Finite Group Databasebeta

← #34010083

Subgroups — #34010083

Order 18688.

490 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 88 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#22625179C212yesmaximal
2#22624871C214yesmaximal, complemented
3C2×#17988453C211yesmaximal, characteristic
4#17989557—22no—
5#17989432—22no—
6#17989432C2214yes—
7C22×C4⋊Dic73C411yescharacteristic
8C22×C4⋊Dic73C2211yescharacteristic
9#17988453—22no—
10#17988453C412yes—
11#17988453C2212yes—
12C2×#14556498—22no—
13C22×C42×C73C411yescharacteristic
14C2×#11924037—42no—
15C22⋊#10596833—24no—
16C22⋊#10596833—42no—
17C23×Dic73—21no—
18C23×Dic73D412yes—
19C2×C4⋊Dic73—21no—
20C2×C4⋊Dic73—41no—
21C2×C4⋊Dic73—41no—
22C2×C4⋊Dic73—41no—
23C2×C4⋊Dic73—41no—
24C2×C4⋊Dic73C2×C412yes—
25C2×C4⋊Dic73C2×C411yescharacteristic
26C2×C4⋊Dic73D412yes—
27C2×C4⋊Dic73C2311yescharacteristic
28#14556498—24no—
29#14556498—24no—
30C23×C4×C73—21no—
31C23×C4×C73C2×C411yescharacteristic
32C2×C42×C73—21no—
33C2×C42×C73—24no—
34C2×C42×C73C2×C412yes—
35#11924037—84no—
36C2×#10596833—44no—
37C22×Dic73—22no—
38C22×Dic73—22no—
39C22×Dic73—22no—
40C22×Dic73—44no—
41C22×Dic73—44no—
42C22×Dic73—41no—
43C22×Dic73—41no—
44C22×Dic73C22⋊C412yes—
45C22×Dic73C2×D412yes—
46C4⋊Dic73—41no—
47C4⋊Dic73—42no—
48C4⋊Dic73—41no—
49C4⋊Dic73—42no—
50C4⋊Dic73—42no—
51C24×C73C22⋊C411yescharacteristic
52C22×C4×C73—21no—
53C22×C4×C73—21no—
54C22×C4×C73—22no—
55C22×C4×C73—22no—
56C22×C4×C73—24no—
57C22×C4×C73—22no—
58C22×C4×C73—42no—
59C22×C4×C73—42no—
60C22×C4×C73C22⋊C411yescharacteristic
61C22×C4×C73C22×C411yescharacteristic
62C42×C73—24no—
63C42×C73—22no—
64C42×C73M1614yescomplemented
65C23×C73—22no—
66C23×C73—22no—
67C23×C73—21no—
68C23×C73—21no—
69C23×C73#9911yescharacteristic
70C23×C73#9911yescharacteristic
71C23×C73C2×C22⋊C411yescharacteristic
72C2×C4×C73—24no—
73C2×C4×C73—24no—
74C2×C4×C73—21no—
75C2×C4×C73—21no—
76C2×C4×C73—21no—
77C2×C4×C73—21no—
78C2×C4×C73—24no—
79C2×C4×C73—48no—
80C2×C4×C73—42no—
81C2×C4×C73—42no—
82C2×C4×C73#10112yes—
83C2×C4×C73C2×M1612yes—
84C2×Dic73—22no—
85C2×Dic73—22no—
86C2×Dic73—44no—
87C2×Dic73—42no—
88C2×Dic73—42no—
89C2×Dic73—44no—
90C2×Dic73—42no—
91C2×Dic73—82no—
92C2×Dic73—82no—
93C2×Dic73—82no—
94#10596833—84no—
95C22×C73—24no—
96C22×C73—24no—
97C22×C73—21no—
98C22×C73—21no—
99C22×C73—21no—
100C22×C73—21no—
101C22×C73—21no—
102C22×C73—21no—
103C22×C73#35412yes—
104C22×C73#35512yes—
105C22×C73#40711yescharacteristic
106C22×C73C2×#9911yescharacteristic
107C22×C73C2×#10111yescharacteristic
108C4×C73—22no—
109C4×C73—22no—
110C4×C73—44no—
111Dic73—42no—
112Dic73—42no—
113Dic73—82no—
114#11511—731nomaximal, Sylow
115C2×C73—21no—
116C2×C73—22no—
117C2×C73—22no—
118C2×C73—21no—
119C2×C73#146311yescharacteristic
120C2×C73C2×#35411yescharacteristic
121C2×C73C2×#35511yescharacteristic
122C2×#382—731no—
123#1463—732no—
124#1324—734no—
125C73#1151111yescharacteristic, complemented, Sylow
126C22×C4⋊C4—731no—
127C22×C4⋊C4—731no—
128C22×C42#6440811yescharacteristic
129#407—1462no—
130#382—732no—
131#382—732no—
132#382—1462no—
133C2×#95—1462no—
134#324—734no—
135#324—1462no—
136C23×C4—21no—
137C23×C4—732no—
138C23×C4—1461no—
139C23×C4C2×#6440811yescharacteristic
140C2×M16—2922no—
141C2×C4⋊C4—2921no—
142C2×C4⋊C4—2921no—
143C2×C4⋊C4—2921no—
144C2×C4⋊C4—2921no—
145C2×C4⋊C4—732no—
146C2×C4⋊C4—731no—
147C2×C4⋊C4—731no—
148C2×C4⋊C4—732no—
149C2×C4⋊C4—1461no—
150C2×C42—21no—
151C2×C42—24no—
152C2×C42C2×#6440812yes—
153C22⋊C8—2922no—
154C22⋊C8—1464no—
155#95—1464no—
156#95—1464no—
157C24C22⋊#6440811yescharacteristic
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4—24no—
163C22×C4—22no—
164C22×C4—42no—
165C22×C4—42no—
166C22×C4—2924no—
167C22×C4—2924no—
168C22×C4—2921no—
169C22×C4—2921no—
170C22×C4—732no—
171C22×C4—732no—
172C22×C4—1462no—
173C22×C4—1462no—
174C22×C4—1462no—
175C22×C4C22⋊#6440811yescharacteristic
176C22×C4C22×#6440811yescharacteristic
177M16—5844no—
178C2×C8—2924no—
179C4⋊C4—2922no—
180C4⋊C4—2922no—
181C4⋊C4—2922no—
182C4⋊C4—2921no—
183C4⋊C4—2921no—
184C42—24no—
185C42—22no—
186C42#1192403714yescomplemented
187C23—22no—
188C23—22no—
189C23—21no—
190C23—21no—
191C23#1455665011yescharacteristic
192C23#1455665011yescharacteristic
193C23C2×C22⋊#6440811yescharacteristic
194C2×C4—24no—
195C2×C4—24no—
196C2×C4—21no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—24no—
201C2×C4—48no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—2924no—
205C2×C4—2922no—
206C2×C4—2922no—
207C2×C4—2924no—
208C2×C4—2922no—
209C2×C4—5842no—
210C2×C4—5842no—
211C2×C4—5842no—
212C2×C4—1462no—
213C2×C4—1462no—
214C2×C4#1455665212yes—
215C2×C4C2×#1192403712yes—
216C8—5844no—
217C22—24no—
218C22—24no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—21no—
223C22—21no—
224C22—21no—
225C22#1798949412yes—
226C22#1798949612yes—
227C22#1798955711yescharacteristic
228C22C2×#1455665011yescharacteristic
229C22C2×#1455665211yescharacteristic
230C4—22no—
231C4—22no—
232C4—44no—
233C4—2922no—
234C4—2922no—
235C4—5842no—
236C2—21no—
237C2—22no—
238C2—22no—
239C2—21no—
240C2#2262517911yescharacteristic
241C2C2×#1798949411yescharacteristic
242C2C2×#1798949611yescharacteristic
subgroupG/H
centerC22C2×#14556650
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
FittingC22×C42×C73C4
socleC22×C73C2×#99
prime radicalC24×C73C22⋊C4
cosocleC2×C4⋊Dic73C23
hypercenterC22×C42#64408
hypocenterC73#11511
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
2. #22624871C2semidirect
64. C42×C73M16semidirect
125. C73#11511semidirect
186. C42#11924037semidirect

Maximal subgroups

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

subgroupindex
#226251792
#226248712
C2×#179884532
#1151173