Finite Group Databasebeta

← #31010471

Subgroups — #31010471

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
1C4×#14524065C211yesmaximal, characteristic
2#22562370C212yesmaximal, complemented
3#22562363C212yesmaximal, complemented
4#22562358C211yesmaximal, characteristic, complemented
5#22562353C211yesmaximal, characteristic, complemented
6#22562348C211yesmaximal, characteristic, complemented
7#22562343C211yesmaximal, characteristic, complemented
8#22561991C212yesmaximal, complemented
9C2×C4⋊(C67⋊C16)C211yesmaximal, characteristic, complemented
10#22561914C212yesmaximal
11C67×#3160C211yesmaximal, characteristic, complemented
12C2×#14524065C2211yescharacteristic
13C4×#11761628—21no—
14C22×C67⋊C16C2212yes—
15C4×Dic268C2211yescharacteristic
16C4×Dic268C2211yescharacteristic
17C4×#11761617—21no—
18C4×#11761617—21no—
19C4×D536C2211yescharacteristic
20C4×D536C2211yescharacteristic
21#16863434C2211yescharacteristic
22#16863393—21no—
23#16863223C2212yes—
24#16863223C2212yes—
25#16863221C2212yes—
26#16863221C2212yes—
27C4⋊(C67⋊C16)C2211yescharacteristic
28C4⋊(C67⋊C16)C2211yescharacteristic
29C4⋊(C67⋊C16)C2212yes—
30#16863199C2212yes—
31#16863199C2212yes—
32#16863196C2212yes—
33#16863196C2212yes—
34#533×C67—21no—
35#500×C67C2211yescharacteristic
36#500×C67C2211yescharacteristic
37#499×C67—21no—
38#499×C67C2212yes—
39#498×C67—21no—
40#427×C67—21no—
41#427×C67C2212yes—
42C2×C4×C8×C67C2211yescharacteristic
43C2×#11761628—21no—
44#14524065—22no—
45#14524065Q814yescomplemented
46C2×Dic268C2311yescharacteristic
47C2×#11761617—21no—
48C2×D536C2311yescharacteristic
49C4×Dic134—21no—
50C4×Dic134—21no—
51C4×D268—21no—
52C4×D268—21no—
53C4×C22⋊D67—41no—
54C4×C22⋊D67—41no—
55#14524002—21no—
56#14524000—41no—
57C42×D67—41no—
58C2×C67⋊C16—24no—
59C2×C67⋊C16C2312yes—
60C2×C67⋊C16C2312yes—
61#14523977C2311yescharacteristic
62#14523977C2311yescharacteristic
63#14523975—21no—
64#14523968—21no—
65#14523968—21no—
66#14523965—21no—
67#14523965—21no—
68C22×C8×C67D411yescharacteristic
69C22×C8×C67C2311yescharacteristic
70C4⋊Q8×C67—21no—
71C4⋊Q8×C67—21no—
72#125×C67—22no—
73C22⋊Q8×C67—41no—
74C22⋊Q8×C67—41no—
75C4×Q8×C67—41no—
76C4×Q8×C67—41no—
77#117×C67—22no—
78C2×C42×C67D411yescharacteristic
79#107×C67—21no—
80#107×C67—21no—
81#107×C67C2312yes—
82#107×C67C2312yes—
83#106×C67—22no—
84#106×C67—22no—
85C4×C8×C67D411yescharacteristic
86C4×C8×C67D411yescharacteristic
87C4×C8×C67C2311yescharacteristic
88C4×C8×C67C2311yescharacteristic
89#11761628—24no—
90#11761628—42no—
91C2×Dic134—21no—
92C2×D268—21no—
93C2×C22⋊D67—41no—
94C2×C4×D67—41no—
95C2×C4×D67—41no—
96C2×C4×D67—41no—
97Dic268—21no—
98Dic268C2×Q812yes—
99#11761617—24no—
100D536—21no—
101D536C2×Q812yes—
102C4⋊Dic67—21no—
103C4⋊Dic67—21no—
104Dic67⋊C4—41no—
105Dic67⋊C4—41no—
106C22⋊Dic67—41no—
107(C2×D67)⋊C4—41no—
108(C2×D67)⋊C4—41no—
109C4×Dic67—41no—
110C4×Dic67—41no—
111C4×Dic67—41no—
112C67⋊C16—22no—
113C67⋊C16—22no—
114C2×Q8×C67—41no—
115C2×Q8×C67—41no—
116C22×C4×C67D811yescharacteristic
117C22×C4×C67D811yescharacteristic
118C22×C4×C67C2×D411yescharacteristic
119C2×C8×C67—22no—
120C2×C8×C67C2×D411yescharacteristic
121C2×C8×C67C2×D411yescharacteristic
122C2×C8×C67C2×D411yescharacteristic
123C2×C8×C67C2×Q812yes—
124C2×C8×C67#4112yes—
125C2×C8×C67C2411yescharacteristic
126C4⋊C4×C67—22no—
127C4⋊C4×C67—22no—
128C4⋊C4×C67—42no—
129C4⋊C4×C67—42no—
130C4⋊C4×C67—41no—
131C4⋊C4×C67—41no—
132C22⋊C4×C67—42no—
133C42×C67—42no—
134C42×C67D811yescharacteristic
135C42×C67D811yescharacteristic
136C42×C67C2×D411yescharacteristic
137C42×C67C2×D411yescharacteristic
138C23×C67C2×D811yescharacteristic
139C22×D67—41no—
140Q8×C67—82no—
141Q8×C67—82no—
142C2×C4×C67—22no—
143C2×C4×C67—41no—
144C2×C4×C67—42no—
145C2×C4×C67—42no—
146C2×C4×C67C22⋊Q812yes—
147C2×C4×C67C2×D811yescharacteristic
148C2×C4×C67C2×D811yescharacteristic
149C2×C4×C67C2×D811yescharacteristic
150C2×C4×C67C2×D811yescharacteristic
151C2×C4×C67C2×D811yescharacteristic
152C2×C4×C67#13712yes—
153C2×C4×C67C22×D411yescharacteristic
154C22⋊D67—44no—
155C2×Dic67—41no—
156C2×Dic67—41no—
157C2×Dic67—41no—
158D268—22no—
159D268—41no—
160C4×D67—44no—
161C4×D67—82no—
162C4×D67—82no—
163Dic134—22no—
164Dic134—41no—
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—82no—
181C4×C67—82no—
182C4×C67#47412yes—
183C4×C67#50812yes—
184C4×C67C2×C22⋊Q811yescharacteristic
185C4×C67C2×#13711yescharacteristic
186Dic67—42no—
187Dic67—81no—
188Dic67—81no—
189#33626—671nomaximal, Sylow
190C2×C67—22no—
191C2×C67C2×#47411yescharacteristic
192C2×C67C2×#50811yescharacteristic
193C2×C67#341411yescharacteristic
194D67—82no—
195#3160D6711yescharacteristic, complemented
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⋊C16—671no—
205#2138—672no—
206C67#3362611yescharacteristic, complemented, Sylow
207C2×#135—671no—
208#533—21no—
209C4×#41—1341no—
210C22×C16—672no—
211#500C2×D6711yescharacteristic
212#500C2×D6711yescharacteristic
213#499—21no—
214#499C2×D6712yes—
215#498—21no—
216C4×Q16—671no—
217C4×Q16—671no—
218C4×SD16—1341no—
219C4×SD16—1341no—
220C4×D8—671no—
221C4×D8—671no—
222#427—21no—
223#427—671no—
224#427C2×D6712yes—
225#416—1341no—
226C2×C4×C8C2×D6711yescharacteristic
227#367—672no—
228#367—672no—
229#366—672no—
230#366—672no—
231C4⋊C16—671no—
232C4⋊C16—671no—
233C4⋊C16—672no—
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×C8C22⋊D6711yescharacteristic
245C22×C8C22×D6711yescharacteristic
246C4⋊Q8—21no—
247C4⋊Q8—21no—
248#125—22no—
249C22⋊Q8—41no—
250C22⋊Q8—41no—
251C4×Q8—41no—
252C4×Q8—41no—
253C4×Q8—1341no—
254C4×Q8—1341no—
255C4×D4—2681no—
256C4×D4—2681no—
257C4×D4—1341no—
258C4×D4—1341no—
259#117—22no—
260#117—2681no—
261#117—1341no—
262C2×C42—2681no—
263C2×C42C22⋊D6711yescharacteristic
264C2×C16—672no—
265C2×C16—672no—
266C2×C16—1344no—
267#107—21no—
268#107—21no—
269#107—671no—
270#107—671no—
271#107C22×D6712yes—
272#107C22×D6712yes—
273#106—22no—
274#106—22no—
275#106—1341no—
276#103—1341no—
277#103—1341no—
278#102—1341no—
279#102—1341no—
280C4×C8C22⋊D6711yescharacteristic
281C4×C8C22⋊D6711yescharacteristic
282C4×C8C22×D6711yescharacteristic
283C4×C8C22×D6711yescharacteristic
284#41—2682no—
285#41—1344no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—1341no—
289C2×D4—2681no—
290C2×D4—1341no—
291C22×C4—2681no—
292C22×C4—2681no—
293C22×C4—2681no—
294C22×C4#1176161311yescharacteristic
295C22×C4#1176161311yescharacteristic
296C22×C4C2×C22⋊D6711yescharacteristic
297Q16—672no—
298Q16—1341no—
299SD16—1344no—
300D8—672no—
301D8—1341no—
302C2×C8—22no—
303C2×C8C2×C22⋊D6711yescharacteristic
304C2×C8C2×C22⋊D6711yescharacteristic
305C2×C8C2×C22⋊D6711yescharacteristic
306C2×C8Q8×D6712yes—
307C2×C8#1176162912yes—
308C2×C8C23×D6711yescharacteristic
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—2681no—
316C4⋊C4—2681no—
317C4⋊C4—1341no—
318C4⋊C4—1341no—
319C22⋊C4—42no—
320C22⋊C4—2681no—
321C22⋊C4—2681no—
322C22⋊C4—2681no—
323C42—42no—
324C42—2681no—
325C42—2681no—
326C42—2681no—
327C42#1176161311yescharacteristic
328C42#1176161311yescharacteristic
329C42C2×C22⋊D6711yescharacteristic
330C42C2×C22⋊D6711yescharacteristic
331C16—1342no—
332C16—1342no—
333C23—2681no—
334C23C2×#1176161311yescharacteristic
335Q8—82no—
336Q8—82no—
337Q8—2681no—
338Q8—1342no—
339D4—2684no—
340D4—2681no—
341D4—1342no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—2681no—
347C2×C4—2684no—
348C2×C4—2681no—
349C2×C4—2681no—
350C2×C4—5362no—
351C2×C4—5362no—
352C2×C4#1452401912yes—
353C2×C4C2×#1176161311yescharacteristic
354C2×C4C2×#1176161311yescharacteristic
355C2×C4C2×#1176161311yescharacteristic
356C2×C4C2×#1176161311yescharacteristic
357C2×C4C2×#1176161311yescharacteristic
358C2×C4#1452407412yes—
359C2×C4C22×C22⋊D6711yescharacteristic
360C8—21no—
361C8—21no—
362C8#1452401912yes—
363C8C2×Q8×D6711yescharacteristic
364C8C2×#1176162911yescharacteristic
365C22—21no—
366C22—21no—
367C22—2682no—
368C22—5361no—
369C22—5361no—
370C22#1686367312yes—
371C22C22×#1176161311yescharacteristic
372C4—21no—
373C4—21no—
374C4—21no—
375C4—82no—
376C4—82no—
377C4—2682no—
378C4—5361no—
379C4—5361no—
380C4#1686367312yes—
381C4#1686382612yes—
382C4C2×#1452401911yescharacteristic
383C4C2×#1452407411yescharacteristic
384C2—22no—
385C2—5362no—
386C2C2×#1686367311yescharacteristic
387C2C2×#1686382611yescharacteristic
388C2#2257334311yescharacteristic
subgroupG/H
centerC2×C4C2×#11761613
commutator, cosocleC2×C8×C67C24
FrattiniC2×C8C23×D67
FittingC67×#3160C2
socleC22×C67C22×D8
prime radicalC2×#14524065C22
hypercenter#3160D67
hypocenterC67#33626
LCSG ▷ C2×C8×C67 ▷ C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3160

Complements

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

NH ≅ G/Ntype
2. #22562370C2semidirect
3. #22562363C2semidirect
4. #22562358C2semidirect
5. #22562353C2semidirect
6. #22562348C2semidirect
7. #22562343C2semidirect
8. #22561991C2semidirect
9. C2×C4⋊(C67⋊C16)C2semidirect
11. C67×#3160C2semidirect
45. #14524065Q8semidirect
195. #3160D67semidirect
206. C67#33626semidirect

Maximal subgroups

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

subgroupindex
C4×#145240652
#225623702
#225623632
#225623582
#225623532
#225623482
#225623432
#225619912
C2×C4⋊(C67⋊C16)2
#225619142
C67×#31602
#3362667