Finite Group Databasebeta

Subgroups — #41008611

Order 26973.

196 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 75 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×(C₃₇⋊C₃)⋊C₂₇	C₃	1	1	yes	maximal, characteristic
2	#22582706	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582706	C₃	1	1	yes	maximal, characteristic, complemented
4	#22582705	C₃	1	1	yes	maximal, characteristic, complemented
5	#22582705	C₃	1	1	yes	maximal, characteristic, complemented
6	#22582704	C₃	1	1	yes	maximal, characteristic, complemented
7	#22582703	C₃	1	1	yes	maximal, characteristic, complemented
8	#22582690	C₃	1	1	yes	maximal, characteristic
9	#22582690	C₃	1	1	yes	maximal, characteristic
10	#22582688	C₃	1	1	yes	maximal, characteristic, complemented
11	#22582688	C₃	1	1	yes	maximal, characteristic, complemented
12	#22582688	C₃	1	1	yes	maximal, characteristic, complemented
13	#22582543	C₃	1	1	yes	maximal, characteristic
14	(C₃₇⋊C₃)⋊C₂₇	—	3	3	no	—
15	C₃×#11758603	C₃²	1	1	yes	characteristic
16	C₃×#11758603	C₃²	1	1	yes	characteristic
17	C₃×#11758603	C₃²	1	1	yes	characteristic
18	C₃×#11758603	C₃²	1	1	yes	characteristic
19	C₃×#11758603	C₃²	1	1	yes	characteristic
20	C₃×#11758603	C₃²	1	1	yes	characteristic
21	C₃×#11758603	C₃²	1	1	yes	characteristic
22	C₃×#11758603	C₃²	1	1	yes	characteristic
23	C₃×#11758603	C₃²	1	1	yes	characteristic
24	C₃×#11758605	C₉	1	1	yes	characteristic
25	C₃×C₉×C₃₇⋊C₃	C₃²	1	1	yes	characteristic
26	#15688506	C₉	1	1	yes	characteristic
27	#15688504	C₉	1	1	yes	characteristic
28	#15688504	C₉	1	1	yes	characteristic
29	#15688504	C₉	1	1	yes	characteristic
30	#15688504	C₉	1	1	yes	characteristic
31	#15688502	C₉	1	1	yes	characteristic
32	#15688502	C₉	1	1	yes	characteristic
33	C₉×#66524	C₃²	1	1	yes	characteristic
34	C₉×#66524	C₃²	1	1	yes	characteristic
35	C₉×#66524	C₃²	1	1	yes	characteristic
36	C₃₇×C₉⋊C₉	C₉	1	1	yes	characteristic
37	C₃²×C₃₇⋊C₃	C₃×C₉	1	1	yes	characteristic
38	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
39	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
40	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
41	C₃×C₉×C₃₇	C₃×C₉	1	1	yes	characteristic
42	C₃×#66524	C₃×C₉	1	1	yes	characteristic
43	C₃×#66524	C₃×C₉	1	1	yes	characteristic
44	C₃×#66524	C₃×C₉	1	1	yes	characteristic
45	C₃×#66524	C₃×C₉	1	1	yes	characteristic
46	C₃×#66524	C₃×C₉	1	1	yes	characteristic
47	C₃×#66524	C₃×C₉	1	1	yes	characteristic
48	C₃×#66524	C₃×C₉	1	1	yes	characteristic
49	C₃×#66524	C₃³	1	1	yes	characteristic
50	#11758605	—	3	3	no	—
51	C₉×C₃₇⋊C₃	—	3	1	no	—
52	C₉×C₃₇⋊C₃	—	3	1	no	—
53	C₉×C₃₇⋊C₃	—	3	1	no	—
54	#11758603	—	3	1	no	—
55	#11758603	—	3	1	no	—
56	#11758603	—	3	1	no	—
57	#11758603	—	3	1	no	—
58	#11758603	—	3	1	no	—
59	#11758603	—	3	1	no	—
60	#11758603	—	3	1	no	—
61	#11758603	—	3	1	no	—
62	#11758603	3₋¹⁺²	1	3	yes	complemented
63	#10624727	—	37	1	no	maximal, Sylow
64	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
65	C₃×C₃₇⋊C₃	—	3	1	no	—
66	C₃×C₃₇⋊C₃	—	3	1	no	—
67	C₃×C₃₇⋊C₃	—	3	1	no	—
68	C₃×C₃₇⋊C₃	C₂₇⋊C₃	1	3	yes	—
69	C₉×C₃₇	—	3	1	no	—
70	C₉×C₃₇	—	3	1	no	—
71	C₉×C₃₇	—	3	1	no	—
72	C₉×C₃₇	—	3	1	no	—
73	#66524	—	3	1	no	—
74	#66524	—	3	1	no	—
75	#66524	—	3	1	no	—
76	#66524	—	3	1	no	—
77	#66524	—	3	1	no	—
78	#66524	—	3	1	no	—
79	#66524	—	3	1	no	—
80	#66524	C₃×3₋¹⁺²	1	1	yes	characteristic
81	#66524	#737	1	1	yes	characteristic
82	#66524	#737	1	1	yes	characteristic
83	C₃²×C₂₇	—	37	1	no	—
84	C₉×3₋¹⁺²	—	37	1	no	—
85	C₉⋊C₂₇	—	37	1	no	—
86	C₉⋊C₂₇	—	37	1	no	—
87	C₉⋊C₂₇	—	37	1	no	—
88	C₉⋊C₂₇	—	37	1	no	—
89	C₉⋊C₂₇	—	37	1	no	—
90	C₉⋊C₂₇	—	37	1	no	—
91	C₃²⋊C₂₇	—	37	1	no	—
92	C₃²⋊C₂₇	—	37	1	no	—
93	#6864	—	37	1	no	—
94	#6864	—	37	1	no	—
95	#6864	—	37	1	no	—
96	C₃×C₃₇	C₉×3₋¹⁺²	1	1	yes	characteristic
97	C₃×C₃₇	C₃×C₂₇⋊C₃	1	1	yes	characteristic
98	C₃×C₃₇	#6903	1	1	yes	characteristic
99	C₃×C₃₇	#6903	1	1	yes	characteristic
100	C₃₇⋊C₃	—	3	3	no	—
101	C₃×3₋¹⁺²	—	37	1	no	—
102	C₃²×C₉	—	37	1	no	—
103	C₃×C₂₇	—	37	1	no	—
104	C₃×C₂₇	—	37	1	no	—
105	C₃×C₂₇	—	37	1	no	—
106	C₃×C₂₇	—	37	1	no	—
107	C₃×C₂₇	—	37	1	no	—
108	C₃×C₂₇	—	37	1	no	—
109	C₃×C₂₇	—	37	1	no	—
110	C₃×C₂₇	—	37	1	no	—
111	C₃×C₂₇	—	37	1	no	—
112	C₃×C₂₇	—	111	3	no	—
113	C₉⋊C₉	—	37	1	no	—
114	C₉⋊C₉	—	37	1	no	—
115	C₉⋊C₉	—	37	1	no	—
116	C₉⋊C₉	—	37	1	no	—
117	C₉⋊C₉	—	37	1	no	—
118	C₉⋊C₉	#66526	1	1	yes	characteristic
119	C₃²⋊C₉	—	37	1	no	—
120	C₃²⋊C₉	—	37	1	no	—
121	C₉²	—	37	1	no	—
122	C₉²	—	37	1	no	—
123	C₉²	—	37	1	no	—
124	C₃₇	#10624727	1	1	yes	characteristic, complemented, Sylow
125	C₃³	—	37	1	no	—
126	3₋¹⁺²	—	111	3	no	—
127	C₃×C₉	—	37	1	no	—
128	C₃×C₉	—	37	1	no	—
129	C₃×C₉	—	37	1	no	—
130	C₃×C₉	—	37	1	no	—
131	C₃×C₉	—	37	1	no	—
132	C₃×C₉	—	37	1	no	—
133	C₃×C₉	—	37	1	no	—
134	C₃×C₉	—	37	1	no	—
135	C₃×C₉	—	111	1	no	—
136	C₃×C₉	—	111	1	no	—
137	C₃×C₉	—	111	1	no	—
138	C₃×C₉	C₃×#66526	1	1	yes	characteristic
139	C₃×C₉	C₃×#66526	1	1	yes	characteristic
140	C₃×C₉	C₃×#66526	1	1	yes	characteristic
141	C₃×C₉	C₃×#66526	1	1	yes	characteristic
142	C₂₇	—	37	3	no	—
143	C₂₇	—	111	1	no	—
144	C₂₇	—	111	1	no	—
145	C₂₇	—	111	1	no	—
146	C₂₇	—	111	1	no	—
147	C₂₇	—	111	1	no	—
148	C₂₇	—	111	1	no	—
149	C₂₇	—	111	1	no	—
150	C₂₇	—	111	1	no	—
151	C₃²	—	37	3	no	—
152	C₃²	—	111	1	no	—
153	C₃²	—	111	1	no	—
154	C₃²	—	111	1	no	—
155	C₃²	C₃²×#66526	1	1	yes	characteristic
156	C₉	—	3	1	no	—
157	C₉	—	3	1	no	—
158	C₉	—	3	1	no	—
159	C₉	—	3	1	no	—
160	C₉	—	37	1	no	—
161	C₉	—	37	1	no	—
162	C₉	—	37	1	no	—
163	C₉	—	111	1	no	—
164	C₉	—	111	1	no	—
165	C₉	—	111	1	no	—
166	C₉	—	111	1	no	—
167	C₉	—	111	1	no	—
168	C₉	—	111	1	no	—
169	C₉	—	111	1	no	—
170	C₃	—	111	3	no	—
171	C₃	3₋¹⁺²×#66526	1	1	yes	characteristic
172	C₃	(C₃₇⋊C₃)⋊(C₂₇⋊C₃)	1	1	yes	characteristic
173	C₃	#22582732	1	1	yes	characteristic
174	C₃	#22582732	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₃²	C₃²×#66526
commutator, socle	C₃²×C₃₇	C₃²×C₉
Fitting	C₃₇×C₉⋊C₉	C₉
prime radical	C₃²×C₃₇⋊C₃	C₃×C₉
cosocle	C₃×#66524	C₃³
hypercenter	C₉⋊C₉	#66526
hypocenter	C₃₇	#10624727
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₉⋊C₉

Complements

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

N	H ≅ G/N	type
2. #22582706	C₃	semidirect
3. #22582706	C₃	semidirect
4. #22582705	C₃	semidirect
5. #22582705	C₃	semidirect
6. #22582704	C₃	semidirect
7. #22582703	C₃	semidirect
10. #22582688	C₃	semidirect
11. #22582688	C₃	semidirect
12. #22582688	C₃	semidirect
62. #11758603	3₋¹⁺²	semidirect
124. C₃₇	#10624727	semidirect

Maximal subgroups

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

subgroup	index
C₃×(C₃₇⋊C₃)⋊C₂₇	3
#22582706	3
#22582705	3
#22582704	3
#22582703	3
#22582690	3
#22582688	3
#22582543	3
#10624727	37