Subgroups — C₂×C₅×#3952 #13102368

Order 1600.

480 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	C₂²×C₄×C₅×D₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅²×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₅×#168	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#3952	C₂	1	4	yes	maximal, direct product factor
5	C₂³×C₅×D₅	C₄	1	1	yes	characteristic
6	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
7	C₂²×C₅×Dic₅	C₄	1	1	yes	characteristic
8	C₂×C₄×C₅×D₅	—	2	2	no	—
9	C₂×C₄×C₅×D₅	—	2	2	no	—
10	C₂×C₄×C₅×D₅	C₄	1	2	yes	—
11	C₂×C₄×C₅×D₅	C₂²	1	2	yes	—
12	C₂×C₅²×C₈	—	2	2	no	—
13	C₂×C₅²×C₈	C₂²	1	2	yes	complemented
14	C₂×C₅×#168	—	2	2	no	—
15	C₂×C₅×#168	C₂²	1	2	yes	complemented
16	C₂×C₅×C₂²⋊C₈	—	5	1	no	maximal
17	C₂×#3952	C₅	1	1	yes	maximal, characteristic, direct product factor
18	C₂³×C₅²	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₅×D₅	—	2	2	no	—
20	C₂²×C₅×D₅	—	2	2	no	—
21	C₂²×C₅×D₅	C₈	1	4	yes	complemented
22	C₂²×C₅×D₅	C₂×C₄	1	2	yes	—
23	C₂×C₄×C₅²	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₅²	D₄	1	2	yes	—
25	C₂×C₄×C₅²	D₄	1	2	yes	—
26	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
27	C₂×C₅×Dic₅	—	2	2	no	—
28	C₂×C₅×Dic₅	C₂×C₄	1	2	yes	—
29	C₄×C₅×D₅	—	2	4	no	—
30	C₄×C₅×D₅	—	2	2	no	—
31	C₄×C₅×D₅	—	2	2	no	—
32	C₅²×C₈	—	2	2	no	—
33	C₅×#168	—	2	2	no	—
34	C₂³×C₄×C₅	—	5	1	no	—
35	C₂²×C₄×D₅	C₂×C₅	1	1	yes	characteristic
36	C₂²×C₅×C₈	—	2	2	no	—
37	C₂²×C₅×C₈	—	5	1	no	—
38	C₂²×C₅×C₈	D₅	1	1	yes	characteristic, complemented
39	C₂²×C₅×C₈	C₂×C₅	1	1	yes	characteristic, complemented
40	C₂²×#168	C₂×C₅	1	1	yes	characteristic, complemented
41	C₅×C₂²⋊C₈	—	5	4	no	—
42	#3952	C₂×C₅	1	4	yes	direct product factor
43	C₂²×C₅²	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₅²	C₂×C₈	1	2	yes	—
45	C₂²×C₅²	M₁₆	1	2	yes	—
46	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₅×D₅	—	2	8	no	—
48	C₂×C₅×D₅	—	2	2	no	—
49	C₂×C₅×D₅	—	2	2	no	—
50	C₂×C₅×D₅	C₂×C₈	1	4	yes	complemented
51	C₄×C₅²	C₂²⋊C₄	1	2	yes	—
52	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
53	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
54	C₅×Dic₅	—	2	2	no	—
55	C₂⁴×C₅	—	5	1	no	—
56	C₂³×D₅	C₄×C₅	1	1	yes	characteristic
57	C₂²×C₄×C₅	—	2	2	no	—
58	C₂²×C₄×C₅	—	5	1	no	—
59	C₂²×C₄×C₅	—	5	2	no	—
60	C₂²×C₄×C₅	—	5	2	no	—
61	C₂²×C₄×C₅	—	10	2	no	—
62	C₂²×C₄×C₅	—	10	2	no	—
63	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂²×C₅	1	1	yes	characteristic
65	C₂²×Dic₅	C₄×C₅	1	1	yes	characteristic
66	C₂×C₄×D₅	—	2	2	no	—
67	C₂×C₄×D₅	—	2	2	no	—
68	C₂×C₄×D₅	C₄×C₅	1	2	yes	—
69	C₂×C₄×D₅	C₂²×C₅	1	2	yes	—
70	C₂×C₅×C₈	—	2	8	no	—
71	C₂×C₅×C₈	—	2	2	no	—
72	C₂×C₅×C₈	—	2	2	no	—
73	C₂×C₅×C₈	—	2	4	no	—
74	C₂×C₅×C₈	—	5	2	no	—
75	C₂×C₅×C₈	—	10	2	no	—
76	C₂×C₅×C₈	C₂×D₅	1	2	yes	complemented
77	C₂×C₅×C₈	C₂²×C₅	1	2	yes	complemented
78	C₂×#168	—	2	2	no	—
79	C₂×#168	C₂²×C₅	1	2	yes	complemented
80	C₂×C₂²⋊C₈	—	5	1	no	Sylow
81	C₂×C₅²	C₂²⋊C₈	1	4	yes	complemented
82	C₂×C₅²	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₅²	C₂²×C₈	1	1	yes	characteristic
84	C₂×C₅²	C₂×M₁₆	1	1	yes	characteristic
85	C₅×D₅	—	2	4	no	—
86	C₂³×C₅	—	2	2	no	—
87	C₂³×C₅	—	5	4	no	—
88	C₂³×C₅	—	5	2	no	—
89	C₂³×C₅	—	10	2	no	—
90	C₂³×C₅	—	10	2	no	—
91	C₂³×C₅	C₄×D₅	1	1	yes	characteristic
92	C₂³×C₅	C₂×C₄×C₅	1	1	yes	characteristic
93	C₂²×D₅	—	2	2	no	—
94	C₂²×D₅	—	2	2	no	—
95	C₂²×D₅	C₅×C₈	1	4	yes	complemented
96	C₂²×D₅	C₂×C₄×C₅	1	2	yes	—
97	C₂×C₄×C₅	—	2	2	no	—
98	C₂×C₄×C₅	—	2	4	no	—
99	C₂×C₄×C₅	—	2	4	no	—
100	C₂×C₄×C₅	—	2	2	no	—
101	C₂×C₄×C₅	—	5	2	no	—
102	C₂×C₄×C₅	—	10	2	no	—
103	C₂×C₄×C₅	—	10	4	no	—
104	C₂×C₄×C₅	—	10	2	no	—
105	C₂×C₄×C₅	—	10	2	no	—
106	C₂×C₄×C₅	C₄×D₅	1	1	yes	characteristic
107	C₂×C₄×C₅	D₂₀	1	2	yes	—
108	C₂×C₄×C₅	C₂²⋊D₅	1	2	yes	—
109	C₂×C₄×C₅	C₂×C₄×C₅	1	1	yes	characteristic
110	C₂×C₄×C₅	C₅×D₄	1	2	yes	—
111	C₂×C₄×C₅	C₅×D₄	1	2	yes	—
112	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
114	C₂×Dic₅	—	2	2	no	—
115	C₂×Dic₅	C₂×C₄×C₅	1	2	yes	—
116	C₄×D₅	—	2	4	no	—
117	C₄×D₅	—	2	2	no	—
118	C₄×D₅	—	2	2	no	—
119	C₅×C₈	—	2	8	no	—
120	C₅×C₈	—	2	2	no	—
121	C₅×C₈	—	2	2	no	—
122	C₅×C₈	—	10	2	no	—
123	#168	—	2	2	no	—
124	C₂³×C₄	—	5	1	no	—
125	C₂²×C₈	—	5	1	no	—
126	C₂²×C₈	C₅×D₅	1	1	yes	characteristic, complemented
127	C₂²⋊C₈	—	5	4	no	—
128	C₅²	C₂×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
129	C₂²×C₅	—	2	2	no	—
130	C₂²×C₅	—	2	4	no	—
131	C₂²×C₅	—	2	4	no	—
132	C₂²×C₅	—	2	4	no	—
133	C₂²×C₅	—	5	4	no	—
134	C₂²×C₅	—	10	8	no	—
135	C₂²×C₅	—	10	2	no	—
136	C₂²×C₅	—	10	2	no	—
137	C₂²×C₅	C₈×D₅	1	2	yes	—
138	C₂²×C₅	#676	1	2	yes	—
139	C₂²×C₅	#685	1	2	yes	—
140	C₂²×C₅	C₅×C₂²⋊C₄	1	2	yes	—
141	C₂²×C₅	C₂×C₅×C₈	1	2	yes	—
142	C₂²×C₅	C₅×M₁₆	1	2	yes	—
143	C₂²×C₅	C₂×C₄×D₅	1	1	yes	characteristic
144	C₂²×C₅	C₂²×C₄×C₅	1	1	yes	characteristic
145	C₂×D₅	—	2	2	no	—
146	C₂×D₅	—	2	2	no	—
147	C₂×D₅	—	2	8	no	—
148	C₂×D₅	C₂×C₅×C₈	1	4	yes	complemented
149	C₄×C₅	—	2	2	no	—
150	C₄×C₅	—	2	2	no	—
151	C₄×C₅	—	2	4	no	—
152	C₄×C₅	—	10	2	no	—
153	C₄×C₅	#685	1	2	yes	—
154	C₄×C₅	C₅×C₂²⋊C₄	1	2	yes	—
155	C₄×C₅	C₂×D₂₀	1	1	yes	characteristic
156	C₄×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
157	C₄×C₅	C₂×C₅×D₄	1	1	yes	characteristic
158	C₄×C₅	C₂×C₅×D₄	1	1	yes	characteristic
159	Dic₅	—	2	2	no	—
160	C₂⁴	—	5	1	no	—
161	C₂²×C₄	—	5	1	no	—
162	C₂²×C₄	—	5	2	no	—
163	C₂²×C₄	—	5	2	no	—
164	C₂²×C₄	—	10	2	no	—
165	C₂²×C₄	—	10	2	no	—
166	C₂²×C₄	C₂×C₅×D₅	1	1	yes	characteristic
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	5	2	no	—
169	C₂×C₈	—	10	2	no	—
170	C₂×C₈	C₂×C₅×D₅	1	2	yes	complemented
171	C₂×C₅	—	2	2	no	—
172	C₂×C₅	—	2	2	no	—
173	C₂×C₅	—	2	8	no	—
174	C₂×C₅	—	2	2	no	—
175	C₂×C₅	—	10	4	no	—
176	C₂×C₅	#3952	1	4	yes	direct product factor
177	C₂×C₅	C₅×C₂²⋊C₈	1	4	yes	complemented
178	C₂×C₅	C₂×C₈×D₅	1	1	yes	characteristic
179	C₂×C₅	C₂×#676	1	1	yes	characteristic
180	C₂×C₅	C₂×#685	1	1	yes	characteristic
181	C₂×C₅	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
182	C₂×C₅	C₂²×C₅×C₈	1	1	yes	characteristic
183	C₂×C₅	C₂×C₅×M₁₆	1	1	yes	characteristic
184	D₅	—	2	4	no	—
185	C₂³	—	5	2	no	—
186	C₂³	—	5	4	no	—
187	C₂³	—	10	2	no	—
188	C₂³	—	10	2	no	—
189	C₂³	C₄×C₅×D₅	1	1	yes	characteristic
190	C₂×C₄	—	5	2	no	—
191	C₂×C₄	—	10	2	no	—
192	C₂×C₄	—	10	2	no	—
193	C₂×C₄	—	10	2	no	—
194	C₂×C₄	—	10	4	no	—
195	C₂×C₄	C₄×C₅×D₅	1	1	yes	characteristic
196	C₂×C₄	C₅×D₂₀	1	2	yes	—
197	C₂×C₄	C₅×C₂²⋊D₅	1	2	yes	—
198	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
199	C₈	—	2	2	no	—
200	C₈	—	10	2	no	—
201	C₅	—	2	2	no	—
202	C₅	C₂×#3952	1	1	yes	characteristic, direct product factor
203	C₅	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic, complemented
204	C₂²	—	5	4	no	—
205	C₂²	—	10	2	no	—
206	C₂²	—	10	8	no	—
207	C₂²	—	10	2	no	—
208	C₂²	C₅×C₈×D₅	1	2	yes	—
209	C₂²	C₅×#676	1	2	yes	—
210	C₂²	C₅×#685	1	2	yes	—
211	C₂²	C₂×C₄×C₅×D₅	1	1	yes	characteristic
212	C₄	—	10	2	no	—
213	C₄	C₅×#685	1	2	yes	—
214	C₄	C₂×C₅×D₂₀	1	1	yes	characteristic
215	C₄	C₂×C₅×C₂²⋊D₅	1	1	yes	characteristic
216	C₂	—	10	4	no	—
217	C₂	C₅×#3952	1	4	yes	direct product factor
218	C₂	C₂×C₅×C₈×D₅	1	1	yes	characteristic
219	C₂	C₂×C₅×#676	1	1	yes	characteristic
220	C₂	C₂×C₅×#685	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₅
commutator	C₂×C₅	C₂²×C₅×C₈
Frattini	C₂×C₄	C₂²×C₅×D₅
Fitting	C₂²×C₅²×C₈	C₂
socle	C₂³×C₅²	C₂×C₄
prime radical	C₂³×C₅×D₅	C₄
cosocle	C₂×C₄×C₅	C₂³×C₅
hypercenter	C₂²×C₅×C₈	D₅
hypocenter	C₅	C₂×C₅×C₂²⋊C₈
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂²×C₅×C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×C₅²×C₈	C₂	semidirect
3. C₂²×C₅×#168	C₂	semidirect
4. C₅×#3952	C₂	direct (N × H)
13. C₂×C₅²×C₈	C₂²	semidirect
15. C₂×C₅×#168	C₂²	semidirect
17. C₂×#3952	C₅	direct (N × H)
21. C₂²×C₅×D₅	C₈	semidirect
38. C₂²×C₅×C₈	D₅	semidirect
39. C₂²×C₅×C₈	C₂×C₅	semidirect
40. C₂²×#168	C₂×C₅	semidirect
42. #3952	C₂×C₅	direct (N × H)
50. C₂×C₅×D₅	C₂×C₈	semidirect
76. C₂×C₅×C₈	C₂×D₅	semidirect
77. C₂×C₅×C₈	C₂²×C₅	semidirect
79. C₂×#168	C₂²×C₅	semidirect
81. C₂×C₅²	C₂²⋊C₈	semidirect
95. C₂²×D₅	C₅×C₈	semidirect
126. C₂²×C₈	C₅×D₅	semidirect
128. C₅²	C₂×C₂²⋊C₈	semidirect
148. C₂×D₅	C₂×C₅×C₈	semidirect
170. C₂×C₈	C₂×C₅×D₅	semidirect
176. C₂×C₅	#3952	direct (N × H)
177. C₂×C₅	C₅×C₂²⋊C₈	semidirect
202. C₅	C₂×#3952	direct (N × H)
203. C₅	C₂×C₅×C₂²⋊C₈	semidirect
217. C₂	C₅×#3952	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×C₄×C₅×D₅	2
C₂²×C₅²×C₈	2
C₂²×C₅×#168	2
C₅×#3952	2
C₂×C₅×C₂²⋊C₈	5
C₂×#3952	5

Subgroups — C2×C5×#3952 #13102368

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#3952 #13102368