Finite Group Databasebeta

← C4×#16863837 #31009160

Subgroups — C4×#16863837 #31009160

Order 17152.

674 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×#16863837C211yesmaximal, characteristic
2C4×#14524065C211yesmaximal, characteristic
3#22562078C212yesmaximal, complemented
4#22562071C212yesmaximal, complemented
5C4×#14523989C212yesmaximal, complemented
6C4×#14523986C212yesmaximal, complemented
7#22561995C211yesmaximal, characteristic, complemented
8#22561929C211yesmaximal, characteristic, complemented
9#22561919C211yesmaximal, characteristic
10C4×#14523983C211yesmaximal, characteristic, complemented
11C2×C4×Q16×C67C211yesmaximal, characteristic, complemented
12C2×#14524065C2211yescharacteristic
13C4×#11761628—21no—
14#16863837C418yesdirect product factor
15C2×#14523989C2212yes—
16C2×#14523986C2212yes—
17C2×#14523983C2211yescharacteristic
18C2×#14523983C2211yescharacteristic
19C4×Dic268C2211yescharacteristic
20C4×Dic268C2211yescharacteristic
21C4×#11761617—21no—
22C4×#11761617—21no—
23C4×D536C2211yescharacteristic
24C4×D536C2211yescharacteristic
25#16863434C2211yescharacteristic
26#16863393—21no—
27#16863223C2212yes—
28#16863221C2212yes—
29#16863200C2212yes—
30#16863200C2212yes—
31#16863199C2212yes—
32#16863196C2212yes—
33#16863172C2212yes—
34C4×C67⋊C16C2212yes—
35C22×Q16×C67C2211yescharacteristic
36C2×C4×Q8×C67—21no—
37C4×Q16×C67—21no—
38C4×Q16×C67—21no—
39C4×Q16×C67C2212yes—
40C4×Q16×C67C2212yes—
41C2×#107×C67C2211yescharacteristic
42C2×#103×C67—21no—
43C2×C4×C8×C67C2211yescharacteristic
44C2×#11761628—21no—
45#14524065—22no—
46#14524065C2×C414yes—
47C2×Dic268C2311yescharacteristic
48C2×#11761617—21no—
49C2×D536C2311yescharacteristic
50C4×Dic134—21no—
51C4×Dic134—21no—
52C4×D268—21no—
53C4×D268—21no—
54C4×C22⋊D67—41no—
55C4×C22⋊D67—41no—
56#14524002—21no—
57#14524000—41no—
58C42×D67—41no—
59#14523989C2×C418yescomplemented
60#14523986C2×C418yescomplemented
61#14523983—22no—
62#14523983C2×C414yescomplemented
63C2×C67⋊C16C2312yes—
64C2×C67⋊C16C2312yes—
65#14523977C2311yescharacteristic
66#14523977C2311yescharacteristic
67#14523975—21no—
68#14523968—21no—
69#14523968—21no—
70#14523965—21no—
71#14523965—21no—
72C22×Q8×C67—21no—
73C2×Q16×C67—21no—
74C2×Q16×C67—22no—
75C2×Q16×C67C2×C414yescomplemented
76C2×Q16×C67C2312yes—
77C22×C8×C67D411yescharacteristic
78C22×C8×C67C2311yescharacteristic
79C4×Q8×C67—22no—
80C4×Q8×C67—22no—
81C4×Q8×C67—41no—
82C4×Q8×C67—41no—
83C2×C4⋊C4×C67—21no—
84C2×C4⋊C4×C67—41no—
85C2×C42×C67—41no—
86C2×C42×C67D411yescharacteristic
87#107×C67—21no—
88#107×C67C2312yes—
89#103×C67—22no—
90#103×C67—22no—
91C4×C8×C67D411yescharacteristic
92C4×C8×C67D411yescharacteristic
93C4×C8×C67C2311yescharacteristic
94C4×C8×C67C2311yescharacteristic
95#11761628—24no—
96#11761628—42no—
97C2×Dic134—21no—
98C2×D268—21no—
99C2×C22⋊D67—41no—
100C2×C4×D67—41no—
101C2×C4×D67—41no—
102C2×C4×D67—41no—
103Dic268—21no—
104Dic268C22×C412yes—
105#11761617—24no—
106D536—21no—
107D536C22×C412yes—
108C4⋊Dic67—21no—
109C4⋊Dic67—21no—
110Dic67⋊C4—41no—
111Dic67⋊C4—41no—
112C22⋊Dic67—41no—
113(C2×D67)⋊C4—41no—
114(C2×D67)⋊C4—41no—
115C4×Dic67—41no—
116C4×Dic67—41no—
117C4×Dic67—41no—
118C67⋊C16—22no—
119C67⋊C16C22×C414yes—
120C2×Q8×C67—24no—
121C2×Q8×C67—22no—
122C2×Q8×C67—41no—
123C2×Q8×C67—42no—
124C22×C4×C67—41no—
125C22×C4×C67—41no—
126C22×C4×C67D811yescharacteristic
127C22×C4×C67D811yescharacteristic
128C22×C4×C67C2×D411yescharacteristic
129Q16×C67—24no—
130Q16×C67—22no—
131Q16×C67C22×C414yes—
132C2×C8×C67—22no—
133C2×C8×C67C22×C412yes—
134C2×C8×C67C2×D411yescharacteristic
135C2×C8×C67C2×D411yescharacteristic
136C2×C8×C67C2×D411yescharacteristic
137C2×C8×C67#4112yes—
138C2×C8×C67C2411yescharacteristic
139C4⋊C4×C67—22no—
140C4⋊C4×C67—41no—
141C4⋊C4×C67—42no—
142C4⋊C4×C67—42no—
143C42×C67—42no—
144C42×C67—42no—
145C42×C67D811yescharacteristic
146C42×C67D811yescharacteristic
147C42×C67C2×D411yescharacteristic
148C42×C67C2×D411yescharacteristic
149C23×C67C2×D811yescharacteristic
150C22×D67—41no—
151Q8×C67—24no—
152Q8×C67—42no—
153Q8×C67—44no—
154C2×C4×C67—22no—
155C2×C4×C67—41no—
156C2×C4×C67—42no—
157C2×C4×C67—44no—
158C2×C4×C67—42no—
159C2×C4×C67—82no—
160C2×C4×C67C4×D412yes—
161C2×C4×C67C2×D811yescharacteristic
162C2×C4×C67C2×D811yescharacteristic
163C2×C4×C67C2×D811yescharacteristic
164C2×C4×C67C2×D811yescharacteristic
165C2×C4×C67C2×D811yescharacteristic
166C2×C4×C67#13512yes—
167C2×C4×C67C22×D411yescharacteristic
168C22⋊D67—44no—
169C2×Dic67—41no—
170C2×Dic67—41no—
171C2×Dic67—41no—
172D268—22no—
173D268—41no—
174C4×D67—44no—
175C4×D67—82no—
176C4×D67—82no—
177Dic134—22no—
178Dic134—41no—
179C8×C67—21no—
180C8×C67—21no—
181C8×C67C4×D412yes—
182C8×C67C23×C411yescharacteristic
183C8×C67C2×#4111yescharacteristic
184C22×C67—21no—
185C22×C67—21no—
186C22×C67C4×D812yes—
187C22×C67C22×D811yescharacteristic
188C2×D67—42no—
189C2×D67—81no—
190C2×D67—81no—
191C4×C67—21no—
192C4×C67—21no—
193C4×C67—21no—
194C4×C67—44no—
195C4×C67—82no—
196C4×C67C4×D812yes—
197C4×C67#51012yescomplemented
198C4×C67C2×C4×D411yescharacteristic
199C4×C67C2×#13511yescharacteristic
200Dic67—42no—
201Dic67—81no—
202Dic67—81no—
203C4×#510—671nomaximal, Sylow
204C2×C67—22no—
205C2×C67C2×C4×D811yescharacteristic
206C2×C67C2×#51011yescharacteristic
207C2×C67#341411yescharacteristic
208D67—82no—
209C2×#510—671no—
210C4×#135—671no—
211C2×C4×Q16D6711yescharacteristic, complemented
212#2176—672no—
213#2175—672no—
214C4×Q32—672no—
215C4×SD32—672no—
216#2159—671no—
217#2141—671no—
218#2139—671no—
219C4×M32—671no—
220C67C4×#51011yescharacteristic, complemented, Sylow
221C2×#135—671no—
222C22×Q16C2×D6711yescharacteristic
223C4×#41—1341no—
224C2×C4×Q8—21no—
225#510—678no—
226C2×Q32—672no—
227C2×SD32—672no—
228C2×M32—671no—
229C2×M32—671no—
230C4×Q16—21no—
231C4×Q16—21no—
232C4×Q16—671no—
233C4×Q16—671no—
234C4×Q16C2×D6712yes—
235C4×Q16C2×D6712yes—
236C4×SD16—1341no—
237C4×SD16—1341no—
238C4×D8—671no—
239C4×D8—671no—
240#427—671no—
241C2×#107C2×D6711yescharacteristic
242#416—1341no—
243C2×#103—21no—
244C2×C4×C8C2×D6711yescharacteristic
245#367—672no—
246#366—672no—
247#358—672no—
248#358—672no—
249#358—672no—
250#357—672no—
251#346—672no—
252C4×C16—672no—
253C2×#41—1341no—
254C22×Q8—21no—
255#135—674no—
256#135—1342no—
257C2×Q16—21no—
258C2×Q16—22no—
259C2×Q16—671no—
260C2×Q16C4×D6714yescomplemented
261C2×Q16C22×D6712yes—
262C2×SD16—1341no—
263C2×D8—671no—
264C22×C8C22⋊D6711yescharacteristic
265C22×C8C22×D6711yescharacteristic
266C4×Q8—22no—
267C4×Q8—22no—
268C4×Q8—41no—
269C4×Q8—41no—
270C4×Q8—1341no—
271C4×Q8—1341no—
272C4×D4—2681no—
273C4×D4—2681no—
274C4×D4—1341no—
275C4×D4—1341no—
276#117—2681no—
277#117—1341no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—41no—
280C2×C42—41no—
281C2×C42—2681no—
282C2×C42C22⋊D6711yescharacteristic
283Q32—678no—
284SD32—678no—
285M32—674no—
286M32—1342no—
287C2×C16—672no—
288C2×C16—672no—
289#107—21no—
290#107—671no—
291#107—671no—
292#107C22×D6712yes—
293#106—1341no—
294#103—22no—
295#103—22no—
296#103—1341no—
297#103—1341no—
298#102—1341no—
299#102—1341no—
300C4×C8C22⋊D6711yescharacteristic
301C4×C8C22⋊D6711yescharacteristic
302C4×C8C22×D6711yescharacteristic
303C4×C8C22×D6711yescharacteristic
304#41—2682no—
305#41—1344no—
306C2×Q8—24no—
307C2×Q8—22no—
308C2×Q8—41no—
309C2×Q8—42no—
310C2×Q8—1341no—
311C2×D4—2681no—
312C2×D4—1341no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—2681no—
316C22×C4—2681no—
317C22×C4—2681no—
318C22×C4#1176161311yescharacteristic
319C22×C4#1176161311yescharacteristic
320C22×C4C2×C22⋊D6711yescharacteristic
321Q16—24no—
322Q16—22no—
323Q16—672no—
324Q16—1341no—
325Q16C2×C4×D6714yes—
326SD16—1344no—
327D8—672no—
328D8—1341no—
329C2×C8—22no—
330C2×C8C2×C4×D6712yes—
331C2×C8C2×C22⋊D6711yescharacteristic
332C2×C8C2×C22⋊D6711yescharacteristic
333C2×C8C2×C22⋊D6711yescharacteristic
334C2×C8#1176162812yes—
335C2×C8C23×D6711yescharacteristic
336C4⋊C4—22no—
337C4⋊C4—41no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—2681no—
341C4⋊C4—2681no—
342C4⋊C4—1341no—
343C4⋊C4—1341no—
344C22⋊C4—2681no—
345C22⋊C4—2681no—
346C22⋊C4—2681no—
347C42—42no—
348C42—42no—
349C42—2681no—
350C42—2681no—
351C42—2681no—
352C42#1176161311yescharacteristic
353C42#1176161311yescharacteristic
354C42C2×C22⋊D6711yescharacteristic
355C42C2×C22⋊D6711yescharacteristic
356C16—674no—
357C16—1342no—
358C23—2681no—
359C23C2×#1176161311yescharacteristic
360Q8—24no—
361Q8—42no—
362Q8—44no—
363Q8—2681no—
364Q8—1342no—
365D4—2684no—
366D4—2681no—
367D4—1342no—
368C2×C4—22no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—82no—
374C2×C4—2681no—
375C2×C4—2684no—
376C2×C4—2681no—
377C2×C4—2681no—
378C2×C4—5362no—
379C2×C4—5362no—
380C2×C4C4×C22⋊D6712yes—
381C2×C4C2×#1176161311yescharacteristic
382C2×C4C2×#1176161311yescharacteristic
383C2×C4C2×#1176161311yescharacteristic
384C2×C4C2×#1176161311yescharacteristic
385C2×C4C2×#1176161311yescharacteristic
386C2×C4#1452406312yes—
387C2×C4C22×C22⋊D6711yescharacteristic
388C8—21no—
389C8—21no—
390C8C4×C22⋊D6712yes—
391C8C22×C4×D6711yescharacteristic
392C8C2×#1176162811yescharacteristic
393C22—21no—
394C22—21no—
395C22—2682no—
396C22—5361no—
397C22—5361no—
398C22C4×#1176161312yes—
399C22C22×#1176161311yescharacteristic
400C4—21no—
401C4—21no—
402C4—21no—
403C4—44no—
404C4—82no—
405C4—2682no—
406C4—5361no—
407C4—5361no—
408C4C4×#1176161312yes—
409C4#1686383712yesdirect product factor
410C4C2×C4×C22⋊D6711yescharacteristic
411C4C2×#1452406311yescharacteristic
412C2—22no—
413C2—5362no—
414C2C2×C4×#1176161311yescharacteristic
415C2C2×#1686383711yescharacteristic
416C2#2257334311yescharacteristic
subgroupG/H
centerC2×C4C2×#11761613
commutatorC8×C67C23×C4
FrattiniC2×C8C23×D67
FittingC2×C4×Q16×C67C2
socleC22×C67C22×D8
prime radicalC2×#14524065C22
cosocleC2×C8×C67C24
hypercenterC2×C4×Q16D67
hypocenterC67C4×#510
LCSG ▷ C8×C67 ▷ C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×Q16

Complements

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

NH ≅ G/Ntype
3. #22562078C2semidirect
4. #22562071C2semidirect
5. C4×#14523989C2semidirect
6. C4×#14523986C2semidirect
7. #22561995C2semidirect
8. #22561929C2semidirect
10. C4×#14523983C2semidirect
11. C2×C4×Q16×C67C2semidirect
14. #16863837C4direct (N × H)
59. #14523989C2×C4semidirect
60. #14523986C2×C4semidirect
62. #14523983C2×C4semidirect
75. C2×Q16×C67C2×C4semidirect
197. C4×C67#510semidirect
211. C2×C4×Q16D67semidirect
220. C67C4×#510semidirect
260. C2×Q16C4×D67semidirect
409. C4#16863837direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168638372
C4×#145240652
#225620782
#225620712
C4×#145239892
C4×#145239862
#225619952
#225619292
#225619192
C4×#145239832
C2×C4×Q16×C672
C4×#51067