Finite Group Databasebeta

← #32983924

Subgroups — #32983924

Order 18688.

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

normal subgroups only

This group has 880 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#22618960C211yesmaximal, characteristic, complemented
2#22609852C212yesmaximal, complemented
3#22608269C211yesmaximal, characteristic, complemented
4#22607837C212yesmaximal, complemented
5C73×#2408C211yesmaximal, characteristic
6#17988659—24no—
7#17988644—21no—
8C2×#14556508—24no—
9#17988551C2212yes—
10#17988551C2212yes—
11C2×C22⋊#10596831—21no—
12#17988313—21no—
13#17988311—21no—
14C23×D4×C73C2211yescharacteristic
15C2×#121×C73—21no—
16C2×#120×C73C412yes—
17C2×C4×D4×C73C2211yescharacteristic
18#394×C73—21no—
19#392×C73—21no—
20#390×C73—21no—
21#380×C73C2211yescharacteristic
22#379×C73C412yes—
23C2×#11924012—22no—
24C2×#11924012D414yescomplemented
25C22×#10596831—22no—
26C4⋊#10596831—22no—
27#14556508—48no—
28C22⋊#10596831—22no—
29C22⋊#10596831—22no—
30C22⋊#10596831—22no—
31C22⋊#10596831—22no—
32C22⋊#10596831—22no—
33#14556502—22no—
34C25×C73C2×C412yes—
35C22×D4×C73—22no—
36C22×D4×C73—21no—
37C22×D4×C73—22no—
38C22×D4×C73—21no—
39C22×D4×C73—22no—
40C22×D4×C73—44no—
41C22×D4×C73C2×C411yescharacteristic
42C23×C4×C73D411yescharacteristic
43C23×C4×C73C2311yescharacteristic
44#121×C73—44no—
45#120×C73—44no—
46C4×D4×C73—22no—
47C4×D4×C73—22no—
48C2×C4⋊C4×C73—21no—
49C2×C4⋊C4×C73D411yescharacteristic
50C2×C22⋊C4×C73—22no—
51C2×C22⋊C4×C73—22no—
52C2×C22⋊C4×C73—22no—
53C2×C22⋊C4×C73C2×C411yescharacteristic
54C2×C22⋊C4×C73C2×C412yes—
55C2×C22⋊C4×C73D411yescharacteristic
56C2×C42×C73D411yescharacteristic
57#95×C73—21no—
58#11924012—44no—
59#11924012—44no—
60C2×#10596831—24no—
61C2×#10596831—42no—
62C2×#10596831—82no—
63C24×C73—22no—
64C24×C73—24no—
65C24×C73—44no—
66C24×C73—42no—
67C24×C73—42no—
68C24×C73—82no—
69C24×C73C22⋊C411yescharacteristic
70C24×C73C22×C411yescharacteristic
71C2×D4×C73—24no—
72C2×D4×C73—28no—
73C2×D4×C73—22no—
74C2×D4×C73—22no—
75C2×D4×C73—41no—
76C2×D4×C73—42no—
77C2×D4×C73—44no—
78C2×D4×C73—44no—
79C2×D4×C73—48no—
80C2×D4×C73—44no—
81C2×D4×C73—44no—
82C2×D4×C73—41no—
83C2×D4×C73—42no—
84C2×D4×C73—41no—
85C2×D4×C73—82no—
86C22×C4×C73—21no—
87C22×C4×C73—21no—
88C22×C4×C73—22no—
89C22×C4×C73—42no—
90C22×C4×C73—42no—
91C22×C4×C73C22⋊C411yescharacteristic
92C22×C4×C73C22⋊C411yescharacteristic
93C22×C4×C73C22⋊C411yescharacteristic
94C22×C4×C73C2×D412yes—
95C22×C4×C73C2×D411yescharacteristic
96C22×C4×C73C2×D411yescharacteristic
97C22×C4×C73#4112yes—
98C4⋊C4×C73—22no—
99C4⋊C4×C73—42no—
100C22⋊C4×C73—22no—
101C22⋊C4×C73—22no—
102C22⋊C4×C73—44no—
103C22⋊C4×C73—44no—
104C22⋊C4×C73—42no—
105C22⋊C4×C73—82no—
106C42×C73—21no—
107C42×C73—21no—
108C23×C73—21no—
109C23×C73—21no—
110C23×C73—21no—
111C23×C73—21no—
112C23×C73—24no—
113C23×C73—28no—
114C23×C73—41no—
115C23×C73—41no—
116C23×C73—41no—
117C23×C73—41no—
118C23×C73—42no—
119C23×C73—42no—
120C23×C73—44no—
121C23×C73—42no—
122C23×C73—42no—
123C23×C73—42no—
124C23×C73—42no—
125C23×C73—84no—
126C23×C73—84no—
127C23×C73—88no—
128C23×C73—88no—
129C23×C73#10012yescomplemented
130C23×C73C2×C22⋊C411yescharacteristic
131C23×C73C4×D412yes—
132D4×C73—48no—
133D4×C73—44no—
134D4×C73—44no—
135D4×C73—44no—
136D4×C73—88no—
137C2×C4×C73—22no—
138C2×C4×C73—24no—
139C2×C4×C73—21no—
140C2×C4×C73—21no—
141C2×C4×C73—21no—
142C2×C4×C73—21no—
143C2×C4×C73—21no—
144C2×C4×C73—41no—
145C2×C4×C73—42no—
146C2×C4×C73—42no—
147C2×C4×C73—41no—
148C2×C4×C73—41no—
149C2×C4×C73—42no—
150C2×C4×C73—42no—
151C2×C4×C73—42no—
152C2×C4×C73—81no—
153C2×C4×C73#12011yescharacteristic
154C2×C4×C73#12112yes—
155C2×C4×C73#12311yescharacteristic
156#10596831—44no—
157#10596831—82no—
158C22×C73—21no—
159C22×C73—21no—
160C22×C73—22no—
161C22×C73—22no—
162C22×C73—22no—
163C22×C73—22no—
164C22×C73—42no—
165C22×C73—42no—
166C22×C73—42no—
167C22×C73—44no—
168C22×C73—44no—
169C22×C73—41no—
170C22×C73—41no—
171C22×C73—44no—
172C22×C73—81no—
173C22×C73—82no—
174C22×C73—82no—
175C22×C73—88no—
176C22×C73—88no—
177C22×C73—82no—
178C22×C73—82no—
179C22×C73#38611yescharacteristic
180C22×C73C2×#10011yescharacteristic
181C22×C73#41311yescharacteristic
182C4×C73—22no—
183C4×C73—21no—
184C4×C73—41no—
185C4×C73—41no—
186C4×C73—41no—
187C4×C73—82no—
188C4×C73B(4,𝔽2)12yes—
189#9503—731nomaximal, Sylow
190C2×C73—21no—
191C2×C73—21no—
192C2×C73—42no—
193C2×C73—42no—
194C2×C73—84no—
195C2×C73#188911yescharacteristic
196C2×C73#190011yescharacteristic
197C2×C73#191411yescharacteristic
198#2973—731no—
199#2408D7311yescharacteristic
200#1783—732no—
201#1465—731no—
202#1305—732no—
203C73#950311yescharacteristic, complemented, Sylow
204C23×D4C2×D7311yescharacteristic
205C2×#121—21no—
206C2×#120Dic7312yes—
207C2×C4×D4C2×D7311yescharacteristic
208#435—1464no—
209#432—1461no—
210C2×#100—1464no—
211#407—732no—
212#407—732no—
213C2×C22⋊C8—1461no—
214#394—21no—
215#392—21no—
216#390—21no—
217#380C2×D7311yescharacteristic
218#379Dic7312yes—
219#324—1461no—
220#323—1461no—
221C25C2×Dic7312yes—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4—21no—
225C22×D4—21no—
226C22×D4—22no—
227C22×D4—44no—
228C22×D4C2×Dic7311yescharacteristic
229C23×C4C22⋊D7311yescharacteristic
230C23×C4C22×D7311yescharacteristic
231C2×M16—734no—
232C2×M16—1462no—
233C22×C8—1462no—
234#121—44no—
235#120—44no—
236C4×D4—22no—
237C4×D4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4C22⋊D7311yescharacteristic
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4C2×Dic7311yescharacteristic
244C2×C22⋊C4C2×Dic7312yes—
245C2×C22⋊C4C22⋊D7311yescharacteristic
246C2×C42C22⋊D7311yescharacteristic
247C4⋊C8—1462no—
248#100—2928no—
249C22⋊C8—1462no—
250C22⋊C8—1462no—
251C22⋊C8—1462no—
252C22⋊C8—1462no—
253C22⋊C8—1462no—
254#97—1462no—
255#95—21no—
256C24—22no—
257C24—24no—
258C24—44no—
259C24—42no—
260C24—42no—
261C24—82no—
262C24C22⋊Dic7311yescharacteristic
263C24C22×Dic7311yescharacteristic
264C2×D4—28no—
265C2×D4—24no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—41no—
269C2×D4—42no—
270C2×D4—44no—
271C2×D4—44no—
272C2×D4—48no—
273C2×D4—44no—
274C2×D4—44no—
275C2×D4—41no—
276C2×D4—42no—
277C2×D4—41no—
278C2×D4—82no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4C22⋊Dic7311yescharacteristic
285C22×C4C22⋊Dic7311yescharacteristic
286C22×C4C22⋊Dic7311yescharacteristic
287C22×C4D4×D7312yes—
288C22×C4C2×C22⋊D7311yescharacteristic
289C22×C4C2×C22⋊D7311yescharacteristic
290C22×C4#1192402712yes—
291M16—2924no—
292M16—2924no—
293C2×C8—2922no—
294C2×C8—5842no—
295C2×C8—1464no—
296C4⋊C4—22no—
297C4⋊C4—42no—
298C22⋊C4—22no—
299C22⋊C4—22no—
300C22⋊C4—44no—
301C22⋊C4—44no—
302C22⋊C4—42no—
303C22⋊C4—82no—
304C42—21no—
305C42—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—24no—
311C23—28no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—42no—
317C23—42no—
318C23—44no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—84no—
324C23—84no—
325C23—88no—
326C23—88no—
327C23#1455650812yescomplemented
328C23C2×C22⋊Dic7311yescharacteristic
329C23D4×Dic7312yes—
330D4—48no—
331D4—44no—
332D4—44no—
333D4—44no—
334D4—88no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—81no—
351C2×C4#1455655911yescharacteristic
352C2×C4#1455656312yes—
353C2×C4#1455657211yescharacteristic
354C8—2924no—
355C8—5842no—
356C22—21no—
357C22—21no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—44no—
366C22—44no—
367C22—41no—
368C22—41no—
369C22—44no—
370C22—81no—
371C22—82no—
372C22—82no—
373C22—88no—
374C22—88no—
375C22—82no—
376C22—82no—
377C22#1798846711yescharacteristic
378C22C2×#1455650811yescharacteristic
379C22#1798857111yescharacteristic
380C4—22no—
381C4—21no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—82no—
386C4#1798877312yes—
387C2—21no—
388C2—21no—
389C2—42no—
390C2—42no—
391C2—84no—
392C2#2261034011yescharacteristic
393C2#2261039511yescharacteristic
394C2#2261047911yescharacteristic
subgroupG/H
centerC22#17988467
commutatorC24×C73C22×C4
FrattiniC23×C4C22×D73
Fitting, prime radicalC73×#2408C2
socleC22×C73#386
cosocleC23×C4×C73C23
hypercenter#2408D73
hypocenterC73#9503
LCSG ▷ C24×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2408

Complements

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

NH ≅ G/Ntype
1. #22618960C2semidirect
2. #22609852C2semidirect
3. #22608269C2semidirect
4. #22607837C2semidirect
24. C2×#11924012D4semidirect
129. C23×C73#100semidirect
203. C73#9503semidirect
327. C23#14556508semidirect

Maximal subgroups

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

subgroupindex
#226189602
#226098522
#226082692
#226078372
C73×#24082
#950373