Subgroups — #13107077

Order 1600.

1176 conjugacy classes of subgroups.

The subgroup lattice has 231 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#87902	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅²×D₄	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×#1039	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₂²⋊(C₅⋊D₅)	C₂	1	2	yes	maximal, complemented
5	C₂×#87806	C₂	1	2	yes	maximal, complemented
6	C₄⋊C₄×C₅⋊D₅	C₂	1	1	yes	maximal, characteristic, complemented
7	#11717207	C₂	1	1	yes	maximal, characteristic, complemented
8	#11717201	C₂	1	2	yes	maximal, complemented
9	#11717197	C₂	1	2	yes	maximal, complemented
10	#11717192	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#6045	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₅²×D₄	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₅²	C₂²	1	2	yes	—
14	C₂×C₂²⋊(C₅⋊D₅)	C₂²	1	2	yes	—
15	C₂²×#1039	C₂²	1	2	yes	—
16	C₂²×#1039	C₂²	1	2	yes	—
17	#87902	C₄	1	8	yes	complemented
18	C₂×C₄×C₅⋊D₅	C₂²	1	2	yes	—
19	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
20	C₂×#6045	C₂²	1	1	yes	characteristic
21	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₅²×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₄²×C₅²	C₂²	1	1	yes	characteristic
24	C₂²⋊#1039	C₂²	1	2	yes	—
25	#87808	C₂²	1	1	yes	characteristic
26	#87808	C₂²	1	1	yes	characteristic
27	C₄⋊#1039	C₂²	1	1	yes	characteristic
28	#87806	C₂²	1	2	yes	—
29	#87806	C₂²	1	1	yes	characteristic
30	#87806	C₂²	1	1	yes	characteristic
31	#87806	C₂²	1	2	yes	—
32	#87806	C₂²	1	2	yes	—
33	C₄×#1039	C₂²	1	2	yes	—
34	C₄×#1039	C₂²	1	1	yes	characteristic
35	C₄×#1039	C₂²	1	2	yes	—
36	#65880	—	5	6	no	maximal
37	C₂³×C₅²	C₂³	1	2	yes	—
38	C₂²×C₅⋊D₅	C₂³	1	1	yes	characteristic
39	C₅²×D₄	C₂×C₄	1	4	yes	complemented
40	C₂×C₄×C₅²	—	4	2	no	—
41	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
43	C₂×C₄×C₅²	C₂³	1	2	yes	—
44	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
45	C₂²⋊(C₅⋊D₅)	C₂×C₄	1	8	yes	complemented
46	C₂×#1039	—	4	2	no	—
47	C₂×#1039	C₂×C₄	1	8	yes	complemented
48	C₂×#1039	C₂³	1	1	yes	characteristic
49	C₂×#1039	C₂³	1	1	yes	characteristic
50	C₂×#1039	C₂³	1	2	yes	—
51	C₂×#1039	C₂³	1	2	yes	—
52	C₂×#1039	C₂³	1	1	yes	characteristic
53	C₄×C₅⋊D₅	—	4	2	no	—
54	C₄×C₅⋊D₅	C₂×C₄	1	4	yes	complemented
55	#6045	C₂×C₄	1	4	yes	complemented
56	C₂×#711	—	5	6	no	—
57	C₄×C₅×D₄	D₅	1	6	yes	complemented
58	D₄×Dic₅	—	5	6	no	—
59	C₄×C₂²⋊D₅	—	5	12	no	—
60	C₂×#683	—	5	12	no	—
61	D₅×C₄⋊C₄	—	5	6	no	—
62	#4033	—	5	6	no	—
63	#4027	—	5	12	no	—
64	#4023	—	5	12	no	—
65	#4018	—	5	6	no	—
66	C₄×Dic₁₀	—	5	6	no	—
67	C₂²×C₅²	—	2	2	no	—
68	C₂²×C₅²	—	2	2	no	—
69	C₂²×C₅²	C₂²×C₄	1	4	yes	—
70	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
71	C₂×C₅⋊D₅	—	2	1	no	—
72	C₂×C₅⋊D₅	—	2	1	no	—
73	C₂×C₅⋊D₅	C₂²×C₄	1	2	yes	—
74	C₄×C₅²	—	2	2	no	—
75	C₄×C₅²	—	2	1	no	—
76	C₄×C₅²	—	2	1	no	—
77	C₄×C₅²	C₂²×C₄	1	2	yes	—
78	#1039	—	2	2	no	—
79	#1039	—	2	1	no	—
80	#1039	—	2	1	no	—
81	#1039	C₂²×C₄	1	4	yes	—
82	#1039	C₂²×C₄	1	2	yes	—
83	C₂×C₅×D₄	C₂×D₅	1	6	yes	—
84	C₂²×C₄×C₅	C₂×D₅	1	12	yes	—
85	C₂×C₂²⋊D₅	—	5	12	no	—
86	C₂²×Dic₅	—	5	12	no	—
87	C₂²×Dic₅	—	5	12	no	—
88	#711	—	5	48	no	—
89	C₂×C₄×D₅	—	5	6	no	—
90	C₂×C₄×D₅	—	5	12	no	—
91	C₂×Dic₁₀	—	5	6	no	—
92	C₅×C₄⋊C₄	C₂×D₅	1	6	yes	—
93	C₅×C₂²⋊C₄	C₂×D₅	1	12	yes	—
94	C₄²×C₅	C₂×D₅	1	6	yes	—
95	C₂²⋊Dic₅	—	5	12	no	—
96	#685	—	5	6	no	—
97	#685	—	5	6	no	—
98	C₄⋊Dic₅	—	5	6	no	—
99	#683	—	5	12	no	—
100	#683	—	5	6	no	—
101	#683	—	5	6	no	—
102	#683	—	5	12	no	—
103	#683	—	5	12	no	—
104	C₄×Dic₅	—	5	12	no	—
105	C₄×Dic₅	—	5	6	no	—
106	C₄×Dic₅	—	5	12	no	—
107	#520	—	25	1	no	Sylow
108	C₂×C₅²	—	2	4	no	—
109	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
111	C₂×C₅²	2₋¹⁺⁴	1	1	yes	characteristic
112	C₅⋊D₅	—	2	2	no	—
113	C₂³×C₅	C₂²×D₅	1	12	yes	—
114	C₂²×D₅	—	5	6	no	—
115	C₅×D₄	C₄×D₅	1	24	yes	complemented
116	C₂×C₄×C₅	—	4	12	no	—
117	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
118	C₂×C₄×C₅	C₂²×D₅	1	12	yes	—
119	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
120	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
121	C₂²⋊D₅	—	5	48	no	—
122	C₂×Dic₅	—	5	48	no	—
123	C₂×Dic₅	—	5	12	no	—
124	C₂×Dic₅	—	5	6	no	—
125	C₂×Dic₅	—	5	6	no	—
126	C₂×Dic₅	—	5	6	no	—
127	C₂×Dic₅	—	5	12	no	—
128	C₂×Dic₅	—	20	12	no	—
129	C₄×D₅	—	5	24	no	—
130	C₄×D₅	—	20	12	no	—
131	Dic₁₀	—	5	24	no	—
132	C₂×#41	—	25	1	no	—
133	C₄×Q₈	—	25	1	no	—
134	C₄×Q₈	—	25	1	no	—
135	C₄×D₄	—	25	1	no	—
136	C₄×D₄	—	25	2	no	—
137	C₄×D₄	—	25	2	no	—
138	C₄×D₄	C₅⋊D₅	1	1	yes	characteristic, complemented
139	#117	—	25	2	no	—
140	#117	—	25	1	no	—
141	C₂×C₄⋊C₄	—	25	2	no	—
142	C₂×C₄⋊C₄	—	25	1	no	—
143	C₅²	#520	1	1	yes	characteristic, complemented, Sylow
144	C₂²×C₅	—	2	12	no	—
145	C₂²×C₅	—	2	12	no	—
146	C₂²×C₅	C₂×C₄×D₅	1	24	yes	—
147	C₂²×C₅	C₂³×D₅	1	6	yes	—
148	C₂×D₅	—	5	12	no	—
149	C₂×D₅	—	10	6	no	—
150	C₂×D₅	—	10	6	no	—
151	C₄×C₅	—	2	12	no	—
152	C₄×C₅	—	2	6	no	—
153	C₄×C₅	—	2	6	no	—
154	C₄×C₅	C₂×C₄×D₅	1	12	yes	—
155	Dic₅	—	5	24	no	—
156	Dic₅	—	5	12	no	—
157	Dic₅	—	10	12	no	—
158	Dic₅	—	10	6	no	—
159	Dic₅	—	10	6	no	—
160	#41	—	25	8	no	—
161	C₂×Q₈	—	25	1	no	—
162	C₂×D₄	—	25	2	no	—
163	C₂×D₄	C₂×C₅⋊D₅	1	1	yes	characteristic
164	C₂²×C₄	—	25	2	no	—
165	C₂²×C₄	—	25	1	no	—
166	C₂²×C₄	—	25	2	no	—
167	C₂²×C₄	—	25	2	no	—
168	C₂²×C₄	C₂×C₅⋊D₅	1	2	yes	—
169	C₄⋊C₄	—	25	2	no	—
170	C₄⋊C₄	—	25	1	no	—
171	C₄⋊C₄	—	25	2	no	—
172	C₄⋊C₄	—	25	2	no	—
173	C₄⋊C₄	—	25	1	no	—
174	C₄⋊C₄	—	25	1	no	—
175	C₄⋊C₄	C₂×C₅⋊D₅	1	1	yes	characteristic
176	C₂²⋊C₄	—	25	2	no	—
177	C₂²⋊C₄	—	25	1	no	—
178	C₂²⋊C₄	—	25	1	no	—
179	C₂²⋊C₄	C₂×C₅⋊D₅	1	2	yes	—
180	C₄²	—	25	2	no	—
181	C₄²	—	25	2	no	—
182	C₄²	—	25	1	no	—
183	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
184	C₂×C₅	—	2	24	no	—
185	C₂×C₅	C₂²×C₄×D₅	1	6	yes	—
186	C₂×C₅	#4144	1	6	yes	—
187	C₂×C₅	#4150	1	6	yes	—
188	D₅	—	10	12	no	—
189	C₂³	—	25	1	no	—
190	C₂³	C₂²×C₅⋊D₅	1	2	yes	—
191	Q₈	—	25	4	no	—
192	D₄	—	25	8	no	—
193	D₄	C₄×C₅⋊D₅	1	4	yes	complemented
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	25	8	no	—
196	C₂×C₄	—	25	2	no	—
197	C₂×C₄	—	25	1	no	—
198	C₂×C₄	—	25	4	no	—
199	C₂×C₄	—	25	1	no	—
200	C₂×C₄	—	25	2	no	—
201	C₂×C₄	—	25	1	no	—
202	C₂×C₄	—	100	2	no	—
203	C₂×C₄	—	100	2	no	—
204	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
205	C₂×C₄	C₂²×C₅⋊D₅	1	2	yes	—
206	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
207	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
208	C₅	#65880	1	6	yes	complemented
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	25	2	no	—
212	C₂²	—	50	1	no	—
213	C₂²	—	50	1	no	—
214	C₂²	C₂×C₄×C₅⋊D₅	1	4	yes	—
215	C₂²	C₂³×C₅⋊D₅	1	1	yes	characteristic
216	C₄	—	2	1	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	1	no	—
219	C₄	—	25	2	no	—
220	C₄	—	25	4	no	—
221	C₄	—	50	2	no	—
222	C₄	—	50	1	no	—
223	C₄	—	50	1	no	—
224	C₄	C₂×C₄×C₅⋊D₅	1	2	yes	—
225	C₂	—	2	4	no	—
226	C₂	—	50	2	no	—
227	C₂	C₂²×C₄×C₅⋊D₅	1	1	yes	characteristic
228	C₂	#11717517	1	1	yes	characteristic
229	C₂	#11717523	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×C₅⋊D₅
commutator	C₂×C₅²	C₂³×C₄
Fitting	C₄×C₅²×D₄	C₂
socle, cosocle	C₂²×C₅²	C₂⁴
prime radical	C₂×#87902	C₂
hypercenter	C₄×D₄	C₅⋊D₅
hypocenter	C₅²	#520
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
2. C₄×C₅²×D₄	C₂	semidirect
3. D₄×#1039	C₂	semidirect
4. C₄×C₂²⋊(C₅⋊D₅)	C₂	semidirect
5. C₂×#87806	C₂	semidirect
6. C₄⋊C₄×C₅⋊D₅	C₂	semidirect
7. #11717207	C₂	semidirect
8. #11717201	C₂	semidirect
9. #11717197	C₂	semidirect
10. #11717192	C₂	semidirect
11. C₄×#6045	C₂	semidirect
17. #87902	C₄	semidirect
39. C₅²×D₄	C₂×C₄	semidirect
45. C₂²⋊(C₅⋊D₅)	C₂×C₄	semidirect
47. C₂×#1039	C₂×C₄	semidirect
54. C₄×C₅⋊D₅	C₂×C₄	semidirect
55. #6045	C₂×C₄	semidirect
57. C₄×C₅×D₄	D₅	semidirect
115. C₅×D₄	C₄×D₅	semidirect
138. C₄×D₄	C₅⋊D₅	semidirect
143. C₅²	#520	semidirect
193. D₄	C₄×C₅⋊D₅	semidirect
208. C₅	#65880	semidirect

Maximal subgroups

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

subgroup	index
C₂×#87902	2
C₄×C₅²×D₄	2
D₄×#1039	2
C₄×C₂²⋊(C₅⋊D₅)	2
C₂×#87806	2
C₄⋊C₄×C₅⋊D₅	2
#11717207	2
#11717201	2
#11717197	2
#11717192	2
C₄×#6045	2
#65880	5