Subgroups — C₃₁×#23352 #21480111

Order 7936.

764 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₃₁×#124	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1985	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1924	C₂	1	4	yes	maximal, complemented
4	C₃₁×#1869	C₂	1	2	yes	maximal
5	C₃₁×#1850	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1831	C₂	1	2	yes	maximal, complemented
7	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
8	C₂×C₃₁×#124	C₄	1	1	yes	characteristic
9	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₃₁×C₂²⋊C₈	C₂²	1	4	yes	—
15	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃₁×#97	C₂²	1	2	yes	—
17	C₃₁×#391	C₄	1	1	yes	characteristic
18	C₃₁×#390	C₄	1	1	yes	characteristic
19	C₃₁×#388	C₄	1	4	yes	—
20	C₃₁×#382	C₄	1	1	yes	characteristic
21	C₄×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
22	C₄³×C₃₁	C₂²	1	1	yes	characteristic
23	C₃₁×#336	C₂²	1	4	yes	—
24	C₃₁×#336	C₂²	1	4	yes	—
25	C₃₁×#336	C₂²	1	4	yes	—
26	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
29	C₂²×C₈×C₃₁	—	4	4	no	—
30	C₂²×C₈×C₃₁	—	4	2	no	—
31	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
32	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
33	C₃₁×#124	—	2	4	no	—
34	C₄×Q₈×C₃₁	—	4	1	no	—
35	C₄×Q₈×C₃₁	—	4	1	no	—
36	C₄×D₄×C₃₁	—	4	1	no	—
37	C₄×D₄×C₃₁	—	4	1	no	—
38	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
41	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
42	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
43	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
48	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
49	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
50	C₃₁×C₄⋊C₈	—	4	2	no	—
51	C₃₁×C₄⋊C₈	—	4	2	no	—
52	C₃₁×C₄⋊C₈	—	4	2	no	—
53	C₃₁×C₂²⋊C₈	—	4	2	no	—
54	C₃₁×C₂²⋊C₈	—	4	4	no	—
55	C₃₁×#97	—	4	2	no	—
56	C₃₁×#95	C₂×C₄	1	4	yes	—
57	C₃₁×#95	C₂×C₄	1	4	yes	—
58	C₂⁴×C₃₁	C₂²×C₄	1	2	yes	—
59	C₂×Q₈×C₃₁	—	2	2	no	—
60	C₂×Q₈×C₃₁	—	4	1	no	—
61	C₂×Q₈×C₃₁	—	4	1	no	—
62	C₂×D₄×C₃₁	—	4	1	no	—
63	C₂×D₄×C₃₁	—	4	1	no	—
64	C₂×D₄×C₃₁	M₁₆	1	4	yes	—
65	C₂²×C₄×C₃₁	—	2	2	no	—
66	C₂²×C₄×C₃₁	—	2	4	no	—
67	C₂²×C₄×C₃₁	—	2	2	no	—
68	C₂²×C₄×C₃₁	—	2	4	no	—
69	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
71	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₃₁	C₂²×C₄	1	2	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	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₃₁	—	2	4	no	—
78	C₂×C₈×C₃₁	—	2	4	no	—
79	C₂×C₈×C₃₁	—	2	4	no	—
80	C₂×C₈×C₃₁	—	2	4	no	—
81	C₂×C₈×C₃₁	—	2	4	no	—
82	C₂×C₈×C₃₁	—	2	4	no	—
83	C₂×C₈×C₃₁	—	4	8	no	—
84	C₄⋊C₄×C₃₁	—	4	1	no	—
85	C₄⋊C₄×C₃₁	—	4	1	no	—
86	C₄⋊C₄×C₃₁	—	4	2	no	—
87	C₂²⋊C₄×C₃₁	—	2	8	no	—
88	C₂²⋊C₄×C₃₁	—	2	8	no	—
89	C₂²⋊C₄×C₃₁	—	4	2	no	—
90	C₄²×C₃₁	—	2	2	no	—
91	C₄²×C₃₁	—	2	4	no	—
92	C₄²×C₃₁	—	2	2	no	—
93	C₄²×C₃₁	—	2	2	no	—
94	C₄²×C₃₁	—	2	2	no	—
95	C₄²×C₃₁	—	2	1	no	—
96	C₄²×C₃₁	—	2	1	no	—
97	C₄²×C₃₁	—	2	2	no	—
98	C₄²×C₃₁	—	4	1	no	—
99	#23352	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
100	C₂³×C₃₁	—	2	2	no	—
101	C₂³×C₃₁	—	2	4	no	—
102	C₂³×C₃₁	—	2	4	no	—
103	C₂³×C₃₁	—	2	2	no	—
104	C₂³×C₃₁	C₂×M₁₆	1	4	yes	—
105	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
106	Q₈×C₃₁	—	4	4	no	—
107	D₄×C₃₁	—	4	4	no	—
108	C₂×C₄×C₃₁	—	2	1	no	—
109	C₂×C₄×C₃₁	—	2	2	no	—
110	C₂×C₄×C₃₁	—	2	2	no	—
111	C₂×C₄×C₃₁	—	2	1	no	—
112	C₂×C₄×C₃₁	—	2	1	no	—
113	C₂×C₄×C₃₁	—	2	2	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	4	no	—
118	C₂×C₄×C₃₁	—	2	2	no	—
119	C₂×C₄×C₃₁	—	2	2	no	—
120	C₂×C₄×C₃₁	—	2	2	no	—
121	C₂×C₄×C₃₁	—	2	2	no	—
122	C₂×C₄×C₃₁	—	2	4	no	—
123	C₂×C₄×C₃₁	—	2	2	no	—
124	C₂×C₄×C₃₁	—	4	2	no	—
125	C₂×C₄×C₃₁	—	4	4	no	—
126	C₂×C₄×C₃₁	—	4	2	no	—
127	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
128	C₂×C₄×C₃₁	#131	1	4	yes	—
129	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
130	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
131	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
133	C₈×C₃₁	—	4	4	no	—
134	C₈×C₃₁	—	4	4	no	—
135	C₄×#124	C₂×C₃₁	1	1	yes	characteristic
136	#1985	C₂×C₃₁	1	2	yes	complemented
137	#1924	C₂×C₃₁	1	4	yes	complemented
138	#1869	C₂×C₃₁	1	2	yes	—
139	#1850	C₂×C₃₁	1	4	yes	complemented
140	#1831	C₂×C₃₁	1	2	yes	complemented
141	C₂²×C₃₁	—	2	4	no	—
142	C₂²×C₃₁	—	2	8	no	—
143	C₂²×C₃₁	—	4	2	no	—
144	C₂²×C₃₁	—	4	2	no	—
145	C₂²×C₃₁	—	4	4	no	—
146	C₂²×C₃₁	#518	1	1	yes	characteristic
147	C₂²×C₃₁	#520	1	2	yes	—
148	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
149	C₂²×C₃₁	C₂×#131	1	2	yes	—
150	C₂²×C₃₁	#568	1	1	yes	characteristic
151	C₄×C₃₁	—	2	2	no	—
152	C₄×C₃₁	—	2	4	no	—
153	C₄×C₃₁	—	4	2	no	—
154	C₄×C₃₁	—	4	2	no	—
155	C₄×C₃₁	—	4	1	no	—
156	C₄×C₃₁	—	4	2	no	—
157	C₄×C₃₁	—	4	2	no	—
158	C₄×C₃₁	H(3,𝔽₄)	1	1	yes	characteristic
159	C₄×C₃₁	#562	1	2	yes	—
160	C₄×C₃₁	#563	1	1	yes	characteristic
161	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
162	C₂×#124	C₄×C₃₁	1	1	yes	characteristic
163	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
164	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
165	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
166	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
167	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
168	C₂×C₂²⋊C₈	C₂²×C₃₁	1	4	yes	—
169	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
170	C₂×#97	C₂²×C₃₁	1	2	yes	—
171	#391	C₄×C₃₁	1	1	yes	characteristic
172	#390	C₄×C₃₁	1	1	yes	characteristic
173	#388	C₄×C₃₁	1	4	yes	—
174	#382	C₄×C₃₁	1	1	yes	characteristic
175	C₄×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
176	C₄³	C₂²×C₃₁	1	1	yes	characteristic
177	#336	C₂²×C₃₁	1	4	yes	—
178	#336	C₂²×C₃₁	1	4	yes	—
179	#336	C₂²×C₃₁	1	4	yes	—
180	C₂×C₃₁	—	4	4	no	—
181	C₂×C₃₁	#2379	1	1	yes	characteristic
182	C₂×C₃₁	#2973	1	1	yes	characteristic
183	C₂×C₃₁	#2979	1	1	yes	characteristic
184	C₂×C₃₁	#2990	1	2	yes	—
185	C₂×C₃₁	#2992	1	2	yes	—
186	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
187	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
188	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
189	C₂²×C₈	—	4	4	no	—
190	C₂²×C₈	—	4	2	no	—
191	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
192	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
193	#124	—	2	4	no	—
194	C₄×Q₈	—	4	1	no	—
195	C₄×Q₈	—	4	1	no	—
196	C₄×D₄	—	4	1	no	—
197	C₄×D₄	—	4	1	no	—
198	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
199	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
200	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
201	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
202	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
203	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
204	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
205	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
206	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
207	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
208	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
209	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
210	C₄⋊C₈	—	4	2	no	—
211	C₄⋊C₈	—	4	2	no	—
212	C₄⋊C₈	—	4	2	no	—
213	C₂²⋊C₈	—	4	2	no	—
214	C₂²⋊C₈	—	4	4	no	—
215	#97	—	4	2	no	—
216	#95	C₂×C₄×C₃₁	1	4	yes	—
217	#95	C₂×C₄×C₃₁	1	4	yes	—
218	C₃₁	#23352	1	1	yes	characteristic, direct product factor, Sylow
219	C₂⁴	C₂²×C₄×C₃₁	1	2	yes	—
220	C₂×Q₈	—	2	2	no	—
221	C₂×Q₈	—	4	1	no	—
222	C₂×Q₈	—	4	1	no	—
223	C₂×D₄	—	4	1	no	—
224	C₂×D₄	—	4	1	no	—
225	C₂×D₄	M₁₆×C₃₁	1	4	yes	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
231	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
232	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
233	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
234	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
235	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
236	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
237	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	4	8	no	—
245	C₄⋊C₄	—	4	1	no	—
246	C₄⋊C₄	—	4	1	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₂²⋊C₄	—	2	8	no	—
249	C₂²⋊C₄	—	2	8	no	—
250	C₂²⋊C₄	—	4	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	4	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	1	no	—
257	C₄²	—	2	1	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	4	1	no	—
260	C₂³	—	2	2	no	—
261	C₂³	—	2	4	no	—
262	C₂³	—	2	4	no	—
263	C₂³	—	2	2	no	—
264	C₂³	C₂×M₁₆×C₃₁	1	4	yes	—
265	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
266	Q₈	—	4	4	no	—
267	D₄	—	4	4	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	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	4	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
288	C₂×C₄	C₃₁×#131	1	4	yes	—
289	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
290	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
291	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
292	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
293	C₈	—	4	4	no	—
294	C₈	—	4	4	no	—
295	C₂²	—	2	4	no	—
296	C₂²	—	2	8	no	—
297	C₂²	—	4	2	no	—
298	C₂²	—	4	2	no	—
299	C₂²	—	4	4	no	—
300	C₂²	C₃₁×#518	1	1	yes	characteristic
301	C₂²	C₃₁×#520	1	2	yes	—
302	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
303	C₂²	C₂×C₃₁×#131	1	2	yes	—
304	C₂²	C₃₁×#568	1	1	yes	characteristic
305	C₄	—	2	2	no	—
306	C₄	—	2	4	no	—
307	C₄	—	4	2	no	—
308	C₄	—	4	2	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	2	no	—
311	C₄	—	4	2	no	—
312	C₄	C₃₁×H(3,𝔽₄)	1	1	yes	characteristic
313	C₄	C₃₁×#562	1	2	yes	—
314	C₄	C₃₁×#563	1	1	yes	characteristic
315	C₂	—	4	4	no	—
316	C₂	C₃₁×#2379	1	1	yes	characteristic
317	C₂	C₃₁×#2973	1	1	yes	characteristic
318	C₂	C₃₁×#2979	1	1	yes	characteristic
319	C₂	C₃₁×#2990	1	2	yes	—
320	C₂	C₃₁×#2992	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	C₂²×D₄×C₃₁	C₂×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
2. C₃₁×#1985	C₂	semidirect
3. C₃₁×#1924	C₂	semidirect
5. C₃₁×#1850	C₂	semidirect
6. C₃₁×#1831	C₂	semidirect
99. #23352	C₃₁	direct (N × H)
136. #1985	C₂×C₃₁	semidirect
137. #1924	C₂×C₃₁	semidirect
139. #1850	C₂×C₃₁	semidirect
140. #1831	C₂×C₃₁	semidirect
218. C₃₁	#23352	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₃₁×#124	2
C₃₁×#1985	2
C₃₁×#1924	2
C₃₁×#1869	2
C₃₁×#1850	2
C₃₁×#1831	2
#23352	31

Subgroups — C31×#23352 #21480111

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23352 #21480111