Finite Group Databasebeta

Subgroups — #41005322

Order 26973.

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

normal subgroups only

This group has 202 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
2	#22582548	C₃	1	1	yes	maximal, characteristic
3	#22582548	C₃	1	1	yes	maximal, characteristic
4	#22582535	C₃	1	1	yes	maximal, characteristic
5	#22582458	C₃	1	1	yes	maximal, characteristic
6	#22582456	C₃	1	1	yes	maximal, characteristic
7	#22582455	C₃	1	1	yes	maximal, characteristic
8	#22582455	C₃	1	1	yes	maximal, characteristic
9	#22582455	C₃	1	1	yes	maximal, characteristic
10	#22582450	C₃	1	1	yes	maximal, characteristic
11	#22582449	C₃	1	1	yes	maximal, characteristic
12	#22582448	C₃	1	1	yes	maximal, characteristic
13	C₃₇×#6861	C₃	1	1	yes	maximal, characteristic
14	C₃×#11758608	C₃²	1	1	yes	characteristic
15	C₃×#11758608	C₃²	1	1	yes	characteristic
16	C₃×#11758608	C₃²	1	1	yes	characteristic
17	C₃²×#66524	C₃²	1	1	yes	characteristic
18	#15688504	—	3	2	no	—
19	#15688504	—	3	2	no	—
20	#15688504	—	3	2	no	—
21	#15688504	—	3	2	no	—
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	C₉×#66524	—	3	1	no	—
28	C₉×#66524	—	3	1	no	—
29	C₉×#66524	—	3	1	no	—
30	C₉×#66524	—	3	1	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	C₃₇×C₃²⋊C₉	C₃²	1	1	yes	characteristic
35	C₃³×C₃₇	C₃³	1	1	yes	characteristic
36	C₃×C₉×C₃₇	—	3	1	no	—
37	C₃×C₉×C₃₇	—	3	1	no	—
38	C₃×C₉×C₃₇	—	3	1	no	—
39	C₃×C₉×C₃₇	—	3	1	no	—
40	#11758608	—	3	3	no	—
41	#11758608	—	3	3	no	—
42	#11758608	—	3	3	no	—
43	C₃×#66524	—	3	1	no	—
44	C₃×#66524	—	3	1	no	—
45	C₃×#66524	—	3	1	no	—
46	C₃×#66524	—	3	3	no	—
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	H(3,𝔽₃)	1	2	yes	—
53	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
54	#10624588	—	37	1	no	maximal, Sylow
55	C₃²×C₃₇	—	3	1	no	—
56	C₃²×C₃₇	—	3	1	no	—
57	C₃²×C₃₇	—	3	1	no	—
58	C₃²×C₃₇	—	3	1	no	—
59	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
60	C₉×C₃₇	—	9	1	no	—
61	C₉×C₃₇	—	9	1	no	—
62	C₉×C₃₇	—	9	1	no	—
63	C₉×C₃₇	—	9	1	no	—
64	#66524	—	3	2	no	—
65	#66524	—	3	3	no	—
66	#66524	—	3	3	no	—
67	#66524	—	3	3	no	—
68	#66524	—	9	1	no	—
69	#66524	—	9	1	no	—
70	#66524	—	9	1	no	—
71	#66524	—	9	1	no	—
72	#66524	—	9	1	no	—
73	#66524	#733	1	3	yes	—
74	C₉×3₋¹⁺²	—	37	1	no	—
75	C₉×3₋¹⁺²	—	37	1	no	—
76	C₉×3₋¹⁺²	—	37	1	no	—
77	#6887	—	37	1	no	—
78	#6861	—	37	1	no	—
79	#6861	—	37	1	no	—
80	#6861	C₃₇⋊C₃	1	1	yes	characteristic
81	#6860	—	37	1	no	—
82	#6860	—	37	1	no	—
83	#6860	—	37	1	no	—
84	#6858	—	37	1	no	—
85	#6858	—	37	1	no	—
86	#6858	—	37	1	no	—
87	C₃×C₃₇	—	9	1	no	—
88	C₃×C₃₇	C₃×#733	1	1	yes	characteristic
89	C₃×C₃₇	#6908	1	1	yes	characteristic
90	C₃×C₃₇	#6908	1	1	yes	characteristic
91	C₃×C₃₇	#6908	1	1	yes	characteristic
92	C₃×3₋¹⁺²	—	37	1	no	—
93	C₃×3₋¹⁺²	—	37	1	no	—
94	C₃×3₋¹⁺²	—	37	1	no	—
95	C₃²×C₉	—	37	1	no	—
96	C₉⋊C₉	—	111	2	no	—
97	C₉⋊C₉	—	111	2	no	—
98	C₉⋊C₉	—	111	2	no	—
99	C₉⋊C₉	—	111	2	no	—
100	C₃²⋊C₉	—	37	1	no	—
101	C₃²⋊C₉	—	37	1	no	—
102	C₃²⋊C₉	—	37	1	no	—
103	C₃²⋊C₉	—	37	1	no	—
104	C₃²⋊C₉	—	37	1	no	—
105	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
106	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
107	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
108	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
109	C₉²	—	111	1	no	—
110	C₉²	—	111	1	no	—
111	C₉²	—	111	1	no	—
112	C₉²	—	111	1	no	—
113	C₃₇	#10624588	1	1	yes	characteristic, complemented, Sylow
114	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
115	3₋¹⁺²	—	111	3	no	—
116	3₋¹⁺²	—	111	3	no	—
117	3₋¹⁺²	—	111	3	no	—
118	C₃×C₉	—	3	1	no	—
119	C₃×C₉	—	3	1	no	—
120	C₃×C₉	—	3	1	no	—
121	C₃×C₉	—	3	1	no	—
122	C₃×C₉	—	37	1	no	—
123	C₃×C₉	—	37	2	no	—
124	C₃×C₉	—	111	3	no	—
125	C₃×C₉	—	111	1	no	—
126	C₃×C₉	—	111	1	no	—
127	C₃×C₉	—	111	1	no	—
128	C₃×C₉	—	111	1	no	—
129	C₃×C₉	—	111	1	no	—
130	C₃×C₉	—	111	1	no	—
131	C₃×C₉	—	111	1	no	—
132	C₃×C₉	—	111	1	no	—
133	C₃²	—	3	1	no	—
134	C₃²	—	3	1	no	—
135	C₃²	—	3	1	no	—
136	C₃²	—	3	1	no	—
137	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
138	C₉	—	9	1	no	—
139	C₉	—	9	1	no	—
140	C₉	—	9	1	no	—
141	C₉	—	9	1	no	—
142	C₉	—	37	3	no	—
143	C₉	—	333	1	no	—
144	C₉	—	333	1	no	—
145	C₉	—	333	1	no	—
146	C₉	—	333	1	no	—
147	C₉	—	333	1	no	—
148	C₉	—	111	2	no	—
149	C₉	—	111	3	no	—
150	C₉	—	111	3	no	—
151	C₉	—	111	3	no	—
152	C₃	—	9	1	no	—
153	C₃	#733×C₃₇⋊C₃	1	1	yes	characteristic
154	C₃	#22582631	1	1	yes	characteristic
155	C₃	#22582632	1	1	yes	characteristic
156	C₃	#22582634	1	1	yes	characteristic

	subgroup	G/H
center	C₃²	H(3,𝔽₃)×C₃₇⋊C₃
commutator, prime radical, cosocle	C₃³×C₃₇	C₃³
Frattini	C₃³	C₃²×C₃₇⋊C₃
Fitting	C₃₇×#6861	C₃
socle	C₃²×C₃₇	C₃×H(3,𝔽₃)
hypercenter	#6861	C₃₇⋊C₃
hypocenter	C₃₇	#10624588
LCS	G ▷ C₃³×C₃₇ ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ #6861

Complements

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

N	H ≅ G/N	type
113. C₃₇	#10624588	semidirect

Maximal subgroups

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

subgroup	index
#22582548	3
#22582535	3
#22582458	3
#22582456	3
#22582455	3
#22582450	3
#22582449	3
#22582448	3
C₃₇×#6861	3
#10624588	37