Finite Group Databasebeta

Subgroups — #41005317

Order 26973.

220 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 52 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	#22582548	C₃	1	1	yes	maximal, characteristic, complemented
2	3₋¹⁺²×#66524	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582539	C₃	1	1	yes	maximal, characteristic
4	#22582535	C₃	1	1	yes	maximal, characteristic, complemented
5	C₃₇⋊#6862	C₃	1	1	yes	maximal, characteristic, complemented
6	#22582458	C₃	1	1	yes	maximal, characteristic, complemented
7	#22582453	C₃	1	1	yes	maximal, characteristic
8	#22582452	C₃	1	1	yes	maximal, characteristic
9	#22582450	C₃	1	1	yes	maximal, characteristic, complemented
10	#22582450	C₃	1	1	yes	maximal, characteristic, complemented
11	#22582447	C₃	1	1	yes	maximal, characteristic, complemented
12	#22582444	C₃	1	1	yes	maximal, characteristic
13	C₃₇×#6858	C₃	1	1	yes	maximal, characteristic, complemented
14	C₃×#11758608	C₃²	1	1	yes	characteristic
15	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	1	yes	characteristic
16	C₃²×#66524	C₃²	1	1	yes	characteristic
17	C₃×C₉×C₃₇⋊C₃	—	3	1	no	—
18	#15688506	—	3	1	no	—
19	#15688505	—	3	1	no	—
20	#15688504	—	3	1	no	—
21	#15688504	—	3	1	no	—
22	#15688504	—	3	1	no	—
23	#15688504	—	3	1	no	—
24	C₃²⋊#66524	C₃²	1	1	yes	characteristic
25	C₃²⋊#66524	C₃²	1	1	yes	characteristic
26	C₃²⋊#66524	C₃²	1	1	yes	characteristic
27	C₃²⋊#66524	C₃²	1	1	yes	characteristic
28	C₃²⋊#66524	C₃²	1	1	yes	characteristic
29	C₃²⋊#66524	C₃²	1	1	yes	characteristic
30	#15688502	—	3	1	no	—
31	#15688502	—	3	1	no	—
32	C₉×#66524	—	3	1	no	—
33	C₉×#66524	—	3	1	no	—
34	C₉×#66524	—	3	1	no	—
35	C₃×3₋¹⁺²×C₃₇	C₃²	1	1	yes	characteristic
36	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
37	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
38	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
39	C₃³×C₃₇	C₃³	1	1	yes	characteristic
40	C₃²×C₃₇⋊C₃	—	3	1	no	—
41	3₋¹⁺²×C₃₇	—	3	3	no	—
42	C₃×C₉×C₃₇	—	3	1	no	—
43	C₃×C₉×C₃₇	—	3	1	no	—
44	C₃×C₉×C₃₇	—	3	1	no	—
45	C₃×C₉×C₃₇	—	3	1	no	—
46	C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
47	#11758608	—	3	3	no	—
48	C₃×#66524	—	3	1	no	—
49	C₃×#66524	—	3	3	no	—
50	C₃×#66524	—	3	1	no	—
51	C₃×#66524	—	3	1	no	—
52	C₃×#66524	—	3	1	no	—
53	C₃×#66524	—	3	1	no	—
54	C₃×#66524	—	3	1	no	—
55	C₃×#66524	—	3	1	no	—
56	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
57	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
58	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
59	C₉×C₃₇⋊C₃	—	9	3	no	—
60	#10624587	—	37	1	no	maximal, Sylow
61	C₃²×C₃₇	—	3	1	no	—
62	C₃²×C₃₇	—	3	1	no	—
63	C₃²×C₃₇	—	3	1	no	—
64	C₃²×C₃₇	—	3	1	no	—
65	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
66	C₃×C₃₇⋊C₃	—	3	3	no	—
67	C₃×C₃₇⋊C₃	—	9	1	no	—
68	C₃×C₃₇⋊C₃	—	9	1	no	—
69	C₃×C₃₇⋊C₃	—	9	1	no	—
70	C₉×C₃₇	—	3	3	no	—
71	C₉×C₃₇	—	9	1	no	—
72	C₉×C₃₇	—	9	1	no	—
73	C₉×C₃₇	—	9	1	no	—
74	#66524	—	3	1	no	—
75	#66524	—	3	1	no	—
76	#66524	—	3	3	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	#731	1	3	yes	—
84	C₉×3₋¹⁺²	—	37	1	no	—
85	C₉×3₋¹⁺²	—	37	1	no	—
86	C₉×H(3,𝔽₃)	—	37	1	no	—
87	#6887	—	37	1	no	—
88	#6862	—	37	1	no	—
89	#6861	—	37	1	no	—
90	#6859	—	37	1	no	—
91	#6859	—	37	1	no	—
92	#6858	—	37	1	no	—
93	#6858	—	37	1	no	—
94	#6858	—	37	1	no	—
95	#6858	C₃₇⋊C₃	1	1	yes	characteristic, complemented
96	#6857	—	37	1	no	—
97	C₃×C₃₇	—	9	1	no	—
98	C₃×C₃₇	C₃×#731	1	1	yes	characteristic
99	C₃×C₃₇	#6908	1	1	yes	characteristic
100	C₃×C₃₇	#6908	1	1	yes	characteristic
101	C₃×C₃₇	#6908	1	1	yes	characteristic
102	C₃₇⋊C₃	—	9	3	no	—
103	C₃×3₋¹⁺²	—	37	1	no	—
104	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
105	C₃×H(3,𝔽₃)	—	37	1	no	—
106	C₃²×C₉	—	37	1	no	—
107	C₃²×C₉	—	111	1	no	—
108	C₉⋊C₉	—	111	1	no	—
109	C₉⋊C₉	—	111	1	no	—
110	C₉⋊C₉	—	111	1	no	—
111	C₉⋊C₉	—	111	1	no	—
112	C₉⋊C₉	—	111	1	no	—
113	C₉⋊C₉	—	111	1	no	—
114	C₃²⋊C₉	—	37	1	no	—
115	C₃²⋊C₉	—	37	1	no	—
116	C₃²⋊C₉	—	37	1	no	—
117	C₃²⋊C₉	—	37	1	no	—
118	C₃²⋊C₉	—	37	1	no	—
119	C₃²⋊C₉	—	37	1	no	—
120	C₃²⋊C₉	—	111	1	no	—
121	C₃²⋊C₉	—	111	1	no	—
122	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
123	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
124	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
125	C₉²	—	111	1	no	—
126	C₉²	—	111	1	no	—
127	C₉²	—	111	1	no	—
128	C₃₇	#10624587	1	1	yes	characteristic, complemented, Sylow
129	C₃³	—	111	1	no	—
130	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
131	3₋¹⁺²	—	3	3	no	—
132	3₋¹⁺²	—	111	3	no	—
133	H(3,𝔽₃)	—	111	3	no	—
134	C₃×C₉	—	3	1	no	—
135	C₃×C₉	—	3	1	no	—
136	C₃×C₉	—	3	1	no	—
137	C₃×C₉	—	3	1	no	—
138	C₃×C₉	—	37	1	no	—
139	C₃×C₉	—	37	1	no	—
140	C₃×C₉	—	37	1	no	—
141	C₃×C₉	—	333	3	no	—
142	C₃×C₉	—	111	3	no	—
143	C₃×C₉	—	111	1	no	—
144	C₃×C₉	—	111	1	no	—
145	C₃×C₉	—	111	1	no	—
146	C₃×C₉	—	111	1	no	—
147	C₃×C₉	—	111	1	no	—
148	C₃×C₉	—	111	1	no	—
149	C₃×C₉	—	111	1	no	—
150	C₃²	—	3	1	no	—
151	C₃²	—	3	1	no	—
152	C₃²	—	3	1	no	—
153	C₃²	—	3	1	no	—
154	C₃²	—	333	1	no	—
155	C₃²	—	333	1	no	—
156	C₃²	—	333	1	no	—
157	C₃²	—	111	3	no	—
158	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
159	C₉	—	3	3	no	—
160	C₉	—	9	1	no	—
161	C₉	—	9	1	no	—
162	C₉	—	9	1	no	—
163	C₉	—	37	3	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	C₉	—	111	3	no	—
173	C₃	—	9	1	no	—
174	C₃	—	333	3	no	—
175	C₃	#731×C₃₇⋊C₃	1	1	yes	characteristic
176	C₃	#22582629	1	1	yes	characteristic
177	C₃	#22582632	1	1	yes	characteristic
178	C₃	#22582634	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₃₇⋊C₃
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6858	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²
hypercenter	#6858	C₃₇⋊C₃
hypocenter	C₃₇	#10624587
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ #6858

Complements

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

N	H ≅ G/N	type
1. #22582548	C₃	semidirect
2. 3₋¹⁺²×#66524	C₃	semidirect
4. #22582535	C₃	semidirect
5. C₃₇⋊#6862	C₃	semidirect
6. #22582458	C₃	semidirect
9. #22582450	C₃	semidirect
10. #22582450	C₃	semidirect
11. #22582447	C₃	semidirect
13. C₃₇×#6858	C₃	semidirect
95. #6858	C₃₇⋊C₃	semidirect
128. C₃₇	#10624587	semidirect

Maximal subgroups

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

subgroup	index
#22582548	3
3₋¹⁺²×#66524	3
#22582539	3
#22582535	3
C₃₇⋊#6862	3
#22582458	3
#22582453	3
#22582452	3
#22582450	3
#22582447	3
#22582444	3
C₃₇×#6858	3
#10624587	37