Finite Group Databasebeta

← #32981214

Subgroups — #32981214

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#22618880C211yesmaximal, characteristic, complemented
2#22610851C211yesmaximal, characteristic
3#22608430C211yesmaximal, characteristic, complemented
4#22608330C211yesmaximal, characteristic
5#22607771C211yesmaximal, characteristic, complemented
6#22607768C211yesmaximal, characteristic, complemented
7C73×#3167C211yesmaximal, characteristic
8#17988718—21no—
9#17988718—22no—
10Q8×#10596831—21no—
11#17988640C2211yescharacteristic
12#17988614—21no—
13C2×C4⋊#10596831C2211yescharacteristic
14C2×#14556518—21no—
15C2×#14556518—21no—
16#17988543—21no—
17C4×#11924012C2211yescharacteristic
18C4×C4⋊Dic73C2211yescharacteristic
19#17988324—22no—
20D4⋊#10596831—21no—
21D4⋊#10596831—21no—
22#533×C73C2211yescharacteristic
23#532×C73C2211yescharacteristic
24#498×C73C411yescharacteristic, complemented
25#492×C73C411yescharacteristic, complemented
26#487×C73C411yescharacteristic, complemented
27#486×C73C411yescharacteristic, complemented
28#465×C73—22no—
29C4×SD16×C73—22no—
30C2×C4×C8×C73C2211yescharacteristic
31C2×#11924012D411yescharacteristic
32C2×#11924012D411yescharacteristic
33C22×#10596831—21no—
34C2×C4⋊Dic73—21no—
35C2×C4×Dic73—21no—
36C4⋊#10596831—22no—
37C4⋊#10596831—22no—
38C4⋊#10596831—21no—
39C4⋊#10596831—21no—
40C4⋊#10596831D411yescharacteristic
41C4⋊#10596831D411yescharacteristic
42#14556518—42no—
43#14556518—42no—
44C22⋊#10596831—21no—
45#14556502—22no—
46#14556502—21no—
47C4×#10596831—22no—
48C4×#10596831—21no—
49C4×#10596831—21no—
50#14556498—21no—
51C2×#41×C73—21no—
52C2×SD16×C73—22no—
53C22×C8×C73—21no—
54C4⋊Q8×C73C2×C411yescharacteristic
55C4⋊Q8×C73C2×C411yescharacteristic
56C4⋊Q8×C73C2×C411yescharacteristic
57#127×C73C2×C411yescharacteristic
58#125×C73—21no—
59#124×C73—21no—
60C22⋊Q8×C73—21no—
61C22⋊Q8×C73D412yes—
62#121×C73—21no—
63#121×C73—21no—
64C4×Q8×C73—21no—
65C4×Q8×C73D412yes—
66C4×D4×C73—21no—
67C4×D4×C73—21no—
68#117×C73—21no—
69C2×C42×C73C2311yescharacteristic
70#106×C73—22no—
71#103×C73—22no—
72#102×C73—22no—
73C4×C8×C73—21no—
74C4×C8×C73C2×C411yescharacteristic
75C4×C8×C73C2×C411yescharacteristic
76C22×Dic73—41no—
77#11924012—21no—
78#11924012—21no—
79#11924012—21no—
80#11924012SD1612yescomplemented
81C2×#10596831—21no—
82C2×#10596831—21no—
83C2×#10596831—21no—
84C2×#10596831—21no—
85C2×#10596831—41no—
86C2×#10596831—41no—
87C2×#10596831—41no—
88C4⋊Dic73—22no—
89C4⋊Dic73—41no—
90C4×Dic73—41no—
91C4×Dic73—41no—
92#41×C73—41no—
93#41×C73—41no—
94#41×C73—41no—
95#41×C73—41no—
96C2×Q8×C73—21no—
97C2×Q8×C73—21no—
98C2×Q8×C73C22⋊C412yes—
99C2×D4×C73—21no—
100C2×D4×C73—21no—
101C2×D4×C73—21no—
102C2×D4×C73—41no—
103C22×C4×C73—41no—
104C22×C4×C73C2×D411yescharacteristic
105C22×C4×C73C2×D411yescharacteristic
106C22×C4×C73#4111yescharacteristic
107SD16×C73—44no—
108C2×C8×C73—21no—
109C2×C8×C73—21no—
110C2×C8×C73—42no—
111C4⋊C4×C73—21no—
112C4⋊C4×C73—21no—
113C4⋊C4×C73—21no—
114C4⋊C4×C73—22no—
115C4⋊C4×C73—22no—
116C4⋊C4×C73—21no—
117C4⋊C4×C73—21no—
118C4⋊C4×C73C22⋊C412yes—
119C22⋊C4×C73—22no—
120C22⋊C4×C73—41no—
121C22⋊C4×C73—41no—
122C42×C73—22no—
123C42×C73C22×C411yescharacteristic
124C42×C73C2×D411yescharacteristic
125C42×C73C2×D411yescharacteristic
126C42×C73#4111yescharacteristic
127C23×C73—41no—
128C23×C73#12111yescharacteristic
129Q8×C73—42no—
130Q8×C73—41no—
131Q8×C73—41no—
132Q8×C73—41no—
133Q8×C73—41no—
134D4×C73—42no—
135D4×C73—41no—
136D4×C73—41no—
137D4×C73—41no—
138D4×C73—41no—
139D4×C73—81no—
140D4×C73—81no—
141C2×C4×C73—21no—
142C2×C4×C73—21no—
143C2×C4×C73—22no—
144C2×C4×C73—22no—
145C2×C4×C73—41no—
146C2×C4×C73—41no—
147C2×C4×C73—81no—
148C2×C4×C73—81no—
149C2×C4×C73C2×C22⋊C411yescharacteristic
150C2×C4×C73C4×D411yescharacteristic
151C2×C4×C73C4×D411yescharacteristic
152C2×C4×C73#12011yescharacteristic
153C2×C4×C73#12111yescharacteristic
154C2×C4×C73#12311yescharacteristic
155C2×C4×C73C2×SD1611yescharacteristic
156C2×C4×C73#13511yescharacteristic
157C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
158C2×C4×C73#13711yescharacteristic
159C2×Dic73—42no—
160C2×Dic73—81no—
161C2×Dic73—81no—
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—42no—
181C4×C73—42no—
182C4×C73—81no—
183C4×C73C4×SD1611yescharacteristic
184C4×C73#44011yescharacteristic
185C4×C73#44311yescharacteristic
186C4×C73#44411yescharacteristic
187C4×C73#44811yescharacteristic
188C4×C73#44911yescharacteristic
189Dic73—82no—
190#9054—731nomaximal, Sylow
191C2×C73—21no—
192C2×C73—21no—
193C2×C73—161no—
194C2×C73#187611yescharacteristic
195C2×C73#188011yescharacteristic
196C2×C73#188411yescharacteristic
197#3167D7311yescharacteristic
198#2962—731no—
199#1980—731no—
200#1503—731no—
201#1480—731no—
202#1279—731no—
203#1278—731no—
204C73#905411yescharacteristic, complemented, Sylow
205#533C2×D7311yescharacteristic
206#532C2×D7311yescharacteristic
207#498Dic7311yescharacteristic, complemented
208#492Dic7311yescharacteristic, complemented
209#487Dic7311yescharacteristic, complemented
210#486Dic7311yescharacteristic, complemented
211#465—22no—
212#446—1461no—
213#446—1462no—
214C8×Q8—1461no—
215C4×SD16—22no—
216#431—731no—
217#424—1461no—
218C2×C4⋊C8—731no—
219C2×#104—1461no—
220C2×#104—1461no—
221#405—1461no—
222C4×M16—731no—
223C2×C4×C8C2×D7311yescharacteristic
224C4×C4⋊C4—731no—
225#328—1462no—
226D4⋊C8—1461no—
227D4⋊C8—1461no—
228C2×#41—21no—
229C2×SD16—22no—
230C2×M16—731no—
231C2×M16—731no—
232C22×C8—21no—
233C22×C8—1461no—
234C4⋊Q8C2×Dic7311yescharacteristic
235C4⋊Q8C2×Dic7311yescharacteristic
236C4⋊Q8C2×Dic7311yescharacteristic
237#127C2×Dic7311yescharacteristic
238#125—21no—
239#124—21no—
240C22⋊Q8—21no—
241C22⋊Q8C22⋊D7312yes—
242#121—21no—
243#121—21no—
244C4×Q8—21no—
245C4×Q8C22⋊D7312yes—
246C4×D4—21no—
247C4×D4—21no—
248#117—21no—
249C2×C4⋊C4—1461no—
250C2×C42—1461no—
251C2×C42C22×D7311yescharacteristic
252#106—22no—
253C4⋊C8—731no—
254C4⋊C8—731no—
255C4⋊C8—1462no—
256C4⋊C8—1462no—
257C4⋊C8—1461no—
258C4⋊C8—1461no—
259#104—2922no—
260#104—2922no—
261#103—22no—
262#102—22no—
263C22⋊C8—1461no—
264#97—1462no—
265#97—1461no—
266C4×C8—21no—
267C4×C8—1462no—
268C4×C8—1461no—
269C4×C8—1461no—
270C4×C8C2×Dic7311yescharacteristic
271C4×C8C2×Dic7311yescharacteristic
272#95—1461no—
273#41—41no—
274#41—41no—
275#41—41no—
276#41—41no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8C22⋊Dic7312yes—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C22×C4—41no—
285C22×C4—2921no—
286C22×C4D4×D7311yescharacteristic
287C22×C4C2×C22⋊D7311yescharacteristic
288C22×C4#1192402711yescharacteristic
289SD16—44no—
290M16—732no—
291M16—1461no—
292M16—1461no—
293M16—1461no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—42no—
297C2×C8—2921no—
298C2×C8—2921no—
299C2×C8—2921no—
300C2×C8—1461no—
301C2×C8—1461no—
302C2×C8—1461no—
303C2×C8—1461no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—2921no—
312C4⋊C4—1462no—
313C4⋊C4C22⋊Dic7312yes—
314C22⋊C4—22no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C42—22no—
318C42—2921no—
319C42—2921no—
320C42C22×Dic7311yescharacteristic
321C42D4×D7311yescharacteristic
322C42C2×C22⋊D7311yescharacteristic
323C42#1192402711yescharacteristic
324C23—41no—
325C23#1455656311yescharacteristic
326Q8—42no—
327Q8—41no—
328Q8—41no—
329Q8—41no—
330Q8—41no—
331D4—42no—
332D4—41no—
333D4—41no—
334D4—41no—
335D4—41no—
336D4—81no—
337D4—81no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—2922no—
347C2×C4—5841no—
348C2×C4—5841no—
349C2×C4C2×C22⋊Dic7311yescharacteristic
350C2×C4D4×Dic7311yescharacteristic
351C2×C4D4×Dic7311yescharacteristic
352C2×C4#1455655911yescharacteristic
353C2×C4#1455656311yescharacteristic
354C2×C4#1455657211yescharacteristic
355C2×C4SD16×D7311yescharacteristic
356C2×C4#1455661211yescharacteristic
357C2×C4#1455661811yescharacteristic
358C2×C4#1455662011yescharacteristic
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#1798846711yescharacteristic
372C22#1798878511yescharacteristic
373C22#1798880711yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—42no—
378C4—42no—
379C4—81no—
380C4—5842no—
381C4SD16×Dic7311yescharacteristic
382C4#1798868811yescharacteristic
383C4#1798870411yescharacteristic
384C4#1798870911yescharacteristic
385C4#1798872411yescharacteristic
386C4#1798873111yescharacteristic
387C2—21no—
388C2—21no—
389C2—161no—
390C2#2261026011yescharacteristic
391C2#2261028811yescharacteristic
392C2#2261031511yescharacteristic
subgroupG/H
centerC2×C4#14556559
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#3167C2
socleC22×C73#386
prime radical#532×C73C22
cosocleC2×C42×C73C23
hypercenter#3167D73
hypocenterC73#9054
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
1. #22618880C2semidirect
3. #22608430C2semidirect
5. #22607771C2semidirect
6. #22607768C2semidirect
24. #498×C73C4semidirect
25. #492×C73C4semidirect
26. #487×C73C4semidirect
27. #486×C73C4semidirect
80. #11924012SD16semidirect
204. C73#9054semidirect
207. #498Dic73semidirect
208. #492Dic73semidirect
209. #487Dic73semidirect
210. #486Dic73semidirect

Maximal subgroups

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

subgroupindex
#226188802
#226108512
#226084302
#226083302
#226077712
#226077682
C73×#31672
#905473