Finite Group Databasebeta

← #31010479

Subgroups — #31010479

Order 17152.

546 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22570723C211yesmaximal, characteristic, complemented
2C4×#14524065C211yesmaximal, characteristic
3#22562374C212yesmaximal, complemented
4#22562367C212yesmaximal, complemented
5#22562362C211yesmaximal, characteristic, complemented
6#22562357C211yesmaximal, characteristic, complemented
7#22562352C211yesmaximal, characteristic, complemented
8#22562347C211yesmaximal, characteristic, complemented
9#22561993C212yesmaximal, complemented
10#22561917C212yesmaximal
11C2×C4⋊C16×C67C211yesmaximal, characteristic, complemented
12C2×#14524065C2211yescharacteristic
13#16863953—21no—
14C4×#11761628—21no—
15#16863801C2211yescharacteristic
16#16863801C2211yescharacteristic
17#16863798—21no—
18#16863798C2212yes—
19#16863794—21no—
20C4×Dic268C2211yescharacteristic
21C4×Dic268C2211yescharacteristic
22C4×#11761617—21no—
23C4×#11761617—21no—
24C4×D536C2211yescharacteristic
25C4×D536C2211yescharacteristic
26#16863434—21no—
27#16863434C2211yescharacteristic
28#16863434C2212yes—
29#16863393—21no—
30#16863224C2212yes—
31#16863224C2212yes—
32#16863222C2212yes—
33#16863222C2212yes—
34#16863201C2212yes—
35#16863201C2212yes—
36#16863198C2212yes—
37#16863198C2212yes—
38C22×C16×C67C2212yes—
39C2×C4×C8×C67C2211yescharacteristic
40C4⋊C16×C67C2211yescharacteristic
41C4⋊C16×C67C2211yescharacteristic
42C4⋊C16×C67C2212yes—
43C2×#11761628—21no—
44#14524065—22no—
45#14524065Q814yescomplemented
46C2×Dic268C2311yescharacteristic
47C2×#11761617—21no—
48C2×D536C2311yescharacteristic
49#14524043—21no—
50#14524043—21no—
51#14524035—22no—
52#14524022—41no—
53#14524022—41no—
54C4×Dic134—21no—
55C4×Dic134—21no—
56C4×Dic134—41no—
57C4×Dic134—41no—
58C4×D268—21no—
59C4×D268—21no—
60C4×C22⋊D67—41no—
61C4×C22⋊D67—41no—
62#14524002—21no—
63#14524002—22no—
64#14524000—41no—
65C42×D67—41no—
66#14523977—21no—
67#14523977—21no—
68#14523977C2311yescharacteristic
69#14523977C2311yescharacteristic
70#14523977C2312yes—
71#14523977C2312yes—
72#14523975—22no—
73#14523975—22no—
74#14523975—21no—
75#14523968—21no—
76#14523968—21no—
77#14523965—21no—
78#14523965—21no—
79C22×C8×C67D411yescharacteristic
80C22×C8×C67C2311yescharacteristic
81C2×C42×C67D411yescharacteristic
82C2×C16×C67—24no—
83C2×C16×C67C2312yes—
84C2×C16×C67C2312yes—
85C4×C8×C67D411yescharacteristic
86C4×C8×C67D411yescharacteristic
87C4×C8×C67C2311yescharacteristic
88C4×C8×C67C2311yescharacteristic
89#11761628—24no—
90#11761628—42no—
91C2×Dic134—21no—
92C2×Dic134—41no—
93C2×Dic134—41no—
94C2×D268—21no—
95C2×C22⋊D67—41no—
96C2×C4×D67—41no—
97C2×C4×D67—41no—
98C2×C4×D67—41no—
99Dic268—21no—
100Dic268C2×Q812yes—
101#11761617—24no—
102D536—21no—
103D536C2×Q812yes—
104C4⋊Dic67—21no—
105C4⋊Dic67—21no—
106C4⋊Dic67—22no—
107C4⋊Dic67—22no—
108C4⋊Dic67—41no—
109C4⋊Dic67—41no—
110Dic67⋊C4—41no—
111Dic67⋊C4—41no—
112Dic67⋊C4—42no—
113Dic67⋊C4—42no—
114C22⋊Dic67—41no—
115C22⋊Dic67—42no—
116(C2×D67)⋊C4—41no—
117(C2×D67)⋊C4—41no—
118C4×Dic67—41no—
119C4×Dic67—41no—
120C4×Dic67—41no—
121C4×Dic67—42no—
122C22×C4×C67D811yescharacteristic
123C22×C4×C67D811yescharacteristic
124C22×C4×C67C2×D411yescharacteristic
125C2×C8×C67—22no—
126C2×C8×C67C2×D411yescharacteristic
127C2×C8×C67C2×D411yescharacteristic
128C2×C8×C67C2×D411yescharacteristic
129C2×C8×C67C2×Q812yes—
130C2×C8×C67#4112yes—
131C2×C8×C67C2411yescharacteristic
132C42×C67D811yescharacteristic
133C42×C67D811yescharacteristic
134C42×C67C2×D411yescharacteristic
135C42×C67C2×D411yescharacteristic
136C16×C67—22no—
137C16×C67—22no—
138C23×C67C2×D811yescharacteristic
139C22×D67—41no—
140C2×C4×C67—22no—
141C2×C4×C67—41no—
142C2×C4×C67C22⋊Q812yes—
143C2×C4×C67C2×D811yescharacteristic
144C2×C4×C67C2×D811yescharacteristic
145C2×C4×C67C2×D811yescharacteristic
146C2×C4×C67C2×D811yescharacteristic
147C2×C4×C67C2×D811yescharacteristic
148C2×C4×C67#13712yes—
149C2×C4×C67C22×D411yescharacteristic
150C22⋊D67—44no—
151C2×Dic67—41no—
152C2×Dic67—41no—
153C2×Dic67—41no—
154C2×Dic67—42no—
155C2×Dic67—42no—
156D268—22no—
157D268—41no—
158C4×D67—44no—
159C4×D67—82no—
160C4×D67—82no—
161Dic134—22no—
162Dic134—41no—
163Dic134—82no—
164Dic134—82no—
165C8×C67—21no—
166C8×C67—21no—
167C8×C67C22⋊Q812yes—
168C8×C67C22×Q811yescharacteristic
169C8×C67C2×#4111yescharacteristic
170C22×C67—21no—
171C22×C67—21no—
172C22×C67#47412yes—
173C22×C67C22×D811yescharacteristic
174C2×D67—42no—
175C2×D67—81no—
176C2×D67—81no—
177C4×C67—21no—
178C4×C67—21no—
179C4×C67—21no—
180C4×C67#47412yes—
181C4×C67#50812yes—
182C4×C67C2×C22⋊Q811yescharacteristic
183C4×C67C2×#13711yescharacteristic
184Dic67—42no—
185Dic67—82no—
186Dic67—81no—
187Dic67—82no—
188Dic67—81no—
189#33626—671nomaximal, Sylow
190C2×C67—22no—
191C2×C67C2×#47411yescharacteristic
192C2×C67C2×#50811yescharacteristic
193C2×C67#341411yescharacteristic
194D67—82no—
195#3160—671no—
196C4×#135—671no—
197#2229—672no—
198#2228—672no—
199#2227—671no—
200#2226—671no—
201#2225—671no—
202#2224—671no—
203#2158—672no—
204C2×C4⋊C16D6711yescharacteristic, complemented
205#2138—672no—
206C67#3362611yescharacteristic, complemented, Sylow
207C2×#135—671no—
208#533—1341no—
209C4×#41—1341no—
210C22×C16C2×D6712yes—
211#500—671no—
212#500—671no—
213#499—672no—
214#499—1341no—
215#498—1341no—
216C4×Q16—671no—
217C4×Q16—671no—
218C4×SD16—1341no—
219C4×SD16—1341no—
220C4×D8—671no—
221C4×D8—671no—
222#427—672no—
223#427—671no—
224#427—1341no—
225#416—1341no—
226C2×C4×C8C2×D6711yescharacteristic
227#367—672no—
228#367—672no—
229#366—672no—
230#366—672no—
231C4⋊C16C2×D6711yescharacteristic
232C4⋊C16C2×D6711yescharacteristic
233C4⋊C16C2×D6712yes—
234#358—672no—
235#358—672no—
236#357—672no—
237#357—672no—
238C2×#41—1341no—
239#135—674no—
240#135—1342no—
241C2×Q16—671no—
242C2×SD16—1341no—
243C2×D8—671no—
244C22×C8D26811yescharacteristic
245C22×C8C22×D6711yescharacteristic
246C4⋊Q8—1341no—
247C4⋊Q8—1341no—
248#125—1342no—
249C22⋊Q8—2681no—
250C22⋊Q8—2681no—
251C4×Q8—2681no—
252C4×Q8—2681no—
253C4×Q8—1341no—
254C4×Q8—1341no—
255C4×D4—2681no—
256C4×D4—2681no—
257C4×D4—1341no—
258C4×D4—1341no—
259#117—2681no—
260#117—1341no—
261#117—1342no—
262C2×C42—2681no—
263C2×C42D26811yescharacteristic
264C2×C16—24no—
265C2×C16C22×D6712yes—
266C2×C16C22×D6712yes—
267#107—672no—
268#107—672no—
269#107—671no—
270#107—671no—
271#107—1341no—
272#107—1341no—
273#106—1341no—
274#106—1342no—
275#106—1342no—
276#103—1341no—
277#103—1341no—
278#102—1341no—
279#102—1341no—
280C4×C8D26811yescharacteristic
281C4×C8D26811yescharacteristic
282C4×C8C22×D6711yescharacteristic
283C4×C8C22×D6711yescharacteristic
284#41—2682no—
285#41—1344no—
286C2×Q8—2681no—
287C2×Q8—2681no—
288C2×Q8—1341no—
289C2×D4—2681no—
290C2×D4—1341no—
291C22×C4—2681no—
292C22×C4—2681no—
293C22×C4—2681no—
294C22×C4D53611yescharacteristic
295C22×C4D53611yescharacteristic
296C22×C4C2×D26811yescharacteristic
297Q16—672no—
298Q16—1341no—
299SD16—1344no—
300D8—672no—
301D8—1341no—
302C2×C8—22no—
303C2×C8C2×D26811yescharacteristic
304C2×C8C2×D26811yescharacteristic
305C2×C8C2×D26811yescharacteristic
306C2×C8Q8×D6712yes—
307C2×C8#1176162712yes—
308C2×C8C23×D6711yescharacteristic
309C4⋊C4—2681no—
310C4⋊C4—2681no—
311C4⋊C4—2682no—
312C4⋊C4—2682no—
313C4⋊C4—2681no—
314C4⋊C4—2681no—
315C4⋊C4—1341no—
316C4⋊C4—1341no—
317C4⋊C4—1342no—
318C4⋊C4—1342no—
319C22⋊C4—2681no—
320C22⋊C4—2682no—
321C22⋊C4—2681no—
322C22⋊C4—2681no—
323C42—2681no—
324C42—2681no—
325C42—2681no—
326C42—2682no—
327C42D53611yescharacteristic
328C42D53611yescharacteristic
329C42C2×D26811yescharacteristic
330C42C2×D26811yescharacteristic
331C16—22no—
332C16—22no—
333C23—2681no—
334C23C2×D53611yescharacteristic
335Q8—2681no—
336Q8—5362no—
337Q8—5362no—
338Q8—1342no—
339D4—2684no—
340D4—2681no—
341D4—1342no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—2681no—
345C2×C4—2684no—
346C2×C4—2681no—
347C2×C4—2681no—
348C2×C4—2682no—
349C2×C4—2682no—
350C2×C4—5362no—
351C2×C4—5362no—
352C2×C4#1452402312yes—
353C2×C4C2×D53611yescharacteristic
354C2×C4C2×D53611yescharacteristic
355C2×C4C2×D53611yescharacteristic
356C2×C4C2×D53611yescharacteristic
357C2×C4C2×D53611yescharacteristic
358C2×C4#1452407612yes—
359C2×C4C22×D26811yescharacteristic
360C8—21no—
361C8—21no—
362C8#1452402312yes—
363C8C2×Q8×D6711yescharacteristic
364C8C2×#1176162711yescharacteristic
365C22—21no—
366C22—21no—
367C22—2682no—
368C22—5361no—
369C22—5361no—
370C22#1686367712yes—
371C22C22×D53611yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—2682no—
376C4—5361no—
377C4—5362no—
378C4—5361no—
379C4—5362no—
380C4#1686367712yes—
381C4#1686382812yes—
382C4C2×#1452402311yescharacteristic
383C4C2×#1452407611yescharacteristic
384C2—22no—
385C2—5362no—
386C2C2×#1686367711yescharacteristic
387C2C2×#1686382811yescharacteristic
388C2#2257334511yescharacteristic
subgroupG/H
centerC2×C4C2×D536
commutator, cosocleC2×C8×C67C24
FrattiniC2×C8C23×D67
FittingC2×C4⋊C16×C67C2
socleC22×C67C22×D8
prime radicalC2×#14524065C22
hypercenterC2×C4⋊C16D67
hypocenterC67#33626
LCSG ▷ C2×C8×C67 ▷ C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4⋊C16

Complements

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

NH ≅ G/Ntype
1. #22570723C2semidirect
3. #22562374C2semidirect
4. #22562367C2semidirect
5. #22562362C2semidirect
6. #22562357C2semidirect
7. #22562352C2semidirect
8. #22562347C2semidirect
9. #22561993C2semidirect
11. C2×C4⋊C16×C67C2semidirect
45. #14524065Q8semidirect
204. C2×C4⋊C16D67semidirect
206. C67#33626semidirect

Maximal subgroups

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

subgroupindex
#225707232
C4×#145240652
#225623742
#225623672
#225623622
#225623572
#225623522
#225623472
#225619932
#225619172
C2×C4⋊C16×C672
#3362667