Subgroups — C₄×#87902 #13107076

Order 1600.

1240 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	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×C₄×#1039	C₂	1	2	yes	maximal, complemented
6	#11717212	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	C₄²×C₅⋊D₅	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	direct product factor
18	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₅⋊D₅	C₂²	1	1	yes	characteristic
21	C₂×#6045	C₂²	1	1	yes	characteristic
22	C₅²×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₅²×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₄²×C₅²	C₂²	1	1	yes	characteristic
25	C₂²⋊#1039	C₂²	1	2	yes	—
26	#87808	C₂²	1	2	yes	—
27	C₄⋊#1039	C₂²	1	1	yes	characteristic
28	#87806	C₂²	1	2	yes	—
29	#87806	C₂²	1	2	yes	—
30	C₄×#1039	C₂²	1	2	yes	—
31	C₄×#1039	C₂²	1	1	yes	characteristic
32	C₄×#1039	C₂²	1	1	yes	characteristic
33	C₄×#1039	C₂²	1	1	yes	characteristic
34	C₄×#1039	C₂²	1	2	yes	—
35	C₄×#1039	C₂²	1	2	yes	—
36	C₄×#711	—	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	—	2	4	no	—
41	C₂×C₄×C₅²	C₂³	1	2	yes	—
42	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
43	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
44	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
45	C₂²⋊(C₅⋊D₅)	C₂×C₄	1	8	yes	complemented
46	C₂×#1039	—	2	4	no	—
47	C₂×#1039	C₂×C₄	1	8	yes	complemented
48	C₂×#1039	C₂³	1	2	yes	—
49	C₂×#1039	C₂³	1	1	yes	characteristic
50	C₂×#1039	C₂³	1	2	yes	—
51	C₂×#1039	C₂³	1	1	yes	characteristic
52	C₂×#1039	C₂³	1	1	yes	characteristic
53	C₄×C₅⋊D₅	—	2	2	no	—
54	C₄×C₅⋊D₅	—	2	2	no	—
55	C₄×C₅⋊D₅	C₂×C₄	1	4	yes	complemented
56	#6045	C₂×C₄	1	4	yes	complemented
57	C₂×#711	—	5	6	no	—
58	C₄×C₅×D₄	D₅	1	6	yes	complemented
59	D₄×Dic₅	—	5	6	no	—
60	C₄×C₂²⋊D₅	—	5	12	no	—
61	C₂×C₄×Dic₅	—	5	12	no	—
62	#4038	—	5	6	no	—
63	#4033	—	5	6	no	—
64	#4027	—	5	12	no	—
65	#4023	—	5	12	no	—
66	C₄²×D₅	—	5	6	no	—
67	C₄×Dic₁₀	—	5	6	no	—
68	C₂²×C₅²	—	2	2	no	—
69	C₂²×C₅²	—	2	2	no	—
70	C₂²×C₅²	C₂²×C₄	1	4	yes	—
71	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
72	C₂×C₅⋊D₅	—	2	1	no	—
73	C₂×C₅⋊D₅	—	2	1	no	—
74	C₂×C₅⋊D₅	C₂²×C₄	1	2	yes	—
75	C₄×C₅²	—	2	1	no	—
76	C₄×C₅²	—	2	2	no	—
77	C₄×C₅²	C₂²×C₄	1	2	yes	—
78	C₄×C₅²	#41	1	2	yes	complemented
79	#1039	—	2	1	no	—
80	#1039	—	2	2	no	—
81	#1039	C₂²×C₄	1	4	yes	—
82	#1039	C₂²×C₄	1	2	yes	—
83	#1039	#41	1	2	yes	complemented
84	C₂×C₅×D₄	C₂×D₅	1	6	yes	—
85	C₂²×C₄×C₅	C₂×D₅	1	12	yes	—
86	C₂×C₂²⋊D₅	—	5	12	no	—
87	C₂²×Dic₅	—	5	12	no	—
88	C₂²×Dic₅	—	5	12	no	—
89	#711	—	5	48	no	—
90	C₂×C₄×D₅	—	5	6	no	—
91	C₂×C₄×D₅	—	5	6	no	—
92	C₂×C₄×D₅	—	5	6	no	—
93	C₂×Dic₁₀	—	5	6	no	—
94	C₅×C₄⋊C₄	C₂×D₅	1	6	yes	—
95	C₅×C₂²⋊C₄	C₂×D₅	1	12	yes	—
96	C₄²×C₅	C₂×D₅	1	6	yes	—
97	C₂²⋊Dic₅	—	5	12	no	—
98	#685	—	5	12	no	—
99	C₄⋊Dic₅	—	5	6	no	—
100	#683	—	5	12	no	—
101	#683	—	5	12	no	—
102	C₄×Dic₅	—	5	12	no	—
103	C₄×Dic₅	—	5	6	no	—
104	C₄×Dic₅	—	5	6	no	—
105	C₄×Dic₅	—	5	6	no	—
106	C₄×Dic₅	—	5	12	no	—
107	C₄×Dic₅	—	5	12	no	—
108	C₄×#41	—	25	1	no	Sylow
109	C₂×C₅²	—	2	4	no	—
110	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₅²	C₂×#41	1	1	yes	characteristic
112	C₂×C₅²	C₂×#41	1	1	yes	characteristic
113	C₅⋊D₅	—	2	2	no	—
114	C₂³×C₅	C₂²×D₅	1	12	yes	—
115	C₂²×D₅	—	5	6	no	—
116	C₅×D₄	C₄×D₅	1	24	yes	complemented
117	C₂×C₄×C₅	—	2	24	no	—
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	C₂×C₄×C₅	C₂²×D₅	1	6	yes	—
122	C₂²⋊D₅	—	5	48	no	—
123	C₂×Dic₅	—	5	48	no	—
124	C₂×Dic₅	—	5	6	no	—
125	C₂×Dic₅	—	5	6	no	—
126	C₂×Dic₅	—	5	12	no	—
127	C₂×Dic₅	—	5	6	no	—
128	C₂×Dic₅	—	5	12	no	—
129	C₂×Dic₅	—	10	24	no	—
130	C₄×D₅	—	5	24	no	—
131	C₄×D₅	—	10	12	no	—
132	C₄×D₅	—	10	12	no	—
133	Dic₁₀	—	5	24	no	—
134	C₂×#41	—	25	1	no	—
135	C₄×Q₈	—	25	1	no	—
136	C₄×Q₈	—	25	1	no	—
137	C₄×D₄	—	25	1	no	—
138	C₄×D₄	—	25	2	no	—
139	C₄×D₄	—	25	2	no	—
140	C₄×D₄	C₅⋊D₅	1	1	yes	characteristic, complemented
141	#117	—	25	1	no	—
142	#117	—	25	2	no	—
143	C₂×C₄²	—	25	2	no	—
144	C₂×C₄²	—	25	1	no	—
145	C₅²	C₄×#41	1	1	yes	characteristic, complemented, Sylow
146	C₂²×C₅	—	2	12	no	—
147	C₂²×C₅	—	2	12	no	—
148	C₂²×C₅	C₂×C₄×D₅	1	24	yes	—
149	C₂²×C₅	C₂³×D₅	1	6	yes	—
150	C₂×D₅	—	5	12	no	—
151	C₂×D₅	—	10	6	no	—
152	C₂×D₅	—	10	6	no	—
153	C₄×C₅	—	2	12	no	—
154	C₄×C₅	—	2	6	no	—
155	C₄×C₅	C₂×C₄×D₅	1	12	yes	—
156	C₄×C₅	#711	1	12	yes	complemented
157	Dic₅	—	5	24	no	—
158	Dic₅	—	5	12	no	—
159	Dic₅	—	5	12	no	—
160	Dic₅	—	10	6	no	—
161	Dic₅	—	10	12	no	—
162	#41	—	25	8	no	—
163	C₂×Q₈	—	25	1	no	—
164	C₂×D₄	—	25	2	no	—
165	C₂×D₄	C₂×C₅⋊D₅	1	1	yes	characteristic
166	C₂²×C₄	—	25	2	no	—
167	C₂²×C₄	—	25	1	no	—
168	C₂²×C₄	—	25	2	no	—
169	C₂²×C₄	—	25	1	no	—
170	C₂²×C₄	—	25	1	no	—
171	C₂²×C₄	C₂×C₅⋊D₅	1	2	yes	—
172	C₄⋊C₄	—	25	2	no	—
173	C₄⋊C₄	—	25	2	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	2	no	—
178	C₂²⋊C₄	C₂×C₅⋊D₅	1	2	yes	—
179	C₄²	—	25	2	no	—
180	C₄²	—	25	1	no	—
181	C₄²	—	25	2	no	—
182	C₄²	—	25	1	no	—
183	C₄²	—	25	1	no	—
184	C₄²	—	25	2	no	—
185	C₄²	C₂×C₅⋊D₅	1	1	yes	characteristic
186	C₂×C₅	—	2	24	no	—
187	C₂×C₅	C₂²×C₄×D₅	1	6	yes	—
188	C₂×C₅	C₂×#711	1	6	yes	—
189	C₂×C₅	D₅×#41	1	6	yes	—
190	D₅	—	10	12	no	—
191	C₂³	—	25	1	no	—
192	C₂³	C₂²×C₅⋊D₅	1	2	yes	—
193	Q₈	—	25	4	no	—
194	D₄	—	25	8	no	—
195	D₄	C₄×C₅⋊D₅	1	4	yes	complemented
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	25	8	no	—
198	C₂×C₄	—	25	1	no	—
199	C₂×C₄	—	25	1	no	—
200	C₂×C₄	—	25	2	no	—
201	C₂×C₄	—	25	4	no	—
202	C₂×C₄	—	25	1	no	—
203	C₂×C₄	—	25	2	no	—
204	C₂×C₄	—	50	2	no	—
205	C₂×C₄	—	50	4	no	—
206	C₂×C₄	—	50	2	no	—
207	C₂×C₄	C₂²×C₅⋊D₅	1	2	yes	—
208	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
209	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
210	C₂×C₄	C₂²×C₅⋊D₅	1	1	yes	characteristic
211	C₅	C₄×#711	1	6	yes	complemented
212	C₂²	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	25	2	no	—
215	C₂²	—	50	1	no	—
216	C₂²	—	50	1	no	—
217	C₂²	C₂×C₄×C₅⋊D₅	1	4	yes	—
218	C₂²	C₂³×C₅⋊D₅	1	1	yes	characteristic
219	C₄	—	2	2	no	—
220	C₄	—	2	1	no	—
221	C₄	—	25	2	no	—
222	C₄	—	25	4	no	—
223	C₄	—	25	2	no	—
224	C₄	—	50	1	no	—
225	C₄	—	50	2	no	—
226	C₄	C₂×C₄×C₅⋊D₅	1	2	yes	—
227	C₄	#87902	1	2	yes	direct product factor
228	C₂	—	2	4	no	—
229	C₂	—	50	2	no	—
230	C₂	C₂²×C₄×C₅⋊D₅	1	1	yes	characteristic
231	C₂	C₂×#87902	1	1	yes	characteristic
232	C₂	#41×C₅⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₅⋊D₅
commutator	C₂×C₅²	C₂³×C₄
Frattini	C₂²	C₂³×C₅⋊D₅
Fitting	C₄×C₅²×D₄	C₂
socle, cosocle	C₂²×C₅²	C₂⁴
prime radical	C₂×#87902	C₂
hypercenter	C₄×D₄	C₅⋊D₅
hypocenter	C₅²	C₄×#41
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂×C₄ ◁ 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×C₄×#1039	C₂	semidirect
6. #11717212	C₂	semidirect
7. #11717207	C₂	semidirect
8. #11717201	C₂	semidirect
9. #11717197	C₂	semidirect
10. C₄²×C₅⋊D₅	C₂	semidirect
11. C₄×#6045	C₂	semidirect
17. #87902	C₄	direct (N × H)
39. C₅²×D₄	C₂×C₄	semidirect
45. C₂²⋊(C₅⋊D₅)	C₂×C₄	semidirect
47. C₂×#1039	C₂×C₄	semidirect
55. C₄×C₅⋊D₅	C₂×C₄	semidirect
56. #6045	C₂×C₄	semidirect
58. C₄×C₅×D₄	D₅	semidirect
78. C₄×C₅²	#41	semidirect
83. #1039	#41	semidirect
116. C₅×D₄	C₄×D₅	semidirect
140. C₄×D₄	C₅⋊D₅	semidirect
145. C₅²	C₄×#41	semidirect
156. C₄×C₅	#711	semidirect
195. D₄	C₄×C₅⋊D₅	semidirect
211. C₅	C₄×#711	semidirect
227. C₄	#87902	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×#87902	2
C₄×C₅²×D₄	2
D₄×#1039	2
C₄×C₂²⋊(C₅⋊D₅)	2
C₂×C₄×#1039	2
#11717212	2
#11717207	2
#11717201	2
#11717197	2
C₄²×C₅⋊D₅	2
C₄×#6045	2
C₄×#711	5

Subgroups — C4×#87902 #13107076

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#87902 #13107076