Finite Group Databasebeta

← #16930617

Subgroups — #16930617

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#14538784C211yesmaximal, characteristic
2#14529612C211yesmaximal, characteristic, complemented
3#14527278C211yesmaximal, characteristic, complemented
4#14527209C211yesmaximal, characteristic, complemented
5#14526687C211yesmaximal, characteristic, complemented
6#14526617C211yesmaximal, characteristic, complemented
7C17×#2293C211yesmaximal, characteristic, complemented
8#11763359C2211yescharacteristic
9#11763122—21no—
10#11763089C412yescomplemented
11#11763004C412yescomplemented
12#11762943—21no—
13C4×#63228—22no—
14#11762862C2211yescharacteristic
15#11762862C2211yescharacteristic
16#11762833C2211yescharacteristic
17C2×#10587476—21no—
18C2×#10587476—21no—
19#11762542—22no—
20D4⋊#3622—22no—
21C2×C4×D4×C17C2211yescharacteristic
22C2×C17×#117—21no—
23C17×#390—21no—
24C17×#389—21no—
25C17×#388—22no—
26C4×C4⋊C4×C17C2211yescharacteristic
27C4×C22⋊C4×C17—21no—
28C43×C17C2211yescharacteristic
29C22×D4×C17D411yescharacteristic
30C23×C4×C17—21no—
31C2×#63252—21no—
32C4×D4×C17—21no—
33C4×D4×C17—21no—
34C4×D4×C17C2×C412yes—
35C4×D4×C17D412yes—
36C17×#117—42no—
37C17×#117—42no—
38C2×C4⋊C4×C17—22no—
39C2×C4⋊C4×C17—21no—
40C2×C4⋊C4×C17D411yescharacteristic
41C2×C22⋊C4×C17—21no—
42C2×C22⋊C4×C17—22no—
43C2×C22⋊C4×C17—22no—
44C2×C42×C17—21no—
45C2×C42×C17—21no—
46C2×C42×C17—21no—
47C2×C42×C17D411yescharacteristic
48C2×C42×C17D411yescharacteristic
49C2×C42×C17C2311yescharacteristic
50C2×#63228—22no—
51#10587616—21no—
52#10587616—21no—
53C2×#63223—21no—
54#10587561—21no—
55#10587560C2×C411yescharacteristic
56C4×D68—21no—
57C4×D68—21no—
58#10587555C2×C411yescharacteristic
59C17×#95—22no—
60C17×#95D412yes—
61#10587503—22no—
62C22⋊#3622—21no—
63C22⋊#3622—21no—
64#10587480—22no—
65#10587478—22no—
66#10587476—42no—
67#10587476—42no—
68C4⋊#3622—21no—
69C4⋊#3622—21no—
70C4⋊#3622—21no—
71C4⋊#3622C2×C412yes—
72C4×#3622—21no—
73C4×#3622—21no—
74C24×C17—21no—
75C2×D4×C17—21no—
76C2×D4×C17—42no—
77C2×D4×C17C22⋊C412yes—
78C22×C4×C17—21no—
79C22×C4×C17—21no—
80C22×C4×C17—21no—
81C22×C4×C17—22no—
82C22×C4×C17—22no—
83C22×C4×C17—22no—
84C22×C4×C17—41no—
85C22×C4×C17—41no—
86C22×C4×C17—41no—
87C22×C4×C17—41no—
88C22×C4×C17C2×D411yescharacteristic
89C22×C4×C17C2×D411yescharacteristic
90C22×C4×C17C2×D411yescharacteristic
91C2×C22⋊D17—41no—
92#63252—41no—
93#63252—41no—
94#63252—41no—
95#63252—41no—
96C2×D68—21no—
97C2×D68—21no—
98C2×D68—21no—
99C2×C4×D17—41no—
100C2×Dic34—21no—
101C4⋊C4×C17—21no—
102C4⋊C4×C17—42no—
103C4⋊C4×C17—44no—
104C4⋊C4×C17C22⋊C412yes—
105C22⋊C4×C17—22no—
106C22⋊C4×C17—22no—
107C22⋊C4×C17—44no—
108C22⋊C4×C17—44no—
109C42×C17—21no—
110C42×C17—21no—
111C42×C17—21no—
112C42×C17—42no—
113C42×C17—42no—
114C42×C17—41no—
115C42×C17—41no—
116C42×C17D812yescomplemented
117C42×C17C22×C411yescharacteristic
118C42×C17C2×D411yescharacteristic
119C42×C17#4111yescharacteristic
120C42×C17#4111yescharacteristic
121#63228—44no—
122#63227—41no—
123#63227—41no—
124C4⋊Dic17—21no—
125C4⋊Dic17—21no—
126#63223—41no—
127#63223—41no—
128C2×#3622—22no—
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#12011yescharacteristic
138C22×D17—41no—
139D4×C17—22no—
140D4×C17—22no—
141D4×C17—42no—
142C2×C4×C17—22no—
143C2×C4×C17—22no—
144C2×C4×C17—22no—
145C2×C4×C17—22no—
146C2×C4×C17—41no—
147C2×C4×C17—41no—
148C2×C4×C17—41no—
149C2×C4×C17—41no—
150C2×C4×C17—41no—
151C2×C4×C17—41no—
152C2×C4×C17—41no—
153C2×C4×C17—42no—
154C2×C4×C17—42no—
155C2×C4×C17—42no—
156C2×C4×C17—42no—
157C2×C4×C17—42no—
158C2×C4×C17—42no—
159C2×C4×C17—42no—
160C2×C4×C17C2×C22⋊C411yescharacteristic
161C2×C4×C17C4×D411yescharacteristic
162C2×C4×C17C4×D411yescharacteristic
163C2×C4×C17#12111yescharacteristic
164C2×C4×C17#12111yescharacteristic
165C2×C4×C17#12311yescharacteristic
166C2×C4×C17C2×D811yescharacteristic
167C2×C4×C17#13511yescharacteristic
168C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
169C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
170C22⋊D17—81no—
171C22⋊D17—81no—
172C2×Dic17—41no—
173D68—42no—
174D68—41no—
175D68—41no—
176D68—81no—
177C4×D17—81no—
178C4×D17—81no—
179Dic34—41no—
180Dic34—41no—
181#3622—42no—
182#3622—81no—
183#3622—81no—
184#3100—171no—
185#2293D1711yescharacteristic, complemented
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#38611yescharacteristic
199C22×C17#44711yescharacteristic
200C22×C17#44911yescharacteristic
201C2×D17—81no—
202C2×D17—81no—
203C2×D17—81no—
204C4×C17—21no—
205C4×C17—21no—
206C4×C17—21no—
207C4×C17—42no—
208C4×C17—42no—
209C4×C17—42no—
210C4×C17—82no—
211C4×C17#42112yes—
212C4×C17C4×D811yescharacteristic
213C4×C17#44211yescharacteristic
214C4×C17#45911yescharacteristic
215C4×C17#46311yescharacteristic
216Dic17—81no—
217#532—171no—
218C2×C4×D4C2×D1711yescharacteristic
219C2×#117—21no—
220#480—341no—
221#474—172no—
222#459—172no—
223#447—341no—
224C4×D8—342no—
225#431—171no—
226#431—171no—
227#423—171no—
228C2×#104—341no—
229C2×#104—341no—
230#390—21no—
231#389—21no—
232#388—22no—
233C4×C4⋊C4C2×D1711yescharacteristic
234C4×C22⋊C4—21no—
235C43C2×D1711yescharacteristic
236#327—342no—
237D4⋊C8—342no—
238C2×C17—21no—
239C2×C17—21no—
240C2×C17—82no—
241C2×C17#187911yescharacteristic
242C2×C17#189711yescharacteristic
243C2×C17#190611yescharacteristic
244D17—161no—
245C2×#41—341no—
246C22×D4C22⋊D1711yescharacteristic
247C23×C4—21no—
248C2×D8—342no—
249C2×M16—341no—
250C4⋊Q8—171no—
251#127—171no—
252#124—341no—
253#121—341no—
254#121—341no—
255C4×D4—21no—
256C4×D4—21no—
257C4×D4—341no—
258C4×D4—341no—
259C4×D4C4×D1712yes—
260C4×D4D6812yes—
261#117—42no—
262#117—42no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4C22⋊D1711yescharacteristic
266C2×C22⋊C4—21no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42C22⋊D1711yescharacteristic
273C2×C42C22⋊D1711yescharacteristic
274C2×C42C22×D1711yescharacteristic
275#107—342no—
276C4⋊C8—172no—
277C4⋊C8—341no—
278C4⋊C8—341no—
279C4⋊C8—341no—
280#104—682no—
281#104—682no—
282#102—342no—
283#102—342no—
284C22⋊C8—341no—
285C22⋊C8—341no—
286C4×C8—341no—
287C4×C8—341no—
288#95—22no—
289#95C22⋊D1712yes—
290C17#923711yescharacteristic, complemented, Sylow
291C24—21no—
292#41—681no—
293#41—681no—
294#41—681no—
295#41—681no—
296C2×Q8—341no—
297C2×D4—21no—
298C2×D4—42no—
299C2×D4—341no—
300C2×D4—341no—
301C2×D4—341no—
302C2×D4—681no—
303C2×D4#6322712yes—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—681no—
315C22×C4C2×C22⋊D1711yescharacteristic
316C22×C4C2×C22⋊D1711yescharacteristic
317C22×C4C2×C22⋊D1711yescharacteristic
318D8—684no—
319M16—681no—
320M16—681no—
321C2×C8—342no—
322C2×C8—681no—
323C2×C8—681no—
324C4⋊C4—21no—
325C4⋊C4—42no—
326C4⋊C4—44no—
327C4⋊C4—341no—
328C4⋊C4—341no—
329C4⋊C4#6322712yes—
330C22⋊C4—22no—
331C22⋊C4—22no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—681no—
335C22⋊C4—681no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—42no—
340C42—42no—
341C42—41no—
342C42—41no—
343C42#6322812yescomplemented
344C42C2×C4×D1711yescharacteristic
345C42C2×D6811yescharacteristic
346C42#6325211yescharacteristic
347C42#6325211yescharacteristic
348C23—22no—
349C23—42no—
350C23—42no—
351C23—41no—
352C23—41no—
353C23—681no—
354C23#1058763911yescharacteristic
355Q8—681no—
356Q8—681no—
357D4—22no—
358D4—22no—
359D4—42no—
360D4—682no—
361D4—681no—
362D4—681no—
363D4—1361no—
364D4—1361no—
365D4—1361no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—681no—
385C2×C4—1361no—
386C2×C4—1361no—
387C2×C4C2×#6322711yescharacteristic
388C2×C4C4×C22⋊D1711yescharacteristic
389C2×C4C4×C22⋊D1711yescharacteristic
390C2×C4#1058761511yescharacteristic
391C2×C4#1058761611yescharacteristic
392C2×C4#1058761611yescharacteristic
393C2×C4C2×#6322811yescharacteristic
394C2×C4#1058761811yescharacteristic
395C2×C4#1058763511yescharacteristic
396C2×C4#1058763611yescharacteristic
397C8—682no—
398C8—1361no—
399C8—1361no—
400C22—21no—
401C22—21no—
402C22—42no—
403C22—42no—
404C22—42no—
405C22—82no—
406C22—82no—
407C22—1361no—
408C22—1361no—
409C22—1361no—
410C22#1176268811yescharacteristic
411C22#1176294311yescharacteristic
412C22#1176295411yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—42no—
417C4—42no—
418C4—42no—
419C4—82no—
420C4—1361no—
421C4#1176282712yes—
422C4C4×#6322811yescharacteristic
423C4#1176291911yescharacteristic
424C4#1176300411yescharacteristic
425C4#1176302411yescharacteristic
426C2—21no—
427C2—21no—
428C2—82no—
429C2—2721no—
430C2#1452913211yescharacteristic
431C2#1452923211yescharacteristic
432C2#1452928511yescharacteristic
subgroupG/H
centerC2×C4#10587616
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#2293C2
socleC22×C17#386
prime radical#14538784C2
cosocleC2×C42×C17C23
hypercenter#2293D17
hypocenterC17#9237
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2293

Complements

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

NH ≅ G/Ntype
2. #14529612C2semidirect
3. #14527278C2semidirect
4. #14527209C2semidirect
5. #14526687C2semidirect
6. #14526617C2semidirect
7. C17×#2293C2semidirect
10. #11763089C4semidirect
11. #11763004C4semidirect
116. C42×C17D8semidirect
185. #2293D17semidirect
290. C17#9237semidirect
343. C42#63228semidirect

Maximal subgroups

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

subgroupindex
#145387842
#145296122
#145272782
#145272092
#145266872
#145266172
C17×#22932
#923717