Subgroups — #14297598

Order 1920.

544 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×C₃×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#91120	C₂	1	2	yes	maximal
3	#11751025	C₂	1	2	yes	maximal
4	#11746351	C₂	1	2	yes	maximal
5	#10602430	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
7	C₃×C₄×C₅×Q₈	—	2	2	no	—
8	C₃×C₄×C₅×Q₈	—	2	2	no	—
9	C₂×C₃×C₅×C₄⋊C₄	C₄	1	2	yes	—
10	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄²×C₅	C₄	1	2	yes	—
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₂×#6710	C₂²	1	4	yes	—
14	#91120	—	2	4	no	—
15	C₄⋊(C₃⋊#168)	—	2	2	no	—
16	C₄⋊(C₃⋊#168)	—	2	2	no	—
17	#71637	—	5	1	no	maximal, Hall
18	C₂×C₄×C₅×Q₈	D₃	1	1	yes	characteristic
19	C₂×#3968	—	3	2	no	—
20	#65213	—	3	2	no	—
21	#64750	—	3	2	no	—
22	C₂×C₃×C₅×Q₈	—	2	2	no	—
23	C₂×C₃×C₅×Q₈	—	2	2	no	—
24	C₂×C₃×C₅×Q₈	D₄	1	2	yes	—
25	C₂×C₃×C₅×Q₈	Q₈	1	2	yes	—
26	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
28	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
29	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
31	C₃×C₅×C₄⋊C₄	—	2	4	no	—
32	C₃×C₅×C₄⋊C₄	—	2	2	no	—
33	C₃×C₄²×C₅	—	2	2	no	—
34	C₃×C₄²×C₅	—	2	2	no	—
35	C₃×C₄²×C₅	D₄	1	2	yes	—
36	C₃×C₄²×C₅	D₄	1	2	yes	—
37	#6710	—	2	4	no	—
38	#6710	—	2	4	no	—
39	C₂×C₃⋊#168	—	2	4	no	—
40	C₂×C₃×C₄×Q₈	D₅	1	1	yes	characteristic
41	C₂×#836	—	5	2	no	—
42	#4924	—	5	2	no	—
43	#4530	—	5	2	no	—
44	C₂²×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
45	C₄×C₅×Q₈	—	2	2	no	—
46	C₄×C₅×Q₈	—	2	2	no	—
47	C₂×C₅×C₄⋊C₄	Dic₃	1	2	yes	—
48	C₂×C₅×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
49	C₂×C₄²×C₅	Dic₃	1	2	yes	—
50	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
51	C₂×#681	—	3	4	no	—
52	#3968	—	6	4	no	—
53	C₄⋊#168	—	6	2	no	—
54	C₄⋊#168	—	6	2	no	—
55	#1801	—	15	1	no	Sylow
56	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
57	C₃×C₅×Q₈	—	2	4	no	—
58	C₃×C₅×Q₈	—	2	4	no	—
59	C₂×C₃×C₄×C₅	—	2	2	no	—
60	C₂×C₃×C₄×C₅	—	2	2	no	—
61	C₂×C₃×C₄×C₅	—	2	2	no	—
62	C₂×C₃×C₄×C₅	—	2	2	no	—
63	C₂×C₃×C₄×C₅	—	4	1	no	—
64	C₂×C₃×C₄×C₅	—	4	2	no	—
65	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
67	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	C₄⋊C₄	1	4	yes	—
69	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
71	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	#41	1	2	yes	—
73	C₃⋊#168	—	4	4	no	—
74	C₂²×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
75	C₃×C₄×Q₈	—	2	2	no	—
76	C₃×C₄×Q₈	—	2	2	no	—
77	C₂×C₃×C₄⋊C₄	Dic₅	1	2	yes	—
78	C₂×C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
79	C₂×C₃×C₄²	Dic₅	1	2	yes	—
80	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
81	C₂×#208	—	5	4	no	—
82	#836	—	10	4	no	—
83	C₄⋊(C₃⋊C₈)	—	10	2	no	—
84	C₄⋊(C₃⋊C₈)	—	10	2	no	—
85	C₂×C₅×Q₈	—	2	2	no	—
86	C₂×C₅×Q₈	—	2	2	no	—
87	C₂×C₅×Q₈	Dic₆	1	2	yes	—
88	C₂×C₅×Q₈	D₁₂	1	2	yes	—
89	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
90	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
91	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
92	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
93	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
94	C₅×C₄⋊C₄	—	2	4	no	—
95	C₅×C₄⋊C₄	—	2	2	no	—
96	C₄²×C₅	—	2	2	no	—
97	C₄²×C₅	—	2	2	no	—
98	C₄²×C₅	#67	1	2	yes	—
99	C₄²×C₅	#67	1	2	yes	—
100	#681	—	6	4	no	—
101	#681	—	6	4	no	—
102	C₂×#168	—	6	4	no	—
103	C₂×C₄×Q₈	D₁₅	1	1	yes	characteristic
104	#423	—	15	2	no	—
105	C₂×#101	—	15	2	no	—
106	#343	—	15	2	no	—
107	C₂²×C₃×C₅	—	2	1	no	—
108	C₂²×C₃×C₅	—	2	1	no	—
109	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂²×C₃×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂²×C₃×C₅	#117	1	1	yes	characteristic
112	C₃×C₄×C₅	—	2	4	no	—
113	C₃×C₄×C₅	—	2	1	no	—
114	C₃×C₄×C₅	—	2	2	no	—
115	C₃×C₄×C₅	—	4	2	no	—
116	C₃×C₄×C₅	#101	1	2	yes	complemented
117	C₃×C₄×C₅	#121	1	2	yes	—
118	C₃×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₃×Q₈	—	2	2	no	—
120	C₂×C₃×Q₈	—	2	2	no	—
121	C₂×C₃×Q₈	Dic₁₀	1	2	yes	—
122	C₂×C₃×Q₈	D₂₀	1	2	yes	—
123	C₂²×C₃×C₄	C₂×Dic₅	1	2	yes	—
124	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
125	C₂²×C₃×C₄	C₂×Dic₅	1	2	yes	—
126	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
127	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
128	C₃×C₄⋊C₄	—	2	4	no	—
129	C₃×C₄⋊C₄	—	2	2	no	—
130	C₃×C₄²	—	2	2	no	—
131	C₃×C₄²	—	2	2	no	—
132	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
133	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
134	#208	—	10	4	no	—
135	#208	—	10	4	no	—
136	C₂×C₃⋊C₈	—	10	4	no	—
137	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
138	C₅×Q₈	—	2	4	no	—
139	C₅×Q₈	—	2	4	no	—
140	C₂×C₄×C₅	—	2	2	no	—
141	C₂×C₄×C₅	—	2	2	no	—
142	C₂×C₄×C₅	—	2	2	no	—
143	C₂×C₄×C₅	—	2	2	no	—
144	C₂×C₄×C₅	—	4	1	no	—
145	C₂×C₄×C₅	—	4	2	no	—
146	C₂×C₄×C₅	C₄⋊Dic₃	1	4	yes	—
147	C₂×C₄×C₅	#217	1	1	yes	characteristic
148	C₂×C₄×C₅	#217	1	2	yes	—
149	C₂×C₄×C₅	#217	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂×Dic₆	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
152	C₂×C₄×C₅	#235	1	2	yes	—
153	C₂×C₄×C₅	C₂×#67	1	2	yes	—
154	#168	—	12	4	no	—
155	C₂²×Q₈	C₂×D₁₅	1	1	yes	characteristic
156	C₂×M₁₆	—	15	4	no	—
157	C₄×Q₈	—	2	2	no	—
158	C₄×Q₈	—	2	2	no	—
159	C₂×C₄⋊C₄	Dic₁₅	1	2	yes	—
160	C₂×C₄⋊C₄	C₂×D₁₅	1	1	yes	characteristic
161	C₂×C₄²	Dic₁₅	1	2	yes	—
162	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
163	C₄⋊C₈	—	30	2	no	—
164	C₄⋊C₈	—	30	2	no	—
165	#101	—	30	4	no	—
166	C₂×C₃×C₅	—	2	1	no	—
167	C₂×C₃×C₅	—	2	1	no	—
168	C₂×C₃×C₅	#380	1	1	yes	characteristic
169	C₂×C₃×C₅	C₂×#101	1	1	yes	characteristic
170	C₂×C₃×C₅	#413	1	1	yes	characteristic
171	C₂³×C₃	C₂²×Dic₅	1	1	yes	characteristic
172	C₃×Q₈	—	2	4	no	—
173	C₃×Q₈	—	2	4	no	—
174	C₂×C₃×C₄	—	2	2	no	—
175	C₂×C₃×C₄	—	2	2	no	—
176	C₂×C₃×C₄	—	2	2	no	—
177	C₂×C₃×C₄	—	2	2	no	—
178	C₂×C₃×C₄	—	4	1	no	—
179	C₂×C₃×C₄	—	4	2	no	—
180	C₂×C₃×C₄	C₄⋊Dic₅	1	4	yes	—
181	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
182	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
183	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
184	C₂×C₃×C₄	C₂×Dic₁₀	1	1	yes	characteristic
185	C₂×C₃×C₄	C₂×D₂₀	1	1	yes	characteristic
186	C₂×C₃×C₄	#709	1	2	yes	—
187	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
188	C₃⋊C₈	—	20	4	no	—
189	C₂²×C₅	—	2	1	no	—
190	C₂²×C₅	—	2	1	no	—
191	C₂²×C₅	C₂×C₄⋊Dic₃	1	1	yes	characteristic
192	C₂²×C₅	#926	1	1	yes	characteristic
193	C₂²×C₅	C₂×#217	1	1	yes	characteristic
194	C₄×C₅	—	2	4	no	—
195	C₄×C₅	—	2	1	no	—
196	C₄×C₅	—	2	2	no	—
197	C₄×C₅	—	4	2	no	—
198	C₄×C₅	#836	1	2	yes	complemented
199	C₄×C₅	#924	1	2	yes	—
200	C₄×C₅	#930	1	2	yes	—
201	C₂×Q₈	—	2	2	no	—
202	C₂×Q₈	—	2	2	no	—
203	C₂×Q₈	Dic₃₀	1	2	yes	—
204	C₂×Q₈	D₆₀	1	2	yes	—
205	C₂²×C₄	C₂×Dic₁₅	1	2	yes	—
206	C₂²×C₄	C₂×Dic₁₅	1	1	yes	characteristic
207	C₂²×C₄	C₂×Dic₁₅	1	2	yes	—
208	C₂²×C₄	C₂×Dic₁₅	1	1	yes	characteristic
209	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
210	M₁₆	—	30	4	no	—
211	M₁₆	—	30	4	no	—
212	C₂×C₈	—	30	4	no	—
213	C₄⋊C₄	—	2	4	no	—
214	C₄⋊C₄	—	2	2	no	—
215	C₄²	—	2	2	no	—
216	C₄²	—	2	2	no	—
217	C₄²	#1220	1	2	yes	—
218	C₄²	#1220	1	2	yes	—
219	C₃×C₅	#1801	1	1	yes	characteristic, complemented, Hall
220	C₂²×C₃	—	2	1	no	—
221	C₂²×C₃	—	2	1	no	—
222	C₂²×C₃	C₂×C₄⋊Dic₅	1	1	yes	characteristic
223	C₂²×C₃	#4072	1	1	yes	characteristic
224	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
225	C₃×C₄	—	2	4	no	—
226	C₃×C₄	—	2	1	no	—
227	C₃×C₄	—	2	2	no	—
228	C₃×C₄	—	4	2	no	—
229	C₃×C₄	#3968	1	2	yes	complemented
230	C₃×C₄	#4070	1	2	yes	—
231	C₃×C₄	#4076	1	2	yes	—
232	C₂×C₅	—	2	1	no	—
233	C₂×C₅	—	2	1	no	—
234	C₂×C₅	#5212	1	1	yes	characteristic
235	C₂×C₅	C₂×#836	1	1	yes	characteristic
236	C₂×C₅	#5237	1	1	yes	characteristic
237	C₂³	C₂²×Dic₁₅	1	1	yes	characteristic
238	Q₈	—	2	4	no	—
239	Q₈	—	2	4	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	C₄⋊Dic₁₅	1	4	yes	—
247	C₂×C₄	#6719	1	1	yes	characteristic
248	C₂×C₄	#6719	1	2	yes	—
249	C₂×C₄	#6719	1	1	yes	characteristic
250	C₂×C₄	C₂×Dic₃₀	1	1	yes	characteristic
251	C₂×C₄	C₂×D₆₀	1	1	yes	characteristic
252	C₂×C₄	#6817	1	2	yes	—
253	C₂×C₄	C₂×#1220	1	2	yes	—
254	C₈	—	60	4	no	—
255	C₂×C₃	—	2	1	no	—
256	C₂×C₃	—	2	1	no	—
257	C₂×C₃	#65501	1	1	yes	characteristic
258	C₂×C₃	C₂×#3968	1	1	yes	characteristic
259	C₂×C₃	#65526	1	1	yes	characteristic
260	C₅	#71637	1	1	yes	characteristic, complemented, Sylow
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	C₂×C₄⋊Dic₁₅	1	1	yes	characteristic
264	C₂²	#91815	1	1	yes	characteristic
265	C₂²	C₂×#6719	1	1	yes	characteristic
266	C₄	—	2	4	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	2	no	—
269	C₄	—	4	2	no	—
270	C₄	#91120	1	2	yes	complemented
271	C₄	#91813	1	2	yes	—
272	C₄	#91819	1	2	yes	—
273	C₃	#10602430	1	1	yes	characteristic, complemented, Sylow
274	C₂	—	2	1	no	—
275	C₂	—	2	1	no	—
276	C₂	#11751313	1	1	yes	characteristic
277	C₂	C₂×#91120	1	1	yes	characteristic
278	C₂	#11751338	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6719
commutator, prime radical	C₂³×C₃×C₅	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅
Fitting	C₂×C₃×C₄×C₅×Q₈	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
cosocle	C₂²×C₃×C₄×C₅	C₂³
hypercenter	C₂×C₄×Q₈	D₁₅
hypocenter	C₃×C₅	#1801
LCS	G ▷ C₂³×C₃×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×Q₈

Complements

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

N	H ≅ G/N	type
116. C₃×C₄×C₅	#101	semidirect
198. C₄×C₅	#836	semidirect
219. C₃×C₅	#1801	semidirect
229. C₃×C₄	#3968	semidirect
260. C₅	#71637	semidirect
270. C₄	#91120	semidirect
273. C₃	#10602430	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₄×C₅×Q₈	2
C₂×#91120	2
#11751025	2
#11746351	2
#10602430	3
#71637	5