Finite Group Databasebeta

← #32995147

Subgroups — #32995147

Order 18688.

434 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 93 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#22612556C211yesmaximal, characteristic
2#22608244C212yesmaximal
3#22607875C211yesmaximal, characteristic, complemented
4#22607874C212yesmaximal, complemented
5C73×#1324C211yesmaximal, characteristic, complemented
6C2×#14556549C2211yescharacteristic
7C2×#14556548C411yescharacteristic
8C2×C4⋊C4×D73C411yescharacteristic
9#17988637—22no—
10#17988484—22no—
11#17988461—22no—
12#17988457C411yescharacteristic
13#17988456C411yescharacteristic
14#17988452C2211yescharacteristic
15#17988313C2212yes—
16#17988310—22no—
17#382×C73C2211yescharacteristic
18#324×C73C2212yes—
19C22×C4×D73—21no—
20C22×C4×D73C2×C411yescharacteristic
21#14556549—42no—
22#14556548—22no—
23#14556548C814yescomplemented
24C2×C4⋊Dic73—22no—
25C2×C4⋊Dic73C2×C411yescharacteristic
26C2×C4⋊Dic73C2×C411yescharacteristic
27C2×#11924007—21no—
28C2×#11924007—21no—
29C2×#11924007—22no—
30C4⋊C4×D73—42no—
31C2×#11924005—21no—
32C2×#11924005D412yes—
33C2×C4×Dic73C2×C411yescharacteristic
34C2×C4×Dic73D412yes—
35#14556520—42no—
36C22⋊#10596831—22no—
37C8×Dic73—42no—
38#14556498—22no—
39C2×C4⋊C4×C73C2×C411yescharacteristic
40C2×C4⋊C4×C73C2311yescharacteristic
41C2×C42×C73C2×C411yescharacteristic
42C22⋊C8×C73—22no—
43#95×C73—22no—
44C23×D73C22⋊C411yescharacteristic
45C22×Dic73—21no—
46C22×Dic73—21no—
47C22×Dic73C22⋊C412yes—
48C22×Dic73C22⋊C411yescharacteristic
49C2×C4×D73—22no—
50C2×C4×D73—22no—
51C2×C4×D73—42no—
52C2×C4×D73—82no—
53C2×C4×D73C2×C812yes—
54C2×C4×D73M1612yes—
55C2×#10596831—42no—
56C4⋊Dic73—41no—
57C4⋊Dic73—41no—
58C4⋊Dic73—82no—
59#11924007—24no—
60#11924007—42no—
61#11924007—82no—
62#11924005—24no—
63#11924005—42no—
64C4×Dic73—21no—
65C4×Dic73—22no—
66C4×Dic73—22no—
67C4×Dic73C2×C812yes—
68C22×C4×C73—21no—
69C22×C4×C73—21no—
70C22×C4×C73C22×C411yescharacteristic
71C22×C4×C73C2×D412yes—
72C2×C8×C73—42no—
73C4⋊C4×C73—41no—
74C4⋊C4×C73—41no—
75C42×C73—21no—
76C42×C73C2×C812yes—
77C23×C73C2×C22⋊C411yescharacteristic
78C22×D73—21no—
79C22×D73—21no—
80C22×D73—41no—
81C22×D73—41no—
82C22×D73C22⋊C812yes—
83C2×C4×C73—21no—
84C2×C4×C73—21no—
85C2×C4×C73—22no—
86C2×C4×C73—22no—
87C2×C4×C73—41no—
88C2×C4×C73—41no—
89C2×C4×C73—42no—
90C2×C4×C73—42no—
91C2×C4×C73—41no—
92C2×C4×C73C22×C811yescharacteristic
93C2×C4×C73C2×M1611yescharacteristic
94C2×Dic73—22no—
95C2×Dic73—22no—
96C2×Dic73—41no—
97C2×Dic73—41no—
98C2×Dic73—42no—
99C2×Dic73—42no—
100C2×Dic73—41no—
101C2×Dic73C22⋊C812yes—
102C2×Dic73#9912yes—
103C2×Dic73#10112yes—
104C4×D73—24no—
105C4×D73—24no—
106C4×D73—84no—
107C8×C73—82no—
108#10596831—82no—
109C22×C73—21no—
110C22×C73—21no—
111C22×C73C2×C22⋊C811yescharacteristic
112C22×C73C2×#9911yescharacteristic
113C22×C73C2×#10111yescharacteristic
114C2×D73—22no—
115C2×D73—42no—
116C2×D73—41no—
117C2×D73—41no—
118C2×D73—42no—
119C4×C73—21no—
120C4×C73—21no—
121C4×C73—42no—
122C4×C73—42no—
123C4×C73—81no—
124Dic73—42no—
125Dic73—42no—
126Dic73—81no—
127Dic73#32414yes—
128#11449—731nomaximal, Sylow
129C2×C73—21no—
130C2×C73—21no—
131C2×C73C2×#32411yescharacteristic
132C2×C73#212911yescharacteristic
133C2×C73#213411yescharacteristic
134D73—42no—
135#2317—731no—
136#1458—732no—
137#1324—731no—
138#1324—732no—
139#1324D7311yescharacteristic, complemented
140C73#1144911yescharacteristic, complemented, Sylow
141C2×#117—731no—
142C2×#117—731no—
143C22×C4⋊C4—731no—
144#431—1462no—
145#389—1462no—
146#385—1462no—
147#384—731no—
148#384—731no—
149#382—731no—
150#382C2×D7311yescharacteristic
151#324—732no—
152#324C2×D7312yes—
153#323—1462no—
154C23×C4—731no—
155C23×C4—1461no—
156#117—2922no—
157#117—734no—
158#117—1462no—
159C2×C4⋊C4—2922no—
160C2×C4⋊C4—731no—
161C2×C4⋊C4—731no—
162C2×C4⋊C4—1461no—
163C2×C4⋊C4—1462no—
164C2×C4⋊C4—1462no—
165C2×C4⋊C4—1461no—
166C2×C4⋊C4C4×D7311yescharacteristic
167C2×C4⋊C4C22×D7311yescharacteristic
168C2×C22⋊C4—732no—
169C2×C22⋊C4—1461no—
170C2×C42—731no—
171C2×C42—732no—
172C2×C42C4×D7311yescharacteristic
173C4⋊C8—2922no—
174C22⋊C8—22no—
175C22⋊C8—1462no—
176C4×C8—2922no—
177#95—22no—
178#95—1462no—
179C24—731no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—2922no—
183C22×C4—5842no—
184C22×C4—732no—
185C22×C4—731no—
186C22×C4—732no—
187C22×C4—732no—
188C22×C4—1462no—
189C22×C4—1462no—
190C22×C4—1461no—
191C22×C4—1461no—
192C22×C4C2×C4×D7311yescharacteristic
193C22×C4D4×D7312yes—
194C2×C8—42no—
195C2×C8—2922no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—2921no—
199C4⋊C4—2921no—
200C4⋊C4—2922no—
201C4⋊C4—5842no—
202C4⋊C4—5842no—
203C4⋊C4—1464no—
204C22⋊C4—2922no—
205C22⋊C4—1464no—
206C42—21no—
207C42—732no—
208C42—1462no—
209C42—1462no—
210C42—1461no—
211C42C8×D7312yes—
212C23—2921no—
213C23—2921no—
214C23—732no—
215C23—1461no—
216C23—1461no—
217C23C22⋊C4×D7311yescharacteristic
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—2921no—
228C2×C4—2921no—
229C2×C4—2922no—
230C2×C4—2922no—
231C2×C4—2921no—
232C2×C4—5844no—
233C2×C4—732no—
234C2×C4—732no—
235C2×C4—732no—
236C2×C4—1464no—
237C2×C4—1464no—
238C2×C4—1462no—
239C2×C4—1462no—
240C2×C4C2×C8×D7311yescharacteristic
241C2×C4M16×D7311yescharacteristic
242C8—82no—
243C8—5842no—
244C22—21no—
245C22—21no—
246C22—2922no—
247C22—2922no—
248C22—2921no—
249C22—2921no—
250C22—1462no—
251C22C22⋊C8×D7311yescharacteristic
252C22#99×D7311yescharacteristic
253C22#101×D7311yescharacteristic
254C4—21no—
255C4—21no—
256C4—42no—
257C4—42no—
258C4—81no—
259C4—2922no—
260C4—2922no—
261C4—5841no—
262C4—734no—
263C2—21no—
264C2—21no—
265C2—2922no—
266C2#324×D7311yescharacteristic
267C2#2261160411yescharacteristic
268C2#2261162111yescharacteristic
subgroupG/H
centerC22#99×D73
commutatorC2×C4×C73C22×C8
FrattiniC2×C4⋊C4C22×D73
FittingC73×#1324C2
socleC22×C73C2×#99
prime radicalC23×D73C22⋊C4
cosocleC2×C4⋊C4×C73C23
hypercenter#1324D73
hypocenterC73#11449
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1324

Complements

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

NH ≅ G/Ntype
3. #22607875C2semidirect
4. #22607874C2semidirect
5. C73×#1324C2semidirect
23. #14556548C8semidirect
139. #1324D73semidirect
140. C73#11449semidirect

Maximal subgroups

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

subgroupindex
#226125562
#226082442
#226078752
#226078742
C73×#13242
#1144973