Finite Group Databasebeta

Subgroups — #41006321

Order 26973.

218 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 94 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×#15688510	C₃	1	1	yes	maximal, characteristic
2	C₃×#15688509	C₃	1	1	yes	maximal, characteristic
3	C₃×C₂₇×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
4	#22582536	C₃	1	1	yes	maximal, characteristic
5	#22582490	C₃	1	1	yes	maximal, characteristic, complemented
6	#22582489	C₃	1	1	yes	maximal, characteristic, complemented
7	#22582488	C₃	1	1	yes	maximal, characteristic, complemented
8	#22582488	C₃	1	1	yes	maximal, characteristic, complemented
9	#22582488	C₃	1	1	yes	maximal, characteristic, complemented
10	#22582488	C₃	1	1	yes	maximal, characteristic, complemented
11	#22582488	C₃	1	1	yes	maximal, characteristic, complemented
12	#22582488	C₃	1	1	yes	maximal, characteristic, complemented
13	C₃₇×C₉⋊C₂₇	C₃	1	1	yes	maximal, characteristic, complemented
14	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
15	#15688510	—	3	3	no	—
16	#15688509	—	3	3	no	—
17	C₃×#11758601	C₃²	1	1	yes	characteristic
18	C₃×#11758601	C₃²	1	1	yes	characteristic
19	C₃×#11758601	C₃²	1	1	yes	characteristic
20	C₃×#11758601	C₃²	1	1	yes	characteristic
21	C₃×#11758601	C₃²	1	1	yes	characteristic
22	C₃×#11758601	C₃²	1	1	yes	characteristic
23	C₂₇×C₃₇⋊C₃	—	3	3	no	—
24	#15688504	C₉	1	2	yes	—
25	#15688504	C₉	1	2	yes	—
26	#15688504	C₉	1	2	yes	—
27	#15688502	C₉	1	1	yes	characteristic
28	#15688502	C₉	1	2	yes	—
29	C₉×#66524	C₃²	1	1	yes	characteristic
30	C₉×#66524	C₃²	1	1	yes	characteristic
31	C₃×C₂₇×C₃₇	C₃²	1	1	yes	characteristic
32	C₃×C₂₇×C₃₇	C₃²	1	1	yes	characteristic
33	C₃×C₂₇×C₃₇	C₃²	1	1	yes	characteristic
34	C₉²×C₃₇	C₃²	1	1	yes	characteristic
35	C₃²×C₃₇⋊C₃	C₃×C₉	1	1	yes	characteristic
36	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
37	C₃×C₉×C₃₇	C₃×C₉	1	2	yes	—
38	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
39	C₃×#66524	C₃×C₉	1	1	yes	characteristic
40	C₃×#66524	C₃×C₉	1	1	yes	characteristic
41	C₃×#66524	C₃×C₉	1	2	yes	—
42	C₃×#66524	C₃×C₉	1	2	yes	—
43	C₃×#66524	C₃×C₉	1	2	yes	—
44	C₉×C₃₇⋊C₃	3₋¹⁺²	1	3	yes	—
45	C₉×C₃₇⋊C₃	3₋¹⁺²	1	3	yes	—
46	C₉×C₃₇⋊C₃	3₋¹⁺²	1	3	yes	—
47	C₂₇×C₃₇	—	3	1	no	—
48	C₂₇×C₃₇	—	3	1	no	—
49	C₂₇×C₃₇	—	3	1	no	—
50	#11758601	—	3	1	no	—
51	#11758601	—	3	1	no	—
52	#11758601	—	3	1	no	—
53	#11758601	—	3	1	no	—
54	#11758601	—	3	1	no	—
55	#11758601	—	3	1	no	—
56	#10624707	—	37	1	no	maximal, Sylow
57	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
58	C₃×C₃₇⋊C₃	—	3	1	no	—
59	C₃×C₃₇⋊C₃	—	3	2	no	—
60	C₃×C₃₇⋊C₃	C₉⋊C₉	1	3	yes	—
61	C₉×C₃₇	—	3	1	no	—
62	C₉×C₃₇	—	3	2	no	—
63	C₉×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
64	C₉×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
65	C₉×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
66	#66524	—	3	1	no	—
67	#66524	—	3	1	no	—
68	#66524	—	3	2	no	—
69	#66524	—	3	2	no	—
70	#66524	C₉⋊C₉	1	6	yes	—
71	C₃×C₂₇⋊C₃	—	37	1	no	—
72	C₃×C₂₇⋊C₃	—	37	1	no	—
73	C₃²×C₂₇	—	37	1	no	—
74	#6887	—	37	1	no	—
75	C₉⋊C₂₇	—	37	1	no	—
76	C₉⋊C₂₇	—	37	1	no	—
77	C₉⋊C₂₇	—	37	1	no	—
78	C₉⋊C₂₇	—	37	1	no	—
79	C₉⋊C₂₇	—	37	1	no	—
80	C₉⋊C₂₇	—	37	1	no	—
81	C₉⋊C₂₇	—	37	1	no	—
82	C₉⋊C₂₇	—	37	1	no	—
83	C₉⋊C₂₇	C₃₇⋊C₃	1	1	yes	characteristic, complemented
84	C₃×C₃₇	C₃×C₉⋊C₉	1	1	yes	characteristic
85	C₃×C₃₇	#6903	1	1	yes	characteristic
86	C₃×C₃₇	#6903	1	2	yes	—
87	C₃₇⋊C₃	—	3	3	no	—
88	C₃²×C₉	—	37	1	no	—
89	C₂₇⋊C₃	—	111	3	no	—
90	C₂₇⋊C₃	—	111	3	no	—
91	C₃×C₂₇	—	37	1	no	—
92	C₃×C₂₇	—	37	1	no	—
93	C₃×C₂₇	—	37	1	no	—
94	C₃×C₂₇	—	37	1	no	—
95	C₃×C₂₇	—	37	1	no	—
96	C₃×C₂₇	—	37	1	no	—
97	C₃×C₂₇	—	111	3	no	—
98	C₃×C₂₇	C₃×C₃₇⋊C₃	1	1	yes	characteristic
99	C₃×C₂₇	C₃×C₃₇⋊C₃	1	1	yes	characteristic
100	C₃×C₂₇	C₃×C₃₇⋊C₃	1	1	yes	characteristic
101	C₉⋊C₉	—	37	2	no	—
102	C₉⋊C₉	—	37	2	no	—
103	C₉⋊C₉	—	37	2	no	—
104	C₃²⋊C₉	—	37	1	no	—
105	C₃²⋊C₉	—	37	2	no	—
106	C₉²	—	37	1	no	—
107	C₉²	—	37	1	no	—
108	C₉²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
109	C₃₇	#10624707	1	1	yes	characteristic, complemented, Sylow
110	C₃³	—	37	1	no	—
111	C₃×C₉	—	37	2	no	—
112	C₃×C₉	—	37	3	no	—
113	C₃×C₉	—	37	3	no	—
114	C₃×C₉	—	37	3	no	—
115	C₃×C₉	—	37	1	no	—
116	C₃×C₉	—	37	1	no	—
117	C₃×C₉	—	37	2	no	—
118	C₃×C₉	—	37	2	no	—
119	C₃×C₉	C₉×C₃₇⋊C₃	1	1	yes	characteristic
120	C₃×C₉	C₉×C₃₇⋊C₃	1	2	yes	—
121	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
122	C₂₇	—	3	1	no	—
123	C₂₇	—	3	1	no	—
124	C₂₇	—	3	1	no	—
125	C₂₇	—	111	1	no	—
126	C₂₇	—	111	1	no	—
127	C₂₇	—	111	1	no	—
128	C₂₇	—	111	1	no	—
129	C₂₇	—	111	1	no	—
130	C₂₇	—	111	1	no	—
131	C₃²	—	37	3	no	—
132	C₃²	—	111	1	no	—
133	C₃²	—	111	2	no	—
134	C₃²	C₃×C₉×C₃₇⋊C₃	1	1	yes	characteristic
135	C₉	—	3	1	no	—
136	C₉	—	3	2	no	—
137	C₉	—	37	6	no	—
138	C₉	—	111	1	no	—
139	C₉	—	111	1	no	—
140	C₉	—	111	2	no	—
141	C₉	—	111	2	no	—
142	C₉	3₋¹⁺²×C₃₇⋊C₃	1	1	yes	characteristic
143	C₉	3₋¹⁺²×C₃₇⋊C₃	1	1	yes	characteristic
144	C₉	3₋¹⁺²×C₃₇⋊C₃	1	1	yes	characteristic
145	C₃	—	111	3	no	—
146	C₃	C₉⋊C₉×C₃₇⋊C₃	1	1	yes	characteristic
147	C₃	#22582612	1	1	yes	characteristic
148	C₃	#22582614	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃×C₉	C₃²×C₃₇⋊C₃
commutator, socle	C₃²×C₃₇	C₃²×C₉
Fitting	C₃₇×C₉⋊C₂₇	C₃
prime radical	C₃²×C₃₇⋊C₃	C₃×C₉
cosocle	C₃×C₉×C₃₇	C₃³
hypercenter	C₉⋊C₂₇	C₃₇⋊C₃
hypocenter	C₃₇	#10624707
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃×C₉ ◁ C₉⋊C₂₇

Complements

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

N	H ≅ G/N	type
5. #22582490	C₃	semidirect
6. #22582489	C₃	semidirect
7. #22582488	C₃	semidirect
8. #22582488	C₃	semidirect
9. #22582488	C₃	semidirect
10. #22582488	C₃	semidirect
11. #22582488	C₃	semidirect
12. #22582488	C₃	semidirect
13. C₃₇×C₉⋊C₂₇	C₃	semidirect
83. C₉⋊C₂₇	C₃₇⋊C₃	semidirect
109. C₃₇	#10624707	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15688510	3
C₃×#15688509	3
C₃×C₂₇×C₃₇⋊C₃	3
#22582536	3
#22582490	3
#22582489	3
#22582488	3
C₃₇×C₉⋊C₂₇	3
#10624707	37