Finite Group Databasebeta

← #34000563

Subgroups — #34000563

Order 18688.

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

normal subgroups only

This group has 336 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#22626159C211yesmaximal, characteristic
2#22625807C211yesmaximal, characteristic
3#341×D73C211yesmaximal, characteristic
4C2×#14556666—21no—
5C2×#14556659C412yes—
6#17989529—21no—
7#17989482C412yescomplemented
8C22⋊#11924031—21no—
9#17989461—21no—
10C2×M16×D73—21no—
11C22×C8×D73C2211yescharacteristic
12#17988352C411yescharacteristic, complemented
13#17988351—21no—
14#341×C73C411yescharacteristic, complemented
15C2×C4⋊#64408—22no—
16C2×C22⋊#64408—41no—
17#14556666—42no—
18C2×#11924031—22no—
19#14556659—42no—
20(C2×#64408)⋊C4—41no—
21C22×C4×D73D411yescharacteristic
22C2×#11924012—21no—
23C2×#11924011—41no—
24M16×D73—24no—
25M16×D73—42no—
26C22×#10596831D411yescharacteristic
27C2×C8×D73—42no—
28C2×C8×D73—41no—
29C2×C8×D73C2×C411yescharacteristic
30C2×C8×D73C2×C411yescharacteristic
31C2×C8×D73D411yescharacteristic
32C2×C8×D73Q811yescharacteristic
33C2×M16×C73—21no—
34C22×C8×C73C2×C411yescharacteristic
35C22×#64408—42no—
36C4⋊#64408—42no—
37C4⋊#64408—42no—
38C22⋊#64408—82no—
39#11924031—42no—
40C23×D73SD1611yescharacteristic
41C22×Dic73Q1611yescharacteristic
42C2×C4×D73—22no—
43C2×C4×D73—22no—
44C2×C4×D73C22⋊C411yescharacteristic
45C2×C4×D73C4⋊C411yescharacteristic
46C2×C4×D73D811yescharacteristic
47C2×C4×D73SD1611yescharacteristic
48#11924012—22no—
49#11924012—41no—
50#11924011—44no—
51C2×#10596831—41no—
52C2×#10596831—41no—
53C2×#10596831C22⋊C411yescharacteristic
54C2×#10596831C4⋊C411yescharacteristic
55C8×D73—22no—
56C8×D73—22no—
57C8×D73—41no—
58C8×D73—41no—
59C8×D73—44no—
60C22×C4×C73C22⋊C411yescharacteristic
61M16×C73—22no—
62M16×C73—41no—
63C2×C8×C73—41no—
64C2×C8×C73—41no—
65C2×C8×C73C4211yescharacteristic
66C2×C8×C73C4⋊C411yescharacteristic
67C23×C73#10311yescharacteristic
68C22×D73—22no—
69C22×D73—22no—
70C22×D73—41no—
71C22×D73#10211yescharacteristic
72C22×D73#10611yescharacteristic
73C2×#64408—82no—
74C2×#64408—82no—
75C2×#64408—82no—
76C2×C4×C73—21no—
77C2×C4×C73—21no—
78C2×C4×C73#9511yescharacteristic
79C2×C4×C73#10211yescharacteristic
80C2×Dic73—22no—
81C2×Dic73#10311yescharacteristic
82C2×Dic73#10711yescharacteristic
83C4×D73—22no—
84C4×D73—21no—
85C4×D73—21no—
86C4×D73—22no—
87C4×D73#9912yes—
88C4×D73#10412yes—
89C8×C73—21no—
90C8×C73—21no—
91C8×C73—42no—
92#10596831—21no—
93#10596831—21no—
94#10596831—42no—
95C22×C73—21no—
96C22×C73—21no—
97C22×C73—21no—
98C22×C73#34011yescharacteristic
99C2×D73—21no—
100C2×D73—21no—
101C2×D73—21no—
102C2×D73—21no—
103C2×D73—44no—
104C2×D73#32712yes—
105C2×D73#36512yes—
106#64408—162no—
107C4×C73—21no—
108C4×C73#33911yescharacteristic
109C4×C73#34211yescharacteristic
110Dic73—21no—
111Dic73#32812yes—
112#7263—731nomaximal, Sylow
113C2×C73—41no—
114C2×C73#129411yescharacteristic
115C2×C73#138311yescharacteristic
116C2×C73#138911yescharacteristic
117D73—22no—
118D73—22no—
119#2112—731no—
120#1807—731no—
121C2×#341—731no—
122C73#726311yescharacteristic, complemented, Sylow
123C22×M16—1461no—
124C23×C8—731no—
125C2×M32—1461no—
126C2×#106—732no—
127#380—1461no—
128C22⋊C16—732no—
129C22⋊C16—1461no—
130#341—731no—
131#341—1461no—
132#341#6440811yescharacteristic, complemented
133#340—1461no—
134C23×C4—731no—
135C2×M16—21no—
136C2×M16—2921no—
137C2×M16—2922no—
138C2×M16—1464no—
139C2×M16—1461no—
140C22×C8—2921no—
141C22×C8—2922no—
142C22×C8—731no—
143C22×C8—731no—
144C22×C8—731no—
145C22×C8—731no—
146C22×C8—731no—
147C22×C8C2×#6440811yescharacteristic
148C2×C4⋊C4—1462no—
149C2×C22⋊C4—2921no—
150M32—2922no—
151C2×C16—1462no—
152#106—2922no—
153#95—2921no—
154C24—731no—
155C22×C4—2922no—
156C22×C4—731no—
157C22×C4—731no—
158C22×C4—731no—
159C22×C4—731no—
160C22×C4—731no—
161C22×C4—1462no—
162C22×C4—1462no—
163C22×C4C22⋊#6440811yescharacteristic
164M16—22no—
165M16—41no—
166M16—2924no—
167M16—2921no—
168M16—1462no—
169C2×C8—41no—
170C2×C8—41no—
171C2×C8—2921no—
172C2×C8—2921no—
173C2×C8—2924no—
174C2×C8—2921no—
175C2×C8—2921no—
176C2×C8—731no—
177C2×C8—731no—
178C2×C8—1462no—
179C2×C8—1462no—
180C2×C8C4×#6440811yescharacteristic
181C2×C8C4⋊#6440811yescharacteristic
182C4⋊C4—2922no—
183C4⋊C4—2922no—
184C22⋊C4—5842no—
185C16—2922no—
186C23—2921no—
187C23—731no—
188C23—731no—
189C23—1462no—
190C23—1462no—
191C23#1455665411yescharacteristic
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—5842no—
195C2×C4—5842no—
196C2×C4—5842no—
197C2×C4—731no—
198C2×C4—732no—
199C2×C4—732no—
200C2×C4—731no—
201C2×C4—1462no—
202C2×C4—1462no—
203C2×C4—1462no—
204C2×C4—1461no—
205C2×C4—1461no—
206C2×C4(C2×#64408)⋊C411yescharacteristic
207C2×C4#1455665311yescharacteristic
208C8—21no—
209C8—21no—
210C8—42no—
211C8—2922no—
212C8—1461no—
213C8—1461no—
214C22—21no—
215C22—21no—
216C22—21no—
217C22—2924no—
218C22—732no—
219C22—732no—
220C22—1461no—
221C22—1461no—
222C22—1461no—
223C22—1461no—
224C22#1798946011yescharacteristic
225C4—21no—
226C4—732no—
227C4—11682no—
228C4—1461no—
229C4#1798945811yescharacteristic
230C4#1798946411yescharacteristic
231C2—41no—
232C2—1462no—
233C2—1462no—
234C2#2262480511yescharacteristic
235C2#2262499711yescharacteristic
236C2#2262501111yescharacteristic
subgroupG/H
centerC22#17989460
commutatorC2×C8×C73C42
FrattiniC22×C8C2×#64408
Fitting#341×C73C4
socleC22×C73#340
prime radicalC23×D73SD16
cosocleC22×C8×D73C22
hypercenter#341#64408
hypocenterC73#7263
LCSG ▷ C2×C8×C73 ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #341

Complements

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

NH ≅ G/Ntype
7. #17989482C4semidirect
12. #17988352C4semidirect
14. #341×C73C4semidirect
122. C73#7263semidirect
132. #341#64408semidirect

Maximal subgroups

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

subgroupindex
#226261592
#226258072
#341×D732
#726373