Finite Group Databasebeta

← #32988732

Subgroups — #32988732

Order 18688.

576 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 113 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#22621075C212yesmaximal
2#22610908C211yesmaximal, characteristic
3#22608603C212yesmaximal
4#22607764C211yesmaximal, characteristic
5C73×#2868C211yesmaximal, characteristic
6#17989144C2211yescharacteristic
7#17988983—22no—
8#17988939—22no—
9#17988917—24no—
10#17988869—24no—
11#17988640C2212yes—
12#17988611C2211yescharacteristic
13#17988347—22no—
14#17988333—24no—
15C2×C4⋊Q8×C73—22no—
16C2×C4⋊Q8×C73C2212yes—
17#384×C73—24no—
18C43×C73C2211yescharacteristic
19C2×#11924012—21no—
20#14556591D412yes—
21#14556583—21no—
22#14556570—22no—
23C4×Dic146—22no—
24#14556550—21no—
25#14556522—28no—
26C4⋊#10596831—22no—
27C4⋊#10596831D414yes—
28#14556514—28no—
29C22⋊#10596831—22no—
30C4×#10596831—22no—
31C22×Q8×C73—22no—
32C4⋊Q8×C73—22no—
33C4⋊Q8×C73—22no—
34C4⋊Q8×C73—22no—
35C4⋊Q8×C73—22no—
36C4⋊Q8×C73—44no—
37C4⋊Q8×C73—44no—
38C2×C4⋊C4×C73—28no—
39C2×C4⋊C4×C73—22no—
40C2×C4⋊C4×C73D414yes—
41C2×C42×C73—22no—
42C2×C42×C73D412yes—
43C2×C42×C73C2311yescharacteristic
44C2×Dic146—22no—
45#11924012—42no—
46C2×#10596831—24no—
47C2×#10596831—41no—
48C4⋊Dic73—22no—
49C4⋊Dic73—22no—
50#11924007—42no—
51C22⋊Dic73—41no—
52C4×Dic73—41no—
53C2×Q8×C73—42no—
54C2×Q8×C73—42no—
55C2×Q8×C73—44no—
56C2×Q8×C73—44no—
57C22×C4×C73—22no—
58C22×C4×C73—24no—
59C22×C4×C73C2×D412yes—
60C22×C4×C73C2×D411yescharacteristic
61C4⋊C4×C73—24no—
62C4⋊C4×C73—24no—
63C4⋊C4×C73—416no—
64C4⋊C4×C73—42no—
65C4⋊C4×C73—42no—
66C42×C73—28no—
67C42×C73—22no—
68C42×C73SD1614yes—
69C42×C73C2×D411yescharacteristic
70C42×C73C2×D412yes—
71C42×C73#4111yescharacteristic
72C23×C73#12011yescharacteristic
73Q8×C73—84no—
74Q8×C73—82no—
75Q8×C73—82no—
76C2×C4×C73—28no—
77C2×C4×C73—24no—
78C2×C4×C73—24no—
79C2×C4×C73—44no—
80C2×C4×C73—48no—
81C2×C4×C73#12012yes—
82C2×C4×C73#12111yescharacteristic
83C2×C4×C73#12112yes—
84C2×C4×C73#12711yescharacteristic
85C2×C4×C73C2×SD1612yes—
86C2×C4×C73AGL(1,ℤ/8)12yes—
87C2×Dic73—42no—
88Dic146—44no—
89#10596831—44no—
90#10596831—81no—
91C22×C73—22no—
92C22×C73#39211yescharacteristic
93C22×C73#45012yes—
94C4×C73—28no—
95C4×C73—22no—
96C4×C73—84no—
97C4×C73#45614yes—
98C4×C73#46012yes—
99C4×C73#49211yescharacteristic
100C4×C73#49611yescharacteristic
101Dic73—82no—
102#10299—731nomaximal, Sylow
103C2×C73—22no—
104C2×C73#200512yes—
105C2×C73#201411yescharacteristic
106#3229—732no—
107#2868D7311yescharacteristic
108#1994—731no—
109#1546—732no—
110#1276—731no—
111C73#1029911yescharacteristic, complemented, Sylow
112#533—731no—
113C2×C4⋊Q8—22no—
114C2×C4⋊Q8C2×D7312yes—
115#498—1462no—
116#487—1462no—
117#483—1464no—
118#475—1464no—
119#431—732no—
120#423—731no—
121#384—24no—
122C43C2×D7311yescharacteristic
123#339—1462no—
124#332—1464no—
125C22×Q8—22no—
126C2×M16—1461no—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—44no—
132C4⋊Q8—44no—
133C4⋊Q8—732no—
134#125—1461no—
135C22⋊Q8—1462no—
136C4×Q8—1462no—
137#117—1461no—
138C2×C4⋊C4—28no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4C22⋊D7314yes—
141C2×C42—22no—
142C2×C42C22⋊D7312yes—
143C2×C42C22×D7311yescharacteristic
144#106—1468no—
145C4⋊C8—734no—
146C4⋊C8—1462no—
147#103—1468no—
148C22⋊C8—1462no—
149C4×C8—1462no—
150C2×Q8—42no—
151C2×Q8—42no—
152C2×Q8—44no—
153C2×Q8—44no—
154C2×Q8—1462no—
155C22×C4—22no—
156C22×C4—24no—
157C22×C4C2×C22⋊D7312yes—
158C22×C4C2×C22⋊D7311yescharacteristic
159M16—2922no—
160C2×C8—2921no—
161C2×C8—1464no—
162C4⋊C4—24no—
163C4⋊C4—24no—
164C4⋊C4—416no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—2922no—
168C4⋊C4—1462no—
169C4⋊C4—1462no—
170C22⋊C4—2921no—
171C42—28no—
172C42—22no—
173C42—2921no—
174C42#1192401614yes—
175C42D4×D7311yescharacteristic
176C42D4×D7312yes—
177C42#1192402711yescharacteristic
178C23#1455656011yescharacteristic
179Q8—82no—
180Q8—82no—
181Q8—84no—
182Q8—2924no—
183C2×C4—28no—
184C2×C4—24no—
185C2×C4—24no—
186C2×C4—48no—
187C2×C4—44no—
188C2×C4—2922no—
189C2×C4#1455655912yes—
190C2×C4#1455656211yescharacteristic
191C2×C4#1455656312yes—
192C2×C4#1455658711yescharacteristic
193C2×C4C2×#1192401612yes—
194C2×C4#1455661712yes—
195C8—2924no—
196C8—5841no—
197C22—22no—
198C22#1798849511yescharacteristic
199C22#1798873812yes—
200C4—28no—
201C4—22no—
202C4—84no—
203C4—5842no—
204C4#1798876914yes—
205C4#1798878412yes—
206C4#1798895911yescharacteristic
207C4#1798897411yescharacteristic
208C2—22no—
209C2#2261096712yes—
210C2#2261101911yescharacteristic
subgroupG/H
centerC22#17988495
commutator, cosocleC2×C42×C73C23
FrattiniC2×C42C22×D73
FittingC73×#2868C2
socleC22×C73#392
prime radicalC23×C73#120
hypercenter#2868D73
hypocenterC73#10299
LCSG ▷ C2×C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2868

Complements

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

NH ≅ G/Ntype
111. C73#10299semidirect

Maximal subgroups

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

subgroupindex
#226210752
#226109082
#226086032
#226077642
C73×#28682
#1029973