Subgroups — #14280403

Order 1920.

448 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	#11750786	C₂	1	1	yes	maximal, characteristic
2	C₄×#6651	C₂	1	1	yes	maximal, characteristic
3	C₄×#6648	C₂	1	1	yes	maximal, characteristic
4	C₅×#4454	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₈×#168	C₂	1	1	yes	maximal, characteristic, complemented
6	#11745831	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃⋊C₈×#168	C₂	1	1	yes	maximal, characteristic, complemented
8	#10598860	—	3	1	no	maximal, Hall
9	C₄²×D₁₅	C₂²	1	1	yes	characteristic
10	C₂×#6651	C₄	1	2	yes	—
11	C₂×#6648	C₄	1	2	yes	—
12	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
13	#91104	C₄	1	2	yes	—
14	#91098	C₄	1	2	yes	—
15	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
16	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₄×C₅×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₃×C₄×#168	C₂²	1	1	yes	characteristic
19	C₃×C₄×#168	C₂²	1	1	yes	characteristic
20	#90954	C₄	1	2	yes	—
21	#90951	C₄	1	2	yes	—
22	C₈×ΓL(1,𝔽₂₅)	—	5	1	no	maximal, Hall
23	C₄×#676	—	3	1	no	—
24	#64974	—	3	1	no	—
25	C₄×C₈×D₅	—	3	1	no	—
26	C₅×#322	—	3	1	no	—
27	#64674	—	3	1	no	—
28	C₈×#168	—	3	1	no	—
29	C₈×#168	D₃	1	1	yes	characteristic, complemented
30	C₂×C₄×D₁₅	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₁₅	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×D₁₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
34	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
35	C₄×Dic₁₅	C₂×C₄	1	1	yes	characteristic
36	C₄×Dic₁₅	C₂×C₄	1	1	yes	characteristic
37	C₄×Dic₁₅	C₂×C₄	1	1	yes	characteristic
38	C₂×C₃⋊#168	C₂×C₄	1	2	yes	—
39	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
40	C₂×C₅×C₃⋊C₈	C₂×C₄	1	2	yes	—
41	C₂×C₃×#168	C₂×C₄	1	2	yes	—
42	C₂×C₃×#168	C₂×C₄	1	2	yes	—
43	#6651	C₈	1	8	yes	complemented
44	#6648	—	2	4	no	—
45	C₄×ΓL(1,𝔽₂₅)	—	5	1	no	—
46	#4685	—	5	1	no	—
47	C₄×D₃×C₈	—	5	1	no	—
48	C₃×C₈²	—	5	1	no	—
49	#4455	—	5	1	no	—
50	#4454	D₅	1	1	yes	characteristic, complemented
51	C₈×C₃⋊C₈	—	5	1	no	—
52	C₂×#676	—	3	2	no	—
53	C₂×C₈×D₅	—	3	2	no	—
54	C₄²×D₅	—	3	1	no	—
55	C₄×C₅×C₈	—	3	1	no	—
56	C₄×C₅×C₈	—	3	1	no	—
57	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
58	#3952	—	3	2	no	—
59	#3947	—	3	2	no	—
60	#3946	—	3	2	no	—
61	C₈×Dic₅	—	3	2	no	—
62	C₄×#168	—	3	1	no	—
63	C₄×#168	C₂×D₃	1	1	yes	characteristic
64	C₄×#168	C₂×D₃	1	1	yes	characteristic
65	C₈×M₁₆	—	15	1	no	Sylow
66	C₂²×D₁₅	C₄²	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
70	C₂×Dic₁₅	C₄²	1	1	yes	characteristic
71	C₂×Dic₁₅	C₄²	1	1	yes	characteristic
72	C₂×Dic₁₅	C₄²	1	1	yes	characteristic
73	C₄×D₁₅	—	2	2	no	—
74	C₄×D₁₅	—	2	2	no	—
75	C₄×D₁₅	C₂×C₈	1	4	yes	—
76	C₃×C₅×C₈	—	2	2	no	—
77	C₃⋊#168	—	2	2	no	—
78	C₃×#168	C₂×C₈	1	4	yes	complemented
79	C₃×#168	M₁₆	1	4	yes	complemented
80	C₅×C₃⋊C₈	—	2	2	no	—
81	C₅×C₃⋊C₈	C₂×C₈	1	4	yes	complemented
82	C₂×ΓL(1,𝔽₂₅)	—	5	2	no	—
83	C₂×D₃×C₈	—	5	2	no	—
84	C₄²×D₃	—	5	1	no	—
85	C₃×C₄×C₈	—	5	1	no	—
86	C₃×C₄×C₈	—	5	1	no	—
87	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
88	(C₂×D₃)⋊C₈	—	5	2	no	—
89	#815	—	5	2	no	—
90	Dic₃⋊C₈	—	5	2	no	—
91	C₈×Dic₃	—	5	2	no	—
92	C₄×C₃⋊C₈	—	5	1	no	—
93	C₄×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
94	C₄×C₃⋊C₈	C₂×D₅	1	1	yes	characteristic
95	C₂×C₄×D₅	—	3	1	no	—
96	C₂×C₄×D₅	—	3	1	no	—
97	C₂×C₄×D₅	—	3	1	no	—
98	C₂×C₅×C₈	—	3	2	no	—
99	C₂×C₅×C₈	—	3	2	no	—
100	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
101	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
102	C₄×Dic₅	—	3	1	no	—
103	C₄×Dic₅	—	3	1	no	—
104	C₄×Dic₅	—	3	1	no	—
105	C₂×#168	—	3	2	no	—
106	C₂×#168	C₄×D₃	1	2	yes	—
107	C₂×#168	C₄×D₃	1	2	yes	—
108	#676	—	3	8	no	—
109	C₈×D₅	—	6	4	no	—
110	#431	—	15	1	no	—
111	C₄×M₁₆	—	15	1	no	—
112	C₂×C₄×C₈	—	15	1	no	—
113	#322	—	3	1	no	—
114	#322	—	15	1	no	—
115	C₈²	—	5	1	no	—
116	C₈²	—	15	1	no	—
117	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
118	C₂×D₁₅	—	2	1	no	—
119	C₂×D₁₅	—	2	1	no	—
120	C₂×D₁₅	C₄×C₈	1	2	yes	—
121	C₃×C₄×C₅	C₂²×C₈	1	2	yes	—
122	C₃×C₄×C₅	C₂×M₁₆	1	2	yes	—
123	C₃×C₄×C₅	#131	1	2	yes	—
124	Dic₁₅	—	2	1	no	—
125	Dic₁₅	—	2	1	no	—
126	Dic₁₅	C₄×C₈	1	2	yes	—
127	C₂×C₄×D₃	—	5	1	no	—
128	C₂×C₄×D₃	—	5	1	no	—
129	C₂×C₄×D₃	—	5	1	no	—
130	C₂×C₃×C₈	—	5	2	no	—
131	C₂×C₃×C₈	—	5	2	no	—
132	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
133	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
134	C₄×Dic₃	—	5	1	no	—
135	C₄×Dic₃	—	5	1	no	—
136	C₄×Dic₃	—	5	1	no	—
137	C₂×C₃⋊C₈	—	5	2	no	—
138	C₂×C₃⋊C₈	C₄×D₅	1	2	yes	—
139	C₂×C₃⋊C₈	C₄×D₅	1	2	yes	—
140	ΓL(1,𝔽₂₅)	—	5	8	no	—
141	D₃×C₈	—	10	4	no	—
142	C₂²×D₅	—	3	1	no	—
143	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
146	C₂×Dic₅	—	3	1	no	—
147	C₂×Dic₅	—	3	1	no	—
148	C₂×Dic₅	—	3	1	no	—
149	C₄×D₅	—	3	4	no	—
150	C₄×D₅	—	6	2	no	—
151	C₄×D₅	—	6	2	no	—
152	C₅×C₈	—	2	2	no	—
153	C₅×C₈	—	3	4	no	—
154	C₅×C₈	—	6	2	no	—
155	#168	—	6	2	no	—
156	#168	D₃×C₈	1	4	yes	complemented
157	#168	ΓL(1,𝔽₂₅)	1	4	yes	complemented
158	C₂×M₁₆	—	15	2	no	—
159	C₂²×C₈	—	15	2	no	—
160	C₂×C₄²	—	15	1	no	—
161	C₄⋊C₈	—	15	2	no	—
162	C₂²⋊C₈	—	15	2	no	—
163	#97	—	15	2	no	—
164	C₄×C₈	—	3	1	no	—
165	C₄×C₈	—	3	1	no	—
166	C₄×C₈	—	5	1	no	—
167	C₄×C₈	—	5	1	no	—
168	C₄×C₈	—	15	2	no	—
169	C₄×C₈	—	15	1	no	—
170	C₄×C₈	D₃×D₅	1	1	yes	characteristic
171	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
172	C₂×C₃×C₅	C₄×M₁₆	1	1	yes	characteristic
173	C₂×C₃×C₅	#405	1	1	yes	characteristic
174	D₁₅	—	2	2	no	—
175	C₂²×D₃	—	5	1	no	—
176	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
177	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
179	C₂×Dic₃	—	5	1	no	—
180	C₂×Dic₃	—	5	1	no	—
181	C₂×Dic₃	—	5	1	no	—
182	C₄×D₃	—	5	4	no	—
183	C₄×D₃	—	10	2	no	—
184	C₄×D₃	—	10	2	no	—
185	C₃×C₈	—	2	2	no	—
186	C₃×C₈	—	5	4	no	—
187	C₃×C₈	—	5	4	no	—
188	C₃⋊C₈	—	2	2	no	—
189	C₃⋊C₈	—	10	2	no	—
190	C₃⋊C₈	C₈×D₅	1	4	yes	complemented
191	C₂²×C₅	C₄²×D₃	1	1	yes	characteristic
192	C₂×D₅	—	3	2	no	—
193	C₂×D₅	—	6	1	no	—
194	C₂×D₅	—	6	1	no	—
195	C₄×C₅	C₂×D₃×C₈	1	2	yes	—
196	C₄×C₅	C₂×ΓL(1,𝔽₂₅)	1	2	yes	—
197	C₄×C₅	#901	1	2	yes	—
198	Dic₅	—	3	2	no	—
199	Dic₅	—	6	1	no	—
200	Dic₅	—	6	1	no	—
201	C₂²×C₄	—	15	1	no	—
202	C₂²×C₄	—	15	1	no	—
203	C₂²×C₄	—	15	1	no	—
204	M₁₆	—	15	8	no	—
205	C₂×C₈	—	3	2	no	—
206	C₂×C₈	—	3	2	no	—
207	C₂×C₈	—	5	2	no	—
208	C₂×C₈	—	5	2	no	—
209	C₂×C₈	—	15	2	no	—
210	C₂×C₈	—	30	4	no	—
211	C₂×C₈	#1200	1	2	yes	—
212	C₄²	—	15	1	no	—
213	C₄²	—	15	1	no	—
214	C₄²	—	15	1	no	—
215	C₄²	C₂×D₃×D₅	1	1	yes	characteristic
216	C₃×C₅	C₈×M₁₆	1	1	yes	characteristic, complemented, Hall
217	C₂²×C₃	C₄²×D₅	1	1	yes	characteristic
218	C₂×D₃	—	5	2	no	—
219	C₂×D₃	—	10	1	no	—
220	C₂×D₃	—	10	1	no	—
221	C₃×C₄	C₂×C₈×D₅	1	2	yes	—
222	C₃×C₄	D₅×M₁₆	1	2	yes	—
223	C₃×C₄	#4053	1	2	yes	—
224	Dic₃	—	5	2	no	—
225	Dic₃	—	10	1	no	—
226	Dic₃	—	10	1	no	—
227	C₂×C₅	C₄×D₃×C₈	1	1	yes	characteristic
228	C₂×C₅	C₄×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
229	C₂×C₅	#4689	1	1	yes	characteristic
230	D₅	—	6	2	no	—
231	C₂³	—	15	1	no	—
232	C₂×C₄	—	15	1	no	—
233	C₂×C₄	—	15	4	no	—
234	C₂×C₄	—	15	1	no	—
235	C₂×C₄	—	15	1	no	—
236	C₂×C₄	—	30	2	no	—
237	C₂×C₄	—	30	2	no	—
238	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
239	C₂×C₄	C₄×D₃×D₅	1	1	yes	characteristic
240	C₂×C₄	C₂×#1200	1	1	yes	characteristic
241	C₈	—	2	2	no	—
242	C₈	—	3	4	no	—
243	C₈	—	5	4	no	—
244	C₈	—	5	4	no	—
245	C₈	—	6	2	no	—
246	C₈	—	30	2	no	—
247	C₂×C₃	C₄×C₈×D₅	1	1	yes	characteristic
248	C₂×C₃	#64996	1	1	yes	characteristic
249	C₂×C₃	#64997	1	1	yes	characteristic
250	D₃	—	10	2	no	—
251	C₅	C₈×ΓL(1,𝔽₂₅)	1	1	yes	characteristic, complemented, Sylow
252	C₂²	—	15	2	no	—
253	C₂²	—	30	1	no	—
254	C₂²	—	30	1	no	—
255	C₂²	C₄×#1200	1	1	yes	characteristic
256	C₄	—	15	2	no	—
257	C₄	—	30	1	no	—
258	C₄	—	30	1	no	—
259	C₄	D₃×C₈×D₅	1	2	yes	—
260	C₄	D₅×ΓL(1,𝔽₂₅)	1	2	yes	—
261	C₄	#91290	1	2	yes	—
262	C₃	#10598860	1	1	yes	characteristic, complemented, Sylow
263	C₂	—	30	2	no	—
264	C₂	#11746981	1	1	yes	characteristic
265	C₂	C₈×#1200	1	1	yes	characteristic
266	C₂	#11747592	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂×D₃×D₅
commutator	C₂×C₃×C₅	C₂×C₄×C₈
Fitting	C₃×C₄×C₅×C₈	C₂²
socle	C₂²×C₃×C₅	C₂×C₄²
prime radical	C₂²×D₁₅	C₄²
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄×C₈	D₃×D₅
hypocenter	C₃×C₅	C₈×M₁₆
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₄² ◁ C₄×C₈

Complements

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

N	H ≅ G/N	type
4. C₅×#4454	C₂	semidirect
5. C₃×C₈×#168	C₂	semidirect
6. #11745831	C₂	semidirect
7. C₃⋊C₈×#168	C₂	semidirect
29. C₈×#168	D₃	semidirect
43. #6651	C₈	semidirect
50. #4454	D₅	semidirect
78. C₃×#168	C₂×C₈	semidirect
79. C₃×#168	M₁₆	semidirect
81. C₅×C₃⋊C₈	C₂×C₈	semidirect
156. #168	D₃×C₈	semidirect
157. #168	ΓL(1,𝔽₂₅)	semidirect
190. C₃⋊C₈	C₈×D₅	semidirect
216. C₃×C₅	C₈×M₁₆	semidirect
251. C₅	C₈×ΓL(1,𝔽₂₅)	semidirect
262. C₃	#10598860	semidirect

Maximal subgroups

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

subgroup	index
#11750786	2
C₄×#6651	2
C₄×#6648	2
C₅×#4454	2
C₃×C₈×#168	2
#11745831	2
C₃⋊C₈×#168	2
#10598860	3
C₈×ΓL(1,𝔽₂₅)	5