Finite Group Databasebeta

← #32983608

Subgroups — #32983608

Order 18688.

434 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 99 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#22620480C211yesmaximal, characteristic
2#22608558C211yesmaximal, characteristic
3#22608442C212yesmaximal
4#22607824C212yesmaximal
5C73×#2371C211yesmaximal, characteristic
6#17989144C2211yescharacteristic
7#17988997C414yescomplemented
8#17988986—22no—
9#17988644C2212yes—
10#17988627—22no—
11C4×#11924012C2211yescharacteristic
12#17988329—22no—
13#17988323—22no—
14#17988323—22no—
15C2×#125×C73C2211yescharacteristic
16#387×C73—22no—
17#387×C73—22no—
18C4×C4⋊C4×C73—22no—
19#376×C73C2212yes—
20C2×#11924012—21no—
21#14556591C2×C412yes—
22#14556583—21no—
23#14556570—22no—
24C4×Dic146—22no—
25#14556550—21no—
26#14556524—24no—
27#14556522—24no—
28C4⋊#10596831—22no—
29C22⋊#10596831—22no—
30#14556502—22no—
31#14556502C2×C412yes—
32C4×#10596831—21no—
33#125×C73—22no—
34#125×C73C2×C412yes—
35#125×C73D412yes—
36C2×C4⋊C4×C73—22no—
37C2×C4⋊C4×C73—22no—
38C2×C4⋊C4×C73D412yes—
39C2×C42×C73—22no—
40C2×C42×C73—22no—
41C2×C42×C73C2311yescharacteristic
42#95×C73—22no—
43#95×C73—22no—
44#95×C73—22no—
45#95×C73D412yes—
46#95×C73D412yes—
47C2×Dic146—22no—
48#11924012—24no—
49C2×#10596831—22no—
50C2×#10596831—42no—
51C4⋊Dic73—22no—
52C4⋊Dic73—22no—
53#11924007—42no—
54C22⋊Dic73—41no—
55C4×Dic73—41no—
56C22×C4×C73—22no—
57C22×C4×C73—22no—
58C22×C4×C73—22no—
59C22×C4×C73C2×D411yescharacteristic
60C22×C4×C73C2×D412yes—
61C4⋊C4×C73—24no—
62C4⋊C4×C73—24no—
63C4⋊C4×C73—22no—
64C4⋊C4×C73—44no—
65C4⋊C4×C73C22⋊C414yes—
66C42×C73—44no—
67C42×C73—44no—
68C42×C73C22×C411yescharacteristic
69C42×C73C2×D411yescharacteristic
70C42×C73#4112yes—
71C23×C73#12011yescharacteristic
72C2×C4×C73—22no—
73C2×C4×C73—24no—
74C2×C4×C73—24no—
75C2×C4×C73—42no—
76C2×C4×C73—42no—
77C2×C4×C73—42no—
78C2×C4×C73—42no—
79C2×C4×C73—42no—
80C2×C4×C73—42no—
81C2×C4×C73—42no—
82C2×C4×C73—42no—
83C2×C4×C73C2×C22⋊C411yescharacteristic
84C2×C4×C73C4×D412yes—
85C2×C4×C73#12112yes—
86C2×C4×C73#12311yescharacteristic
87C2×Dic73—42no—
88Dic146—82no—
89#10596831—24no—
90#10596831—82no—
91C22×C73—22no—
92C22×C73#38611yescharacteristic
93C22×C73#45612yes—
94C4×C73—21no—
95C4×C73—21no—
96C4×C73—22no—
97C4×C73—22no—
98C4×C73—44no—
99C4×C73—44no—
100C4×C73—82no—
101Dic73—82no—
102#9440—731nomaximal, Sylow
103C2×C73—22no—
104C2×C73#188611yescharacteristic
105C2×C73#190612yes—
106#3166—731no—
107#2371D7311yescharacteristic
108#1534—731no—
109#1506—732no—
110#1300—732no—
111C73#944011yescharacteristic, complemented, Sylow
112#533—731no—
113C2×#125C2×D7311yescharacteristic
114#501—734no—
115#499—1462no—
116#432—732no—
117#428—1462no—
118C4×M16—731no—
119#387—22no—
120#387—22no—
121C4×C4⋊C4—22no—
122#376C2×D7312yes—
123#330—1462no—
124#328—1462no—
125#328—1462no—
126C2×M16—1461no—
127C4⋊Q8—732no—
128#125—22no—
129#125—1461no—
130#125C4×D7312yes—
131#125D29212yes—
132C22⋊Q8—1462no—
133C4×Q8—1462no—
134#117—1461no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4C22⋊D7312yes—
138C2×C42—22no—
139C2×C42—22no—
140C2×C42C22×D7311yescharacteristic
141#107—1464no—
142#106—1464no—
143C4⋊C8—1462no—
144C22⋊C8—1462no—
145#97—732no—
146#97—1462no—
147C4×C8—1461no—
148#95—22no—
149#95—22no—
150#95—22no—
151#95C22⋊D7312yes—
152#95C22⋊D7312yes—
153C2×Q8—1462no—
154C22×C4—22no—
155C22×C4—22no—
156C22×C4—22no—
157C22×C4C2×C22⋊D7311yescharacteristic
158C22×C4C2×C22⋊D7312yes—
159M16—1464no—
160C2×C8—2922no—
161C2×C8—1462no—
162C4⋊C4—24no—
163C4⋊C4—24no—
164C4⋊C4—22no—
165C4⋊C4—44no—
166C4⋊C4—2922no—
167C4⋊C4—1462no—
168C4⋊C4—1462no—
169C4⋊C4#1192400514yes—
170C22⋊C4—2921no—
171C42—44no—
172C42—44no—
173C42—2921no—
174C42C2×C4×D7311yescharacteristic
175C42C2×D29211yescharacteristic
176C42#1192402812yes—
177C23#1455656011yescharacteristic
178Q8—5842no—
179C2×C4—22no—
180C2×C4—24no—
181C2×C4—24no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—2922no—
191C2×C4C2×#1192400511yescharacteristic
192C2×C4C4×C22⋊D7312yes—
193C2×C4#1455656512yes—
194C2×C4#1455657411yescharacteristic
195C8—5842no—
196C8—1464no—
197C22—22no—
198C22#1798846611yescharacteristic
199C22#1798877012yes—
200C4—22no—
201C4—22no—
202C4—21no—
203C4—21no—
204C4—44no—
205C4—44no—
206C4—82no—
207C4—5842no—
208C2—22no—
209C2#2261032411yescharacteristic
210C2#2261043212yes—
subgroupG/H
centerC22#17988466
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#2371C2
socleC22×C73#386
prime radicalC23×C73#120
cosocleC2×C42×C73C23
hypercenter#2371D73
hypocenterC73#9440
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2371

Complements

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

NH ≅ G/Ntype
7. #17988997C4semidirect
111. C73#9440semidirect

Maximal subgroups

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

subgroupindex
#226204802
#226085582
#226084422
#226078242
C73×#23712
#944073