Finite Group Databasebeta

← #34013228

Subgroups — #34013228

Order 18688.

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

normal subgroups only

This group has 638 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#22625765C212yesmaximal
2#22625028C214yesmaximal, complemented
3C2×#17988493C211yesmaximal, characteristic
4C2×#14556652—22no—
5C2×#14556652C2214yes—
6#17989557—22no—
7C23×Dic146C2211yescharacteristic
8C22×C4⋊Dic73C411yescharacteristic
9#17988493—22no—
10#17988493C412yes—
11#17988493C2212yes—
12C2×#14556498—22no—
13C22×C42×C73C411yescharacteristic
14C2×#11924037—24no—
15#14556652—28no—
16#14556652—44no—
17C22⋊#10596833—42no—
18C22⋊#10596833—42no—
19C22×Dic146—24no—
20C22×Dic146—41no—
21C22×Dic146—41no—
22C22×Dic146—42no—
23C22×Dic146C2×C411yescharacteristic
24C22×Dic146D412yes—
25C22×Dic146C2311yescharacteristic
26C23×Dic73—21no—
27C23×Dic73D412yes—
28C2×C4⋊Dic73—21no—
29C2×C4⋊Dic73—41no—
30C2×C4⋊Dic73—41no—
31C2×C4⋊Dic73C2×C412yes—
32#14556498—24no—
33#14556498—24no—
34C23×C4×C73—21no—
35C23×C4×C73C2×C411yescharacteristic
36C2×C42×C73—21no—
37C2×C42×C73—42no—
38C2×C42×C73C2×C412yes—
39#11924037—44no—
40#11924037—44no—
41C2×#10596833—44no—
42C2×Dic146—24no—
43C2×Dic146—22no—
44C2×Dic146—24no—
45C2×Dic146—41no—
46C2×Dic146—41no—
47C2×Dic146—42no—
48C2×Dic146—41no—
49C2×Dic146—41no—
50C2×Dic146—48no—
51C2×Dic146—48no—
52C2×Dic146#4114yes—
53C22×Dic73—22no—
54C22×Dic73—22no—
55C22×Dic73—24no—
56C22×Dic73—22no—
57C22×Dic73—42no—
58C22×Dic73—44no—
59C22×Dic73—41no—
60C22×Dic73—41no—
61C22×Dic73C22⋊C412yes—
62C22×Dic73C2×D412yes—
63C4⋊Dic73—42no—
64C4⋊Dic73—42no—
65C24×C73C22⋊C411yescharacteristic
66C22×C4×C73—21no—
67C22×C4×C73—21no—
68C22×C4×C73—22no—
69C22×C4×C73—22no—
70C22×C4×C73—24no—
71C22×C4×C73—22no—
72C22×C4×C73—42no—
73C22×C4×C73—42no—
74C22×C4×C73C22⋊C411yescharacteristic
75C22×C4×C73C22×C411yescharacteristic
76C42×C73—42no—
77C42×C73—42no—
78C23×C73—22no—
79C23×C73—22no—
80C23×C73—21no—
81C23×C73—21no—
82C23×C73#9911yescharacteristic
83C23×C73#9911yescharacteristic
84C23×C73C2×C22⋊C411yescharacteristic
85C2×C4×C73—24no—
86C2×C4×C73—24no—
87C2×C4×C73—21no—
88C2×C4×C73—21no—
89C2×C4×C73—21no—
90C2×C4×C73—21no—
91C2×C4×C73—24no—
92C2×C4×C73—48no—
93C2×C4×C73—42no—
94C2×C4×C73—42no—
95C2×C4×C73#9912yes—
96C2×C4×C73#11712yes—
97C2×Dic73—22no—
98C2×Dic73—22no—
99C2×Dic73—22no—
100C2×Dic73—22no—
101C2×Dic73—22no—
102C2×Dic73—44no—
103C2×Dic73—44no—
104C2×Dic73—44no—
105C2×Dic73—42no—
106C2×Dic73—82no—
107C2×Dic73—82no—
108C2×Dic73#12314yes—
109Dic146—44no—
110Dic146—48no—
111Dic146—44no—
112#10596833—84no—
113C22×C73—24no—
114C22×C73—24no—
115C22×C73—21no—
116C22×C73—21no—
117C22×C73—21no—
118C22×C73—21no—
119C22×C73—21no—
120C22×C73—21no—
121C22×C73#35512yes—
122C22×C73#35612yes—
123C22×C73#38111yescharacteristic
124C22×C73C2×#9911yescharacteristic
125C22×C73C2×#9911yescharacteristic
126C4×C73—22no—
127C4×C73—22no—
128C4×C73—44no—
129Dic73—24no—
130Dic73—42no—
131Dic73—82no—
132#12860—731nomaximal, Sylow
133C2×C73—21no—
134C2×C73—22no—
135C2×C73—22no—
136C2×C73—21no—
137C2×C73#181911yescharacteristic
138C2×C73C2×#35511yescharacteristic
139C2×C73C2×#35611yescharacteristic
140C2×#391—731no—
141#1784—732no—
142#1394—734no—
143C73#1286011yescharacteristic, complemented, Sylow
144C23×Q8—731no—
145C22×C4⋊C4—731no—
146C22×C42#6440811yescharacteristic
147C2×#101—734no—
148C2×#101—1462no—
149#407—1462no—
150#391—732no—
151#391—732no—
152#391—1462no—
153C2×#95—1462no—
154C22×Q8—2922no—
155C22×Q8—2921no—
156C22×Q8—2921no—
157C22×Q8—732no—
158C22×Q8—731no—
159C22×Q8—731no—
160C22×Q8—1464no—
161C23×C4—21no—
162C23×C4—732no—
163C23×C4—1461no—
164C23×C4C2×#6440811yescharacteristic
165C2×M16—1464no—
166C2×C4⋊C4—2921no—
167C2×C4⋊C4—2921no—
168C2×C4⋊C4—732no—
169C2×C4⋊C4—1461no—
170C2×C42—21no—
171C2×C42—42no—
172C2×C42C2×#6440812yes—
173#101—2924no—
174#101—1468no—
175C22⋊C8—2922no—
176C22⋊C8—2922no—
177#95—1464no—
178#95—1464no—
179C24C22⋊#6440811yescharacteristic
180C2×Q8—2928no—
181C2×Q8—2928no—
182C2×Q8—2921no—
183C2×Q8—2921no—
184C2×Q8—2922no—
185C2×Q8—2921no—
186C2×Q8—2921no—
187C2×Q8—734no—
188C2×Q8—1464no—
189C2×Q8—1464no—
190C2×Q8—1462no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—22no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—2922no—
200C22×C4—2924no—
201C22×C4—2921no—
202C22×C4—2921no—
203C22×C4—732no—
204C22×C4—732no—
205C22×C4—1462no—
206C22×C4—1464no—
207C22×C4—1462no—
208C22×C4—1462no—
209C22×C4C22⋊#6440811yescharacteristic
210C22×C4C22×#6440811yescharacteristic
211M16—2924no—
212M16—2924no—
213C2×C8—2924no—
214C4⋊C4—2922no—
215C4⋊C4—2922no—
216C42—42no—
217C42—42no—
218C23—22no—
219C23—22no—
220C23—21no—
221C23—21no—
222C23#1455665011yescharacteristic
223C23#1455665011yescharacteristic
224C23C2×C22⋊#6440811yescharacteristic
225Q8—2924no—
226Q8—2928no—
227Q8—2924no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—24no—
235C2×C4—48no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4—2924no—
239C2×C4—2924no—
240C2×C4—2924no—
241C2×C4—2922no—
242C2×C4—5842no—
243C2×C4—5842no—
244C2×C4—734no—
245C2×C4—1462no—
246C2×C4—1462no—
247C2×C4—1462no—
248C2×C4—1462no—
249C2×C4—1462no—
250C2×C4#1455664912yes—
251C2×C4#1455667012yes—
252C8—5844no—
253C22—24no—
254C22—24no—
255C22—21no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22#1798949612yes—
262C22#1798949812yes—
263C22#1798953011yescharacteristic
264C22C2×#1455664911yescharacteristic
265C22C2×#1455665011yescharacteristic
266C4—22no—
267C4—22no—
268C4—44no—
269C4—2922no—
270C4—5842no—
271C4—1464no—
272C2—21no—
273C2—22no—
274C2—22no—
275C2—21no—
276C2#2262582611yescharacteristic
277C2C2×#1798949611yescharacteristic
278C2C2×#1798949811yescharacteristic
subgroupG/H
centerC22C2×#14556650
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
FittingC22×C42×C73C4
socleC22×C73C2×#99
prime radicalC24×C73C22⋊C4
cosocleC22×Dic146C23
hypercenterC22×C42#64408
hypocenterC73#12860
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. #22625028C2semidirect
143. C73#12860semidirect

Maximal subgroups

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

subgroupindex
#226257652
#226250282
C2×#179884932
#1286073