Finite Group Databasebeta

← #16930217

Subgroups — #16930217

Order 4352.

562 conjugacy classes of subgroups.

The subgroup lattice has 434 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×#2906C211yesmaximal, 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—
23C2×C17×#131—21no—
24C4×#41×C17C2211yescharacteristic
25C17×#424—21no—
26C17×#424—22no—
27C17×#423—21no—
28C17×#423C2211yescharacteristic
29C2×C17×C4⋊C8—21no—
30C2×C17×C4⋊C8C2211yescharacteristic
31C2×#41×C17D411yescharacteristic
32C22×D68—21no—
33C22×D68—21no—
34C22×D68—21no—
35C17×#131—42no—
36C17×#131—42no—
37C2×M16×C17—22no—
38C2×M16×C17—21no—
39C22×C8×C17—22no—
40C22×C8×C17—21no—
41C4×Q8×C17C2×C411yescharacteristic
42C4×Q8×C17D411yescharacteristic
43C4×D4×C17—21no—
44C4×D4×C17—21no—
45C4×D4×C17C2×C411yescharacteristic
46C4×D4×C17D411yescharacteristic
47C17×#117—21no—
48C17×#117D411yescharacteristic
49C2×C42×C17—21no—
50C2×C42×C17C2311yescharacteristic
51#10587635—42no—
52#10587635—42no—
53C2×#63230—22no—
54C2×#63228—22no—
55C2×#63223—21no—
56C22×#3622—21no—
57#10587560—21no—
58#10587560—21no—
59#10587560—21no—
60#10587560C2×C412yes—
61C17×C4⋊C8—22no—
62C17×C4⋊C8—22no—
63C17×C4⋊C8—21no—
64C17×C4⋊C8—21no—
65C17×C4⋊C8D411yescharacteristic
66C17×C4⋊C8D411yescharacteristic
67C17×C4⋊C8D412yes—
68C22⋊#3622—21no—
69#10587480—22no—
70#10587480—22no—
71C4⋊#3622—21no—
72C4⋊#3622—21no—
73C4⋊#3622—21no—
74C4⋊#3622—21no—
75C4⋊#3622C2×C412yes—
76C4×#3622—21no—
77C23×D17—41no—
78#41×C17—21no—
79#41×C17—21no—
80#41×C17D812yes—
81#41×C17SD1612yes—
82C2×Q8×C17C22⋊C411yescharacteristic
83C2×D4×C17—21no—
84C2×D4×C17C22⋊C411yescharacteristic
85C22×C4×C17—21no—
86C22×C4×C17—21no—
87C22×C4×C17—21no—
88C22×C4×C17C2×D411yescharacteristic
89C22×C4×C17#4111yescharacteristic
90C22×C4×C17#4111yescharacteristic
91C2×D68—22no—
92C2×D68—22no—
93C2×D68—22no—
94C2×D68—41no—
95C2×D68—41no—
96C2×D68—41no—
97C2×D68—41no—
98C2×D68—41no—
99C2×D68—41no—
100C2×D68—41no—
101C2×D68—82no—
102C2×D68—81no—
103C2×D68—81no—
104M16×C17—44no—
105M16×C17—41no—
106M16×C17—41no—
107C2×C8×C17—21no—
108C2×C8×C17—22no—
109C2×C8×C17—21no—
110C2×C8×C17—44no—
111C2×C8×C17—41no—
112C2×C8×C17—41no—
113C4⋊C4×C17—21no—
114C4⋊C4×C17—21no—
115C4⋊C4×C17C22⋊C411yescharacteristic
116C4⋊C4×C17C22⋊C411yescharacteristic
117C22⋊C4×C17—21no—
118C22⋊C4×C17—21no—
119C22⋊C4×C17—21no—
120C42×C17—21no—
121C42×C17—21no—
122C42×C17—21no—
123C42×C17C22×C411yescharacteristic
124C42×C17C2×D411yescharacteristic
125C42×C17C2×D411yescharacteristic
126C42×C17C2×D411yescharacteristic
127#63230—44no—
128#63228—44no—
129#63223—41no—
130#63223—41no—
131C2×#3622—22no—
132C2×#3622—41no—
133C2×#3622—41no—
134C2×#3622—81no—
135#9173—171nomaximal, Sylow
136C23×C17—21no—
137C23×C17#12311yescharacteristic
138C22×D17—42no—
139C22×D17—82no—
140C22×D17—82no—
141C22×D17—81no—
142C22×D17—81no—
143Q8×C17—21no—
144Q8×C17#10212yes—
145D4×C17—22no—
146D4×C17—22no—
147D4×C17—21no—
148D4×C17#10212yes—
149C2×C4×C17—21no—
150C2×C4×C17—22no—
151C2×C4×C17—22no—
152C2×C4×C17—21no—
153C2×C4×C17—21no—
154C2×C4×C17—21no—
155C2×C4×C17—21no—
156C2×C4×C17—41no—
157C2×C4×C17—41no—
158C2×C4×C17—41no—
159C2×C4×C17—81no—
160C2×C4×C17C2×C22⋊C411yescharacteristic
161C2×C4×C17C4×D411yescharacteristic
162C2×C4×C17C4×D411yescharacteristic
163C2×C4×C17#12011yescharacteristic
164C2×C4×C17#12111yescharacteristic
165C2×C4×C17#12111yescharacteristic
166C2×C4×C17C2×D811yescharacteristic
167C2×C4×C17C2×SD1611yescharacteristic
168C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C17#13711yescharacteristic
170D68—42no—
171D68—42no—
172D68—82no—
173D68—82no—
174D68—82no—
175D68—81no—
176D68—81no—
177D68—81no—
178D68—81no—
179C8×C17—41no—
180C8×C17—42no—
181C8×C17—41no—
182#3622—42no—
183#3622—81no—
184#3622—81no—
185#3038—171no—
186#2906D1711yescharacteristic, complemented
187#1545—171no—
188C2×#331—171no—
189#1477—171no—
190#1301—171no—
191#1292—171no—
192C22×C17—21no—
193C22×C17—21no—
194C22×C17—22no—
195C22×C17—41no—
196C22×C17—41no—
197C22×C17#38611yescharacteristic
198C22×C17#48011yescharacteristic
199C22×C17#48111yescharacteristic
200C2×D17—82no—
201C2×D17—82no—
202C2×D17—82no—
203C2×D17—162no—
204C2×D17—162no—
205C4×C17—22no—
206C4×C17—21no—
207C4×C17—21no—
208C4×C17—21no—
209C4×C17—41no—
210C4×C17—41no—
211C4×C17C2×#10211yescharacteristic
212C4×C17#41711yescharacteristic
213C4×C17#44711yescharacteristic
214C4×C17#44811yescharacteristic
215C4×C17#47111yescharacteristic
216C4×C17#47211yescharacteristic
217C2×AGL(1,ℤ/8)—341no—
218C2×#131—21no—
219C2×#127—171no—
220C4×#41C2×D1711yescharacteristic
221#460—172no—
222#460—341no—
223#459—172no—
224#459—341no—
225#431—171no—
226#424—21no—
227#424—22no—
228#423—21no—
229#423—171no—
230#423C2×D1711yescharacteristic
231C2×C4⋊C8—21no—
232C2×C4⋊C8—171no—
233C2×C4⋊C8C2×D1711yescharacteristic
234#418—341no—
235#331—341no—
236#331—341no—
237#331—342no—
238Q8⋊C8—341no—
239D4⋊C8—341no—
240C2×C17—21no—
241C2×C17—21no—
242C2×C17—42no—
243C2×C17#187411yescharacteristic
244C2×C17#190311yescharacteristic
245C2×C17#190711yescharacteristic
246D17—162no—
247C2×#41C22⋊D1711yescharacteristic
248C22×D4—341no—
249C22×D4—341no—
250C22×D4—341no—
251AGL(1,ℤ/8)—682no—
252AGL(1,ℤ/8)—682no—
253C2×SD16—342no—
254C2×D8—342no—
255#131—42no—
256#131—42no—
257C2×M16—22no—
258C2×M16—21no—
259C2×M16—341no—
260C22×C8—22no—
261C22×C8—21no—
262C22×C8—341no—
263#127—172no—
264#127—341no—
265#127—341no—
266#127—341no—
267C4×Q8C4×D1711yescharacteristic
268C4×Q8D6811yescharacteristic
269C4×D4—21no—
270C4×D4—21no—
271C4×D4C4×D1711yescharacteristic
272C4×D4D6811yescharacteristic
273#117—21no—
274#117C22⋊D1711yescharacteristic
275C2×C42—21no—
276C2×C42C22×D1711yescharacteristic
277C4⋊C8—22no—
278C4⋊C8—22no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—172no—
282C4⋊C8—341no—
283C4⋊C8—341no—
284C4⋊C8—341no—
285C4⋊C8—341no—
286C4⋊C8C22⋊D1711yescharacteristic
287C4⋊C8C22⋊D1711yescharacteristic
288C4⋊C8C22⋊D1712yes—
289#102—342no—
290#102—342no—
291C22⋊C8—341no—
292C4×C8—341no—
293C17#917311yescharacteristic, complemented, Sylow
294C24—681no—
295#41—21no—
296#41—21no—
297#41#6322812yes—
298#41#6323012yes—
299C2×Q8#6322711yescharacteristic
300C2×D4—21no—
301C2×D4—342no—
302C2×D4—342no—
303C2×D4—342no—
304C2×D4—681no—
305C2×D4—681no—
306C2×D4—681no—
307C2×D4—681no—
308C2×D4—681no—
309C2×D4—681no—
310C2×D4—681no—
311C2×D4—1361no—
312C2×D4—1361no—
313C2×D4—1362no—
314C2×D4#6322711yescharacteristic
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4#6325211yescharacteristic
319C22×C4#6325211yescharacteristic
320C22×C4C2×C22⋊D1711yescharacteristic
321SD16—684no—
322D8—684no—
323M16—44no—
324M16—41no—
325M16—41no—
326M16—681no—
327M16—681no—
328C2×C8—21no—
329C2×C8—22no—
330C2×C8—21no—
331C2×C8—44no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—342no—
335C2×C8—681no—
336C2×C8—681no—
337C2×C8—1361no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4#6322711yescharacteristic
341C4⋊C4#6322711yescharacteristic
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—21no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42C2×C4×D1711yescharacteristic
349C42C2×D6811yescharacteristic
350C42C2×C22⋊D1711yescharacteristic
351C42C2×C22⋊D1711yescharacteristic
352C23—21no—
353C23—682no—
354C23—1362no—
355C23—1362no—
356C23—1361no—
357C23—1361no—
358C23#1058761511yescharacteristic
359Q8—21no—
360Q8#1058748012yes—
361D4—22no—
362D4—22no—
363D4—21no—
364D4—682no—
365D4—682no—
366D4—1362no—
367D4—1362no—
368D4—1362no—
369D4—1361no—
370D4—1361no—
371D4—1361no—
372D4—1361no—
373D4#1058748012yes—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—81no—
385C2×C4C2×#6322711yescharacteristic
386C2×C4C4×C22⋊D1711yescharacteristic
387C2×C4C4×C22⋊D1711yescharacteristic
388C2×C4#1058761611yescharacteristic
389C2×C4#1058761611yescharacteristic
390C2×C4C2×#6322811yescharacteristic
391C2×C4#1058761811yescharacteristic
392C2×C4C2×#6323011yescharacteristic
393C2×C4#1058762811yescharacteristic
394C2×C4#1058763911yescharacteristic
395C8—41no—
396C8—42no—
397C8—41no—
398C8—682no—
399C8—1361no—
400C8—1361no—
401C22—21no—
402C22—21no—
403C22—22no—
404C22—41no—
405C22—41no—
406C22—2722no—
407C22—2722no—
408C22—1362no—
409C22—1362no—
410C22—1362no—
411C22#1176268811yescharacteristic
412C22#1176312211yescharacteristic
413C22#1176312711yescharacteristic
414C4—21no—
415C4—22no—
416C4—21no—
417C4—21no—
418C4—41no—
419C4—41no—
420C4C2×#1058748011yescharacteristic
421C4#1176280811yescharacteristic
422C4#1176294311yescharacteristic
423C4#1176294811yescharacteristic
424C4#1176307211yescharacteristic
425C4#1176307911yescharacteristic
426C2—21no—
427C2—21no—
428C2—42no—
429C2—2722no—
430C2#1452910111yescharacteristic
431C2#1452926411yescharacteristic
432C2#1452929211yescharacteristic
subgroupG/H
centerC22#11762688
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#2906C2
socleC22×C17#386
prime radical#14538264C2
cosocleC2×C42×C17C23
hypercenter#2906D17
hypocenterC17#9173
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2906

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×#2906C2semidirect
186. #2906D17semidirect
293. C17#9173semidirect

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×#29062
#917317