Subgroups — #14294869

Order 1920.

592 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄³×C₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₃⋊#168	C₂	1	2	yes	maximal
3	#11746345	C₂	1	4	yes	maximal
4	#10600181	—	3	1	no	maximal, Hall
5	C₂×C₃×C₄²×C₅	C₄	1	2	yes	—
6	C₂×C₃×C₄²×C₅	C₄	1	2	yes	—
7	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
8	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₂²×C₃⋊#168	C₂²	1	4	yes	—
10	C₄×C₃⋊#168	C₄	1	8	yes	complemented
11	#69388	—	5	1	no	maximal, Hall
12	C₄³×C₅	D₃	1	1	yes	characteristic
13	C₂×C₄×#168	—	3	2	no	—
14	#64744	—	3	4	no	—
15	C₂²×C₃×C₄×C₅	C₂×C₄	1	4	yes	—
16	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
19	C₃×C₄²×C₅	—	2	4	no	—
20	C₃×C₄²×C₅	—	2	4	no	—
21	C₃×C₄²×C₅	C₈	1	8	yes	complemented
22	C₃×C₄²×C₅	C₂×C₄	1	4	yes	—
23	C₂×C₃⋊#168	—	2	8	no	—
24	C₂×C₃⋊#168	C₂×C₄	1	8	yes	—
25	C₃×C₄³	D₅	1	1	yes	characteristic
26	C₂×C₄×C₃⋊C₈	—	5	2	no	—
27	#4524	—	5	4	no	—
28	C₂×C₄²×C₅	Dic₃	1	2	yes	—
29	C₂×C₄²×C₅	Dic₃	1	2	yes	—
30	C₂×C₄²×C₅	C₂×D₃	1	2	yes	—
31	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
32	C₂²×#168	—	3	4	no	—
33	C₄×#168	—	3	8	no	—
34	#1744	—	15	1	no	Sylow
35	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₃×C₄×C₅	—	2	4	no	—
37	C₂×C₃×C₄×C₅	—	2	4	no	—
38	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
39	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
40	C₂×C₃×C₄×C₅	C₂×C₈	1	8	yes	—
41	C₂×C₃×C₄×C₅	C₂×C₈	1	4	yes	—
42	C₂×C₃×C₄×C₅	M₁₆	1	4	yes	—
43	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
44	C₂×C₃×C₄×C₅	#41	1	4	yes	—
45	C₃⋊#168	—	2	8	no	—
46	C₂×C₃×C₄²	Dic₅	1	2	yes	—
47	C₂×C₃×C₄²	Dic₅	1	2	yes	—
48	C₂×C₃×C₄²	C₂×D₅	1	2	yes	—
49	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
50	C₂²×C₃⋊C₈	—	5	4	no	—
51	C₄×C₃⋊C₈	—	5	8	no	—
52	C₂²×C₄×C₅	C₂×Dic₃	1	4	yes	—
53	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
56	C₄²×C₅	—	2	4	no	—
57	C₄²×C₅	—	2	4	no	—
58	C₄²×C₅	C₃⋊C₈	1	8	yes	complemented
59	C₄²×C₅	C₄×D₃	1	4	yes	—
60	C₂×#168	—	3	8	no	—
61	C₂×#168	—	6	8	no	—
62	C₂×C₄×C₈	—	15	2	no	—
63	C₄³	D₁₅	1	1	yes	characteristic
64	#336	—	15	4	no	—
65	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
66	C₂²×C₃×C₅	#117	1	2	yes	—
67	C₂²×C₃×C₅	C₂²×C₈	1	2	yes	—
68	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
69	C₃×C₄×C₅	—	2	8	no	—
70	C₃×C₄×C₅	C₄×C₈	1	4	yes	—
71	C₃×C₄×C₅	#97	1	4	yes	—
72	C₃×C₄×C₅	#117	1	4	yes	—
73	C₂²×C₃×C₄	C₂×Dic₅	1	4	yes	—
74	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
75	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
76	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
77	C₃×C₄²	—	2	4	no	—
78	C₃×C₄²	—	2	4	no	—
79	C₃×C₄²	#168	1	8	yes	complemented
80	C₃×C₄²	C₄×D₅	1	4	yes	—
81	C₂×C₃⋊C₈	—	5	8	no	—
82	C₂×C₃⋊C₈	—	10	8	no	—
83	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
84	C₂×C₄×C₅	—	2	4	no	—
85	C₂×C₄×C₅	—	2	4	no	—
86	C₂×C₄×C₅	C₂×C₃⋊C₈	1	8	yes	—
87	C₂×C₄×C₅	C₂×C₃⋊C₈	1	4	yes	—
88	C₂×C₄×C₅	#208	1	4	yes	—
89	C₂×C₄×C₅	C₄×Dic₃	1	2	yes	—
90	C₂×C₄×C₅	C₄×Dic₃	1	2	yes	—
91	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
92	C₂×C₄×C₅	#235	1	4	yes	—
93	#168	—	6	8	no	—
94	C₂²×C₈	—	15	4	no	—
95	C₂×C₄²	Dic₁₅	1	2	yes	—
96	C₂×C₄²	Dic₁₅	1	2	yes	—
97	C₂×C₄²	C₂×D₁₅	1	2	yes	—
98	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
99	C₄×C₈	—	15	8	no	—
100	C₂×C₃×C₅	#376	1	1	yes	characteristic
101	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
102	C₂×C₃×C₅	C₂×#97	1	1	yes	characteristic
103	C₂×C₃×C₅	#431	1	4	yes	—
104	C₂³×C₃	C₂²×Dic₅	1	1	yes	characteristic
105	C₂×C₃×C₄	—	2	4	no	—
106	C₂×C₃×C₄	—	2	4	no	—
107	C₂×C₃×C₄	C₂×#168	1	8	yes	—
108	C₂×C₃×C₄	C₂×#168	1	4	yes	—
109	C₂×C₃×C₄	#681	1	4	yes	—
110	C₂×C₃×C₄	C₄×Dic₅	1	2	yes	—
111	C₂×C₃×C₄	C₄×Dic₅	1	2	yes	—
112	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
113	C₂×C₃×C₄	#709	1	4	yes	—
114	C₃⋊C₈	—	10	8	no	—
115	C₂²×C₅	C₂²×C₃⋊C₈	1	2	yes	—
116	C₂²×C₅	C₂×#208	1	2	yes	—
117	C₂²×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
118	C₂²×C₅	#926	1	2	yes	—
119	C₄×C₅	—	2	8	no	—
120	C₄×C₅	C₄×C₃⋊C₈	1	4	yes	—
121	C₄×C₅	#803	1	4	yes	—
122	C₄×C₅	#872	1	4	yes	—
123	C₂²×C₄	C₂×Dic₁₅	1	4	yes	—
124	C₂²×C₄	C₂×Dic₁₅	1	1	yes	characteristic
125	C₂²×C₄	C₂×Dic₁₅	1	1	yes	characteristic
126	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
127	C₂×C₈	—	15	8	no	—
128	C₂×C₈	—	30	8	no	—
129	C₄²	—	2	4	no	—
130	C₄²	—	2	4	no	—
131	C₄²	C₃⋊#168	1	8	yes	complemented
132	C₄²	C₄×D₁₅	1	4	yes	—
133	C₃×C₅	#1744	1	1	yes	characteristic, complemented, Hall
134	C₂²×C₃	C₂²×#168	1	2	yes	—
135	C₂²×C₃	C₂×#681	1	2	yes	—
136	C₂²×C₃	C₂×C₄×Dic₅	1	1	yes	characteristic
137	C₂²×C₃	#4072	1	2	yes	—
138	C₃×C₄	—	2	8	no	—
139	C₃×C₄	C₄×#168	1	4	yes	—
140	C₃×C₄	#3935	1	4	yes	—
141	C₃×C₄	#4018	1	4	yes	—
142	C₂×C₅	C₂×C₄×C₃⋊C₈	1	1	yes	characteristic
143	C₂×C₅	C₂×#803	1	1	yes	characteristic
144	C₂×C₅	#4925	1	4	yes	—
145	C₂×C₅	#4932	1	1	yes	characteristic
146	C₂³	C₂²×Dic₁₅	1	1	yes	characteristic
147	C₂×C₄	—	2	4	no	—
148	C₂×C₄	—	2	4	no	—
149	C₂×C₄	C₂×C₃⋊#168	1	8	yes	—
150	C₂×C₄	C₂×C₃⋊#168	1	4	yes	—
151	C₂×C₄	#6710	1	4	yes	—
152	C₂×C₄	C₄×Dic₁₅	1	2	yes	—
153	C₂×C₄	C₄×Dic₁₅	1	2	yes	—
154	C₂×C₄	C₂×C₄×D₁₅	1	2	yes	—
155	C₂×C₄	#6817	1	4	yes	—
156	C₈	—	30	8	no	—
157	C₂×C₃	C₂×C₄×#168	1	1	yes	characteristic
158	C₂×C₃	C₂×#3935	1	1	yes	characteristic
159	C₂×C₃	#65214	1	4	yes	—
160	C₂×C₃	#65221	1	1	yes	characteristic
161	C₅	#69388	1	1	yes	characteristic, complemented, Sylow
162	C₂²	C₂²×C₃⋊#168	1	2	yes	—
163	C₂²	C₂×#6710	1	2	yes	—
164	C₂²	C₂×C₄×Dic₁₅	1	1	yes	characteristic
165	C₂²	#91815	1	2	yes	—
166	C₄	—	2	8	no	—
167	C₄	C₄×C₃⋊#168	1	4	yes	—
168	C₄	#91087	1	4	yes	—
169	C₄	#91761	1	4	yes	—
170	C₃	#10600181	1	1	yes	characteristic, complemented, Sylow
171	C₂	C₂×C₄×C₃⋊#168	1	1	yes	characteristic
172	C₂	C₂×#91087	1	1	yes	characteristic
173	C₂	#11751026	1	4	yes	—
174	C₂	#11751033	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄²	C₂×D₁₅
commutator	C₂×C₃×C₅	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₃×C₄³×C₅	C₂
socle, prime radical	C₂³×C₃×C₅	C₂²×C₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₄³	D₁₅
hypocenter	C₃×C₅	#1744
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄² ◁ C₄³

Complements

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

N	H ≅ G/N	type
10. C₄×C₃⋊#168	C₄	semidirect
21. C₃×C₄²×C₅	C₈	semidirect
58. C₄²×C₅	C₃⋊C₈	semidirect
79. C₃×C₄²	#168	semidirect
131. C₄²	C₃⋊#168	semidirect
133. C₃×C₅	#1744	semidirect
161. C₅	#69388	semidirect
170. C₃	#10600181	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄³×C₅	2
C₂×C₄×C₃⋊#168	2
#11746345	2
#10600181	3
#69388	5