Finite Group Databasebeta

Subgroups — #41008229

Order 26973.

948 conjugacy classes of subgroups; only the 268 normal subgroups are shown.

normal subgroups only

This group has 948 conjugacy classes of subgroups — too many to draw. Only its 268 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	#22582679	C₃	1	3	yes	maximal, complemented
2	C₃×#15688532	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582663	C₃	1	9	yes	maximal, complemented
4	#22582662	C₃	1	9	yes	maximal, complemented
5	#22582652	C₃	1	1	yes	maximal, characteristic, complemented
6	#22582648	C₃	1	1	yes	maximal, characteristic, complemented
7	#22582639	C₃	1	3	yes	maximal, complemented
8	#22582634	C₃	1	9	yes	maximal, complemented
9	C₃×#15688517	C₃	1	3	yes	maximal, complemented
10	C₃₇×#6912	C₃	1	1	yes	maximal, characteristic, complemented
11	#15688532	—	3	9	no	—
12	#15688532	—	3	9	no	—
13	#15688532	C₃²	1	27	yes	complemented
14	#15688531	—	3	9	no	—
15	C₃×#11758608	C₃²	1	3	yes	—
16	C₃×#11758608	C₃²	1	1	yes	characteristic
17	C₃×#11758605	C₃²	1	3	yes	complemented
18	C₃×#11758606	C₃²	1	3	yes	complemented
19	3₋¹⁺²×C₃₇⋊C₃	—	3	9	no	—
20	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	1	yes	characteristic
21	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	3	yes	—
22	C₃²×#66524	C₃²	1	1	yes	characteristic
23	C₃×C₉×C₃₇⋊C₃	C₃²	1	3	yes	complemented
24	#15688521	C₃²	1	9	yes	complemented
25	#15688519	C₃²	1	9	yes	—
26	#15688517	C₃²	1	9	yes	complemented
27	#15688517	C₃²	1	9	yes	complemented
28	#15688517	C₃²	1	9	yes	complemented
29	#15688514	C₃²	1	27	yes	complemented
30	C₃×3₋¹⁺²×C₃₇	C₃²	1	3	yes	—
31	C₃×3₋¹⁺²×C₃₇	C₃²	1	1	yes	characteristic
32	C₃₇×#733	C₃²	1	9	yes	complemented
33	C₃³×C₃₇	C₃³	1	1	yes	characteristic
34	C₃²×C₃₇⋊C₃	—	3	3	no	—
35	C₃²×C₃₇⋊C₃	C₃³	1	3	yes	—
36	3₋¹⁺²×C₃₇	—	3	3	no	—
37	3₋¹⁺²×C₃₇	—	3	3	no	—
38	3₋¹⁺²×C₃₇	—	3	3	no	—
39	3₋¹⁺²×C₃₇	C₃³	1	9	yes	—
40	C₃×C₉×C₃₇	—	3	3	no	—
41	C₃×C₉×C₃₇	C₃³	1	3	yes	—
42	C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
43	C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
44	C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
45	C₃²⋊(C₃₇⋊C₃)	C₃³	1	9	yes	—
46	#11758608	—	3	3	no	—
47	#11758608	—	3	3	no	—
48	#11758608	H(3,𝔽₃)	1	9	yes	—
49	#11758608	C₃³	1	9	yes	—
50	C₃×#66524	—	3	3	no	—
51	C₃×#66524	—	3	3	no	—
52	C₃×#66524	C₃³	1	1	yes	characteristic
53	C₃×#66524	C₃³	1	1	yes	characteristic
54	C₃×#66524	C₃³	1	1	yes	characteristic
55	C₃×#66524	C₃³	1	3	yes	—
56	#11758606	—	3	27	no	—
57	#11758606	—	3	9	no	—
58	#11758605	—	3	27	no	—
59	#11758605	—	3	9	no	—
60	C₉×C₃₇⋊C₃	—	3	9	no	—
61	C₉×C₃₇⋊C₃	—	3	27	no	—
62	#10624935	—	37	1	no	maximal, Sylow
63	C₃²×C₃₇	—	9	1	no	—
64	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	3	yes	—
65	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
66	C₃×C₃₇⋊C₃	—	3	3	no	—
67	C₃×C₃₇⋊C₃	—	3	9	no	—
68	C₃×C₃₇⋊C₃	—	9	3	no	—
69	C₃×C₃₇⋊C₃	—	27	3	no	—
70	C₉×C₃₇	—	3	9	no	—
71	C₉×C₃₇	—	3	3	no	—
72	#66524	—	3	1	no	—
73	#66524	—	3	9	no	—
74	#66524	—	3	1	no	—
75	#66524	—	3	1	no	—
76	#66524	C₃×H(3,𝔽₃)	1	9	yes	—
77	3₋¹⁺⁴	—	37	3	no	—
78	C₃×#737	—	37	1	no	—
79	#6913	—	37	9	no	—
80	#6913	—	37	9	no	—
81	#6912	—	37	1	no	—
82	#6912	C₃₇⋊C₃	1	1	yes	characteristic, complemented
83	#6911	—	37	1	no	—
84	#6909	—	37	3	no	—
85	#6908	—	37	9	no	—
86	C₃×#731	—	37	3	no	—
87	C₃×C₃₇	—	3	1	no	—
88	C₃×C₃₇	—	3	3	no	—
89	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
90	C₃₇⋊C₃	—	9	3	no	—
91	C₃₇⋊C₃	—	9	9	no	—
92	#737	—	37	27	no	—
93	#737	—	111	9	no	—
94	#737	—	111	9	no	—
95	#737	—	111	9	no	—
96	C₃×3₋¹⁺²	—	37	1	no	—
97	C₃×3₋¹⁺²	—	37	3	no	—
98	C₃×3₋¹⁺²	—	37	3	no	—
99	C₃×3₋¹⁺²	—	37	3	no	—
100	C₃×3₋¹⁺²	—	111	9	no	—
101	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	3	yes	—
102	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
103	C₃×H(3,𝔽₃)	—	37	3	no	—
104	C₃×H(3,𝔽₃)	—	37	1	no	—
105	C₃²×C₉	—	37	1	no	—
106	C₃²×C₉	—	37	3	no	—
107	#733	—	37	9	no	—
108	#733	C₃×C₃₇⋊C₃	1	9	yes	complemented
109	#732	—	37	9	no	—
110	#731	—	37	9	no	—
111	#731	—	37	9	no	—
112	#731	—	37	9	no	—
113	AΣL(1,𝔽₂₇)	—	37	27	no	—
114	C₃₇	#10624935	1	1	yes	characteristic, complemented, Sylow
115	C₃³	—	37	3	no	—
116	C₃³	—	111	3	no	—
117	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
118	3₋¹⁺²	—	3	3	no	—
119	3₋¹⁺²	—	3	3	no	—
120	3₋¹⁺²	—	3	3	no	—
121	3₋¹⁺²	—	37	9	no	—
122	3₋¹⁺²	—	37	9	no	—
123	3₋¹⁺²	—	111	9	no	—
124	3₋¹⁺²	—	111	27	no	—
125	3₋¹⁺²	—	111	9	no	—
126	3₋¹⁺²	—	111	27	no	—
127	3₋¹⁺²	—	111	3	no	—
128	3₋¹⁺²	—	111	3	no	—
129	3₋¹⁺²	C₃²×C₃₇⋊C₃	1	9	yes	—
130	H(3,𝔽₃)	—	37	9	no	—
131	H(3,𝔽₃)	—	111	3	no	—
132	H(3,𝔽₃)	—	111	3	no	—
133	H(3,𝔽₃)	—	111	3	no	—
134	C₃×C₉	—	3	3	no	—
135	C₃×C₉	—	37	3	no	—
136	C₃×C₉	—	37	1	no	—
137	C₃×C₉	—	37	1	no	—
138	C₃×C₉	—	37	1	no	—
139	C₃×C₉	—	111	9	no	—
140	C₃×C₉	—	111	27	no	—
141	C₃×C₉	—	111	3	no	—
142	C₃×C₉	—	111	3	no	—
143	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
144	C₃²	—	9	1	no	—
145	C₃²	—	333	3	no	—
146	C₃²	—	111	9	no	—
147	C₃²	—	111	3	no	—
148	C₃²	—	999	3	no	—
149	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	3	yes	—
150	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
151	C₉	—	3	9	no	—
152	C₉	—	3	3	no	—
153	C₉	—	37	9	no	—
154	C₉	—	111	1	no	—
155	C₉	—	111	1	no	—
156	C₉	—	111	1	no	—
157	C₉	—	111	9	no	—
158	C₃	—	3	1	no	—
159	C₃	—	3	3	no	—
160	C₃	—	333	3	no	—
161	C₃	—	333	9	no	—
162	C₃	C₃×H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₃	C₃×H(3,𝔽₃)×C₃₇⋊C₃
commutator, cosocle	C₃²×C₃₇	C₃⁴
Frattini	C₃²	C₃³×C₃₇⋊C₃
Fitting	C₃₇×#6912	C₃
socle	C₃×C₃₇	C₃²×H(3,𝔽₃)
hypercenter	#6912	C₃₇⋊C₃
hypocenter	C₃₇	#10624935
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ #6912

Complements

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

N	H ≅ G/N	type
1. #22582679	C₃	semidirect
2. C₃×#15688532	C₃	semidirect
3. #22582663	C₃	semidirect
4. #22582662	C₃	semidirect
5. #22582652	C₃	semidirect
6. #22582648	C₃	semidirect
7. #22582639	C₃	semidirect
8. #22582634	C₃	semidirect
9. C₃×#15688517	C₃	semidirect
10. C₃₇×#6912	C₃	semidirect
13. #15688532	C₃²	semidirect
17. C₃×#11758605	C₃²	semidirect
18. C₃×#11758606	C₃²	semidirect
23. C₃×C₉×C₃₇⋊C₃	C₃²	semidirect
24. #15688521	C₃²	semidirect
26. #15688517	C₃²	semidirect
27. #15688517	C₃²	semidirect
28. #15688517	C₃²	semidirect
29. #15688514	C₃²	semidirect
32. C₃₇×#733	C₃²	semidirect
82. #6912	C₃₇⋊C₃	semidirect
108. #733	C₃×C₃₇⋊C₃	semidirect
114. C₃₇	#10624935	semidirect

Maximal subgroups

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

subgroup	index
#22582679	3
C₃×#15688532	3
#22582663	3
#22582662	3
#22582652	3
#22582648	3
#22582639	3
#22582634	3
C₃×#15688517	3
C₃₇×#6912	3
#10624935	37