Subgroups — #14299144

Order 1920.

512 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91223	C₂	1	2	yes	maximal
2	C₂×#91222	C₂	1	2	yes	maximal
3	#11752348	C₂	1	1	yes	maximal, characteristic
4	#11752347	C₂	1	1	yes	maximal, characteristic
5	C₃×D₅×#97	C₂	1	1	yes	maximal, characteristic
6	#10603883	—	3	1	no	maximal, Hall
7	C₂×C₄⋊ΓL(1,𝔽₁₆)	—	2	2	no	—
8	C₂×C₄⋊ΓL(1,𝔽₁₆)	C₂²	1	2	yes	—
9	C₂×C₄⋊ΓL(1,𝔽₁₆)	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊ΓL(1,𝔽₁₆)	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₈×D₅	C₂²	1	2	yes	—
12	C₃×C₄²×D₅	C₂²	1	1	yes	characteristic
13	#91235	—	2	2	no	—
14	#91223	—	2	4	no	—
15	#91222	—	2	4	no	—
16	C₃×C₅×#97	C₄	1	1	yes	characteristic, complemented
17	C₃×#3947	C₄	1	2	yes	complemented
18	C₃×#3935	C₄	1	1	yes	characteristic, complemented
19	#69816	—	5	1	no	maximal, Hall
20	C₂×#3994	—	3	2	no	—
21	C₂×#3993	—	3	2	no	—
22	#65689	—	3	1	no	—
23	#65688	—	3	1	no	—
24	D₅×#97	D₃	1	1	yes	characteristic
25	C₂²×ΓL(1,𝔽₁₆)	—	2	2	no	—
26	C₂²×ΓL(1,𝔽₁₆)	—	2	1	no	—
27	C₂²×ΓL(1,𝔽₁₆)	—	2	1	no	—
28	C₂×C₃×C₄×D₅	D₄	1	2	yes	—
29	C₂×C₃×C₄×D₅	C₂³	1	1	yes	characteristic
30	C₄⋊ΓL(1,𝔽₁₆)	—	2	2	no	—
31	C₄⋊ΓL(1,𝔽₁₆)	—	2	2	no	—
32	C₄⋊ΓL(1,𝔽₁₆)	—	2	2	no	—
33	C₄⋊ΓL(1,𝔽₁₆)	—	2	2	no	—
34	C₄⋊ΓL(1,𝔽₁₆)	—	4	4	no	—
35	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
36	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
38	C₃×C₄×Dic₅	Q₈	1	2	yes	—
39	C₂×C₃×#168	C₂×C₄	1	2	yes	—
40	C₃×C₈×D₅	D₄	1	4	yes	—
41	C₃×C₈×D₅	Q₈	1	4	yes	—
42	C₂×C₃×#97	—	5	1	no	—
43	C₂×#818	—	5	2	no	—
44	C₂×#817	—	5	2	no	—
45	#4936	—	5	1	no	—
46	#4935	—	5	1	no	—
47	C₂×C₄⋊AGL(1,𝔽₅)	—	3	2	no	—
48	C₂×C₄⋊AGL(1,𝔽₅)	—	3	1	no	—
49	C₂×C₄⋊AGL(1,𝔽₅)	—	3	1	no	—
50	C₂×C₄⋊AGL(1,𝔽₅)	—	6	2	no	—
51	C₂×C₈×D₅	C₂×D₃	1	2	yes	—
52	C₄²×D₅	C₂×D₃	1	1	yes	characteristic
53	(C₂×AGL(1,𝔽₅))⋊C₄	—	6	2	no	—
54	#3994	—	6	4	no	—
55	#3993	—	6	4	no	—
56	C₅×#97	Dic₃	1	1	yes	characteristic, complemented
57	#3947	Dic₃	1	2	yes	complemented
58	#3935	Dic₃	1	1	yes	characteristic, complemented
59	#1847	—	15	1	no	Sylow
60	C₂²×C₃×D₅	C₂×D₄	1	1	yes	characteristic
61	C₂×ΓL(1,𝔽₁₆)	—	4	2	no	—
62	C₂×ΓL(1,𝔽₁₆)	—	4	1	no	—
63	C₂×ΓL(1,𝔽₁₆)	—	4	1	no	—
64	C₂×ΓL(1,𝔽₁₆)	—	4	4	no	—
65	C₂×ΓL(1,𝔽₁₆)	—	4	2	no	—
66	C₂×ΓL(1,𝔽₁₆)	—	4	2	no	—
67	C₂×C₃×C₄×C₅	C₄⋊C₄	1	2	yes	—
68	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₃×Dic₅	C₄⋊C₄	1	2	yes	—
70	C₂×C₃×Dic₅	C₂×Q₈	1	1	yes	characteristic
71	C₃×C₄×D₅	—	2	2	no	—
72	C₃×C₄×D₅	—	2	2	no	—
73	C₃×C₄×D₅	C₂×D₄	1	2	yes	—
74	C₃×C₄×D₅	C₂×Q₈	1	2	yes	—
75	C₃×C₅×C₈	C₄⋊C₄	1	4	yes	complemented
76	C₃×#168	C₄⋊C₄	1	4	yes	complemented
77	C₂²×C₃×C₈	—	5	2	no	—
78	C₂×C₃×C₄²	—	5	1	no	—
79	C₂×C₄⋊Dic₃	—	5	2	no	—
80	C₂×C₄⋊Dic₃	—	5	1	no	—
81	C₂×C₄⋊Dic₃	—	5	1	no	—
82	C₂×C₄⋊Dic₃	—	10	2	no	—
83	C₃×#97	—	5	2	no	—
84	C₃×#97	—	5	1	no	—
85	C₃×#97	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
86	C₃⋊#95	—	10	2	no	—
87	#818	—	10	4	no	—
88	#817	—	10	4	no	—
89	C₂²×AGL(1,𝔽₅)	—	6	2	no	—
90	C₂²×AGL(1,𝔽₅)	—	6	1	no	—
91	C₂²×AGL(1,𝔽₅)	—	6	1	no	—
92	C₂×C₄×D₅	D₁₂	1	2	yes	—
93	C₂×C₄×D₅	C₂²×D₃	1	1	yes	characteristic
94	C₄⋊AGL(1,𝔽₅)	—	6	2	no	—
95	C₄⋊AGL(1,𝔽₅)	—	6	2	no	—
96	C₄⋊AGL(1,𝔽₅)	—	6	2	no	—
97	C₄⋊AGL(1,𝔽₅)	—	6	2	no	—
98	C₄⋊AGL(1,𝔽₅)	—	12	4	no	—
99	C₂×C₅×C₈	C₂×Dic₃	1	2	yes	—
100	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
101	C₄×Dic₅	Dic₆	1	2	yes	—
102	C₄×Dic₅	C₂×Dic₃	1	1	yes	characteristic
103	C₂×#168	C₂×Dic₃	1	2	yes	—
104	C₈×D₅	Dic₆	1	4	yes	—
105	C₈×D₅	D₁₂	1	4	yes	—
106	C₂×#107	—	15	2	no	—
107	C₂×#106	—	15	2	no	—
108	C₂×#97	—	5	1	no	—
109	#384	—	15	1	no	—
110	#382	—	15	1	no	—
111	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₃×D₅	#127	1	1	yes	characteristic
113	C₂×C₃×D₅	C₄⋊Q₈	1	1	yes	characteristic
114	C₂×C₃×D₅	AGL(1,ℤ/8)	1	2	yes	—
115	C₂×C₃×D₅	#137	1	2	yes	—
116	ΓL(1,𝔽₁₆)	—	8	2	no	—
117	ΓL(1,𝔽₁₆)	—	8	1	no	—
118	ΓL(1,𝔽₁₆)	—	8	1	no	—
119	C₃×C₄×C₅	—	2	2	no	—
120	C₃×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
121	C₃×Dic₅	—	2	2	no	—
122	C₃×Dic₅	C₄⋊Q₈	1	2	yes	—
123	C₂²×C₃×C₄	—	5	1	no	—
124	C₂²×C₃×C₄	—	5	2	no	—
125	C₂²×Dic₃	—	10	1	no	—
126	C₂²×Dic₃	—	10	1	no	—
127	C₂²×Dic₃	—	10	2	no	—
128	C₂×C₃×C₈	—	5	2	no	—
129	C₂×C₃×C₈	—	5	4	no	—
130	C₂×C₃×C₈	—	5	4	no	—
131	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	2	yes	—
132	C₃×C₄²	—	5	2	no	—
133	C₃×C₄²	—	5	1	no	—
134	C₃×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
135	C₄⋊Dic₃	—	10	2	no	—
136	C₄⋊Dic₃	—	10	2	no	—
137	C₄⋊Dic₃	—	10	2	no	—
138	C₄⋊Dic₃	—	10	2	no	—
139	C₄⋊Dic₃	—	20	4	no	—
140	C₂²×D₅	C₂×D₁₂	1	1	yes	characteristic
141	C₂×AGL(1,𝔽₅)	—	12	2	no	—
142	C₂×AGL(1,𝔽₅)	—	12	1	no	—
143	C₂×AGL(1,𝔽₅)	—	12	1	no	—
144	C₂×AGL(1,𝔽₅)	—	12	4	no	—
145	C₂×AGL(1,𝔽₅)	—	12	2	no	—
146	C₂×AGL(1,𝔽₅)	—	12	2	no	—
147	C₂×C₄×C₅	C₄⋊Dic₃	1	2	yes	—
148	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
149	C₂×Dic₅	C₄⋊Dic₃	1	2	yes	—
150	C₂×Dic₅	C₂×Dic₆	1	1	yes	characteristic
151	C₄×D₅	—	2	2	no	—
152	C₄×D₅	—	2	2	no	—
153	C₄×D₅	C₂×Dic₆	1	2	yes	—
154	C₄×D₅	C₂×D₁₂	1	2	yes	—
155	C₅×C₈	C₄⋊Dic₃	1	4	yes	complemented
156	#168	C₄⋊Dic₃	1	4	yes	complemented
157	C₂²×C₈	—	5	2	no	—
158	C₂×C₄⋊C₄	—	15	1	no	—
159	C₂×C₄⋊C₄	—	15	1	no	—
160	C₂×C₄⋊C₄	—	15	2	no	—
161	C₂×C₄⋊C₄	—	30	2	no	—
162	C₂×C₄²	—	5	1	no	—
163	#107	—	30	4	no	—
164	#106	—	30	4	no	—
165	#97	—	5	2	no	—
166	#97	—	5	1	no	—
167	#97	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
168	#95	—	30	2	no	—
169	C₂×C₃×C₅	#384	1	1	yes	characteristic
170	C₂×C₃×C₅	#428	1	2	yes	—
171	C₃×D₅	#496	1	1	yes	characteristic
172	C₃×D₅	#497	1	1	yes	characteristic
173	C₃×D₅	#501	1	2	yes	—
174	C₂³×C₃	—	5	1	no	—
175	C₂×C₃×C₄	—	5	1	no	—
176	C₂×C₃×C₄	—	5	2	no	—
177	C₂×C₃×C₄	—	5	2	no	—
178	C₂×C₃×C₄	—	5	2	no	—
179	C₂×C₃×C₄	—	10	2	no	—
180	C₂×C₃×C₄	—	10	2	no	—
181	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	2	yes	—
182	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
183	C₂×Dic₃	—	20	1	no	—
184	C₂×Dic₃	—	20	2	no	—
185	C₂×Dic₃	—	20	1	no	—
186	C₂×Dic₃	—	20	4	no	—
187	C₂×Dic₃	—	20	2	no	—
188	C₂×Dic₃	—	20	2	no	—
189	C₃×C₈	—	5	4	no	—
190	C₃×C₈	C₄⋊AGL(1,𝔽₅)	1	4	yes	complemented
191	C₂²×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
192	C₂×D₅	#869	1	1	yes	characteristic
193	C₂×D₅	#874	1	1	yes	characteristic
194	C₂×D₅	#908	1	2	yes	—
195	C₂×D₅	#909	1	2	yes	—
196	AGL(1,𝔽₅)	—	24	2	no	—
197	AGL(1,𝔽₅)	—	24	1	no	—
198	AGL(1,𝔽₅)	—	24	1	no	—
199	C₄×C₅	—	2	2	no	—
200	C₄×C₅	C₂×C₄⋊Dic₃	1	2	yes	—
201	Dic₅	—	2	2	no	—
202	Dic₅	#869	1	2	yes	—
203	C₂²×C₄	—	5	1	no	—
204	C₂²×C₄	—	5	2	no	—
205	C₂²×C₄	—	30	1	no	—
206	C₂²×C₄	—	30	1	no	—
207	C₂²×C₄	—	30	2	no	—
208	C₂×C₈	—	5	2	no	—
209	C₂×C₈	—	5	4	no	—
210	C₂×C₈	—	5	4	no	—
211	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	2	yes	—
212	C₄⋊C₄	—	30	2	no	—
213	C₄⋊C₄	—	30	2	no	—
214	C₄⋊C₄	—	30	2	no	—
215	C₄⋊C₄	—	30	2	no	—
216	C₄⋊C₄	—	60	4	no	—
217	C₄²	—	5	2	no	—
218	C₄²	—	5	1	no	—
219	C₄²	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
220	C₃×C₅	#1847	1	1	yes	characteristic, complemented, Hall
221	C₂²×C₃	—	5	1	no	—
222	C₂²×C₃	—	5	1	no	—
223	C₂²×C₃	—	5	2	no	—
224	C₂²×C₃	—	5	2	no	—
225	C₂²×C₃	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
226	C₃×C₄	—	2	2	no	—
227	C₃×C₄	—	5	2	no	—
228	C₃×C₄	—	10	2	no	—
229	C₃×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	2	yes	—
230	Dic₃	—	40	2	no	—
231	Dic₃	—	40	1	no	—
232	Dic₃	—	40	1	no	—
233	C₂×C₅	#4935	1	1	yes	characteristic
234	C₂×C₅	#5121	1	2	yes	—
235	D₅	#4702	1	2	yes	—
236	D₅	#4712	1	1	yes	characteristic
237	D₅	#4715	1	1	yes	characteristic
238	C₂³	—	5	1	no	—
239	C₂×C₄	—	5	1	no	—
240	C₂×C₄	—	5	2	no	—
241	C₂×C₄	—	5	2	no	—
242	C₂×C₄	—	5	2	no	—
243	C₂×C₄	—	10	2	no	—
244	C₂×C₄	—	10	2	no	—
245	C₂×C₄	—	60	1	no	—
246	C₂×C₄	—	60	2	no	—
247	C₂×C₄	—	60	1	no	—
248	C₂×C₄	—	60	4	no	—
249	C₂×C₄	—	60	2	no	—
250	C₂×C₄	—	60	2	no	—
251	C₂×C₄	C₄⋊ΓL(1,𝔽₁₆)	1	2	yes	—
252	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
253	C₈	—	5	4	no	—
254	C₈	C₄⋊ΓL(1,𝔽₁₆)	1	4	yes	complemented
255	C₂×C₃	—	5	1	no	—
256	C₂×C₃	—	5	1	no	—
257	C₂×C₃	—	5	2	no	—
258	C₂×C₃	#65688	1	1	yes	characteristic
259	C₂×C₃	#65727	1	2	yes	—
260	C₅	#69816	1	1	yes	characteristic, complemented, Sylow
261	C₂²	—	5	1	no	—
262	C₂²	—	5	1	no	—
263	C₂²	—	5	2	no	—
264	C₂²	—	5	2	no	—
265	C₂²	C₂×C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
266	C₄	—	2	2	no	—
267	C₄	—	5	2	no	—
268	C₄	—	10	2	no	—
269	C₄	—	120	2	no	—
270	C₄	—	120	1	no	—
271	C₄	—	120	1	no	—
272	C₄	C₂×C₄⋊ΓL(1,𝔽₁₆)	1	2	yes	—
273	C₃	#10603883	1	1	yes	characteristic, complemented, Sylow
274	C₂	—	5	1	no	—
275	C₂	—	5	1	no	—
276	C₂	—	5	2	no	—
277	C₂	#11752347	1	1	yes	characteristic
278	C₂	#11752386	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₄⋊ΓL(1,𝔽₁₆)
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)
Fitting	C₃×C₅×#97	C₄
socle	C₂²×C₃×C₅	C₂×C₄⋊C₄
prime radical	C₂²×C₃×D₅	C₂×D₄
cosocle	C₂×C₃×C₄×D₅	C₂³
hypercenter	#97	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#1847
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
16. C₃×C₅×#97	C₄	semidirect
17. C₃×#3947	C₄	semidirect
18. C₃×#3935	C₄	semidirect
56. C₅×#97	Dic₃	semidirect
57. #3947	Dic₃	semidirect
58. #3935	Dic₃	semidirect
75. C₃×C₅×C₈	C₄⋊C₄	semidirect
76. C₃×#168	C₄⋊C₄	semidirect
85. C₃×#97	AGL(1,𝔽₅)	semidirect
155. C₅×C₈	C₄⋊Dic₃	semidirect
156. #168	C₄⋊Dic₃	semidirect
167. #97	ΓL(1,𝔽₁₆)	semidirect
190. C₃×C₈	C₄⋊AGL(1,𝔽₅)	semidirect
220. C₃×C₅	#1847	semidirect
254. C₈	C₄⋊ΓL(1,𝔽₁₆)	semidirect
260. C₅	#69816	semidirect
273. C₃	#10603883	semidirect

Maximal subgroups

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

subgroup	index
C₂×#91223	2
C₂×#91222	2
#11752348	2
#11752347	2
C₃×D₅×#97	2
#10603883	3
#69816	5