Finite Group Databasebeta

Subgroups — #41006865

Order 26973.

264 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 61 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×#15688513	C₃	1	1	yes	maximal, characteristic
2	#22582573	C₃	1	2	yes	maximal, complemented
3	#22582560	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₃²⋊#66524	C₃	1	1	yes	maximal, characteristic, complemented
5	#22582481	C₃	1	2	yes	maximal, complemented
6	#22582478	C₃	1	2	yes	maximal, complemented
7	#22582475	C₃	1	2	yes	maximal, complemented
8	#22582468	C₃	1	1	yes	maximal, characteristic, complemented
9	C₃₇×#6866	C₃	1	1	yes	maximal, characteristic, complemented
10	C₃×#11758608	C₃²	1	2	yes	—
11	C₃×#11758606	—	3	1	no	—
12	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	1	yes	characteristic
13	C₃²×#66524	C₃²	1	2	yes	—
14	C₃²×#66524	C₃²	1	1	yes	characteristic
15	#15688513	—	3	3	no	—
16	#15688506	—	3	2	no	—
17	#15688504	—	3	2	no	—
18	#15688504	—	3	2	no	—
19	C₃²⋊#66524	—	3	2	no	—
20	C₃²⋊#66524	—	3	1	no	—
21	C₃²⋊#66524	C₃²	1	1	yes	characteristic
22	C₃²⋊#66524	C₃²	1	2	yes	—
23	#15688502	—	3	2	no	—
24	C₃⁴×C₃₇	C₃²	1	1	yes	characteristic
25	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
26	C₃₇×C₃²⋊C₉	C₃²	1	2	yes	—
27	C₃³×C₃₇	—	3	1	no	—
28	C₃³×C₃₇	—	3	2	no	—
29	C₃³×C₃₇	—	9	2	no	—
30	C₃³×C₃₇	—	9	1	no	—
31	C₃³×C₃₇	3₋¹⁺²	1	3	yes	complemented
32	C₃³×C₃₇	C₃³	1	1	yes	characteristic
33	C₃²×C₃₇⋊C₃	—	3	1	no	—
34	C₃×C₉×C₃₇	—	3	1	no	—
35	C₃×C₉×C₃₇	—	3	2	no	—
36	C₃²⋊(C₃₇⋊C₃)	—	3	1	no	—
37	C₃²⋊(C₃₇⋊C₃)	—	3	2	no	—
38	#11758608	—	3	2	no	—
39	#11758608	—	3	4	no	—
40	C₃×#66524	—	3	1	no	—
41	C₃×#66524	—	3	2	no	—
42	C₃×#66524	—	3	2	no	—
43	C₃×#66524	—	3	2	no	—
44	C₃×#66524	—	9	1	no	—
45	C₃×#66524	—	9	2	no	—
46	C₃×#66524	H(3,𝔽₃)	1	2	yes	—
47	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
48	#11758606	—	9	3	no	—
49	#10624788	—	37	1	no	maximal, Sylow
50	C₃²×C₃₇	—	3	2	no	—
51	C₃²×C₃₇	—	3	1	no	—
52	C₃²×C₃₇	—	3	3	no	—
53	C₃²×C₃₇	—	9	3	no	—
54	C₃²×C₃₇	—	9	6	no	—
55	C₃²×C₃₇	—	9	2	no	—
56	C₃²×C₃₇	—	9	1	no	—
57	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
58	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
59	C₃²×C₃₇	#737	1	2	yes	—
60	C₃×C₃₇⋊C₃	—	9	2	no	—
61	C₃×C₃₇⋊C₃	—	9	1	no	—
62	C₃×C₃₇⋊C₃	—	9	1	no	—
63	C₉×C₃₇	—	9	2	no	—
64	C₉×C₃₇	—	9	1	no	—
65	#66524	—	3	1	no	—
66	#66524	—	3	2	no	—
67	#66524	—	9	2	no	—
68	#66524	—	9	2	no	—
69	#66524	—	9	1	no	—
70	#66524	—	9	2	no	—
71	C₃×AΣL(1,𝔽₂₇)	—	37	1	no	—
72	#6894	—	37	2	no	—
73	C₉⋊H(3,𝔽₃)	—	37	1	no	—
74	C₃×C₃²⋊C₉	—	37	1	no	—
75	3₋¹⁺²⋊C₉	—	37	2	no	—
76	3₋¹⁺²⋊C₉	—	37	2	no	—
77	H(3,𝔽₃)⋊C₉	—	37	2	no	—
78	#6866	—	37	1	no	—
79	#6866	C₃₇⋊C₃	1	1	yes	characteristic, complemented
80	C₃×C₃₇	—	3	1	no	—
81	C₃×C₃₇	—	3	2	no	—
82	C₃×C₃₇	—	9	3	no	—
83	C₃×C₃₇	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
84	C₃×C₃₇	#6909	1	1	yes	characteristic
85	C₃×C₃₇	#6913	1	2	yes	—
86	C₃₇⋊C₃	—	27	1	no	—
87	C₃⁴	C₃×C₃₇⋊C₃	1	1	yes	characteristic
88	C₃×3₋¹⁺²	—	37	2	no	—
89	C₃×3₋¹⁺²	—	111	1	no	—
90	C₃×H(3,𝔽₃)	—	37	1	no	—
91	C₃²×C₉	—	37	1	no	—
92	C₃²×C₉	—	37	2	no	—
93	AΣL(1,𝔽₂₇)	—	111	3	no	—
94	C₉⋊C₉	—	111	2	no	—
95	C₉⋊C₉	—	111	2	no	—
96	C₉⋊C₉	—	111	2	no	—
97	C₃²⋊C₉	—	37	2	no	—
98	C₃²⋊C₉	—	37	1	no	—
99	C₃²⋊C₉	—	111	2	no	—
100	C₃²⋊C₉	—	111	2	no	—
101	C₃²⋊C₉	—	111	1	no	—
102	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	2	yes	—
103	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
104	C₃₇	#10624788	1	1	yes	characteristic, complemented, Sylow
105	C₃³	—	3	1	no	—
106	C₃³	—	3	2	no	—
107	C₃³	—	9	1	no	—
108	C₃³	—	9	2	no	—
109	C₃³	—	111	1	no	—
110	C₃³	#11758606	1	3	yes	complemented
111	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
112	3₋¹⁺²	—	333	3	no	—
113	3₋¹⁺²	—	111	2	no	—
114	3₋¹⁺²	—	111	4	no	—
115	H(3,𝔽₃)	—	111	1	no	—
116	H(3,𝔽₃)	—	111	2	no	—
117	C₃×C₉	—	3	1	no	—
118	C₃×C₉	—	3	2	no	—
119	C₃×C₉	—	37	2	no	—
120	C₃×C₉	—	37	1	no	—
121	C₃×C₉	—	333	2	no	—
122	C₃×C₉	—	333	1	no	—
123	C₃×C₉	—	111	1	no	—
124	C₃×C₉	—	111	2	no	—
125	C₃×C₉	—	111	2	no	—
126	C₃×C₉	—	111	2	no	—
127	C₃²	—	3	1	no	—
128	C₃²	—	3	2	no	—
129	C₃²	—	3	3	no	—
130	C₃²	—	9	3	no	—
131	C₃²	—	9	6	no	—
132	C₃²	—	9	1	no	—
133	C₃²	—	9	2	no	—
134	C₃²	—	333	1	no	—
135	C₃²	—	333	1	no	—
136	C₃²	—	333	2	no	—
137	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
138	C₃²	C₃×#11758606	1	1	yes	characteristic
139	C₃²	#15688532	1	2	yes	—
140	C₉	—	9	1	no	—
141	C₉	—	9	2	no	—
142	C₉	—	333	2	no	—
143	C₉	—	333	2	no	—
144	C₉	—	333	1	no	—
145	C₉	—	333	2	no	—
146	C₉	—	111	1	no	—
147	C₉	—	111	2	no	—
148	C₃	—	3	1	no	—
149	C₃	—	3	2	no	—
150	C₃	—	9	3	no	—
151	C₃	—	999	1	no	—
152	C₃	#22582560	1	1	yes	characteristic
153	C₃	#22582636	1	1	yes	characteristic
154	C₃	#22582654	1	2	yes	—

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₃₇⋊C₃
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6866	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃×#15688513	C₃
hypercenter	#6866	C₃₇⋊C₃
hypocenter	C₃₇	#10624788
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ #6866

Complements

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

N	H ≅ G/N	type
2. #22582573	C₃	semidirect
3. #22582560	C₃	semidirect
4. C₃×C₃²⋊#66524	C₃	semidirect
5. #22582481	C₃	semidirect
6. #22582478	C₃	semidirect
7. #22582475	C₃	semidirect
8. #22582468	C₃	semidirect
9. C₃₇×#6866	C₃	semidirect
31. C₃³×C₃₇	3₋¹⁺²	semidirect
79. #6866	C₃₇⋊C₃	semidirect
104. C₃₇	#10624788	semidirect
110. C₃³	#11758606	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15688513	3
#22582573	3
#22582560	3
C₃×C₃²⋊#66524	3
#22582481	3
#22582478	3
#22582475	3
#22582468	3
C₃₇×#6866	3
#10624788	37