Subgroups — C₂×C₄×#6648 #14411177

Order 1920.

1520 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×C₄²×D₁₅	C₂	1	1	yes	maximal, characteristic
2	C₂²×#6648	C₂	1	2	yes	maximal
3	C₂×C₄×C₅×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×C₄×#168	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#90951	C₂	1	2	yes	maximal, complemented
6	C₄×#6648	C₂	1	8	yes	maximal, direct product factor
7	C₂×C₄×C₈×D₅	—	3	1	no	maximal, Hall
8	C₂²×C₄×D₁₅	C₄	1	2	yes	—
9	C₂²×C₄×D₁₅	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₁₅	C₄	1	2	yes	—
12	C₂×C₄×Dic₁₅	C₂²	1	1	yes	characteristic
13	C₄²×D₁₅	C₄	1	4	yes	—
14	C₄²×D₁₅	C₂²	1	4	yes	—
15	C₂²×C₅×C₃⋊C₈	C₂²	1	2	yes	—
16	C₂²×C₃×#168	C₂²	1	2	yes	—
17	C₂×#6648	C₄	1	16	yes	direct product factor
18	C₂×#6648	C₂²	1	8	yes	—
19	C₄×C₅×C₃⋊C₈	C₂²	1	4	yes	complemented
20	C₃×C₄×#168	C₂²	1	4	yes	complemented
21	#90951	C₂²	1	8	yes	complemented
22	C₂×C₄×D₃×C₈	—	5	1	no	maximal, Hall
23	C₂²×C₈×D₅	—	3	2	no	—
24	C₂×C₄²×D₅	—	3	1	no	—
25	C₂×C₄×C₅×C₈	—	3	1	no	—
26	C₂×C₈×Dic₅	—	3	2	no	—
27	C₂×C₄×#168	D₃	1	1	yes	characteristic, complemented
28	C₄×C₈×D₅	—	3	8	no	—
29	C₂³×D₁₅	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
31	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
32	C₂²×Dic₁₅	C₂×C₄	1	1	yes	characteristic
33	C₂²×Dic₁₅	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₁₅	C₈	1	16	yes	complemented
35	C₂×C₄×D₁₅	C₂×C₄	1	2	yes	—
36	C₂×C₄×D₁₅	C₂×C₄	1	8	yes	—
37	C₂×C₄×D₁₅	C₂×C₄	1	8	yes	—
38	C₂×C₄×D₁₅	C₂³	1	2	yes	—
39	C₃×C₄²×C₅	C₂×C₄	1	2	yes	—
40	C₃×C₄²×C₅	C₂³	1	2	yes	—
41	C₄×Dic₁₅	C₂×C₄	1	2	yes	—
42	C₄×Dic₁₅	C₂×C₄	1	8	yes	—
43	C₄×Dic₁₅	C₂³	1	2	yes	—
44	C₂×C₅×C₃⋊C₈	C₂×C₄	1	8	yes	complemented
45	C₂×C₅×C₃⋊C₈	C₂³	1	4	yes	—
46	C₂×C₃×#168	C₂×C₄	1	8	yes	complemented
47	C₂×C₃×#168	C₂³	1	4	yes	—
48	#6648	C₂×C₄	1	32	yes	direct product factor
49	C₂²×D₃×C₈	—	5	2	no	—
50	C₂×C₄²×D₃	—	5	1	no	—
51	C₂×C₃×C₄×C₈	—	5	1	no	—
52	C₂×C₈×Dic₃	—	5	2	no	—
53	C₂×C₄×C₃⋊C₈	D₅	1	1	yes	characteristic, complemented
54	C₄×D₃×C₈	—	5	8	no	—
55	C₂²×C₄×D₅	—	3	2	no	—
56	C₂²×C₄×D₅	—	3	1	no	—
57	C₂²×C₅×C₈	—	3	2	no	—
58	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
59	C₂×C₄×Dic₅	—	3	2	no	—
60	C₂×C₄×Dic₅	—	3	1	no	—
61	C₂²×#168	C₂×D₃	1	2	yes	—
62	C₂×C₈×D₅	—	3	16	no	—
63	C₂×C₈×D₅	—	3	8	no	—
64	C₄²×D₅	—	3	4	no	—
65	C₄²×D₅	—	3	4	no	—
66	C₄×C₅×C₈	—	3	4	no	—
67	C₈×Dic₅	—	3	8	no	—
68	C₄×#168	C₂×D₃	1	4	yes	complemented
69	C₂²×C₄×C₈	—	15	1	no	Sylow
70	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
71	C₂²×D₁₅	C₄²	1	4	yes	—
72	C₂²×D₁₅	C₂×C₈	1	8	yes	—
73	C₂²×D₁₅	C₂²×C₄	1	2	yes	—
74	C₂×C₃×C₄×C₅	C₂×C₈	1	8	yes	complemented
75	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
77	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
78	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
79	C₂×Dic₁₅	C₄²	1	4	yes	—
80	C₂×Dic₁₅	C₂×C₈	1	8	yes	complemented
81	C₂×Dic₁₅	C₂²×C₄	1	2	yes	—
82	C₂×Dic₁₅	C₂²×C₄	1	4	yes	—
83	C₄×D₁₅	C₄²	1	8	yes	—
84	C₄×D₁₅	C₂×C₈	1	32	yes	complemented
85	C₄×D₁₅	C₂²×C₄	1	8	yes	—
86	C₃×#168	C₂²×C₄	1	8	yes	complemented
87	C₅×C₃⋊C₈	C₂²×C₄	1	8	yes	complemented
88	C₂²×C₄×D₃	—	5	2	no	—
89	C₂²×C₄×D₃	—	5	1	no	—
90	C₂²×C₃×C₈	—	5	2	no	—
91	C₂×C₃×C₄²	C₂×D₅	1	1	yes	characteristic
92	C₂×C₄×Dic₃	—	5	2	no	—
93	C₂×C₄×Dic₃	—	5	1	no	—
94	C₂²×C₃⋊C₈	C₂×D₅	1	2	yes	—
95	C₂×D₃×C₈	—	5	16	no	—
96	C₂×D₃×C₈	—	5	8	no	—
97	C₄²×D₃	—	5	4	no	—
98	C₄²×D₃	—	5	4	no	—
99	C₃×C₄×C₈	—	5	4	no	—
100	C₈×Dic₃	—	5	8	no	—
101	C₄×C₃⋊C₈	C₂×D₅	1	4	yes	complemented
102	C₂³×D₅	—	3	1	no	—
103	C₂²×C₄×C₅	C₄×D₃	1	2	yes	—
104	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
105	C₂²×Dic₅	—	3	1	no	—
106	C₂²×Dic₅	—	3	2	no	—
107	C₂×C₄×D₅	—	3	16	no	—
108	C₂×C₄×D₅	—	3	2	no	—
109	C₂×C₄×D₅	—	3	8	no	—
110	C₂×C₄×D₅	—	3	8	no	—
111	C₂×C₄×D₅	—	3	2	no	—
112	C₂×C₅×C₈	—	3	8	no	—
113	C₂×C₅×C₈	—	3	4	no	—
114	C₄²×C₅	C₄×D₃	1	2	yes	—
115	C₄²×C₅	C₂²×D₃	1	2	yes	—
116	C₄×Dic₅	—	3	2	no	—
117	C₄×Dic₅	—	3	8	no	—
118	C₄×Dic₅	—	3	2	no	—
119	C₂×#168	C₄×D₃	1	8	yes	complemented
120	C₂×#168	C₂²×D₃	1	4	yes	—
121	C₈×D₅	—	3	32	no	—
122	C₂³×C₈	—	15	2	no	—
123	C₂²×C₄²	—	15	1	no	—
124	C₂×C₄×C₈	—	3	1	no	—
125	C₂×C₄×C₈	—	5	1	no	—
126	C₂×C₄×C₈	—	15	8	no	—
127	C₂×C₄×C₈	—	15	2	no	—
128	C₂²×C₃×C₅	C₂×C₄²	1	2	yes	—
129	C₂²×C₃×C₅	C₂²×C₈	1	4	yes	—
130	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
131	C₂×D₁₅	C₄×C₈	1	16	yes	complemented
132	C₂×D₁₅	C₂×C₄²	1	4	yes	—
133	C₂×D₁₅	C₂²×C₈	1	8	yes	—
134	C₃×C₄×C₅	C₂×C₄²	1	2	yes	—
135	C₃×C₄×C₅	C₂²×C₈	1	8	yes	complemented
136	C₃×C₄×C₅	C₂³×C₄	1	2	yes	—
137	Dic₁₅	C₂×C₄²	1	4	yes	—
138	Dic₁₅	C₂²×C₈	1	8	yes	complemented
139	C₂³×D₃	—	5	1	no	—
140	C₂²×C₃×C₄	C₄×D₅	1	2	yes	—
141	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
142	C₂²×Dic₃	—	5	1	no	—
143	C₂²×Dic₃	—	5	2	no	—
144	C₂×C₄×D₃	—	5	16	no	—
145	C₂×C₄×D₃	—	5	2	no	—
146	C₂×C₄×D₃	—	5	8	no	—
147	C₂×C₄×D₃	—	5	8	no	—
148	C₂×C₄×D₃	—	5	2	no	—
149	C₂×C₃×C₈	—	5	8	no	—
150	C₂×C₃×C₈	—	5	4	no	—
151	C₃×C₄²	C₄×D₅	1	2	yes	—
152	C₃×C₄²	C₂²×D₅	1	2	yes	—
153	C₄×Dic₃	—	5	2	no	—
154	C₄×Dic₃	—	5	8	no	—
155	C₄×Dic₃	—	5	2	no	—
156	C₂×C₃⋊C₈	C₄×D₅	1	8	yes	complemented
157	C₂×C₃⋊C₈	C₂²×D₅	1	4	yes	—
158	D₃×C₈	—	5	32	no	—
159	C₂³×C₅	C₂×C₄×D₃	1	1	yes	characteristic
160	C₂²×D₅	—	3	4	no	—
161	C₂²×D₅	—	3	8	no	—
162	C₂²×D₅	—	3	2	no	—
163	C₂×C₄×C₅	D₃×C₈	1	8	yes	complemented
164	C₂×C₄×C₅	C₂×C₄×D₃	1	4	yes	—
165	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
166	C₂×C₄×C₅	C₂×C₄×D₃	1	4	yes	—
167	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
168	C₂×Dic₅	—	3	4	no	—
169	C₂×Dic₅	—	3	8	no	—
170	C₂×Dic₅	—	3	2	no	—
171	C₂×Dic₅	—	3	4	no	—
172	C₄×D₅	—	3	8	no	—
173	C₄×D₅	—	3	32	no	—
174	C₄×D₅	—	3	8	no	—
175	C₅×C₈	—	3	8	no	—
176	#168	C₂×C₄×D₃	1	8	yes	complemented
177	C₂³×C₄	—	15	2	no	—
178	C₂³×C₄	—	15	1	no	—
179	C₂²×C₈	—	3	2	no	—
180	C₂²×C₈	—	5	2	no	—
181	C₂²×C₈	—	15	8	no	—
182	C₂²×C₈	—	15	16	no	—
183	C₂×C₄²	—	15	2	no	—
184	C₂×C₄²	—	15	1	no	—
185	C₂×C₄²	—	15	4	no	—
186	C₂×C₄²	—	15	4	no	—
187	C₂×C₄²	D₃×D₅	1	1	yes	characteristic
188	C₄×C₈	—	3	4	no	—
189	C₄×C₈	—	5	4	no	—
190	C₄×C₈	—	15	8	no	—
191	C₂×C₃×C₅	C₂×C₄×C₈	1	4	yes	complemented
192	C₂×C₃×C₅	C₂²×C₄²	1	1	yes	characteristic
193	C₂×C₃×C₅	C₂³×C₈	1	2	yes	—
194	D₁₅	C₂×C₄×C₈	1	8	yes	complemented
195	C₂³×C₃	C₂×C₄×D₅	1	1	yes	characteristic
196	C₂²×D₃	—	5	2	no	—
197	C₂²×D₃	—	5	4	no	—
198	C₂²×D₃	—	5	8	no	—
199	C₂×C₃×C₄	C₈×D₅	1	8	yes	complemented
200	C₂×C₃×C₄	C₂×C₄×D₅	1	4	yes	—
201	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
202	C₂×C₃×C₄	C₂×C₄×D₅	1	4	yes	—
203	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
204	C₂×Dic₃	—	5	2	no	—
205	C₂×Dic₃	—	5	4	no	—
206	C₂×Dic₃	—	5	4	no	—
207	C₂×Dic₃	—	5	8	no	—
208	C₄×D₃	—	5	8	no	—
209	C₄×D₃	—	5	8	no	—
210	C₄×D₃	—	5	32	no	—
211	C₃×C₈	—	5	8	no	—
212	C₃⋊C₈	C₂×C₄×D₅	1	8	yes	complemented
213	C₂²×C₅	C₄²×D₃	1	2	yes	—
214	C₂²×C₅	C₂×D₃×C₈	1	4	yes	—
215	C₂²×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
216	C₂×D₅	—	3	16	no	—
217	C₂×D₅	—	3	4	no	—
218	C₂×D₅	—	3	8	no	—
219	C₄×C₅	C₄²×D₃	1	2	yes	—
220	C₄×C₅	C₂×D₃×C₈	1	8	yes	complemented
221	C₄×C₅	C₂²×C₄×D₃	1	2	yes	—
222	Dic₅	—	3	4	no	—
223	Dic₅	—	3	8	no	—
224	C₂⁴	—	15	1	no	—
225	C₂²×C₄	—	15	1	no	—
226	C₂²×C₄	—	15	2	no	—
227	C₂²×C₄	—	15	8	no	—
228	C₂²×C₄	—	15	2	no	—
229	C₂²×C₄	—	15	8	no	—
230	C₂²×C₄	—	15	2	no	—
231	C₂²×C₄	—	15	16	no	—
232	C₂²×C₄	#1200	1	2	yes	—
233	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
234	C₂×C₈	—	3	8	no	—
235	C₂×C₈	—	3	4	no	—
236	C₂×C₈	—	5	8	no	—
237	C₂×C₈	—	5	4	no	—
238	C₂×C₈	—	15	32	no	—
239	C₄²	—	15	2	no	—
240	C₄²	—	15	8	no	—
241	C₄²	—	15	2	no	—
242	C₄²	#1200	1	2	yes	—
243	C₄²	C₂×D₃×D₅	1	2	yes	—
244	C₃×C₅	C₂²×C₄×C₈	1	1	yes	characteristic, complemented, Hall
245	C₂²×C₃	C₄²×D₅	1	2	yes	—
246	C₂²×C₃	C₂×C₈×D₅	1	4	yes	—
247	C₂²×C₃	C₂²×C₄×D₅	1	1	yes	characteristic
248	C₂×D₃	—	5	4	no	—
249	C₂×D₃	—	5	8	no	—
250	C₂×D₃	—	5	16	no	—
251	C₃×C₄	C₄²×D₅	1	2	yes	—
252	C₃×C₄	C₂×C₈×D₅	1	8	yes	complemented
253	C₃×C₄	C₂²×C₄×D₅	1	2	yes	—
254	Dic₃	—	5	4	no	—
255	Dic₃	—	5	8	no	—
256	C₂×C₅	C₄×D₃×C₈	1	4	yes	complemented
257	C₂×C₅	C₂×C₄²×D₃	1	1	yes	characteristic
258	C₂×C₅	C₂²×D₃×C₈	1	2	yes	—
259	D₅	—	3	8	no	—
260	C₂³	—	15	4	no	—
261	C₂³	—	15	2	no	—
262	C₂³	—	15	8	no	—
263	C₂³	C₂×#1200	1	1	yes	characteristic
264	C₂×C₄	—	15	4	no	—
265	C₂×C₄	—	15	2	no	—
266	C₂×C₄	—	15	4	no	—
267	C₂×C₄	—	15	8	no	—
268	C₂×C₄	—	15	8	no	—
269	C₂×C₄	—	15	8	no	—
270	C₂×C₄	—	15	32	no	—
271	C₂×C₄	#6648	1	8	yes	direct product factor
272	C₂×C₄	C₄×D₃×D₅	1	4	yes	—
273	C₂×C₄	C₂×#1200	1	1	yes	characteristic
274	C₂×C₄	C₂×#1200	1	4	yes	—
275	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
276	C₈	—	3	8	no	—
277	C₈	—	5	8	no	—
278	C₂×C₃	C₄×C₈×D₅	1	4	yes	complemented
279	C₂×C₃	C₂×C₄²×D₅	1	1	yes	characteristic
280	C₂×C₃	C₂²×C₈×D₅	1	2	yes	—
281	D₃	—	5	8	no	—
282	C₅	C₂×C₄×D₃×C₈	1	1	yes	characteristic, complemented, Sylow
283	C₂²	—	15	4	no	—
284	C₂²	—	15	8	no	—
285	C₂²	—	15	16	no	—
286	C₂²	C₂×#6648	1	4	yes	—
287	C₂²	C₄×#1200	1	2	yes	—
288	C₂²	C₂²×#1200	1	1	yes	characteristic
289	C₄	—	15	4	no	—
290	C₄	—	15	8	no	—
291	C₄	C₂×#6648	1	8	yes	direct product factor
292	C₄	C₄×#1200	1	2	yes	—
293	C₄	C₂×C₄×D₃×D₅	1	2	yes	—
294	C₃	C₂×C₄×C₈×D₅	1	1	yes	characteristic, complemented, Sylow
295	C₂	—	15	8	no	—
296	C₂	C₄×#6648	1	4	yes	direct product factor
297	C₂	C₂²×#6648	1	2	yes	—
298	C₂	C₂×C₄×#1200	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₄²	D₃×D₅
commutator, hypocenter	C₃×C₅	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂²×D₃×D₅
Fitting	C₂×C₃×C₄²×C₅	C₂²
socle	C₂³×C₃×C₅	C₂²×C₄
prime radical	C₂³×D₁₅	C₂×C₄
cosocle	C₂×C₃×C₄×C₅	C₂⁴
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
3. C₂×C₄×C₅×C₃⋊C₈	C₂	semidirect
4. C₂×C₃×C₄×#168	C₂	semidirect
5. C₂×#90951	C₂	semidirect
6. C₄×#6648	C₂	direct (N × H)
17. C₂×#6648	C₄	direct (N × H)
19. C₄×C₅×C₃⋊C₈	C₂²	semidirect
20. C₃×C₄×#168	C₂²	semidirect
21. #90951	C₂²	semidirect
27. C₂×C₄×#168	D₃	semidirect
34. C₂×C₄×D₁₅	C₈	semidirect
44. C₂×C₅×C₃⋊C₈	C₂×C₄	semidirect
46. C₂×C₃×#168	C₂×C₄	semidirect
48. #6648	C₂×C₄	direct (N × H)
53. C₂×C₄×C₃⋊C₈	D₅	semidirect
68. C₄×#168	C₂×D₃	semidirect
74. C₂×C₃×C₄×C₅	C₂×C₈	semidirect
80. C₂×Dic₁₅	C₂×C₈	semidirect
84. C₄×D₁₅	C₂×C₈	semidirect
86. C₃×#168	C₂²×C₄	semidirect
87. C₅×C₃⋊C₈	C₂²×C₄	semidirect
101. C₄×C₃⋊C₈	C₂×D₅	semidirect
119. C₂×#168	C₄×D₃	semidirect
131. C₂×D₁₅	C₄×C₈	semidirect
135. C₃×C₄×C₅	C₂²×C₈	semidirect
138. Dic₁₅	C₂²×C₈	semidirect
156. C₂×C₃⋊C₈	C₄×D₅	semidirect
163. C₂×C₄×C₅	D₃×C₈	semidirect
176. #168	C₂×C₄×D₃	semidirect
191. C₂×C₃×C₅	C₂×C₄×C₈	semidirect
194. D₁₅	C₂×C₄×C₈	semidirect
199. C₂×C₃×C₄	C₈×D₅	semidirect
212. C₃⋊C₈	C₂×C₄×D₅	semidirect
220. C₄×C₅	C₂×D₃×C₈	semidirect
244. C₃×C₅	C₂²×C₄×C₈	semidirect
252. C₃×C₄	C₂×C₈×D₅	semidirect
256. C₂×C₅	C₄×D₃×C₈	semidirect
271. C₂×C₄	#6648	direct (N × H)
278. C₂×C₃	C₄×C₈×D₅	semidirect
282. C₅	C₂×C₄×D₃×C₈	semidirect
291. C₄	C₂×#6648	direct (N × H)
294. C₃	C₂×C₄×C₈×D₅	semidirect
296. C₂	C₄×#6648	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×C₄²×D₁₅	2
C₂²×#6648	2
C₂×C₄×C₅×C₃⋊C₈	2
C₂×C₃×C₄×#168	2
C₂×#90951	2
C₄×#6648	2
C₂×C₄×C₈×D₅	3
C₂×C₄×D₃×C₈	5

Subgroups — C2×C4×#6648 #14411177

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×#6648 #14411177