Subgroups — C₂×#11716933 #13105904

Order 1600.

608 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#709	C₂	1	1	yes	maximal, characteristic
2	C₂²×#6029	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#681	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#87768	C₂	1	2	yes	maximal, complemented
5	#11716933	C₂	1	8	yes	maximal, direct product factor
6	C₂×D₅×#168	C₂	1	2	yes	maximal, complemented
7	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
8	C₂×C₅×C₂²⋊D₅	C₄	1	2	yes	—
9	C₅×#709	C₄	1	4	yes	—
10	C₅×#709	C₂²	1	4	yes	—
11	C₂×C₅×D₂₀	C₄	1	1	yes	characteristic
12	C₂×C₄×C₅×D₅	C₂²	1	2	yes	—
13	C₂×C₅×Dic₁₀	C₄	1	1	yes	characteristic
14	C₂×#6029	C₂²	1	1	yes	characteristic
15	C₂×#6029	C₂²	1	1	yes	characteristic
16	C₂×#6029	C₂²	1	4	yes	complemented
17	C₅×#681	C₂²	1	4	yes	complemented
18	C₂×C₅×#168	C₂²	1	2	yes	—
19	#87768	C₂²	1	8	yes	complemented
20	D₅×#168	C₂²	1	8	yes	complemented
21	C₂×#4094	—	5	1	no	maximal
22	C₂×#4053	—	5	1	no	maximal
23	C₂³×C₅²	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₅×D₅	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×C₅²	C₂×C₄	1	2	yes	—
27	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₅²	C₂³	1	2	yes	—
29	C₅×C₂²⋊D₅	C₂×C₄	1	8	yes	—
30	C₂×C₅×Dic₅	C₂×C₄	1	2	yes	—
31	C₅×D₂₀	C₂×C₄	1	4	yes	—
32	C₄×C₅×D₅	C₂×C₄	1	4	yes	—
33	C₄×C₅×D₅	C₂³	1	4	yes	—
34	C₅×Dic₁₀	C₂×C₄	1	4	yes	—
35	#6029	C₂³	1	2	yes	—
36	#6029	C₂³	1	2	yes	—
37	C₅×#168	C₂³	1	4	yes	—
38	C₂×C₅×#41	—	5	1	no	—
39	C₂×#709	D₅	1	1	yes	characteristic
40	C₂×C₅×M₁₆	—	5	1	no	—
41	#4094	—	5	8	no	—
42	C₂×#681	—	5	2	no	—
43	C₂×#681	D₅	1	1	yes	characteristic, complemented
44	C₂²×#168	—	5	1	no	—
45	C₂²×#168	—	5	1	no	—
46	C₂²×#168	—	5	2	no	—
47	C₂²×#168	—	10	2	no	—
48	#4053	—	5	8	no	—
49	C₂×#676	—	5	2	no	—
50	C₂×C₈×D₅	—	5	2	no	—
51	C₂²×C₅²	—	2	2	no	—
52	C₂²×C₅²	C₂²×C₄	1	2	yes	—
53	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₅×D₅	—	2	4	no	—
55	C₂×C₅×D₅	C₂²×C₄	1	4	yes	—
56	C₄×C₅²	C₂²×C₄	1	2	yes	—
57	C₄×C₅²	C₂²×C₄	1	1	yes	characteristic
58	C₄×C₅²	C₂⁴	1	1	yes	characteristic
59	C₅×Dic₅	C₂²×C₄	1	4	yes	—
60	C₅×#41	—	5	4	no	—
61	C₅×#41	—	5	4	no	—
62	C₂×C₅×Q₈	—	5	1	no	—
63	C₂×C₅×D₄	—	5	2	no	—
64	C₂×C₅×D₄	—	5	1	no	—
65	C₂²×C₄×C₅	—	2	2	no	—
66	C₂²×C₄×C₅	—	5	2	no	—
67	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
69	C₂×C₂²⋊D₅	Dic₅	1	2	yes	—
70	#709	Dic₅	1	4	yes	—
71	#709	C₂×D₅	1	4	yes	—
72	C₂×D₂₀	Dic₅	1	1	yes	characteristic
73	C₂×C₄×D₅	C₂×D₅	1	2	yes	—
74	C₂×Dic₁₀	Dic₅	1	1	yes	characteristic
75	C₅×M₁₆	—	5	4	no	—
76	C₂×C₅×C₈	—	5	2	no	—
77	#681	—	5	8	no	—
78	#681	C₂×D₅	1	4	yes	complemented
79	C₂×#168	—	5	1	no	—
80	C₂×#168	—	5	1	no	—
81	C₂×#168	—	5	4	no	—
82	C₂×#168	—	5	1	no	—
83	C₂×#168	—	5	1	no	—
84	C₂×#168	—	5	4	no	—
85	C₂×#168	—	5	8	no	—
86	C₂×#168	—	10	2	no	—
87	C₂×#168	—	10	2	no	—
88	C₂×#168	—	10	8	no	—
89	C₂×#168	C₂×D₅	1	2	yes	—
90	#676	—	5	8	no	—
91	C₈×D₅	—	5	8	no	—
92	C₂×#131	—	25	1	no	Sylow
93	C₂×C₅²	—	2	2	no	—
94	C₂×C₅²	#131	1	2	yes	complemented
95	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
96	C₅×D₅	—	2	4	no	—
97	C₂³×C₅	—	2	2	no	—
98	C₂³×C₅	—	5	2	no	—
99	C₂³×C₅	C₄×D₅	1	1	yes	characteristic
100	C₂³×C₅	C₂×Dic₅	1	1	yes	characteristic
101	C₂²×D₅	C₂×Dic₅	1	2	yes	—
102	C₅×Q₈	—	5	4	no	—
103	C₅×D₄	—	5	8	no	—
104	C₅×D₄	—	5	4	no	—
105	C₂×C₄×C₅	—	2	2	no	—
106	C₂×C₄×C₅	—	2	4	no	—
107	C₂×C₄×C₅	—	2	4	no	—
108	C₂×C₄×C₅	—	2	2	no	—
109	C₂×C₄×C₅	—	5	2	no	—
110	C₂×C₄×C₅	—	5	4	no	—
111	C₂×C₄×C₅	—	5	4	no	—
112	C₂×C₄×C₅	C₄×D₅	1	1	yes	characteristic
113	C₂×C₄×C₅	C₄×D₅	1	2	yes	—
114	C₂×C₄×C₅	C₂×Dic₅	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂×Dic₅	1	2	yes	—
116	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
117	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
118	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
119	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
120	C₂²⋊D₅	C₂×Dic₅	1	8	yes	—
121	C₂×Dic₅	C₂×Dic₅	1	2	yes	—
122	D₂₀	C₂×Dic₅	1	4	yes	—
123	C₄×D₅	C₂×Dic₅	1	4	yes	—
124	C₄×D₅	C₂²×D₅	1	4	yes	—
125	Dic₁₀	C₂×Dic₅	1	4	yes	—
126	C₅×C₈	—	5	4	no	—
127	#168	—	5	2	no	—
128	#168	—	5	2	no	—
129	#168	—	5	2	no	—
130	#168	—	5	2	no	—
131	#168	—	10	4	no	—
132	#168	—	10	4	no	—
133	#168	C₂²×D₅	1	4	yes	—
134	C₂×#41	—	5	1	no	—
135	#131	—	25	8	no	—
136	C₂×M₁₆	—	5	1	no	—
137	C₂×M₁₆	—	25	2	no	—
138	C₂²×C₈	—	25	2	no	—
139	C₂²×C₈	—	25	1	no	—
140	C₅²	C₂×#131	1	1	yes	characteristic, complemented, Sylow
141	C₂²×C₅	—	2	8	no	—
142	C₂²×C₅	—	2	2	no	—
143	C₂²×C₅	—	2	2	no	—
144	C₂²×C₅	—	2	4	no	—
145	C₂²×C₅	—	2	2	no	—
146	C₂²×C₅	—	5	4	no	—
147	C₂²×C₅	—	10	4	no	—
148	C₂²×C₅	C₂×C₄×D₅	1	2	yes	—
149	C₂²×C₅	C₂×C₄×D₅	1	1	yes	characteristic
150	C₂²×C₅	C₂²×Dic₅	1	2	yes	—
151	C₂²×C₅	C₂²×Dic₅	1	1	yes	characteristic
152	C₂×D₅	—	2	4	no	—
153	C₂×D₅	C₂²×Dic₅	1	4	yes	—
154	C₄×C₅	—	2	2	no	—
155	C₄×C₅	—	2	4	no	—
156	C₄×C₅	—	2	2	no	—
157	C₄×C₅	—	5	4	no	—
158	C₄×C₅	C₂×C₄×D₅	1	1	yes	characteristic
159	C₄×C₅	C₂×C₄×D₅	1	2	yes	—
160	C₄×C₅	C₂²×Dic₅	1	2	yes	—
161	C₄×C₅	C₂²×Dic₅	1	1	yes	characteristic
162	C₄×C₅	C₂³×D₅	1	1	yes	characteristic
163	C₄×C₅	C₂³×D₅	1	1	yes	characteristic
164	Dic₅	C₂²×Dic₅	1	4	yes	—
165	#41	—	5	4	no	—
166	#41	—	5	4	no	—
167	C₂×Q₈	—	5	1	no	—
168	C₂×D₄	—	5	2	no	—
169	C₂×D₄	—	5	1	no	—
170	C₂²×C₄	—	5	2	no	—
171	C₂²×C₄	D₅²	1	1	yes	characteristic
172	M₁₆	—	5	4	no	—
173	M₁₆	—	25	8	no	—
174	C₂×C₈	—	5	2	no	—
175	C₂×C₈	—	25	8	no	—
176	C₂×C₈	—	25	4	no	—
177	C₂×C₈	—	25	1	no	—
178	C₂×C₈	—	25	1	no	—
179	C₂×C₅	—	2	8	no	—
180	C₂×C₅	—	2	2	no	—
181	C₂×C₅	—	2	2	no	—
182	C₂×C₅	—	2	4	no	—
183	C₂×C₅	—	2	2	no	—
184	C₂×C₅	—	10	4	no	—
185	C₂×C₅	#4053	1	2	yes	complemented
186	C₂×C₅	#4094	1	2	yes	complemented
187	C₂×C₅	C₂²×C₄×D₅	1	1	yes	characteristic
188	C₂×C₅	C₂³×Dic₅	1	1	yes	characteristic
189	D₅	—	2	4	no	—
190	C₂³	—	5	2	no	—
191	C₂³	D₅×Dic₅	1	1	yes	characteristic
192	Q₈	—	5	4	no	—
193	D₄	—	5	8	no	—
194	D₄	—	5	4	no	—
195	C₂×C₄	—	5	2	no	—
196	C₂×C₄	—	5	4	no	—
197	C₂×C₄	—	5	4	no	—
198	C₂×C₄	D₅×Dic₅	1	1	yes	characteristic
199	C₂×C₄	D₅×Dic₅	1	2	yes	—
200	C₂×C₄	C₂×D₅²	1	2	yes	—
201	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
202	C₈	—	5	4	no	—
203	C₈	—	25	2	no	—
204	C₈	—	25	2	no	—
205	C₅	—	2	2	no	—
206	C₅	C₂×#4053	1	1	yes	characteristic, complemented
207	C₅	C₂×#4094	1	1	yes	characteristic, complemented
208	C₂²	—	2	2	no	—
209	C₂²	—	5	4	no	—
210	C₂²	—	10	4	no	—
211	C₂²	C₂×D₅×Dic₅	1	2	yes	—
212	C₂²	C₂×D₅×Dic₅	1	1	yes	characteristic
213	C₄	—	5	4	no	—
214	C₄	C₂×D₅×Dic₅	1	1	yes	characteristic
215	C₄	C₂×D₅×Dic₅	1	2	yes	—
216	C₄	C₂²×D₅²	1	1	yes	characteristic
217	C₂	—	2	2	no	—
218	C₂	—	10	4	no	—
219	C₂	#11716933	1	2	yes	direct product factor
220	C₂	C₂²×D₅×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₅²
commutator	C₂×C₅²	C₂³×C₄
Frattini	C₄	C₂²×D₅²
Fitting	C₂²×C₄×C₅²	C₂²
socle	C₂²×C₅²	C₂²×C₄
prime radical	C₂×C₅×#709	C₂
cosocle	C₄×C₅²	C₂⁴
hypercenter	C₂²×C₄	D₅²
hypocenter	C₅²	C₂×#131
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₂²×#6029	C₂	semidirect
3. C₂×C₅×#681	C₂	semidirect
4. C₂×#87768	C₂	semidirect
5. #11716933	C₂	direct (N × H)
6. C₂×D₅×#168	C₂	semidirect
16. C₂×#6029	C₂²	semidirect
17. C₅×#681	C₂²	semidirect
19. #87768	C₂²	semidirect
20. D₅×#168	C₂²	semidirect
43. C₂×#681	D₅	semidirect
78. #681	C₂×D₅	semidirect
94. C₂×C₅²	#131	semidirect
140. C₅²	C₂×#131	semidirect
185. C₂×C₅	#4053	semidirect
186. C₂×C₅	#4094	semidirect
206. C₅	C₂×#4053	semidirect
207. C₅	C₂×#4094	semidirect
219. C₂	#11716933	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×C₅×#709	2
C₂²×#6029	2
C₂×C₅×#681	2
C₂×#87768	2
#11716933	2
C₂×D₅×#168	2
C₂×#4094	5
C₂×#4053	5

Subgroups — C2×#11716933 #13105904

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11716933 #13105904