Subgroups — C₂×C₅×#2219 #11956322

Order 1280.

682 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#467	C₂	1	2	yes	maximal
4	C₅×#2219	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×#367	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×#358	C₂	1	2	yes	maximal, complemented
7	C₂²×C₅×Q₁₆	C₂²	1	2	yes	—
8	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×C₅×C₂²⋊Q₈	—	2	2	no	—
10	C₂×C₅×Q₃₂	—	2	4	no	—
11	C₂×C₅×Q₃₂	C₂²	1	4	yes	complemented
12	C₂×C₅×M₃₂	—	2	4	no	—
13	C₂×C₅×M₃₂	C₂²	1	4	yes	—
14	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
16	C₅×#467	—	2	4	no	—
17	C₅×#467	C₂²	1	8	yes	—
18	C₂×C₅×#107	C₂²	1	2	yes	—
19	C₂×C₅×#103	—	2	2	no	—
20	C₅×#367	C₂²	1	4	yes	complemented
21	C₅×#358	C₂²	1	8	yes	complemented
22	C₂×#2219	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂²×C₅×Q₈	—	2	2	no	—
24	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
25	C₂×C₅×Q₁₆	—	2	4	no	—
26	C₂×C₅×Q₁₆	—	2	2	no	—
27	C₂×C₅×Q₁₆	—	2	4	no	—
28	C₂×C₅×Q₁₆	C₂³	1	4	yes	—
29	C₂²×C₅×C₈	—	2	2	no	—
30	C₂²×C₅×C₈	—	2	2	no	—
31	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
32	C₂²×C₅×C₈	D₄	1	2	yes	—
33	C₂²×C₅×C₈	C₂³	1	2	yes	—
34	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
35	C₅×C₂²⋊Q₈	—	2	8	no	—
36	C₅×C₂²⋊Q₈	—	4	4	no	—
37	C₂×C₅×C₄⋊C₄	—	2	2	no	—
38	C₂×C₅×C₄⋊C₄	—	4	2	no	—
39	C₂×C₅×C₂²⋊C₄	—	4	2	no	—
40	C₅×Q₃₂	—	2	8	no	—
41	C₅×M₃₂	—	2	8	no	—
42	C₂×C₅×C₁₆	—	2	2	no	—
43	C₂×C₅×C₁₆	D₄	1	4	yes	—
44	C₂×C₅×C₁₆	C₂³	1	2	yes	—
45	C₂×C₅×C₁₆	C₂³	1	2	yes	—
46	C₅×#107	—	2	2	no	—
47	C₅×#107	C₂³	1	4	yes	—
48	C₅×#103	—	2	8	no	—
49	C₂²×Q₃₂	C₂×C₅	1	1	yes	characteristic, complemented
50	C₂²×M₃₂	C₂×C₅	1	1	yes	characteristic
51	C₂×#467	C₂×C₅	1	2	yes	—
52	#2219	C₂×C₅	1	8	yes	direct product factor
53	C₂×#367	C₂×C₅	1	1	yes	characteristic, complemented
54	C₂×#358	C₂×C₅	1	2	yes	complemented
55	C₂⁴×C₅	D₈	1	1	yes	characteristic
56	C₂×C₅×Q₈	—	2	4	no	—
57	C₂×C₅×Q₈	—	4	2	no	—
58	C₂×C₅×Q₈	—	4	4	no	—
59	C₂×C₅×Q₈	—	4	4	no	—
60	C₂²×C₄×C₅	—	2	2	no	—
61	C₂²×C₄×C₅	—	2	2	no	—
62	C₂²×C₄×C₅	—	4	2	no	—
63	C₂²×C₄×C₅	—	4	2	no	—
64	C₂²×C₄×C₅	D₈	1	2	yes	—
65	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
67	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₅×Q₁₆	—	2	4	no	—
69	C₅×Q₁₆	—	2	4	no	—
70	C₅×Q₁₆	—	4	4	no	—
71	C₂×C₅×C₈	—	2	2	no	—
72	C₂×C₅×C₈	—	2	4	no	—
73	C₂×C₅×C₈	—	2	2	no	—
74	C₂×C₅×C₈	—	2	2	no	—
75	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
76	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
77	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
78	C₂×C₅×C₈	#41	1	2	yes	—
79	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
80	C₅×C₄⋊C₄	—	2	4	no	—
81	C₅×C₄⋊C₄	—	4	2	no	—
82	C₅×C₄⋊C₄	—	4	8	no	—
83	C₅×C₂²⋊C₄	—	4	8	no	—
84	C₅×C₁₆	—	2	2	no	—
85	C₅×C₁₆	C₂×D₄	1	4	yes	—
86	C₂²×Q₁₆	C₂²×C₅	1	2	yes	—
87	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
88	C₂×C₂²⋊Q₈	—	2	2	no	—
89	C₂×Q₃₂	—	2	4	no	—
90	C₂×Q₃₂	C₂²×C₅	1	4	yes	complemented
91	C₂×M₃₂	—	2	4	no	—
92	C₂×M₃₂	C₂²×C₅	1	4	yes	—
93	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
94	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
95	#467	—	2	4	no	—
96	#467	C₂²×C₅	1	8	yes	—
97	C₂×#107	C₂²×C₅	1	2	yes	—
98	C₂×#103	—	2	2	no	—
99	#367	C₂²×C₅	1	4	yes	complemented
100	#358	C₂²×C₅	1	8	yes	complemented
101	C₂³×C₅	—	2	2	no	—
102	C₂³×C₅	—	2	4	no	—
103	C₂³×C₅	C₂×D₈	1	2	yes	—
104	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
105	C₅×Q₈	—	4	4	no	—
106	C₅×Q₈	—	4	4	no	—
107	C₅×Q₈	—	8	4	no	—
108	C₂×C₄×C₅	—	2	2	no	—
109	C₂×C₄×C₅	—	2	2	no	—
110	C₂×C₄×C₅	—	2	2	no	—
111	C₂×C₄×C₅	—	2	4	no	—
112	C₂×C₄×C₅	—	4	4	no	—
113	C₂×C₄×C₅	—	4	4	no	—
114	C₂×C₄×C₅	—	8	4	no	—
115	C₂×C₄×C₅	—	8	4	no	—
116	C₂×C₄×C₅	#121	1	2	yes	—
117	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
118	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₅	#135	1	2	yes	—
120	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
121	C₅×C₈	—	2	2	no	—
122	C₅×C₈	#121	1	2	yes	—
123	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
124	C₅×C₈	C₂×#41	1	1	yes	characteristic
125	C₂²×Q₈	—	2	2	no	—
126	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
127	C₂×Q₁₆	—	2	4	no	—
128	C₂×Q₁₆	—	2	2	no	—
129	C₂×Q₁₆	—	2	4	no	—
130	C₂×Q₁₆	C₂³×C₅	1	4	yes	—
131	C₂²×C₈	—	2	2	no	—
132	C₂²×C₈	—	2	2	no	—
133	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
134	C₂²×C₈	C₅×D₄	1	2	yes	—
135	C₂²×C₈	C₂³×C₅	1	2	yes	—
136	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
137	C₂²⋊Q₈	—	2	8	no	—
138	C₂²⋊Q₈	—	4	4	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	—	4	2	no	—
141	C₂×C₂²⋊C₄	—	4	2	no	—
142	Q₃₂	—	2	8	no	—
143	M₃₂	—	2	8	no	—
144	C₂×C₁₆	—	2	2	no	—
145	C₂×C₁₆	C₅×D₄	1	4	yes	—
146	C₂×C₁₆	C₂³×C₅	1	2	yes	—
147	C₂×C₁₆	C₂³×C₅	1	2	yes	—
148	#107	—	2	2	no	—
149	#107	C₂³×C₅	1	4	yes	—
150	#103	—	2	8	no	—
151	C₂²×C₅	—	2	4	no	—
152	C₂²×C₅	—	2	2	no	—
153	C₂²×C₅	—	4	4	no	—
154	C₂²×C₅	#466	1	2	yes	—
155	C₂²×C₅	#510	1	4	yes	—
156	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
157	C₄×C₅	—	2	2	no	—
158	C₄×C₅	—	8	4	no	—
159	C₄×C₅	—	8	4	no	—
160	C₄×C₅	#466	1	2	yes	—
161	C₄×C₅	C₂×#121	1	1	yes	characteristic
162	C₄×C₅	C₂×#135	1	1	yes	characteristic
163	C₂⁴	C₅×D₈	1	1	yes	characteristic
164	C₂×Q₈	—	2	4	no	—
165	C₂×Q₈	—	4	2	no	—
166	C₂×Q₈	—	4	4	no	—
167	C₂×Q₈	—	4	4	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	4	2	no	—
171	C₂²×C₄	—	4	2	no	—
172	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
173	C₂²×C₄	C₅×D₈	1	2	yes	—
174	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
175	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
176	Q₁₆	—	2	4	no	—
177	Q₁₆	—	2	4	no	—
178	Q₁₆	—	4	4	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	4	no	—
181	C₂×C₈	—	2	2	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
184	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
185	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
186	C₂×C₈	C₅×#41	1	2	yes	—
187	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	—	4	8	no	—
191	C₂²⋊C₄	—	4	8	no	—
192	C₁₆	—	2	2	no	—
193	C₁₆	C₂×C₅×D₄	1	4	yes	—
194	C₂×C₅	—	4	2	no	—
195	C₂×C₅	#2219	1	4	yes	direct product factor
196	C₂×C₅	C₂×#466	1	1	yes	characteristic
197	C₂×C₅	C₂×#510	1	2	yes	—
198	C₂³	—	2	2	no	—
199	C₂³	—	2	4	no	—
200	C₂³	C₂×C₅×D₈	1	2	yes	—
201	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
202	Q₈	—	4	4	no	—
203	Q₈	—	4	4	no	—
204	Q₈	—	8	4	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	8	4	no	—
212	C₂×C₄	—	8	4	no	—
213	C₂×C₄	C₅×#121	1	2	yes	—
214	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
215	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
216	C₂×C₄	C₅×#135	1	2	yes	—
217	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
218	C₈	—	2	2	no	—
219	C₈	C₅×#121	1	2	yes	—
220	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
221	C₈	C₂×C₅×#41	1	1	yes	characteristic
222	C₅	C₂×#2219	1	1	yes	characteristic, direct product factor, Sylow
223	C₂²	—	2	2	no	—
224	C₂²	—	2	4	no	—
225	C₂²	—	4	4	no	—
226	C₂²	C₅×#466	1	2	yes	—
227	C₂²	C₅×#510	1	4	yes	—
228	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
229	C₄	—	2	2	no	—
230	C₄	—	8	4	no	—
231	C₄	—	8	4	no	—
232	C₄	C₅×#466	1	2	yes	—
233	C₄	C₂×C₅×#121	1	1	yes	characteristic
234	C₄	C₂×C₅×#135	1	1	yes	characteristic
235	C₂	—	4	2	no	—
236	C₂	C₅×#2219	1	4	yes	direct product factor
237	C₂	C₂×C₅×#466	1	1	yes	characteristic
238	C₂	C₂×C₅×#510	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂⁴×C₅	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×Q₃₂	C₂	semidirect
4. C₅×#2219	C₂	direct (N × H)
5. C₂×C₅×#367	C₂	semidirect
6. C₂×C₅×#358	C₂	semidirect
11. C₂×C₅×Q₃₂	C₂²	semidirect
20. C₅×#367	C₂²	semidirect
21. C₅×#358	C₂²	semidirect
22. C₂×#2219	C₅	direct (N × H)
49. C₂²×Q₃₂	C₂×C₅	semidirect
52. #2219	C₂×C₅	direct (N × H)
53. C₂×#367	C₂×C₅	semidirect
54. C₂×#358	C₂×C₅	semidirect
90. C₂×Q₃₂	C₂²×C₅	semidirect
99. #367	C₂²×C₅	semidirect
100. #358	C₂²×C₅	semidirect
195. C₂×C₅	#2219	direct (N × H)
222. C₅	C₂×#2219	direct (N × H)
236. C₂	C₅×#2219	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×C₅×Q₃₂	2
C₂²×C₅×M₃₂	2
C₂×C₅×#467	2
C₅×#2219	2
C₂×C₅×#367	2
C₂×C₅×#358	2
C₂×#2219	5

Subgroups — C2×C5×#2219 #11956322

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2219 #11956322