Finite Group Databasebeta

← #34004764

Subgroups — #34004764

Order 18688.

1026 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 1026 conjugacy classes of subgroups — too many to draw. Only its 124 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#22625765C212yesmaximal
2#22625179C214yesmaximal
3C22×#14556591C211yesmaximal, characteristic
4C2×#14556652—24no—
5#17989557C2214yes—
6#17989432—28no—
7C23×Dic146C411yescharacteristic
8C23×Dic146C2211yescharacteristic
9C2×#14556591—24no—
10C2×#14556591—28no—
11C22×C4⋊Dic73C412yes—
12C22×C4⋊Dic73C2212yes—
13C22×C42×C73C411yescharacteristic
14C2×#11924037—24no—
15#14556652—48no—
16C22⋊#10596833—28no—
17C22×Dic146—24no—
18C22×Dic146—24no—
19C22×Dic146—22no—
20C22×Dic146—44no—
21C22×Dic146—44no—
22C22×Dic146D414yes—
23C23×Dic73C2×C412yes—
24C23×Dic73C2×C411yescharacteristic
25C23×Dic73C2311yescharacteristic
26#14556591—232no—
27C2×C4⋊Dic73—24no—
28C2×C4⋊Dic73—24no—
29C2×C4⋊Dic73—24no—
30C2×C4⋊Dic73—24no—
31C2×C4⋊Dic73—24no—
32C2×C4⋊Dic73D418yes—
33C23×C4×C73C2×C411yescharacteristic
34C23×C4×C73C2×C412yes—
35C2×C42×C73—22no—
36C2×C42×C73—24no—
37#11924037—48no—
38C2×#10596833—44no—
39C2×Dic146—28no—
40C2×Dic146—28no—
41C2×Dic146—28no—
42C2×Dic146—44no—
43C2×Dic146—416no—
44C2×Dic146—416no—
45C2×Dic146—44no—
46C2×Dic146—44no—
47C22×Dic73—22no—
48C22×Dic73—22no—
49C22×Dic73—22no—
50C22×Dic73—42no—
51C22×Dic73—42no—
52C22×Dic73—44no—
53C22×Dic73C22⋊C412yes—
54C22×Dic73C22⋊C414yes—
55C22×Dic73C2×D412yes—
56C22×Dic73C2×D414yes—
57C4⋊Dic73—216no—
58C4⋊Dic73—216no—
59C24×C73C22×C411yescharacteristic
60C22×C4×C73—22no—
61C22×C4×C73—22no—
62C22×C4×C73—22no—
63C22×C4×C73—24no—
64C22×C4×C73—28no—
65C22×C4×C73C22⋊C412yes—
66C22×C4×C73C22⋊C414yes—
67C42×C73—28no—
68C23×C73—22no—
69C23×C73—24no—
70C23×C73C2×C22⋊C411yescharacteristic
71C23×C73C2×C22⋊C412yes—
72C2×C4×C73—28no—
73C2×C4×C73—28no—
74C2×C4×C73—28no—
75C2×C4×C73—24no—
76C2×C4×C73—22no—
77C2×C4×C73—22no—
78C2×C4×C73—24no—
79C2×C4×C73#9914yes—
80C2×C4×C73#10118yescomplemented
81C2×Dic73—24no—
82C2×Dic73—22no—
83C2×Dic73—24no—
84C2×Dic73—24no—
85C2×Dic73—44no—
86C2×Dic73—44no—
87C2×Dic73—44no—
88C2×Dic73—44no—
89C2×Dic73—44no—
90C2×Dic73#12014yes—
91Dic146—416no—
92Dic146—416no—
93#10596833—84no—
94C22×C73—28no—
95C22×C73—22no—
96C22×C73—22no—
97C22×C73—22no—
98C22×C73#37911yescharacteristic
99C22×C73C2×#9912yes—
100C22×C73C2×#10114yes—
101C4×C73—24no—
102C4×C73—28no—
103Dic73—24no—
104Dic73—42no—
105Dic73—44no—
106#8551—731nomaximal, Sylow
107C2×C73—22no—
108C2×C73—24no—
109C2×C73#178111yescharacteristic
110C2×C73#178412yes—
111C22×C4⋊Q8—731no—
112#1784—732no—
113#1463—734no—
114C73#855111yescharacteristic, complemented, Sylow
115C23×Q8—731no—
116C23×Q8—731no—
117C2×C4⋊Q8—1468no—
118C2×C4⋊Q8—1464no—
119C22×C4⋊C4—732no—
120C22×C4⋊C4—732no—
121C22×C42#6440811yescharacteristic
122C2×#101—1464no—
123#407—734no—
124#324—1468no—
125C22×Q8—2924no—
126C22×Q8—2924no—
127C22×Q8—734no—
128C22×Q8—1464no—
129C22×Q8—1464no—
130C22×Q8—1462no—
131C23×C4—731no—
132C23×C4—732no—
133C23×C4—731no—
134C23×C4C2×#6440812yes—
135C23×C4C2×#6440811yescharacteristic
136C2×M16—1464no—
137C4⋊Q8—14632no—
138C2×C4⋊C4—738no—
139C2×C4⋊C4—1464no—
140C2×C4⋊C4—1464no—
141C2×C4⋊C4—1464no—
142C2×C4⋊C4—1464no—
143C2×C4⋊C4—1464no—
144C2×C42—22no—
145C2×C42—24no—
146#101—2928no—
147C22⋊C8—1468no—
148C24C22×#6440811yescharacteristic
149C2×Q8—29216no—
150C2×Q8—29216no—
151C2×Q8—2924no—
152C2×Q8—2924no—
153C2×Q8—2924no—
154C2×Q8—1468no—
155C2×Q8—1468no—
156C2×Q8—1468no—
157C22×C4—22no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—24no—
161C22×C4—28no—
162C22×C4—2924no—
163C22×C4—2922no—
164C22×C4—2922no—
165C22×C4—734no—
166C22×C4—734no—
167C22×C4—732no—
168C22×C4—732no—
169C22×C4—1462no—
170C22×C4—1462no—
171C22×C4—1462no—
172C22×C4C22⋊#6440812yes—
173C22×C4C22⋊#6440814yes—
174M16—2928no—
175C2×C8—2924no—
176C4⋊C4—14616no—
177C4⋊C4—14616no—
178C42—28no—
179C23—22no—
180C23—24no—
181C23C2×C22⋊#6440811yescharacteristic
182C23C2×C22⋊#6440812yes—
183Q8—29216no—
184Q8—29216no—
185C2×C4—28no—
186C2×C4—28no—
187C2×C4—28no—
188C2×C4—24no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—24no—
192C2×C4—2924no—
193C2×C4—2924no—
194C2×C4—2924no—
195C2×C4—2924no—
196C2×C4—2924no—
197C2×C4—734no—
198C2×C4—1464no—
199C2×C4—1462no—
200C2×C4—1464no—
201C2×C4—1464no—
202C2×C4#1455665014yes—
203C2×C4#1455665218yescomplemented
204C8—5844no—
205C22—28no—
206C22—22no—
207C22—22no—
208C22—22no—
209C22#1798952811yescharacteristic
210C22C2×#1455665012yes—
211C22C2×#1455665214yes—
212C4—24no—
213C4—28no—
214C4—2924no—
215C4—2922no—
216C4—1464no—
217C2—22no—
218C2—24no—
219C2#2262575911yescharacteristic
220C2#2262576512yes—
subgroupG/H
centerC22#17989528
commutator, prime radicalC24×C73C22×C4
FrattiniC24C22×#64408
FittingC22×C42×C73C4
socleC22×C73#379
cosocleC23×Dic73C23
hypercenterC22×C42#64408
hypocenterC73#8551
LCSG ▷ C24×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C24 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
80. C2×C4×C73#101semidirect
114. C73#8551semidirect
203. C2×C4#14556652semidirect

Maximal subgroups

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

subgroupindex
#226257652
#226251792
C22×#145565912
#855173