Finite Group Databasebeta

Subgroups — #41007425

Order 26973.

188 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 43 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	#22582663	C₃	1	1	yes	maximal, characteristic, complemented
2	#22582658	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582650	C₃	1	1	yes	maximal, characteristic, complemented
4	#22582603	C₃	1	1	yes	maximal, characteristic, complemented
5	#22582524	C₃	1	1	yes	maximal, characteristic, complemented
6	#22582521	C₃	1	1	yes	maximal, characteristic, complemented
7	#22582519	C₃	1	1	yes	maximal, characteristic, complemented
8	#22582516	C₃	1	1	yes	maximal, characteristic, complemented
9	#22582516	C₃	1	1	yes	maximal, characteristic, complemented
10	#22582511	C₃	1	1	yes	maximal, characteristic, complemented
11	#22582510	C₃	1	1	yes	maximal, characteristic, complemented
12	#22582503	C₃	1	1	yes	maximal, characteristic, complemented
13	C₃₇×#6878	C₃	1	1	yes	maximal, characteristic, complemented
14	#15688532	—	3	1	no	—
15	#15688530	—	3	1	no	—
16	C₃×#11758606	—	3	1	no	—
17	C₃×#11758606	C₃²	1	1	yes	characteristic
18	3₋¹⁺²×C₃₇⋊C₃	—	3	1	no	—
19	#15688521	C₃²	1	1	yes	characteristic
20	#15688521	C₃²	1	1	yes	characteristic
21	#15688521	C₃²	1	1	yes	characteristic
22	#15688521	C₃²	1	1	yes	characteristic
23	#15688521	C₃²	1	1	yes	characteristic
24	#15688520	—	3	1	no	—
25	#15688520	—	3	1	no	—
26	#15688519	C₃²	1	1	yes	characteristic
27	#15688517	—	3	1	no	—
28	#15688515	—	3	1	no	—
29	#15688514	—	3	1	no	—
30	#15688512	—	3	1	no	—
31	#15688506	—	3	1	no	—
32	#15688506	—	3	1	no	—
33	#15688506	C₃²	1	1	yes	characteristic
34	#15688506	C₃²	1	1	yes	characteristic
35	C₃₇×#733	C₃²	1	1	yes	characteristic
36	C₃₇×#733	C₃²	1	1	yes	characteristic
37	C₃₇×#732	C₃²	1	1	yes	characteristic
38	C₉²×C₃₇	C₃²	1	1	yes	characteristic
39	C₃²×C₃₇⋊C₃	H(3,𝔽₃)	1	1	yes	characteristic
40	3₋¹⁺²×C₃₇	—	3	1	no	—
41	3₋¹⁺²×C₃₇	—	3	1	no	—
42	H(3,𝔽₃)×C₃₇	—	3	1	no	—
43	C₃×C₉×C₃₇	—	3	1	no	—
44	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
45	C₃²⋊(C₃₇⋊C₃)	—	3	1	no	—
46	#11758608	—	3	1	no	—
47	#11758608	—	3	1	no	—
48	#11758608	—	3	1	no	—
49	#11758608	—	3	1	no	—
50	#11758608	—	3	1	no	—
51	#11758608	—	9	2	no	—
52	C₃×#66524	—	3	1	no	—
53	C₃×#66524	—	3	1	no	—
54	C₃×#66524	—	9	1	no	—
55	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
56	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
57	#11758606	—	9	1	no	—
58	#11758606	—	9	1	no	—
59	#11758606	—	9	3	no	—
60	#11758606	—	9	1	no	—
61	#11758605	—	9	1	no	—
62	#11758605	—	9	1	no	—
63	C₉×C₃₇⋊C₃	—	9	1	no	—
64	C₉×C₃₇⋊C₃	—	9	1	no	—
65	#10624852	—	37	1	no	maximal, Sylow
66	C₃²×C₃₇	—	9	1	no	—
67	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
68	C₃×C₃₇⋊C₃	—	3	1	no	—
69	C₃×C₃₇⋊C₃	—	9	1	no	—
70	C₃×C₃₇⋊C₃	—	9	1	no	—
71	C₉×C₃₇	—	3	3	no	—
72	C₉×C₃₇	—	3	1	no	—
73	C₉×C₃₇	—	9	1	no	—
74	C₉×C₃₇	—	9	1	no	—
75	#66524	—	3	1	no	—
76	#66524	—	3	1	no	—
77	#66524	—	9	1	no	—
78	#66524	—	9	1	no	—
79	#66524	—	9	1	no	—
80	#66524	—	9	1	no	—
81	#66524	—	9	1	no	—
82	#66524	—	9	1	no	—
83	#66524	—	9	1	no	—
84	#6913	—	37	1	no	—
85	#6913	—	37	1	no	—
86	#6912	—	37	1	no	—
87	#6900	—	37	1	no	—
88	#6883	—	37	1	no	—
89	#6883	—	37	1	no	—
90	#6882	—	37	1	no	—
91	#6882	—	37	1	no	—
92	#6882	—	37	1	no	—
93	#6881	—	37	1	no	—
94	#6880	—	37	1	no	—
95	#6878	—	37	1	no	—
96	#6878	C₃₇⋊C₃	1	1	yes	characteristic, complemented
97	C₃×C₃₇	—	3	1	no	—
98	C₃×C₃₇	—	27	1	no	—
99	C₃×C₃₇	#6911	1	1	yes	characteristic
100	C₃₇⋊C₃	—	9	1	no	—
101	C₃₇⋊C₃	—	27	1	no	—
102	#737	—	111	1	no	—
103	#737	—	111	1	no	—
104	C₃×3₋¹⁺²	—	37	1	no	—
105	C₃×3₋¹⁺²	—	111	1	no	—
106	C₃×3₋¹⁺²	—	111	1	no	—
107	#733	—	37	1	no	—
108	#733	—	37	1	no	—
109	#733	—	37	1	no	—
110	#733	—	37	1	no	—
111	#733	—	37	1	no	—
112	#733	—	111	1	no	—
113	#733	—	111	1	no	—
114	#733	C₃×C₃₇⋊C₃	1	1	yes	characteristic
115	#733	C₃×C₃₇⋊C₃	1	1	yes	characteristic
116	#732	—	37	1	no	—
117	#732	C₃×C₃₇⋊C₃	1	1	yes	characteristic
118	#731	—	111	1	no	—
119	#731	—	111	1	no	—
120	AΣL(1,𝔽₂₇)	—	111	1	no	—
121	AΣL(1,𝔽₂₇)	—	111	1	no	—
122	C₉⋊C₉	—	37	1	no	—
123	C₉⋊C₉	—	37	1	no	—
124	C₉⋊C₉	—	111	1	no	—
125	C₉⋊C₉	—	111	1	no	—
126	C₉²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
127	C₃₇	#10624852	1	1	yes	characteristic, complemented, Sylow
128	C₃³	—	37	1	no	—
129	3₋¹⁺²	—	3	1	no	—
130	3₋¹⁺²	—	3	1	no	—
131	3₋¹⁺²	—	333	3	no	—
132	3₋¹⁺²	—	333	1	no	—
133	3₋¹⁺²	—	333	1	no	—
134	3₋¹⁺²	—	333	1	no	—
135	3₋¹⁺²	—	333	1	no	—
136	3₋¹⁺²	—	333	1	no	—
137	3₋¹⁺²	—	333	2	no	—
138	3₋¹⁺²	—	111	1	no	—
139	3₋¹⁺²	—	111	1	no	—
140	3₋¹⁺²	—	111	1	no	—
141	3₋¹⁺²	—	111	1	no	—
142	3₋¹⁺²	—	111	1	no	—
143	H(3,𝔽₃)	—	3	1	no	—
144	H(3,𝔽₃)	—	111	1	no	—
145	C₃×C₉	—	3	1	no	—
146	C₃×C₉	—	37	1	no	—
147	C₃×C₉	—	37	1	no	—
148	C₃×C₉	—	333	1	no	—
149	C₃×C₉	—	333	1	no	—
150	C₃×C₉	—	333	1	no	—
151	C₃×C₉	—	111	1	no	—
152	C₃×C₉	—	111	1	no	—
153	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
154	C₃²	—	9	1	no	—
155	C₃²	—	333	1	no	—
156	C₃²	—	333	1	no	—
157	C₃²	—	111	1	no	—
158	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
159	C₉	—	3	3	no	—
160	C₉	—	3	1	no	—
161	C₉	—	9	1	no	—
162	C₉	—	9	1	no	—
163	C₉	—	333	1	no	—
164	C₉	—	333	1	no	—
165	C₉	—	333	1	no	—
166	C₉	—	333	1	no	—
167	C₉	—	333	1	no	—
168	C₉	—	333	1	no	—
169	C₉	—	333	1	no	—
170	C₉	—	111	1	no	—
171	C₉	—	111	1	no	—
172	C₃	—	3	1	no	—
173	C₃	—	27	1	no	—
174	C₃	—	333	1	no	—
175	C₃	—	999	1	no	—
176	C₃	#22582646	1	1	yes	characteristic

	subgroup	G/H
center	C₃	#22582646
commutator, cosocle	C₃×C₉×C₃₇	C₃³
Frattini	C₃×C₉	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6878	C₃
socle	C₃×C₃₇	#6911
hypercenter	#6878	C₃₇⋊C₃
hypocenter	C₃₇	#10624852
LCS	G ▷ C₃×C₉×C₃₇ ▷ C₃²×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃×C₉ ◁ #6878

Complements

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

N	H ≅ G/N	type
1. #22582663	C₃	semidirect
2. #22582658	C₃	semidirect
3. #22582650	C₃	semidirect
4. #22582603	C₃	semidirect
5. #22582524	C₃	semidirect
6. #22582521	C₃	semidirect
7. #22582519	C₃	semidirect
8. #22582516	C₃	semidirect
9. #22582516	C₃	semidirect
10. #22582511	C₃	semidirect
11. #22582510	C₃	semidirect
12. #22582503	C₃	semidirect
13. C₃₇×#6878	C₃	semidirect
96. #6878	C₃₇⋊C₃	semidirect
127. C₃₇	#10624852	semidirect

Maximal subgroups

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

subgroup	index
#22582663	3
#22582658	3
#22582650	3
#22582603	3
#22582524	3
#22582521	3
#22582519	3
#22582516	3
#22582511	3
#22582510	3
#22582503	3
C₃₇×#6878	3
#10624852	37