Finite Group Databasebeta

Subgroups — #41006692

Order 26973.

402 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 61 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	#22582679	C₃	1	1	yes	maximal, characteristic, complemented
2	#22582651	C₃	1	2	yes	maximal, complemented
3	#22582651	C₃	1	2	yes	maximal, complemented
4	#22582643	C₃	1	1	yes	maximal, characteristic, complemented
5	#22582642	C₃	1	2	yes	maximal, complemented
6	#22582638	C₃	1	1	yes	maximal, characteristic
7	C₃×#15688521	C₃	1	2	yes	maximal, complemented
8	C₃×#15688516	C₃	1	1	yes	maximal, characteristic, complemented
9	C₃×C₃₇×#731	C₃	1	1	yes	maximal, characteristic, complemented
10	#15688532	—	3	6	no	—
11	#15688531	—	3	3	no	—
12	C₃×#11758608	C₃²	1	2	yes	—
13	C₃×#11758608	C₃²	1	2	yes	—
14	C₃×#11758608	C₃²	1	2	yes	—
15	C₃×#11758608	C₃²	1	2	yes	—
16	3₋¹⁺²×C₃₇⋊C₃	—	3	3	no	—
17	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	1	yes	characteristic
18	#15688521	—	3	6	no	—
19	#15688520	—	3	6	no	—
20	#15688520	—	3	6	no	—
21	#15688516	—	3	3	no	—
22	#15688516	—	3	3	no	—
23	#15688515	—	3	6	no	—
24	#15688513	—	3	3	no	—
25	C₃×3₋¹⁺²×C₃₇	C₃²	1	1	yes	characteristic
26	C₃×3₋¹⁺²×C₃₇	C₃²	1	1	yes	characteristic
27	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
28	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
29	C₃₇×#731	—	3	3	no	—
30	C₃³×C₃₇	—	3	1	no	—
31	C₃³×C₃₇	C₃³	1	1	yes	characteristic
32	C₃²×C₃₇⋊C₃	—	3	1	no	—
33	3₋¹⁺²×C₃₇	—	3	2	no	—
34	3₋¹⁺²×C₃₇	—	3	1	no	—
35	3₋¹⁺²×C₃₇	—	3	2	no	—
36	3₋¹⁺²×C₃₇	H(3,𝔽₃)	1	3	yes	—
37	H(3,𝔽₃)×C₃₇	—	3	1	no	—
38	H(3,𝔽₃)×C₃₇	—	3	2	no	—
39	C₃×C₉×C₃₇	—	3	1	no	—
40	C₃×C₉×C₃₇	—	3	1	no	—
41	C₃×C₉×C₃₇	—	3	1	no	—
42	C₃×C₉×C₃₇	—	3	2	no	—
43	C₃×C₉×C₃₇	—	3	1	no	—
44	C₃²⋊(C₃₇⋊C₃)	—	3	2	no	—
45	C₃²⋊(C₃₇⋊C₃)	H(3,𝔽₃)	1	3	yes	—
46	#11758608	—	3	2	no	—
47	#11758608	—	3	2	no	—
48	#11758608	—	3	2	no	—
49	#11758608	—	3	2	no	—
50	#11758608	—	3	2	no	—
51	#11758608	—	3	2	no	—
52	#11758608	—	3	2	no	—
53	#11758608	—	3	2	no	—
54	#11758608	—	3	2	no	—
55	#11758608	—	3	2	no	—
56	#11758608	—	3	2	no	—
57	#11758608	H(3,𝔽₃)	1	6	yes	—
58	C₃×#66524	—	3	2	no	—
59	C₃×#66524	—	3	2	no	—
60	C₃×#66524	—	3	2	no	—
61	C₃×#66524	—	3	2	no	—
62	#11758606	—	3	9	no	—
63	#11758605	—	3	9	no	—
64	C₉×C₃₇⋊C₃	—	3	9	no	—
65	#10624764	—	37	1	no	maximal, Sylow
66	C₃²×C₃₇	—	9	1	no	—
67	C₃²×C₃₇	—	9	1	no	—
68	C₃²×C₃₇	—	27	1	no	—
69	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	2	yes	—
70	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
71	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
72	C₃×C₃₇⋊C₃	—	3	3	no	—
73	C₃×C₃₇⋊C₃	—	27	1	no	—
74	C₉×C₃₇	—	3	3	no	—
75	C₉×C₃₇	—	9	1	no	—
76	C₉×C₃₇	—	9	1	no	—
77	#66524	—	3	6	no	—
78	#66524	—	9	2	no	—
79	#66524	—	9	2	no	—
80	#66524	—	9	2	no	—
81	3₋¹⁺⁴	—	37	1	no	—
82	#6912	—	37	2	no	—
83	#6912	—	37	2	no	—
84	#6910	—	37	1	no	—
85	#6910	—	37	2	no	—
86	#6909	—	37	1	no	—
87	C₃×#733	—	37	2	no	—
88	C₃×#731	—	37	1	no	—
89	C₃×#731	C₃₇⋊C₃	1	1	yes	characteristic, complemented
90	C₃×C₃₇	—	3	1	no	—
91	C₃×C₃₇	—	3	1	no	—
92	C₃×C₃₇	—	3	2	no	—
93	C₃×C₃₇	—	27	1	no	—
94	C₃×C₃₇	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
95	C₃₇⋊C₃	—	9	3	no	—
96	#737	—	111	3	no	—
97	#737	—	111	6	no	—
98	C₃×3₋¹⁺²	—	37	2	no	—
99	C₃×3₋¹⁺²	—	37	2	no	—
100	C₃×3₋¹⁺²	—	37	2	no	—
101	C₃×3₋¹⁺²	—	37	2	no	—
102	C₃×3₋¹⁺²	—	111	3	no	—
103	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
104	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
105	C₃×H(3,𝔽₃)	—	37	1	no	—
106	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
107	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
108	#733	—	111	6	no	—
109	#733	—	111	6	no	—
110	#733	—	111	6	no	—
111	#731	—	3	3	no	—
112	#731	—	111	3	no	—
113	#731	—	111	3	no	—
114	#731	—	111	6	no	—
115	AΣL(1,𝔽₂₇)	—	111	3	no	—
116	C₃₇	#10624764	1	1	yes	characteristic, complemented, Sylow
117	C₃³	—	3	1	no	—
118	C₃³	—	111	1	no	—
119	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
120	3₋¹⁺²	—	3	2	no	—
121	3₋¹⁺²	—	3	2	no	—
122	3₋¹⁺²	—	3	1	no	—
123	3₋¹⁺²	—	37	6	no	—
124	3₋¹⁺²	—	111	9	no	—
125	3₋¹⁺²	—	111	9	no	—
126	3₋¹⁺²	—	111	2	no	—
127	3₋¹⁺²	—	111	2	no	—
128	3₋¹⁺²	—	111	2	no	—
129	3₋¹⁺²	—	111	2	no	—
130	3₋¹⁺²	—	111	2	no	—
131	3₋¹⁺²	—	111	2	no	—
132	3₋¹⁺²	—	111	2	no	—
133	3₋¹⁺²	—	111	2	no	—
134	3₋¹⁺²	—	111	2	no	—
135	3₋¹⁺²	—	111	2	no	—
136	3₋¹⁺²	—	111	2	no	—
137	3₋¹⁺²	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
138	H(3,𝔽₃)	—	3	2	no	—
139	H(3,𝔽₃)	—	3	1	no	—
140	H(3,𝔽₃)	—	37	3	no	—
141	H(3,𝔽₃)	—	111	2	no	—
142	C₃×C₉	—	3	1	no	—
143	C₃×C₉	—	3	1	no	—
144	C₃×C₉	—	3	1	no	—
145	C₃×C₉	—	3	2	no	—
146	C₃×C₉	—	3	1	no	—
147	C₃×C₉	—	111	9	no	—
148	C₃×C₉	—	111	2	no	—
149	C₃×C₉	—	111	2	no	—
150	C₃×C₉	—	111	2	no	—
151	C₃×C₉	—	111	2	no	—
152	C₃²	—	9	1	no	—
153	C₃²	—	9	1	no	—
154	C₃²	—	27	1	no	—
155	C₃²	—	111	3	no	—
156	C₃²	—	999	1	no	—
157	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
158	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	2	yes	—
159	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
160	C₉	—	3	3	no	—
161	C₉	—	9	1	no	—
162	C₉	—	9	1	no	—
163	C₉	—	333	2	no	—
164	C₉	—	333	2	no	—
165	C₉	—	333	2	no	—
166	C₉	—	111	6	no	—
167	C₃	—	3	1	no	—
168	C₃	—	3	1	no	—
169	C₃	—	3	2	no	—
170	C₃	—	27	1	no	—
171	C₃	—	333	3	no	—
172	C₃	(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₃	(C₃×C₃₇⋊C₃)⋊H(3,𝔽₃)
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃×C₃₇×#731	C₃
socle	C₃×C₃₇	C₃²⋊H(3,𝔽₃)
prime radical	#22582638	C₃
hypercenter	C₃×#731	C₃₇⋊C₃
hypocenter	C₃₇	#10624764
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃×#731

Complements

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

N	H ≅ G/N	type
1. #22582679	C₃	semidirect
2. #22582651	C₃	semidirect
3. #22582651	C₃	semidirect
4. #22582643	C₃	semidirect
5. #22582642	C₃	semidirect
7. C₃×#15688521	C₃	semidirect
8. C₃×#15688516	C₃	semidirect
9. C₃×C₃₇×#731	C₃	semidirect
89. C₃×#731	C₃₇⋊C₃	semidirect
116. C₃₇	#10624764	semidirect

Maximal subgroups

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

subgroup	index
#22582679	3
#22582651	3
#22582643	3
#22582642	3
#22582638	3
C₃×#15688521	3
C₃×#15688516	3
C₃×C₃₇×#731	3
#10624764	37