Finite Group Databasebeta

← #32972666

Subgroups — #32972666

Order 18688.

218 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22607855C211yesmaximal, characteristic
2#22607799C211yesmaximal, characteristic
3C73×#2075C211yesmaximal, characteristic
4#17988447—21no—
5C22⋊#11924003—21no—
6C2×#107×C73C411yescharacteristic, complemented
7C2×C22⋊C8×C73C2211yescharacteristic
8#380×C73—21no—
9#340×C73—21no—
10#340×C73C411yescharacteristic, complemented
11C2×C4⋊Dic73—41no—
12C2×C22⋊Dic73—41no—
13C2×#11924003—41no—
14#14556498—22no—
15C23×C4×C73D411yescharacteristic
16C22×C8×C73C2×C411yescharacteristic
17C22×C8×C73D411yescharacteristic
18C2×C4⋊C4×C73—21no—
19C2×C4⋊C4×C73—21no—
20C2×C22⋊C4×C73—41no—
21#107×C73—41no—
22C22⋊C8×C73—41no—
23#95×C73—41no—
24C22×Dic73—42no—
25C4⋊Dic73—81no—
26C4⋊Dic73—81no—
27C22⋊Dic73—81no—
28C22⋊Dic73—81no—
29#11924003—81no—
30C24×C73D811yescharacteristic
31C22×C4×C73—21no—
32C22×C4×C73—21no—
33C22×C4×C73—41no—
34C22×C4×C73—41no—
35C22×C4×C73—41no—
36C22×C4×C73—41no—
37C22×C4×C73C22⋊C411yescharacteristic
38C22×C4×C73SD1611yescharacteristic
39C2×C8×C73—21no—
40C2×C8×C73—21no—
41C2×C8×C73—41no—
42C2×C8×C73—41no—
43C4⋊C4×C73—41no—
44C4⋊C4×C73—41no—
45C4⋊C4×C73—41no—
46C4⋊C4×C73—41no—
47C22⋊C4×C73—81no—
48C22⋊C4×C73—81no—
49C23×C73—21no—
50C23×C73—21no—
51C23×C73—41no—
52C23×C73—41no—
53C23×C73#10211yescharacteristic
54C23×C73D1611yescharacteristic
55C23×C73SD3211yescharacteristic
56C2×C4×C73—21no—
57C2×C4×C73—21no—
58C2×C4×C73—21no—
59C2×C4×C73—41no—
60C2×C4×C73—41no—
61C2×C4×C73—41no—
62C2×C4×C73—81no—
63C2×C4×C73—81no—
64C2×C4×C73—81no—
65C2×C4×C73—81no—
66C2×C4×C73—81no—
67C2×C4×C73—81no—
68C2×C4×C73—81no—
69C2×C4×C73#9911yescharacteristic
70C2×C4×C73#10411yescharacteristic
71C2×Dic73—82no—
72C2×Dic73—82no—
73C2×Dic73—82no—
74C8×C73—41no—
75C8×C73—41no—
76C22×C73—21no—
77C22×C73—21no—
78C22×C73—21no—
79C22×C73—21no—
80C22×C73—41no—
81C22×C73—41no—
82C22×C73—41no—
83C22×C73—41no—
84C22×C73—81no—
85C22×C73#32711yescharacteristic
86C22×C73#35711yescharacteristic
87C22×C73#36011yescharacteristic
88C4×C73—21no—
89C4×C73—41no—
90C4×C73—161no—
91C4×C73—161no—
92C4×C73#35211yescharacteristic
93C4×C73#35411yescharacteristic
94Dic73—162no—
95#7079—731nomaximal, Sylow
96C2×C73—21no—
97C2×C73—21no—
98C2×C73—41no—
99C2×C73—41no—
100C2×C73#134011yescharacteristic
101C2×C73#134411yescharacteristic
102C2×C73#134711yescharacteristic
103#2075D7311yescharacteristic
104#1314—731no—
105#1289—731no—
106C73#707911yescharacteristic, complemented, Sylow
107C2×#107Dic7311yescharacteristic, complemented
108C2×C22⋊C8C2×D7311yescharacteristic
109#380—21no—
110#380—1461no—
111C22⋊C16—1461no—
112#340—21no—
113#340Dic7311yescharacteristic, complemented
114C23×C4C22⋊D7311yescharacteristic
115C22×C8C2×Dic7311yescharacteristic
116C22×C8C22⋊D7311yescharacteristic
117C2×C4⋊C4—21no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—2921no—
120C2×C22⋊C4—41no—
121C2×C22⋊C4—2921no—
122C2×C16—2921no—
123#107—41no—
124C22⋊C8—41no—
125#95—41no—
126#95—1462no—
127C24#1192401311yescharacteristic
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—41no—
131C22×C4—41no—
132C22×C4—41no—
133C22×C4—41no—
134C22×C4—2922no—
135C22×C4C22⋊Dic7311yescharacteristic
136C22×C4#1192401611yescharacteristic
137C2×C8—21no—
138C2×C8—21no—
139C2×C8—41no—
140C2×C8—41no—
141C4⋊C4—41no—
142C4⋊C4—41no—
143C4⋊C4—41no—
144C4⋊C4—41no—
145C4⋊C4—5841no—
146C4⋊C4—5841no—
147C22⋊C4—81no—
148C22⋊C4—81no—
149C22⋊C4—5841no—
150C22⋊C4—5841no—
151C16—5841no—
152C23—21no—
153C23—21no—
154C23—41no—
155C23—41no—
156C23#1455651211yescharacteristic
157C23#1455653211yescharacteristic
158C23#1455653511yescharacteristic
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—21no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—81no—
166C2×C4—81no—
167C2×C4—81no—
168C2×C4—81no—
169C2×C4—81no—
170C2×C4—81no—
171C2×C4—81no—
172C2×C4—5842no—
173C2×C4—5842no—
174C2×C4—5842no—
175C2×C4#1455650611yescharacteristic
176C2×C4#1455651811yescharacteristic
177C8—41no—
178C8—41no—
179C22—21no—
180C22—21no—
181C22—21no—
182C22—21no—
183C22—41no—
184C22—41no—
185C22—41no—
186C22—41no—
187C22—81no—
188C22#1798832111yescharacteristic
189C22#1798838811yescharacteristic
190C22#1798839711yescharacteristic
191C4—21no—
192C4—41no—
193C4—161no—
194C4—161no—
195C4—11682no—
196C4#1798837711yescharacteristic
197C4#1798838211yescharacteristic
198C2—21no—
199C2—21no—
200C2—41no—
201C2—41no—
202C2#2260791911yescharacteristic
203C2#2260793111yescharacteristic
204C2#2260794011yescharacteristic
subgroupG/H
centerC22#17988321
commutatorC22×C8×C73C2×C4
FrattiniC2×C22⋊C8C2×D73
FittingC73×#2075C2
socleC22×C73#327
prime radicalC24×C73D8
cosocleC2×C22⋊C8×C73C22
hypercenter#2075D73
hypocenterC73#7079
LCSG ▷ C22×C8×C73 ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C8 ◁ #2075

Complements

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

NH ≅ G/Ntype
6. C2×#107×C73C4semidirect
10. #340×C73C4semidirect
106. C73#7079semidirect
107. C2×#107Dic73semidirect
113. #340Dic73semidirect

Maximal subgroups

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

subgroupindex
#226078552
#226077992
C73×#20752
#707973