Finite Group Databasebeta

Subgroups — #41006913

Order 26973.

268 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 64 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	#22582556	C₃	1	2	yes	maximal, complemented
2	H(3,𝔽₃)×#66524	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×#15688504	C₃	1	1	yes	maximal, characteristic, complemented
4	H(3,𝔽₃)⋊#66524	C₃	1	2	yes	maximal, complemented
5	H(3,𝔽₃)⋊#66524	C₃	1	2	yes	maximal, complemented
6	H(3,𝔽₃)⋊#66524	C₃	1	1	yes	maximal, characteristic, complemented
7	H(3,𝔽₃)⋊#66524	C₃	1	1	yes	maximal, characteristic, complemented
8	#22582472	C₃	1	2	yes	maximal, complemented
9	C₃×C₃₇×#732	C₃	1	1	yes	maximal, characteristic
10	C₃×#11758608	—	3	2	no	—
11	C₃²×#66524	—	3	1	no	—
12	C₃²×#66524	C₃²	1	1	yes	characteristic
13	C₃²×#66524	C₃²	1	1	yes	characteristic
14	C₃²×#66524	C₃²	1	1	yes	characteristic
15	#15688504	—	3	2	no	—
16	#15688504	—	3	1	no	—
17	C₃²⋊#66524	—	3	2	no	—
18	C₃²⋊#66524	—	3	4	no	—
19	C₃²⋊#66524	C₃²	1	2	yes	—
20	C₃²⋊#66524	C₃²	1	2	yes	—
21	C₃²⋊#66524	C₃²	1	2	yes	—
22	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	2	yes	—
23	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
24	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
25	C₃₇×#732	—	3	3	no	—
26	C₃³×C₃₇	—	3	1	no	—
27	C₃³×C₃₇	—	3	2	no	—
28	C₃³×C₃₇	C₃³	1	1	yes	characteristic
29	H(3,𝔽₃)×C₃₇	—	3	2	no	—
30	H(3,𝔽₃)×C₃₇	—	3	2	no	—
31	H(3,𝔽₃)×C₃₇	—	3	4	no	—
32	H(3,𝔽₃)×C₃₇	3₋¹⁺²	1	3	yes	complemented
33	C₃×C₉×C₃₇	—	3	1	no	—
34	C₃×C₉×C₃₇	—	3	1	no	—
35	C₃×C₉×C₃₇	—	3	2	no	—
36	#11758608	—	9	6	no	—
37	C₃×#66524	—	3	3	no	—
38	C₃×#66524	—	3	2	no	—
39	C₃×#66524	—	3	2	no	—
40	C₃×#66524	—	3	2	no	—
41	C₃×#66524	—	3	1	no	—
42	C₃×#66524	—	3	1	no	—
43	C₃×#66524	—	9	3	no	—
44	C₃×#66524	—	9	1	no	—
45	C₃×#66524	—	9	1	no	—
46	C₃×#66524	H(3,𝔽₃)	1	2	yes	—
47	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
48	#10624792	—	37	1	no	maximal, Sylow
49	C₃²×C₃₇	—	3	1	no	—
50	C₃²×C₃₇	—	3	1	no	—
51	C₃²×C₃₇	—	3	1	no	—
52	C₃²×C₃₇	—	3	3	no	—
53	C₃²×C₃₇	—	9	1	no	—
54	C₃²×C₃₇	—	9	1	no	—
55	C₃²×C₃₇	—	9	1	no	—
56	C₃²×C₃₇	—	9	2	no	—
57	C₃²×C₃₇	—	9	2	no	—
58	C₃²×C₃₇	—	9	2	no	—
59	C₃²×C₃₇	—	9	2	no	—
60	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
61	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
62	C₃²×C₃₇	#737	1	2	yes	—
63	C₉×C₃₇	—	9	1	no	—
64	#66524	—	3	2	no	—
65	#66524	—	9	1	no	—
66	#66524	—	9	1	no	—
67	#66524	—	9	2	no	—
68	#66524	—	9	2	no	—
69	#66524	—	9	2	no	—
70	#66524	#732	1	3	yes	complemented
71	C₃×#732	C₃₇⋊C₃	1	1	yes	characteristic
72	C₉⋊H(3,𝔽₃)	—	37	2	no	—
73	C₉×H(3,𝔽₃)	—	37	1	no	—
74	C₃×C₉⋊C₉	—	37	1	no	—
75	H(3,𝔽₃)⋊C₉	—	37	2	no	—
76	H(3,𝔽₃)⋊C₉	—	37	2	no	—
77	H(3,𝔽₃)⋊C₉	—	37	1	no	—
78	H(3,𝔽₃)⋊C₉	—	37	1	no	—
79	#6868	—	37	2	no	—
80	C₃×C₃₇	—	3	1	no	—
81	C₃×C₃₇	—	3	2	no	—
82	C₃×C₃₇	—	9	3	no	—
83	C₃×C₃₇	—	27	2	no	—
84	C₃×C₃₇	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
85	C₃×C₃₇	C₃×#732	1	1	yes	characteristic
86	C₃×C₃₇	#6913	1	1	yes	characteristic
87	C₃×C₃₇	#6913	1	1	yes	characteristic
88	C₃×3₋¹⁺²	—	111	2	no	—
89	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	2	yes	—
90	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
91	C₃²×C₉	—	37	1	no	—
92	C₃²×C₉	—	37	1	no	—
93	C₃²×C₉	—	37	1	no	—
94	C₃²×C₉	—	111	1	no	—
95	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
96	#732	—	3	3	no	—
97	C₉⋊C₉	—	111	1	no	—
98	C₉⋊C₉	—	111	2	no	—
99	C₃²⋊C₉	—	37	2	no	—
100	C₃²⋊C₉	—	37	2	no	—
101	C₃²⋊C₉	—	37	2	no	—
102	C₃²⋊C₉	—	111	2	no	—
103	C₃²⋊C₉	—	111	4	no	—
104	C₃₇	#10624792	1	1	yes	characteristic, complemented, Sylow
105	C₃³	—	3	1	no	—
106	C₃³	—	3	2	no	—
107	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
108	3₋¹⁺²	—	333	6	no	—
109	H(3,𝔽₃)	—	3	2	no	—
110	H(3,𝔽₃)	—	3	2	no	—
111	H(3,𝔽₃)	—	3	4	no	—
112	H(3,𝔽₃)	#11758608	1	3	yes	complemented
113	C₃×C₉	—	3	1	no	—
114	C₃×C₉	—	3	1	no	—
115	C₃×C₉	—	3	2	no	—
116	C₃×C₉	—	37	1	no	—
117	C₃×C₉	—	37	2	no	—
118	C₃×C₉	—	333	3	no	—
119	C₃×C₉	—	333	1	no	—
120	C₃×C₉	—	333	1	no	—
121	C₃×C₉	—	111	3	no	—
122	C₃×C₉	—	111	2	no	—
123	C₃×C₉	—	111	2	no	—
124	C₃×C₉	—	111	2	no	—
125	C₃×C₉	—	111	1	no	—
126	C₃×C₉	—	111	1	no	—
127	C₃²	—	3	1	no	—
128	C₃²	—	3	1	no	—
129	C₃²	—	3	1	no	—
130	C₃²	—	3	3	no	—
131	C₃²	—	9	1	no	—
132	C₃²	—	9	1	no	—
133	C₃²	—	9	1	no	—
134	C₃²	—	9	2	no	—
135	C₃²	—	9	2	no	—
136	C₃²	—	9	2	no	—
137	C₃²	—	9	2	no	—
138	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
139	C₃²	C₃×#11758608	1	1	yes	characteristic
140	C₃²	#15688530	1	2	yes	—
141	C₉	—	9	1	no	—
142	C₉	—	37	3	no	—
143	C₉	—	333	2	no	—
144	C₉	—	333	2	no	—
145	C₉	—	333	2	no	—
146	C₉	—	333	1	no	—
147	C₉	—	333	1	no	—
148	C₉	—	111	2	no	—
149	C₃	—	3	1	no	—
150	C₃	—	3	2	no	—
151	C₃	—	9	3	no	—
152	C₃	—	27	2	no	—
153	C₃	#22582556	1	1	yes	characteristic
154	C₃	#732×C₃₇⋊C₃	1	1	yes	characteristic
155	C₃	#22582657	1	1	yes	characteristic
156	C₃	#22582658	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₃₇⋊C₃
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting, prime radical	C₃×C₃₇×#732	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
hypercenter	C₃×#732	C₃₇⋊C₃
hypocenter	C₃₇	#10624792
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃×#732

Complements

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

N	H ≅ G/N	type
1. #22582556	C₃	semidirect
2. H(3,𝔽₃)×#66524	C₃	semidirect
3. C₃×#15688504	C₃	semidirect
4. H(3,𝔽₃)⋊#66524	C₃	semidirect
5. H(3,𝔽₃)⋊#66524	C₃	semidirect
6. H(3,𝔽₃)⋊#66524	C₃	semidirect
7. H(3,𝔽₃)⋊#66524	C₃	semidirect
8. #22582472	C₃	semidirect
32. H(3,𝔽₃)×C₃₇	3₋¹⁺²	semidirect
70. #66524	#732	semidirect
104. C₃₇	#10624792	semidirect
112. H(3,𝔽₃)	#11758608	semidirect

Maximal subgroups

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

subgroup	index
#22582556	3
H(3,𝔽₃)×#66524	3
C₃×#15688504	3
H(3,𝔽₃)⋊#66524	3
#22582472	3
C₃×C₃₇×#732	3
#10624792	37