Finite Group Databasebeta

Subgroups — #41007332

Order 26973.

226 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22582634	C₃	1	1	yes	maximal, characteristic, complemented
2	#22582634	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582631	C₃	1	1	yes	maximal, characteristic, complemented
4	C₉²×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
5	#22582509	C₃	1	1	yes	maximal, characteristic, complemented
6	#22582508	C₃	1	1	yes	maximal, characteristic, complemented
7	#22582506	C₃	1	1	yes	maximal, characteristic, complemented
8	#22582505	C₃	1	1	yes	maximal, characteristic, complemented
9	#22582504	C₃	1	1	yes	maximal, characteristic, complemented
10	#22582503	C₃	1	1	yes	maximal, characteristic, complemented
11	#22582502	C₃	1	1	yes	maximal, characteristic, complemented
12	#22582501	C₃	1	1	yes	maximal, characteristic, complemented
13	C₃₇×#6878	C₃	1	1	yes	maximal, characteristic, complemented
14	#15688532	—	3	1	no	—
15	#15688532	—	3	1	no	—
16	#15688530	—	3	1	no	—
17	C₃×C₉×C₃₇⋊C₃	—	3	1	no	—
18	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
19	#15688521	C₃²	1	1	yes	characteristic
20	#15688521	C₃²	1	1	yes	characteristic
21	#15688521	C₃²	1	1	yes	characteristic
22	#15688521	C₃²	1	1	yes	characteristic
23	#15688519	C₃²	1	1	yes	characteristic
24	#15688519	C₃²	1	1	yes	characteristic
25	#15688517	—	3	1	no	—
26	#15688517	—	3	1	no	—
27	#15688515	—	3	1	no	—
28	#15688514	—	3	1	no	—
29	#15688514	—	3	1	no	—
30	#15688512	—	3	1	no	—
31	C₉×#66524	—	3	1	no	—
32	C₉×#66524	—	3	1	no	—
33	C₉×#66524	C₃²	1	1	yes	characteristic
34	C₉×#66524	C₃²	1	1	yes	characteristic
35	C₃₇×#733	C₃²	1	1	yes	characteristic
36	C₃₇×#733	C₃²	1	1	yes	characteristic
37	C₃₇×#732	C₃²	1	1	yes	characteristic
38	C₉²×C₃₇	C₃²	1	1	yes	characteristic
39	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
40	3₋¹⁺²×C₃₇	—	3	1	no	—
41	3₋¹⁺²×C₃₇	—	3	1	no	—
42	H(3,𝔽₃)×C₃₇	—	3	1	no	—
43	C₃×C₉×C₃₇	—	3	1	no	—
44	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
45	C₃²⋊(C₃₇⋊C₃)	—	3	1	no	—
46	C₃²⋊(C₃₇⋊C₃)	—	3	1	no	—
47	#11758608	—	3	1	no	—
48	#11758608	—	3	1	no	—
49	#11758608	—	3	1	no	—
50	#11758608	—	3	1	no	—
51	#11758608	—	9	1	no	—
52	#11758608	—	9	1	no	—
53	C₃×#66524	—	3	1	no	—
54	C₃×#66524	—	3	1	no	—
55	C₃×#66524	—	9	1	no	—
56	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
57	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
58	#11758606	—	9	1	no	—
59	#11758606	—	9	1	no	—
60	#11758605	—	9	1	no	—
61	#11758605	—	9	1	no	—
62	C₉×C₃₇⋊C₃	—	3	9	no	—
63	C₉×C₃₇⋊C₃	—	3	1	no	—
64	C₉×C₃₇⋊C₃	—	3	1	no	—
65	C₉×C₃₇⋊C₃	—	3	1	no	—
66	C₉×C₃₇⋊C₃	—	9	1	no	—
67	C₉×C₃₇⋊C₃	—	9	1	no	—
68	#10624837	—	37	1	no	maximal, Sylow
69	C₃²×C₃₇	—	9	1	no	—
70	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
71	C₃×C₃₇⋊C₃	—	3	1	no	—
72	C₃×C₃₇⋊C₃	—	3	3	no	—
73	C₃×C₃₇⋊C₃	—	9	1	no	—
74	C₃×C₃₇⋊C₃	—	9	1	no	—
75	C₉×C₃₇	—	3	3	no	—
76	C₉×C₃₇	—	3	1	no	—
77	C₉×C₃₇	—	9	1	no	—
78	C₉×C₃₇	—	9	1	no	—
79	#66524	—	3	3	no	—
80	#66524	—	3	3	no	—
81	#66524	—	3	1	no	—
82	#66524	—	9	1	no	—
83	#66524	—	9	1	no	—
84	#66524	—	9	1	no	—
85	#66524	—	9	1	no	—
86	#66524	#732	1	1	yes	characteristic
87	#66524	#732	1	1	yes	characteristic
88	#66524	#732	1	1	yes	characteristic
89	#6908	—	37	1	no	—
90	#6908	—	37	1	no	—
91	#6908	—	37	1	no	—
92	C₃×C₉²	—	37	1	no	—
93	#6880	—	37	1	no	—
94	#6880	—	37	1	no	—
95	#6879	—	37	1	no	—
96	#6878	—	37	1	no	—
97	#6878	—	37	1	no	—
98	#6878	—	37	1	no	—
99	#6878	—	37	1	no	—
100	#6878	—	37	1	no	—
101	#6878	C₃₇⋊C₃	1	1	yes	characteristic, complemented
102	C₃×C₃₇	—	3	1	no	—
103	C₃×C₃₇	—	27	1	no	—
104	C₃×C₃₇	C₃×#732	1	1	yes	characteristic
105	C₃₇⋊C₃	—	3	3	no	—
106	C₃₇⋊C₃	—	27	1	no	—
107	C₃₇⋊C₃	—	27	1	no	—
108	#737	—	111	1	no	—
109	#737	—	111	1	no	—
110	#737	—	111	1	no	—
111	C₃²×C₉	—	37	1	no	—
112	C₃²×C₉	—	111	1	no	—
113	#733	—	37	1	no	—
114	#733	—	37	1	no	—
115	#733	—	37	1	no	—
116	#733	—	37	1	no	—
117	#733	C₃×C₃₇⋊C₃	1	1	yes	characteristic
118	#733	C₃×C₃₇⋊C₃	1	1	yes	characteristic
119	#732	—	37	1	no	—
120	#732	—	37	1	no	—
121	#732	C₃×C₃₇⋊C₃	1	1	yes	characteristic
122	#731	—	111	1	no	—
123	#731	—	111	1	no	—
124	#731	—	111	1	no	—
125	AΣL(1,𝔽₂₇)	—	111	1	no	—
126	AΣL(1,𝔽₂₇)	—	111	1	no	—
127	AΣL(1,𝔽₂₇)	—	111	1	no	—
128	C₉²	—	37	1	no	—
129	C₉²	—	37	1	no	—
130	C₉²	—	111	1	no	—
131	C₉²	—	111	1	no	—
132	C₉²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
133	C₃₇	#10624837	1	1	yes	characteristic, complemented, Sylow
134	C₃³	—	37	1	no	—
135	3₋¹⁺²	—	3	1	no	—
136	3₋¹⁺²	—	3	1	no	—
137	3₋¹⁺²	—	333	1	no	—
138	3₋¹⁺²	—	333	1	no	—
139	3₋¹⁺²	—	333	1	no	—
140	3₋¹⁺²	—	333	1	no	—
141	3₋¹⁺²	—	333	1	no	—
142	3₋¹⁺²	—	333	1	no	—
143	3₋¹⁺²	—	111	1	no	—
144	3₋¹⁺²	—	111	1	no	—
145	3₋¹⁺²	—	111	1	no	—
146	3₋¹⁺²	—	111	1	no	—
147	H(3,𝔽₃)	—	3	1	no	—
148	H(3,𝔽₃)	—	111	1	no	—
149	H(3,𝔽₃)	—	111	1	no	—
150	C₃×C₉	—	3	1	no	—
151	C₃×C₉	—	37	1	no	—
152	C₃×C₉	—	37	1	no	—
153	C₃×C₉	—	333	1	no	—
154	C₃×C₉	—	333	1	no	—
155	C₃×C₉	—	333	1	no	—
156	C₃×C₉	—	111	1	no	—
157	C₃×C₉	—	111	1	no	—
158	C₃×C₉	—	111	1	no	—
159	C₃×C₉	—	111	9	no	—
160	C₃×C₉	—	111	1	no	—
161	C₃×C₉	—	111	1	no	—
162	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
163	C₃²	—	9	1	no	—
164	C₃²	—	333	1	no	—
165	C₃²	—	333	1	no	—
166	C₃²	—	111	3	no	—
167	C₃²	—	111	1	no	—
168	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
169	C₉	—	3	3	no	—
170	C₉	—	3	1	no	—
171	C₉	—	9	1	no	—
172	C₉	—	9	1	no	—
173	C₉	—	37	1	no	—
174	C₉	—	37	1	no	—
175	C₉	—	37	1	no	—
176	C₉	—	333	1	no	—
177	C₉	—	333	1	no	—
178	C₉	—	333	1	no	—
179	C₉	—	333	1	no	—
180	C₉	—	111	3	no	—
181	C₉	—	111	3	no	—
182	C₉	—	111	1	no	—
183	C₃	—	3	1	no	—
184	C₃	—	27	1	no	—
185	C₃	—	111	3	no	—
186	C₃	—	999	1	no	—
187	C₃	—	999	1	no	—
188	C₃	#732×C₃₇⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#732×C₃₇⋊C₃
commutator, cosocle	C₃×C₉×C₃₇	C₃³
Frattini	C₃×C₉	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6878	C₃
socle	C₃×C₃₇	C₃×#732
hypercenter	#6878	C₃₇⋊C₃
hypocenter	C₃₇	#10624837
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃×C₉ ◁ #6878

Complements

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

N	H ≅ G/N	type
1. #22582634	C₃	semidirect
2. #22582634	C₃	semidirect
3. #22582631	C₃	semidirect
4. C₉²×C₃₇⋊C₃	C₃	semidirect
5. #22582509	C₃	semidirect
6. #22582508	C₃	semidirect
7. #22582506	C₃	semidirect
8. #22582505	C₃	semidirect
9. #22582504	C₃	semidirect
10. #22582503	C₃	semidirect
11. #22582502	C₃	semidirect
12. #22582501	C₃	semidirect
13. C₃₇×#6878	C₃	semidirect
101. #6878	C₃₇⋊C₃	semidirect
133. C₃₇	#10624837	semidirect

Maximal subgroups

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

subgroup	index
#22582634	3
#22582631	3
C₉²×C₃₇⋊C₃	3
#22582509	3
#22582508	3
#22582506	3
#22582505	3
#22582504	3
#22582503	3
#22582502	3
#22582501	3
C₃₇×#6878	3
#10624837	37