Finite Group Databasebeta

← #32981209

Subgroups — #32981209

Order 18688.

470 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620486C211yesmaximal, characteristic
2#22618878C211yesmaximal, characteristic, complemented
3#22608429C211yesmaximal, characteristic, complemented
4#22608330C211yesmaximal, characteristic
5#22607772C211yesmaximal, characteristic, complemented
6#22607769C211yesmaximal, characteristic, complemented
7C73×#1980C211yesmaximal, characteristic
8#17989144C2211yescharacteristic
9#17988983C411yescharacteristic, complemented
10#17988959C411yescharacteristic, complemented
11#17988939C411yescharacteristic, complemented
12#17988933C411yescharacteristic, complemented
13#17988814—22no—
14#17988717—21no—
15#17988715—22no—
16C8×Dic146—21no—
17C4×#11924016—22no—
18#17988640C2211yescharacteristic
19#17988615—21no—
20C2×C4⋊#10596831C2211yescharacteristic
21#17988545—21no—
22C2×C4×#10596831C2211yescharacteristic
23#17988323—22no—
24D4⋊#10596831—21no—
25D4⋊#10596831—21no—
26#532×C73C2211yescharacteristic
27C2×#104×C73—21no—
28C2×#104×C73—21no—
29C4×M16×C73C2211yescharacteristic
30C4×C4⋊C4×C73C2211yescharacteristic
31C2×#11924016—22no—
32C22×#10596831—21no—
33C22×#10596831—21no—
34#14556591C2×C411yescharacteristic
35#14556591C2×C411yescharacteristic
36#14556583—21no—
37#14556570—21no—
38#14556570D412yes—
39C4×Dic146—21no—
40C4×Dic146D412yes—
41#14556550—21no—
42#14556522—22no—
43C4⋊#10596831—21no—
44C4⋊#10596831—21no—
45C4⋊#10596831D411yescharacteristic
46C4⋊#10596831D411yescharacteristic
47#14556520—22no—
48#14556520—22no—
49#14556514—22no—
50#14556512—22no—
51C22⋊#10596831—21no—
52#14556501—22no—
53C4×#10596831—21no—
54C4×#10596831—21no—
55C4×#10596831C2×C411yescharacteristic
56C4×#10596831C2×C411yescharacteristic
57C8×Dic73—22no—
58C2×#41×C73—21no—
59C2×M16×C73D411yescharacteristic
60C2×M16×C73D411yescharacteristic
61C4⋊Q8×C73C2×C411yescharacteristic
62#127×C73C2×C411yescharacteristic
63#124×C73—21no—
64#121×C73—21no—
65#121×C73—21no—
66C4×D4×C73—21no—
67C4×D4×C73—21no—
68C2×C4⋊C4×C73—21no—
69C2×C42×C73—21no—
70C2×C42×C73C2311yescharacteristic
71#104×C73—42no—
72#104×C73—42no—
73#97×C73—21no—
74C4×C8×C73—21no—
75#95×C73—21no—
76C2×Dic146—21no—
77C2×Dic146C22⋊C412yes—
78#11924016—44no—
79C2×#10596831—21no—
80C2×#10596831—21no—
81C2×#10596831—21no—
82C2×#10596831—21no—
83C2×#10596831—42no—
84C2×#10596831—41no—
85C2×#10596831—41no—
86C2×#10596831—41no—
87C4⋊Dic73—21no—
88C4⋊Dic73—21no—
89C4⋊Dic73—21no—
90C4⋊Dic73C22⋊C412yes—
91#11924007—22no—
92#11924007—22no—
93C22⋊Dic73—22no—
94C4×Dic73—22no—
95#41×C73—41no—
96#41×C73—41no—
97#41×C73—41no—
98#41×C73—41no—
99C2×Q8×C73—21no—
100C2×D4×C73—21no—
101C2×D4×C73—21no—
102C2×D4×C73—21no—
103C2×D4×C73—41no—
104C22×C4×C73—41no—
105C22×C4×C73—41no—
106C22×C4×C73C2×D411yescharacteristic
107C22×C4×C73C2×D411yescharacteristic
108C22×C4×C73#4111yescharacteristic
109M16×C73—21no—
110M16×C73—21no—
111M16×C73—21no—
112M16×C73SD1612yescomplemented
113C2×C8×C73—21no—
114C2×C8×C73—21no—
115C4⋊C4×C73—22no—
116C4⋊C4×C73—21no—
117C4⋊C4×C73—21no—
118C4⋊C4×C73—41no—
119C22⋊C4×C73—41no—
120C22⋊C4×C73—41no—
121C42×C73—41no—
122C42×C73—41no—
123C42×C73C22×C411yescharacteristic
124C42×C73C2×D411yescharacteristic
125C42×C73C2×D411yescharacteristic
126C42×C73#4111yescharacteristic
127C23×C73—41no—
128C23×C73#12111yescharacteristic
129Q8×C73—41no—
130Q8×C73—41no—
131D4×C73—42no—
132D4×C73—41no—
133D4×C73—41no—
134D4×C73—41no—
135D4×C73—41no—
136D4×C73—81no—
137D4×C73—81no—
138C2×C4×C73—21no—
139C2×C4×C73—21no—
140C2×C4×C73—41no—
141C2×C4×C73—42no—
142C2×C4×C73—41no—
143C2×C4×C73—81no—
144C2×C4×C73—81no—
145C2×C4×C73—81no—
146C2×C4×C73—81no—
147C2×C4×C73C2×C22⋊C411yescharacteristic
148C2×C4×C73C4×D411yescharacteristic
149C2×C4×C73C4×D411yescharacteristic
150C2×C4×C73#12011yescharacteristic
151C2×C4×C73#12111yescharacteristic
152C2×C4×C73#12311yescharacteristic
153C2×C4×C73C2×SD1611yescharacteristic
154C2×C4×C73#13511yescharacteristic
155C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C73#13711yescharacteristic
157C2×Dic73—22no—
158C2×Dic73—22no—
159Dic146—41no—
160Dic146—41no—
161Dic146—42no—
162C8×C73—22no—
163C8×C73—41no—
164#10596831—22no—
165#10596831—41no—
166#10596831—41no—
167#10596831—41no—
168#10596831—81no—
169C22×C73—21no—
170C22×C73—21no—
171C22×C73—81no—
172C22×C73—81no—
173C22×C73—81no—
174C22×C73#38611yescharacteristic
175C22×C73#46011yescharacteristic
176C22×C73#46411yescharacteristic
177C4×C73—21no—
178C4×C73—21no—
179C4×C73—21no—
180C4×C73—82no—
181C4×C73—81no—
182C4×C73C4×SD1611yescharacteristic
183C4×C73#44011yescharacteristic
184C4×C73#44311yescharacteristic
185C4×C73#44411yescharacteristic
186C4×C73#44811yescharacteristic
187C4×C73#44911yescharacteristic
188Dic73—42no—
189Dic73—42no—
190#9054—731nomaximal, Sylow
191C2×C73—21no—
192C2×C73—21no—
193C2×C73—161no—
194C2×C73#187611yescharacteristic
195C2×C73#188011yescharacteristic
196C2×C73#188411yescharacteristic
197#3167—731no—
198#2962—731no—
199#1980D7311yescharacteristic
200#1503—731no—
201#1480—731no—
202#1279—731no—
203#1278—731no—
204C73#905411yescharacteristic, complemented, Sylow
205#533—731no—
206#532C2×D7311yescharacteristic
207#498—731no—
208#492—731no—
209#487—731no—
210#486—731no—
211#465—1462no—
212#446—1461no—
213#446—1462no—
214C8×Q8—1461no—
215C4×SD16—1462no—
216#431—731no—
217#424—1461no—
218C2×C4⋊C8—731no—
219C2×#104—21no—
220C2×#104—21no—
221#405—1461no—
222C4×M16C2×D7311yescharacteristic
223C2×C4×C8—731no—
224C4×C4⋊C4C2×D7311yescharacteristic
225#328—1462no—
226D4⋊C8—1461no—
227D4⋊C8—1461no—
228C2×#41—21no—
229C2×SD16—1462no—
230C2×M16C22⋊D7311yescharacteristic
231C2×M16C22⋊D7311yescharacteristic
232C22×C8—1461no—
233C22×C8—1461no—
234C4⋊Q8—731no—
235C4⋊Q8—731no—
236C4⋊Q8C4×D7311yescharacteristic
237#127C4×D7311yescharacteristic
238#125—1461no—
239#124—21no—
240C22⋊Q8—732no—
241C22⋊Q8—1461no—
242#121—21no—
243#121—21no—
244C4×Q8—732no—
245C4×Q8—1461no—
246C4×D4—21no—
247C4×D4—21no—
248#117—1461no—
249C2×C4⋊C4—21no—
250C2×C42—21no—
251C2×C42C22×D7311yescharacteristic
252#106—1462no—
253C4⋊C8—731no—
254C4⋊C8—731no—
255C4⋊C8—1462no—
256C4⋊C8—1462no—
257C4⋊C8—1461no—
258C4⋊C8—1461no—
259#104—42no—
260#104—42no—
261#103—1462no—
262#102—1462no—
263C22⋊C8—1461no—
264#97—21no—
265#97—1462no—
266C4×C8—21no—
267C4×C8—731no—
268C4×C8—731no—
269C4×C8—1462no—
270C4×C8—1461no—
271C4×C8—1461no—
272#95—21no—
273#41—41no—
274#41—41no—
275#41—41no—
276#41—41no—
277C2×Q8—21no—
278C2×Q8—732no—
279C2×Q8—1461no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4D4×D7311yescharacteristic
287C22×C4C2×C22⋊D7311yescharacteristic
288C22×C4#1192402711yescharacteristic
289SD16—2924no—
290M16—21no—
291M16—21no—
292M16—21no—
293M16#1192401612yescomplemented
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—2921no—
297C2×C8—2921no—
298C2×C8—2922no—
299C2×C8—2921no—
300C2×C8—1461no—
301C2×C8—1461no—
302C2×C8—1461no—
303C2×C8—1461no—
304C4⋊C4—22no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—41no—
308C4⋊C4—732no—
309C4⋊C4—1461no—
310C4⋊C4—1461no—
311C4⋊C4—1461no—
312C4⋊C4—1462no—
313C4⋊C4—1462no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—1462no—
317C42—41no—
318C42—41no—
319C42—1462no—
320C42C2×C4×D7311yescharacteristic
321C42D4×D7311yescharacteristic
322C42D4×D7311yescharacteristic
323C42#1192402811yescharacteristic
324C23—41no—
325C23#1455656211yescharacteristic
326Q8—41no—
327Q8—41no—
328Q8—2922no—
329Q8—2921no—
330Q8—2921no—
331D4—42no—
332D4—41no—
333D4—41no—
334D4—41no—
335D4—41no—
336D4—81no—
337D4—81no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—1462no—
348C2×C4—1462no—
349C2×C4C22⋊C4×D7311yescharacteristic
350C2×C4#1455655111yescharacteristic
351C2×C4C4×C22⋊D7311yescharacteristic
352C2×C4#1455655911yescharacteristic
353C2×C4#1455656411yescharacteristic
354C2×C4#1455657311yescharacteristic
355C2×C4C2×#1192401611yescharacteristic
356C2×C4#1455661111yescharacteristic
357C2×C4#1455661711yescharacteristic
358C2×C4#1455662111yescharacteristic
359C8—22no—
360C8—41no—
361C8—2921no—
362C8—2921no—
363C8—2921no—
364C8—5841no—
365C8—1462no—
366C22—21no—
367C22—21no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22#1798846311yescharacteristic
372C22#1798878411yescharacteristic
373C22#1798880611yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—82no—
378C4—81no—
379C4—2922no—
380C4—2922no—
381C4C4×#1192401611yescharacteristic
382C4#1798868711yescharacteristic
383C4#1798870111yescharacteristic
384C4#1798870611yescharacteristic
385C4#1798872511yescharacteristic
386C4#1798873011yescharacteristic
387C2—21no—
388C2—21no—
389C2—161no—
390C2#2261025511yescharacteristic
391C2#2261028911yescharacteristic
392C2#2261031111yescharacteristic
subgroupG/H
centerC2×C4#14556559
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#1980C2
socleC22×C73#386
prime radical#532×C73C22
cosocleC2×C42×C73C23
hypercenter#1980D73
hypocenterC73#9054
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1980

Complements

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

NH ≅ G/Ntype
2. #22618878C2semidirect
3. #22608429C2semidirect
5. #22607772C2semidirect
6. #22607769C2semidirect
9. #17988983C4semidirect
10. #17988959C4semidirect
11. #17988939C4semidirect
12. #17988933C4semidirect
112. M16×C73SD16semidirect
204. C73#9054semidirect
293. M16#11924016semidirect

Maximal subgroups

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

subgroupindex
#226204862
#226188782
#226084292
#226083302
#226077722
#226077692
C73×#19802
#905473