Subgroups — C₂×C₃₁×#1566 #21474549

Order 7936.

854 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#423	C₂	1	2	yes	maximal
2	C₂²×C₄×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1566	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃₁×#334	C₂	1	4	yes	maximal, complemented
5	C₂²×M₁₆×C₃₁	—	2	2	no	—
6	C₂³×C₈×C₃₁	C₄	1	2	yes	—
7	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
8	C₃₁×#423	—	2	4	no	—
9	C₃₁×#423	C₂²	1	8	yes	—
10	C₂×C₃₁×C₄⋊C₈	—	2	2	no	—
11	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
12	C₂×C₄×C₈×C₃₁	C₄	1	1	yes	characteristic
13	C₂×C₄×C₈×C₃₁	C₄	1	4	yes	—
14	C₂×C₄×C₈×C₃₁	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₃₁	C₂²	1	4	yes	—
17	C₃₁×#334	C₂²	1	16	yes	complemented
18	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
19	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
20	C₂³×C₄×C₃₁	Q₈	1	1	yes	characteristic
21	C₂×M₁₆×C₃₁	—	2	8	no	—
22	C₂×M₁₆×C₃₁	—	4	8	no	—
23	C₂²×C₈×C₃₁	—	2	8	no	—
24	C₂²×C₈×C₃₁	—	2	4	no	—
25	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₃₁	C₂×C₄	1	8	yes	—
27	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄²×C₃₁	D₄	1	2	yes	—
31	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₃₁	Q₈	1	1	yes	characteristic
33	C₂×C₄²×C₃₁	Q₈	1	2	yes	—
34	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
35	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
36	C₃₁×C₄⋊C₈	—	2	8	no	—
37	C₃₁×C₄⋊C₈	—	2	4	no	—
38	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
39	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
40	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
41	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	—
42	C₄×C₈×C₃₁	C₂³	1	4	yes	—
43	C₂⁴×C₃₁	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₃₁	—	2	4	no	—
45	C₂²×C₄×C₃₁	—	2	4	no	—
46	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₃₁	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₃₁	SD₁₆	1	8	yes	—
53	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
54	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
58	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
59	M₁₆×C₃₁	—	4	16	no	—
60	C₂×C₈×C₃₁	—	2	4	no	—
61	C₂×C₈×C₃₁	—	2	16	no	—
62	C₂×C₈×C₃₁	—	2	8	no	—
63	C₂×C₈×C₃₁	—	4	8	no	—
64	C₂×C₈×C₃₁	M₁₆	1	8	yes	complemented
65	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
66	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
67	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
68	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
69	C₄²×C₃₁	C₄⋊C₄	1	2	yes	—
70	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
71	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
72	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₃₁	C₂×D₄	1	2	yes	—
74	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
75	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
76	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
77	C₂×#1566	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂³×C₃₁	—	2	2	no	—
79	C₂³×C₃₁	—	2	2	no	—
80	C₂³×C₃₁	#106	1	4	yes	—
81	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
82	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₃₁	—	2	2	no	—
84	C₂×C₄×C₃₁	—	2	2	no	—
85	C₂×C₄×C₃₁	—	2	4	no	—
86	C₂×C₄×C₃₁	—	2	4	no	—
87	C₂×C₄×C₃₁	—	2	4	no	—
88	C₂×C₄×C₃₁	—	2	4	no	—
89	C₂×C₄×C₃₁	#106	1	8	yes	—
90	C₂×C₄×C₃₁	#106	1	4	yes	—
91	C₂×C₄×C₃₁	#108	1	4	yes	—
92	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
94	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
96	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₃₁	C₂×M₁₆	1	4	yes	—
99	C₂×C₄×C₃₁	C₂×SD₁₆	1	4	yes	—
100	C₂×C₄×C₃₁	C₂×SD₁₆	1	8	yes	—
101	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
104	C₈×C₃₁	—	2	4	no	—
105	C₈×C₃₁	—	4	8	no	—
106	C₈×C₃₁	C₂×M₁₆	1	8	yes	complemented
107	C₂×#423	C₂×C₃₁	1	2	yes	—
108	C₂²×C₄×C₈	C₂×C₃₁	1	1	yes	characteristic
109	#1566	C₂×C₃₁	1	8	yes	direct product factor
110	C₂×#334	C₂×C₃₁	1	4	yes	complemented
111	C₂²×C₃₁	—	2	8	no	—
112	C₂²×C₃₁	—	2	2	no	—
113	C₂²×C₃₁	—	2	2	no	—
114	C₂²×C₃₁	#423	1	2	yes	—
115	C₂²×C₃₁	C₂×#106	1	4	yes	—
116	C₂²×C₃₁	C₂×#106	1	2	yes	—
117	C₂²×C₃₁	C₂×#108	1	2	yes	—
118	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
119	C₄×C₃₁	—	2	2	no	—
120	C₄×C₃₁	—	2	1	no	—
121	C₄×C₃₁	—	2	1	no	—
122	C₄×C₃₁	#423	1	2	yes	—
123	C₄×C₃₁	C₂×#106	1	4	yes	—
124	C₄×C₃₁	C₂×#106	1	2	yes	—
125	C₄×C₃₁	C₂×#108	1	4	yes	—
126	C₄×C₃₁	C₂²×M₁₆	1	2	yes	—
127	C₄×C₃₁	C₂²×SD₁₆	1	2	yes	—
128	C₂²×M₁₆	—	2	2	no	—
129	C₂³×C₈	C₄×C₃₁	1	2	yes	—
130	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
131	#423	—	2	4	no	—
132	#423	C₂²×C₃₁	1	8	yes	—
133	C₂×C₄⋊C₈	—	2	2	no	—
134	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
135	C₂×C₄×C₈	C₄×C₃₁	1	1	yes	characteristic
136	C₂×C₄×C₈	C₄×C₃₁	1	4	yes	—
137	C₂×C₄×C₈	C₄×C₃₁	1	1	yes	characteristic
138	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
139	C₂×C₄×C₈	C₂²×C₃₁	1	4	yes	—
140	#334	C₂²×C₃₁	1	16	yes	complemented
141	C₂×C₃₁	—	2	4	no	—
142	C₂×C₃₁	#1566	1	4	yes	direct product factor
143	C₂×C₃₁	C₂×#423	1	1	yes	characteristic
144	C₂×C₃₁	C₂²×#106	1	1	yes	characteristic
145	C₂×C₃₁	C₂²×#108	1	1	yes	characteristic
146	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
147	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
148	C₂³×C₄	Q₈×C₃₁	1	1	yes	characteristic
149	C₂×M₁₆	—	2	8	no	—
150	C₂×M₁₆	—	4	8	no	—
151	C₂²×C₈	—	2	8	no	—
152	C₂²×C₈	—	2	4	no	—
153	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
154	C₂²×C₈	C₂×C₄×C₃₁	1	8	yes	—
155	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
156	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
157	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
158	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
159	C₂×C₄²	D₄×C₃₁	1	2	yes	—
160	C₂×C₄²	Q₈×C₃₁	1	1	yes	characteristic
161	C₂×C₄²	Q₈×C₃₁	1	2	yes	—
162	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
163	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
164	C₄⋊C₈	—	2	8	no	—
165	C₄⋊C₈	—	2	4	no	—
166	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
167	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
168	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
169	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	—
170	C₄×C₈	C₂³×C₃₁	1	4	yes	—
171	C₃₁	C₂×#1566	1	1	yes	characteristic, direct product factor, Sylow
172	C₂⁴	C₄⋊C₄×C₃₁	1	1	yes	characteristic
173	C₂²×C₄	—	2	4	no	—
174	C₂²×C₄	—	2	4	no	—
175	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
176	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
177	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
178	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
179	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
180	C₂²×C₄	C₄⋊C₄×C₃₁	1	1	yes	characteristic
181	C₂²×C₄	SD₁₆×C₃₁	1	8	yes	—
182	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
183	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
184	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
185	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
186	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
187	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
188	M₁₆	—	4	16	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	16	no	—
191	C₂×C₈	—	2	8	no	—
192	C₂×C₈	—	4	8	no	—
193	C₂×C₈	M₁₆×C₃₁	1	8	yes	complemented
194	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
195	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
196	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
197	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
198	C₄²	C₄⋊C₄×C₃₁	1	2	yes	—
199	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
200	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
201	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
202	C₄²	C₂×D₄×C₃₁	1	2	yes	—
203	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
204	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
205	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
206	C₂³	—	2	2	no	—
207	C₂³	—	2	2	no	—
208	C₂³	C₃₁×#106	1	4	yes	—
209	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
210	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	C₃₁×#106	1	8	yes	—
218	C₂×C₄	C₃₁×#106	1	4	yes	—
219	C₂×C₄	C₃₁×#108	1	4	yes	—
220	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
221	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
222	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
223	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
224	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
225	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
226	C₂×C₄	C₂×M₁₆×C₃₁	1	4	yes	—
227	C₂×C₄	C₂×SD₁₆×C₃₁	1	4	yes	—
228	C₂×C₄	C₂×SD₁₆×C₃₁	1	8	yes	—
229	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
230	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
231	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
232	C₈	—	2	4	no	—
233	C₈	—	4	8	no	—
234	C₈	C₂×M₁₆×C₃₁	1	8	yes	complemented
235	C₂²	—	2	8	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	2	2	no	—
238	C₂²	C₃₁×#423	1	2	yes	—
239	C₂²	C₂×C₃₁×#106	1	4	yes	—
240	C₂²	C₂×C₃₁×#106	1	2	yes	—
241	C₂²	C₂×C₃₁×#108	1	2	yes	—
242	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
243	C₄	—	2	2	no	—
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	C₃₁×#423	1	2	yes	—
247	C₄	C₂×C₃₁×#106	1	2	yes	—
248	C₄	C₂×C₃₁×#106	1	4	yes	—
249	C₄	C₂×C₃₁×#108	1	4	yes	—
250	C₄	C₂²×M₁₆×C₃₁	1	2	yes	—
251	C₄	C₂²×SD₁₆×C₃₁	1	2	yes	—
252	C₂	—	2	4	no	—
253	C₂	C₃₁×#1566	1	4	yes	direct product factor
254	C₂	C₂×C₃₁×#423	1	1	yes	characteristic
255	C₂	C₂²×C₃₁×#106	1	1	yes	characteristic
256	C₂	C₂²×C₃₁×#108	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1566	C₂	direct (N × H)
4. C₂×C₃₁×#334	C₂	semidirect
17. C₃₁×#334	C₂²	semidirect
64. C₂×C₈×C₃₁	M₁₆	semidirect
77. C₂×#1566	C₃₁	direct (N × H)
106. C₈×C₃₁	C₂×M₁₆	semidirect
109. #1566	C₂×C₃₁	direct (N × H)
110. C₂×#334	C₂×C₃₁	semidirect
140. #334	C₂²×C₃₁	semidirect
142. C₂×C₃₁	#1566	direct (N × H)
171. C₃₁	C₂×#1566	direct (N × H)
193. C₂×C₈	M₁₆×C₃₁	semidirect
234. C₈	C₂×M₁₆×C₃₁	semidirect
253. C₂	C₃₁×#1566	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×C₃₁×#423	2
C₂²×C₄×C₈×C₃₁	2
C₃₁×#1566	2
C₂×C₃₁×#334	2
C₂×#1566	31

Subgroups — C2×C31×#1566 #21474549

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1566 #21474549