Subgroups — #14312652

Order 1920.

368 conjugacy classes of subgroups.

The subgroup lattice has 252 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	C₂×D₅⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	C₃×D₅×M₃₂	C₂	1	1	yes	maximal, characteristic
4	#11746715	C₂	1	2	yes	maximal, complemented
5	C₁₆×ΓL(1,𝔽₁₆)	C₂	1	2	yes	maximal, complemented
6	#10604113	—	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	C₂×C₃⋊#674	C₄	1	2	yes	complemented
11	#91221	C₄	1	2	yes	—
12	C₈×ΓL(1,𝔽₁₆)	C₂²	1	2	yes	—
13	D₅⋊(C₃⋊C₁₆)	C₄	1	2	yes	complemented
14	D₅⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
15	D₅⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
16	C₃×C₅×M₃₂	C₄	1	1	yes	characteristic, complemented
17	C₃×#3944	C₄	1	1	yes	characteristic, complemented
18	C₃×#3930	C₄	1	2	yes	complemented
19	C₃×D₅×C₁₆	C₂²	1	2	yes	—
20	#71419	—	5	1	no	maximal, Hall
21	#65718	—	3	1	no	—
22	C₂×D₅⋊C₁₆	—	3	1	no	—
23	D₅×M₃₂	D₃	1	1	yes	characteristic
24	#64844	—	3	2	no	—
25	C₁₆×AGL(1,𝔽₅)	—	3	2	no	—
26	C₂×C₃×C₄×D₅	C₂×C₄	1	1	yes	characteristic
27	#6763	C₈	1	2	yes	—
28	#6762	C₈	1	2	yes	—
29	C₄⋊ΓL(1,𝔽₁₆)	C₈	1	2	yes	—
30	C₄×ΓL(1,𝔽₁₆)	C₂×C₄	1	2	yes	—
31	#6758	C₈	1	2	yes	—
32	D₅⋊(C₃⋊C₈)	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
35	C₃×C₈×D₅	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	C₂³	1	1	yes	characteristic
39	C₃⋊#674	C₂×C₄	1	2	yes	—
40	C₃⋊#674	C₂×C₄	1	2	yes	—
41	C₃×C₅×C₁₆	C₂×C₄	1	2	yes	—
42	C₃×#672	C₂×C₄	1	2	yes	—
43	C₂×C₃×M₃₂	—	5	1	no	—
44	#5101	—	5	1	no	—
45	C₂²×C₃⋊C₁₆	—	5	1	no	—
46	#4502	—	5	2	no	—
47	Dic₃×C₁₆	—	5	2	no	—
48	#4130	—	3	1	no	—
49	D₅⋊M₁₆	—	3	1	no	—
50	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
51	C₂×#674	—	3	2	no	—
52	#3992	—	3	2	no	—
53	C₈×AGL(1,𝔽₅)	—	3	2	no	—
54	D₅⋊C₁₆	—	3	2	no	—
55	D₅⋊C₁₆	—	3	1	no	—
56	D₅⋊C₁₆	—	3	1	no	—
57	C₅×M₃₂	Dic₃	1	1	yes	characteristic, complemented
58	#3944	Dic₃	1	1	yes	characteristic, complemented
59	#3930	Dic₃	1	2	yes	complemented
60	D₅×C₁₆	C₂×D₃	1	2	yes	—
61	#2108	—	15	1	no	Sylow
62	C₂²×C₃×D₅	C₂×C₈	1	1	yes	characteristic
63	C₂×ΓL(1,𝔽₁₆)	C₂×C₈	1	4	yes	—
64	C₂×C₃×C₄×C₅	C₄²	1	1	yes	characteristic
65	C₂×C₃×Dic₅	C₂×C₈	1	1	yes	characteristic
66	C₃×C₄×D₅	C₄²	1	1	yes	characteristic
67	C₃×C₄×D₅	C₂×C₈	1	1	yes	characteristic
68	C₃×C₄×D₅	C₂×C₈	1	1	yes	characteristic
69	C₃×C₄×D₅	C₂²×C₄	1	1	yes	characteristic
70	C₃⋊#170	C₂×C₈	1	4	yes	—
71	C₃×C₅×C₈	C₄²	1	1	yes	characteristic
72	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
73	C₃×#168	C₄²	1	1	yes	characteristic
74	C₃×#168	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₃×C₈	—	5	1	no	—
76	#926	—	5	1	no	—
77	C₂×#208	—	5	1	no	—
78	C₃×M₃₂	—	5	1	no	—
79	C₃×M₃₂	—	5	2	no	—
80	C₃×M₃₂	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
81	C₂×C₃×C₁₆	—	5	2	no	—
82	#815	—	5	2	no	—
83	C₈×Dic₃	—	5	2	no	—
84	C₂×C₃⋊C₁₆	—	5	2	no	—
85	C₂×C₃⋊C₁₆	—	5	1	no	—
86	C₂×C₃⋊C₁₆	—	5	1	no	—
87	C₂×C₃⋊C₁₆	—	5	2	no	—
88	C₂×C₄×D₅	C₄×D₃	1	1	yes	characteristic
89	C₂²⋊AGL(1,𝔽₅)	—	3	2	no	—
90	#704	—	3	2	no	—
91	C₄⋊AGL(1,𝔽₅)	—	3	2	no	—
92	C₄×AGL(1,𝔽₅)	—	3	2	no	—
93	#700	—	3	2	no	—
94	D₅⋊C₈	—	3	2	no	—
95	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
96	C₂×#168	C₄×D₃	1	1	yes	characteristic
97	C₈×D₅	C₄×D₃	1	1	yes	characteristic
98	C₈×D₅	C₄×D₃	1	1	yes	characteristic
99	C₈×D₅	C₂×Dic₃	1	1	yes	characteristic
100	C₈×D₅	C₂²×D₃	1	1	yes	characteristic
101	#674	—	3	2	no	—
102	#674	—	3	2	no	—
103	C₅×C₁₆	C₂×Dic₃	1	2	yes	—
104	#672	C₂×Dic₃	1	2	yes	—
105	C₂×M₃₂	—	5	1	no	—
106	C₂²×C₁₆	—	15	1	no	—
107	#405	—	15	1	no	—
108	#346	—	15	2	no	—
109	C₄×C₁₆	—	15	2	no	—
110	C₂²×C₃×C₅	C₄×C₈	1	1	yes	characteristic
111	C₂×C₃×D₅	—	2	2	no	—
112	C₂×C₃×D₅	C₄×C₈	1	1	yes	characteristic
113	C₂×C₃×D₅	C₂²×C₈	1	1	yes	characteristic
114	ΓL(1,𝔽₁₆)	—	2	4	no	—
115	C₃×C₄×C₅	C₄×C₈	1	1	yes	characteristic
116	C₃×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
117	C₃×Dic₅	C₄×C₈	1	1	yes	characteristic
118	C₃×Dic₅	C₂²×C₈	1	1	yes	characteristic
119	C₂²×C₃×C₄	—	5	1	no	—
120	C₂×C₃×C₈	—	5	1	no	—
121	C₂×C₃×C₈	—	5	1	no	—
122	C₂×C₃×C₈	—	5	1	no	—
123	C₂×C₃×C₈	—	5	1	no	—
124	C₂×C₃×C₈	—	5	1	no	—
125	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
126	#217	—	5	2	no	—
127	C₄⋊Dic₃	—	5	2	no	—
128	C₄×Dic₃	—	5	2	no	—
129	#208	—	5	2	no	—
130	#208	—	5	2	no	—
131	C₂×C₃⋊C₈	—	5	2	no	—
132	C₃×C₁₆	—	5	2	no	—
133	C₃×C₁₆	C₂×AGL(1,𝔽₅)	1	2	yes	—
134	C₃⋊C₁₆	—	5	2	no	—
135	C₃⋊C₁₆	—	5	2	no	—
136	C₂²×D₅	D₃×C₈	1	1	yes	characteristic
137	C₂×AGL(1,𝔽₅)	—	3	4	no	—
138	C₂×C₄×C₅	C₄×Dic₃	1	1	yes	characteristic
139	C₂×Dic₅	D₃×C₈	1	1	yes	characteristic
140	C₄×D₅	D₃×C₈	1	1	yes	characteristic
141	C₄×D₅	D₃×C₈	1	1	yes	characteristic
142	C₄×D₅	C₄×Dic₃	1	1	yes	characteristic
143	C₄×D₅	C₂×C₄×D₃	1	1	yes	characteristic
144	#170	—	3	4	no	—
145	C₅×C₈	C₄×Dic₃	1	1	yes	characteristic
146	C₅×C₈	C₂²×Dic₃	1	1	yes	characteristic
147	#168	C₄×Dic₃	1	1	yes	characteristic
148	#168	C₂×C₄×D₃	1	1	yes	characteristic
149	C₂×M₁₆	—	15	1	no	—
150	C₂²×C₈	—	5	1	no	—
151	#117	—	15	1	no	—
152	M₃₂	—	5	1	no	—
153	M₃₂	—	5	2	no	—
154	M₃₂	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
155	C₂×C₁₆	—	5	2	no	—
156	C₂×C₁₆	—	15	1	no	—
157	C₂×C₁₆	—	15	1	no	—
158	C₂×C₁₆	—	15	2	no	—
159	C₂×C₁₆	—	15	2	no	—
160	#97	—	15	2	no	—
161	C₄×C₈	—	15	2	no	—
162	C₂×C₃×C₅	—	2	1	no	—
163	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
164	C₃×D₅	—	2	1	no	—
165	C₃×D₅	#504	1	2	yes	—
166	C₂³×C₃	—	5	1	no	—
167	C₂×C₃×C₄	—	5	1	no	—
168	C₂×C₃×C₄	—	5	1	no	—
169	C₂×C₃×C₄	—	5	1	no	—
170	C₂×C₃×C₄	—	5	1	no	—
171	C₂×C₃×C₄	—	5	1	no	—
172	C₂×C₃×C₄	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
173	C₂×Dic₃	—	5	4	no	—
174	C₃×C₈	—	5	1	no	—
175	C₃×C₈	—	5	1	no	—
176	C₃×C₈	C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
177	C₃×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
178	C₃⋊C₈	—	5	4	no	—
179	C₂²×C₅	C₈×Dic₃	1	1	yes	characteristic
180	C₂×D₅	—	2	2	no	—
181	C₂×D₅	C₈×Dic₃	1	1	yes	characteristic
182	C₂×D₅	C₂×D₃×C₈	1	1	yes	characteristic
183	AGL(1,𝔽₅)	—	6	4	no	—
184	C₄×C₅	C₈×Dic₃	1	1	yes	characteristic
185	C₄×C₅	C₂×C₄×Dic₃	1	1	yes	characteristic
186	Dic₅	C₈×Dic₃	1	1	yes	characteristic
187	Dic₅	C₂×D₃×C₈	1	1	yes	characteristic
188	C₂²×C₄	—	5	1	no	—
189	M₁₆	—	15	2	no	—
190	M₁₆	—	15	2	no	—
191	C₂×C₈	—	5	1	no	—
192	C₂×C₈	—	5	1	no	—
193	C₂×C₈	—	5	1	no	—
194	C₂×C₈	—	5	1	no	—
195	C₂×C₈	—	5	1	no	—
196	C₂×C₈	—	15	2	no	—
197	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
198	C₄⋊C₄	—	15	2	no	—
199	C₂²⋊C₄	—	15	2	no	—
200	C₄²	—	15	2	no	—
201	C₁₆	—	5	2	no	—
202	C₁₆	—	15	2	no	—
203	C₁₆	—	15	2	no	—
204	C₁₆	C₂×ΓL(1,𝔽₁₆)	1	2	yes	—
205	C₃×C₅	#2108	1	1	yes	characteristic, complemented, Hall
206	C₂²×C₃	—	5	1	no	—
207	C₂²×C₃	—	5	1	no	—
208	C₂²×C₃	—	10	2	no	—
209	C₂²×C₃	C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
210	C₃×C₄	—	5	1	no	—
211	C₃×C₄	—	5	1	no	—
212	C₃×C₄	C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
213	C₃×C₄	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
214	Dic₃	—	10	4	no	—
215	C₂×C₅	—	2	1	no	—
216	C₂×C₅	C₂×C₈×Dic₃	1	1	yes	characteristic
217	D₅	—	2	1	no	—
218	D₅	#4907	1	2	yes	—
219	C₂³	—	5	1	no	—
220	C₂×C₄	—	5	1	no	—
221	C₂×C₄	—	5	1	no	—
222	C₂×C₄	—	5	1	no	—
223	C₂×C₄	—	5	1	no	—
224	C₂×C₄	—	5	1	no	—
225	C₂×C₄	—	15	4	no	—
226	C₂×C₄	C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
227	C₈	—	5	1	no	—
228	C₈	—	5	1	no	—
229	C₈	—	15	4	no	—
230	C₈	C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
231	C₈	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
232	C₂×C₃	—	2	1	no	—
233	C₂×C₃	—	5	2	no	—
234	C₂×C₃	—	10	1	no	—
235	C₂×C₃	C₂×C₈×AGL(1,𝔽₅)	1	1	yes	characteristic
236	C₅	#71419	1	1	yes	characteristic, complemented, Sylow
237	C₂²	—	5	1	no	—
238	C₂²	—	5	1	no	—
239	C₂²	—	10	2	no	—
240	C₂²	C₈×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
241	C₄	—	5	1	no	—
242	C₄	—	5	1	no	—
243	C₄	—	30	4	no	—
244	C₄	C₈×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
245	C₄	C₂×C₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
246	C₃	#10604113	1	1	yes	characteristic, complemented, Sylow
247	C₂	—	2	1	no	—
248	C₂	—	5	2	no	—
249	C₂	—	10	1	no	—
250	C₂	C₂×C₈×ΓL(1,𝔽₁₆)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂²×ΓL(1,𝔽₁₆)
commutator, socle	C₂×C₃×C₅	C₂×C₄×C₈
Fitting	C₃×C₅×M₃₂	C₄
prime radical	C₂²×C₃×D₅	C₂×C₈
cosocle	C₃×C₈×D₅	C₂³
hypercenter	M₃₂	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#2108
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
4. #11746715	C₂	semidirect
5. C₁₆×ΓL(1,𝔽₁₆)	C₂	semidirect
10. C₂×C₃⋊#674	C₄	semidirect
13. D₅⋊(C₃⋊C₁₆)	C₄	semidirect
16. C₃×C₅×M₃₂	C₄	semidirect
17. C₃×#3944	C₄	semidirect
18. C₃×#3930	C₄	semidirect
57. C₅×M₃₂	Dic₃	semidirect
58. #3944	Dic₃	semidirect
59. #3930	Dic₃	semidirect
80. C₃×M₃₂	AGL(1,𝔽₅)	semidirect
154. M₃₂	ΓL(1,𝔽₁₆)	semidirect
205. C₃×C₅	#2108	semidirect
236. C₅	#71419	semidirect
246. C₃	#10604113	semidirect

Maximal subgroups

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

subgroup	index
#11752377	2
C₂×D₅⋊(C₃⋊C₁₆)	2
C₃×D₅×M₃₂	2
#11746715	2
C₁₆×ΓL(1,𝔽₁₆)	2
#10604113	3
#71419	5