Finite Group Databasebeta

Subgroups — #41006112

Order 26973.

184 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 49 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	#22582576	C₃	1	1	yes	maximal, characteristic, complemented
2	#22582573	C₃	1	1	yes	maximal, characteristic, complemented
3	#22582568	C₃	1	1	yes	maximal, characteristic, complemented
4	#22582561	C₃	1	1	yes	maximal, characteristic
5	C₃₇⋊#6862	C₃	1	1	yes	maximal, characteristic, complemented
6	#22582456	C₃	1	1	yes	maximal, characteristic, complemented
7	#22582453	C₃	1	1	yes	maximal, characteristic
8	#22582453	C₃	1	1	yes	maximal, characteristic
9	#22582449	C₃	1	1	yes	maximal, characteristic, complemented
10	#22582448	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×#11758605	—	3	1	no	—
16	C₃×C₃²⋊(C₃₇⋊C₃)	C₃²	1	1	yes	characteristic
17	C₃²×#66524	C₃²	1	1	yes	characteristic
18	#15688506	—	3	1	no	—
19	#15688505	—	3	1	no	—
20	#15688504	—	3	2	no	—
21	#15688504	—	3	1	no	—
22	#15688504	—	3	2	no	—
23	#15688504	—	3	2	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	2	no	—
31	C₃×3₋¹⁺²×C₃₇	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	C₃²×C₃₇⋊C₃	—	3	1	no	—
37	3₋¹⁺²×C₃₇	—	9	1	no	—
38	C₃×C₉×C₃₇	—	3	1	no	—
39	C₃×C₉×C₃₇	—	3	1	no	—
40	C₃×C₉×C₃₇	—	3	1	no	—
41	C₃×C₉×C₃₇	—	3	1	no	—
42	C₃²⋊(C₃₇⋊C₃)	—	9	1	no	—
43	#11758608	—	9	1	no	—
44	C₃×#66524	—	3	1	no	—
45	C₃×#66524	—	3	1	no	—
46	C₃×#66524	—	3	1	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	—	9	1	no	—
52	C₃×#66524	H(3,𝔽₃)	1	2	yes	—
53	C₃×#66524	H(3,𝔽₃)	1	1	yes	characteristic
54	#11758605	—	9	3	no	—
55	#10624672	—	37	1	no	maximal, Sylow
56	C₃²×C₃₇	—	3	1	no	—
57	C₃²×C₃₇	—	3	1	no	—
58	C₃²×C₃₇	—	3	1	no	—
59	C₃²×C₃₇	—	3	1	no	—
60	C₃²×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
61	C₃×C₃₇⋊C₃	—	9	1	no	—
62	C₃×C₃₇⋊C₃	—	9	1	no	—
63	C₃×C₃₇⋊C₃	—	9	1	no	—
64	C₃×C₃₇⋊C₃	—	9	1	no	—
65	C₉×C₃₇	—	9	1	no	—
66	C₉×C₃₇	—	9	1	no	—
67	C₉×C₃₇	—	9	1	no	—
68	C₉×C₃₇	—	9	1	no	—
69	#66524	—	3	1	no	—
70	#66524	—	3	2	no	—
71	#66524	—	9	1	no	—
72	#66524	—	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	#6895	—	37	1	no	—
79	#6894	—	37	1	no	—
80	#6894	—	37	1	no	—
81	C₉⋊H(3,𝔽₃)	—	37	1	no	—
82	#6862	—	37	1	no	—
83	#6861	—	37	1	no	—
84	#6859	—	37	1	no	—
85	#6859	—	37	1	no	—
86	#6858	—	37	1	no	—
87	#6858	—	37	1	no	—
88	#6858	—	37	1	no	—
89	#6858	C₃₇⋊C₃	1	1	yes	characteristic, complemented
90	#6857	—	37	1	no	—
91	C₃×C₃₇	—	9	1	no	—
92	C₃×C₃₇	#6910	1	1	yes	characteristic
93	C₃×C₃₇	#6913	1	1	yes	characteristic
94	C₃×C₃₇	#6913	1	1	yes	characteristic
95	C₃×C₃₇	#6913	1	1	yes	characteristic
96	C₃₇⋊C₃	—	27	1	no	—
97	C₃×3₋¹⁺²	—	37	1	no	—
98	C₃×3₋¹⁺²	—	111	1	no	—
99	C₃×3₋¹⁺²	C₃×C₃₇⋊C₃	1	1	yes	characteristic
100	C₃×H(3,𝔽₃)	—	37	1	no	—
101	C₃²×C₉	—	37	1	no	—
102	C₉⋊C₉	—	111	2	no	—
103	C₉⋊C₉	—	111	2	no	—
104	C₉⋊C₉	—	111	1	no	—
105	C₉⋊C₉	—	111	2	no	—
106	C₉⋊C₉	—	111	1	no	—
107	C₉⋊C₉	—	111	1	no	—
108	C₃²⋊C₉	—	37	1	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	—	111	2	no	—
115	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
116	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
117	C₃²⋊C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
118	C₃₇	#10624672	1	1	yes	characteristic, complemented, Sylow
119	C₃³	—	111	1	no	—
120	C₃³	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
121	3₋¹⁺²	—	9	1	no	—
122	3₋¹⁺²	—	333	3	no	—
123	3₋¹⁺²	—	333	1	no	—
124	H(3,𝔽₃)	—	333	1	no	—
125	C₃×C₉	—	3	1	no	—
126	C₃×C₉	—	3	1	no	—
127	C₃×C₉	—	3	1	no	—
128	C₃×C₉	—	3	1	no	—
129	C₃×C₉	—	37	2	no	—
130	C₃×C₉	—	37	1	no	—
131	C₃×C₉	—	333	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	C₃²	—	3	1	no	—
140	C₃²	—	3	1	no	—
141	C₃²	—	3	1	no	—
142	C₃²	—	3	1	no	—
143	C₃²	—	333	1	no	—
144	C₃²	—	333	1	no	—
145	C₃²	—	333	1	no	—
146	C₃²	—	333	1	no	—
147	C₃²	H(3,𝔽₃)×C₃₇⋊C₃	1	1	yes	characteristic
148	C₉	—	9	1	no	—
149	C₉	—	9	1	no	—
150	C₉	—	9	1	no	—
151	C₉	—	9	1	no	—
152	C₉	—	333	1	no	—
153	C₉	—	333	1	no	—
154	C₉	—	333	1	no	—
155	C₉	—	333	1	no	—
156	C₉	—	333	1	no	—
157	C₉	—	333	1	no	—
158	C₉	—	333	1	no	—
159	C₉	—	111	1	no	—
160	C₉	—	111	2	no	—
161	C₃	—	9	1	no	—
162	C₃	—	999	1	no	—
163	C₃	#22582641	1	1	yes	characteristic
164	C₃	#22582654	1	1	yes	characteristic
165	C₃	#22582659	1	1	yes	characteristic
166	C₃	#22582662	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	#10624672
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. #22582576	C₃	semidirect
2. #22582573	C₃	semidirect
3. #22582568	C₃	semidirect
5. C₃₇⋊#6862	C₃	semidirect
6. #22582456	C₃	semidirect
9. #22582449	C₃	semidirect
10. #22582448	C₃	semidirect
11. #22582447	C₃	semidirect
13. C₃₇×#6858	C₃	semidirect
89. #6858	C₃₇⋊C₃	semidirect
118. C₃₇	#10624672	semidirect

Maximal subgroups

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

subgroup	index
#22582576	3
#22582573	3
#22582568	3
#22582561	3
C₃₇⋊#6862	3
#22582456	3
#22582453	3
#22582449	3
#22582448	3
#22582447	3
#22582444	3
C₃₇×#6858	3
#10624672	37