Subgroups — C₃₁×#22264 #21479023

Order 7936.

1018 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#436	C₂	1	2	yes	maximal, complemented
2	C₄×C₃₁×#127	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1954	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1870	C₂	1	8	yes	maximal, complemented
5	C₃₁×#1831	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1744	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×M₁₆×C₃₁	C₂²	1	4	yes	—
8	C₂×C₃₁×#127	C₄	1	1	yes	characteristic
9	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
10	C₂×C₄×D₄×C₃₁	C₂²	1	4	yes	—
11	C₃₁×#436	—	2	8	no	—
12	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₃₁×C₂²⋊C₈	C₂²	1	4	yes	—
14	C₂×C₃₁×C₂²⋊C₈	C₂²	1	8	yes	—
15	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#390	C₄	1	2	yes	—
18	C₃₁×#390	C₄	1	4	yes	—
19	C₃₁×#384	C₄	1	1	yes	characteristic
20	C₄³×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#336	C₂²	1	4	yes	—
22	C₃₁×#336	C₂²	1	4	yes	—
23	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
24	C₂²×D₄×C₃₁	C₂×C₄	1	4	yes	—
25	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
26	C₂×M₁₆×C₃₁	—	2	8	no	—
27	C₂×M₁₆×C₃₁	—	4	8	no	—
28	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
29	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
30	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
31	C₃₁×#127	—	2	4	no	—
32	C₄×D₄×C₃₁	—	2	4	no	—
33	C₄×D₄×C₃₁	—	2	4	no	—
34	C₄×D₄×C₃₁	—	4	8	no	—
35	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
37	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	4	yes	—
38	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	8	yes	—
39	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
43	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
44	C₃₁×C₄⋊C₈	—	2	4	no	—
45	C₃₁×C₂²⋊C₈	—	2	8	no	—
46	C₃₁×C₂²⋊C₈	—	4	8	no	—
47	C₄×C₈×C₃₁	—	2	2	no	—
48	C₄×C₈×C₃₁	D₄	1	4	yes	complemented
49	C₂⁴×C₃₁	C₂²×C₄	1	4	yes	—
50	C₂×D₄×C₃₁	—	2	4	no	—
51	C₂×D₄×C₃₁	—	2	8	no	—
52	C₂×D₄×C₃₁	—	2	4	no	—
53	C₂×D₄×C₃₁	—	2	4	no	—
54	C₂×D₄×C₃₁	—	4	4	no	—
55	C₂×D₄×C₃₁	—	4	4	no	—
56	C₂²×C₄×C₃₁	—	2	4	no	—
57	C₂²×C₄×C₃₁	—	2	4	no	—
58	C₂²×C₄×C₃₁	—	2	8	no	—
59	C₂²×C₄×C₃₁	—	2	8	no	—
60	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
62	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
66	M₁₆×C₃₁	—	4	16	no	—
67	C₂×C₈×C₃₁	—	2	4	no	—
68	C₂×C₈×C₃₁	—	2	4	no	—
69	C₂×C₈×C₃₁	—	2	2	no	—
70	C₂×C₈×C₃₁	—	2	4	no	—
71	C₂×C₈×C₃₁	—	2	8	no	—
72	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
73	C₄⋊C₄×C₃₁	—	2	4	no	—
74	C₄⋊C₄×C₃₁	—	4	4	no	—
75	C₂²⋊C₄×C₃₁	—	2	8	no	—
76	C₂²⋊C₄×C₃₁	—	4	8	no	—
77	C₄²×C₃₁	—	2	2	no	—
78	C₄²×C₃₁	—	2	4	no	—
79	C₄²×C₃₁	—	4	2	no	—
80	C₄²×C₃₁	M₁₆	1	4	yes	complemented
81	C₄²×C₃₁	C₂×D₄	1	2	yes	—
82	C₄²×C₃₁	#41	1	2	yes	—
83	#22264	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
84	C₂³×C₃₁	—	2	4	no	—
85	C₂³×C₃₁	—	2	8	no	—
86	C₂³×C₃₁	—	2	4	no	—
87	C₂³×C₃₁	—	2	4	no	—
88	C₂³×C₃₁	—	2	8	no	—
89	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
90	D₄×C₃₁	—	4	16	no	—
91	D₄×C₃₁	—	4	8	no	—
92	C₂×C₄×C₃₁	—	2	2	no	—
93	C₂×C₄×C₃₁	—	2	2	no	—
94	C₂×C₄×C₃₁	—	2	2	no	—
95	C₂×C₄×C₃₁	—	2	4	no	—
96	C₂×C₄×C₃₁	—	2	2	no	—
97	C₂×C₄×C₃₁	—	2	8	no	—
98	C₂×C₄×C₃₁	—	2	2	no	—
99	C₂×C₄×C₃₁	—	2	2	no	—
100	C₂×C₄×C₃₁	—	4	4	no	—
101	C₂×C₄×C₃₁	—	4	4	no	—
102	C₂×C₄×C₃₁	—	4	8	no	—
103	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₃₁	C₂×M₁₆	1	4	yes	—
106	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
107	C₂×C₄×C₃₁	#131	1	2	yes	—
108	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
110	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
111	C₈×C₃₁	—	2	4	no	—
112	C₈×C₃₁	—	4	4	no	—
113	C₈×C₃₁	—	4	2	no	—
114	C₂×#436	C₂×C₃₁	1	2	yes	complemented
115	C₄×#127	C₂×C₃₁	1	1	yes	characteristic
116	#1954	C₂×C₃₁	1	2	yes	complemented
117	#1870	C₂×C₃₁	1	8	yes	complemented
118	#1831	C₂×C₃₁	1	1	yes	characteristic, complemented
119	#1744	C₂×C₃₁	1	1	yes	characteristic, complemented
120	C₂²×C₃₁	—	4	4	no	—
121	C₂²×C₃₁	—	4	4	no	—
122	C₂²×C₃₁	—	4	4	no	—
123	C₂²×C₃₁	—	4	8	no	—
124	C₂²×C₃₁	—	4	8	no	—
125	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
126	C₂²×C₃₁	#518	1	2	yes	—
127	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
128	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
129	C₂²×C₃₁	#568	1	2	yes	—
130	C₄×C₃₁	—	2	2	no	—
131	C₄×C₃₁	—	2	4	no	—
132	C₄×C₃₁	—	4	4	no	—
133	C₄×C₃₁	—	4	2	no	—
134	C₄×C₃₁	#436	1	4	yes	—
135	C₄×C₃₁	#535	1	2	yes	—
136	C₄×C₃₁	#540	1	2	yes	—
137	C₂²×M₁₆	C₂²×C₃₁	1	4	yes	—
138	C₂×#127	C₄×C₃₁	1	1	yes	characteristic
139	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
140	C₂×C₄×D₄	C₂²×C₃₁	1	4	yes	—
141	#436	—	2	8	no	—
142	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
143	C₂×C₂²⋊C₈	C₂²×C₃₁	1	4	yes	—
144	C₂×C₂²⋊C₈	C₂²×C₃₁	1	8	yes	—
145	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
146	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
147	#390	C₄×C₃₁	1	2	yes	—
148	#390	C₄×C₃₁	1	4	yes	—
149	#384	C₄×C₃₁	1	1	yes	characteristic
150	C₄³	C₂²×C₃₁	1	1	yes	characteristic
151	#336	C₂²×C₃₁	1	4	yes	—
152	#336	C₂²×C₃₁	1	4	yes	—
153	C₂×C₃₁	—	8	4	no	—
154	C₂×C₃₁	#2328	1	1	yes	characteristic
155	C₂×C₃₁	C₂×#436	1	1	yes	characteristic
156	C₂×C₃₁	#2932	1	1	yes	characteristic
157	C₂×C₃₁	#2975	1	2	yes	—
158	C₂×C₃₁	#2981	1	2	yes	—
159	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
160	C₂²×D₄	C₂×C₄×C₃₁	1	4	yes	—
161	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
162	C₂×M₁₆	—	2	8	no	—
163	C₂×M₁₆	—	4	8	no	—
164	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
165	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
166	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
167	#127	—	2	4	no	—
168	C₄×D₄	—	2	4	no	—
169	C₄×D₄	—	2	4	no	—
170	C₄×D₄	—	4	8	no	—
171	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
172	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
173	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	4	yes	—
174	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	8	yes	—
175	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
176	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
177	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
178	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
179	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
180	C₄⋊C₈	—	2	4	no	—
181	C₂²⋊C₈	—	2	8	no	—
182	C₂²⋊C₈	—	4	8	no	—
183	C₄×C₈	—	2	2	no	—
184	C₄×C₈	D₄×C₃₁	1	4	yes	complemented
185	C₃₁	#22264	1	1	yes	characteristic, direct product factor, Sylow
186	C₂⁴	C₂²×C₄×C₃₁	1	4	yes	—
187	C₂×D₄	—	2	4	no	—
188	C₂×D₄	—	2	8	no	—
189	C₂×D₄	—	2	4	no	—
190	C₂×D₄	—	2	4	no	—
191	C₂×D₄	—	4	4	no	—
192	C₂×D₄	—	4	4	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	8	no	—
196	C₂²×C₄	—	2	8	no	—
197	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
198	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
199	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
200	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
201	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
202	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
203	M₁₆	—	4	16	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	4	no	—
208	C₂×C₈	—	2	8	no	—
209	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	4	4	no	—
212	C₂²⋊C₄	—	2	8	no	—
213	C₂²⋊C₄	—	4	8	no	—
214	C₄²	—	2	2	no	—
215	C₄²	—	2	4	no	—
216	C₄²	—	4	2	no	—
217	C₄²	M₁₆×C₃₁	1	4	yes	complemented
218	C₄²	C₂×D₄×C₃₁	1	2	yes	—
219	C₄²	#41×C₃₁	1	2	yes	—
220	C₂³	—	2	4	no	—
221	C₂³	—	2	8	no	—
222	C₂³	—	2	4	no	—
223	C₂³	—	2	4	no	—
224	C₂³	—	2	8	no	—
225	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
226	D₄	—	4	16	no	—
227	D₄	—	4	8	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	8	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	8	no	—
239	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
240	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
241	C₂×C₄	C₂×M₁₆×C₃₁	1	4	yes	—
242	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
243	C₂×C₄	C₃₁×#131	1	2	yes	—
244	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
245	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
246	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
247	C₈	—	2	4	no	—
248	C₈	—	4	4	no	—
249	C₈	—	4	2	no	—
250	C₂²	—	4	4	no	—
251	C₂²	—	4	4	no	—
252	C₂²	—	4	4	no	—
253	C₂²	—	4	8	no	—
254	C₂²	—	4	8	no	—
255	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
256	C₂²	C₃₁×#518	1	2	yes	—
257	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
258	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
259	C₂²	C₃₁×#568	1	2	yes	—
260	C₄	—	2	2	no	—
261	C₄	—	2	4	no	—
262	C₄	—	4	4	no	—
263	C₄	—	4	2	no	—
264	C₄	C₃₁×#436	1	4	yes	—
265	C₄	C₃₁×#535	1	2	yes	—
266	C₄	C₃₁×#540	1	2	yes	—
267	C₂	—	8	4	no	—
268	C₂	C₃₁×#2328	1	1	yes	characteristic
269	C₂	C₂×C₃₁×#436	1	1	yes	characteristic
270	C₂	C₃₁×#2932	1	1	yes	characteristic
271	C₂	C₃₁×#2975	1	2	yes	—
272	C₂	C₃₁×#2981	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	C₂×C₃₁×#127	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
1. C₂×C₃₁×#436	C₂	semidirect
3. C₃₁×#1954	C₂	semidirect
4. C₃₁×#1870	C₂	semidirect
5. C₃₁×#1831	C₂	semidirect
6. C₃₁×#1744	C₂	semidirect
48. C₄×C₈×C₃₁	D₄	semidirect
80. C₄²×C₃₁	M₁₆	semidirect
83. #22264	C₃₁	direct (N × H)
114. C₂×#436	C₂×C₃₁	semidirect
116. #1954	C₂×C₃₁	semidirect
117. #1870	C₂×C₃₁	semidirect
118. #1831	C₂×C₃₁	semidirect
119. #1744	C₂×C₃₁	semidirect
184. C₄×C₈	D₄×C₃₁	semidirect
185. C₃₁	#22264	direct (N × H)
217. 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×#436	2
C₄×C₃₁×#127	2
C₃₁×#1954	2
C₃₁×#1870	2
C₃₁×#1831	2
C₃₁×#1744	2
#22264	31

Subgroups — C31×#22264 #21479023

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22264 #21479023