Subgroups — C₅×#10844 #11933564

Order 1280.

814 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#423	C₂	1	2	yes	maximal
2	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#341	C₂	1	4	yes	maximal
4	C₂²×C₅×M₁₆	C₂²	1	4	yes	—
5	C₂³×C₅×C₈	C₂²	1	2	yes	—
6	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₅×#423	—	2	4	no	—
8	C₅×#423	C₄	1	8	yes	—
9	C₂×C₅×C₄⋊C₈	—	2	4	no	—
10	C₂×C₄×C₅×C₈	—	2	4	no	—
11	C₂×C₄×C₅×C₈	C₄	1	2	yes	—
12	C₂×C₄×C₅×C₈	C₄	1	2	yes	—
13	C₅×#341	—	2	8	no	—
14	#10844	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
15	C₂³×C₄×C₅	Q₈	1	2	yes	—
16	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
17	C₂×C₅×M₁₆	—	2	4	no	—
18	C₂×C₅×M₁₆	—	2	8	no	—
19	C₂×C₅×M₁₆	—	2	8	no	—
20	C₂×C₅×M₁₆	C₂×C₄	1	8	yes	—
21	C₂²×C₅×C₈	—	2	4	no	—
22	C₂²×C₅×C₈	—	2	4	no	—
23	C₂²×C₅×C₈	—	2	4	no	—
24	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
25	C₂²×C₅×C₈	D₄	1	4	yes	—
26	C₂²×C₅×C₈	Q₈	1	4	yes	—
27	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₅	D₄	1	4	yes	—
31	C₂×C₄²×C₅	D₄	1	2	yes	—
32	C₅×C₄⋊C₈	—	2	16	no	—
33	C₄×C₅×C₈	—	2	4	no	—
34	C₄×C₅×C₈	—	2	4	no	—
35	C₂×#423	C₂×C₅	1	2	yes	—
36	C₂²×C₄×C₈	C₂×C₅	1	1	yes	characteristic
37	C₂×#341	C₂×C₅	1	4	yes	—
38	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
39	C₂²×C₄×C₅	—	2	8	no	—
40	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
41	C₂²×C₄×C₅	C₄⋊C₄	1	8	yes	—
42	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
43	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
45	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
46	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
47	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
49	C₂²×C₄×C₅	#41	1	4	yes	—
50	C₅×M₁₆	—	2	16	no	—
51	C₅×M₁₆	—	4	8	no	—
52	C₂×C₅×C₈	—	2	4	no	—
53	C₂×C₅×C₈	—	2	4	no	—
54	C₂×C₅×C₈	—	2	8	no	—
55	C₂×C₅×C₈	—	2	8	no	—
56	C₂×C₅×C₈	—	2	8	no	—
57	C₂×C₅×C₈	—	4	4	no	—
58	C₂×C₅×C₈	—	4	4	no	—
59	C₂×C₅×C₈	C₄⋊C₄	1	8	yes	—
60	C₄²×C₅	C₄²	1	2	yes	—
61	C₄²×C₅	C₄²	1	2	yes	—
62	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
63	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
64	C₄²×C₅	C₂²⋊C₄	1	8	yes	—
65	C₂²×M₁₆	C₂²×C₅	1	4	yes	—
66	C₂³×C₈	C₂²×C₅	1	2	yes	—
67	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
68	#423	—	2	4	no	—
69	#423	C₄×C₅	1	8	yes	—
70	C₂×C₄⋊C₈	—	2	4	no	—
71	C₂×C₄×C₈	—	2	4	no	—
72	C₂×C₄×C₈	C₄×C₅	1	2	yes	—
73	C₂×C₄×C₈	C₄×C₅	1	2	yes	—
74	#341	—	2	8	no	—
75	C₂³×C₅	—	2	4	no	—
76	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
77	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂³×C₅	#125	1	2	yes	—
79	C₂³×C₅	C₄⋊Q₈	1	2	yes	—
80	C₂×C₄×C₅	—	2	4	no	—
81	C₂×C₄×C₅	—	2	16	no	—
82	C₂×C₄×C₅	#95	1	4	yes	—
83	C₂×C₄×C₅	#95	1	4	yes	—
84	C₂×C₄×C₅	#108	1	8	yes	—
85	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
86	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	2	yes	—
88	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
89	C₂×C₄×C₅	#117	1	2	yes	—
90	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
91	C₂×C₄×C₅	C₄×Q₈	1	4	yes	—
92	C₂×C₄×C₅	#121	1	4	yes	—
93	C₂×C₄×C₅	C₂²⋊Q₈	1	4	yes	—
94	C₂×C₄×C₅	#124	1	2	yes	—
95	C₂×C₄×C₅	#127	1	1	yes	characteristic
96	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
97	C₅×C₈	—	2	8	no	—
98	C₅×C₈	—	4	8	no	—
99	C₂³×C₄	C₅×Q₈	1	2	yes	—
100	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
101	C₂×M₁₆	—	2	4	no	—
102	C₂×M₁₆	—	2	8	no	—
103	C₂×M₁₆	—	2	8	no	—
104	C₂×M₁₆	C₂×C₄×C₅	1	8	yes	—
105	C₂²×C₈	—	2	4	no	—
106	C₂²×C₈	—	2	4	no	—
107	C₂²×C₈	—	2	4	no	—
108	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
109	C₂²×C₈	C₅×D₄	1	4	yes	—
110	C₂²×C₈	C₅×Q₈	1	4	yes	—
111	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
112	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
113	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
114	C₂×C₄²	C₅×D₄	1	4	yes	—
115	C₂×C₄²	C₅×D₄	1	2	yes	—
116	C₄⋊C₈	—	2	16	no	—
117	C₄×C₈	—	2	4	no	—
118	C₄×C₈	—	2	4	no	—
119	C₂²×C₅	—	2	4	no	—
120	C₂²×C₅	—	2	4	no	—
121	C₂²×C₅	—	2	4	no	—
122	C₂²×C₅	C₂×#95	1	1	yes	characteristic
123	C₂²×C₅	#382	1	1	yes	characteristic
124	C₂²×C₅	#384	1	1	yes	characteristic
125	C₂²×C₅	#389	1	4	yes	—
126	C₂²×C₅	C₂×#108	1	4	yes	—
127	C₄×C₅	—	2	4	no	—
128	C₄×C₅	#341	1	8	yes	complemented
129	C₄×C₅	C₄×C₄⋊C₄	1	2	yes	—
130	C₄×C₅	#380	1	2	yes	—
131	C₄×C₅	#390	1	2	yes	—
132	C₄×C₅	#391	1	2	yes	—
133	C₂⁴	C₂×C₅×Q₈	1	1	yes	characteristic
134	C₂²×C₄	—	2	8	no	—
135	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
136	C₂²×C₄	C₅×C₄⋊C₄	1	8	yes	—
137	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
138	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
139	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
140	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
141	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
142	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
143	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
144	C₂²×C₄	C₅×#41	1	4	yes	—
145	M₁₆	—	2	16	no	—
146	M₁₆	—	4	8	no	—
147	C₂×C₈	—	2	4	no	—
148	C₂×C₈	—	2	4	no	—
149	C₂×C₈	—	2	8	no	—
150	C₂×C₈	—	2	8	no	—
151	C₂×C₈	—	2	8	no	—
152	C₂×C₈	—	4	4	no	—
153	C₂×C₈	—	4	4	no	—
154	C₂×C₈	C₅×C₄⋊C₄	1	8	yes	—
155	C₄²	C₄²×C₅	1	2	yes	—
156	C₄²	C₄²×C₅	1	2	yes	—
157	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
158	C₄²	C₅×C₂²⋊C₄	1	2	yes	—
159	C₄²	C₅×C₂²⋊C₄	1	8	yes	—
160	C₂×C₅	—	2	4	no	—
161	C₂×C₅	C₄⋊#95	1	1	yes	characteristic
162	C₂×C₅	C₂×#341	1	2	yes	—
163	C₂×C₅	#1837	1	2	yes	—
164	C₂×C₅	#1848	1	2	yes	—
165	C₂³	—	2	4	no	—
166	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
167	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
168	C₂³	C₅×#125	1	2	yes	—
169	C₂³	C₅×C₄⋊Q₈	1	2	yes	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	16	no	—
172	C₂×C₄	C₅×#95	1	4	yes	—
173	C₂×C₄	C₅×#95	1	4	yes	—
174	C₂×C₄	C₅×#108	1	8	yes	—
175	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
176	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
177	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	2	yes	—
178	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
179	C₂×C₄	C₅×#117	1	2	yes	—
180	C₂×C₄	C₄×C₅×D₄	1	4	yes	—
181	C₂×C₄	C₄×C₅×Q₈	1	4	yes	—
182	C₂×C₄	C₅×#121	1	4	yes	—
183	C₂×C₄	C₅×C₂²⋊Q₈	1	4	yes	—
184	C₂×C₄	C₅×#124	1	2	yes	—
185	C₂×C₄	C₅×#127	1	1	yes	characteristic
186	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
187	C₈	—	2	8	no	—
188	C₈	—	4	8	no	—
189	C₅	#10844	1	1	yes	characteristic, direct product factor, Sylow
190	C₂²	—	2	4	no	—
191	C₂²	—	2	4	no	—
192	C₂²	—	2	4	no	—
193	C₂²	C₂×C₅×#95	1	1	yes	characteristic
194	C₂²	C₅×#382	1	1	yes	characteristic
195	C₂²	C₅×#384	1	1	yes	characteristic
196	C₂²	C₅×#389	1	4	yes	—
197	C₂²	C₂×C₅×#108	1	4	yes	—
198	C₄	—	2	4	no	—
199	C₄	C₅×#341	1	8	yes	complemented
200	C₄	C₄×C₅×C₄⋊C₄	1	2	yes	—
201	C₄	C₅×#380	1	2	yes	—
202	C₄	C₅×#390	1	2	yes	—
203	C₄	C₅×#391	1	2	yes	—
204	C₂	—	2	4	no	—
205	C₂	C₅×C₄⋊#95	1	1	yes	characteristic
206	C₂	C₂×C₅×#341	1	2	yes	—
207	C₂	C₅×#1837	1	2	yes	—
208	C₂	C₅×#1848	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₅
Frattini, cosocle	C₂³×C₄	C₂³×C₅
socle	C₂³×C₅	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ 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
14. #10844	C₅	direct (N × H)
128. C₄×C₅	#341	semidirect
189. C₅	#10844	direct (N × H)
199. C₄	C₅×#341	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#423	2
C₂²×C₄×C₅×C₈	2
C₂×C₅×#341	2
#10844	5

Subgroups — C5×#10844 #11933564

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#10844 #11933564