Subgroups — #14312756

Order 1920.

320 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11752377	C₂	1	1	yes	maximal, characteristic
2	D₅⋊#812	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#3930	C₂	1	1	yes	maximal, characteristic
4	#11746717	C₂	1	2	yes	maximal, complemented
5	#11746716	C₂	1	2	yes	maximal, complemented
6	#10604101	—	3	1	no	maximal, Hall
7	#91989	C₄	1	1	yes	characteristic
8	D₅⋊#208	C₄	1	1	yes	characteristic
9	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
10	#91227	C₄	1	2	yes	complemented
11	#91221	C₂²	1	2	yes	—
12	C₈×ΓL(1,𝔽₁₆)	C₄	1	2	yes	—
13	#91219	C₂²	1	2	yes	—
14	D₅⋊(C₃⋊C₁₆)	C₄	1	2	yes	complemented
15	C₂×C₃×C₅×C₁₆	C₄	1	1	yes	characteristic, complemented
16	C₂×C₃×#672	C₄	1	1	yes	characteristic, complemented
17	C₃×#3930	C₄	1	1	yes	characteristic, complemented
18	C₃×#3930	C₄	1	1	yes	characteristic, complemented
19	C₃×#3930	C₂²	1	1	yes	characteristic
20	C₃×#3930	C₂²	1	1	yes	characteristic
21	#71425	—	5	1	no	maximal, Hall
22	#65718	—	3	1	no	—
23	D₅⋊M₃₂	—	3	1	no	—
24	C₂×#3930	D₃	1	1	yes	characteristic
25	#64846	—	3	2	no	—
26	#64845	—	3	2	no	—
27	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
28	#6763	—	2	1	no	—
29	#6762	—	2	1	no	—
30	C₄⋊ΓL(1,𝔽₁₆)	—	2	1	no	—
31	C₄×ΓL(1,𝔽₁₆)	C₂×C₄	1	2	yes	—
32	#6758	—	2	1	no	—
33	D₅⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
34	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
36	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
37	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
38	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
39	C₃×C₈×D₅	C₂³	1	1	yes	characteristic
40	C₃⋊#674	C₂×C₄	1	4	yes	—
41	C₃×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
42	C₃×C₅×C₁₆	C₂×C₄	1	1	yes	characteristic
43	C₃×#672	C₂×C₄	1	1	yes	characteristic
44	C₃×#672	C₂×C₄	1	1	yes	characteristic
45	C₂×C₃×M₃₂	—	5	1	no	—
46	#5101	—	5	1	no	—
47	C₂×#812	—	5	1	no	—
48	#4512	—	5	2	no	—
49	#4512	—	5	2	no	—
50	#4130	—	3	1	no	—
51	D₅⋊M₁₆	—	3	1	no	—
52	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
53	#3998	—	3	2	no	—
54	#3992	—	3	2	no	—
55	C₈×AGL(1,𝔽₅)	—	3	2	no	—
56	#3990	—	3	2	no	—
57	D₅⋊C₁₆	—	3	2	no	—
58	C₂×C₅×C₁₆	Dic₃	1	1	yes	characteristic, complemented
59	C₂×#672	Dic₃	1	1	yes	characteristic, complemented
60	#3930	Dic₃	1	1	yes	characteristic, complemented
61	#3930	Dic₃	1	1	yes	characteristic, complemented
62	#3930	C₂×D₃	1	1	yes	characteristic
63	#3930	C₂×D₃	1	1	yes	characteristic
64	#2110	—	15	1	no	Sylow
65	C₂²×C₃×D₅	M₁₆	1	1	yes	characteristic
66	C₂×ΓL(1,𝔽₁₆)	—	2	2	no	—
67	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
68	C₂×C₃×Dic₅	M₁₆	1	1	yes	characteristic
69	C₃×C₄×D₅	C₄²	1	1	yes	characteristic
70	C₃×C₄×D₅	M₁₆	1	1	yes	characteristic
71	C₃×C₄×D₅	M₁₆	1	1	yes	characteristic
72	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
73	C₃⋊#170	—	2	2	no	—
74	C₃×C₅×C₈	C₄²	1	1	yes	characteristic
75	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
76	C₃×#168	C₄²	1	1	yes	characteristic
77	C₃×#168	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₃×C₈	—	5	1	no	—
79	#926	—	5	1	no	—
80	C₂×#208	—	5	1	no	—
81	C₃×M₃₂	—	5	1	no	—
82	C₃×M₃₂	—	5	1	no	—
83	C₃×M₃₂	—	5	1	no	—
84	C₃×M₃₂	—	5	1	no	—
85	C₂×C₃×C₁₆	—	5	1	no	—
86	C₂×C₃×C₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
87	#815	—	5	2	no	—
88	C₈×Dic₃	—	5	2	no	—
89	#812	—	5	2	no	—
90	#812	—	5	2	no	—
91	C₂×C₃⋊C₁₆	—	5	2	no	—
92	C₂×C₄×D₅	C₄×D₃	1	1	yes	characteristic
93	C₂²⋊AGL(1,𝔽₅)	—	6	1	no	—
94	#704	—	6	1	no	—
95	C₄⋊AGL(1,𝔽₅)	—	6	1	no	—
96	C₄×AGL(1,𝔽₅)	—	3	2	no	—
97	#700	—	6	1	no	—
98	D₅⋊C₈	—	3	2	no	—
99	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
100	C₂×#168	C₄×D₃	1	1	yes	characteristic
101	C₈×D₅	C₄×D₃	1	1	yes	characteristic
102	C₈×D₅	C₄×D₃	1	1	yes	characteristic
103	C₈×D₅	C₂×Dic₃	1	1	yes	characteristic
104	C₈×D₅	C₂²×D₃	1	1	yes	characteristic
105	#674	—	3	4	no	—
106	C₅×C₁₆	C₂×Dic₃	1	1	yes	characteristic
107	C₅×C₁₆	C₂×Dic₃	1	1	yes	characteristic
108	#672	C₂×Dic₃	1	1	yes	characteristic
109	#672	C₂×Dic₃	1	1	yes	characteristic
110	C₂×M₃₂	—	5	1	no	—
111	C₂×M₃₂	—	15	1	no	—
112	#405	—	15	1	no	—
113	#347	—	15	2	no	—
114	#347	—	15	2	no	—
115	C₂²×C₃×C₅	#97	1	1	yes	characteristic
116	C₂×C₃×D₅	—	4	1	no	—
117	C₂×C₃×D₅	#97	1	1	yes	characteristic
118	C₂×C₃×D₅	C₂×M₁₆	1	1	yes	characteristic
119	ΓL(1,𝔽₁₆)	—	4	2	no	—
120	C₃×C₄×C₅	#97	1	1	yes	characteristic
121	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
122	C₃×Dic₅	#97	1	1	yes	characteristic
123	C₃×Dic₅	C₂×M₁₆	1	1	yes	characteristic
124	C₂²×C₃×C₄	—	5	1	no	—
125	C₂×C₃×C₈	—	5	1	no	—
126	C₂×C₃×C₈	—	5	1	no	—
127	C₂×C₃×C₈	—	5	1	no	—
128	C₂×C₃×C₈	—	5	1	no	—
129	C₂×C₃×C₈	—	5	1	no	—
130	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
131	#217	—	10	1	no	—
132	C₄⋊Dic₃	—	10	1	no	—
133	C₄×Dic₃	—	5	2	no	—
134	#208	—	10	1	no	—
135	#208	—	10	1	no	—
136	C₂×C₃⋊C₈	—	5	2	no	—
137	C₃×C₁₆	—	5	1	no	—
138	C₃×C₁₆	—	5	1	no	—
139	C₃×C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
140	C₃×C₁₆	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
141	C₃⋊C₁₆	—	5	4	no	—
142	C₂²×D₅	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
143	C₂×AGL(1,𝔽₅)	—	6	2	no	—
144	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
145	C₂×Dic₅	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
146	C₄×D₅	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
147	C₄×D₅	ΓL(1,𝔽₂₅)	1	1	yes	characteristic
148	C₄×D₅	C₄×Dic₃	1	1	yes	characteristic
149	C₄×D₅	C₂×C₄×D₃	1	1	yes	characteristic
150	#170	—	6	2	no	—
151	C₅×C₈	C₄×Dic₃	1	1	yes	characteristic
152	C₅×C₈	C₂²×Dic₃	1	1	yes	characteristic
153	#168	C₄×Dic₃	1	1	yes	characteristic
154	#168	C₂×C₄×D₃	1	1	yes	characteristic
155	C₂×M₁₆	—	15	1	no	—
156	C₂²×C₈	—	5	1	no	—
157	#117	—	15	1	no	—
158	M₃₂	—	5	1	no	—
159	M₃₂	—	5	1	no	—
160	M₃₂	—	5	1	no	—
161	M₃₂	—	5	1	no	—
162	M₃₂	—	15	2	no	—
163	M₃₂	—	15	2	no	—
164	C₂×C₁₆	—	5	1	no	—
165	C₂×C₁₆	—	15	2	no	—
166	C₂×C₁₆	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
167	#97	—	15	2	no	—
168	C₄×C₈	—	15	2	no	—
169	C₂×C₃×C₅	—	2	1	no	—
170	C₂×C₃×C₅	C₂×#97	1	1	yes	characteristic
171	C₃×D₅	—	2	1	no	—
172	C₃×D₅	—	2	1	no	—
173	C₂³×C₃	—	5	1	no	—
174	C₂×C₃×C₄	—	5	1	no	—
175	C₂×C₃×C₄	—	5	1	no	—
176	C₂×C₃×C₄	—	5	1	no	—
177	C₂×C₃×C₄	—	5	1	no	—
178	C₂×C₃×C₄	—	5	1	no	—
179	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
180	C₂×Dic₃	—	10	2	no	—
181	C₃×C₈	—	5	1	no	—
182	C₃×C₈	—	5	1	no	—
183	C₃×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
184	C₃×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
185	C₃⋊C₈	—	10	2	no	—
186	C₂²×C₅	#815	1	1	yes	characteristic
187	C₂×D₅	—	4	1	no	—
188	C₂×D₅	#815	1	1	yes	characteristic
189	C₂×D₅	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
190	AGL(1,𝔽₅)	—	12	2	no	—
191	C₄×C₅	#815	1	1	yes	characteristic
192	C₄×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
193	Dic₅	#815	1	1	yes	characteristic
194	Dic₅	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
195	C₂²×C₄	—	5	1	no	—
196	M₁₆	—	30	1	no	—
197	M₁₆	—	30	1	no	—
198	C₂×C₈	—	5	1	no	—
199	C₂×C₈	—	5	1	no	—
200	C₂×C₈	—	5	1	no	—
201	C₂×C₈	—	5	1	no	—
202	C₂×C₈	—	5	1	no	—
203	C₂×C₈	—	15	2	no	—
204	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
205	C₄⋊C₄	—	30	1	no	—
206	C₂²⋊C₄	—	30	1	no	—
207	C₄²	—	15	2	no	—
208	C₁₆	—	5	1	no	—
209	C₁₆	—	5	1	no	—
210	C₁₆	—	15	4	no	—
211	C₁₆	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
212	C₁₆	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
213	C₃×C₅	#2110	1	1	yes	characteristic, complemented, Hall
214	C₂²×C₃	—	5	1	no	—
215	C₂²×C₃	—	5	1	no	—
216	C₂²×C₃	—	20	1	no	—
217	C₂²×C₃	#3992	1	1	yes	characteristic
218	C₃×C₄	—	5	1	no	—
219	C₃×C₄	—	5	1	no	—
220	C₃×C₄	#3992	1	1	yes	characteristic
221	C₃×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
222	Dic₃	—	20	2	no	—
223	C₂×C₅	—	2	1	no	—
224	C₂×C₅	C₂×#815	1	1	yes	characteristic
225	D₅	—	2	1	no	—
226	D₅	—	2	1	no	—
227	C₂³	—	5	1	no	—
228	C₂×C₄	—	5	1	no	—
229	C₂×C₄	—	5	1	no	—
230	C₂×C₄	—	5	1	no	—
231	C₂×C₄	—	5	1	no	—
232	C₂×C₄	—	5	1	no	—
233	C₂×C₄	—	30	2	no	—
234	C₂×C₄	C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
235	C₈	—	5	1	no	—
236	C₈	—	5	1	no	—
237	C₈	—	30	2	no	—
238	C₈	C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
239	C₈	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
240	C₂×C₃	—	2	1	no	—
241	C₂×C₃	—	10	1	no	—
242	C₂×C₃	—	10	1	no	—
243	C₂×C₃	C₂×#3992	1	1	yes	characteristic
244	C₅	#71425	1	1	yes	characteristic, complemented, Sylow
245	C₂²	—	5	1	no	—
246	C₂²	—	5	1	no	—
247	C₂²	—	20	1	no	—
248	C₂²	#91221	1	1	yes	characteristic
249	C₄	—	5	1	no	—
250	C₄	—	5	1	no	—
251	C₄	—	60	2	no	—
252	C₄	#91221	1	1	yes	characteristic
253	C₄	C₂×C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
254	C₃	#10604101	1	1	yes	characteristic, complemented, Sylow
255	C₂	—	2	1	no	—
256	C₂	—	10	1	no	—
257	C₂	—	10	1	no	—
258	C₂	C₂×#91221	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₄×ΓL(1,𝔽₁₆)
commutator	C₃×C₄×C₅	C₂×C₄²
Frattini	C₈	C₂²×ΓL(1,𝔽₁₆)
Fitting	C₂×C₃×C₅×C₁₆	C₄
socle	C₂×C₃×C₅	C₂×#97
prime radical	C₂²×C₃×D₅	M₁₆
cosocle	C₃×C₈×D₅	C₂³
hypercenter	C₂×C₁₆	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#2110
LCS	G ▷ C₃×C₄×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
4. #11746717	C₂	semidirect
5. #11746716	C₂	semidirect
10. #91227	C₄	semidirect
14. D₅⋊(C₃⋊C₁₆)	C₄	semidirect
15. C₂×C₃×C₅×C₁₆	C₄	semidirect
16. C₂×C₃×#672	C₄	semidirect
17. C₃×#3930	C₄	semidirect
18. C₃×#3930	C₄	semidirect
58. C₂×C₅×C₁₆	Dic₃	semidirect
59. C₂×#672	Dic₃	semidirect
60. #3930	Dic₃	semidirect
61. #3930	Dic₃	semidirect
86. C₂×C₃×C₁₆	AGL(1,𝔽₅)	semidirect
166. C₂×C₁₆	ΓL(1,𝔽₁₆)	semidirect
213. C₃×C₅	#2110	semidirect
244. C₅	#71425	semidirect
254. C₃	#10604101	semidirect

Maximal subgroups

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

subgroup	index
#11752377	2
D₅⋊#812	2
C₂×C₃×#3930	2
#11746717	2
#11746716	2
#10604101	3
#71425	5