Subgroups — C₂×C₅×#4092 #13106967

Order 1600.

800 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅²×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×C₅×D₅	C₂	1	1	yes	maximal, characteristic
3	C₅×#4092	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅×#685	C₂	1	2	yes	maximal
5	C₂×C₅×C₄⋊Dic₅	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×#683	C₂	1	2	yes	maximal, complemented
7	C₂³×C₅×D₅	C₂²	1	1	yes	characteristic
8	C₂×C₅²×Q₈	—	2	4	no	—
9	C₂×C₅²×Q₈	C₂²	1	4	yes	complemented
10	C₂²×C₄×C₅²	C₂²	1	2	yes	—
11	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
12	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
13	C₂²×C₅×Dic₅	C₂²	1	2	yes	—
14	C₂×C₄×C₅×D₅	—	2	4	no	—
15	C₂×C₄×C₅×D₅	C₂²	1	4	yes	—
16	C₅×#685	C₂²	1	8	yes	—
17	C₅×C₄⋊Dic₅	C₂²	1	4	yes	complemented
18	C₅×#683	C₂²	1	8	yes	complemented
19	C₂×C₅×C₂²⋊Q₈	—	5	1	no	maximal
20	C₂×#4092	C₅	1	1	yes	maximal, characteristic, direct product factor
21	C₂³×C₅²	C₂³	1	1	yes	characteristic
22	C₂²×C₅×D₅	—	2	2	no	—
23	C₂²×C₅×D₅	—	2	2	no	—
24	C₂²×C₅×D₅	Q₈	1	4	yes	complemented
25	C₂²×C₅×D₅	C₂³	1	2	yes	—
26	C₅²×Q₈	—	2	8	no	—
27	C₂×C₄×C₅²	—	2	4	no	—
28	C₂×C₄×C₅²	D₄	1	4	yes	—
29	C₂×C₄×C₅²	C₂³	1	4	yes	—
30	C₂×C₄×C₅²	C₂³	1	2	yes	—
31	C₂×C₅×Dic₅	—	2	2	no	—
32	C₂×C₅×Dic₅	—	2	4	no	—
33	C₂×C₅×Dic₅	C₂³	1	2	yes	—
34	C₂×C₅×Dic₅	C₂³	1	4	yes	—
35	C₄×C₅×D₅	—	2	8	no	—
36	C₂²×C₅×Q₈	—	2	2	no	—
37	C₂²×C₅×Q₈	D₅	1	1	yes	characteristic, complemented
38	C₂²×C₅×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
39	C₂³×C₄×C₅	—	5	1	no	—
40	C₂²×C₄×D₅	C₂×C₅	1	1	yes	characteristic
41	C₅×C₂²⋊Q₈	—	5	8	no	—
42	C₂×C₅×C₄⋊C₄	—	5	2	no	—
43	C₂×C₅×C₄⋊C₄	—	5	1	no	—
44	C₂×C₅×C₂²⋊C₄	—	5	2	no	—
45	#4092	C₂×C₅	1	8	yes	direct product factor
46	C₂×#685	C₂×C₅	1	2	yes	—
47	C₂×C₄⋊Dic₅	C₂×C₅	1	1	yes	characteristic, complemented
48	C₂×#683	C₂×C₅	1	2	yes	complemented
49	C₂²×C₅²	C₂×D₄	1	2	yes	—
50	C₂²×C₅²	C₂×Q₈	1	2	yes	—
51	C₂²×C₅²	#41	1	2	yes	—
52	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
53	C₂×C₅×D₅	—	2	8	no	—
54	C₂×C₅×D₅	—	2	2	no	—
55	C₂×C₅×D₅	—	2	2	no	—
56	C₂×C₅×D₅	C₂×Q₈	1	4	yes	complemented
57	C₄×C₅²	—	2	4	no	—
58	C₄×C₅²	C₂×D₄	1	4	yes	—
59	C₅×Dic₅	—	2	2	no	—
60	C₅×Dic₅	—	2	4	no	—
61	C₂⁴×C₅	—	5	1	no	—
62	C₂³×D₅	C₂²×C₅	1	1	yes	characteristic
63	C₂×C₅×Q₈	—	2	16	no	—
64	C₂×C₅×Q₈	—	2	4	no	—
65	C₂×C₅×Q₈	—	2	4	no	—
66	C₂×C₅×Q₈	—	2	8	no	—
67	C₂×C₅×Q₈	C₂×D₅	1	4	yes	complemented
68	C₂×C₅×Q₈	C₂²×C₅	1	4	yes	complemented
69	C₂²×C₄×C₅	—	2	4	no	—
70	C₂²×C₄×C₅	—	2	2	no	—
71	C₂²×C₄×C₅	—	5	2	no	—
72	C₂²×C₄×C₅	—	5	1	no	—
73	C₂²×C₄×C₅	—	5	4	no	—
74	C₂²×C₄×C₅	—	10	4	no	—
75	C₂²×C₄×C₅	C₂×D₅	1	2	yes	—
76	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂²×C₅	1	2	yes	—
78	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
79	C₂²×Dic₅	C₂²×C₅	1	1	yes	characteristic
80	C₂²×Dic₅	C₂²×C₅	1	2	yes	—
81	C₂×C₄×D₅	—	2	4	no	—
82	C₂×C₄×D₅	C₂²×C₅	1	4	yes	—
83	C₅×C₄⋊C₄	—	5	8	no	—
84	C₅×C₄⋊C₄	—	5	4	no	—
85	C₅×C₂²⋊C₄	—	5	8	no	—
86	#685	C₂²×C₅	1	8	yes	—
87	C₄⋊Dic₅	C₂²×C₅	1	4	yes	complemented
88	#683	C₂²×C₅	1	8	yes	complemented
89	C₂×C₂²⋊Q₈	—	5	1	no	Sylow
90	C₂×C₅²	C₂²⋊Q₈	1	4	yes	complemented
91	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₅²	C₂²×Q₈	1	1	yes	characteristic
93	C₂×C₅²	C₂×#41	1	1	yes	characteristic
94	C₅×D₅	—	2	4	no	—
95	C₂³×C₅	—	2	2	no	—
96	C₂³×C₅	—	5	4	no	—
97	C₂³×C₅	—	5	2	no	—
98	C₂³×C₅	—	10	2	no	—
99	C₂³×C₅	—	10	2	no	—
100	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
101	C₂³×C₅	C₂³×C₅	1	1	yes	characteristic
102	C₂²×D₅	—	2	2	no	—
103	C₂²×D₅	—	2	2	no	—
104	C₂²×D₅	C₅×Q₈	1	4	yes	complemented
105	C₂²×D₅	C₂³×C₅	1	2	yes	—
106	C₅×Q₈	—	2	32	no	—
107	C₅×Q₈	—	2	8	no	—
108	C₅×Q₈	—	2	8	no	—
109	C₂×C₄×C₅	—	2	16	no	—
110	C₂×C₄×C₅	—	2	4	no	—
111	C₂×C₄×C₅	—	2	4	no	—
112	C₂×C₄×C₅	—	2	8	no	—
113	C₂×C₄×C₅	—	2	8	no	—
114	C₂×C₄×C₅	—	2	4	no	—
115	C₂×C₄×C₅	—	5	4	no	—
116	C₂×C₄×C₅	—	5	2	no	—
117	C₂×C₄×C₅	—	10	4	no	—
118	C₂×C₄×C₅	—	10	2	no	—
119	C₂×C₄×C₅	—	10	8	no	—
120	C₂×C₄×C₅	C₂²⋊D₅	1	4	yes	—
121	C₂×C₄×C₅	C₅×D₄	1	4	yes	—
122	C₂×C₄×C₅	C₂²×D₅	1	4	yes	—
123	C₂×C₄×C₅	C₂²×D₅	1	2	yes	—
124	C₂×C₄×C₅	C₂³×C₅	1	4	yes	—
125	C₂×C₄×C₅	C₂³×C₅	1	2	yes	—
126	C₂×Dic₅	—	2	2	no	—
127	C₂×Dic₅	—	2	4	no	—
128	C₂×Dic₅	C₂³×C₅	1	2	yes	—
129	C₂×Dic₅	C₂³×C₅	1	4	yes	—
130	C₄×D₅	—	2	8	no	—
131	C₂²×Q₈	C₅×D₅	1	1	yes	characteristic, complemented
132	C₂³×C₄	—	5	1	no	—
133	C₂²⋊Q₈	—	5	8	no	—
134	C₂×C₄⋊C₄	—	5	2	no	—
135	C₂×C₄⋊C₄	—	5	1	no	—
136	C₂×C₂²⋊C₄	—	5	2	no	—
137	C₅²	C₂×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
138	C₂²×C₅	—	2	4	no	—
139	C₂²×C₅	—	2	4	no	—
140	C₂²×C₅	—	2	4	no	—
141	C₂²×C₅	—	2	2	no	—
142	C₂²×C₅	—	5	4	no	—
143	C₂²×C₅	—	10	2	no	—
144	C₂²×C₅	—	10	2	no	—
145	C₂²×C₅	—	10	8	no	—
146	C₂²×C₅	Q₈×D₅	1	2	yes	—
147	C₂²×C₅	#713	1	2	yes	—
148	C₂²×C₅	C₂×C₂²⋊D₅	1	2	yes	—
149	C₂²×C₅	C₂×C₅×D₄	1	2	yes	—
150	C₂²×C₅	C₂×C₅×Q₈	1	2	yes	—
151	C₂²×C₅	C₅×#41	1	2	yes	—
152	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
153	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
154	C₂×D₅	—	2	8	no	—
155	C₂×D₅	—	2	2	no	—
156	C₂×D₅	—	2	2	no	—
157	C₂×D₅	C₂×C₅×Q₈	1	4	yes	complemented
158	C₄×C₅	—	2	16	no	—
159	C₄×C₅	—	2	4	no	—
160	C₄×C₅	—	2	4	no	—
161	C₄×C₅	—	2	8	no	—
162	C₄×C₅	—	10	4	no	—
163	C₄×C₅	—	10	2	no	—
164	C₄×C₅	C₂×C₂²⋊D₅	1	4	yes	—
165	C₄×C₅	C₂×C₅×D₄	1	4	yes	—
166	Dic₅	—	2	2	no	—
167	Dic₅	—	2	4	no	—
168	C₂⁴	—	5	1	no	—
169	C₂×Q₈	—	2	4	no	—
170	C₂×Q₈	C₂×C₅×D₅	1	4	yes	complemented
171	C₂²×C₄	—	5	2	no	—
172	C₂²×C₄	—	5	1	no	—
173	C₂²×C₄	—	5	4	no	—
174	C₂²×C₄	—	10	4	no	—
175	C₂²×C₄	C₂×C₅×D₅	1	2	yes	—
176	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
177	C₄⋊C₄	—	5	8	no	—
178	C₄⋊C₄	—	5	4	no	—
179	C₂²⋊C₄	—	5	8	no	—
180	C₂×C₅	—	2	8	no	—
181	C₂×C₅	—	2	2	no	—
182	C₂×C₅	—	2	2	no	—
183	C₂×C₅	—	2	2	no	—
184	C₂×C₅	—	10	4	no	—
185	C₂×C₅	#4092	1	4	yes	direct product factor
186	C₂×C₅	C₅×C₂²⋊Q₈	1	4	yes	complemented
187	C₂×C₅	C₂×Q₈×D₅	1	1	yes	characteristic
188	C₂×C₅	C₂×#713	1	1	yes	characteristic
189	C₂×C₅	C₂²×C₂²⋊D₅	1	1	yes	characteristic
190	C₂×C₅	C₂²×C₅×D₄	1	1	yes	characteristic
191	C₂×C₅	C₂²×C₅×Q₈	1	1	yes	characteristic
192	C₂×C₅	C₂×C₅×#41	1	1	yes	characteristic
193	D₅	—	2	4	no	—
194	C₂³	—	5	4	no	—
195	C₂³	—	5	2	no	—
196	C₂³	—	10	2	no	—
197	C₂³	—	10	2	no	—
198	C₂³	C₂²×C₅×D₅	1	1	yes	characteristic
199	Q₈	—	2	8	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	5	4	no	—
202	C₂×C₄	—	5	2	no	—
203	C₂×C₄	—	10	4	no	—
204	C₂×C₄	—	10	2	no	—
205	C₂×C₄	—	10	8	no	—
206	C₂×C₄	C₅×C₂²⋊D₅	1	4	yes	—
207	C₂×C₄	C₂²×C₅×D₅	1	4	yes	—
208	C₂×C₄	C₂²×C₅×D₅	1	2	yes	—
209	C₅	—	2	2	no	—
210	C₅	C₂×#4092	1	1	yes	characteristic, direct product factor
211	C₅	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic, complemented
212	C₂²	—	5	4	no	—
213	C₂²	—	10	8	no	—
214	C₂²	—	10	2	no	—
215	C₂²	—	10	2	no	—
216	C₂²	C₅×Q₈×D₅	1	2	yes	—
217	C₂²	C₅×#713	1	2	yes	—
218	C₂²	C₂×C₅×C₂²⋊D₅	1	2	yes	—
219	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
220	C₄	—	2	4	no	—
221	C₄	—	10	4	no	—
222	C₄	—	10	2	no	—
223	C₄	C₂×C₅×C₂²⋊D₅	1	4	yes	—
224	C₂	—	10	4	no	—
225	C₂	C₅×#4092	1	4	yes	direct product factor
226	C₂	C₂×C₅×Q₈×D₅	1	1	yes	characteristic
227	C₂	C₂×C₅×#713	1	1	yes	characteristic
228	C₂	C₂²×C₅×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₅	C₂²×D₅
commutator, cosocle	C₂²×C₅	C₂⁴×C₅
Frattini	C₂²	C₂³×C₅×D₅
Fitting	C₂²×C₅²×Q₈	C₂
socle	C₂³×C₅²	C₂³
prime radical	C₂³×C₅×D₅	C₂²
hypercenter	C₂²×C₅×Q₈	D₅
hypocenter	C₅	C₂×C₅×C₂²⋊Q₈
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂³×C₅ ◁ C₂²×C₅×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅²×Q₈	C₂	semidirect
3. C₅×#4092	C₂	direct (N × H)
5. C₂×C₅×C₄⋊Dic₅	C₂	semidirect
6. C₂×C₅×#683	C₂	semidirect
9. C₂×C₅²×Q₈	C₂²	semidirect
17. C₅×C₄⋊Dic₅	C₂²	semidirect
18. C₅×#683	C₂²	semidirect
20. C₂×#4092	C₅	direct (N × H)
24. C₂²×C₅×D₅	Q₈	semidirect
37. C₂²×C₅×Q₈	D₅	semidirect
38. C₂²×C₅×Q₈	C₂×C₅	semidirect
45. #4092	C₂×C₅	direct (N × H)
47. C₂×C₄⋊Dic₅	C₂×C₅	semidirect
48. C₂×#683	C₂×C₅	semidirect
56. C₂×C₅×D₅	C₂×Q₈	semidirect
67. C₂×C₅×Q₈	C₂×D₅	semidirect
68. C₂×C₅×Q₈	C₂²×C₅	semidirect
87. C₄⋊Dic₅	C₂²×C₅	semidirect
88. #683	C₂²×C₅	semidirect
90. C₂×C₅²	C₂²⋊Q₈	semidirect
104. C₂²×D₅	C₅×Q₈	semidirect
131. C₂²×Q₈	C₅×D₅	semidirect
137. C₅²	C₂×C₂²⋊Q₈	semidirect
157. C₂×D₅	C₂×C₅×Q₈	semidirect
170. C₂×Q₈	C₂×C₅×D₅	semidirect
185. C₂×C₅	#4092	direct (N × H)
186. C₂×C₅	C₅×C₂²⋊Q₈	semidirect
210. C₅	C₂×#4092	direct (N × H)
211. C₅	C₂×C₅×C₂²⋊Q₈	semidirect
225. C₂	C₅×#4092	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×C₅²×Q₈	2
C₂²×C₄×C₅×D₅	2
C₅×#4092	2
C₂×C₅×#685	2
C₂×C₅×C₄⋊Dic₅	2
C₂×C₅×#683	2
C₂×C₅×C₂²⋊Q₈	5
C₂×#4092	5

Subgroups — C2×C5×#4092 #13106967

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#4092 #13106967