Subgroups — #13143795

Order 1728.

650 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#88523	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×C₃⋊D₉	C₂	1	1	yes	maximal, characteristic
3	#11721709	C₂	1	1	yes	maximal, characteristic
4	C₃×C₉×#97	C₂	1	1	yes	maximal, characteristic, complemented
5	#11720538	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×C₃⋊Dic₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₃⋊(C₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
8	#10592407	—	3	1	no	maximal
9	#10588389	—	3	3	no	maximal
10	C₂×C₄×C₃⋊D₉	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
13	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
14	#88536	C₄	1	2	yes	—
15	#88534	C₄	1	2	yes	—
16	C₄×C₃⋊Dic₉	C₂²	1	1	yes	characteristic
17	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
18	C₂×C₃⋊(C₉⋊C₈)	C₂²	1	1	yes	characteristic
19	#88523	C₄	1	4	yes	complemented
20	C₈×C₃⋊D₉	C₄	1	4	yes	complemented
21	C₂×#3743	—	3	1	no	—
22	C₂×C₈×C₃⋊D₃	—	3	1	no	—
23	#64081	—	3	1	no	—
24	C₃²×#97	D₃	1	1	yes	characteristic
25	#63642	—	3	1	no	—
26	C₈×C₃⋊Dic₃	—	3	1	no	—
27	C₄×C₃⋊(C₃⋊C₈)	—	3	1	no	—
28	C₂×#3663	—	3	3	no	—
29	C₂×C₈×D₉	—	3	3	no	—
30	#63434	—	3	3	no	—
31	C₉×#97	D₃	1	3	yes	complemented
32	#63375	—	3	3	no	—
33	C₈×Dic₉	—	3	3	no	—
34	C₄×C₉⋊C₈	—	3	3	no	—
35	C₂²×C₃⋊D₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×C₄×C₉	C₂×C₄	1	2	yes	—
37	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
38	C₂×C₃⋊Dic₉	C₂×C₄	1	2	yes	—
39	C₂×C₃⋊Dic₉	C₂×C₄	1	1	yes	characteristic
40	C₄×C₃⋊D₉	C₂×C₄	1	2	yes	—
41	C₄×C₃⋊D₉	C₂×C₄	1	2	yes	—
42	C₃×C₈×C₉	C₂×C₄	1	2	yes	—
43	C₃×C₈×C₉	C₂×C₄	1	2	yes	—
44	C₃⋊(C₉⋊C₈)	C₂×C₄	1	2	yes	—
45	C₃⋊(C₉⋊C₈)	C₂×C₄	1	2	yes	—
46	#4708	—	9	3	no	—
47	#4708	—	9	1	no	—
48	C₂×C₄×C₃⋊D₃	—	3	1	no	—
49	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
50	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
51	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
52	#3752	—	3	2	no	—
53	#3750	—	3	2	no	—
54	C₄×C₃⋊Dic₃	—	3	1	no	—
55	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
56	C₂×C₃⋊(C₃⋊C₈)	—	3	1	no	—
57	#3743	—	3	4	no	—
58	C₈×C₃⋊D₃	—	3	4	no	—
59	C₂×C₄×D₉	—	3	3	no	—
60	C₂×C₈×C₉	C₂×D₃	1	3	yes	—
61	C₂×C₈×C₉	C₂×D₃	1	3	yes	—
62	C₄²×C₉	C₂×D₃	1	3	yes	—
63	(C₂×D₉)⋊C₄	—	3	6	no	—
64	Dic₉⋊C₄	—	3	6	no	—
65	C₄×Dic₉	—	3	3	no	—
66	C₂×C₉⋊C₈	—	3	3	no	—
67	C₂×C₉⋊C₈	—	3	3	no	—
68	#3663	—	3	12	no	—
69	C₈×D₉	—	3	12	no	—
70	C₂²×C₃×C₉	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₃⋊D₉	—	2	2	no	—
72	C₂×C₃⋊D₉	C₄²	1	2	yes	—
73	C₃×C₄×C₉	—	2	2	no	—
74	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
75	C₃×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
76	C₃⋊Dic₉	—	2	2	no	—
77	C₃⋊Dic₉	C₄²	1	2	yes	—
78	C₂×ΓL(1,𝔽₂₅)	—	9	3	no	—
79	C₂×ΓL(1,𝔽₂₅)	—	9	1	no	—
80	C₂×D₃×C₈	—	9	3	no	—
81	C₂×D₃×C₈	—	9	1	no	—
82	#872	—	9	3	no	—
83	#872	—	9	1	no	—
84	C₃×#97	D₉	1	3	yes	complemented
85	C₃×#97	C₃⋊D₃	1	1	yes	characteristic
86	#815	—	9	3	no	—
87	#815	—	9	1	no	—
88	C₈×Dic₃	—	9	3	no	—
89	C₈×Dic₃	—	9	1	no	—
90	C₄×C₃⋊C₈	—	9	3	no	—
91	C₄×C₃⋊C₈	—	9	1	no	—
92	C₂²×C₃⋊D₃	—	3	1	no	—
93	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
94	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
95	C₂×C₃⋊Dic₃	—	3	1	no	—
96	C₂×C₃⋊Dic₃	—	3	2	no	—
97	C₄×C₃⋊D₃	—	3	2	no	—
98	C₄×C₃⋊D₃	—	3	2	no	—
99	C₂²×D₉	—	3	3	no	—
100	C₃²×C₈	C₄×D₃	1	2	yes	—
101	C₃²×C₈	C₄×D₃	1	2	yes	—
102	C₃⋊(C₃⋊C₈)	—	3	2	no	—
103	C₃⋊(C₃⋊C₈)	—	3	2	no	—
104	C₂×C₄×C₉	C₄×D₃	1	6	yes	—
105	C₂×C₄×C₉	C₂²×D₃	1	3	yes	—
106	C₂×Dic₉	—	3	3	no	—
107	C₂×Dic₉	—	3	6	no	—
108	C₄×D₉	—	3	6	no	—
109	C₄×D₉	—	3	6	no	—
110	C₈×C₉	C₄×D₃	1	6	yes	—
111	C₈×C₉	C₄×D₃	1	6	yes	—
112	C₉⋊C₈	—	3	6	no	—
113	C₉⋊C₈	—	3	6	no	—
114	#405	—	27	1	no	Sylow
115	C₂×C₃×C₉	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₃×C₉	#131	1	2	yes	—
117	C₃⋊D₉	—	2	2	no	—
118	C₂×C₄×D₃	—	9	3	no	—
119	C₂×C₄×D₃	—	9	1	no	—
120	C₂×C₃×C₈	C₂×D₉	1	3	yes	—
121	C₂×C₃×C₈	C₂×D₉	1	3	yes	—
122	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
123	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
124	C₃×C₄²	C₂×D₉	1	3	yes	—
125	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
126	(C₂×D₃)⋊C₄	—	9	6	no	—
127	(C₂×D₃)⋊C₄	—	9	2	no	—
128	Dic₃⋊C₄	—	9	6	no	—
129	Dic₃⋊C₄	—	9	2	no	—
130	C₄×Dic₃	—	9	3	no	—
131	C₄×Dic₃	—	9	1	no	—
132	C₂×C₃⋊C₈	—	9	3	no	—
133	C₂×C₃⋊C₈	—	9	3	no	—
134	C₂×C₃⋊C₈	—	9	1	no	—
135	C₂×C₃⋊C₈	—	9	1	no	—
136	ΓL(1,𝔽₂₅)	—	9	12	no	—
137	ΓL(1,𝔽₂₅)	—	9	4	no	—
138	D₃×C₈	—	9	12	no	—
139	D₃×C₈	—	9	4	no	—
140	C₂²×C₃²	C₂×C₄×D₃	1	1	yes	characteristic
141	C₂×C₃⋊D₃	—	3	2	no	—
142	C₂×C₃⋊D₃	—	6	2	no	—
143	C₃²×C₄	—	2	2	no	—
144	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
145	C₃²×C₄	C₂×C₄×D₃	1	1	yes	characteristic
146	C₃⋊Dic₃	—	3	2	no	—
147	C₃⋊Dic₃	—	6	2	no	—
148	C₂²×C₉	C₂×C₄×D₃	1	3	yes	—
149	C₂×D₉	—	3	6	no	—
150	C₂×D₉	—	6	6	no	—
151	C₄×C₉	—	2	6	no	—
152	C₄×C₉	C₂×C₄×D₃	1	3	yes	—
153	C₄×C₉	C₂×C₄×D₃	1	3	yes	—
154	Dic₉	—	3	6	no	—
155	Dic₉	—	6	6	no	—
156	C₂×M₁₆	—	27	1	no	—
157	C₂²×C₈	—	27	1	no	—
158	#117	—	27	1	no	—
159	#97	—	27	1	no	—
160	#97	C₃⋊D₉	1	1	yes	characteristic, complemented
161	C₄×C₈	—	27	1	no	—
162	C₄×C₈	—	27	1	no	—
163	C₃×C₉	#405	1	1	yes	characteristic, complemented, Sylow
164	C₂²×D₃	—	9	3	no	—
165	C₂²×D₃	—	9	1	no	—
166	C₂×C₃×C₄	C₄×D₉	1	6	yes	—
167	C₂×C₃×C₄	C₂²×D₉	1	3	yes	—
168	C₂×C₃×C₄	C₄×C₃⋊D₃	1	2	yes	—
169	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
170	C₂×Dic₃	—	9	3	no	—
171	C₂×Dic₃	—	9	1	no	—
172	C₂×Dic₃	—	9	6	no	—
173	C₂×Dic₃	—	9	2	no	—
174	C₄×D₃	—	9	6	no	—
175	C₄×D₃	—	9	6	no	—
176	C₄×D₃	—	9	2	no	—
177	C₄×D₃	—	9	2	no	—
178	C₃×C₈	C₄×D₉	1	6	yes	—
179	C₃×C₈	C₄×D₉	1	6	yes	—
180	C₃×C₈	C₄×C₃⋊D₃	1	2	yes	—
181	C₃×C₈	C₄×C₃⋊D₃	1	2	yes	—
182	C₃⋊C₈	—	9	6	no	—
183	C₃⋊C₈	—	9	6	no	—
184	C₃⋊C₈	—	9	2	no	—
185	C₃⋊C₈	—	9	2	no	—
186	C₂×C₃²	C₄²×D₃	1	1	yes	characteristic
187	C₂×C₃²	#907	1	2	yes	—
188	C₃⋊D₃	—	6	2	no	—
189	C₂×C₉	C₄²×D₃	1	3	yes	—
190	C₂×C₉	#907	1	6	yes	—
191	D₉	—	6	6	no	—
192	C₂²×C₄	—	27	1	no	—
193	M₁₆	—	27	4	no	—
194	C₂×C₈	—	27	1	no	—
195	C₂×C₈	—	27	1	no	—
196	C₂×C₈	—	27	4	no	—
197	C₂×C₈	C₂×C₃⋊D₉	1	1	yes	characteristic
198	C₂×C₈	C₂×C₃⋊D₉	1	1	yes	characteristic
199	C₄⋊C₄	—	27	2	no	—
200	C₂²⋊C₄	—	27	2	no	—
201	C₄²	—	27	1	no	—
202	C₄²	C₂×C₃⋊D₉	1	1	yes	characteristic
203	C₂²×C₃	C₂×C₄×D₉	1	3	yes	—
204	C₂²×C₃	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
205	C₂×D₃	—	9	2	no	—
206	C₂×D₃	—	9	6	no	—
207	C₂×D₃	—	18	2	no	—
208	C₂×D₃	—	18	6	no	—
209	C₃×C₄	—	2	2	no	—
210	C₃×C₄	—	2	6	no	—
211	C₃×C₄	C₂×C₄×D₉	1	3	yes	—
212	C₃×C₄	C₂×C₄×D₉	1	3	yes	—
213	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
214	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
215	Dic₃	—	9	2	no	—
216	Dic₃	—	9	6	no	—
217	Dic₃	—	18	2	no	—
218	Dic₃	—	18	6	no	—
219	C₃²	#4708	1	1	yes	characteristic
220	C₉	#4708	1	3	yes	complemented
221	C₂³	—	27	1	no	—
222	C₂×C₄	—	27	1	no	—
223	C₂×C₄	—	27	2	no	—
224	C₂×C₄	—	27	2	no	—
225	C₂×C₄	—	27	2	no	—
226	C₂×C₄	C₄×C₃⋊D₉	1	2	yes	—
227	C₂×C₄	C₂²×C₃⋊D₉	1	1	yes	characteristic
228	C₈	—	27	2	no	—
229	C₈	—	27	2	no	—
230	C₈	C₄×C₃⋊D₉	1	2	yes	—
231	C₈	C₄×C₃⋊D₉	1	2	yes	—
232	C₂×C₃	C₄²×D₉	1	3	yes	—
233	C₂×C₃	#63469	1	6	yes	—
234	C₂×C₃	C₄²×C₃⋊D₃	1	1	yes	characteristic
235	C₂×C₃	#64116	1	2	yes	—
236	D₃	—	18	2	no	—
237	D₃	—	18	6	no	—
238	C₂²	—	27	2	no	—
239	C₂²	—	54	2	no	—
240	C₂²	C₂×C₄×C₃⋊D₉	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	27	2	no	—
243	C₄	—	54	2	no	—
244	C₄	C₂×C₄×C₃⋊D₉	1	1	yes	characteristic
245	C₄	C₂×C₄×C₃⋊D₉	1	1	yes	characteristic
246	C₃	#10588389	1	3	yes	complemented
247	C₃	#10592407	1	1	yes	characteristic
248	C₂	—	54	2	no	—
249	C₂	C₄²×C₃⋊D₉	1	1	yes	characteristic
250	C₂	#11721782	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×C₃⋊D₉
commutator	C₂×C₃×C₉	C₂×C₄²
Frattini	C₂×C₃×C₄	C₂²×C₃⋊D₃
Fitting	C₃×C₉×#97	C₂
socle	C₂²×C₃²	C₂×C₄×D₃
prime radical	C₂²×C₃⋊D₉	C₂×C₄
cosocle	C₂×C₃×C₄×C₉	C₂³
hypercenter	#97	C₃⋊D₉
hypocenter	C₃×C₉	#405
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂×C₄ ◁ #97

Complements

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

N	H ≅ G/N	type
4. C₃×C₉×#97	C₂	semidirect
5. #11720538	C₂	semidirect
6. C₈×C₃⋊Dic₉	C₂	semidirect
7. C₄×C₃⋊(C₉⋊C₈)	C₂	semidirect
19. #88523	C₄	semidirect
20. C₈×C₃⋊D₉	C₄	semidirect
31. C₉×#97	D₃	semidirect
84. C₃×#97	D₉	semidirect
160. #97	C₃⋊D₉	semidirect
163. C₃×C₉	#405	semidirect
220. C₉	#4708	semidirect
246. C₃	#10588389	semidirect

Maximal subgroups

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

subgroup	index
C₂×#88523	2
C₂×C₈×C₃⋊D₉	2
#11721709	2
C₃×C₉×#97	2
#11720538	2
C₈×C₃⋊Dic₉	2
C₄×C₃⋊(C₉⋊C₈)	2
#10592407	3
#10588389	3