Finite Group Databasebeta

Subgroups — #41008232

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	#22582680	C₃	1	3	yes	maximal, complemented
2	C₃×#15688531	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582658	C₃	1	9	yes	maximal, complemented
4	#22582657	C₃	1	9	yes	maximal, complemented
5	#22582642	C₃	1	3	yes	maximal, complemented
6	#22582637	C₃	1	3	yes	maximal, complemented
7	#22582631	C₃	1	9	yes	maximal, complemented
8	C₃×#15688521	C₃	1	2	yes	maximal, complemented
9	C₃×C₃₇×#732	C₃	1	1	yes	maximal, characteristic, complemented
10	#15688531	—	3	3	no	—
11	#15688531	—	3	6	no	—
12	#15688530	—	3	18	no	—
13	#15688530	C₃²	1	27	yes	complemented
14	C₃×#11758608	—	3	9	no	—
15	C₃×#11758608	C₃²	1	6	yes	—
16	C₃×#11758608	C₃²	1	1	yes	characteristic
17	C₃×#11758608	C₃²	1	1	yes	characteristic
18	C₃×#11758605	C₃²	1	3	yes	complemented
19	C₃×#11758606	C₃²	1	3	yes	complemented
20	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	1	yes	characteristic
21	C₃×C₉×C₃₇⋊C₃	C₃²	1	3	yes	complemented
22	#15688521	C₃²	1	6	yes	complemented
23	#15688521	C₃²	1	6	yes	complemented
24	#15688521	C₃²	1	6	yes	complemented
25	#15688515	C₃²	1	27	yes	complemented
26	#15688512	C₃²	1	27	yes	complemented
27	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	3	yes	—
28	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
29	C₃₇×#732	C₃²	1	3	yes	—
30	C₃₇×#732	C₃²	1	3	yes	—
31	C₃₇×#732	C₃²	1	3	yes	—
32	C₃³×C₃₇	—	3	3	no	—
33	C₃³×C₃₇	C₃³	1	1	yes	characteristic
34	C₃²×C₃₇⋊C₃	C₃³	1	3	yes	—
35	H(3,𝔽₃)×C₃₇	—	3	6	no	—
36	H(3,𝔽₃)×C₃₇	C₃³	1	9	yes	—
37	C₃×C₉×C₃₇	—	3	2	no	—
38	C₃×C₉×C₃₇	—	3	1	no	—
39	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
40	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
41	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
42	C₃²⋊(C₃₇⋊C₃)	—	3	2	no	—
43	C₃²⋊(C₃₇⋊C₃)	—	3	1	no	—
44	#11758608	—	3	6	no	—
45	#11758608	—	3	6	no	—
46	#11758608	—	3	2	no	—
47	#11758608	—	3	1	no	—
48	#11758608	—	3	54	no	—
49	#11758608	H(3,𝔽₃)	1	3	yes	—
50	#11758608	H(3,𝔽₃)	1	6	yes	—
51	#11758608	C₃³	1	18	yes	—
52	C₃×#66524	—	3	6	no	—
53	C₃×#66524	—	3	27	no	—
54	C₃×#66524	C₃³	1	3	yes	—
55	C₃×#66524	C₃³	1	3	yes	—
56	#11758606	—	3	9	no	—
57	#11758605	—	3	9	no	—
58	C₉×C₃₇⋊C₃	—	3	9	no	—
59	#10624936	—	37	1	no	maximal, Sylow
60	C₃²×C₃₇	—	3	9	no	—
61	C₃²×C₃₇	—	9	1	no	—
62	C₃²×C₃₇	—	27	3	no	—
63	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	2	yes	—
64	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
65	C₃²×C₃₇	C₃⁴	1	1	yes	characteristic
66	C₃×C₃₇⋊C₃	—	3	3	no	—
67	C₃×C₃₇⋊C₃	—	9	3	no	—
68	C₉×C₃₇	—	3	1	no	—
69	C₉×C₃₇	—	3	1	no	—
70	C₉×C₃₇	—	3	1	no	—
71	#66524	—	3	3	no	—
72	#66524	—	3	18	no	—
73	#66524	C₃×H(3,𝔽₃)	1	9	yes	—
74	3₋¹⁺⁴	—	37	3	no	—
75	C₃×#737	—	37	1	no	—
76	#6913	—	37	9	no	—
77	#6913	—	37	9	no	—
78	#6910	—	37	3	no	—
79	#6909	—	37	3	no	—
80	#6908	—	37	9	no	—
81	C₃×#733	—	37	2	no	—
82	C₃×#732	C₃₇⋊C₃	1	1	yes	characteristic, complemented
83	C₃×C₃₇	—	3	1	no	—
84	C₃×C₃₇	—	3	1	no	—
85	C₃×C₃₇	—	3	2	no	—
86	C₃×C₃₇	—	9	9	no	—
87	C₃×C₃₇	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
88	C₃₇⋊C₃	—	9	3	no	—
89	#737	—	37	27	no	—
90	#737	—	111	3	no	—
91	#737	—	111	6	no	—
92	#737	—	111	18	no	—
93	C₃×3₋¹⁺²	—	37	1	no	—
94	C₃×3₋¹⁺²	—	37	1	no	—
95	C₃×3₋¹⁺²	—	37	3	no	—
96	C₃×3₋¹⁺²	—	37	3	no	—
97	C₃×3₋¹⁺²	—	37	6	no	—
98	C₃×3₋¹⁺²	—	111	9	no	—
99	C₃×H(3,𝔽₃)	—	37	1	no	—
100	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	3	yes	—
101	C₃²×C₉	—	37	3	no	—
102	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
103	#733	—	37	6	no	—
104	#733	—	37	6	no	—
105	#733	—	37	6	no	—
106	#732	C₃×C₃₇⋊C₃	1	3	yes	—
107	#732	C₃×C₃₇⋊C₃	1	3	yes	—
108	#732	C₃×C₃₇⋊C₃	1	3	yes	—
109	#731	—	37	27	no	—
110	AΣL(1,𝔽₂₇)	—	37	27	no	—
111	C₃₇	#10624936	1	1	yes	characteristic, complemented, Sylow
112	C₃³	—	3	3	no	—
113	C₃³	—	37	3	no	—
114	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
115	3₋¹⁺²	—	37	3	no	—
116	3₋¹⁺²	—	37	6	no	—
117	3₋¹⁺²	—	37	18	no	—
118	3₋¹⁺²	—	111	9	no	—
119	3₋¹⁺²	—	111	9	no	—
120	3₋¹⁺²	—	111	54	no	—
121	3₋¹⁺²	—	111	6	no	—
122	3₋¹⁺²	—	111	2	no	—
123	3₋¹⁺²	—	111	1	no	—
124	3₋¹⁺²	—	111	6	no	—
125	H(3,𝔽₃)	—	3	6	no	—
126	H(3,𝔽₃)	—	111	2	no	—
127	H(3,𝔽₃)	—	111	1	no	—
128	H(3,𝔽₃)	C₃²×C₃₇⋊C₃	1	9	yes	—
129	C₃×C₉	—	3	2	no	—
130	C₃×C₉	—	3	1	no	—
131	C₃×C₉	—	37	3	no	—
132	C₃×C₉	—	37	3	no	—
133	C₃×C₉	—	111	9	no	—
134	C₃×C₉	—	111	27	no	—
135	C₃×C₉	—	111	6	no	—
136	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
137	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
138	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
139	C₃²	—	3	9	no	—
140	C₃²	—	9	1	no	—
141	C₃²	—	27	3	no	—
142	C₃²	—	333	3	no	—
143	C₃²	—	111	3	no	—
144	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	2	yes	—
145	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
146	C₃²	C₃³×C₃₇⋊C₃	1	1	yes	characteristic
147	C₉	—	3	1	no	—
148	C₉	—	3	1	no	—
149	C₉	—	3	1	no	—
150	C₉	—	37	9	no	—
151	C₉	—	111	3	no	—
152	C₉	—	111	18	no	—
153	C₃	—	3	1	no	—
154	C₃	—	3	1	no	—
155	C₃	—	3	2	no	—
156	C₃	—	9	9	no	—
157	C₃	—	333	3	no	—
158	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	C₃×C₃₇×#732	C₃
socle	C₃×C₃₇	C₃²×H(3,𝔽₃)
hypercenter	C₃×#732	C₃₇⋊C₃
hypocenter	C₃₇	#10624936
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃×#732

Complements

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

N	H ≅ G/N	type
1. #22582680	C₃	semidirect
2. C₃×#15688531	C₃	semidirect
3. #22582658	C₃	semidirect
4. #22582657	C₃	semidirect
5. #22582642	C₃	semidirect
6. #22582637	C₃	semidirect
7. #22582631	C₃	semidirect
8. C₃×#15688521	C₃	semidirect
9. C₃×C₃₇×#732	C₃	semidirect
13. #15688530	C₃²	semidirect
18. C₃×#11758605	C₃²	semidirect
19. C₃×#11758606	C₃²	semidirect
21. C₃×C₉×C₃₇⋊C₃	C₃²	semidirect
22. #15688521	C₃²	semidirect
23. #15688521	C₃²	semidirect
24. #15688521	C₃²	semidirect
25. #15688515	C₃²	semidirect
26. #15688512	C₃²	semidirect
82. C₃×#732	C₃₇⋊C₃	semidirect
111. C₃₇	#10624936	semidirect

Maximal subgroups

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

subgroup	index
#22582680	3
C₃×#15688531	3
#22582658	3
#22582657	3
#22582642	3
#22582637	3
#22582631	3
C₃×#15688521	3
C₃×C₃₇×#732	3
#10624936	37