Subgroups — C₃×C₄×D₇₂ #13141418

Order 1728.

446 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×D₇₂	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×D₃₆	C₂	1	2	yes	maximal, complemented
3	C₃×C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#63380	C₂	1	2	yes	maximal, complemented
5	C₃×#63378	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×C₄×D₂₄	—	3	1	no	maximal
7	C₄×D₇₂	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂×C₃×D₃₆	C₂²	1	2	yes	—
9	C₂×C₃×C₄×D₉	—	2	2	no	—
10	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
13	C₃×(C₂×D₉)⋊C₄	—	2	2	no	—
14	C₃×C₄⋊Dic₉	C₂²	1	2	yes	—
15	C₃×D₇₂	C₄	1	4	yes	direct product factor
16	C₂×C₃×D₂₄	—	3	1	no	—
17	C₃×C₄×D₁₂	—	3	2	no	—
18	C₃²×C₄×C₈	D₃	1	1	yes	characteristic
19	C₃×#821	—	3	2	no	—
20	C₃×#818	—	3	1	no	—
21	C₂×D₇₂	C₂×C₃	1	1	yes	characteristic
22	C₄×D₃₆	C₂×C₃	1	2	yes	complemented
23	C₄×C₈×C₉	—	2	1	no	—
24	C₄×C₈×C₉	C₂×C₃	1	1	yes	characteristic, complemented
25	#63380	C₂×C₃	1	2	yes	complemented
26	#63378	C₂×C₃	1	1	yes	characteristic, complemented
27	C₂²×C₃×D₉	—	2	2	no	—
28	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
29	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
30	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
31	C₂×C₃×Dic₉	—	2	2	no	—
32	C₃×D₃₆	—	2	2	no	—
33	C₃×D₃₆	C₂×C₄	1	4	yes	complemented
34	C₃×C₄×D₉	—	4	4	no	—
35	C₃×C₈×C₉	—	2	1	no	—
36	C₃×C₈×C₉	C₂×C₄	1	2	yes	complemented
37	C₃×C₄×D₈	—	9	1	no	—
38	C₄×D₂₄	—	3	1	no	—
39	C₂×C₃×D₁₂	—	3	2	no	—
40	C₂×C₃×C₄×D₃	—	6	2	no	—
41	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
42	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
43	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
44	C₃×(C₂×D₃)⋊C₄	—	6	2	no	—
45	C₃×C₄⋊Dic₃	—	3	2	no	—
46	C₃×D₂₄	—	3	4	no	—
47	C₂×D₃₆	C₂²×C₃	1	2	yes	—
48	C₂×C₄×D₉	—	2	2	no	—
49	C₂×C₈×C₉	—	2	1	no	—
50	C₂×C₈×C₉	—	2	1	no	—
51	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
52	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
53	C₄²×C₉	—	2	1	no	—
54	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
55	(C₂×D₉)⋊C₄	—	2	2	no	—
56	C₄⋊Dic₉	C₂²×C₃	1	2	yes	—
57	D₇₂	C₃×C₄	1	4	yes	direct product factor
58	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
59	C₂×C₃×D₉	—	2	4	no	—
60	C₂×C₃×D₉	—	4	2	no	—
61	C₂×C₃×D₉	—	4	2	no	—
62	C₃×C₄×C₉	—	2	1	no	—
63	C₃×C₄×C₉	D₈	1	2	yes	complemented
64	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
65	C₃×C₄×C₉	#41	1	1	yes	characteristic
66	C₃×Dic₉	—	4	2	no	—
67	C₂×C₃×D₈	—	9	1	no	—
68	C₃×C₄×D₄	—	9	2	no	—
69	C₂×D₂₄	—	3	1	no	—
70	C₄×D₁₂	—	3	2	no	—
71	C₃×#107	—	9	1	no	—
72	C₃×#102	—	9	2	no	—
73	C₃×C₄×C₈	—	2	1	no	—
74	C₃×C₄×C₈	D₉	1	1	yes	characteristic, complemented
75	C₃×C₄×C₈	C₃×D₃	1	1	yes	characteristic
76	#821	—	3	2	no	—
77	#818	—	3	1	no	—
78	C₂²×C₃×D₃	—	6	2	no	—
79	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
80	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
81	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
82	C₂×C₃×Dic₃	—	6	2	no	—
83	C₃×D₁₂	—	3	4	no	—
84	C₃×D₁₂	—	6	2	no	—
85	C₃×C₄×D₃	—	12	4	no	—
86	C₂²×D₉	—	2	2	no	—
87	C₃²×C₈	—	2	1	no	—
88	C₃²×C₈	C₄×D₃	1	2	yes	—
89	C₂×C₄×C₉	—	2	1	no	—
90	C₂×C₄×C₉	—	2	1	no	—
91	C₂×C₄×C₉	—	2	1	no	—
92	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
93	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
94	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
95	C₂×Dic₉	—	2	2	no	—
96	D₃₆	—	2	2	no	—
97	D₃₆	C₂×C₃×C₄	1	4	yes	complemented
98	C₄×D₉	—	4	4	no	—
99	C₈×C₉	—	2	2	no	—
100	C₈×C₉	—	2	1	no	—
101	C₈×C₉	—	2	2	no	—
102	C₈×C₉	C₂×C₃×C₄	1	2	yes	complemented
103	C₄×D₈	—	9	1	no	Sylow
104	C₂×C₃×C₉	C₄×D₄	1	1	yes	characteristic
105	C₂×C₃×C₉	C₂×D₈	1	1	yes	characteristic
106	C₂×C₃×C₉	#135	1	1	yes	characteristic
107	C₃×D₉	—	4	4	no	—
108	C₂×C₃×D₄	—	9	2	no	—
109	C₂²×C₃×C₄	—	18	2	no	—
110	C₂×D₁₂	—	3	2	no	—
111	C₂×C₄×D₃	—	6	2	no	—
112	C₃×D₈	—	9	4	no	—
113	C₂×C₃×C₈	—	2	1	no	—
114	C₂×C₃×C₈	—	2	1	no	—
115	C₂×C₃×C₈	C₂×D₉	1	1	yes	characteristic
116	C₂×C₃×C₈	C₂×D₉	1	1	yes	characteristic
117	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
118	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
119	C₃×C₄⋊C₄	—	9	2	no	—
120	C₃×C₂²⋊C₄	—	18	2	no	—
121	C₃×C₄²	—	2	1	no	—
122	C₃×C₄²	C₂×D₉	1	1	yes	characteristic
123	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
124	(C₂×D₃)⋊C₄	—	6	2	no	—
125	C₄⋊Dic₃	—	3	2	no	—
126	D₂₄	—	3	4	no	—
127	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
128	C₂×C₃×D₃	—	6	4	no	—
129	C₂×C₃×D₃	—	12	2	no	—
130	C₂×C₃×D₃	—	12	2	no	—
131	C₃²×C₄	—	2	1	no	—
132	C₃²×C₄	D₂₄	1	2	yes	—
133	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
134	C₃²×C₄	#235	1	1	yes	characteristic
135	C₃×Dic₃	—	12	2	no	—
136	C₂²×C₉	—	2	1	no	—
137	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
138	C₂×D₉	—	2	4	no	—
139	C₂×D₉	—	4	2	no	—
140	C₂×D₉	—	4	2	no	—
141	C₄×C₉	—	2	2	no	—
142	C₄×C₉	—	2	1	no	—
143	C₄×C₉	—	2	2	no	—
144	C₄×C₉	—	2	1	no	—
145	C₄×C₉	—	2	1	no	—
146	C₄×C₉	C₃×D₈	1	2	yes	complemented
147	C₄×C₉	C₂²×C₃×C₄	1	1	yes	characteristic
148	C₄×C₉	C₃×#41	1	1	yes	characteristic
149	Dic₉	—	4	2	no	—
150	C₂×D₈	—	9	1	no	—
151	C₄×D₄	—	9	2	no	—
152	#107	—	9	1	no	—
153	#102	—	9	2	no	—
154	C₄×C₈	C₃×D₉	1	1	yes	characteristic, complemented
155	C₃×C₉	C₄×D₈	1	1	yes	characteristic, complemented, Sylow
156	C₂³×C₃	—	18	2	no	—
157	C₂²×D₃	—	6	2	no	—
158	C₃×D₄	—	9	4	no	—
159	C₃×D₄	—	18	2	no	—
160	C₂×C₃×C₄	—	2	1	no	—
161	C₂×C₃×C₄	—	2	1	no	—
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	18	2	no	—
164	C₂×C₃×C₄	—	36	4	no	—
165	C₂×C₃×C₄	D₃₆	1	1	yes	characteristic
166	C₂×C₃×C₄	D₃₆	1	1	yes	characteristic
167	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
168	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
169	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
171	C₂×Dic₃	—	6	2	no	—
172	D₁₂	—	3	4	no	—
173	D₁₂	—	6	2	no	—
174	C₄×D₃	—	12	4	no	—
175	C₃×C₈	—	2	2	no	—
176	C₃×C₈	—	2	1	no	—
177	C₃×C₈	—	2	1	no	—
178	C₃×C₈	—	2	2	no	—
179	C₃×C₈	C₄×D₉	1	2	yes	complemented
180	C₃×C₈	C₃×C₄×D₃	1	2	yes	—
181	C₂×C₃²	C₄×D₁₂	1	1	yes	characteristic
182	C₂×C₃²	C₂×D₂₄	1	1	yes	characteristic
183	C₂×C₃²	#904	1	1	yes	characteristic
184	C₃×D₃	—	12	4	no	—
185	C₂×C₉	—	2	1	no	—
186	C₂×C₉	—	2	1	no	—
187	C₂×C₉	—	2	1	no	—
188	C₂×C₉	C₃×C₄×D₄	1	1	yes	characteristic
189	C₂×C₉	C₂×C₃×D₈	1	1	yes	characteristic
190	C₂×C₉	C₃×#135	1	1	yes	characteristic
191	D₉	—	4	4	no	—
192	C₂×D₄	—	9	2	no	—
193	C₂²×C₄	—	18	2	no	—
194	D₈	—	9	4	no	—
195	C₂×C₈	C₂×C₃×D₉	1	1	yes	characteristic
196	C₂×C₈	C₂×C₃×D₉	1	1	yes	characteristic
197	C₄⋊C₄	—	9	2	no	—
198	C₂²⋊C₄	—	18	2	no	—
199	C₄²	C₂×C₃×D₉	1	1	yes	characteristic
200	C₂²×C₃	—	2	1	no	—
201	C₂²×C₃	—	18	4	no	—
202	C₂²×C₃	—	36	2	no	—
203	C₂²×C₃	—	36	2	no	—
204	C₂²×C₃	C₂×D₃₆	1	1	yes	characteristic
205	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
206	C₂×D₃	—	6	4	no	—
207	C₂×D₃	—	12	2	no	—
208	C₂×D₃	—	12	2	no	—
209	C₃×C₄	—	2	2	no	—
210	C₃×C₄	—	2	1	no	—
211	C₃×C₄	—	2	1	no	—
212	C₃×C₄	—	2	1	no	—
213	C₃×C₄	—	2	1	no	—
214	C₃×C₄	—	2	2	no	—
215	C₃×C₄	—	36	2	no	—
216	C₃×C₄	D₇₂	1	2	yes	direct product factor
217	C₃×C₄	C₂×C₄×D₉	1	1	yes	characteristic
218	C₃×C₄	#3697	1	1	yes	characteristic
219	C₃×C₄	C₃×D₂₄	1	2	yes	—
220	C₃×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
221	C₃×C₄	C₃×#235	1	1	yes	characteristic
222	Dic₃	—	12	2	no	—
223	C₃²	C₄×D₂₄	1	1	yes	characteristic
224	C₉	—	2	1	no	—
225	C₉	C₃×C₄×D₈	1	1	yes	characteristic, complemented
226	C₂³	—	18	2	no	—
227	D₄	—	9	4	no	—
228	D₄	—	18	2	no	—
229	C₂×C₄	—	18	2	no	—
230	C₂×C₄	—	36	4	no	—
231	C₂×C₄	C₃×D₃₆	1	1	yes	characteristic
232	C₂×C₄	C₃×D₃₆	1	1	yes	characteristic
233	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
234	C₈	—	2	1	no	—
235	C₈	C₃×C₄×D₉	1	2	yes	complemented
236	C₂×C₃	—	2	1	no	—
237	C₂×C₃	—	2	1	no	—
238	C₂×C₃	—	2	1	no	—
239	C₂×C₃	—	36	4	no	—
240	C₂×C₃	C₄×D₃₆	1	1	yes	characteristic
241	C₂×C₃	C₂×D₇₂	1	1	yes	characteristic
242	C₂×C₃	#63467	1	1	yes	characteristic
243	C₂×C₃	C₃×C₄×D₁₂	1	1	yes	characteristic
244	C₂×C₃	C₂×C₃×D₂₄	1	1	yes	characteristic
245	C₂×C₃	C₃×#904	1	1	yes	characteristic
246	D₃	—	12	4	no	—
247	C₂²	—	18	4	no	—
248	C₂²	—	36	2	no	—
249	C₂²	—	36	2	no	—
250	C₂²	C₂×C₃×D₃₆	1	1	yes	characteristic
251	C₄	—	2	1	no	—
252	C₄	—	36	2	no	—
253	C₄	C₃×D₇₂	1	2	yes	direct product factor
254	C₄	C₂×C₃×C₄×D₉	1	1	yes	characteristic
255	C₄	C₃×#3697	1	1	yes	characteristic
256	C₃	—	2	1	no	—
257	C₃	C₄×D₇₂	1	1	yes	characteristic, direct product factor
258	C₃	C₃×C₄×D₂₄	1	1	yes	characteristic
259	C₂	—	36	4	no	—
260	C₂	C₃×C₄×D₃₆	1	1	yes	characteristic
261	C₂	C₂×C₃×D₇₂	1	1	yes	characteristic
262	C₂	C₃×#63467	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	D₃₆
commutator	C₄×C₉	C₂²×C₃×C₄
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₃×C₄×C₈×C₉	C₂
socle	C₂²×C₃²	C₂×D₁₂
prime radical	C₂×C₃×D₇₂	C₂
cosocle	C₂×C₄×C₉	C₂³×C₃
hypercenter	C₃×C₄×C₈	D₉
hypocenter	C₉	C₃×C₄×D₈
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄² ◁ C₃×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₃×C₄×D₃₆	C₂	semidirect
3. C₃×C₄×C₈×C₉	C₂	semidirect
4. C₃×#63380	C₂	semidirect
5. C₃×#63378	C₂	semidirect
7. C₄×D₇₂	C₃	direct (N × H)
15. C₃×D₇₂	C₄	direct (N × H)
22. C₄×D₃₆	C₂×C₃	semidirect
24. C₄×C₈×C₉	C₂×C₃	semidirect
25. #63380	C₂×C₃	semidirect
26. #63378	C₂×C₃	semidirect
33. C₃×D₃₆	C₂×C₄	semidirect
36. C₃×C₈×C₉	C₂×C₄	semidirect
57. D₇₂	C₃×C₄	direct (N × H)
63. C₃×C₄×C₉	D₈	semidirect
74. C₃×C₄×C₈	D₉	semidirect
97. D₃₆	C₂×C₃×C₄	semidirect
102. C₈×C₉	C₂×C₃×C₄	semidirect
146. C₄×C₉	C₃×D₈	semidirect
154. C₄×C₈	C₃×D₉	semidirect
155. C₃×C₉	C₄×D₈	semidirect
179. C₃×C₈	C₄×D₉	semidirect
216. C₃×C₄	D₇₂	direct (N × H)
225. C₉	C₃×C₄×D₈	semidirect
235. C₈	C₃×C₄×D₉	semidirect
253. C₄	C₃×D₇₂	direct (N × H)
257. C₃	C₄×D₇₂	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×D₇₂	2
C₃×C₄×D₃₆	2
C₃×C₄×C₈×C₉	2
C₃×#63380	2
C₃×#63378	2
C₃×C₄×D₂₄	3
C₄×D₇₂	3

Subgroups — C3×C4×D72 #13141418

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₄×D₇₂ #13141418