Subgroups — #14315113

Order 1920.

328 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#3998	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₃×#674	C₂	1	1	yes	maximal, characteristic, complemented
3	#67×#168	C₂	1	1	yes	maximal, characteristic
4	#11746782	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₃⋊#674	C₂	1	1	yes	maximal, characteristic, complemented
6	(C₂×D₃)⋊#674	C₂	1	1	yes	maximal, characteristic, complemented
7	#11746542	C₂	1	1	yes	maximal, characteristic, complemented
8	#10604413	—	3	1	no	maximal, Hall
9	C₄×C₅×#67	C₄	1	1	yes	characteristic
10	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
11	C₂×D₃×#168	C₂²	1	1	yes	characteristic
12	C₂×C₃⋊#674	C₂²	1	1	yes	characteristic
13	C₂×C₃⋊#674	C₂²	1	1	yes	characteristic
14	C₃×#3998	—	2	2	no	—
15	C₂×C₃×#674	C₂²	1	1	yes	characteristic
16	C₂×C₃×#674	C₂²	1	1	yes	characteristic
17	D₃×#674	—	2	2	no	—
18	#91114	C₄	1	1	yes	characteristic
19	Dic₃⋊#168	C₄	1	1	yes	characteristic
20	(C₂×D₃)⋊#168	C₄	1	1	yes	characteristic
21	Dic₃×#168	C₂²	1	1	yes	characteristic
22	#71433	—	5	1	no	maximal, Hall
23	C₂×#3998	D₃	1	1	yes	characteristic, complemented
24	C₂²×#674	—	3	1	no	—
25	D₄×#168	—	3	1	no	—
26	C₂²⋊#674	—	3	1	no	—
27	C₂²⋊#674	—	3	1	no	—
28	#64859	—	3	1	no	—
29	C₄⋊#674	—	3	1	no	—
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
31	C₂×C₅×#67	C₈	1	1	yes	characteristic
32	C₂×C₄×C₅×D₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
34	C₂×C₃⋊#168	C₂×C₄	1	1	yes	characteristic
35	C₅×#217	C₈	1	1	yes	characteristic
36	C₅×(C₂×D₃)⋊C₄	C₈	1	1	yes	characteristic
37	C₅×Dic₃⋊C₄	C₈	1	1	yes	characteristic
38	C₄×C₅×Dic₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×#168	—	2	1	no	—
40	C₂×C₃×#168	—	2	1	no	—
41	C₂×C₃×#168	C₂×C₄	1	1	yes	characteristic
42	C₂×C₃×#168	C₂³	1	1	yes	characteristic
43	D₃×#168	—	2	1	no	—
44	D₃×#168	—	2	1	no	—
45	C₃⋊#674	—	2	1	no	—
46	C₃⋊#674	—	2	1	no	—
47	C₃×#674	—	2	1	no	—
48	C₃×#674	D₄	1	2	yes	complemented
49	C₂×C₃×M₃₂	—	5	1	no	—
50	C₈×#67	—	5	1	no	—
51	C₂×D₃×C₁₆	—	5	1	no	—
52	#4549	—	5	1	no	—
53	(C₂×D₃)⋊C₁₆	—	5	1	no	—
54	#4502	—	5	1	no	—
55	Dic₃⋊C₁₆	—	5	1	no	—
56	C₄×C₅×D₄	—	3	1	no	—
57	C₂²×#168	—	3	1	no	—
58	C₂²×#168	C₂×D₃	1	1	yes	characteristic
59	#3998	—	2	2	no	—
60	C₂×#674	—	3	1	no	—
61	C₂×#674	—	3	1	no	—
62	C₂×#674	—	6	2	no	—
63	C₂×#674	C₂×D₃	1	1	yes	characteristic
64	C₂×#674	C₂×D₃	1	1	yes	characteristic
65	C₂²⋊#168	—	3	1	no	—
66	C₂²⋊#168	—	3	1	no	—
67	C₄⋊#168	—	3	1	no	—
68	C₄×#168	—	3	1	no	—
69	#2168	—	15	1	no	Sylow
70	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
71	C₂²×C₅×D₃	C₂×C₈	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	—	2	1	no	—
73	C₂×C₃×C₄×C₅	—	2	1	no	—
74	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₅×#67	—	2	2	no	—
77	C₂×C₅×Dic₃	C₂×C₈	1	1	yes	characteristic
78	C₂×C₅×Dic₃	C₂×C₈	1	1	yes	characteristic
79	C₂×C₅×Dic₃	C₂×C₈	1	1	yes	characteristic
80	C₄×C₅×D₃	—	2	1	no	—
81	C₄×C₅×D₃	—	2	1	no	—
82	C₃⋊#168	—	2	1	no	—
83	C₃⋊#168	—	2	1	no	—
84	C₃×#168	—	2	1	no	—
85	C₃×#168	C₂×D₄	1	1	yes	characteristic
86	C₃×#168	#41	1	1	yes	characteristic
87	C₂²×C₃×C₈	—	5	1	no	—
88	C₄×#67	AGL(1,𝔽₅)	1	1	yes	characteristic
89	C₂×D₃×C₈	—	5	1	no	—
90	C₃×M₃₂	—	10	2	no	—
91	C₂×C₃×C₁₆	—	5	1	no	—
92	C₂×C₃×C₁₆	—	5	1	no	—
93	#830	—	5	1	no	—
94	(C₂×D₃)⋊C₈	—	5	1	no	—
95	Dic₃⋊C₈	—	5	1	no	—
96	C₈×Dic₃	—	5	1	no	—
97	C₂×C₃⋊C₁₆	—	5	1	no	—
98	C₂×C₃⋊C₁₆	—	5	1	no	—
99	D₃×C₁₆	—	10	2	no	—
100	C₂×C₅×D₄	—	3	1	no	—
101	C₂²×C₄×C₅	—	3	1	no	—
102	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
103	C₅×C₄⋊C₄	—	3	1	no	—
104	C₅×C₂²⋊C₄	—	3	1	no	—
105	C₅×C₂²⋊C₄	—	3	1	no	—
106	C₄²×C₅	—	3	1	no	—
107	C₂×#168	—	2	1	no	—
108	C₂×#168	—	2	1	no	—
109	C₂×#168	—	3	1	no	—
110	C₂×#168	—	3	1	no	—
111	C₂×#168	—	6	1	no	—
112	C₂×#168	—	6	1	no	—
113	C₂×#168	C₄×D₃	1	1	yes	characteristic
114	C₂×#168	C₂²×D₃	1	1	yes	characteristic
115	#674	—	2	1	no	—
116	#674	—	6	1	no	—
117	#674	—	6	1	no	—
118	#674	#67	1	2	yes	complemented
119	C₂×M₃₂	—	5	1	no	—
120	C₂²×C₁₆	—	15	1	no	—
121	C₈×D₄	—	15	1	no	—
122	C₄⋊C₁₆	—	15	1	no	—
123	C₂²⋊C₁₆	—	15	1	no	—
124	C₂²⋊C₁₆	—	15	1	no	—
125	#346	—	15	1	no	—
126	C₂²×C₃×C₅	—	2	1	no	—
127	C₂²×C₃×C₅	—	2	1	no	—
128	C₂²×C₃×C₅	—	2	1	no	—
129	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₅×D₃	—	2	1	no	—
131	C₂×C₅×D₃	—	2	1	no	—
132	C₂×C₅×D₃	M₃₂	1	2	yes	complemented
133	C₃×C₄×C₅	—	2	1	no	—
134	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
135	C₃×C₄×C₅	#131	1	1	yes	characteristic
136	C₅×Dic₃	—	2	1	no	—
137	C₅×Dic₃	—	2	1	no	—
138	C₅×Dic₃	—	2	1	no	—
139	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
140	C₂×#67	#170	1	1	yes	characteristic
141	C₂×C₄×D₃	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
142	C₂×C₃×C₈	—	5	1	no	—
143	C₂×C₃×C₈	—	5	1	no	—
144	C₂×C₃×C₈	—	10	1	no	—
145	C₂×C₃×C₈	—	10	1	no	—
146	#217	#170	1	1	yes	characteristic
147	(C₂×D₃)⋊C₄	#170	1	1	yes	characteristic
148	Dic₃⋊C₄	#170	1	1	yes	characteristic
149	C₄×Dic₃	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
150	C₂×C₃⋊C₈	—	5	1	no	—
151	C₂×C₃⋊C₈	—	5	1	no	—
152	D₃×C₈	—	10	1	no	—
153	D₃×C₈	—	10	1	no	—
154	C₃×C₁₆	—	5	2	no	—
155	C₃×C₁₆	—	10	1	no	—
156	C₃⋊C₁₆	—	10	1	no	—
157	C₃⋊C₁₆	—	10	1	no	—
158	C₂³×C₅	—	3	1	no	—
159	C₂³×C₅	D₃×C₈	1	1	yes	characteristic
160	C₅×D₄	—	6	2	no	—
161	C₂×C₄×C₅	—	2	1	no	—
162	C₂×C₄×C₅	—	2	1	no	—
163	C₂×C₄×C₅	—	3	1	no	—
164	C₂×C₄×C₅	—	3	1	no	—
165	C₂×C₄×C₅	—	3	1	no	—
166	C₂×C₄×C₅	—	6	1	no	—
167	C₂×C₄×C₅	—	6	1	no	—
168	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
169	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
170	#168	—	2	1	no	—
171	#168	—	6	1	no	—
172	#168	—	6	1	no	—
173	#168	#235	1	1	yes	characteristic
174	#168	C₂×#67	1	1	yes	characteristic
175	C₂²×C₈	—	5	1	no	—
176	C₂²×C₈	—	15	1	no	—
177	C₄×D₄	—	3	1	no	—
178	M₃₂	—	10	2	no	—
179	C₂×C₁₆	—	5	1	no	—
180	C₂×C₁₆	—	5	1	no	—
181	C₂×C₁₆	—	15	1	no	—
182	C₂×C₁₆	—	15	1	no	—
183	C₂×C₁₆	—	30	2	no	—
184	C₄⋊C₈	—	15	1	no	—
185	C₂²⋊C₈	—	15	1	no	—
186	C₂²⋊C₈	—	15	1	no	—
187	C₄×C₈	—	15	1	no	—
188	C₂×C₃×C₅	—	4	1	no	—
189	C₂×C₃×C₅	C₈×D₄	1	1	yes	characteristic
190	C₂×C₃×C₅	C₂×M₃₂	1	1	yes	characteristic
191	C₂×C₃×C₅	#504	1	1	yes	characteristic
192	C₅×D₃	—	2	2	no	—
193	C₂³×C₃	C₂×#170	1	1	yes	characteristic
194	C₂²×D₃	C₂×#170	1	1	yes	characteristic
195	C₂×C₃×C₄	—	2	1	no	—
196	C₂×C₃×C₄	—	2	1	no	—
197	C₂×C₃×C₄	C₂×#170	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
199	#67	—	2	2	no	—
200	C₂×Dic₃	C₂×#170	1	1	yes	characteristic
201	C₂×Dic₃	C₂×#170	1	1	yes	characteristic
202	C₂×Dic₃	C₂×#170	1	1	yes	characteristic
203	C₄×D₃	—	2	1	no	—
204	C₄×D₃	—	2	1	no	—
205	C₃×C₈	—	5	1	no	—
206	C₃×C₈	—	5	1	no	—
207	C₃×C₈	—	10	1	no	—
208	C₃⋊C₈	—	10	1	no	—
209	C₃⋊C₈	—	10	1	no	—
210	C₂²×C₅	—	2	1	no	—
211	C₂²×C₅	—	2	1	no	—
212	C₂²×C₅	—	2	1	no	—
213	C₂²×C₅	—	3	2	no	—
214	C₂²×C₅	—	6	1	no	—
215	C₂²×C₅	—	6	1	no	—
216	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
217	C₄×C₅	—	2	1	no	—
218	C₄×C₅	—	6	1	no	—
219	C₄×C₅	—	6	1	no	—
220	C₄×C₅	—	6	1	no	—
221	C₄×C₅	#901	1	1	yes	characteristic
222	C₄×C₅	C₄×#67	1	1	yes	characteristic
223	C₂×D₄	—	3	1	no	—
224	C₂²×C₄	—	3	1	no	—
225	C₂²×C₄	D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
226	C₂×C₈	—	5	1	no	—
227	C₂×C₈	—	5	1	no	—
228	C₂×C₈	—	10	1	no	—
229	C₂×C₈	—	10	1	no	—
230	C₂×C₈	—	15	1	no	—
231	C₂×C₈	—	15	1	no	—
232	C₂×C₈	—	30	1	no	—
233	C₂×C₈	—	30	1	no	—
234	C₄⋊C₄	—	3	1	no	—
235	C₂²⋊C₄	—	3	1	no	—
236	C₂²⋊C₄	—	3	1	no	—
237	C₄²	—	3	1	no	—
238	C₁₆	—	5	2	no	—
239	C₁₆	—	10	1	no	—
240	C₁₆	—	30	1	no	—
241	C₁₆	—	30	1	no	—
242	C₃×C₅	#2168	1	1	yes	characteristic, complemented, Hall
243	C₂²×C₃	—	2	1	no	—
244	C₂²×C₃	—	2	1	no	—
245	C₂²×C₃	—	2	1	no	—
246	C₂²×C₃	C₂²×#170	1	1	yes	characteristic
247	C₂×D₃	—	2	1	no	—
248	C₂×D₃	—	2	1	no	—
249	C₂×D₃	#3998	1	2	yes	complemented
250	C₃×C₄	—	2	1	no	—
251	C₃×C₄	#4131	1	1	yes	characteristic
252	C₃×C₄	D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
253	Dic₃	—	2	1	no	—
254	Dic₃	—	2	1	no	—
255	Dic₃	—	2	1	no	—
256	C₂×C₅	—	4	1	no	—
257	C₂×C₅	—	6	2	no	—
258	C₂×C₅	D₃×M₃₂	1	1	yes	characteristic
259	C₂×C₅	#4907	1	1	yes	characteristic
260	C₂×C₅	C₈×#67	1	1	yes	characteristic
261	C₂³	—	3	1	no	—
262	C₂³	D₃×#170	1	1	yes	characteristic
263	D₄	—	6	2	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	3	1	no	—
267	C₂×C₄	—	3	1	no	—
268	C₂×C₄	—	3	1	no	—
269	C₂×C₄	—	6	1	no	—
270	C₂×C₄	—	6	1	no	—
271	C₂×C₄	D₃×#170	1	1	yes	characteristic
272	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)	1	1	yes	characteristic
273	C₈	—	5	1	no	—
274	C₈	—	5	1	no	—
275	C₈	—	10	1	no	—
276	C₈	—	30	1	no	—
277	C₈	—	30	1	no	—
278	C₂×C₃	—	4	1	no	—
279	C₂×C₃	C₂×#3998	1	1	yes	characteristic
280	C₂×C₃	D₄×#170	1	1	yes	characteristic
281	C₂×C₃	#65791	1	1	yes	characteristic
282	D₃	—	2	2	no	—
283	C₅	#71433	1	1	yes	characteristic, complemented, Sylow
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	3	2	no	—
288	C₂²	—	6	1	no	—
289	C₂²	—	6	1	no	—
290	C₂²	C₂×D₃×#170	1	1	yes	characteristic
291	C₄	—	2	1	no	—
292	C₄	—	6	1	no	—
293	C₄	—	6	1	no	—
294	C₄	—	6	1	no	—
295	C₄	#91927	1	1	yes	characteristic
296	C₄	AGL(1,𝔽₅)×#67	1	1	yes	characteristic
297	C₃	#10604413	1	1	yes	characteristic, complemented, Sylow
298	C₂	—	4	1	no	—
299	C₂	—	6	2	no	—
300	C₂	D₃×#3998	1	1	yes	characteristic
301	C₂	#11751738	1	1	yes	characteristic
302	C₂	#67×#170	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×D₃×AGL(1,𝔽₅)
commutator, socle	C₂²×C₃×C₅	C₂²×C₈
Fitting	C₂²×C₃×C₄×C₅	C₂×C₄
prime radical	C₂×C₅×#67	C₈
cosocle	C₂×C₃×#168	C₂³
hypercenter	C₂²×C₄	D₃×AGL(1,𝔽₅)
hypocenter	C₃×C₅	#2168
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×#3998	C₂	semidirect
2. C₂×D₃×#674	C₂	semidirect
4. #11746782	C₂	semidirect
5. Dic₃⋊#674	C₂	semidirect
6. (C₂×D₃)⋊#674	C₂	semidirect
7. #11746542	C₂	semidirect
23. C₂×#3998	D₃	semidirect
48. C₃×#674	D₄	semidirect
118. #674	#67	semidirect
132. C₂×C₅×D₃	M₃₂	semidirect
242. C₃×C₅	#2168	semidirect
249. C₂×D₃	#3998	semidirect
283. C₅	#71433	semidirect
297. C₃	#10604413	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#3998	2
C₂×D₃×#674	2
#67×#168	2
#11746782	2
Dic₃⋊#674	2
(C₂×D₃)⋊#674	2
#11746542	2
#10604413	3
#71433	5