Subgroups — C₃×C₄×#3968 #14295460

Order 1920.

592 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×#3968	C₂	1	2	yes	maximal
3	C₃×C₄×#681	C₂	1	2	yes	maximal
4	C₃×#64750	C₂	1	2	yes	maximal
5	C₄×#3968	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
6	C₂²×C₃×C₅×Q₈	C₂²	1	1	yes	characteristic
7	C₃×C₄×C₅×Q₈	—	2	2	no	—
8	C₃×C₄×C₅×Q₈	—	2	2	no	—
9	C₂×C₃×C₅×C₄⋊C₄	C₄	1	2	yes	—
10	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄²×C₅	C₄	1	2	yes	—
12	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
13	C₂×C₃×#681	C₂²	1	4	yes	—
14	C₃×#3968	C₄	1	8	yes	direct product factor
15	C₃×#3935	—	2	2	no	—
16	C₃×C₄×#168	—	2	2	no	—
17	C₃×C₄×#101	—	5	1	no	maximal, Hall
18	C₂×C₄×C₅×Q₈	C₂×C₃	1	1	yes	characteristic
19	C₂×#3968	C₂×C₃	1	2	yes	—
20	C₄×#681	C₂×C₃	1	2	yes	—
21	#64750	C₂×C₃	1	2	yes	—
22	C₂×C₃×C₅×Q₈	—	2	2	no	—
23	C₂×C₃×C₅×Q₈	—	2	2	no	—
24	C₂×C₃×C₅×Q₈	C₂×C₄	1	4	yes	—
25	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
26	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
28	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
30	C₃×C₅×C₄⋊C₄	—	2	4	no	—
31	C₃×C₅×C₄⋊C₄	—	2	2	no	—
32	C₃×C₄²×C₅	—	2	2	no	—
33	C₃×C₄²×C₅	—	2	2	no	—
34	C₃×C₄²×C₅	D₄	1	2	yes	—
35	C₃×C₄²×C₅	D₄	1	2	yes	—
36	C₃×#681	—	2	4	no	—
37	C₃×#681	C₂×C₄	1	8	yes	—
38	C₂×C₃×#168	—	2	4	no	—
39	C₂×C₃×C₄×Q₈	D₅	1	1	yes	characteristic
40	C₂×C₃×#101	—	5	2	no	—
41	C₃×C₄×M₁₆	—	5	2	no	—
42	C₃×#343	—	5	2	no	—
43	C₂²×C₅×Q₈	C₂²×C₃	1	1	yes	characteristic
44	C₄×C₅×Q₈	—	2	2	no	—
45	C₄×C₅×Q₈	—	2	2	no	—
46	C₂×C₅×C₄⋊C₄	C₃×C₄	1	2	yes	—
47	C₂×C₅×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
48	C₂×C₄²×C₅	C₃×C₄	1	2	yes	—
49	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
50	C₂×#681	C₂²×C₃	1	4	yes	—
51	#3968	C₃×C₄	1	8	yes	direct product factor
52	#3935	—	2	2	no	—
53	C₄×#168	—	2	2	no	—
54	C₄×#101	—	5	1	no	Sylow
55	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
56	C₃×C₅×Q₈	—	2	8	no	—
57	C₂×C₃×C₄×C₅	—	2	2	no	—
58	C₂×C₃×C₄×C₅	—	2	2	no	—
59	C₂×C₃×C₄×C₅	—	2	2	no	—
60	C₂×C₃×C₄×C₅	—	2	2	no	—
61	C₂×C₃×C₄×C₅	—	4	1	no	—
62	C₂×C₃×C₄×C₅	—	4	2	no	—
63	C₂×C₃×C₄×C₅	C₄²	1	4	yes	—
64	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	2	yes	—
66	C₂×C₃×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
68	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
69	C₂×C₃×C₄×C₅	#41	1	2	yes	—
70	C₃×#168	—	2	8	no	—
71	C₂²×C₃×Q₈	C₂×D₅	1	1	yes	characteristic
72	C₂×C₃×M₁₆	—	5	4	no	—
73	C₃×C₄×Q₈	—	2	2	no	—
74	C₃×C₄×Q₈	—	2	2	no	—
75	C₂×C₃×C₄⋊C₄	Dic₅	1	2	yes	—
76	C₂×C₃×C₄⋊C₄	C₂×D₅	1	1	yes	characteristic
77	C₂×C₃×C₄²	Dic₅	1	2	yes	—
78	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
79	C₃×#101	—	5	8	no	—
80	C₃×#97	—	10	2	no	—
81	C₃×C₄×C₈	—	10	2	no	—
82	C₂×C₅×Q₈	—	2	2	no	—
83	C₂×C₅×Q₈	—	2	2	no	—
84	C₂×C₅×Q₈	C₂×C₃×C₄	1	4	yes	—
85	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
86	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
88	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
90	C₅×C₄⋊C₄	—	2	4	no	—
91	C₅×C₄⋊C₄	—	2	2	no	—
92	C₄²×C₅	—	2	2	no	—
93	C₄²×C₅	—	2	2	no	—
94	C₄²×C₅	C₃×D₄	1	2	yes	—
95	C₄²×C₅	C₃×D₄	1	2	yes	—
96	#681	—	2	4	no	—
97	#681	C₂×C₃×C₄	1	8	yes	—
98	C₂×#168	—	2	4	no	—
99	C₂×C₄×Q₈	C₃×D₅	1	1	yes	characteristic
100	C₂×#101	—	5	2	no	—
101	C₄×M₁₆	—	5	2	no	—
102	#343	—	5	2	no	—
103	C₂²×C₃×C₅	—	2	1	no	—
104	C₂²×C₃×C₅	—	2	1	no	—
105	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
106	C₂²×C₃×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
107	C₂²×C₃×C₅	#117	1	1	yes	characteristic
108	C₃×C₄×C₅	—	2	4	no	—
109	C₃×C₄×C₅	—	2	1	no	—
110	C₃×C₄×C₅	—	2	2	no	—
111	C₃×C₄×C₅	—	4	2	no	—
112	C₃×C₄×C₅	#101	1	2	yes	complemented
113	C₃×C₄×C₅	C₄×D₄	1	4	yes	—
114	C₂×C₃×Q₈	—	2	2	no	—
115	C₂×C₃×Q₈	—	2	2	no	—
116	C₂×C₃×Q₈	C₄×D₅	1	4	yes	—
117	C₂²×C₃×C₄	C₂×Dic₅	1	2	yes	—
118	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
119	C₂²×C₃×C₄	C₂×Dic₅	1	2	yes	—
120	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
121	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
122	C₃×M₁₆	—	5	8	no	—
123	C₃×M₁₆	—	10	4	no	—
124	C₂×C₃×C₈	—	10	4	no	—
125	C₃×C₄⋊C₄	—	2	4	no	—
126	C₃×C₄⋊C₄	—	2	2	no	—
127	C₃×C₄²	—	2	2	no	—
128	C₃×C₄²	—	2	2	no	—
129	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
130	C₃×C₄²	C₂²⋊D₅	1	2	yes	—
131	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
132	C₅×Q₈	—	2	8	no	—
133	C₂×C₄×C₅	—	2	2	no	—
134	C₂×C₄×C₅	—	2	2	no	—
135	C₂×C₄×C₅	—	2	2	no	—
136	C₂×C₄×C₅	—	2	2	no	—
137	C₂×C₄×C₅	—	4	1	no	—
138	C₂×C₄×C₅	—	4	2	no	—
139	C₂×C₄×C₅	C₃×C₄²	1	4	yes	—
140	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	2	yes	—
142	C₂×C₄×C₅	C₃×C₂²⋊C₄	1	1	yes	characteristic
143	C₂×C₄×C₅	C₂²×C₃×C₄	1	2	yes	—
144	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
145	C₂×C₄×C₅	C₃×#41	1	2	yes	—
146	#168	—	2	8	no	—
147	C₂²×Q₈	C₂×C₃×D₅	1	1	yes	characteristic
148	C₂×M₁₆	—	5	4	no	—
149	C₄×Q₈	—	2	2	no	—
150	C₄×Q₈	—	2	2	no	—
151	C₂×C₄⋊C₄	C₃×Dic₅	1	2	yes	—
152	C₂×C₄⋊C₄	C₂×C₃×D₅	1	1	yes	characteristic
153	C₂×C₄²	C₃×Dic₅	1	2	yes	—
154	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
155	#101	—	5	8	no	—
156	#97	—	10	2	no	—
157	C₄×C₈	—	10	2	no	—
158	C₂×C₃×C₅	—	2	1	no	—
159	C₂×C₃×C₅	—	2	1	no	—
160	C₂×C₃×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
161	C₂×C₃×C₅	C₂×#101	1	1	yes	characteristic
162	C₂×C₃×C₅	#413	1	1	yes	characteristic
163	C₂³×C₃	C₂²×Dic₅	1	1	yes	characteristic
164	C₃×Q₈	—	2	8	no	—
165	C₂×C₃×C₄	—	2	2	no	—
166	C₂×C₃×C₄	—	2	2	no	—
167	C₂×C₃×C₄	—	2	2	no	—
168	C₂×C₃×C₄	—	2	2	no	—
169	C₂×C₃×C₄	—	4	1	no	—
170	C₂×C₃×C₄	—	4	2	no	—
171	C₂×C₃×C₄	C₄×Dic₅	1	4	yes	—
172	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
173	C₂×C₃×C₄	C₂²⋊Dic₅	1	2	yes	—
174	C₂×C₃×C₄	C₂²⋊Dic₅	1	1	yes	characteristic
175	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
176	C₂×C₃×C₄	#709	1	2	yes	—
177	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
178	C₃×C₈	—	10	8	no	—
179	C₂²×C₅	—	2	1	no	—
180	C₂²×C₅	—	2	1	no	—
181	C₂²×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
182	C₂²×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
183	C₂²×C₅	C₃×#117	1	1	yes	characteristic
184	C₄×C₅	—	2	4	no	—
185	C₄×C₅	—	2	1	no	—
186	C₄×C₅	—	2	2	no	—
187	C₄×C₅	—	4	2	no	—
188	C₄×C₅	C₃×#101	1	2	yes	complemented
189	C₄×C₅	C₃×C₄×D₄	1	4	yes	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×Q₈	—	2	2	no	—
192	C₂×Q₈	C₃×C₄×D₅	1	4	yes	—
193	C₂²×C₄	C₂×C₃×Dic₅	1	2	yes	—
194	C₂²×C₄	C₂×C₃×Dic₅	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₃×Dic₅	1	2	yes	—
196	C₂²×C₄	C₂×C₃×Dic₅	1	1	yes	characteristic
197	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
198	M₁₆	—	5	8	no	—
199	M₁₆	—	10	4	no	—
200	C₂×C₈	—	10	4	no	—
201	C₄⋊C₄	—	2	4	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄²	—	2	2	no	—
204	C₄²	—	2	2	no	—
205	C₄²	C₃×C₂²⋊D₅	1	2	yes	—
206	C₄²	C₃×C₂²⋊D₅	1	2	yes	—
207	C₃×C₅	C₄×#101	1	1	yes	characteristic, complemented, Hall
208	C₂²×C₃	—	2	1	no	—
209	C₂²×C₃	—	2	1	no	—
210	C₂²×C₃	C₂×C₄×Dic₅	1	1	yes	characteristic
211	C₂²×C₃	#4072	1	1	yes	characteristic
212	C₂²×C₃	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
213	C₃×C₄	—	2	4	no	—
214	C₃×C₄	—	2	1	no	—
215	C₃×C₄	—	2	2	no	—
216	C₃×C₄	—	4	2	no	—
217	C₃×C₄	#3968	1	2	yes	direct product factor
218	C₃×C₄	C₄×C₂²⋊D₅	1	4	yes	—
219	C₂×C₅	—	2	1	no	—
220	C₂×C₅	—	2	1	no	—
221	C₂×C₅	C₃×C₄×C₂²⋊C₄	1	1	yes	characteristic
222	C₂×C₅	C₂×C₃×#101	1	1	yes	characteristic
223	C₂×C₅	C₃×#413	1	1	yes	characteristic
224	C₂³	C₂²×C₃×Dic₅	1	1	yes	characteristic
225	Q₈	—	2	8	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	C₃×C₄×Dic₅	1	4	yes	—
233	C₂×C₄	C₃×C₂²⋊Dic₅	1	1	yes	characteristic
234	C₂×C₄	C₃×C₂²⋊Dic₅	1	2	yes	—
235	C₂×C₄	C₃×C₂²⋊Dic₅	1	1	yes	characteristic
236	C₂×C₄	C₂×C₃×C₄×D₅	1	2	yes	—
237	C₂×C₄	C₃×#709	1	2	yes	—
238	C₂×C₄	C₂×C₃×C₂²⋊D₅	1	2	yes	—
239	C₈	—	10	8	no	—
240	C₂×C₃	—	2	1	no	—
241	C₂×C₃	—	2	1	no	—
242	C₂×C₃	C₄×C₂²⋊Dic₅	1	1	yes	characteristic
243	C₂×C₃	C₂×#3968	1	1	yes	characteristic
244	C₂×C₃	#65526	1	1	yes	characteristic
245	C₅	C₃×C₄×#101	1	1	yes	characteristic, complemented, Sylow
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	C₂×C₃×C₄×Dic₅	1	1	yes	characteristic
249	C₂²	C₃×#4072	1	1	yes	characteristic
250	C₂²	C₂×C₃×C₂²⋊Dic₅	1	1	yes	characteristic
251	C₄	—	2	4	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	2	no	—
254	C₄	—	4	2	no	—
255	C₄	C₃×#3968	1	2	yes	direct product factor
256	C₄	C₃×C₄×C₂²⋊D₅	1	4	yes	—
257	C₃	C₄×#3968	1	1	yes	characteristic, direct product factor, Sylow
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	C₃×C₄×C₂²⋊Dic₅	1	1	yes	characteristic
261	C₂	C₂×C₃×#3968	1	1	yes	characteristic
262	C₂	C₃×#65526	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²⋊Dic₅
commutator	C₂²×C₅	C₂×C₃×C₄²
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₂×C₃×C₄×C₅×Q₈	C₂
socle	C₂²×C₃×C₅	C₂×C₂²⋊C₄
prime radical	C₂³×C₃×C₅	C₂²×C₄
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₂×C₃×C₄×Q₈	D₅
hypocenter	C₅	C₃×C₄×#101
LCS	G ▷ C₂²×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×Q₈

Complements

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

N	H ≅ G/N	type
5. C₄×#3968	C₃	direct (N × H)
14. C₃×#3968	C₄	direct (N × H)
51. #3968	C₃×C₄	direct (N × H)
112. C₃×C₄×C₅	#101	semidirect
188. C₄×C₅	C₃×#101	semidirect
207. C₃×C₅	C₄×#101	semidirect
217. C₃×C₄	#3968	direct (N × H)
245. C₅	C₃×C₄×#101	semidirect
255. C₄	C₃×#3968	direct (N × H)
257. C₃	C₄×#3968	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×C₄×C₅×Q₈	2
C₂×C₃×#3968	2
C₃×C₄×#681	2
C₃×#64750	2
C₄×#3968	3
C₃×C₄×#101	5

Subgroups — C3×C4×#3968 #14295460

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×#3968 #14295460