Subgroups — C₃₁×#2422 #16842355

Order 3968.

652 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#123	C₂	1	2	yes	maximal, complemented
2	C₂×C₃₁×#121	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#397	C₂	1	2	yes	maximal, complemented
5	C₃₁×#394	C₂	1	2	yes	maximal, complemented
6	C₃₁×#390	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#389	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#385	C₂	1	2	yes	maximal, complemented
9	C₃₁×#384	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#380	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₂²⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄×C₃₁	C₂²	1	2	yes	—
13	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₃₁	C₂²	1	2	yes	—
16	C₃₁×#123	—	2	8	no	—
17	C₃₁×#121	—	2	4	no	—
18	C₄×D₄×C₃₁	—	2	4	no	—
19	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
23	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
25	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
27	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
28	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
29	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
31	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
33	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
34	C₃₁×#95	C₂²	1	1	yes	characteristic
35	C₃₁×#95	C₂²	1	2	yes	—
36	C₃₁×#95	C₂²	1	2	yes	—
37	C₃₁×#95	C₂²	1	1	yes	characteristic
38	C₂⁴×C₃₁	C₂³	1	2	yes	—
39	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
40	C₂×D₄×C₃₁	—	2	4	no	—
41	C₂×D₄×C₃₁	—	2	4	no	—
42	C₂×D₄×C₃₁	—	2	2	no	—
43	C₂×D₄×C₃₁	—	2	2	no	—
44	C₂×D₄×C₃₁	—	4	2	no	—
45	C₂×D₄×C₃₁	D₄	1	4	yes	—
46	C₂²×C₄×C₃₁	—	2	4	no	—
47	C₂²×C₄×C₃₁	—	2	2	no	—
48	C₂²×C₄×C₃₁	—	2	4	no	—
49	C₂²×C₄×C₃₁	—	2	2	no	—
50	C₂²×C₄×C₃₁	—	2	2	no	—
51	C₂²×C₄×C₃₁	—	4	2	no	—
52	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
55	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
56	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
57	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₃₁	—	2	2	no	—
62	C₄⋊C₄×C₃₁	—	2	2	no	—
63	C₄⋊C₄×C₃₁	—	2	4	no	—
64	C₄⋊C₄×C₃₁	—	2	4	no	—
65	C₄⋊C₄×C₃₁	—	2	2	no	—
66	C₄⋊C₄×C₃₁	—	4	2	no	—
67	C₂²⋊C₄×C₃₁	—	2	4	no	—
68	C₂²⋊C₄×C₃₁	—	2	4	no	—
69	C₂²⋊C₄×C₃₁	—	2	4	no	—
70	C₂²⋊C₄×C₃₁	—	2	4	no	—
71	C₂²⋊C₄×C₃₁	—	2	2	no	—
72	C₂²⋊C₄×C₃₁	—	2	4	no	—
73	C₂²⋊C₄×C₃₁	D₄	1	4	yes	complemented
74	C₄²×C₃₁	—	2	2	no	—
75	C₄²×C₃₁	—	2	2	no	—
76	C₂³×C₃₁	—	2	2	no	—
77	C₂³×C₃₁	—	2	1	no	—
78	C₂³×C₃₁	—	2	1	no	—
79	C₂³×C₃₁	—	2	2	no	—
80	C₂³×C₃₁	—	2	1	no	—
81	C₂³×C₃₁	—	2	1	no	—
82	C₂³×C₃₁	—	2	2	no	—
83	C₂³×C₃₁	—	2	2	no	—
84	C₂³×C₃₁	—	2	2	no	—
85	C₂³×C₃₁	—	2	1	no	—
86	C₂³×C₃₁	—	2	1	no	—
87	C₂³×C₃₁	—	2	2	no	—
88	C₂³×C₃₁	C₂×D₄	1	2	yes	—
89	C₂³×C₃₁	C₂×D₄	1	4	yes	—
90	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
91	D₄×C₃₁	—	2	8	no	—
92	D₄×C₃₁	—	4	8	no	—
93	C₂×C₄×C₃₁	—	2	1	no	—
94	C₂×C₄×C₃₁	—	2	1	no	—
95	C₂×C₄×C₃₁	—	2	1	no	—
96	C₂×C₄×C₃₁	—	2	1	no	—
97	C₂×C₄×C₃₁	—	2	1	no	—
98	C₂×C₄×C₃₁	—	2	1	no	—
99	C₂×C₄×C₃₁	—	2	1	no	—
100	C₂×C₄×C₃₁	—	2	2	no	—
101	C₂×C₄×C₃₁	—	2	1	no	—
102	C₂×C₄×C₃₁	—	2	2	no	—
103	C₂×C₄×C₃₁	—	2	2	no	—
104	C₂×C₄×C₃₁	—	2	2	no	—
105	C₂×C₄×C₃₁	—	2	2	no	—
106	C₂×C₄×C₃₁	—	2	2	no	—
107	C₂×C₄×C₃₁	—	2	1	no	—
108	C₂×C₄×C₃₁	—	2	2	no	—
109	C₂×C₄×C₃₁	—	2	1	no	—
110	C₂×C₄×C₃₁	—	2	1	no	—
111	C₂×C₄×C₃₁	—	2	2	no	—
112	C₂×C₄×C₃₁	—	2	2	no	—
113	C₂×C₄×C₃₁	—	2	1	no	—
114	C₂×C₄×C₃₁	—	4	8	no	—
115	C₂×C₄×C₃₁	—	4	4	no	—
116	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
117	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
118	C₂×C₄×C₃₁	C₂×D₄	1	2	yes	—
119	C₂×C₄×C₃₁	#41	1	2	yes	—
120	C₂×C₄×C₃₁	#41	1	2	yes	—
121	C₂×C₄×C₃₁	#41	1	2	yes	—
122	#2422	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
123	C₂²×C₃₁	—	2	4	no	—
124	C₂²×C₃₁	—	2	2	no	—
125	C₂²×C₃₁	—	2	4	no	—
126	C₂²×C₃₁	—	2	2	no	—
127	C₂²×C₃₁	—	2	2	no	—
128	C₂²×C₃₁	—	2	4	no	—
129	C₂²×C₃₁	—	4	1	no	—
130	C₂²×C₃₁	—	4	2	no	—
131	C₂²×C₃₁	—	4	1	no	—
132	C₂²×C₃₁	—	4	2	no	—
133	C₂²×C₃₁	—	4	2	no	—
134	C₂²×C₃₁	—	4	1	no	—
135	C₂²×C₃₁	—	4	1	no	—
136	C₂²×C₃₁	—	4	2	no	—
137	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
138	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
139	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
140	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
141	C₂²×C₃₁	C₂×#41	1	1	yes	characteristic
142	C₂²×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
143	C₂²×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
144	C₄×C₃₁	—	2	2	no	—
145	C₄×C₃₁	—	2	2	no	—
146	C₄×C₃₁	—	2	2	no	—
147	C₄×C₃₁	—	4	1	no	—
148	C₄×C₃₁	—	4	2	no	—
149	C₄×C₃₁	—	4	2	no	—
150	C₄×C₃₁	—	4	2	no	—
151	C₄×C₃₁	—	4	1	no	—
152	C₄×C₃₁	#123	1	4	yes	complemented
153	C₂×#123	C₂×C₃₁	1	2	yes	complemented
154	C₂×#121	C₂×C₃₁	1	1	yes	characteristic
155	C₂×C₄×D₄	C₂×C₃₁	1	1	yes	characteristic, complemented
156	#397	C₂×C₃₁	1	2	yes	complemented
157	#394	C₂×C₃₁	1	2	yes	complemented
158	#390	C₂×C₃₁	1	1	yes	characteristic, complemented
159	#389	C₂×C₃₁	1	1	yes	characteristic, complemented
160	#385	C₂×C₃₁	1	2	yes	complemented
161	#384	C₂×C₃₁	1	1	yes	characteristic, complemented
162	#380	C₂×C₃₁	1	1	yes	characteristic, complemented
163	C₄×C₂²⋊C₄	C₂×C₃₁	1	1	yes	characteristic, complemented
164	C₂×C₃₁	—	4	4	no	—
165	C₂×C₃₁	—	4	2	no	—
166	C₂×C₃₁	C₂×#123	1	1	yes	characteristic
167	C₂×C₃₁	#532	1	1	yes	characteristic
168	C₂×C₃₁	#542	1	1	yes	characteristic
169	C₂×C₃₁	D₄²	1	1	yes	characteristic
170	C₂×C₃₁	#547	1	1	yes	characteristic
171	C₂×C₃₁	#547	1	1	yes	characteristic
172	C₂×C₃₁	#553	1	1	yes	characteristic
173	C₂²×D₄	C₂²×C₃₁	1	2	yes	—
174	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
175	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
176	C₂³×C₄	C₂²×C₃₁	1	2	yes	—
177	#123	—	2	8	no	—
178	#121	—	2	4	no	—
179	C₄×D₄	—	2	4	no	—
180	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
181	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
182	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
183	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
184	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
185	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
186	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
187	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
188	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
189	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
190	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
191	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
192	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
193	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
194	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
195	#95	C₂²×C₃₁	1	1	yes	characteristic
196	#95	C₂²×C₃₁	1	1	yes	characteristic
197	#95	C₂²×C₃₁	1	2	yes	—
198	#95	C₂²×C₃₁	1	2	yes	—
199	C₃₁	#2422	1	1	yes	characteristic, direct product factor, Sylow
200	C₂⁴	C₂³×C₃₁	1	2	yes	—
201	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
202	C₂×D₄	—	2	4	no	—
203	C₂×D₄	—	2	4	no	—
204	C₂×D₄	—	2	2	no	—
205	C₂×D₄	—	2	2	no	—
206	C₂×D₄	—	4	2	no	—
207	C₂×D₄	D₄×C₃₁	1	4	yes	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	4	2	no	—
214	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
215	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
216	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
217	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
218	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
219	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
220	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
221	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
222	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	4	2	no	—
229	C₂²⋊C₄	—	2	4	no	—
230	C₂²⋊C₄	—	2	4	no	—
231	C₂²⋊C₄	—	2	4	no	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₂²⋊C₄	—	2	2	no	—
234	C₂²⋊C₄	—	2	4	no	—
235	C₂²⋊C₄	D₄×C₃₁	1	4	yes	complemented
236	C₄²	—	2	2	no	—
237	C₄²	—	2	2	no	—
238	C₂³	—	2	2	no	—
239	C₂³	—	2	1	no	—
240	C₂³	—	2	1	no	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	1	no	—
243	C₂³	—	2	1	no	—
244	C₂³	—	2	2	no	—
245	C₂³	—	2	2	no	—
246	C₂³	—	2	2	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	2	1	no	—
249	C₂³	—	2	2	no	—
250	C₂³	C₂×D₄×C₃₁	1	2	yes	—
251	C₂³	C₂×D₄×C₃₁	1	4	yes	—
252	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
253	D₄	—	2	8	no	—
254	D₄	—	4	8	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	1	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	1	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	1	no	—
276	C₂×C₄	—	4	8	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
279	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
280	C₂×C₄	C₂×D₄×C₃₁	1	2	yes	—
281	C₂×C₄	#41×C₃₁	1	2	yes	—
282	C₂×C₄	#41×C₃₁	1	2	yes	—
283	C₂×C₄	#41×C₃₁	1	2	yes	—
284	C₂²	—	2	4	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	4	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	4	no	—
290	C₂²	—	4	1	no	—
291	C₂²	—	4	2	no	—
292	C₂²	—	4	1	no	—
293	C₂²	—	4	2	no	—
294	C₂²	—	4	2	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	—	4	2	no	—
298	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
299	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
300	C₂²	C₂×#41×C₃₁	1	1	yes	characteristic
301	C₂²	C₂×#41×C₃₁	1	1	yes	characteristic
302	C₂²	C₂×#41×C₃₁	1	1	yes	characteristic
303	C₂²	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
304	C₂²	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
305	C₄	—	2	2	no	—
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	2	no	—
310	C₄	—	4	2	no	—
311	C₄	—	4	2	no	—
312	C₄	—	4	1	no	—
313	C₄	C₃₁×#123	1	4	yes	complemented
314	C₂	—	4	4	no	—
315	C₂	—	4	2	no	—
316	C₂	C₂×C₃₁×#123	1	1	yes	characteristic
317	C₂	C₃₁×#532	1	1	yes	characteristic
318	C₂	C₃₁×#542	1	1	yes	characteristic
319	C₂	D₄²×C₃₁	1	1	yes	characteristic
320	C₂	C₃₁×#547	1	1	yes	characteristic
321	C₂	C₃₁×#547	1	1	yes	characteristic
322	C₂	C₃₁×#553	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#121	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#123	C₂	semidirect
3. C₂×C₄×D₄×C₃₁	C₂	semidirect
4. C₃₁×#397	C₂	semidirect
5. C₃₁×#394	C₂	semidirect
6. C₃₁×#390	C₂	semidirect
7. C₃₁×#389	C₂	semidirect
8. C₃₁×#385	C₂	semidirect
9. C₃₁×#384	C₂	semidirect
10. C₃₁×#380	C₂	semidirect
11. C₄×C₂²⋊C₄×C₃₁	C₂	semidirect
73. C₂²⋊C₄×C₃₁	D₄	semidirect
122. #2422	C₃₁	direct (N × H)
152. C₄×C₃₁	#123	semidirect
153. C₂×#123	C₂×C₃₁	semidirect
155. C₂×C₄×D₄	C₂×C₃₁	semidirect
156. #397	C₂×C₃₁	semidirect
157. #394	C₂×C₃₁	semidirect
158. #390	C₂×C₃₁	semidirect
159. #389	C₂×C₃₁	semidirect
160. #385	C₂×C₃₁	semidirect
161. #384	C₂×C₃₁	semidirect
162. #380	C₂×C₃₁	semidirect
163. C₄×C₂²⋊C₄	C₂×C₃₁	semidirect
199. C₃₁	#2422	direct (N × H)
235. C₂²⋊C₄	D₄×C₃₁	semidirect
313. C₄	C₃₁×#123	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#123	2
C₂×C₃₁×#121	2
C₂×C₄×D₄×C₃₁	2
C₃₁×#397	2
C₃₁×#394	2
C₃₁×#390	2
C₃₁×#389	2
C₃₁×#385	2
C₃₁×#384	2
C₃₁×#380	2
C₄×C₂²⋊C₄×C₃₁	2
#2422	31

Subgroups — C31×#2422 #16842355

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#2422 #16842355