Finite Group Databasebeta

← #16930219

Subgroups — #16930219

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#14537352C211yesmaximal, characteristic, complemented
2#14527452C211yesmaximal, characteristic, complemented
3C2×#11762553C211yesmaximal, characteristic, complemented
4#14527171C211yesmaximal, characteristic, complemented
5#14526679C211yesmaximal, characteristic, complemented
6#14526658C211yesmaximal, characteristic, complemented
7C17×#3038C211yesmaximal, characteristic
8C2×#10587634—21no—
9#11762862C2211yescharacteristic
10#11762836—21no—
11#11762836—22no—
12#11762833—21no—
13#11762833C2211yescharacteristic
14C2×C4⋊#3622—21no—
15C2×C4⋊#3622C2211yescharacteristic
16C2×C4⋊#3622C2211yescharacteristic
17#11762553—21no—
18#11762553—21no—
19#11762553—22no—
20Q8⋊#3622—21no—
21D4⋊#3622—21no—
22C2×C17×AGL(1,ℤ/8)—21no—
23C2×C17×#127C2211yescharacteristic
24C4×#41×C17C2211yescharacteristic
25C17×#460—21no—
26C17×#460C412yes—
27C17×#459—21no—
28C17×#459C412yes—
29C17×#423C2211yescharacteristic
30C17×#418—21no—
31C2×#41×C17D411yescharacteristic
32C22×D4×C17—21no—
33C22×D4×C17—21no—
34C22×D4×C17—21no—
35C17×AGL(1,ℤ/8)—42no—
36C17×AGL(1,ℤ/8)—42no—
37C2×SD16×C17—22no—
38C2×D8×C17—22no—
39C2×M16×C17—21no—
40C17×#127—21no—
41C17×#127—21no—
42C17×#127—21no—
43C17×#127C2×C412yes—
44C4×Q8×C17C2×C411yescharacteristic
45C4×Q8×C17D411yescharacteristic
46C4×D4×C17—21no—
47C4×D4×C17—21no—
48C4×D4×C17C2×C411yescharacteristic
49C4×D4×C17D411yescharacteristic
50C17×#117—21no—
51C17×#117D411yescharacteristic
52C2×C42×C17—21no—
53C2×C42×C17C2311yescharacteristic
54#10587634—42no—
55#10587634—42no—
56C2×#63223—22no—
57C2×#63223—21no—
58C22×#3622—22no—
59C22×#3622—21no—
60C22×#3622—21no—
61C17×C4⋊C8—21no—
62C17×C4⋊C8C2×C412yes—
63C17×#102—22no—
64C17×#102—22no—
65C22⋊#3622—21no—
66C4⋊#3622—22no—
67C4⋊#3622—22no—
68C4⋊#3622—21no—
69C4⋊#3622—21no—
70C4⋊#3622—21no—
71C4⋊#3622—21no—
72C4⋊#3622—21no—
73C4⋊#3622D411yescharacteristic
74C4⋊#3622D411yescharacteristic
75C4⋊#3622D412yes—
76C4×#3622—21no—
77C24×C17—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×C17—22no—
85C2×D4×C17—22no—
86C2×D4×C17—22no—
87C2×D4×C17—41no—
88C2×D4×C17—41no—
89C2×D4×C17—41no—
90C2×D4×C17—41no—
91C2×D4×C17—41no—
92C2×D4×C17—41no—
93C2×D4×C17—41no—
94C2×D4×C17—81no—
95C2×D4×C17—81no—
96C2×D4×C17—82no—
97C2×D4×C17C22⋊C411yescharacteristic
98C22×C4×C17—21no—
99C22×C4×C17—21no—
100C22×C4×C17—21no—
101C22×C4×C17C2×D411yescharacteristic
102C22×C4×C17#4111yescharacteristic
103C22×C4×C17#4111yescharacteristic
104SD16×C17—44no—
105D8×C17—44no—
106M16×C17—41no—
107M16×C17—41no—
108C2×C8×C17—22no—
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#63223—44no—
124#63223—41no—
125#63223—41no—
126C2×#3622—21no—
127C2×#3622—22no—
128C2×#3622—21no—
129C2×#3622—44no—
130C2×#3622—41no—
131C2×#3622—41no—
132C2×#3622—41no—
133C2×#3622—41no—
134C2×#3622—81no—
135#9173—171nomaximal, Sylow
136C23×C17—21no—
137C23×C17—42no—
138C23×C17—82no—
139C23×C17—82no—
140C23×C17—81no—
141C23×C17—81no—
142C23×C17#12311yescharacteristic
143Q8×C17—21no—
144Q8×C17#10212yes—
145D4×C17—22no—
146D4×C17—22no—
147D4×C17—21no—
148D4×C17—42no—
149D4×C17—42no—
150D4×C17—82no—
151D4×C17—82no—
152D4×C17—82no—
153D4×C17—81no—
154D4×C17—81no—
155D4×C17—81no—
156D4×C17—81no—
157D4×C17#10212yes—
158C2×C4×C17—21no—
159C2×C4×C17—22no—
160C2×C4×C17—22no—
161C2×C4×C17—21no—
162C2×C4×C17—21no—
163C2×C4×C17—21no—
164C2×C4×C17—21no—
165C2×C4×C17—41no—
166C2×C4×C17—41no—
167C2×C4×C17—41no—
168C2×C4×C17—81no—
169C2×C4×C17C2×C22⋊C411yescharacteristic
170C2×C4×C17C4×D411yescharacteristic
171C2×C4×C17C4×D411yescharacteristic
172C2×C4×C17#12011yescharacteristic
173C2×C4×C17#12111yescharacteristic
174C2×C4×C17#12111yescharacteristic
175C2×C4×C17C2×D811yescharacteristic
176C2×C4×C17C2×SD1611yescharacteristic
177C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
178C2×C4×C17#13711yescharacteristic
179C8×C17—42no—
180#3622—41no—
181#3622—42no—
182#3622—41no—
183#3622—81no—
184#3622—81no—
185#3038D1711yescharacteristic
186#2906—171no—
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—82no—
198C22×C17—82no—
199C22×C17—82no—
200C22×C17—162no—
201C22×C17—162no—
202C22×C17#38611yescharacteristic
203C22×C17#48011yescharacteristic
204C22×C17#48111yescharacteristic
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)—21no—
218C2×#131—341no—
219C2×#127C2×D1711yescharacteristic
220C4×#41C2×D1711yescharacteristic
221#460—21no—
222#460Dic1712yes—
223#459—21no—
224#459Dic1712yes—
225#431—171no—
226#424—342no—
227#424—341no—
228#423—171no—
229#423—341no—
230#423C2×D1711yescharacteristic
231C2×C4⋊C8—171no—
232C2×C4⋊C8—171no—
233C2×C4⋊C8—341no—
234#418—21no—
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—162no—
244C2×C17#187411yescharacteristic
245C2×C17#190311yescharacteristic
246C2×C17#190711yescharacteristic
247C2×#41C22⋊D1711yescharacteristic
248C22×D4—21no—
249C22×D4—21no—
250C22×D4—21no—
251AGL(1,ℤ/8)—42no—
252AGL(1,ℤ/8)—42no—
253C2×SD16—22no—
254C2×D8—22no—
255#131—682no—
256#131—682no—
257C2×M16—21no—
258C2×M16—342no—
259C2×M16—341no—
260C22×C8—342no—
261C22×C8—341no—
262C22×C8—341no—
263#127—21no—
264#127—21no—
265#127—21no—
266#127C2×Dic1712yes—
267C4×Q8C2×Dic1711yescharacteristic
268C4×Q8C22⋊D1711yescharacteristic
269C4×D4—21no—
270C4×D4—21no—
271C4×D4C2×Dic1711yescharacteristic
272C4×D4C22⋊D1711yescharacteristic
273#117—21no—
274#117C22⋊D1711yescharacteristic
275C2×C42—21no—
276C2×C42C22×D1711yescharacteristic
277C4⋊C8—21no—
278C4⋊C8—171no—
279C4⋊C8—171no—
280C4⋊C8—172no—
281C4⋊C8—342no—
282C4⋊C8—342no—
283C4⋊C8—341no—
284C4⋊C8—341no—
285C4⋊C8—341no—
286C4⋊C8—341no—
287C4⋊C8—341no—
288C4⋊C8C2×Dic1712yes—
289#102—22no—
290#102—22no—
291C22⋊C8—341no—
292C4×C8—341no—
293C17#917311yescharacteristic, complemented, Sylow
294C24—41no—
295#41—21no—
296#41—21no—
297#41#6322812yes—
298#41#6322912yes—
299C2×Q8C22⋊Dic1711yescharacteristic
300C2×D4—21no—
301C2×D4—22no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C2×D4—81no—
312C2×D4—81no—
313C2×D4—82no—
314C2×D4C22⋊Dic1711yescharacteristic
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4#6325411yescharacteristic
319C22×C4#6325411yescharacteristic
320C22×C4C2×C22⋊D1711yescharacteristic
321SD16—44no—
322D8—44no—
323M16—41no—
324M16—41no—
325M16—684no—
326M16—681no—
327M16—681no—
328C2×C8—22no—
329C2×C8—341no—
330C2×C8—341no—
331C2×C8—342no—
332C2×C8—684no—
333C2×C8—681no—
334C2×C8—681no—
335C2×C8—681no—
336C2×C8—681no—
337C2×C8—1361no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4C22⋊Dic1711yescharacteristic
341C4⋊C4C22⋊Dic1711yescharacteristic
342C22⋊C4—21no—
343C22⋊C4—21no—
344C22⋊C4—21no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42D4×D1711yescharacteristic
349C42D4×D1711yescharacteristic
350C42C22×Dic1711yescharacteristic
351C42C2×C22⋊D1711yescharacteristic
352C23—21no—
353C23—42no—
354C23—82no—
355C23—82no—
356C23—81no—
357C23—81no—
358C23#1058762111yescharacteristic
359Q8—21no—
360Q8#1058750312yes—
361D4—22no—
362D4—22no—
363D4—21no—
364D4—42no—
365D4—42no—
366D4—82no—
367D4—82no—
368D4—82no—
369D4—81no—
370D4—81no—
371D4—81no—
372D4—81no—
373D4#1058750312yes—
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×#6322811yescharacteristic
386C2×C4C2×#6322911yescharacteristic
387C2×C4D4×Dic1711yescharacteristic
388C2×C4D4×Dic1711yescharacteristic
389C2×C4#1058762311yescharacteristic
390C2×C4#1058762511yescharacteristic
391C2×C4#1058762511yescharacteristic
392C2×C4#1058763511yescharacteristic
393C2×C4#1058763711yescharacteristic
394C2×C4C2×C22⋊Dic1711yescharacteristic
395C8—42no—
396C8—681no—
397C8—682no—
398C8—681no—
399C8—1361no—
400C8—1361no—
401C22—21no—
402C22—21no—
403C22—22no—
404C22—41no—
405C22—41no—
406C22—82no—
407C22—82no—
408C22—82no—
409C22—162no—
410C22—162no—
411C22#1176268911yescharacteristic
412C22#1176312011yescharacteristic
413C22#1176312511yescharacteristic
414C4—21no—
415C4—22no—
416C4—21no—
417C4—21no—
418C4—41no—
419C4—41no—
420C4C2×#1058750311yescharacteristic
421C4#1176281011yescharacteristic
422C4#1176294211yescharacteristic
423C4#1176294711yescharacteristic
424C4#1176307011yescharacteristic
425C4#1176307711yescharacteristic
426C2—21no—
427C2—21no—
428C2—42no—
429C2—162no—
430C2#1452910211yescharacteristic
431C2#1452926611yescharacteristic
432C2#1452929411yescharacteristic
subgroupG/H
centerC22#11762689
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
Fitting, prime radicalC17×#3038C2
socleC22×C17#386
cosocleC2×C42×C17C23
hypercenter#3038D17
hypocenterC17#9173
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3038

Complements

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

NH ≅ G/Ntype
1. #14537352C2semidirect
2. #14527452C2semidirect
3. C2×#11762553C2semidirect
4. #14527171C2semidirect
5. #14526679C2semidirect
6. #14526658C2semidirect
293. C17#9173semidirect

Maximal subgroups

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

subgroupindex
#145373522
#145274522
C2×#117625532
#145271712
#145266792
#145266582
C17×#30382
#917317