Subgroups — C₂×C₅×#2161 #11956133

Order 1280.

642 conjugacy classes of subgroups.

The subgroup lattice has 216 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	C₂²×C₅×#108	C₂	1	2	yes	maximal
3	C₅×#2161	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅×#368	C₂	1	4	yes	maximal, complemented
5	C₂²×C₅×M₁₆	—	2	2	no	—
6	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂×C₅×M₃₂	C₄	1	4	yes	—
8	C₂×C₅×M₃₂	C₄	1	4	yes	—
9	C₂×C₅×M₃₂	C₂²	1	4	yes	—
10	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
11	C₂×C₅×#108	—	2	4	no	—
12	C₂×C₅×#108	—	2	2	no	—
13	C₂×C₅×#108	C₂²	1	4	yes	—
14	C₂×C₅×#108	C₂²	1	8	yes	—
15	C₅×#368	C₂²	1	16	yes	complemented
16	C₂×#2161	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
18	C₂×C₅×M₁₆	—	2	8	no	—
19	C₂×C₅×M₁₆	—	2	4	no	—
20	C₂×C₅×M₁₆	—	4	4	no	—
21	C₂×C₅×M₁₆	—	4	2	no	—
22	C₂²×C₅×C₈	—	4	2	no	—
23	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
24	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
25	C₂²×C₅×C₈	D₄	1	2	yes	—
26	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
27	C₂²×C₅×C₈	Q₈	1	2	yes	—
28	C₂²×C₅×C₈	Q₈	1	2	yes	—
29	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
30	C₂²×C₅×C₈	C₂³	1	2	yes	—
31	C₅×M₃₂	C₂×C₄	1	16	yes	—
32	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
33	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
34	C₂×C₅×C₁₆	C₂³	1	4	yes	—
35	C₅×#108	—	2	4	no	—
36	C₅×#108	—	2	8	no	—
37	C₅×#108	C₂³	1	8	yes	—
38	C₂²×M₃₂	C₂×C₅	1	1	yes	characteristic
39	C₂²×#108	C₂×C₅	1	2	yes	—
40	#2161	C₂×C₅	1	8	yes	direct product factor
41	C₂×#368	C₂×C₅	1	4	yes	complemented
42	C₂⁴×C₅	Q₁₆	1	1	yes	characteristic
43	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
44	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
45	C₂²×C₄×C₅	D₈	1	2	yes	—
46	C₂²×C₄×C₅	D₈	1	2	yes	—
47	C₂²×C₄×C₅	Q₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₅	Q₁₆	1	2	yes	—
49	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
50	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
51	C₅×M₁₆	—	2	8	no	—
52	C₅×M₁₆	—	4	8	no	—
53	C₅×M₁₆	—	4	4	no	—
54	C₂×C₅×C₈	—	4	8	no	—
55	C₂×C₅×C₈	—	4	4	no	—
56	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
57	C₂×C₅×C₈	C₄⋊C₄	1	4	yes	—
58	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
59	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
60	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
63	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
64	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
65	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
66	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
68	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
69	C₅×C₁₆	C₂²×C₄	1	8	yes	—
70	C₂²×M₁₆	—	2	2	no	—
71	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
72	C₂×M₃₂	C₄×C₅	1	4	yes	—
73	C₂×M₃₂	C₄×C₅	1	4	yes	—
74	C₂×M₃₂	C₂²×C₅	1	4	yes	—
75	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
76	C₂×#108	—	2	4	no	—
77	C₂×#108	—	2	2	no	—
78	C₂×#108	C₂²×C₅	1	4	yes	—
79	C₂×#108	C₂²×C₅	1	8	yes	—
80	#368	C₂²×C₅	1	16	yes	complemented
81	C₂³×C₅	—	4	2	no	—
82	C₂³×C₅	#107	1	2	yes	—
83	C₂³×C₅	#107	1	2	yes	—
84	C₂³×C₅	C₂×Q₁₆	1	1	yes	characteristic
85	C₂³×C₅	C₂×Q₁₆	1	2	yes	—
86	C₂×C₄×C₅	#107	1	2	yes	—
87	C₂×C₄×C₅	#107	1	4	yes	—
88	C₂×C₄×C₅	#107	1	2	yes	—
89	C₂×C₄×C₅	#107	1	4	yes	—
90	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
91	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₅	C₂×D₈	1	4	yes	—
94	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
97	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
98	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
99	C₅×C₈	—	4	8	no	—
100	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₅×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₅×C₈	C₂×C₄⋊C₄	1	4	yes	—
103	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
104	C₅×C₈	C₂²×Q₈	1	1	yes	characteristic
105	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
106	C₂×M₁₆	—	2	4	no	—
107	C₂×M₁₆	—	2	8	no	—
108	C₂×M₁₆	—	4	2	no	—
109	C₂×M₁₆	—	4	4	no	—
110	C₂²×C₈	—	4	2	no	—
111	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
112	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
113	C₂²×C₈	C₅×D₄	1	2	yes	—
114	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
115	C₂²×C₈	C₅×Q₈	1	2	yes	—
116	C₂²×C₈	C₅×Q₈	1	2	yes	—
117	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
118	C₂²×C₈	C₂³×C₅	1	2	yes	—
119	M₃₂	C₂×C₄×C₅	1	16	yes	—
120	C₂×C₁₆	C₂×C₄×C₅	1	4	yes	—
121	C₂×C₁₆	C₂×C₄×C₅	1	4	yes	—
122	C₂×C₁₆	C₂³×C₅	1	4	yes	—
123	#108	—	2	4	no	—
124	#108	—	2	8	no	—
125	#108	C₂³×C₅	1	8	yes	—
126	C₂²×C₅	—	2	4	no	—
127	C₂²×C₅	—	2	2	no	—
128	C₂²×C₅	—	4	2	no	—
129	C₂²×C₅	—	4	2	no	—
130	C₂²×C₅	C₂×#107	1	4	yes	—
131	C₂²×C₅	C₂×#107	1	1	yes	characteristic
132	C₂²×C₅	C₂×#107	1	1	yes	characteristic
133	C₂²×C₅	C₂²×Q₁₆	1	1	yes	characteristic
134	C₄×C₅	C₂×#107	1	4	yes	—
135	C₄×C₅	C₂×#107	1	1	yes	characteristic
136	C₄×C₅	C₂×#107	1	1	yes	characteristic
137	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
138	C₄×C₅	C₂²×D₈	1	1	yes	characteristic
139	C₂⁴	C₅×Q₁₆	1	1	yes	characteristic
140	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
141	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
142	C₂²×C₄	C₅×D₈	1	2	yes	—
143	C₂²×C₄	C₅×D₈	1	2	yes	—
144	C₂²×C₄	C₅×Q₁₆	1	1	yes	characteristic
145	C₂²×C₄	C₅×Q₁₆	1	2	yes	—
146	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
147	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
148	M₁₆	—	2	8	no	—
149	M₁₆	—	4	8	no	—
150	M₁₆	—	4	4	no	—
151	C₂×C₈	—	4	4	no	—
152	C₂×C₈	—	4	8	no	—
153	C₂×C₈	C₅×C₄⋊C₄	1	4	yes	—
154	C₂×C₈	C₅×C₄⋊C₄	1	4	yes	—
155	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
156	C₂×C₈	C₅×C₄⋊C₄	1	2	yes	—
157	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
158	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
159	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
160	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
161	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
162	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
163	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
164	C₂×C₈	C₂×C₅×Q₈	1	4	yes	—
165	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
166	C₁₆	C₂²×C₄×C₅	1	8	yes	—
167	C₂×C₅	—	2	1	no	—
168	C₂×C₅	—	2	4	no	—
169	C₂×C₅	—	2	1	no	—
170	C₂×C₅	#2161	1	2	yes	direct product factor
171	C₂×C₅	C₂²×#107	1	1	yes	characteristic
172	C₂³	—	4	2	no	—
173	C₂³	C₅×#107	1	2	yes	—
174	C₂³	C₅×#107	1	2	yes	—
175	C₂³	C₂×C₅×Q₁₆	1	1	yes	characteristic
176	C₂³	C₂×C₅×Q₁₆	1	2	yes	—
177	C₂×C₄	C₅×#107	1	4	yes	—
178	C₂×C₄	C₅×#107	1	2	yes	—
179	C₂×C₄	C₅×#107	1	4	yes	—
180	C₂×C₄	C₅×#107	1	2	yes	—
181	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
182	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
183	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
184	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
185	C₂×C₄	C₂×C₅×D₈	1	4	yes	—
186	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
187	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
188	C₂×C₄	C₂×C₅×Q₁₆	1	2	yes	—
189	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
190	C₈	—	4	8	no	—
191	C₈	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
192	C₈	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
193	C₈	C₂×C₅×C₄⋊C₄	1	4	yes	—
194	C₈	C₂³×C₄×C₅	1	1	yes	characteristic
195	C₈	C₂²×C₅×Q₈	1	1	yes	characteristic
196	C₅	C₂×#2161	1	1	yes	characteristic, direct product factor, Sylow
197	C₂²	—	2	2	no	—
198	C₂²	—	2	4	no	—
199	C₂²	—	4	2	no	—
200	C₂²	—	4	2	no	—
201	C₂²	C₂×C₅×#107	1	4	yes	—
202	C₂²	C₂×C₅×#107	1	1	yes	characteristic
203	C₂²	C₂×C₅×#107	1	1	yes	characteristic
204	C₂²	C₂²×C₅×Q₁₆	1	1	yes	characteristic
205	C₄	C₂×C₅×#107	1	1	yes	characteristic
206	C₄	C₂×C₅×#107	1	4	yes	—
207	C₄	C₂×C₅×#107	1	1	yes	characteristic
208	C₄	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
209	C₄	C₂²×C₅×D₈	1	1	yes	characteristic
210	C₂	—	2	1	no	—
211	C₂	—	2	4	no	—
212	C₂	—	2	1	no	—
213	C₂	C₅×#2161	1	2	yes	direct product factor
214	C₂	C₂²×C₅×#107	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₈
commutator	C₈	C₂³×C₄×C₅
Frattini, cosocle	C₂×C₈	C₂⁴×C₅
socle	C₂²×C₅	C₂×#107
prime radical	C₂⁴×C₅	Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#2161	C₂	direct (N × H)
4. C₂×C₅×#368	C₂	semidirect
15. C₅×#368	C₂²	semidirect
16. C₂×#2161	C₅	direct (N × H)
40. #2161	C₂×C₅	direct (N × H)
41. C₂×#368	C₂×C₅	semidirect
80. #368	C₂²×C₅	semidirect
170. C₂×C₅	#2161	direct (N × H)
196. C₅	C₂×#2161	direct (N × H)
213. C₂	C₅×#2161	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
C₂²×C₅×#108	2
C₅×#2161	2
C₂×C₅×#368	2
C₂×#2161	5

Subgroups — C2×C5×#2161 #11956133

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2161 #11956133