Subgroups — C₂×C₅×#2151 #11956116

Order 1280.

734 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂³×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#405	C₂	1	1	yes	maximal, characteristic
4	C₅×#2151	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×C₄⋊C₁₆	C₂	1	4	yes	maximal, complemented
6	C₂²×C₅×M₁₆	C₄	1	1	yes	characteristic
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₅×#117	C₄	1	1	yes	characteristic
9	C₂×C₅×M₃₂	—	2	4	no	—
10	C₂×C₅×M₃₂	C₂²	1	4	yes	—
11	C₂²×C₅×C₁₆	—	2	4	no	—
12	C₂²×C₅×C₁₆	C₂²	1	4	yes	—
13	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
14	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
15	C₅×#405	C₄	1	4	yes	—
16	C₅×#405	C₂²	1	4	yes	—
17	C₂×C₅×#97	C₄	1	2	yes	—
18	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
19	C₅×C₄⋊C₁₆	C₂²	1	16	yes	complemented
20	C₂×#2151	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
22	C₂×C₅×M₁₆	—	2	4	no	—
23	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
24	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
27	C₂²×C₅×C₈	D₄	1	4	yes	—
28	C₂²×C₅×C₈	Q₈	1	4	yes	—
29	C₂²×C₅×C₈	C₂³	1	2	yes	—
30	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
31	C₅×#117	C₈	1	4	yes	—
32	C₅×#117	C₂×C₄	1	4	yes	—
33	C₂×C₅×C₄⋊C₄	C₈	1	2	yes	—
34	C₂×C₅×C₂²⋊C₄	C₈	1	2	yes	—
35	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
36	C₅×M₃₂	—	2	8	no	—
37	C₂×C₅×C₁₆	—	2	8	no	—
38	C₂×C₅×C₁₆	—	2	4	no	—
39	C₂×C₅×C₁₆	—	2	4	no	—
40	C₂×C₅×C₁₆	C₂³	1	4	yes	—
41	C₂×C₅×C₁₆	C₂³	1	4	yes	—
42	C₅×#97	C₂×C₄	1	8	yes	—
43	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
44	C₄×C₅×C₈	C₂³	1	4	yes	—
45	C₂²×M₃₂	C₂×C₅	1	1	yes	characteristic
46	C₂³×C₁₆	C₂×C₅	1	1	yes	characteristic
47	C₂×#405	C₂×C₅	1	1	yes	characteristic
48	#2151	C₂×C₅	1	8	yes	direct product factor
49	C₂×C₄⋊C₁₆	C₂×C₅	1	4	yes	complemented
50	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
52	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
53	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₅	M₁₆	1	4	yes	—
56	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
58	C₅×M₁₆	—	2	8	no	—
59	C₂×C₅×C₈	—	2	4	no	—
60	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
61	C₂×C₅×C₈	C₄⋊C₄	1	8	yes	—
62	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
63	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
64	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
66	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
67	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
68	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
69	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
70	C₅×C₄⋊C₄	C₂×C₈	1	8	yes	—
71	C₅×C₂²⋊C₄	C₂×C₈	1	8	yes	—
72	C₄²×C₅	C₂×C₈	1	4	yes	—
73	C₄²×C₅	C₂²×C₄	1	4	yes	—
74	C₅×C₁₆	—	2	4	no	—
75	C₅×C₁₆	—	2	4	no	—
76	C₂²×M₁₆	C₄×C₅	1	1	yes	characteristic
77	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
78	C₂×#117	C₄×C₅	1	1	yes	characteristic
79	C₂×M₃₂	—	2	4	no	—
80	C₂×M₃₂	C₂²×C₅	1	4	yes	—
81	C₂²×C₁₆	—	2	4	no	—
82	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
83	C₂²×C₁₆	C₂²×C₅	1	4	yes	—
84	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
85	#405	C₄×C₅	1	4	yes	—
86	#405	C₂²×C₅	1	4	yes	—
87	C₂×#97	C₄×C₅	1	2	yes	—
88	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
89	C₄⋊C₁₆	C₂²×C₅	1	16	yes	complemented
90	C₂³×C₅	—	2	4	no	—
91	C₂³×C₅	C₄⋊C₈	1	4	yes	—
92	C₂³×C₅	C₂²×C₈	1	2	yes	—
93	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₅	—	2	8	no	—
95	C₂×C₄×C₅	C₄⋊C₈	1	8	yes	—
96	C₂×C₄×C₅	C₄⋊C₈	1	4	yes	—
97	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
98	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₅	C₂²×C₈	1	8	yes	—
100	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
102	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
103	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
104	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
105	C₅×C₈	—	2	4	no	—
106	C₅×C₈	C₂×C₄⋊C₄	1	4	yes	—
107	C₅×C₈	C₂×C₄⋊C₄	1	2	yes	—
108	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
109	C₅×C₈	C₂²×Q₈	1	1	yes	characteristic
110	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
111	C₂×M₁₆	—	2	4	no	—
112	C₂×M₁₆	C₂×C₄×C₅	1	4	yes	—
113	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
114	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
115	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
116	C₂²×C₈	C₅×D₄	1	4	yes	—
117	C₂²×C₈	C₅×Q₈	1	4	yes	—
118	C₂²×C₈	C₂³×C₅	1	2	yes	—
119	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
120	#117	C₅×C₈	1	4	yes	—
121	#117	C₂×C₄×C₅	1	4	yes	—
122	C₂×C₄⋊C₄	C₅×C₈	1	2	yes	—
123	C₂×C₂²⋊C₄	C₅×C₈	1	2	yes	—
124	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
125	M₃₂	—	2	8	no	—
126	C₂×C₁₆	—	2	4	no	—
127	C₂×C₁₆	—	2	8	no	—
128	C₂×C₁₆	—	2	4	no	—
129	C₂×C₁₆	C₂³×C₅	1	4	yes	—
130	C₂×C₁₆	C₂³×C₅	1	4	yes	—
131	#97	C₂×C₄×C₅	1	8	yes	—
132	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
133	C₄×C₈	C₂³×C₅	1	4	yes	—
134	C₂²×C₅	—	2	4	no	—
135	C₂²×C₅	—	2	8	no	—
136	C₂²×C₅	C₂×C₄⋊C₈	1	4	yes	—
137	C₂²×C₅	C₂×C₄⋊C₈	1	2	yes	—
138	C₂²×C₅	#504	1	4	yes	—
139	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
140	C₄×C₅	—	2	8	no	—
141	C₄×C₅	C₂×C₄⋊C₈	1	2	yes	—
142	C₄×C₅	C₂×C₄⋊C₈	1	4	yes	—
143	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
145	C₂⁴	C₂×C₅×C₈	1	1	yes	characteristic
146	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
147	C₂²×C₄	C₂×C₅×C₈	1	4	yes	—
148	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
149	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
150	C₂²×C₄	C₅×M₁₆	1	4	yes	—
151	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
152	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
153	M₁₆	—	2	8	no	—
154	C₂×C₈	—	2	4	no	—
155	C₂×C₈	C₅×C₄⋊C₄	1	4	yes	—
156	C₂×C₈	C₅×C₄⋊C₄	1	8	yes	—
157	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
158	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
159	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
160	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
161	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
162	C₂×C₈	C₂×C₅×Q₈	1	4	yes	—
163	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
164	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
165	C₄⋊C₄	C₂×C₅×C₈	1	8	yes	—
166	C₂²⋊C₄	C₂×C₅×C₈	1	8	yes	—
167	C₄²	C₂×C₅×C₈	1	4	yes	—
168	C₄²	C₂²×C₄×C₅	1	4	yes	—
169	C₁₆	—	2	4	no	—
170	C₁₆	—	2	4	no	—
171	C₂×C₅	—	2	4	no	—
172	C₂×C₅	#2151	1	4	yes	direct product factor
173	C₂×C₅	C₂²×C₄⋊C₈	1	1	yes	characteristic
174	C₂×C₅	C₂×#504	1	2	yes	—
175	C₂³	—	2	4	no	—
176	C₂³	C₅×C₄⋊C₈	1	4	yes	—
177	C₂³	C₂²×C₅×C₈	1	2	yes	—
178	C₂³	C₂²×C₅×C₈	1	1	yes	characteristic
179	C₂×C₄	—	2	8	no	—
180	C₂×C₄	C₅×C₄⋊C₈	1	8	yes	—
181	C₂×C₄	C₅×C₄⋊C₈	1	4	yes	—
182	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
183	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
184	C₂×C₄	C₂²×C₅×C₈	1	8	yes	—
185	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
186	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
187	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
188	C₂×C₄	C₂×C₅×M₁₆	1	4	yes	—
189	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
190	C₈	—	2	4	no	—
191	C₈	C₂×C₅×C₄⋊C₄	1	2	yes	—
192	C₈	C₂×C₅×C₄⋊C₄	1	4	yes	—
193	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
194	C₈	C₂²×C₅×Q₈	1	1	yes	characteristic
195	C₅	C₂×#2151	1	1	yes	characteristic, direct product factor, Sylow
196	C₂²	—	2	8	no	—
197	C₂²	—	2	4	no	—
198	C₂²	C₂×C₅×C₄⋊C₈	1	4	yes	—
199	C₂²	C₂×C₅×C₄⋊C₈	1	2	yes	—
200	C₂²	C₅×#504	1	4	yes	—
201	C₂²	C₂³×C₅×C₈	1	1	yes	characteristic
202	C₄	—	2	8	no	—
203	C₄	C₂×C₅×C₄⋊C₈	1	2	yes	—
204	C₄	C₂×C₅×C₄⋊C₈	1	4	yes	—
205	C₄	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
206	C₄	C₂²×C₅×M₁₆	1	1	yes	characteristic
207	C₂	—	2	4	no	—
208	C₂	C₅×#2151	1	4	yes	direct product factor
209	C₂	C₂²×C₅×C₄⋊C₈	1	1	yes	characteristic
210	C₂	C₂×C₅×#504	1	2	yes	—

	subgroup	G/H
center	C₂²×C₅×C₈	C₂³
commutator	C₂²	C₂³×C₅×C₈
Frattini, cosocle	C₂×C₈	C₂⁴×C₅
socle	C₂³×C₅	C₂²×C₈
prime radical	C₂⁴×C₅	C₂×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#2151	C₂	direct (N × H)
5. C₂×C₅×C₄⋊C₁₆	C₂	semidirect
19. C₅×C₄⋊C₁₆	C₂²	semidirect
20. C₂×#2151	C₅	direct (N × H)
48. #2151	C₂×C₅	direct (N × H)
49. C₂×C₄⋊C₁₆	C₂×C₅	semidirect
89. C₄⋊C₁₆	C₂²×C₅	semidirect
172. C₂×C₅	#2151	direct (N × H)
195. C₅	C₂×#2151	direct (N × H)
208. C₂	C₅×#2151	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×M₃₂	2
C₂³×C₅×C₁₆	2
C₂×C₅×#405	2
C₅×#2151	2
C₂×C₅×C₄⋊C₁₆	2
C₂×#2151	5

Subgroups — C2×C5×#2151 #11956116

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2151 #11956116