Finite Group Databasebeta

← #32983305

Subgroups — #32983305

Order 18688.

640 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618955C211yesmaximal, characteristic, complemented
2#22609919C211yesmaximal, characteristic, complemented
3#22608499C212yesmaximal, complemented
4#22607810C212yesmaximal, complemented
5C73×#2543C211yesmaximal, characteristic
6D4×#10596831—24no—
7#17988640—21no—
8C2×#14556506—21no—
9C2×C22⋊#10596831—21no—
10C2×C22⋊#10596831C2212yes—
11#17988447C2212yes—
12#17988354—21no—
13#17988321—24no—
14C2×#121×C73—21no—
15C2×#121×C73C2211yescharacteristic
16C2×C4×D4×C73C2211yescharacteristic
17#398×C73C411yescharacteristic, complemented
18#397×C73C411yescharacteristic, complemented
19#394×C73C411yescharacteristic, complemented
20#392×C73C411yescharacteristic, complemented
21#388×C73—21no—
22#386×C73—21no—
23#386×C73—21no—
24C2×#95×C73C2211yescharacteristic
25C22×#10596831—22no—
26C22×#10596831—22no—
27C2×C4⋊Dic73—22no—
28C2×C22⋊Dic73—22no—
29C4⋊#10596831—22no—
30#14556506—42no—
31C22⋊#10596831—22no—
32C22⋊#10596831—22no—
33C22⋊#10596831D414yescomplemented
34C4×#10596831—22no—
35#14556498—22no—
36C22×D4×C73—21no—
37C22×D4×C73—21no—
38C22×D4×C73C2×C411yescharacteristic
39C22×D4×C73D411yescharacteristic
40C23×C4×C73—21no—
41C23×C4×C73D411yescharacteristic
42C23×C4×C73C2311yescharacteristic
43#121×C73—22no—
44#121×C73—22no—
45#121×C73—42no—
46#121×C73—42no—
47C4×D4×C73—22no—
48C4×D4×C73—22no—
49C2×C4⋊C4×C73—21no—
50C2×C4⋊C4×C73C2×C411yescharacteristic
51C2×C4⋊C4×C73C2×C411yescharacteristic
52C2×C22⋊C4×C73—21no—
53C2×C22⋊C4×C73—21no—
54C2×C22⋊C4×C73—21no—
55C2×C22⋊C4×C73—21no—
56C2×C22⋊C4×C73—21no—
57C2×C22⋊C4×C73C2×C411yescharacteristic
58C2×C22⋊C4×C73D411yescharacteristic
59C2×C42×C73D411yescharacteristic
60#95×C73—21no—
61#95×C73—21no—
62#95×C73—21no—
63#95×C73C2×C411yescharacteristic
64#95×C73C2×C411yescharacteristic
65C22×Dic73—42no—
66C2×#10596831—24no—
67C2×#10596831—82no—
68C2×#10596831—82no—
69C4⋊Dic73—82no—
70C22⋊Dic73—42no—
71C22⋊Dic73—42no—
72C24×C73—21no—
73C24×C73C22⋊C411yescharacteristic
74C24×C73C2×D411yescharacteristic
75C2×D4×C73—22no—
76C2×D4×C73—22no—
77C2×D4×C73—22no—
78C2×D4×C73—42no—
79C2×D4×C73—42no—
80C2×D4×C73—42no—
81C2×D4×C73—41no—
82C2×D4×C73—41no—
83C2×D4×C73—42no—
84C2×D4×C73—81no—
85C2×D4×C73—81no—
86C2×D4×C73D812yes—
87C2×D4×C73SD1612yes—
88C22×C4×C73—21no—
89C22×C4×C73—21no—
90C22×C4×C73—21no—
91C22×C4×C73—21no—
92C22×C4×C73—21no—
93C22×C4×C73—42no—
94C22×C4×C73—41no—
95C22×C4×C73—41no—
96C22×C4×C73—81no—
97C22×C4×C73—81no—
98C22×C4×C73—81no—
99C22×C4×C73C22⋊C411yescharacteristic
100C22×C4×C73C22⋊C411yescharacteristic
101C22×C4×C73C22⋊C411yescharacteristic
102C22×C4×C73C22×C411yescharacteristic
103C22×C4×C73C2×D412yes—
104C22×C4×C73C2×D411yescharacteristic
105C22×C4×C73#4112yes—
106C4⋊C4×C73—22no—
107C4⋊C4×C73—22no—
108C4⋊C4×C73—42no—
109C22⋊C4×C73—22no—
110C22⋊C4×C73—22no—
111C22⋊C4×C73—42no—
112C22⋊C4×C73—42no—
113C22⋊C4×C73—42no—
114C22⋊C4×C73—41no—
115C22⋊C4×C73—41no—
116C22⋊C4×C73—81no—
117C42×C73—21no—
118C42×C73—21no—
119C23×C73—21no—
120C23×C73—21no—
121C23×C73—41no—
122C23×C73—41no—
123C23×C73—41no—
124C23×C73—41no—
125C23×C73—41no—
126C23×C73—41no—
127C23×C73—41no—
128C23×C73—41no—
129C23×C73—41no—
130C23×C73—41no—
131C23×C73—41no—
132C23×C73#10212yes—
133C23×C73C2×C22⋊C411yescharacteristic
134C23×C73#12112yes—
135C23×C73C2×D811yescharacteristic
136C23×C73C2×SD1611yescharacteristic
137C23×C73#13512yes—
138D4×C73—24no—
139D4×C73—24no—
140D4×C73—82no—
141D4×C73—84no—
142D4×C73—82no—
143D4×C73—82no—
144C2×C4×C73—21no—
145C2×C4×C73—21no—
146C2×C4×C73—22no—
147C2×C4×C73—21no—
148C2×C4×C73—21no—
149C2×C4×C73—41no—
150C2×C4×C73—41no—
151C2×C4×C73—42no—
152C2×C4×C73—42no—
153C2×C4×C73—42no—
154C2×C4×C73—41no—
155C2×C4×C73—41no—
156C2×C4×C73—41no—
157C2×C4×C73—41no—
158C2×C4×C73—41no—
159C2×C4×C73—41no—
160C2×C4×C73—41no—
161C2×C4×C73—41no—
162C2×C4×C73—41no—
163C2×C4×C73—41no—
164C2×C4×C73—81no—
165C2×C4×C73—82no—
166C2×C4×C73—82no—
167C2×C4×C73#10212yes—
168C2×C4×C73C4×D412yes—
169C2×C4×C73#12011yescharacteristic
170C2×C4×C73#12311yescharacteristic
171C2×Dic73—82no—
172C2×Dic73—82no—
173C2×Dic73—82no—
174#10596831—44no—
175C22×C73—22no—
176C22×C73—21no—
177C22×C73—22no—
178C22×C73—21no—
179C22×C73—22no—
180C22×C73—41no—
181C22×C73—41no—
182C22×C73—42no—
183C22×C73—42no—
184C22×C73—41no—
185C22×C73—41no—
186C22×C73—81no—
187C22×C73—81no—
188C22×C73—81no—
189C22×C73—81no—
190C22×C73—81no—
191C22×C73—81no—
192C22×C73—81no—
193C22×C73—81no—
194C22×C73#38611yescharacteristic
195C22×C73C2×#10211yescharacteristic
196C22×C73#41611yescharacteristic
197C22×C73#46512yes—
198C22×C73#46612yes—
199C4×C73—22no—
200C4×C73—21no—
201C4×C73—21no—
202C4×C73—22no—
203C4×C73—42no—
204C4×C73—41no—
205C4×C73—41no—
206C4×C73—81no—
207C4×C73—81no—
208Dic73—162no—
209#9384—731nomaximal, Sylow
210C2×C73—22no—
211C2×C73—21no—
212C2×C73—21no—
213C2×C73—22no—
214C2×C73—42no—
215C2×C73—161no—
216C2×C73#188411yescharacteristic
217C2×C73#189311yescharacteristic
218C2×C73#191311yescharacteristic
219#2972—731no—
220#2543D7311yescharacteristic
221#1799—731no—
222#1519—732no—
223#1294—732no—
224C73#938411yescharacteristic, complemented, Sylow
225C2×#121—21no—
226C2×#121C2×D7311yescharacteristic
227C2×C4×D4C2×D7311yescharacteristic
228C8×D4—1464no—
229#431—1461no—
230C2×#99—1461no—
231C2×C22⋊C8—732no—
232C2×C22⋊C8—1461no—
233#398Dic7311yescharacteristic, complemented
234#397Dic7311yescharacteristic, complemented
235#394Dic7311yescharacteristic, complemented
236#392Dic7311yescharacteristic, complemented
237#388—21no—
238#386—21no—
239#386—21no—
240#380—732no—
241C2×#95C2×D7311yescharacteristic
242#342—1461no—
243#327—1464no—
244C22×D4—21no—
245C22×D4—21no—
246C22×D4C2×Dic7311yescharacteristic
247C22×D4C22⋊D7311yescharacteristic
248C23×C4—21no—
249C23×C4C22⋊D7311yescharacteristic
250C23×C4C22×D7311yescharacteristic
251C22×C8—1462no—
252C22×C8—1462no—
253#121—22no—
254#121—22no—
255#121—42no—
256#121—42no—
257C4×D4—22no—
258C4×D4—22no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—1462no—
261C2×C4⋊C4C2×Dic7311yescharacteristic
262C2×C4⋊C4C2×Dic7311yescharacteristic
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—1462no—
269C2×C22⋊C4C2×Dic7311yescharacteristic
270C2×C22⋊C4C22⋊D7311yescharacteristic
271C2×C42C22⋊D7311yescharacteristic
272C4⋊C8—1462no—
273#99—2922no—
274C22⋊C8—734no—
275C22⋊C8—1462no—
276C22⋊C8—1462no—
277C4×C8—1462no—
278#95—21no—
279#95—21no—
280#95—21no—
281#95—1462no—
282#95C2×Dic7311yescharacteristic
283#95C2×Dic7311yescharacteristic
284C24—21no—
285C24C22⋊Dic7311yescharacteristic
286C24C2×C22⋊D7311yescharacteristic
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—42no—
296C2×D4—81no—
297C2×D4—81no—
298C2×D4#1192401312yes—
299C2×D4#1192401612yes—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—81no—
309C22×C4—81no—
310C22×C4—81no—
311C22×C4—2922no—
312C22×C4C22⋊Dic7311yescharacteristic
313C22×C4C22⋊Dic7311yescharacteristic
314C22×C4C22⋊Dic7311yescharacteristic
315C22×C4C22×Dic7311yescharacteristic
316C22×C4D4×D7312yes—
317C22×C4C2×C22⋊D7311yescharacteristic
318C22×C4#1192402712yes—
319C2×C8—5842no—
320C2×C8—5842no—
321C2×C8—1464no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—42no—
325C4⋊C4—5842no—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—81no—
334C22⋊C4—2922no—
335C22⋊C4—2922no—
336C42—21no—
337C42—21no—
338C23—21no—
339C23—21no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23#1455651212yes—
352C23C2×C22⋊Dic7311yescharacteristic
353C23#1455656312yes—
354C23C2×#1192401311yescharacteristic
355C23C2×#1192401611yescharacteristic
356C23#1455661112yes—
357D4—24no—
358D4—24no—
359D4—82no—
360D4—84no—
361D4—82no—
362D4—82no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—81no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—5842no—
387C2×C4—5842no—
388C2×C4—5842no—
389C2×C4#1455651212yes—
390C2×C4D4×Dic7312yes—
391C2×C4#1455655911yescharacteristic
392C2×C4#1455657211yescharacteristic
393C8—2924no—
394C22—21no—
395C22—22no—
396C22—21no—
397C22—22no—
398C22—22no—
399C22—41no—
400C22—41no—
401C22—42no—
402C22—42no—
403C22—41no—
404C22—41no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22—81no—
411C22—81no—
412C22—81no—
413C22#1798846711yescharacteristic
414C22C2×#1455651211yescharacteristic
415C22#1798858311yescharacteristic
416C22#1798881412yes—
417C22#1798882112yes—
418C4—21no—
419C4—21no—
420C4—22no—
421C4—22no—
422C4—42no—
423C4—41no—
424C4—41no—
425C4—81no—
426C4—81no—
427C4—11682no—
428C2—21no—
429C2—22no—
430C2—21no—
431C2—22no—
432C2—42no—
433C2—161no—
434C2#2261031511yescharacteristic
435C2#2261036011yescharacteristic
436C2#2261047411yescharacteristic
subgroupG/H
centerC22#17988467
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
Fitting, prime radicalC73×#2543C2
socleC22×C73#386
cosocleC23×C4×C73C23
hypercenter#2543D73
hypocenterC73#9384
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2543

Complements

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

NH ≅ G/Ntype
1. #22618955C2semidirect
2. #22609919C2semidirect
3. #22608499C2semidirect
4. #22607810C2semidirect
17. #398×C73C4semidirect
18. #397×C73C4semidirect
19. #394×C73C4semidirect
20. #392×C73C4semidirect
33. C22⋊#10596831D4semidirect
224. C73#9384semidirect
233. #398Dic73semidirect
234. #397Dic73semidirect
235. #394Dic73semidirect
236. #392Dic73semidirect

Maximal subgroups

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

subgroupindex
#226189552
#226099192
#226084992
#226078102
C73×#25432
#938473