Subgroups — C₂×C₅×#2225 #11956343

Order 1280.

574 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#500	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic
3	C₅×#2225	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₅×#367	C₂	1	2	yes	maximal
5	C₂×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×#358	C₂	1	2	yes	maximal
7	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₅×C₄⋊Q₈	—	2	1	no	—
9	C₂×C₄×C₅×Q₈	—	2	1	no	—
10	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
11	C₅×#500	—	2	2	no	—
12	C₅×#500	C₂²	1	4	yes	—
13	C₄×C₅×Q₁₆	—	2	2	no	—
14	C₄×C₅×Q₁₆	C₂²	1	4	yes	—
15	C₂×C₅×#107	—	2	1	no	—
16	C₂×C₅×#107	C₂²	1	2	yes	—
17	C₂×C₅×#107	C₂²	1	1	yes	characteristic
18	C₂×C₅×#103	—	2	1	no	—
19	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₅×#367	C₂²	1	8	yes	—
21	C₅×C₄⋊C₁₆	C₂²	1	4	yes	—
22	C₅×#358	C₂²	1	8	yes	—
23	C₂×#2225	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂²×C₅×Q₈	—	2	1	no	—
25	C₂²×C₅×Q₈	—	4	1	no	—
26	C₂×C₅×Q₁₆	—	2	1	no	—
27	C₂×C₅×Q₁₆	—	2	2	no	—
28	C₂×C₅×Q₁₆	Q₈	1	4	yes	complemented
29	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
30	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
31	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
32	C₅×C₄⋊Q₈	—	2	4	no	—
33	C₅×C₄⋊Q₈	—	4	2	no	—
34	C₄×C₅×Q₈	—	2	4	no	—
35	C₄×C₅×Q₈	—	4	2	no	—
36	C₂×C₅×C₄⋊C₄	—	2	1	no	—
37	C₂×C₅×C₄⋊C₄	—	2	2	no	—
38	C₂×C₅×C₄⋊C₄	—	4	1	no	—
39	C₂×C₅×C₄⋊C₄	—	4	1	no	—
40	C₂×C₄²×C₅	—	4	1	no	—
41	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₂×C₅×C₁₆	—	2	4	no	—
43	C₂×C₅×C₁₆	C₂³	1	4	yes	—
44	C₅×#107	—	2	4	no	—
45	C₅×#107	—	2	1	no	—
46	C₅×#107	—	2	2	no	—
47	C₅×#107	C₂³	1	4	yes	—
48	C₅×#107	C₂³	1	2	yes	—
49	C₅×#103	—	2	4	no	—
50	C₄×C₅×C₈	D₄	1	2	yes	—
51	C₄×C₅×C₈	C₂³	1	2	yes	—
52	C₂×#500	C₂×C₅	1	1	yes	characteristic
53	C₂×C₄×Q₁₆	C₂×C₅	1	1	yes	characteristic
54	#2225	C₂×C₅	1	8	yes	direct product factor
55	C₂×#367	C₂×C₅	1	2	yes	—
56	C₂×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic
57	C₂×#358	C₂×C₅	1	2	yes	—
58	C₂×C₅×Q₈	—	2	4	no	—
59	C₂×C₅×Q₈	—	2	2	no	—
60	C₂×C₅×Q₈	—	4	1	no	—
61	C₂×C₅×Q₈	—	4	4	no	—
62	C₂×C₅×Q₈	—	4	2	no	—
63	C₂×C₅×Q₈	—	8	4	no	—
64	C₂²×C₄×C₅	—	4	1	no	—
65	C₂²×C₄×C₅	—	4	1	no	—
66	C₂²×C₄×C₅	—	4	2	no	—
67	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
70	C₅×Q₁₆	—	2	4	no	—
71	C₅×Q₁₆	—	2	2	no	—
72	C₅×Q₁₆	C₂×Q₈	1	4	yes	complemented
73	C₂×C₅×C₈	—	2	2	no	—
74	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
75	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
76	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
77	C₂×C₅×C₈	#41	1	2	yes	—
78	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
79	C₅×C₄⋊C₄	—	2	2	no	—
80	C₅×C₄⋊C₄	—	2	4	no	—
81	C₅×C₄⋊C₄	—	4	1	no	—
82	C₅×C₄⋊C₄	—	4	2	no	—
83	C₅×C₄⋊C₄	—	4	4	no	—
84	C₅×C₄⋊C₄	—	4	4	no	—
85	C₄²×C₅	—	4	4	no	—
86	C₄²×C₅	D₈	1	2	yes	—
87	C₄²×C₅	C₂×D₄	1	2	yes	—
88	C₅×C₁₆	—	2	4	no	—
89	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic
90	C₂×C₄⋊Q₈	—	2	1	no	—
91	C₂×C₄×Q₈	—	2	1	no	—
92	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
93	#500	—	2	2	no	—
94	#500	C₂²×C₅	1	4	yes	—
95	C₄×Q₁₆	—	2	2	no	—
96	C₄×Q₁₆	C₂²×C₅	1	4	yes	—
97	C₂×#107	—	2	1	no	—
98	C₂×#107	C₂²×C₅	1	2	yes	—
99	C₂×#107	C₂²×C₅	1	1	yes	characteristic
100	C₂×#103	—	2	1	no	—
101	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
102	#367	C₂²×C₅	1	8	yes	—
103	C₄⋊C₁₆	C₂²×C₅	1	4	yes	—
104	#358	C₂²×C₅	1	8	yes	—
105	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
106	C₅×Q₈	—	2	4	no	—
107	C₅×Q₈	—	4	2	no	—
108	C₅×Q₈	—	4	4	no	—
109	C₅×Q₈	—	8	8	no	—
110	C₂×C₄×C₅	—	2	2	no	—
111	C₂×C₄×C₅	—	4	2	no	—
112	C₂×C₄×C₅	—	4	4	no	—
113	C₂×C₄×C₅	—	4	2	no	—
114	C₂×C₄×C₅	—	4	4	no	—
115	C₂×C₄×C₅	—	8	4	no	—
116	C₂×C₄×C₅	—	8	2	no	—
117	C₂×C₄×C₅	SD₃₂	1	4	yes	—
118	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
120	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
122	C₂×C₄×C₅	#137	1	2	yes	—
123	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
124	C₅×C₈	—	2	2	no	—
125	C₅×C₈	C₂²⋊Q₈	1	2	yes	—
126	C₅×C₈	C₂²×Q₈	1	1	yes	characteristic
127	C₅×C₈	C₂×#41	1	1	yes	characteristic
128	C₂²×Q₈	—	2	1	no	—
129	C₂²×Q₈	—	4	1	no	—
130	C₂×Q₁₆	—	2	1	no	—
131	C₂×Q₁₆	—	2	2	no	—
132	C₂×Q₁₆	C₅×Q₈	1	4	yes	complemented
133	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
134	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
135	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
136	C₄⋊Q₈	—	2	4	no	—
137	C₄⋊Q₈	—	4	2	no	—
138	C₄×Q₈	—	2	4	no	—
139	C₄×Q₈	—	4	2	no	—
140	C₂×C₄⋊C₄	—	2	1	no	—
141	C₂×C₄⋊C₄	—	2	2	no	—
142	C₂×C₄⋊C₄	—	4	1	no	—
143	C₂×C₄⋊C₄	—	4	1	no	—
144	C₂×C₄²	—	4	1	no	—
145	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
146	C₂×C₁₆	—	2	4	no	—
147	C₂×C₁₆	C₂³×C₅	1	4	yes	—
148	#107	—	2	4	no	—
149	#107	—	2	1	no	—
150	#107	—	2	2	no	—
151	#107	C₂³×C₅	1	4	yes	—
152	#107	C₂³×C₅	1	2	yes	—
153	#103	—	2	4	no	—
154	C₄×C₈	C₅×D₄	1	2	yes	—
155	C₄×C₈	C₂³×C₅	1	2	yes	—
156	C₂²×C₅	#474	1	2	yes	—
157	C₂²×C₅	C₂×SD₃₂	1	2	yes	—
158	C₂²×C₅	#510	1	2	yes	—
159	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
160	C₄×C₅	—	2	2	no	—
161	C₄×C₅	—	4	4	no	—
162	C₄×C₅	—	8	4	no	—
163	C₄×C₅	—	8	2	no	—
164	C₄×C₅	#474	1	2	yes	—
165	C₄×C₅	C₂×SD₃₂	1	4	yes	—
166	C₄×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
167	C₄×C₅	C₂×#137	1	1	yes	characteristic
168	C₂×Q₈	—	2	4	no	—
169	C₂×Q₈	—	2	2	no	—
170	C₂×Q₈	—	4	1	no	—
171	C₂×Q₈	—	4	4	no	—
172	C₂×Q₈	—	4	2	no	—
173	C₂×Q₈	—	8	4	no	—
174	C₂²×C₄	—	4	1	no	—
175	C₂²×C₄	—	4	1	no	—
176	C₂²×C₄	—	4	2	no	—
177	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
178	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
179	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
180	Q₁₆	—	2	4	no	—
181	Q₁₆	—	2	2	no	—
182	Q₁₆	C₂×C₅×Q₈	1	4	yes	complemented
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
185	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
186	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
187	C₂×C₈	C₅×#41	1	2	yes	—
188	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
189	C₄⋊C₄	—	2	2	no	—
190	C₄⋊C₄	—	2	4	no	—
191	C₄⋊C₄	—	4	1	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	—	4	4	no	—
194	C₄⋊C₄	—	4	4	no	—
195	C₄²	—	4	4	no	—
196	C₄²	C₅×D₈	1	2	yes	—
197	C₄²	C₂×C₅×D₄	1	2	yes	—
198	C₁₆	—	2	4	no	—
199	C₂×C₅	#2225	1	4	yes	direct product factor
200	C₂×C₅	C₂×#474	1	1	yes	characteristic
201	C₂×C₅	C₂²×SD₃₂	1	1	yes	characteristic
202	C₂×C₅	C₂×#510	1	1	yes	characteristic
203	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
204	Q₈	—	2	4	no	—
205	Q₈	—	4	4	no	—
206	Q₈	—	4	2	no	—
207	Q₈	—	8	8	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	4	no	—
213	C₂×C₄	—	8	4	no	—
214	C₂×C₄	—	8	2	no	—
215	C₂×C₄	C₅×SD₃₂	1	4	yes	—
216	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
217	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
218	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
219	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
220	C₂×C₄	C₅×#137	1	2	yes	—
221	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
222	C₈	—	2	2	no	—
223	C₈	C₅×C₂²⋊Q₈	1	2	yes	—
224	C₈	C₂²×C₅×Q₈	1	1	yes	characteristic
225	C₈	C₂×C₅×#41	1	1	yes	characteristic
226	C₅	C₂×#2225	1	1	yes	characteristic, direct product factor, Sylow
227	C₂²	C₅×#474	1	2	yes	—
228	C₂²	C₂×C₅×SD₃₂	1	2	yes	—
229	C₂²	C₅×#510	1	2	yes	—
230	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	4	4	no	—
233	C₄	—	8	4	no	—
234	C₄	—	8	2	no	—
235	C₄	C₅×#474	1	2	yes	—
236	C₄	C₂×C₅×SD₃₂	1	4	yes	—
237	C₄	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic
238	C₄	C₂×C₅×#137	1	1	yes	characteristic
239	C₂	C₅×#2225	1	4	yes	direct product factor
240	C₂	C₂×C₅×#474	1	1	yes	characteristic
241	C₂	C₂²×C₅×SD₃₂	1	1	yes	characteristic
242	C₂	C₂×C₅×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#2225	C₂	direct (N × H)
23. C₂×#2225	C₅	direct (N × H)
28. C₂×C₅×Q₁₆	Q₈	semidirect
54. #2225	C₂×C₅	direct (N × H)
72. C₅×Q₁₆	C₂×Q₈	semidirect
132. C₂×Q₁₆	C₅×Q₈	semidirect
182. Q₁₆	C₂×C₅×Q₈	semidirect
199. C₂×C₅	#2225	direct (N × H)
226. C₅	C₂×#2225	direct (N × H)
239. C₂	C₅×#2225	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×C₅×#500	2
C₂×C₄×C₅×Q₁₆	2
C₅×#2225	2
C₂×C₅×#367	2
C₂×C₅×C₄⋊C₁₆	2
C₂×C₅×#358	2
C₂×#2225	5

Subgroups — C2×C5×#2225 #11956343

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#2225 #11956343