Subgroups — #12014710

Order 1280.

358 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#366	C₂	1	1	yes	maximal, characteristic
2	C₂×#64725	C₂	1	1	yes	maximal, characteristic
3	C₂×#64722	C₂	1	1	yes	maximal, characteristic
4	Dic₅×#107	C₂	1	2	yes	maximal
5	#10598213	C₂	1	2	yes	maximal
6	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₅×#107	C₂²	1	2	yes	—
8	C₂×#3950	C₂²	1	2	yes	—
9	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
10	C₂²×#672	C₂²	1	1	yes	characteristic
11	C₄⋊C₄×Dic₅	—	2	2	no	—
12	C₅×#366	C₄	1	4	yes	complemented
13	#64749	—	2	2	no	—
14	#64725	C₄	1	4	yes	complemented
15	#64722	C₄	1	4	yes	complemented
16	#12674	—	5	1	no	maximal, Sylow
17	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
18	C₂×C₅×C₄⋊C₄	—	2	2	no	—
19	C₂×C₄⋊Dic₅	—	2	2	no	—
20	C₂×C₄×Dic₅	—	4	2	no	—
21	C₂×C₄×Dic₅	D₄	1	1	yes	characteristic
22	C₂²×#168	D₄	1	1	yes	characteristic
23	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
25	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
26	C₅×#107	—	2	2	no	—
27	C₅×#107	C₂×C₄	1	4	yes	—
28	#3963	—	4	2	no	—
29	#3950	—	2	2	no	—
30	#3950	C₂×C₄	1	4	yes	—
31	C₈×Dic₅	C₂×C₄	1	2	yes	—
32	C₈×Dic₅	Q₈	1	2	yes	—
33	C₂×#672	—	2	1	no	—
34	C₂×#672	—	2	1	no	—
35	C₂×#672	C₂×C₄	1	2	yes	—
36	C₂×#367	—	5	1	no	—
37	C₂×#366	D₅	1	1	yes	characteristic
38	C₂×#346	—	5	1	no	—
39	C₄×#107	—	5	2	no	—
40	#1380	—	5	2	no	—
41	C₂²×C₄×C₅	—	4	2	no	—
42	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₅	—	4	2	no	—
44	C₂²×Dic₅	D₈	1	2	yes	—
45	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₅×C₈	#41	1	1	yes	characteristic
50	C₂×C₅×C₈	#41	1	1	yes	characteristic
51	C₅×C₄⋊C₄	—	2	4	no	—
52	C₅×C₄⋊C₄	—	4	2	no	—
53	C₄⋊Dic₅	—	2	4	no	—
54	C₄⋊Dic₅	—	4	2	no	—
55	C₄×Dic₅	—	8	4	no	—
56	C₄×Dic₅	C₄⋊C₄	1	2	yes	—
57	C₄×Dic₅	Q₁₆	1	2	yes	—
58	C₂×#168	—	2	1	no	—
59	C₂×#168	—	2	1	no	—
60	C₂×#168	C₄⋊C₄	1	2	yes	—
61	C₅×C₁₆	C₄²	1	4	yes	complemented
62	#672	—	2	2	no	—
63	C₂²×C₁₆	—	5	1	no	—
64	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
65	C₂×#107	—	5	2	no	—
66	C₂×#107	C₂×D₅	1	2	yes	—
67	C₂×C₄×C₈	—	5	1	no	—
68	C₄×C₄⋊C₄	—	10	2	no	—
69	#367	—	5	4	no	—
70	#366	Dic₅	1	4	yes	complemented
71	#346	—	5	4	no	—
72	#340	—	10	2	no	—
73	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₅	—	4	4	no	—
75	C₂×C₄×C₅	—	8	2	no	—
76	C₂×C₄×C₅	—	8	2	no	—
77	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
79	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
80	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
81	C₂×C₄×C₅	#135	1	1	yes	characteristic
82	C₂×C₄×C₅	#135	1	1	yes	characteristic
83	C₂×Dic₅	—	2	2	no	—
84	C₂×Dic₅	—	2	2	no	—
85	C₂×Dic₅	—	4	4	no	—
86	C₂×Dic₅	—	8	2	no	—
87	C₂×Dic₅	—	8	2	no	—
88	C₂×Dic₅	#107	1	4	yes	—
89	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
90	C₅×C₈	#117	1	1	yes	characteristic
91	C₅×C₈	C₄×Q₈	1	1	yes	characteristic
92	C₅×C₈	C₄×Q₈	1	1	yes	characteristic
93	#168	—	2	2	no	—
94	C₂²×C₈	—	5	1	no	—
95	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
96	C₂×C₄⋊C₄	—	2	2	no	—
97	C₂×C₄⋊C₄	—	10	2	no	—
98	C₂×C₄²	—	5	1	no	—
99	C₂×C₄²	—	20	2	no	—
100	C₂×C₁₆	—	5	2	no	—
101	C₂×C₁₆	—	10	1	no	—
102	C₂×C₁₆	—	10	1	no	—
103	C₂×C₁₆	C₄×D₅	1	2	yes	—
104	C₂×C₁₆	C₄×D₅	1	2	yes	—
105	C₂×C₁₆	C₂×Dic₅	1	2	yes	—
106	#107	—	2	2	no	—
107	#107	—	5	4	no	—
108	#107	—	10	2	no	—
109	#107	C₂×Dic₅	1	4	yes	—
110	C₄×C₈	—	5	2	no	—
111	C₄×C₈	—	5	2	no	—
112	#95	—	20	2	no	—
113	C₂²×C₅	C₂×#107	1	1	yes	characteristic
114	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
115	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
116	C₂²×C₅	#509	1	2	yes	—
117	C₂²×C₅	#510	1	2	yes	—
118	C₄×C₅	—	8	4	no	—
119	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
120	C₄×C₅	#427	1	1	yes	characteristic
121	C₄×C₅	C₄×Q₁₆	1	1	yes	characteristic
122	C₄×C₅	C₄×Q₁₆	1	1	yes	characteristic
123	Dic₅	—	2	4	no	—
124	Dic₅	—	8	4	no	—
125	C₂²×C₄	—	4	2	no	—
126	C₂²×C₄	—	5	2	no	—
127	C₂²×C₄	—	20	2	no	—
128	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
129	C₂×C₈	—	5	2	no	—
130	C₂×C₈	—	10	1	no	—
131	C₂×C₈	—	10	1	no	—
132	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
133	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
134	C₂×C₈	#711	1	1	yes	characteristic
135	C₂×C₈	Q₈×D₅	1	1	yes	characteristic
136	C₂×C₈	#713	1	1	yes	characteristic
137	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
138	C₄⋊C₄	—	2	4	no	—
139	C₄⋊C₄	—	4	2	no	—
140	C₄⋊C₄	—	10	4	no	—
141	C₄⋊C₄	—	20	2	no	—
142	C₄²	—	5	2	no	—
143	C₄²	—	5	2	no	—
144	C₄²	—	40	4	no	—
145	C₁₆	—	10	2	no	—
146	C₁₆	C₄×Dic₅	1	4	yes	complemented
147	C₂×C₅	C₄×#107	1	1	yes	characteristic
148	C₂×C₅	#2159	1	2	yes	—
149	C₂×C₅	#2175	1	2	yes	—
150	C₂×C₅	#2176	1	1	yes	characteristic
151	C₂×C₅	#2177	1	1	yes	characteristic
152	C₂³	D₅×D₈	1	1	yes	characteristic
153	C₂×C₄	—	4	4	no	—
154	C₂×C₄	—	5	4	no	—
155	C₂×C₄	—	8	2	no	—
156	C₂×C₄	—	8	2	no	—
157	C₂×C₄	—	10	2	no	—
158	C₂×C₄	—	10	2	no	—
159	C₂×C₄	—	20	4	no	—
160	C₂×C₄	—	40	2	no	—
161	C₂×C₄	—	40	2	no	—
162	C₂×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₄	#4039	1	1	yes	characteristic
164	C₂×C₄	#4058	1	1	yes	characteristic
165	C₂×C₄	D₅×Q₁₆	1	1	yes	characteristic
166	C₂×C₄	#4065	1	1	yes	characteristic
167	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
168	C₈	—	10	2	no	—
169	C₈	#4033	1	1	yes	characteristic
170	C₈	#4038	1	1	yes	characteristic
171	C₈	C₂×C₄×Dic₅	1	1	yes	characteristic
172	C₈	Q₈×Dic₅	1	1	yes	characteristic
173	C₅	#12674	1	1	yes	characteristic, complemented, Sylow
174	C₂²	#65161	1	1	yes	characteristic
175	C₂²	D₅×#107	1	1	yes	characteristic
176	C₂²	#65200	1	2	yes	—
177	C₂²	#65207	1	2	yes	—
178	C₂²	D₈×Dic₅	1	1	yes	characteristic
179	C₄	—	8	4	no	—
180	C₄	—	10	4	no	—
181	C₄	—	40	4	no	—
182	C₄	#65162	1	1	yes	characteristic
183	C₄	#65169	1	1	yes	characteristic
184	C₄	C₄⋊C₄×Dic₅	1	1	yes	characteristic
185	C₄	Q₁₆×Dic₅	1	1	yes	characteristic
186	C₂	#10600112	1	2	yes	—
187	C₂	#10600119	1	2	yes	—
188	C₂	Dic₅×#107	1	1	yes	characteristic
189	C₂	#10602254	1	1	yes	characteristic
190	C₂	#10602289	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₅×D₈
commutator	C₅×C₈	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂×C₅×#366	C₂
socle, prime radical	C₂³×C₅	C₂×D₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×#366	D₅
hypocenter	C₅	#12674
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
12. C₅×#366	C₄	semidirect
14. #64725	C₄	semidirect
15. #64722	C₄	semidirect
61. C₅×C₁₆	C₄²	semidirect
70. #366	Dic₅	semidirect
146. C₁₆	C₄×Dic₅	semidirect
173. C₅	#12674	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#366	2
C₂×#64725	2
C₂×#64722	2
Dic₅×#107	2
#10598213	2
#12674	5