Finite Group Databasebeta

Subgroups — #41006656

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	C₃×#15688530	C₃	1	1	yes	maximal, characteristic
2	#22582652	C₃	1	2	yes	maximal, complemented
3	#22582651	C₃	1	2	yes	maximal, complemented
4	#22582651	C₃	1	2	yes	maximal, complemented
5	#22582642	C₃	1	1	yes	maximal, characteristic
6	#22582642	C₃	1	1	yes	maximal, characteristic
7	C₃×#15688520	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃×#15688520	C₃	1	2	yes	maximal, complemented
9	C₃₇×#6910	C₃	1	1	yes	maximal, characteristic
10	#15688530	—	3	3	no	—
11	#15688530	—	3	6	no	—
12	C₃×#11758608	—	3	2	no	—
13	C₃×#11758608	C₃²	1	1	yes	characteristic
14	C₃×#11758608	C₃²	1	2	yes	—
15	C₃×#11758608	C₃²	1	2	yes	—
16	C₃×#11758608	C₃²	1	1	yes	characteristic
17	C₃×#11758608	C₃²	1	2	yes	—
18	C₃²×#66524	—	3	1	no	—
19	C₃²×#66524	C₃²	1	1	yes	characteristic
20	#15688521	—	3	6	no	—
21	#15688520	—	3	6	no	—
22	#15688520	—	3	6	no	—
23	#15688520	—	3	2	no	—
24	#15688520	—	3	1	no	—
25	#15688520	—	3	4	no	—
26	#15688520	—	3	2	no	—
27	#15688515	—	3	3	no	—
28	#15688515	—	3	3	no	—
29	C₃×3₋¹⁺²×C₃₇	C₃²	1	1	yes	characteristic
30	C₃×3₋¹⁺²×C₃₇	C₃²	1	2	yes	—
31	C₃×H(3,𝔽₃)×C₃₇	C₃²	1	1	yes	characteristic
32	C₃₇×#731	—	3	3	no	—
33	C₃³×C₃₇	—	3	1	no	—
34	C₃³×C₃₇	C₃³	1	1	yes	characteristic
35	3₋¹⁺²×C₃₇	—	3	2	no	—
36	3₋¹⁺²×C₃₇	—	3	2	no	—
37	3₋¹⁺²×C₃₇	—	3	2	no	—
38	3₋¹⁺²×C₃₇	—	3	1	no	—
39	3₋¹⁺²×C₃₇	—	3	2	no	—
40	H(3,𝔽₃)×C₃₇	—	3	2	no	—
41	H(3,𝔽₃)×C₃₇	H(3,𝔽₃)	1	3	yes	—
42	C₃×C₉×C₃₇	—	3	1	no	—
43	C₃×C₉×C₃₇	—	3	2	no	—
44	#11758608	—	3	18	no	—
45	#11758608	—	3	2	no	—
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	1	no	—
52	#11758608	—	3	2	no	—
53	#11758608	—	3	2	no	—
54	#11758608	—	3	1	no	—
55	#11758608	—	3	2	no	—
56	#11758608	—	3	2	no	—
57	#11758608	H(3,𝔽₃)	1	6	yes	—
58	C₃×#66524	—	3	9	no	—
59	C₃×#66524	—	3	2	no	—
60	C₃×#66524	—	3	2	no	—
61	C₃×#66524	—	3	1	no	—
62	C₃×#66524	—	3	1	no	—
63	C₃×#66524	—	3	1	no	—
64	C₃×#66524	—	3	2	no	—
65	C₃×#66524	—	3	2	no	—
66	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
67	C₃×#66524	H(3,𝔽₃)	1	2	yes	—
68	#10624758	—	37	1	no	maximal, Sylow
69	C₃²×C₃₇	—	3	3	no	—
70	C₃²×C₃₇	—	9	1	no	—
71	C₃²×C₃₇	—	9	3	no	—
72	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	2	yes	—
73	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
74	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
75	C₉×C₃₇	—	9	1	no	—
76	C₉×C₃₇	—	9	2	no	—
77	#66524	—	3	1	no	—
78	#66524	—	3	6	no	—
79	#66524	—	3	2	no	—
80	#66524	—	9	1	no	—
81	#66524	—	9	1	no	—
82	#66524	—	9	2	no	—
83	#66524	—	9	2	no	—
84	C₃×#737	—	37	1	no	—
85	#6912	—	37	2	no	—
86	#6912	—	37	2	no	—
87	#6912	—	37	2	no	—
88	#6910	—	37	1	no	—
89	#6910	—	37	1	no	—
90	#6910	C₃₇⋊C₃	1	1	yes	characteristic
91	C₃×#733	—	37	2	no	—
92	C₃×#733	—	37	1	no	—
93	C₃×C₃₇	—	3	1	no	—
94	C₃×C₃₇	—	3	2	no	—
95	C₃×C₃₇	—	3	1	no	—
96	C₃×C₃₇	—	9	3	no	—
97	C₃×C₃₇	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
98	#737	—	111	6	no	—
99	#737	—	111	3	no	—
100	C₃×3₋¹⁺²	—	37	2	no	—
101	C₃×3₋¹⁺²	—	37	2	no	—
102	C₃×3₋¹⁺²	—	37	2	no	—
103	C₃×3₋¹⁺²	—	37	1	no	—
104	C₃×3₋¹⁺²	—	37	1	no	—
105	C₃×3₋¹⁺²	—	111	2	no	—
106	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	2	yes	—
107	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
108	C₃×H(3,𝔽₃)	C₃×C₃₇⋊C₃	1	1	yes	characteristic
109	C₃²×C₉	—	37	1	no	—
110	C₃²×C₉	—	111	1	no	—
111	#733	—	111	6	no	—
112	#733	—	111	2	no	—
113	#733	—	111	1	no	—
114	#733	—	111	4	no	—
115	#733	—	111	2	no	—
116	#733	—	111	6	no	—
117	#733	—	111	6	no	—
118	#731	—	3	3	no	—
119	#731	—	111	3	no	—
120	#731	—	111	3	no	—
121	C₃₇	#10624758	1	1	yes	characteristic, complemented, Sylow
122	C₃³	—	3	1	no	—
123	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
124	3₋¹⁺²	—	3	2	no	—
125	3₋¹⁺²	—	3	2	no	—
126	3₋¹⁺²	—	3	2	no	—
127	3₋¹⁺²	—	3	1	no	—
128	3₋¹⁺²	—	3	2	no	—
129	3₋¹⁺²	—	37	6	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	1	no	—
137	3₋¹⁺²	—	111	1	no	—
138	3₋¹⁺²	—	111	2	no	—
139	3₋¹⁺²	—	111	2	no	—
140	3₋¹⁺²	—	111	18	no	—
141	3₋¹⁺²	—	111	2	no	—
142	3₋¹⁺²	—	111	2	no	—
143	H(3,𝔽₃)	—	3	2	no	—
144	H(3,𝔽₃)	C₃²⋊(C₃₇⋊C₃)	1	3	yes	—
145	C₃×C₉	—	3	2	no	—
146	C₃×C₉	—	3	1	no	—
147	C₃×C₉	—	37	1	no	—
148	C₃×C₉	—	37	2	no	—
149	C₃×C₉	—	111	2	no	—
150	C₃×C₉	—	111	2	no	—
151	C₃×C₉	—	111	1	no	—
152	C₃×C₉	—	111	1	no	—
153	C₃×C₉	—	111	9	no	—
154	C₃×C₉	—	111	2	no	—
155	C₃×C₉	—	111	2	no	—
156	C₃×C₉	—	111	1	no	—
157	C₃²	—	3	3	no	—
158	C₃²	—	9	1	no	—
159	C₃²	—	9	3	no	—
160	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	2	yes	—
161	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
162	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
163	C₉	—	9	2	no	—
164	C₉	—	9	1	no	—
165	C₉	—	333	2	no	—
166	C₉	—	333	2	no	—
167	C₉	—	333	1	no	—
168	C₉	—	333	1	no	—
169	C₉	—	111	1	no	—
170	C₉	—	111	2	no	—
171	C₉	—	111	6	no	—
172	C₃	—	3	1	no	—
173	C₃	—	3	2	no	—
174	C₃	—	3	1	no	—
175	C₃	—	9	3	no	—
176	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	C₃₇×#6910	C₃
socle	C₃×C₃₇	C₃²⋊H(3,𝔽₃)
prime radical	C₃×H(3,𝔽₃)×C₃₇	C₃²
hypercenter	#6910	C₃₇⋊C₃
hypocenter	C₃₇	#10624758
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ #6910

Complements

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

N	H ≅ G/N	type
2. #22582652	C₃	semidirect
3. #22582651	C₃	semidirect
4. #22582651	C₃	semidirect
7. C₃×#15688520	C₃	semidirect
8. C₃×#15688520	C₃	semidirect
121. C₃₇	#10624758	semidirect

Maximal subgroups

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

subgroup	index
C₃×#15688530	3
#22582652	3
#22582651	3
#22582642	3
C₃×#15688520	3
C₃₇×#6910	3
#10624758	37