Subgroups — C₅×#22300 #11945020

Order 1280.

618 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2402	C₂	1	1	yes	maximal, characteristic
2	C₅×#1940	C₂	1	2	yes	maximal
3	C₅×#1939	C₂	1	4	yes	maximal
4	C₅×#1916	C₂	1	4	yes	maximal
5	C₅×#1843	C₂	1	1	yes	maximal, characteristic
6	C₅×C₈×C₄⋊C₄	C₂	1	2	yes	maximal
7	C₅×#1746	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
10	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
11	C₂×C₅×#97	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₅×C₈	C₂²	1	4	yes	—
13	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₅×#400	C₄	1	4	yes	—
15	C₅×#398	C₄	1	4	yes	—
16	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
17	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
18	C₅×#376	C₂²	1	1	yes	characteristic
19	C₅×#336	C₂²	1	4	yes	—
20	C₅×#336	C₂²	1	4	yes	—
21	C₅×#336	C₂²	1	2	yes	—
22	C₅×#336	C₂²	1	2	yes	—
23	#22300	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
24	C₂²×C₅×C₈	C₂³	1	4	yes	—
25	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
26	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
27	C₂²×C₅×C₈	C₂³	1	2	yes	—
28	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
29	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
30	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₅	C₂³	1	2	yes	—
32	C₂×C₄²×C₅	C₂³	1	4	yes	—
33	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
34	C₅×C₄⋊C₈	—	2	4	no	—
35	C₅×C₄⋊C₈	—	4	4	no	—
36	C₅×C₄⋊C₈	—	4	4	no	—
37	C₅×#97	—	2	2	no	—
38	C₄×C₅×C₈	—	4	4	no	—
39	C₄×C₅×C₈	Q₈	1	4	yes	—
40	C₅×#95	C₂×C₄	1	4	yes	—
41	C₅×#95	C₂×C₄	1	2	yes	—
42	C₅×#95	C₂×C₄	1	4	yes	—
43	C₅×#95	C₂×C₄	1	4	yes	—
44	C₅×#95	C₂×C₄	1	2	yes	—
45	#2402	C₂×C₅	1	1	yes	characteristic
46	#1940	C₂×C₅	1	2	yes	—
47	#1939	C₂×C₅	1	4	yes	—
48	#1916	C₂×C₅	1	4	yes	—
49	#1843	C₂×C₅	1	1	yes	characteristic
50	C₈×C₄⋊C₄	C₂×C₅	1	2	yes	—
51	#1746	C₂×C₅	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
53	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
54	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
56	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
57	C₂×C₅×C₈	—	2	2	no	—
58	C₂×C₅×C₈	—	2	2	no	—
59	C₂×C₅×C₈	—	2	1	no	—
60	C₂×C₅×C₈	—	2	1	no	—
61	C₂×C₅×C₈	—	2	4	no	—
62	C₂×C₅×C₈	—	2	4	no	—
63	C₂×C₅×C₈	—	2	4	no	—
64	C₂×C₅×C₈	—	2	2	no	—
65	C₂×C₅×C₈	—	2	2	no	—
66	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
67	C₅×C₄⋊C₄	—	2	8	no	—
68	C₅×C₄⋊C₄	—	2	8	no	—
69	C₅×C₄⋊C₄	—	2	8	no	—
70	C₄²×C₅	—	2	4	no	—
71	C₄²×C₅	—	2	2	no	—
72	C₄²×C₅	—	2	4	no	—
73	C₄²×C₅	—	2	2	no	—
74	C₄²×C₅	C₂×Q₈	1	2	yes	—
75	C₄²×C₅	#41	1	2	yes	—
76	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
77	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
78	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
79	C₂×#97	C₂²×C₅	1	1	yes	characteristic
80	C₂×C₄×C₈	C₂²×C₅	1	4	yes	—
81	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
82	#400	C₄×C₅	1	4	yes	—
83	#398	C₄×C₅	1	4	yes	—
84	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
85	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
86	#376	C₂²×C₅	1	1	yes	characteristic
87	#336	C₂²×C₅	1	4	yes	—
88	#336	C₂²×C₅	1	2	yes	—
89	#336	C₂²×C₅	1	4	yes	—
90	#336	C₂²×C₅	1	2	yes	—
91	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
92	C₂×C₄×C₅	—	2	4	no	—
93	C₂×C₄×C₅	—	2	4	no	—
94	C₂×C₄×C₅	—	2	4	no	—
95	C₂×C₄×C₅	—	2	4	no	—
96	C₂×C₄×C₅	—	2	4	no	—
97	C₂×C₄×C₅	—	2	4	no	—
98	C₂×C₄×C₅	—	2	4	no	—
99	C₂×C₄×C₅	—	2	4	no	—
100	C₂×C₄×C₅	—	2	4	no	—
101	C₂×C₄×C₅	C₄×Q₈	1	4	yes	—
102	C₂×C₄×C₅	#131	1	8	yes	—
103	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
104	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
105	C₂×C₄×C₅	H(5,𝔽₂)	1	2	yes	—
106	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
107	C₅×C₈	—	2	4	no	—
108	C₅×C₈	—	4	1	no	—
109	C₅×C₈	—	4	4	no	—
110	C₅×C₈	—	4	1	no	—
111	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
112	C₂²×C₈	C₂³×C₅	1	4	yes	—
113	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
114	C₂²×C₈	C₂³×C₅	1	2	yes	—
115	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
116	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
117	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
118	C₂×C₄²	C₂³×C₅	1	2	yes	—
119	C₂×C₄²	C₂³×C₅	1	4	yes	—
120	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
121	C₄⋊C₈	—	2	4	no	—
122	C₄⋊C₈	—	4	4	no	—
123	C₄⋊C₈	—	4	4	no	—
124	#97	—	2	2	no	—
125	C₄×C₈	—	4	4	no	—
126	C₄×C₈	C₅×Q₈	1	4	yes	—
127	#95	C₂×C₄×C₅	1	4	yes	—
128	#95	C₂×C₄×C₅	1	2	yes	—
129	#95	C₂×C₄×C₅	1	4	yes	—
130	#95	C₂×C₄×C₅	1	4	yes	—
131	#95	C₂×C₄×C₅	1	2	yes	—
132	C₂²×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
133	C₂²×C₅	#520	1	2	yes	—
134	C₂²×C₅	C₂×#131	1	4	yes	—
135	C₄×C₅	—	2	4	no	—
136	C₄×C₅	—	2	8	no	—
137	C₄×C₅	—	4	4	no	—
138	C₄×C₅	—	4	4	no	—
139	C₄×C₅	#540	1	1	yes	characteristic
140	C₄×C₅	#541	1	1	yes	characteristic
141	C₄×C₅	#544	1	2	yes	—
142	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
143	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
144	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
145	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
146	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
147	C₂×C₈	—	2	2	no	—
148	C₂×C₈	—	2	2	no	—
149	C₂×C₈	—	2	1	no	—
150	C₂×C₈	—	2	1	no	—
151	C₂×C₈	—	2	4	no	—
152	C₂×C₈	—	2	4	no	—
153	C₂×C₈	—	2	4	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	C₂×C₅×Q₈	1	4	yes	—
157	C₄⋊C₄	—	2	8	no	—
158	C₄⋊C₄	—	2	8	no	—
159	C₄⋊C₄	—	2	8	no	—
160	C₄²	—	2	4	no	—
161	C₄²	—	2	2	no	—
162	C₄²	—	2	4	no	—
163	C₄²	—	2	2	no	—
164	C₄²	C₂×C₅×Q₈	1	2	yes	—
165	C₄²	C₅×#41	1	2	yes	—
166	C₂×C₅	#2330	1	1	yes	characteristic
167	C₂×C₅	#2960	1	2	yes	—
168	C₂×C₅	#2994	1	4	yes	—
169	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	4	no	—
177	C₂×C₄	—	2	4	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	C₄×C₅×Q₈	1	4	yes	—
180	C₂×C₄	C₅×#131	1	8	yes	—
181	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
182	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
183	C₂×C₄	C₅×H(5,𝔽₂)	1	2	yes	—
184	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
185	C₈	—	2	4	no	—
186	C₈	—	4	1	no	—
187	C₈	—	4	4	no	—
188	C₈	—	4	1	no	—
189	C₅	#22300	1	1	yes	characteristic, direct product factor, Sylow
190	C₂²	C₂×C₄×C₅×Q₈	1	1	yes	characteristic
191	C₂²	C₅×#520	1	2	yes	—
192	C₂²	C₂×C₅×#131	1	4	yes	—
193	C₄	—	2	8	no	—
194	C₄	—	2	4	no	—
195	C₄	—	4	4	no	—
196	C₄	—	4	4	no	—
197	C₄	C₅×#540	1	1	yes	characteristic
198	C₄	C₅×#541	1	1	yes	characteristic
199	C₄	C₅×#544	1	2	yes	—
200	C₂	C₅×#2330	1	1	yes	characteristic
201	C₂	C₅×#2960	1	2	yes	—
202	C₂	C₅×#2994	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
23. #22300	C₅	direct (N × H)
189. C₅	#22300	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₅×#2402	2
C₅×#1940	2
C₅×#1939	2
C₅×#1916	2
C₅×#1843	2
C₅×C₈×C₄⋊C₄	2
C₅×#1746	2
#22300	5

Subgroups — C5×#22300 #11945020

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22300 #11945020