Finite Group Databasebeta

← #30754221

Subgroups — #30754221

Order 17152.

454 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#22561136C211yesmaximal, characteristic
2#22561015C211yesmaximal, characteristic
3#22560972C211yesmaximal, characteristic, complemented
4C2×#16863139C211yesmaximal, characteristic
5#22558093C211yesmaximal, characteristic, complemented
6#22558044C211yesmaximal, characteristic, complemented
7C67×#1770C211yesmaximal, characteristic, complemented
8C2×#14524040C2211yescharacteristic
9C2×#14523974—21no—
10C2×C4⋊(C67⋊C8)C2211yescharacteristic
11C2×C4⋊(C67⋊C8)C2211yescharacteristic
12C2×#14523965—21no—
13C2×#14523965—21no—
14C2×#14523963—21no—
15C2×#14523963—21no—
16#16863266C2211yescharacteristic
17#16863158—21no—
18#16863139C414yescomplemented
19C2×C4⋊C8×C67C2211yescharacteristic
20C2×#97×C67—21no—
21C2×C4×C8×C67C2211yescharacteristic
22C4×C4⋊C4×C67C2211yescharacteristic
23#336×C67—21no—
24C22×D268—21no—
25C22×D268—21no—
26C22×D268—21no—
27C22×C67⋊C8—21no—
28C22×C67⋊C8—21no—
29#14524040—21no—
30#14524040—21no—
31#14524040—21no—
32#14524040C2×C412yes—
33C2×C4⋊Dic67—21no—
34C2×C4⋊Dic67—21no—
35C2×C4⋊Dic67—21no—
36#14523974—42no—
37C4⋊(C67⋊C8)—21no—
38C4⋊(C67⋊C8)—21no—
39C4⋊(C67⋊C8)—21no—
40C4⋊(C67⋊C8)C2×C412yes—
41#14523965—42no—
42#14523965—42no—
43#14523963—42no—
44#14523963—42no—
45C22×C8×C67—21no—
46C22×C8×C67D411yescharacteristic
47C22×C8×C67D411yescharacteristic
48C2×C4⋊C4×C67D411yescharacteristic
49C2×C4⋊C4×C67D411yescharacteristic
50C2×C42×C67—21no—
51C2×C42×C67C2311yescharacteristic
52C4⋊C8×C67—21no—
53C4⋊C8×C67C2×C412yes—
54#97×C67—42no—
55C4×C8×C67—21no—
56C4×C8×C67—21no—
57#95×C67—21no—
58C23×D67—41no—
59C2×D268—22no—
60C2×D268—22no—
61C2×D268—22no—
62C2×D268—41no—
63C2×D268—41no—
64C2×D268—41no—
65C2×D268—41no—
66C2×D268—41no—
67C2×D268—41no—
68C2×D268—41no—
69C2×D268—41no—
70C2×D268—41no—
71C2×D268—41no—
72C2×D268—41no—
73C2×D268—81no—
74C2×D268—81no—
75C22×Dic67—41no—
76C2×C67⋊C8—22no—
77C2×C67⋊C8—41no—
78C2×C67⋊C8—41no—
79C2×C67⋊C8—41no—
80C2×C67⋊C8—41no—
81C2×C67⋊C8—41no—
82C4⋊Dic67—41no—
83C4⋊Dic67—41no—
84C4⋊Dic67—41no—
85C4⋊Dic67—41no—
86C4⋊Dic67—81no—
87C22×C4×C67—21no—
88C22×C4×C67—21no—
89C22×C4×C67C2×D411yescharacteristic
90C22×C4×C67C2×D411yescharacteristic
91C22×C4×C67#4111yescharacteristic
92C2×C8×C67—21no—
93C2×C8×C67—21no—
94C2×C8×C67—22no—
95C2×C8×C67—21no—
96C2×C8×C67—21no—
97C2×C8×C67—21no—
98C2×C8×C67—41no—
99C2×C8×C67—41no—
100C2×C8×C67SD1612yes—
101C4⋊C4×C67—21no—
102C4⋊C4×C67—21no—
103C4⋊C4×C67—21no—
104C4⋊C4×C67—21no—
105C42×C67—41no—
106C42×C67—41no—
107C42×C67C22×C411yescharacteristic
108C42×C67#4111yescharacteristic
109C42×C67#4111yescharacteristic
110C42×C67#4111yescharacteristic
111C23×C67#12111yescharacteristic
112C22×D67—42no—
113C22×D67—81no—
114C22×D67—81no—
115C22×D67—81no—
116C22×D67—81no—
117C22×D67—81no—
118C22×D67—81no—
119C2×C4×C67—21no—
120C2×C4×C67—21no—
121C2×C4×C67—41no—
122C2×C4×C67—41no—
123C2×C4×C67—41no—
124C2×C4×C67—41no—
125C2×C4×C67—41no—
126C2×C4×C67—41no—
127C2×C4×C67#11711yescharacteristic
128C2×C4×C67C4×D411yescharacteristic
129C2×C4×C67C4×D411yescharacteristic
130C2×C4×C67#12311yescharacteristic
131C2×C4×C67#12411yescharacteristic
132C2×C4×C67#12611yescharacteristic
133C2×C4×C67C2×SD1611yescharacteristic
134C2×C4×C67#13511yescharacteristic
135C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
139C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
140C2×C4×C67#13711yescharacteristic
141C2×Dic67—81no—
142C2×Dic67—81no—
143C2×Dic67—81no—
144D268—42no—
145D268—42no—
146D268—42no—
147D268—42no—
148D268—82no—
149D268—82no—
150D268—81no—
151D268—81no—
152D268—81no—
153D268—81no—
154C8×C67—22no—
155C8×C67—42no—
156C8×C67—41no—
157C67⋊C8—42no—
158C67⋊C8—81no—
159C22×C67#38811yescharacteristic
160C22×C67#46011yescharacteristic
161C22×C67#46311yescharacteristic
162C22×C67#46811yescharacteristic
163C22×C67#46911yescharacteristic
164C22×C67#47111yescharacteristic
165C22×C67#47211yescharacteristic
166C2×D67—82no—
167C2×D67—82no—
168C2×D67—82no—
169C2×D67—161no—
170C2×D67—161no—
171C2×D67—161no—
172C2×D67—161no—
173C4×C67—21no—
174C4×C67—21no—
175C4×C67—81no—
176C4×C67—81no—
177C4×C67C4×SD1611yescharacteristic
178C4×C67#44011yescharacteristic
179C4×C67#44211yescharacteristic
180C4×C67#44211yescharacteristic
181C4×C67#48111yescharacteristic
182C4×C67#48211yescharacteristic
183C4×C67#48911yescharacteristic
184C4×C67#49011yescharacteristic
185Dic67—161no—
186#9572—671nomaximal, Sylow
187C2×C67#168611yescharacteristic
188C2×C67#168811yescharacteristic
189C2×C67#169711yescharacteristic
190C2×C67#170111yescharacteristic
191C2×C67#192611yescharacteristic
192C2×C67#193111yescharacteristic
193C2×C67#193611yescharacteristic
194D67—162no—
195#1982—671no—
196#1957—671no—
197#1948—671no—
198#1770D6711yescharacteristic, complemented
199C2×#331—671no—
200#1296—671no—
201#1277—671no—
202C67#957211yescharacteristic, complemented, Sylow
203C2×#127—671no—
204C2×#106—1341no—
205C2×C4⋊C8—671no—
206C2×C4⋊C8—671no—
207C2×C4⋊C8C2×D6711yescharacteristic
208C2×#102—1341no—
209C2×#102—1341no—
210C2×#102—1341no—
211C2×#102—1341no—
212C2×#97—21no—
213C2×C4×C8C2×D6711yescharacteristic
214#384—671no—
215C4×C4⋊C4C2×D6711yescharacteristic
216#340—1341no—
217#336—21no—
218#331—674no—
219C22×D4—1341no—
220C22×D4—1341no—
221C22×D4—1341no—
222C22×C8—21no—
223C22×C8—1341no—
224C22×C8—1341no—
225C22×C8C22⋊D6711yescharacteristic
226C22×C8C22⋊D6711yescharacteristic
227#127—672no—
228#127—1341no—
229#127—1341no—
230#127—1341no—
231C2×C4⋊C4—1341no—
232C2×C4⋊C4—1341no—
233C2×C4⋊C4—1341no—
234C2×C4⋊C4D26811yescharacteristic
235C2×C4⋊C4D26811yescharacteristic
236C2×C42—21no—
237C2×C42C22×D6711yescharacteristic
238#106—2682no—
239C4⋊C8—21no—
240C4⋊C8—672no—
241C4⋊C8—1341no—
242C4⋊C8—1341no—
243C4⋊C8—1341no—
244C4⋊C8C4×D6712yes—
245#102—2682no—
246#102—2682no—
247#102—2682no—
248#102—2682no—
249#97—42no—
250C4×C8—21no—
251C4×C8—21no—
252#95—21no—
253C24—2681no—
254C2×D4—2681no—
255C2×D4—2681no—
256C2×D4—2681no—
257C2×D4—2681no—
258C2×D4—2681no—
259C2×D4—2681no—
260C2×D4—2681no—
261C2×D4—2681no—
262C2×D4—2681no—
263C2×D4—2681no—
264C2×D4—2681no—
265C2×D4—5361no—
266C2×D4—5361no—
267C2×D4—1342no—
268C2×D4—1342no—
269C2×D4—1342no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—2681no—
273C22×C4C2×C22⋊D6711yescharacteristic
274C22×C4C2×D26811yescharacteristic
275C22×C4#1176162811yescharacteristic
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—22no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—2681no—
285C2×C8—2681no—
286C2×C8—2681no—
287C2×C8—2681no—
288C2×C8—2681no—
289C2×C8—1342no—
290C2×C8#1176161512yes—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—2681no—
296C4⋊C4—2681no—
297C4⋊C4—2681no—
298C4⋊C4—2681no—
299C4⋊C4—5361no—
300C42—41no—
301C42—41no—
302C42C2×C4×D6711yescharacteristic
303C42#1176162811yescharacteristic
304C42#1176162811yescharacteristic
305C42#1176162811yescharacteristic
306C23—2682no—
307C23—5361no—
308C23—5361no—
309C23—5361no—
310C23—5361no—
311C23—5361no—
312C23—5361no—
313C23#1452401711yescharacteristic
314D4—2682no—
315D4—2682no—
316D4—2682no—
317D4—2682no—
318D4—5362no—
319D4—5362no—
320D4—5361no—
321D4—5361no—
322D4—5361no—
323D4—5361no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—5361no—
333C2×C4—5361no—
334C2×C4—5361no—
335C2×C4#1452400011yescharacteristic
336C2×C4C4×C22⋊D6711yescharacteristic
337C2×C4C4×D26811yescharacteristic
338C2×C4#1452402611yescharacteristic
339C2×C4#1452403111yescharacteristic
340C2×C4#1452403711yescharacteristic
341C2×C4C2×#1176161511yescharacteristic
342C2×C4#1452406311yescharacteristic
343C2×C4#1452406811yescharacteristic
344C2×C4#1452406911yescharacteristic
345C2×C4#1452407111yescharacteristic
346C2×C4#1452407111yescharacteristic
347C2×C4#1452407111yescharacteristic
348C2×C4#1452407611yescharacteristic
349C8—22no—
350C8—42no—
351C8—41no—
352C8—2682no—
353C8—5361no—
354C22—5362no—
355C22—5362no—
356C22—5362no—
357C22—10721no—
358C22—10721no—
359C22—10721no—
360C22—10721no—
361C22#1686328711yescharacteristic
362C22#1686359611yescharacteristic
363C22#1686361111yescharacteristic
364C22#1686364211yescharacteristic
365C22#1686364911yescharacteristic
366C22#1686365911yescharacteristic
367C22#1686366611yescharacteristic
368C4—21no—
369C4—21no—
370C4—81no—
371C4—81no—
372C4—10721no—
373C4C4×#1176161511yescharacteristic
374C4#1686349811yescharacteristic
375C4#1686350611yescharacteristic
376C4#1686350911yescharacteristic
377C4#1686371411yescharacteristic
378C4#1686372011yescharacteristic
379C4#1686375611yescharacteristic
380C4#1686376111yescharacteristic
381C2—10722no—
382C2#2255968111yescharacteristic
383C2#2255969311yescharacteristic
384C2#2255974211yescharacteristic
385C2#2255977011yescharacteristic
386C2#2256083611yescharacteristic
387C2#2256087111yescharacteristic
388C2#2256090611yescharacteristic
subgroupG/H
centerC23#14524017
commutatorC42×C67C22×C4
FrattiniC2×C42C22×D67
FittingC67×#1770C2
socleC23×C67#121
prime radicalC2×#14524040C22
cosocleC2×C42×C67C23
hypercenter#1770D67
hypocenterC67#9572
LCSG ▷ C42×C67 ▷ C22×C67 ▷ C67
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1770

Complements

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

NH ≅ G/Ntype
3. #22560972C2semidirect
5. #22558093C2semidirect
6. #22558044C2semidirect
7. C67×#1770C2semidirect
18. #16863139C4semidirect
198. #1770D67semidirect
202. C67#9572semidirect

Maximal subgroups

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

subgroupindex
#225611362
#225610152
#225609722
C2×#168631392
#225580932
#225580442
C67×#17702
#957267