Subgroups — C₂×#11716849 #13104648

Order 1600.

687 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#1042	C₂	1	2	yes	maximal
2	C₂²×C₄×C₅×D₅	C₂	1	1	yes	maximal, characteristic
3	#11716849	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₅×D₅	C₂²	1	1	yes	characteristic
5	C₂²×#1042	—	2	4	no	—
6	C₂²×#1042	—	2	4	no	—
7	C₂²×#1042	C₄	1	8	yes	complemented
8	C₂²×#1042	C₂²	1	4	yes	—
9	C₂²×C₄×C₅²	C₄	1	1	yes	characteristic, complemented
10	C₂²×C₅×Dic₅	C₄	1	1	yes	characteristic, complemented
11	C₂×C₄×C₅×D₅	—	2	4	no	—
12	C₂×C₄×C₅×D₅	C₄	1	2	yes	complemented
13	C₂×C₄×C₅×D₅	C₂²	1	2	yes	—
14	C₂×(C₂×AGL(1,𝔽₅))⋊C₄	—	5	1	no	maximal
15	C₂×#3963	—	5	1	no	maximal
16	C₂³×C₅²	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₅×D₅	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₅×D₅	C₂×C₄	1	4	yes	—
19	C₂²×C₅×D₅	D₄	1	2	yes	—
20	C₂²×C₅×D₅	D₄	1	4	yes	—
21	C₂²×C₅×D₅	Q₈	1	2	yes	—
22	C₂²×C₅×D₅	C₂³	1	1	yes	characteristic
23	C₂×#1042	—	2	16	no	—
24	C₂×#1042	—	2	4	no	—
25	C₂×#1042	—	2	4	no	—
26	C₂×#1042	C₂×C₄	1	8	yes	complemented
27	C₂×C₄×C₅²	—	2	2	no	—
28	C₂×C₄×C₅²	C₂×C₄	1	2	yes	complemented
29	C₂×C₅×Dic₅	—	2	2	no	—
30	C₂×C₅×Dic₅	C₂×C₄	1	2	yes	complemented
31	C₄×C₅×D₅	—	2	4	no	—
32	C₄×C₅×D₅	—	2	4	no	—
33	C₂³×AGL(1,𝔽₅)	—	5	2	no	—
34	C₂³×C₄×C₅	—	5	1	no	—
35	C₂³×Dic₅	—	5	2	no	—
36	C₂²×C₄×D₅	D₅	1	1	yes	characteristic
37	(C₂×AGL(1,𝔽₅))⋊C₄	—	5	4	no	—
38	#3963	—	5	4	no	—
39	C₂²×C₅²	C₄²	1	2	yes	—
40	C₂²×C₅²	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₅²	C₄⋊C₄	1	2	yes	—
42	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₅×D₅	C₄²	1	2	yes	—
44	C₂×C₅×D₅	C₂²⋊C₄	1	8	yes	—
45	C₂×C₅×D₅	C₂²⋊C₄	1	2	yes	—
46	C₂×C₅×D₅	C₄⋊C₄	1	8	yes	—
47	C₂×C₅×D₅	C₄⋊C₄	1	2	yes	—
48	C₂×C₅×D₅	C₂²×C₄	1	2	yes	—
49	C₂×C₅×D₅	C₂×D₄	1	2	yes	—
50	C₂×C₅×D₅	C₂×D₄	1	1	yes	characteristic
51	C₂×C₅×D₅	C₂×Q₈	1	1	yes	characteristic
52	#1042	—	2	8	no	—
53	C₄×C₅²	—	2	2	no	—
54	C₅×Dic₅	—	2	2	no	—
55	C₂⁴×C₅	—	5	1	no	—
56	C₂³×D₅	C₂×D₅	1	1	yes	characteristic
57	C₂²×AGL(1,𝔽₅)	—	5	8	no	—
58	C₂²×AGL(1,𝔽₅)	—	5	4	no	—
59	C₂²×AGL(1,𝔽₅)	—	10	4	no	—
60	C₂²×AGL(1,𝔽₅)	—	10	4	no	—
61	C₂²×C₄×C₅	—	4	1	no	—
62	C₂²×C₄×C₅	—	5	1	no	—
63	C₂²×C₄×C₅	—	5	2	no	—
64	C₂²×C₄×C₅	—	5	2	no	—
65	C₂²×C₄×C₅	—	10	4	no	—
66	C₂²×C₄×C₅	Dic₅	1	1	yes	characteristic, complemented
67	C₂²×C₄×C₅	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
68	C₂²×Dic₅	—	5	8	no	—
69	C₂²×Dic₅	—	5	4	no	—
70	C₂²×Dic₅	—	10	4	no	—
71	C₂²×Dic₅	—	10	4	no	—
72	C₂²×Dic₅	Dic₅	1	1	yes	characteristic, complemented
73	C₂×C₄×D₅	—	2	4	no	—
74	C₂×C₄×D₅	Dic₅	1	2	yes	complemented
75	C₂×C₄×D₅	C₂×D₅	1	2	yes	—
76	C₂×#95	—	25	1	no	Sylow
77	C₂×C₅²	#95	1	4	yes	complemented
78	C₂×C₅²	C₂×C₄²	1	1	yes	characteristic
79	C₂×C₅²	C₂×C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₅²	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₅×D₅	#95	1	4	yes	complemented
82	C₅×D₅	C₂×C₂²⋊C₄	1	2	yes	—
83	C₅×D₅	C₂×C₄⋊C₄	1	2	yes	—
84	C₂³×C₅	—	4	1	no	—
85	C₂³×C₅	—	5	4	no	—
86	C₂³×C₅	—	5	2	no	—
87	C₂³×C₅	—	5	2	no	—
88	C₂³×C₅	—	5	4	no	—
89	C₂³×C₅	—	5	1	no	—
90	C₂³×C₅	—	5	1	no	—
91	C₂³×C₅	C₂×Dic₅	1	1	yes	characteristic
92	C₂³×C₅	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
93	C₂²×D₅	Dic₁₀	1	2	yes	—
94	C₂²×D₅	C₄×D₅	1	4	yes	—
95	C₂²×D₅	D₂₀	1	2	yes	—
96	C₂²×D₅	C₂×Dic₅	1	1	yes	characteristic
97	C₂²×D₅	C₂²⋊D₅	1	4	yes	—
98	C₂²×D₅	C₂²×D₅	1	1	yes	characteristic
99	C₂×AGL(1,𝔽₅)	—	5	8	no	—
100	C₂×AGL(1,𝔽₅)	—	10	16	no	—
101	C₂×AGL(1,𝔽₅)	—	10	4	no	—
102	C₂×AGL(1,𝔽₅)	—	10	4	no	—
103	C₂×C₄×C₅	—	2	2	no	—
104	C₂×C₄×C₅	—	2	2	no	—
105	C₂×C₄×C₅	—	4	4	no	—
106	C₂×C₄×C₅	—	4	2	no	—
107	C₂×C₄×C₅	—	5	2	no	—
108	C₂×C₄×C₅	—	10	4	no	—
109	C₂×C₄×C₅	—	10	4	no	—
110	C₂×C₄×C₅	—	10	2	no	—
111	C₂×C₄×C₅	C₂×Dic₅	1	2	yes	complemented
112	C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	1	2	yes	complemented
113	C₂×Dic₅	—	2	2	no	—
114	C₂×Dic₅	—	5	8	no	—
115	C₂×Dic₅	—	10	4	no	—
116	C₂×Dic₅	—	10	4	no	—
117	C₂×Dic₅	—	10	16	no	—
118	C₂×Dic₅	C₂×Dic₅	1	2	yes	complemented
119	C₄×D₅	—	2	4	no	—
120	C₄×D₅	—	2	4	no	—
121	C₂³×C₄	—	5	1	no	—
122	C₂³×C₄	—	25	2	no	—
123	#95	—	25	4	no	—
124	C₅²	C₂×#95	1	1	yes	characteristic, complemented, Sylow
125	C₂²×C₅	—	4	2	no	—
126	C₂²×C₅	—	4	2	no	—
127	C₂²×C₅	—	4	2	no	—
128	C₂²×C₅	—	4	1	no	—
129	C₂²×C₅	—	5	2	no	—
130	C₂²×C₅	—	5	2	no	—
131	C₂²×C₅	—	5	2	no	—
132	C₂²×C₅	—	5	1	no	—
133	C₂²×C₅	—	5	1	no	—
134	C₂²×C₅	—	5	2	no	—
135	C₂²×C₅	—	5	2	no	—
136	C₂²×C₅	—	5	8	no	—
137	C₂²×C₅	—	5	8	no	—
138	C₂²×C₅	C₄×Dic₅	1	2	yes	—
139	C₂²×C₅	C₄⋊Dic₅	1	2	yes	—
140	C₂²×C₅	C₂²⋊Dic₅	1	2	yes	—
141	C₂²×C₅	C₄×AGL(1,𝔽₅)	1	2	yes	—
142	C₂²×C₅	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
143	C₂²×C₅	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
144	C₂²×C₅	C₂²×Dic₅	1	1	yes	characteristic
145	C₂²×C₅	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
146	C₂×D₅	C₄×Dic₅	1	2	yes	—
147	C₂×D₅	#683	1	8	yes	—
148	C₂×D₅	C₄⋊Dic₅	1	2	yes	—
149	C₂×D₅	#685	1	8	yes	—
150	C₂×D₅	C₂²⋊Dic₅	1	2	yes	—
151	C₂×D₅	C₂×Dic₁₀	1	1	yes	characteristic
152	C₂×D₅	C₂×C₄×D₅	1	2	yes	—
153	C₂×D₅	C₂×D₂₀	1	1	yes	characteristic
154	C₂×D₅	C₂×C₂²⋊D₅	1	2	yes	—
155	AGL(1,𝔽₅)	—	10	8	no	—
156	C₄×C₅	—	2	2	no	—
157	C₄×C₅	—	2	2	no	—
158	C₄×C₅	—	4	4	no	—
159	C₄×C₅	—	10	2	no	—
160	Dic₅	—	2	2	no	—
161	Dic₅	—	10	8	no	—
162	C₂⁴	—	5	1	no	—
163	C₂²×C₄	—	5	1	no	—
164	C₂²×C₄	—	5	2	no	—
165	C₂²×C₄	—	5	2	no	—
166	C₂²×C₄	—	10	4	no	—
167	C₂²×C₄	—	25	4	no	—
168	C₂²×C₄	—	25	8	no	—
169	C₂²×C₄	—	50	4	no	—
170	C₂²×C₄	—	50	4	no	—
171	C₂²×C₄	#1042	1	1	yes	characteristic, complemented
172	C₂×C₅	—	4	1	no	—
173	C₂×C₅	—	4	1	no	—
174	C₂×C₅	—	4	1	no	—
175	C₂×C₅	—	4	4	no	—
176	C₂×C₅	—	5	2	no	—
177	C₂×C₅	—	5	4	no	—
178	C₂×C₅	—	5	2	no	—
179	C₂×C₅	#3963	1	4	yes	complemented
180	C₂×C₅	(C₂×AGL(1,𝔽₅))⋊C₄	1	4	yes	complemented
181	C₂×C₅	C₂×C₄×Dic₅	1	1	yes	characteristic
182	C₂×C₅	C₂×C₄⋊Dic₅	1	1	yes	characteristic
183	C₂×C₅	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
184	C₂×C₅	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
185	C₂×C₅	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
186	C₂×C₅	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
187	D₅	#3963	1	4	yes	complemented
188	D₅	C₂×#683	1	2	yes	—
189	D₅	C₂×#685	1	2	yes	—
190	C₂³	—	5	4	no	—
191	C₂³	—	5	2	no	—
192	C₂³	—	5	2	no	—
193	C₂³	—	5	4	no	—
194	C₂³	—	5	1	no	—
195	C₂³	—	5	1	no	—
196	C₂³	C₂×#1042	1	1	yes	characteristic
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	5	2	no	—
199	C₂×C₄	—	10	4	no	—
200	C₂×C₄	—	10	4	no	—
201	C₂×C₄	—	10	2	no	—
202	C₂×C₄	—	25	8	no	—
203	C₂×C₄	—	50	4	no	—
204	C₂×C₄	—	50	4	no	—
205	C₂×C₄	—	50	16	no	—
206	C₂×C₄	C₂×#1042	1	2	yes	complemented
207	C₅	—	4	1	no	—
208	C₅	C₂×#3963	1	1	yes	characteristic, complemented
209	C₅	C₂×(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic, complemented
210	C₂²	—	5	2	no	—
211	C₂²	—	5	2	no	—
212	C₂²	—	5	1	no	—
213	C₂²	—	5	2	no	—
214	C₂²	—	5	1	no	—
215	C₂²	—	5	2	no	—
216	C₂²	—	5	2	no	—
217	C₂²	—	5	8	no	—
218	C₂²	—	5	8	no	—
219	C₂²	C₄×#1042	1	2	yes	—
220	C₂²	C₄⋊#1042	1	2	yes	—
221	C₂²	C₂²⋊#1042	1	2	yes	—
222	C₂²	C₂²×#1042	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	10	2	no	—
225	C₄	—	50	8	no	—
226	C₂	—	5	2	no	—
227	C₂	—	5	4	no	—
228	C₂	—	5	2	no	—
229	C₂	#11716849	1	4	yes	direct product factor
230	C₂	C₂×C₄×#1042	1	1	yes	characteristic
231	C₂	C₂×C₄⋊#1042	1	1	yes	characteristic
232	C₂	C₂×C₂²⋊#1042	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#1042
commutator	C₂×C₅²	C₂×C₄²
Frattini	C₂²	C₂²×#1042
Fitting	C₂²×C₄×C₅²	C₄
socle	C₂³×C₅²	C₂×C₄
prime radical	C₂³×C₅×D₅	C₂²
cosocle	C₂²×C₅×D₅	C₂³
hypercenter	C₂²×C₄	#1042
hypocenter	C₅²	C₂×#95
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
3. #11716849	C₂	direct (N × H)
7. C₂²×#1042	C₄	semidirect
9. C₂²×C₄×C₅²	C₄	semidirect
10. C₂²×C₅×Dic₅	C₄	semidirect
12. C₂×C₄×C₅×D₅	C₄	semidirect
26. C₂×#1042	C₂×C₄	semidirect
28. C₂×C₄×C₅²	C₂×C₄	semidirect
30. C₂×C₅×Dic₅	C₂×C₄	semidirect
66. C₂²×C₄×C₅	Dic₅	semidirect
67. C₂²×C₄×C₅	AGL(1,𝔽₅)	semidirect
72. C₂²×Dic₅	Dic₅	semidirect
74. C₂×C₄×D₅	Dic₅	semidirect
77. C₂×C₅²	#95	semidirect
81. C₅×D₅	#95	semidirect
111. C₂×C₄×C₅	C₂×Dic₅	semidirect
112. C₂×C₄×C₅	C₂×AGL(1,𝔽₅)	semidirect
118. C₂×Dic₅	C₂×Dic₅	semidirect
124. C₅²	C₂×#95	semidirect
171. C₂²×C₄	#1042	semidirect
179. C₂×C₅	#3963	semidirect
180. C₂×C₅	(C₂×AGL(1,𝔽₅))⋊C₄	semidirect
187. D₅	#3963	semidirect
206. C₂×C₄	C₂×#1042	semidirect
208. C₅	C₂×#3963	semidirect
209. C₅	C₂×(C₂×AGL(1,𝔽₅))⋊C₄	semidirect
229. C₂	#11716849	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#1042	2
C₂²×C₄×C₅×D₅	2
#11716849	2
C₂×(C₂×AGL(1,𝔽₅))⋊C₄	5
C₂×#3963	5

Subgroups — C2×#11716849 #13104648

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11716849 #13104648