Subgroups — C₃×#10602577 #14301487

Order 1920.

648 conjugacy classes of subgroups.

The subgroup lattice has 300 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	C₃×#66146	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#3968	C₂	1	1	yes	maximal, characteristic
4	C₃×#65514	C₂	1	2	yes	maximal
5	C₃×#64743	C₂	1	2	yes	maximal
6	#10602577	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
9	C₃×C₅×C₄⋊Q₈	—	2	2	no	—
10	C₃×C₅×C₄⋊Q₈	C₄	1	4	yes	complemented
11	C₂×C₃×C₅×C₄⋊C₄	—	2	2	no	—
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₃×Q₈×Dic₅	—	2	4	no	—
14	C₃×#4072	—	2	2	no	—
15	C₃×#4072	C₂²	1	2	yes	—
16	C₂×C₃×#681	C₂²	1	2	yes	—
17	C₃×#3968	—	2	2	no	—
18	C₃×#3967	—	2	4	no	—
19	C₃×#1886	—	5	1	no	maximal, Hall
20	C₂×C₅×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
21	#66146	C₂×C₃	1	1	yes	characteristic
22	C₂×#3968	C₂×C₃	1	1	yes	characteristic
23	#65514	C₂×C₃	1	2	yes	—
24	#64743	C₂×C₃	1	2	yes	—
25	C₂×C₃×C₅×Q₈	—	2	2	no	—
26	C₂×C₃×C₅×Q₈	—	2	1	no	—
27	C₂×C₃×C₅×Q₈	—	2	2	no	—
28	C₂×C₃×C₅×Q₈	—	2	2	no	—
29	C₂×C₃×C₅×Q₈	—	2	2	no	—
30	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₅×Q₈	C₂×C₄	1	2	yes	—
32	C₂×C₃×C₅×Q₈	D₄	1	2	yes	—
33	C₂²×C₃×C₄×C₅	—	2	1	no	—
34	C₂²×C₃×C₄×C₅	—	2	1	no	—
35	C₂²×C₃×C₄×C₅	D₄	1	2	yes	—
36	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
37	C₃×C₅×C₄⋊C₄	—	2	8	no	—
38	C₃×C₄²×C₅	—	2	1	no	—
39	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
40	C₃×C₂²⋊Dic₅	—	2	2	no	—
41	C₃×C₄⋊Dic₅	—	2	4	no	—
42	C₃×C₄⋊Dic₅	D₄	1	4	yes	—
43	C₃×C₄×Dic₅	—	2	4	no	—
44	C₃×C₄×Dic₅	—	2	2	no	—
45	C₃×#681	—	2	2	no	—
46	C₃×#681	—	2	2	no	—
47	C₂×C₃×#168	—	2	2	no	—
48	C₂×C₃×C₄⋊Q₈	D₅	1	1	yes	characteristic
49	C₃×#519	—	5	1	no	—
50	C₃×#419	—	5	2	no	—
51	C₂×C₃×#101	—	5	1	no	—
52	C₃×#339	—	5	2	no	—
53	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
54	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
55	C₅×C₄⋊Q₈	—	2	2	no	—
56	C₅×C₄⋊Q₈	C₃×C₄	1	4	yes	complemented
57	C₂×C₅×C₄⋊C₄	—	2	2	no	—
58	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
59	Q₈×Dic₅	—	2	4	no	—
60	#4072	—	2	2	no	—
61	#4072	C₂²×C₃	1	2	yes	—
62	C₂×#681	C₂²×C₃	1	2	yes	—
63	#3968	—	2	2	no	—
64	#3967	—	2	4	no	—
65	#1886	—	5	1	no	Sylow
66	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
67	C₃×C₅×Q₈	—	2	4	no	—
68	C₃×C₅×Q₈	—	2	4	no	—
69	C₃×C₅×Q₈	—	2	4	no	—
70	C₃×C₅×Q₈	—	4	2	no	—
71	C₂×C₃×C₄×C₅	—	2	4	no	—
72	C₂×C₃×C₄×C₅	—	2	2	no	—
73	C₂×C₃×C₄×C₅	—	2	2	no	—
74	C₂×C₃×C₄×C₅	—	2	2	no	—
75	C₂×C₃×C₄×C₅	—	2	2	no	—
76	C₂×C₃×C₄×C₅	—	4	1	no	—
77	C₂×C₃×C₄×C₅	—	4	1	no	—
78	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	4	yes	—
79	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
81	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃×C₄×C₅	#41	1	2	yes	—
83	C₂×C₃×Dic₅	—	2	4	no	—
84	C₃×#168	—	4	2	no	—
85	C₂²×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
86	C₂²×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
87	C₂×C₃×M₁₆	—	5	2	no	—
88	C₃×C₄⋊Q₈	—	2	2	no	—
89	C₃×C₄⋊Q₈	Dic₅	1	4	yes	complemented
90	C₃×C₄×Q₈	—	10	4	no	—
91	C₃×#117	—	5	2	no	—
92	C₃×#117	—	10	2	no	—
93	C₂×C₃×C₄⋊C₄	—	2	2	no	—
94	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
95	C₃×#103	—	10	4	no	—
96	C₃×#101	—	10	2	no	—
97	C₂×C₅×Q₈	—	2	2	no	—
98	C₂×C₅×Q₈	—	2	1	no	—
99	C₂×C₅×Q₈	—	2	2	no	—
100	C₂×C₅×Q₈	—	2	2	no	—
101	C₂×C₅×Q₈	—	2	2	no	—
102	C₂×C₅×Q₈	C₂×C₃×C₄	1	2	yes	—
103	C₂×C₅×Q₈	C₂×C₃×C₄	1	2	yes	—
104	C₂×C₅×Q₈	C₃×D₄	1	2	yes	—
105	C₂²×C₄×C₅	—	2	1	no	—
106	C₂²×C₄×C₅	—	2	1	no	—
107	C₂²×C₄×C₅	C₃×D₄	1	2	yes	—
108	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
109	C₅×C₄⋊C₄	—	2	8	no	—
110	C₄²×C₅	—	2	1	no	—
111	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
112	C₂²⋊Dic₅	—	2	2	no	—
113	C₄⋊Dic₅	—	2	4	no	—
114	C₄⋊Dic₅	C₃×D₄	1	4	yes	—
115	C₄×Dic₅	—	2	4	no	—
116	C₄×Dic₅	—	2	2	no	—
117	#681	—	2	2	no	—
118	#681	—	2	2	no	—
119	C₂×#168	—	2	2	no	—
120	C₂×C₄⋊Q₈	C₃×D₅	1	1	yes	characteristic
121	#519	—	5	1	no	—
122	#419	—	5	2	no	—
123	C₂×#101	—	5	1	no	—
124	#339	—	5	2	no	—
125	C₂²×C₃×C₅	—	2	2	no	—
126	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂²×C₃×C₅	#120	1	1	yes	characteristic
128	C₂²×C₃×C₅	#123	1	1	yes	characteristic
129	C₃×C₄×C₅	—	2	4	no	—
130	C₃×C₄×C₅	—	2	4	no	—
131	C₃×C₄×C₅	—	4	2	no	—
132	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
133	C₃×C₄×C₅	#121	1	2	yes	—
134	C₃×Dic₅	—	4	4	no	—
135	C₂×C₃×Q₈	—	2	2	no	—
136	C₂×C₃×Q₈	—	2	2	no	—
137	C₂×C₃×Q₈	—	2	1	no	—
138	C₂×C₃×Q₈	—	2	2	no	—
139	C₂×C₃×Q₈	—	2	2	no	—
140	C₂×C₃×Q₈	C₂×Dic₅	1	2	yes	—
141	C₂×C₃×Q₈	C₂×Dic₅	1	2	yes	—
142	C₂×C₃×Q₈	C₂²⋊D₅	1	2	yes	—
143	C₂²×C₃×C₄	—	2	1	no	—
144	C₂²×C₃×C₄	—	2	1	no	—
145	C₂²×C₃×C₄	C₂²⋊D₅	1	2	yes	—
146	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
147	C₃×M₁₆	—	10	2	no	—
148	C₃×M₁₆	—	10	2	no	—
149	C₂×C₃×C₈	—	10	2	no	—
150	C₃×C₄⋊C₄	—	2	8	no	—
151	C₃×C₄⋊C₄	—	5	4	no	—
152	C₃×C₄⋊C₄	—	10	4	no	—
153	C₃×C₂²⋊C₄	—	10	2	no	—
154	C₃×C₄²	—	2	1	no	—
155	C₃×C₄²	—	10	4	no	—
156	C₃×C₄²	—	10	2	no	—
157	C₃×C₄²	C₂×Dic₅	1	2	yes	—
158	C₂³×C₅	C₂×C₃×D₄	1	1	yes	characteristic
159	C₅×Q₈	—	2	4	no	—
160	C₅×Q₈	—	2	4	no	—
161	C₅×Q₈	—	2	4	no	—
162	C₅×Q₈	—	4	2	no	—
163	C₂×C₄×C₅	—	2	4	no	—
164	C₂×C₄×C₅	—	2	2	no	—
165	C₂×C₄×C₅	—	2	2	no	—
166	C₂×C₄×C₅	—	2	2	no	—
167	C₂×C₄×C₅	—	2	2	no	—
168	C₂×C₄×C₅	—	4	1	no	—
169	C₂×C₄×C₅	—	4	1	no	—
170	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	4	yes	—
171	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
172	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
173	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
174	C₂×C₄×C₅	C₃×#41	1	2	yes	—
175	C₂×Dic₅	—	2	4	no	—
176	#168	—	4	2	no	—
177	C₂²×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
178	C₂²×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
179	C₂×M₁₆	—	5	2	no	—
180	C₄⋊Q₈	—	2	2	no	—
181	C₄⋊Q₈	C₃×Dic₅	1	4	yes	complemented
182	C₄×Q₈	—	10	4	no	—
183	#117	—	5	2	no	—
184	#117	—	10	2	no	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
187	#103	—	10	4	no	—
188	#101	—	10	2	no	—
189	C₂×C₃×C₅	—	2	1	no	—
190	C₂×C₃×C₅	—	2	1	no	—
191	C₂×C₃×C₅	#386	1	1	yes	characteristic
192	C₂×C₃×C₅	#456	1	2	yes	—
193	C₂³×C₃	C₂×C₂²⋊D₅	1	1	yes	characteristic
194	C₃×Q₈	—	2	4	no	—
195	C₃×Q₈	—	2	4	no	—
196	C₃×Q₈	—	2	4	no	—
197	C₃×Q₈	—	4	2	no	—
198	C₂×C₃×C₄	—	2	4	no	—
199	C₂×C₃×C₄	—	2	2	no	—
200	C₂×C₃×C₄	—	2	2	no	—
201	C₂×C₃×C₄	—	2	2	no	—
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	—	4	1	no	—
204	C₂×C₃×C₄	—	4	1	no	—
205	C₂×C₃×C₄	—	10	4	no	—
206	C₂×C₃×C₄	C₂²⋊Dic₅	1	4	yes	—
207	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
208	C₂×C₃×C₄	#711	1	2	yes	—
209	C₂×C₃×C₄	C₂²×Dic₅	1	1	yes	characteristic
210	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
211	C₃×C₈	—	20	2	no	—
212	C₂²×C₅	—	2	2	no	—
213	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
214	C₂²×C₅	C₃×#120	1	1	yes	characteristic
215	C₂²×C₅	C₃×#123	1	1	yes	characteristic
216	C₄×C₅	—	2	4	no	—
217	C₄×C₅	—	2	4	no	—
218	C₄×C₅	—	4	2	no	—
219	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
220	C₄×C₅	C₃×#121	1	2	yes	—
221	Dic₅	—	4	4	no	—
222	C₂×Q₈	—	2	2	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	2	1	no	—
225	C₂×Q₈	—	2	2	no	—
226	C₂×Q₈	—	2	2	no	—
227	C₂×Q₈	C₂×C₃×Dic₅	1	2	yes	—
228	C₂×Q₈	C₂×C₃×Dic₅	1	2	yes	—
229	C₂×Q₈	C₃×C₂²⋊D₅	1	2	yes	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	C₃×C₂²⋊D₅	1	2	yes	—
233	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
234	M₁₆	—	10	2	no	—
235	M₁₆	—	10	2	no	—
236	C₂×C₈	—	10	2	no	—
237	C₄⋊C₄	—	2	8	no	—
238	C₄⋊C₄	—	5	4	no	—
239	C₄⋊C₄	—	10	4	no	—
240	C₂²⋊C₄	—	10	2	no	—
241	C₄²	—	2	1	no	—
242	C₄²	—	10	4	no	—
243	C₄²	—	10	2	no	—
244	C₄²	C₂×C₃×Dic₅	1	2	yes	—
245	C₃×C₅	#1886	1	1	yes	characteristic, complemented, Hall
246	C₂²×C₃	—	2	2	no	—
247	C₂²×C₃	#4081	1	1	yes	characteristic
248	C₂²×C₃	#4083	1	1	yes	characteristic
249	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
250	C₃×C₄	—	2	4	no	—
251	C₃×C₄	—	2	4	no	—
252	C₃×C₄	—	4	2	no	—
253	C₃×C₄	—	20	4	no	—
254	C₃×C₄	D₄×Dic₅	1	2	yes	—
255	C₃×C₄	#4085	1	2	yes	—
256	C₂×C₅	—	2	1	no	—
257	C₂×C₅	—	2	1	no	—
258	C₂×C₅	C₃×#386	1	1	yes	characteristic
259	C₂×C₅	C₃×#456	1	2	yes	—
260	C₂³	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
261	Q₈	—	2	4	no	—
262	Q₈	—	2	4	no	—
263	Q₈	—	2	4	no	—
264	Q₈	—	4	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	10	4	no	—
273	C₂×C₄	C₃×C₂²⋊Dic₅	1	4	yes	—
274	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
275	C₂×C₄	C₃×#711	1	2	yes	—
276	C₂×C₄	C₂²×C₃×Dic₅	1	1	yes	characteristic
277	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	1	yes	characteristic
278	C₈	—	20	2	no	—
279	C₂×C₃	—	2	1	no	—
280	C₂×C₃	—	2	1	no	—
281	C₂×C₃	#65384	1	2	yes	—
282	C₂×C₃	#65509	1	1	yes	characteristic
283	C₅	C₃×#1886	1	1	yes	characteristic, complemented, Sylow
284	C₂²	—	2	2	no	—
285	C₂²	C₃×#4081	1	1	yes	characteristic
286	C₂²	C₃×#4083	1	1	yes	characteristic
287	C₂²	C₂×C₃×C₂²⋊Dic₅	1	1	yes	characteristic
288	C₄	—	2	4	no	—
289	C₄	—	2	4	no	—
290	C₄	—	4	2	no	—
291	C₄	—	20	4	no	—
292	C₄	C₃×D₄×Dic₅	1	2	yes	—
293	C₄	C₃×#4085	1	2	yes	—
294	C₃	#10602577	1	1	yes	characteristic, direct product factor, Sylow
295	C₂	—	2	1	no	—
296	C₂	—	2	1	no	—
297	C₂	C₃×#65384	1	2	yes	—
298	C₂	C₃×#65509	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	#4083
commutator	C₂×C₄×C₅	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₂×C₃×C₅×C₄⋊Q₈	C₂
socle	C₂²×C₃×C₅	#120
prime radical	C₂³×C₃×C₅	C₂×D₄
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₂×C₃×C₄⋊Q₈	D₅
hypocenter	C₅	C₃×#1886
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
6. #10602577	C₃	direct (N × H)
10. C₃×C₅×C₄⋊Q₈	C₄	semidirect
56. C₅×C₄⋊Q₈	C₃×C₄	semidirect
89. C₃×C₄⋊Q₈	Dic₅	semidirect
181. C₄⋊Q₈	C₃×Dic₅	semidirect
245. C₃×C₅	#1886	semidirect
283. C₅	C₃×#1886	semidirect
294. C₃	#10602577	direct (N × H)

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
C₃×#66146	2
C₂×C₃×#3968	2
C₃×#65514	2
C₃×#64743	2
#10602577	3
C₃×#1886	5

Subgroups — C3×#10602577 #14301487

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10602577 #14301487