Subgroups — C₂×C₃₁×#1580 #21474607

Order 7936.

894 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×D₄×C₃₁	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1580	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃₁×#334	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₁×D₄⋊C₈	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#322	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×D₈×C₃₁	C₄	1	1	yes	characteristic
8	C₂³×C₈×C₃₁	—	2	2	no	—
9	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
10	C₄×D₈×C₃₁	C₄	1	4	yes	—
11	C₄×D₈×C₃₁	C₂²	1	4	yes	—
12	C₈×D₄×C₃₁	—	2	4	no	—
13	C₈×D₄×C₃₁	C₂²	1	8	yes	complemented
14	C₂×C₃₁×#107	C₄	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃₁×#102	C₄	1	2	yes	—
17	C₂×C₃₁×C₂²⋊C₈	—	2	2	no	—
18	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#334	C₂²	1	4	yes	complemented
22	C₃₁×D₄⋊C₈	C₂²	1	8	yes	complemented
23	C₃₁×#322	C₂²	1	4	yes	complemented
24	C₂²×D₄×C₃₁	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₃₁	—	2	2	no	—
26	C₂×D₈×C₃₁	C₈	1	8	yes	complemented
27	C₂×D₈×C₃₁	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₃₁	—	2	2	no	—
29	C₂²×C₈×C₃₁	—	2	1	no	—
30	C₂²×C₈×C₃₁	—	2	8	no	—
31	C₂²×C₈×C₃₁	—	4	8	no	—
32	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₃₁	D₄	1	2	yes	—
35	C₄×D₄×C₃₁	—	2	2	no	—
36	C₄×D₄×C₃₁	—	2	2	no	—
37	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
38	C₄×D₄×C₃₁	C₂³	1	4	yes	—
39	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
40	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
41	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
42	C₃₁×#107	C₂×C₄	1	4	yes	—
43	C₃₁×C₄⋊C₈	—	2	2	no	—
44	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
45	C₃₁×#102	C₂×C₄	1	8	yes	—
46	C₃₁×C₂²⋊C₈	—	2	8	no	—
47	C₄×C₈×C₃₁	—	2	1	no	—
48	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₃₁	D₄	1	2	yes	—
50	C₄×C₈×C₃₁	C₂³	1	2	yes	—
51	C₄×C₈×C₃₁	C₂³	1	2	yes	—
52	C₄×C₈×C₃₁	C₂³	1	2	yes	—
53	C₂⁴×C₃₁	—	2	2	no	—
54	C₂×D₄×C₃₁	—	2	2	no	—
55	C₂×D₄×C₃₁	—	2	4	no	—
56	C₂×D₄×C₃₁	C₂×C₈	1	8	yes	complemented
57	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
58	C₂²×C₄×C₃₁	—	2	2	no	—
59	C₂²×C₄×C₃₁	—	2	4	no	—
60	C₂²×C₄×C₃₁	—	2	4	no	—
61	C₂²×C₄×C₃₁	—	4	4	no	—
62	C₂²×C₄×C₃₁	—	4	4	no	—
63	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
66	D₈×C₃₁	C₂×C₈	1	16	yes	complemented
67	C₂×C₈×C₃₁	—	2	2	no	—
68	C₂×C₈×C₃₁	—	2	4	no	—
69	C₂×C₈×C₃₁	—	2	4	no	—
70	C₂×C₈×C₃₁	—	2	2	no	—
71	C₂×C₈×C₃₁	—	4	16	no	—
72	C₂×C₈×C₃₁	—	4	2	no	—
73	C₂×C₈×C₃₁	—	4	4	no	—
74	C₂×C₈×C₃₁	C₂×C₈	1	4	yes	complemented
75	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
78	C₄⋊C₄×C₃₁	—	2	2	no	—
79	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
80	C₂²⋊C₄×C₃₁	—	2	4	no	—
81	C₂²⋊C₄×C₃₁	—	2	4	no	—
82	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
84	C₄²×C₃₁	#41	1	1	yes	characteristic
85	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
86	C₂×#1580	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
87	C₂³×C₃₁	—	2	4	no	—
88	C₂³×C₃₁	—	2	8	no	—
89	C₂³×C₃₁	—	4	4	no	—
90	C₂³×C₃₁	—	4	4	no	—
91	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
92	D₄×C₃₁	—	2	4	no	—
93	D₄×C₃₁	—	2	8	no	—
94	D₄×C₃₁	C₂²×C₈	1	8	yes	complemented
95	C₂×C₄×C₃₁	—	2	2	no	—
96	C₂×C₄×C₃₁	—	2	4	no	—
97	C₂×C₄×C₃₁	—	4	4	no	—
98	C₂×C₄×C₃₁	—	4	4	no	—
99	C₂×C₄×C₃₁	—	4	4	no	—
100	C₂×C₄×C₃₁	—	4	8	no	—
101	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
105	C₂×C₄×C₃₁	#131	1	2	yes	—
106	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
107	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
108	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
109	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
110	C₈×C₃₁	—	2	2	no	—
111	C₈×C₃₁	—	2	4	no	—
112	C₈×C₃₁	—	4	2	no	—
113	C₈×C₃₁	—	4	4	no	—
114	C₈×C₃₁	C₂²×C₈	1	4	yes	complemented
115	C₂×C₄×D₈	C₂×C₃₁	1	1	yes	characteristic
116	C₂×C₈×D₄	C₂×C₃₁	1	2	yes	complemented
117	#1580	C₂×C₃₁	1	8	yes	direct product factor
118	C₂×#334	C₂×C₃₁	1	1	yes	characteristic, complemented
119	C₂×D₄⋊C₈	C₂×C₃₁	1	2	yes	complemented
120	C₂×#322	C₂×C₃₁	1	1	yes	characteristic, complemented
121	C₂²×C₃₁	—	2	8	no	—
122	C₂²×C₃₁	—	4	16	no	—
123	C₂²×C₃₁	—	4	4	no	—
124	C₂²×C₃₁	—	4	4	no	—
125	C₂²×C₃₁	C₈×D₄	1	2	yes	—
126	C₂²×C₃₁	#442	1	2	yes	—
127	C₂²×C₃₁	#444	1	2	yes	—
128	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
129	C₄×C₃₁	—	2	1	no	—
130	C₄×C₃₁	—	2	1	no	—
131	C₄×C₃₁	—	4	4	no	—
132	C₄×C₃₁	C₈×D₄	1	2	yes	—
133	C₄×C₃₁	#442	1	2	yes	—
134	C₄×C₃₁	C₂³×C₈	1	1	yes	characteristic
135	C₄×C₃₁	C₂×#131	1	1	yes	characteristic
136	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
137	C₂²×D₈	C₄×C₃₁	1	1	yes	characteristic
138	C₂³×C₈	—	2	2	no	—
139	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
140	C₄×D₈	C₄×C₃₁	1	4	yes	—
141	C₄×D₈	C₂²×C₃₁	1	4	yes	—
142	C₈×D₄	—	2	4	no	—
143	C₈×D₄	C₂²×C₃₁	1	8	yes	complemented
144	C₂×#107	C₄×C₃₁	1	1	yes	characteristic
145	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
146	C₂×#102	C₄×C₃₁	1	2	yes	—
147	C₂×C₂²⋊C₈	—	2	2	no	—
148	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
149	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
150	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
151	#334	C₂²×C₃₁	1	4	yes	complemented
152	D₄⋊C₈	C₂²×C₃₁	1	8	yes	complemented
153	#322	C₂²×C₃₁	1	4	yes	complemented
154	C₂×C₃₁	—	4	8	no	—
155	C₂×C₃₁	#1580	1	4	yes	direct product factor
156	C₂×C₃₁	C₂×C₈×D₄	1	1	yes	characteristic
157	C₂×C₃₁	C₂×#442	1	1	yes	characteristic
158	C₂×C₃₁	C₂×#444	1	1	yes	characteristic
159	C₂²×D₄	C₂×C₄×C₃₁	1	2	yes	—
160	C₂³×C₄	—	2	2	no	—
161	C₂×D₈	C₈×C₃₁	1	8	yes	complemented
162	C₂×D₈	C₂×C₄×C₃₁	1	4	yes	—
163	C₂²×C₈	—	2	2	no	—
164	C₂²×C₈	—	2	1	no	—
165	C₂²×C₈	—	2	8	no	—
166	C₂²×C₈	—	4	8	no	—
167	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
168	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
169	C₂²×C₈	D₄×C₃₁	1	2	yes	—
170	C₄×D₄	—	2	2	no	—
171	C₄×D₄	—	2	2	no	—
172	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
173	C₄×D₄	C₂³×C₃₁	1	4	yes	—
174	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
175	C₂×C₂²⋊C₄	—	2	2	no	—
176	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
177	#107	C₂×C₄×C₃₁	1	4	yes	—
178	C₄⋊C₈	—	2	2	no	—
179	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
180	#102	C₂×C₄×C₃₁	1	8	yes	—
181	C₂²⋊C₈	—	2	8	no	—
182	C₄×C₈	—	2	1	no	—
183	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
184	C₄×C₈	D₄×C₃₁	1	2	yes	—
185	C₄×C₈	C₂³×C₃₁	1	2	yes	—
186	C₄×C₈	C₂³×C₃₁	1	2	yes	—
187	C₄×C₈	C₂³×C₃₁	1	2	yes	—
188	C₃₁	C₂×#1580	1	1	yes	characteristic, direct product factor, Sylow
189	C₂⁴	—	2	2	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	2	4	no	—
192	C₂×D₄	C₂×C₈×C₃₁	1	8	yes	complemented
193	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	4	4	no	—
198	C₂²×C₄	—	4	4	no	—
199	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
200	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
201	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
202	D₈	C₂×C₈×C₃₁	1	16	yes	complemented
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	4	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	4	16	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	—	4	4	no	—
210	C₂×C₈	C₂×C₈×C₃₁	1	4	yes	complemented
211	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
212	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
213	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
214	C₄⋊C₄	—	2	2	no	—
215	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
216	C₂²⋊C₄	—	2	4	no	—
217	C₂²⋊C₄	—	2	4	no	—
218	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
219	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
220	C₄²	#41×C₃₁	1	1	yes	characteristic
221	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
222	C₂³	—	2	8	no	—
223	C₂³	—	2	4	no	—
224	C₂³	—	4	4	no	—
225	C₂³	—	4	4	no	—
226	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
227	D₄	—	2	8	no	—
228	D₄	—	2	4	no	—
229	D₄	C₂²×C₈×C₃₁	1	8	yes	complemented
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	8	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
237	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
238	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
239	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
240	C₂×C₄	C₃₁×#131	1	2	yes	—
241	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
242	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
243	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
244	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
245	C₈	—	2	2	no	—
246	C₈	—	2	4	no	—
247	C₈	—	4	2	no	—
248	C₈	—	4	4	no	—
249	C₈	C₂²×C₈×C₃₁	1	4	yes	complemented
250	C₂²	—	2	8	no	—
251	C₂²	—	4	16	no	—
252	C₂²	—	4	4	no	—
253	C₂²	—	4	4	no	—
254	C₂²	C₈×D₄×C₃₁	1	2	yes	—
255	C₂²	C₃₁×#442	1	2	yes	—
256	C₂²	C₃₁×#444	1	2	yes	—
257	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	4	4	no	—
261	C₄	C₈×D₄×C₃₁	1	2	yes	—
262	C₄	C₃₁×#442	1	2	yes	—
263	C₄	C₂³×C₈×C₃₁	1	1	yes	characteristic
264	C₄	C₂×C₃₁×#131	1	1	yes	characteristic
265	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
266	C₂	—	4	8	no	—
267	C₂	C₃₁×#1580	1	4	yes	direct product factor
268	C₂	C₂×C₈×D₄×C₃₁	1	1	yes	characteristic
269	C₂	C₂×C₃₁×#442	1	1	yes	characteristic
270	C₂	C₂×C₃₁×#444	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₄	C₂³×C₈×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₄×D₄
prime radical	C₂²×D₈×C₃₁	C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₂×C₄×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×D₄×C₃₁	C₂	semidirect
3. C₃₁×#1580	C₂	direct (N × H)
4. C₂×C₃₁×#334	C₂	semidirect
5. C₂×C₃₁×D₄⋊C₈	C₂	semidirect
6. C₂×C₃₁×#322	C₂	semidirect
13. C₈×D₄×C₃₁	C₂²	semidirect
21. C₃₁×#334	C₂²	semidirect
22. C₃₁×D₄⋊C₈	C₂²	semidirect
23. C₃₁×#322	C₂²	semidirect
26. C₂×D₈×C₃₁	C₈	semidirect
56. C₂×D₄×C₃₁	C₂×C₈	semidirect
66. D₈×C₃₁	C₂×C₈	semidirect
74. C₂×C₈×C₃₁	C₂×C₈	semidirect
86. C₂×#1580	C₃₁	direct (N × H)
94. D₄×C₃₁	C₂²×C₈	semidirect
114. C₈×C₃₁	C₂²×C₈	semidirect
116. C₂×C₈×D₄	C₂×C₃₁	semidirect
117. #1580	C₂×C₃₁	direct (N × H)
118. C₂×#334	C₂×C₃₁	semidirect
119. C₂×D₄⋊C₈	C₂×C₃₁	semidirect
120. C₂×#322	C₂×C₃₁	semidirect
143. C₈×D₄	C₂²×C₃₁	semidirect
151. #334	C₂²×C₃₁	semidirect
152. D₄⋊C₈	C₂²×C₃₁	semidirect
153. #322	C₂²×C₃₁	semidirect
155. C₂×C₃₁	#1580	direct (N × H)
161. C₂×D₈	C₈×C₃₁	semidirect
188. C₃₁	C₂×#1580	direct (N × H)
192. C₂×D₄	C₂×C₈×C₃₁	semidirect
202. D₈	C₂×C₈×C₃₁	semidirect
210. C₂×C₈	C₂×C₈×C₃₁	semidirect
229. D₄	C₂²×C₈×C₃₁	semidirect
249. C₈	C₂²×C₈×C₃₁	semidirect
267. C₂	C₃₁×#1580	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×D₈×C₃₁	2
C₂×C₈×D₄×C₃₁	2
C₃₁×#1580	2
C₂×C₃₁×#334	2
C₂×C₃₁×D₄⋊C₈	2
C₂×C₃₁×#322	2
C₂×#1580	31

Subgroups — C2×C31×#1580 #21474607

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1580 #21474607