Subgroups — #14286124

Order 1920.

360 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×#6717	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₃⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
3	#11750935	C₂	1	1	yes	maximal, characteristic
4	C₃×C₅×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#11746301	C₂	1	1	yes	maximal, characteristic, complemented
6	#11746280	C₂	1	1	yes	maximal, characteristic
7	C₈×C₃⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
8	#10599005	—	3	1	no	maximal, Hall
9	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
10	C₂×#6717	C₄	1	1	yes	characteristic
11	C₂×C₈×D₁₅	—	2	1	no	—
12	C₄×D₆₀	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	#91119	C₄	1	1	yes	characteristic
16	#91104	—	2	1	no	—
17	#91093	C₄	1	1	yes	characteristic
18	#91092	C₄	1	1	yes	characteristic
19	C₄⋊(C₃⋊#168)	—	2	1	no	—
20	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
21	C₄×C₃⋊#168	—	2	1	no	—
22	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
23	C₄×C₃⋊#168	C₂²	1	1	yes	characteristic
24	#68212	—	5	1	no	maximal, Hall
25	C₄×#688	—	3	1	no	—
26	Q₈×#168	—	3	1	no	—
27	#65123	—	3	1	no	—
28	C₅×Q₈⋊C₈	D₃	1	1	yes	characteristic, complemented
29	#64700	—	3	1	no	—
30	#64679	—	3	1	no	—
31	C₈×#168	—	3	1	no	—
32	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
33	C₂×D₆₀	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×D₁₅	—	2	1	no	—
35	C₂×C₃×C₅×C₈	—	2	1	no	—
36	C₂×C₃×C₅×C₈	—	2	1	no	—
37	C₃×C₅×C₄⋊C₄	—	2	1	no	—
38	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
39	C₃×C₄²×C₅	—	2	1	no	—
40	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
41	#6717	C₈	1	4	yes	complemented
42	#6714	—	2	1	no	—
43	C₄⋊Dic₁₅	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃⋊#168	—	2	1	no	—
45	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
47	C₂×C₃⋊#168	D₄	1	1	yes	characteristic
48	C₂×C₃⋊#168	D₄	1	1	yes	characteristic
49	C₈×D₁₅	—	4	2	no	—
50	C₄×#215	—	5	1	no	—
51	Q₈×C₃⋊C₈	—	5	1	no	—
52	#4834	—	5	1	no	—
53	C₃×Q₈⋊C₈	D₅	1	1	yes	characteristic, complemented
54	#4480	—	5	1	no	—
55	#4459	—	5	1	no	—
56	C₈×C₃⋊C₈	—	5	1	no	—
57	C₄×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
58	C₂×#688	—	3	1	no	—
59	C₂×C₈×D₅	—	6	1	no	—
60	C₄×D₂₀	—	3	1	no	—
61	C₅×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
62	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
63	#3967	—	3	1	no	—
64	#3952	—	6	1	no	—
65	#3941	—	3	1	no	—
66	#3940	—	3	1	no	—
67	C₄⋊#168	—	3	1	no	—
68	C₄⋊#168	—	6	1	no	—
69	C₄×#168	—	3	1	no	—
70	C₄×#168	—	3	1	no	—
71	C₄×#168	—	6	1	no	—
72	C₈×SD₁₆	—	15	1	no	Sylow
73	C₂²×D₁₅	—	2	1	no	—
74	C₃×C₅×Q₈	—	2	1	no	—
75	C₃×C₅×Q₈	C₂×C₈	1	2	yes	complemented
76	C₂×C₃×C₄×C₅	—	2	1	no	—
77	C₂×C₃×C₄×C₅	—	2	1	no	—
78	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
81	C₂×Dic₁₅	—	2	1	no	—
82	D₆₀	—	2	1	no	—
83	D₆₀	C₂×C₈	1	2	yes	—
84	C₄×D₁₅	—	4	1	no	—
85	C₄×D₁₅	—	4	1	no	—
86	C₃×C₅×C₈	—	2	2	no	—
87	C₃×C₅×C₈	—	4	1	no	—
88	C₃⋊#168	—	2	1	no	—
89	C₃⋊#168	—	2	1	no	—
90	C₃⋊#168	—	4	1	no	—
91	C₃⋊#168	C₂×C₈	1	2	yes	complemented
92	C₃⋊#168	SD₁₆	1	2	yes	complemented
93	C₃×C₄×Q₈	C₂×D₅	1	1	yes	characteristic
94	C₂×#215	—	5	1	no	—
95	C₂×D₃×C₈	—	10	1	no	—
96	C₄×D₁₂	—	5	1	no	—
97	C₃×C₄⋊C₈	C₂×D₅	1	1	yes	characteristic
98	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
99	#835	—	5	1	no	—
100	(C₂×D₃)⋊C₈	—	10	1	no	—
101	#809	—	5	1	no	—
102	#808	—	5	1	no	—
103	C₄⋊(C₃⋊C₈)	—	5	1	no	—
104	C₄⋊(C₃⋊C₈)	—	10	1	no	—
105	C₄×C₃⋊C₈	—	5	1	no	—
106	C₄×C₃⋊C₈	—	5	1	no	—
107	C₄×C₃⋊C₈	—	10	1	no	—
108	C₂×C₅×Q₈	C₄×D₃	1	1	yes	characteristic
109	C₂×D₂₀	—	3	1	no	—
110	C₂×C₄×D₅	—	6	1	no	—
111	C₂×C₅×C₈	—	2	1	no	—
112	C₂×C₅×C₈	—	2	1	no	—
113	C₅×C₄⋊C₄	—	2	1	no	—
114	C₅×C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
115	C₄²×C₅	—	2	1	no	—
116	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
117	#688	—	3	4	no	—
118	#685	—	6	1	no	—
119	C₄⋊Dic₅	—	3	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	C₈×D₅	—	12	2	no	—
126	C₈×Q₈	—	15	1	no	—
127	C₄×SD₁₆	—	15	1	no	—
128	C₈×D₄	—	15	1	no	—
129	#334	—	15	1	no	—
130	Q₈⋊C₈	D₁₅	1	1	yes	characteristic, complemented
131	D₄⋊C₈	—	15	1	no	—
132	C₈²	—	15	1	no	—
133	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
134	C₂×D₁₅	—	2	2	no	—
135	C₂×D₁₅	—	4	1	no	—
136	C₂×D₁₅	—	4	1	no	—
137	C₃×C₄×C₅	—	2	2	no	—
138	C₃×C₄×C₅	—	2	1	no	—
139	C₃×C₄×C₅	—	4	1	no	—
140	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
141	C₃×C₄×C₅	#131	1	1	yes	characteristic
142	C₃×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
143	C₃×C₄×C₅	#135	1	1	yes	characteristic
144	Dic₁₅	—	4	1	no	—
145	C₂×C₃×Q₈	C₄×D₅	1	1	yes	characteristic
146	C₂×D₁₂	—	5	1	no	—
147	C₂×C₄×D₃	—	10	1	no	—
148	C₂×C₃×C₈	—	2	1	no	—
149	C₂×C₃×C₈	—	2	1	no	—
150	C₃×C₄⋊C₄	—	2	1	no	—
151	C₃×C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
152	C₃×C₄²	—	2	1	no	—
153	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
154	#215	—	5	4	no	—
155	(C₂×D₃)⋊C₄	—	10	1	no	—
156	C₄⋊Dic₃	—	5	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	D₃×C₈	—	20	2	no	—
163	C₂²×D₅	—	6	1	no	—
164	C₅×Q₈	—	2	1	no	—
165	C₅×Q₈	D₃×C₈	1	2	yes	complemented
166	C₂×C₄×C₅	—	2	1	no	—
167	C₂×C₄×C₅	—	2	1	no	—
168	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
169	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
170	C₂×C₄×C₅	#237	1	1	yes	characteristic
171	C₂×Dic₅	—	6	1	no	—
172	D₂₀	—	3	2	no	—
173	D₂₀	—	6	1	no	—
174	C₄×D₅	—	12	1	no	—
175	C₄×D₅	—	12	1	no	—
176	C₅×C₈	—	2	2	no	—
177	C₅×C₈	—	4	1	no	—
178	#168	—	3	2	no	—
179	#168	—	3	2	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	C₂×D₁₅	1	1	yes	characteristic
186	C₄×D₄	—	15	1	no	—
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	C₈×D₄	1	1	yes	characteristic
199	C₂×C₃×C₅	C₄×SD₁₆	1	1	yes	characteristic
200	C₂×C₃×C₅	#443	1	1	yes	characteristic
201	D₁₅	—	4	2	no	—
202	C₂²×D₃	—	10	1	no	—
203	C₃×Q₈	—	2	1	no	—
204	C₃×Q₈	C₈×D₅	1	2	yes	complemented
205	C₂×C₃×C₄	—	2	1	no	—
206	C₂×C₃×C₄	—	2	1	no	—
207	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
208	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
209	C₂×C₃×C₄	#711	1	1	yes	characteristic
210	C₂×Dic₃	—	10	1	no	—
211	D₁₂	—	5	2	no	—
212	D₁₂	—	10	1	no	—
213	C₄×D₃	—	20	1	no	—
214	C₄×D₃	—	20	1	no	—
215	C₃×C₈	—	2	2	no	—
216	C₃×C₈	—	4	1	no	—
217	C₃⋊C₈	—	5	2	no	—
218	C₃⋊C₈	—	5	2	no	—
219	C₃⋊C₈	—	10	1	no	—
220	C₃⋊C₈	—	10	1	no	—
221	C₃⋊C₈	—	20	1	no	—
222	C₂²×C₅	#67⋊C₄	1	1	yes	characteristic
223	C₂×D₅	—	6	2	no	—
224	C₂×D₅	—	12	1	no	—
225	C₂×D₅	—	12	1	no	—
226	C₄×C₅	—	2	2	no	—
227	C₄×C₅	—	2	1	no	—
228	C₄×C₅	—	4	1	no	—
229	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
230	C₄×C₅	#907	1	1	yes	characteristic
231	C₄×C₅	D₃×SD₁₆	1	1	yes	characteristic
232	C₄×C₅	#919	1	1	yes	characteristic
233	Dic₅	—	12	1	no	—
234	C₂×Q₈	C₄×D₁₅	1	1	yes	characteristic
235	C₂×D₄	—	15	1	no	—
236	C₂²×C₄	—	30	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	—	2	1	no	—
247	C₄⋊C₄	—	15	1	no	—
248	C₄⋊C₄	C₄×D₁₅	1	1	yes	characteristic
249	C₂²⋊C₄	—	30	1	no	—
250	C₄²	—	2	1	no	—
251	C₄²	C₂²×D₁₅	1	1	yes	characteristic
252	C₃×C₅	C₈×SD₁₆	1	1	yes	characteristic, complemented, Hall
253	C₂²×C₃	#4027	1	1	yes	characteristic
254	C₂×D₃	—	10	2	no	—
255	C₂×D₃	—	20	1	no	—
256	C₂×D₃	—	20	1	no	—
257	C₃×C₄	—	2	1	no	—
258	C₃×C₄	—	2	2	no	—
259	C₃×C₄	—	4	1	no	—
260	C₃×C₄	C₂×C₈×D₅	1	1	yes	characteristic
261	C₃×C₄	#4053	1	1	yes	characteristic
262	C₃×C₄	D₅×SD₁₆	1	1	yes	characteristic
263	C₃×C₄	#4065	1	1	yes	characteristic
264	Dic₃	—	20	1	no	—
265	C₂×C₅	#67⋊C₈	1	1	yes	characteristic
266	C₂×C₅	#4788	1	1	yes	characteristic
267	C₂×C₅	#4824	1	1	yes	characteristic
268	D₅	—	12	2	no	—
269	C₂³	—	30	1	no	—
270	Q₈	—	2	1	no	—
271	Q₈	C₈×D₁₅	1	2	yes	complemented
272	D₄	—	15	2	no	—
273	D₄	—	30	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	30	1	no	—
277	C₂×C₄	—	60	1	no	—
278	C₂×C₄	—	60	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	—	2	2	no	—
283	C₈	—	4	1	no	—
284	C₈	—	15	2	no	—
285	C₈	—	15	2	no	—
286	C₈	—	30	1	no	—
287	C₈	—	30	1	no	—
288	C₈	—	60	1	no	—
289	C₂×C₃	#65014	1	1	yes	characteristic
290	C₂×C₃	#65077	1	1	yes	characteristic
291	C₂×C₃	#65113	1	1	yes	characteristic
292	D₃	—	20	2	no	—
293	C₅	#68212	1	1	yes	characteristic, complemented, Sylow
294	C₂²	—	30	2	no	—
295	C₂²	—	60	1	no	—
296	C₂²	—	60	1	no	—
297	C₂²	#91770	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	2	no	—
300	C₄	—	4	1	no	—
301	C₄	—	60	1	no	—
302	C₄	C₂×C₈×D₁₅	1	1	yes	characteristic
303	C₄	#91796	1	1	yes	characteristic
304	C₄	SD₁₆×D₁₅	1	1	yes	characteristic
305	C₄	#91808	1	1	yes	characteristic
306	C₃	#10599005	1	1	yes	characteristic, complemented, Sylow
307	C₂	—	60	2	no	—
308	C₂	#11750826	1	1	yes	characteristic
309	C₂	#11750889	1	1	yes	characteristic
310	C₂	#11750925	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×Q₈⋊C₈	C₂
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×D₆₀	C₂×C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	Q₈⋊C₈	D₁₅
hypocenter	C₃×C₅	C₈×SD₁₆
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ Q₈⋊C₈

Complements

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

N	H ≅ G/N	type
2. Q₈×C₃⋊#168	C₂	semidirect
4. C₃×C₅×Q₈⋊C₈	C₂	semidirect
5. #11746301	C₂	semidirect
7. C₈×C₃⋊#168	C₂	semidirect
28. C₅×Q₈⋊C₈	D₃	semidirect
41. #6717	C₈	semidirect
53. C₃×Q₈⋊C₈	D₅	semidirect
75. C₃×C₅×Q₈	C₂×C₈	semidirect
91. C₃⋊#168	C₂×C₈	semidirect
92. C₃⋊#168	SD₁₆	semidirect
130. Q₈⋊C₈	D₁₅	semidirect
165. C₅×Q₈	D₃×C₈	semidirect
204. C₃×Q₈	C₈×D₅	semidirect
252. C₃×C₅	C₈×SD₁₆	semidirect
271. Q₈	C₈×D₁₅	semidirect
293. C₅	#68212	semidirect
306. C₃	#10599005	semidirect

Maximal subgroups

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

subgroup	index
C₄×#6717	2
Q₈×C₃⋊#168	2
#11750935	2
C₃×C₅×Q₈⋊C₈	2
#11746301	2
#11746280	2
C₈×C₃⋊#168	2
#10599005	3
#68212	5