Subgroups — C₅×#9875 #11932595

Order 1280.

434 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₅×#125	C₂	1	1	yes	maximal, characteristic
2	C₅×#1831	C₂	1	1	yes	maximal, characteristic
3	C₅×#1744	C₂	1	1	yes	maximal, characteristic
4	C₅×#1279	C₂	1	4	yes	maximal
5	C₂×C₅×#125	C₄	1	1	yes	characteristic
6	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
8	C₅×#389	C₄	1	2	yes	—
9	C₅×#387	—	2	2	no	—
10	C₅×#382	C₄	1	1	yes	characteristic
11	C₄×C₅×C₄⋊C₄	—	2	2	no	—
12	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
13	C₄³×C₅	C₂²	1	1	yes	characteristic
14	C₅×#336	—	2	2	no	—
15	C₅×#336	—	2	2	no	—
16	#9875	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
17	C₂²×C₅×C₈	—	2	2	no	—
18	C₂²×C₅×C₈	—	2	2	no	—
19	C₅×#125	—	2	4	no	—
20	C₂×C₅×C₄⋊C₄	—	2	2	no	—
21	C₂×C₅×C₄⋊C₄	—	2	2	no	—
22	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
23	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₅	—	2	2	no	—
25	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄²×C₅	D₄	1	2	yes	—
28	C₂×C₄²×C₅	D₄	1	2	yes	—
29	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
30	C₅×C₄⋊C₈	—	2	4	no	—
31	C₄×C₅×C₈	—	2	2	no	—
32	C₄×C₅×C₈	Q₈	1	4	yes	—
33	C₅×#95	—	2	2	no	—
34	C₅×#95	—	2	2	no	—
35	C₅×#95	—	2	2	no	—
36	C₄×#125	C₂×C₅	1	1	yes	characteristic
37	#1831	C₂×C₅	1	1	yes	characteristic
38	#1744	C₂×C₅	1	1	yes	characteristic
39	#1279	C₂×C₅	1	4	yes	—
40	C₂²×C₄×C₅	—	2	2	no	—
41	C₂²×C₄×C₅	—	2	2	no	—
42	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
43	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
44	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
46	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
47	C₂×C₅×C₈	—	2	4	no	—
48	C₂×C₅×C₈	—	4	4	no	—
49	C₂×C₅×C₈	—	4	2	no	—
50	C₂×C₅×C₈	—	4	4	no	—
51	C₅×C₄⋊C₄	—	2	8	no	—
52	C₅×C₄⋊C₄	—	2	2	no	—
53	C₅×C₄⋊C₄	—	2	4	no	—
54	C₅×C₄⋊C₄	—	4	4	no	—
55	C₅×C₄⋊C₄	M₁₆	1	4	yes	—
56	C₄²×C₅	—	2	2	no	—
57	C₄²×C₅	—	2	2	no	—
58	C₄²×C₅	—	2	4	no	—
59	C₄²×C₅	—	4	2	no	—
60	C₄²×C₅	—	4	4	no	—
61	C₄²×C₅	C₂×Q₈	1	2	yes	—
62	C₄²×C₅	#41	1	2	yes	—
63	C₂×#125	C₄×C₅	1	1	yes	characteristic
64	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
65	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
66	#389	C₄×C₅	1	2	yes	—
67	#387	—	2	2	no	—
68	#382	C₄×C₅	1	1	yes	characteristic
69	C₄×C₄⋊C₄	—	2	2	no	—
70	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
71	C₄³	C₂²×C₅	1	1	yes	characteristic
72	#336	—	2	2	no	—
73	#336	—	2	2	no	—
74	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₅	—	2	2	no	—
76	C₂×C₄×C₅	—	2	2	no	—
77	C₂×C₄×C₅	—	2	4	no	—
78	C₂×C₄×C₅	—	2	2	no	—
79	C₂×C₄×C₅	—	2	2	no	—
80	C₂×C₄×C₅	—	2	2	no	—
81	C₂×C₄×C₅	—	2	2	no	—
82	C₂×C₄×C₅	—	2	2	no	—
83	C₂×C₄×C₅	—	4	2	no	—
84	C₂×C₄×C₅	—	4	2	no	—
85	C₂×C₄×C₅	—	4	2	no	—
86	C₂×C₄×C₅	—	4	2	no	—
87	C₂×C₄×C₅	—	4	2	no	—
88	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
89	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
90	C₂×C₄×C₅	#124	1	1	yes	characteristic
91	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
92	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
93	C₂×C₄×C₅	#131	1	2	yes	—
94	C₂×C₄×C₅	#135	1	4	yes	—
95	C₅×C₈	—	4	4	no	—
96	C₅×C₈	—	8	2	no	—
97	C₂²×C₈	—	2	2	no	—
98	C₂²×C₈	—	2	2	no	—
99	#125	—	2	4	no	—
100	C₂×C₄⋊C₄	—	2	2	no	—
101	C₂×C₄⋊C₄	—	2	2	no	—
102	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
103	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
104	C₂×C₄²	—	2	2	no	—
105	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
106	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
107	C₂×C₄²	C₅×D₄	1	2	yes	—
108	C₂×C₄²	C₅×D₄	1	2	yes	—
109	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
110	C₄⋊C₈	—	2	4	no	—
111	C₄×C₈	—	2	2	no	—
112	C₄×C₈	C₅×Q₈	1	4	yes	—
113	#95	—	2	2	no	—
114	#95	—	2	2	no	—
115	#95	—	2	2	no	—
116	C₂²×C₅	#391	1	1	yes	characteristic
117	C₂²×C₅	#407	1	1	yes	characteristic
118	C₂²×C₅	#408	1	1	yes	characteristic
119	C₂²×C₅	#416	1	2	yes	—
120	C₂²×C₅	#421	1	2	yes	—
121	C₄×C₅	—	2	2	no	—
122	C₄×C₅	—	4	2	no	—
123	C₄×C₅	—	4	2	no	—
124	C₄×C₅	—	4	4	no	—
125	C₄×C₅	—	8	2	no	—
126	C₄×C₅	#446	1	4	yes	—
127	C₄×C₅	#488	1	2	yes	—
128	C₄×C₅	#499	1	2	yes	—
129	C₂²×C₄	—	2	2	no	—
130	C₂²×C₄	—	2	2	no	—
131	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
132	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
133	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
134	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
135	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
136	C₂×C₈	—	2	4	no	—
137	C₂×C₈	—	4	4	no	—
138	C₂×C₈	—	4	2	no	—
139	C₂×C₈	—	4	4	no	—
140	C₄⋊C₄	—	2	8	no	—
141	C₄⋊C₄	—	2	2	no	—
142	C₄⋊C₄	—	2	4	no	—
143	C₄⋊C₄	—	4	4	no	—
144	C₄⋊C₄	C₅×M₁₆	1	4	yes	—
145	C₄²	—	2	2	no	—
146	C₄²	—	2	2	no	—
147	C₄²	—	2	4	no	—
148	C₄²	—	4	2	no	—
149	C₄²	—	4	4	no	—
150	C₄²	C₂×C₅×Q₈	1	2	yes	—
151	C₄²	C₅×#41	1	2	yes	—
152	C₂×C₅	#1482	1	2	yes	—
153	C₂×C₅	#1488	1	2	yes	—
154	C₂×C₅	#1985	1	1	yes	characteristic
155	C₂×C₅	#1991	1	1	yes	characteristic
156	C₂×C₅	#1995	1	1	yes	characteristic
157	C₂³	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	4	2	no	—
167	C₂×C₄	—	4	2	no	—
168	C₂×C₄	—	4	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
172	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
173	C₂×C₄	C₅×#124	1	1	yes	characteristic
174	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
175	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
176	C₂×C₄	C₅×#131	1	2	yes	—
177	C₂×C₄	C₅×#135	1	4	yes	—
178	C₈	—	4	4	no	—
179	C₈	—	8	2	no	—
180	C₅	#9875	1	1	yes	characteristic, direct product factor, Sylow
181	C₂²	C₅×#391	1	1	yes	characteristic
182	C₂²	C₅×#407	1	1	yes	characteristic
183	C₂²	C₅×#408	1	1	yes	characteristic
184	C₂²	C₅×#416	1	2	yes	—
185	C₂²	C₅×#421	1	2	yes	—
186	C₄	—	2	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	4	2	no	—
189	C₄	—	4	4	no	—
190	C₄	—	8	2	no	—
191	C₄	C₅×#446	1	4	yes	—
192	C₄	C₅×#488	1	2	yes	—
193	C₄	C₅×#499	1	2	yes	—
194	C₂	C₅×#1482	1	2	yes	—
195	C₂	C₅×#1488	1	2	yes	—
196	C₂	C₅×#1985	1	1	yes	characteristic
197	C₂	C₅×#1991	1	1	yes	characteristic
198	C₂	C₅×#1995	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle, prime radical	C₂³×C₅	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₄³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
16. #9875	C₅	direct (N × H)
180. C₅	#9875	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₅×#125	2
C₅×#1831	2
C₅×#1744	2
C₅×#1279	2
#9875	5

Subgroups — C5×#9875 #11932595

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#9875 #11932595