Subgroups — C₃₁×#21350 #21478109

Order 7936.

954 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#431	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#1940	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1851	C₂	1	4	yes	maximal
6	C₄×C₃₁×C₄⋊C₈	C₂	1	4	yes	maximal, complemented
7	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
8	C₂×C₃₁×#117	C₄	1	2	yes	—
9	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
10	C₃₁×#431	—	2	4	no	—
11	C₂×C₃₁×C₄⋊C₈	C₂²	1	8	yes	—
12	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
14	C₄×M₁₆×C₃₁	—	2	4	no	—
15	C₂×C₃₁×#97	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
18	C₄×C₄⋊C₄×C₃₁	C₄	1	2	yes	—
19	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
20	C₄×C₂²⋊C₄×C₃₁	C₄	1	2	yes	—
21	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
22	C₃₁×#376	C₄	1	1	yes	characteristic
23	C₃₁×#376	C₄	1	1	yes	characteristic
24	C₄³×C₃₁	C₂²	1	2	yes	—
25	C₃₁×#336	C₂²	1	8	yes	—
26	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
29	C₂×M₁₆×C₃₁	—	2	4	no	—
30	C₂×M₁₆×C₃₁	—	2	8	no	—
31	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
32	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
33	C₃₁×#117	—	2	8	no	—
34	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
35	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
38	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₃₁	D₄	1	4	yes	—
42	C₂×C₄²×C₃₁	Q₈	1	4	yes	—
43	C₂×C₄²×C₃₁	C₂³	1	4	yes	—
44	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
46	C₃₁×C₄⋊C₈	—	2	16	no	—
47	C₃₁×C₄⋊C₈	—	2	4	no	—
48	C₃₁×C₂²⋊C₈	—	2	4	no	—
49	C₃₁×#97	—	2	4	no	—
50	C₄×C₈×C₃₁	—	2	4	no	—
51	C₄×C₈×C₃₁	—	2	4	no	—
52	C₃₁×#95	C₂×C₄	1	2	yes	—
53	C₃₁×#95	C₂×C₄	1	2	yes	—
54	C₃₁×#95	C₂×C₄	1	2	yes	—
55	C₃₁×#95	C₂×C₄	1	2	yes	—
56	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	—	2	4	no	—
58	C₂²×C₄×C₃₁	—	2	4	no	—
59	C₂²×C₄×C₃₁	—	2	4	no	—
60	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
61	C₂²×C₄×C₃₁	C₄⋊C₄	1	4	yes	—
62	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
63	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
72	M₁₆×C₃₁	—	2	16	no	—
73	C₂×C₈×C₃₁	—	2	4	no	—
74	C₂×C₈×C₃₁	—	2	4	no	—
75	C₂×C₈×C₃₁	—	2	4	no	—
76	C₂×C₈×C₃₁	—	2	8	no	—
77	C₂×C₈×C₃₁	#41	1	8	yes	—
78	C₄⋊C₄×C₃₁	—	2	8	no	—
79	C₂²⋊C₄×C₃₁	—	2	8	no	—
80	C₄²×C₃₁	—	2	4	no	—
81	C₄²×C₃₁	—	2	8	no	—
82	C₄²×C₃₁	—	2	8	no	—
83	C₄²×C₃₁	C₄⋊C₄	1	4	yes	—
84	C₄²×C₃₁	C₄⋊C₄	1	4	yes	—
85	C₄²×C₃₁	M₁₆	1	8	yes	complemented
86	C₄²×C₃₁	C₂×D₄	1	2	yes	—
87	C₄²×C₃₁	C₂×D₄	1	2	yes	—
88	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
89	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
90	#21350	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₃₁	—	2	2	no	—
92	C₂³×C₃₁	—	2	2	no	—
93	C₂³×C₃₁	—	2	2	no	—
94	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
95	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
96	C₂×C₄×C₃₁	—	2	4	no	—
97	C₂×C₄×C₃₁	—	2	2	no	—
98	C₂×C₄×C₃₁	—	2	2	no	—
99	C₂×C₄×C₃₁	—	2	4	no	—
100	C₂×C₄×C₃₁	—	2	2	no	—
101	C₂×C₄×C₃₁	—	2	4	no	—
102	C₂×C₄×C₃₁	—	2	4	no	—
103	C₂×C₄×C₃₁	—	2	8	no	—
104	C₂×C₄×C₃₁	—	2	8	no	—
105	C₂×C₄×C₃₁	—	2	4	no	—
106	C₂×C₄×C₃₁	—	2	4	no	—
107	C₂×C₄×C₃₁	—	2	4	no	—
108	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₃₁	C₂×M₁₆	1	8	yes	—
114	C₂×C₄×C₃₁	C₂×M₁₆	1	4	yes	—
115	C₂×C₄×C₃₁	#131	1	4	yes	—
116	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
117	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
118	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
119	C₈×C₃₁	—	2	8	no	—
120	C₈×C₃₁	—	4	4	no	—
121	C₂×#431	C₂×C₃₁	1	1	yes	characteristic
122	C₂×C₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic
123	C₄×#117	C₂×C₃₁	1	1	yes	characteristic
124	#1940	C₂×C₃₁	1	4	yes	complemented
125	#1851	C₂×C₃₁	1	4	yes	—
126	C₄×C₄⋊C₈	C₂×C₃₁	1	4	yes	complemented
127	C₂²×C₃₁	—	2	2	no	—
128	C₂²×C₃₁	—	2	2	no	—
129	C₂²×C₃₁	—	2	2	no	—
130	C₂²×C₃₁	—	4	4	no	—
131	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
132	C₂²×C₃₁	C₄×#41	1	2	yes	—
133	C₂²×C₃₁	C₂²×M₁₆	1	2	yes	—
134	C₂²×C₃₁	C₂×#131	1	2	yes	—
135	C₄×C₃₁	—	2	2	no	—
136	C₄×C₃₁	—	2	2	no	—
137	C₄×C₃₁	—	2	8	no	—
138	C₄×C₃₁	—	2	2	no	—
139	C₄×C₃₁	—	4	4	no	—
140	C₄×C₃₁	#423	1	4	yes	—
141	C₄×C₃₁	#424	1	4	yes	—
142	C₄×C₃₁	#525	1	2	yes	—
143	C₄×C₃₁	#533	1	2	yes	—
144	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
145	C₂×#117	C₄×C₃₁	1	2	yes	—
146	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
147	#431	—	2	4	no	—
148	C₂×C₄⋊C₈	C₂²×C₃₁	1	8	yes	—
149	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
150	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
151	C₄×M₁₆	—	2	4	no	—
152	C₂×#97	C₂²×C₃₁	1	2	yes	—
153	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
154	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
155	C₄×C₄⋊C₄	C₄×C₃₁	1	2	yes	—
156	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
157	C₄×C₂²⋊C₄	C₄×C₃₁	1	2	yes	—
158	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
159	#376	C₄×C₃₁	1	1	yes	characteristic
160	#376	C₄×C₃₁	1	1	yes	characteristic
161	C₄³	C₂²×C₃₁	1	2	yes	—
162	#336	C₂²×C₃₁	1	8	yes	—
163	C₂×C₃₁	—	4	2	no	—
164	C₂×C₃₁	#2296	1	1	yes	characteristic
165	C₂×C₃₁	C₂×#423	1	1	yes	characteristic
166	C₂×C₃₁	C₂×#424	1	1	yes	characteristic
167	C₂×C₃₁	#2965	1	4	yes	—
168	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
169	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
170	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
171	C₂×M₁₆	—	2	4	no	—
172	C₂×M₁₆	—	2	8	no	—
173	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
174	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
175	#117	—	2	8	no	—
176	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
177	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
178	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
179	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
180	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
181	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
182	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
183	C₂×C₄²	D₄×C₃₁	1	4	yes	—
184	C₂×C₄²	Q₈×C₃₁	1	4	yes	—
185	C₂×C₄²	C₂³×C₃₁	1	4	yes	—
186	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
187	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
188	C₄⋊C₈	—	2	16	no	—
189	C₄⋊C₈	—	2	4	no	—
190	C₂²⋊C₈	—	2	4	no	—
191	#97	—	2	4	no	—
192	C₄×C₈	—	2	4	no	—
193	C₄×C₈	—	2	4	no	—
194	#95	C₂×C₄×C₃₁	1	2	yes	—
195	#95	C₂×C₄×C₃₁	1	2	yes	—
196	#95	C₂×C₄×C₃₁	1	2	yes	—
197	#95	C₂×C₄×C₃₁	1	2	yes	—
198	C₃₁	#21350	1	1	yes	characteristic, direct product factor, Sylow
199	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	2	4	no	—
203	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
204	C₂²×C₄	C₄⋊C₄×C₃₁	1	4	yes	—
205	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
206	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
207	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
208	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
209	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
210	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
211	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
213	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
214	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
215	M₁₆	—	2	16	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	2	4	no	—
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	8	no	—
220	C₂×C₈	#41×C₃₁	1	8	yes	—
221	C₄⋊C₄	—	2	8	no	—
222	C₂²⋊C₄	—	2	8	no	—
223	C₄²	—	2	4	no	—
224	C₄²	—	2	8	no	—
225	C₄²	—	2	8	no	—
226	C₄²	C₄⋊C₄×C₃₁	1	4	yes	—
227	C₄²	C₄⋊C₄×C₃₁	1	4	yes	—
228	C₄²	M₁₆×C₃₁	1	8	yes	complemented
229	C₄²	C₂×D₄×C₃₁	1	2	yes	—
230	C₄²	C₂×D₄×C₃₁	1	2	yes	—
231	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
232	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
233	C₂³	—	2	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
237	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	2	8	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
251	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
252	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
253	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
254	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
255	C₂×C₄	C₂×M₁₆×C₃₁	1	8	yes	—
256	C₂×C₄	C₂×M₁₆×C₃₁	1	4	yes	—
257	C₂×C₄	C₃₁×#131	1	4	yes	—
258	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
259	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
260	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
261	C₈	—	2	8	no	—
262	C₈	—	4	4	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	4	4	no	—
267	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
268	C₂²	C₄×#41×C₃₁	1	2	yes	—
269	C₂²	C₂²×M₁₆×C₃₁	1	2	yes	—
270	C₂²	C₂×C₃₁×#131	1	2	yes	—
271	C₄	—	2	2	no	—
272	C₄	—	2	2	no	—
273	C₄	—	2	8	no	—
274	C₄	—	2	2	no	—
275	C₄	—	4	4	no	—
276	C₄	C₃₁×#423	1	4	yes	—
277	C₄	C₃₁×#424	1	4	yes	—
278	C₄	C₃₁×#525	1	2	yes	—
279	C₄	C₃₁×#533	1	2	yes	—
280	C₂	—	4	2	no	—
281	C₂	C₃₁×#2296	1	1	yes	characteristic
282	C₂	C₂×C₃₁×#423	1	1	yes	characteristic
283	C₂	C₂×C₃₁×#424	1	1	yes	characteristic
284	C₂	C₃₁×#2965	1	4	yes	—

	subgroup	G/H
center	C₂×C₄²×C₃₁	C₂³
commutator	C₂³	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂³×C₄
prime radical	C₂⁴×C₃₁	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₃₁×#1940	C₂	semidirect
6. C₄×C₃₁×C₄⋊C₈	C₂	semidirect
85. C₄²×C₃₁	M₁₆	semidirect
90. #21350	C₃₁	direct (N × H)
124. #1940	C₂×C₃₁	semidirect
126. C₄×C₄⋊C₈	C₂×C₃₁	semidirect
198. C₃₁	#21350	direct (N × H)
228. C₄²	M₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#431	2
C₂×C₄×M₁₆×C₃₁	2
C₄×C₃₁×#117	2
C₃₁×#1940	2
C₃₁×#1851	2
C₄×C₃₁×C₄⋊C₈	2
#21350	31

Subgroups — C31×#21350 #21478109

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21350 #21478109