Finite Group Databasebeta

← #34010046

Subgroups — #34010046

Order 18688.

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

normal subgroups only

This group has 706 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#22625176C212yesmaximal
2#22624857C214yesmaximal, complemented
3C2×#17988450C211yesmaximal, characteristic
4#17989557—22no—
5#17989430—22no—
6#17989430C2214yes—
7C22×C22⋊Dic73C411yescharacteristic
8C22×C22⋊Dic73C2211yescharacteristic
9#17988450—22no—
10#17988450C412yes—
11#17988450C2212yes—
12C2×#14556498—22no—
13C24×C4×C73C411yescharacteristic
14C2×#11924037—42no—
15C22⋊#10596833—24no—
16C22⋊#10596833—42no—
17C23×Dic73—21no—
18C23×Dic73D412yes—
19C2×C22⋊Dic73—21no—
20C2×C22⋊Dic73—42no—
21C2×C22⋊Dic73—42no—
22C2×C22⋊Dic73C2×C412yes—
23C2×C22⋊Dic73C2×C411yescharacteristic
24C2×C22⋊Dic73D412yes—
25C2×C22⋊Dic73C2311yescharacteristic
26#14556498—24no—
27#14556498—24no—
28C25×C73C2×C411yescharacteristic
29C23×C4×C73—21no—
30C23×C4×C73—21no—
31C23×C4×C73—24no—
32C23×C4×C73—42no—
33C23×C4×C73—42no—
34C23×C4×C73C2×C412yes—
35#11924037—84no—
36C2×#10596833—44no—
37C22×Dic73—22no—
38C22×Dic73—22no—
39C22×Dic73—22no—
40C22×Dic73—44no—
41C22×Dic73—42no—
42C22×Dic73—44no—
43C22×Dic73C22⋊C412yes—
44C22×Dic73C2×D412yes—
45C22⋊Dic73—41no—
46C22⋊Dic73—42no—
47C22⋊Dic73—41no—
48C22⋊Dic73—42no—
49C22⋊Dic73—42no—
50C24×C73—22no—
51C24×C73—22no—
52C24×C73—22no—
53C24×C73—44no—
54C24×C73C22⋊C411yescharacteristic
55C24×C73C22⋊C411yescharacteristic
56C24×C73C22×C411yescharacteristic
57C22×C4×C73—24no—
58C22×C4×C73—24no—
59C22×C4×C73—22no—
60C22×C4×C73—22no—
61C22×C4×C73—416no—
62C22×C4×C73—42no—
63C22×C4×C73—42no—
64C22×C4×C73—42no—
65C22×C4×C73—42no—
66C22×C4×C73—42no—
67C22×C4×C73—42no—
68C22×C4×C73M1614yescomplemented
69C23×C73—24no—
70C23×C73—21no—
71C23×C73—21no—
72C23×C73—28no—
73C23×C73—21no—
74C23×C73—21no—
75C23×C73—21no—
76C23×C73—21no—
77C23×C73—42no—
78C23×C73—42no—
79C23×C73—48no—
80C23×C73—48no—
81C23×C73—44no—
82C23×C73—44no—
83C23×C73#9912yes—
84C23×C73#10012yes—
85C23×C73C2×C22⋊C411yescharacteristic
86C23×C73C2×M1612yes—
87C2×C4×C73—24no—
88C2×C4×C73—48no—
89C2×C4×C73—48no—
90C2×C4×C73—44no—
91C2×C4×C73—42no—
92C2×C4×C73—42no—
93C2×C4×C73—44no—
94C2×Dic73—22no—
95C2×Dic73—22no—
96C2×Dic73—44no—
97C2×Dic73—42no—
98C2×Dic73—42no—
99C2×Dic73—44no—
100C2×Dic73—42no—
101C2×Dic73—82no—
102C2×Dic73—84no—
103#10596833—84no—
104C22×C73—24no—
105C22×C73—24no—
106C22×C73—22no—
107C22×C73—22no—
108C22×C73—22no—
109C22×C73—22no—
110C22×C73—22no—
111C22×C73—42no—
112C22×C73—42no—
113C22×C73—42no—
114C22×C73—416no—
115C22×C73—42no—
116C22×C73—42no—
117C22×C73—42no—
118C22×C73#35214yes—
119C22×C73#40711yescharacteristic
120C22×C73C2×#9911yescharacteristic
121C22×C73C2×#10011yescharacteristic
122C4×C73—44no—
123Dic73—42no—
124Dic73—42no—
125Dic73—82no—
126#11500—731nomaximal, Sylow
127C2×C73—21no—
128C2×C73—24no—
129C2×C73—21no—
130C2×C73—42no—
131C2×C73—42no—
132C2×C73#146211yescharacteristic
133C2×C73C2×#35212yes—
134C2×#381—731no—
135#1462—732no—
136#1317—734no—
137C73#1150011yescharacteristic, complemented, Sylow
138C24×C4#6440811yescharacteristic
139C22×C22⋊C4—731no—
140C22×C22⋊C4—731no—
141#407—1462no—
142#381—732no—
143#381—732no—
144#381—1462no—
145C2×#95—1462no—
146#323—734no—
147#323—1462no—
148C25C2×#6440811yescharacteristic
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—24no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—732no—
155C23×C4—1461no—
156C23×C4C2×#6440812yes—
157C2×M16—2922no—
158C2×C22⋊C4—2922no—
159C2×C22⋊C4—2922no—
160C2×C22⋊C4—732no—
161C2×C22⋊C4—731no—
162C2×C22⋊C4—731no—
163C2×C22⋊C4—732no—
164C2×C22⋊C4—1461no—
165C22⋊C8—2922no—
166C22⋊C8—1464no—
167#95—1464no—
168#95—1464no—
169C24—22no—
170C24—22no—
171C24—22no—
172C24—44no—
173C24C22⋊#6440811yescharacteristic
174C24C22⋊#6440811yescharacteristic
175C24C22×#6440811yescharacteristic
176C22×C4—24no—
177C22×C4—24no—
178C22×C4—22no—
179C22×C4—22no—
180C22×C4—416no—
181C22×C4—42no—
182C22×C4—42no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4—42no—
187C22×C4—2924no—
188C22×C4—2922no—
189C22×C4—2924no—
190C22×C4—732no—
191C22×C4—732no—
192C22×C4—1462no—
193C22×C4—1462no—
194C22×C4—1462no—
195C22×C4#1192403714yescomplemented
196M16—5844no—
197C2×C8—2924no—
198C22⋊C4—2922no—
199C22⋊C4—2922no—
200C22⋊C4—2921no—
201C22⋊C4—2922no—
202C22⋊C4—2921no—
203C23—24no—
204C23—21no—
205C23—21no—
206C23—28no—
207C23—21no—
208C23—21no—
209C23—21no—
210C23—21no—
211C23—42no—
212C23—42no—
213C23—48no—
214C23—48no—
215C23—44no—
216C23—44no—
217C23#1455664912yes—
218C23#1455665112yes—
219C23C2×C22⋊#6440811yescharacteristic
220C23C2×#1192403712yes—
221C2×C4—24no—
222C2×C4—48no—
223C2×C4—44no—
224C2×C4—48no—
225C2×C4—42no—
226C2×C4—42no—
227C2×C4—44no—
228C2×C4—2924no—
229C2×C4—2922no—
230C2×C4—2922no—
231C2×C4—2924no—
232C2×C4—2922no—
233C2×C4—5842no—
234C2×C4—5844no—
235C2×C4—1462no—
236C2×C4—1462no—
237C8—5844no—
238C22—24no—
239C22—24no—
240C22—22no—
241C22—22no—
242C22—22no—
243C22—22no—
244C22—22no—
245C22—42no—
246C22—42no—
247C22—42no—
248C22—416no—
249C22—42no—
250C22—42no—
251C22—42no—
252C22#1798948914yes—
253C22#1798955711yescharacteristic
254C22C2×#1455664911yescharacteristic
255C22C2×#1455665111yescharacteristic
256C4—44no—
257C4—2922no—
258C4—2922no—
259C4—5842no—
260C2—21no—
261C2—24no—
262C2—21no—
263C2—42no—
264C2—42no—
265C2#2262517611yescharacteristic
266C2C2×#1798948912yes—
subgroupG/H
centerC22C2×#14556649
commutatorC24×C73C22×C4
FrattiniC24C22×#64408
FittingC24×C4×C73C4
socleC22×C73C2×#99
prime radicalC25×C73C2×C4
cosocleC2×C22⋊Dic73C23
hypercenterC24×C4#64408
hypocenterC73#11500
LCSG ▷ C24×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C24 ◁ C25 ◁ C24×C4

Complements

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

NH ≅ G/Ntype
2. #22624857C2semidirect
68. C22×C4×C73M16semidirect
137. C73#11500semidirect
195. C22×C4#11924037semidirect

Maximal subgroups

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

subgroupindex
#226251762
#226248572
C2×#179884502
#1150073