Subgroups — #11790086

Order 1152.

372 conjugacy classes of subgroups.

The subgroup lattice has 297 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#389	C₂	1	1	yes	maximal, characteristic
2	C₂×#63367	C₂	1	1	yes	maximal, characteristic
3	C₂×#63367	C₂	1	1	yes	maximal, characteristic
4	#10588608	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₉⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#10588206	C₂	1	1	yes	maximal, characteristic
7	#10588206	C₂	1	1	yes	maximal, characteristic
8	#69618	—	3	1	no	maximal
9	C₂×C₉×C₄⋊C₄	—	2	1	no	—
10	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
11	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₉	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
15	C₂×Dic₉⋊C₄	—	2	1	no	—
16	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
18	C₉×#95	—	2	1	no	—
19	C₉⋊#95	—	2	1	no	—
20	#63367	—	2	2	no	—
21	#63367	C₄	1	4	yes	complemented
22	C₄×C₉⋊C₈	—	2	2	no	—
23	C₃×#389	D₃	1	1	yes	characteristic
24	C₂×#808	—	3	1	no	—
25	C₂×#808	—	3	1	no	—
26	#4928	—	3	1	no	—
27	C₂×C₄×C₃⋊C₈	—	3	1	no	—
28	#4529	—	3	1	no	—
29	#4529	—	3	1	no	—
30	C₂²×C₄×C₉	—	2	1	no	—
31	C₂²×C₄×C₉	—	2	1	no	—
32	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₉	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
35	C₂²×Dic₉	—	2	1	no	—
36	C₂²×Dic₉	—	2	1	no	—
37	C₉×C₄⋊C₄	—	2	1	no	—
38	C₉×C₄⋊C₄	—	2	1	no	—
39	C₉×C₄⋊C₄	—	2	1	no	—
40	C₉×C₄⋊C₄	—	4	2	no	—
41	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
42	C₄²×C₉	—	2	1	no	—
43	C₄²×C₉	—	2	1	no	—
44	C₄⋊Dic₉	—	2	1	no	—
45	C₄⋊Dic₉	—	2	1	no	—
46	C₄⋊Dic₉	—	2	1	no	—
47	C₄⋊Dic₉	C₂×C₄	1	2	yes	—
48	Dic₉⋊C₄	—	4	2	no	—
49	C₂×C₉⋊C₈	—	2	1	no	—
50	C₂×C₉⋊C₈	—	2	1	no	—
51	C₂×C₉⋊C₈	C₂×C₄	1	2	yes	—
52	C₂×C₉⋊C₈	D₄	1	2	yes	—
53	C₂×C₉⋊C₈	Q₈	1	2	yes	—
54	C₂×C₉⋊C₈	Q₈	1	2	yes	—
55	#1941	—	9	1	no	Sylow
56	C₂×C₃×C₄⋊C₄	—	2	1	no	—
57	C₂×C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
58	C₂×C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
59	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
60	C₂×C₄⋊Dic₃	—	3	1	no	—
61	C₂×C₄⋊Dic₃	—	3	1	no	—
62	C₂×Dic₃⋊C₄	—	6	1	no	—
63	C₂²×C₃⋊C₈	—	3	1	no	—
64	C₂²×C₃⋊C₈	—	3	1	no	—
65	C₃×#95	—	2	1	no	—
66	C₃⋊#95	—	6	1	no	—
67	#808	—	3	4	no	—
68	#808	—	6	2	no	—
69	C₄×C₃⋊C₈	—	6	2	no	—
70	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₄×C₉	—	2	1	no	—
72	C₂×C₄×C₉	—	2	1	no	—
73	C₂×C₄×C₉	—	2	1	no	—
74	C₂×C₄×C₉	—	2	1	no	—
75	C₂×C₄×C₉	—	2	2	no	—
76	C₂×C₄×C₉	—	4	1	no	—
77	C₂×C₄×C₉	—	4	1	no	—
78	C₂×C₄×C₉	—	4	1	no	—
79	C₂×C₄×C₉	—	4	1	no	—
80	C₂×C₄×C₉	—	4	1	no	—
81	C₂×C₄×C₉	SD₁₆	1	2	yes	—
82	C₂×C₄×C₉	SD₁₆	1	2	yes	—
83	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
86	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
87	C₂×C₄×C₉	#41	1	1	yes	characteristic
88	C₂×C₄×C₉	#41	1	1	yes	characteristic
89	C₂×Dic₉	—	2	2	no	—
90	C₂×Dic₉	—	4	1	no	—
91	C₂×Dic₉	—	4	1	no	—
92	C₂×Dic₉	—	4	1	no	—
93	C₂×Dic₉	—	4	1	no	—
94	C₂×Dic₉	—	4	1	no	—
95	C₉⋊C₈	—	2	2	no	—
96	C₉⋊C₈	C₄⋊C₄	1	4	yes	complemented
97	C₂×#106	—	9	1	no	—
98	C₂×#106	—	9	1	no	—
99	C₂×C₄×C₈	—	9	1	no	—
100	#389	—	9	1	no	—
101	#389	D₉	1	1	yes	characteristic
102	#340	—	9	1	no	—
103	#340	—	9	1	no	—
104	C₂²×C₃×C₄	—	2	1	no	—
105	C₂²×C₃×C₄	—	2	1	no	—
106	C₂²×C₃×C₄	#67	1	1	yes	characteristic
107	C₂²×C₃×C₄	#67	1	1	yes	characteristic
108	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
109	C₂²×Dic₃	—	6	1	no	—
110	C₂²×Dic₃	—	6	1	no	—
111	C₃×C₄⋊C₄	—	2	1	no	—
112	C₃×C₄⋊C₄	—	2	1	no	—
113	C₃×C₄⋊C₄	—	2	1	no	—
114	C₃×C₄⋊C₄	—	4	2	no	—
115	C₃×C₄⋊C₄	C₄×D₃	1	2	yes	—
116	C₃×C₄²	—	2	1	no	—
117	C₃×C₄²	—	2	1	no	—
118	C₄⋊Dic₃	—	3	2	no	—
119	C₄⋊Dic₃	—	6	1	no	—
120	C₄⋊Dic₃	—	6	1	no	—
121	C₄⋊Dic₃	—	6	1	no	—
122	Dic₃⋊C₄	—	12	2	no	—
123	C₂×C₃⋊C₈	—	3	2	no	—
124	C₂×C₃⋊C₈	—	3	2	no	—
125	C₂×C₃⋊C₈	—	3	2	no	—
126	C₂×C₃⋊C₈	—	3	2	no	—
127	C₂×C₃⋊C₈	—	6	1	no	—
128	C₂×C₃⋊C₈	—	6	1	no	—
129	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
130	C₂²×C₉	#121	1	1	yes	characteristic
131	C₂²×C₉	C₄⋊Q₈	1	1	yes	characteristic
132	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
133	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
134	C₂²×C₉	#135	1	1	yes	characteristic
135	C₂²×C₉	#135	1	1	yes	characteristic
136	C₄×C₉	—	2	2	no	—
137	C₄×C₉	—	2	2	no	—
138	C₄×C₉	—	4	2	no	—
139	C₄×C₉	—	8	1	no	—
140	C₄×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₄×C₉	C₄×Q₈	1	1	yes	characteristic
142	C₄×C₉	C₂²⋊Q₈	1	1	yes	characteristic
143	C₄×C₉	#125	1	1	yes	characteristic
144	Dic₉	—	4	2	no	—
145	Dic₉	—	8	1	no	—
146	C₂²×C₈	—	9	1	no	—
147	C₂²×C₈	—	9	1	no	—
148	C₂×C₄⋊C₄	—	2	1	no	—
149	C₂×C₄⋊C₄	—	9	1	no	—
150	C₂×C₄⋊C₄	—	9	1	no	—
151	C₂×C₄⋊C₄	—	18	1	no	—
152	C₂×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
153	C₂×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
154	C₂×C₄²	C₂×D₉	1	1	yes	characteristic
155	#106	—	9	4	no	—
156	#106	—	18	2	no	—
157	C₄×C₈	—	18	2	no	—
158	#95	—	2	1	no	—
159	#95	—	18	1	no	—
160	C₂³×C₃	C₂×#67	1	1	yes	characteristic
161	C₂×C₃×C₄	—	2	1	no	—
162	C₂×C₃×C₄	—	2	1	no	—
163	C₂×C₃×C₄	—	2	1	no	—
164	C₂×C₃×C₄	—	2	1	no	—
165	C₂×C₃×C₄	—	2	2	no	—
166	C₂×C₃×C₄	—	4	1	no	—
167	C₂×C₃×C₄	—	4	1	no	—
168	C₂×C₃×C₄	—	4	1	no	—
169	C₂×C₃×C₄	—	4	1	no	—
170	C₂×C₃×C₄	—	4	1	no	—
171	C₂×C₃×C₄	#214	1	2	yes	—
172	C₂×C₃×C₄	#215	1	2	yes	—
173	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
174	C₂×C₃×C₄	#235	1	1	yes	characteristic
175	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
176	C₂×C₃×C₄	#237	1	1	yes	characteristic
177	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
178	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
179	C₂×Dic₃	—	6	2	no	—
180	C₂×Dic₃	—	12	1	no	—
181	C₂×Dic₃	—	12	1	no	—
182	C₂×Dic₃	—	12	1	no	—
183	C₂×Dic₃	—	12	1	no	—
184	C₂×Dic₃	—	12	1	no	—
185	C₃⋊C₈	—	3	4	no	—
186	C₃⋊C₈	—	6	2	no	—
187	C₂×C₉	#389	1	1	yes	characteristic
188	C₂×C₉	C₄×SD₁₆	1	1	yes	characteristic
189	C₂×C₉	C₄×SD₁₆	1	1	yes	characteristic
190	C₂×C₉	#465	1	1	yes	characteristic
191	C₂×C₉	#465	1	1	yes	characteristic
192	C₂×C₉	#498	1	1	yes	characteristic
193	C₂×C₉	#499	1	1	yes	characteristic
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	18	1	no	—
197	C₂²×C₄	—	18	1	no	—
198	C₂²×C₄	#611	1	1	yes	characteristic
199	C₂²×C₄	#611	1	1	yes	characteristic
200	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
201	C₂×C₈	—	9	2	no	—
202	C₂×C₈	—	9	2	no	—
203	C₂×C₈	—	9	2	no	—
204	C₂×C₈	—	9	2	no	—
205	C₂×C₈	—	18	1	no	—
206	C₂×C₈	—	18	1	no	—
207	C₄⋊C₄	—	2	1	no	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₄⋊C₄	—	9	2	no	—
212	C₄⋊C₄	—	18	1	no	—
213	C₄⋊C₄	—	18	1	no	—
214	C₄⋊C₄	—	18	1	no	—
215	C₄⋊C₄	—	36	2	no	—
216	C₄⋊C₄	C₄×D₉	1	2	yes	—
217	C₄²	—	2	1	no	—
218	C₄²	—	2	1	no	—
219	C₂²×C₃	C₄×#67	1	1	yes	characteristic
220	C₂²×C₃	C₂×#214	1	1	yes	characteristic
221	C₂²×C₃	#939	1	1	yes	characteristic
222	C₂²×C₃	C₂×#215	1	1	yes	characteristic
223	C₂²×C₃	#944	1	1	yes	characteristic
224	C₂²×C₃	#950	1	1	yes	characteristic
225	C₂²×C₃	#950	1	1	yes	characteristic
226	C₃×C₄	—	2	2	no	—
227	C₃×C₄	—	2	2	no	—
228	C₃×C₄	—	4	2	no	—
229	C₃×C₄	—	8	1	no	—
230	C₃×C₄	#887	1	1	yes	characteristic
231	C₃×C₄	#889	1	1	yes	characteristic
232	C₃×C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
233	C₃×C₄	#896	1	1	yes	characteristic
234	Dic₃	—	12	2	no	—
235	Dic₃	—	24	1	no	—
236	C₉	#1941	1	1	yes	characteristic, complemented, Sylow
237	C₂³	C₂×#611	1	1	yes	characteristic
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	1	no	—
248	C₂×C₄	—	18	2	no	—
249	C₂×C₄	—	36	1	no	—
250	C₂×C₄	—	36	1	no	—
251	C₂×C₄	—	36	1	no	—
252	C₂×C₄	—	36	1	no	—
253	C₂×C₄	—	36	1	no	—
254	C₂×C₄	#3672	1	2	yes	—
255	C₂×C₄	#3675	1	2	yes	—
256	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
257	C₂×C₄	#3697	1	1	yes	characteristic
258	C₂×C₄	D₄×D₉	1	1	yes	characteristic
259	C₂×C₄	#3699	1	1	yes	characteristic
260	C₂×C₄	Q₈×D₉	1	1	yes	characteristic
261	C₂×C₄	Q₈×D₉	1	1	yes	characteristic
262	C₈	—	9	4	no	—
263	C₈	—	18	2	no	—
264	C₂×C₃	#4976	1	1	yes	characteristic
265	C₂×C₃	C₄×#214	1	1	yes	characteristic
266	C₂×C₃	C₄×#215	1	1	yes	characteristic
267	C₂×C₃	#5041	1	1	yes	characteristic
268	C₂×C₃	#5048	1	1	yes	characteristic
269	C₂×C₃	#5065	1	1	yes	characteristic
270	C₂×C₃	#5080	1	1	yes	characteristic
271	C₂²	C₄×#611	1	1	yes	characteristic
272	C₂²	C₂×#3672	1	1	yes	characteristic
273	C₂²	#63501	1	1	yes	characteristic
274	C₂²	C₂×#3675	1	1	yes	characteristic
275	C₂²	#63506	1	1	yes	characteristic
276	C₂²	#63513	1	1	yes	characteristic
277	C₂²	#63513	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	4	2	no	—
281	C₄	—	8	1	no	—
282	C₄	—	36	2	no	—
283	C₄	—	72	1	no	—
284	C₄	#63449	1	1	yes	characteristic
285	C₄	#63451	1	1	yes	characteristic
286	C₄	C₄⋊C₄×D₉	1	1	yes	characteristic
287	C₄	#63458	1	1	yes	characteristic
288	C₃	#69618	1	1	yes	characteristic
289	C₂	#10588656	1	1	yes	characteristic
290	C₂	C₄×#3672	1	1	yes	characteristic
291	C₂	C₄×#3675	1	1	yes	characteristic
292	C₂	#10588717	1	1	yes	characteristic
293	C₂	#10588731	1	1	yes	characteristic
294	C₂	#10588744	1	1	yes	characteristic
295	C₂	#10588760	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×#611
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₉×#389	C₂
socle, prime radical	C₂³×C₃	C₂×#67
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	#389	D₉
hypocenter	C₉	#1941
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

Complements

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

N	H ≅ G/N	type
21. #63367	C₄	semidirect
96. C₉⋊C₈	C₄⋊C₄	semidirect
236. C₉	#1941	semidirect

Maximal subgroups

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

subgroup	index
C₉×#389	2
C₂×#63367	2
#10588608	2
C₂×C₄×C₉⋊C₈	2
#10588206	2
#69618	3