Finite Group Databasebeta

← #30856192

Subgroups — #30856192

Order 17152.

616 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22568631C211yesmaximal, characteristic
2#22560357C212yesmaximal, complemented
3#22560333C211yesmaximal, characteristic, complemented
4#22560311C212yesmaximal
5#22560256C211yesmaximal, characteristic
6#22560252C211yesmaximal, characteristic
7#22560091C211yesmaximal, characteristic, complemented
8#22560090C211yesmaximal, characteristic, complemented
9#22560008C212yesmaximal
10#22559963C212yesmaximal, complemented
11C67×#1833C211yesmaximal, characteristic, complemented
12C2×#14524051C2211yescharacteristic
13C2×#14524049C2211yescharacteristic
14#16863864C2211yescharacteristic
15C2×#14523979C2211yescharacteristic
16#16863434C2211yescharacteristic
17#16863378C2212yes—
18#16863369C2212yes—
19#16863354C2212yes—
20#16863353C412yescomplemented
21#16863352C2212yes—
22#16863352C2211yescharacteristic
23#16863350C412yescomplemented
24#16863348C412yescomplemented
25M16×Dic67C2211yescharacteristic
26C2×C8×Dic67C2211yescharacteristic
27C4×C8×D67C412yescomplemented
28#16863167C2212yes—
29#16863157C2212yes—
30#16863155C2212yes—
31C67⋊#338C2212yes—
32#16863152C2212yes—
33#16863152C2212yes—
34C2×#108×C67C2211yescharacteristic
35#427×C67C2211yescharacteristic
36#405×C67C2211yescharacteristic
37#344×C67C2212yes—
38H(3,ℤ/4)×C67C2212yes—
39C2×#11761628C2311yescharacteristic
40#14524051—21no—
41#14524051—21no—
42#14524051—21no—
43#14524051—21no—
44#14524049—21no—
45#14524049—21no—
46#14524049—21no—
47#14524049—21no—
48C2×#11761612C2312yes—
49C2×#11761612C2311yescharacteristic
50C2×#11761611C2×C412yes—
51C2×#11761611C2×C412yes—
52C22×C67⋊C8C2311yescharacteristic
53C2×C8×D67C2×C412yes—
54C2×C8×D67C2×C412yes—
55#14524008—21no—
56#14524005—21no—
57(C22⋊D67)⋊C4—22no—
58#14524002C2312yes—
59#14524000C2×C412yes—
60C2×C4×Dic67C2311yescharacteristic
61C42×D67C2×C412yes—
62#14523979—21no—
63#14523979—21no—
64#14523977—21no—
65#14523975—21no—
66#14523961—22no—
67#14523959—22no—
68#14523957—24no—
69#14523954—22no—
70#14523954C2×C412yes—
71#14523953—22no—
72#14523953C2×C412yes—
73C4×C67⋊C8—22no—
74C4×C67⋊C8C2×C412yes—
75C8×Dic67—22no—
76C8×Dic67C2×C411yescharacteristic
77C8×Dic67C2×C412yes—
78C8×Dic67C2×C411yescharacteristic
79C8×Dic67C2×C411yescharacteristic
80C8×Dic67C2×C411yescharacteristic
81C2×M16×C67C2312yes—
82C2×M16×C67C2311yescharacteristic
83C22×C8×C67C2311yescharacteristic
84#117×C67C2312yes—
85#117×C67C2311yescharacteristic
86#108×C67—21no—
87#108×C67—21no—
88#107×C67—21no—
89#106×C67—21no—
90#97×C67C2×C412yes—
91C4×C8×C67C2×C412yes—
92#11761628—21no—
93#11761628—21no—
94#11761628—21no—
95#11761628—21no—
96C2×Dic134#4111yescharacteristic
97C2×D268#4111yescharacteristic
98C2×C22⋊D67#4112yes—
99C22×Dic67—21no—
100C2×C4×D67—22no—
101C2×C4×D67C22×C412yes—
102#11761612—21no—
103#11761612—21no—
104#11761612—22no—
105#11761612—22no—
106#11761611—22no—
107#11761611—22no—
108#11761611—22no—
109#11761611—22no—
110C2×C67⋊C8—22no—
111C2×C67⋊C8C22×C412yes—
112C2×C67⋊C8C22×C411yescharacteristic
113C2×C67⋊C8C22×C411yescharacteristic
114C2×C67⋊C8#4111yescharacteristic
115C2×C67⋊C8#4111yescharacteristic
116C2×C67⋊C8#4111yescharacteristic
117C2×C67⋊C8#4111yescharacteristic
118C8×D67—22no—
119C8×D67—22no—
120C8×D67—22no—
121C8×D67—22no—
122C4⋊Dic67—22no—
123Dic67⋊C4—22no—
124C22⋊Dic67—22no—
125(C2×D67)⋊C4—22no—
126C4×Dic67—21no—
127C4×Dic67—22no—
128C4×Dic67—22no—
129C4×Dic67C22×C411yescharacteristic
130C4×Dic67C22×C411yescharacteristic
131C22×C4×C67C2411yescharacteristic
132M16×C67—21no—
133M16×C67—21no—
134M16×C67—22no—
135M16×C67—22no—
136C2×C8×C67—22no—
137C2×C8×C67C22×C412yes—
138C2×C8×C67C22×C411yescharacteristic
139C2×C8×C67C22×C411yescharacteristic
140C2×C8×C67#4111yescharacteristic
141C2×C8×C67#4111yescharacteristic
142C2×C8×C67#4111yescharacteristic
143C2×C8×C67#4111yescharacteristic
144C4⋊C4×C67—21no—
145C4⋊C4×C67—22no—
146C22⋊C4×C67—21no—
147C22⋊C4×C67—22no—
148C42×C67—22no—
149C42×C67C22×C412yes—
150C23×C67C2×#4111yescharacteristic
151C22×D67#11712yes—
152C2×C4×C67—22no—
153C2×C4×C67—42no—
154C2×C4×C67C23×C411yescharacteristic
155C2×C4×C67C2×#4111yescharacteristic
156C2×C4×C67C2×#4111yescharacteristic
157C2×C4×C67C2×#4111yescharacteristic
158C2×C4×C67C2×#4111yescharacteristic
159C2×C4×C67C2×#4111yescharacteristic
160C22⋊D67—22no—
161C22⋊D67—22no—
162C2×Dic67—21no—
163C2×Dic67—21no—
164C2×Dic67—42no—
165C2×Dic67—42no—
166C2×Dic67#11712yes—
167D268—21no—
168D268—21no—
169C4×D67—22no—
170C4×D67—22no—
171C4×D67—44no—
172Dic134—21no—
173Dic134—21no—
174C8×C67—22no—
175C8×C67—42no—
176C8×C67#11711yescharacteristic
177C8×C67#11711yescharacteristic
178C8×C67#11711yescharacteristic
179C8×C67#11711yescharacteristic
180C67⋊C8—22no—
181C67⋊C8—42no—
182C67⋊C8#11711yescharacteristic
183C67⋊C8#11711yescharacteristic
184C67⋊C8#11711yescharacteristic
185C67⋊C8#11711yescharacteristic
186C22×C67—41no—
187C22×C67C2×#11711yescharacteristic
188C22×C67#52911yescharacteristic
189C22×C67#52911yescharacteristic
190C2×D67—22no—
191C2×D67—42no—
192C4×C67—42no—
193C4×C67—82no—
194C4×C67C2×#11711yescharacteristic
195C4×C67C2×#11711yescharacteristic
196C4×C67#52911yescharacteristic
197C4×C67#52911yescharacteristic
198Dic67—22no—
199Dic67—22no—
200Dic67—41no—
201Dic67—82no—
202#20787—671nomaximal, Sylow
203C2×C67—21no—
204C2×C67—21no—
205C2×C67—21no—
206C2×C67#228311yescharacteristic
207D67—42no—
208C4×#131—671no—
209#1841—672no—
210#1833—671no—
211#1833D6711yescharacteristic, complemented
212#1827—672no—
213#1813—671no—
214#1812—671no—
215#1772—671no—
216#1772—671no—
217#1757—672no—
218#1747—672no—
219C67#2078711yescharacteristic, complemented, Sylow
220C2×#131—671no—
221C2×#131—671no—
222C4×#41—671no—
223C2×#108—671no—
224C2×#108C2×D6711yescharacteristic
225#427—671no—
226#427C2×D6711yescharacteristic
227#413—672no—
228#410—672no—
229#405—671no—
230#405—672no—
231#405—672no—
232#405—672no—
233#405—672no—
234#405C2×D6711yescharacteristic
235C4×M16—671no—
236C4×M16—672no—
237C2×C4×C8—671no—
238C2×C4×C8—672no—
239#344—672no—
240#344C2×D6712yes—
241#339—672no—
242#339—672no—
243#338—672no—
244H(3,ℤ/4)—672no—
245H(3,ℤ/4)—672no—
246H(3,ℤ/4)C2×D6712yes—
247C2×#41—671no—
248#131—1341no—
249#131—1341no—
250#131—1341no—
251#131—1341no—
252#131—1341no—
253#131—1341no—
254#131—1341no—
255#131—1341no—
256C2×M16—671no—
257C2×M16—672no—
258C2×M16—672no—
259C2×M16—672no—
260C2×M16C22×D6712yes—
261C2×M16C22×D6711yescharacteristic
262C22×C8—671no—
263C22×C8—672no—
264C22×C8—672no—
265C22×C8C22×D6711yescharacteristic
266C4×Q8—1341no—
267C4×D4—1342no—
268C4×D4—1341no—
269#117—672no—
270#117—672no—
271#117C22×D6712yes—
272#117C22×D6711yescharacteristic
273C2×C42—671no—
274C2×C42—672no—
275#108—21no—
276#108—21no—
277#108—1341no—
278#108—1341no—
279#107—21no—
280#107—1341no—
281#106—21no—
282#106—1341no—
283#101—1342no—
284#100—1342no—
285#99—1344no—
286#97—672no—
287#97—672no—
288#97—1342no—
289#97—1342no—
290#97C4×D6712yes—
291C4×C8—671no—
292C4×C8—671no—
293C4×C8—672no—
294C4×C8—671no—
295C4×C8—671no—
296C4×C8—672no—
297C4×C8—1342no—
298C4×C8—1342no—
299C4×C8C4×D6712yes—
300#41—1341no—
301#41—1341no—
302#41—1341no—
303#41—1341no—
304C2×Q8—671no—
305C2×D4—672no—
306C2×D4—671no—
307C22×C4—672no—
308C22×C4—1342no—
309C22×C4—1341no—
310C22×C4C23×D6711yescharacteristic
311M16—21no—
312M16—21no—
313M16—22no—
314M16—22no—
315M16—1342no—
316M16—1342no—
317M16—1342no—
318M16—1342no—
319M16—1341no—
320M16—1341no—
321M16—1342no—
322M16—1342no—
323C2×C8—22no—
324C2×C8—671no—
325C2×C8—671no—
326C2×C8—671no—
327C2×C8—671no—
328C2×C8—671no—
329C2×C8—671no—
330C2×C8—672no—
331C2×C8—1342no—
332C2×C8—1342no—
333C2×C8—1342no—
334C2×C8—1342no—
335C2×C8—1342no—
336C2×C8C2×C4×D6712yes—
337C2×C8C2×C4×D6711yescharacteristic
338C2×C8C2×C4×D6711yescharacteristic
339C2×C8#1176162711yescharacteristic
340C2×C8#1176162711yescharacteristic
341C2×C8#1176162911yescharacteristic
342C2×C8#1176162911yescharacteristic
343C4⋊C4—21no—
344C4⋊C4—22no—
345C4⋊C4—1342no—
346C4⋊C4—1342no—
347C22⋊C4—21no—
348C22⋊C4—22no—
349C22⋊C4—1342no—
350C22⋊C4—1342no—
351C42—22no—
352C42—671no—
353C42—671no—
354C42—1342no—
355C42—1342no—
356C42—1341no—
357C42C2×C4×D6712yes—
358C23—672no—
359C23#41×D6711yescharacteristic
360Q8—1341no—
361Q8—1341no—
362D4—1342no—
363D4—1342no—
364D4—1341no—
365D4—1341no—
366C2×C4—22no—
367C2×C4—42no—
368C2×C4—2684no—
369C2×C4—2682no—
370C2×C4—2682no—
371C2×C4—672no—
372C2×C4—1341no—
373C2×C4—1341no—
374C2×C4—1342no—
375C2×C4—1342no—
376C2×C4C22×C4×D6711yescharacteristic
377C2×C4#41×D6711yescharacteristic
378C2×C4#41×D6711yescharacteristic
379C2×C4#41×D6711yescharacteristic
380C2×C4C2×#1176162711yescharacteristic
381C2×C4C2×#1176162911yescharacteristic
382C8—22no—
383C8—42no—
384C8—2682no—
385C8—671no—
386C8—671no—
387C8—671no—
388C8—671no—
389C8—1342no—
390C8#1452400111yescharacteristic
391C8#1452400111yescharacteristic
392C8#1452400111yescharacteristic
393C8#1452400111yescharacteristic
394C22—41no—
395C22—2682no—
396C22—1342no—
397C22#117×D6711yescharacteristic
398C22#1686392511yescharacteristic
399C22#1686393111yescharacteristic
400C4—42no—
401C4—82no—
402C4—2681no—
403C4—5362no—
404C4—1342no—
405C4—1342no—
406C4#117×D6711yescharacteristic
407C4C2×#1452400111yescharacteristic
408C4#1686392511yescharacteristic
409C4#1686393111yescharacteristic
410C2—21no—
411C2—21no—
412C2—21no—
413C2—2682no—
414C2#2256260211yescharacteristic
subgroupG/H
centerC4#117×D67
commutatorC2×C4×C67C23×C4
FrattiniC22×C4C23×D67
FittingC67×#1833C2
socleC2×C67#2283
prime radicalC2×#11761628C23
cosocleC22×C4×C67C24
hypercenter#1833D67
hypocenterC67#20787
LCSG ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C4 ◁ #117 ◁ #1833

Complements

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

NH ≅ G/Ntype
2. #22560357C2semidirect
3. #22560333C2semidirect
7. #22560091C2semidirect
8. #22560090C2semidirect
10. #22559963C2semidirect
11. C67×#1833C2semidirect
20. #16863353C4semidirect
23. #16863350C4semidirect
24. #16863348C4semidirect
27. C4×C8×D67C4semidirect
211. #1833D67semidirect
219. C67#20787semidirect

Maximal subgroups

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

subgroupindex
#225686312
#225603572
#225603332
#225603112
#225602562
#225602522
#225600912
#225600902
#225600082
#225599632
C67×#18332
#2078767