Subgroups — C₅×#13275 #11935995

Order 1280.

398 conjugacy classes of subgroups.

The subgroup lattice has 196 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×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₄×C₅×#107	C₂	1	1	yes	maximal, characteristic
4	C₅×#1380	C₂	1	4	yes	maximal
5	C₂×C₅×C₄⋊Q₈	—	2	1	no	—
6	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
7	C₅×#500	—	2	2	no	—
8	C₅×#500	C₄	1	4	yes	complemented
9	C₂×C₅×#107	—	2	1	no	—
10	C₂×C₅×#107	C₂²	1	2	yes	—
11	C₂×C₅×#107	C₂²	1	2	yes	—
12	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₄×C₅×C₄⋊C₄	—	2	1	no	—
14	C₅×C₄⋊C₁₆	—	2	2	no	—
15	C₅×#340	—	2	1	no	—
16	#13275	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂²×C₅×Q₈	—	4	1	no	—
18	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
19	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
20	C₅×C₄⋊Q₈	—	2	4	no	—
21	C₅×C₄⋊Q₈	—	4	2	no	—
22	C₂×C₅×C₄⋊C₄	—	2	2	no	—
23	C₂×C₅×C₄⋊C₄	—	2	2	no	—
24	C₂×C₅×C₄⋊C₄	—	4	1	no	—
25	C₂×C₄²×C₅	—	4	1	no	—
26	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
27	C₂×C₅×C₁₆	—	2	2	no	—
28	C₂×C₅×C₁₆	—	2	4	no	—
29	C₅×#107	—	2	4	no	—
30	C₅×#107	—	2	2	no	—
31	C₅×#107	—	2	2	no	—
32	C₅×#107	C₂×C₄	1	4	yes	—
33	C₅×#107	Q₈	1	4	yes	complemented
34	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
35	C₄×C₅×C₈	D₄	1	2	yes	—
36	C₅×#95	—	4	1	no	—
37	C₂×#500	C₂×C₅	1	1	yes	characteristic
38	C₂×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic
39	C₄×#107	C₂×C₅	1	1	yes	characteristic
40	#1380	C₂×C₅	1	4	yes	—
41	C₂×C₅×Q₈	—	4	4	no	—
42	C₂×C₅×Q₈	—	8	2	no	—
43	C₂×C₅×Q₈	—	8	2	no	—
44	C₂²×C₄×C₅	—	4	2	no	—
45	C₂²×C₄×C₅	—	4	2	no	—
46	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
47	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
48	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
49	C₂×C₅×C₈	—	2	2	no	—
50	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
51	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
53	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
54	C₂×C₅×C₈	#41	1	2	yes	—
55	C₅×C₄⋊C₄	—	2	4	no	—
56	C₅×C₄⋊C₄	—	2	4	no	—
57	C₅×C₄⋊C₄	—	4	4	no	—
58	C₅×C₄⋊C₄	—	4	2	no	—
59	C₅×C₄⋊C₄	—	4	2	no	—
60	C₄²×C₅	—	8	2	no	—
61	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
62	C₄²×C₅	SD₁₆	1	2	yes	—
63	C₅×C₁₆	—	4	2	no	—
64	C₂×C₄⋊Q₈	—	2	1	no	—
65	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
66	#500	—	2	2	no	—
67	#500	C₄×C₅	1	4	yes	complemented
68	C₂×#107	—	2	1	no	—
69	C₂×#107	C₂²×C₅	1	2	yes	—
70	C₂×#107	C₂²×C₅	1	2	yes	—
71	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
72	C₄×C₄⋊C₄	—	2	1	no	—
73	C₄⋊C₁₆	—	2	2	no	—
74	#340	—	2	1	no	—
75	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
76	C₅×Q₈	—	8	8	no	—
77	C₂×C₄×C₅	—	2	2	no	—
78	C₂×C₄×C₅	—	4	4	no	—
79	C₂×C₄×C₅	—	4	4	no	—
80	C₂×C₄×C₅	—	8	2	no	—
81	C₂×C₄×C₅	—	8	2	no	—
82	C₂×C₄×C₅	—	8	2	no	—
83	C₂×C₄×C₅	—	8	2	no	—
84	C₂×C₄×C₅	#102	1	2	yes	—
85	C₂×C₄×C₅	#102	1	2	yes	—
86	C₂×C₄×C₅	D₁₆	1	2	yes	—
87	C₂×C₄×C₅	SD₃₂	1	2	yes	—
88	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
90	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
91	C₂×C₄×C₅	#137	1	2	yes	—
92	C₅×C₈	—	2	2	no	—
93	C₅×C₈	C₄×Q₈	1	2	yes	—
94	C₅×C₈	#124	1	1	yes	characteristic
95	C₅×C₈	C₄⋊Q₈	1	1	yes	characteristic
96	C₂²×Q₈	—	4	1	no	—
97	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
98	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
99	C₄⋊Q₈	—	2	4	no	—
100	C₄⋊Q₈	—	4	2	no	—
101	C₂×C₄⋊C₄	—	2	2	no	—
102	C₂×C₄⋊C₄	—	2	2	no	—
103	C₂×C₄⋊C₄	—	4	1	no	—
104	C₂×C₄²	—	4	1	no	—
105	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
106	C₂×C₁₆	—	2	2	no	—
107	C₂×C₁₆	—	2	4	no	—
108	#107	—	2	4	no	—
109	#107	—	2	2	no	—
110	#107	—	2	2	no	—
111	#107	C₂×C₄×C₅	1	4	yes	—
112	#107	C₅×Q₈	1	4	yes	complemented
113	C₄×C₈	C₂×C₄×C₅	1	2	yes	—
114	C₄×C₈	C₅×D₄	1	2	yes	—
115	#95	—	4	1	no	—
116	C₂²×C₅	C₂×#102	1	1	yes	characteristic
117	C₂²×C₅	#474	1	2	yes	—
118	C₂²×C₅	C₂×D₁₆	1	1	yes	characteristic
119	C₂²×C₅	C₂×SD₃₂	1	1	yes	characteristic
120	C₂²×C₅	#510	1	2	yes	—
121	C₄×C₅	—	2	2	no	—
122	C₄×C₅	—	8	4	no	—
123	C₄×C₅	—	8	4	no	—
124	C₄×C₅	#357	1	4	yes	—
125	C₄×C₅	#391	1	1	yes	characteristic
126	C₄×C₅	#419	1	1	yes	characteristic
127	C₄×C₅	#475	1	2	yes	—
128	C₂×Q₈	—	4	4	no	—
129	C₂×Q₈	—	8	2	no	—
130	C₂×Q₈	—	8	2	no	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	4	2	no	—
133	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
134	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
135	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
136	C₂×C₈	—	2	2	no	—
137	C₂×C₈	C₅×C₂²⋊C₄	1	2	yes	—
138	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
139	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
140	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
141	C₂×C₈	C₅×#41	1	2	yes	—
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	—	2	4	no	—
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	4	4	no	—
147	C₄²	—	8	2	no	—
148	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
149	C₄²	C₅×SD₁₆	1	2	yes	—
150	C₁₆	—	4	2	no	—
151	C₂×C₅	#1988	1	1	yes	characteristic
152	C₂×C₅	C₂×#357	1	1	yes	characteristic
153	C₂×C₅	#2141	1	1	yes	characteristic
154	C₂×C₅	#2224	1	2	yes	—
155	C₂×C₅	#2225	1	2	yes	—
156	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
157	Q₈	—	8	8	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	4	4	no	—
160	C₂×C₄	—	4	4	no	—
161	C₂×C₄	—	8	2	no	—
162	C₂×C₄	—	8	2	no	—
163	C₂×C₄	—	8	2	no	—
164	C₂×C₄	—	8	2	no	—
165	C₂×C₄	C₅×#102	1	2	yes	—
166	C₂×C₄	C₅×#102	1	2	yes	—
167	C₂×C₄	C₅×D₁₆	1	2	yes	—
168	C₂×C₄	C₅×SD₃₂	1	2	yes	—
169	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
170	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
171	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
172	C₂×C₄	C₅×#137	1	2	yes	—
173	C₈	—	2	2	no	—
174	C₈	C₄×C₅×Q₈	1	2	yes	—
175	C₈	C₅×#124	1	1	yes	characteristic
176	C₈	C₅×C₄⋊Q₈	1	1	yes	characteristic
177	C₅	#13275	1	1	yes	characteristic, direct product factor, Sylow
178	C₂²	C₂×C₅×#102	1	1	yes	characteristic
179	C₂²	C₅×#474	1	2	yes	—
180	C₂²	C₂×C₅×D₁₆	1	1	yes	characteristic
181	C₂²	C₂×C₅×SD₃₂	1	1	yes	characteristic
182	C₂²	C₅×#510	1	2	yes	—
183	C₄	—	2	2	no	—
184	C₄	—	8	4	no	—
185	C₄	—	8	4	no	—
186	C₄	C₅×#357	1	4	yes	—
187	C₄	C₅×#391	1	1	yes	characteristic
188	C₄	C₅×#419	1	1	yes	characteristic
189	C₄	C₅×#475	1	2	yes	—
190	C₂	C₅×#1988	1	1	yes	characteristic
191	C₂	C₂×C₅×#357	1	1	yes	characteristic
192	C₂	C₅×#2141	1	1	yes	characteristic
193	C₂	C₅×#2224	1	2	yes	—
194	C₂	C₅×#2225	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₂²×C₈	C₂³×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
8. C₅×#500	C₄	semidirect
16. #13275	C₅	direct (N × H)
33. C₅×#107	Q₈	semidirect
67. #500	C₄×C₅	semidirect
112. #107	C₅×Q₈	semidirect
177. C₅	#13275	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×C₅×C₄⋊C₁₆	2
C₄×C₅×#107	2
C₅×#1380	2
#13275	5

Subgroups — C5×#13275 #11935995

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#13275 #11935995