Finite Group Databasebeta

← #32989863

Subgroups — #32989863

Order 18688.

490 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988864C211yesmaximal, characteristic
2#22610734C211yesmaximal, characteristic, complemented
3#22610724C211yesmaximal, characteristic
4#22609932C211yesmaximal, characteristic
5C2×#17988339C211yesmaximal, characteristic, complemented
6#22607815C211yesmaximal, characteristic, complemented
7C2×#332×C73C211yesmaximal, characteristic, complemented
8C2×C4×D292C2211yescharacteristic
9#17988864—22no—
10#17988864—22no—
11C2×#14556524—21no—
12C2×C4⋊#10596831C2211yescharacteristic
13C2×#14556515—21no—
14C2×#14556515—21no—
15C2×#14556513—21no—
16C2×#14556511—21no—
17C2×#14556503—21no—
18C2×#14556503—21no—
19C2×C4×#10596831C2211yescharacteristic
20#17988457C2211yescharacteristic
21#17988350—21no—
22#17988339—22no—
23C2×C4⋊Q8×C73C2211yescharacteristic
24C2×C4⋊C8×C73C2211yescharacteristic
25C2×C4⋊C8×C73C2211yescharacteristic
26#332×C73—22no—
27C22×D292D411yescharacteristic
28C22×C4×D73—21no—
29C22×#10596831—21no—
30C22×#10596831D411yescharacteristic
31C22×#10596831D411yescharacteristic
32C4×D292—21no—
33C4×D292—21no—
34C4×D292—21no—
35C4×D292—21no—
36C2×C4⋊Dic73—21no—
37C2×C4⋊Dic73—21no—
38C2×C4⋊Dic73—21no—
39C2×C4⋊Dic73D411yescharacteristic
40C2×#11924005—21no—
41#14556524—24no—
42C4⋊#10596831—21no—
43C4⋊#10596831D412yes—
44#14556515—42no—
45#14556515—42no—
46#14556513—42no—
47#14556511—24no—
48#14556503—42no—
49#14556503—42no—
50C4×#10596831—21no—
51C4×#10596831—21no—
52C22×Q8×C73—21no—
53C22×C8×C73—21no—
54C22×C8×C73—21no—
55C4⋊Q8×C73—21no—
56C4⋊Q8×C73—21no—
57C4⋊Q8×C73—21no—
58C4⋊Q8×C73D412yes—
59C2×C4⋊C4×C73—21no—
60C2×C4⋊C4×C73—21no—
61C2×C42×C73C2311yescharacteristic
62C4⋊C8×C73—21no—
63C4⋊C8×C73—21no—
64C4⋊C8×C73—21no—
65C4⋊C8×C73—21no—
66C23×D73—21no—
67C2×D292—21no—
68C2×D292—21no—
69C2×D292—21no—
70C2×D292—21no—
71C2×D292—21no—
72C2×D292Q1612yes—
73C22×Dic73—21no—
74C22×Dic73—41no—
75C2×C4×D73—41no—
76C2×C4×D73—41no—
77C2×C4×D73—41no—
78C2×C4×D73—41no—
79C2×C4×D73—41no—
80C2×C4×D73—41no—
81C2×#10596831—21no—
82C2×#10596831—21no—
83C2×#10596831—22no—
84C2×#10596831—21no—
85C2×#10596831—21no—
86C2×#10596831—21no—
87C2×#10596831—41no—
88C2×#10596831—41no—
89C2×#10596831D812yes—
90C4⋊Dic73—21no—
91C4⋊Dic73—21no—
92C4⋊Dic73—41no—
93C4⋊Dic73—41no—
94C4⋊Dic73—41no—
95C4⋊Dic73—41no—
96C4⋊Dic73—81no—
97#11924005—41no—
98#11924005—41no—
99C2×Q8×C73—22no—
100C2×Q8×C73—41no—
101C2×Q8×C73—41no—
102C2×Q8×C73—41no—
103C2×Q8×C73—41no—
104C2×Q8×C73—41no—
105C22×C4×C73—41no—
106C22×C4×C73C2×D411yescharacteristic
107C22×C4×C73C2×D411yescharacteristic
108C22×C4×C73#4111yescharacteristic
109C2×C8×C73—41no—
110C2×C8×C73—41no—
111C2×C8×C73—41no—
112C2×C8×C73—41no—
113C2×C8×C73—41no—
114C2×C8×C73—41no—
115C4⋊C4×C73—22no—
116C4⋊C4×C73—22no—
117C4⋊C4×C73—41no—
118C4⋊C4×C73—41no—
119C42×C73C2×D411yescharacteristic
120C42×C73C2×D411yescharacteristic
121C42×C73#4111yescharacteristic
122C42×C73#4111yescharacteristic
123C23×C73#12111yescharacteristic
124C22×D73—22no—
125C22×D73—41no—
126C22×D73—41no—
127C22×D73—41no—
128C22×D73—41no—
129C22×D73—41no—
130C22×D73—41no—
131Q8×C73—42no—
132Q8×C73—42no—
133Q8×C73—81no—
134Q8×C73—81no—
135C2×C4×C73—21no—
136C2×C4×C73—21no—
137C2×C4×C73—42no—
138C2×C4×C73—81no—
139C2×C4×C73—81no—
140C2×C4×C73#12011yescharacteristic
141C2×C4×C73#12111yescharacteristic
142C2×C4×C73#12111yescharacteristic
143C2×C4×C73#12311yescharacteristic
144C2×C4×C73#12311yescharacteristic
145C2×C4×C73#12411yescharacteristic
146C2×C4×C73C2×D811yescharacteristic
147C2×C4×C73C2×Q1611yescharacteristic
148C2×C4×C73#13511yescharacteristic
149C2×C4×C73#13511yescharacteristic
150C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
151C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
152C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C73#13711yescharacteristic
154C2×Dic73—41no—
155C2×Dic73—41no—
156C2×Dic73—41no—
157C2×Dic73—81no—
158C2×Dic73—81no—
159C2×Dic73—81no—
160D292—22no—
161D292—22no—
162D292—41no—
163D292—41no—
164C4×D73—81no—
165C4×D73—81no—
166C4×D73—81no—
167C4×D73—81no—
168C8×C73—81no—
169C8×C73—81no—
170#10596831—22no—
171#10596831—42no—
172#10596831—41no—
173C22×C73#39411yescharacteristic
174C22×C73#45911yescharacteristic
175C22×C73#46311yescharacteristic
176C22×C73#46711yescharacteristic
177C22×C73#46811yescharacteristic
178C22×C73#47111yescharacteristic
179C22×C73#47311yescharacteristic
180C2×D73—42no—
181C2×D73—42no—
182C2×D73—42no—
183C2×D73—81no—
184C2×D73—81no—
185C2×D73—81no—
186C2×D73—81no—
187C4×C73—21no—
188C4×C73—21no—
189C4×C73—82no—
190C4×C73#44911yescharacteristic
191C4×C73#45111yescharacteristic
192C4×C73#45911yescharacteristic
193C4×C73#46311yescharacteristic
194C4×C73#46611yescharacteristic
195C4×C73#46911yescharacteristic
196C4×C73#48411yescharacteristic
197C4×C73#48511yescharacteristic
198Dic73—81no—
199Dic73—161no—
200#10485—731nomaximal, Sylow
201C2×C73#166911yescharacteristic
202C2×C73#168311yescharacteristic
203C2×C73#168511yescharacteristic
204C2×C73#169911yescharacteristic
205C2×C73#203611yescharacteristic
206C2×C73#205211yescharacteristic
207C2×C73#205411yescharacteristic
208D73—82no—
209C2×#474—731no—
210#1956—731no—
211#1954—731no—
212#1802—731no—
213C2×#335—731no—
214C2×#332D7311yescharacteristic, complemented
215#1296—731no—
216C73#1048511yescharacteristic, complemented, Sylow
217C2×C4⋊Q8C2×D7311yescharacteristic
218C2×C4×D4—731no—
219#474—1462no—
220#474—1462no—
221C2×#107—1461no—
222C2×C4⋊C8—731no—
223C2×C4⋊C8C2×D7311yescharacteristic
224C2×C4⋊C8C2×D7311yescharacteristic
225C2×#103—1461no—
226C2×#103—1461no—
227C2×#102—1461no—
228C2×#102—1461no—
229C2×C22⋊C8—1461no—
230C2×C22⋊C8—1461no—
231C2×C4×C8—731no—
232#384—731no—
233#340—1461no—
234#335—1462no—
235#332—22no—
236C22×Q8—21no—
237C22×D4—731no—
238C23×C4—1461no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—731no—
242C22×C8—731no—
243C22×C8—1461no—
244C4⋊Q8—21no—
245C4⋊Q8—21no—
246C4⋊Q8—21no—
247C4⋊Q8C22⋊D7312yes—
248C4×D4—1461no—
249C4×D4—1461no—
250C4×D4—1461no—
251C4×D4—1461no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—731no—
255C2×C4⋊C4—1461no—
256C2×C4⋊C4—1461no—
257C2×C4⋊C4—1461no—
258C2×C22⋊C4—1461no—
259C2×C42C22×D7311yescharacteristic
260#107—1464no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—732no—
266C4⋊C8—1461no—
267#103—2922no—
268#103—2922no—
269#102—2922no—
270#102—1464no—
271C22⋊C8—2922no—
272C22⋊C8—2922no—
273C4×C8—1461no—
274C4×C8—1461no—
275C24—1461no—
276C2×Q8—22no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×D4—732no—
283C2×D4—1461no—
284C2×D4—1461no—
285C2×D4—1461no—
286C2×D4—1461no—
287C2×D4—1461no—
288C22×C4—41no—
289C22×C4—2921no—
290C22×C4—2921no—
291C22×C4—2921no—
292C22×C4—2921no—
293C22×C4—2921no—
294C22×C4—2921no—
295C22×C4—2921no—
296C22×C4—1461no—
297C22×C4D4×D7311yescharacteristic
298C22×C4D4×D7311yescharacteristic
299C22×C4#1192402811yescharacteristic
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—2921no—
307C2×C8—2921no—
308C2×C8—732no—
309C2×C8—1462no—
310C2×C8—1461no—
311C2×C8—1461no—
312C2×C8—1461no—
313C2×C8—1461no—
314C2×C8—1461no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—2921no—
320C4⋊C4—2921no—
321C4⋊C4—2921no—
322C4⋊C4—2921no—
323C4⋊C4—5841no—
324C4⋊C4—1461no—
325C4⋊C4—1461no—
326C22⋊C4—2921no—
327C22⋊C4—2921no—
328C42D4×D7311yescharacteristic
329C42C2×C22⋊D7311yescharacteristic
330C42#1192402711yescharacteristic
331C42#1192402811yescharacteristic
332C23—2921no—
333C23—2921no—
334C23—2921no—
335C23—2921no—
336C23—2921no—
337C23—2921no—
338C23—1462no—
339C23#1455656411yescharacteristic
340Q8—42no—
341Q8—42no—
342Q8—81no—
343Q8—81no—
344D4—2921no—
345D4—2921no—
346D4—1462no—
347D4—1462no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—42no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—2921no—
354C2×C4—2921no—
355C2×C4—2921no—
356C2×C4—5841no—
357C2×C4—5841no—
358C2×C4—5841no—
359C2×C4—5841no—
360C2×C4—5841no—
361C2×C4—5841no—
362C2×C4—5841no—
363C2×C4#1455655911yescharacteristic
364C2×C4#1455656311yescharacteristic
365C2×C4#1455656411yescharacteristic
366C2×C4#1455657311yescharacteristic
367C2×C4#1455657411yescharacteristic
368C2×C4#1455657711yescharacteristic
369C2×C4D8×D7311yescharacteristic
370C2×C4Q16×D7311yescharacteristic
371C2×C4#1455661211yescharacteristic
372C2×C4#1455661211yescharacteristic
373C2×C4#1455661511yescharacteristic
374C2×C4#1455661811yescharacteristic
375C2×C4#1455661811yescharacteristic
376C2×C4#1455662311yescharacteristic
377C8—81no—
378C8—81no—
379C8—2922no—
380C8—2921no—
381C8—1462no—
382C22—2922no—
383C22—2922no—
384C22—2922no—
385C22—5841no—
386C22—5841no—
387C22—5841no—
388C22—5841no—
389C22#1798850211yescharacteristic
390C22#1798878111yescharacteristic
391C22#1798880111yescharacteristic
392C22#1798882711yescharacteristic
393C22#1798883611yescharacteristic
394C22#1798885311yescharacteristic
395C22#1798886111yescharacteristic
396C4—21no—
397C4—21no—
398C4—82no—
399C4—5841no—
400C4—11681no—
401C4#1798873111yescharacteristic
402C4#1798874111yescharacteristic
403C4#1798878011yescharacteristic
404C4#1798880411yescharacteristic
405C4#1798882211yescharacteristic
406C4#1798883911yescharacteristic
407C4#1798892111yescharacteristic
408C4#1798892611yescharacteristic
409C2—5842no—
410C2#2260929511yescharacteristic
411C2#2260938111yescharacteristic
412C2#2260939911yescharacteristic
413C2#2260947711yescharacteristic
414C2#2261113211yescharacteristic
415C2#2261123011yescharacteristic
416C2#2261124011yescharacteristic
subgroupG/H
centerC23#14556564
commutator, cosocleC2×C42×C73C23
FrattiniC2×C42C22×D73
FittingC2×#332×C73C2
socleC23×C73#121
prime radicalC22×D292D4
hypercenterC2×#332D73
hypocenterC73#10485
LCSG ▷ C2×C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#332

Complements

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

NH ≅ G/Ntype
2. #22610734C2semidirect
5. C2×#17988339C2semidirect
6. #22607815C2semidirect
7. C2×#332×C73C2semidirect
214. C2×#332D73semidirect
216. C73#10485semidirect

Maximal subgroups

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

subgroupindex
C2×#179888642
#226107342
#226107242
#226099322
C2×#179883392
#226078152
C2×#332×C732
#1048573