Subgroups — C₂×C₃₁×#1450 #21474072

Order 7936.

982 conjugacy classes of subgroups.

The subgroup lattice has 210 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	2	yes	maximal
2	C₂²×C₄×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1450	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₃₁×#322	C₂	1	2	yes	maximal, complemented
5	C₂×C₈²×C₃₁	C₂	1	2	yes	maximal, complemented
6	C₂³×C₈×C₃₁	C₄	1	2	yes	—
7	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
8	C₃₁×#431	C₄	1	8	yes	—
9	C₃₁×#431	C₂²	1	8	yes	—
10	C₂×C₃₁×C₄⋊C₈	C₄	1	4	yes	—
11	C₂×C₃₁×C₂²⋊C₈	C₄	1	4	yes	—
12	C₂×C₄×C₈×C₃₁	C₄	1	2	yes	—
13	C₂×C₄×C₈×C₃₁	C₄	1	4	yes	—
14	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₃₁	C₂²	1	4	yes	—
16	C₂×C₄×C₈×C₃₁	C₂²	1	4	yes	—
17	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#322	C₂²	1	8	yes	complemented
19	C₈²×C₃₁	C₂²	1	8	yes	complemented
20	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
21	C₂³×C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₃₁	C₈	1	16	yes	complemented
23	C₂²×C₈×C₃₁	C₂×C₄	1	8	yes	—
24	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₃₁	C₂×C₄	1	8	yes	—
27	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
33	C₃₁×C₄⋊C₈	C₂×C₄	1	16	yes	—
34	C₃₁×C₂²⋊C₈	C₂×C₄	1	16	yes	—
35	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₃₁	C₂×C₄	1	8	yes	—
37	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₃₁	C₂×C₄	1	8	yes	—
39	C₄×C₈×C₃₁	C₂³	1	2	yes	—
40	C₄×C₈×C₃₁	C₂³	1	2	yes	—
41	C₄×C₈×C₃₁	C₂³	1	8	yes	—
42	C₂⁴×C₃₁	C₄²	1	1	yes	characteristic
43	C₂²×C₄×C₃₁	—	2	8	no	—
44	C₂²×C₄×C₃₁	C₄²	1	2	yes	—
45	C₂²×C₄×C₃₁	C₄²	1	4	yes	—
46	C₂²×C₄×C₃₁	C₄²	1	2	yes	—
47	C₂²×C₄×C₃₁	C₄²	1	1	yes	characteristic
48	C₂²×C₄×C₃₁	C₂×C₈	1	8	yes	—
49	C₂²×C₄×C₃₁	C₂²×C₄	1	4	yes	—
50	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
53	C₂×C₈×C₃₁	—	2	16	no	—
54	C₂×C₈×C₃₁	C₂×C₈	1	8	yes	—
55	C₂×C₈×C₃₁	C₂×C₈	1	8	yes	—
56	C₂×C₈×C₃₁	C₂×C₈	1	32	yes	complemented
57	C₂×C₈×C₃₁	C₂²×C₄	1	16	yes	—
58	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
59	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
60	C₄²×C₃₁	C₄²	1	2	yes	—
61	C₄²×C₃₁	C₄²	1	4	yes	—
62	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
63	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
64	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
65	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
66	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
67	C₂×#1450	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂³×C₃₁	—	2	4	no	—
69	C₂³×C₃₁	C₄×C₈	1	4	yes	—
70	C₂³×C₃₁	C₂×C₄²	1	2	yes	—
71	C₂³×C₃₁	C₂×C₄²	1	1	yes	characteristic
72	C₂×C₄×C₃₁	—	2	4	no	—
73	C₂×C₄×C₃₁	—	2	8	no	—
74	C₂×C₄×C₃₁	—	2	8	no	—
75	C₂×C₄×C₃₁	C₄×C₈	1	8	yes	—
76	C₂×C₄×C₃₁	C₄×C₈	1	4	yes	—
77	C₂×C₄×C₃₁	C₂×C₄²	1	4	yes	—
78	C₂×C₄×C₃₁	C₂×C₄²	1	1	yes	characteristic
79	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
80	C₂×C₄×C₃₁	C₂×C₄²	1	2	yes	—
81	C₂×C₄×C₃₁	C₂²×C₈	1	4	yes	—
82	C₂×C₄×C₃₁	C₂²×C₈	1	8	yes	—
83	C₂×C₄×C₃₁	#131	1	8	yes	—
84	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
85	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
86	C₈×C₃₁	—	2	16	no	—
87	C₈×C₃₁	C₂²×C₈	1	8	yes	—
88	C₈×C₃₁	C₂²×C₈	1	8	yes	—
89	C₂×#431	C₂×C₃₁	1	2	yes	—
90	C₂²×C₄×C₈	C₂×C₃₁	1	1	yes	characteristic
91	#1450	C₂×C₃₁	1	8	yes	direct product factor
92	C₂×#322	C₂×C₃₁	1	2	yes	complemented
93	C₂×C₈²	C₂×C₃₁	1	2	yes	complemented
94	C₂²×C₃₁	—	2	4	no	—
95	C₂²×C₃₁	—	2	8	no	—
96	C₂²×C₃₁	C₂×C₄×C₈	1	4	yes	—
97	C₂²×C₃₁	C₂×C₄×C₈	1	2	yes	—
98	C₂²×C₃₁	#405	1	4	yes	—
99	C₂²×C₃₁	C₂²×C₄²	1	1	yes	characteristic
100	C₄×C₃₁	—	2	4	no	—
101	C₄×C₃₁	C₂×C₄×C₈	1	2	yes	—
102	C₄×C₃₁	C₂×C₄×C₈	1	4	yes	—
103	C₄×C₃₁	#405	1	4	yes	—
104	C₄×C₃₁	C₂³×C₈	1	2	yes	—
105	C₄×C₃₁	C₂×#131	1	4	yes	—
106	C₂³×C₈	C₄×C₃₁	1	2	yes	—
107	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
108	#431	C₄×C₃₁	1	8	yes	—
109	#431	C₂²×C₃₁	1	8	yes	—
110	C₂×C₄⋊C₈	C₄×C₃₁	1	4	yes	—
111	C₂×C₂²⋊C₈	C₄×C₃₁	1	4	yes	—
112	C₂×C₄×C₈	C₄×C₃₁	1	2	yes	—
113	C₂×C₄×C₈	C₄×C₃₁	1	4	yes	—
114	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
115	C₂×C₄×C₈	C₂²×C₃₁	1	4	yes	—
116	C₂×C₄×C₈	C₂²×C₃₁	1	4	yes	—
117	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
118	#322	C₂²×C₃₁	1	8	yes	complemented
119	C₈²	C₂²×C₃₁	1	8	yes	complemented
120	C₂×C₃₁	—	2	4	no	—
121	C₂×C₃₁	#1450	1	4	yes	direct product factor
122	C₂×C₃₁	C₂²×C₄×C₈	1	1	yes	characteristic
123	C₂×C₃₁	C₂×#405	1	2	yes	—
124	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
125	C₂³×C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
126	C₂²×C₈	C₈×C₃₁	1	16	yes	complemented
127	C₂²×C₈	C₂×C₄×C₃₁	1	8	yes	—
128	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
129	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
130	C₂²×C₈	C₂×C₄×C₃₁	1	8	yes	—
131	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
132	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
133	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
134	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
135	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
136	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
137	C₄⋊C₈	C₂×C₄×C₃₁	1	16	yes	—
138	C₂²⋊C₈	C₂×C₄×C₃₁	1	16	yes	—
139	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
140	C₄×C₈	C₂×C₄×C₃₁	1	8	yes	—
141	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
142	C₄×C₈	C₂×C₄×C₃₁	1	8	yes	—
143	C₄×C₈	C₂³×C₃₁	1	2	yes	—
144	C₄×C₈	C₂³×C₃₁	1	8	yes	—
145	C₄×C₈	C₂³×C₃₁	1	2	yes	—
146	C₃₁	C₂×#1450	1	1	yes	characteristic, direct product factor, Sylow
147	C₂⁴	C₄²×C₃₁	1	1	yes	characteristic
148	C₂²×C₄	—	2	8	no	—
149	C₂²×C₄	C₄²×C₃₁	1	2	yes	—
150	C₂²×C₄	C₄²×C₃₁	1	4	yes	—
151	C₂²×C₄	C₄²×C₃₁	1	2	yes	—
152	C₂²×C₄	C₄²×C₃₁	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₈×C₃₁	1	8	yes	—
154	C₂²×C₄	C₂²×C₄×C₃₁	1	4	yes	—
155	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
156	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
157	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
158	C₂×C₈	—	2	16	no	—
159	C₂×C₈	C₂×C₈×C₃₁	1	8	yes	—
160	C₂×C₈	C₂×C₈×C₃₁	1	8	yes	—
161	C₂×C₈	C₂×C₈×C₃₁	1	32	yes	complemented
162	C₂×C₈	C₂²×C₄×C₃₁	1	16	yes	—
163	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
164	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
165	C₄²	C₄²×C₃₁	1	2	yes	—
166	C₄²	C₄²×C₃₁	1	4	yes	—
167	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
168	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
169	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
170	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
171	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
172	C₂³	—	2	4	no	—
173	C₂³	C₄×C₈×C₃₁	1	4	yes	—
174	C₂³	C₂×C₄²×C₃₁	1	2	yes	—
175	C₂³	C₂×C₄²×C₃₁	1	1	yes	characteristic
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	2	8	no	—
179	C₂×C₄	C₄×C₈×C₃₁	1	8	yes	—
180	C₂×C₄	C₄×C₈×C₃₁	1	4	yes	—
181	C₂×C₄	C₂×C₄²×C₃₁	1	4	yes	—
182	C₂×C₄	C₂×C₄²×C₃₁	1	1	yes	characteristic
183	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
184	C₂×C₄	C₂×C₄²×C₃₁	1	2	yes	—
185	C₂×C₄	C₂²×C₈×C₃₁	1	4	yes	—
186	C₂×C₄	C₂²×C₈×C₃₁	1	8	yes	—
187	C₂×C₄	C₃₁×#131	1	8	yes	—
188	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
189	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
190	C₈	—	2	16	no	—
191	C₈	C₂²×C₈×C₃₁	1	8	yes	—
192	C₈	C₂²×C₈×C₃₁	1	8	yes	—
193	C₂²	—	2	4	no	—
194	C₂²	—	2	8	no	—
195	C₂²	C₂×C₄×C₈×C₃₁	1	4	yes	—
196	C₂²	C₂×C₄×C₈×C₃₁	1	2	yes	—
197	C₂²	C₃₁×#405	1	4	yes	—
198	C₂²	C₂²×C₄²×C₃₁	1	1	yes	characteristic
199	C₄	—	2	4	no	—
200	C₄	C₂×C₄×C₈×C₃₁	1	2	yes	—
201	C₄	C₂×C₄×C₈×C₃₁	1	4	yes	—
202	C₄	C₃₁×#405	1	4	yes	—
203	C₄	C₂³×C₈×C₃₁	1	2	yes	—
204	C₄	C₂×C₃₁×#131	1	4	yes	—
205	C₂	—	2	4	no	—
206	C₂	C₃₁×#1450	1	4	yes	direct product factor
207	C₂	C₂²×C₄×C₈×C₃₁	1	1	yes	characteristic
208	C₂	C₂×C₃₁×#405	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1450	C₂	direct (N × H)
4. C₂×C₃₁×#322	C₂	semidirect
5. C₂×C₈²×C₃₁	C₂	semidirect
18. C₃₁×#322	C₂²	semidirect
19. C₈²×C₃₁	C₂²	semidirect
22. C₂²×C₈×C₃₁	C₈	semidirect
56. C₂×C₈×C₃₁	C₂×C₈	semidirect
67. C₂×#1450	C₃₁	direct (N × H)
91. #1450	C₂×C₃₁	direct (N × H)
92. C₂×#322	C₂×C₃₁	semidirect
93. C₂×C₈²	C₂×C₃₁	semidirect
118. #322	C₂²×C₃₁	semidirect
119. C₈²	C₂²×C₃₁	semidirect
121. C₂×C₃₁	#1450	direct (N × H)
126. C₂²×C₈	C₈×C₃₁	semidirect
146. C₃₁	C₂×#1450	direct (N × H)
161. C₂×C₈	C₂×C₈×C₃₁	semidirect
206. C₂	C₃₁×#1450	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×#431	2
C₂²×C₄×C₈×C₃₁	2
C₃₁×#1450	2
C₂×C₃₁×#322	2
C₂×C₈²×C₃₁	2
C₂×#1450	31

Subgroups — C2×C31×#1450 #21474072

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1450 #21474072