Subgroups — #11786122

Order 1152.

360 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	C₄²×Dic₉	C₂	1	1	yes	maximal, characteristic
2	C₉×#336	C₂	1	1	yes	maximal, characteristic
3	#10588228	C₂	1	2	yes	maximal
4	#10588228	C₂	1	2	yes	maximal
5	#10588200	C₂	1	1	yes	maximal, characteristic
6	#68607	—	3	1	no	maximal
7	C₂²×C₈×C₉	C₂²	1	2	yes	—
8	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₄×Dic₉	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₉	C₄	1	1	yes	characteristic
11	C₂×C₄×Dic₉	C₄	1	1	yes	characteristic
12	C₂×C₄×Dic₉	C₄	1	1	yes	characteristic
13	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
15	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
16	C₄²×Dic₃	—	3	1	no	—
17	C₃×#336	D₃	1	1	yes	characteristic
18	#4550	—	3	2	no	—
19	#4550	—	3	2	no	—
20	#4524	—	3	1	no	—
21	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
24	C₂²×Dic₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×Dic₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×Dic₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×Dic₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₉	—	2	2	no	—
29	C₂×C₈×C₉	—	2	2	no	—
30	C₂×C₈×C₉	—	2	2	no	—
31	C₄²×C₉	—	2	2	no	—
32	C₄×Dic₉	—	2	2	no	—
33	C₄×Dic₉	—	2	2	no	—
34	C₄×Dic₉	—	2	1	no	—
35	C₄×Dic₉	—	2	1	no	—
36	C₄×Dic₉	—	2	2	no	—
37	C₄×Dic₉	—	2	2	no	—
38	C₄×Dic₉	C₈	1	4	yes	complemented
39	C₂×C₉⋊C₈	—	2	2	no	—
40	C₂×C₉⋊C₈	—	2	2	no	—
41	C₂×C₉⋊C₈	—	2	2	no	—
42	#1839	—	9	1	no	Sylow
43	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
44	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
45	C₂×C₄×Dic₃	—	3	1	no	—
46	C₂×C₄×Dic₃	—	3	1	no	—
47	C₂×C₄×Dic₃	—	3	1	no	—
48	C₂×C₄×Dic₃	—	3	1	no	—
49	C₂×C₄×Dic₃	—	3	1	no	—
50	C₂×C₄×Dic₃	—	3	1	no	—
51	C₂²×C₃⋊C₈	—	3	2	no	—
52	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₉	—	2	1	no	—
54	C₂×C₄×C₉	—	2	1	no	—
55	C₂×C₄×C₉	—	2	1	no	—
56	C₂×C₄×C₉	—	2	1	no	—
57	C₂×C₄×C₉	C₂×C₈	1	2	yes	—
58	C₂×C₄×C₉	M₁₆	1	2	yes	—
59	C₂×C₄×C₉	#41	1	2	yes	—
60	C₂×C₄×C₉	#41	1	2	yes	—
61	C₂×C₄×C₉	#41	1	1	yes	characteristic
62	C₂×C₄×C₉	#41	1	1	yes	characteristic
63	C₂×Dic₉	—	2	1	no	—
64	C₂×Dic₉	—	2	1	no	—
65	C₂×Dic₉	—	2	1	no	—
66	C₂×Dic₉	—	2	1	no	—
67	C₂×Dic₉	—	2	1	no	—
68	C₂×Dic₉	—	2	1	no	—
69	C₂×Dic₉	—	2	1	no	—
70	C₂×Dic₉	—	2	1	no	—
71	C₂×Dic₉	C₂×C₈	1	2	yes	—
72	C₂×Dic₉	C₂×C₈	1	2	yes	—
73	C₂×Dic₉	M₁₆	1	2	yes	—
74	C₂×Dic₉	M₁₆	1	2	yes	—
75	C₈×C₉	—	4	2	no	—
76	C₉⋊C₈	—	4	2	no	—
77	C₄³	—	9	1	no	—
78	#336	—	9	2	no	—
79	#336	—	9	2	no	—
80	#336	—	9	1	no	—
81	#336	D₉	1	1	yes	characteristic
82	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
83	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
84	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
85	C₂²×Dic₃	—	3	1	no	—
86	C₂²×Dic₃	—	3	1	no	—
87	C₂²×Dic₃	—	3	1	no	—
88	C₂²×Dic₃	—	3	1	no	—
89	C₂×C₃×C₈	—	2	2	no	—
90	C₂×C₃×C₈	—	2	2	no	—
91	C₂×C₃×C₈	—	2	2	no	—
92	C₃×C₄²	—	2	2	no	—
93	C₄×Dic₃	—	3	4	no	—
94	C₄×Dic₃	—	6	2	no	—
95	C₄×Dic₃	—	6	2	no	—
96	C₄×Dic₃	—	6	2	no	—
97	C₄×Dic₃	—	6	2	no	—
98	C₄×Dic₃	—	6	1	no	—
99	C₄×Dic₃	—	6	1	no	—
100	C₂×C₃⋊C₈	—	6	2	no	—
101	C₂×C₃⋊C₈	—	6	2	no	—
102	C₂×C₃⋊C₈	—	6	2	no	—
103	C₂²×C₉	#117	1	1	yes	characteristic
104	C₂²×C₉	#117	1	1	yes	characteristic
105	C₂²×C₉	#117	1	1	yes	characteristic
106	C₂²×C₉	C₂²×C₈	1	1	yes	characteristic
107	C₂²×C₉	C₂×M₁₆	1	1	yes	characteristic
108	C₂²×C₉	C₂×M₁₆	1	1	yes	characteristic
109	C₂²×C₉	C₂×M₁₆	1	1	yes	characteristic
110	C₄×C₉	—	2	2	no	—
111	C₄×C₉	—	2	2	no	—
112	C₄×C₉	#126	1	2	yes	—
113	C₄×C₉	#126	1	2	yes	—
114	Dic₉	—	2	2	no	—
115	Dic₉	—	2	2	no	—
116	Dic₉	—	2	2	no	—
117	Dic₉	—	2	2	no	—
118	C₂²×C₈	—	9	2	no	—
119	C₂²×C₈	C₂×D₉	1	2	yes	—
120	C₂×C₄²	—	9	1	no	—
121	C₂×C₄²	—	9	1	no	—
122	C₂×C₄²	—	9	1	no	—
123	C₂×C₄²	—	9	1	no	—
124	C₂×C₄²	—	9	1	no	—
125	C₂×C₄²	—	9	1	no	—
126	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
127	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
128	C₂×C₃×C₄	—	2	1	no	—
129	C₂×C₃×C₄	—	2	1	no	—
130	C₂×C₃×C₄	—	2	1	no	—
131	C₂×C₃×C₄	—	2	1	no	—
132	C₂×C₃×C₄	D₃×C₈	1	2	yes	—
133	C₂×C₃×C₄	ΓL(1,𝔽₂₅)	1	2	yes	—
134	C₂×C₃×C₄	#235	1	2	yes	—
135	C₂×C₃×C₄	#237	1	1	yes	characteristic
136	C₂×C₃×C₄	#237	1	2	yes	—
137	C₂×C₃×C₄	#239	1	1	yes	characteristic
138	C₂×Dic₃	—	3	2	no	—
139	C₂×Dic₃	—	3	2	no	—
140	C₂×Dic₃	—	3	2	no	—
141	C₂×Dic₃	—	3	2	no	—
142	C₂×Dic₃	—	6	1	no	—
143	C₂×Dic₃	—	6	1	no	—
144	C₂×Dic₃	—	6	1	no	—
145	C₂×Dic₃	—	6	1	no	—
146	C₂×Dic₃	—	6	1	no	—
147	C₂×Dic₃	—	6	1	no	—
148	C₂×Dic₃	—	6	1	no	—
149	C₂×Dic₃	—	6	1	no	—
150	C₃×C₈	—	4	2	no	—
151	C₃⋊C₈	—	12	2	no	—
152	C₂×C₉	#383	1	1	yes	characteristic
153	C₂×C₉	#431	1	1	yes	characteristic
154	C₂×C₉	#431	1	1	yes	characteristic
155	C₂×C₉	#431	1	1	yes	characteristic
156	C₂×C₉	#432	1	1	yes	characteristic
157	C₂×C₉	#432	1	1	yes	characteristic
158	C₂×C₉	#432	1	1	yes	characteristic
159	C₂²×C₄	—	9	1	no	—
160	C₂²×C₄	—	9	1	no	—
161	C₂²×C₄	—	9	1	no	—
162	C₂²×C₄	—	9	1	no	—
163	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
164	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
165	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	18	2	no	—
170	C₂×C₈	—	18	2	no	—
171	C₂×C₈	—	18	2	no	—
172	C₄²	—	2	2	no	—
173	C₄²	—	9	4	no	—
174	C₄²	—	18	2	no	—
175	C₄²	—	18	2	no	—
176	C₄²	—	18	2	no	—
177	C₄²	—	18	2	no	—
178	C₄²	—	18	1	no	—
179	C₄²	—	18	1	no	—
180	C₂²×C₃	#872	1	1	yes	characteristic
181	C₂²×C₃	#877	1	1	yes	characteristic
182	C₂²×C₃	#892	1	1	yes	characteristic
183	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
184	C₂²×C₃	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
185	C₂²×C₃	D₃×M₁₆	1	1	yes	characteristic
186	C₂²×C₃	D₃×M₁₆	1	1	yes	characteristic
187	C₃×C₄	—	2	2	no	—
188	C₃×C₄	—	2	2	no	—
189	C₃×C₄	#879	1	2	yes	—
190	C₃×C₄	#898	1	2	yes	—
191	Dic₃	—	6	2	no	—
192	Dic₃	—	6	2	no	—
193	Dic₃	—	6	2	no	—
194	Dic₃	—	6	2	no	—
195	C₉	#1839	1	1	yes	characteristic, complemented, Sylow
196	C₂³	C₂×C₄×D₉	1	1	yes	characteristic
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	9	2	no	—
202	C₂×C₄	—	9	2	no	—
203	C₂×C₄	—	9	2	no	—
204	C₂×C₄	—	9	2	no	—
205	C₂×C₄	—	18	1	no	—
206	C₂×C₄	—	18	1	no	—
207	C₂×C₄	—	18	1	no	—
208	C₂×C₄	—	18	1	no	—
209	C₂×C₄	—	18	1	no	—
210	C₂×C₄	—	18	1	no	—
211	C₂×C₄	—	18	1	no	—
212	C₂×C₄	—	18	1	no	—
213	C₂×C₄	C₈×D₉	1	2	yes	—
214	C₂×C₄	#3663	1	2	yes	—
215	C₂×C₄	#3697	1	2	yes	—
216	C₂×C₄	#3699	1	1	yes	characteristic
217	C₂×C₄	#3699	1	2	yes	—
218	C₂×C₄	#3701	1	1	yes	characteristic
219	C₈	—	4	2	no	—
220	C₈	—	36	2	no	—
221	C₂×C₃	#4651	1	1	yes	characteristic
222	C₂×C₃	#4685	1	1	yes	characteristic
223	C₂×C₃	#4705	1	1	yes	characteristic
224	C₂×C₃	#4718	1	1	yes	characteristic
225	C₂×C₃	#4719	1	1	yes	characteristic
226	C₂×C₃	#4833	1	1	yes	characteristic
227	C₂×C₃	#4835	1	1	yes	characteristic
228	C₂²	#63434	1	1	yes	characteristic
229	C₂²	#63439	1	1	yes	characteristic
230	C₂²	#63454	1	1	yes	characteristic
231	C₂²	C₂×C₈×D₉	1	1	yes	characteristic
232	C₂²	C₂×#3663	1	1	yes	characteristic
233	C₂²	M₁₆×D₉	1	1	yes	characteristic
234	C₂²	M₁₆×D₉	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	2	2	no	—
237	C₄	—	18	2	no	—
238	C₄	—	18	2	no	—
239	C₄	—	18	2	no	—
240	C₄	—	18	2	no	—
241	C₄	#63441	1	2	yes	—
242	C₄	#63460	1	2	yes	—
243	C₃	#68607	1	1	yes	characteristic
244	C₂	#10588331	1	1	yes	characteristic
245	C₂	#10588367	1	1	yes	characteristic
246	C₂	#10588386	1	1	yes	characteristic
247	C₂	#10588398	1	1	yes	characteristic
248	C₂	#10588399	1	1	yes	characteristic
249	C₂	#10588517	1	1	yes	characteristic
250	C₂	#10588519	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₉
commutator	C₂²×C₉	C₂²×C₈
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₉×#336	C₂
socle, prime radical	C₂³×C₃	C₂×C₄×D₃
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	#336	D₉
hypocenter	C₉	#1839
LCS	G ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₄ ◁ #336

Complements

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

N	H ≅ G/N	type
38. C₄×Dic₉	C₈	semidirect
195. C₉	#1839	semidirect

Maximal subgroups

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

subgroup	index
C₄²×Dic₉	2
C₉×#336	2
#10588228	2
#10588200	2
#68607	3