Subgroups — C₅×#17619 #11940339

Order 1280.

938 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3502	C₂	1	1	yes	maximal, characteristic
2	C₅×#2982	C₂	1	1	yes	maximal, characteristic
3	C₅×#2351	C₂	1	1	yes	maximal, characteristic
4	C₅×#1518	C₂	1	6	yes	maximal
5	C₅×#1503	C₂	1	6	yes	maximal
6	C₅×Q₈²	C₄	1	4	yes	complemented
7	C₅×#557	—	2	4	no	—
8	C₅×#557	C₄	1	4	yes	complemented
9	C₅×#544	—	2	3	no	—
10	C₅×#533	C₂²	1	6	yes	—
11	C₅×#519	—	2	1	no	—
12	C₂×C₄×C₅×Q₈	—	2	1	no	—
13	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
14	C₅×C₈×Q₈	—	2	4	no	—
15	C₅×#431	C₂²	1	6	yes	—
16	C₅×#389	—	2	3	no	—
17	C₅×#382	C₂²	1	3	yes	—
18	C₄×C₅×C₄⋊C₄	C₂²	1	3	yes	—
19	C₅×#328	C₂²	1	4	yes	—
20	C₅×#328	C₂²	1	12	yes	—
21	#17619	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂²×C₅×Q₈	D₄	1	1	yes	characteristic
23	C₅×C₄⋊Q₈	—	2	12	no	—
24	C₅×C₄⋊Q₈	C₂×C₄	1	6	yes	—
25	C₅×C₄⋊Q₈	C₂×C₄	1	6	yes	—
26	C₅×#125	—	2	12	no	—
27	C₅×#125	—	2	6	no	—
28	C₅×C₂²⋊Q₈	—	2	6	no	—
29	C₅×C₂²⋊Q₈	C₂³	1	4	yes	—
30	C₄×C₅×Q₈	—	2	4	no	—
31	C₄×C₅×Q₈	—	2	4	no	—
32	C₄×C₅×Q₈	—	2	6	no	—
33	C₄×C₅×Q₈	—	4	4	no	—
34	C₄×C₅×Q₈	C₂×C₄	1	12	yes	—
35	C₄×C₅×Q₈	C₂×C₄	1	4	yes	—
36	C₄×C₅×Q₈	D₄	1	4	yes	—
37	C₅×#117	—	2	6	no	—
38	C₂×C₅×C₄⋊C₄	—	2	3	no	—
39	C₂×C₅×C₄⋊C₄	—	2	3	no	—
40	C₂×C₅×C₄⋊C₄	D₄	1	3	yes	—
41	C₂×C₄²×C₅	—	2	3	no	—
42	C₂×C₄²×C₅	C₂³	1	3	yes	—
43	C₅×C₄⋊C₈	—	2	6	no	—
44	C₅×C₂²⋊C₈	C₂³	1	4	yes	—
45	C₄×C₅×C₈	—	2	6	no	—
46	C₅×#95	—	2	3	no	—
47	C₅×#95	C₂³	1	1	yes	characteristic
48	C₅×#95	C₂³	1	3	yes	—
49	#3502	C₂×C₅	1	1	yes	characteristic
50	#2982	C₂×C₅	1	1	yes	characteristic
51	#2351	C₂×C₅	1	1	yes	characteristic
52	#1518	C₂×C₅	1	6	yes	—
53	#1503	C₂×C₅	1	6	yes	—
54	C₂×C₅×Q₈	—	2	6	no	—
55	C₂×C₅×Q₈	C₂²⋊C₄	1	4	yes	—
56	C₂×C₅×Q₈	SD₁₆	1	4	yes	—
57	C₂×C₅×Q₈	Q₁₆	1	4	yes	—
58	C₂×C₅×Q₈	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₅	—	2	1	no	—
60	C₂²×C₄×C₅	—	2	3	no	—
61	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
62	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
63	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
64	C₂×C₅×C₈	—	2	4	no	—
65	C₅×C₄⋊C₄	—	2	6	no	—
66	C₅×C₄⋊C₄	—	2	6	no	—
67	C₅×C₄⋊C₄	—	2	6	no	—
68	C₅×C₄⋊C₄	—	2	12	no	—
69	C₅×C₄⋊C₄	—	2	4	no	—
70	C₅×C₄⋊C₄	—	4	6	no	—
71	C₅×C₄⋊C₄	—	4	6	no	—
72	C₅×C₄⋊C₄	C₂²⋊C₄	1	12	yes	—
73	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
74	C₅×C₂²⋊C₄	—	2	4	no	—
75	C₄²×C₅	—	2	12	no	—
76	C₄²×C₅	—	4	6	no	—
77	C₄²×C₅	C₂²×C₄	1	6	yes	—
78	C₄²×C₅	C₂×D₄	1	6	yes	—
79	Q₈²	C₄×C₅	1	4	yes	complemented
80	#557	—	2	4	no	—
81	#557	C₄×C₅	1	4	yes	complemented
82	#544	—	2	3	no	—
83	#533	C₂²×C₅	1	6	yes	—
84	#519	—	2	1	no	—
85	C₂×C₄×Q₈	—	2	1	no	—
86	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
87	C₈×Q₈	—	2	4	no	—
88	#431	C₂²×C₅	1	6	yes	—
89	#389	—	2	3	no	—
90	#382	C₂²×C₅	1	3	yes	—
91	C₄×C₄⋊C₄	C₂²×C₅	1	3	yes	—
92	#328	C₂²×C₅	1	4	yes	—
93	#328	C₂²×C₅	1	12	yes	—
94	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
95	C₅×Q₈	—	4	4	no	—
96	C₅×Q₈	—	8	6	no	—
97	C₅×Q₈	#103	1	16	yes	—
98	C₂×C₄×C₅	—	2	2	no	—
99	C₂×C₄×C₅	—	2	4	no	—
100	C₂×C₄×C₅	—	2	4	no	—
101	C₂×C₄×C₅	—	4	3	no	—
102	C₂×C₄×C₅	—	4	1	no	—
103	C₂×C₄×C₅	—	4	1	no	—
104	C₂×C₄×C₅	—	4	1	no	—
105	C₂×C₄×C₅	—	4	3	no	—
106	C₂×C₄×C₅	—	4	3	no	—
107	C₂×C₄×C₅	—	4	3	no	—
108	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	6	yes	—
109	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	6	yes	—
110	C₂×C₄×C₅	C₂×SD₁₆	1	6	yes	—
111	C₂×C₄×C₅	C₂×Q₁₆	1	6	yes	—
112	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
114	C₅×C₈	—	4	4	no	—
115	C₂²×Q₈	C₅×D₄	1	1	yes	characteristic
116	C₄⋊Q₈	—	2	12	no	—
117	C₄⋊Q₈	C₂×C₄×C₅	1	6	yes	—
118	C₄⋊Q₈	C₂×C₄×C₅	1	6	yes	—
119	#125	—	2	12	no	—
120	#125	—	2	6	no	—
121	C₂²⋊Q₈	—	2	6	no	—
122	C₂²⋊Q₈	C₂³×C₅	1	4	yes	—
123	C₄×Q₈	—	2	4	no	—
124	C₄×Q₈	—	2	4	no	—
125	C₄×Q₈	—	2	6	no	—
126	C₄×Q₈	—	4	4	no	—
127	C₄×Q₈	C₂×C₄×C₅	1	12	yes	—
128	C₄×Q₈	C₂×C₄×C₅	1	4	yes	—
129	C₄×Q₈	C₅×D₄	1	4	yes	—
130	#117	—	2	6	no	—
131	C₂×C₄⋊C₄	—	2	3	no	—
132	C₂×C₄⋊C₄	—	2	3	no	—
133	C₂×C₄⋊C₄	C₅×D₄	1	3	yes	—
134	C₂×C₄²	—	2	3	no	—
135	C₂×C₄²	C₂³×C₅	1	3	yes	—
136	C₄⋊C₈	—	2	6	no	—
137	C₂²⋊C₈	C₂³×C₅	1	4	yes	—
138	C₄×C₈	—	2	6	no	—
139	#95	—	2	3	no	—
140	#95	C₂³×C₅	1	1	yes	characteristic
141	#95	C₂³×C₅	1	3	yes	—
142	C₂²×C₅	—	2	1	no	—
143	C₂²×C₅	—	2	1	no	—
144	C₂²×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
145	C₂²×C₅	C₂²×SD₁₆	1	1	yes	characteristic
146	C₂²×C₅	C₂²×Q₁₆	1	1	yes	characteristic
147	C₄×C₅	—	2	1	no	—
148	C₄×C₅	—	2	1	no	—
149	C₄×C₅	—	2	6	no	—
150	C₄×C₅	—	4	2	no	—
151	C₄×C₅	—	4	4	no	—
152	C₄×C₅	—	4	4	no	—
153	C₄×C₅	—	8	3	no	—
154	C₄×C₅	C₂×#103	1	12	yes	—
155	C₂×Q₈	—	2	6	no	—
156	C₂×Q₈	C₅×C₂²⋊C₄	1	4	yes	—
157	C₂×Q₈	C₅×SD₁₆	1	4	yes	—
158	C₂×Q₈	C₅×Q₁₆	1	4	yes	—
159	C₂×Q₈	C₂²×C₄×C₅	1	4	yes	—
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	2	3	no	—
162	C₂²×C₄	C₂×C₅×D₄	1	3	yes	—
163	C₂²×C₄	C₂×C₅×D₄	1	3	yes	—
164	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
165	C₂×C₈	—	2	4	no	—
166	C₄⋊C₄	—	2	6	no	—
167	C₄⋊C₄	—	2	6	no	—
168	C₄⋊C₄	—	2	6	no	—
169	C₄⋊C₄	—	2	12	no	—
170	C₄⋊C₄	—	2	4	no	—
171	C₄⋊C₄	—	4	6	no	—
172	C₄⋊C₄	—	4	6	no	—
173	C₄⋊C₄	C₅×C₂²⋊C₄	1	12	yes	—
174	C₄⋊C₄	C₂²×C₄×C₅	1	4	yes	—
175	C₂²⋊C₄	—	2	4	no	—
176	C₄²	—	2	12	no	—
177	C₄²	—	4	6	no	—
178	C₄²	C₂²×C₄×C₅	1	6	yes	—
179	C₄²	C₂×C₅×D₄	1	6	yes	—
180	C₂×C₅	—	2	1	no	—
181	C₂×C₅	—	2	1	no	—
182	C₂×C₅	#2884	1	1	yes	characteristic
183	C₂×C₅	C₂²×#103	1	1	yes	characteristic
184	C₂×C₅	#2901	1	1	yes	characteristic
185	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
186	Q₈	—	4	4	no	—
187	Q₈	—	8	6	no	—
188	Q₈	C₅×#103	1	16	yes	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	4	3	no	—
193	C₂×C₄	—	4	1	no	—
194	C₂×C₄	—	4	1	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	3	no	—
197	C₂×C₄	—	4	3	no	—
198	C₂×C₄	—	4	3	no	—
199	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	6	yes	—
200	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	6	yes	—
201	C₂×C₄	C₂×C₅×SD₁₆	1	6	yes	—
202	C₂×C₄	C₂×C₅×Q₁₆	1	6	yes	—
203	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
204	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
205	C₈	—	4	4	no	—
206	C₅	#17619	1	1	yes	characteristic, direct product factor, Sylow
207	C₂²	—	2	1	no	—
208	C₂²	—	2	1	no	—
209	C₂²	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
210	C₂²	C₂²×C₅×SD₁₆	1	1	yes	characteristic
211	C₂²	C₂²×C₅×Q₁₆	1	1	yes	characteristic
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	2	6	no	—
215	C₄	—	4	2	no	—
216	C₄	—	4	4	no	—
217	C₄	—	4	4	no	—
218	C₄	—	8	3	no	—
219	C₄	C₂×C₅×#103	1	12	yes	—
220	C₂	—	2	1	no	—
221	C₂	—	2	1	no	—
222	C₂	C₅×#2884	1	1	yes	characteristic
223	C₂	C₂²×C₅×#103	1	1	yes	characteristic
224	C₂	C₅×#2901	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₂³×C₅	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄×C₅×Q₈ ◁ G

Complements

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

N	H ≅ G/N	type
6. C₅×Q₈²	C₄	semidirect
8. C₅×#557	C₄	semidirect
21. #17619	C₅	direct (N × H)
79. Q₈²	C₄×C₅	semidirect
81. #557	C₄×C₅	semidirect
206. C₅	#17619	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×#3502	2
C₅×#2982	2
C₅×#2351	2
C₅×#1518	2
C₅×#1503	2
#17619	5

Subgroups — C5×#17619 #11940339

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#17619 #11940339