Finite Group Databasebeta

Subgroups — #41005215

Order 26973.

524 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 115 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	C₃²×#11758605	C₃	1	1	yes	maximal, characteristic
2	C₃²×#11758606	C₃	1	1	yes	maximal, characteristic
3	C₃×#15688502	C₃	1	2	yes	maximal
4	#22582441	C₃	1	4	yes	maximal, complemented
5	#22582441	C₃	1	2	yes	maximal, complemented
6	#22582441	C₃	1	2	yes	maximal, complemented
7	C₃₇×#6855	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
9	C₃×#11758605	—	3	3	no	—
10	C₃×#11758605	—	3	3	no	—
11	C₃×#11758605	—	3	6	no	—
12	C₃×#11758606	—	3	3	no	—
13	C₃×#11758606	—	3	3	no	—
14	C₃×#11758606	—	3	6	no	—
15	C₃²×#66524	C₃²	1	4	yes	—
16	C₃²×#66524	C₃²	1	2	yes	—
17	C₃²×#66524	C₃²	1	2	yes	—
18	#15688502	—	3	6	no	—
19	C₃²×C₉×C₃₇	C₃²	1	2	yes	—
20	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
21	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
22	C₃³×C₃₇	C₃³	1	1	yes	characteristic
23	C₃²×C₃₇⋊C₃	—	3	2	no	—
24	C₃²×C₃₇⋊C₃	—	3	1	no	—
25	C₃²×C₃₇⋊C₃	—	3	2	no	—
26	C₃²×C₃₇⋊C₃	—	3	1	no	—
27	C₃²×C₃₇⋊C₃	—	3	2	no	—
28	C₃²×C₃₇⋊C₃	—	3	4	no	—
29	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	3	yes	—
30	C₃×C₉×C₃₇	—	3	1	no	—
31	C₃×C₉×C₃₇	—	3	1	no	—
32	C₃×C₉×C₃₇	—	3	2	no	—
33	C₃×C₉×C₃₇	—	3	2	no	—
34	C₃×C₉×C₃₇	—	3	2	no	—
35	C₃×C₉×C₃₇	—	3	2	no	—
36	C₃×C₉×C₃₇	—	3	4	no	—
37	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	complemented
38	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	complemented
39	C₃×#66524	—	3	2	no	—
40	C₃×#66524	—	3	2	no	—
41	C₃×#66524	—	3	2	no	—
42	C₃×#66524	—	3	2	no	—
43	C₃×#66524	—	3	2	no	—
44	C₃×#66524	—	3	2	no	—
45	C₃×#66524	—	3	4	no	—
46	C₃×#66524	—	3	4	no	—
47	C₃×#66524	—	3	8	no	—
48	C₃×#66524	3₋¹⁺²	1	6	yes	complemented
49	C₃×#66524	3₋¹⁺²	1	6	yes	complemented
50	#11758606	—	3	27	no	—
51	#11758605	—	3	27	no	—
52	#10624567	—	37	1	no	maximal, Sylow
53	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
54	C₃²×C₃₇	C₃×3₋¹⁺²	1	2	yes	—
55	C₃²×C₃₇	C₃×3₋¹⁺²	1	2	yes	—
56	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
57	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
58	C₃²×C₃₇	#737	1	2	yes	—
59	C₃²×C₃₇	#737	1	4	yes	—
60	C₃×C₃₇⋊C₃	—	3	3	no	—
61	C₃×C₃₇⋊C₃	—	3	3	no	—
62	C₃×C₃₇⋊C₃	—	3	6	no	—
63	C₃×C₃₇⋊C₃	—	9	1	no	—
64	C₃×C₃₇⋊C₃	—	9	4	no	—
65	C₃×C₃₇⋊C₃	—	9	2	no	—
66	C₃×C₃₇⋊C₃	—	9	2	no	—
67	C₉×C₃₇	—	3	3	no	—
68	C₉×C₃₇	—	3	3	no	—
69	C₉×C₃₇	—	9	2	no	—
70	#66524	—	3	6	no	—
71	#66524	—	3	6	no	—
72	#66524	—	9	4	no	—
73	C₃²×3₋¹⁺²	—	37	1	no	—
74	C₃²×3₋¹⁺²	—	37	1	no	—
75	C₃×C₃²⋊C₉	—	37	2	no	—
76	#6855	—	37	4	no	—
77	#6855	—	37	2	no	—
78	#6855	—	37	2	no	—
79	#6855	C₃₇⋊C₃	1	1	yes	characteristic, complemented
80	C₃×C₃₇	#6891	1	1	yes	characteristic
81	C₃×C₃₇	#6891	1	1	yes	characteristic
82	C₃×C₃₇	#6893	1	2	yes	—
83	C₃×C₃₇	#6896	1	1	yes	characteristic
84	C₃×C₃₇	#6896	1	4	yes	—
85	C₃×C₃₇	#6896	1	2	yes	—
86	C₃×C₃₇	#6896	1	2	yes	—
87	C₃₇⋊C₃	—	9	3	no	—
88	C₃⁴	—	37	1	no	—
89	C₃×3₋¹⁺²	—	111	6	no	—
90	C₃×3₋¹⁺²	—	111	6	no	—
91	C₃×3₋¹⁺²	—	111	3	no	—
92	C₃×3₋¹⁺²	—	111	3	no	—
93	C₃×3₋¹⁺²	—	111	3	no	—
94	C₃×3₋¹⁺²	—	111	3	no	—
95	C₃²×C₉	—	37	2	no	—
96	C₃²×C₉	—	37	2	no	—
97	C₃²×C₉	—	37	4	no	—
98	C₃²×C₉	C₃×C₃₇⋊C₃	1	2	yes	—
99	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
100	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
101	C₃²⋊C₉	—	111	6	no	—
102	C₃₇	#10624567	1	1	yes	characteristic, complemented, Sylow
103	C₃³	—	37	3	no	—
104	C₃³	—	111	2	no	—
105	C₃³	—	111	2	no	—
106	C₃³	—	111	1	no	—
107	C₃³	—	111	1	no	—
108	C₃³	—	111	2	no	—
109	C₃³	—	111	4	no	—
110	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
111	3₋¹⁺²	—	111	27	no	—
112	3₋¹⁺²	—	111	27	no	—
113	C₃×C₉	—	3	1	no	—
114	C₃×C₉	—	3	1	no	—
115	C₃×C₉	—	3	2	no	—
116	C₃×C₉	—	3	2	no	—
117	C₃×C₉	—	3	2	no	—
118	C₃×C₉	—	3	2	no	—
119	C₃×C₉	—	3	4	no	—
120	C₃×C₉	—	37	6	no	—
121	C₃×C₉	—	37	6	no	—
122	C₃×C₉	—	111	4	no	—
123	C₃×C₉	—	111	4	no	—
124	C₃×C₉	—	111	8	no	—
125	C₃×C₉	—	111	2	no	—
126	C₃×C₉	—	111	2	no	—
127	C₃×C₉	—	111	2	no	—
128	C₃×C₉	—	111	2	no	—
129	C₃×C₉	—	111	2	no	—
130	C₃×C₉	—	111	2	no	—
131	C₃×C₉	#11758605	1	3	yes	complemented
132	C₃×C₉	#11758606	1	3	yes	complemented
133	C₃²	—	333	1	no	—
134	C₃²	—	333	4	no	—
135	C₃²	—	333	2	no	—
136	C₃²	—	333	2	no	—
137	C₃²	—	111	3	no	—
138	C₃²	—	111	3	no	—
139	C₃²	—	111	6	no	—
140	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
141	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	2	yes	—
142	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	2	yes	—
143	C₃²	C₃×#11758606	1	1	yes	characteristic
144	C₃²	C₃×#11758605	1	1	yes	characteristic
145	C₃²	#15688531	1	2	yes	—
146	C₃²	#15688532	1	4	yes	—
147	C₉	—	3	3	no	—
148	C₉	—	3	3	no	—
149	C₉	—	9	2	no	—
150	C₉	—	333	4	no	—
151	C₉	—	111	6	no	—
152	C₉	—	111	6	no	—
153	C₃	—	333	3	no	—
154	C₃	#22582552	1	1	yes	characteristic
155	C₃	#22582553	1	1	yes	characteristic
156	C₃	#22582563	1	2	yes	—
157	C₃	#22582581	1	1	yes	characteristic
158	C₃	#22582582	1	4	yes	—
159	C₃	#22582583	1	2	yes	—
160	C₃	#22582584	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃³	C₃²×C₃₇⋊C₃
commutator, socle, cosocle	C₃³×C₃₇	C₃³
Fitting	C₃₇×#6855	C₃
prime radical	C₃³×C₃₇⋊C₃	C₃²
hypercenter	#6855	C₃₇⋊C₃
hypocenter	C₃₇	#10624567
LCS	G ▷ C₃³×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃³ ◁ #6855

Complements

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

N	H ≅ G/N	type
4. #22582441	C₃	semidirect
5. #22582441	C₃	semidirect
6. #22582441	C₃	semidirect
7. C₃₇×#6855	C₃	semidirect
37. C₃×C₉×C₃₇	3₋¹⁺²	semidirect
38. C₃×C₉×C₃₇	3₋¹⁺²	semidirect
48. C₃×#66524	3₋¹⁺²	semidirect
49. C₃×#66524	3₋¹⁺²	semidirect
79. #6855	C₃₇⋊C₃	semidirect
102. C₃₇	#10624567	semidirect
131. C₃×C₉	#11758605	semidirect
132. C₃×C₉	#11758606	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#11758605	3
C₃²×#11758606	3
C₃×#15688502	3
#22582441	3
C₃₇×#6855	3
#10624567	37