Finite Group Databasebeta

Subgroups — #41006009

Order 26973.

236 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 49 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	#22582553	C₃	1	1	yes	maximal, characteristic
2	#22582552	C₃	1	1	yes	maximal, characteristic
3	#22582552	C₃	1	1	yes	maximal, characteristic
4	#22582551	C₃	1	1	yes	maximal, characteristic
5	#22582450	C₃	1	2	yes	maximal, complemented
6	#22582449	C₃	1	2	yes	maximal, complemented
7	#22582448	C₃	1	2	yes	maximal, complemented
8	#22582447	C₃	1	2	yes	maximal, complemented
9	C₃₇×#6858	C₃	1	1	yes	maximal, characteristic, complemented
10	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
11	C₃×#11758608	C₃²	1	2	yes	—
12	C₃×#11758605	—	3	1	no	—
13	C₃×#11758606	—	3	1	no	—
14	C₃×#11758606	—	3	1	no	—
15	C₃²⋊#66524	C₃²	1	2	yes	—
16	C₃²⋊#66524	C₃²	1	2	yes	—
17	C₃²⋊#66524	C₃²	1	2	yes	—
18	#15688502	—	3	2	no	—
19	#15688502	—	3	1	no	—
20	#15688502	—	3	2	no	—
21	#15688502	—	3	2	no	—
22	#15688502	—	3	2	no	—
23	C₃×3₋¹⁺²×C₃₇	C₃²	1	1	yes	characteristic
24	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
25	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
26	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
27	C₃³×C₃₇	C₃³	1	1	yes	characteristic
28	C₃²×C₃₇⋊C₃	—	9	1	no	—
29	C₃²×C₃₇⋊C₃	—	9	1	no	—
30	C₃²×C₃₇⋊C₃	—	9	1	no	—
31	C₃²×C₃₇⋊C₃	—	9	1	no	—
32	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	2	yes	—
33	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
34	3₋¹⁺²×C₃₇	—	9	1	no	—
35	C₃×C₉×C₃₇	—	3	1	no	—
36	C₃×C₉×C₃₇	—	3	1	no	—
37	C₃×C₉×C₃₇	—	3	1	no	—
38	C₃×C₉×C₃₇	—	3	1	no	—
39	#11758608	—	9	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	#11758606	—	9	3	no	—
45	#11758606	—	9	3	no	—
46	#11758605	—	9	3	no	—
47	#10624660	—	37	1	no	maximal, Sylow
48	C₃²×C₃₇	—	3	1	no	—
49	C₃²×C₃₇	—	3	1	no	—
50	C₃²×C₃₇	—	3	1	no	—
51	C₃²×C₃₇	—	3	1	no	—
52	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
53	C₃×C₃₇⋊C₃	—	3	2	no	—
54	C₃×C₃₇⋊C₃	—	3	1	no	—
55	C₃×C₃₇⋊C₃	—	3	1	no	—
56	C₃×C₃₇⋊C₃	—	3	2	no	—
57	C₃×C₃₇⋊C₃	—	3	1	no	—
58	C₃×C₃₇⋊C₃	—	3	1	no	—
59	C₃×C₃₇⋊C₃	—	3	2	no	—
60	C₃×C₃₇⋊C₃	—	3	2	no	—
61	C₃×C₃₇⋊C₃	—	9	9	no	—
62	C₉×C₃₇	—	9	1	no	—
63	C₉×C₃₇	—	9	1	no	—
64	C₉×C₃₇	—	9	1	no	—
65	C₉×C₃₇	—	9	1	no	—
66	#66524	—	9	2	no	—
67	#66524	—	9	2	no	—
68	#66524	—	9	2	no	—
69	#66524	—	9	2	no	—
70	#6891	—	37	1	no	—
71	#6891	—	37	1	no	—
72	#6891	—	37	1	no	—
73	#6891	—	37	1	no	—
74	#6858	—	37	2	no	—
75	#6858	—	37	2	no	—
76	#6858	—	37	2	no	—
77	#6858	—	37	2	no	—
78	#6858	C₃₇⋊C₃	1	1	yes	characteristic, complemented
79	C₃×C₃₇	—	9	1	no	—
80	C₃×C₃₇	#6909	1	1	yes	characteristic
81	C₃×C₃₇	#6910	1	1	yes	characteristic
82	C₃×C₃₇	#6910	1	1	yes	characteristic
83	C₃×C₃₇	#6912	1	1	yes	characteristic
84	C₃₇⋊C₃	—	9	1	no	—
85	C₃₇⋊C₃	—	9	2	no	—
86	C₃⁴	—	37	1	no	—
87	C₃×3₋¹⁺²	—	37	2	no	—
88	C₃×3₋¹⁺²	—	111	1	no	—
89	C₃×3₋¹⁺²	—	111	1	no	—
90	C₃×3₋¹⁺²	—	111	1	no	—
91	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
92	C₃²⋊C₉	—	37	2	no	—
93	C₃²⋊C₉	—	37	2	no	—
94	C₃²⋊C₉	—	37	2	no	—
95	C₃²⋊C₉	—	111	2	no	—
96	C₃²⋊C₉	—	111	1	no	—
97	C₃²⋊C₉	—	111	2	no	—
98	C₃²⋊C₉	—	111	2	no	—
99	C₃²⋊C₉	—	111	2	no	—
100	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
101	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
102	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
103	C₃₇	#10624660	1	1	yes	characteristic, complemented, Sylow
104	C₃³	—	37	2	no	—
105	C₃³	—	37	1	no	—
106	C₃³	—	333	1	no	—
107	C₃³	—	333	1	no	—
108	C₃³	—	333	1	no	—
109	C₃³	—	333	1	no	—
110	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
111	3₋¹⁺²	—	9	1	no	—
112	3₋¹⁺²	—	333	3	no	—
113	3₋¹⁺²	—	333	3	no	—
114	3₋¹⁺²	—	333	3	no	—
115	3₋¹⁺²	—	333	2	no	—
116	C₃×C₉	—	3	1	no	—
117	C₃×C₉	—	3	1	no	—
118	C₃×C₉	—	3	1	no	—
119	C₃×C₉	—	3	1	no	—
120	C₃×C₉	—	111	2	no	—
121	C₃×C₉	—	111	2	no	—
122	C₃×C₉	—	111	2	no	—
123	C₃×C₉	—	111	2	no	—
124	C₃²	—	3	1	no	—
125	C₃²	—	3	1	no	—
126	C₃²	—	3	1	no	—
127	C₃²	—	3	1	no	—
128	C₃²	—	333	9	no	—
129	C₃²	—	111	1	no	—
130	C₃²	—	111	2	no	—
131	C₃²	—	111	1	no	—
132	C₃²	—	111	2	no	—
133	C₃²	—	111	1	no	—
134	C₃²	—	111	2	no	—
135	C₃²	—	111	2	no	—
136	C₃²	—	111	1	no	—
137	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
138	C₉	—	9	1	no	—
139	C₉	—	9	1	no	—
140	C₉	—	9	1	no	—
141	C₉	—	9	1	no	—
142	C₉	—	333	2	no	—
143	C₉	—	333	2	no	—
144	C₉	—	333	2	no	—
145	C₉	—	333	2	no	—
146	C₃	—	9	1	no	—
147	C₃	—	333	1	no	—
148	C₃	—	333	2	no	—
149	C₃	#22582635	1	1	yes	characteristic
150	C₃	#22582640	1	1	yes	characteristic
151	C₃	#22582641	1	1	yes	characteristic
152	C₃	#22582650	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	C₃₇×#6858	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃³×C₃₇⋊C₃	C₃²
hypercenter	#6858	C₃₇⋊C₃
hypocenter	C₃₇	#10624660
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ #6858

Complements

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

N	H ≅ G/N	type
5. #22582450	C₃	semidirect
6. #22582449	C₃	semidirect
7. #22582448	C₃	semidirect
8. #22582447	C₃	semidirect
9. C₃₇×#6858	C₃	semidirect
78. #6858	C₃₇⋊C₃	semidirect
103. C₃₇	#10624660	semidirect

Maximal subgroups

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

subgroup	index
#22582553	3
#22582552	3
#22582551	3
#22582450	3
#22582449	3
#22582448	3
#22582447	3
C₃₇×#6858	3
#10624660	37