Subgroups — #14286278

Order 1920.

352 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#6716	C₂	1	1	yes	maximal, characteristic
2	D₄×C₃⋊#168	C₂	1	1	yes	maximal, characteristic
3	#11750931	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₅×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#11746300	C₂	1	1	yes	maximal, characteristic, complemented
6	#11746277	C₂	1	1	yes	maximal, characteristic, complemented
7	#11746275	C₂	1	1	yes	maximal, characteristic, complemented
8	#10598943	—	3	1	no	maximal, Hall
9	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
10	C₂×#6716	C₄	1	1	yes	characteristic
11	C₂²×C₃⋊#168	—	2	1	no	—
12	C₄×Dic₃₀	C₂²	1	1	yes	characteristic
13	C₃×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
15	#91116	C₄	1	1	yes	characteristic
16	#91114	—	2	1	no	—
17	#91098	—	2	1	no	—
18	C₈×Dic₁₅	—	2	1	no	—
19	#91094	C₄	1	1	yes	characteristic
20	#91092	C₄	1	1	yes	characteristic
21	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
22	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
23	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
24	#68150	—	5	1	no	maximal, Hall
25	C₄×#687	—	3	1	no	—
26	D₄×#168	—	3	1	no	—
27	#65119	—	3	1	no	—
28	C₅×D₄⋊C₈	D₃	1	1	yes	characteristic
29	#64699	—	3	1	no	—
30	#64676	—	3	1	no	—
31	#64674	—	3	1	no	—
32	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₃×C₄×C₅	—	2	1	no	—
34	C₂×Dic₃₀	C₂×C₄	1	1	yes	characteristic
35	C₂×C₃×C₅×C₈	—	2	1	no	—
36	C₂×C₃×C₅×C₈	—	2	1	no	—
37	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
38	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
39	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
40	#6716	C₈	1	4	yes	complemented
41	C₄⋊Dic₁₅	C₂×C₄	1	1	yes	characteristic
42	#6712	—	2	1	no	—
43	C₄×Dic₁₅	—	2	1	no	—
44	C₂×C₃⋊#168	—	2	1	no	—
45	C₂×C₃⋊#168	—	4	2	no	—
46	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
47	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
48	C₂×C₃⋊#168	D₄	1	1	yes	characteristic
49	C₂×C₃⋊#168	D₄	1	1	yes	characteristic
50	C₄×#214	—	5	1	no	—
51	D₄×C₃⋊C₈	—	5	1	no	—
52	#4830	—	5	1	no	—
53	C₃×D₄⋊C₈	D₅	1	1	yes	characteristic
54	#4479	—	5	1	no	—
55	#4456	—	5	1	no	—
56	#4454	—	5	1	no	—
57	C₄×C₅×D₄	C₂×D₃	1	1	yes	characteristic
58	C₂×#687	—	3	1	no	—
59	C₂²×#168	—	6	1	no	—
60	C₄×Dic₁₀	—	3	1	no	—
61	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
62	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
63	#3964	—	3	1	no	—
64	C₂²⋊#168	—	6	1	no	—
65	#3946	—	6	1	no	—
66	C₈×Dic₅	—	6	1	no	—
67	#3942	—	3	1	no	—
68	#3940	—	3	1	no	—
69	C₄⋊#168	—	3	1	no	—
70	C₄×#168	—	3	1	no	—
71	C₄×#168	—	3	1	no	—
72	SD₁₆⋊C₈	—	15	1	no	Sylow
73	C₂³×C₃×C₅	—	2	1	no	—
74	C₃×C₅×D₄	—	2	1	no	—
75	C₃×C₅×D₄	C₂×C₈	1	2	yes	—
76	C₂×C₃×C₄×C₅	—	2	1	no	—
77	C₂×C₃×C₄×C₅	—	4	1	no	—
78	C₂×C₃×C₄×C₅	—	4	1	no	—
79	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
82	C₂×Dic₁₅	—	2	1	no	—
83	C₂×Dic₁₅	—	2	1	no	—
84	Dic₃₀	—	2	1	no	—
85	Dic₃₀	C₂×C₈	1	2	yes	complemented
86	C₃×C₅×C₈	—	4	1	no	—
87	C₃×C₅×C₈	—	4	1	no	—
88	C₃⋊#168	—	2	1	no	—
89	C₃⋊#168	—	2	1	no	—
90	C₃⋊#168	—	2	1	no	—
91	C₃⋊#168	—	4	1	no	—
92	C₃⋊#168	C₂×C₈	1	2	yes	complemented
93	C₃×C₄×D₄	C₂×D₅	1	1	yes	characteristic
94	C₂×#214	—	5	1	no	—
95	C₂²×C₃⋊C₈	—	10	1	no	—
96	C₄×Dic₆	—	5	1	no	—
97	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
98	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
99	#832	—	5	1	no	—
100	#830	—	10	1	no	—
101	Dic₃⋊C₈	—	10	1	no	—
102	C₈×Dic₃	—	10	1	no	—
103	#810	—	5	1	no	—
104	#808	—	5	1	no	—
105	C₄⋊(C₃⋊C₈)	—	5	1	no	—
106	C₄×C₃⋊C₈	—	5	1	no	—
107	C₄×C₃⋊C₈	—	5	1	no	—
108	C₂×C₅×D₄	C₄×D₃	1	1	yes	characteristic
109	C₂²×C₄×C₅	—	2	1	no	—
110	C₂×Dic₁₀	—	3	1	no	—
111	C₂×C₅×C₈	—	2	1	no	—
112	C₂×C₅×C₈	—	2	1	no	—
113	C₅×C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
114	C₅×C₂²⋊C₄	—	2	1	no	—
115	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
116	#687	—	3	4	no	—
117	C₄⋊Dic₅	—	3	1	no	—
118	#683	—	6	1	no	—
119	C₄×Dic₅	—	6	1	no	—
120	C₂×#168	—	3	1	no	—
121	C₂×#168	—	3	1	no	—
122	C₂×#168	—	3	1	no	—
123	C₂×#168	—	3	1	no	—
124	C₂×#168	—	6	1	no	—
125	C₂×#168	—	12	2	no	—
126	C₈×Q₈	—	15	1	no	—
127	C₄×SD₁₆	—	15	1	no	—
128	C₈×D₄	—	15	1	no	—
129	#335	—	15	1	no	—
130	Q₈⋊C₈	—	15	1	no	—
131	D₄⋊C₈	D₁₅	1	1	yes	characteristic
132	#322	—	15	1	no	—
133	C₂²×C₃×C₅	—	2	2	no	—
134	C₂²×C₃×C₅	—	4	1	no	—
135	C₂²×C₃×C₅	—	4	1	no	—
136	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
137	C₃×C₄×C₅	—	2	1	no	—
138	C₃×C₄×C₅	—	4	1	no	—
139	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
140	C₃×C₄×C₅	#131	1	1	yes	characteristic
141	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₃×C₄×C₅	#137	1	1	yes	characteristic
143	Dic₁₅	—	2	2	no	—
144	Dic₁₅	—	4	1	no	—
145	C₂×C₃×D₄	C₄×D₅	1	1	yes	characteristic
146	C₂²×C₃×C₄	—	2	1	no	—
147	C₂×Dic₆	—	5	1	no	—
148	C₂×C₃×C₈	—	2	1	no	—
149	C₂×C₃×C₈	—	2	1	no	—
150	C₃×C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
151	C₃×C₂²⋊C₄	—	2	1	no	—
152	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
153	#214	—	5	4	no	—
154	C₄⋊Dic₃	—	5	1	no	—
155	Dic₃⋊C₄	—	10	1	no	—
156	C₄×Dic₃	—	10	1	no	—
157	C₂×C₃⋊C₈	—	5	1	no	—
158	C₂×C₃⋊C₈	—	5	1	no	—
159	C₂×C₃⋊C₈	—	5	1	no	—
160	C₂×C₃⋊C₈	—	5	1	no	—
161	C₂×C₃⋊C₈	—	10	1	no	—
162	C₂×C₃⋊C₈	—	20	2	no	—
163	C₂³×C₅	—	2	1	no	—
164	C₅×D₄	—	2	1	no	—
165	C₅×D₄	D₃×C₈	1	2	yes	—
166	C₂×C₄×C₅	—	2	1	no	—
167	C₂×C₄×C₅	—	4	1	no	—
168	C₂×C₄×C₅	—	4	1	no	—
169	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
170	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
171	C₂×C₄×C₅	#237	1	1	yes	characteristic
172	C₂×Dic₅	—	6	1	no	—
173	C₂×Dic₅	—	6	1	no	—
174	Dic₁₀	—	3	2	no	—
175	Dic₁₀	—	6	1	no	—
176	C₅×C₈	—	4	1	no	—
177	C₅×C₈	—	4	1	no	—
178	#168	—	3	2	no	—
179	#168	—	6	1	no	—
180	#168	—	6	1	no	—
181	#168	—	6	1	no	—
182	#168	—	12	1	no	—
183	C₂×SD₁₆	—	15	1	no	—
184	C₂²×C₈	—	30	1	no	—
185	C₄×Q₈	—	15	1	no	—
186	C₄×D₄	C₂×D₁₅	1	1	yes	characteristic
187	#106	—	15	1	no	—
188	C₄⋊C₈	—	15	1	no	—
189	C₄⋊C₈	—	30	1	no	—
190	C₄⋊C₈	C₂×D₁₅	1	1	yes	characteristic
191	#103	—	15	1	no	—
192	#102	—	15	1	no	—
193	C₂²⋊C₈	—	30	1	no	—
194	C₄×C₈	—	15	1	no	—
195	C₄×C₈	—	15	1	no	—
196	C₄×C₈	—	30	1	no	—
197	C₄×C₈	C₂×D₁₅	1	1	yes	characteristic
198	C₂×C₃×C₅	—	4	2	no	—
199	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
200	C₂×C₃×C₅	#440	1	1	yes	characteristic
201	C₂×C₃×C₅	#444	1	1	yes	characteristic
202	C₂³×C₃	—	2	1	no	—
203	C₃×D₄	—	2	1	no	—
204	C₃×D₄	C₈×D₅	1	2	yes	—
205	C₂×C₃×C₄	—	2	1	no	—
206	C₂×C₃×C₄	—	4	1	no	—
207	C₂×C₃×C₄	—	4	1	no	—
208	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
209	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
210	C₂×C₃×C₄	#711	1	1	yes	characteristic
211	C₂×Dic₃	—	10	1	no	—
212	C₂×Dic₃	—	10	1	no	—
213	Dic₆	—	5	2	no	—
214	Dic₆	—	10	1	no	—
215	C₃×C₈	—	4	1	no	—
216	C₃×C₈	—	4	1	no	—
217	C₃⋊C₈	—	5	2	no	—
218	C₃⋊C₈	—	10	1	no	—
219	C₃⋊C₈	—	10	1	no	—
220	C₃⋊C₈	—	10	1	no	—
221	C₃⋊C₈	—	20	1	no	—
222	C₂²×C₅	—	2	2	no	—
223	C₂²×C₅	—	4	1	no	—
224	C₂²×C₅	—	4	1	no	—
225	C₂²×C₅	#67⋊C₄	1	1	yes	characteristic
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	—	4	1	no	—
228	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
229	C₄×C₅	#907	1	1	yes	characteristic
230	C₄×C₅	#911	1	1	yes	characteristic
231	C₄×C₅	#915	1	1	yes	characteristic
232	Dic₅	—	6	2	no	—
233	Dic₅	—	12	1	no	—
234	C₂×Q₈	—	15	1	no	—
235	C₂×D₄	C₄×D₁₅	1	1	yes	characteristic
236	C₂²×C₄	—	2	1	no	—
237	SD₁₆	—	15	4	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	15	1	no	—
241	C₂×C₈	—	15	1	no	—
242	C₂×C₈	—	15	1	no	—
243	C₂×C₈	—	15	1	no	—
244	C₂×C₈	—	30	1	no	—
245	C₂×C₈	—	60	2	no	—
246	C₄⋊C₄	—	15	1	no	—
247	C₄⋊C₄	—	30	1	no	—
248	C₄⋊C₄	C₄×D₁₅	1	1	yes	characteristic
249	C₂²⋊C₄	—	2	1	no	—
250	C₄²	—	30	1	no	—
251	C₄²	C₂²×D₁₅	1	1	yes	characteristic
252	C₃×C₅	SD₁₆⋊C₈	1	1	yes	characteristic, complemented, Hall
253	C₂²×C₃	—	2	2	no	—
254	C₂²×C₃	—	4	1	no	—
255	C₂²×C₃	—	4	1	no	—
256	C₂²×C₃	#4027	1	1	yes	characteristic
257	C₃×C₄	—	2	1	no	—
258	C₃×C₄	—	4	1	no	—
259	C₃×C₄	C₂×C₈×D₅	1	1	yes	characteristic
260	C₃×C₄	#4053	1	1	yes	characteristic
261	C₃×C₄	#4057	1	1	yes	characteristic
262	C₃×C₄	#4061	1	1	yes	characteristic
263	Dic₃	—	10	2	no	—
264	Dic₃	—	20	1	no	—
265	C₂×C₅	—	4	2	no	—
266	C₂×C₅	#67⋊C₈	1	1	yes	characteristic
267	C₂×C₅	#4757	1	1	yes	characteristic
268	C₂×C₅	#4823	1	1	yes	characteristic
269	C₂³	—	2	1	no	—
270	Q₈	—	15	2	no	—
271	Q₈	—	30	1	no	—
272	D₄	—	2	1	no	—
273	D₄	C₈×D₁₅	1	2	yes	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	4	1	no	—
276	C₂×C₄	—	4	1	no	—
277	C₂×C₄	—	30	1	no	—
278	C₂×C₄	—	30	1	no	—
279	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
280	C₂×C₄	D₄×D₁₅	1	1	yes	characteristic
281	C₂×C₄	#6819	1	1	yes	characteristic
282	C₈	—	4	1	no	—
283	C₈	—	4	1	no	—
284	C₈	—	15	2	no	—
285	C₈	—	30	1	no	—
286	C₈	—	30	1	no	—
287	C₈	—	30	1	no	—
288	C₈	—	60	1	no	—
289	C₂×C₃	—	4	2	no	—
290	C₂×C₃	#65014	1	1	yes	characteristic
291	C₂×C₃	#65046	1	1	yes	characteristic
292	C₂×C₃	#65112	1	1	yes	characteristic
293	C₅	#68150	1	1	yes	characteristic, complemented, Sylow
294	C₂²	—	2	2	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	#91770	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	4	1	no	—
300	C₄	—	30	2	no	—
301	C₄	—	60	1	no	—
302	C₄	C₂×C₈×D₁₅	1	1	yes	characteristic
303	C₄	#91796	1	1	yes	characteristic
304	C₄	#91800	1	1	yes	characteristic
305	C₄	#91804	1	1	yes	characteristic
306	C₃	#10598943	1	1	yes	characteristic, complemented, Sylow
307	C₂	—	4	2	no	—
308	C₂	#11750826	1	1	yes	characteristic
309	C₂	#11750858	1	1	yes	characteristic
310	C₂	#11750924	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₅
commutator	C₃×C₄×C₅	C₂²×C₈
Frattini	C₄²	C₂²×D₁₅
Fitting	C₃×C₅×D₄⋊C₈	C₂
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×C₃×C₅×D₄	C₂×C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	D₄⋊C₈	D₁₅
hypocenter	C₃×C₅	SD₁₆⋊C₈
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ D₄⋊C₈

Complements

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

N	H ≅ G/N	type
3. #11750931	C₂	semidirect
5. #11746300	C₂	semidirect
6. #11746277	C₂	semidirect
7. #11746275	C₂	semidirect
40. #6716	C₈	semidirect
85. Dic₃₀	C₂×C₈	semidirect
92. C₃⋊#168	C₂×C₈	semidirect
252. C₃×C₅	SD₁₆⋊C₈	semidirect
293. C₅	#68150	semidirect
306. C₃	#10598943	semidirect

Maximal subgroups

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

subgroup	index
C₄×#6716	2
D₄×C₃⋊#168	2
#11750931	2
C₃×C₅×D₄⋊C₈	2
#11746300	2
#11746277	2
#11746275	2
#10598943	3
#68150	5