Finite Group Databasebeta

← #33002250

Subgroups — #33002250

Order 18688.

638 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 115 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#22609855C212yesmaximal
2#22608058C214yesmaximal, complemented
3C2×#391×C73C211yesmaximal, characteristic
4C2×#14556510—22no—
5C2×#14556510C2214yes—
6#17988551—22no—
7C23×Q8×C73C2211yescharacteristic
8C22×C4⋊C4×C73C411yescharacteristic
9C22×C42×C73C411yescharacteristic
10#391×C73—22no—
11#391×C73C412yes—
12#391×C73C2212yes—
13C2×#95×C73—22no—
14C2×#11924012—24no—
15#14556510—28no—
16#14556510—44no—
17C22⋊#10596831—42no—
18C22⋊#10596831—42no—
19C22×Q8×C73—24no—
20C22×Q8×C73—41no—
21C22×Q8×C73—41no—
22C22×Q8×C73—42no—
23C22×Q8×C73C2×C411yescharacteristic
24C22×Q8×C73D412yes—
25C22×Q8×C73C2311yescharacteristic
26C23×C4×C73—21no—
27C23×C4×C73—21no—
28C23×C4×C73C2×C411yescharacteristic
29C23×C4×C73D412yes—
30C2×C4⋊C4×C73—21no—
31C2×C4⋊C4×C73—41no—
32C2×C4⋊C4×C73—41no—
33C2×C4⋊C4×C73C2×C412yes—
34C2×C42×C73—21no—
35C2×C42×C73—42no—
36C2×C42×C73C2×C412yes—
37#95×C73—24no—
38#95×C73—24no—
39#11924012—44no—
40#11924012—44no—
41C2×#10596831—44no—
42C24×C73C22⋊C411yescharacteristic
43C2×Q8×C73—24no—
44C2×Q8×C73—22no—
45C2×Q8×C73—24no—
46C2×Q8×C73—41no—
47C2×Q8×C73—41no—
48C2×Q8×C73—42no—
49C2×Q8×C73—41no—
50C2×Q8×C73—41no—
51C2×Q8×C73—48no—
52C2×Q8×C73—48no—
53C2×Q8×C73#4114yes—
54C22×C4×C73—21no—
55C22×C4×C73—21no—
56C22×C4×C73—22no—
57C22×C4×C73—22no—
58C22×C4×C73—24no—
59C22×C4×C73—22no—
60C22×C4×C73—22no—
61C22×C4×C73—22no—
62C22×C4×C73—24no—
63C22×C4×C73—22no—
64C22×C4×C73—42no—
65C22×C4×C73—42no—
66C22×C4×C73—42no—
67C22×C4×C73—44no—
68C22×C4×C73—41no—
69C22×C4×C73—41no—
70C22×C4×C73C22⋊C412yes—
71C22×C4×C73C22⋊C411yescharacteristic
72C22×C4×C73C22×C411yescharacteristic
73C22×C4×C73C2×D412yes—
74C4⋊C4×C73—42no—
75C4⋊C4×C73—42no—
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
85Q8×C73—44no—
86Q8×C73—48no—
87Q8×C73—44no—
88C2×C4×C73—24no—
89C2×C4×C73—24no—
90C2×C4×C73—21no—
91C2×C4×C73—21no—
92C2×C4×C73—21no—
93C2×C4×C73—21no—
94C2×C4×C73—22no—
95C2×C4×C73—22no—
96C2×C4×C73—24no—
97C2×C4×C73—22no—
98C2×C4×C73—22no—
99C2×C4×C73—22no—
100C2×C4×C73—44no—
101C2×C4×C73—44no—
102C2×C4×C73—48no—
103C2×C4×C73—42no—
104C2×C4×C73—42no—
105C2×C4×C73—44no—
106C2×C4×C73—42no—
107C2×C4×C73—82no—
108C2×C4×C73—82no—
109C2×C4×C73#9912yes—
110C2×C4×C73#11712yes—
111C2×C4×C73#12314yes—
112#10596831—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—24no—
127C4×C73—22no—
128C4×C73—22no—
129C4×C73—44no—
130C4×C73—42no—
131C4×C73—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×#391D7311yescharacteristic
141#1784—732no—
142#1394—734no—
143C73#1286011yescharacteristic, complemented, Sylow
144C23×Q8C2×D7311yescharacteristic
145C22×C4⋊C4Dic7311yescharacteristic
146C22×C42Dic7311yescharacteristic
147C2×#101—734no—
148C2×#101—1462no—
149#407—1462no—
150#391—22no—
151#391Dic7312yes—
152#391C2×D7312yes—
153C2×#95—22no—
154C22×Q8—24no—
155C22×Q8—41no—
156C22×Q8—41no—
157C22×Q8—42no—
158C22×Q8C2×Dic7311yescharacteristic
159C22×Q8C22⋊D7312yes—
160C22×Q8C22×D7311yescharacteristic
161C23×C4—21no—
162C23×C4—21no—
163C23×C4C2×Dic7311yescharacteristic
164C23×C4C22⋊D7312yes—
165C2×M16—1464no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4C2×Dic7312yes—
170C2×C42—21no—
171C2×C42—42no—
172C2×C42C2×Dic7312yes—
173#101—2924no—
174#101—1468no—
175C22⋊C8—2922no—
176C22⋊C8—2922no—
177#95—24no—
178#95—24no—
179C24C22⋊Dic7311yescharacteristic
180C2×Q8—24no—
181C2×Q8—22no—
182C2×Q8—24no—
183C2×Q8—41no—
184C2×Q8—41no—
185C2×Q8—42no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8—48no—
189C2×Q8—48no—
190C2×Q8#1192402814yes—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—24no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—24no—
200C22×C4—22no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—44no—
205C22×C4—41no—
206C22×C4—41no—
207C22×C4C22⋊Dic7312yes—
208C22×C4C22⋊Dic7311yescharacteristic
209C22×C4C22×Dic7311yescharacteristic
210C22×C4C2×C22⋊D7312yes—
211M16—2924no—
212M16—2924no—
213C2×C8—2924no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C42—42no—
217C42—42no—
218C23—22no—
219C23—22no—
220C23—21no—
221C23—21no—
222C23#1455650611yescharacteristic
223C23#1455650611yescharacteristic
224C23C2×C22⋊Dic7311yescharacteristic
225Q8—44no—
226Q8—48no—
227Q8—44no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—22no—
236C2×C4—24no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—44no—
241C2×C4—44no—
242C2×C4—48no—
243C2×C4—42no—
244C2×C4—42no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—82no—
248C2×C4—82no—
249C2×C4#1455650612yes—
250C2×C4#1455655012yes—
251C2×C4#1455657414yes—
252C8—5844no—
253C22—24no—
254C22—24no—
255C22—21no—
256C22—21no—
257C22—21no—
258C22—21no—
259C22—21no—
260C22—21no—
261C22#1798838412yes—
262C22#1798838612yes—
263C22#1798845011yescharacteristic
264C22C2×#1455650611yescharacteristic
265C22C2×#1455650611yescharacteristic
266C4—24no—
267C4—22no—
268C4—22no—
269C4—44no—
270C4—42no—
271C4—82no—
272C2—21no—
273C2—22no—
274C2—22no—
275C2—21no—
276C2#2261000511yescharacteristic
277C2C2×#1798838411yescharacteristic
278C2C2×#1798838611yescharacteristic
subgroupG/H
centerC22C2×#14556506
commutatorC22×C4×C73C22×C4
FrattiniC22×Q8C22×D73
FittingC2×#391×C73C2
socleC22×C73C2×#99
prime radicalC24×C73C22⋊C4
cosocleC22×Q8×C73C23
hypercenterC2×#391D73
hypocenterC73#12860
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
2. #22608058C2semidirect
143. C73#12860semidirect

Maximal subgroups

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

subgroupindex
#226098552
#226080582
C2×#391×C732
#1286073