Subgroups — C₅×#8028 #11930748

Order 1280.

778 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×M₁₆	C₂	1	2	yes	maximal
2	C₅×#2285	C₂	1	1	yes	maximal, characteristic
3	C₅×#1295	C₂	1	4	yes	maximal, complemented
4	C₂²×C₅×M₁₆	—	2	2	no	—
5	C₂×C₅×#117	C₄	1	2	yes	—
6	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₄×C₅×M₁₆	—	2	4	no	—
8	C₄×C₅×M₁₆	C₄	1	8	yes	complemented
9	C₂×C₅×#97	C₂²	1	4	yes	—
10	C₂×C₄×C₅×C₈	—	2	2	no	—
11	C₅×#384	C₄	1	2	yes	—
12	C₅×#382	C₂²	1	2	yes	—
13	C₅×#380	—	2	2	no	—
14	C₄×C₅×C₄⋊C₄	—	2	2	no	—
15	#8028	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
17	C₂³×C₄×C₅	D₄	1	2	yes	—
18	C₂×C₅×M₁₆	—	2	16	no	—
19	C₂×C₅×M₁₆	—	2	8	no	—
20	C₂²×C₅×C₈	—	2	4	no	—
21	C₅×#117	—	4	4	no	—
22	C₂×C₅×C₄⋊C₄	—	2	4	no	—
23	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
24	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
25	C₂×C₄²×C₅	—	2	2	no	—
26	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₅	D₄	1	2	yes	—
29	C₂×C₄²×C₅	D₄	1	2	yes	—
30	C₂×C₄²×C₅	Q₈	1	2	yes	—
31	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
32	C₅×#97	—	2	4	no	—
33	C₅×#97	C₂×C₄	1	8	yes	—
34	C₄×C₅×C₈	—	2	8	no	—
35	C₅×#95	—	2	4	no	—
36	C₂×C₄×M₁₆	C₂×C₅	1	2	yes	—
37	#2285	C₂×C₅	1	1	yes	characteristic
38	#1295	C₂×C₅	1	4	yes	complemented
39	C₂⁴×C₅	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅	—	2	2	no	—
41	C₂²×C₄×C₅	—	2	2	no	—
42	C₂²×C₄×C₅	—	2	8	no	—
43	C₂²×C₄×C₅	—	2	4	no	—
44	C₂²×C₄×C₅	C₄²	1	2	yes	—
45	C₂²×C₄×C₅	C₂²⋊C₄	1	4	yes	—
46	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
49	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
52	C₅×M₁₆	—	2	32	no	—
53	C₂×C₅×C₈	—	2	16	no	—
54	C₂×C₅×C₈	—	2	8	no	—
55	C₅×C₄⋊C₄	—	4	4	no	—
56	C₅×C₄⋊C₄	—	4	8	no	—
57	C₅×C₂²⋊C₄	—	4	4	no	—
58	C₄²×C₅	—	4	4	no	—
59	C₄²×C₅	C₄²	1	2	yes	—
60	C₄²×C₅	C₂²⋊C₄	1	4	yes	—
61	C₄²×C₅	C₄⋊C₄	1	2	yes	—
62	C₄²×C₅	C₄⋊C₄	1	4	yes	—
63	C₄²×C₅	C₂²×C₄	1	2	yes	—
64	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
65	C₄²×C₅	C₂×Q₈	1	1	yes	characteristic
66	C₂²×M₁₆	—	2	2	no	—
67	C₂×#117	C₄×C₅	1	2	yes	—
68	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
69	C₄×M₁₆	—	2	4	no	—
70	C₄×M₁₆	C₄×C₅	1	8	yes	complemented
71	C₂×#97	C₂²×C₅	1	4	yes	—
72	C₂×C₄×C₈	—	2	2	no	—
73	#384	C₄×C₅	1	2	yes	—
74	#382	C₂²×C₅	1	2	yes	—
75	#380	—	2	2	no	—
76	C₄×C₄⋊C₄	—	2	2	no	—
77	C₂³×C₅	—	2	1	no	—
78	C₂³×C₅	—	2	1	no	—
79	C₂³×C₅	—	2	4	no	—
80	C₂³×C₅	#95	1	2	yes	—
81	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₄×C₅	—	2	1	no	—
83	C₂×C₄×C₅	—	2	1	no	—
84	C₂×C₄×C₅	—	2	4	no	—
85	C₂×C₄×C₅	—	2	8	no	—
86	C₂×C₄×C₅	—	2	8	no	—
87	C₂×C₄×C₅	—	2	4	no	—
88	C₂×C₄×C₅	—	2	4	no	—
89	C₂×C₄×C₅	—	4	4	no	—
90	C₂×C₄×C₅	—	4	8	no	—
91	C₂×C₄×C₅	#95	1	4	yes	—
92	C₂×C₄×C₅	#95	1	2	yes	—
93	C₂×C₄×C₅	#100	1	2	yes	—
94	C₂×C₄×C₅	#101	1	2	yes	—
95	C₂×C₄×C₅	#104	1	8	yes	—
96	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
97	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
98	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₅×C₈	—	2	16	no	—
101	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
102	C₂³×C₄	C₅×D₄	1	2	yes	—
103	C₂×M₁₆	—	2	16	no	—
104	C₂×M₁₆	—	2	8	no	—
105	C₂²×C₈	—	2	4	no	—
106	#117	—	4	4	no	—
107	C₂×C₄⋊C₄	—	2	4	no	—
108	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
109	C₂×C₂²⋊C₄	—	2	2	no	—
110	C₂×C₄²	—	2	2	no	—
111	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
112	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
113	C₂×C₄²	C₅×D₄	1	2	yes	—
114	C₂×C₄²	C₅×D₄	1	2	yes	—
115	C₂×C₄²	C₅×Q₈	1	2	yes	—
116	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
117	#97	—	2	4	no	—
118	#97	C₂×C₄×C₅	1	8	yes	—
119	C₄×C₈	—	2	8	no	—
120	#95	—	2	4	no	—
121	C₂²×C₅	—	2	4	no	—
122	C₂²×C₅	—	2	2	no	—
123	C₂²×C₅	—	4	2	no	—
124	C₂²×C₅	—	4	2	no	—
125	C₂²×C₅	C₂×#95	1	1	yes	characteristic
126	C₂²×C₅	C₂×#100	1	1	yes	characteristic
127	C₂²×C₅	C₂×#101	1	1	yes	characteristic
128	C₂²×C₅	C₂×#104	1	4	yes	—
129	C₄×C₅	—	2	2	no	—
130	C₄×C₅	—	2	2	no	—
131	C₄×C₅	—	2	1	no	—
132	C₄×C₅	—	2	1	no	—
133	C₄×C₅	—	8	4	no	—
134	C₄×C₅	#339	1	8	yes	—
135	C₄×C₅	#343	1	4	yes	—
136	C₂⁴	C₅×C₂²⋊C₄	1	1	yes	characteristic
137	C₂²×C₄	—	2	2	no	—
138	C₂²×C₄	—	2	2	no	—
139	C₂²×C₄	—	2	8	no	—
140	C₂²×C₄	—	2	4	no	—
141	C₂²×C₄	C₄²×C₅	1	2	yes	—
142	C₂²×C₄	C₅×C₂²⋊C₄	1	4	yes	—
143	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
144	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
145	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
146	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
147	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
148	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
149	M₁₆	—	2	32	no	—
150	C₂×C₈	—	2	16	no	—
151	C₂×C₈	—	2	8	no	—
152	C₄⋊C₄	—	4	4	no	—
153	C₄⋊C₄	—	4	8	no	—
154	C₂²⋊C₄	—	4	4	no	—
155	C₄²	—	4	4	no	—
156	C₄²	C₄²×C₅	1	2	yes	—
157	C₄²	C₅×C₂²⋊C₄	1	4	yes	—
158	C₄²	C₅×C₄⋊C₄	1	4	yes	—
159	C₄²	C₅×C₄⋊C₄	1	2	yes	—
160	C₄²	C₂²×C₄×C₅	1	2	yes	—
161	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
162	C₄²	C₂×C₅×Q₈	1	1	yes	characteristic
163	C₂×C₅	—	4	2	no	—
164	C₂×C₅	#1525	1	2	yes	—
165	C₂×C₅	#1526	1	2	yes	—
166	C₂×C₅	C₂×#339	1	2	yes	—
167	C₂×C₅	C₂×#343	1	1	yes	characteristic
168	C₂³	—	2	1	no	—
169	C₂³	—	2	1	no	—
170	C₂³	—	2	4	no	—
171	C₂³	C₅×#95	1	2	yes	—
172	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
173	C₂×C₄	—	2	1	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	4	no	—
176	C₂×C₄	—	2	8	no	—
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	2	4	no	—
179	C₂×C₄	—	2	4	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	—	4	8	no	—
182	C₂×C₄	C₅×#95	1	4	yes	—
183	C₂×C₄	C₅×#95	1	2	yes	—
184	C₂×C₄	C₅×#100	1	2	yes	—
185	C₂×C₄	C₅×#101	1	2	yes	—
186	C₂×C₄	C₅×#104	1	8	yes	—
187	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
188	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
189	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
190	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
191	C₈	—	2	16	no	—
192	C₅	#8028	1	1	yes	characteristic, direct product factor, Sylow
193	C₂²	—	2	4	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	4	2	no	—
196	C₂²	—	4	2	no	—
197	C₂²	C₂×C₅×#95	1	1	yes	characteristic
198	C₂²	C₂×C₅×#100	1	1	yes	characteristic
199	C₂²	C₂×C₅×#101	1	1	yes	characteristic
200	C₂²	C₂×C₅×#104	1	4	yes	—
201	C₄	—	2	2	no	—
202	C₄	—	2	2	no	—
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	8	4	no	—
206	C₄	C₅×#339	1	8	yes	—
207	C₄	C₅×#343	1	4	yes	—
208	C₂	—	4	2	no	—
209	C₂	C₅×#1525	1	2	yes	—
210	C₂	C₅×#1526	1	2	yes	—
211	C₂	C₂×C₅×#339	1	2	yes	—
212	C₂	C₂×C₅×#343	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle	C₂³×C₅	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₅	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#1295	C₂	semidirect
8. C₄×C₅×M₁₆	C₄	semidirect
15. #8028	C₅	direct (N × H)
38. #1295	C₂×C₅	semidirect
70. C₄×M₁₆	C₄×C₅	semidirect
192. C₅	#8028	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×C₄×C₅×M₁₆	2
C₅×#2285	2
C₅×#1295	2
#8028	5

Subgroups — C5×#8028 #11930748

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8028 #11930748