Subgroups — C₅×#12025 #11934745

Order 1280.

224 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₅×#1568	C₂	1	1	yes	maximal, characteristic
3	C₅×#1567	C₂	1	1	yes	maximal, characteristic
4	C₅×#1372	C₂	1	2	yes	maximal, complemented
5	C₅×#1371	C₂	1	2	yes	maximal, complemented
6	C₂×C₅×M₃₂	C₄	1	2	yes	—
7	C₅×#432	—	2	1	no	—
8	C₅×#423	—	2	1	no	—
9	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₅×#346	C₄	1	1	yes	characteristic
11	C₅×#346	C₄	1	1	yes	characteristic
12	C₄×C₅×C₁₆	C₂²	1	2	yes	—
13	C₅×#335	—	2	1	no	—
14	C₅×#335	C₂²	1	1	yes	characteristic
15	C₅×#335	C₂²	1	2	yes	—
16	C₅×#335	C₂²	1	1	yes	characteristic
17	C₅×#334	—	2	1	no	—
18	#12025	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂×C₅×M₁₆	—	4	1	no	—
20	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₅×C₈	Q₈	1	1	yes	characteristic
22	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
23	C₅×M₃₂	—	2	4	no	—
24	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
25	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
26	C₅×C₄⋊C₈	—	2	1	no	—
27	C₅×C₄⋊C₈	—	2	1	no	—
28	C₅×C₄⋊C₈	—	2	2	no	—
29	C₅×C₄⋊C₈	—	4	1	no	—
30	C₅×C₂²⋊C₈	—	4	1	no	—
31	C₅×#97	—	4	1	no	—
32	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
33	C₄×C₅×C₈	D₄	1	1	yes	characteristic
34	C₄×C₅×C₈	Q₈	1	1	yes	characteristic
35	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
36	C₄×M₃₂	C₂×C₅	1	1	yes	characteristic
37	#1568	C₂×C₅	1	1	yes	characteristic
38	#1567	C₂×C₅	1	1	yes	characteristic
39	#1372	C₂×C₅	1	2	yes	complemented
40	#1371	C₂×C₅	1	2	yes	complemented
41	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
43	C₂²×C₄×C₅	Q₁₆	1	1	yes	characteristic
44	C₅×M₁₆	—	8	2	no	—
45	C₂×C₅×C₈	—	2	2	no	—
46	C₂×C₅×C₈	—	2	2	no	—
47	C₂×C₅×C₈	—	4	2	no	—
48	C₂×C₅×C₈	—	4	1	no	—
49	C₂×C₅×C₈	—	4	1	no	—
50	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
54	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
55	C₄²×C₅	D₈	1	1	yes	characteristic
56	C₄²×C₅	Q₁₆	1	1	yes	characteristic
57	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
58	C₅×C₁₆	—	2	2	no	—
59	C₅×C₁₆	M₁₆	1	4	yes	complemented
60	C₂×M₃₂	C₄×C₅	1	2	yes	—
61	#432	—	2	1	no	—
62	#423	—	2	1	no	—
63	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
64	#346	C₄×C₅	1	1	yes	characteristic
65	#346	C₄×C₅	1	1	yes	characteristic
66	C₄×C₁₆	C₂²×C₅	1	2	yes	—
67	#335	—	2	1	no	—
68	#335	C₂²×C₅	1	2	yes	—
69	#335	C₂²×C₅	1	1	yes	characteristic
70	#335	C₂²×C₅	1	1	yes	characteristic
71	#334	—	2	1	no	—
72	C₂³×C₅	#107	1	1	yes	characteristic
73	C₂×C₄×C₅	—	2	1	no	—
74	C₂×C₄×C₅	—	2	1	no	—
75	C₂×C₄×C₅	—	2	1	no	—
76	C₂×C₄×C₅	—	2	1	no	—
77	C₂×C₄×C₅	—	4	1	no	—
78	C₂×C₄×C₅	#107	1	1	yes	characteristic
79	C₂×C₄×C₅	#107	1	1	yes	characteristic
80	C₂×C₄×C₅	#107	1	1	yes	characteristic
81	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
83	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
84	C₅×C₈	—	2	1	no	—
85	C₅×C₈	—	2	1	no	—
86	C₅×C₈	—	8	1	no	—
87	C₅×C₈	—	8	2	no	—
88	C₅×C₈	—	8	1	no	—
89	C₅×C₈	#108	1	2	yes	—
90	C₅×C₈	C₂×M₁₆	1	2	yes	—
91	C₂×M₁₆	—	4	1	no	—
92	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
93	C₂²×C₈	C₅×Q₈	1	1	yes	characteristic
94	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
95	M₃₂	—	2	4	no	—
96	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
97	C₂×C₁₆	C₂×C₄×C₅	1	2	yes	—
98	C₄⋊C₈	—	2	1	no	—
99	C₄⋊C₈	—	2	1	no	—
100	C₄⋊C₈	—	2	2	no	—
101	C₄⋊C₈	—	4	1	no	—
102	C₂²⋊C₈	—	4	1	no	—
103	#97	—	4	1	no	—
104	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
105	C₄×C₈	C₅×D₄	1	1	yes	characteristic
106	C₄×C₈	C₅×Q₈	1	1	yes	characteristic
107	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
108	C₂²×C₅	—	2	1	no	—
109	C₂²×C₅	—	2	1	no	—
110	C₂²×C₅	—	2	1	no	—
111	C₂²×C₅	C₂×#107	1	1	yes	characteristic
112	C₄×C₅	—	2	1	no	—
113	C₄×C₅	—	2	1	no	—
114	C₄×C₅	—	2	1	no	—
115	C₄×C₅	—	2	1	no	—
116	C₄×C₅	#423	1	1	yes	characteristic
117	C₄×C₅	C₂×#108	1	1	yes	characteristic
118	C₄×C₅	#509	1	1	yes	characteristic
119	C₄×C₅	#510	1	1	yes	characteristic
120	C₂²×C₄	C₅×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
122	C₂²×C₄	C₅×Q₁₆	1	1	yes	characteristic
123	M₁₆	—	8	2	no	—
124	C₂×C₈	—	2	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	4	1	no	—
127	C₂×C₈	—	4	1	no	—
128	C₂×C₈	—	4	2	no	—
129	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₈	C₅×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
132	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
133	C₄²	C₅×C₄⋊C₄	1	1	yes	characteristic
134	C₄²	C₅×D₈	1	1	yes	characteristic
135	C₄²	C₅×Q₁₆	1	1	yes	characteristic
136	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
137	C₁₆	—	2	2	no	—
138	C₁₆	C₅×M₁₆	1	4	yes	complemented
139	C₂×C₅	—	4	1	no	—
140	C₂×C₅	#1567	1	1	yes	characteristic
141	C₂×C₅	#2159	1	1	yes	characteristic
142	C₂×C₅	#2161	1	1	yes	characteristic
143	C₂³	C₅×#107	1	1	yes	characteristic
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	—	2	1	no	—
148	C₂×C₄	—	4	1	no	—
149	C₂×C₄	C₅×#107	1	1	yes	characteristic
150	C₂×C₄	C₅×#107	1	1	yes	characteristic
151	C₂×C₄	C₅×#107	1	1	yes	characteristic
152	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
153	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
154	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
155	C₈	—	2	1	no	—
156	C₈	—	2	1	no	—
157	C₈	—	8	2	no	—
158	C₈	—	8	1	no	—
159	C₈	—	8	1	no	—
160	C₈	C₅×#108	1	2	yes	—
161	C₈	C₂×C₅×M₁₆	1	2	yes	—
162	C₅	#12025	1	1	yes	characteristic, direct product factor, Sylow
163	C₂²	—	2	1	no	—
164	C₂²	—	2	1	no	—
165	C₂²	—	2	1	no	—
166	C₂²	C₂×C₅×#107	1	1	yes	characteristic
167	C₄	—	2	1	no	—
168	C₄	—	2	1	no	—
169	C₄	—	2	1	no	—
170	C₄	—	2	1	no	—
171	C₄	C₅×#423	1	1	yes	characteristic
172	C₄	C₂×C₅×#108	1	1	yes	characteristic
173	C₄	C₅×#509	1	1	yes	characteristic
174	C₄	C₅×#510	1	1	yes	characteristic
175	C₂	—	4	1	no	—
176	C₂	C₅×#1567	1	1	yes	characteristic
177	C₂	C₅×#2159	1	1	yes	characteristic
178	C₂	C₅×#2161	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₄×C₈	C₂³×C₅
socle	C₂²×C₅	C₂×#107
prime radical	C₂³×C₅	#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×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
4. C₅×#1372	C₂	semidirect
5. C₅×#1371	C₂	semidirect
18. #12025	C₅	direct (N × H)
39. #1372	C₂×C₅	semidirect
40. #1371	C₂×C₅	semidirect
59. C₅×C₁₆	M₁₆	semidirect
138. C₁₆	C₅×M₁₆	semidirect
162. C₅	#12025	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₅×M₃₂	2
C₅×#1568	2
C₅×#1567	2
C₅×#1372	2
C₅×#1371	2
#12025	5

Subgroups — C5×#12025 #11934745

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#12025 #11934745