Finite Group Databasebeta

← #33240879

Subgroups — #33240879

Order 18688.

556 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619536C211yesmaximal, characteristic
2C2×#17988634C211yesmaximal, characteristic, complemented
3#22612149C214yesmaximal, complemented
4#22612144C214yesmaximal, complemented
5#22611705C211yesmaximal, characteristic, complemented
6#22611704C211yesmaximal, characteristic, complemented
7#22611680C211yesmaximal, characteristic, complemented
8C2×#17988368C211yesmaximal, characteristic, complemented
9C73×#2148C211yesmaximal, characteristic, complemented
10C22×#11924011C2211yescharacteristic
11C2×M16×D73—21no—
12#17989094—21no—
13C2×#14556531C2211yescharacteristic
14#17988842C2212yes—
15#17988837C2212yes—
16#17988830—22no—
17#17988634—22no—
18#17988634C2214yes—
19C2×#14556527C2211yescharacteristic
20C2×#14556526—21no—
21#17988625C2211yescharacteristic
22#17988583—21no—
23#17988411C2214yescomplemented
24#17988410C2214yescomplemented
25#17988403C2212yescomplemented
26#17988400C2212yescomplemented
27#17988368C2214yes—
28C2×#135×C73C2211yescharacteristic
29C2×M32×C73C2211yescharacteristic
30C2×#108×C73C2211yescharacteristic
31#362×C73C2212yescomplemented
32#361×C73C2212yescomplemented
33C22×C4×D73D411yescharacteristic
34C2×#11924012—21no—
35C2×#11924011—21no—
36C2×#11924011—21no—
37C2×#11924011—21no—
38C2×#11924011—21no—
39C2×#11924011—41no—
40C2×#11924011D412yes—
41C2×#11924011C2312yes—
42M16×D73—24no—
43M16×D73—42no—
44C22×#10596831D411yescharacteristic
45C2×C8×D73—41no—
46#14556572—41no—
47#14556567—22no—
48#14556562—22no—
49#14556559—41no—
50C4×C22⋊D73—42no—
51#14556550—21no—
52#14556531—21no—
53#14556531C2312yes—
54C2×#11924003—21no—
55#14556527—21no—
56#14556527C2312yes—
57#14556526—24no—
58#14556525—21no—
59#14556523C2312yes—
60#14556515—22no—
61#14556512—22no—
62C2×#41×C73—21no—
63#135×C73—21no—
64#135×C73—21no—
65#135×C73—21no—
66#135×C73—21no—
67C2×Q16×C73C2311yescharacteristic
68C2×SD16×C73—21no—
69C2×D8×C73C2311yescharacteristic
70C2×M16×C73—21no—
71C22×C8×C73C2311yescharacteristic
72M32×C73—21no—
73M32×C73C2312yes—
74C2×C16×C73—21no—
75#108×C73—21no—
76#108×C73C2312yes—
77C23×D73SD1611yescharacteristic
78C2×C22⋊D73—42no—
79C22×Dic73SD1611yescharacteristic
80C2×C4×D73—21no—
81C2×C4×D73—21no—
82C2×C4×D73—21no—
83C2×C4×D73—21no—
84C2×C4×D73SD1612yes—
85C2×C4×D73C2×D412yes—
86#11924012—22no—
87#11924012—41no—
88#11924011—22no—
89#11924011—22no—
90#11924011—22no—
91#11924011—22no—
92#11924011—44no—
93C2×#10596831—21no—
94C2×#10596831—21no—
95C2×#10596831—41no—
96C2×#10596831C2×D412yes—
97C8×D73—44no—
98C4⋊Dic73—22no—
99#11924007—42no—
100C22⋊Dic73—41no—
101C22⋊Dic73—42no—
102#11924005—42no—
103C4×Dic73—41no—
104#11924003—22no—
105#41×C73—41no—
106#41×C73—41no—
107#41×C73—41no—
108#41×C73—41no—
109C2×Q8×C73—21no—
110C2×D4×C73—21no—
111C2×D4×C73—41no—
112C22×C4×C73—41no—
113C22×C4×C73C2×D411yescharacteristic
114Q16×C73—21no—
115Q16×C73—21no—
116SD16×C73—41no—
117SD16×C73—41no—
118D8×C73—21no—
119D8×C73—21no—
120M16×C73—22no—
121M16×C73—41no—
122C2×C8×C73—41no—
123C2×C8×C73C2×D411yescharacteristic
124C2×C8×C73#4111yescharacteristic
125C2×C8×C73#4111yescharacteristic
126C2×C8×C73#4111yescharacteristic
127C2×C8×C73#4111yescharacteristic
128C2×C8×C73C2411yescharacteristic
129C16×C73—22no—
130C23×C73—41no—
131C23×C73C2×SD1611yescharacteristic
132C22×D73—21no—
133C22×D73—21no—
134C22×D73—81no—
135C22×D73C2×SD1612yes—
136Q8×C73—41no—
137Q8×C73—41no—
138D4×C73—41no—
139D4×C73—41no—
140D4×C73—81no—
141D4×C73—81no—
142C2×C4×C73—41no—
143C2×C4×C73—81no—
144C2×C4×C73—81no—
145C2×C4×C73#12311yescharacteristic
146C2×C4×C73#12311yescharacteristic
147C2×C4×C73C2×SD1611yescharacteristic
148C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C73C22×D411yescharacteristic
151C22⋊D73—84no—
152C2×Dic73—21no—
153C2×Dic73—21no—
154C2×Dic73—42no—
155C2×Dic73C2×SD1612yes—
156C4×D73—22no—
157C4×D73—22no—
158C4×D73—22no—
159C4×D73—22no—
160C8×C73—42no—
161C8×C73#12311yescharacteristic
162C8×C73#12311yescharacteristic
163C8×C73C2×#4111yescharacteristic
164C8×C73C2×#4111yescharacteristic
165#10596831—22no—
166#10596831—42no—
167C22×C73—41no—
168C22×C73—82no—
169C22×C73—81no—
170C22×C73#48111yescharacteristic
171C22×C73#48111yescharacteristic
172C22×C73C22×SD1611yescharacteristic
173C2×D73—22no—
174C2×D73—42no—
175C2×D73—42no—
176C2×D73—81no—
177C4×C73—81no—
178C4×C73#48111yescharacteristic
179C4×C73#48111yescharacteristic
180C4×C73C2×#12311yescharacteristic
181C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
182Dic73—22no—
183Dic73—82no—
184#33656—731nomaximal, Sylow
185C2×C73—21no—
186C2×C73—21no—
187C2×C73—21no—
188C2×C73—161no—
189C2×C73C2×#48111yescharacteristic
190D73—42no—
191#3054—731no—
192C2×#430—731no—
193#2239—734no—
194#2238—734no—
195#2155—731no—
196#2155—731no—
197#2148—731no—
198#2148D7311yescharacteristic, complemented
199C2×#349—731no—
200C73#3365611yescharacteristic, complemented, Sylow
201C2×#135C2×D7311yescharacteristic
202C22×M16—731no—
203C22×M16—1461no—
204#525—1461no—
205C2×M32—731no—
206C2×M32C2×D7311yescharacteristic
207#470—732no—
208#469—732no—
209#468—1462no—
210#430—734no—
211#430—1462no—
212C2×#108—731no—
213C2×#108—1461no—
214C2×#108C2×D7311yescharacteristic
215#427—731no—
216#416—1461no—
217#365—734no—
218#365—734no—
219#362—732no—
220#362C2×D7312yescomplemented
221#361—732no—
222#361C2×D7312yescomplemented
223#349—734no—
224C2×#41—21no—
225C23×C4—731no—
226#135—21no—
227#135—21no—
228#135—21no—
229#135—21no—
230C2×Q16C22×D7311yescharacteristic
231C2×SD16—21no—
232C2×D8C22×D7311yescharacteristic
233C2×M16—21no—
234C2×M16—2921no—
235C2×M16—2922no—
236C2×M16—732no—
237C2×M16—732no—
238C2×M16—1464no—
239C2×M16—1461no—
240C2×M16—1461no—
241C2×M16—1461no—
242C2×M16—1461no—
243C2×M16—1461no—
244C22×C8—2921no—
245C22×C8—731no—
246C22×C8C22×D7311yescharacteristic
247#123—2921no—
248C22⋊Q8—1462no—
249#121—1462no—
250#120—2921no—
251C4×D4—2922no—
252#117—1461no—
253M32—21no—
254M32—732no—
255M32—1461no—
256M32C22×D7312yes—
257C2×C16—21no—
258C2×C16—1461no—
259#108—21no—
260#108—732no—
261#108—1464no—
262#108—1461no—
263#108C22×D7312yes—
264#107—1461no—
265#106—732no—
266#103—1462no—
267#102—1462no—
268C24—731no—
269#41—41no—
270#41—41no—
271#41—41no—
272#41—41no—
273C2×Q8—21no—
274C2×D4—21no—
275C2×D4—41no—
276C2×D4—2922no—
277C22×C4—41no—
278C22×C4—731no—
279C22×C4—732no—
280C22×C4—732no—
281C22×C4—1461no—
282C22×C4—1461no—
283C22×C4—1461no—
284C22×C4—1461no—
285C22×C4D4×D7311yescharacteristic
286Q16—21no—
287Q16—21no—
288SD16—41no—
289SD16—41no—
290D8—21no—
291D8—21no—
292M16—22no—
293M16—41no—
294M16—2924no—
295M16—2921no—
296M16—1462no—
297M16—1462no—
298M16—1462no—
299M16—1462no—
300M16—1462no—
301C2×C8—41no—
302C2×C8—2924no—
303C2×C8—2921no—
304C2×C8—732no—
305C2×C8—1461no—
306C2×C8—1461no—
307C2×C8D4×D7311yescharacteristic
308C2×C8#1192402711yescharacteristic
309C2×C8#1192402711yescharacteristic
310C2×C8#1192402811yescharacteristic
311C2×C8#1192402811yescharacteristic
312C2×C8C23×D7311yescharacteristic
313C4⋊C4—2922no—
314C4⋊C4—1462no—
315C22⋊C4—2922no—
316C22⋊C4—2921no—
317C22⋊C4—2922no—
318C42—2921no—
319C16—22no—
320C16—1462no—
321C23—41no—
322C23—5841no—
323C23—732no—
324C23—1461no—
325C23—1461no—
326C23SD16×D7311yescharacteristic
327Q8—41no—
328Q8—41no—
329D4—41no—
330D4—41no—
331D4—81no—
332D4—81no—
333D4—5844no—
334C2×C4—41no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—2922no—
338C2×C4—732no—
339C2×C4—1461no—
340C2×C4—1461no—
341C2×C4—1462no—
342C2×C4—1462no—
343C2×C4—1462no—
344C2×C4—1462no—
345C2×C4#1455657311yescharacteristic
346C2×C4#1455657311yescharacteristic
347C2×C4SD16×D7311yescharacteristic
348C2×C4#1455661511yescharacteristic
349C2×C4#1455661511yescharacteristic
350C2×C4C2×D4×D7311yescharacteristic
351C8—42no—
352C8—2922no—
353C8—1462no—
354C8#1455657311yescharacteristic
355C8#1455657311yescharacteristic
356C8C2×#1192402711yescharacteristic
357C8C2×#1192402811yescharacteristic
358C22—41no—
359C22—82no—
360C22—81no—
361C22—2922no—
362C22—2922no—
363C22—5841no—
364C22—1462no—
365C22#1798890411yescharacteristic
366C22#1798890411yescharacteristic
367C22C2×SD16×D7311yescharacteristic
368C4—81no—
369C4—5842no—
370C4—1462no—
371C4#1798890411yescharacteristic
372C4#1798890411yescharacteristic
373C4C2×#1455657311yescharacteristic
374C4C2×#1455661511yescharacteristic
375C2—21no—
376C2—21no—
377C2—21no—
378C2—161no—
379C2—2922no—
380C2C2×#1798890411yescharacteristic
subgroupG/H
centerC4#17988904
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#2148C2
socleC2×C73C2×#481
prime radical#22619536C2
hypercenter#2148D73
hypocenterC73#33656
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ #2148

Complements

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

NH ≅ G/Ntype
2. C2×#17988634C2semidirect
3. #22612149C2semidirect
4. #22612144C2semidirect
5. #22611705C2semidirect
6. #22611704C2semidirect
7. #22611680C2semidirect
8. C2×#17988368C2semidirect
9. C73×#2148C2semidirect
23. #17988411C22semidirect
24. #17988410C22semidirect
25. #17988403C22semidirect
26. #17988400C22semidirect
31. #362×C73C22semidirect
32. #361×C73C22semidirect
198. #2148D73semidirect
200. C73#33656semidirect
220. #362C2×D73semidirect
222. #361C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226195362
C2×#179886342
#226121492
#226121442
#226117052
#226117042
#226116802
C2×#179883682
C73×#21482
#3365673