Subgroups — #12014689

Order 1280.

318 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#64726	C₂	1	1	yes	maximal, characteristic
3	#10602197	C₂	1	1	yes	maximal, characteristic
4	#10600572	C₂	1	2	yes	maximal
5	#10598298	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×#3951	C₂²	1	2	yes	—
10	#65395	C₂²	1	2	yes	—
11	#65390	—	2	2	no	—
12	C₄×C₅×C₁₆	C₄	1	4	yes	complemented
13	#64743	—	2	2	no	—
14	#64726	C₄	1	4	yes	complemented
15	#64725	C₄	1	2	yes	complemented
16	#64724	C₄	1	2	yes	complemented
17	#12670	—	5	1	no	maximal, Sylow
18	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
19	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
20	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
21	#4072	—	2	2	no	—
22	C₂×#681	—	2	2	no	—
23	C₂×C₅×C₁₆	—	2	1	no	—
24	C₂×C₅×C₁₆	—	2	1	no	—
25	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
26	C₂×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
28	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
29	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
30	C₄×C₅×C₈	C₂×C₄	1	2	yes	—
31	C₄×C₅×C₈	Q₈	1	2	yes	—
32	#3951	—	2	2	no	—
33	#3951	C₂×C₄	1	4	yes	—
34	#3950	C₂×C₄	1	4	yes	—
35	#3949	—	2	2	no	—
36	#3947	—	4	2	no	—
37	C₈×Dic₅	—	4	2	no	—
38	C₂×#368	—	5	1	no	—
39	#2158	—	5	1	no	—
40	C₂×C₄×C₁₆	D₅	1	1	yes	characteristic
41	#1772	—	5	2	no	—
42	#1382	—	5	2	no	—
43	C₂²×C₄×C₅	—	2	1	no	—
44	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
45	C₂×C₅×C₈	C₄⋊C₄	1	2	yes	—
46	C₂×C₅×C₈	D₈	1	1	yes	characteristic
47	C₂×C₅×C₈	D₈	1	1	yes	characteristic
48	C₂×C₅×C₈	Q₁₆	1	2	yes	—
49	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₅×C₈	#41	1	1	yes	characteristic
54	C₂×C₅×C₈	#41	1	1	yes	characteristic
55	C₄²×C₅	—	2	1	no	—
56	C₄²×C₅	C₄⋊C₄	1	2	yes	—
57	C₂²⋊Dic₅	—	4	2	no	—
58	C₄⋊Dic₅	—	2	4	no	—
59	C₄×Dic₅	—	4	2	no	—
60	#681	—	2	4	no	—
61	#681	—	4	2	no	—
62	C₂×#168	—	4	2	no	—
63	C₅×C₁₆	—	2	2	no	—
64	C₅×C₁₆	C₄²	1	2	yes	—
65	C₅×C₁₆	C₄²	1	2	yes	—
66	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
67	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
68	C₂×#108	—	5	2	no	—
69	#427	—	5	2	no	—
70	#405	—	10	2	no	—
71	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
72	#368	—	5	4	no	—
73	#367	—	5	2	no	—
74	#366	—	5	2	no	—
75	C₄×C₁₆	Dic₅	1	4	yes	complemented
76	#339	—	10	2	no	—
77	C₂³×C₅	#135	1	1	yes	characteristic
78	C₂×C₄×C₅	—	2	4	no	—
79	C₂×C₄×C₅	—	2	2	no	—
80	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
82	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
83	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
84	C₂×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
85	C₂×C₄×C₅	#135	1	1	yes	characteristic
86	C₂×Dic₅	—	4	4	no	—
87	C₅×C₈	#107	1	2	yes	—
88	C₅×C₈	#107	1	2	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	—	4	4	no	—
94	C₂×M₁₆	—	10	2	no	—
95	C₂²×C₈	D₂₀	1	1	yes	characteristic
96	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
97	#117	—	10	2	no	—
98	C₂×C₄²	D₂₀	1	1	yes	characteristic
99	C₂×C₁₆	—	2	1	no	—
100	C₂×C₁₆	—	2	1	no	—
101	C₂×C₁₆	C₄×D₅	1	1	yes	characteristic
102	C₂×C₁₆	C₄×D₅	1	1	yes	characteristic
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	#108	—	5	4	no	—
107	#108	—	10	2	no	—
108	#107	—	5	4	no	—
109	#106	—	10	2	no	—
110	#97	—	20	2	no	—
111	C₄×C₈	—	20	2	no	—
112	C₄×C₈	Dic₁₀	1	2	yes	—
113	C₄×C₈	C₂×Dic₅	1	2	yes	—
114	C₂²×C₅	—	2	2	no	—
115	C₂²×C₅	#427	1	1	yes	characteristic
116	C₂²×C₅	C₄×D₈	1	1	yes	characteristic
117	C₂²×C₅	C₄×Q₁₆	1	1	yes	characteristic
118	C₄×C₅	—	2	4	no	—
119	C₄×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
120	C₄×C₅	C₂×#107	1	1	yes	characteristic
121	C₄×C₅	C₄×D₈	1	1	yes	characteristic
122	C₄×C₅	C₄×Q₁₆	1	1	yes	characteristic
123	Dic₅	—	8	4	no	—
124	C₂²×C₄	—	2	1	no	—
125	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
126	M₁₆	—	10	4	no	—
127	M₁₆	—	20	2	no	—
128	C₂×C₈	—	20	2	no	—
129	C₂×C₈	D₄₀	1	1	yes	characteristic
130	C₂×C₈	D₄₀	1	1	yes	characteristic
131	C₂×C₈	Dic₂₀	1	2	yes	—
132	C₂×C₈	C₄⋊Dic₅	1	2	yes	—
133	C₂×C₈	C₂×Dic₁₀	1	1	yes	characteristic
134	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
135	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
136	C₂×C₈	#709	1	1	yes	characteristic
137	C₂×C₈	#709	1	1	yes	characteristic
138	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
139	C₄⋊C₄	—	10	4	no	—
140	C₂²⋊C₄	—	20	2	no	—
141	C₄²	—	2	1	no	—
142	C₄²	—	20	2	no	—
143	C₄²	C₄⋊Dic₅	1	2	yes	—
144	C₁₆	—	2	2	no	—
145	C₁₆	C₄×Dic₅	1	2	yes	—
146	C₁₆	C₄×Dic₅	1	2	yes	—
147	C₂×C₅	—	2	1	no	—
148	C₂×C₅	—	2	1	no	—
149	C₂×C₅	C₄×#107	1	1	yes	characteristic
150	C₂×C₅	#2178	1	2	yes	—
151	C₂³	#4050	1	1	yes	characteristic
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	20	4	no	—
155	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
156	C₂×C₄	C₄×D₂₀	1	1	yes	characteristic
157	C₂×C₄	C₂×D₄₀	1	1	yes	characteristic
158	C₂×C₄	#4050	1	1	yes	characteristic
159	C₂×C₄	C₂×Dic₂₀	1	1	yes	characteristic
160	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
161	C₈	—	20	4	no	—
162	C₈	#3950	1	2	yes	—
163	C₈	#3950	1	2	yes	—
164	C₈	C₄×Dic₁₀	1	1	yes	characteristic
165	C₈	C₄×Dic₁₀	1	1	yes	characteristic
166	C₈	C₂×C₄×Dic₅	1	1	yes	characteristic
167	C₈	#4072	1	1	yes	characteristic
168	C₅	#12670	1	1	yes	characteristic, complemented, Sylow
169	C₂²	—	2	2	no	—
170	C₂²	C₄×D₄₀	1	1	yes	characteristic
171	C₂²	C₄×Dic₂₀	1	1	yes	characteristic
172	C₂²	#65395	1	1	yes	characteristic
173	C₄	—	2	4	no	—
174	C₄	—	40	4	no	—
175	C₄	C₄×D₄₀	1	1	yes	characteristic
176	C₄	C₄×Dic₂₀	1	1	yes	characteristic
177	C₄	C₄×C₄⋊Dic₅	1	1	yes	characteristic
178	C₄	C₂×#3950	1	1	yes	characteristic
179	C₂	—	2	1	no	—
180	C₂	—	2	1	no	—
181	C₂	#10599860	1	2	yes	—
182	C₂	C₄×#3950	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄₀
commutator	C₅×C₈	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅
Fitting	C₂×C₄×C₅×C₁₆	C₂
socle	C₂²×C₅	C₄×D₈
prime radical	C₂³×C₅	#135
cosocle	C₂²×C₅×C₈	C₂³
hypercenter	C₂×C₄×C₁₆	D₅
hypocenter	C₅	#12670
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
12. C₄×C₅×C₁₆	C₄	semidirect
14. #64726	C₄	semidirect
15. #64725	C₄	semidirect
16. #64724	C₄	semidirect
75. C₄×C₁₆	Dic₅	semidirect
168. C₅	#12670	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅×C₁₆	2
C₂×#64726	2
#10602197	2
#10600572	2
#10598298	2
#12670	5