Subgroups — #41007715

Order 26973.

620 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582600	C₃	1	4	yes	maximal, complemented
2	#22582598	C₃	1	4	yes	maximal, complemented
3	#22582598	C₃	1	4	yes	maximal, complemented
4	#22582598	C₃	1	4	yes	maximal, complemented
5	#22582595	C₃	1	2	yes	maximal, complemented
6	#22582590	C₃	1	2	yes	maximal, complemented
7	#22582590	C₃	1	4	yes	maximal, complemented
8	#22582590	C₃	1	2	yes	maximal, complemented
9	#22582570	C₃	1	4	yes	maximal, complemented
10	#22582567	C₃	1	2	yes	maximal, complemented
11	#22582553	C₃	1	1	yes	maximal, characteristic
12	#22582552	C₃	1	2	yes	maximal
13	C₉×#11758605	C₃	1	1	yes	maximal, characteristic, complemented
14	#22582543	C₃	1	2	yes	maximal, complemented
15	C₃²⋊C₉×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
16	C₃₇×#6898	C₃	1	1	yes	maximal, characteristic, complemented
17	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
18	C₃×#11758605	C₃²	1	2	yes	—
19	C₃×#11758605	C₃²	1	1	yes	characteristic
20	C₃×#11758606	C₃²	1	2	yes	—
21	C₃×#11758606	C₃²	1	4	yes	—
22	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
23	C₃×C₉×C₃₇⋊C₃	C₃²	1	2	yes	—
24	#15688506	C₃²	1	2	yes	complemented
25	#15688506	C₃²	1	2	yes	complemented
26	#15688506	C₃²	1	4	yes	complemented
27	#15688505	C₃²	1	2	yes	complemented
28	#15688505	C₃²	1	2	yes	complemented
29	#15688505	C₃²	1	4	yes	complemented
30	#15688504	C₃²	1	4	yes	complemented
31	#15688504	C₃²	1	4	yes	complemented
32	#15688504	C₃²	1	4	yes	complemented
33	#15688504	C₃²	1	4	yes	complemented
34	#15688504	C₃²	1	4	yes	complemented
35	#15688504	C₃²	1	4	yes	complemented
36	#15688504	C₃²	1	4	yes	complemented
37	#15688504	C₃²	1	4	yes	complemented
38	#15688504	C₃²	1	8	yes	complemented
39	#15688504	C₃²	1	4	yes	complemented
40	#15688504	C₃²	1	4	yes	complemented
41	C₃²⋊#66524	C₃²	1	2	yes	—
42	C₃²⋊#66524	C₃²	1	4	yes	—
43	C₃²⋊#66524	C₃²	1	2	yes	—
44	#15688502	C₃²	1	2	yes	—
45	#15688502	C₃²	1	4	yes	—
46	#15688502	C₃²	1	4	yes	—
47	#15688502	C₃²	1	2	yes	—
48	#15688502	C₃²	1	2	yes	—
49	#15688502	C₃²	1	4	yes	—
50	#15688502	C₃²	1	2	yes	—
51	#15688502	C₃²	1	4	yes	—
52	C₉×#66524	C₃²	1	2	yes	complemented
53	C₉×#66524	C₃²	1	4	yes	complemented
54	C₉×#66524	C₃²	1	2	yes	complemented
55	C₃₇×C₉⋊C₉	C₃²	1	2	yes	complemented
56	C₃₇×C₉⋊C₉	C₃²	1	2	yes	complemented
57	C₃₇×C₉⋊C₉	C₃²	1	4	yes	complemented
58	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
59	C₃₇×C₃²⋊C₉	C₃²	1	2	yes	—
60	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
61	C₉²×C₃₇	C₃²	1	1	yes	characteristic, complemented
62	C₃³×C₃₇	C₃³	1	1	yes	characteristic
63	C₃²×C₃₇⋊C₃	—	3	3	no	—
64	C₃²×C₃₇⋊C₃	—	9	1	no	—
65	C₃²×C₃₇⋊C₃	—	9	2	no	—
66	C₃²×C₃₇⋊C₃	C₃³	1	2	yes	—
67	C₃²×C₃₇⋊C₃	C₃³	1	1	yes	characteristic
68	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
69	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
70	C₃×C₉×C₃₇	C₃³	1	2	yes	—
71	C₃×C₉×C₃₇	C₃³	1	4	yes	—
72	C₃×C₉×C₃₇	C₃³	1	2	yes	—
73	C₃×C₉×C₃₇	C₃³	1	2	yes	—
74	C₃×#66524	C₃³	1	2	yes	—
75	C₃×#66524	C₃³	1	4	yes	—
76	C₃×#66524	C₃³	1	4	yes	—
77	C₃×#66524	C₃³	1	4	yes	—
78	C₃×#66524	C₃³	1	2	yes	—
79	C₃×#66524	C₃³	1	4	yes	—
80	C₃×#66524	C₃³	1	4	yes	—
81	#11758606	—	9	2	no	—
82	#11758606	—	9	4	no	—
83	#11758605	—	3	6	no	—
84	#11758605	—	3	3	no	—
85	C₉×C₃₇⋊C₃	—	9	1	no	—
86	C₉×C₃₇⋊C₃	—	9	2	no	—
87	#10624890	—	37	1	no	maximal, Sylow
88	C₃²×C₃₇	—	3	1	no	—
89	C₃²×C₃₇	—	3	2	no	—
90	C₃²×C₃₇	—	3	1	no	—
91	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
92	C₃×C₃₇⋊C₃	—	3	1	no	—
93	C₃×C₃₇⋊C₃	—	3	2	no	—
94	C₃×C₃₇⋊C₃	—	3	2	no	—
95	C₃×C₃₇⋊C₃	—	3	2	no	—
96	C₃×C₃₇⋊C₃	—	3	2	no	—
97	C₃×C₃₇⋊C₃	—	3	2	no	—
98	C₃×C₃₇⋊C₃	—	9	9	no	—
99	C₃×C₃₇⋊C₃	#737	1	3	yes	—
100	C₉×C₃₇	—	3	1	no	—
101	C₉×C₃₇	—	3	2	no	—
102	C₉×C₃₇	—	3	1	no	—
103	C₉×C₃₇	—	3	2	no	—
104	C₉×C₃₇	—	3	2	no	—
105	C₉×C₃₇	—	3	4	no	—
106	#66524	—	3	4	no	—
107	#66524	—	3	2	no	—
108	#66524	—	3	2	no	—
109	#66524	—	3	4	no	—
110	#66524	—	3	4	no	—
111	#66524	—	3	4	no	—
112	#66524	—	3	4	no	—
113	#6899	—	37	4	no	—
114	#6899	—	37	4	no	—
115	#6899	—	37	4	no	—
116	#6899	—	37	4	no	—
117	#6899	—	37	2	no	—
118	#6898	—	37	4	no	—
119	#6898	—	37	2	no	—
120	#6898	—	37	2	no	—
121	#6898	C₃₇⋊C₃	1	1	yes	characteristic, complemented
122	#6894	—	37	2	no	—
123	#6894	—	37	4	no	—
124	#6891	—	37	2	no	—
125	#6891	—	37	1	no	—
126	C₉×3₋¹⁺²	—	37	2	no	—
127	C₉×3₋¹⁺²	—	37	1	no	—
128	C₃×C₃²⋊C₉	—	37	1	no	—
129	C₃×C₃₇	—	9	1	no	—
130	C₃×C₃₇	C₃×#737	1	1	yes	characteristic
131	C₃×C₃₇	3₋¹⁺⁴	1	2	yes	—
132	C₃×C₃₇	3₋¹⁺⁴	1	1	yes	characteristic
133	C₃₇⋊C₃	—	3	3	no	—
134	C₃₇⋊C₃	—	9	2	no	—
135	C₃⁴	—	37	1	no	—
136	C₃×3₋¹⁺²	—	37	4	no	—
137	C₃×3₋¹⁺²	—	37	2	no	—
138	C₃×3₋¹⁺²	—	37	2	no	—
139	C₃×3₋¹⁺²	—	37	1	no	—
140	C₃²×C₉	—	37	1	no	—
141	C₃²×C₉	—	37	2	no	—
142	C₉⋊C₉	—	37	4	no	—
143	C₉⋊C₉	—	37	4	no	—
144	C₉⋊C₉	—	37	4	no	—
145	C₉⋊C₉	—	37	4	no	—
146	C₉⋊C₉	—	37	4	no	—
147	C₉⋊C₉	—	37	4	no	—
148	C₉⋊C₉	—	37	8	no	—
149	C₉⋊C₉	—	37	4	no	—
150	C₉⋊C₉	—	37	4	no	—
151	C₉⋊C₉	—	37	4	no	—
152	C₉⋊C₉	—	37	4	no	—
153	C₉⋊C₉	—	37	2	no	—
154	C₉⋊C₉	—	37	2	no	—
155	C₉⋊C₉	—	37	4	no	—
156	C₉⋊C₉	—	37	2	no	—
157	C₉⋊C₉	—	37	2	no	—
158	C₉⋊C₉	—	37	4	no	—
159	C₉⋊C₉	C₃×C₃₇⋊C₃	1	2	yes	complemented
160	C₉⋊C₉	C₃×C₃₇⋊C₃	1	2	yes	complemented
161	C₉⋊C₉	C₃×C₃₇⋊C₃	1	4	yes	complemented
162	C₃²⋊C₉	—	37	2	no	—
163	C₃²⋊C₉	—	37	4	no	—
164	C₃²⋊C₉	—	37	2	no	—
165	C₃²⋊C₉	—	37	4	no	—
166	C₃²⋊C₉	—	37	2	no	—
167	C₃²⋊C₉	—	37	4	no	—
168	C₃²⋊C₉	—	37	2	no	—
169	C₃²⋊C₉	—	37	4	no	—
170	C₃²⋊C₉	—	37	4	no	—
171	C₃²⋊C₉	—	37	2	no	—
172	C₃²⋊C₉	—	37	2	no	—
173	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
174	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	2	yes	—
175	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
176	C₉²	—	37	2	no	—
177	C₉²	—	37	4	no	—
178	C₉²	—	37	2	no	—
179	C₉²	C₃×C₃₇⋊C₃	1	1	yes	characteristic, complemented
180	C₃₇	#10624890	1	1	yes	characteristic, complemented, Sylow
181	C₃³	—	37	1	no	—
182	C₃³	—	37	2	no	—
183	C₃³	—	333	2	no	—
184	C₃³	—	333	1	no	—
185	C₃³	—	111	3	no	—
186	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
187	3₋¹⁺²	—	333	4	no	—
188	3₋¹⁺²	—	333	2	no	—
189	3₋¹⁺²	—	111	6	no	—
190	3₋¹⁺²	—	111	3	no	—
191	C₃×C₉	—	37	2	no	—
192	C₃×C₉	—	37	4	no	—
193	C₃×C₉	—	37	2	no	—
194	C₃×C₉	—	37	4	no	—
195	C₃×C₉	—	37	4	no	—
196	C₃×C₉	—	37	4	no	—
197	C₃×C₉	—	37	4	no	—
198	C₃×C₉	—	333	1	no	—
199	C₃×C₉	—	333	2	no	—
200	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
201	C₃×C₉	C₃²×C₃₇⋊C₃	1	2	yes	—
202	C₃×C₉	C₃²×C₃₇⋊C₃	1	4	yes	—
203	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
204	C₃×C₉	C₃²×C₃₇⋊C₃	1	2	yes	—
205	C₃×C₉	C₃²×C₃₇⋊C₃	1	2	yes	—
206	C₃²	—	3	1	no	—
207	C₃²	—	3	2	no	—
208	C₃²	—	3	1	no	—
209	C₃²	—	37	3	no	—
210	C₃²	—	333	9	no	—
211	C₃²	—	111	2	no	—
212	C₃²	—	111	2	no	—
213	C₃²	—	111	2	no	—
214	C₃²	—	111	2	no	—
215	C₃²	—	111	2	no	—
216	C₃²	—	111	1	no	—
217	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
218	C₉	—	3	2	no	—
219	C₉	—	3	1	no	—
220	C₉	—	3	4	no	—
221	C₉	—	3	2	no	—
222	C₉	—	3	1	no	—
223	C₉	—	3	2	no	—
224	C₉	—	111	4	no	—
225	C₉	—	111	2	no	—
226	C₉	—	111	4	no	—
227	C₉	—	111	4	no	—
228	C₉	—	111	4	no	—
229	C₉	—	111	2	no	—
230	C₉	—	111	4	no	—
231	C₃	—	9	1	no	—
232	C₃	—	333	2	no	—
233	C₃	—	111	3	no	—
234	C₃	#737×C₃₇⋊C₃	1	1	yes	characteristic
235	C₃	#22582677	1	2	yes	—
236	C₃	#22582678	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃³×C₃₇⋊C₃
commutator, socle, cosocle	C₃²×C₃₇	C₃⁴
Fitting	C₃₇×#6898	C₃
prime radical	C₃³×C₃₇⋊C₃	C₃²
hypercenter	#6898	C₃₇⋊C₃
hypocenter	C₃₇	#10624890
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ #6898

Complements

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

N	H ≅ G/N	type
1. #22582600	C₃	semidirect
2. #22582598	C₃	semidirect
3. #22582598	C₃	semidirect
4. #22582598	C₃	semidirect
5. #22582595	C₃	semidirect
6. #22582590	C₃	semidirect
7. #22582590	C₃	semidirect
8. #22582590	C₃	semidirect
9. #22582570	C₃	semidirect
10. #22582567	C₃	semidirect
13. C₉×#11758605	C₃	semidirect
14. #22582543	C₃	semidirect
16. C₃₇×#6898	C₃	semidirect
24. #15688506	C₃²	semidirect
25. #15688506	C₃²	semidirect
26. #15688506	C₃²	semidirect
27. #15688505	C₃²	semidirect
28. #15688505	C₃²	semidirect
29. #15688505	C₃²	semidirect
30. #15688504	C₃²	semidirect
31. #15688504	C₃²	semidirect
32. #15688504	C₃²	semidirect
33. #15688504	C₃²	semidirect
34. #15688504	C₃²	semidirect
35. #15688504	C₃²	semidirect
36. #15688504	C₃²	semidirect
37. #15688504	C₃²	semidirect
38. #15688504	C₃²	semidirect
39. #15688504	C₃²	semidirect
40. #15688504	C₃²	semidirect
52. C₉×#66524	C₃²	semidirect
53. C₉×#66524	C₃²	semidirect
54. C₉×#66524	C₃²	semidirect
55. C₃₇×C₉⋊C₉	C₃²	semidirect
56. C₃₇×C₉⋊C₉	C₃²	semidirect
57. C₃₇×C₉⋊C₉	C₃²	semidirect
61. C₉²×C₃₇	C₃²	semidirect
121. #6898	C₃₇⋊C₃	semidirect
159. C₉⋊C₉	C₃×C₃₇⋊C₃	semidirect
160. C₉⋊C₉	C₃×C₃₇⋊C₃	semidirect
161. C₉⋊C₉	C₃×C₃₇⋊C₃	semidirect
179. C₉²	C₃×C₃₇⋊C₃	semidirect
180. C₃₇	#10624890	semidirect

Maximal subgroups

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

subgroup	index
#22582600	3
#22582598	3
#22582595	3
#22582590	3
#22582570	3
#22582567	3
#22582553	3
#22582552	3
C₉×#11758605	3
#22582543	3
C₃²⋊C₉×C₃₇⋊C₃	3
C₃₇×#6898	3
#10624890	37