Subgroups — #11787011

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×#63464	C₂	1	1	yes	maximal, characteristic
2	C₉×#424	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#63384	C₂	1	1	yes	maximal, characteristic
4	C₂×#63381	C₂	1	1	yes	maximal, characteristic, complemented
5	#10588683	C₂	1	1	yes	maximal, characteristic
6	#10588228	C₂	1	1	yes	maximal, characteristic, complemented
7	#10588208	C₂	1	1	yes	maximal, characteristic, complemented
8	#70545	—	3	1	no	maximal
9	C₂×#3697	C₂²	1	1	yes	characteristic
10	C₂×C₉×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₈×C₉	C₂²	1	1	yes	characteristic
12	C₉×#117	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
14	C₂×#3667	C₂²	1	1	yes	characteristic
15	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
16	#63464	—	2	2	no	—
17	#63464	—	2	2	no	—
18	C₂×#3663	—	2	1	no	—
19	C₂×C₈×D₉	—	2	1	no	—
20	C₉×C₄⋊C₈	—	2	1	no	—
21	C₉×C₄⋊C₈	—	2	1	no	—
22	#63384	—	2	2	no	—
23	#63381	—	2	2	no	—
24	#63369	C₄	1	1	yes	characteristic, complemented
25	#63369	C₄	1	1	yes	characteristic, complemented
26	#63368	C₄	1	1	yes	characteristic, complemented
27	#63368	C₄	1	1	yes	characteristic, complemented
28	C₂×#901	—	3	1	no	—
29	C₃×#424	D₃	1	1	yes	characteristic
30	C₂×#825	—	3	1	no	—
31	C₂×#822	—	3	1	no	—
32	#5002	—	3	1	no	—
33	#4550	—	3	1	no	—
34	#4532	—	3	1	no	—
35	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
36	C₂×#611	—	2	1	no	—
37	C₂²×Dic₉	—	2	1	no	—
38	#3697	—	2	1	no	—
39	#3697	—	2	1	no	—
40	#3697	D₄	1	2	yes	—
41	#3697	Q₈	1	2	yes	—
42	C₂×D₃₆	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₉	—	2	1	no	—
44	C₂×Dic₁₈	C₂×C₄	1	1	yes	characteristic
45	C₉×M₁₆	—	2	1	no	—
46	C₉×M₁₆	—	2	1	no	—
47	C₂×C₈×C₉	—	2	1	no	—
48	C₂×C₈×C₉	—	2	1	no	—
49	C₂×C₈×C₉	—	2	1	no	—
50	C₂×C₈×C₉	D₄	1	1	yes	characteristic
51	C₂×C₈×C₉	D₄	1	1	yes	characteristic
52	C₉×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
53	C₉×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
54	C₉×C₂²⋊C₄	—	2	1	no	—
55	C₄²×C₉	—	2	1	no	—
56	C₄×Dic₉	—	2	1	no	—
57	C₄×Dic₉	—	2	1	no	—
58	#3667	—	2	1	no	—
59	#3667	D₄	1	2	yes	—
60	C₂×C₉⋊C₈	—	2	1	no	—
61	C₂×C₉⋊C₈	—	2	1	no	—
62	C₂×C₉⋊C₈	—	2	1	no	—
63	C₂×C₉⋊C₈	C₂×C₄	1	1	yes	characteristic
64	C₂×C₉⋊C₈	C₂×C₄	1	1	yes	characteristic
65	#3663	—	2	4	no	—
66	C₈×D₉	—	2	4	no	—
67	#1866	—	9	1	no	Sylow
68	C₂×#235	—	3	1	no	—
69	C₂×C₃×M₁₆	C₂×D₃	1	1	yes	characteristic
70	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
71	C₃×#117	C₂×D₃	1	1	yes	characteristic
72	C₂×C₄×Dic₃	—	3	1	no	—
73	C₂×#208	—	3	1	no	—
74	C₂²×C₃⋊C₈	—	3	1	no	—
75	#901	—	6	2	no	—
76	#901	—	6	2	no	—
77	C₂×ΓL(1,𝔽₂₅)	—	6	1	no	—
78	C₂×D₃×C₈	—	6	1	no	—
79	C₃×C₄⋊C₈	—	2	1	no	—
80	C₃×C₄⋊C₈	—	2	1	no	—
81	#825	—	6	2	no	—
82	#822	—	6	2	no	—
83	#810	—	3	1	no	—
84	#810	—	3	1	no	—
85	#809	—	3	1	no	—
86	#809	—	3	1	no	—
87	C₂³×C₉	#41	1	1	yes	characteristic
88	C₂²×D₉	—	2	1	no	—
89	C₂×C₄×C₉	—	2	1	no	—
90	C₂×C₄×C₉	—	2	1	no	—
91	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
96	C₂×C₄×C₉	#41	1	1	yes	characteristic
97	#611	—	2	2	no	—
98	#611	—	2	2	no	—
99	C₂×Dic₉	—	2	1	no	—
100	C₂×Dic₉	—	2	2	no	—
101	C₂×Dic₉	—	4	1	no	—
102	C₂×Dic₉	—	4	1	no	—
103	D₃₆	—	2	1	no	—
104	D₃₆	C₄⋊C₄	1	2	yes	—
105	C₄×D₉	—	2	2	no	—
106	C₄×D₉	—	2	2	no	—
107	Dic₁₈	—	2	1	no	—
108	Dic₁₈	C₄⋊C₄	1	2	yes	—
109	C₈×C₉	—	2	1	no	—
110	C₈×C₉	—	2	1	no	—
111	C₈×C₉	—	4	1	no	—
112	C₉⋊C₈	—	2	1	no	—
113	C₉⋊C₈	—	2	1	no	—
114	C₉⋊C₈	—	2	2	no	—
115	C₂×#131	—	9	1	no	—
116	C₂×#108	—	9	1	no	—
117	#424	D₉	1	1	yes	characteristic, complemented
118	C₂×#104	—	9	1	no	—
119	#416	—	9	1	no	—
120	#339	—	9	1	no	—
121	#336	—	9	1	no	—
122	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
123	C₂×#67	—	6	1	no	—
124	C₂²×Dic₃	—	6	1	no	—
125	#235	—	3	2	no	—
126	#235	—	3	2	no	—
127	#235	—	6	1	no	—
128	#235	—	6	1	no	—
129	C₂×D₁₂	—	3	1	no	—
130	C₂×C₄×D₃	—	6	1	no	—
131	C₂×Dic₆	—	3	1	no	—
132	C₃×M₁₆	—	2	1	no	—
133	C₃×M₁₆	—	2	1	no	—
134	C₂×C₃×C₈	—	2	1	no	—
135	C₂×C₃×C₈	—	2	1	no	—
136	C₂×C₃×C₈	—	2	1	no	—
137	C₂×C₃×C₈	#67	1	1	yes	characteristic
138	C₂×C₃×C₈	#67	1	1	yes	characteristic
139	C₃×C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
140	C₃×C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
141	C₃×C₂²⋊C₄	—	2	1	no	—
142	C₃×C₄²	—	2	1	no	—
143	C₄×Dic₃	—	6	1	no	—
144	C₄×Dic₃	—	6	1	no	—
145	#208	—	3	2	no	—
146	#208	—	6	1	no	—
147	C₂×C₃⋊C₈	—	3	1	no	—
148	C₂×C₃⋊C₈	—	3	1	no	—
149	C₂×C₃⋊C₈	—	6	1	no	—
150	C₂×C₃⋊C₈	—	6	1	no	—
151	C₂×C₃⋊C₈	—	6	1	no	—
152	ΓL(1,𝔽₂₅)	—	6	4	no	—
153	D₃×C₈	—	6	4	no	—
154	C₂²×C₉	—	2	1	no	—
155	C₂²×C₉	—	2	1	no	—
156	C₂²×C₉	C₄×D₄	1	1	yes	characteristic
157	C₂²×C₉	C₂²⋊Q₈	1	1	yes	characteristic
158	C₂²×C₉	#123	1	1	yes	characteristic
159	C₂×D₉	—	2	2	no	—
160	C₂×D₉	—	4	1	no	—
161	C₂×D₉	—	4	1	no	—
162	C₄×C₉	—	4	1	no	—
163	C₄×C₉	—	4	1	no	—
164	C₄×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
165	C₄×C₉	C₄×D₄	1	1	yes	characteristic
166	C₄×C₉	#120	1	1	yes	characteristic
167	C₄×C₉	C₂²⋊Q₈	1	1	yes	characteristic
168	Dic₉	—	2	2	no	—
169	Dic₉	—	4	2	no	—
170	C₂×#41	—	9	1	no	—
171	#131	—	18	2	no	—
172	#131	—	18	2	no	—
173	C₂×M₁₆	—	9	1	no	—
174	C₂×M₁₆	—	18	1	no	—
175	C₂×M₁₆	C₂×D₉	1	1	yes	characteristic
176	C₂²×C₈	—	9	1	no	—
177	C₂²×C₈	—	18	1	no	—
178	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
179	#117	C₂×D₉	1	1	yes	characteristic
180	C₂×C₄²	—	9	1	no	—
181	#108	—	18	2	no	—
182	C₄⋊C₈	—	2	1	no	—
183	C₄⋊C₈	—	2	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	#235	1	1	yes	characteristic
190	C₂²×D₃	—	6	1	no	—
191	C₂×C₃×C₄	—	2	1	no	—
192	C₂×C₃×C₄	—	2	1	no	—
193	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
194	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
195	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
196	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
197	C₂×C₃×C₄	#239	1	1	yes	characteristic
198	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
199	#67	—	6	2	no	—
200	#67	—	6	2	no	—
201	C₂×Dic₃	—	6	1	no	—
202	C₂×Dic₃	—	6	2	no	—
203	C₂×Dic₃	—	12	1	no	—
204	C₂×Dic₃	—	12	1	no	—
205	D₁₂	—	3	2	no	—
206	D₁₂	—	6	1	no	—
207	C₄×D₃	—	6	2	no	—
208	C₄×D₃	—	6	2	no	—
209	Dic₆	—	3	2	no	—
210	Dic₆	—	6	1	no	—
211	C₃×C₈	—	2	1	no	—
212	C₃×C₈	—	2	1	no	—
213	C₃×C₈	—	4	1	no	—
214	C₃⋊C₈	—	6	1	no	—
215	C₃⋊C₈	—	6	1	no	—
216	C₃⋊C₈	—	6	2	no	—
217	C₂×C₉	—	2	1	no	—
218	C₂×C₉	—	2	1	no	—
219	C₂×C₉	#385	1	1	yes	characteristic
220	C₂×C₉	#443	1	1	yes	characteristic
221	C₂×C₉	#444	1	1	yes	characteristic
222	D₉	—	4	2	no	—
223	#41	—	9	2	no	—
224	#41	—	9	2	no	—
225	#41	—	18	1	no	—
226	#41	—	18	1	no	—
227	C₂×Q₈	—	9	1	no	—
228	C₂×D₄	—	9	1	no	—
229	C₂×D₄	—	18	1	no	—
230	C₂²×C₄	—	18	1	no	—
231	C₂²×C₄	—	18	1	no	—
232	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
233	M₁₆	—	2	1	no	—
234	M₁₆	—	2	1	no	—
235	M₁₆	—	9	2	no	—
236	M₁₆	—	18	4	no	—
237	M₁₆	—	18	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	9	1	no	—
242	C₂×C₈	—	9	1	no	—
243	C₂×C₈	—	18	4	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	C₄×D₉	1	1	yes	characteristic
250	C₄⋊C₄	C₄×D₉	1	1	yes	characteristic
251	C₂²⋊C₄	—	2	1	no	—
252	C₄²	—	2	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	#894	1	1	yes	characteristic
258	C₂²×C₃	#896	1	1	yes	characteristic
259	C₂²×C₃	C₄×#67	1	1	yes	characteristic
260	C₂×D₃	—	6	2	no	—
261	C₂×D₃	—	12	1	no	—
262	C₂×D₃	—	12	1	no	—
263	C₃×C₄	—	4	1	no	—
264	C₃×C₄	—	4	1	no	—
265	C₃×C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
266	C₃×C₄	#893	1	1	yes	characteristic
267	C₃×C₄	#937	1	1	yes	characteristic
268	C₃×C₄	#946	1	1	yes	characteristic
269	Dic₃	—	6	2	no	—
270	Dic₃	—	12	2	no	—
271	C₉	#1866	1	1	yes	characteristic, complemented, Sylow
272	C₂³	—	18	1	no	—
273	C₂³	#3697	1	1	yes	characteristic
274	Q₈	—	9	2	no	—
275	Q₈	—	18	1	no	—
276	D₄	—	9	2	no	—
277	D₄	—	18	2	no	—
278	D₄	—	18	2	no	—
279	D₄	—	18	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	18	1	no	—
283	C₂×C₄	—	18	2	no	—
284	C₂×C₄	—	18	2	no	—
285	C₂×C₄	—	18	2	no	—
286	C₂×C₄	—	36	1	no	—
287	C₂×C₄	—	36	1	no	—
288	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
289	C₂×C₄	D₄×D₉	1	1	yes	characteristic
290	C₂×C₄	D₄×D₉	1	1	yes	characteristic
291	C₂×C₄	Q₈×D₉	1	1	yes	characteristic
292	C₂×C₄	#3701	1	1	yes	characteristic
293	C₂×C₄	C₂×#611	1	1	yes	characteristic
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	—	4	1	no	—
297	C₈	—	18	2	no	—
298	C₈	—	18	1	no	—
299	C₈	—	18	1	no	—
300	C₂×C₃	—	2	1	no	—
301	C₂×C₃	—	2	1	no	—
302	C₂×C₃	#4989	1	1	yes	characteristic
303	C₂×C₃	#5144	1	1	yes	characteristic
304	C₂×C₃	#5145	1	1	yes	characteristic
305	D₃	—	12	2	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	1	no	—
308	C₂²	—	18	2	no	—
309	C₂²	—	36	1	no	—
310	C₂²	—	36	1	no	—
311	C₂²	#63456	1	1	yes	characteristic
312	C₂²	#63458	1	1	yes	characteristic
313	C₂²	C₄×#611	1	1	yes	characteristic
314	C₄	—	4	1	no	—
315	C₄	—	4	1	no	—
316	C₄	—	18	2	no	—
317	C₄	—	36	2	no	—
318	C₄	C₄⋊C₄×D₉	1	1	yes	characteristic
319	C₄	#63455	1	1	yes	characteristic
320	C₄	#63499	1	1	yes	characteristic
321	C₄	#63508	1	1	yes	characteristic
322	C₃	#70545	1	1	yes	characteristic
323	C₂	—	2	1	no	—
324	C₂	—	2	1	no	—
325	C₂	—	36	2	no	—
326	C₂	#10588669	1	1	yes	characteristic
327	C₂	#10588824	1	1	yes	characteristic
328	C₂	#10588825	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	C₉×#424	C₂
socle	C₂²×C₃	C₄×#67
prime radical	C₂×#3697	C₂²
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	#424	D₉
hypocenter	C₉	#1866
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #424

Complements

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

N	H ≅ G/N	type
2. C₉×#424	C₂	semidirect
4. C₂×#63381	C₂	semidirect
6. #10588228	C₂	semidirect
7. #10588208	C₂	semidirect
24. #63369	C₄	semidirect
25. #63369	C₄	semidirect
26. #63368	C₄	semidirect
27. #63368	C₄	semidirect
117. #424	D₉	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×#63464	2
C₉×#424	2
C₂×#63384	2
C₂×#63381	2
#10588683	2
#10588228	2
#10588208	2
#70545	3