Subgroups — #14298776

Order 1920.

344 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₅⋊#819	C₂	1	1	yes	maximal, characteristic
2	#11752380	C₂	1	1	yes	maximal, characteristic
3	C₃×#64978	C₂	1	1	yes	maximal, characteristic
4	#11746765	C₂	1	2	yes	maximal
5	#11746734	C₂	1	2	yes	maximal
6	#10603852	—	3	1	no	maximal, Hall
7	#91989	C₂²	1	2	yes	—
8	D₅⋊#208	C₂²	1	2	yes	—
9	C₂×C₃×#676	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
11	C₃×#4018	C₂²	1	1	yes	characteristic
12	#91225	—	2	1	no	—
13	#91224	—	2	1	no	—
14	#91223	—	2	1	no	—
15	#91222	—	2	1	no	—
16	C₃×C₄×C₅×C₈	C₄	1	1	yes	characteristic, complemented
17	C₃×#3947	C₄	1	1	yes	characteristic, complemented
18	C₃×C₈×Dic₅	C₄	1	1	yes	characteristic, complemented
19	C₃×#3935	C₄	1	1	yes	characteristic, complemented
20	#69896	—	5	1	no	maximal, Hall
21	D₅⋊#108	—	3	1	no	—
22	#65721	—	3	1	no	—
23	#64978	D₃	1	1	yes	characteristic
24	#64894	—	3	2	no	—
25	#64863	—	3	2	no	—
26	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
27	#6763	—	2	2	no	—
28	#6762	—	2	2	no	—
29	C₄⋊ΓL(1,𝔽₁₆)	—	2	2	no	—
30	C₄×ΓL(1,𝔽₁₆)	—	2	2	no	—
31	#6758	—	2	2	no	—
32	D₅⋊(C₃⋊C₈)	—	2	2	no	—
33	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
35	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
36	C₃×#685	—	2	1	no	—
37	C₃×#683	—	2	1	no	—
38	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
40	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
41	C₃×#676	—	2	1	no	—
42	C₃×#676	—	2	1	no	—
43	C₃×C₈×D₅	D₄	1	1	yes	characteristic
44	C₃×C₈×D₅	D₄	1	1	yes	characteristic
45	C₃×C₈×D₅	Q₈	1	1	yes	characteristic
46	C₃×C₈×D₅	Q₈	1	1	yes	characteristic
47	C₃×#405	—	5	1	no	—
48	C₂×#819	—	5	1	no	—
49	#5106	—	5	1	no	—
50	#4556	—	5	2	no	—
51	#4531	—	5	2	no	—
52	#4130	—	3	2	no	—
53	D₅⋊M₁₆	—	3	2	no	—
54	C₂×#676	C₂×D₃	1	1	yes	characteristic
55	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
56	#4018	C₂×D₃	1	1	yes	characteristic
57	#3996	—	6	1	no	—
58	#3995	—	6	1	no	—
59	#3994	—	6	1	no	—
60	#3993	—	6	1	no	—
61	C₄×C₅×C₈	Dic₃	1	1	yes	characteristic, complemented
62	#3947	Dic₃	1	1	yes	characteristic, complemented
63	C₈×Dic₅	Dic₃	1	1	yes	characteristic, complemented
64	#3935	Dic₃	1	1	yes	characteristic, complemented
65	#1833	—	15	1	no	Sylow
66	C₂²×C₃×D₅	#41	1	1	yes	characteristic
67	C₂×ΓL(1,𝔽₁₆)	—	4	2	no	—
68	C₂×C₃×C₄×C₅	—	2	1	no	—
69	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₃×Dic₅	—	2	1	no	—
71	C₂×C₃×Dic₅	#41	1	1	yes	characteristic
72	C₃×C₄×D₅	C₂×D₄	1	1	yes	characteristic
73	C₃×C₄×D₅	C₂×Q₈	1	1	yes	characteristic
74	C₃×C₄×D₅	#41	1	1	yes	characteristic
75	C₃×C₄×D₅	#41	1	1	yes	characteristic
76	C₃⋊#170	—	4	2	no	—
77	C₃×C₅×C₈	—	2	1	no	—
78	C₃×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
79	C₃×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
80	C₃×#168	—	2	1	no	—
81	C₃×#168	C₄⋊C₄	1	1	yes	characteristic
82	C₃×#168	C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₃×M₁₆	—	5	1	no	—
84	C₂²×C₃×C₈	—	5	1	no	—
85	C₃×#117	—	5	1	no	—
86	#926	—	5	2	no	—
87	C₂×#208	—	5	2	no	—
88	C₃×#97	—	5	1	no	—
89	C₃×#97	—	5	1	no	—
90	C₃×C₄×C₈	—	5	1	no	—
91	C₃×C₄×C₈	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
92	#819	—	10	1	no	—
93	#819	—	10	1	no	—
94	#818	—	10	1	no	—
95	#817	—	10	1	no	—
96	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
97	C₂²⋊AGL(1,𝔽₅)	—	6	2	no	—
98	#704	—	6	2	no	—
99	C₄⋊AGL(1,𝔽₅)	—	6	2	no	—
100	C₄×AGL(1,𝔽₅)	—	6	2	no	—
101	#700	—	6	2	no	—
102	D₅⋊C₈	—	6	2	no	—
103	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
104	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
105	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
106	#685	—	2	1	no	—
107	#683	—	2	1	no	—
108	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic
109	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
110	C₂×#168	C₂×Dic₃	1	1	yes	characteristic
111	#676	—	2	1	no	—
112	#676	—	2	1	no	—
113	C₈×D₅	Dic₆	1	1	yes	characteristic
114	C₈×D₅	Dic₆	1	1	yes	characteristic
115	C₈×D₅	D₁₂	1	1	yes	characteristic
116	C₈×D₅	D₁₂	1	1	yes	characteristic
117	C₂×#108	—	15	1	no	—
118	#427	—	15	1	no	—
119	#405	—	5	1	no	—
120	#344	—	15	2	no	—
121	H(3,ℤ/4)	—	15	2	no	—
122	C₂²×C₃×C₅	#117	1	1	yes	characteristic
123	C₂×C₃×D₅	—	4	1	no	—
124	C₂×C₃×D₅	#124	1	1	yes	characteristic
125	C₂×C₃×D₅	#125	1	1	yes	characteristic
126	ΓL(1,𝔽₁₆)	—	8	2	no	—
127	C₃×C₄×C₅	—	4	1	no	—
128	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₃×C₄×C₅	#117	1	1	yes	characteristic
130	C₃×Dic₅	—	4	1	no	—
131	C₃×Dic₅	#124	1	1	yes	characteristic
132	C₃×Dic₅	#125	1	1	yes	characteristic
133	C₂²×C₃×C₄	—	5	1	no	—
134	C₃×M₁₆	—	10	1	no	—
135	C₃×M₁₆	—	10	1	no	—
136	C₂×C₃×C₈	—	5	1	no	—
137	C₂×C₃×C₈	—	5	1	no	—
138	C₂×C₃×C₈	—	5	1	no	—
139	C₂×C₃×C₈	—	5	1	no	—
140	C₂×C₃×C₈	—	5	1	no	—
141	C₂×C₃×C₈	—	5	1	no	—
142	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
143	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
144	C₃×C₄⋊C₄	—	10	1	no	—
145	C₃×C₂²⋊C₄	—	10	1	no	—
146	C₃×C₄²	—	5	1	no	—
147	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
148	#217	—	10	2	no	—
149	C₄⋊Dic₃	—	10	2	no	—
150	C₄×Dic₃	—	10	2	no	—
151	#208	—	10	2	no	—
152	#208	—	10	2	no	—
153	C₂×C₃⋊C₈	—	10	2	no	—
154	C₂²×D₅	#235	1	1	yes	characteristic
155	C₂×AGL(1,𝔽₅)	—	12	2	no	—
156	C₂×C₄×C₅	—	2	1	no	—
157	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
158	C₂×Dic₅	—	2	1	no	—
159	C₂×Dic₅	#235	1	1	yes	characteristic
160	C₄×D₅	C₂×Dic₆	1	1	yes	characteristic
161	C₄×D₅	C₂×D₁₂	1	1	yes	characteristic
162	C₄×D₅	#235	1	1	yes	characteristic
163	C₄×D₅	#235	1	1	yes	characteristic
164	#170	—	12	2	no	—
165	C₅×C₈	—	2	1	no	—
166	C₅×C₈	C₄⋊Dic₃	1	1	yes	characteristic
167	C₅×C₈	C₄⋊Dic₃	1	1	yes	characteristic
168	#168	—	2	1	no	—
169	#168	C₄⋊Dic₃	1	1	yes	characteristic
170	#168	C₄⋊Dic₃	1	1	yes	characteristic
171	C₂×M₁₆	—	5	1	no	—
172	C₂×M₁₆	—	15	2	no	—
173	C₂²×C₈	—	5	1	no	—
174	#117	—	5	1	no	—
175	#117	—	15	2	no	—
176	#108	—	30	1	no	—
177	#108	—	30	1	no	—
178	#107	—	30	1	no	—
179	#106	—	30	1	no	—
180	#97	—	5	1	no	—
181	#97	—	5	1	no	—
182	C₄×C₈	—	5	1	no	—
183	C₄×C₈	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
184	C₂×C₃×C₅	—	2	1	no	—
185	C₂×C₃×C₅	#382	1	1	yes	characteristic
186	C₃×D₅	—	2	1	no	—
187	C₃×D₅	—	2	1	no	—
188	C₂³×C₃	—	5	1	no	—
189	C₂×C₃×C₄	—	2	1	no	—
190	C₂×C₃×C₄	—	5	1	no	—
191	C₂×C₃×C₄	—	5	1	no	—
192	C₂×C₃×C₄	—	5	1	no	—
193	C₂×C₃×C₄	—	5	1	no	—
194	C₂×C₃×C₄	—	5	1	no	—
195	C₂×C₃×C₄	—	10	1	no	—
196	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
197	C₂×Dic₃	—	20	2	no	—
198	C₃×C₈	—	2	1	no	—
199	C₃×C₈	—	5	1	no	—
200	C₃×C₈	—	5	1	no	—
201	C₃×C₈	—	10	1	no	—
202	C₃×C₈	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
203	C₃×C₈	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
204	C₃⋊C₈	—	20	2	no	—
205	C₂²×C₅	#926	1	1	yes	characteristic
206	C₂×D₅	—	4	1	no	—
207	C₂×D₅	#870	1	1	yes	characteristic
208	C₂×D₅	#875	1	1	yes	characteristic
209	AGL(1,𝔽₅)	—	24	2	no	—
210	C₄×C₅	—	4	1	no	—
211	C₄×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
212	C₄×C₅	#926	1	1	yes	characteristic
213	Dic₅	—	4	1	no	—
214	Dic₅	#870	1	1	yes	characteristic
215	Dic₅	#875	1	1	yes	characteristic
216	C₂²×C₄	—	5	1	no	—
217	M₁₆	—	10	1	no	—
218	M₁₆	—	10	1	no	—
219	M₁₆	—	30	2	no	—
220	M₁₆	—	30	2	no	—
221	C₂×C₈	—	5	1	no	—
222	C₂×C₈	—	5	1	no	—
223	C₂×C₈	—	5	1	no	—
224	C₂×C₈	—	5	1	no	—
225	C₂×C₈	—	5	1	no	—
226	C₂×C₈	—	5	1	no	—
227	C₂×C₈	—	30	2	no	—
228	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
229	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
230	C₄⋊C₄	—	10	1	no	—
231	C₄⋊C₄	—	30	2	no	—
232	C₂²⋊C₄	—	10	1	no	—
233	C₂²⋊C₄	—	30	2	no	—
234	C₄²	—	5	1	no	—
235	C₄²	—	30	2	no	—
236	C₄²	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
237	C₃×C₅	#1833	1	1	yes	characteristic, complemented, Hall
238	C₂²×C₃	—	5	1	no	—
239	C₂²×C₃	—	5	1	no	—
240	C₂²×C₃	—	20	1	no	—
241	C₂²×C₃	#4130	1	1	yes	characteristic
242	C₃×C₄	—	4	1	no	—
243	C₃×C₄	—	5	1	no	—
244	C₃×C₄	—	5	1	no	—
245	C₃×C₄	—	20	1	no	—
246	C₃×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
247	C₃×C₄	#4130	1	1	yes	characteristic
248	Dic₃	—	40	2	no	—
249	C₂×C₅	—	2	1	no	—
250	C₂×C₅	#4936	1	1	yes	characteristic
251	D₅	—	2	1	no	—
252	D₅	—	2	1	no	—
253	C₂³	—	5	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	5	1	no	—
256	C₂×C₄	—	5	1	no	—
257	C₂×C₄	—	5	1	no	—
258	C₂×C₄	—	5	1	no	—
259	C₂×C₄	—	5	1	no	—
260	C₂×C₄	—	10	1	no	—
261	C₂×C₄	—	60	2	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	—	10	1	no	—
267	C₈	—	60	2	no	—
268	C₈	C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
269	C₈	C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
270	C₂×C₃	—	2	1	no	—
271	C₂×C₃	—	10	1	no	—
272	C₂×C₃	—	10	1	no	—
273	C₂×C₃	#65689	1	1	yes	characteristic
274	C₅	#69896	1	1	yes	characteristic, complemented, Sylow
275	C₂²	—	5	1	no	—
276	C₂²	—	5	1	no	—
277	C₂²	—	20	1	no	—
278	C₂²	#91989	1	1	yes	characteristic
279	C₄	—	4	1	no	—
280	C₄	—	5	1	no	—
281	C₄	—	5	1	no	—
282	C₄	—	20	1	no	—
283	C₄	—	120	2	no	—
284	C₄	C₂×C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
285	C₄	#91989	1	1	yes	characteristic
286	C₃	#10603852	1	1	yes	characteristic, complemented, Sylow
287	C₂	—	2	1	no	—
288	C₂	—	10	1	no	—
289	C₂	—	10	1	no	—
290	C₂	#11752348	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#91989
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)
Fitting	C₃×C₄×C₅×C₈	C₄
socle	C₂×C₃×C₅	#382
prime radical	C₂²×C₃×D₅	#41
cosocle	C₂×C₃×C₄×D₅	C₂³
hypercenter	C₄×C₈	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#1833
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
16. C₃×C₄×C₅×C₈	C₄	semidirect
17. C₃×#3947	C₄	semidirect
18. C₃×C₈×Dic₅	C₄	semidirect
19. C₃×#3935	C₄	semidirect
61. C₄×C₅×C₈	Dic₃	semidirect
62. #3947	Dic₃	semidirect
63. C₈×Dic₅	Dic₃	semidirect
64. #3935	Dic₃	semidirect
91. C₃×C₄×C₈	AGL(1,𝔽₅)	semidirect
183. C₄×C₈	ΓL(1,𝔽₁₆)	semidirect
237. C₃×C₅	#1833	semidirect
274. C₅	#69896	semidirect
286. C₃	#10603852	semidirect

Maximal subgroups

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

subgroup	index
D₅⋊#819	2
#11752380	2
C₃×#64978	2
#11746765	2
#11746734	2
#10603852	3
#69896	5