Subgroups — C₃×C₂²⋊C₄×#168 #14295149

Order 1920.

768 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	C₂³×C₃×#168	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₅×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×C₂²⋊#168	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×C₄×#168	C₂	1	2	yes	maximal, complemented
5	C₃×#64744	C₂	1	2	yes	maximal, complemented
6	C₂²⋊C₄×#168	C₃	1	1	yes	maximal, characteristic, direct product factor, Hall
7	C₂³×C₃×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₅×C₂²⋊C₄	C₄	1	2	yes	—
9	C₂×C₃×C₄²×C₅	C₂²	1	2	yes	—
10	C₂²×C₃×#168	—	2	4	no	—
11	C₂²×C₃×#168	C₄	1	4	yes	complemented
12	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
13	C₂²×C₃×#168	C₂²	1	2	yes	—
14	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
15	C₃×C₅×#95	C₄	1	2	yes	—
16	C₃×C₂²⋊#168	C₄	1	4	yes	complemented
17	C₃×C₄×#168	—	2	4	no	—
18	C₃×C₈×C₂²⋊C₄	—	5	1	no	maximal, Hall
19	C₂³×#168	C₂×C₃	1	1	yes	characteristic
20	C₄×C₅×C₂²⋊C₄	C₂×C₃	1	1	yes	characteristic
21	C₂×C₂²⋊#168	C₂×C₃	1	1	yes	characteristic
22	C₂×C₄×#168	C₂×C₃	1	2	yes	complemented
23	#64744	C₂×C₃	1	2	yes	complemented
24	C₂⁴×C₃×C₅	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃×C₄×C₅	—	2	2	no	—
26	C₂²×C₃×C₄×C₅	—	2	2	no	—
27	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
29	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	4	yes	—
31	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₃×C₅×C₂²⋊C₄	C₈	1	8	yes	complemented
33	C₃×C₄²×C₅	—	2	2	no	—
34	C₃×C₄²×C₅	—	2	2	no	—
35	C₂×C₃×#168	—	2	2	no	—
36	C₂×C₃×#168	—	2	2	no	—
37	C₂×C₃×#168	—	2	2	no	—
38	C₂×C₃×#168	—	2	8	no	—
39	C₂×C₃×#168	C₂×C₄	1	4	yes	—
40	C₂×C₃×#168	C₂×C₄	1	2	yes	—
41	C₂×C₃×#168	C₂×C₄	1	2	yes	—
42	C₂×C₃×#168	D₄	1	4	yes	—
43	C₂³×C₃×C₈	—	5	1	no	—
44	C₂×C₃×C₂²⋊C₈	—	5	1	no	—
45	C₂×C₃×C₄×C₈	—	5	2	no	—
46	C₃×C₄×C₂²⋊C₄	D₅	1	1	yes	characteristic
47	C₃×#336	—	5	2	no	—
48	C₂³×C₄×C₅	C₂²×C₃	1	1	yes	characteristic
49	C₂×C₅×C₂²⋊C₄	C₃×C₄	1	2	yes	—
50	C₂×C₄²×C₅	C₂²×C₃	1	2	yes	—
51	C₂²×#168	—	2	4	no	—
52	C₂²×#168	C₃×C₄	1	4	yes	complemented
53	C₂²×#168	C₂²×C₃	1	1	yes	characteristic
54	C₂²×#168	C₂²×C₃	1	2	yes	—
55	C₂²×#168	C₂²×C₃	1	1	yes	characteristic
56	C₅×#95	C₃×C₄	1	2	yes	—
57	C₂²⋊#168	C₃×C₄	1	4	yes	complemented
58	C₄×#168	—	2	4	no	—
59	C₈×C₂²⋊C₄	—	5	1	no	Sylow
60	C₂³×C₃×C₅	—	2	2	no	—
61	C₂³×C₃×C₅	—	2	2	no	—
62	C₂³×C₃×C₅	C₄²	1	2	yes	—
63	C₂³×C₃×C₅	C₂×C₈	1	4	yes	—
64	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₃×C₄×C₅	—	2	4	no	—
66	C₂×C₃×C₄×C₅	—	2	2	no	—
67	C₂×C₃×C₄×C₅	—	2	4	no	—
68	C₂×C₃×C₄×C₅	—	2	2	no	—
69	C₂×C₃×C₄×C₅	—	2	2	no	—
70	C₂×C₃×C₄×C₅	—	2	4	no	—
71	C₂×C₃×C₄×C₅	C₄²	1	2	yes	—
72	C₂×C₃×C₄×C₅	C₂×C₈	1	8	yes	complemented
73	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
76	C₂×C₃×C₄×C₅	#41	1	2	yes	—
77	C₃×#168	—	2	2	no	—
78	C₃×#168	—	2	4	no	—
79	C₃×#168	C₂²⋊C₄	1	4	yes	direct product factor
80	C₂³×C₃×C₄	C₂×D₅	1	1	yes	characteristic
81	C₂²×C₃×C₈	—	5	1	no	—
82	C₂²×C₃×C₈	—	5	4	no	—
83	C₂²×C₃×C₈	—	5	1	no	—
84	C₂²×C₃×C₈	—	5	2	no	—
85	C₂²×C₃×C₈	—	10	4	no	—
86	C₂×C₃×C₂²⋊C₄	Dic₅	1	2	yes	—
87	C₂×C₃×C₄²	C₂×D₅	1	2	yes	—
88	C₃×C₂²⋊C₈	—	5	4	no	—
89	C₃×C₄×C₈	—	10	4	no	—
90	C₃×#95	Dic₅	1	2	yes	—
91	C₂⁴×C₅	C₂×C₃×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₅	—	2	2	no	—
93	C₂²×C₄×C₅	—	2	2	no	—
94	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
95	C₂²×C₄×C₅	C₂×C₃×C₄	1	2	yes	—
96	C₂²×C₄×C₅	C₂×C₃×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₅	C₂×C₃×C₄	1	4	yes	—
98	C₂²×C₄×C₅	C₂³×C₃	1	1	yes	characteristic
99	C₅×C₂²⋊C₄	C₃×C₈	1	8	yes	complemented
100	C₄²×C₅	—	2	2	no	—
101	C₄²×C₅	—	2	2	no	—
102	C₂×#168	—	2	2	no	—
103	C₂×#168	—	2	2	no	—
104	C₂×#168	—	2	2	no	—
105	C₂×#168	—	2	8	no	—
106	C₂×#168	C₂×C₃×C₄	1	4	yes	—
107	C₂×#168	C₂×C₃×C₄	1	2	yes	—
108	C₂×#168	C₂×C₃×C₄	1	2	yes	—
109	C₂×#168	C₃×D₄	1	4	yes	—
110	C₂³×C₈	—	5	1	no	—
111	C₂×C₂²⋊C₈	—	5	1	no	—
112	C₂×C₄×C₈	—	5	2	no	—
113	C₄×C₂²⋊C₄	C₃×D₅	1	1	yes	characteristic
114	#336	—	5	2	no	—
115	C₂²×C₃×C₅	—	2	2	no	—
116	C₂²×C₃×C₅	—	2	2	no	—
117	C₂²×C₃×C₅	—	2	8	no	—
118	C₂²×C₃×C₅	C₄×C₈	1	4	yes	complemented
119	C₂²×C₃×C₅	C₂×C₄²	1	1	yes	characteristic
120	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
121	C₂²×C₃×C₅	C₂²×C₈	1	2	yes	—
122	C₂²×C₃×C₅	#131	1	2	yes	—
123	C₃×C₄×C₅	—	2	2	no	—
124	C₃×C₄×C₅	—	2	8	no	—
125	C₃×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₃×C₄×C₅	#117	1	1	yes	characteristic
127	C₃×C₄×C₅	C₄×D₄	1	2	yes	—
128	C₂⁴×C₃	C₂×Dic₅	1	1	yes	characteristic
129	C₂²×C₃×C₄	—	2	2	no	—
130	C₂²×C₃×C₄	—	2	2	no	—
131	C₂²×C₃×C₄	C₄×D₅	1	2	yes	—
132	C₂²×C₃×C₄	C₄×D₅	1	2	yes	—
133	C₂²×C₃×C₄	C₂×Dic₅	1	4	yes	—
134	C₂²×C₃×C₄	C₂×Dic₅	1	1	yes	characteristic
135	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
136	C₂×C₃×C₈	—	5	2	no	—
137	C₂×C₃×C₈	—	5	4	no	—
138	C₂×C₃×C₈	—	5	4	no	—
139	C₂×C₃×C₈	—	5	2	no	—
140	C₂×C₃×C₈	—	10	8	no	—
141	C₂×C₃×C₈	—	10	2	no	—
142	C₂×C₃×C₈	—	10	2	no	—
143	C₂×C₃×C₈	—	10	2	no	—
144	C₃×C₂²⋊C₄	#168	1	8	yes	direct product factor
145	C₃×C₄²	—	2	2	no	—
146	C₃×C₄²	—	2	2	no	—
147	C₂³×C₅	—	2	2	no	—
148	C₂³×C₅	—	2	2	no	—
149	C₂³×C₅	C₃×C₄²	1	2	yes	—
150	C₂³×C₅	C₂×C₃×C₈	1	4	yes	—
151	C₂³×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
152	C₂×C₄×C₅	—	2	4	no	—
153	C₂×C₄×C₅	—	2	2	no	—
154	C₂×C₄×C₅	—	2	4	no	—
155	C₂×C₄×C₅	—	2	2	no	—
156	C₂×C₄×C₅	—	2	2	no	—
157	C₂×C₄×C₅	—	2	4	no	—
158	C₂×C₄×C₅	C₃×C₄²	1	2	yes	—
159	C₂×C₄×C₅	C₂×C₃×C₈	1	8	yes	complemented
160	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
161	C₂×C₄×C₅	C₂²×C₃×C₄	1	1	yes	characteristic
162	C₂×C₄×C₅	C₂×C₃×D₄	1	2	yes	—
163	C₂×C₄×C₅	C₃×#41	1	2	yes	—
164	#168	—	2	2	no	—
165	#168	—	2	4	no	—
166	#168	C₃×C₂²⋊C₄	1	4	yes	direct product factor
167	C₂³×C₄	C₂×C₃×D₅	1	1	yes	characteristic
168	C₂²×C₈	—	5	1	no	—
169	C₂²×C₈	—	5	4	no	—
170	C₂²×C₈	—	5	1	no	—
171	C₂²×C₈	—	5	2	no	—
172	C₂²×C₈	—	10	4	no	—
173	C₂×C₂²⋊C₄	C₃×Dic₅	1	2	yes	—
174	C₂×C₄²	C₂×C₃×D₅	1	2	yes	—
175	C₂²⋊C₈	—	5	4	no	—
176	C₄×C₈	—	10	4	no	—
177	#95	C₃×Dic₅	1	2	yes	—
178	C₂×C₃×C₅	—	2	4	no	—
179	C₂×C₃×C₅	C₄×C₂²⋊C₄	1	1	yes	characteristic
180	C₂×C₃×C₅	C₂×C₄×C₈	1	1	yes	characteristic
181	C₂×C₃×C₅	#405	1	1	yes	characteristic
182	C₂×C₃×C₅	C₈×D₄	1	4	yes	—
183	C₂³×C₃	—	2	2	no	—
184	C₂³×C₃	—	2	2	no	—
185	C₂³×C₃	C₂×#168	1	4	yes	—
186	C₂³×C₃	C₄×Dic₅	1	2	yes	—
187	C₂³×C₃	C₂²×Dic₅	1	1	yes	characteristic
188	C₂×C₃×C₄	—	2	4	no	—
189	C₂×C₃×C₄	—	2	2	no	—
190	C₂×C₃×C₄	—	2	4	no	—
191	C₂×C₃×C₄	—	2	2	no	—
192	C₂×C₃×C₄	—	2	2	no	—
193	C₂×C₃×C₄	—	2	4	no	—
194	C₂×C₃×C₄	C₂×#168	1	8	yes	complemented
195	C₂×C₃×C₄	C₄×Dic₅	1	2	yes	—
196	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
197	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
198	C₂×C₃×C₄	D₄×D₅	1	2	yes	—
199	C₂×C₃×C₄	#711	1	2	yes	—
200	C₃×C₈	—	5	4	no	—
201	C₃×C₈	—	10	2	no	—
202	C₃×C₈	—	10	4	no	—
203	C₂²×C₅	—	2	2	no	—
204	C₂²×C₅	—	2	2	no	—
205	C₂²×C₅	—	2	8	no	—
206	C₂²×C₅	C₃×C₄×C₈	1	4	yes	complemented
207	C₂²×C₅	C₂×C₃×C₄²	1	1	yes	characteristic
208	C₂²×C₅	C₃×C₄×D₄	1	2	yes	—
209	C₂²×C₅	C₂²×C₃×C₈	1	2	yes	—
210	C₂²×C₅	C₃×#131	1	2	yes	—
211	C₄×C₅	—	2	2	no	—
212	C₄×C₅	—	2	8	no	—
213	C₄×C₅	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
214	C₄×C₅	C₃×#117	1	1	yes	characteristic
215	C₄×C₅	C₃×C₄×D₄	1	2	yes	—
216	C₂⁴	C₂×C₃×Dic₅	1	1	yes	characteristic
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	C₃×C₄×D₅	1	2	yes	—
220	C₂²×C₄	C₃×C₄×D₅	1	2	yes	—
221	C₂²×C₄	C₂×C₃×Dic₅	1	4	yes	—
222	C₂²×C₄	C₂×C₃×Dic₅	1	1	yes	characteristic
223	C₂²×C₄	C₂²×C₃×D₅	1	1	yes	characteristic
224	C₂×C₈	—	5	2	no	—
225	C₂×C₈	—	5	4	no	—
226	C₂×C₈	—	5	4	no	—
227	C₂×C₈	—	5	2	no	—
228	C₂×C₈	—	10	8	no	—
229	C₂×C₈	—	10	2	no	—
230	C₂×C₈	—	10	2	no	—
231	C₂×C₈	—	10	2	no	—
232	C₂²⋊C₄	C₃×#168	1	8	yes	direct product factor
233	C₄²	—	2	2	no	—
234	C₄²	—	2	2	no	—
235	C₃×C₅	C₈×C₂²⋊C₄	1	1	yes	characteristic, complemented, Hall
236	C₂²×C₃	—	2	2	no	—
237	C₂²×C₃	—	2	8	no	—
238	C₂²×C₃	—	2	2	no	—
239	C₂²×C₃	C₄×#168	1	4	yes	complemented
240	C₂²×C₃	C₂²×#168	1	2	yes	—
241	C₂²×C₃	C₂×C₄×Dic₅	1	1	yes	characteristic
242	C₂²×C₃	D₄×Dic₅	1	2	yes	—
243	C₂²×C₃	#4094	1	2	yes	—
244	C₃×C₄	—	2	8	no	—
245	C₃×C₄	—	2	2	no	—
246	C₃×C₄	#4023	1	1	yes	characteristic
247	C₃×C₄	D₅×C₂²⋊C₄	1	1	yes	characteristic
248	C₃×C₄	#4027	1	2	yes	—
249	C₂×C₅	—	2	4	no	—
250	C₂×C₅	C₃×C₄×C₂²⋊C₄	1	1	yes	characteristic
251	C₂×C₅	C₂×C₃×C₄×C₈	1	1	yes	characteristic
252	C₂×C₅	C₃×#405	1	1	yes	characteristic
253	C₂×C₅	C₃×C₈×D₄	1	4	yes	—
254	C₂³	—	2	2	no	—
255	C₂³	—	2	2	no	—
256	C₂³	C₂×C₃×#168	1	4	yes	—
257	C₂³	C₃×C₄×Dic₅	1	2	yes	—
258	C₂³	C₂²×C₃×Dic₅	1	1	yes	characteristic
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	C₂×C₃×#168	1	8	yes	complemented
266	C₂×C₄	C₃×C₄×Dic₅	1	2	yes	—
267	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
268	C₂×C₄	C₂×C₃×C₄×D₅	1	1	yes	characteristic
269	C₂×C₄	C₃×D₄×D₅	1	2	yes	—
270	C₂×C₄	C₃×#711	1	2	yes	—
271	C₈	—	5	4	no	—
272	C₈	—	10	2	no	—
273	C₈	—	10	4	no	—
274	C₂×C₃	—	2	4	no	—
275	C₂×C₃	C₂×C₄×#168	1	1	yes	characteristic
276	C₂×C₃	C₂²⋊C₄×Dic₅	1	1	yes	characteristic
277	C₂×C₃	#65286	1	1	yes	characteristic
278	C₂×C₃	D₄×#168	1	4	yes	—
279	C₅	C₃×C₈×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
280	C₂²	—	2	2	no	—
281	C₂²	—	2	8	no	—
282	C₂²	—	2	2	no	—
283	C₂²	C₃×C₄×#168	1	4	yes	complemented
284	C₂²	C₂²×C₃×#168	1	2	yes	—
285	C₂²	C₂×C₃×C₄×Dic₅	1	1	yes	characteristic
286	C₂²	C₃×D₄×Dic₅	1	2	yes	—
287	C₂²	C₃×#4094	1	2	yes	—
288	C₄	—	2	8	no	—
289	C₄	—	2	2	no	—
290	C₄	C₃×#4023	1	1	yes	characteristic
291	C₄	C₃×D₅×C₂²⋊C₄	1	1	yes	characteristic
292	C₄	C₃×#4027	1	2	yes	—
293	C₃	C₂²⋊C₄×#168	1	1	yes	characteristic, direct product factor, Sylow
294	C₂	—	2	4	no	—
295	C₂	C₂×C₃×C₄×#168	1	1	yes	characteristic
296	C₂	C₃×C₂²⋊C₄×Dic₅	1	1	yes	characteristic
297	C₂	C₃×#65286	1	1	yes	characteristic
298	C₂	C₃×D₄×#168	1	4	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×C₂²⋊C₄	C₂
socle	C₂³×C₃×C₅	C₂²×C₄
prime radical	C₂⁴×C₃×C₅	C₂×C₄
cosocle	C₂²×C₄×C₅	C₂³×C₃
hypercenter	C₃×C₄×C₂²⋊C₄	D₅
hypocenter	C₅	C₃×C₈×C₂²⋊C₄
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ C₃×C₄×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
4. C₂×C₃×C₄×#168	C₂	semidirect
5. C₃×#64744	C₂	semidirect
6. C₂²⋊C₄×#168	C₃	direct (N × H)
11. C₂²×C₃×#168	C₄	semidirect
16. C₃×C₂²⋊#168	C₄	semidirect
22. C₂×C₄×#168	C₂×C₃	semidirect
23. #64744	C₂×C₃	semidirect
32. C₃×C₅×C₂²⋊C₄	C₈	semidirect
52. C₂²×#168	C₃×C₄	semidirect
57. C₂²⋊#168	C₃×C₄	semidirect
72. C₂×C₃×C₄×C₅	C₂×C₈	semidirect
79. C₃×#168	C₂²⋊C₄	direct (N × H)
99. C₅×C₂²⋊C₄	C₃×C₈	semidirect
118. C₂²×C₃×C₅	C₄×C₈	semidirect
144. C₃×C₂²⋊C₄	#168	direct (N × H)
159. C₂×C₄×C₅	C₂×C₃×C₈	semidirect
166. #168	C₃×C₂²⋊C₄	direct (N × H)
194. C₂×C₃×C₄	C₂×#168	semidirect
206. C₂²×C₅	C₃×C₄×C₈	semidirect
232. C₂²⋊C₄	C₃×#168	direct (N × H)
235. C₃×C₅	C₈×C₂²⋊C₄	semidirect
239. C₂²×C₃	C₄×#168	semidirect
265. C₂×C₄	C₂×C₃×#168	semidirect
279. C₅	C₃×C₈×C₂²⋊C₄	semidirect
283. C₂²	C₃×C₄×#168	semidirect
293. C₃	C₂²⋊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
C₂³×C₃×#168	2
C₃×C₄×C₅×C₂²⋊C₄	2
C₂×C₃×C₂²⋊#168	2
C₂×C₃×C₄×#168	2
C₃×#64744	2
C₂²⋊C₄×#168	3
C₃×C₈×C₂²⋊C₄	5

Subgroups — C3×C22⋊C4×#168 #14295149

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₂²⋊C₄×#168 #14295149