Finite Group Databasebeta

Subgroups — #41008523

Order 26973.

312 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 84 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×#15688542	C₃	1	1	yes	maximal, characteristic, complemented
2	(C₃₇⋊C₃)⋊(C₂₇⋊C₃)	C₃	1	2	yes	maximal, complemented
3	#22582702	C₃	1	6	yes	maximal, complemented
4	#22582701	C₃	1	3	yes	maximal, complemented
5	#737×C₃₇⋊C₃	C₃	1	1	yes	maximal, characteristic
6	#15688543	—	3	2	no	—
7	#15688542	—	3	2	no	—
8	#15688542	—	3	1	no	—
9	#15688542	—	3	2	no	—
10	#15688541	C₃²	1	6	yes	complemented
11	(C₃₇⋊C₃)⋊C₂₇	—	3	2	no	—
12	C₃×#11758603	C₃²	1	3	yes	complemented
13	#15688532	—	3	2	no	—
14	#15688532	—	3	2	no	—
15	#15688532	C₉	1	2	yes	—
16	#15688532	C₃²	1	2	yes	—
17	#15688531	—	3	2	no	—
18	#15688531	C₉	1	2	yes	—
19	#15688530	C₉	1	1	yes	characteristic
20	#15688530	C₃²	1	1	yes	characteristic
21	3₋¹⁺²×C₃₇⋊C₃	—	3	2	no	—
22	3₋¹⁺²×C₃₇⋊C₃	C₉	1	2	yes	—
23	H(3,𝔽₃)×C₃₇⋊C₃	C₉	1	1	yes	characteristic
24	C₃×C₉×C₃₇⋊C₃	—	3	1	no	—
25	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
26	C₃₇×#737	C₉	1	1	yes	characteristic
27	C₃²×C₃₇⋊C₃	—	3	1	no	—
28	C₃²×C₃₇⋊C₃	C₃×C₉	1	1	yes	characteristic
29	3₋¹⁺²×C₃₇	—	3	2	no	—
30	3₋¹⁺²×C₃₇	C₃×C₉	1	2	yes	—
31	H(3,𝔽₃)×C₃₇	C₃×C₉	1	1	yes	characteristic
32	C₃×C₉×C₃₇	—	3	1	no	—
33	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
34	C₃²⋊(C₃₇⋊C₃)	—	3	2	no	—
35	C₃²⋊(C₃₇⋊C₃)	C₃×C₉	1	2	yes	—
36	#11758608	—	3	2	no	—
37	#11758608	—	3	2	no	—
38	#11758608	C₃×C₉	1	2	yes	—
39	#11758608	C₃×C₉	1	2	yes	—
40	C₃×#66524	—	3	1	no	—
41	C₃×#66524	—	3	1	no	—
42	C₃×#66524	C₃×C₉	1	1	yes	characteristic
43	C₃×#66524	C₃³	1	1	yes	characteristic
44	#11758606	—	3	2	no	—
45	#11758606	—	3	2	no	—
46	#11758606	—	3	2	no	—
47	#11758606	—	3	2	no	—
48	#11758605	—	3	2	no	—
49	#11758605	—	3	2	no	—
50	#11758605	—	3	2	no	—
51	#11758605	—	3	2	no	—
52	C₉×C₃₇⋊C₃	—	3	2	no	—
53	C₉×C₃₇⋊C₃	—	3	2	no	—
54	C₉×C₃₇⋊C₃	—	3	2	no	—
55	C₉×C₃₇⋊C₃	—	3	2	no	—
56	C₉×C₃₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
57	C₉×C₃₇⋊C₃	H(3,𝔽₃)	1	2	yes	—
58	C₉×C₃₇⋊C₃	3₋¹⁺²	1	2	yes	—
59	C₉×C₃₇⋊C₃	3₋¹⁺²	1	2	yes	—
60	C₉×C₃₇⋊C₃	3₋¹⁺²	1	2	yes	—
61	#11758603	—	3	3	no	—
62	#11758603	—	3	6	no	—
63	#10624704	—	37	1	no	maximal, Sylow
64	C₃²×C₃₇	—	3	1	no	—
65	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
66	C₃×C₃₇⋊C₃	—	3	2	no	—
67	C₃×C₃₇⋊C₃	—	9	1	no	—
68	C₃×C₃₇⋊C₃	—	9	3	no	—
69	C₃×C₃₇⋊C₃	C₃²⋊C₉	1	1	yes	characteristic
70	C₃×C₃₇⋊C₃	C₃²⋊C₉	1	2	yes	—
71	C₉×C₃₇	—	3	2	no	—
72	C₉×C₃₇	C₃²⋊C₉	1	1	yes	characteristic
73	C₉×C₃₇	C₃²⋊C₉	1	2	yes	—
74	#66524	—	3	2	no	—
75	#66524	—	3	2	no	—
76	#66524	C₃²⋊C₉	1	1	yes	characteristic
77	#66524	C₃²⋊C₉	1	2	yes	—
78	#66524	C₃×H(3,𝔽₃)	1	1	yes	characteristic
79	#66524	C₃×3₋¹⁺²	1	2	yes	—
80	C₃×#737	—	37	1	no	—
81	C₃×C₂₇⋊C₃	—	37	1	no	—
82	C₃×C₂₇⋊C₃	—	37	2	no	—
83	#6873	—	37	3	no	—
84	#6873	—	37	6	no	—
85	C₃×C₃₇	—	3	1	no	—
86	C₃×C₃₇	—	9	1	no	—
87	C₃×C₃₇	C₃×C₃²⋊C₉	1	1	yes	characteristic
88	C₃₇⋊C₃	—	3	1	no	—
89	C₃₇⋊C₃	—	3	2	no	—
90	C₃₇⋊C₃	—	9	2	no	—
91	#737	—	37	2	no	—
92	#737	—	37	2	no	—
93	#737	—	37	1	no	—
94	#737	—	37	1	no	—
95	#737	—	37	2	no	—
96	#737	—	111	2	no	—
97	#737	—	111	2	no	—
98	#737	—	111	2	no	—
99	#737	#66526	1	1	yes	characteristic
100	C₃×3₋¹⁺²	—	37	2	no	—
101	C₃×3₋¹⁺²	—	111	2	no	—
102	C₃×H(3,𝔽₃)	—	37	1	no	—
103	C₃²×C₉	—	37	1	no	—
104	C₃²×C₉	—	111	1	no	—
105	C₂₇⋊C₃	—	37	6	no	—
106	C₂₇⋊C₃	—	111	2	no	—
107	C₂₇⋊C₃	—	111	1	no	—
108	C₂₇⋊C₃	—	111	2	no	—
109	C₂₇⋊C₃	—	111	2	no	—
110	C₃×C₂₇	—	37	3	no	—
111	C₃×C₂₇	—	111	2	no	—
112	C₃₇	#10624704	1	1	yes	characteristic, complemented, Sylow
113	C₃³	—	37	1	no	—
114	C₃³	—	111	1	no	—
115	3₋¹⁺²	—	3	2	no	—
116	3₋¹⁺²	—	37	2	no	—
117	3₋¹⁺²	—	37	2	no	—
118	3₋¹⁺²	—	111	2	no	—
119	3₋¹⁺²	—	111	2	no	—
120	3₋¹⁺²	—	111	2	no	—
121	3₋¹⁺²	—	111	2	no	—
122	3₋¹⁺²	—	111	2	no	—
123	3₋¹⁺²	—	111	2	no	—
124	3₋¹⁺²	—	111	2	no	—
125	3₋¹⁺²	—	111	2	no	—
126	3₋¹⁺²	—	111	2	no	—
127	3₋¹⁺²	—	111	2	no	—
128	3₋¹⁺²	C₃×#66526	1	2	yes	—
129	H(3,𝔽₃)	—	37	2	no	—
130	H(3,𝔽₃)	—	111	2	no	—
131	H(3,𝔽₃)	C₃×#66526	1	1	yes	characteristic
132	C₃×C₉	—	3	1	no	—
133	C₃×C₉	—	37	1	no	—
134	C₃×C₉	—	37	1	no	—
135	C₃×C₉	—	37	2	no	—
136	C₃×C₉	—	37	1	no	—
137	C₃×C₉	—	37	2	no	—
138	C₃×C₉	—	37	2	no	—
139	C₃×C₉	—	37	2	no	—
140	C₃×C₉	—	111	2	no	—
141	C₃×C₉	—	111	2	no	—
142	C₃×C₉	—	111	2	no	—
143	C₃×C₉	—	111	1	no	—
144	C₃×C₉	—	111	1	no	—
145	C₃×C₉	—	111	2	no	—
146	C₃×C₉	C₃×#66526	1	1	yes	characteristic
147	C₂₇	—	111	3	no	—
148	C₂₇	—	111	6	no	—
149	C₃²	—	3	1	no	—
150	C₃²	—	37	1	no	—
151	C₃²	—	37	2	no	—
152	C₃²	—	333	1	no	—
153	C₃²	—	333	3	no	—
154	C₃²	—	111	2	no	—
155	C₃²	C₃²×#66526	1	1	yes	characteristic
156	C₉	—	3	2	no	—
157	C₉	—	37	1	no	—
158	C₉	—	37	1	no	—
159	C₉	—	37	2	no	—
160	C₉	—	37	2	no	—
161	C₉	—	111	2	no	—
162	C₉	—	111	2	no	—
163	C₉	C₃²⋊#66526	1	1	yes	characteristic
164	C₉	C₃²⋊#66526	1	2	yes	—
165	C₃	—	3	1	no	—
166	C₃	—	9	1	no	—
167	C₃	—	333	2	no	—
168	C₃	—	111	1	no	—
169	C₃	—	111	2	no	—
170	C₃	C₃×C₃²⋊#66526	1	1	yes	characteristic

	subgroup	G/H
center	C₃	C₃×C₃²⋊#66526
commutator	C₃²×C₃₇	C₃²×C₉
Frattini	C₃²	C₃²×#66526
Fitting	C₃₇×#737	C₉
socle	C₃×C₃₇	C₃×C₃²⋊C₉
prime radical	H(3,𝔽₃)×C₃₇⋊C₃	C₉
cosocle	C₃×#66524	C₃³
hypercenter	#737	#66526
hypocenter	C₃₇	#10624704
LCS	G ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃×C₉ ◁ #737

Complements

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

N	H ≅ G/N	type
1. C₃×#15688542	C₃	semidirect
2. (C₃₇⋊C₃)⋊(C₂₇⋊C₃)	C₃	semidirect
3. #22582702	C₃	semidirect
4. #22582701	C₃	semidirect
10. #15688541	C₃²	semidirect
12. C₃×#11758603	C₃²	semidirect
112. C₃₇	#10624704	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15688542	3
(C₃₇⋊C₃)⋊(C₂₇⋊C₃)	3
#22582702	3
#22582701	3
#737×C₃₇⋊C₃	3
#10624704	37