Subgroups — C₃×C₄⋊C₄×#168 #14296239

Order 1920.

592 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄⋊#168	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₄×#168	C₂	1	2	yes	maximal
4	C₂×C₃×C₄×#168	C₂	1	1	yes	maximal, characteristic
5	C₃×#64744	C₂	1	2	yes	maximal
6	C₄⋊C₄×#168	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂×C₃×C₅×C₄⋊C₄	C₄	1	2	yes	—
8	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
9	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
10	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
11	C₂²×C₃×#168	C₂²	1	2	yes	—
12	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
13	C₃×C₅×#95	C₄	1	2	yes	—
14	C₃×C₄⋊#168	C₄	1	4	yes	complemented
15	C₃×C₄×#168	—	2	4	no	—
16	C₃×C₄×#168	C₄	1	4	yes	complemented
17	C₃×C₈×C₄⋊C₄	—	5	1	no	maximal, Hall
18	C₄×C₅×C₄⋊C₄	C₂×C₃	1	1	yes	characteristic
19	C₂×C₄⋊#168	C₂×C₃	1	1	yes	characteristic
20	C₂×C₄×#168	C₂×C₃	1	2	yes	—
21	C₂×C₄×#168	C₂×C₃	1	1	yes	characteristic
22	#64744	C₂×C₃	1	2	yes	—
23	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃×C₄×C₅	C₂×C₄	1	4	yes	—
25	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
27	C₃×C₅×C₄⋊C₄	C₈	1	8	yes	complemented
28	C₃×C₄²×C₅	—	2	2	no	—
29	C₃×C₄²×C₅	—	2	2	no	—
30	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
31	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
32	C₂×C₃×#168	—	2	2	no	—
33	C₂×C₃×#168	—	2	1	no	—
34	C₂×C₃×#168	—	2	1	no	—
35	C₂×C₃×#168	—	2	2	no	—
36	C₂×C₃×#168	C₂×C₄	1	2	yes	—
37	C₂×C₃×#168	C₂×C₄	1	4	yes	—
38	C₂×C₃×#168	C₂×C₄	1	2	yes	—
39	C₂×C₃×#168	D₄	1	2	yes	—
40	C₂×C₃×#168	Q₈	1	2	yes	—
41	C₂×C₃×C₄⋊C₈	—	5	1	no	—
42	C₂×C₃×C₄×C₈	—	5	1	no	—
43	C₂×C₃×C₄×C₈	—	5	2	no	—
44	C₃×C₄×C₄⋊C₄	D₅	1	1	yes	characteristic
45	C₃×#336	—	5	2	no	—
46	C₂×C₅×C₄⋊C₄	C₃×C₄	1	2	yes	—
47	C₂×C₄²×C₅	C₂²×C₃	1	2	yes	—
48	C₂×C₄²×C₅	C₂²×C₃	1	1	yes	characteristic
49	C₂²×#168	C₂²×C₃	1	2	yes	—
50	C₂²×#168	C₂²×C₃	1	1	yes	characteristic
51	C₂²×#168	C₂²×C₃	1	1	yes	characteristic
52	C₅×#95	C₃×C₄	1	2	yes	—
53	C₄⋊#168	C₃×C₄	1	4	yes	complemented
54	C₄×#168	—	2	4	no	—
55	C₄×#168	C₃×C₄	1	4	yes	complemented
56	C₈×C₄⋊C₄	—	5	1	no	Sylow
57	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₃×C₄×C₅	—	2	4	no	—
59	C₂×C₃×C₄×C₅	—	2	2	no	—
60	C₂×C₃×C₄×C₅	—	2	4	no	—
61	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
62	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
63	C₂×C₃×C₄×C₅	C₂×C₈	1	8	yes	—
64	C₂×C₃×C₄×C₅	C₂×C₈	1	4	yes	—
65	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
69	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
70	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
71	C₃×#168	—	2	2	no	—
72	C₃×#168	—	2	4	no	—
73	C₃×#168	C₄⋊C₄	1	4	yes	direct product factor
74	C₂²×C₃×C₈	—	5	1	no	—
75	C₂²×C₃×C₈	—	5	2	no	—
76	C₂²×C₃×C₈	—	5	1	no	—
77	C₂×C₃×C₄⋊C₄	Dic₅	1	2	yes	—
78	C₂×C₃×C₄²	C₂×D₅	1	2	yes	—
79	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
80	C₃×C₄⋊C₈	—	5	4	no	—
81	C₃×C₄×C₈	—	5	4	no	—
82	C₃×C₄×C₈	—	10	4	no	—
83	C₃×#95	Dic₅	1	2	yes	—
84	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₂×C₃×C₄	1	4	yes	—
86	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
88	C₅×C₄⋊C₄	C₃×C₈	1	8	yes	complemented
89	C₄²×C₅	—	2	2	no	—
90	C₄²×C₅	—	2	2	no	—
91	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
92	C₄²×C₅	C₂×C₃×C₄	1	2	yes	—
93	C₂×#168	—	2	2	no	—
94	C₂×#168	—	2	1	no	—
95	C₂×#168	—	2	1	no	—
96	C₂×#168	—	2	2	no	—
97	C₂×#168	C₂×C₃×C₄	1	2	yes	—
98	C₂×#168	C₂×C₃×C₄	1	4	yes	—
99	C₂×#168	C₂×C₃×C₄	1	2	yes	—
100	C₂×#168	C₃×D₄	1	2	yes	—
101	C₂×#168	C₃×Q₈	1	2	yes	—
102	C₂×C₄⋊C₈	—	5	1	no	—
103	C₂×C₄×C₈	—	5	1	no	—
104	C₂×C₄×C₈	—	5	2	no	—
105	C₄×C₄⋊C₄	C₃×D₅	1	1	yes	characteristic
106	#336	—	5	2	no	—
107	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
108	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
109	C₂²×C₃×C₅	C₄×Q₈	1	1	yes	characteristic
110	C₂²×C₃×C₅	C₂²×C₈	1	2	yes	—
111	C₂²×C₃×C₅	#131	1	2	yes	—
112	C₃×C₄×C₅	—	2	2	no	—
113	C₃×C₄×C₅	—	2	8	no	—
114	C₃×C₄×C₅	C₄×C₈	1	4	yes	complemented
115	C₃×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₃×C₄×C₅	#117	1	1	yes	characteristic
117	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
118	C₃×C₄×C₅	C₄×Q₈	1	1	yes	characteristic
119	C₂²×C₃×C₄	C₂×Dic₅	1	4	yes	—
120	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
121	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
122	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
123	C₂×C₃×C₈	—	5	2	no	—
124	C₂×C₃×C₈	—	5	2	no	—
125	C₂×C₃×C₈	—	5	2	no	—
126	C₂×C₃×C₈	—	5	2	no	—
127	C₂×C₃×C₈	—	5	4	no	—
128	C₂×C₃×C₈	—	10	2	no	—
129	C₂×C₃×C₈	—	10	1	no	—
130	C₂×C₃×C₈	—	10	1	no	—
131	C₂×C₃×C₈	—	10	2	no	—
132	C₃×C₄⋊C₄	#168	1	8	yes	direct product factor
133	C₃×C₄²	—	2	2	no	—
134	C₃×C₄²	—	2	2	no	—
135	C₃×C₄²	C₄×D₅	1	2	yes	—
136	C₃×C₄²	C₄×D₅	1	2	yes	—
137	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
138	C₂×C₄×C₅	—	2	4	no	—
139	C₂×C₄×C₅	—	2	2	no	—
140	C₂×C₄×C₅	—	2	4	no	—
141	C₂×C₄×C₅	C₃×C₄²	1	2	yes	—
142	C₂×C₄×C₅	C₃×C₄²	1	2	yes	—
143	C₂×C₄×C₅	C₂×C₃×C₈	1	8	yes	—
144	C₂×C₄×C₅	C₂×C₃×C₈	1	4	yes	—
145	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂×C₃×D₄	1	1	yes	characteristic
148	C₂×C₄×C₅	C₂×C₃×Q₈	1	1	yes	characteristic
149	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
150	C₂×C₄×C₅	C₃×#41	1	1	yes	characteristic
151	#168	—	2	2	no	—
152	#168	—	2	4	no	—
153	#168	C₃×C₄⋊C₄	1	4	yes	direct product factor
154	C₂²×C₈	—	5	1	no	—
155	C₂²×C₈	—	5	1	no	—
156	C₂²×C₈	—	5	2	no	—
157	C₂×C₄⋊C₄	C₃×Dic₅	1	2	yes	—
158	C₂×C₄²	C₂×C₃×D₅	1	2	yes	—
159	C₂×C₄²	C₂×C₃×D₅	1	1	yes	characteristic
160	C₄⋊C₈	—	5	4	no	—
161	C₄×C₈	—	5	4	no	—
162	C₄×C₈	—	10	4	no	—
163	#95	C₃×Dic₅	1	2	yes	—
164	C₂×C₃×C₅	C₄×C₄⋊C₄	1	1	yes	characteristic
165	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
166	C₂×C₃×C₅	#405	1	1	yes	characteristic
167	C₂×C₃×C₅	C₈×D₄	1	2	yes	—
168	C₂×C₃×C₅	C₈×Q₈	1	2	yes	—
169	C₂³×C₃	C₂²×Dic₅	1	1	yes	characteristic
170	C₂×C₃×C₄	—	2	4	no	—
171	C₂×C₃×C₄	—	2	2	no	—
172	C₂×C₃×C₄	—	2	4	no	—
173	C₂×C₃×C₄	C₂×#168	1	8	yes	—
174	C₂×C₃×C₄	C₂×#168	1	4	yes	—
175	C₂×C₃×C₄	C₄×Dic₅	1	2	yes	—
176	C₂×C₃×C₄	C₄×Dic₅	1	2	yes	—
177	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
178	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
179	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
180	C₂×C₃×C₄	#711	1	1	yes	characteristic
181	C₂×C₃×C₄	Q₈×D₅	1	1	yes	characteristic
182	C₂×C₃×C₄	#713	1	1	yes	characteristic
183	C₃×C₈	—	5	4	no	—
184	C₃×C₈	—	10	2	no	—
185	C₃×C₈	—	10	4	no	—
186	C₂²×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
187	C₂²×C₅	C₃×C₄×D₄	1	1	yes	characteristic
188	C₂²×C₅	C₃×C₄×Q₈	1	1	yes	characteristic
189	C₂²×C₅	C₂²×C₃×C₈	1	2	yes	—
190	C₂²×C₅	C₃×#131	1	2	yes	—
191	C₄×C₅	—	2	2	no	—
192	C₄×C₅	—	2	8	no	—
193	C₄×C₅	C₃×C₄×C₈	1	4	yes	complemented
194	C₄×C₅	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
195	C₄×C₅	C₃×#117	1	1	yes	characteristic
196	C₄×C₅	C₃×C₄×D₄	1	1	yes	characteristic
197	C₄×C₅	C₃×C₄×Q₈	1	1	yes	characteristic
198	C₂²×C₄	C₂×C₃×Dic₅	1	4	yes	—
199	C₂²×C₄	C₂×C₃×Dic₅	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₃×Dic₅	1	1	yes	characteristic
201	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
202	C₂×C₈	—	5	2	no	—
203	C₂×C₈	—	5	2	no	—
204	C₂×C₈	—	5	2	no	—
205	C₂×C₈	—	5	2	no	—
206	C₂×C₈	—	5	4	no	—
207	C₂×C₈	—	10	2	no	—
208	C₂×C₈	—	10	1	no	—
209	C₂×C₈	—	10	1	no	—
210	C₂×C₈	—	10	2	no	—
211	C₄⋊C₄	C₃×#168	1	8	yes	direct product factor
212	C₄²	—	2	2	no	—
213	C₄²	—	2	2	no	—
214	C₄²	C₃×C₄×D₅	1	2	yes	—
215	C₄²	C₃×C₄×D₅	1	2	yes	—
216	C₃×C₅	C₈×C₄⋊C₄	1	1	yes	characteristic, complemented, Hall
217	C₂²×C₃	C₂²×#168	1	2	yes	—
218	C₂²×C₃	C₂×C₄×Dic₅	1	1	yes	characteristic
219	C₂²×C₃	D₄×Dic₅	1	1	yes	characteristic
220	C₂²×C₃	Q₈×Dic₅	1	1	yes	characteristic
221	C₂²×C₃	#4094	1	2	yes	—
222	C₃×C₄	—	2	2	no	—
223	C₃×C₄	—	2	8	no	—
224	C₃×C₄	C₄×#168	1	4	yes	complemented
225	C₃×C₄	#4033	1	1	yes	characteristic
226	C₃×C₄	D₅×C₄⋊C₄	1	1	yes	characteristic
227	C₃×C₄	#4038	1	1	yes	characteristic
228	C₃×C₄	#4039	1	1	yes	characteristic
229	C₂×C₅	C₃×C₄×C₄⋊C₄	1	1	yes	characteristic
230	C₂×C₅	C₂×C₃×C₄×C₈	1	1	yes	characteristic
231	C₂×C₅	C₃×#405	1	1	yes	characteristic
232	C₂×C₅	C₃×C₈×D₄	1	2	yes	—
233	C₂×C₅	C₃×C₈×Q₈	1	2	yes	—
234	C₂³	C₂²×C₃×Dic₅	1	1	yes	characteristic
235	C₂×C₄	—	2	4	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	C₂×C₃×#168	1	8	yes	—
239	C₂×C₄	C₂×C₃×#168	1	4	yes	—
240	C₂×C₄	C₃×C₄×Dic₅	1	2	yes	—
241	C₂×C₄	C₃×C₄×Dic₅	1	2	yes	—
242	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
243	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
244	C₂×C₄	C₃×D₄×D₅	1	1	yes	characteristic
245	C₂×C₄	C₃×#711	1	1	yes	characteristic
246	C₂×C₄	C₃×Q₈×D₅	1	1	yes	characteristic
247	C₂×C₄	C₃×#713	1	1	yes	characteristic
248	C₈	—	5	4	no	—
249	C₈	—	10	2	no	—
250	C₈	—	10	4	no	—
251	C₂×C₃	C₂×C₄×#168	1	1	yes	characteristic
252	C₂×C₃	C₄⋊C₄×Dic₅	1	1	yes	characteristic
253	C₂×C₃	#65286	1	1	yes	characteristic
254	C₂×C₃	D₄×#168	1	2	yes	—
255	C₂×C₃	Q₈×#168	1	2	yes	—
256	C₅	C₃×C₈×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
257	C₂²	C₂²×C₃×#168	1	2	yes	—
258	C₂²	C₂×C₃×C₄×Dic₅	1	1	yes	characteristic
259	C₂²	C₃×D₄×Dic₅	1	1	yes	characteristic
260	C₂²	C₃×Q₈×Dic₅	1	1	yes	characteristic
261	C₂²	C₃×#4094	1	2	yes	—
262	C₄	—	2	2	no	—
263	C₄	—	2	8	no	—
264	C₄	C₃×C₄×#168	1	4	yes	complemented
265	C₄	C₃×#4033	1	1	yes	characteristic
266	C₄	C₃×D₅×C₄⋊C₄	1	1	yes	characteristic
267	C₄	C₃×#4038	1	1	yes	characteristic
268	C₄	C₃×#4039	1	1	yes	characteristic
269	C₃	C₄⋊C₄×#168	1	1	yes	characteristic, direct product factor, Sylow
270	C₂	C₂×C₃×C₄×#168	1	1	yes	characteristic
271	C₂	C₃×C₄⋊C₄×Dic₅	1	1	yes	characteristic
272	C₂	C₃×#65286	1	1	yes	characteristic
273	C₂	C₃×D₄×#168	1	2	yes	—
274	C₂	C₃×Q₈×#168	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄	C₂²×D₅
commutator	C₂×C₅	C₂×C₃×C₄×C₈
Frattini	C₂²×C₄	C₂²×C₃×D₅
Fitting	C₃×C₄×C₅×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₃×C₅	C₂²×C₄
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₃×C₄×C₄⋊C₄	D₅
hypocenter	C₅	C₃×C₈×C₄⋊C₄
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₃×C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
6. C₄⋊C₄×#168	C₃	direct (N × H)
14. C₃×C₄⋊#168	C₄	semidirect
16. C₃×C₄×#168	C₄	semidirect
27. C₃×C₅×C₄⋊C₄	C₈	semidirect
53. C₄⋊#168	C₃×C₄	semidirect
55. C₄×#168	C₃×C₄	semidirect
73. C₃×#168	C₄⋊C₄	direct (N × H)
88. C₅×C₄⋊C₄	C₃×C₈	semidirect
114. C₃×C₄×C₅	C₄×C₈	semidirect
132. C₃×C₄⋊C₄	#168	direct (N × H)
153. #168	C₃×C₄⋊C₄	direct (N × H)
193. C₄×C₅	C₃×C₄×C₈	semidirect
211. C₄⋊C₄	C₃×#168	direct (N × H)
216. C₃×C₅	C₈×C₄⋊C₄	semidirect
224. C₃×C₄	C₄×#168	semidirect
256. C₅	C₃×C₈×C₄⋊C₄	semidirect
264. C₄	C₃×C₄×#168	semidirect
269. C₃	C₄⋊C₄×#168	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₄×C₅×C₄⋊C₄	2
C₂×C₃×C₄⋊#168	2
C₂×C₃×C₄×#168	2
C₃×#64744	2
C₄⋊C₄×#168	3
C₃×C₈×C₄⋊C₄	5

Subgroups — C3×C4⋊C4×#168 #14296239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄⋊C₄×#168 #14296239