Subgroups — C₅×#22257 #11944977

Order 1280.

586 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3162	C₂	1	1	yes	maximal, characteristic
2	C₅×#2960	C₂	1	2	yes	maximal
3	C₅×#1995	C₂	1	2	yes	maximal
4	C₅×#1922	C₂	1	8	yes	maximal
5	C₅×#1838	C₂	1	1	yes	maximal, characteristic
6	C₅×#1746	C₂	1	1	yes	maximal, characteristic
7	C₅×#533	C₂²	1	2	yes	—
8	C₅×#501	C₄	1	4	yes	complemented
9	C₅×#501	C₄	1	4	yes	complemented
10	C₅×#446	—	2	2	no	—
11	C₅×#446	—	2	2	no	—
12	C₅×C₈×Q₈	—	2	2	no	—
13	C₅×C₈×Q₈	—	2	2	no	—
14	C₅×#427	C₂²	1	4	yes	—
15	C₅×#424	C₂²	1	4	yes	—
16	C₅×#423	C₂²	1	2	yes	—
17	C₅×#405	C₂²	1	4	yes	—
18	C₂×C₅×#97	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	C₅×#376	C₂²	1	1	yes	characteristic
21	C₅×#341	C₂²	1	4	yes	—
22	C₅×#339	C₂²	1	8	yes	—
23	C₅×#336	C₂²	1	4	yes	—
24	#22257	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂×C₅×M₁₆	C₂³	1	4	yes	—
26	C₂²×C₅×C₈	C₂³	1	2	yes	—
27	C₂²×C₅×C₈	C₂³	1	2	yes	—
28	C₅×C₄⋊Q₈	C₂×C₄	1	2	yes	—
29	C₅×C₄⋊Q₈	C₂×C₄	1	2	yes	—
30	C₅×#125	C₂×C₄	1	4	yes	—
31	C₅×C₂²⋊Q₈	—	2	2	no	—
32	C₅×C₂²⋊Q₈	—	2	2	no	—
33	C₄×C₅×Q₈	—	2	2	no	—
34	C₄×C₅×Q₈	—	2	2	no	—
35	C₅×#117	C₂³	1	4	yes	—
36	C₂×C₄²×C₅	C₂³	1	2	yes	—
37	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
38	C₅×#107	C₂×C₄	1	8	yes	—
39	C₅×#106	C₂×C₄	1	4	yes	—
40	C₅×#106	C₂×C₄	1	4	yes	—
41	C₅×C₄⋊C₈	—	2	4	no	—
42	C₅×C₄⋊C₈	—	2	4	no	—
43	C₅×C₄⋊C₈	—	2	2	no	—
44	C₅×C₄⋊C₈	—	2	2	no	—
45	C₅×#97	—	2	4	no	—
46	C₅×#97	C₂×C₄	1	4	yes	—
47	C₄×C₅×C₈	—	2	4	no	—
48	C₄×C₅×C₈	D₄	1	1	yes	characteristic
49	C₄×C₅×C₈	D₄	1	2	yes	—
50	C₄×C₅×C₈	D₄	1	1	yes	characteristic
51	C₅×#95	—	2	2	no	—
52	#3162	C₂×C₅	1	1	yes	characteristic
53	#2960	C₂×C₅	1	2	yes	—
54	#1995	C₂×C₅	1	2	yes	—
55	#1922	C₂×C₅	1	8	yes	—
56	#1838	C₂×C₅	1	1	yes	characteristic
57	#1746	C₂×C₅	1	1	yes	characteristic
58	C₂×C₅×Q₈	—	2	2	no	—
59	C₂×C₅×Q₈	—	2	2	no	—
60	C₂²×C₄×C₅	—	2	2	no	—
61	C₂²×C₄×C₅	#41	1	2	yes	—
62	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
63	C₅×M₁₆	—	2	8	no	—
64	C₂×C₅×C₈	—	2	4	no	—
65	C₂×C₅×C₈	—	2	4	no	—
66	C₂×C₅×C₈	—	2	4	no	—
67	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
68	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
69	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
70	C₅×C₄⋊C₄	—	2	4	no	—
71	C₅×C₄⋊C₄	—	2	4	no	—
72	C₅×C₄⋊C₄	—	2	2	no	—
73	C₅×C₄⋊C₄	—	2	2	no	—
74	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
75	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
76	C₅×C₂²⋊C₄	—	2	4	no	—
77	C₄²×C₅	—	2	4	no	—
78	C₄²×C₅	—	2	4	no	—
79	C₄²×C₅	C₂²×C₄	1	2	yes	—
80	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
82	#533	C₂²×C₅	1	2	yes	—
83	#501	C₄×C₅	1	4	yes	complemented
84	#501	C₄×C₅	1	4	yes	complemented
85	#446	—	2	2	no	—
86	#446	—	2	2	no	—
87	C₈×Q₈	—	2	2	no	—
88	C₈×Q₈	—	2	2	no	—
89	#427	C₂²×C₅	1	4	yes	—
90	#424	C₂²×C₅	1	4	yes	—
91	#423	C₂²×C₅	1	2	yes	—
92	#405	C₂²×C₅	1	4	yes	—
93	C₂×#97	C₂²×C₅	1	1	yes	characteristic
94	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
95	#376	C₂²×C₅	1	1	yes	characteristic
96	#341	C₂²×C₅	1	4	yes	—
97	#339	C₂²×C₅	1	8	yes	—
98	#336	C₂²×C₅	1	4	yes	—
99	C₂³×C₅	C₂×#41	1	1	yes	characteristic
100	C₅×Q₈	—	4	4	no	—
101	C₅×Q₈	—	4	4	no	—
102	C₂×C₄×C₅	—	2	4	no	—
103	C₂×C₄×C₅	—	2	4	no	—
104	C₂×C₄×C₅	—	4	2	no	—
105	C₂×C₄×C₅	—	4	4	no	—
106	C₂×C₄×C₅	—	4	2	no	—
107	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
111	C₂×C₄×C₅	2₋¹⁺⁴	1	4	yes	—
112	C₅×C₈	—	2	4	no	—
113	C₅×C₈	—	2	2	no	—
114	C₅×C₈	—	2	2	no	—
115	C₅×C₈	—	4	4	no	—
116	C₂×M₁₆	C₂³×C₅	1	4	yes	—
117	C₂²×C₈	C₂³×C₅	1	2	yes	—
118	C₂²×C₈	C₂³×C₅	1	2	yes	—
119	C₄⋊Q₈	C₂×C₄×C₅	1	2	yes	—
120	C₄⋊Q₈	C₂×C₄×C₅	1	2	yes	—
121	#125	C₂×C₄×C₅	1	4	yes	—
122	C₂²⋊Q₈	—	2	2	no	—
123	C₂²⋊Q₈	—	2	2	no	—
124	C₄×Q₈	—	2	2	no	—
125	C₄×Q₈	—	2	2	no	—
126	#117	C₂³×C₅	1	4	yes	—
127	C₂×C₄²	C₂³×C₅	1	2	yes	—
128	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
129	#107	C₂×C₄×C₅	1	8	yes	—
130	#106	C₂×C₄×C₅	1	4	yes	—
131	#106	C₂×C₄×C₅	1	4	yes	—
132	C₄⋊C₈	—	2	4	no	—
133	C₄⋊C₈	—	2	4	no	—
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	2	2	no	—
136	#97	—	2	4	no	—
137	#97	C₂×C₄×C₅	1	4	yes	—
138	C₄×C₈	—	2	4	no	—
139	C₄×C₈	C₅×D₄	1	1	yes	characteristic
140	C₄×C₈	C₅×D₄	1	2	yes	—
141	C₄×C₈	C₅×D₄	1	1	yes	characteristic
142	#95	—	2	2	no	—
143	C₂²×C₅	—	2	2	no	—
144	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
145	C₂²×C₅	#541	1	2	yes	—
146	C₄×C₅	—	2	2	no	—
147	C₄×C₅	—	2	2	no	—
148	C₄×C₅	—	4	4	no	—
149	C₄×C₅	—	4	4	no	—
150	C₄×C₅	—	8	2	no	—
151	C₄×C₅	#519	1	2	yes	—
152	C₄×C₅	#536	1	2	yes	—
153	C₂×Q₈	—	2	2	no	—
154	C₂×Q₈	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	C₅×#41	1	2	yes	—
157	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
158	M₁₆	—	2	8	no	—
159	C₂×C₈	—	2	4	no	—
160	C₂×C₈	—	2	4	no	—
161	C₂×C₈	—	2	4	no	—
162	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
163	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
164	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
165	C₄⋊C₄	—	2	4	no	—
166	C₄⋊C₄	—	2	4	no	—
167	C₄⋊C₄	—	2	2	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	C₂²×C₄×C₅	1	4	yes	—
170	C₄⋊C₄	C₂²×C₄×C₅	1	4	yes	—
171	C₂²⋊C₄	—	2	4	no	—
172	C₄²	—	2	4	no	—
173	C₄²	—	2	4	no	—
174	C₄²	C₂²×C₄×C₅	1	2	yes	—
175	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
176	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
177	C₂×C₅	—	2	2	no	—
178	C₂×C₅	#2327	1	1	yes	characteristic
179	C₂×C₅	#2955	1	2	yes	—
180	C₂³	C₂×C₅×#41	1	1	yes	characteristic
181	Q₈	—	4	4	no	—
182	Q₈	—	4	4	no	—
183	C₂×C₄	—	2	4	no	—
184	C₂×C₄	—	2	4	no	—
185	C₂×C₄	—	4	2	no	—
186	C₂×C₄	—	4	4	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
189	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
190	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
191	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
192	C₂×C₄	C₅×2₋¹⁺⁴	1	4	yes	—
193	C₈	—	2	4	no	—
194	C₈	—	2	2	no	—
195	C₈	—	2	2	no	—
196	C₈	—	4	4	no	—
197	C₅	#22257	1	1	yes	characteristic, direct product factor, Sylow
198	C₂²	—	2	2	no	—
199	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
200	C₂²	C₅×#541	1	2	yes	—
201	C₄	—	2	2	no	—
202	C₄	—	2	2	no	—
203	C₄	—	4	4	no	—
204	C₄	—	4	4	no	—
205	C₄	—	8	2	no	—
206	C₄	C₅×#519	1	2	yes	—
207	C₄	C₅×#536	1	2	yes	—
208	C₂	—	2	2	no	—
209	C₂	C₅×#2327	1	1	yes	characteristic
210	C₂	C₅×#2955	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₂³×C₅	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
8. C₅×#501	C₄	semidirect
9. C₅×#501	C₄	semidirect
24. #22257	C₅	direct (N × H)
83. #501	C₄×C₅	semidirect
84. #501	C₄×C₅	semidirect
197. C₅	#22257	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×#3162	2
C₅×#2960	2
C₅×#1995	2
C₅×#1922	2
C₅×#1838	2
C₅×#1746	2
#22257	5

Subgroups — C5×#22257 #11944977

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22257 #11944977