Subgroups — C₅×#11968 #11934688

Order 1280.

296 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3160	C₂	1	1	yes	maximal, characteristic
2	C₅×#2162	C₂	1	1	yes	maximal, characteristic
3	C₅×#1566	C₂	1	1	yes	maximal, characteristic
4	C₅×#1363	C₂	1	4	yes	maximal, complemented
5	C₅×#533	—	2	1	no	—
6	C₅×#500	C₄	1	1	yes	characteristic
7	C₅×#500	C₄	1	1	yes	characteristic
8	C₅×#499	—	2	1	no	—
9	C₅×#499	C₂²	1	2	yes	—
10	C₅×#498	—	2	1	no	—
11	C₅×#427	—	2	1	no	—
12	C₅×#427	C₄	1	2	yes	—
13	C₅×#423	—	2	1	no	—
14	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₅×C₄⋊C₁₆	C₂²	1	2	yes	—
16	C₅×C₂²⋊C₁₆	—	2	1	no	—
17	C₄×C₅×C₁₆	—	2	1	no	—
18	C₅×#334	—	2	1	no	—
19	C₅×#334	C₂²	1	2	yes	—
20	#11968	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂×C₅×M₁₆	—	4	1	no	—
22	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
24	C₅×C₄⋊Q₈	—	2	1	no	—
25	C₅×C₄⋊Q₈	—	2	1	no	—
26	C₅×#125	—	2	2	no	—
27	C₅×C₂²⋊Q₈	—	4	1	no	—
28	C₅×C₂²⋊Q₈	—	4	1	no	—
29	C₄×C₅×Q₈	—	4	1	no	—
30	C₄×C₅×Q₈	—	4	1	no	—
31	C₅×#117	—	2	2	no	—
32	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
33	C₂×C₅×C₁₆	—	2	2	no	—
34	C₅×#107	—	2	1	no	—
35	C₅×#107	—	2	1	no	—
36	C₅×#107	C₂×C₄	1	2	yes	—
37	C₅×#107	C₂×C₄	1	2	yes	—
38	C₅×#106	—	2	2	no	—
39	C₅×#106	—	2	2	no	—
40	C₅×C₄⋊C₈	—	2	2	no	—
41	C₅×C₄⋊C₈	—	4	1	no	—
42	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
43	C₄×C₅×C₈	D₄	1	1	yes	characteristic
44	C₄×C₅×C₈	D₄	1	1	yes	characteristic
45	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
46	#3160	C₂×C₅	1	1	yes	characteristic
47	#2162	C₂×C₅	1	1	yes	characteristic
48	#1566	C₂×C₅	1	1	yes	characteristic
49	#1363	C₂×C₅	1	4	yes	complemented
50	C₂×C₅×Q₈	—	4	1	no	—
51	C₂×C₅×Q₈	—	4	1	no	—
52	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
55	C₅×M₁₆	—	8	2	no	—
56	C₂×C₅×C₈	—	2	2	no	—
57	C₂×C₅×C₈	—	2	2	no	—
58	C₂×C₅×C₈	—	4	2	no	—
59	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
63	C₅×C₄⋊C₄	—	2	2	no	—
64	C₅×C₄⋊C₄	—	2	2	no	—
65	C₅×C₄⋊C₄	—	4	1	no	—
66	C₅×C₄⋊C₄	—	4	1	no	—
67	C₅×C₄⋊C₄	—	4	2	no	—
68	C₅×C₄⋊C₄	—	4	2	no	—
69	C₅×C₂²⋊C₄	—	4	2	no	—
70	C₄²×C₅	—	4	2	no	—
71	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₅	D₈	1	1	yes	characteristic
73	C₄²×C₅	SD₁₆	1	1	yes	characteristic
74	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
75	C₅×C₁₆	—	4	2	no	—
76	#533	—	2	1	no	—
77	#500	C₄×C₅	1	1	yes	characteristic
78	#500	C₄×C₅	1	1	yes	characteristic
79	#499	—	2	1	no	—
80	#499	C₂²×C₅	1	2	yes	—
81	#498	—	2	1	no	—
82	#427	—	2	1	no	—
83	#427	C₄×C₅	1	2	yes	—
84	#423	—	2	1	no	—
85	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
86	C₄⋊C₁₆	C₂²×C₅	1	2	yes	—
87	C₂²⋊C₁₆	—	2	1	no	—
88	C₄×C₁₆	—	2	1	no	—
89	#334	—	2	1	no	—
90	#334	C₂²×C₅	1	2	yes	—
91	C₂³×C₅	#102	1	1	yes	characteristic
92	C₅×Q₈	—	8	2	no	—
93	C₅×Q₈	—	8	2	no	—
94	C₂×C₄×C₅	—	2	2	no	—
95	C₂×C₄×C₅	—	2	2	no	—
96	C₂×C₄×C₅	—	2	1	no	—
97	C₂×C₄×C₅	—	2	1	no	—
98	C₂×C₄×C₅	—	4	2	no	—
99	C₂×C₄×C₅	—	4	2	no	—
100	C₂×C₄×C₅	#102	1	1	yes	characteristic
101	C₂×C₄×C₅	#102	1	1	yes	characteristic
102	C₂×C₄×C₅	#102	1	1	yes	characteristic
103	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
106	C₅×C₈	—	2	1	no	—
107	C₅×C₈	—	2	1	no	—
108	C₅×C₈	—	8	2	no	—
109	C₅×C₈	#101	1	2	yes	—
110	C₅×C₈	#137	1	2	yes	—
111	C₂×M₁₆	—	4	1	no	—
112	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
113	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
114	C₄⋊Q₈	—	2	1	no	—
115	C₄⋊Q₈	—	2	1	no	—
116	#125	—	2	2	no	—
117	C₂²⋊Q₈	—	4	1	no	—
118	C₂²⋊Q₈	—	4	1	no	—
119	C₄×Q₈	—	4	1	no	—
120	C₄×Q₈	—	4	1	no	—
121	#117	—	2	2	no	—
122	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
123	C₂×C₁₆	—	2	2	no	—
124	#107	—	2	1	no	—
125	#107	—	2	1	no	—
126	#107	C₂×C₄×C₅	1	2	yes	—
127	#107	C₂×C₄×C₅	1	2	yes	—
128	#106	—	2	2	no	—
129	#106	—	2	2	no	—
130	C₄⋊C₈	—	2	2	no	—
131	C₄⋊C₈	—	4	1	no	—
132	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
133	C₄×C₈	C₅×D₄	1	1	yes	characteristic
134	C₄×C₈	C₅×D₄	1	1	yes	characteristic
135	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
136	C₂²×C₅	—	2	1	no	—
137	C₂²×C₅	—	2	1	no	—
138	C₂²×C₅	—	2	1	no	—
139	C₂²×C₅	C₂×#102	1	1	yes	characteristic
140	C₄×C₅	—	2	1	no	—
141	C₄×C₅	—	2	1	no	—
142	C₄×C₅	—	2	1	no	—
143	C₄×C₅	—	8	2	no	—
144	C₄×C₅	—	8	2	no	—
145	C₄×C₅	#359	1	2	yes	—
146	C₄×C₅	#360	1	2	yes	—
147	C₄×C₅	C₂×#101	1	1	yes	characteristic
148	C₄×C₅	#419	1	1	yes	characteristic
149	C₂×Q₈	—	4	1	no	—
150	C₂×Q₈	—	4	1	no	—
151	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
153	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
154	M₁₆	—	8	2	no	—
155	C₂×C₈	—	2	2	no	—
156	C₂×C₈	—	2	2	no	—
157	C₂×C₈	—	4	2	no	—
158	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
159	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
160	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
161	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
162	C₄⋊C₄	—	2	2	no	—
163	C₄⋊C₄	—	2	2	no	—
164	C₄⋊C₄	—	4	2	no	—
165	C₄⋊C₄	—	4	2	no	—
166	C₄⋊C₄	—	4	1	no	—
167	C₄⋊C₄	—	4	1	no	—
168	C₂²⋊C₄	—	4	2	no	—
169	C₄²	—	4	2	no	—
170	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
171	C₄²	C₅×D₈	1	1	yes	characteristic
172	C₄²	C₅×SD₁₆	1	1	yes	characteristic
173	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
174	C₁₆	—	4	2	no	—
175	C₂×C₅	—	4	1	no	—
176	C₂×C₅	#1546	1	1	yes	characteristic
177	C₂×C₅	C₂×#359	1	1	yes	characteristic
178	C₂×C₅	C₂×#360	1	1	yes	characteristic
179	C₂³	C₅×#102	1	1	yes	characteristic
180	Q₈	—	8	2	no	—
181	Q₈	—	8	2	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	4	2	no	—
187	C₂×C₄	—	4	2	no	—
188	C₂×C₄	C₅×#102	1	1	yes	characteristic
189	C₂×C₄	C₅×#102	1	1	yes	characteristic
190	C₂×C₄	C₅×#102	1	1	yes	characteristic
191	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
192	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
193	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
194	C₈	—	2	1	no	—
195	C₈	—	2	1	no	—
196	C₈	—	8	2	no	—
197	C₈	C₅×#101	1	2	yes	—
198	C₈	C₅×#137	1	2	yes	—
199	C₅	#11968	1	1	yes	characteristic, direct product factor, Sylow
200	C₂²	—	2	1	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	C₂×C₅×#102	1	1	yes	characteristic
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	2	1	no	—
207	C₄	—	8	2	no	—
208	C₄	—	8	2	no	—
209	C₄	C₅×#359	1	2	yes	—
210	C₄	C₅×#360	1	2	yes	—
211	C₄	C₂×C₅×#101	1	1	yes	characteristic
212	C₄	C₅×#419	1	1	yes	characteristic
213	C₂	—	4	1	no	—
214	C₂	C₅×#1546	1	1	yes	characteristic
215	C₂	C₂×C₅×#359	1	1	yes	characteristic
216	C₂	C₂×C₅×#360	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₅
Frattini, cosocle	C₄×C₈	C₂³×C₅
prime radical	C₂³×C₅	#102
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×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×#1363	C₂	semidirect
20. #11968	C₅	direct (N × H)
49. #1363	C₂×C₅	semidirect
199. C₅	#11968	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×#3160	2
C₅×#2162	2
C₅×#1566	2
C₅×#1363	2
#11968	5

Subgroups — C5×#11968 #11934688

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#11968 #11934688