Subgroups — #14312659

Order 1920.

368 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#4689	C₂	1	1	yes	maximal, characteristic
2	C₂×#90935	C₂	1	1	yes	maximal, characteristic
3	C₂×D₃×#672	C₂	1	1	yes	maximal, characteristic
4	C₄×C₃⋊#672	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#64681	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745874	C₂	1	1	yes	maximal, characteristic, complemented
7	Dic₃×#672	C₂	1	1	yes	maximal, characteristic, complemented
8	#10600674	—	3	1	no	maximal, Hall
9	C₂×C₅×ΓL(1,𝔽₂₅)	C₄	1	1	yes	characteristic
10	C₂×C₅×D₃×C₈	C₂²	1	1	yes	characteristic
11	C₅×#872	C₄	1	1	yes	characteristic
12	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₃⋊#672	C₂²	1	1	yes	characteristic
14	C₂×C₃⋊#672	C₂²	1	1	yes	characteristic
15	C₅×#815	C₄	1	1	yes	characteristic
16	C₅×C₈×Dic₃	C₂²	1	1	yes	characteristic
17	C₅×#803	C₄	1	1	yes	characteristic
18	C₂×C₃×#672	C₂²	1	1	yes	characteristic
19	C₂×C₃×#672	C₂²	1	1	yes	characteristic
20	#90935	C₄	1	4	yes	complemented
21	D₃×#672	C₄	1	4	yes	complemented
22	#69072	—	5	1	no	maximal, Hall
23	C₅×#405	—	3	1	no	—
24	C₂×#3944	—	3	1	no	—
25	C₂²×#672	—	3	1	no	—
26	#64681	—	3	1	no	—
27	#64681	D₃	1	1	yes	characteristic, complemented
28	C₄×#672	—	3	1	no	—
29	C₄×#672	—	3	1	no	—
30	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
33	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
34	C₅×(C₂×D₃)⋊C₄	C₈	1	2	yes	—
35	C₅×Dic₃⋊C₄	C₈	1	2	yes	—
36	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
37	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
38	C₂×C₅×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
39	C₅×ΓL(1,𝔽₂₅)	C₈	1	4	yes	—
40	C₅×D₃×C₈	C₂×C₄	1	2	yes	—
41	C₅×D₃×C₈	C₂×C₄	1	2	yes	—
42	C₃⋊#672	C₂×C₄	1	2	yes	—
43	C₃⋊#672	C₂×C₄	1	2	yes	—
44	C₃×#672	C₂×C₄	1	2	yes	—
45	C₃×#672	C₂×C₄	1	2	yes	—
46	C₂×#798	—	5	1	no	—
47	C₂×D₃×C₁₆	—	5	1	no	—
48	#4689	D₅	1	1	yes	characteristic
49	C₃×#346	—	5	1	no	—
50	#4502	—	5	1	no	—
51	Dic₃×C₁₆	—	5	1	no	—
52	C₄×C₃⋊C₁₆	—	5	1	no	—
53	C₂×C₅×M₁₆	—	3	1	no	—
54	C₂²×C₅×C₈	—	3	1	no	—
55	C₅×#117	—	3	1	no	—
56	C₅×#97	—	3	1	no	—
57	C₅×#97	—	3	1	no	—
58	C₄×C₅×C₈	—	3	1	no	—
59	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
60	#3944	—	3	4	no	—
61	C₂×#672	—	3	1	no	—
62	C₂×#672	—	3	1	no	—
63	C₂×#672	—	3	4	no	—
64	C₂×#672	C₂×D₃	1	1	yes	characteristic
65	C₂×#672	C₂×D₃	1	1	yes	characteristic
66	#2108	—	15	1	no	Sylow
67	C₂²×C₅×D₃	C₂×C₈	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
69	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₅×Dic₃	C₂×C₈	1	2	yes	—
71	C₂×C₅×Dic₃	C₂×C₈	1	1	yes	characteristic
72	C₄×C₅×D₃	C₄²	1	2	yes	—
73	C₄×C₅×D₃	C₂×C₈	1	2	yes	—
74	C₃×C₅×C₈	C₂×C₈	1	2	yes	—
75	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
76	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
77	C₅×C₃⋊C₈	C₄²	1	2	yes	—
78	C₅×C₃⋊C₈	C₂×C₈	1	2	yes	—
79	C₂×ΓL(1,𝔽₂₅)	Dic₅	1	1	yes	characteristic
80	C₂×D₃×C₈	C₂×D₅	1	1	yes	characteristic
81	#872	Dic₅	1	1	yes	characteristic
82	C₂×C₃×C₁₆	—	5	1	no	—
83	C₂×C₃×C₁₆	—	5	1	no	—
84	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
85	#815	Dic₅	1	1	yes	characteristic
86	C₈×Dic₃	C₂×D₅	1	1	yes	characteristic
87	C₂×C₃⋊C₁₆	—	5	1	no	—
88	C₂×C₃⋊C₁₆	—	5	1	no	—
89	#803	Dic₅	1	1	yes	characteristic
90	#798	—	5	4	no	—
91	D₃×C₁₆	—	5	4	no	—
92	C₂²×C₄×C₅	—	3	1	no	—
93	C₅×M₁₆	—	3	4	no	—
94	C₂×C₅×C₈	—	3	1	no	—
95	C₂×C₅×C₈	—	3	1	no	—
96	C₂×C₅×C₈	—	3	2	no	—
97	C₂×C₅×C₈	—	3	2	no	—
98	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
99	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
100	C₅×C₄⋊C₄	—	3	2	no	—
101	C₅×C₂²⋊C₄	—	3	2	no	—
102	C₄²×C₅	—	3	1	no	—
103	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
104	#672	—	3	2	no	—
105	#672	—	3	2	no	—
106	#672	C₄×D₃	1	2	yes	—
107	#672	C₄×D₃	1	2	yes	—
108	C₂×M₃₂	—	15	1	no	—
109	C₂²×C₁₆	—	15	1	no	—
110	#405	—	3	1	no	—
111	#346	—	5	1	no	—
112	#346	—	15	1	no	—
113	C₄×C₁₆	—	15	1	no	—
114	C₄×C₁₆	—	15	1	no	—
115	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
116	C₂×C₅×D₃	—	2	2	no	—
117	C₂×C₅×D₃	C₄×C₈	1	2	yes	—
118	C₃×C₄×C₅	—	2	2	no	—
119	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
120	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
121	C₅×Dic₃	—	2	2	no	—
122	C₅×Dic₃	C₄×C₈	1	2	yes	—
123	C₂×C₄×D₃	C₂×Dic₅	1	1	yes	characteristic
124	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
125	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
126	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
127	(C₂×D₃)⋊C₄	#168	1	2	yes	—
128	Dic₃⋊C₄	#168	1	2	yes	—
129	C₄×Dic₃	C₂×Dic₅	1	1	yes	characteristic
130	C₂×C₃⋊C₈	C₂×Dic₅	1	1	yes	characteristic
131	C₂×C₃⋊C₈	C₂×Dic₅	1	1	yes	characteristic
132	ΓL(1,𝔽₂₅)	#168	1	4	yes	—
133	D₃×C₈	C₄×D₅	1	2	yes	—
134	D₃×C₈	C₄×D₅	1	2	yes	—
135	C₃×C₁₆	—	5	2	no	—
136	C₃×C₁₆	—	5	2	no	—
137	C₃⋊C₁₆	—	5	2	no	—
138	C₃⋊C₁₆	—	5	2	no	—
139	C₂³×C₅	—	3	1	no	—
140	C₂×C₄×C₅	—	3	1	no	—
141	C₂×C₄×C₅	—	3	2	no	—
142	C₂×C₄×C₅	—	3	2	no	—
143	C₂×C₄×C₅	—	3	2	no	—
144	C₂×C₄×C₅	D₃×C₈	1	2	yes	—
145	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
146	C₅×C₈	—	3	2	no	—
147	C₅×C₈	—	3	2	no	—
148	C₅×C₈	D₃×C₈	1	2	yes	—
149	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
150	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
151	C₂×M₁₆	—	3	1	no	—
152	C₂²×C₈	—	3	1	no	—
153	#117	—	3	1	no	—
154	M₃₂	—	15	4	no	—
155	C₂×C₁₆	—	5	1	no	—
156	C₂×C₁₆	—	5	1	no	—
157	C₂×C₁₆	—	15	4	no	—
158	C₂×C₁₆	—	15	1	no	—
159	C₂×C₁₆	—	15	1	no	—
160	#97	—	3	1	no	—
161	#97	—	3	1	no	—
162	C₄×C₈	—	3	1	no	—
163	C₄×C₈	D₃×D₅	1	1	yes	characteristic
164	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
165	C₂×C₃×C₅	#504	1	2	yes	—
166	C₅×D₃	—	2	2	no	—
167	C₂²×D₃	C₂×#168	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂×#168	1	2	yes	—
169	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
170	C₂×Dic₃	C₂×#168	1	2	yes	—
171	C₂×Dic₃	C₂×#168	1	1	yes	characteristic
172	C₄×D₃	C₂×#168	1	2	yes	—
173	C₄×D₃	C₄×Dic₅	1	2	yes	—
174	C₃×C₈	C₂×#168	1	2	yes	—
175	C₃×C₈	C₂×C₄×D₅	1	1	yes	characteristic
176	C₃×C₈	C₂×C₄×D₅	1	1	yes	characteristic
177	C₃⋊C₈	C₂×#168	1	2	yes	—
178	C₃⋊C₈	C₄×Dic₅	1	2	yes	—
179	C₂²×C₅	—	3	2	no	—
180	C₂²×C₅	—	6	2	no	—
181	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
182	C₄×C₅	—	2	2	no	—
183	C₄×C₅	—	3	2	no	—
184	C₄×C₅	—	6	2	no	—
185	C₄×C₅	C₄²×D₃	1	1	yes	characteristic
186	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
187	C₂²×C₄	—	3	1	no	—
188	M₁₆	—	3	4	no	—
189	C₂×C₈	—	3	1	no	—
190	C₂×C₈	—	3	1	no	—
191	C₂×C₈	—	3	2	no	—
192	C₂×C₈	—	3	2	no	—
193	C₂×C₈	D₃×Dic₅	1	1	yes	characteristic
194	C₂×C₈	C₂×D₃×D₅	1	1	yes	characteristic
195	C₄⋊C₄	—	3	2	no	—
196	C₂²⋊C₄	—	3	2	no	—
197	C₄²	—	3	1	no	—
198	C₄²	D₃×Dic₅	1	1	yes	characteristic
199	C₁₆	—	5	2	no	—
200	C₁₆	—	5	2	no	—
201	C₁₆	—	15	2	no	—
202	C₁₆	—	15	2	no	—
203	C₃×C₅	#2108	1	1	yes	characteristic, complemented, Hall
204	C₂²×C₃	C₂²×#168	1	1	yes	characteristic
205	C₂×D₃	—	2	2	no	—
206	C₂×D₃	C₄×#168	1	2	yes	—
207	C₃×C₄	—	2	2	no	—
208	C₃×C₄	C₂²×#168	1	1	yes	characteristic
209	C₃×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
210	Dic₃	—	2	2	no	—
211	Dic₃	C₄×#168	1	2	yes	—
212	C₂×C₅	—	6	2	no	—
213	C₂×C₅	C₄×D₃×C₈	1	1	yes	characteristic
214	C₂×C₅	#4907	1	2	yes	—
215	C₂³	—	3	1	no	—
216	C₂×C₄	—	3	1	no	—
217	C₂×C₄	—	3	2	no	—
218	C₂×C₄	—	3	2	no	—
219	C₂×C₄	—	3	2	no	—
220	C₂×C₄	D₃×#168	1	2	yes	—
221	C₂×C₄	C₂×D₃×Dic₅	1	1	yes	characteristic
222	C₈	—	3	2	no	—
223	C₈	—	3	2	no	—
224	C₈	D₃×#168	1	2	yes	—
225	C₈	C₄×D₃×D₅	1	1	yes	characteristic
226	C₈	C₄×D₃×D₅	1	1	yes	characteristic
227	C₂×C₃	C₂×C₄×#168	1	1	yes	characteristic
228	C₂×C₃	#65429	1	2	yes	—
229	D₃	—	2	2	no	—
230	C₅	#69072	1	1	yes	characteristic, complemented, Sylow
231	C₂²	—	3	2	no	—
232	C₂²	—	6	2	no	—
233	C₂²	C₂×D₃×#168	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	3	2	no	—
236	C₄	—	6	2	no	—
237	C₄	C₂×D₃×#168	1	1	yes	characteristic
238	C₄	C₄×D₃×Dic₅	1	1	yes	characteristic
239	C₃	#10600674	1	1	yes	characteristic, complemented, Sylow
240	C₂	—	6	2	no	—
241	C₂	C₄×D₃×#168	1	1	yes	characteristic
242	C₂	#11747529	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×D₃×D₅
commutator	C₂×C₃×C₅	C₂×C₄×C₈
Fitting	C₃×C₄×C₅×C₈	C₂²
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂²×C₅×D₃	C₂×C₈
cosocle	C₂×C₃×C₅×C₈	C₂³
hypercenter	C₄×C₈	D₃×D₅
hypocenter	C₃×C₅	#2108
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
4. C₄×C₃⋊#672	C₂	semidirect
5. C₃×#64681	C₂	semidirect
6. #11745874	C₂	semidirect
7. Dic₃×#672	C₂	semidirect
20. #90935	C₄	semidirect
21. D₃×#672	C₄	semidirect
27. #64681	D₃	semidirect
203. C₃×C₅	#2108	semidirect
230. C₅	#69072	semidirect
239. C₃	#10600674	semidirect

Maximal subgroups

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

subgroup	index
C₅×#4689	2
C₂×#90935	2
C₂×D₃×#672	2
C₄×C₃⋊#672	2
C₃×#64681	2
#11745874	2
Dic₃×#672	2
#10600674	3
#69072	5