Subgroups — #12099451

Order 1280.

1158 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄³×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂×#65213	C₂	1	2	yes	maximal
3	#10602320	C₂	1	4	yes	maximal
4	#10600185	C₂	1	8	yes	maximal, complemented
5	C₂²×C₄²×C₅	C₄	1	4	yes	—
6	C₂²×C₄²×C₅	C₂²	1	2	yes	—
7	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₂²×#681	C₂²	1	4	yes	—
9	C₄³×C₅	C₄	1	2	yes	—
10	C₄³×C₅	C₄	1	2	yes	—
11	C₄³×C₅	C₂²	1	4	yes	—
12	#65213	—	2	8	no	—
13	C₂×C₄⋊#168	C₂²	1	8	yes	—
14	#64744	C₂²	1	16	yes	—
15	#20752	—	5	1	no	maximal, Sylow
16	C₂³×C₄×C₅	C₂×C₄	1	4	yes	—
17	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
18	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
19	C₂×C₄²×C₅	—	2	8	no	—
20	C₂×C₄²×C₅	—	2	16	no	—
21	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
22	C₂×C₄²×C₅	C₂×C₄	1	8	yes	—
23	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₅	D₄	1	4	yes	—
28	C₂×C₄²×C₅	Q₈	1	4	yes	—
29	C₂×C₄²×C₅	C₂³	1	4	yes	—
30	C₂×C₄²×C₅	C₂³	1	2	yes	—
31	C₂×#681	—	2	8	no	—
32	C₂×#681	—	2	16	no	—
33	C₂²×#168	C₂³	1	8	yes	—
34	C₄⋊#168	—	2	16	no	—
35	C₂×#423	—	5	2	no	—
36	C₂×C₄³	D₅	1	1	yes	characteristic
37	#1831	—	5	8	no	—
38	#1728	—	5	4	no	—
39	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅	—	2	8	no	—
41	C₂²×C₄×C₅	—	2	16	no	—
42	C₂²×C₄×C₅	—	2	8	no	—
43	C₂²×C₄×C₅	C₄⋊C₄	1	8	yes	—
44	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
45	C₂²×C₄×C₅	C₂²×C₄	1	8	yes	—
46	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
48	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
50	C₂²×C₄×C₅	#41	1	8	yes	—
51	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
52	C₄²×C₅	—	2	8	no	—
53	C₄²×C₅	—	2	16	no	—
54	C₄²×C₅	—	2	8	no	—
55	C₄²×C₅	—	2	8	no	—
56	C₄²×C₅	C₄⋊C₄	1	4	yes	—
57	C₄²×C₅	C₄⋊C₄	1	4	yes	—
58	C₄²×C₅	M₁₆	1	16	yes	complemented
59	C₄²×C₅	C₂×D₄	1	4	yes	—
60	C₄²×C₅	C₂×Q₈	1	4	yes	—
61	#681	—	4	16	no	—
62	C₂×#168	—	2	8	no	—
63	C₂×#168	—	2	16	no	—
64	C₂²×M₁₆	—	5	4	no	—
65	C₂²×C₄²	Dic₅	1	4	yes	—
66	C₂²×C₄²	C₂×D₅	1	2	yes	—
67	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
68	#423	—	10	8	no	—
69	C₂×C₄⋊C₈	—	5	8	no	—
70	C₄³	Dic₅	1	2	yes	—
71	C₄³	Dic₅	1	2	yes	—
72	C₄³	C₂×D₅	1	4	yes	—
73	#336	—	5	16	no	—
74	C₂³×C₅	—	2	4	no	—
75	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
76	C₂³×C₅	#117	1	4	yes	—
77	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
78	C₂×C₄×C₅	—	2	32	no	—
79	C₂×C₄×C₅	—	2	8	no	—
80	C₂×C₄×C₅	—	2	4	no	—
81	C₂×C₄×C₅	—	2	8	no	—
82	C₂×C₄×C₅	—	2	16	no	—
83	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
84	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
85	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
86	C₂×C₄×C₅	#117	1	4	yes	—
87	C₂×C₄×C₅	#124	1	8	yes	—
88	C₂×C₄×C₅	#125	1	8	yes	—
89	C₂×C₄×C₅	C₂×M₁₆	1	16	yes	—
90	C₂×C₄×C₅	C₂×M₁₆	1	8	yes	—
91	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
92	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
93	C₂×C₄×C₅	C₂×#41	1	4	yes	—
94	#168	—	4	8	no	—
95	C₂³×C₄	C₂×Dic₅	1	4	yes	—
96	C₂³×C₄	C₂×Dic₅	1	2	yes	—
97	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
98	C₂×M₁₆	—	10	8	no	—
99	C₂×M₁₆	—	10	16	no	—
100	C₂²×C₈	—	5	8	no	—
101	C₂×C₄²	—	2	8	no	—
102	C₂×C₄²	—	2	16	no	—
103	C₂×C₄²	Dic₁₀	1	4	yes	—
104	C₂×C₄²	D₂₀	1	4	yes	—
105	C₂×C₄²	C₂×Dic₅	1	4	yes	—
106	C₂×C₄²	C₂×Dic₅	1	8	yes	—
107	C₂×C₄²	C₂×Dic₅	1	4	yes	—
108	C₂×C₄²	C₂×Dic₅	1	4	yes	—
109	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
110	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
111	C₂×C₄²	C₂²×D₅	1	4	yes	—
112	C₂×C₄²	C₂²×D₅	1	2	yes	—
113	C₄⋊C₈	—	10	16	no	—
114	C₂²×C₅	—	2	4	no	—
115	C₂²×C₅	—	2	8	no	—
116	C₂²×C₅	#382	1	4	yes	—
117	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
118	C₂²×C₅	C₂×#117	1	2	yes	—
119	C₂²×C₅	C₂²×M₁₆	1	4	yes	—
120	C₄×C₅	—	2	4	no	—
121	C₄×C₅	—	2	16	no	—
122	C₄×C₅	#382	1	4	yes	—
123	C₄×C₅	#423	1	8	yes	—
124	C₄×C₅	C₂×#124	1	2	yes	—
125	C₄×C₅	C₂×#125	1	2	yes	—
126	C₂⁴	C₂²×Dic₅	1	1	yes	characteristic
127	C₂²×C₄	—	2	8	no	—
128	C₂²×C₄	—	2	16	no	—
129	C₂²×C₄	—	2	8	no	—
130	C₂²×C₄	C₄⋊Dic₅	1	8	yes	—
131	C₂²×C₄	C₂×Dic₁₀	1	2	yes	—
132	C₂²×C₄	C₂×D₂₀	1	2	yes	—
133	C₂²×C₄	#709	1	8	yes	—
134	C₂²×C₄	C₂²×Dic₅	1	8	yes	—
135	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
136	C₂²×C₄	C₂²×Dic₅	1	2	yes	—
137	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
138	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
139	M₁₆	—	20	16	no	—
140	C₂×C₈	—	10	8	no	—
141	C₂×C₈	—	10	16	no	—
142	C₄²	—	2	8	no	—
143	C₄²	—	2	16	no	—
144	C₄²	—	2	8	no	—
145	C₄²	—	2	8	no	—
146	C₄²	#681	1	16	yes	complemented
147	C₄²	C₄⋊Dic₅	1	4	yes	—
148	C₄²	C₄⋊Dic₅	1	4	yes	—
149	C₄²	C₂×Dic₁₀	1	4	yes	—
150	C₄²	C₂×D₂₀	1	4	yes	—
151	C₂×C₅	—	2	4	no	—
152	C₂×C₅	C₂×#382	1	1	yes	characteristic
153	C₂×C₅	C₂×#423	1	2	yes	—
154	C₂×C₅	#2920	1	4	yes	—
155	C₂³	—	2	4	no	—
156	C₂³	C₂×C₄⋊Dic₅	1	2	yes	—
157	C₂³	#4072	1	4	yes	—
158	C₂³	C₂³×Dic₅	1	1	yes	characteristic
159	C₂×C₄	—	2	32	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	2	8	no	—
162	C₂×C₄	—	2	8	no	—
163	C₂×C₄	—	2	16	no	—
164	C₂×C₄	#4016	1	8	yes	—
165	C₂×C₄	#4021	1	8	yes	—
166	C₂×C₄	C₂×#681	1	16	yes	—
167	C₂×C₄	C₂×#681	1	8	yes	—
168	C₂×C₄	C₂×C₄⋊Dic₅	1	4	yes	—
169	C₂×C₄	C₂×C₄⋊Dic₅	1	4	yes	—
170	C₂×C₄	C₂×C₄⋊Dic₅	1	2	yes	—
171	C₂×C₄	#4072	1	4	yes	—
172	C₂×C₄	C₂²×Dic₁₀	1	1	yes	characteristic
173	C₂×C₄	C₂²×D₂₀	1	1	yes	characteristic
174	C₂×C₄	C₂×#709	1	4	yes	—
175	C₈	—	20	8	no	—
176	C₅	#20752	1	1	yes	characteristic, complemented, Sylow
177	C₂²	—	2	4	no	—
178	C₂²	—	2	8	no	—
179	C₂²	#65225	1	4	yes	—
180	C₂²	C₂²×#681	1	4	yes	—
181	C₂²	C₂²×C₄⋊Dic₅	1	1	yes	characteristic
182	C₂²	C₂×#4072	1	2	yes	—
183	C₄	—	2	4	no	—
184	C₄	—	2	16	no	—
185	C₄	#65213	1	8	yes	—
186	C₄	#65225	1	4	yes	—
187	C₄	C₂×#4016	1	2	yes	—
188	C₄	C₂×#4021	1	2	yes	—
189	C₂	—	2	4	no	—
190	C₂	C₂×#65213	1	2	yes	—
191	C₂	#10613780	1	4	yes	—
192	C₂	C₂×#65225	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅
commutator, socle	C₂³×C₅	C₂³×C₄
Fitting	C₂×C₄³×C₅	C₂
prime radical	C₂⁴×C₅	C₂²×C₄
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	C₂×C₄³	D₅
hypocenter	C₅	#20752
LCS	G ▷ C₂³×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
4. #10600185	C₂	semidirect
58. C₄²×C₅	M₁₆	semidirect
146. C₄²	#681	semidirect
176. C₅	#20752	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄³×C₅	2
C₂×#65213	2
#10602320	2
#10600185	2
#20752	5