Subgroups — C₅×#22270 #11944990

Order 1280.

886 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#494	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅×C₄⋊Q₈	C₂	1	2	yes	maximal
3	C₅×#1956	C₂	1	2	yes	maximal
4	C₅×#1878	C₂	1	8	yes	maximal
5	C₅×#1834	C₂	1	1	yes	maximal, characteristic
6	C₅×#1744	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₅×Q₁₆	C₂²	1	4	yes	—
8	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂×C₄×C₅×Q₈	C₂²	1	4	yes	—
10	C₅×#494	C₄	1	8	yes	complemented
11	C₂×C₅×#103	C₂²	1	8	yes	—
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₅×#391	C₂²	1	4	yes	—
15	C₅×#389	—	2	2	no	—
16	C₅×#389	—	2	2	no	—
17	C₅×#384	C₂²	1	2	yes	—
18	C₄×C₅×C₄⋊C₄	—	2	2	no	—
19	C₄×C₅×C₄⋊C₄	—	2	2	no	—
20	C₄³×C₅	C₂²	1	1	yes	characteristic
21	C₅×#340	C₂²	1	4	yes	—
22	C₅×#336	C₂²	1	4	yes	—
23	#22270	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂²×C₅×Q₈	C₂³	1	4	yes	—
25	C₂×C₅×Q₁₆	—	4	8	no	—
26	C₂×C₅×Q₁₆	C₂×C₄	1	16	yes	—
27	C₂²×C₅×C₈	C₂³	1	2	yes	—
28	C₂²×C₅×C₈	C₂³	1	2	yes	—
29	C₅×C₄⋊Q₈	—	2	4	no	—
30	C₅×C₄⋊Q₈	C₂×C₄	1	8	yes	—
31	C₄×C₅×Q₈	—	4	8	no	—
32	C₂×C₅×C₄⋊C₄	—	2	2	no	—
33	C₂×C₅×C₄⋊C₄	—	2	2	no	—
34	C₂×C₅×C₄⋊C₄	—	2	2	no	—
35	C₂×C₅×C₄⋊C₄	—	2	2	no	—
36	C₂×C₅×C₄⋊C₄	—	2	4	no	—
37	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
38	C₂×C₄²×C₅	—	2	4	no	—
39	C₂×C₄²×C₅	D₄	1	2	yes	—
40	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅	C₂³	1	2	yes	—
43	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
44	C₅×#103	—	4	8	no	—
45	C₄×C₅×C₈	—	2	2	no	—
46	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
47	C₅×#95	—	2	4	no	—
48	C₅×#95	—	2	4	no	—
49	C₂×#494	C₂×C₅	1	1	yes	characteristic
50	C₄×C₄⋊Q₈	C₂×C₅	1	2	yes	—
51	#1956	C₂×C₅	1	2	yes	—
52	#1878	C₂×C₅	1	8	yes	—
53	#1834	C₂×C₅	1	1	yes	characteristic
54	#1744	C₂×C₅	1	1	yes	characteristic
55	C₂×C₅×Q₈	—	2	4	no	—
56	C₂×C₅×Q₈	—	2	8	no	—
57	C₂×C₅×Q₈	—	2	8	no	—
58	C₂×C₅×Q₈	C₂²×C₄	1	8	yes	—
59	C₂²×C₄×C₅	—	2	4	no	—
60	C₂²×C₄×C₅	—	2	4	no	—
61	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
66	C₅×Q₁₆	—	4	16	no	—
67	C₂×C₅×C₈	—	2	4	no	—
68	C₂×C₅×C₈	—	2	2	no	—
69	C₂×C₅×C₈	—	2	4	no	—
70	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
71	C₅×C₄⋊C₄	—	2	8	no	—
72	C₅×C₄⋊C₄	—	2	8	no	—
73	C₅×C₄⋊C₄	—	2	8	no	—
74	C₅×C₄⋊C₄	—	4	8	no	—
75	C₅×C₄⋊C₄	—	4	4	no	—
76	C₅×C₄⋊C₄	—	4	4	no	—
77	C₄²×C₅	—	2	2	no	—
78	C₄²×C₅	—	2	4	no	—
79	C₄²×C₅	—	2	4	no	—
80	C₄²×C₅	—	4	8	no	—
81	C₄²×C₅	Q₁₆	1	4	yes	complemented
82	C₄²×C₅	C₂²×C₄	1	2	yes	—
83	C₄²×C₅	#41	1	2	yes	—
84	C₂²×Q₁₆	C₂²×C₅	1	4	yes	—
85	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
86	C₂×C₄×Q₈	C₂²×C₅	1	4	yes	—
87	#494	C₄×C₅	1	8	yes	complemented
88	C₂×#103	C₂²×C₅	1	8	yes	—
89	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
90	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
91	#391	C₂²×C₅	1	4	yes	—
92	#389	—	2	2	no	—
93	#389	—	2	2	no	—
94	#384	C₂²×C₅	1	2	yes	—
95	C₄×C₄⋊C₄	—	2	2	no	—
96	C₄×C₄⋊C₄	—	2	2	no	—
97	C₄³	C₂²×C₅	1	1	yes	characteristic
98	#340	C₂²×C₅	1	4	yes	—
99	#336	C₂²×C₅	1	4	yes	—
100	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
101	C₅×Q₈	—	2	16	no	—
102	C₅×Q₈	—	4	8	no	—
103	C₂×C₄×C₅	—	2	2	no	—
104	C₂×C₄×C₅	—	2	2	no	—
105	C₂×C₄×C₅	—	2	8	no	—
106	C₂×C₄×C₅	—	2	2	no	—
107	C₂×C₄×C₅	—	2	2	no	—
108	C₂×C₄×C₅	—	2	2	no	—
109	C₂×C₄×C₅	—	4	4	no	—
110	C₂×C₄×C₅	—	4	4	no	—
111	C₂×C₄×C₅	—	4	4	no	—
112	C₂×C₄×C₅	—	4	4	no	—
113	C₂×C₄×C₅	—	4	4	no	—
114	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₅	C₂×Q₁₆	1	4	yes	—
117	C₂×C₄×C₅	C₂×Q₁₆	1	2	yes	—
118	C₂×C₄×C₅	#135	1	2	yes	—
119	C₂×C₄×C₅	#137	1	4	yes	—
120	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
122	C₂×C₄×C₅	H(5,𝔽₂)	1	4	yes	—
123	C₅×C₈	—	2	4	no	—
124	C₅×C₈	—	4	2	no	—
125	C₂²×Q₈	C₂³×C₅	1	4	yes	—
126	C₂×Q₁₆	—	4	8	no	—
127	C₂×Q₁₆	C₂×C₄×C₅	1	16	yes	—
128	C₂²×C₈	C₂³×C₅	1	2	yes	—
129	C₂²×C₈	C₂³×C₅	1	2	yes	—
130	C₄⋊Q₈	—	2	4	no	—
131	C₄⋊Q₈	C₂×C₄×C₅	1	8	yes	—
132	C₄×Q₈	—	4	8	no	—
133	C₂×C₄⋊C₄	—	2	2	no	—
134	C₂×C₄⋊C₄	—	2	2	no	—
135	C₂×C₄⋊C₄	—	2	4	no	—
136	C₂×C₄⋊C₄	—	2	2	no	—
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
139	C₂×C₄²	—	2	4	no	—
140	C₂×C₄²	C₅×D₄	1	2	yes	—
141	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
142	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
143	C₂×C₄²	C₂³×C₅	1	2	yes	—
144	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
145	#103	—	4	8	no	—
146	C₄×C₈	—	2	2	no	—
147	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
148	#95	—	2	4	no	—
149	#95	—	2	4	no	—
150	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
151	C₂²×C₅	#535	1	2	yes	—
152	C₂²×C₅	C₂²×Q₁₆	1	1	yes	characteristic
153	C₂²×C₅	C₂×#135	1	1	yes	characteristic
154	C₂²×C₅	C₂×#137	1	2	yes	—
155	C₄×C₅	—	2	4	no	—
156	C₄×C₅	—	4	8	no	—
157	C₄×C₅	—	4	2	no	—
158	C₄×C₅	—	8	4	no	—
159	C₄×C₅	C₄×Q₁₆	1	4	yes	—
160	C₄×C₅	#441	1	4	yes	—
161	C₄×C₅	#518	1	2	yes	—
162	C₄×C₅	#540	1	2	yes	—
163	C₂×Q₈	—	2	4	no	—
164	C₂×Q₈	—	2	8	no	—
165	C₂×Q₈	—	2	8	no	—
166	C₂×Q₈	C₂²×C₄×C₅	1	8	yes	—
167	C₂²×C₄	—	2	4	no	—
168	C₂²×C₄	—	2	4	no	—
169	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
170	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
171	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
172	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
173	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
174	Q₁₆	—	4	16	no	—
175	C₂×C₈	—	2	4	no	—
176	C₂×C₈	—	2	2	no	—
177	C₂×C₈	—	2	4	no	—
178	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
179	C₄⋊C₄	—	2	8	no	—
180	C₄⋊C₄	—	2	8	no	—
181	C₄⋊C₄	—	2	8	no	—
182	C₄⋊C₄	—	4	8	no	—
183	C₄⋊C₄	—	4	4	no	—
184	C₄⋊C₄	—	4	4	no	—
185	C₄²	—	2	2	no	—
186	C₄²	—	2	4	no	—
187	C₄²	—	2	4	no	—
188	C₄²	—	4	8	no	—
189	C₄²	C₅×Q₁₆	1	4	yes	complemented
190	C₄²	C₂²×C₄×C₅	1	2	yes	—
191	C₄²	C₅×#41	1	2	yes	—
192	C₂×C₅	#2328	1	1	yes	characteristic
193	C₂×C₅	C₂×C₄×Q₁₆	1	1	yes	characteristic
194	C₂×C₅	C₂×#441	1	1	yes	characteristic
195	C₂×C₅	#3209	1	2	yes	—
196	C₂×C₅	#3210	1	2	yes	—
197	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
198	Q₈	—	2	16	no	—
199	Q₈	—	4	8	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	8	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	4	4	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
212	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
213	C₂×C₄	C₂×C₅×Q₁₆	1	4	yes	—
214	C₂×C₄	C₂×C₅×Q₁₆	1	2	yes	—
215	C₂×C₄	C₅×#135	1	2	yes	—
216	C₂×C₄	C₅×#137	1	4	yes	—
217	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
218	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
219	C₂×C₄	C₅×H(5,𝔽₂)	1	4	yes	—
220	C₈	—	2	4	no	—
221	C₈	—	4	2	no	—
222	C₅	#22270	1	1	yes	characteristic, direct product factor, Sylow
223	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
224	C₂²	C₅×#535	1	2	yes	—
225	C₂²	C₂²×C₅×Q₁₆	1	1	yes	characteristic
226	C₂²	C₂×C₅×#135	1	1	yes	characteristic
227	C₂²	C₂×C₅×#137	1	2	yes	—
228	C₄	—	2	4	no	—
229	C₄	—	4	8	no	—
230	C₄	—	4	2	no	—
231	C₄	—	8	4	no	—
232	C₄	C₄×C₅×Q₁₆	1	4	yes	—
233	C₄	C₅×#441	1	4	yes	—
234	C₄	C₅×#518	1	2	yes	—
235	C₄	C₅×#540	1	2	yes	—
236	C₂	C₅×#2328	1	1	yes	characteristic
237	C₂	C₂×C₄×C₅×Q₁₆	1	1	yes	characteristic
238	C₂	C₂×C₅×#441	1	1	yes	characteristic
239	C₂	C₅×#3209	1	2	yes	—
240	C₂	C₅×#3210	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₄³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
10. C₅×#494	C₄	semidirect
23. #22270	C₅	direct (N × H)
81. C₄²×C₅	Q₁₆	semidirect
87. #494	C₄×C₅	semidirect
189. C₄²	C₅×Q₁₆	semidirect
222. C₅	#22270	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×#494	2
C₄×C₅×C₄⋊Q₈	2
C₅×#1956	2
C₅×#1878	2
C₅×#1834	2
C₅×#1744	2
#22270	5

Subgroups — C5×#22270 #11944990

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22270 #11944990