Finite Group Databasebeta

Subgroups — #41008188

Order 26973.

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

normal subgroups only

This group has 960 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	C₃×#15688532	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×#15688530	C₃	1	3	yes	maximal, complemented
3	#22582658	C₃	1	9	yes	maximal, complemented
4	#22582657	C₃	1	9	yes	maximal, complemented
5	#22582652	C₃	1	2	yes	maximal, complemented
6	#22582642	C₃	1	3	yes	maximal, complemented
7	#22582637	C₃	1	3	yes	maximal, complemented
8	#22582631	C₃	1	9	yes	maximal, complemented
9	C₃₇×#6911	C₃	1	1	yes	maximal, characteristic, complemented
10	#15688532	—	3	3	no	—
11	#15688532	—	3	6	no	—
12	#15688530	—	3	6	no	—
13	#15688530	—	3	6	no	—
14	#15688530	—	3	6	no	—
15	#15688530	C₃²	1	9	yes	complemented
16	#15688530	C₃²	1	9	yes	complemented
17	#15688530	C₃²	1	9	yes	complemented
18	C₃×#11758608	—	3	6	no	—
19	C₃×#11758608	C₃²	1	6	yes	—
20	C₃×#11758608	C₃²	1	1	yes	characteristic
21	C₃×#11758605	C₃²	1	3	yes	complemented
22	C₃×#11758606	C₃²	1	3	yes	complemented
23	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	1	yes	characteristic
24	C₃²×#66524	—	3	3	no	—
25	C₃²×#66524	C₃²	1	1	yes	characteristic
26	C₃×C₉×C₃₇⋊C₃	C₃²	1	3	yes	complemented
27	#15688521	C₃²	1	18	yes	complemented
28	#15688515	C₃²	1	27	yes	complemented
29	#15688512	C₃²	1	27	yes	complemented
30	C₃×3₋¹⁺²×C₃₇	C₃²	1	1	yes	characteristic
31	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	3	yes	—
32	C₃₇×#732	C₃²	1	9	yes	—
33	C₃³×C₃₇	—	3	3	no	—
34	C₃³×C₃₇	C₃³	1	1	yes	characteristic
35	C₃²×C₃₇⋊C₃	C₃³	1	3	yes	—
36	3₋¹⁺²×C₃₇	—	3	3	no	—
37	H(3,𝔽₃)×C₃₇	—	3	6	no	—
38	H(3,𝔽₃)×C₃₇	C₃³	1	9	yes	—
39	C₃×C₉×C₃₇	C₃³	1	3	yes	—
40	C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
41	#11758608	—	3	12	no	—
42	#11758608	—	3	3	no	—
43	#11758608	—	3	18	no	—
44	#11758608	—	3	18	no	—
45	#11758608	—	3	18	no	—
46	#11758608	C₃³	1	18	yes	—
47	C₃×#66524	—	3	6	no	—
48	C₃×#66524	—	3	9	no	—
49	C₃×#66524	—	3	9	no	—
50	C₃×#66524	—	3	9	no	—
51	C₃×#66524	H(3,𝔽₃)	1	3	yes	—
52	C₃×#66524	H(3,𝔽₃)	1	6	yes	—
53	C₃×#66524	C₃³	1	3	yes	—
54	C₃×#66524	C₃³	1	1	yes	characteristic
55	C₃×#66524	C₃³	1	1	yes	characteristic
56	C₃×#66524	C₃³	1	1	yes	characteristic
57	#11758606	—	3	9	no	—
58	#11758605	—	3	9	no	—
59	C₉×C₃₇⋊C₃	—	3	3	no	—
60	C₉×C₃₇⋊C₃	—	3	6	no	—
61	#10624929	—	37	1	no	maximal, Sylow
62	C₃²×C₃₇	—	3	9	no	—
63	C₃²×C₃₇	—	9	1	no	—
64	C₃²×C₃₇	—	9	9	no	—
65	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	3	yes	—
66	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
67	C₃×C₃₇⋊C₃	—	3	3	no	—
68	C₃×C₃₇⋊C₃	—	9	3	no	—
69	C₉×C₃₇	—	3	3	no	—
70	#66524	—	3	3	no	—
71	#66524	—	3	18	no	—
72	#66524	C₃×H(3,𝔽₃)	1	1	yes	characteristic
73	#66524	C₃×H(3,𝔽₃)	1	2	yes	—
74	#66524	C₃×H(3,𝔽₃)	1	3	yes	—
75	#66524	C₃×H(3,𝔽₃)	1	3	yes	—
76	C₃×#737	—	37	1	no	—
77	C₃×#737	—	37	3	no	—
78	#6913	—	37	9	no	—
79	#6913	—	37	9	no	—
80	#6912	—	37	2	no	—
81	#6911	C₃₇⋊C₃	1	1	yes	characteristic, complemented
82	#6910	—	37	3	no	—
83	#6909	—	37	3	no	—
84	#6908	—	37	9	no	—
85	C₃×C₃₇	—	3	1	no	—
86	C₃×C₃₇	—	3	3	no	—
87	C₃×C₃₇	—	9	9	no	—
88	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
89	C₃₇⋊C₃	—	9	3	no	—
90	#737	—	37	9	no	—
91	#737	—	37	9	no	—
92	#737	—	37	9	no	—
93	#737	—	111	6	no	—
94	#737	—	111	6	no	—
95	#737	—	111	6	no	—
96	#737	—	111	3	no	—
97	#737	—	111	6	no	—
98	C₃×3₋¹⁺²	—	37	1	no	—
99	C₃×3₋¹⁺²	—	37	3	no	—
100	C₃×3₋¹⁺²	—	37	3	no	—
101	C₃×3₋¹⁺²	—	37	6	no	—
102	C₃×3₋¹⁺²	—	111	6	no	—
103	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
104	C₃×H(3,𝔽₃)	—	37	1	no	—
105	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	3	yes	—
106	C₃²×C₉	—	37	1	no	—
107	C₃²×C₉	—	37	3	no	—
108	C₃²×C₉	—	111	3	no	—
109	#733	—	37	18	no	—
110	#732	C₃×C₃₇⋊C₃	1	9	yes	—
111	#731	—	37	27	no	—
112	AΣL(1,𝔽₂₇)	—	37	27	no	—
113	C₃₇	#10624929	1	1	yes	characteristic, complemented, Sylow
114	C₃³	—	3	3	no	—
115	C₃³	—	37	3	no	—
116	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
117	3₋¹⁺²	—	3	3	no	—
118	3₋¹⁺²	—	37	18	no	—
119	3₋¹⁺²	—	111	9	no	—
120	3₋¹⁺²	—	111	9	no	—
121	3₋¹⁺²	—	111	18	no	—
122	3₋¹⁺²	—	111	18	no	—
123	3₋¹⁺²	—	111	18	no	—
124	3₋¹⁺²	—	111	3	no	—
125	3₋¹⁺²	—	111	12	no	—
126	H(3,𝔽₃)	—	3	6	no	—
127	H(3,𝔽₃)	—	111	3	no	—
128	H(3,𝔽₃)	C₃²×C₃₇⋊C₃	1	9	yes	—
129	C₃×C₉	—	37	3	no	—
130	C₃×C₉	—	37	6	no	—
131	C₃×C₉	—	37	1	no	—
132	C₃×C₉	—	37	1	no	—
133	C₃×C₉	—	37	1	no	—
134	C₃×C₉	—	37	3	no	—
135	C₃×C₉	—	111	3	no	—
136	C₃×C₉	—	111	6	no	—
137	C₃×C₉	—	111	9	no	—
138	C₃×C₉	—	111	9	no	—
139	C₃×C₉	—	111	9	no	—
140	C₃×C₉	—	111	6	no	—
141	C₃×C₉	C₃²×C₃₇⋊C₃	1	3	yes	—
142	C₃²	—	3	9	no	—
143	C₃²	—	9	1	no	—
144	C₃²	—	9	9	no	—
145	C₃²	—	333	3	no	—
146	C₃²	—	111	3	no	—
147	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	3	yes	—
148	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
149	C₉	—	3	3	no	—
150	C₉	—	37	1	no	—
151	C₉	—	37	3	no	—
152	C₉	—	37	3	no	—
153	C₉	—	37	2	no	—
154	C₉	—	111	3	no	—
155	C₉	—	111	18	no	—
156	C₃	—	3	3	no	—
157	C₃	—	3	1	no	—
158	C₃	—	9	9	no	—
159	C₃	—	333	3	no	—
160	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×#6911	C₃
socle	C₃×C₃₇	C₃²×H(3,𝔽₃)
hypercenter	#6911	C₃₇⋊C₃
hypocenter	C₃₇	#10624929
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ #6911

Complements

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

N	H ≅ G/N	type
1. C₃×#15688532	C₃	semidirect
2. C₃×#15688530	C₃	semidirect
3. #22582658	C₃	semidirect
4. #22582657	C₃	semidirect
5. #22582652	C₃	semidirect
6. #22582642	C₃	semidirect
7. #22582637	C₃	semidirect
8. #22582631	C₃	semidirect
9. C₃₇×#6911	C₃	semidirect
15. #15688530	C₃²	semidirect
16. #15688530	C₃²	semidirect
17. #15688530	C₃²	semidirect
21. C₃×#11758605	C₃²	semidirect
22. C₃×#11758606	C₃²	semidirect
26. C₃×C₉×C₃₇⋊C₃	C₃²	semidirect
27. #15688521	C₃²	semidirect
28. #15688515	C₃²	semidirect
29. #15688512	C₃²	semidirect
81. #6911	C₃₇⋊C₃	semidirect
113. C₃₇	#10624929	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
C₃×#15688530	3
#22582658	3
#22582657	3
#22582652	3
#22582642	3
#22582637	3
#22582631	3
C₃₇×#6911	3
#10624929	37