Subgroups — #14280437

Order 1920.

424 conjugacy classes of subgroups.

The subgroup lattice has 172 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	#11751026	C₂	1	2	yes	maximal
3	#11746276	C₂	1	2	yes	maximal, complemented
4	C₈×C₃⋊#168	C₂	1	2	yes	maximal, complemented
5	#10600517	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₅×C₈	C₄	1	2	yes	—
7	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
8	C₃×C₄×C₅×C₈	C₄	1	2	yes	—
9	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
11	#91114	C₄	1	4	yes	—
12	C₄⋊(C₃⋊#168)	C₄	1	4	yes	—
13	C₄×C₃⋊#168	C₂²	1	4	yes	—
14	#69724	—	5	1	no	maximal, Hall
15	C₂×C₄×C₅×C₈	D₃	1	1	yes	characteristic
16	#65214	—	3	2	no	—
17	#64675	—	3	2	no	—
18	C₈×#168	—	3	2	no	—
19	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
20	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
21	C₂×C₃×C₅×C₈	C₈	1	8	yes	complemented
22	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
23	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
24	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
25	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
26	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
27	C₂×C₃⋊#168	C₂×C₄	1	8	yes	—
28	C₂×C₃×C₄×C₈	D₅	1	1	yes	characteristic
29	#4925	—	5	2	no	—
30	#4455	—	5	2	no	—
31	C₈×C₃⋊C₈	—	5	2	no	—
32	C₂²×C₅×C₈	Dic₃	1	2	yes	—
33	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
34	C₄×C₅×C₈	Dic₃	1	2	yes	—
35	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
36	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
37	C₂²⋊#168	—	3	4	no	—
38	C₄⋊#168	—	3	4	no	—
39	C₄×#168	—	3	4	no	—
40	#1450	—	15	1	no	Sylow
41	C₂³×C₃×C₅	C₄²	1	1	yes	characteristic
42	C₂×C₃×C₄×C₅	—	2	4	no	—
43	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
44	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
45	C₂×C₃×C₄×C₅	C₂×C₈	1	4	yes	—
46	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
48	C₃×C₅×C₈	C₂×C₈	1	4	yes	—
49	C₃×C₅×C₈	C₂×C₈	1	4	yes	—
50	C₃⋊#168	—	2	8	no	—
51	C₂²×C₃×C₈	Dic₅	1	2	yes	—
52	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
53	C₃×C₄×C₈	Dic₅	1	2	yes	—
54	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
55	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
56	#830	—	5	4	no	—
57	C₄⋊(C₃⋊C₈)	—	5	4	no	—
58	C₄×C₃⋊C₈	—	5	4	no	—
59	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
60	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
61	C₂×C₅×C₈	C₃⋊C₈	1	8	yes	complemented
62	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
63	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
64	C₄²×C₅	C₄×D₃	1	2	yes	—
65	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
66	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
67	C₂×#168	—	3	8	no	—
68	#431	—	15	2	no	—
69	C₂×C₄×C₈	D₁₅	1	1	yes	characteristic
70	#322	—	15	2	no	—
71	C₈²	—	15	2	no	—
72	C₂²×C₃×C₅	—	2	2	no	—
73	C₂²×C₃×C₅	C₄×C₈	1	2	yes	—
74	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
75	C₃×C₄×C₅	—	2	2	no	—
76	C₃×C₄×C₅	C₄×C₈	1	2	yes	—
77	C₃×C₄×C₅	C₂²×C₈	1	2	yes	—
78	C₃×C₄×C₅	#131	1	4	yes	—
79	C₂²×C₃×C₄	C₄×D₅	1	2	yes	—
80	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
81	C₂×C₃×C₈	#168	1	8	yes	complemented
82	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
83	C₂×C₃×C₈	C₂×Dic₅	1	2	yes	—
84	C₃×C₄²	C₄×D₅	1	2	yes	—
85	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
86	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
87	C₂×C₃⋊C₈	—	5	8	no	—
88	C₂³×C₅	C₄×Dic₃	1	1	yes	characteristic
89	C₂×C₄×C₅	—	2	4	no	—
90	C₂×C₄×C₅	C₂×C₃⋊C₈	1	4	yes	—
91	C₂×C₄×C₅	C₄×Dic₃	1	2	yes	—
92	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
93	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
94	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
95	C₅×C₈	C₂×C₃⋊C₈	1	4	yes	—
96	C₅×C₈	C₂×C₃⋊C₈	1	4	yes	—
97	#168	—	6	8	no	—
98	C₂²×C₈	Dic₁₅	1	2	yes	—
99	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
100	C₄⋊C₈	—	15	4	no	—
101	C₂²⋊C₈	—	15	4	no	—
102	C₄×C₈	—	15	4	no	—
103	C₄×C₈	Dic₁₅	1	2	yes	—
104	C₄×C₈	C₂×D₁₅	1	1	yes	characteristic
105	C₄×C₈	C₂×D₁₅	1	1	yes	characteristic
106	C₂×C₃×C₅	—	2	2	no	—
107	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
108	C₂×C₃×C₅	#405	1	2	yes	—
109	C₂³×C₃	C₄×Dic₅	1	1	yes	characteristic
110	C₂×C₃×C₄	—	2	4	no	—
111	C₂×C₃×C₄	C₂×#168	1	4	yes	—
112	C₂×C₃×C₄	C₄×Dic₅	1	2	yes	—
113	C₂×C₃×C₄	C₄×Dic₅	1	1	yes	characteristic
114	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
115	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
116	C₃×C₈	C₂×#168	1	4	yes	—
117	C₃×C₈	C₂×#168	1	4	yes	—
118	C₃⋊C₈	—	10	8	no	—
119	C₂²×C₅	—	2	2	no	—
120	C₂²×C₅	C₄×C₃⋊C₈	1	2	yes	—
121	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
122	C₄×C₅	—	2	2	no	—
123	C₄×C₅	C₄×C₃⋊C₈	1	2	yes	—
124	C₄×C₅	#901	1	4	yes	—
125	C₄×C₅	C₂²×C₃⋊C₈	1	2	yes	—
126	C₂²×C₄	C₄×D₁₅	1	2	yes	—
127	C₂²×C₄	C₂×Dic₁₅	1	1	yes	characteristic
128	C₂×C₈	—	15	8	no	—
129	C₂×C₈	C₃⋊#168	1	8	yes	complemented
130	C₂×C₈	C₂×Dic₁₅	1	2	yes	—
131	C₂×C₈	C₂×Dic₁₅	1	2	yes	—
132	C₄²	C₄×D₁₅	1	2	yes	—
133	C₄²	C₂×Dic₁₅	1	1	yes	characteristic
134	C₄²	C₂²×D₁₅	1	1	yes	characteristic
135	C₃×C₅	#1450	1	1	yes	characteristic, complemented, Hall
136	C₂²×C₃	—	2	2	no	—
137	C₂²×C₃	C₄×#168	1	2	yes	—
138	C₂²×C₃	C₂×C₄×Dic₅	1	1	yes	characteristic
139	C₃×C₄	—	2	2	no	—
140	C₃×C₄	C₄×#168	1	2	yes	—
141	C₃×C₄	#4047	1	4	yes	—
142	C₃×C₄	C₂²×#168	1	2	yes	—
143	C₂×C₅	—	2	2	no	—
144	C₂×C₅	C₂×C₄×C₃⋊C₈	1	1	yes	characteristic
145	C₂×C₅	#5101	1	2	yes	—
146	C₂³	C₄×Dic₁₅	1	1	yes	characteristic
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	C₂×C₃⋊#168	1	4	yes	—
149	C₂×C₄	C₄×Dic₁₅	1	2	yes	—
150	C₂×C₄	C₄×Dic₁₅	1	1	yes	characteristic
151	C₂×C₄	C₂×C₄×D₁₅	1	2	yes	—
152	C₂×C₄	C₂²×Dic₁₅	1	1	yes	characteristic
153	C₈	—	30	8	no	—
154	C₈	C₂×C₃⋊#168	1	4	yes	—
155	C₈	C₂×C₃⋊#168	1	4	yes	—
156	C₂×C₃	—	2	2	no	—
157	C₂×C₃	C₂×C₄×#168	1	1	yes	characteristic
158	C₂×C₃	#65390	1	2	yes	—
159	C₅	#69724	1	1	yes	characteristic, complemented, Sylow
160	C₂²	—	2	2	no	—
161	C₂²	C₄×C₃⋊#168	1	2	yes	—
162	C₂²	C₂×C₄×Dic₁₅	1	1	yes	characteristic
163	C₄	—	2	2	no	—
164	C₄	C₄×C₃⋊#168	1	2	yes	—
165	C₄	#91790	1	4	yes	—
166	C₄	C₂²×C₃⋊#168	1	2	yes	—
167	C₃	#10600517	1	1	yes	characteristic, complemented, Sylow
168	C₂	—	2	2	no	—
169	C₂	C₂×C₄×C₃⋊#168	1	1	yes	characteristic
170	C₂	#11751202	1	2	yes	—

	subgroup	G/H
center	C₄×C₈	C₂×D₁₅
commutator	C₂×C₃×C₅	C₂×C₄×C₈
Frattini	C₄²	C₂²×D₁₅
Fitting	C₂×C₃×C₄×C₅×C₈	C₂
socle	C₂²×C₃×C₅	C₂×C₄²
prime radical	C₂³×C₃×C₅	C₄²
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₂×C₄×C₈	D₁₅
hypocenter	C₃×C₅	#1450
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #11746276	C₂	semidirect
4. C₈×C₃⋊#168	C₂	semidirect
21. C₂×C₃×C₅×C₈	C₈	semidirect
61. C₂×C₅×C₈	C₃⋊C₈	semidirect
81. C₂×C₃×C₈	#168	semidirect
129. C₂×C₈	C₃⋊#168	semidirect
135. C₃×C₅	#1450	semidirect
159. C₅	#69724	semidirect
167. C₃	#10600517	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₅×C₈	2
#11751026	2
#11746276	2
C₈×C₃⋊#168	2
#10600517	3
#69724	5