Subgroups — C₅×#8278 #11930998

Order 1280.

394 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#1767	C₂	1	1	yes	maximal, characteristic
2	C₄×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₅×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#335	C₂	1	2	yes	maximal
5	C₅×#1277	C₂	1	2	yes	maximal
6	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
7	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
8	C₂×C₅×#97	—	2	1	no	—
9	C₂×C₄×C₅×C₈	—	2	1	no	—
10	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₄³×C₅	C₂²	1	1	yes	characteristic
13	C₅×#336	—	2	1	no	—
14	C₅×#336	—	2	1	no	—
15	C₅×#336	C₄	1	4	yes	—
16	C₅×#335	C₄	1	8	yes	complemented
17	#8278	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
18	C₂²×C₅×C₈	—	2	2	no	—
19	C₂²×C₅×C₈	—	2	2	no	—
20	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
24	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₅	Q₈	1	2	yes	—
26	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
27	C₅×C₄⋊C₈	—	2	2	no	—
28	C₅×C₄⋊C₈	—	2	2	no	—
29	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	—
30	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	—
31	C₅×#97	—	4	2	no	—
32	C₄×C₅×C₈	—	2	2	no	—
33	C₄×C₅×C₈	—	4	2	no	—
34	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
35	#1767	C₂×C₅	1	1	yes	characteristic
36	C₄×C₄⋊C₈	C₂×C₅	1	1	yes	characteristic
37	#1744	C₂×C₅	1	1	yes	characteristic
38	C₂×#335	C₂×C₅	1	2	yes	—
39	#1277	C₂×C₅	1	2	yes	—
40	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
41	C₂²×C₄×C₅	C₄⋊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×Q₈	1	1	yes	characteristic
45	C₂×C₅×C₈	—	2	4	no	—
46	C₂×C₅×C₈	—	2	2	no	—
47	C₂×C₅×C₈	—	2	4	no	—
48	C₂×C₅×C₈	—	2	4	no	—
49	C₂×C₅×C₈	—	2	4	no	—
50	C₂×C₅×C₈	—	4	2	no	—
51	C₂×C₅×C₈	—	4	2	no	—
52	C₂×C₅×C₈	—	4	2	no	—
53	C₂×C₅×C₈	—	4	2	no	—
54	C₂×C₅×C₈	C₄²	1	4	yes	—
55	C₄²×C₅	—	2	4	no	—
56	C₄²×C₅	—	2	2	no	—
57	C₄²×C₅	—	4	2	no	—
58	C₄²×C₅	D₈	1	2	yes	—
59	C₄²×C₅	Q₁₆	1	2	yes	—
60	C₄²×C₅	C₂²×C₄	1	2	yes	—
61	C₄²×C₅	#41	1	2	yes	—
62	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
63	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
64	C₂×#97	—	2	1	no	—
65	C₂×C₄×C₈	—	2	1	no	—
66	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
67	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
68	C₄³	C₂²×C₅	1	1	yes	characteristic
69	#336	—	2	1	no	—
70	#336	—	2	1	no	—
71	#336	C₄×C₅	1	4	yes	—
72	#335	C₄×C₅	1	8	yes	complemented
73	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₄×C₅	—	2	2	no	—
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	#107	1	4	yes	—
81	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₅	#117	1	1	yes	characteristic
83	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
84	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
85	C₂×C₄×C₅	#131	1	4	yes	—
86	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
88	C₂×C₄×C₅	#135	1	2	yes	—
89	C₅×C₈	—	2	4	no	—
90	C₅×C₈	—	4	2	no	—
91	C₅×C₈	—	4	4	no	—
92	C₅×C₈	—	4	4	no	—
93	C₂²×C₈	—	2	2	no	—
94	C₂²×C₈	—	2	2	no	—
95	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
96	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
97	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
98	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
99	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
100	C₂×C₄²	C₅×Q₈	1	2	yes	—
101	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
102	C₄⋊C₈	—	2	2	no	—
103	C₄⋊C₈	—	2	2	no	—
104	C₄⋊C₈	C₂×C₄×C₅	1	4	yes	—
105	C₄⋊C₈	C₂×C₄×C₅	1	4	yes	—
106	#97	—	4	2	no	—
107	C₄×C₈	—	2	2	no	—
108	C₄×C₈	—	4	2	no	—
109	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
110	C₂²×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
111	C₂²×C₅	#424	1	2	yes	—
112	C₂²×C₅	C₂×#107	1	1	yes	characteristic
113	C₂²×C₅	#427	1	1	yes	characteristic
114	C₂²×C₅	#430	1	2	yes	—
115	C₄×C₅	—	2	4	no	—
116	C₄×C₅	—	2	2	no	—
117	C₄×C₅	—	4	2	no	—
118	C₄×C₅	#405	1	2	yes	—
119	C₄×C₅	#433	1	2	yes	—
120	C₄×C₅	C₄×D₈	1	2	yes	—
121	C₄×C₅	C₄×Q₁₆	1	2	yes	—
122	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
123	C₂²×C₄	C₅×C₄⋊C₄	1	2	yes	—
124	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
125	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
126	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
127	C₂×C₈	—	2	4	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	4	no	—
130	C₂×C₈	—	2	4	no	—
131	C₂×C₈	—	2	4	no	—
132	C₂×C₈	—	4	2	no	—
133	C₂×C₈	—	4	2	no	—
134	C₂×C₈	—	4	2	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	C₄²×C₅	1	4	yes	—
137	C₄²	—	2	4	no	—
138	C₄²	—	2	2	no	—
139	C₄²	—	4	2	no	—
140	C₄²	C₅×D₈	1	2	yes	—
141	C₄²	C₅×Q₁₆	1	2	yes	—
142	C₄²	C₂²×C₄×C₅	1	2	yes	—
143	C₄²	C₅×#41	1	2	yes	—
144	C₂×C₅	#1571	1	2	yes	—
145	C₂×C₅	#1572	1	2	yes	—
146	C₂×C₅	#1768	1	1	yes	characteristic
147	C₂×C₅	C₄×#107	1	1	yes	characteristic
148	C₂×C₅	#1778	1	1	yes	characteristic
149	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	C₅×#107	1	4	yes	—
157	C₂×C₄	C₂×C₄²×C₅	1	1	yes	characteristic
158	C₂×C₄	C₅×#117	1	1	yes	characteristic
159	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
160	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
161	C₂×C₄	C₅×#131	1	4	yes	—
162	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
163	C₂×C₄	C₂×C₅×Q₁₆	1	1	yes	characteristic
164	C₂×C₄	C₅×#135	1	2	yes	—
165	C₈	—	2	4	no	—
166	C₈	—	4	2	no	—
167	C₈	—	4	4	no	—
168	C₈	—	4	4	no	—
169	C₅	#8278	1	1	yes	characteristic, direct product factor, Sylow
170	C₂²	C₄×C₅×C₄⋊C₄	1	1	yes	characteristic
171	C₂²	C₅×#424	1	2	yes	—
172	C₂²	C₂×C₅×#107	1	1	yes	characteristic
173	C₂²	C₅×#427	1	1	yes	characteristic
174	C₂²	C₅×#430	1	2	yes	—
175	C₄	—	2	2	no	—
176	C₄	—	2	4	no	—
177	C₄	—	4	2	no	—
178	C₄	C₅×#405	1	2	yes	—
179	C₄	C₅×#433	1	2	yes	—
180	C₄	C₄×C₅×D₈	1	2	yes	—
181	C₄	C₄×C₅×Q₁₆	1	2	yes	—
182	C₂	C₅×#1571	1	2	yes	—
183	C₂	C₅×#1572	1	2	yes	—
184	C₂	C₅×#1768	1	1	yes	characteristic
185	C₂	C₄×C₅×#107	1	1	yes	characteristic
186	C₂	C₅×#1778	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₂×C₄	C₂×C₄²×C₅
Frattini, cosocle	C₂×C₄²	C₂³×C₅
socle, prime radical	C₂³×C₅	C₂×C₄⋊C₄
LCS	G ▷ C₂×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. C₅×#335	C₄	semidirect
17. #8278	C₅	direct (N × H)
72. #335	C₄×C₅	semidirect
169. C₅	#8278	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×#1767	2
C₄×C₅×C₄⋊C₈	2
C₅×#1744	2
C₂×C₅×#335	2
C₅×#1277	2
#8278	5

Subgroups — C5×#8278 #11930998

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#8278 #11930998