Finite Group Databasebeta

← #30859255

Subgroups — #30859255

Order 17152.

890 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16863443C211yesmaximal, characteristic
2#22568797C211yesmaximal, characteristic
3#424×D67C212yesmaximal
4C2×#16863350C211yesmaximal, characteristic
5#22560963C211yesmaximal, characteristic, complemented
6#22560961C212yesmaximal, complemented
7#22560960C211yesmaximal, characteristic, complemented
8#22560957C211yesmaximal, characteristic, complemented
9#22560954C212yesmaximal, complemented
10#22559917C212yesmaximal
11C67×#1945C211yesmaximal, characteristic, complemented
12C22×#11761611C2211yescharacteristic
13C2×M16×D67C2212yes—
14C22×C8×D67C2211yescharacteristic
15C2×#14524000C2211yescharacteristic
16#117×D67C2212yes—
17#16863443—24no—
18C2×#14523979C2211yescharacteristic
19C2×#14523978C2212yes—
20#16863437C414yescomplemented
21#16863435C414yescomplemented
22#16863434C2211yescharacteristic
23#16863432C2212yes—
24#16863424C2212yes—
25#16863423C2212yes—
26#16863422C2212yes—
27#16863421—24no—
28C4⋊C8×D67—24no—
29#16863350—24no—
30C2×#14523954C2211yescharacteristic
31C2×#14523953C2211yescharacteristic
32C2×C8×Dic67C2211yescharacteristic
33#16863157C2212yes—
34#16863156C2212yes—
35#16863155C2212yes—
36C2×#108×C67C2211yescharacteristic
37#427×C67C2211yescharacteristic
38#424×C67C2212yes—
39C2×C4×C8×C67C2211yescharacteristic
40#339×C67C2212yes—
41C22×C4×D67C2311yescharacteristic
42C2×#11761612C2312yes—
43C2×#11761611—22no—
44C2×#11761611C2×C412yes—
45C2×#11761611C2×C412yes—
46C2×#11761611D414yes—
47C2×#11761611Q814yes—
48M16×D67—24no—
49M16×D67—24no—
50C22×C67⋊C8C2311yescharacteristic
51C22×C67⋊C8C2311yescharacteristic
52C2×C8×D67—22no—
53C2×C8×D67—22no—
54C2×C8×D67—22no—
55C2×C8×D67D414yes—
56#14524002C2312yes—
57#14524001C2×C414yes—
58#14524000—22no—
59#14524000—22no—
60#14524000Q814yes—
61#14523999—22no—
62C2×Dic67⋊C4—21no—
63C4⋊C4×D67C2×C414yes—
64C2×(C2×D67)⋊C4—21no—
65C22⋊C4×D67—22no—
66C2×C4×Dic67C2311yescharacteristic
67C42×D67—22no—
68#14523979—22no—
69#14523978—24no—
70#14523977C2×C412yes—
71#14523976C2×C414yes—
72#14523975C2×C412yes—
73#14523974C2×C414yes—
74C4⋊(C67⋊C8)—24no—
75Dic67⋊C8—24no—
76Dic67⋊C8—24no—
77#14523954—22no—
78#14523953—22no—
79C8×Dic67—22no—
80C2×M16×C67C2312yes—
81C22×C8×C67C2311yescharacteristic
82C22×C8×C67C2311yescharacteristic
83#117×C67C2312yes—
84C2×C42×C67C2311yescharacteristic
85#108×C67—22no—
86#107×C67C2×C412yes—
87#106×C67C2×C412yes—
88C4⋊C8×C67—24no—
89C4×C8×C67—22no—
90C23×D67#4111yescharacteristic
91C22×Dic67—21no—
92C22×Dic67#4111yescharacteristic
93C2×C4×D67—24no—
94C2×C4×D67C22×C412yes—
95C2×C4×D67C2×D412yes—
96C2×C4×D67C2×D412yes—
97C2×C4×D67C2×Q812yes—
98C2×C4×D67C2×Q812yes—
99C2×C4×D67#4112yes—
100#11761612—22no—
101#11761612—22no—
102#11761611—44no—
103#11761611C4⋊C418yes—
104#11761611C4⋊C418yes—
105C2×C67⋊C8—21no—
106C2×C67⋊C8—21no—
107C2×C67⋊C8—22no—
108C2×C67⋊C8C22×C412yes—
109C2×C67⋊C8C2×D412yes—
110C2×C67⋊C8C2×D412yes—
111C2×C67⋊C8C2×Q812yes—
112C8×D67—28no—
113C8×D67—44no—
114C4⋊Dic67C22×C414yes—
115Dic67⋊C4—24no—
116Dic67⋊C4—24no—
117C22⋊Dic67—22no—
118(C2×D67)⋊C4—24no—
119(C2×D67)⋊C4—24no—
120C4×Dic67—21no—
121C4×Dic67—22no—
122C4×Dic67—24no—
123C4×Dic67C2×Q812yes—
124C22×C4×C67—21no—
125C22×C4×C67C2411yescharacteristic
126M16×C67—22no—
127M16×C67—22no—
128C2×C8×C67—21no—
129C2×C8×C67—21no—
130C2×C8×C67—22no—
131C2×C8×C67C22×C411yescharacteristic
132C2×C8×C67C22×C411yescharacteristic
133C2×C8×C67C2×D412yes—
134C2×C8×C67C2×D412yes—
135C2×C8×C67C2×Q812yes—
136C4⋊C4×C67C22×C414yes—
137C22⋊C4×C67—22no—
138C42×C67—22no—
139C42×C67—21no—
140C42×C67C2×Q812yes—
141C23×C67C2×#4111yescharacteristic
142C22×D67—42no—
143C22×D67C4×D412yes—
144C22×D67C22⋊Q812yes—
145C22×D67#12512yes—
146C2×C4×C67—24no—
147C2×C4×C67—22no—
148C2×C4×C67—42no—
149C2×C4×C67C23×C411yescharacteristic
150C2×C4×C67C22×D411yescharacteristic
151C2×C4×C67C22×D411yescharacteristic
152C2×C4×C67C22×Q811yescharacteristic
153C2×C4×C67C22×Q811yescharacteristic
154C2×C4×C67C2×#4111yescharacteristic
155C2×Dic67—24no—
156C2×Dic67—22no—
157C2×Dic67—42no—
158C2×Dic67C4×D412yes—
159C2×Dic67C22⋊Q812yes—
160C2×Dic67#12512yes—
161C4×D67—48no—
162C4×D67C2×C4⋊C414yes—
163C4×D67C4×Q814yes—
164C4×D67#12114yes—
165C4×D67C4⋊Q814yes—
166C8×C67—22no—
167C8×C67—42no—
168C8×C67C2×C4⋊C412yes—
169C8×C67C2×C4⋊C412yes—
170C67⋊C8—22no—
171C67⋊C8—42no—
172C67⋊C8C2×C4⋊C414yes—
173C22×C67—22no—
174C22×C67C2×C4×D411yescharacteristic
175C22×C67C2×C22⋊Q811yescharacteristic
176C22×C67C2×#12511yescharacteristic
177C2×D67—24no—
178C2×D67—42no—
179C2×D67—42no—
180C2×D67#38914yes—
181C4×C67—42no—
182C4×C67—44no—
183C4×C67C22×C4⋊C411yescharacteristic
184C4×C67C2×C4×Q811yescharacteristic
185C4×C67C2×#12111yescharacteristic
186C4×C67C2×C4⋊Q811yescharacteristic
187Dic67—42no—
188Dic67—44no—
189Dic67#38914yes—
190#21164—671nomaximal, Sylow
191C2×C67—21no—
192C2×C67—21no—
193C2×C67C2×#38911yescharacteristic
194C2×C67#295512yes—
195D67—24no—
196C2×#430—671no—
197#2911—671no—
198C2×#424—672no—
199C2×#405—671no—
200#1946—671no—
201#1946—671no—
202#1946—672no—
203#1945—672no—
204#1945—671no—
205#1945D6711yescharacteristic, complemented
206#1733—672no—
207C67#2116411yescharacteristic, complemented, Sylow
208C22×M16—671no—
209C22×M16—672no—
210C23×C8—671no—
211C2×#117—671no—
212C2×#117—672no—
213#430—1344no—
214C2×#108—672no—
215C2×#108—671no—
216C2×#108C2×D6711yescharacteristic
217#428—674no—
218#428—674no—
219#427—671no—
220#427—672no—
221#427C2×D6711yescharacteristic
222#424—672no—
223#424—672no—
224#424—672no—
225#424—1344no—
226#424C2×D6712yes—
227C2×C4⋊C8—1344no—
228#405—1344no—
229C2×#97—671no—
230C2×#97—671no—
231C2×C4×C8—671no—
232C2×C4×C8C2×D6711yescharacteristic
233#339—672no—
234#339—672no—
235#339—672no—
236#339C2×D6712yes—
237C23×C4—671no—
238C2×M16—672no—
239C2×M16—672no—
240C2×M16—674no—
241C2×M16—674no—
242C2×M16—672no—
243C2×M16—1342no—
244C2×M16—1344no—
245C2×M16—1344no—
246C2×M16C22×D6712yes—
247C22×C8—671no—
248C22×C8—671no—
249C22×C8—674no—
250C22×C8—1342no—
251C22×C8—1342no—
252C22×C8—1342no—
253C22×C8C22×D6711yescharacteristic
254C22×C8C22×D6711yescharacteristic
255#117—674no—
256#117—672no—
257#117—674no—
258#117—1342no—
259#117—1342no—
260#117—1342no—
261#117C22×D6712yes—
262C2×C4⋊C4—674no—
263C2×C4⋊C4—1341no—
264C2×C22⋊C4—1342no—
265C2×C22⋊C4—1341no—
266C2×C42—671no—
267C2×C42—1342no—
268C2×C42C22×D6711yescharacteristic
269#108—22no—
270#108—1344no—
271#108—1342no—
272#107—672no—
273#107—674no—
274#107C4×D6712yes—
275#106—672no—
276#106—674no—
277#106C4×D6712yes—
278C4⋊C8—24no—
279C4⋊C8—1344no—
280C4⋊C8—1344no—
281C4⋊C8—1344no—
282#97—1342no—
283#97—1342no—
284C4×C8—22no—
285C4×C8—1342no—
286C24—671no—
287C22×C4—21no—
288C22×C4—671no—
289C22×C4—672no—
290C22×C4—672no—
291C22×C4—672no—
292C22×C4—672no—
293C22×C4—672no—
294C22×C4—672no—
295C22×C4—1344no—
296C22×C4—1341no—
297C22×C4C23×D6711yescharacteristic
298M16—22no—
299M16—22no—
300M16—2684no—
301M16—678no—
302M16—678no—
303M16—1342no—
304M16—1342no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—22no—
308C2×C8—2684no—
309C2×C8—672no—
310C2×C8—672no—
311C2×C8—672no—
312C2×C8—672no—
313C2×C8—1348no—
314C2×C8—1342no—
315C2×C8—1341no—
316C2×C8—1341no—
317C2×C8C2×C4×D6711yescharacteristic
318C2×C8C2×C4×D6711yescharacteristic
319C2×C8D4×D6712yes—
320C2×C8D4×D6712yes—
321C2×C8Q8×D6712yes—
322C4⋊C4—674no—
323C4⋊C4—1344no—
324C4⋊C4—1344no—
325C4⋊C4C2×C4×D6714yes—
326C22⋊C4—22no—
327C22⋊C4—1344no—
328C22⋊C4—1344no—
329C22⋊C4—1342no—
330C42—21no—
331C42—22no—
332C42—672no—
333C42—1344no—
334C42—1341no—
335C42—1342no—
336C42Q8×D6712yes—
337C23—2682no—
338C23—672no—
339C23—672no—
340C23—672no—
341C23#41×D6711yescharacteristic
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—42no—
345C2×C4—2688no—
346C2×C4—2682no—
347C2×C4—672no—
348C2×C4—672no—
349C2×C4—672no—
350C2×C4—674no—
351C2×C4—674no—
352C2×C4—674no—
353C2×C4—674no—
354C2×C4—1344no—
355C2×C4—1342no—
356C2×C4C22×C4×D6711yescharacteristic
357C2×C4C2×D4×D6711yescharacteristic
358C2×C4C2×D4×D6711yescharacteristic
359C2×C4C2×Q8×D6711yescharacteristic
360C2×C4C2×Q8×D6711yescharacteristic
361C2×C4#41×D6711yescharacteristic
362C8—22no—
363C8—42no—
364C8—2682no—
365C8—674no—
366C8—1342no—
367C8C4⋊C4×D6712yes—
368C8C4⋊C4×D6712yes—
369C22—22no—
370C22—2682no—
371C22—2682no—
372C22—674no—
373C22—1344no—
374C22C4×D4×D6711yescharacteristic
375C22C22⋊Q8×D6711yescharacteristic
376C22#125×D6711yescharacteristic
377C4—42no—
378C4—44no—
379C4—2682no—
380C4—2684no—
381C4—674no—
382C4C2×C4⋊C4×D6711yescharacteristic
383C4C4×Q8×D6711yescharacteristic
384C4#121×D6711yescharacteristic
385C4C4⋊Q8×D6711yescharacteristic
386C2—21no—
387C2—21no—
388C2—1344no—
389C2#389×D6711yescharacteristic
390C2#2256910712yes—
subgroupG/H
centerC2×C4C2×D4×D67
commutatorC2×C4×C67C23×C4
FrattiniC22×C4C23×D67
FittingC67×#1945C2
socleC22×C67C2×C4×D4
prime radicalC23×D67#41
cosocleC22×C4×C67C24
hypercenter#1945D67
hypocenterC67#21164
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1945

Complements

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

NH ≅ G/Ntype
5. #22560963C2semidirect
6. #22560961C2semidirect
7. #22560960C2semidirect
8. #22560957C2semidirect
9. #22560954C2semidirect
11. C67×#1945C2semidirect
20. #16863437C4semidirect
21. #16863435C4semidirect
205. #1945D67semidirect
207. C67#21164semidirect

Maximal subgroups

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

subgroupindex
C2×#168634432
#225687972
#424×D672
C2×#168633502
#225609632
#225609612
#225609602
#225609572
#225609542
#225599172
C67×#19452
#2116467