Subgroups — C₅×#9902 #11932622

Order 1280.

438 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	C₅×#2859	C₂	1	1	yes	maximal, characteristic
2	C₅×#1842	C₂	1	1	yes	maximal, characteristic
3	C₅×#1767	C₂	1	1	yes	maximal, characteristic
4	C₅×#1296	C₂	1	4	yes	maximal
5	C₂×C₅×#125	—	2	1	no	—
6	C₂×C₅×C₄⋊C₈	—	2	2	no	—
7	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₅×#97	—	2	1	no	—
10	C₅×#389	—	2	2	no	—
11	C₅×#389	C₄	1	4	yes	—
12	C₅×#384	C₂²	1	2	yes	—
13	C₅×#382	—	2	1	no	—
14	C₄³×C₅	C₂²	1	1	yes	characteristic
15	C₅×#336	—	2	1	no	—
16	#9902	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	—	4	4	no	—
20	C₂×C₅×C₄⋊C₄	—	2	4	no	—
21	C₂×C₅×C₄⋊C₄	—	2	2	no	—
22	C₂×C₅×C₄⋊C₄	—	2	2	no	—
23	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₅	D₄	1	2	yes	—
26	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
27	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
28	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
29	C₅×C₄⋊C₈	—	2	2	no	—
30	C₅×C₄⋊C₈	—	2	4	no	—
31	C₅×C₄⋊C₈	—	4	4	no	—
32	C₅×C₄⋊C₈	Q₈	1	4	yes	—
33	C₅×#97	—	4	2	no	—
34	C₅×#95	—	2	4	no	—
35	#2859	C₂×C₅	1	1	yes	characteristic
36	#1842	C₂×C₅	1	1	yes	characteristic
37	#1767	C₂×C₅	1	1	yes	characteristic
38	#1296	C₂×C₅	1	4	yes	—
39	C₂²×C₄×C₅	—	2	4	no	—
40	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
45	C₂×C₅×C₈	—	2	4	no	—
46	C₂×C₅×C₈	—	4	2	no	—
47	C₂×C₅×C₈	—	4	2	no	—
48	C₂×C₅×C₈	—	4	2	no	—
49	C₂×C₅×C₈	—	4	4	no	—
50	C₅×C₄⋊C₄	—	4	4	no	—
51	C₅×C₄⋊C₄	—	4	8	no	—
52	C₅×C₄⋊C₄	—	4	2	no	—
53	C₅×C₄⋊C₄	—	4	4	no	—
54	C₅×C₄⋊C₄	—	4	2	no	—
55	C₄²×C₅	—	2	2	no	—
56	C₄²×C₅	—	2	2	no	—
57	C₄²×C₅	—	2	2	no	—
58	C₄²×C₅	—	4	2	no	—
59	C₄²×C₅	SD₁₆	1	2	yes	—
60	C₄²×C₅	Q₁₆	1	2	yes	—
61	C₄²×C₅	C₂×Q₈	1	2	yes	—
62	C₄²×C₅	#41	1	2	yes	—
63	C₂×#125	—	2	1	no	—
64	C₂×C₄⋊C₈	—	2	2	no	—
65	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
66	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
67	C₂×#97	—	2	1	no	—
68	#389	—	2	2	no	—
69	#389	C₄×C₅	1	4	yes	—
70	#384	C₂²×C₅	1	2	yes	—
71	#382	—	2	1	no	—
72	C₄³	C₂²×C₅	1	1	yes	characteristic
73	#336	—	2	1	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	2	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	—	4	4	no	—
82	C₂×C₄×C₅	—	4	4	no	—
83	C₂×C₄×C₅	—	4	4	no	—
84	C₂×C₄×C₅	#103	1	4	yes	—
85	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
86	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
87	C₂×C₄×C₅	#124	1	1	yes	characteristic
88	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
89	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
90	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
91	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
92	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
93	C₂×C₄×C₅	#137	1	2	yes	—
94	C₅×C₈	—	4	4	no	—
95	C₅×C₈	—	8	2	no	—
96	C₂²×C₈	—	2	2	no	—
97	C₂²×C₈	—	2	2	no	—
98	#125	—	4	4	no	—
99	C₂×C₄⋊C₄	—	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	4	yes	—
103	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
104	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
105	C₂×C₄²	C₅×D₄	1	2	yes	—
106	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
107	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
108	C₄⋊C₈	—	2	2	no	—
109	C₄⋊C₈	—	2	4	no	—
110	C₄⋊C₈	—	4	4	no	—
111	C₄⋊C₈	C₅×Q₈	1	4	yes	—
112	#97	—	4	2	no	—
113	#95	—	2	4	no	—
114	C₂²×C₅	#391	1	1	yes	characteristic
115	C₂²×C₅	#413	1	2	yes	—
116	C₂²×C₅	C₂×#103	1	1	yes	characteristic
117	C₂²×C₅	#417	1	2	yes	—
118	C₂²×C₅	#418	1	1	yes	characteristic
119	C₄×C₅	—	2	4	no	—
120	C₄×C₅	—	2	2	no	—
121	C₄×C₅	—	4	2	no	—
122	C₄×C₅	—	8	4	no	—
123	C₄×C₅	#476	1	2	yes	—
124	C₄×C₅	#477	1	2	yes	—
125	C₄×C₅	#490	1	1	yes	characteristic
126	C₄×C₅	#491	1	1	yes	characteristic
127	C₄×C₅	#501	1	2	yes	—
128	C₂²×C₄	—	2	4	no	—
129	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
130	C₂²×C₄	C₅×C₂²⋊C₄	1	2	yes	—
131	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
132	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
133	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
134	C₂×C₈	—	2	4	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	4	2	no	—
137	C₂×C₈	—	4	2	no	—
138	C₂×C₈	—	4	4	no	—
139	C₄⋊C₄	—	4	4	no	—
140	C₄⋊C₄	—	4	8	no	—
141	C₄⋊C₄	—	4	4	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₄⋊C₄	—	4	2	no	—
144	C₄²	—	2	2	no	—
145	C₄²	—	2	2	no	—
146	C₄²	—	2	2	no	—
147	C₄²	—	4	2	no	—
148	C₄²	C₅×SD₁₆	1	2	yes	—
149	C₄²	C₅×Q₁₆	1	2	yes	—
150	C₄²	C₂×C₅×Q₈	1	2	yes	—
151	C₄²	C₅×#41	1	2	yes	—
152	C₂×C₅	#1550	1	2	yes	—
153	C₂×C₅	#1557	1	2	yes	—
154	C₂×C₅	#1987	1	1	yes	characteristic
155	C₂×C₅	#1992	1	1	yes	characteristic
156	C₂×C₅	#1998	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	2	no	—
161	C₂×C₄	—	2	2	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	4	4	no	—
165	C₂×C₄	—	4	4	no	—
166	C₂×C₄	—	4	4	no	—
167	C₂×C₄	C₅×#103	1	4	yes	—
168	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
169	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
170	C₂×C₄	C₅×#124	1	1	yes	characteristic
171	C₂×C₄	C₅×C₄⋊Q₈	1	1	yes	characteristic
172	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
173	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
174	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
175	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
176	C₂×C₄	C₅×#137	1	2	yes	—
177	C₈	—	4	4	no	—
178	C₈	—	8	2	no	—
179	C₅	#9902	1	1	yes	characteristic, direct product factor, Sylow
180	C₂²	C₅×#391	1	1	yes	characteristic
181	C₂²	C₅×#413	1	2	yes	—
182	C₂²	C₂×C₅×#103	1	1	yes	characteristic
183	C₂²	C₅×#417	1	2	yes	—
184	C₂²	C₅×#418	1	1	yes	characteristic
185	C₄	—	2	4	no	—
186	C₄	—	2	2	no	—
187	C₄	—	4	2	no	—
188	C₄	—	8	4	no	—
189	C₄	C₅×#476	1	2	yes	—
190	C₄	C₅×#477	1	2	yes	—
191	C₄	C₅×#490	1	1	yes	characteristic
192	C₄	C₅×#491	1	1	yes	characteristic
193	C₄	C₅×#501	1	2	yes	—
194	C₂	C₅×#1550	1	2	yes	—
195	C₂	C₅×#1557	1	2	yes	—
196	C₂	C₅×#1987	1	1	yes	characteristic
197	C₂	C₅×#1992	1	1	yes	characteristic
198	C₂	C₅×#1998	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
16. #9902	C₅	direct (N × H)
179. C₅	#9902	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×#2859	2
C₅×#1842	2
C₅×#1767	2
C₅×#1296	2
#9902	5

Subgroups — C5×#9902 #11932622

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#9902 #11932622