Subgroups — #41007631

Order 26973.

758 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688532	C₃	1	1	yes	maximal, characteristic
2	#22582603	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582602	C₃	1	1	yes	maximal, characteristic, complemented
4	#22582600	C₃	1	2	yes	maximal, complemented
5	#22582599	C₃	1	2	yes	maximal, complemented
6	#22582594	C₃	1	2	yes	maximal, complemented
7	#22582584	C₃	1	3	yes	maximal, complemented
8	#22582583	C₃	1	3	yes	maximal, complemented
9	#22582580	C₃	1	3	yes	maximal, complemented
10	#22582567	C₃	1	3	yes	maximal, complemented
11	#22582566	C₃	1	3	yes	maximal, complemented
12	#22582555	C₃	1	3	yes	maximal
13	3₋¹⁺²×#66524	C₃	1	3	yes	maximal, complemented
14	#22582543	C₃	1	3	yes	maximal, complemented
15	#22582542	C₃	1	3	yes	maximal, complemented
16	#22582537	C₃	1	2	yes	maximal, complemented
17	C₉²×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
18	C₃₇×#6900	C₃	1	1	yes	maximal, characteristic, complemented
19	#15688532	—	3	9	no	—
20	C₃×#11758608	C₃²	1	1	yes	characteristic
21	C₃×#11758605	C₃²	1	3	yes	—
22	C₃×#11758605	C₃²	1	1	yes	characteristic
23	C₃×#11758605	C₃²	1	1	yes	characteristic
24	C₃×#11758605	C₃²	1	1	yes	characteristic
25	C₃×#11758606	C₃²	1	3	yes	—
26	C₃×#11758606	C₃²	1	1	yes	characteristic
27	C₃×#11758606	C₃²	1	1	yes	characteristic
28	C₃×#11758606	C₃²	1	1	yes	characteristic
29	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	1	yes	characteristic
30	C₃²×#66524	C₃²	1	1	yes	characteristic
31	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
32	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
33	C₃×C₉×C₃₇⋊C₃	C₃²	1	3	yes	—
34	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
35	#15688506	C₃²	1	2	yes	—
36	#15688506	C₃²	1	3	yes	—
37	#15688506	C₃²	1	6	yes	—
38	#15688506	C₃²	1	3	yes	—
39	#15688505	C₃²	1	2	yes	—
40	#15688505	C₃²	1	3	yes	—
41	#15688505	C₃²	1	6	yes	—
42	#15688505	C₃²	1	3	yes	—
43	#15688504	C₃²	1	6	yes	—
44	#15688504	C₃²	1	6	yes	—
45	#15688504	C₃²	1	6	yes	—
46	#15688504	C₃²	1	6	yes	—
47	#15688504	C₃²	1	6	yes	—
48	C₃²⋊#66524	C₃²	1	6	yes	—
49	#15688502	C₃²	1	6	yes	—
50	#15688502	C₃²	1	6	yes	—
51	#15688502	C₃²	1	6	yes	—
52	C₉×#66524	C₃²	1	6	yes	—
53	C₉×#66524	C₃²	1	3	yes	—
54	C₉×#66524	C₃²	1	3	yes	—
55	C₉×#66524	C₃²	1	2	yes	—
56	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
57	C₃×3₋¹⁺²×C₃₇	C₃²	1	1	yes	characteristic
58	C₃₇×C₉⋊C₉	C₃²	1	2	yes	—
59	C₃₇×C₉⋊C₉	C₃²	1	3	yes	—
60	C₃₇×C₉⋊C₉	C₃²	1	3	yes	—
61	C₉²×C₃₇	C₃²	1	1	yes	characteristic
62	C₃³×C₃₇	C₃³	1	1	yes	characteristic
63	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
64	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
65	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
66	3₋¹⁺²×C₃₇	—	3	3	no	—
67	3₋¹⁺²×C₃₇	—	3	9	no	—
68	C₃×C₉×C₃₇	C₃³	1	6	yes	—
69	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
70	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
71	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
72	C₃×C₉×C₃₇	C₃³	1	3	yes	—
73	C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
74	#11758608	—	3	3	no	—
75	C₃×#66524	—	3	3	no	—
76	C₃×#66524	C₃³	1	6	yes	—
77	C₃×#66524	C₃³	1	6	yes	—
78	C₃×#66524	C₃³	1	6	yes	—
79	C₃×#66524	C₃³	1	1	yes	characteristic
80	C₃×#66524	C₃³	1	1	yes	characteristic
81	C₃×#66524	C₃³	1	1	yes	characteristic
82	C₃×#66524	C₃³	1	1	yes	characteristic
83	C₃×#66524	C₃³	1	1	yes	characteristic
84	C₃×#66524	C₃³	1	1	yes	characteristic
85	#11758606	—	3	3	no	—
86	#11758606	—	3	3	no	—
87	#11758606	—	3	9	no	—
88	#11758606	—	3	3	no	—
89	#11758605	—	3	3	no	—
90	#11758605	—	3	3	no	—
91	#11758605	—	3	9	no	—
92	#11758605	—	3	3	no	—
93	C₉×C₃₇⋊C₃	—	3	9	no	—
94	C₉×C₃₇⋊C₃	—	3	3	no	—
95	C₉×C₃₇⋊C₃	—	3	3	no	—
96	C₉×C₃₇⋊C₃	3₋¹⁺²	1	9	yes	complemented
97	#10624879	—	37	1	no	maximal, Sylow
98	C₃²×C₃₇	—	3	1	no	—
99	C₃²×C₃₇	—	3	3	no	—
100	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
101	C₃×C₃₇⋊C₃	—	3	3	no	—
102	C₃×C₃₇⋊C₃	—	3	1	no	—
103	C₃×C₃₇⋊C₃	—	3	1	no	—
104	C₃×C₃₇⋊C₃	—	3	3	no	—
105	C₃×C₃₇⋊C₃	—	3	3	no	—
106	C₃×C₃₇⋊C₃	C₃×3₋¹⁺²	1	3	yes	—
107	C₉×C₃₇	—	3	1	no	—
108	C₉×C₃₇	—	3	1	no	—
109	C₉×C₃₇	—	3	6	no	—
110	C₉×C₃₇	C₃×3₋¹⁺²	1	3	yes	—
111	C₉×C₃₇	#737	1	9	yes	complemented
112	#66524	—	3	1	no	—
113	#66524	—	3	1	no	—
114	#66524	—	3	1	no	—
115	#66524	—	3	1	no	—
116	#66524	—	3	6	no	—
117	#66524	—	3	6	no	—
118	#66524	—	3	6	no	—
119	#66524	C₃×3₋¹⁺²	1	3	yes	—
120	#66524	C₃×3₋¹⁺²	1	3	yes	—
121	C₃×#737	—	37	1	no	—
122	#6900	—	37	1	no	—
123	#6900	—	37	1	no	—
124	#6900	C₃₇⋊C₃	1	1	yes	characteristic, complemented
125	#6899	—	37	2	no	—
126	#6899	—	37	2	no	—
127	#6899	—	37	2	no	—
128	#6896	—	37	3	no	—
129	#6896	—	37	3	no	—
130	#6896	—	37	3	no	—
131	#6894	—	37	3	no	—
132	#6894	—	37	3	no	—
133	C₉⋊H(3,𝔽₃)	—	37	3	no	—
134	C₉×3₋¹⁺²	—	37	3	no	—
135	C₉×3₋¹⁺²	—	37	3	no	—
136	C₉×3₋¹⁺²	—	37	3	no	—
137	#6887	—	37	2	no	—
138	C₃×C₉²	—	37	1	no	—
139	C₃×C₃₇	—	9	1	no	—
140	C₃×C₃₇	C₃²×3₋¹⁺²	1	1	yes	characteristic
141	C₃×C₃₇	C₃×#737	1	3	yes	—
142	C₃₇⋊C₃	—	3	3	no	—
143	C₃₇⋊C₃	—	9	1	no	—
144	C₃₇⋊C₃	—	9	1	no	—
145	#737	—	111	9	no	—
146	C₃×3₋¹⁺²	—	37	1	no	—
147	C₃×3₋¹⁺²	—	37	1	no	—
148	C₃×3₋¹⁺²	—	37	1	no	—
149	C₃×3₋¹⁺²	—	37	1	no	—
150	C₃×3₋¹⁺²	—	37	1	no	—
151	C₃×3₋¹⁺²	—	37	1	no	—
152	C₃×3₋¹⁺²	—	37	1	no	—
153	C₃×3₋¹⁺²	—	37	3	no	—
154	C₃×3₋¹⁺²	—	37	3	no	—
155	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
156	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	3	yes	—
157	C₃×H(3,𝔽₃)	—	37	1	no	—
158	C₃²×C₉	—	37	1	no	—
159	C₃²×C₉	—	37	1	no	—
160	C₃²×C₉	—	37	1	no	—
161	C₃²×C₉	—	37	1	no	—
162	C₃²×C₉	—	37	3	no	—
163	C₉⋊C₉	—	37	6	no	—
164	C₉⋊C₉	—	37	6	no	—
165	C₉⋊C₉	—	37	6	no	—
166	C₉⋊C₉	—	37	6	no	—
167	C₉⋊C₉	—	37	6	no	—
168	C₉⋊C₉	—	37	2	no	—
169	C₉⋊C₉	—	37	6	no	—
170	C₉⋊C₉	—	37	3	no	—
171	C₉⋊C₉	—	37	3	no	—
172	C₉⋊C₉	—	37	2	no	—
173	C₉⋊C₉	—	37	6	no	—
174	C₉⋊C₉	—	37	3	no	—
175	C₉⋊C₉	—	37	3	no	—
176	C₉⋊C₉	C₃×C₃₇⋊C₃	1	2	yes	—
177	C₉⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
178	C₉⋊C₉	C₃×C₃₇⋊C₃	1	3	yes	—
179	C₃²⋊C₉	—	37	6	no	—
180	C₃²⋊C₉	—	37	6	no	—
181	C₃²⋊C₉	—	37	6	no	—
182	C₃²⋊C₉	—	37	6	no	—
183	C₉²	—	37	3	no	—
184	C₉²	—	37	3	no	—
185	C₉²	—	37	2	no	—
186	C₉²	—	37	6	no	—
187	C₉²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
188	C₃₇	#10624879	1	1	yes	characteristic, complemented, Sylow
189	C₃³	—	37	1	no	—
190	C₃³	—	37	1	no	—
191	C₃³	—	37	1	no	—
192	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
193	3₋¹⁺²	—	3	3	no	—
194	3₋¹⁺²	—	3	9	no	—
195	3₋¹⁺²	—	111	3	no	—
196	3₋¹⁺²	—	111	3	no	—
197	3₋¹⁺²	—	111	3	no	—
198	3₋¹⁺²	—	111	3	no	—
199	3₋¹⁺²	—	111	3	no	—
200	3₋¹⁺²	—	111	3	no	—
201	3₋¹⁺²	—	111	3	no	—
202	3₋¹⁺²	—	111	9	no	—
203	3₋¹⁺²	—	111	9	no	—
204	H(3,𝔽₃)	—	111	3	no	—
205	C₃×C₉	—	37	1	no	—
206	C₃×C₉	—	37	1	no	—
207	C₃×C₉	—	37	9	no	—
208	C₃×C₉	—	37	6	no	—
209	C₃×C₉	—	37	6	no	—
210	C₃×C₉	—	37	6	no	—
211	C₃×C₉	—	37	1	no	—
212	C₃×C₉	—	37	1	no	—
213	C₃×C₉	—	37	1	no	—
214	C₃×C₉	—	37	1	no	—
215	C₃×C₉	—	111	3	no	—
216	C₃×C₉	—	111	3	no	—
217	C₃×C₉	—	111	3	no	—
218	C₃×C₉	—	111	9	no	—
219	C₃×C₉	C₃²×C₃₇⋊C₃	1	6	yes	—
220	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
221	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
222	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
223	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
224	C₃²	—	3	1	no	—
225	C₃²	—	3	3	no	—
226	C₃²	—	37	3	no	—
227	C₃²	—	111	1	no	—
228	C₃²	—	111	1	no	—
229	C₃²	—	111	3	no	—
230	C₃²	—	111	3	no	—
231	C₃²	—	111	3	no	—
232	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
233	C₉	—	3	1	no	—
234	C₉	—	3	1	no	—
235	C₉	—	3	6	no	—
236	C₉	—	37	3	no	—
237	C₉	—	37	3	no	—
238	C₉	—	111	1	no	—
239	C₉	—	111	1	no	—
240	C₉	—	111	1	no	—
241	C₉	—	111	1	no	—
242	C₉	—	111	6	no	—
243	C₉	—	111	6	no	—
244	C₉	—	111	6	no	—
245	C₉	3₋¹⁺²×C₃₇⋊C₃	1	3	yes	—
246	C₉	#15688532	1	9	yes	complemented
247	C₃	—	9	1	no	—
248	C₃	—	333	1	no	—
249	C₃	—	333	1	no	—
250	C₃	—	111	3	no	—
251	C₃	C₃×3₋¹⁺²×C₃₇⋊C₃	1	1	yes	characteristic
252	C₃	C₃×#15688532	1	3	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₇⋊C₃
commutator, socle, cosocle	C₃²×C₃₇	C₃⁴
Fitting	C₃₇×#6900	C₃
prime radical	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²
hypercenter	#6900	C₃₇⋊C₃
hypocenter	C₃₇	#10624879
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ #6900

Complements

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

N	H ≅ G/N	type
2. #22582603	C₃	semidirect
3. #22582602	C₃	semidirect
4. #22582600	C₃	semidirect
5. #22582599	C₃	semidirect
6. #22582594	C₃	semidirect
7. #22582584	C₃	semidirect
8. #22582583	C₃	semidirect
9. #22582580	C₃	semidirect
10. #22582567	C₃	semidirect
11. #22582566	C₃	semidirect
13. 3₋¹⁺²×#66524	C₃	semidirect
14. #22582543	C₃	semidirect
15. #22582542	C₃	semidirect
16. #22582537	C₃	semidirect
17. C₉²×C₃₇⋊C₃	C₃	semidirect
18. C₃₇×#6900	C₃	semidirect
96. C₉×C₃₇⋊C₃	3₋¹⁺²	semidirect
111. C₉×C₃₇	#737	semidirect
124. #6900	C₃₇⋊C₃	semidirect
188. C₃₇	#10624879	semidirect
246. C₉	#15688532	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15688532	3
#22582603	3
#22582602	3
#22582600	3
#22582599	3
#22582594	3
#22582584	3
#22582583	3
#22582580	3
#22582567	3
#22582566	3
#22582555	3
3₋¹⁺²×#66524	3
#22582543	3
#22582542	3
#22582537	3
C₉²×C₃₇⋊C₃	3
C₃₇×#6900	3
#10624879	37