Subgroups — #14314632

Order 1920.

312 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11752380	C₂	1	1	yes	maximal, characteristic
2	C₂×#91223	C₂	1	1	yes	maximal, characteristic
3	C₃×D₅×M₃₂	C₂	1	1	yes	maximal, characteristic
4	#11746721	C₂	1	2	yes	maximal, complemented
5	#11746720	C₂	1	2	yes	maximal, complemented
6	#10604118	—	3	1	no	maximal, Hall
7	#91989	—	2	1	no	—
8	C₂×C₄⋊ΓL(1,𝔽₁₆)	—	2	1	no	—
9	C₂×C₃×C₈×D₅	C₂²	1	1	yes	characteristic
10	#91223	—	2	1	no	—
11	#91223	C₂²	1	1	yes	characteristic
12	#91223	C₂²	1	2	yes	—
13	#91223	C₂²	1	1	yes	characteristic
14	#91222	—	2	1	no	—
15	C₃×C₅×M₃₂	C₄	1	1	yes	characteristic, complemented
16	C₃×#3944	C₄	1	1	yes	characteristic, complemented
17	C₃×#3930	C₄	1	2	yes	complemented
18	C₃×D₅×C₁₆	C₂²	1	2	yes	—
19	#71428	—	5	1	no	maximal, Hall
20	#65721	—	3	1	no	—
21	C₂×#3994	—	3	1	no	—
22	D₅×M₃₂	D₃	1	1	yes	characteristic
23	#64850	—	3	2	no	—
24	#64849	—	3	2	no	—
25	C₂²×ΓL(1,𝔽₁₆)	—	4	1	no	—
26	C₂×C₃×C₄×D₅	D₄	1	1	yes	characteristic
27	#6763	—	4	1	no	—
28	C₄⋊ΓL(1,𝔽₁₆)	—	2	2	no	—
29	C₄⋊ΓL(1,𝔽₁₆)	—	2	1	no	—
30	C₄⋊ΓL(1,𝔽₁₆)	—	2	1	no	—
31	C₄⋊ΓL(1,𝔽₁₆)	—	4	1	no	—
32	C₄×ΓL(1,𝔽₁₆)	—	4	1	no	—
33	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃×#168	Q₈	1	1	yes	characteristic
35	C₃×C₈×D₅	C₂×C₄	1	1	yes	characteristic
36	C₃×C₈×D₅	D₄	1	1	yes	characteristic
37	C₃×C₈×D₅	Q₈	1	1	yes	characteristic
38	C₃×C₈×D₅	C₂³	1	1	yes	characteristic
39	C₃×C₅×C₁₆	C₂×C₄	1	2	yes	complemented
40	C₃×#672	C₂×C₄	1	2	yes	complemented
41	C₂×C₃×M₃₂	—	5	1	no	—
42	#5106	—	5	1	no	—
43	C₂×#818	—	5	1	no	—
44	#4505	—	5	2	no	—
45	#4504	—	5	2	no	—
46	#4130	—	6	1	no	—
47	C₂×C₄⋊AGL(1,𝔽₅)	—	6	1	no	—
48	C₂×C₈×D₅	C₂×D₃	1	1	yes	characteristic
49	#3994	—	3	1	no	—
50	#3994	—	3	2	no	—
51	#3994	—	3	1	no	—
52	#3994	—	6	1	no	—
53	#3993	—	6	1	no	—
54	C₅×M₃₂	Dic₃	1	1	yes	characteristic, complemented
55	#3944	Dic₃	1	1	yes	characteristic, complemented
56	#3930	Dic₃	1	2	yes	complemented
57	D₅×C₁₆	C₂×D₃	1	2	yes	—
58	#2159	—	15	1	no	Sylow
59	C₂²×C₃×D₅	D₈	1	1	yes	characteristic
60	C₂×ΓL(1,𝔽₁₆)	—	4	1	no	—
61	C₂×ΓL(1,𝔽₁₆)	—	4	1	no	—
62	C₂×ΓL(1,𝔽₁₆)	—	4	2	no	—
63	C₂×ΓL(1,𝔽₁₆)	—	8	2	no	—
64	C₂×C₃×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
65	C₂×C₃×Dic₅	Q₁₆	1	1	yes	characteristic
66	C₃×C₄×D₅	C₄⋊C₄	1	1	yes	characteristic
67	C₃×C₄×D₅	D₈	1	1	yes	characteristic
68	C₃×C₄×D₅	Q₁₆	1	1	yes	characteristic
69	C₃×C₄×D₅	C₂×D₄	1	1	yes	characteristic
70	C₃×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
71	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
72	C₃×#168	C₄⋊C₄	1	1	yes	characteristic
73	C₃×#168	C₂×Q₈	1	1	yes	characteristic
74	C₂²×C₃×C₈	—	5	1	no	—
75	#926	—	10	1	no	—
76	C₂×C₄⋊Dic₃	—	10	1	no	—
77	C₃×M₃₂	—	5	1	no	—
78	C₃×M₃₂	—	5	2	no	—
79	C₃×M₃₂	AGL(1,𝔽₅)	1	1	yes	characteristic, complemented
80	C₂×C₃×C₁₆	—	5	2	no	—
81	#818	—	5	1	no	—
82	#818	—	5	2	no	—
83	#818	—	5	1	no	—
84	#818	—	10	1	no	—
85	#817	—	10	1	no	—
86	C₂²×AGL(1,𝔽₅)	—	12	1	no	—
87	C₂×C₄×D₅	D₁₂	1	1	yes	characteristic
88	C₂²⋊AGL(1,𝔽₅)	—	12	1	no	—
89	C₄⋊AGL(1,𝔽₅)	—	6	2	no	—
90	C₄⋊AGL(1,𝔽₅)	—	6	1	no	—
91	C₄⋊AGL(1,𝔽₅)	—	6	1	no	—
92	C₄⋊AGL(1,𝔽₅)	—	12	1	no	—
93	C₄×AGL(1,𝔽₅)	—	12	1	no	—
94	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
95	C₂×#168	Dic₆	1	1	yes	characteristic
96	C₈×D₅	Dic₆	1	1	yes	characteristic
97	C₈×D₅	D₁₂	1	1	yes	characteristic
98	C₈×D₅	C₂×Dic₃	1	1	yes	characteristic
99	C₈×D₅	C₂²×D₃	1	1	yes	characteristic
100	C₅×C₁₆	C₂×Dic₃	1	2	yes	complemented
101	#672	C₂×Dic₃	1	2	yes	complemented
102	C₂×M₃₂	—	5	1	no	—
103	#427	—	15	1	no	—
104	C₂×#107	—	15	1	no	—
105	#367	—	15	2	no	—
106	#366	—	15	2	no	—
107	C₂²×C₃×C₅	#107	1	1	yes	characteristic
108	C₂×C₃×D₅	—	2	1	no	—
109	C₂×C₃×D₅	—	2	1	no	—
110	C₂×C₃×D₅	#107	1	1	yes	characteristic
111	C₂×C₃×D₅	C₂×D₈	1	1	yes	characteristic
112	ΓL(1,𝔽₁₆)	—	8	1	no	—
113	ΓL(1,𝔽₁₆)	—	8	2	no	—
114	ΓL(1,𝔽₁₆)	—	8	1	no	—
115	C₃×C₄×C₅	#107	1	1	yes	characteristic
116	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₃×Dic₅	#107	1	1	yes	characteristic
118	C₃×Dic₅	C₂×Q₁₆	1	1	yes	characteristic
119	C₂²×C₃×C₄	—	5	1	no	—
120	C₂²×Dic₃	—	20	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	—	5	1	no	—
126	C₂×C₃×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
127	#217	—	20	1	no	—
128	C₄⋊Dic₃	—	10	1	no	—
129	C₄⋊Dic₃	—	10	1	no	—
130	C₄⋊Dic₃	—	10	2	no	—
131	C₄⋊Dic₃	—	20	1	no	—
132	C₄×Dic₃	—	20	1	no	—
133	C₃×C₁₆	—	5	2	no	—
134	C₃×C₁₆	C₂×AGL(1,𝔽₅)	1	2	yes	complemented
135	C₂²×D₅	D₂₄	1	1	yes	characteristic
136	C₂×AGL(1,𝔽₅)	—	12	1	no	—
137	C₂×AGL(1,𝔽₅)	—	12	1	no	—
138	C₂×AGL(1,𝔽₅)	—	12	2	no	—
139	C₂×AGL(1,𝔽₅)	—	24	2	no	—
140	C₂×C₄×C₅	C₄⋊Dic₃	1	1	yes	characteristic
141	C₂×Dic₅	Dic₁₂	1	1	yes	characteristic
142	C₄×D₅	D₂₄	1	1	yes	characteristic
143	C₄×D₅	Dic₁₂	1	1	yes	characteristic
144	C₄×D₅	C₄⋊Dic₃	1	1	yes	characteristic
145	C₄×D₅	C₂×D₁₂	1	1	yes	characteristic
146	C₅×C₈	C₄⋊Dic₃	1	1	yes	characteristic
147	C₅×C₈	C₂²×Dic₃	1	1	yes	characteristic
148	#168	C₄⋊Dic₃	1	1	yes	characteristic
149	#168	C₂×Dic₆	1	1	yes	characteristic
150	C₂²×C₈	—	5	1	no	—
151	#117	—	30	1	no	—
152	C₂×C₄⋊C₄	—	30	1	no	—
153	M₃₂	—	5	1	no	—
154	M₃₂	—	5	2	no	—
155	M₃₂	ΓL(1,𝔽₁₆)	1	1	yes	characteristic, complemented
156	C₂×C₁₆	—	5	2	no	—
157	#107	—	15	1	no	—
158	#107	—	15	2	no	—
159	#107	—	15	1	no	—
160	#107	—	30	1	no	—
161	#106	—	30	1	no	—
162	C₂×C₃×C₅	—	2	1	no	—
163	C₂×C₃×C₅	C₂×#107	1	1	yes	characteristic
164	C₃×D₅	—	2	1	no	—
165	C₃×D₅	#509	1	1	yes	characteristic
166	C₃×D₅	#510	1	1	yes	characteristic
167	C₂³×C₃	—	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	—	5	1	no	—
173	C₂×C₃×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
174	C₂×Dic₃	—	20	1	no	—
175	C₂×Dic₃	—	20	1	no	—
176	C₂×Dic₃	—	20	2	no	—
177	C₂×Dic₃	—	40	2	no	—
178	C₃×C₈	—	5	1	no	—
179	C₃×C₈	—	5	1	no	—
180	C₃×C₈	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
181	C₃×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
182	C₂²×C₅	#818	1	1	yes	characteristic
183	C₂×D₅	—	2	1	no	—
184	C₂×D₅	—	2	1	no	—
185	C₂×D₅	#818	1	1	yes	characteristic
186	C₂×D₅	C₂×D₂₄	1	1	yes	characteristic
187	AGL(1,𝔽₅)	—	24	1	no	—
188	AGL(1,𝔽₅)	—	24	2	no	—
189	AGL(1,𝔽₅)	—	24	1	no	—
190	C₄×C₅	#818	1	1	yes	characteristic
191	C₄×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
192	Dic₅	#818	1	1	yes	characteristic
193	Dic₅	C₂×Dic₁₂	1	1	yes	characteristic
194	C₂²×C₄	—	5	1	no	—
195	C₂²×C₄	—	60	1	no	—
196	C₂×C₈	—	5	1	no	—
197	C₂×C₈	—	5	1	no	—
198	C₂×C₈	—	5	1	no	—
199	C₂×C₈	—	5	1	no	—
200	C₂×C₈	—	5	1	no	—
201	C₂×C₈	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
202	C₄⋊C₄	—	30	1	no	—
203	C₄⋊C₄	—	30	1	no	—
204	C₄⋊C₄	—	30	2	no	—
205	C₄⋊C₄	—	60	1	no	—
206	C₂²⋊C₄	—	60	1	no	—
207	C₄²	—	60	1	no	—
208	C₁₆	—	5	2	no	—
209	C₁₆	C₂×ΓL(1,𝔽₁₆)	1	2	yes	complemented
210	C₃×C₅	#2159	1	1	yes	characteristic, complemented, Hall
211	C₂²×C₃	—	5	1	no	—
212	C₂²×C₃	—	5	1	no	—
213	C₂²×C₃	—	10	1	no	—
214	C₂²×C₃	—	10	1	no	—
215	C₂²×C₃	#3994	1	1	yes	characteristic
216	C₃×C₄	—	5	1	no	—
217	C₃×C₄	—	5	1	no	—
218	C₃×C₄	#3994	1	1	yes	characteristic
219	C₃×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
220	Dic₃	—	40	1	no	—
221	Dic₃	—	40	1	no	—
222	Dic₃	—	40	2	no	—
223	C₂×C₅	—	2	1	no	—
224	C₂×C₅	C₂×#818	1	1	yes	characteristic
225	D₅	—	2	1	no	—
226	D₅	#4908	1	1	yes	characteristic
227	D₅	#4909	1	1	yes	characteristic
228	C₂³	—	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	—	5	1	no	—
234	C₂×C₄	—	60	1	no	—
235	C₂×C₄	—	60	1	no	—
236	C₂×C₄	—	60	2	no	—
237	C₂×C₄	—	120	2	no	—
238	C₂×C₄	C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
239	C₈	—	5	1	no	—
240	C₈	—	5	1	no	—
241	C₈	C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
242	C₈	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
243	C₂×C₃	—	2	1	no	—
244	C₂×C₃	—	5	1	no	—
245	C₂×C₃	—	5	1	no	—
246	C₂×C₃	—	10	1	no	—
247	C₂×C₃	C₂×#3994	1	1	yes	characteristic
248	C₅	#71428	1	1	yes	characteristic, complemented, Sylow
249	C₂²	—	5	1	no	—
250	C₂²	—	5	1	no	—
251	C₂²	—	10	1	no	—
252	C₂²	—	10	1	no	—
253	C₂²	#91223	1	1	yes	characteristic
254	C₄	—	5	1	no	—
255	C₄	—	5	1	no	—
256	C₄	—	120	2	no	—
257	C₄	—	120	1	no	—
258	C₄	—	120	1	no	—
259	C₄	#91223	1	1	yes	characteristic
260	C₄	C₂×C₄⋊ΓL(1,𝔽₁₆)	1	1	yes	characteristic
261	C₃	#10604118	1	1	yes	characteristic, complemented, Sylow
262	C₂	—	2	1	no	—
263	C₂	—	5	1	no	—
264	C₂	—	5	1	no	—
265	C₂	—	10	1	no	—
266	C₂	C₂×#91223	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#91223
commutator	C₃×C₅×C₈	C₂²×C₄
Frattini	C₈	C₂²×ΓL(1,𝔽₁₆)
Fitting	C₃×C₅×M₃₂	C₄
socle	C₂×C₃×C₅	C₂×#107
prime radical	C₂²×C₃×D₅	D₈
cosocle	C₃×C₈×D₅	C₂³
hypercenter	M₃₂	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	#2159
LCS	G ▷ C₃×C₅×C₈ ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
4. #11746721	C₂	semidirect
5. #11746720	C₂	semidirect
15. C₃×C₅×M₃₂	C₄	semidirect
16. C₃×#3944	C₄	semidirect
17. C₃×#3930	C₄	semidirect
39. C₃×C₅×C₁₆	C₂×C₄	semidirect
40. C₃×#672	C₂×C₄	semidirect
54. C₅×M₃₂	Dic₃	semidirect
55. #3944	Dic₃	semidirect
56. #3930	Dic₃	semidirect
79. C₃×M₃₂	AGL(1,𝔽₅)	semidirect
100. C₅×C₁₆	C₂×Dic₃	semidirect
101. #672	C₂×Dic₃	semidirect
134. C₃×C₁₆	C₂×AGL(1,𝔽₅)	semidirect
155. M₃₂	ΓL(1,𝔽₁₆)	semidirect
209. C₁₆	C₂×ΓL(1,𝔽₁₆)	semidirect
210. C₃×C₅	#2159	semidirect
248. C₅	#71428	semidirect
261. C₃	#10604118	semidirect

Maximal subgroups

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

subgroup	index
#11752380	2
C₂×#91223	2
C₃×D₅×M₃₂	2
#11746721	2
#11746720	2
#10604118	3
#71428	5