Subgroups — #11857221

Order 1152.

846 conjugacy classes of subgroups.

The subgroup lattice has 273 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₉×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#63488	C₂	1	1	yes	maximal, characteristic
3	C₂²×#3667	C₂	1	1	yes	maximal, characteristic
4	M₁₆×Dic₉	C₂	1	4	yes	maximal, complemented
5	#10588797	C₂	1	4	yes	maximal, complemented
6	#10588782	C₂	1	4	yes	maximal, complemented
7	#83747	—	3	1	no	maximal
8	C₂³×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₂×C₉×M₁₆	C₄	1	8	yes	complemented
10	C₂×C₉×M₁₆	C₂²	1	2	yes	—
11	C₂×C₉×M₁₆	C₂²	1	2	yes	—
12	C₂²×C₈×C₉	C₂²	1	2	yes	—
13	C₂×#3676	C₄	1	2	yes	—
14	#63488	C₄	1	2	yes	—
15	#63488	C₄	1	2	yes	—
16	#63488	C₂²	1	2	yes	—
17	#63488	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₉	C₄	1	2	yes	—
19	C₂×C₄×Dic₉	C₂²	1	2	yes	—
20	C₂×#3667	C₄	1	8	yes	complemented
21	C₂×#3667	C₂²	1	2	yes	—
22	C₂×#3667	C₂²	1	2	yes	—
23	C₂²×C₉⋊C₈	C₂²	1	2	yes	—
24	#63375	C₂²	1	8	yes	complemented
25	C₈×Dic₉	C₂²	1	8	yes	complemented
26	C₂²×C₃×M₁₆	D₃	1	1	yes	characteristic
27	C₂×#926	—	3	1	no	—
28	C₂²×#208	—	3	1	no	—
29	#5122	—	3	4	no	—
30	Dic₃×M₁₆	—	3	4	no	—
31	#5101	—	3	4	no	—
32	C₂⁴×C₉	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₉	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₉	C₂×C₄	1	4	yes	—
37	C₂²×C₄×C₉	C₂×C₄	1	4	yes	—
38	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
39	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₉	C₂³	1	1	yes	characteristic
41	C₂²×Dic₉	C₂×C₄	1	4	yes	—
42	C₉×M₁₆	C₂×C₄	1	8	yes	—
43	C₉×M₁₆	C₂×C₄	1	8	yes	complemented
44	C₂×C₈×C₉	C₂×C₄	1	8	yes	complemented
45	C₂×C₈×C₉	C₂³	1	4	yes	—
46	#3676	C₂×C₄	1	4	yes	—
47	#3676	C₂×C₄	1	4	yes	—
48	C₄⋊Dic₉	C₂×C₄	1	4	yes	—
49	C₄⋊Dic₉	C₂×C₄	1	4	yes	—
50	C₄×Dic₉	C₂×C₄	1	4	yes	—
51	C₄×Dic₉	C₂³	1	4	yes	—
52	#3667	C₂×C₄	1	8	yes	—
53	#3667	C₂×C₄	1	8	yes	complemented
54	C₂×C₉⋊C₈	C₂×C₄	1	8	yes	complemented
55	C₂×C₉⋊C₈	C₂³	1	4	yes	—
56	#2873	—	9	1	no	Sylow
57	C₂³×C₃×C₄	C₂×D₃	1	1	yes	characteristic
58	C₂×C₃×M₁₆	Dic₃	1	8	yes	—
59	C₂×C₃×M₁₆	C₂×D₃	1	2	yes	—
60	C₂×C₃×M₁₆	C₂×D₃	1	2	yes	—
61	C₂²×C₃×C₈	C₂×D₃	1	2	yes	—
62	C₂×#217	—	3	2	no	—
63	#926	—	3	2	no	—
64	#926	—	3	2	no	—
65	#926	—	3	2	no	—
66	#926	—	3	2	no	—
67	C₂×C₄⋊Dic₃	—	3	2	no	—
68	C₂×C₄×Dic₃	—	3	2	no	—
69	C₂×#208	—	3	8	no	—
70	C₂×#208	—	3	2	no	—
71	C₂×#208	—	3	2	no	—
72	C₂²×C₃⋊C₈	—	3	2	no	—
73	#815	—	3	8	no	—
74	C₈×Dic₃	—	3	8	no	—
75	C₂³×C₉	—	4	2	no	—
76	C₂³×C₉	C₄²	1	4	yes	—
77	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
78	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
79	C₂³×C₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₉	C₄²	1	4	yes	—
81	C₂×C₄×C₉	C₄²	1	4	yes	—
82	C₂×C₄×C₉	C₄²	1	4	yes	—
83	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₉	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
87	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
88	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₉	C₂²×C₄	1	2	yes	—
92	C₂×C₄×C₉	C₂⁴	1	1	yes	characteristic
93	C₂×Dic₉	—	4	4	no	—
94	C₂×Dic₉	C₂²×C₄	1	8	yes	—
95	C₈×C₉	C₂²×C₄	1	8	yes	—
96	C₉⋊C₈	C₂²×C₄	1	8	yes	—
97	C₂²×M₁₆	—	9	1	no	—
98	C₂²×M₁₆	D₉	1	1	yes	characteristic
99	C₂×#117	—	9	1	no	—
100	#405	—	9	4	no	—
101	#405	—	9	4	no	—
102	C₄×M₁₆	—	9	4	no	—
103	C₂⁴×C₃	C₄×D₃	1	1	yes	characteristic
104	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
105	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
106	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
107	C₂²×C₃×C₄	C₄×D₃	1	4	yes	—
108	C₂²×C₃×C₄	C₂×Dic₃	1	4	yes	—
109	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
110	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
111	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
112	C₂²×Dic₃	—	3	4	no	—
113	C₃×M₁₆	C₂×Dic₃	1	8	yes	—
114	C₃×M₁₆	C₂×Dic₃	1	8	yes	—
115	C₂×C₃×C₈	C₂×Dic₃	1	8	yes	—
116	C₂×C₃×C₈	C₂²×D₃	1	4	yes	—
117	#217	—	3	4	no	—
118	#217	—	3	4	no	—
119	C₄⋊Dic₃	—	3	4	no	—
120	C₄⋊Dic₃	—	3	4	no	—
121	C₄×Dic₃	—	3	4	no	—
122	C₄×Dic₃	—	3	4	no	—
123	#208	—	3	8	no	—
124	#208	—	3	8	no	—
125	C₂×C₃⋊C₈	—	3	8	no	—
126	C₂×C₃⋊C₈	—	3	4	no	—
127	C₂²×C₉	—	2	2	no	—
128	C₂²×C₉	—	2	2	no	—
129	C₂²×C₉	—	2	2	no	—
130	C₂²×C₉	—	4	4	no	—
131	C₂²×C₉	C₂×C₄²	1	2	yes	—
132	C₂²×C₉	C₂×C₄²	1	2	yes	—
133	C₂²×C₉	C₂×C₄²	1	2	yes	—
134	C₂²×C₉	C₂³×C₄	1	1	yes	characteristic
135	C₄×C₉	C₂×C₄²	1	2	yes	—
136	C₄×C₉	C₂×C₄²	1	2	yes	—
137	C₄×C₉	C₂×C₄²	1	2	yes	—
138	C₄×C₉	C₂³×C₄	1	1	yes	characteristic
139	C₄×C₉	C₂³×C₄	1	1	yes	characteristic
140	Dic₉	—	2	8	no	—
141	C₂³×C₄	C₂×D₉	1	1	yes	characteristic
142	C₂×M₁₆	—	9	8	no	—
143	C₂×M₁₆	—	9	2	no	—
144	C₂×M₁₆	—	9	2	no	—
145	C₂×M₁₆	Dic₉	1	8	yes	complemented
146	C₂×M₁₆	C₂×D₉	1	2	yes	—
147	C₂×M₁₆	C₂×D₉	1	2	yes	—
148	C₂²×C₈	—	9	2	no	—
149	C₂²×C₈	C₂×D₉	1	2	yes	—
150	#117	—	9	2	no	—
151	#117	—	9	2	no	—
152	#117	—	9	2	no	—
153	#117	—	9	2	no	—
154	C₂×C₄⋊C₄	—	9	2	no	—
155	C₂×C₂²⋊C₄	—	9	2	no	—
156	C₂×C₄²	—	9	2	no	—
157	#97	—	9	8	no	—
158	C₄×C₈	—	9	8	no	—
159	C₂³×C₃	—	4	2	no	—
160	C₂³×C₃	C₄×Dic₃	1	4	yes	—
161	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
162	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
163	C₂³×C₃	C₂×C₄×D₃	1	1	yes	characteristic
164	C₂×C₃×C₄	C₄×Dic₃	1	4	yes	—
165	C₂×C₃×C₄	C₄×Dic₃	1	4	yes	—
166	C₂×C₃×C₄	C₄×Dic₃	1	4	yes	—
167	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
168	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
169	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
170	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
171	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
172	C₂×C₃×C₄	C₂×C₄×D₃	1	2	yes	—
173	C₂×C₃×C₄	C₂²×Dic₃	1	2	yes	—
174	C₂×C₃×C₄	C₂²×Dic₃	1	2	yes	—
175	C₂×C₃×C₄	C₂²×Dic₃	1	2	yes	—
176	C₂×C₃×C₄	C₂³×D₃	1	1	yes	characteristic
177	C₂×Dic₃	—	3	8	no	—
178	C₂×Dic₃	—	12	4	no	—
179	C₃×C₈	C₂²×Dic₃	1	8	yes	—
180	C₃⋊C₈	—	3	8	no	—
181	C₂×C₉	—	2	2	no	—
182	C₂×C₉	—	2	2	no	—
183	C₂×C₉	—	2	2	no	—
184	C₂×C₉	C₂²×C₄²	1	1	yes	characteristic
185	C₂×C₉	#568	1	2	yes	—
186	C₂⁴	C₄×D₉	1	1	yes	characteristic
187	C₂²×C₄	—	9	4	no	—
188	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
189	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
190	C₂²×C₄	C₄×D₉	1	1	yes	characteristic
191	C₂²×C₄	C₄×D₉	1	4	yes	—
192	C₂²×C₄	C₂×Dic₉	1	4	yes	—
193	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
194	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
195	C₂²×C₄	C₂²×D₉	1	1	yes	characteristic
196	M₁₆	—	9	8	no	—
197	M₁₆	—	9	8	no	—
198	M₁₆	C₂×Dic₉	1	8	yes	—
199	M₁₆	C₂×Dic₉	1	8	yes	complemented
200	C₂×C₈	—	9	8	no	—
201	C₂×C₈	—	9	4	no	—
202	C₂×C₈	C₂×Dic₉	1	8	yes	complemented
203	C₂×C₈	C₂²×D₉	1	4	yes	—
204	C₄⋊C₄	—	9	4	no	—
205	C₄⋊C₄	—	9	4	no	—
206	C₂²⋊C₄	—	9	4	no	—
207	C₂²⋊C₄	—	9	4	no	—
208	C₄²	—	9	4	no	—
209	C₄²	—	9	4	no	—
210	C₂²×C₃	—	2	2	no	—
211	C₂²×C₃	—	2	2	no	—
212	C₂²×C₃	—	2	2	no	—
213	C₂²×C₃	—	4	4	no	—
214	C₂²×C₃	C₂×C₄×Dic₃	1	2	yes	—
215	C₂²×C₃	C₂×C₄×Dic₃	1	2	yes	—
216	C₂²×C₃	C₂×C₄×Dic₃	1	2	yes	—
217	C₂²×C₃	C₂²×C₄×D₃	1	1	yes	characteristic
218	C₃×C₄	C₂×C₄×Dic₃	1	2	yes	—
219	C₃×C₄	C₂×C₄×Dic₃	1	2	yes	—
220	C₃×C₄	C₂×C₄×Dic₃	1	2	yes	—
221	C₃×C₄	C₂²×C₄×D₃	1	1	yes	characteristic
222	C₃×C₄	C₂³×Dic₃	1	1	yes	characteristic
223	Dic₃	—	6	8	no	—
224	C₉	#2873	1	1	yes	characteristic, complemented, Sylow
225	C₂³	—	4	2	no	—
226	C₂³	C₄×Dic₉	1	4	yes	—
227	C₂³	C₂×C₄×D₉	1	1	yes	characteristic
228	C₂³	C₂×C₄×D₉	1	1	yes	characteristic
229	C₂³	C₂×C₄×D₉	1	1	yes	characteristic
230	C₂×C₄	—	9	8	no	—
231	C₂×C₄	—	36	4	no	—
232	C₂×C₄	C₄×Dic₉	1	4	yes	—
233	C₂×C₄	C₄×Dic₉	1	4	yes	—
234	C₂×C₄	C₄×Dic₉	1	4	yes	—
235	C₂×C₄	C₂×C₄×D₉	1	2	yes	—
236	C₂×C₄	C₂×C₄×D₉	1	2	yes	—
237	C₂×C₄	C₂×C₄×D₉	1	2	yes	—
238	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
239	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
240	C₂×C₄	C₂×C₄×D₉	1	1	yes	characteristic
241	C₂×C₄	C₂²×Dic₉	1	2	yes	—
242	C₂×C₄	C₂²×Dic₉	1	2	yes	—
243	C₂×C₄	C₂²×Dic₉	1	2	yes	—
244	C₂×C₄	C₂³×D₉	1	1	yes	characteristic
245	C₈	—	9	8	no	—
246	C₈	C₂²×Dic₉	1	8	yes	—
247	C₂×C₃	—	2	2	no	—
248	C₂×C₃	—	2	2	no	—
249	C₂×C₃	—	2	2	no	—
250	C₂×C₃	#5745	1	2	yes	—
251	C₂×C₃	C₂²×C₄×Dic₃	1	1	yes	characteristic
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	4	4	no	—
256	C₂²	C₂×C₄×Dic₉	1	2	yes	—
257	C₂²	C₂×C₄×Dic₉	1	2	yes	—
258	C₂²	C₂×C₄×Dic₉	1	2	yes	—
259	C₂²	C₂²×C₄×D₉	1	1	yes	characteristic
260	C₄	—	18	8	no	—
261	C₄	C₂×C₄×Dic₉	1	2	yes	—
262	C₄	C₂×C₄×Dic₉	1	2	yes	—
263	C₄	C₂×C₄×Dic₉	1	2	yes	—
264	C₄	C₂²×C₄×D₉	1	1	yes	characteristic
265	C₄	C₂³×Dic₉	1	1	yes	characteristic
266	C₃	#83747	1	1	yes	characteristic
267	C₂	—	2	2	no	—
268	C₂	—	2	2	no	—
269	C₂	—	2	2	no	—
270	C₂	#10589847	1	2	yes	—
271	C₂	C₂²×C₄×Dic₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₉
commutator	C₂×C₉	C₂²×C₄²
Frattini	C₂×C₃×C₄	C₂³×D₃
Fitting	C₂²×C₉×M₁₆	C₂
socle	C₂²×C₃	C₂²×C₄×D₃
prime radical	C₂⁴×C₃	C₄×D₃
cosocle	C₂×C₄×C₉	C₂⁴
hypercenter	C₂²×M₁₆	D₉
hypocenter	C₉	#2873
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
4. M₁₆×Dic₉	C₂	semidirect
5. #10588797	C₂	semidirect
6. #10588782	C₂	semidirect
9. C₂×C₉×M₁₆	C₄	semidirect
20. C₂×#3667	C₄	semidirect
24. #63375	C₂²	semidirect
25. C₈×Dic₉	C₂²	semidirect
43. C₉×M₁₆	C₂×C₄	semidirect
44. C₂×C₈×C₉	C₂×C₄	semidirect
53. #3667	C₂×C₄	semidirect
54. C₂×C₉⋊C₈	C₂×C₄	semidirect
145. C₂×M₁₆	Dic₉	semidirect
199. M₁₆	C₂×Dic₉	semidirect
202. C₂×C₈	C₂×Dic₉	semidirect
224. C₉	#2873	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₉×M₁₆	2
C₂×#63488	2
C₂²×#3667	2
M₁₆×Dic₉	2
#10588797	2
#10588782	2
#83747	3