Finite Group Databasebeta

Subgroups — #41005148

Order 26973.

332 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 124 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×#15688506	C₃	1	1	yes	maximal, characteristic
2	C₃×#15688504	C₃	1	2	yes	maximal
3	C₃×C₉×#66524	C₃	1	1	yes	maximal, characteristic
4	#22582441	C₃	1	1	yes	maximal, characteristic
5	#22582441	C₃	1	1	yes	maximal, characteristic
6	#22582441	C₃	1	2	yes	maximal
7	#22582441	C₃	1	2	yes	maximal
8	#22582441	C₃	1	2	yes	maximal
9	C₃₇×#6855	C₃	1	1	yes	maximal, characteristic
10	C₃²×#66524	C₃²	1	2	yes	—
11	C₃²×#66524	C₃²	1	2	yes	—
12	C₃²×#66524	C₃²	1	2	yes	—
13	C₃²×#66524	C₃²	1	1	yes	characteristic
14	C₃²×#66524	C₃²	1	2	yes	—
15	#15688506	C₉	1	9	yes	complemented
16	#15688504	—	3	6	no	—
17	C₉×#66524	—	3	3	no	—
18	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
19	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
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	C₃×C₉×C₃₇	—	3	1	no	—
24	C₃×C₉×C₃₇	—	3	1	no	—
25	C₃×C₉×C₃₇	—	3	2	no	—
26	C₃×C₉×C₃₇	—	3	2	no	—
27	C₃×C₉×C₃₇	—	3	2	no	—
28	C₃×C₉×C₃₇	—	3	1	no	—
29	C₃×C₉×C₃₇	—	3	1	no	—
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	C₃×C₉	1	3	yes	—
34	C₃×C₉×C₃₇	3₋¹⁺²	1	3	yes	—
35	C₃×#66524	—	3	1	no	—
36	C₃×#66524	—	3	2	no	—
37	C₃×#66524	—	3	2	no	—
38	C₃×#66524	—	3	1	no	—
39	C₃×#66524	—	3	1	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	2	no	—
46	C₃×#66524	—	3	2	no	—
47	C₃×#66524	—	3	2	no	—
48	C₃×#66524	—	3	2	no	—
49	C₃×#66524	—	3	2	no	—
50	C₃×#66524	—	3	2	no	—
51	C₃×#66524	—	3	2	no	—
52	C₃×#66524	C₃×C₉	1	6	yes	—
53	C₃×#66524	C₃×C₉	1	3	yes	—
54	C₃×#66524	3₋¹⁺²	1	6	yes	—
55	#10624559	—	37	1	no	maximal, Sylow
56	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
57	C₃²×C₃₇	C₃×3₋¹⁺²	1	2	yes	—
58	C₃²×C₃₇	C₃×3₋¹⁺²	1	1	yes	characteristic
59	C₃²×C₃₇	#737	1	1	yes	characteristic
60	C₃²×C₃₇	#737	1	1	yes	characteristic
61	C₃²×C₃₇	#737	1	1	yes	characteristic
62	C₃²×C₃₇	#737	1	2	yes	—
63	C₃²×C₃₇	#737	1	2	yes	—
64	C₃²×C₃₇	#737	1	2	yes	—
65	C₉×C₃₇	—	3	3	no	—
66	C₉×C₃₇	—	3	3	no	—
67	C₉×C₃₇	—	9	1	no	—
68	C₉×C₃₇	—	9	1	no	—
69	#66524	—	3	6	no	—
70	#66524	—	3	3	no	—
71	#66524	—	3	6	no	—
72	#66524	—	9	2	no	—
73	#66524	—	9	2	no	—
74	C₃×C₉⋊C₉	—	37	1	no	—
75	C₃×C₉⋊C₉	—	37	2	no	—
76	C₃×C₉²	—	37	1	no	—
77	#6855	—	37	2	no	—
78	#6855	—	37	1	no	—
79	#6855	—	37	2	no	—
80	#6855	—	37	1	no	—
81	#6855	—	37	2	no	—
82	#6855	C₃₇⋊C₃	1	1	yes	characteristic
83	C₃×C₃₇	#6887	1	1	yes	characteristic
84	C₃×C₃₇	C₉×3₋¹⁺²	1	2	yes	—
85	C₃×C₃₇	C₉×3₋¹⁺²	1	1	yes	characteristic
86	C₃×C₃₇	#6894	1	2	yes	—
87	C₃×C₃₇	#6895	1	1	yes	characteristic
88	C₃×C₃₇	#6896	1	1	yes	characteristic
89	C₃×C₃₇	#6897	1	2	yes	—
90	C₃×C₃₇	#6898	1	1	yes	characteristic
91	C₃×C₃₇	#6899	1	2	yes	—
92	C₃²×C₉	—	37	1	no	—
93	C₃²×C₉	—	37	2	no	—
94	C₃²×C₉	—	37	2	no	—
95	C₃²×C₉	—	37	2	no	—
96	C₃²×C₉	—	37	2	no	—
97	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
98	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
99	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
100	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
101	C₉⋊C₉	—	37	9	no	—
102	C₉⋊C₉	—	111	6	no	—
103	C₉²	—	111	3	no	—
104	C₃₇	#10624559	1	1	yes	characteristic, complemented, Sylow
105	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
106	C₃×C₉	—	3	1	no	—
107	C₃×C₉	—	3	1	no	—
108	C₃×C₉	—	3	2	no	—
109	C₃×C₉	—	3	2	no	—
110	C₃×C₉	—	3	2	no	—
111	C₃×C₉	—	3	1	no	—
112	C₃×C₉	—	3	1	no	—
113	C₃×C₉	—	3	1	no	—
114	C₃×C₉	—	3	1	no	—
115	C₃×C₉	—	3	2	no	—
116	C₃×C₉	—	37	6	no	—
117	C₃×C₉	—	37	3	no	—
118	C₃×C₉	—	37	6	no	—
119	C₃×C₉	—	111	2	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	C₃×C₉	—	111	2	no	—
125	C₃×C₉	—	111	2	no	—
126	C₃×C₉	—	111	2	no	—
127	C₃×C₉	—	111	2	no	—
128	C₃×C₉	—	111	1	no	—
129	C₃×C₉	—	111	2	no	—
130	C₃×C₉	—	111	1	no	—
131	C₃×C₉	—	111	1	no	—
132	C₃×C₉	—	111	2	no	—
133	C₃×C₉	—	111	2	no	—
134	C₃×C₉	—	111	2	no	—
135	C₃×C₉	—	111	2	no	—
136	C₃×C₉	C₉×C₃₇⋊C₃	1	3	yes	—
137	C₃×C₉	#11758606	1	3	yes	—
138	C₃²	C₃×C₉×C₃₇⋊C₃	1	1	yes	characteristic
139	C₃²	3₋¹⁺²×C₃₇⋊C₃	1	2	yes	—
140	C₃²	C₃×#11758606	1	1	yes	characteristic
141	C₃²	#15688530	1	1	yes	characteristic
142	C₃²	#15688531	1	1	yes	characteristic
143	C₃²	#15688531	1	1	yes	characteristic
144	C₃²	#15688532	1	2	yes	—
145	C₃²	#15688532	1	2	yes	—
146	C₃²	#15688532	1	2	yes	—
147	C₉	—	3	3	no	—
148	C₉	—	3	3	no	—
149	C₉	—	9	1	no	—
150	C₉	—	9	1	no	—
151	C₉	—	333	2	no	—
152	C₉	—	333	2	no	—
153	C₉	—	111	6	no	—
154	C₉	—	111	3	no	—
155	C₉	—	111	6	no	—
156	C₃	#22582535	1	1	yes	characteristic
157	C₃	#22582542	1	2	yes	—
158	C₃	C₉×#11758606	1	1	yes	characteristic
159	C₃	#22582566	1	2	yes	—
160	C₃	#22582575	1	1	yes	characteristic
161	C₃	#22582579	1	1	yes	characteristic
162	C₃	#22582589	1	2	yes	—
163	C₃	#22582590	1	1	yes	characteristic
164	C₃	#22582597	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃³	C₃²×C₃₇⋊C₃
commutator	C₃²×C₃₇	C₃²×C₉
Fitting	C₃₇×#6855	C₃
socle, prime radical, cosocle	C₃³×C₃₇	C₃³
hypercenter	#6855	C₃₇⋊C₃
hypocenter	C₃₇	#10624559
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃³ ◁ #6855

Complements

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

N	H ≅ G/N	type
15. #15688506	C₉	semidirect
104. C₃₇	#10624559	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15688506	3
C₃×#15688504	3
C₃×C₉×#66524	3
#22582441	3
C₃₇×#6855	3
#10624559	37