Finite Group Databasebeta

← #16930224

Subgroups — #16930224

Order 4352.

516 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14538264C211yesmaximal, characteristic
2#14527453C211yesmaximal, characteristic, complemented
3C2×#11762552C211yesmaximal, characteristic, complemented
4#14527171C211yesmaximal, characteristic, complemented
5#14526681C211yesmaximal, characteristic, complemented
6#14526658C211yesmaximal, characteristic, complemented
7C17×#2925C211yesmaximal, characteristic, complemented
8C2×#10587635—21no—
9C2×#10587560C2211yescharacteristic
10#11763009—21no—
11#11763009C412yes—
12#11763004—21no—
13#11763004C412yes—
14#11762862C2211yescharacteristic
15#11762833C2211yescharacteristic
16C2×C4⋊#3622C2211yescharacteristic
17#11762813—21no—
18#11762552—21no—
19#11762552—21no—
20#11762552—22no—
21Q8⋊#3622—21no—
22D4⋊#3622—21no—
23C4×#41×C17C2211yescharacteristic
24C17×#433—22no—
25C17×#432—21no—
26C17×#431—21no—
27C17×#423C2211yescharacteristic
28C2×C17×C4⋊C8C2211yescharacteristic
29C17×#408—21no—
30C17×#405—21no—
31C2×#41×C17D411yescharacteristic
32C22×D68—21no—
33C22×D68—21no—
34C22×D68—21no—
35C2×M16×C17—21no—
36C22×C8×C17—21no—
37C4×Q8×C17C2×C411yescharacteristic
38C4×Q8×C17D411yescharacteristic
39C4×D4×C17—21no—
40C4×D4×C17—21no—
41C4×D4×C17C2×C411yescharacteristic
42C4×D4×C17D411yescharacteristic
43C17×#117—21no—
44C17×#117D411yescharacteristic
45C2×C42×C17—21no—
46C2×C42×C17C2311yescharacteristic
47#10587635—42no—
48#10587635—42no—
49C2×#63230—22no—
50C2×#63228—22no—
51C2×#63223—21no—
52C22×#3622—21no—
53#10587560—21no—
54#10587560—21no—
55#10587560—21no—
56#10587560C2×C412yes—
57C17×C4⋊C8—21no—
58C17×C4⋊C8—21no—
59C17×C4⋊C8D411yescharacteristic
60C17×C4⋊C8D411yescharacteristic
61C17×C4⋊C8D412yes—
62C17×C22⋊C8—22no—
63C17×C22⋊C8—22no—
64C17×#97—22no—
65C4×C8×C17—22no—
66C22⋊#3622—21no—
67#10587480—22no—
68#10587480—22no—
69C4⋊#3622—21no—
70C4⋊#3622—21no—
71C4⋊#3622—21no—
72C4⋊#3622—21no—
73C4⋊#3622C2×C412yes—
74C4×#3622—21no—
75C23×D17—41no—
76#41×C17—21no—
77#41×C17—21no—
78#41×C17—21no—
79#41×C17—21no—
80C2×Q8×C17C22⋊C411yescharacteristic
81C2×D4×C17—21no—
82C2×D4×C17C22⋊C411yescharacteristic
83C22×C4×C17—21no—
84C22×C4×C17—21no—
85C22×C4×C17—21no—
86C22×C4×C17C2×D411yescharacteristic
87C22×C4×C17#4111yescharacteristic
88C22×C4×C17#4111yescharacteristic
89C2×D68—22no—
90C2×D68—22no—
91C2×D68—22no—
92C2×D68—41no—
93C2×D68—41no—
94C2×D68—41no—
95C2×D68—41no—
96C2×D68—41no—
97C2×D68—41no—
98C2×D68—41no—
99C2×D68—82no—
100C2×D68—81no—
101C2×D68—81no—
102M16×C17—41no—
103M16×C17—41no—
104C2×C8×C17—21no—
105C2×C8×C17—21no—
106C2×C8×C17—22no—
107C2×C8×C17—41no—
108C2×C8×C17—41no—
109C4⋊C4×C17—21no—
110C4⋊C4×C17—21no—
111C4⋊C4×C17C22⋊C411yescharacteristic
112C4⋊C4×C17C22⋊C411yescharacteristic
113C22⋊C4×C17—21no—
114C22⋊C4×C17—21no—
115C22⋊C4×C17—21no—
116C42×C17—21no—
117C42×C17—21no—
118C42×C17—21no—
119C42×C17C22×C411yescharacteristic
120C42×C17C2×D411yescharacteristic
121C42×C17C2×D411yescharacteristic
122C42×C17C2×D411yescharacteristic
123#63230—44no—
124#63228—44no—
125#63223—41no—
126#63223—41no—
127C2×#3622—22no—
128C2×#3622—41no—
129C2×#3622—41no—
130C2×#3622—81no—
131#9174—171nomaximal, Sylow
132C23×C17—21no—
133C23×C17#12311yescharacteristic
134C22×D17—42no—
135C22×D17—82no—
136C22×D17—82no—
137C22×D17—81no—
138C22×D17—81no—
139Q8×C17—21no—
140Q8×C17—21no—
141D4×C17—21no—
142D4×C17—21no—
143D4×C17—41no—
144D4×C17—41no—
145C2×C4×C17—21no—
146C2×C4×C17—21no—
147C2×C4×C17—21no—
148C2×C4×C17—21no—
149C2×C4×C17—21no—
150C2×C4×C17—41no—
151C2×C4×C17—41no—
152C2×C4×C17—41no—
153C2×C4×C17—41no—
154C2×C4×C17—41no—
155C2×C4×C17—81no—
156C2×C4×C17C2×C22⋊C411yescharacteristic
157C2×C4×C17C4×D411yescharacteristic
158C2×C4×C17C4×D411yescharacteristic
159C2×C4×C17#12011yescharacteristic
160C2×C4×C17#12111yescharacteristic
161C2×C4×C17#12111yescharacteristic
162C2×C4×C17#13511yescharacteristic
163C2×C4×C17#13511yescharacteristic
164C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
166D68—42no—
167D68—42no—
168D68—82no—
169D68—82no—
170D68—82no—
171D68—81no—
172D68—81no—
173D68—81no—
174D68—81no—
175C8×C17—41no—
176C8×C17—41no—
177C8×C17—42no—
178#3622—42no—
179#3622—81no—
180#3622—81no—
181#3038—171no—
182#2925D1711yescharacteristic, complemented
183#1545—171no—
184C2×#331—171no—
185#1477—171no—
186#1301—171no—
187#1292—171no—
188C22×C17—21no—
189C22×C17—21no—
190C22×C17—41no—
191C22×C17—41no—
192C22×C17—41no—
193C22×C17#38611yescharacteristic
194C22×C17#48211yescharacteristic
195C22×C17#48211yescharacteristic
196C2×D17—82no—
197C2×D17—82no—
198C2×D17—82no—
199C2×D17—162no—
200C2×D17—162no—
201C4×C17—21no—
202C4×C17—21no—
203C4×C17—21no—
204C4×C17—41no—
205C4×C17—41no—
206C4×C17—41no—
207C4×C17#41611yescharacteristic
208C4×C17#41811yescharacteristic
209C4×C17#44911yescharacteristic
210C4×C17#44911yescharacteristic
211C4×C17#47111yescharacteristic
212C4×C17#47211yescharacteristic
213C2×AGL(1,ℤ/8)—341no—
214C2×#127—171no—
215C4×#41C2×D1711yescharacteristic
216#460—172no—
217#460—341no—
218#459—172no—
219#459—341no—
220#433—22no—
221#432—21no—
222#431—21no—
223#431—171no—
224#423—171no—
225#423C2×D1711yescharacteristic
226C2×C4⋊C8—171no—
227C2×C4⋊C8C2×D1711yescharacteristic
228#418—341no—
229#408—21no—
230#405—21no—
231#331—341no—
232#331—341no—
233#331—342no—
234Q8⋊C8—341no—
235D4⋊C8—341no—
236C2×C17—21no—
237C2×C17—21no—
238C2×C17—81no—
239C2×C17#187511yescharacteristic
240C2×C17#190311yescharacteristic
241C2×C17#190711yescharacteristic
242D17—162no—
243C2×#41C22⋊D1711yescharacteristic
244C22×D4—341no—
245C22×D4—341no—
246C22×D4—341no—
247AGL(1,ℤ/8)—682no—
248AGL(1,ℤ/8)—682no—
249C2×SD16—342no—
250C2×D8—342no—
251C2×M16—21no—
252C2×M16—341no—
253C22×C8—21no—
254C22×C8—341no—
255#127—172no—
256#127—341no—
257#127—341no—
258#127—341no—
259C4×Q8C4×D1711yescharacteristic
260C4×Q8D6811yescharacteristic
261C4×D4—21no—
262C4×D4—21no—
263C4×D4C4×D1711yescharacteristic
264C4×D4D6811yescharacteristic
265#117—21no—
266#117C22⋊D1711yescharacteristic
267C2×C42—21no—
268C2×C42C22×D1711yescharacteristic
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—172no—
272C4⋊C8—341no—
273C4⋊C8—341no—
274C4⋊C8—341no—
275C4⋊C8—341no—
276C4⋊C8C22⋊D1711yescharacteristic
277C4⋊C8C22⋊D1711yescharacteristic
278C4⋊C8C22⋊D1712yes—
279#102—342no—
280#102—342no—
281C22⋊C8—22no—
282C22⋊C8—22no—
283C22⋊C8—341no—
284#97—22no—
285C4×C8—22no—
286C4×C8—341no—
287C17#917411yescharacteristic, complemented, Sylow
288C24—681no—
289#41—21no—
290#41—21no—
291#41—21no—
292#41—21no—
293C2×Q8#6322711yescharacteristic
294C2×D4—21no—
295C2×D4—342no—
296C2×D4—342no—
297C2×D4—342no—
298C2×D4—681no—
299C2×D4—681no—
300C2×D4—681no—
301C2×D4—681no—
302C2×D4—681no—
303C2×D4—681no—
304C2×D4—681no—
305C2×D4—1361no—
306C2×D4—1361no—
307C2×D4—1362no—
308C2×D4#6322711yescharacteristic
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4#6325211yescharacteristic
313C22×C4#6325211yescharacteristic
314C22×C4C2×C22⋊D1711yescharacteristic
315SD16—684no—
316D8—684no—
317M16—41no—
318M16—41no—
319M16—681no—
320M16—681no—
321C2×C8—21no—
322C2×C8—22no—
323C2×C8—21no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—342no—
327C2×C8—681no—
328C2×C8—681no—
329C2×C8—1361no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4#6322711yescharacteristic
333C4⋊C4#6322711yescharacteristic
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—21no—
337C42—21no—
338C42—21no—
339C42—21no—
340C42C2×C4×D1711yescharacteristic
341C42C2×D6811yescharacteristic
342C42C2×C22⋊D1711yescharacteristic
343C42C2×C22⋊D1711yescharacteristic
344C23—21no—
345C23—682no—
346C23—1362no—
347C23—1362no—
348C23—1361no—
349C23—1361no—
350C23#1058761511yescharacteristic
351Q8—21no—
352Q8—21no—
353D4—21no—
354D4—21no—
355D4—41no—
356D4—41no—
357D4—682no—
358D4—682no—
359D4—1362no—
360D4—1362no—
361D4—1362no—
362D4—1361no—
363D4—1361no—
364D4—1361no—
365D4—1361no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—81no—
377C2×C4C2×#6322711yescharacteristic
378C2×C4C4×C22⋊D1711yescharacteristic
379C2×C4C4×C22⋊D1711yescharacteristic
380C2×C4#1058761611yescharacteristic
381C2×C4#1058761611yescharacteristic
382C2×C4#1058763511yescharacteristic
383C2×C4#1058763511yescharacteristic
384C2×C4#1058763611yescharacteristic
385C2×C4#1058763611yescharacteristic
386C2×C4#1058763911yescharacteristic
387C8—41no—
388C8—41no—
389C8—42no—
390C8—682no—
391C8—1361no—
392C8—1361no—
393C22—21no—
394C22—21no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—2722no—
399C22—2722no—
400C22—1362no—
401C22—1362no—
402C22—1362no—
403C22#1176268811yescharacteristic
404C22#1176313311yescharacteristic
405C22#1176313311yescharacteristic
406C4—21no—
407C4—21no—
408C4—21no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4#1176280311yescharacteristic
413C4#1176281311yescharacteristic
414C4#1176295411yescharacteristic
415C4#1176295411yescharacteristic
416C4#1176307211yescharacteristic
417C4#1176307911yescharacteristic
418C2—21no—
419C2—21no—
420C2—81no—
421C2—2722no—
422C2#1452910611yescharacteristic
423C2#1452926411yescharacteristic
424C2#1452929211yescharacteristic
subgroupG/H
centerC22#11762688
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#2925C2
socleC22×C17#386
prime radical#14538264C2
cosocleC2×C42×C17C23
hypercenter#2925D17
hypocenterC17#9174
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2925

Complements

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

NH ≅ G/Ntype
2. #14527453C2semidirect
3. C2×#11762552C2semidirect
4. #14527171C2semidirect
5. #14526681C2semidirect
6. #14526658C2semidirect
7. C17×#2925C2semidirect
182. #2925D17semidirect
287. C17#9174semidirect

Maximal subgroups

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

subgroupindex
#145382642
#145274532
C2×#117625522
#145271712
#145266812
#145266582
C17×#29252
#917417