Subgroups — C₃₁×#21912 #21478671

Order 7936.

866 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1916	C₂	1	8	yes	maximal, complemented
3	C₃₁×#1817	C₂	1	2	yes	maximal
4	C₃₁×#1797	C₂	1	2	yes	maximal
5	C₄×C₃₁×C₂²⋊C₈	C₂	1	2	yes	maximal
6	C₂²×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
7	C₂²×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
8	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
9	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
12	C₂×C₃₁×C₂²⋊C₈	C₂²	1	4	yes	—
13	C₂×C₄×C₈×C₃₁	C₂²	1	4	yes	—
14	C₄×C₄⋊C₄×C₃₁	C₄	1	4	yes	—
15	C₄×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
16	C₂×C₃₁×#95	C₄	1	1	yes	characteristic
17	C₂×C₃₁×#95	C₄	1	1	yes	characteristic
18	C₃₁×#336	C₂²	1	4	yes	—
19	C₃₁×#336	C₂²	1	8	yes	—
20	C₃₁×#336	C₂²	1	4	yes	—
21	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
28	C₂²×C₈×C₃₁	C₂³	1	8	yes	—
29	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
30	C₂×C₄⋊C₄×C₃₁	C₈	1	8	yes	complemented
31	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
32	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₃₁	—	2	4	no	—
34	C₂×C₄²×C₃₁	—	4	2	no	—
35	C₂×C₄²×C₃₁	—	4	2	no	—
36	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
39	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
40	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
41	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
42	C₃₁×C₄⋊C₈	—	4	4	no	—
43	C₃₁×C₂²⋊C₈	—	2	8	no	—
44	C₄×C₈×C₃₁	—	4	4	no	—
45	C₃₁×#95	C₂×C₄	1	4	yes	—
46	C₃₁×#95	C₂×C₄	1	4	yes	—
47	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
48	C₂²×C₄×C₃₁	—	2	2	no	—
49	C₂²×C₄×C₃₁	—	2	2	no	—
50	C₂²×C₄×C₃₁	—	2	2	no	—
51	C₂²×C₄×C₃₁	—	2	2	no	—
52	C₂²×C₄×C₃₁	—	2	2	no	—
53	C₂²×C₄×C₃₁	—	2	2	no	—
54	C₂²×C₄×C₃₁	—	4	1	no	—
55	C₂²×C₄×C₃₁	—	4	1	no	—
56	C₂²×C₄×C₃₁	—	4	1	no	—
57	C₂²×C₄×C₃₁	—	4	1	no	—
58	C₂²×C₄×C₃₁	—	4	1	no	—
59	C₂²×C₄×C₃₁	—	4	1	no	—
60	C₂²×C₄×C₃₁	—	4	1	no	—
61	C₂²×C₄×C₃₁	—	4	1	no	—
62	C₂²×C₄×C₃₁	—	4	1	no	—
63	C₂²×C₄×C₃₁	—	4	1	no	—
64	C₂²×C₄×C₃₁	C₂×C₈	1	4	yes	—
65	C₂²×C₄×C₃₁	C₂×C₈	1	4	yes	—
66	C₂²×C₄×C₃₁	C₂×C₈	1	4	yes	—
67	C₂²×C₄×C₃₁	M₁₆	1	4	yes	—
68	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
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	2	yes	—
74	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₃₁	#41	1	4	yes	—
76	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₃₁	—	2	8	no	—
78	C₂×C₈×C₃₁	—	2	8	no	—
79	C₂×C₈×C₃₁	—	2	8	no	—
80	C₄⋊C₄×C₃₁	—	2	8	no	—
81	C₄⋊C₄×C₃₁	C₂×C₈	1	16	yes	—
82	C₄²×C₃₁	—	2	4	no	—
83	C₄²×C₃₁	—	2	4	no	—
84	C₄²×C₃₁	—	2	4	no	—
85	C₄²×C₃₁	—	2	4	no	—
86	C₄²×C₃₁	—	4	2	no	—
87	C₄²×C₃₁	—	4	2	no	—
88	#21912	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂³×C₃₁	—	2	1	no	—
90	C₂³×C₃₁	—	2	1	no	—
91	C₂³×C₃₁	—	2	1	no	—
92	C₂³×C₃₁	—	2	1	no	—
93	C₂³×C₃₁	#117	1	2	yes	—
94	C₂³×C₃₁	C₂²×C₈	1	2	yes	—
95	C₂³×C₃₁	C₂×M₁₆	1	2	yes	—
96	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₃₁	—	2	2	no	—
98	C₂×C₄×C₃₁	—	2	2	no	—
99	C₂×C₄×C₃₁	—	2	2	no	—
100	C₂×C₄×C₃₁	—	2	2	no	—
101	C₂×C₄×C₃₁	—	2	1	no	—
102	C₂×C₄×C₃₁	—	2	1	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	8	no	—
108	C₂×C₄×C₃₁	—	2	8	no	—
109	C₂×C₄×C₃₁	—	2	2	no	—
110	C₂×C₄×C₃₁	—	2	2	no	—
111	C₂×C₄×C₃₁	—	2	2	no	—
112	C₂×C₄×C₃₁	—	2	2	no	—
113	C₂×C₄×C₃₁	—	2	2	no	—
114	C₂×C₄×C₃₁	—	2	2	no	—
115	C₂×C₄×C₃₁	—	4	4	no	—
116	C₂×C₄×C₃₁	—	4	4	no	—
117	C₂×C₄×C₃₁	—	4	4	no	—
118	C₂×C₄×C₃₁	—	4	4	no	—
119	C₂×C₄×C₃₁	—	4	4	no	—
120	C₂×C₄×C₃₁	#117	1	2	yes	—
121	C₂×C₄×C₃₁	C₂²×C₈	1	4	yes	—
122	C₂×C₄×C₃₁	C₂²×C₈	1	4	yes	—
123	C₂×C₄×C₃₁	C₂²×C₈	1	4	yes	—
124	C₂×C₄×C₃₁	C₂×M₁₆	1	4	yes	—
125	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
126	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
127	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
128	C₈×C₃₁	—	4	8	no	—
129	C₂×C₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic
130	#1916	C₂×C₃₁	1	8	yes	complemented
131	#1817	C₂×C₃₁	1	2	yes	—
132	#1797	C₂×C₃₁	1	2	yes	—
133	C₄×C₂²⋊C₈	C₂×C₃₁	1	2	yes	—
134	C₂²×C₃₁	—	2	2	no	—
135	C₂²×C₃₁	—	2	2	no	—
136	C₂²×C₃₁	—	2	2	no	—
137	C₂²×C₃₁	—	2	2	no	—
138	C₂²×C₃₁	—	2	2	no	—
139	C₂²×C₃₁	—	2	2	no	—
140	C₂²×C₃₁	#431	1	4	yes	—
141	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
142	C₂²×C₃₁	#518	1	1	yes	characteristic
143	C₂²×C₃₁	#519	1	1	yes	characteristic
144	C₂²×C₃₁	C₂³×C₈	1	1	yes	characteristic
145	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
146	C₂²×C₃₁	#568	1	1	yes	characteristic
147	C₂²×C₃₁	#568	1	1	yes	characteristic
148	C₄×C₃₁	—	2	2	no	—
149	C₄×C₃₁	—	2	4	no	—
150	C₄×C₃₁	—	2	4	no	—
151	C₄×C₃₁	—	2	4	no	—
152	C₄×C₃₁	—	2	4	no	—
153	C₄×C₃₁	—	4	2	no	—
154	C₄×C₃₁	—	4	2	no	—
155	C₄×C₃₁	#539	1	4	yes	—
156	C₂²×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
157	C₂²×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
158	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
159	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
160	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
161	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
162	C₂×C₂²⋊C₈	C₂²×C₃₁	1	4	yes	—
163	C₂×C₄×C₈	C₂²×C₃₁	1	4	yes	—
164	C₄×C₄⋊C₄	C₄×C₃₁	1	4	yes	—
165	C₄×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
166	C₂×#95	C₄×C₃₁	1	1	yes	characteristic
167	C₂×#95	C₄×C₃₁	1	1	yes	characteristic
168	#336	C₂²×C₃₁	1	4	yes	—
169	#336	C₂²×C₃₁	1	8	yes	—
170	#336	C₂²×C₃₁	1	4	yes	—
171	C₂×C₃₁	—	2	4	no	—
172	C₂×C₃₁	#2313	1	1	yes	characteristic
173	C₂×C₃₁	C₂×#431	1	1	yes	characteristic
174	C₂×C₃₁	#2921	1	1	yes	characteristic
175	C₂×C₃₁	#2972	1	1	yes	characteristic
176	C₂×C₃₁	#2973	1	1	yes	characteristic
177	C₂×C₃₁	#2982	1	1	yes	characteristic
178	C₂×C₃₁	#2983	1	1	yes	characteristic
179	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
180	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
181	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
182	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
183	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
184	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
185	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
186	C₂²×C₈	C₂³×C₃₁	1	8	yes	—
187	C₂×C₄⋊C₄	—	2	4	no	—
188	C₂×C₄⋊C₄	C₈×C₃₁	1	8	yes	complemented
189	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
190	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
191	C₂×C₄²	—	2	4	no	—
192	C₂×C₄²	—	4	2	no	—
193	C₂×C₄²	—	4	2	no	—
194	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
195	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
196	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
197	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
198	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
199	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
200	C₄⋊C₈	—	4	4	no	—
201	C₂²⋊C₈	—	2	8	no	—
202	C₄×C₈	—	4	4	no	—
203	#95	C₂×C₄×C₃₁	1	4	yes	—
204	#95	C₂×C₄×C₃₁	1	4	yes	—
205	C₃₁	#21912	1	1	yes	characteristic, direct product factor, Sylow
206	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	—	4	1	no	—
216	C₂²×C₄	—	4	1	no	—
217	C₂²×C₄	—	4	1	no	—
218	C₂²×C₄	—	4	1	no	—
219	C₂²×C₄	—	4	1	no	—
220	C₂²×C₄	—	4	1	no	—
221	C₂²×C₄	—	4	1	no	—
222	C₂²×C₄	—	4	1	no	—
223	C₂²×C₄	C₂×C₈×C₃₁	1	4	yes	—
224	C₂²×C₄	C₂×C₈×C₃₁	1	4	yes	—
225	C₂²×C₄	C₂×C₈×C₃₁	1	4	yes	—
226	C₂²×C₄	M₁₆×C₃₁	1	4	yes	—
227	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
228	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
229	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
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	#41×C₃₁	1	4	yes	—
235	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
236	C₂×C₈	—	2	8	no	—
237	C₂×C₈	—	2	8	no	—
238	C₂×C₈	—	2	8	no	—
239	C₄⋊C₄	—	2	8	no	—
240	C₄⋊C₄	C₂×C₈×C₃₁	1	16	yes	—
241	C₄²	—	2	4	no	—
242	C₄²	—	2	4	no	—
243	C₄²	—	2	4	no	—
244	C₄²	—	2	4	no	—
245	C₄²	—	4	2	no	—
246	C₄²	—	4	2	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	2	1	no	—
249	C₂³	—	2	1	no	—
250	C₂³	—	2	1	no	—
251	C₂³	C₃₁×#117	1	2	yes	—
252	C₂³	C₂²×C₈×C₃₁	1	2	yes	—
253	C₂³	C₂×M₁₆×C₃₁	1	2	yes	—
254	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	1	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	8	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	4	4	no	—
277	C₂×C₄	—	4	4	no	—
278	C₂×C₄	C₃₁×#117	1	2	yes	—
279	C₂×C₄	C₂²×C₈×C₃₁	1	4	yes	—
280	C₂×C₄	C₂²×C₈×C₃₁	1	4	yes	—
281	C₂×C₄	C₂²×C₈×C₃₁	1	4	yes	—
282	C₂×C₄	C₂×M₁₆×C₃₁	1	4	yes	—
283	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
284	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
285	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
286	C₈	—	4	8	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	2	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	C₃₁×#431	1	4	yes	—
294	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
295	C₂²	C₃₁×#518	1	1	yes	characteristic
296	C₂²	C₃₁×#519	1	1	yes	characteristic
297	C₂²	C₂³×C₈×C₃₁	1	1	yes	characteristic
298	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
299	C₂²	C₃₁×#568	1	1	yes	characteristic
300	C₂²	C₃₁×#568	1	1	yes	characteristic
301	C₄	—	2	2	no	—
302	C₄	—	2	4	no	—
303	C₄	—	2	4	no	—
304	C₄	—	2	4	no	—
305	C₄	—	2	4	no	—
306	C₄	—	4	2	no	—
307	C₄	—	4	2	no	—
308	C₄	C₃₁×#539	1	4	yes	—
309	C₂	—	2	4	no	—
310	C₂	C₃₁×#2313	1	1	yes	characteristic
311	C₂	C₂×C₃₁×#431	1	1	yes	characteristic
312	C₂	C₃₁×#2921	1	1	yes	characteristic
313	C₂	C₃₁×#2972	1	1	yes	characteristic
314	C₂	C₃₁×#2973	1	1	yes	characteristic
315	C₂	C₃₁×#2982	1	1	yes	characteristic
316	C₂	C₃₁×#2983	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator	C₂²	C₂³×C₈×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂³×C₄
prime radical	C₂⁴×C₃₁	C₂²×C₄
LCS	G ▷ C₂² ▷ 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×#1916	C₂	semidirect
30. C₂×C₄⋊C₄×C₃₁	C₈	semidirect
88. #21912	C₃₁	direct (N × H)
130. #1916	C₂×C₃₁	semidirect
188. C₂×C₄⋊C₄	C₈×C₃₁	semidirect
205. C₃₁	#21912	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×C₄×C₄⋊C₄×C₃₁	2
C₃₁×#1916	2
C₃₁×#1817	2
C₃₁×#1797	2
C₄×C₃₁×C₂²⋊C₈	2
#21912	31

Subgroups — C31×#21912 #21478671

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21912 #21478671