	—	2	8	no	—
68	C₂×D₄×C₃₁	—	2	4	no	—
69	C₂×D₄×C₃₁	—	2	4	no	—
70	C₂×D₄×C₃₁	—	2	2	no	—
71	C₂×D₄×C₃₁	—	4	4	no	—
72	C₂×D₄×C₃₁	—	4	4	no	—
73	C₂×D₄×C₃₁	#41	1	4	yes	—
74	C₂²×C₄×C₃₁	—	2	2	no	—
75	C₂²×C₄×C₃₁	—	2	2	no	—
76	C₂²×C₄×C₃₁	—	2	2	no	—
77	C₂²×C₄×C₃₁	—	4	4	no	—
78	C₂²×C₄×C₃₁	—	4	8	no	—
79	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
82	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
84	SD₁₆×C₃₁	—	2	32	no	—
85	C₂×C₈×C₃₁	—	2	4	no	—
86	C₂×C₈×C₃₁	—	2	4	no	—
87	C₂×C₈×C₃₁	C₂×D₄	1	8	yes	—
88	C₄⋊C₄×C₃₁	—	2	8	no	—
89	C₄⋊C₄×C₃₁	—	2	4	no	—
90	C₄⋊C₄×C₃₁	—	2	4	no	—
91	C₄⋊C₄×C₃₁	—	4	4	no	—
92	C₄⋊C₄×C₃₁	—	4	4	no	—
93	C₄⋊C₄×C₃₁	—	4	4	no	—
94	C₂²⋊C₄×C₃₁	—	4	4	no	—
95	C₂²⋊C₄×C₃₁	—	4	8	no	—
96	C₄²×C₃₁	—	2	2	no	—
97	C₄²×C₃₁	—	2	2	no	—
98	C₄²×C₃₁	—	4	2	no	—
99	C₄²×C₃₁	—	4	4	no	—
100	C₄²×C₃₁	SD₁₆	1	8	yes	complemented
101	C₄²×C₃₁	C₂×D₄	1	4	yes	—
102	#23799	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂³×C₃₁	—	2	4	no	—
104	C₂³×C₃₁	—	4	2	no	—
105	C₂³×C₃₁	—	4	2	no	—
106	C₂³×C₃₁	—	4	4	no	—
107	C₂³×C₃₁	—	4	4	no	—
108	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
109	Q₈×C₃₁	—	2	8	no	—
110	Q₈×C₃₁	—	4	4	no	—
111	D₄×C₃₁	—	2	8	no	—
112	D₄×C₃₁	—	4	16	no	—
113	D₄×C₃₁	—	4	4	no	—
114	C₂×C₄×C₃₁	—	2	2	no	—
115	C₂×C₄×C₃₁	—	2	2	no	—
116	C₂×C₄×C₃₁	—	2	2	no	—
117	C₂×C₄×C₃₁	—	2	2	no	—
118	C₂×C₄×C₃₁	—	2	2	no	—
119	C₂×C₄×C₃₁	—	2	4	no	—
120	C₂×C₄×C₃₁	—	4	2	no	—
121	C₂×C₄×C₃₁	—	4	2	no	—
122	C₂×C₄×C₃₁	—	4	2	no	—
123	C₂×C₄×C₃₁	—	4	2	no	—
124	C₂×C₄×C₃₁	—	4	4	no	—
125	C₂×C₄×C₃₁	—	4	4	no	—
126	C₂×C₄×C₃₁	—	8	8	no	—
127	C₂×C₄×C₃₁	#127	1	2	yes	—
128	C₂×C₄×C₃₁	#127	1	2	yes	—
129	C₂×C₄×C₃₁	C₂×SD₁₆	1	4	yes	—
130	C₂×C₄×C₃₁	C₂×SD₁₆	1	4	yes	—
131	C₂×C₄×C₃₁	C₂×SD₁₆	1	4	yes	—
132	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
133	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
134	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
135	C₈×C₃₁	—	2	8	no	—
136	C₂×C₄×SD₁₆	C₂×C₃₁	1	4	yes	complemented
137	C₄×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
138	C₄×#127	C₂×C₃₁	1	1	yes	characteristic
139	#2096	C₂×C₃₁	1	2	yes	complemented
140	#2094	C₂×C₃₁	1	2	yes	complemented
141	#2014	C₂×C₃₁	1	2	yes	—
142	#2013	C₂×C₃₁	1	2	yes	complemented
143	#1744	C₂×C₃₁	1	1	yes	characteristic, complemented
144	C₂²×C₃₁	—	4	8	no	—
145	C₂²×C₃₁	—	4	4	no	—
146	C₂²×C₃₁	—	8	4	no	—
147	C₂²×C₃₁	—	8	2	no	—
148	C₂²×C₃₁	—	8	2	no	—
149	C₂²×C₃₁	#525	1	1	yes	characteristic
150	C₂²×C₃₁	#525	1	1	yes	characteristic
151	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
152	C₂²×C₃₁	C₂²×SD₁₆	1	2	yes	—
153	C₂²×C₃₁	#575	1	2	yes	—
154	C₄×C₃₁	—	2	4	no	—
155	C₄×C₃₁	—	2	4	no	—
156	C₄×C₃₁	—	2	2	no	—
157	C₄×C₃₁	—	4	4	no	—
158	C₄×C₃₁	—	8	2	no	—
159	C₄×C₃₁	—	8	2	no	—
160	C₄×C₃₁	#492	1	4	yes	—
161	C₄×C₃₁	#525	1	4	yes	—
162	C₂²×SD₁₆	C₂²×C₃₁	1	4	yes	—
163	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
164	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
165	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
166	C₂×C₄×D₄	—	2	2	no	—
167	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
168	C₄×SD₁₆	—	2	16	no	—
169	C₂×#106	C₂²×C₃₁	1	4	yes	—
170	C₂×#103	C₂²×C₃₁	1	4	yes	—
171	C₂×#102	C₂²×C₃₁	1	4	yes	—
172	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
173	#391	C₂²×C₃₁	1	2	yes	—
174	#390	—	2	2	no	—
175	#390	C₂²×C₃₁	1	2	yes	—
176	#389	—	2	2	no	—
177	#384	C₂²×C₃₁	1	1	yes	characteristic
178	#384	C₂²×C₃₁	1	1	yes	characteristic
179	C₄×C₄⋊C₄	—	2	2	no	—
180	C₄³	C₂²×C₃₁	1	1	yes	characteristic
181	#336	C₂²×C₃₁	1	2	yes	—
182	#336	C₂²×C₃₁	1	2	yes	—
183	C₂×C₃₁	—	8	4	no	—
184	C₂×C₃₁	#2397	1	1	yes	characteristic
185	C₂×C₃₁	#3101	1	2	yes	—
186	C₂×C₃₁	#3103	1	2	yes	—
187	C₂×C₃₁	C₂×#492	1	1	yes	characteristic
188	C₂×C₃₁	#3150	1	1	yes	characteristic
189	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
190	C₂²×D₄	—	2	2	no	—
191	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
192	C₂³×C₄	—	2	2	no	—
193	C₂×SD₁₆	—	2	8	no	—
194	C₂×SD₁₆	—	2	8	no	—
195	C₂×SD₁₆	—	2	8	no	—
196	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
197	C₄⋊Q₈	—	2	4	no	—
198	#127	—	2	4	no	—
199	C₄×Q₈	—	2	8	no	—
200	C₄×D₄	—	2	8	no	—
201	C₄×D₄	—	4	8	no	—
202	C₂×C₄⋊C₄	—	2	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
207	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
208	C₂×C₂²⋊C₄	—	2	4	no	—
209	C₂×C₂²⋊C₄	—	2	2	no	—
210	C₂×C₄²	—	2	2	no	—
211	C₂×C₄²	D₄×C₃₁	1	2	yes	—
212	C₂×C₄²	D₄×C₃₁	1	2	yes	—
213	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
214	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
215	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
216	#106	—	2	8	no	—
217	#103	—	2	8	no	—
218	#102	—	2	8	no	—
219	C₄×C₈	D₄×C₃₁	1	8	yes	complemented
220	#95	—	2	4	no	—
221	C₃₁	#23799	1	1	yes	characteristic, direct product factor, Sylow
222	C₂⁴	—	2	2	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	2	4	no	—
225	C₂×Q₈	—	2	4	no	—
226	C₂×Q₈	#41×C₃₁	1	4	yes	—
227	C₂×D₄	—	2	8	no	—
228	C₂×D₄	—	2	4	no	—
229	C₂×D₄	—	2	4	no	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	4	4	no	—
232	C₂×D₄	—	4	4	no	—
233	C₂×D₄	#41×C₃₁	1	4	yes	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	4	4	no	—
238	C₂²×C₄	—	4	8	no	—
239	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
240	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
241	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
242	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
243	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
244	SD₁₆	—	2	32	no	—
245	C₂×C₈	—	2	4	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	C₂×D₄×C₃₁	1	8	yes	—
248	C₄⋊C₄	—	2	8	no	—
249	C₄⋊C₄	—	2	4	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	4	4	no	—
252	C₄⋊C₄	—	4	4	no	—
253	C₄⋊C₄	—	4	4	no	—
254	C₂²⋊C₄	—	4	4	no	—
255	C₂²⋊C₄	—	4	8	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	4	2	no	—
259	C₄²	—	4	4	no	—
260	C₄²	SD₁₆×C₃₁	1	8	yes	complemented
261	C₄²	C₂×D₄×C₃₁	1	4	yes	—
262	C₂³	—	2	4	no	—
263	C₂³	—	4	2	no	—
264	C₂³	—	4	2	no	—
265	C₂³	—	4	4	no	—
266	C₂³	—	4	4	no	—
267	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
268	Q₈	—	2	8	no	—
269	Q₈	—	4	4	no	—
270	D₄	—	2	8	no	—
271	D₄	—	4	16	no	—
272	D₄	—	4	4	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	8	8	no	—
286	C₂×C₄	C₃₁×#127	1	2	yes	—
287	C₂×C₄	C₃₁×#127	1	2	yes	—
288	C₂×C₄	C₂×SD₁₆×C₃₁	1	4	yes	—
289	C₂×C₄	C₂×SD₁₆×C₃₁	1	4	yes	—
290	C₂×C₄	C₂×SD₁₆×C₃₁	1	4	yes	—
291	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
292	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
293	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
294	C₈	—	2	8	no	—
295	C₂²	—	4	8	no	—
296	C₂²	—	4	4	no	—
297	C₂²	—	8	4	no	—
298	C₂²	—	8	2	no	—
299	C₂²	—	8	2	no	—
300	C₂²	C₃₁×#525	1	1	yes	characteristic
301	C₂²	C₃₁×#525	1	1	yes	characteristic
302	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
303	C₂²	C₂²×SD₁₆×C₃₁	1	2	yes	—
304	C₂²	C₃₁×#575	1	2	yes	—
305	C₄	—	2	4	no	—
306	C₄	—	2	4	no	—
307	C₄	—	2	2	no	—
308	C₄	—	4	4	no	—
309	C₄	—	8	2	no	—
310	C₄	—	8	2	no	—
311	C₄	C₃₁×#492	1	4	yes	—
312	C₄	C₃₁×#525	1	4	yes	—
313	C₂	—	8	4	no	—
314	C₂	C₃₁×#2397	1	1	yes	characteristic
315	C₂	C₃₁×#3101	1	2	yes	—
316	C₂	C₃₁×#3103	1	2	yes	—
317	C₂	C₂×C₃₁×#492	1	1	yes	characteristic
318	C₂	C₃₁×#3150	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	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₄×SD₁₆×C₃₁	C₂	semidirect
2. C₄×C₃₁×C₄⋊Q₈	C₂	semidirect
4. C₃₁×#2096	C₂	semidirect
5. C₃₁×#2094	C₂	semidirect
7. C₃₁×#2013	C₂	semidirect
8. C₃₁×#1744	C₂	semidirect
60. C₄×C₈×C₃₁	D₄	semidirect
100. C₄²×C₃₁	SD₁₆	semidirect
102. #23799	C₃₁	direct (N × H)
136. C₂×C₄×SD₁₆	C₂×C₃₁	semidirect
137. C₄×C₄⋊Q₈	C₂×C₃₁	semidirect
139. #2096	C₂×C₃₁	semidirect
140. #2094	C₂×C₃₁	semidirect
142. #2013	C₂×C₃₁	semidirect
143. #1744	C₂×C₃₁	semidirect
219. C₄×C₈	D₄×C₃₁	semidirect
221. C₃₁	#23799	direct (N × H)
260. C₄²	SD₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×SD₁₆×C₃₁	2
C₄×C₃₁×C₄⋊Q₈	2
C₄×C₃₁×#127	2
C₃₁×#2096	2
C₃₁×#2094	2
C₃₁×#2014	2
C₃₁×#2013	2
C₃₁×#1744	2
#23799	31

Subgroups — C31×#23799 #21480558

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23799 #21480558