Subgroups — #14312681

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	C₂×#91165	C₂	1	1	yes	maximal, characteristic
2	C₂×D₃×#674	C₂	1	1	yes	maximal, characteristic
3	#11746977	C₂	1	1	yes	maximal, characteristic
4	#11746730	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₄×#674	C₂	1	1	yes	maximal, characteristic, complemented
6	#11746542	C₂	1	1	yes	maximal, characteristic, complemented
7	Dic₃×#674	C₂	1	1	yes	maximal, characteristic, complemented
8	#10604366	—	3	1	no	maximal, Hall
9	C₅×#872	C₄	1	1	yes	characteristic
10	C₂×#6650	C₄	1	1	yes	characteristic
11	C₂×D₃×#168	C₂²	1	1	yes	characteristic
12	C₂×C₃⋊#674	C₂²	1	1	yes	characteristic
13	C₂×C₃⋊#674	C₂²	1	1	yes	characteristic
14	C₂×C₃×#674	C₂²	1	1	yes	characteristic
15	C₂×C₃×#674	C₂²	1	1	yes	characteristic
16	#91165	C₄	1	4	yes	complemented
17	D₃×#674	C₄	1	4	yes	complemented
18	#91087	C₄	1	1	yes	characteristic
19	C₃×C₄×#168	C₂²	1	1	yes	characteristic
20	#90953	C₄	1	1	yes	characteristic
21	Dic₃×#168	C₂²	1	1	yes	characteristic
22	#69051	—	5	1	no	maximal, Hall
23	C₂×#3998	—	3	1	no	—
24	C₂²×#674	—	3	1	no	—
25	#65286	—	3	1	no	—
26	#64859	—	3	1	no	—
27	#64859	—	3	1	no	—
28	C₄×#674	—	3	1	no	—
29	C₄×#674	D₃	1	1	yes	characteristic, complemented
30	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
31	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃⋊#168	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×C₃×#168	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃×#168	C₂³	1	1	yes	characteristic
39	#6650	C₈	1	4	yes	—
40	D₃×#168	C₂×C₄	1	2	yes	—
41	D₃×#168	C₂×C₄	1	2	yes	—
42	C₃⋊#674	C₂×C₄	1	2	yes	—
43	C₃⋊#674	C₂×C₄	1	2	yes	—
44	C₃×#674	C₂×C₄	1	2	yes	—
45	C₃×#674	C₂×C₄	1	2	yes	—
46	C₂×#798	—	5	1	no	—
47	C₂×D₃×C₁₆	—	5	1	no	—
48	#4689	—	5	1	no	—
49	C₃×C₄×C₁₆	—	5	1	no	—
50	#4502	—	5	1	no	—
51	Dic₃×C₁₆	—	5	1	no	—
52	#4461	—	5	1	no	—
53	C₅×#117	—	3	1	no	—
54	C₂×#681	—	3	1	no	—
55	C₂²×#168	—	3	1	no	—
56	#3998	—	3	4	no	—
57	C₂×#674	—	3	1	no	—
58	C₂×#674	—	3	1	no	—
59	C₂×#674	—	3	4	no	—
60	C₂×#674	C₂×D₃	1	1	yes	characteristic
61	C₂×#674	C₂×D₃	1	1	yes	characteristic
62	#3935	—	3	1	no	—
63	#3935	—	3	1	no	—
64	C₄×#168	—	3	1	no	—
65	C₄×#168	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⋊#168	C₄²	1	2	yes	—
75	C₃⋊#168	C₂×C₈	1	2	yes	—
76	C₃×#168	C₂×C₈	1	2	yes	—
77	C₃×#168	C₂²×C₄	1	1	yes	characteristic
78	C₃×#168	C₂²×C₄	1	1	yes	characteristic
79	C₂×ΓL(1,𝔽₂₅)	—	5	1	no	—
80	C₂×D₃×C₈	—	5	1	no	—
81	#872	AGL(1,𝔽₅)	1	1	yes	characteristic
82	C₂×C₃×C₁₆	—	5	1	no	—
83	C₂×C₃×C₁₆	—	5	1	no	—
84	C₃×C₄×C₈	—	5	1	no	—
85	#815	—	5	1	no	—
86	C₈×Dic₃	—	5	1	no	—
87	C₂×C₃⋊C₁₆	—	5	1	no	—
88	C₂×C₃⋊C₁₆	—	5	1	no	—
89	#803	—	5	1	no	—
90	#798	—	5	4	no	—
91	D₃×C₁₆	—	5	4	no	—
92	C₂²×C₄×C₅	—	3	1	no	—
93	C₅×C₄⋊C₄	—	3	2	no	—
94	C₅×C₂²⋊C₄	—	3	2	no	—
95	C₄²×C₅	—	3	1	no	—
96	C₄²×C₅	C₄×D₃	1	1	yes	characteristic
97	#681	—	3	4	no	—
98	C₂×#168	—	3	1	no	—
99	C₂×#168	—	3	1	no	—
100	C₂×#168	—	3	2	no	—
101	C₂×#168	—	3	2	no	—
102	C₂×#168	C₄×D₃	1	1	yes	characteristic
103	C₂×#168	C₂²×D₃	1	1	yes	characteristic
104	#674	—	3	2	no	—
105	#674	—	3	2	no	—
106	#674	C₄×D₃	1	2	yes	—
107	#674	C₄×D₃	1	2	yes	—
108	C₂×M₃₂	—	15	1	no	—
109	C₂²×C₁₆	—	15	1	no	—
110	#405	—	15	1	no	—
111	#346	—	15	1	no	—
112	#346	—	15	1	no	—
113	C₄×C₁₆	—	5	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×AGL(1,𝔽₅)	1	1	yes	characteristic
124	C₂×C₃×C₈	—	5	1	no	—
125	C₂×C₃×C₈	—	5	1	no	—
126	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
127	(C₂×D₃)⋊C₄	#170	1	2	yes	—
128	Dic₃⋊C₄	#170	1	2	yes	—
129	C₄×Dic₃	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
130	C₂×C₃⋊C₈	—	5	1	no	—
131	C₂×C₃⋊C₈	—	5	1	no	—
132	ΓL(1,𝔽₂₅)	—	5	4	no	—
133	D₃×C₈	—	5	2	no	—
134	D₃×C₈	—	5	2	no	—
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	#168	—	3	2	no	—
147	#168	—	3	2	no	—
148	#168	D₃×C₈	1	2	yes	—
149	#168	C₂×C₄×D₃	1	1	yes	characteristic
150	#168	C₂×C₄×D₃	1	1	yes	characteristic
151	C₂×M₁₆	—	15	1	no	—
152	C₂²×C₈	—	15	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	—	15	1	no	—
161	#97	—	15	1	no	—
162	C₄×C₈	—	5	1	no	—
163	C₄×C₈	—	15	1	no	—
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×#170	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂×#170	1	2	yes	—
169	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
170	C₂×Dic₃	C₂×#170	1	2	yes	—
171	C₂×Dic₃	C₂×#170	1	1	yes	characteristic
172	C₄×D₃	D₅⋊C₈	1	2	yes	—
173	C₄×D₃	C₄×AGL(1,𝔽₅)	1	2	yes	—
174	C₃×C₈	—	5	2	no	—
175	C₃×C₈	—	5	1	no	—
176	C₃×C₈	—	5	1	no	—
177	C₃⋊C₈	—	5	2	no	—
178	C₃⋊C₈	—	5	2	no	—
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	—	15	4	no	—
189	C₂×C₈	—	5	1	no	—
190	C₂×C₈	—	5	1	no	—
191	C₂×C₈	—	15	2	no	—
192	C₂×C₈	—	15	2	no	—
193	C₂×C₈	—	15	1	no	—
194	C₂×C₈	—	15	1	no	—
195	C₄⋊C₄	—	3	2	no	—
196	C₂²⋊C₄	—	3	2	no	—
197	C₄²	—	3	1	no	—
198	C₄²	D₃×AGL(1,𝔽₅)	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×#170	1	1	yes	characteristic
205	C₂×D₃	—	2	2	no	—
206	C₂×D₃	C₄×#170	1	2	yes	—
207	C₃×C₄	—	2	2	no	—
208	C₃×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
209	C₃×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
210	Dic₃	—	2	2	no	—
211	Dic₃	C₄×#170	1	2	yes	—
212	C₂×C₅	—	6	2	no	—
213	C₂×C₅	C₄×D₃×C₈	1	1	yes	characteristic
214	C₂×C₅	#4901	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×#170	1	2	yes	—
221	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
222	C₈	—	5	2	no	—
223	C₈	—	5	1	no	—
224	C₈	—	5	1	no	—
225	C₈	—	15	2	no	—
226	C₈	—	15	2	no	—
227	C₂×C₃	C₂×C₄×#170	1	1	yes	characteristic
228	C₂×C₃	#65791	1	2	yes	—
229	D₃	—	2	2	no	—
230	C₅	#69051	1	1	yes	characteristic, complemented, Sylow
231	C₂²	—	3	2	no	—
232	C₂²	—	6	2	no	—
233	C₂²	C₂×D₃×#170	1	1	yes	characteristic
234	C₄	—	2	2	no	—
235	C₄	—	3	2	no	—
236	C₄	—	6	2	no	—
237	C₄	D₃×D₅⋊C₈	1	1	yes	characteristic
238	C₄	C₄×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
239	C₃	#10604366	1	1	yes	characteristic, complemented, Sylow
240	C₂	—	6	2	no	—
241	C₂	#11751736	1	2	yes	—
242	C₂	C₄×D₃×#170	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)
commutator	C₂×C₃×C₅	C₂×C₄×C₈
Fitting	C₃×C₄²×C₅	C₂×C₄
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂²×C₅×D₃	C₂×C₈
cosocle	C₂×C₃×#168	C₂³
hypercenter	C₄²	D₃×AGL(1,𝔽₅)
hypocenter	C₃×C₅	#2108
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
4. #11746730	C₂	semidirect
5. C₃×C₄×#674	C₂	semidirect
6. #11746542	C₂	semidirect
7. Dic₃×#674	C₂	semidirect
16. #91165	C₄	semidirect
17. D₃×#674	C₄	semidirect
29. C₄×#674	D₃	semidirect
203. C₃×C₅	#2108	semidirect
230. C₅	#69051	semidirect
239. C₃	#10604366	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91165	2
C₂×D₃×#674	2
#11746977	2
#11746730	2
C₃×C₄×#674	2
#11746542	2
Dic₃×#674	2
#10604366	3
#69051	5