Subgroups — #11789737

Order 1152.

384 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#103	C₂	1	1	yes	maximal, characteristic
2	#10588909	C₂	1	1	yes	maximal, characteristic
3	C₂×#63395	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×Dic₉	C₂	1	1	yes	maximal, characteristic
5	#10588652	C₂	1	1	yes	maximal, characteristic
6	#10588229	C₂	1	1	yes	maximal, characteristic
7	#10588206	C₂	1	1	yes	maximal, characteristic
8	#70254	—	3	1	no	maximal
9	C₂²×Q₈×C₉	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
14	C₂×Dic₉⋊C₄	—	2	1	no	—
15	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
16	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₉×#103	—	2	2	no	—
18	#63395	C₄	1	4	yes	complemented
19	C₉⋊#95	—	2	1	no	—
20	C₉⋊#95	—	2	1	no	—
21	C₉⋊#95	—	2	1	no	—
22	C₈×Dic₉	—	2	2	no	—
23	C₂×C₃×#103	D₃	1	1	yes	characteristic
24	#5229	—	3	1	no	—
25	C₂×#835	—	3	1	no	—
26	C₂×C₈×Dic₃	—	3	1	no	—
27	#4972	—	3	1	no	—
28	#4551	—	3	1	no	—
29	#4529	—	3	1	no	—
30	C₂×Q₈×C₉	—	2	1	no	—
31	C₂×Q₈×C₉	—	2	1	no	—
32	C₂×Q₈×C₉	—	2	1	no	—
33	C₂×Q₈×C₉	—	2	1	no	—
34	C₂×Q₈×C₉	—	2	1	no	—
35	C₂×Q₈×C₉	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₉	—	2	1	no	—
37	C₂²×C₄×C₉	—	2	1	no	—
38	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
39	C₂²×Dic₉	—	2	1	no	—
40	C₂²×Dic₉	—	2	1	no	—
41	C₂²×Dic₉	D₄	1	1	yes	characteristic
42	C₂²×Dic₉	D₄	1	1	yes	characteristic
43	C₂×C₈×C₉	—	2	1	no	—
44	C₂×C₈×C₉	—	2	1	no	—
45	C₂×C₈×C₉	D₄	1	2	yes	—
46	C₉×C₄⋊C₄	—	2	1	no	—
47	C₉×C₄⋊C₄	—	2	1	no	—
48	C₄⋊Dic₉	—	2	1	no	—
49	C₄⋊Dic₉	—	2	1	no	—
50	C₄⋊Dic₉	—	2	1	no	—
51	C₄⋊Dic₉	C₂×C₄	1	2	yes	—
52	Dic₉⋊C₄	—	4	2	no	—
53	C₄×Dic₉	—	2	1	no	—
54	C₄×Dic₉	—	2	1	no	—
55	C₂×C₉⋊C₈	—	2	1	no	—
56	C₂×C₉⋊C₈	—	2	1	no	—
57	C₂×C₉⋊C₈	C₂×C₄	1	2	yes	—
58	#1933	—	9	1	no	Sylow
59	C₂²×C₃×Q₈	C₂×D₃	1	1	yes	characteristic
60	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
61	C₂×C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
62	C₂×C₄⋊Dic₃	—	3	1	no	—
63	C₂×C₄⋊Dic₃	—	3	1	no	—
64	C₂×Dic₃⋊C₄	—	6	1	no	—
65	C₂×C₄×Dic₃	—	3	1	no	—
66	C₂²×C₃⋊C₈	—	3	1	no	—
67	C₃×#103	—	2	2	no	—
68	#835	—	3	4	no	—
69	C₃⋊#95	—	6	1	no	—
70	C₃⋊#95	—	6	1	no	—
71	C₃⋊#95	—	6	1	no	—
72	C₈×Dic₃	—	6	2	no	—
73	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
74	Q₈×C₉	—	2	2	no	—
75	Q₈×C₉	—	2	2	no	—
76	Q₈×C₉	—	4	1	no	—
77	Q₈×C₉	—	4	1	no	—
78	C₂×C₄×C₉	—	2	2	no	—
79	C₂×C₄×C₉	—	4	1	no	—
80	C₂×C₄×C₉	—	4	1	no	—
81	C₂×C₄×C₉	—	4	1	no	—
82	C₂×C₄×C₉	—	4	1	no	—
83	C₂×C₄×C₉	—	4	1	no	—
84	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₉	#41	1	1	yes	characteristic
87	C₂×C₄×C₉	#41	1	1	yes	characteristic
88	C₂×C₄×C₉	#41	1	1	yes	characteristic
89	C₂×C₄×C₉	#41	1	1	yes	characteristic
90	C₂×Dic₉	—	2	1	no	—
91	C₂×Dic₉	—	2	1	no	—
92	C₂×Dic₉	—	2	1	no	—
93	C₂×Dic₉	—	2	1	no	—
94	C₂×Dic₉	—	2	2	no	—
95	C₂×Dic₉	—	4	1	no	—
96	C₂×Dic₉	—	4	1	no	—
97	C₂×Dic₉	—	4	1	no	—
98	C₂×Dic₉	—	4	1	no	—
99	C₂×Dic₉	—	4	1	no	—
100	C₂×Dic₉	SD₁₆	1	2	yes	—
101	C₂×Dic₉	Q₁₆	1	2	yes	—
102	C₈×C₉	—	2	2	no	—
103	C₉⋊C₈	—	2	2	no	—
104	C₂×#103	—	9	1	no	—
105	C₂×#103	D₉	1	1	yes	characteristic
106	C₂×C₄×C₈	—	9	1	no	—
107	#391	—	9	1	no	—
108	#389	—	9	1	no	—
109	#340	—	9	1	no	—
110	#340	—	9	1	no	—
111	C₂×C₃×Q₈	—	2	1	no	—
112	C₂×C₃×Q₈	—	2	1	no	—
113	C₂×C₃×Q₈	—	2	1	no	—
114	C₂×C₃×Q₈	—	2	1	no	—
115	C₂×C₃×Q₈	—	2	1	no	—
116	C₂×C₃×Q₈	C₄×D₃	1	2	yes	—
117	C₂²×C₃×C₄	—	2	1	no	—
118	C₂²×C₃×C₄	—	2	1	no	—
119	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
120	C₂²×Dic₃	—	3	1	no	—
121	C₂²×Dic₃	—	3	1	no	—
122	C₂²×Dic₃	—	6	1	no	—
123	C₂²×Dic₃	—	6	1	no	—
124	C₂×C₃×C₈	—	2	1	no	—
125	C₂×C₃×C₈	—	2	1	no	—
126	C₂×C₃×C₈	#67	1	2	yes	—
127	C₃×C₄⋊C₄	—	2	1	no	—
128	C₃×C₄⋊C₄	—	2	1	no	—
129	C₄⋊Dic₃	—	3	2	no	—
130	C₄⋊Dic₃	—	6	1	no	—
131	C₄⋊Dic₃	—	6	1	no	—
132	C₄⋊Dic₃	—	6	1	no	—
133	Dic₃⋊C₄	—	12	2	no	—
134	C₄×Dic₃	—	6	1	no	—
135	C₄×Dic₃	—	6	1	no	—
136	C₂×C₃⋊C₈	—	3	2	no	—
137	C₂×C₃⋊C₈	—	6	1	no	—
138	C₂×C₃⋊C₈	—	6	1	no	—
139	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
140	C₂²×C₉	#121	1	1	yes	characteristic
141	C₂²×C₉	#124	1	1	yes	characteristic
142	C₂²×C₉	C₂×SD₁₆	1	1	yes	characteristic
143	C₂²×C₉	C₂×Q₁₆	1	1	yes	characteristic
144	C₂²×C₉	#135	1	1	yes	characteristic
145	C₂²×C₉	#135	1	1	yes	characteristic
146	C₄×C₉	—	4	2	no	—
147	C₄×C₉	—	8	1	no	—
148	C₄×C₉	#117	1	1	yes	characteristic
149	C₄×C₉	C₄×D₄	1	1	yes	characteristic
150	C₄×C₉	#123	1	1	yes	characteristic
151	C₄×C₉	#126	1	1	yes	characteristic
152	Dic₉	—	2	2	no	—
153	Dic₉	—	2	2	no	—
154	Dic₉	—	4	2	no	—
155	Dic₉	—	8	1	no	—
156	C₂²×Q₈	C₂×D₉	1	1	yes	characteristic
157	C₂²×C₈	—	9	1	no	—
158	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
159	C₂×C₄⋊C₄	—	9	1	no	—
160	C₂×C₄⋊C₄	—	9	1	no	—
161	C₂×C₄⋊C₄	—	18	1	no	—
162	C₂×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
163	C₂×C₄²	—	9	1	no	—
164	#103	—	2	2	no	—
165	#103	—	9	4	no	—
166	C₄×C₈	—	18	2	no	—
167	#95	—	18	1	no	—
168	#95	—	18	1	no	—
169	#95	—	18	1	no	—
170	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
171	C₃×Q₈	—	2	2	no	—
172	C₃×Q₈	—	2	2	no	—
173	C₃×Q₈	—	4	1	no	—
174	C₃×Q₈	—	4	1	no	—
175	C₂×C₃×C₄	—	2	2	no	—
176	C₂×C₃×C₄	—	4	1	no	—
177	C₂×C₃×C₄	—	4	1	no	—
178	C₂×C₃×C₄	—	4	1	no	—
179	C₂×C₃×C₄	—	4	1	no	—
180	C₂×C₃×C₄	—	4	1	no	—
181	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
182	C₂×C₃×C₄	#235	1	1	yes	characteristic
183	C₂×C₃×C₄	#237	1	1	yes	characteristic
184	C₂×C₃×C₄	#237	1	1	yes	characteristic
185	C₂×C₃×C₄	#237	1	1	yes	characteristic
186	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
187	C₂×Dic₃	—	3	2	no	—
188	C₂×Dic₃	—	3	2	no	—
189	C₂×Dic₃	—	6	1	no	—
190	C₂×Dic₃	—	6	1	no	—
191	C₂×Dic₃	—	6	1	no	—
192	C₂×Dic₃	—	6	1	no	—
193	C₂×Dic₃	—	6	2	no	—
194	C₂×Dic₃	—	12	1	no	—
195	C₂×Dic₃	—	12	1	no	—
196	C₂×Dic₃	—	12	1	no	—
197	C₂×Dic₃	—	12	1	no	—
198	C₂×Dic₃	—	12	1	no	—
199	C₃×C₈	—	2	2	no	—
200	C₃⋊C₈	—	6	2	no	—
201	C₂×C₉	#388	1	1	yes	characteristic
202	C₂×C₉	C₄×SD₁₆	1	1	yes	characteristic
203	C₂×C₉	C₄×Q₁₆	1	1	yes	characteristic
204	C₂×C₉	#465	1	1	yes	characteristic
205	C₂×C₉	#467	1	1	yes	characteristic
206	C₂×C₉	#487	1	1	yes	characteristic
207	C₂×C₉	#488	1	1	yes	characteristic
208	C₂×Q₈	—	2	1	no	—
209	C₂×Q₈	—	2	1	no	—
210	C₂×Q₈	—	2	1	no	—
211	C₂×Q₈	—	2	1	no	—
212	C₂×Q₈	—	2	1	no	—
213	C₂×Q₈	C₄×D₉	1	2	yes	—
214	C₂²×C₄	—	2	1	no	—
215	C₂²×C₄	—	2	1	no	—
216	C₂²×C₄	—	9	1	no	—
217	C₂²×C₄	—	9	1	no	—
218	C₂²×C₄	—	18	1	no	—
219	C₂²×C₄	—	18	1	no	—
220	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
221	C₂×C₈	—	2	1	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	9	2	no	—
224	C₂×C₈	—	18	1	no	—
225	C₂×C₈	—	18	1	no	—
226	C₂×C₈	#611	1	2	yes	—
227	C₄⋊C₄	—	2	1	no	—
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	9	2	no	—
230	C₄⋊C₄	—	18	1	no	—
231	C₄⋊C₄	—	18	1	no	—
232	C₄⋊C₄	—	18	1	no	—
233	C₄⋊C₄	—	36	2	no	—
234	C₄²	—	18	1	no	—
235	C₄²	—	18	1	no	—
236	C₂²×C₃	#67⋊C₄	1	1	yes	characteristic
237	C₂²×C₃	#885	1	1	yes	characteristic
238	C₂²×C₃	D₃×SD₁₆	1	1	yes	characteristic
239	C₂²×C₃	#916	1	1	yes	characteristic
240	C₂²×C₃	D₃×Q₁₆	1	1	yes	characteristic
241	C₂²×C₃	#919	1	1	yes	characteristic
242	C₂²×C₃	#938	1	1	yes	characteristic
243	C₃×C₄	—	4	2	no	—
244	C₃×C₄	—	8	1	no	—
245	C₃×C₄	#877	1	1	yes	characteristic
246	C₃×C₄	#879	1	1	yes	characteristic
247	C₃×C₄	C₄×#67	1	1	yes	characteristic
248	C₃×C₄	#935	1	1	yes	characteristic
249	Dic₃	—	6	2	no	—
250	Dic₃	—	6	2	no	—
251	Dic₃	—	12	2	no	—
252	Dic₃	—	24	1	no	—
253	C₉	#1933	1	1	yes	characteristic, complemented, Sylow
254	C₂³	D₄×D₉	1	1	yes	characteristic
255	Q₈	—	2	2	no	—
256	Q₈	—	2	2	no	—
257	Q₈	—	4	1	no	—
258	Q₈	—	4	1	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	4	1	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	1	no	—
265	C₂×C₄	—	9	2	no	—
266	C₂×C₄	—	9	2	no	—
267	C₂×C₄	—	18	1	no	—
268	C₂×C₄	—	18	1	no	—
269	C₂×C₄	—	18	1	no	—
270	C₂×C₄	—	18	1	no	—
271	C₂×C₄	—	18	2	no	—
272	C₂×C₄	—	36	1	no	—
273	C₂×C₄	—	36	1	no	—
274	C₂×C₄	—	36	1	no	—
275	C₂×C₄	—	36	1	no	—
276	C₂×C₄	—	36	1	no	—
277	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
278	C₂×C₄	#3697	1	1	yes	characteristic
279	C₂×C₄	#3699	1	1	yes	characteristic
280	C₂×C₄	#3699	1	1	yes	characteristic
281	C₂×C₄	#3699	1	1	yes	characteristic
282	C₂×C₄	C₂×#611	1	1	yes	characteristic
283	C₈	—	2	2	no	—
284	C₈	—	18	2	no	—
285	C₂×C₃	#4788	1	1	yes	characteristic
286	C₂×C₃	#4790	1	1	yes	characteristic
287	C₂×C₃	#4792	1	1	yes	characteristic
288	C₂×C₃	#4800	1	1	yes	characteristic
289	C₂×C₃	#4944	1	1	yes	characteristic
290	C₂×C₃	#5171	1	1	yes	characteristic
291	C₂×C₃	#5188	1	1	yes	characteristic
292	C₂²	#611⋊C₄	1	1	yes	characteristic
293	C₂²	#63447	1	1	yes	characteristic
294	C₂²	SD₁₆×D₉	1	1	yes	characteristic
295	C₂²	#63478	1	1	yes	characteristic
296	C₂²	Q₁₆×D₉	1	1	yes	characteristic
297	C₂²	#63481	1	1	yes	characteristic
298	C₂²	#63500	1	1	yes	characteristic
299	C₄	—	4	2	no	—
300	C₄	—	8	1	no	—
301	C₄	—	18	2	no	—
302	C₄	—	18	2	no	—
303	C₄	—	36	2	no	—
304	C₄	—	72	1	no	—
305	C₄	#63439	1	1	yes	characteristic
306	C₄	#63441	1	1	yes	characteristic
307	C₄	C₄×#611	1	1	yes	characteristic
308	C₄	#63497	1	1	yes	characteristic
309	C₃	#70254	1	1	yes	characteristic
310	C₂	#10588468	1	1	yes	characteristic
311	C₂	#10588470	1	1	yes	characteristic
312	C₂	#10588472	1	1	yes	characteristic
313	C₂	#10588480	1	1	yes	characteristic
314	C₂	#10588624	1	1	yes	characteristic
315	C₂	#10588851	1	1	yes	characteristic
316	C₂	#10588868	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×D₉
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂×C₉×#103	C₂
socle, prime radical	C₂³×C₃	D₃×D₄
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂×#103	D₉
hypocenter	C₉	#1933
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#103

Complements

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

N	H ≅ G/N	type
18. #63395	C₄	semidirect
253. C₉	#1933	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×#103	2
#10588909	2
C₂×#63395	2
C₂×C₈×Dic₉	2
#10588652	2
#10588229	2
#10588206	2
#70254	3