Subgroups — #12014707

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×#346	C₂	1	1	yes	maximal, characteristic
2	C₂×#64725	C₂	1	1	yes	maximal, characteristic
3	C₂×#64724	C₂	1	1	yes	maximal, characteristic
4	C₄×#3950	C₂	1	2	yes	maximal
5	#10598297	C₂	1	2	yes	maximal
6	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂×#3950	C₂²	1	2	yes	—
10	C₂×#3950	C₂²	1	2	yes	—
11	C₄×C₄⋊Dic₅	—	2	2	no	—
12	C₅×#346	C₄	1	4	yes	complemented
13	#64771	—	2	2	no	—
14	#64725	C₄	1	4	yes	complemented
15	#64724	C₄	1	4	yes	complemented
16	#12674	—	5	1	no	maximal, Sylow
17	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
18	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
19	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
20	C₂×C₄⋊Dic₅	—	2	2	no	—
21	C₂×C₄⋊Dic₅	—	2	2	no	—
22	C₂×C₄×Dic₅	—	4	2	no	—
23	C₂×C₅×C₁₆	—	2	1	no	—
24	C₂×C₅×C₁₆	—	2	1	no	—
25	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
26	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
28	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
29	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
30	C₄×C₅×C₈	Q₈	1	2	yes	—
31	#3963	—	4	2	no	—
32	#3950	—	2	2	no	—
33	#3950	—	2	2	no	—
34	#3950	C₂×C₄	1	4	yes	—
35	#3950	C₂×C₄	1	4	yes	—
36	C₂×#367	—	5	1	no	—
37	C₂×#366	—	5	1	no	—
38	C₂×#346	D₅	1	1	yes	characteristic
39	C₄×#107	—	5	2	no	—
40	#1380	—	5	2	no	—
41	C₂²×C₄×C₅	D₈	1	2	yes	—
42	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
43	C₂²×Dic₅	—	4	2	no	—
44	C₂²×Dic₅	—	4	2	no	—
45	C₂×C₅×C₈	—	2	1	no	—
46	C₂×C₅×C₈	—	2	1	no	—
47	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
48	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₅×C₈	#41	1	1	yes	characteristic
53	C₂×C₅×C₈	#41	1	1	yes	characteristic
54	C₄²×C₅	C₄⋊C₄	1	2	yes	—
55	C₄²×C₅	Q₁₆	1	2	yes	—
56	C₄⋊Dic₅	—	2	4	no	—
57	C₄⋊Dic₅	—	2	4	no	—
58	C₄⋊Dic₅	—	4	2	no	—
59	C₄⋊Dic₅	—	4	2	no	—
60	C₄×Dic₅	—	8	4	no	—
61	C₅×C₁₆	—	2	2	no	—
62	C₅×C₁₆	C₄²	1	4	yes	complemented
63	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
64	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
65	C₂×#107	—	5	2	no	—
66	C₂×#107	—	5	2	no	—
67	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
68	C₄×C₄⋊C₄	—	10	2	no	—
69	#367	—	5	4	no	—
70	#366	—	5	4	no	—
71	#346	Dic₅	1	4	yes	complemented
72	#340	—	10	2	no	—
73	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
74	C₂×C₄×C₅	—	2	2	no	—
75	C₂×C₄×C₅	—	2	2	no	—
76	C₂×C₄×C₅	#107	1	4	yes	—
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	—	4	4	no	—
84	C₂×Dic₅	—	4	4	no	—
85	C₂×Dic₅	—	8	2	no	—
86	C₂×Dic₅	—	8	2	no	—
87	C₂×Dic₅	—	8	2	no	—
88	C₂×Dic₅	—	8	2	no	—
89	C₅×C₈	—	2	2	no	—
90	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
91	C₅×C₈	#117	1	1	yes	characteristic
92	C₅×C₈	C₄×Q₈	1	1	yes	characteristic
93	C₅×C₈	C₄×Q₈	1	1	yes	characteristic
94	C₂²×C₈	D₂₀	1	1	yes	characteristic
95	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
96	C₂×C₄⋊C₄	—	10	2	no	—
97	C₂×C₄⋊C₄	—	10	2	no	—
98	C₂×C₄²	—	20	2	no	—
99	C₂×C₄²	D₂₀	1	1	yes	characteristic
100	C₂×C₁₆	—	2	1	no	—
101	C₂×C₁₆	—	2	1	no	—
102	C₂×C₁₆	C₄×D₅	1	2	yes	—
103	C₂×C₁₆	C₄×D₅	1	2	yes	—
104	C₂×C₁₆	C₂×Dic₅	1	2	yes	—
105	C₂×C₁₆	C₂×Dic₅	1	2	yes	—
106	#107	—	5	4	no	—
107	#107	—	5	4	no	—
108	#107	—	10	2	no	—
109	#107	—	10	2	no	—
110	C₄×C₈	Dic₁₀	1	2	yes	—
111	C₄×C₈	C₂×Dic₅	1	2	yes	—
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	—	2	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	—	8	4	no	—
124	Dic₅	—	8	4	no	—
125	C₂²×C₄	—	20	2	no	—
126	C₂²×C₄	—	20	2	no	—
127	C₂²×C₄	D₄₀	1	2	yes	—
128	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
129	C₂×C₈	—	2	1	no	—
130	C₂×C₈	—	2	1	no	—
131	C₂×C₈	C₄⋊Dic₅	1	2	yes	—
132	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
133	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
134	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
135	C₂×C₈	#709	1	1	yes	characteristic
136	C₂×C₈	#709	1	1	yes	characteristic
137	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
138	C₄⋊C₄	—	10	4	no	—
139	C₄⋊C₄	—	10	4	no	—
140	C₄⋊C₄	—	20	2	no	—
141	C₄⋊C₄	—	20	2	no	—
142	C₄²	—	40	4	no	—
143	C₄²	Dic₂₀	1	2	yes	—
144	C₄²	C₄⋊Dic₅	1	2	yes	—
145	C₁₆	—	2	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	C₂×D₄₀	1	1	yes	characteristic
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	20	4	no	—
156	C₂×C₄	—	20	4	no	—
157	C₂×C₄	—	40	2	no	—
158	C₂×C₄	—	40	2	no	—
159	C₂×C₄	—	40	2	no	—
160	C₂×C₄	—	40	2	no	—
161	C₂×C₄	#3950	1	4	yes	—
162	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
163	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
164	C₂×C₄	#4050	1	1	yes	characteristic
165	C₂×C₄	#4050	1	1	yes	characteristic
166	C₂×C₄	C₂×Dic₂₀	1	1	yes	characteristic
167	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
168	C₈	—	2	2	no	—
169	C₈	C₄×Dic₁₀	1	1	yes	characteristic
170	C₈	C₄×Dic₁₀	1	1	yes	characteristic
171	C₈	C₂×C₄×Dic₅	1	1	yes	characteristic
172	C₈	#4072	1	1	yes	characteristic
173	C₅	#12674	1	1	yes	characteristic, complemented, Sylow
174	C₂²	C₄×D₄₀	1	1	yes	characteristic
175	C₂²	C₄×D₄₀	1	1	yes	characteristic
176	C₂²	#65197	1	2	yes	—
177	C₂²	#65198	1	2	yes	—
178	C₂²	C₂×#3950	1	1	yes	characteristic
179	C₄	—	2	4	no	—
180	C₄	—	40	4	no	—
181	C₄	—	40	4	no	—
182	C₄	C₄×Dic₂₀	1	1	yes	characteristic
183	C₄	C₄×Dic₂₀	1	1	yes	characteristic
184	C₄	C₄×C₄⋊Dic₅	1	1	yes	characteristic
185	C₄	#65395	1	1	yes	characteristic
186	C₂	#10599870	1	2	yes	—
187	C₂	#10599871	1	1	yes	characteristic
188	C₂	#10599875	1	1	yes	characteristic
189	C₂	C₄×#3950	1	1	yes	characteristic
190	C₂	#10602221	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄₀
commutator	C₅×C₈	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂×C₅×#346	C₂
socle, prime radical	C₂³×C₅	C₂×D₈
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×#346	D₅
hypocenter	C₅	#12674
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
12. C₅×#346	C₄	semidirect
14. #64725	C₄	semidirect
15. #64724	C₄	semidirect
62. C₅×C₁₆	C₄²	semidirect
71. #346	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×#346	2
C₂×#64725	2
C₂×#64724	2
C₄×#3950	2
#10598297	2
#12674	5