Subgroups — #14315196

Order 1920.

292 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11752450	C₂	1	1	yes	maximal, characteristic, complemented
2	#11752384	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×#65113	C₂	1	1	yes	maximal, characteristic
4	#11746803	C₂	1	1	yes	maximal, characteristic, complemented
5	#11746769	C₂	1	1	yes	maximal, characteristic, complemented
6	#11746732	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₃⋊#674	C₂	1	1	yes	maximal, characteristic, complemented
8	#10603999	—	3	1	no	maximal, Hall
9	C₃×#4096	C₄	1	1	yes	characteristic
10	C₃×#4094	C₂²	1	1	yes	characteristic
11	C₃×#4053	C₂²	1	1	yes	characteristic
12	#91227	—	2	1	no	—
13	#91227	C₂²	1	1	yes	characteristic
14	#91227	C₂²	1	1	yes	characteristic
15	C₂×C₃⋊#674	—	2	1	no	—
16	C₂×C₃⋊#674	C₂²	1	1	yes	characteristic
17	C₂×C₃⋊#674	C₂²	1	1	yes	characteristic
18	#91219	—	2	1	no	—
19	D₅⋊(C₃⋊C₁₆)	—	2	1	no	—
20	C₃×C₅×#104	C₄	1	1	yes	characteristic
21	C₃×#3954	C₄	1	1	yes	characteristic
22	C₃×#3937	C₄	1	1	yes	characteristic
23	C₃×C₄×#168	C₂²	1	1	yes	characteristic
24	#70923	—	5	1	no	maximal, Hall
25	#65791	—	3	1	no	—
26	#65725	—	3	1	no	—
27	#65113	D₃	1	1	yes	characteristic
28	#64932	—	3	1	no	—
29	#64898	—	3	1	no	—
30	#64861	—	3	1	no	—
31	C₄×#674	—	3	1	no	—
32	C₃×C₅×#41	C₂×C₄	1	1	yes	characteristic
33	C₃×#709	C₂×C₄	1	1	yes	characteristic
34	C₃×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
35	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₃×#689	C₈	1	1	yes	characteristic
37	C₃×#688	C₈	1	1	yes	characteristic
38	C₃×#687	C₈	1	1	yes	characteristic
39	C₃×#686	C₈	1	1	yes	characteristic
40	C₃×#681	—	2	1	no	—
41	C₃×#681	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃×#168	—	2	1	no	—
43	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃×#168	C₂³	1	1	yes	characteristic
45	C₃×#676	—	2	1	no	—
46	C₃×C₈×D₅	—	2	1	no	—
47	C₃⋊#674	—	2	1	no	—
48	C₃⋊#674	—	2	1	no	—
49	C₃⋊#674	—	2	1	no	—
50	C₃⋊#674	D₄	1	1	yes	characteristic
51	C₃⋊#674	D₄	1	1	yes	characteristic
52	C₃×#443	—	5	1	no	—
53	#5140	—	5	1	no	—
54	#5140	—	5	1	no	—
55	#4561	—	5	1	no	—
56	#4561	—	5	1	no	—
57	#4463	—	5	1	no	—
58	C₄×C₃⋊C₁₆	—	5	1	no	—
59	#4096	Dic₃	1	1	yes	characteristic
60	#4094	C₂×D₃	1	1	yes	characteristic
61	#4053	C₂×D₃	1	1	yes	characteristic
62	#3998	—	3	1	no	—
63	#3998	—	3	1	no	—
64	#3998	—	6	1	no	—
65	C₂×#674	—	3	1	no	—
66	C₂×#674	—	3	1	no	—
67	C₂×#674	—	6	1	no	—
68	#3990	—	6	1	no	—
69	D₅⋊C₁₆	—	6	1	no	—
70	C₅×#104	Dic₃	1	1	yes	characteristic
71	#3954	Dic₃	1	1	yes	characteristic
72	#3937	Dic₃	1	1	yes	characteristic
73	C₄×#168	C₂×D₃	1	1	yes	characteristic
74	#2170	—	15	1	no	Sylow
75	C₃×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
76	C₃×C₅×D₄	—	2	1	no	—
77	C₃×C₅×D₄	C₂×C₈	1	1	yes	characteristic
78	C₂×C₃×C₄×C₅	—	2	1	no	—
79	C₂×C₃×C₄×C₅	—	2	1	no	—
80	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
81	C₃×C₂²⋊D₅	—	2	1	no	—
82	C₃×D₂₀	C₂×C₈	1	1	yes	characteristic
83	C₃×C₄×D₅	—	2	1	no	—
84	C₃×Dic₁₀	C₂×C₈	1	1	yes	characteristic
85	C₃×C₅×C₈	—	2	1	no	—
86	C₃×#168	—	2	1	no	—
87	C₃×#168	C₂×C₈	1	1	yes	characteristic
88	C₃×#168	C₂×C₈	1	1	yes	characteristic
89	C₃×#168	C₂×D₄	1	1	yes	characteristic
90	C₃×#168	#41	1	1	yes	characteristic
91	C₃×#135	—	5	1	no	—
92	C₃×#131	—	5	1	no	—
93	C₃×#131	—	5	1	no	—
94	C₃×#108	—	5	1	no	—
95	C₃×#104	—	5	1	no	—
96	C₃×#104	AGL(1,𝔽₅)	1	1	yes	characteristic
97	C₃×C₄×C₈	—	5	1	no	—
98	#812	—	5	1	no	—
99	#812	—	5	1	no	—
100	#812	—	10	1	no	—
101	#812	—	10	1	no	—
102	C₂×C₃⋊C₁₆	—	5	1	no	—
103	C₂×C₃⋊C₁₆	—	5	1	no	—
104	C₂×C₃⋊C₁₆	—	10	1	no	—
105	C₂×C₃⋊C₁₆	—	10	1	no	—
106	C₅×#41	C₂×Dic₃	1	1	yes	characteristic
107	#709	C₂×Dic₃	1	1	yes	characteristic
108	C₅×M₁₆	C₂×Dic₃	1	1	yes	characteristic
109	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
110	#689	C₃⋊C₈	1	1	yes	characteristic
111	#688	C₃⋊C₈	1	1	yes	characteristic
112	#687	C₃⋊C₈	1	1	yes	characteristic
113	#686	C₃⋊C₈	1	1	yes	characteristic
114	#681	—	2	1	no	—
115	#681	C₂×Dic₃	1	1	yes	characteristic
116	C₂×#168	—	2	1	no	—
117	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
118	C₂×#168	C₂²×D₃	1	1	yes	characteristic
119	#676	—	2	1	no	—
120	C₈×D₅	—	2	1	no	—
121	#674	—	3	1	no	—
122	#674	—	3	1	no	—
123	#674	—	6	1	no	—
124	#674	—	6	1	no	—
125	#674	—	6	1	no	—
126	#504	—	15	1	no	—
127	#504	—	15	1	no	—
128	#443	—	5	1	no	—
129	#364	—	15	1	no	—
130	#350	—	15	1	no	—
131	#350	—	15	1	no	—
132	C₄×C₁₆	—	15	1	no	—
133	C₂²×C₃×C₅	—	2	1	no	—
134	C₂²×C₃×C₅	#131	1	1	yes	characteristic
135	C₂×C₃×D₅	—	2	1	no	—
136	C₃×C₄×C₅	—	2	2	no	—
137	C₃×C₄×C₅	—	2	1	no	—
138	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
139	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
140	C₃×Dic₅	—	2	1	no	—
141	C₃×#41	—	5	1	no	—
142	C₃×#41	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
143	C₃×Q₁₆	—	5	1	no	—
144	C₃×SD₁₆	—	5	1	no	—
145	C₃×SD₁₆	—	5	1	no	—
146	C₃×D₈	—	5	1	no	—
147	C₃×M₁₆	—	5	1	no	—
148	C₃×M₁₆	—	10	1	no	—
149	C₃×M₁₆	—	10	1	no	—
150	C₃×M₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
151	C₂×C₃×C₈	—	5	1	no	—
152	C₂×C₃×C₈	—	5	1	no	—
153	C₂×C₃×C₈	—	10	1	no	—
154	C₂×C₃×C₈	—	10	1	no	—
155	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
156	C₃⋊C₁₆	—	5	1	no	—
157	C₃⋊C₁₆	—	5	1	no	—
158	C₃⋊C₁₆	—	10	1	no	—
159	C₃⋊C₁₆	—	10	1	no	—
160	C₃⋊C₁₆	—	10	1	no	—
161	C₅×Q₈	C₂×C₃⋊C₈	1	1	yes	characteristic
162	C₅×D₄	—	2	1	no	—
163	C₅×D₄	C₂×C₃⋊C₈	1	1	yes	characteristic
164	C₂×C₄×C₅	—	2	1	no	—
165	C₂×C₄×C₅	—	2	1	no	—
166	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
167	C₂²⋊D₅	—	2	1	no	—
168	D₂₀	C₂×C₃⋊C₈	1	1	yes	characteristic
169	C₄×D₅	—	2	1	no	—
170	Dic₁₀	C₂×C₃⋊C₈	1	1	yes	characteristic
171	C₅×C₈	—	2	1	no	—
172	#168	—	2	1	no	—
173	#168	C₂×C₃⋊C₈	1	1	yes	characteristic
174	#168	C₂×C₃⋊C₈	1	1	yes	characteristic
175	#168	D₃×D₄	1	1	yes	characteristic
176	#168	#237	1	1	yes	characteristic
177	#135	—	5	1	no	—
178	#131	—	5	1	no	—
179	#131	—	5	1	no	—
180	M₃₂	—	15	1	no	—
181	M₃₂	—	15	1	no	—
182	M₃₂	—	30	1	no	—
183	M₃₂	—	30	1	no	—
184	C₂×C₁₆	—	15	1	no	—
185	C₂×C₁₆	—	15	1	no	—
186	C₂×C₁₆	—	30	1	no	—
187	C₂×C₁₆	—	30	1	no	—
188	#108	—	5	1	no	—
189	#104	—	5	1	no	—
190	#104	ΓL(1,𝔽₁₆)	1	1	yes	characteristic
191	C₄×C₈	—	5	1	no	—
192	C₂×C₃×C₅	—	2	1	no	—
193	C₂×C₃×C₅	—	4	1	no	—
194	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
195	C₃×D₅	—	4	1	no	—
196	C₃×Q₈	—	5	1	no	—
197	C₃×Q₈	D₅⋊C₈	1	1	yes	characteristic
198	C₃×D₄	—	2	1	no	—
199	C₃×D₄	—	5	1	no	—
200	C₃×D₄	—	10	1	no	—
201	C₃×D₄	D₅⋊C₈	1	1	yes	characteristic
202	C₂×C₃×C₄	—	2	1	no	—
203	C₂×C₃×C₄	—	2	1	no	—
204	C₂×C₃×C₄	—	10	1	no	—
205	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
206	C₃×C₈	—	2	1	no	—
207	C₃×C₈	—	5	1	no	—
208	C₃×C₈	—	5	1	no	—
209	C₃×C₈	—	5	1	no	—
210	C₃×C₈	—	5	1	no	—
211	C₃×C₈	—	10	1	no	—
212	C₂²×C₅	—	2	1	no	—
213	C₂²×C₅	#948	1	1	yes	characteristic
214	C₂×D₅	—	2	1	no	—
215	C₄×C₅	—	2	2	no	—
216	C₄×C₅	—	2	1	no	—
217	C₄×C₅	C₂²×C₃⋊C₈	1	1	yes	characteristic
218	C₄×C₅	D₄×Dic₃	1	1	yes	characteristic
219	Dic₅	—	2	1	no	—
220	#41	—	5	1	no	—
221	#41	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
222	Q₁₆	—	5	1	no	—
223	SD₁₆	—	5	1	no	—
224	SD₁₆	—	5	1	no	—
225	D₈	—	5	1	no	—
226	M₁₆	—	5	1	no	—
227	M₁₆	—	10	1	no	—
228	M₁₆	—	10	1	no	—
229	M₁₆	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
230	C₂×C₈	—	5	1	no	—
231	C₂×C₈	—	5	1	no	—
232	C₂×C₈	—	10	1	no	—
233	C₂×C₈	—	10	1	no	—
234	C₄²	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
235	C₁₆	—	15	1	no	—
236	C₁₆	—	15	1	no	—
237	C₁₆	—	30	1	no	—
238	C₁₆	—	30	1	no	—
239	C₁₆	—	30	1	no	—
240	C₃×C₅	#2170	1	1	yes	characteristic, complemented, Hall
241	C₂²×C₃	—	2	1	no	—
242	C₂²×C₃	—	10	1	no	—
243	C₂²×C₃	#4131	1	1	yes	characteristic
244	C₃×C₄	—	2	2	no	—
245	C₃×C₄	—	2	1	no	—
246	C₃×C₄	—	10	1	no	—
247	C₃×C₄	C₂×D₅⋊C₈	1	1	yes	characteristic
248	C₃×C₄	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
249	C₂×C₅	—	2	1	no	—
250	C₂×C₅	—	4	1	no	—
251	C₂×C₅	D₄×C₃⋊C₈	1	1	yes	characteristic
252	D₅	—	4	1	no	—
253	Q₈	—	5	1	no	—
254	Q₈	D₅⋊(C₃⋊C₈)	1	1	yes	characteristic
255	D₄	—	2	1	no	—
256	D₄	—	5	1	no	—
257	D₄	—	10	1	no	—
258	D₄	D₅⋊(C₃⋊C₈)	1	1	yes	characteristic
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	10	1	no	—
262	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
263	C₈	—	2	1	no	—
264	C₈	—	5	1	no	—
265	C₈	—	5	1	no	—
266	C₈	—	5	1	no	—
267	C₈	—	5	1	no	—
268	C₈	—	10	1	no	—
269	C₂×C₃	—	2	1	no	—
270	C₂×C₃	—	4	1	no	—
271	C₂×C₃	—	20	1	no	—
272	C₂×C₃	#65692	1	1	yes	characteristic
273	C₅	#70923	1	1	yes	characteristic, complemented, Sylow
274	C₂²	—	2	1	no	—
275	C₂²	—	10	1	no	—
276	C₂²	#91990	1	1	yes	characteristic
277	C₄	—	2	2	no	—
278	C₄	—	2	1	no	—
279	C₄	—	10	1	no	—
280	C₄	C₂×D₅⋊(C₃⋊C₈)	1	1	yes	characteristic
281	C₄	D₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
282	C₃	#10603999	1	1	yes	characteristic, complemented, Sylow
283	C₂	—	2	1	no	—
284	C₂	—	4	1	no	—
285	C₂	—	20	1	no	—
286	C₂	#11752351	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×ΓL(1,𝔽₁₆)
commutator	C₃×C₄×C₅	C₂²×C₈
Frattini	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)
Fitting	C₃×C₅×#104	C₄
socle	C₂×C₃×C₅	C₈×D₄
prime radical	C₃×#4096	C₄
cosocle	C₂×C₃×#168	C₂³
hypercenter	#104	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#2170
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #104

Complements

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

N	H ≅ G/N	type
1. #11752450	C₂	semidirect
2. #11752384	C₂	semidirect
4. #11746803	C₂	semidirect
5. #11746769	C₂	semidirect
6. #11746732	C₂	semidirect
7. C₄×C₃⋊#674	C₂	semidirect
240. C₃×C₅	#2170	semidirect
273. C₅	#70923	semidirect
282. C₃	#10603999	semidirect

Maximal subgroups

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

subgroup	index
#11752450	2
#11752384	2
C₃×#65113	2
#11746803	2
#11746769	2
#11746732	2
C₄×C₃⋊#674	2
#10603999	3
#70923	5