Finite Group Databasebeta

← #16930619

Subgroups — #16930619

Order 4352.

588 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#14531249C211yesmaximal, characteristic, complemented
2#14529613C211yesmaximal, characteristic, complemented
3#14527279C211yesmaximal, characteristic, complemented
4#14527209C211yesmaximal, characteristic, complemented
5#14526686C211yesmaximal, characteristic, complemented
6#14526616C211yesmaximal, characteristic, complemented
7C17×#3100C211yesmaximal, characteristic
8C2×#10587612—21no—
9#11762862C2211yescharacteristic
10#11762833C2211yescharacteristic
11C2×#10587508—21no—
12C2×#10587508—21no—
13#11762712—21no—
14#11762708—21no—
15#11762702—22no—
16C4×C4⋊Dic17C2211yescharacteristic
17C4×C22⋊Dic17—21no—
18C42×Dic17C2211yescharacteristic
19#11762543—22no—
20D4⋊#3622—22no—
21C17×#532C2211yescharacteristic
22C2×C4×D4×C17C2211yescharacteristic
23C17×#480—21no—
24C17×#474C412yescomplemented
25C17×#459C412yescomplemented
26C17×#447—21no—
27C4×D8×C17—22no—
28C17×#431C2211yescharacteristic
29C2×#41×C17—21no—
30C22×D4×C17D411yescharacteristic
31C23×C4×C17—21no—
32C2×D8×C17—22no—
33C17×C4⋊Q8C2×C411yescharacteristic
34C17×#127C2×C411yescharacteristic
35C17×#124—21no—
36C17×#121—21no—
37C17×#121—21no—
38C4×D4×C17—21no—
39C4×D4×C17—21no—
40C4×D4×C17—21no—
41C4×D4×C17—21no—
42C4×D4×C17C2×C412yes—
43C4×D4×C17D412yes—
44C2×C4⋊C4×C17D411yescharacteristic
45C2×C22⋊C4×C17—21no—
46C2×C42×C17C2311yescharacteristic
47C2×C22⋊Dic17—22no—
48C2×C22⋊Dic17—22no—
49#10587612—42no—
50#10587612—42no—
51C2×C4⋊Dic17—22no—
52C2×C4⋊Dic17—21no—
53C2×C4×Dic17—21no—
54C2×C4×Dic17—21no—
55C2×C4×Dic17—21no—
56C2×C4×Dic17D411yescharacteristic
57C2×C4×Dic17D411yescharacteristic
58C2×#63223—21no—
59C17×#107—22no—
60C17×C4⋊C8C2×C412yes—
61C17×#102—22no—
62C17×#102—22no—
63C17×C22⋊C8—21no—
64C4×C8×C17—21no—
65#10587508—42no—
66#10587508—42no—
67#10587502—22no—
68#10587502D412yes—
69C22⋊#3622—21no—
70C4⋊#3622—21no—
71C4⋊#3622—21no—
72C4⋊#3622—21no—
73C4×#3622—21no—
74C24×C17—21no—
75#41×C17—41no—
76#41×C17—41no—
77#41×C17—41no—
78#41×C17—41no—
79C2×Q8×C17—21no—
80C2×D4×C17—21no—
81C2×D4×C17—21no—
82C2×D4×C17—21no—
83C2×D4×C17—21no—
84C2×D4×C17—42no—
85C2×D4×C17—41no—
86C2×D4×C17C22⋊C412yes—
87C22×C4×C17—21no—
88C22×C4×C17—22no—
89C22×C4×C17—22no—
90C22×C4×C17—41no—
91C22×C4×C17—41no—
92C22×C4×C17—41no—
93C22×C4×C17—41no—
94C22×C4×C17—41no—
95C22×C4×C17C2×D411yescharacteristic
96C22×C4×C17C2×D411yescharacteristic
97C22×C4×C17C2×D411yescharacteristic
98C22×Dic17—21no—
99C22×Dic17—21no—
100C22×Dic17—22no—
101D8×C17—44no—
102C2×C8×C17—22no—
103C4⋊C4×C17—21no—
104C4⋊C4×C17—21no—
105C4⋊C4×C17—21no—
106C4⋊C4×C17C22⋊C412yes—
107C22⋊C4×C17—22no—
108C22⋊C4×C17—22no—
109C22⋊C4×C17—41no—
110C22⋊C4×C17—41no—
111C42×C17C22×C411yescharacteristic
112C42×C17C2×D411yescharacteristic
113C42×C17#4111yescharacteristic
114C42×C17#4111yescharacteristic
115C22⋊Dic17—44no—
116C22⋊Dic17—44no—
117C4⋊Dic17—44no—
118C4⋊Dic17—42no—
119C4×Dic17—21no—
120C4×Dic17—21no—
121C4×Dic17—21no—
122C4×Dic17—42no—
123C4×Dic17—42no—
124C4×Dic17—41no—
125C4×Dic17—41no—
126C4×Dic17D812yescomplemented
127#63223—41no—
128#63223—41no—
129C2×#3622—41no—
130C2×#3622—41no—
131#9237—171nomaximal, Sylow
132C23×C17—22no—
133C23×C17—42no—
134C23×C17—42no—
135C23×C17—41no—
136C23×C17—41no—
137C23×C17—41no—
138C23×C17#12011yescharacteristic
139Q8×C17—41no—
140Q8×C17—41no—
141D4×C17—22no—
142D4×C17—22no—
143D4×C17—42no—
144D4×C17—42no—
145D4×C17—41no—
146D4×C17—41no—
147D4×C17—81no—
148D4×C17—81no—
149D4×C17—81no—
150C2×C4×C17—22no—
151C2×C4×C17—22no—
152C2×C4×C17—41no—
153C2×C4×C17—41no—
154C2×C4×C17—41no—
155C2×C4×C17—42no—
156C2×C4×C17—42no—
157C2×C4×C17—42no—
158C2×C4×C17—42no—
159C2×C4×C17—41no—
160C2×C4×C17—81no—
161C2×C4×C17—81no—
162C2×C4×C17C2×C22⋊C411yescharacteristic
163C2×C4×C17C4×D411yescharacteristic
164C2×C4×C17C4×D411yescharacteristic
165C2×C4×C17#12111yescharacteristic
166C2×C4×C17#12111yescharacteristic
167C2×C4×C17#12311yescharacteristic
168C2×C4×C17C2×D811yescharacteristic
169C2×C4×C17#13511yescharacteristic
170C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
171C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
172C2×Dic17—22no—
173C2×Dic17—22no—
174C2×Dic17—41no—
175C2×Dic17—41no—
176C2×Dic17—41no—
177C2×Dic17—41no—
178C2×Dic17—42no—
179C2×Dic17—42no—
180C2×Dic17—42no—
181C8×C17—42no—
182#3622—81no—
183#3622—81no—
184#3100D1711yescharacteristic
185#2293—171no—
186#1962—171no—
187#1502—171no—
188#1486—171no—
189#1303—171no—
190#1276—171no—
191C22×C17—21no—
192C22×C17—21no—
193C22×C17—42no—
194C22×C17—42no—
195C22×C17—42no—
196C22×C17—82no—
197C22×C17—82no—
198C22×C17—81no—
199C22×C17—81no—
200C22×C17—81no—
201C22×C17#38611yescharacteristic
202C22×C17#44711yescharacteristic
203C22×C17#44911yescharacteristic
204C4×C17—21no—
205C4×C17—21no—
206C4×C17—21no—
207C4×C17—42no—
208C4×C17—81no—
209C4×C17#42112yes—
210C4×C17C4×D811yescharacteristic
211C4×C17#44211yescharacteristic
212C4×C17#45911yescharacteristic
213C4×C17#46311yescharacteristic
214Dic17—42no—
215Dic17—42no—
216Dic17—82no—
217#532C2×D1711yescharacteristic
218C2×C4×D4C2×D1711yescharacteristic
219C2×#117—341no—
220#480—21no—
221#474Dic1712yescomplemented
222#459Dic1712yescomplemented
223#447—21no—
224C4×D8—22no—
225#431—171no—
226#431C2×D1711yescharacteristic
227#423—171no—
228C2×#104—341no—
229C2×#104—341no—
230#390—341no—
231#389—341no—
232#388—342no—
233C4×C4⋊C4—171no—
234C4×C22⋊C4—341no—
235C43—171no—
236#327—342no—
237D4⋊C8—342no—
238C2×C17—21no—
239C2×C17—21no—
240C2×C17—82no—
241C2×C17—161no—
242C2×C17#187911yescharacteristic
243C2×C17#189711yescharacteristic
244C2×C17#190611yescharacteristic
245C2×#41—21no—
246C22×D4C22⋊D1711yescharacteristic
247C23×C4—21no—
248C2×D8—22no—
249C2×M16—341no—
250C4⋊Q8C2×Dic1711yescharacteristic
251#127C2×Dic1711yescharacteristic
252#124—21no—
253#121—21no—
254#121—21no—
255C4×D4—21no—
256C4×D4—21no—
257C4×D4—21no—
258C4×D4—21no—
259C4×D4C2×Dic1712yes—
260C4×D4C22⋊D1712yes—
261#117—682no—
262#117—682no—
263C2×C4⋊C4—342no—
264C2×C4⋊C4—341no—
265C2×C4⋊C4C22⋊D1711yescharacteristic
266C2×C22⋊C4—21no—
267C2×C22⋊C4—342no—
268C2×C22⋊C4—342no—
269C2×C42—171no—
270C2×C42—171no—
271C2×C42—341no—
272C2×C42—341no—
273C2×C42—341no—
274C2×C42C22×D1711yescharacteristic
275#107—22no—
276C4⋊C8—341no—
277C4⋊C8—341no—
278C4⋊C8—341no—
279C4⋊C8C2×Dic1712yes—
280#104—682no—
281#104—682no—
282#102—22no—
283#102—22no—
284C22⋊C8—21no—
285C22⋊C8—341no—
286C4×C8—21no—
287C4×C8—341no—
288#95—172no—
289#95—342no—
290C17#923711yescharacteristic, complemented, Sylow
291C24—21no—
292#41—41no—
293#41—41no—
294#41—41no—
295#41—41no—
296C2×Q8—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—42no—
302C2×D4—41no—
303C2×D4C22⋊Dic1712yes—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—341no—
313C22×C4—341no—
314C22×C4—342no—
315C22×C4D4×D1711yescharacteristic
316C22×C4D4×D1711yescharacteristic
317C22×C4C2×C22⋊D1711yescharacteristic
318D8—44no—
319M16—681no—
320M16—681no—
321C2×C8—22no—
322C2×C8—681no—
323C2×C8—681no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—682no—
328C4⋊C4—684no—
329C4⋊C4C22⋊Dic1712yes—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—684no—
335C22⋊C4—684no—
336C42—172no—
337C42—341no—
338C42—341no—
339C42—341no—
340C42—682no—
341C42—682no—
342C42—681no—
343C42—681no—
344C42#6325411yescharacteristic
345C42#6325411yescharacteristic
346C42C22×Dic1711yescharacteristic
347C42C2×C22⋊D1711yescharacteristic
348C23—22no—
349C23—42no—
350C23—42no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23#1058762311yescharacteristic
355Q8—41no—
356Q8—41no—
357D4—22no—
358D4—22no—
359D4—42no—
360D4—42no—
361D4—41no—
362D4—41no—
363D4—81no—
364D4—81no—
365D4—81no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—342no—
379C2×C4—342no—
380C2×C4—681no—
381C2×C4—681no—
382C2×C4—681no—
383C2×C4—681no—
384C2×C4—682no—
385C2×C4—682no—
386C2×C4—682no—
387C2×C4D8×D1711yescharacteristic
388C2×C4#1058759711yescharacteristic
389C2×C4#1058759711yescharacteristic
390C2×C4#1058759811yescharacteristic
391C2×C4D4×Dic1711yescharacteristic
392C2×C4D4×Dic1711yescharacteristic
393C2×C4#1058762111yescharacteristic
394C2×C4#1058762511yescharacteristic
395C2×C4#1058762511yescharacteristic
396C2×C4C2×C22⋊Dic1711yescharacteristic
397C8—42no—
398C8—1361no—
399C8—1361no—
400C22—21no—
401C22—21no—
402C22—42no—
403C22—42no—
404C22—42no—
405C22—82no—
406C22—82no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22#1176268911yescharacteristic
411C22#1176294111yescharacteristic
412C22#1176295111yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—42no—
417C4—81no—
418C4—682no—
419C4—682no—
420C4—1362no—
421C4#1176282612yes—
422C4D8×Dic1711yescharacteristic
423C4#1176292111yescharacteristic
424C4#1176300211yescharacteristic
425C4#1176302211yescharacteristic
426C2—21no—
427C2—21no—
428C2—82no—
429C2—161no—
430C2#1452913411yescharacteristic
431C2#1452923311yescharacteristic
432C2#1452928711yescharacteristic
subgroupG/H
centerC2×C4#10587625
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
Fitting, prime radicalC17×#3100C2
socleC22×C17#386
cosocleC2×C42×C17C23
hypercenter#3100D17
hypocenterC17#9237
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
1. #14531249C2semidirect
2. #14529613C2semidirect
3. #14527279C2semidirect
4. #14527209C2semidirect
5. #14526686C2semidirect
6. #14526616C2semidirect
24. C17×#474C4semidirect
25. C17×#459C4semidirect
126. C4×Dic17D8semidirect
221. #474Dic17semidirect
222. #459Dic17semidirect
290. C17#9237semidirect

Maximal subgroups

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

subgroupindex
#145312492
#145296132
#145272792
#145272092
#145266862
#145266162
C17×#31002
#923717