Subgroups — #11785270

Order 1152.

438 conjugacy classes of subgroups.

The subgroup lattice has 327 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#63449	C₂	1	1	yes	maximal, characteristic
2	C₂×#63376	C₂	1	1	yes	maximal, characteristic
3	C₂×Dic₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#10588657	C₂	1	1	yes	maximal, characteristic
5	C₂×#63368	C₂	1	1	yes	maximal, characteristic
6	C₉×#340	C₂	1	1	yes	maximal, characteristic
7	#10588206	C₂	1	1	yes	maximal, characteristic
8	#68905	—	3	1	no	maximal
9	C₂²×Dic₁₈	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×C₉×C₄⋊C₄	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	—	2	1	no	—
16	C₂×C₄×Dic₉	C₂²	1	1	yes	characteristic
17	C₂²×C₉⋊C₈	C₂²	1	1	yes	characteristic
18	#63449	—	2	2	no	—
19	#63449	C₄	1	4	yes	complemented
20	C₉⋊#95	—	2	1	no	—
21	C₉⋊#95	—	2	1	no	—
22	#63376	—	2	2	no	—
23	Dic₉⋊C₈	—	2	2	no	—
24	#63368	—	2	2	no	—
25	C₂×#887	—	3	1	no	—
26	C₂×#816	—	3	1	no	—
27	C₂×Dic₃⋊C₈	—	3	1	no	—
28	#4977	—	3	1	no	—
29	C₂×#810	—	3	1	no	—
30	C₃×#340	D₃	1	1	yes	characteristic
31	#4529	—	3	1	no	—
32	C₂²×C₄×C₉	—	2	1	no	—
33	C₂²×C₄×C₉	—	2	1	no	—
34	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
35	C₂²×Dic₉	—	2	1	no	—
36	C₂²×Dic₉	—	2	1	no	—
37	C₂²×Dic₉	D₄	1	1	yes	characteristic
38	C₂²×Dic₉	D₄	1	1	yes	characteristic
39	C₂×Dic₁₈	—	2	1	no	—
40	C₂×Dic₁₈	—	2	1	no	—
41	C₂×Dic₁₈	—	2	1	no	—
42	C₂×Dic₁₈	C₂×C₄	1	2	yes	—
43	C₂×Dic₁₈	D₄	1	2	yes	—
44	C₂×Dic₁₈	Q₈	1	2	yes	—
45	C₂×C₈×C₉	—	2	1	no	—
46	C₂×C₈×C₉	—	2	1	no	—
47	C₂×C₈×C₉	—	2	1	no	—
48	C₉×C₄⋊C₄	—	2	1	no	—
49	C₉×C₄⋊C₄	—	2	1	no	—
50	C₉×C₄⋊C₄	—	2	1	no	—
51	C₉×C₄⋊C₄	C₂×C₄	1	2	yes	—
52	C₄⋊Dic₉	—	2	1	no	—
53	C₄⋊Dic₉	—	2	1	no	—
54	Dic₉⋊C₄	—	2	4	no	—
55	C₄×Dic₉	—	2	4	no	—
56	C₄×Dic₉	C₂×C₄	1	2	yes	—
57	C₄×Dic₉	D₄	1	2	yes	—
58	C₂×C₉⋊C₈	—	2	1	no	—
59	C₂×C₉⋊C₈	—	2	1	no	—
60	C₂×C₉⋊C₈	—	2	1	no	—
61	#1805	—	9	1	no	Sylow
62	C₂²×Dic₆	—	3	1	no	—
63	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
64	C₂×C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
65	C₂×C₃×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
66	C₂×C₄⋊Dic₃	—	3	1	no	—
67	C₂×Dic₃⋊C₄	—	6	1	no	—
68	C₂×C₄×Dic₃	—	3	1	no	—
69	C₂×C₄×Dic₃	—	6	1	no	—
70	C₂²×C₃⋊C₈	—	3	1	no	—
71	#887	—	3	4	no	—
72	#887	—	6	2	no	—
73	C₃⋊#95	—	6	1	no	—
74	C₃⋊#95	—	6	1	no	—
75	#816	—	6	2	no	—
76	Dic₃⋊C₈	—	6	2	no	—
77	#810	—	6	2	no	—
78	C₂³×C₉	C₂×D₄	1	1	yes	characteristic
79	C₂×C₄×C₉	—	2	2	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	—	4	1	no	—
85	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×C₉	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₄×C₉	#41	1	1	yes	characteristic
90	C₂×C₄×C₉	#41	1	1	yes	characteristic
91	C₂×Dic₉	—	2	2	no	—
92	C₂×Dic₉	—	2	2	no	—
93	C₂×Dic₉	—	2	2	no	—
94	C₂×Dic₉	—	2	1	no	—
95	C₂×Dic₉	—	2	1	no	—
96	C₂×Dic₉	—	2	1	no	—
97	C₂×Dic₉	—	2	1	no	—
98	C₂×Dic₉	—	4	1	no	—
99	C₂×Dic₉	—	4	1	no	—
100	C₂×Dic₉	—	4	1	no	—
101	C₂×Dic₉	C₂²⋊C₄	1	2	yes	—
102	C₂×Dic₉	C₂²⋊C₄	1	2	yes	—
103	Dic₁₈	—	2	2	no	—
104	Dic₁₈	—	2	2	no	—
105	Dic₁₈	—	2	2	no	—
106	Dic₁₈	C₄⋊C₄	1	4	yes	—
107	C₈×C₉	—	4	1	no	—
108	C₉⋊C₈	—	4	1	no	—
109	C₂×C₄×Q₈	—	9	1	no	—
110	C₂×C₄⋊C₈	—	9	1	no	—
111	C₂×#103	—	9	1	no	—
112	C₂×#103	—	9	1	no	—
113	#382	—	9	1	no	—
114	#340	—	9	1	no	—
115	#340	D₉	1	1	yes	characteristic
116	C₂²×C₃×C₄	—	2	1	no	—
117	C₂²×C₃×C₄	—	2	1	no	—
118	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
119	C₂²×Dic₃	—	3	1	no	—
120	C₂²×Dic₃	—	3	1	no	—
121	C₂²×Dic₃	—	6	1	no	—
122	C₂²×Dic₃	—	6	1	no	—
123	C₂×Dic₆	—	3	2	no	—
124	C₂×Dic₆	—	3	2	no	—
125	C₂×Dic₆	—	3	2	no	—
126	C₂×Dic₆	—	6	1	no	—
127	C₂×Dic₆	—	6	1	no	—
128	C₂×Dic₆	—	6	1	no	—
129	C₂×C₃×C₈	—	2	1	no	—
130	C₂×C₃×C₈	—	2	1	no	—
131	C₂×C₃×C₈	—	2	1	no	—
132	C₃×C₄⋊C₄	—	2	1	no	—
133	C₃×C₄⋊C₄	—	2	1	no	—
134	C₃×C₄⋊C₄	—	2	1	no	—
135	C₃×C₄⋊C₄	C₄×D₃	1	2	yes	—
136	C₄⋊Dic₃	—	6	1	no	—
137	C₄⋊Dic₃	—	6	1	no	—
138	Dic₃⋊C₄	—	6	4	no	—
139	C₄×Dic₃	—	3	2	no	—
140	C₄×Dic₃	—	3	2	no	—
141	C₄×Dic₃	—	6	4	no	—
142	C₂×C₃⋊C₈	—	6	1	no	—
143	C₂×C₃⋊C₈	—	6	1	no	—
144	C₂×C₃⋊C₈	—	6	1	no	—
145	C₂²×C₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂²×C₉	#121	1	1	yes	characteristic
147	C₂²×C₉	C₂²⋊Q₈	1	1	yes	characteristic
148	C₂²×C₉	#135	1	1	yes	characteristic
149	C₂²×C₉	#135	1	1	yes	characteristic
150	C₂²×C₉	#137	1	1	yes	characteristic
151	C₂²×C₉	#137	1	1	yes	characteristic
152	C₄×C₉	—	4	2	no	—
153	C₄×C₉	—	8	1	no	—
154	C₄×C₉	C₂×C₄⋊C₄	1	1	yes	characteristic
155	C₄×C₉	#117	1	1	yes	characteristic
156	C₄×C₉	#121	1	1	yes	characteristic
157	C₄×C₉	C₂²⋊Q₈	1	1	yes	characteristic
158	Dic₉	—	2	4	no	—
159	Dic₉	—	2	2	no	—
160	Dic₉	—	2	2	no	—
161	Dic₉	—	8	1	no	—
162	C₂²×Q₈	—	9	1	no	—
163	C₂²×C₈	—	9	1	no	—
164	C₂²×C₈	C₂×D₉	1	1	yes	characteristic
165	C₄×Q₈	—	9	4	no	—
166	C₄×Q₈	—	18	2	no	—
167	C₂×C₄⋊C₄	—	9	1	no	—
168	C₂×C₄⋊C₄	—	18	1	no	—
169	C₂×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
170	C₂×C₄⋊C₄	C₂×D₉	1	1	yes	characteristic
171	C₂×C₄²	—	9	1	no	—
172	C₂×C₄²	—	18	1	no	—
173	C₄⋊C₈	—	18	2	no	—
174	#103	—	18	2	no	—
175	#103	—	18	2	no	—
176	#95	—	18	1	no	—
177	#95	—	18	1	no	—
178	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
179	C₂×C₃×C₄	—	2	2	no	—
180	C₂×C₃×C₄	—	4	1	no	—
181	C₂×C₃×C₄	—	4	1	no	—
182	C₂×C₃×C₄	—	4	1	no	—
183	C₂×C₃×C₄	—	4	1	no	—
184	C₂×C₃×C₄	—	4	1	no	—
185	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
186	C₂×C₃×C₄	#235	1	1	yes	characteristic
187	C₂×C₃×C₄	#235	1	1	yes	characteristic
188	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
189	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
190	C₂×C₃×C₄	D₃×Q₈	1	1	yes	characteristic
191	C₂×Dic₃	—	3	2	no	—
192	C₂×Dic₃	—	3	2	no	—
193	C₂×Dic₃	—	6	2	no	—
194	C₂×Dic₃	—	6	2	no	—
195	C₂×Dic₃	—	6	2	no	—
196	C₂×Dic₃	—	6	1	no	—
197	C₂×Dic₃	—	6	1	no	—
198	C₂×Dic₃	—	6	1	no	—
199	C₂×Dic₃	—	6	1	no	—
200	C₂×Dic₃	—	12	1	no	—
201	C₂×Dic₃	—	12	1	no	—
202	C₂×Dic₃	—	12	1	no	—
203	Dic₆	—	3	4	no	—
204	Dic₆	—	6	2	no	—
205	Dic₆	—	6	2	no	—
206	Dic₆	—	6	2	no	—
207	C₃×C₈	—	4	1	no	—
208	C₃⋊C₈	—	12	1	no	—
209	C₂×C₉	#380	1	1	yes	characteristic
210	C₂×C₉	#416	1	1	yes	characteristic
211	C₂×C₉	#419	1	1	yes	characteristic
212	C₂×C₉	#464	1	1	yes	characteristic
213	C₂×C₉	#464	1	1	yes	characteristic
214	C₂×C₉	#479	1	1	yes	characteristic
215	C₂×C₉	#479	1	1	yes	characteristic
216	C₂×Q₈	—	9	2	no	—
217	C₂×Q₈	—	9	2	no	—
218	C₂×Q₈	—	9	2	no	—
219	C₂×Q₈	—	18	1	no	—
220	C₂×Q₈	—	18	1	no	—
221	C₂×Q₈	—	18	1	no	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	2	1	no	—
224	C₂²×C₄	—	9	1	no	—
225	C₂²×C₄	—	9	1	no	—
226	C₂²×C₄	—	18	1	no	—
227	C₂²×C₄	—	18	1	no	—
228	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
229	C₂×C₈	—	2	1	no	—
230	C₂×C₈	—	2	1	no	—
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	18	1	no	—
233	C₂×C₈	—	18	1	no	—
234	C₂×C₈	—	18	1	no	—
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	18	4	no	—
239	C₄⋊C₄	—	18	1	no	—
240	C₄⋊C₄	—	18	1	no	—
241	C₄⋊C₄	C₄×D₉	1	2	yes	—
242	C₄²	—	9	2	no	—
243	C₄²	—	9	2	no	—
244	C₄²	—	18	4	no	—
245	C₂²×C₃	D₃×C₂²⋊C₄	1	1	yes	characteristic
246	C₂²×C₃	#884	1	1	yes	characteristic
247	C₂²×C₃	#896	1	1	yes	characteristic
248	C₂²×C₃	#912	1	1	yes	characteristic
249	C₂²×C₃	#915	1	1	yes	characteristic
250	C₂²×C₃	#916	1	1	yes	characteristic
251	C₂²×C₃	#918	1	1	yes	characteristic
252	C₃×C₄	—	4	2	no	—
253	C₃×C₄	—	8	1	no	—
254	C₃×C₄	#872	1	1	yes	characteristic
255	C₃×C₄	#884	1	1	yes	characteristic
256	C₃×C₄	D₃×C₄⋊C₄	1	1	yes	characteristic
257	C₃×C₄	#896	1	1	yes	characteristic
258	Dic₃	—	6	4	no	—
259	Dic₃	—	6	2	no	—
260	Dic₃	—	6	2	no	—
261	Dic₃	—	24	1	no	—
262	C₉	#1805	1	1	yes	characteristic, complemented, Sylow
263	C₂³	D₄×D₉	1	1	yes	characteristic
264	Q₈	—	9	4	no	—
265	Q₈	—	18	2	no	—
266	Q₈	—	18	2	no	—
267	Q₈	—	18	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	4	1	no	—
270	C₂×C₄	—	4	1	no	—
271	C₂×C₄	—	4	1	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	4	1	no	—
274	C₂×C₄	—	9	2	no	—
275	C₂×C₄	—	9	2	no	—
276	C₂×C₄	—	18	2	no	—
277	C₂×C₄	—	18	2	no	—
278	C₂×C₄	—	18	2	no	—
279	C₂×C₄	—	18	1	no	—
280	C₂×C₄	—	18	1	no	—
281	C₂×C₄	—	18	1	no	—
282	C₂×C₄	—	18	1	no	—
283	C₂×C₄	—	36	1	no	—
284	C₂×C₄	—	36	1	no	—
285	C₂×C₄	—	36	1	no	—
286	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
287	C₂×C₄	#3697	1	1	yes	characteristic
288	C₂×C₄	#3697	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	C₈	—	4	1	no	—
293	C₈	—	36	1	no	—
294	C₂×C₃	#4667	1	1	yes	characteristic
295	C₂×C₃	#4767	1	1	yes	characteristic
296	C₂×C₃	#4772	1	1	yes	characteristic
297	C₂×C₃	#4798	1	1	yes	characteristic
298	C₂×C₃	#4802	1	1	yes	characteristic
299	C₂×C₃	#4857	1	1	yes	characteristic
300	C₂×C₃	#4877	1	1	yes	characteristic
301	C₂²	C₂²⋊C₄×D₉	1	1	yes	characteristic
302	C₂²	#63446	1	1	yes	characteristic
303	C₂²	#63458	1	1	yes	characteristic
304	C₂²	#63474	1	1	yes	characteristic
305	C₂²	#63477	1	1	yes	characteristic
306	C₂²	#63478	1	1	yes	characteristic
307	C₂²	#63480	1	1	yes	characteristic
308	C₄	—	4	2	no	—
309	C₄	—	8	1	no	—
310	C₄	—	18	2	no	—
311	C₄	—	18	2	no	—
312	C₄	—	18	4	no	—
313	C₄	—	72	1	no	—
314	C₄	#63434	1	1	yes	characteristic
315	C₄	#63446	1	1	yes	characteristic
316	C₄	C₄⋊C₄×D₉	1	1	yes	characteristic
317	C₄	#63458	1	1	yes	characteristic
318	C₃	#68905	1	1	yes	characteristic
319	C₂	#10588347	1	1	yes	characteristic
320	C₂	#10588447	1	1	yes	characteristic
321	C₂	#10588452	1	1	yes	characteristic
322	C₂	#10588478	1	1	yes	characteristic
323	C₂	#10588482	1	1	yes	characteristic
324	C₂	#10588537	1	1	yes	characteristic
325	C₂	#10588557	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×D₉
commutator	C₂×C₄×C₉	C₂²×C₄
Frattini	C₂²×C₃×C₄	C₂²×D₃
Fitting	C₉×#340	C₂
socle, prime radical	C₂³×C₃	D₃×D₄
cosocle	C₂²×C₄×C₉	C₂³
hypercenter	#340	D₉
hypocenter	C₉	#1805
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

Complements

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

N	H ≅ G/N	type
19. #63449	C₄	semidirect
262. C₉	#1805	semidirect

Maximal subgroups

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

subgroup	index
C₂×#63449	2
C₂×#63376	2
C₂×Dic₉⋊C₈	2
#10588657	2
C₂×#63368	2
C₉×#340	2
#10588206	2
#68905	3