Finite Group Databasebeta

← #32983424

Subgroups — #32983424

Order 18688.

618 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 111 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#22615311C211yesmaximal, characteristic
2#22608450C212yesmaximal
3#22608261C211yesmaximal, characteristic
4#22607817C212yesmaximal
5C73×#2274C211yesmaximal, characteristic
6C2×#14556570C2211yescharacteristic
7#17988551C2212yes—
8#17988517C412yescomplemented
9#17988507C412yescomplemented
10#17988482—22no—
11#17988460—21no—
12#17988460—21no—
13C4×C22⋊Dic73C2211yescharacteristic
14#17988323—24no—
15#17988313—22no—
16C2×#117×C73—21no—
17C22×C4⋊C4×C73C2211yescharacteristic
18C4×C4⋊C4×C73—24no—
19C4×C22⋊C4×C73—21no—
20C4×C22⋊C4×C73C2212yes—
21C22×Dic146C2×C411yescharacteristic
22C2×#11924012—22no—
23#14556570—22no—
24#14556570—22no—
25C2×C4⋊Dic73C2×C411yescharacteristic
26C2×#11924007—22no—
27C2×#11924007—22no—
28C2×#11924007—21no—
29C2×C22⋊Dic73—21no—
30C2×C22⋊Dic73—21no—
31C2×C4×Dic73—21no—
32C22⋊#10596831—22no—
33C22⋊#10596831—22no—
34#14556498—22no—
35#14556498—22no—
36#14556498C2×C412yes—
37C23×C4×C73D411yescharacteristic
38C23×C4×C73D411yescharacteristic
39C23×C4×C73C2311yescharacteristic
40#117×C73—28no—
41C2×C4⋊C4×C73—22no—
42C2×C4⋊C4×C73—22no—
43C2×C4⋊C4×C73—22no—
44C2×C4⋊C4×C73C2×C412yes—
45C2×C4⋊C4×C73D412yes—
46C2×C22⋊C4×C73—22no—
47C2×C42×C73—22no—
48C2×C42×C73—22no—
49C2×C42×C73—22no—
50#95×C73—22no—
51#95×C73D414yes—
52C2×Dic146—22no—
53C2×Dic146—81no—
54C22×Dic73—22no—
55C22×Dic73—21no—
56C22×Dic73—21no—
57#11924012—82no—
58C2×#10596831—42no—
59C4⋊Dic73—22no—
60#11924007—44no—
61#11924007—42no—
62#11924007—82no—
63C22⋊Dic73—24no—
64C22⋊Dic73—41no—
65C22⋊Dic73—41no—
66C4×Dic73—81no—
67C24×C73C2×D411yescharacteristic
68C22×C4×C73—21no—
69C22×C4×C73—21no—
70C22×C4×C73—24no—
71C22×C4×C73—21no—
72C22×C4×C73—24no—
73C22×C4×C73—21no—
74C22×C4×C73—81no—
75C22×C4×C73—81no—
76C22×C4×C73C22⋊C412yes—
77C22×C4×C73C22⋊C412yes—
78C22×C4×C73C22×C411yescharacteristic
79C22×C4×C73C2×D412yes—
80C22×C4×C73C2×D411yescharacteristic
81C22×C4×C73#4112yes—
82C4⋊C4×C73—28no—
83C4⋊C4×C73—24no—
84C4⋊C4×C73—24no—
85C22⋊C4×C73—28no—
86C42×C73—28no—
87C42×C73—82no—
88C42×C73—82no—
89C23×C73—22no—
90C23×C73—22no—
91C23×C73—41no—
92C23×C73C2×C22⋊C411yescharacteristic
93C23×C73#12011yescharacteristic
94C23×C73#12311yescharacteristic
95C2×C4×C73—22no—
96C2×C4×C73—22no—
97C2×C4×C73—24no—
98C2×C4×C73—22no—
99C2×C4×C73—22no—
100C2×C4×C73—28no—
101C2×C4×C73—22no—
102C2×C4×C73—42no—
103C2×C4×C73—41no—
104C2×C4×C73—41no—
105C2×C4×C73—42no—
106C2×C4×C73—44no—
107C2×C4×C73—42no—
108C2×C4×C73—44no—
109C2×C4×C73—42no—
110C2×C4×C73—81no—
111C2×C4×C73#10414yes—
112C2×C4×C73C4×D412yes—
113C2×C4×C73#12112yes—
114C2×Dic73—24no—
115C2×Dic73—42no—
116C2×Dic73—42no—
117C2×Dic73—42no—
118C2×Dic73—41no—
119C2×Dic73—41no—
120C2×Dic73—42no—
121Dic146—82no—
122#10596831—82no—
123C22×C73—21no—
124C22×C73—21no—
125C22×C73—21no—
126C22×C73—21no—
127C22×C73—21no—
128C22×C73—21no—
129C22×C73—21no—
130C22×C73—44no—
131C22×C73#38611yescharacteristic
132C22×C73C2×#10412yes—
133C22×C73#45612yes—
134C4×C73—24no—
135C4×C73—22no—
136C4×C73—22no—
137C4×C73—48no—
138C4×C73—42no—
139Dic73—44no—
140Dic73—81no—
141Dic73—81no—
142#9408—731nomaximal, Sylow
143C2×C73—22no—
144C2×C73—22no—
145C2×C73—41no—
146C2×C73#188611yescharacteristic
147C2×C73#189611yescharacteristic
148C2×C73#191311yescharacteristic
149#2583—731no—
150#2274D7311yescharacteristic
151#1508—732no—
152#1463—731no—
153#1297—732no—
154C73#940811yescharacteristic, complemented, Sylow
155C2×C22⋊Q8—731no—
156C2×#117—21no—
157C22×C4⋊C4C2×D7311yescharacteristic
158#407—732no—
159#398—732no—
160#395—732no—
161#389—1462no—
162#385—1461no—
163#385—1461no—
164C4×C4⋊C4—24no—
165C4×C22⋊C4—21no—
166C4×C22⋊C4—731no—
167C4×C22⋊C4C2×D7312yes—
168#328—1464no—
169#324—1462no—
170C22×Q8—731no—
171C23×C4C22⋊D7311yescharacteristic
172C23×C4C22⋊D7311yescharacteristic
173C23×C4C22×D7311yescharacteristic
174C2×M16—1462no—
175C22⋊Q8—1462no—
176C22⋊Q8—1462no—
177#117—28no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—731no—
182C2×C4⋊C4—1462no—
183C2×C4⋊C4—1461no—
184C2×C4⋊C4—1462no—
185C2×C4⋊C4C4×D7312yes—
186C2×C4⋊C4D29212yes—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—1461no—
189C2×C22⋊C4—1461no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—1461no—
194C22⋊C8—1462no—
195C22⋊C8—1462no—
196#95—22no—
197#95—732no—
198#95—1462no—
199#95—1462no—
200#95C22⋊D7314yes—
201C24C2×C22⋊D7311yescharacteristic
202C2×Q8—5841no—
203C2×Q8—1462no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—24no—
207C22×C4—21no—
208C22×C4—24no—
209C22×C4—21no—
210C22×C4—81no—
211C22×C4—81no—
212C22×C4—1461no—
213C22×C4—1461no—
214C22×C4—1462no—
215C22×C4#1192400512yes—
216C22×C4#1192400512yes—
217C22×C4C2×C4×D7311yescharacteristic
218C22×C4C2×C22⋊D7312yes—
219C22×C4C2×D29211yescharacteristic
220C22×C4#1192402812yes—
221M16—5842no—
222C2×C8—2922no—
223C4⋊C4—28no—
224C4⋊C4—24no—
225C4⋊C4—24no—
226C4⋊C4—2924no—
227C4⋊C4—2922no—
228C4⋊C4—5842no—
229C4⋊C4—1462no—
230C22⋊C4—28no—
231C22⋊C4—2921no—
232C22⋊C4—2921no—
233C22⋊C4—1464no—
234C42—28no—
235C42—82no—
236C42—82no—
237C42—5841no—
238C23—22no—
239C23—22no—
240C23—41no—
241C23C2×#1192400511yescharacteristic
242C23#1455656011yescharacteristic
243C23#1455657411yescharacteristic
244Q8—5842no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—28no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—41no—
256C2×C4—44no—
257C2×C4—42no—
258C2×C4—44no—
259C2×C4—42no—
260C2×C4—81no—
261C2×C4—2922no—
262C2×C4—2922no—
263C2×C4—2922no—
264C2×C4—2921no—
265C2×C4—2921no—
266C2×C4—2922no—
267C2×C4—1464no—
268C2×C4#1455651914yes—
269C2×C4C4×C22⋊D7312yes—
270C2×C4#1455656512yes—
271C8—5842no—
272C22—21no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—44no—
280C22#1798846611yescharacteristic
281C22C2×#1455651912yes—
282C22#1798877012yes—
283C4—22no—
284C4—22no—
285C4—24no—
286C4—48no—
287C4—42no—
288C4—2924no—
289C4—5841no—
290C4—5841no—
291C2—22no—
292C2—22no—
293C2—41no—
294C2#2261032411yescharacteristic
295C2#2261037411yescharacteristic
296C2#2261047311yescharacteristic
subgroupG/H
centerC22#17988466
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC73×#2274C2
socleC22×C73#386
prime radicalC24×C73C2×D4
cosocleC23×C4×C73C23
hypercenter#2274D73
hypocenterC73#9408
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2274

Complements

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

NH ≅ G/Ntype
8. #17988517C4semidirect
9. #17988507C4semidirect
154. C73#9408semidirect

Maximal subgroups

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

subgroupindex
#226153112
#226084502
#226082612
#226078172
C73×#22742
#940873