Subgroups — #11787014

Order 1152.

405 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₉×#131	C₂	1	1	yes	maximal, characteristic
2	#10588917	C₂	1	1	yes	maximal, characteristic
3	C₂×#63396	C₂	1	1	yes	maximal, characteristic
4	C₂×#63381	C₂	1	1	yes	maximal, characteristic, complemented
5	#10588783	C₂	1	1	yes	maximal, characteristic, complemented
6	#10588235	C₂	1	1	yes	maximal, characteristic, complemented
7	#10588228	C₂	1	1	yes	maximal, characteristic, complemented
8	#72017	—	3	1	no	maximal
9	C₂×C₉×#41	C₂²	1	1	yes	characteristic
10	C₉×#131	—	2	2	no	—
11	C₉×#131	—	2	2	no	—
12	C₂×C₉×M₁₆	—	2	1	no	—
13	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₉	—	2	1	no	—
15	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
16	#63488	C₂²	1	1	yes	characteristic
17	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
18	C₂×#3667	C₂²	1	1	yes	characteristic
19	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
20	#63396	—	2	2	no	—
21	#63395	C₄	1	1	yes	characteristic, complemented
22	#63395	C₄	1	1	yes	characteristic, complemented
23	#63392	C₄	1	1	yes	characteristic, complemented
24	#63392	C₄	1	1	yes	characteristic, complemented
25	#63381	—	2	2	no	—
26	Dic₉⋊C₈	—	2	1	no	—
27	Dic₉⋊C₈	—	2	1	no	—
28	C₂×C₃×#131	D₃	1	1	yes	characteristic
29	C₂×#837	—	3	1	no	—
30	#5233	—	3	1	no	—
31	C₂×#822	—	3	1	no	—
32	#5102	—	3	1	no	—
33	#4555	—	3	1	no	—
34	#4550	—	3	1	no	—
35	C₉×#41	—	2	1	no	—
36	C₉×#41	—	2	1	no	—
37	C₉×#41	D₄	1	2	yes	—
38	C₉×#41	Q₈	1	2	yes	—
39	C₂×Q₈×C₉	C₂×C₄	1	1	yes	characteristic
40	C₂×D₄×C₉	—	2	1	no	—
41	C₂×D₄×C₉	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₄×C₉	—	2	1	no	—
43	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
44	C₂²×Dic₉	—	2	1	no	—
45	C₉×M₁₆	—	2	4	no	—
46	C₉×M₁₆	—	2	1	no	—
47	C₉×M₁₆	D₄	1	2	yes	—
48	C₂×C₈×C₉	—	2	4	no	—
49	C₂×C₈×C₉	—	2	1	no	—
50	C₂×C₈×C₉	—	2	1	no	—
51	C₂×C₈×C₉	—	2	1	no	—
52	C₂×C₈×C₉	D₄	1	1	yes	characteristic
53	C₂×C₈×C₉	D₄	1	1	yes	characteristic
54	#3676	—	2	1	no	—
55	C₄⋊Dic₉	C₂×C₄	1	1	yes	characteristic
56	C₄⋊Dic₉	C₂×C₄	1	1	yes	characteristic
57	C₄×Dic₉	—	2	1	no	—
58	C₄×Dic₉	—	2	1	no	—
59	C₄×Dic₉	—	2	1	no	—
60	#3667	—	2	1	no	—
61	#3667	—	2	1	no	—
62	C₂×C₉⋊C₈	—	2	1	no	—
63	C₂×C₉⋊C₈	—	2	1	no	—
64	C₂×C₉⋊C₈	—	2	1	no	—
65	C₂×C₉⋊C₈	C₂×C₄	1	1	yes	characteristic
66	C₂×C₉⋊C₈	C₂×C₄	1	1	yes	characteristic
67	#1866	—	9	1	no	Sylow
68	C₂×C₃×#41	C₂×D₃	1	1	yes	characteristic
69	C₃×#131	—	2	2	no	—
70	C₃×#131	—	2	2	no	—
71	C₂×C₃×M₁₆	—	2	1	no	—
72	C₂×C₃×M₁₆	C₂×D₃	1	1	yes	characteristic
73	C₂²×C₃×C₈	—	2	1	no	—
74	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
75	#926	—	3	1	no	—
76	C₂×C₄×Dic₃	—	3	1	no	—
77	C₂×#208	—	3	1	no	—
78	C₂²×C₃⋊C₈	—	3	1	no	—
79	#837	—	6	2	no	—
80	#835	—	3	1	no	—
81	#835	—	3	1	no	—
82	#832	—	3	1	no	—
83	#832	—	3	1	no	—
84	#822	—	6	2	no	—
85	Dic₃⋊C₈	—	6	1	no	—
86	Dic₃⋊C₈	—	6	1	no	—
87	C₂³×C₉	—	2	1	no	—
88	C₂³×C₉	#41	1	1	yes	characteristic
89	Q₈×C₉	—	2	1	no	—
90	Q₈×C₉	C₄⋊C₄	1	2	yes	—
91	D₄×C₉	—	2	2	no	—
92	D₄×C₉	—	2	2	no	—
93	D₄×C₉	—	2	1	no	—
94	D₄×C₉	C₄⋊C₄	1	2	yes	—
95	C₂×C₄×C₉	—	2	1	no	—
96	C₂×C₄×C₉	—	2	2	no	—
97	C₂×C₄×C₉	—	2	2	no	—
98	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
103	C₂×C₄×C₉	#41	1	1	yes	characteristic
104	C₂×Dic₉	—	2	2	no	—
105	C₂×Dic₉	—	2	1	no	—
106	C₂×Dic₉	—	2	1	no	—
107	C₂×Dic₉	—	4	1	no	—
108	C₂×Dic₉	—	4	1	no	—
109	C₈×C₉	—	2	1	no	—
110	C₈×C₉	—	2	2	no	—
111	C₈×C₉	—	2	1	no	—
112	C₉⋊C₈	—	2	1	no	—
113	C₉⋊C₈	—	2	1	no	—
114	C₉⋊C₈	—	4	1	no	—
115	C₂×#131	D₉	1	1	yes	characteristic
116	C₂×#108	—	9	1	no	—
117	#424	—	9	1	no	—
118	C₂×#104	—	9	1	no	—
119	#416	—	9	1	no	—
120	#339	—	9	1	no	—
121	#336	—	9	1	no	—
122	C₃×#41	—	2	1	no	—
123	C₃×#41	—	2	1	no	—
124	C₃×#41	Dic₆	1	2	yes	—
125	C₃×#41	#67	1	2	yes	—
126	C₂×C₃×Q₈	C₄×D₃	1	1	yes	characteristic
127	C₂×C₃×D₄	—	2	1	no	—
128	C₂×C₃×D₄	C₄×D₃	1	1	yes	characteristic
129	C₂²×C₃×C₄	—	2	1	no	—
130	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
131	C₂²×Dic₃	—	6	1	no	—
132	C₃×M₁₆	—	2	4	no	—
133	C₃×M₁₆	—	2	1	no	—
134	C₃×M₁₆	#67	1	2	yes	—
135	C₂×C₃×C₈	—	2	4	no	—
136	C₂×C₃×C₈	—	2	1	no	—
137	C₂×C₃×C₈	—	2	1	no	—
138	C₂×C₃×C₈	—	2	1	no	—
139	C₂×C₃×C₈	#67	1	1	yes	characteristic
140	C₂×C₃×C₈	#67	1	1	yes	characteristic
141	#217	—	6	1	no	—
142	C₄⋊Dic₃	—	3	1	no	—
143	C₄⋊Dic₃	—	3	1	no	—
144	C₄×Dic₃	—	6	1	no	—
145	C₄×Dic₃	—	6	1	no	—
146	C₄×Dic₃	—	6	1	no	—
147	#208	—	6	1	no	—
148	#208	—	6	1	no	—
149	C₂×C₃⋊C₈	—	3	1	no	—
150	C₂×C₃⋊C₈	—	3	1	no	—
151	C₂×C₃⋊C₈	—	6	1	no	—
152	C₂×C₃⋊C₈	—	6	1	no	—
153	C₂×C₃⋊C₈	—	6	1	no	—
154	C₂²×C₉	—	2	1	no	—
155	C₂²×C₉	—	2	1	no	—
156	C₂²×C₉	—	2	2	no	—
157	C₂²×C₉	—	4	1	no	—
158	C₂²×C₉	—	4	1	no	—
159	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
160	C₂²×C₉	C₂²⋊Q₈	1	1	yes	characteristic
161	C₂²×C₉	#123	1	1	yes	characteristic
162	C₄×C₉	—	2	2	no	—
163	C₄×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₄×C₉	C₄×D₄	1	1	yes	characteristic
165	C₄×C₉	#120	1	1	yes	characteristic
166	C₄×C₉	C₂²⋊Q₈	1	1	yes	characteristic
167	Dic₉	—	4	2	no	—
168	Dic₉	—	4	1	no	—
169	Dic₉	—	4	1	no	—
170	C₂×#41	C₂×D₉	1	1	yes	characteristic
171	#131	—	2	2	no	—
172	#131	—	2	2	no	—
173	C₂×M₁₆	—	2	1	no	—
174	C₂×M₁₆	—	9	1	no	—
175	C₂×M₁₆	C₂×D₉	1	1	yes	characteristic
176	C₂²×C₈	—	2	1	no	—
177	C₂²×C₈	—	9	1	no	—
178	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
179	#117	—	9	1	no	—
180	C₂×C₄²	—	9	1	no	—
181	#108	—	18	2	no	—
182	C₄⋊C₈	—	18	1	no	—
183	C₄⋊C₈	—	18	1	no	—
184	#104	—	18	2	no	—
185	#103	—	9	1	no	—
186	#103	—	9	1	no	—
187	#102	—	9	1	no	—
188	#102	—	9	1	no	—
189	C₂³×C₃	—	2	1	no	—
190	C₂³×C₃	#235	1	1	yes	characteristic
191	C₃×Q₈	—	2	1	no	—
192	C₃×Q₈	Dic₃⋊C₄	1	2	yes	—
193	C₃×D₄	—	2	2	no	—
194	C₃×D₄	—	2	2	no	—
195	C₃×D₄	—	2	1	no	—
196	C₃×D₄	Dic₃⋊C₄	1	2	yes	—
197	C₂×C₃×C₄	—	2	1	no	—
198	C₂×C₃×C₄	—	2	2	no	—
199	C₂×C₃×C₄	—	2	2	no	—
200	C₂×C₃×C₄	C₂×Dic₆	1	1	yes	characteristic
201	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
202	C₂×C₃×C₄	#235	1	1	yes	characteristic
203	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
206	C₂×Dic₃	—	6	2	no	—
207	C₂×Dic₃	—	6	1	no	—
208	C₂×Dic₃	—	6	1	no	—
209	C₂×Dic₃	—	12	1	no	—
210	C₂×Dic₃	—	12	1	no	—
211	C₃×C₈	—	2	1	no	—
212	C₃×C₈	—	2	2	no	—
213	C₃×C₈	—	2	1	no	—
214	C₃⋊C₈	—	6	1	no	—
215	C₃⋊C₈	—	6	1	no	—
216	C₃⋊C₈	—	12	1	no	—
217	C₂×C₉	—	2	1	no	—
218	C₂×C₉	—	2	1	no	—
219	C₂×C₉	—	4	2	no	—
220	C₂×C₉	#385	1	1	yes	characteristic
221	C₂×C₉	#443	1	1	yes	characteristic
222	C₂×C₉	#444	1	1	yes	characteristic
223	#41	—	2	1	no	—
224	#41	—	2	1	no	—
225	#41	Dic₁₈	1	2	yes	—
226	#41	#611	1	2	yes	—
227	C₂×Q₈	C₄×D₉	1	1	yes	characteristic
228	C₂×D₄	—	2	1	no	—
229	C₂×D₄	C₄×D₉	1	1	yes	characteristic
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	18	1	no	—
232	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
233	M₁₆	—	2	4	no	—
234	M₁₆	—	2	1	no	—
235	M₁₆	—	18	1	no	—
236	M₁₆	—	18	1	no	—
237	M₁₆	#611	1	2	yes	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	9	1	no	—
243	C₂×C₈	—	9	1	no	—
244	C₂×C₈	—	18	1	no	—
245	C₂×C₈	—	18	1	no	—
246	C₂×C₈	—	18	1	no	—
247	C₂×C₈	#611	1	1	yes	characteristic
248	C₂×C₈	#611	1	1	yes	characteristic
249	C₄⋊C₄	—	9	1	no	—
250	C₄⋊C₄	—	9	1	no	—
251	C₂²⋊C₄	—	18	1	no	—
252	C₄²	—	18	1	no	—
253	C₄²	—	18	1	no	—
254	C₄²	—	18	1	no	—
255	C₂²×C₃	—	2	1	no	—
256	C₂²×C₃	—	2	1	no	—
257	C₂²×C₃	—	2	2	no	—
258	C₂²×C₃	—	4	1	no	—
259	C₂²×C₃	—	4	1	no	—
260	C₂²×C₃	#924	1	1	yes	characteristic
261	C₂²×C₃	C₄×#67	1	1	yes	characteristic
262	C₂²×C₃	#929	1	1	yes	characteristic
263	C₃×C₄	—	2	2	no	—
264	C₃×C₄	C₂×Dic₃⋊C₄	1	1	yes	characteristic
265	C₃×C₄	#924	1	1	yes	characteristic
266	C₃×C₄	C₄×#67	1	1	yes	characteristic
267	C₃×C₄	#953	1	1	yes	characteristic
268	Dic₃	—	12	2	no	—
269	Dic₃	—	12	1	no	—
270	Dic₃	—	12	1	no	—
271	C₉	#1866	1	1	yes	characteristic, complemented, Sylow
272	C₂³	—	2	1	no	—
273	C₂³	#3697	1	1	yes	characteristic
274	Q₈	—	2	1	no	—
275	Q₈	Dic₉⋊C₄	1	2	yes	—
276	D₄	—	2	2	no	—
277	D₄	—	2	2	no	—
278	D₄	—	2	1	no	—
279	D₄	Dic₉⋊C₄	1	2	yes	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	18	2	no	—
284	C₂×C₄	—	18	1	no	—
285	C₂×C₄	—	18	1	no	—
286	C₂×C₄	—	36	1	no	—
287	C₂×C₄	—	36	1	no	—
288	C₂×C₄	C₂×Dic₁₈	1	1	yes	characteristic
289	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
290	C₂×C₄	#3697	1	1	yes	characteristic
291	C₂×C₄	C₂×#611	1	1	yes	characteristic
292	C₂×C₄	C₂×#611	1	1	yes	characteristic
293	C₂×C₄	C₂×#611	1	1	yes	characteristic
294	C₈	—	2	1	no	—
295	C₈	—	2	2	no	—
296	C₈	—	2	1	no	—
297	C₈	—	18	1	no	—
298	C₈	—	18	1	no	—
299	C₈	—	36	1	no	—
300	C₂×C₃	—	2	1	no	—
301	C₂×C₃	—	2	1	no	—
302	C₂×C₃	—	4	2	no	—
303	C₂×C₃	#5144	1	1	yes	characteristic
304	C₂×C₃	#5145	1	1	yes	characteristic
305	C₂×C₃	#5210	1	1	yes	characteristic
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	4	1	no	—
310	C₂²	—	4	1	no	—
311	C₂²	#63486	1	1	yes	characteristic
312	C₂²	C₄×#611	1	1	yes	characteristic
313	C₂²	#63491	1	1	yes	characteristic
314	C₄	—	2	2	no	—
315	C₄	—	36	2	no	—
316	C₄	—	36	1	no	—
317	C₄	—	36	1	no	—
318	C₄	C₂×Dic₉⋊C₄	1	1	yes	characteristic
319	C₄	#63486	1	1	yes	characteristic
320	C₄	C₄×#611	1	1	yes	characteristic
321	C₄	#63515	1	1	yes	characteristic
322	C₃	#72017	1	1	yes	characteristic
323	C₂	—	2	1	no	—
324	C₂	—	2	1	no	—
325	C₂	—	4	2	no	—
326	C₂	#10588824	1	1	yes	characteristic
327	C₂	#10588825	1	1	yes	characteristic
328	C₂	#10588890	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#611
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₂×C₉×#131	C₂
socle	C₂²×C₃	C₄×#67
prime radical	C₂×C₃×#41	C₂×D₃
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	C₂×#131	D₉
hypocenter	C₉	#1866
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#131

Complements

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

N	H ≅ G/N	type
4. C₂×#63381	C₂	semidirect
5. #10588783	C₂	semidirect
6. #10588235	C₂	semidirect
7. #10588228	C₂	semidirect
21. #63395	C₄	semidirect
22. #63395	C₄	semidirect
23. #63392	C₄	semidirect
24. #63392	C₄	semidirect
271. C₉	#1866	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₉×#131	2
#10588917	2
C₂×#63396	2
C₂×#63381	2
#10588783	2
#10588235	2
#10588228	2
#72017	3