Subgroups — #14284898

Order 1920.

480 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#11751026	C₂	1	1	yes	maximal, characteristic
3	#11751025	C₂	1	1	yes	maximal, characteristic
4	#11746367	C₂	1	4	yes	maximal, complemented
5	#10602567	—	3	1	no	maximal, Hall
6	C₂²×C₃×C₅×Q₈	C₄	1	2	yes	—
7	C₃×C₅×C₄⋊Q₈	—	2	1	no	—
8	C₃×C₅×C₄⋊Q₈	—	2	1	no	—
9	C₃×C₅×C₄⋊Q₈	C₄	1	2	yes	—
10	C₃×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
12	C₂×C₃×C₅×C₄⋊C₄	—	2	1	no	—
13	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
14	C₂×#6710	—	2	1	no	—
15	#91114	—	2	1	no	—
16	C₄⋊(C₃⋊#168)	—	2	1	no	—
17	C₄⋊(C₃⋊#168)	C₂²	1	2	yes	—
18	C₄⋊(C₃⋊#168)	C₂²	1	2	yes	—
19	C₄×C₃⋊#168	—	2	1	no	—
20	#71774	—	5	1	no	maximal, Hall
21	C₂×C₅×C₄⋊Q₈	D₃	1	1	yes	characteristic
22	#65214	—	3	1	no	—
23	#65213	—	3	1	no	—
24	#64766	—	3	4	no	—
25	C₂×C₃×C₅×Q₈	—	2	2	no	—
26	C₂×C₃×C₅×Q₈	—	4	4	no	—
27	C₂×C₃×C₅×Q₈	C₂×C₄	1	4	yes	—
28	C₂²×C₃×C₄×C₅	—	2	2	no	—
29	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
31	C₂²×C₃×C₄×C₅	D₄	1	1	yes	characteristic
32	C₃×C₅×C₄⋊C₄	—	2	2	no	—
33	C₃×C₅×C₄⋊C₄	—	2	2	no	—
34	C₃×C₅×C₄⋊C₄	—	2	2	no	—
35	C₃×C₅×C₄⋊C₄	—	2	2	no	—
36	C₃×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
37	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
38	C₃×C₄²×C₅	D₄	1	1	yes	characteristic
39	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
40	#6710	—	4	2	no	—
41	C₂×C₃⋊#168	—	2	2	no	—
42	C₂×C₃⋊#168	—	2	2	no	—
43	C₂×C₃×C₄⋊Q₈	D₅	1	1	yes	characteristic
44	#4925	—	5	1	no	—
45	#4924	—	5	1	no	—
46	#4546	—	5	4	no	—
47	C₂²×C₅×Q₈	Dic₃	1	2	yes	—
48	C₅×C₄⋊Q₈	—	2	1	no	—
49	C₅×C₄⋊Q₈	—	2	1	no	—
50	C₅×C₄⋊Q₈	Dic₃	1	2	yes	—
51	C₅×C₄⋊Q₈	C₂×D₃	1	2	yes	—
52	C₂×C₅×C₄⋊C₄	—	2	1	no	—
53	C₂×C₅×C₄⋊C₄	—	2	1	no	—
54	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
55	C₂×#681	—	6	1	no	—
56	C₂²⋊#168	—	6	1	no	—
57	C₄⋊#168	—	3	2	no	—
58	C₄⋊#168	—	3	2	no	—
59	C₄⋊#168	—	6	1	no	—
60	C₄×#168	—	6	1	no	—
61	#1548	—	15	1	no	Sylow
62	C₂³×C₃×C₅	C₂²⋊C₄	1	1	yes	characteristic
63	C₃×C₅×Q₈	—	4	4	no	—
64	C₃×C₅×Q₈	—	4	4	no	—
65	C₂×C₃×C₄×C₅	—	2	2	no	—
66	C₂×C₃×C₄×C₅	—	2	2	no	—
67	C₂×C₃×C₄×C₅	—	2	4	no	—
68	C₂×C₃×C₄×C₅	—	4	4	no	—
69	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	SD₁₆	1	2	yes	—
73	C₂×C₃×C₄×C₅	Q₁₆	1	2	yes	—
74	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₃⋊#168	—	4	2	no	—
78	C₃⋊#168	—	4	2	no	—
79	C₂²×C₃×Q₈	Dic₅	1	2	yes	—
80	C₃×C₄⋊Q₈	—	2	1	no	—
81	C₃×C₄⋊Q₈	—	2	1	no	—
82	C₃×C₄⋊Q₈	Dic₅	1	2	yes	—
83	C₃×C₄⋊Q₈	C₂×D₅	1	2	yes	—
84	C₂×C₃×C₄⋊C₄	—	2	1	no	—
85	C₂×C₃×C₄⋊C₄	—	2	1	no	—
86	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
87	C₂×#208	—	10	1	no	—
88	#830	—	10	1	no	—
89	C₄⋊(C₃⋊C₈)	—	5	2	no	—
90	C₄⋊(C₃⋊C₈)	—	5	2	no	—
91	C₄⋊(C₃⋊C₈)	—	10	1	no	—
92	C₄×C₃⋊C₈	—	10	1	no	—
93	C₂×C₅×Q₈	—	2	2	no	—
94	C₂×C₅×Q₈	—	4	4	no	—
95	C₂×C₅×Q₈	C₂×Dic₃	1	4	yes	—
96	C₂²×C₄×C₅	—	2	2	no	—
97	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
98	C₂²×C₄×C₅	#67	1	1	yes	characteristic
99	C₂²×C₄×C₅	#67	1	1	yes	characteristic
100	C₅×C₄⋊C₄	—	2	2	no	—
101	C₅×C₄⋊C₄	—	2	2	no	—
102	C₅×C₄⋊C₄	—	2	2	no	—
103	C₅×C₄⋊C₄	—	2	2	no	—
104	C₄²×C₅	C₂×Dic₃	1	1	yes	characteristic
105	C₄²×C₅	#67	1	1	yes	characteristic
106	C₄²×C₅	#67	1	1	yes	characteristic
107	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
108	#681	—	12	2	no	—
109	C₂×#168	—	6	2	no	—
110	C₂×#168	—	6	2	no	—
111	C₂×C₄⋊Q₈	D₁₅	1	1	yes	characteristic
112	#431	—	15	1	no	—
113	#423	—	15	1	no	—
114	#333	—	15	4	no	—
115	C₂²×C₃×C₅	—	2	1	no	—
116	C₂²×C₃×C₅	—	2	1	no	—
117	C₂²×C₃×C₅	#103	1	2	yes	—
118	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₃×C₄×C₅	—	2	1	no	—
120	C₃×C₄×C₅	—	2	1	no	—
121	C₃×C₄×C₅	—	4	4	no	—
122	C₃×C₄×C₅	#100	1	2	yes	—
123	C₃×C₄×C₅	#103	1	2	yes	—
124	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
125	C₃×C₄×C₅	C₂×Q₁₆	1	1	yes	characteristic
126	C₃×C₄×C₅	#137	1	2	yes	—
127	C₂×C₃×Q₈	—	2	2	no	—
128	C₂×C₃×Q₈	—	4	4	no	—
129	C₂×C₃×Q₈	C₂×Dic₅	1	4	yes	—
130	C₂²×C₃×C₄	—	2	2	no	—
131	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
132	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
133	C₂²×C₃×C₄	C₂²⋊D₅	1	1	yes	characteristic
134	C₃×C₄⋊C₄	—	2	2	no	—
135	C₃×C₄⋊C₄	—	2	2	no	—
136	C₃×C₄⋊C₄	—	2	2	no	—
137	C₃×C₄⋊C₄	—	2	2	no	—
138	C₃×C₄²	C₂×Dic₅	1	1	yes	characteristic
139	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
140	C₃×C₄²	C₂²⋊D₅	1	1	yes	characteristic
141	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
142	#208	—	20	2	no	—
143	C₂×C₃⋊C₈	—	10	2	no	—
144	C₂×C₃⋊C₈	—	10	2	no	—
145	C₂³×C₅	#217	1	1	yes	characteristic
146	C₅×Q₈	—	4	4	no	—
147	C₅×Q₈	—	4	4	no	—
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	—	2	2	no	—
150	C₂×C₄×C₅	—	2	4	no	—
151	C₂×C₄×C₅	—	4	4	no	—
152	C₂×C₄×C₅	#215	1	2	yes	—
153	C₂×C₄×C₅	#216	1	2	yes	—
154	C₂×C₄×C₅	#217	1	1	yes	characteristic
155	C₂×C₄×C₅	#217	1	1	yes	characteristic
156	C₂×C₄×C₅	#217	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
158	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
159	C₂×C₄×C₅	C₂×#67	1	1	yes	characteristic
160	#168	—	12	2	no	—
161	#168	—	12	2	no	—
162	C₂²×Q₈	Dic₁₅	1	2	yes	—
163	C₂×M₁₆	—	30	1	no	—
164	C₄⋊Q₈	—	2	1	no	—
165	C₄⋊Q₈	—	2	1	no	—
166	C₄⋊Q₈	Dic₁₅	1	2	yes	—
167	C₄⋊Q₈	C₂×D₁₅	1	2	yes	—
168	C₂×C₄⋊C₄	—	2	1	no	—
169	C₂×C₄⋊C₄	—	2	1	no	—
170	C₂×C₄²	C₂×D₁₅	1	1	yes	characteristic
171	C₄⋊C₈	—	15	2	no	—
172	C₄⋊C₈	—	15	2	no	—
173	C₄⋊C₈	—	30	1	no	—
174	C₂²⋊C₈	—	30	1	no	—
175	C₄×C₈	—	30	1	no	—
176	C₂×C₃×C₅	—	2	2	no	—
177	C₂×C₃×C₅	C₂×#100	1	1	yes	characteristic
178	C₂×C₃×C₅	C₂×#103	1	1	yes	characteristic
179	C₂×C₃×C₅	#419	1	1	yes	characteristic
180	C₂³×C₃	C₂²⋊Dic₅	1	1	yes	characteristic
181	C₃×Q₈	—	4	4	no	—
182	C₃×Q₈	—	4	4	no	—
183	C₂×C₃×C₄	—	2	2	no	—
184	C₂×C₃×C₄	—	2	2	no	—
185	C₂×C₃×C₄	—	2	4	no	—
186	C₂×C₃×C₄	—	4	4	no	—
187	C₂×C₃×C₄	#688	1	2	yes	—
188	C₂×C₃×C₄	#689	1	2	yes	—
189	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
190	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
192	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
193	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
194	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
195	C₃⋊C₈	—	20	2	no	—
196	C₃⋊C₈	—	20	2	no	—
197	C₂²×C₅	—	2	1	no	—
198	C₂²×C₅	—	2	1	no	—
199	C₂²×C₅	#835	1	2	yes	—
200	C₂²×C₅	C₂×#217	1	1	yes	characteristic
201	C₄×C₅	—	2	1	no	—
202	C₄×C₅	—	2	1	no	—
203	C₄×C₅	—	4	4	no	—
204	C₄×C₅	#833	1	2	yes	—
205	C₄×C₅	#835	1	2	yes	—
206	C₄×C₅	C₂×#215	1	1	yes	characteristic
207	C₄×C₅	#942	1	2	yes	—
208	C₄×C₅	C₂×#216	1	1	yes	characteristic
209	C₂×Q₈	—	2	2	no	—
210	C₂×Q₈	—	4	4	no	—
211	C₂×Q₈	C₂×Dic₁₅	1	4	yes	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	C₂×Dic₁₅	1	1	yes	characteristic
214	C₂²×C₄	#1220	1	1	yes	characteristic
215	C₂²×C₄	#1220	1	1	yes	characteristic
216	M₁₆	—	60	2	no	—
217	C₂×C₈	—	30	2	no	—
218	C₂×C₈	—	30	2	no	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	2	2	no	—
223	C₄²	C₂×Dic₁₅	1	1	yes	characteristic
224	C₄²	#1220	1	1	yes	characteristic
225	C₄²	#1220	1	1	yes	characteristic
226	C₄²	C₂²×D₁₅	1	1	yes	characteristic
227	C₃×C₅	#1548	1	1	yes	characteristic, complemented, Hall
228	C₂²×C₃	—	2	1	no	—
229	C₂²×C₃	—	2	1	no	—
230	C₂²×C₃	#3967	1	2	yes	—
231	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
232	C₃×C₄	—	2	1	no	—
233	C₃×C₄	—	2	1	no	—
234	C₃×C₄	—	4	4	no	—
235	C₃×C₄	#3965	1	2	yes	—
236	C₃×C₄	#3967	1	2	yes	—
237	C₃×C₄	C₂×#688	1	1	yes	characteristic
238	C₃×C₄	#4088	1	2	yes	—
239	C₃×C₄	C₂×#689	1	1	yes	characteristic
240	C₂×C₅	—	2	2	no	—
241	C₂×C₅	C₂×#833	1	1	yes	characteristic
242	C₂×C₅	C₂×#835	1	1	yes	characteristic
243	C₂×C₅	#5225	1	1	yes	characteristic
244	C₂³	#6719	1	1	yes	characteristic
245	Q₈	—	4	4	no	—
246	Q₈	—	4	4	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	#6717	1	2	yes	—
252	C₂×C₄	#6718	1	2	yes	—
253	C₂×C₄	#6719	1	1	yes	characteristic
254	C₂×C₄	#6719	1	1	yes	characteristic
255	C₂×C₄	#6719	1	1	yes	characteristic
256	C₂×C₄	C₂²×Dic₁₅	1	1	yes	characteristic
257	C₂×C₄	C₂×#1220	1	1	yes	characteristic
258	C₂×C₄	C₂×#1220	1	1	yes	characteristic
259	C₈	—	60	2	no	—
260	C₈	—	60	2	no	—
261	C₂×C₃	—	2	2	no	—
262	C₂×C₃	C₂×#3965	1	1	yes	characteristic
263	C₂×C₃	C₂×#3967	1	1	yes	characteristic
264	C₂×C₃	#65514	1	1	yes	characteristic
265	C₅	#71774	1	1	yes	characteristic, complemented, Sylow
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	#91119	1	2	yes	—
269	C₂²	C₂×#6719	1	1	yes	characteristic
270	C₄	—	2	1	no	—
271	C₄	—	2	1	no	—
272	C₄	—	4	4	no	—
273	C₄	#91117	1	2	yes	—
274	C₄	#91119	1	2	yes	—
275	C₄	C₂×#6717	1	1	yes	characteristic
276	C₄	#91831	1	2	yes	—
277	C₄	C₂×#6718	1	1	yes	characteristic
278	C₃	#10602567	1	1	yes	characteristic, complemented, Sylow
279	C₂	—	2	2	no	—
280	C₂	C₂×#91117	1	1	yes	characteristic
281	C₂	C₂×#91119	1	1	yes	characteristic
282	C₂	#11751326	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6719
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂²×D₁₅
Fitting	C₂×C₃×C₅×C₄⋊Q₈	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂³×C₃×C₅	C₂²⋊C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₅
hypocenter	C₃×C₅	#1548
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂²×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
4. #11746367	C₂	semidirect
227. C₃×C₅	#1548	semidirect
265. C₅	#71774	semidirect
278. C₃	#10602567	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₅×C₄⋊Q₈	2
#11751026	2
#11751025	2
#11746367	2
#10602567	3
#71774	5