Subgroups — #14287168

Order 1920.

332 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	C₃×C₄×C₅×SD₁₆	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₃⋊#168	C₂	1	1	yes	maximal, characteristic, complemented
3	#11751113	C₂	1	1	yes	maximal, characteristic
4	#11746356	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄⋊(C₃⋊#168)	C₂	1	1	yes	maximal, characteristic
6	#11746279	C₂	1	1	yes	maximal, characteristic, complemented
7	#11746275	C₂	1	1	yes	maximal, characteristic, complemented
8	#10601691	—	3	1	no	maximal, Hall
9	C₂×C₃×C₅×SD₁₆	C₄	1	1	yes	characteristic
10	C₃×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
11	C₃×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
12	C₂×#6710	—	2	1	no	—
13	C₃×C₅×#106	C₄	1	1	yes	characteristic
14	C₃×C₅×#103	C₄	1	1	yes	characteristic
15	C₃×C₅×#102	C₄	1	1	yes	characteristic
16	C₃×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
17	#91114	—	2	1	no	—
18	C₄⋊(C₃⋊#168)	—	2	1	no	—
19	C₄⋊(C₃⋊#168)	C₂²	1	1	yes	characteristic
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	#70898	—	5	1	no	maximal, Hall
25	C₄×C₅×SD₁₆	D₃	1	1	yes	characteristic
26	Q₈×#168	—	3	1	no	—
27	#65301	—	3	1	no	—
28	Q₈⋊#168	—	3	1	no	—
29	D₄⋊#168	—	3	1	no	—
30	#64678	—	3	1	no	—
31	#64674	—	3	1	no	—
32	C₂×C₃×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃×C₅×D₄	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₃×C₄×C₅	—	2	1	no	—
35	C₃×C₅×SD₁₆	—	2	2	no	—
36	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
38	C₃×C₅×C₄⋊C₄	—	2	1	no	—
39	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
40	C₃×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
41	C₃×C₅×C₂²⋊C₄	—	2	1	no	—
42	C₃×C₄²×C₅	—	2	1	no	—
43	C₃×C₄²×C₅	C₂³	1	1	yes	characteristic
44	#6710	—	4	2	no	—
45	C₂×C₃⋊#168	—	2	1	no	—
46	C₂×C₃⋊#168	—	2	1	no	—
47	C₂×C₃⋊#168	—	2	1	no	—
48	C₂×C₃⋊#168	D₄	1	1	yes	characteristic
49	C₂×C₃⋊#168	D₄	1	1	yes	characteristic
50	C₃×C₄×SD₁₆	D₅	1	1	yes	characteristic
51	Q₈×C₃⋊C₈	—	5	1	no	—
52	#5012	—	5	1	no	—
53	#4535	—	5	1	no	—
54	D₄⋊(C₃⋊C₈)	—	5	1	no	—
55	#4458	—	5	1	no	—
56	#4454	—	5	1	no	—
57	C₂×C₅×SD₁₆	Dic₃	1	1	yes	characteristic
58	C₄×C₅×Q₈	C₂×D₃	1	1	yes	characteristic
59	C₄×C₅×D₄	C₂×D₃	1	1	yes	characteristic
60	C₂×#681	—	6	1	no	—
61	C₅×#106	Dic₃	1	1	yes	characteristic
62	C₅×#103	Dic₃	1	1	yes	characteristic
63	C₅×#102	Dic₃	1	1	yes	characteristic
64	C₄×C₅×C₈	C₂×D₃	1	1	yes	characteristic
65	C₂²⋊#168	—	6	1	no	—
66	C₄⋊#168	—	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	#1588	—	15	1	no	Sylow
73	C₂³×C₃×C₅	—	2	1	no	—
74	C₃×C₅×Q₈	—	2	1	no	—
75	C₃×C₅×Q₈	M₁₆	1	2	yes	complemented
76	C₃×C₅×D₄	—	2	1	no	—
77	C₃×C₅×D₄	—	2	1	no	—
78	C₂×C₃×C₄×C₅	—	2	1	no	—
79	C₂×C₃×C₄×C₅	—	2	1	no	—
80	C₂×C₃×C₄×C₅	—	2	1	no	—
81	C₂×C₃×C₄×C₅	—	4	1	no	—
82	C₂×C₃×C₄×C₅	—	4	1	no	—
83	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
86	C₃×C₅×C₈	—	2	1	no	—
87	C₃×C₅×C₈	—	2	1	no	—
88	C₃⋊#168	—	2	2	no	—
89	C₃⋊#168	—	2	1	no	—
90	C₃⋊#168	—	2	1	no	—
91	C₃⋊#168	—	4	1	no	—
92	C₃⋊#168	—	4	1	no	—
93	C₂×C₃×SD₁₆	Dic₅	1	1	yes	characteristic
94	C₃×C₄×Q₈	C₂×D₅	1	1	yes	characteristic
95	C₃×C₄×D₄	C₂×D₅	1	1	yes	characteristic
96	C₂×#208	—	10	1	no	—
97	C₃×#106	Dic₅	1	1	yes	characteristic
98	C₃×#103	Dic₅	1	1	yes	characteristic
99	C₃×#102	Dic₅	1	1	yes	characteristic
100	C₃×C₄×C₈	C₂×D₅	1	1	yes	characteristic
101	#830	—	10	1	no	—
102	C₄⋊(C₃⋊C₈)	—	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	C₂×Dic₃	1	1	yes	characteristic
109	C₂×C₅×D₄	C₂×Dic₃	1	1	yes	characteristic
110	C₂²×C₄×C₅	—	2	1	no	—
111	C₅×SD₁₆	—	2	2	no	—
112	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
113	C₂×C₅×C₈	C₂×Dic₃	1	1	yes	characteristic
114	C₅×C₄⋊C₄	—	2	1	no	—
115	C₅×C₄⋊C₄	C₂×Dic₃	1	1	yes	characteristic
116	C₅×C₄⋊C₄	C₂×Dic₃	1	1	yes	characteristic
117	C₅×C₂²⋊C₄	—	2	1	no	—
118	C₄²×C₅	—	2	1	no	—
119	C₄²×C₅	C₂²×D₃	1	1	yes	characteristic
120	#681	—	12	2	no	—
121	C₂×#168	—	3	1	no	—
122	C₂×#168	—	3	1	no	—
123	C₂×#168	—	6	1	no	—
124	C₂×#168	—	6	1	no	—
125	C₂×#168	—	6	1	no	—
126	C₈×Q₈	—	15	1	no	—
127	C₄×SD₁₆	D₁₅	1	1	yes	characteristic
128	#436	—	15	1	no	—
129	#335	—	15	1	no	—
130	Q₈⋊C₈	—	15	1	no	—
131	D₄⋊C₈	—	15	1	no	—
132	#322	—	15	1	no	—
133	C₂²×C₃×C₅	—	4	1	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	2	no	—
138	C₃×C₄×C₅	—	2	1	no	—
139	C₃×C₄×C₅	—	4	1	no	—
140	C₃×C₄×C₅	—	4	1	no	—
141	C₃×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
142	C₃×C₄×C₅	#131	1	1	yes	characteristic
143	C₃×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
144	C₃×C₄×C₅	#137	1	1	yes	characteristic
145	C₂×C₃×Q₈	C₂×Dic₅	1	1	yes	characteristic
146	C₂×C₃×D₄	C₂×Dic₅	1	1	yes	characteristic
147	C₂²×C₃×C₄	—	2	1	no	—
148	C₃×SD₁₆	—	2	2	no	—
149	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
150	C₂×C₃×C₈	C₂×Dic₅	1	1	yes	characteristic
151	C₃×C₄⋊C₄	—	2	1	no	—
152	C₃×C₄⋊C₄	C₂×Dic₅	1	1	yes	characteristic
153	C₃×C₄⋊C₄	C₂×Dic₅	1	1	yes	characteristic
154	C₃×C₂²⋊C₄	—	2	1	no	—
155	C₃×C₄²	—	2	1	no	—
156	C₃×C₄²	C₂²×D₅	1	1	yes	characteristic
157	#208	—	20	2	no	—
158	C₂×C₃⋊C₈	—	5	1	no	—
159	C₂×C₃⋊C₈	—	5	1	no	—
160	C₂×C₃⋊C₈	—	10	1	no	—
161	C₂×C₃⋊C₈	—	10	1	no	—
162	C₂×C₃⋊C₈	—	10	1	no	—
163	C₂³×C₅	—	2	1	no	—
164	C₅×Q₈	—	2	1	no	—
165	C₅×Q₈	#208	1	2	yes	complemented
166	C₅×D₄	—	2	1	no	—
167	C₅×D₄	—	2	1	no	—
168	C₂×C₄×C₅	—	2	1	no	—
169	C₂×C₄×C₅	—	2	1	no	—
170	C₂×C₄×C₅	—	2	1	no	—
171	C₂×C₄×C₅	—	4	1	no	—
172	C₂×C₄×C₅	—	4	1	no	—
173	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
174	C₂×C₄×C₅	#237	1	1	yes	characteristic
175	C₂×C₄×C₅	C₂²×Dic₃	1	1	yes	characteristic
176	C₅×C₈	—	2	1	no	—
177	C₅×C₈	—	2	1	no	—
178	#168	—	6	2	no	—
179	#168	—	6	1	no	—
180	#168	—	6	1	no	—
181	#168	—	12	1	no	—
182	#168	—	12	1	no	—
183	C₂×SD₁₆	Dic₁₅	1	1	yes	characteristic
184	C₂×M₁₆	—	30	1	no	—
185	C₄×Q₈	C₂×D₁₅	1	1	yes	characteristic
186	C₄×D₄	C₂×D₁₅	1	1	yes	characteristic
187	#106	Dic₁₅	1	1	yes	characteristic
188	C₄⋊C₈	—	15	1	no	—
189	C₄⋊C₈	—	15	1	no	—
190	C₄⋊C₈	—	30	1	no	—
191	#103	Dic₁₅	1	1	yes	characteristic
192	#102	Dic₁₅	1	1	yes	characteristic
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	—	8	1	no	—
199	C₂×C₃×C₅	#435	1	1	yes	characteristic
200	C₂×C₃×C₅	#440	1	1	yes	characteristic
201	C₂×C₃×C₅	#443	1	1	yes	characteristic
202	C₂³×C₃	—	2	1	no	—
203	C₃×Q₈	—	2	1	no	—
204	C₃×Q₈	#681	1	2	yes	complemented
205	C₃×D₄	—	2	1	no	—
206	C₃×D₄	—	2	1	no	—
207	C₂×C₃×C₄	—	2	1	no	—
208	C₂×C₃×C₄	—	2	1	no	—
209	C₂×C₃×C₄	—	2	1	no	—
210	C₂×C₃×C₄	—	4	1	no	—
211	C₂×C₃×C₄	—	4	1	no	—
212	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
213	C₂×C₃×C₄	#711	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
215	C₃×C₈	—	2	1	no	—
216	C₃×C₈	—	2	1	no	—
217	C₃⋊C₈	—	10	2	no	—
218	C₃⋊C₈	—	10	1	no	—
219	C₃⋊C₈	—	10	1	no	—
220	C₃⋊C₈	—	20	1	no	—
221	C₃⋊C₈	—	20	1	no	—
222	C₂²×C₅	—	4	1	no	—
223	C₂²×C₅	—	4	1	no	—
224	C₂²×C₅	—	4	1	no	—
225	C₂²×C₅	D₄×Dic₃	1	1	yes	characteristic
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	—	2	2	no	—
228	C₄×C₅	—	4	1	no	—
229	C₄×C₅	—	4	1	no	—
230	C₄×C₅	#914	1	1	yes	characteristic
231	C₄×C₅	#915	1	1	yes	characteristic
232	C₄×C₅	C₂×#208	1	1	yes	characteristic
233	C₄×C₅	#948	1	1	yes	characteristic
234	C₂×Q₈	C₂×Dic₁₅	1	1	yes	characteristic
235	C₂×D₄	C₂×Dic₁₅	1	1	yes	characteristic
236	C₂²×C₄	—	2	1	no	—
237	SD₁₆	—	2	2	no	—
238	M₁₆	—	60	2	no	—
239	C₂×C₈	—	15	1	no	—
240	C₂×C₈	—	15	1	no	—
241	C₂×C₈	—	30	1	no	—
242	C₂×C₈	—	30	1	no	—
243	C₂×C₈	—	30	1	no	—
244	C₂×C₈	C₂×Dic₁₅	1	1	yes	characteristic
245	C₂×C₈	C₂×Dic₁₅	1	1	yes	characteristic
246	C₄⋊C₄	—	2	1	no	—
247	C₄⋊C₄	C₂×Dic₁₅	1	1	yes	characteristic
248	C₄⋊C₄	C₂×Dic₁₅	1	1	yes	characteristic
249	C₂²⋊C₄	—	2	1	no	—
250	C₄²	—	2	1	no	—
251	C₄²	C₂²×D₁₅	1	1	yes	characteristic
252	C₃×C₅	#1588	1	1	yes	characteristic, complemented, Hall
253	C₂²×C₃	—	4	1	no	—
254	C₂²×C₃	—	4	1	no	—
255	C₂²×C₃	—	4	1	no	—
256	C₂²×C₃	D₄×Dic₅	1	1	yes	characteristic
257	C₃×C₄	—	2	2	no	—
258	C₃×C₄	—	2	1	no	—
259	C₃×C₄	—	4	1	no	—
260	C₃×C₄	—	4	1	no	—
261	C₃×C₄	#4060	1	1	yes	characteristic
262	C₃×C₄	#4061	1	1	yes	characteristic
263	C₃×C₄	C₂×#681	1	1	yes	characteristic
264	C₃×C₄	#4094	1	1	yes	characteristic
265	C₂×C₅	—	8	1	no	—
266	C₂×C₅	#5011	1	1	yes	characteristic
267	C₂×C₅	#5164	1	1	yes	characteristic
268	C₂×C₅	#5196	1	1	yes	characteristic
269	C₂³	—	2	1	no	—
270	Q₈	—	2	1	no	—
271	Q₈	#6710	1	2	yes	complemented
272	D₄	—	2	1	no	—
273	D₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	4	1	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	D₄×D₁₅	1	1	yes	characteristic
280	C₂×C₄	#6819	1	1	yes	characteristic
281	C₂×C₄	C₂²×Dic₁₅	1	1	yes	characteristic
282	C₈	—	2	1	no	—
283	C₈	—	2	1	no	—
284	C₈	—	30	2	no	—
285	C₈	—	30	1	no	—
286	C₈	—	30	1	no	—
287	C₈	—	60	1	no	—
288	C₈	—	60	1	no	—
289	C₂×C₃	—	8	1	no	—
290	C₂×C₃	#65300	1	1	yes	characteristic
291	C₂×C₃	#65453	1	1	yes	characteristic
292	C₂×C₃	#65485	1	1	yes	characteristic
293	C₅	#70898	1	1	yes	characteristic, complemented, Sylow
294	C₂²	—	4	1	no	—
295	C₂²	—	4	1	no	—
296	C₂²	—	4	1	no	—
297	C₂²	D₄×Dic₁₅	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	2	no	—
300	C₄	—	4	1	no	—
301	C₄	—	4	1	no	—
302	C₄	#91803	1	1	yes	characteristic
303	C₄	#91804	1	1	yes	characteristic
304	C₄	C₂×#6710	1	1	yes	characteristic
305	C₄	#91837	1	1	yes	characteristic
306	C₃	#10601691	1	1	yes	characteristic, complemented, Sylow
307	C₂	—	8	1	no	—
308	C₂	#11751112	1	1	yes	characteristic
309	C₂	#11751265	1	1	yes	characteristic
310	C₂	#11751297	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₁₅
commutator	C₂×C₃×C₄×C₅	C₂²×C₄
Frattini	C₄²	C₂²×D₁₅
Fitting	C₃×C₄×C₅×SD₁₆	C₂
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×C₃×C₅×D₄	C₂×C₄
cosocle	C₃×C₄²×C₅	C₂³
hypercenter	C₄×SD₁₆	D₁₅
hypocenter	C₃×C₅	#1588
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×SD₁₆

Complements

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

N	H ≅ G/N	type
2. Q₈×C₃⋊#168	C₂	semidirect
4. #11746356	C₂	semidirect
6. #11746279	C₂	semidirect
7. #11746275	C₂	semidirect
75. C₃×C₅×Q₈	M₁₆	semidirect
165. C₅×Q₈	#208	semidirect
204. C₃×Q₈	#681	semidirect
252. C₃×C₅	#1588	semidirect
271. Q₈	#6710	semidirect
293. C₅	#70898	semidirect
306. C₃	#10601691	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄×C₅×SD₁₆	2
Q₈×C₃⋊#168	2
#11751113	2
#11746356	2
D₄⋊(C₃⋊#168)	2
#11746279	2
#11746275	2
#10601691	3
#70898	5