Finite Group Databasebeta

Subgroups — #41007159

Order 26973.

238 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 58 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	#22582578	C₃	1	1	yes	maximal, characteristic, complemented
2	#22582578	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×#15688502	C₃	1	1	yes	maximal, characteristic
4	#22582471	C₃	1	1	yes	maximal, characteristic, complemented
5	#22582471	C₃	1	1	yes	maximal, characteristic, complemented
6	#22582471	C₃	1	1	yes	maximal, characteristic, complemented
7	#22582469	C₃	1	1	yes	maximal, characteristic, complemented
8	#22582469	C₃	1	1	yes	maximal, characteristic, complemented
9	#22582469	C₃	1	1	yes	maximal, characteristic, complemented
10	#22582467	C₃	1	1	yes	maximal, characteristic
11	#22582467	C₃	1	1	yes	maximal, characteristic
12	#22582467	C₃	1	1	yes	maximal, characteristic
13	C₃₇×#6887	C₃	1	1	yes	maximal, characteristic, complemented
14	C₃³×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
15	C₃²×#66524	C₃²	1	1	yes	characteristic
16	C₃²×#66524	C₃²	1	1	yes	characteristic
17	#15688504	—	3	1	no	—
18	#15688504	—	3	1	no	—
19	#15688504	—	3	2	no	—
20	#15688504	—	3	2	no	—
21	C₃²⋊#66524	C₃²	1	1	yes	characteristic
22	C₃²⋊#66524	C₃²	1	1	yes	characteristic
23	C₃²⋊#66524	C₃²	1	1	yes	characteristic
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	#15688502	—	3	1	no	—
28	#15688502	—	3	2	no	—
29	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
30	C₃₇×C₉⋊C₉	—	3	2	no	—
31	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
32	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
33	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
34	C₉²×C₃₇	—	3	1	no	—
35	C₃³×C₃₇	C₃³	1	1	yes	characteristic
36	C₃²×C₃₇⋊C₃	—	3	1	no	—
37	C₃²×C₃₇⋊C₃	—	3	3	no	—
38	C₃²×C₃₇⋊C₃	—	3	3	no	—
39	C₃²×C₃₇⋊C₃	—	9	1	no	—
40	C₃²×C₃₇⋊C₃	—	9	1	no	—
41	C₃×C₉×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	H(3,𝔽₃)	1	1	yes	characteristic
46	C₃×C₉×C₃₇	H(3,𝔽₃)	1	2	yes	—
47	C₃×#66524	—	3	1	no	—
48	C₃×#66524	—	3	1	no	—
49	C₃×#66524	—	3	1	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	3	no	—
56	C₃×#66524	—	3	3	no	—
57	#10624810	—	37	1	no	maximal, Sylow
58	C₃²×C₃₇	—	3	1	no	—
59	C₃²×C₃₇	—	3	1	no	—
60	C₃²×C₃₇	—	3	1	no	—
61	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
62	C₃²×C₃₇	#737	1	3	yes	—
63	C₃×C₃₇⋊C₃	—	3	3	no	—
64	C₃×C₃₇⋊C₃	—	9	1	no	—
65	C₃×C₃₇⋊C₃	—	9	1	no	—
66	C₃×C₃₇⋊C₃	—	9	1	no	—
67	C₃×C₃₇⋊C₃	—	9	9	no	—
68	C₉×C₃₇	—	3	1	no	—
69	C₉×C₃₇	—	3	2	no	—
70	C₉×C₃₇	—	9	1	no	—
71	C₉×C₃₇	—	9	1	no	—
72	C₉×C₃₇	—	9	1	no	—
73	#66524	—	9	1	no	—
74	#66524	—	9	1	no	—
75	#66524	—	9	1	no	—
76	#66524	—	9	1	no	—
77	#66524	—	9	1	no	—
78	#66524	—	9	1	no	—
79	#66524	—	9	1	no	—
80	#66524	—	9	1	no	—
81	#6895	—	37	1	no	—
82	#6895	—	37	1	no	—
83	#6887	C₃₇⋊C₃	1	1	yes	characteristic, complemented
84	C₃×C₃²⋊C₉	—	37	1	no	—
85	#6868	—	37	1	no	—
86	#6868	—	37	1	no	—
87	#6868	—	37	1	no	—
88	#6867	—	37	1	no	—
89	#6867	—	37	1	no	—
90	#6867	—	37	1	no	—
91	#6866	—	37	1	no	—
92	#6866	—	37	1	no	—
93	#6866	—	37	1	no	—
94	C₃×C₃₇	—	3	3	no	—
95	C₃×C₃₇	#6893	1	1	yes	characteristic
96	C₃×C₃₇	#6908	1	1	yes	characteristic
97	C₃×C₃₇	#6913	1	1	yes	characteristic
98	C₃×C₃₇	#6913	1	1	yes	characteristic
99	C₃₇⋊C₃	—	9	3	no	—
100	C₃⁴	—	37	1	no	—
101	C₃²×C₉	—	37	1	no	—
102	C₃²×C₉	—	37	1	no	—
103	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
104	C₉⋊C₉	—	3	2	no	—
105	C₉⋊C₉	—	111	1	no	—
106	C₉⋊C₉	—	111	1	no	—
107	C₉⋊C₉	—	111	2	no	—
108	C₉⋊C₉	—	111	2	no	—
109	C₃²⋊C₉	—	37	1	no	—
110	C₃²⋊C₉	—	37	1	no	—
111	C₃²⋊C₉	—	37	1	no	—
112	C₃²⋊C₉	—	37	1	no	—
113	C₃²⋊C₉	—	37	1	no	—
114	C₃²⋊C₉	—	37	1	no	—
115	C₃²⋊C₉	—	111	1	no	—
116	C₃²⋊C₉	—	111	2	no	—
117	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
118	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
119	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
120	C₉²	—	3	1	no	—
121	C₃₇	#10624810	1	1	yes	characteristic, complemented, Sylow
122	C₃³	—	333	1	no	—
123	C₃³	—	333	1	no	—
124	C₃³	—	111	3	no	—
125	C₃³	—	111	3	no	—
126	C₃³	—	111	1	no	—
127	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
128	C₃×C₉	—	3	3	no	—
129	C₃×C₉	—	3	1	no	—
130	C₃×C₉	—	3	1	no	—
131	C₃×C₉	—	3	1	no	—
132	C₃×C₉	—	111	1	no	—
133	C₃×C₉	—	111	1	no	—
134	C₃×C₉	—	111	1	no	—
135	C₃×C₉	—	111	1	no	—
136	C₃×C₉	—	111	1	no	—
137	C₃×C₉	—	111	1	no	—
138	C₃×C₉	—	111	1	no	—
139	C₃×C₉	—	111	1	no	—
140	C₃×C₉	—	111	3	no	—
141	C₃×C₉	—	111	3	no	—
142	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
143	C₃×C₉	C₃²⋊(C₃₇⋊C₃)	1	2	yes	—
144	C₃²	—	3	1	no	—
145	C₃²	—	3	1	no	—
146	C₃²	—	3	1	no	—
147	C₃²	—	333	9	no	—
148	C₃²	—	333	1	no	—
149	C₃²	—	333	1	no	—
150	C₃²	—	333	1	no	—
151	C₃²	—	111	3	no	—
152	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
153	C₃²	#15688531	1	3	yes	—
154	C₉	—	3	1	no	—
155	C₉	—	3	2	no	—
156	C₉	—	9	1	no	—
157	C₉	—	9	1	no	—
158	C₉	—	9	1	no	—
159	C₉	—	333	1	no	—
160	C₉	—	333	1	no	—
161	C₉	—	333	1	no	—
162	C₉	—	333	1	no	—
163	C₉	—	333	1	no	—
164	C₉	—	333	1	no	—
165	C₉	—	333	1	no	—
166	C₉	—	333	1	no	—
167	C₃	—	3	3	no	—
168	C₃	—	333	3	no	—
169	C₃	#22582564	1	1	yes	characteristic
170	C₃	#22582630	1	1	yes	characteristic
171	C₃	#22582655	1	1	yes	characteristic
172	C₃	#22582656	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	C₃×C₃²⋊(C₃₇⋊C₃)
commutator, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6887	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
prime radical	C₃³×C₃₇⋊C₃	C₃²
hypercenter	#6887	C₃₇⋊C₃
hypocenter	C₃₇	#10624810
LCS	G ▷ C₃³×C₃₇ ▷ C₃×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃²×C₉ ◁ #6887

Complements

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

N	H ≅ G/N	type
1. #22582578	C₃	semidirect
2. #22582578	C₃	semidirect
4. #22582471	C₃	semidirect
5. #22582471	C₃	semidirect
6. #22582471	C₃	semidirect
7. #22582469	C₃	semidirect
8. #22582469	C₃	semidirect
9. #22582469	C₃	semidirect
13. C₃₇×#6887	C₃	semidirect
83. #6887	C₃₇⋊C₃	semidirect
121. C₃₇	#10624810	semidirect

Maximal subgroups

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

subgroup	index
#22582578	3
C₃×#15688502	3
#22582471	3
#22582469	3
#22582467	3
C₃₇×#6887	3
#10624810	37