Subgroups — #12014310

Order 1280.

354 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Dic₅×M₃₂	C₂	1	2	yes	maximal
2	Dic₅×#97	C₂	1	1	yes	maximal, characteristic
3	C₅×#1375	C₂	1	1	yes	maximal, characteristic
4	#10598302	C₂	1	2	yes	maximal
5	#10598154	C₂	1	1	yes	maximal, characteristic
6	C₂×C₅×M₃₂	C₂²	1	2	yes	—
7	C₂×C₅×#97	C₂²	1	1	yes	characteristic
8	C₂×#3947	C₄	1	1	yes	characteristic
9	C₂×#3947	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₅	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
12	C₂×#3944	C₂²	1	2	yes	—
13	C₄²×Dic₅	C₄	1	1	yes	characteristic
14	C₂×#3935	C₄	1	1	yes	characteristic
15	#64722	C₄	1	4	yes	complemented
16	C₁₆×Dic₅	C₄	1	4	yes	complemented
17	#12594	—	5	1	no	maximal, Sylow
18	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
20	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₅	—	2	2	no	—
22	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
24	C₂²×#168	C₂×C₄	1	1	yes	characteristic
25	C₂²×#168	C₂×C₄	1	1	yes	characteristic
26	C₅×M₃₂	—	2	2	no	—
27	C₅×M₃₂	—	2	2	no	—
28	C₂×C₅×C₁₆	C₂×C₄	1	4	yes	—
29	C₅×#97	—	2	2	no	—
30	#3947	—	2	1	no	—
31	#3947	—	2	1	no	—
32	#3947	C₈	1	4	yes	—
33	C₈×Dic₅	—	2	2	no	—
34	C₈×Dic₅	C₂×C₄	1	2	yes	—
35	C₈×Dic₅	C₂×C₄	1	2	yes	—
36	#3944	—	2	2	no	—
37	#3944	—	2	2	no	—
38	C₂×#672	C₂×C₄	1	4	yes	—
39	#3935	C₈	1	4	yes	—
40	C₄×M₃₂	—	5	2	no	—
41	C₄×#97	—	5	1	no	—
42	#1375	—	5	2	no	—
43	#1375	—	5	1	no	—
44	#1375	D₅	1	1	yes	characteristic
45	C₂²×C₄×C₅	—	2	1	no	—
46	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
47	C₂²×Dic₅	—	2	1	no	—
48	C₂²×Dic₅	M₁₆	1	2	yes	—
49	C₂×C₅×C₈	—	2	1	no	—
50	C₂×C₅×C₈	—	2	1	no	—
51	C₂×C₅×C₈	C₂×C₈	1	2	yes	—
52	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
53	C₂×C₅×C₈	#41	1	2	yes	—
54	C₂×C₅×C₈	#41	1	1	yes	characteristic
55	C₂×C₅×C₈	#41	1	1	yes	characteristic
56	C₄²×C₅	—	2	1	no	—
57	C₄²×C₅	C₂×C₈	1	2	yes	—
58	C₄×Dic₅	—	2	1	no	—
59	C₄×Dic₅	—	4	4	no	—
60	C₄×Dic₅	C₄²	1	1	yes	characteristic
61	C₄×Dic₅	C₄²	1	1	yes	characteristic
62	C₄×Dic₅	C₂×C₈	1	2	yes	—
63	C₄×Dic₅	M₁₆	1	2	yes	—
64	C₂×#168	—	2	1	no	—
65	C₂×#168	—	2	1	no	—
66	C₂×#168	C₄²	1	2	yes	—
67	C₂×#168	C₂×C₈	1	2	yes	—
68	C₂×#168	C₂×C₈	1	2	yes	—
69	C₂×#168	C₂×C₈	1	2	yes	—
70	C₅×C₁₆	—	2	4	no	—
71	#672	—	2	4	no	—
72	C₂×M₃₂	—	5	2	no	—
73	C₂×M₃₂	C₂×D₅	1	2	yes	—
74	C₂×#97	—	5	1	no	—
75	C₂×#97	—	5	1	no	—
76	C₂×#97	—	5	1	no	—
77	C₂×#97	C₂×D₅	1	1	yes	characteristic
78	C₂×C₄×C₈	—	5	1	no	—
79	C₂×C₄×C₈	—	5	1	no	—
80	C₄³	—	5	1	no	—
81	#346	—	5	4	no	—
82	C₄×C₁₆	—	5	4	no	—
83	C₂³×C₅	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₅	—	2	2	no	—
85	C₂×C₄×C₅	—	4	2	no	—
86	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
87	C₂×C₄×C₅	#117	1	1	yes	characteristic
88	C₂×C₄×C₅	#117	1	1	yes	characteristic
89	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
91	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
92	C₂×Dic₅	—	2	4	no	—
93	C₂×Dic₅	—	2	2	no	—
94	C₂×Dic₅	—	4	2	no	—
95	C₂×Dic₅	#97	1	2	yes	—
96	C₂×Dic₅	#97	1	2	yes	—
97	C₅×C₈	—	2	2	no	—
98	C₅×C₈	#117	1	2	yes	—
99	C₅×C₈	#117	1	2	yes	—
100	#168	—	2	2	no	—
101	#168	C₄×C₈	1	4	yes	—
102	C₂²×C₈	—	5	1	no	—
103	C₂²×C₈	—	5	1	no	—
104	C₂²×C₈	C₄×D₅	1	1	yes	characteristic
105	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
106	C₂×C₄²	—	5	1	no	—
107	C₂×C₄²	—	5	1	no	—
108	C₂×C₄²	—	10	2	no	—
109	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
110	M₃₂	—	2	2	no	—
111	M₃₂	—	2	2	no	—
112	M₃₂	—	10	2	no	—
113	M₃₂	—	10	2	no	—
114	C₂×C₁₆	—	5	4	no	—
115	C₂×C₁₆	C₄×D₅	1	4	yes	—
116	#97	—	2	2	no	—
117	#97	—	5	4	no	—
118	#97	—	5	4	no	—
119	#97	—	10	1	no	—
120	#97	—	10	1	no	—
121	C₄×C₈	—	5	2	no	—
122	C₄×C₈	—	5	2	no	—
123	C₄×C₈	—	10	2	no	—
124	C₂²×C₅	—	2	2	no	—
125	C₂²×C₅	C₂×#97	1	1	yes	characteristic
126	C₂²×C₅	#431	1	1	yes	characteristic
127	C₂²×C₅	#431	1	1	yes	characteristic
128	C₄×C₅	—	4	2	no	—
129	C₄×C₅	#376	1	1	yes	characteristic
130	C₄×C₅	C₂×C₄×C₈	1	1	yes	characteristic
131	C₄×C₅	#431	1	1	yes	characteristic
132	C₄×C₅	#431	1	1	yes	characteristic
133	Dic₅	—	2	2	no	—
134	Dic₅	—	4	2	no	—
135	Dic₅	#347	1	4	yes	—
136	C₂²×C₄	—	2	1	no	—
137	C₂²×C₄	—	5	2	no	—
138	C₂²×C₄	—	10	1	no	—
139	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	5	2	no	—
143	C₂×C₈	—	5	2	no	—
144	C₂×C₈	—	5	2	no	—
145	C₂×C₈	—	5	2	no	—
146	C₂×C₈	—	10	1	no	—
147	C₂×C₈	—	10	1	no	—
148	C₂×C₈	C₈×D₅	1	2	yes	—
149	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
150	C₂×C₈	#711	1	1	yes	characteristic
151	C₂×C₈	#711	1	2	yes	—
152	C₂×C₈	#713	1	1	yes	characteristic
153	C₄²	—	2	1	no	—
154	C₄²	—	5	1	no	—
155	C₄²	—	5	1	no	—
156	C₄²	—	5	2	no	—
157	C₄²	—	5	2	no	—
158	C₄²	—	10	1	no	—
159	C₄²	—	20	4	no	—
160	C₄²	C₈×D₅	1	2	yes	—
161	C₁₆	—	2	4	no	—
162	C₁₆	—	10	4	no	—
163	C₂×C₅	—	2	1	no	—
164	C₂×C₅	—	2	1	no	—
165	C₂×C₅	#1744	1	1	yes	characteristic
166	C₂×C₅	C₂×#347	1	2	yes	—
167	C₂³	D₅×M₁₆	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	5	2	no	—
171	C₂×C₄	—	5	2	no	—
172	C₂×C₄	—	10	4	no	—
173	C₂×C₄	—	10	2	no	—
174	C₂×C₄	—	20	2	no	—
175	C₂×C₄	C₄²×D₅	1	1	yes	characteristic
176	C₂×C₄	#4023	1	1	yes	characteristic
177	C₂×C₄	#4038	1	1	yes	characteristic
178	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
179	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
180	C₂×C₄	D₅×M₁₆	1	1	yes	characteristic
181	C₈	—	2	2	no	—
182	C₈	—	5	4	no	—
183	C₈	—	10	2	no	—
184	C₈	#4023	1	2	yes	—
185	C₈	#4038	1	2	yes	—
186	C₅	#12594	1	1	yes	characteristic, complemented, Sylow
187	C₂²	—	2	2	no	—
188	C₂²	D₅×#97	1	1	yes	characteristic
189	C₂²	#65007	1	1	yes	characteristic
190	C₂²	#65122	1	1	yes	characteristic
191	C₄	—	4	2	no	—
192	C₄	—	5	4	no	—
193	C₄	—	10	2	no	—
194	C₄	—	20	2	no	—
195	C₄	#64937	1	1	yes	characteristic
196	C₄	C₄×C₈×D₅	1	1	yes	characteristic
197	C₄	#65007	1	1	yes	characteristic
198	C₄	#65122	1	1	yes	characteristic
199	C₂	—	2	1	no	—
200	C₂	—	2	1	no	—
201	C₂	#10599389	1	1	yes	characteristic
202	C₂	D₅×#347	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₄²×D₅
commutator	C₄×C₅	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₅×#1375	C₂
socle	C₂²×C₅	C₂×#97
prime radical	C₂³×C₅	C₂×M₁₆
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	#1375	D₅
hypocenter	C₅	#12594
LCS	G ▷ C₄×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ #1375

Complements

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

N	H ≅ G/N	type
15. #64722	C₄	semidirect
16. C₁₆×Dic₅	C₄	semidirect
186. C₅	#12594	semidirect

Maximal subgroups

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

subgroup	index
Dic₅×M₃₂	2
Dic₅×#97	2
C₅×#1375	2
#10598302	2
#10598154	2
#12594	5