Finite Group Databasebeta

← #34000466

Subgroups — #34000466

Order 18688.

292 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 68 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#22625810C211yesmaximal, characteristic
2#22625807C211yesmaximal, characteristic
3C2×#17988366C211yesmaximal, characteristic
4C2×#14556661—21no—
5C2×#14556659—21no—
6#17989558—21no—
7#17989529—21no—
8#17989463C412yescomplemented
9#17989461C412yes—
10C22×C8×D73C2211yescharacteristic
11C22×#11924003C411yescharacteristic, complemented
12#17988366—21no—
13#17988366—21no—
14C22×C16×C73C411yescharacteristic, complemented
15C2×#11924038—22no—
16C2×C4⋊#64408—22no—
17C2×C22⋊#64408—41no—
18#14556661—41no—
19#14556661—41no—
20#14556659—41no—
21#14556659—41no—
22#14556648—41no—
23C22⋊#10596833—41no—
24(C2×#64408)⋊C4—41no—
25C22×C4×D73D411yescharacteristic
26C22×#10596831Q811yescharacteristic
27C2×C8×D73—41no—
28C2×C8×D73—41no—
29C2×C8×D73C2×C411yescharacteristic
30C2×C8×D73C2×C411yescharacteristic
31C2×C8×D73D411yescharacteristic
32C2×C8×D73D411yescharacteristic
33C2×#11924003—21no—
34C2×#11924003—41no—
35C22×C8×C73C2×C411yescharacteristic
36C2×C16×C73—21no—
37C2×C16×C73—41no—
38C22×#64408—42no—
39#11924038—42no—
40#11924038—42no—
41C2×#10596833—42no—
42C4⋊#64408—42no—
43C4⋊#64408—42no—
44C22⋊#64408—81no—
45C22⋊#64408—81no—
46C23×D73SD1611yescharacteristic
47C22×Dic73SD1611yescharacteristic
48C2×C4×D73—41no—
49C2×C4×D73—41no—
50C2×C4×D73C22⋊C411yescharacteristic
51C2×C4×D73C22⋊C411yescharacteristic
52C2×C4×D73D811yescharacteristic
53C2×C4×D73Q1611yescharacteristic
54C2×#10596831—41no—
55C2×#10596831C4⋊C411yescharacteristic
56C2×#10596831C4⋊C411yescharacteristic
57C8×D73—21no—
58C8×D73—21no—
59C8×D73—21no—
60C8×D73—21no—
61C8×D73—41no—
62C8×D73—41no—
63#11924003—41no—
64C22×C4×C73C4⋊C411yescharacteristic
65C2×C8×C73—21no—
66C2×C8×C73—21no—
67C2×C8×C73C4211yescharacteristic
68C2×C8×C73C22⋊C411yescharacteristic
69C16×C73—41no—
70C23×C73#10611yescharacteristic
71C22×D73—21no—
72C22×D73—21no—
73C22×D73—22no—
74C22×D73—41no—
75C22×D73#10211yescharacteristic
76C22×D73#10311yescharacteristic
77C2×#64408—82no—
78C2×#64408—82no—
79C2×#64408—82no—
80C2×C4×C73—21no—
81C2×C4×C73—21no—
82C2×C4×C73#9511yescharacteristic
83C2×C4×C73#10711yescharacteristic
84C2×Dic73—41no—
85C2×Dic73#10211yescharacteristic
86C2×Dic73#10311yescharacteristic
87C4×D73—21no—
88C4×D73—21no—
89C4×D73—21no—
90C4×D73—21no—
91C4×D73—41no—
92C4×D73—41no—
93#10596833—82no—
94C8×C73—21no—
95C8×C73#9911yescharacteristic
96C8×C73#9911yescharacteristic
97#10596831—21no—
98#10596831—21no—
99C22×C73—21no—
100C22×C73—21no—
101C22×C73#34011yescharacteristic
102C22×C73#36511yescharacteristic
103C22×C73#36511yescharacteristic
104C2×D73—21no—
105C2×D73—21no—
106C2×D73—21no—
107C2×D73—21no—
108C2×D73—21no—
109C2×D73—42no—
110C2×D73—42no—
111C2×D73#35912yes—
112#64408—162no—
113C4×C73—21no—
114C4×C73H(3,ℤ/4)11yescharacteristic
115C4×C73#34211yescharacteristic
116Dic73—21no—
117Dic73—21no—
118#7232—731nomaximal, Sylow
119C2×C73—21no—
120C2×C73—21no—
121C2×C73#128911yescharacteristic
122C2×C73#138211yescharacteristic
123C2×C73#138311yescharacteristic
124D73—22no—
125D73—41no—
126C2×C22⋊C16—731no—
127#1810—731no—
128#1807—731no—
129C73#723211yescharacteristic, complemented, Sylow
130C23×C8—731no—
131C22×C16—731no—
132C22×C16#6440811yescharacteristic, complemented
133C2×#108—1461no—
134C2×#106—1461no—
135#407—1461no—
136#380—1461no—
137C22⋊C16—1461no—
138C22⋊C16—1461no—
139#341—732no—
140#340—732no—
141C23×C4—731no—
142C2×M16—1462no—
143C22×C8—2921no—
144C22×C8—2921no—
145C22×C8—731no—
146C22×C8—731no—
147C22×C8—731no—
148C22×C8—731no—
149C22×C8—731no—
150C22×C8C2×#6440811yescharacteristic
151C2×C4⋊C4—1462no—
152C2×C22⋊C4—2921no—
153C2×C16—21no—
154C2×C16—41no—
155C2×C16—2921no—
156C2×C16—1461no—
157#108—2921no—
158#108—2921no—
159#106—2921no—
160#106—2921no—
161C22⋊C8—2921no—
162C22⋊C8—2921no—
163#95—2921no—
164C24—731no—
165C22×C4—2921no—
166C22×C4—2921no—
167C22×C4—2922no—
168C22×C4—731no—
169C22×C4—731no—
170C22×C4—731no—
171C22×C4—731no—
172C22×C4—731no—
173C22×C4C4⋊#6440811yescharacteristic
174M16—2922no—
175M16—2922no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—2921no—
179C2×C8—2921no—
180C2×C8—2922no—
181C2×C8—2921no—
182C2×C8—731no—
183C2×C8—731no—
184C2×C8—1461no—
185C2×C8—1461no—
186C2×C8—1461no—
187C2×C8—1461no—
188C2×C8C4×#6440811yescharacteristic
189C2×C8C22⋊#6440811yescharacteristic
190C4⋊C4—2922no—
191C4⋊C4—2922no—
192C22⋊C4—5841no—
193C22⋊C4—5841no—
194C16—41no—
195C16—2921no—
196C23—2921no—
197C23—731no—
198C23—731no—
199C23—1462no—
200C23—1461no—
201C23—1461no—
202C23#1455665911yescharacteristic
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—2921no—
206C2×C4—2921no—
207C2×C4—2921no—
208C2×C4—5842no—
209C2×C4—5842no—
210C2×C4—5842no—
211C2×C4—731no—
212C2×C4—731no—
213C2×C4—1461no—
214C2×C4—1461no—
215C2×C4—1461no—
216C2×C4—1461no—
217C2×C4(C2×#64408)⋊C411yescharacteristic
218C2×C4#1455666011yescharacteristic
219C8—21no—
220C8—5842no—
221C8—1461no—
222C8—1461no—
223C8#1455664911yescharacteristic
224C8#1455665011yescharacteristic
225C22—21no—
226C22—21no—
227C22—2922no—
228C22—2922no—
229C22—732no—
230C22—1461no—
231C22—1461no—
232C22—1461no—
233C22—1461no—
234C22—1461no—
235C22#1798946111yescharacteristic
236C22#1798951111yescharacteristic
237C22#1798951211yescharacteristic
238C4—21no—
239C4—11682no—
240C4—1461no—
241C4—1461no—
242C4#1798945511yescharacteristic
243C4#1798946511yescharacteristic
244C2—21no—
245C2—21no—
246C2—2921no—
247C2—1462no—
248C2#2262479411yescharacteristic
249C2#2262499311yescharacteristic
250C2#2262499611yescharacteristic
subgroupG/H
centerC22#17989461
commutatorC2×C8×C73C42
FrattiniC22×C8C2×#64408
FittingC22×C16×C73C4
socleC22×C73#340
prime radicalC23×D73SD16
cosocleC22×C8×D73C22
hypercenterC22×C16#64408
hypocenterC73#7232
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16

Complements

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

NH ≅ G/Ntype
8. #17989463C4semidirect
11. C22×#11924003C4semidirect
14. C22×C16×C73C4semidirect
129. C73#7232semidirect
132. C22×C16#64408semidirect

Maximal subgroups

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

subgroupindex
#226258102
#226258072
C2×#179883662
#723273