Finite Group Databasebeta

← #16930624

Subgroups — #16930624

Order 4352.

512 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14538815C211yesmaximal, characteristic
2#14529612C211yesmaximal, characteristic
3#14527278C211yesmaximal, characteristic
4#14527213C211yesmaximal, characteristic, complemented
5#14526687C211yesmaximal, characteristic, complemented
6#14526617C211yesmaximal, characteristic, complemented
7C17×#2295C211yesmaximal, characteristic, complemented
8#11763359C2211yescharacteristic
9#11763127—21no—
10#11763094C412yescomplemented
11#11763009C412yescomplemented
12#11762948—21no—
13C4×#63230—22no—
14#11762862C2211yescharacteristic
15#11762862C2211yescharacteristic
16#11762833C2211yescharacteristic
17C2×#10587476—21no—
18C2×#10587476—21no—
19#11762542—22no—
20Q8⋊#3622—22no—
21C2×C4×Q8×C17C2211yescharacteristic
22C17×#391—21no—
23C17×#389—21no—
24C17×#387—22no—
25C4×C4⋊C4×C17—21no—
26C4×C4⋊C4×C17C2211yescharacteristic
27C17×#376—21no—
28C43×C17C2211yescharacteristic
29C22×Q8×C17D411yescharacteristic
30C2×#63252—21no—
31C4×Q8×C17—21no—
32C4×Q8×C17—21no—
33C4×Q8×C17C2×C412yes—
34C4×Q8×C17D412yes—
35C2×C4⋊C4×C17—21no—
36C2×C4⋊C4×C17—22no—
37C2×C4⋊C4×C17—21no—
38C2×C4⋊C4×C17D411yescharacteristic
39C2×C42×C17—21no—
40C2×C42×C17—21no—
41C2×C42×C17—21no—
42C2×C42×C17—21no—
43C2×C42×C17D411yescharacteristic
44C2×C42×C17D411yescharacteristic
45C2×C42×C17C2311yescharacteristic
46C2×#63230—22no—
47#10587616—21no—
48#10587616—21no—
49C2×#63223—21no—
50#10587561—21no—
51#10587560C2×C411yescharacteristic
52C4×D68—21no—
53C4×D68—21no—
54#10587555C2×C411yescharacteristic
55C17×#95—22no—
56C17×#95—22no—
57C17×#95—22no—
58C17×#95D412yes—
59#10587506—22no—
60C22⋊#3622—21no—
61C22⋊#3622—21no—
62#10587480—22no—
63#10587479—22no—
64#10587476—42no—
65#10587476—42no—
66C4⋊#3622—21no—
67C4⋊#3622—21no—
68C4⋊#3622—21no—
69C4⋊#3622C2×C412yes—
70C4×#3622—21no—
71C4×#3622—21no—
72C2×Q8×C17—21no—
73C2×Q8×C17—42no—
74C2×Q8×C17C22⋊C412yes—
75C22×C4×C17—21no—
76C22×C4×C17—21no—
77C22×C4×C17—21no—
78C22×C4×C17—21no—
79C22×C4×C17—22no—
80C22×C4×C17C2×D411yescharacteristic
81C22×C4×C17C2×D411yescharacteristic
82C22×C4×C17C2×D411yescharacteristic
83C2×C22⋊D17—41no—
84#63252—41no—
85#63252—41no—
86#63252—41no—
87#63252—41no—
88C2×D68—21no—
89C2×D68—21no—
90C2×D68—21no—
91C2×C4×D17—41no—
92C2×Dic34—21no—
93C4⋊C4×C17—22no—
94C4⋊C4×C17—22no—
95C4⋊C4×C17—21no—
96C4⋊C4×C17—42no—
97C4⋊C4×C17—44no—
98C4⋊C4×C17C22⋊C412yes—
99C42×C17—22no—
100C42×C17—22no—
101C42×C17—21no—
102C42×C17—21no—
103C42×C17—21no—
104C42×C17—42no—
105C42×C17—42no—
106C42×C17—41no—
107C42×C17—41no—
108C42×C17SD1612yescomplemented
109C42×C17C22×C411yescharacteristic
110C42×C17C2×D411yescharacteristic
111C42×C17#4111yescharacteristic
112C42×C17#4111yescharacteristic
113#63230—44no—
114#63227—41no—
115#63227—41no—
116C4⋊Dic17—21no—
117C4⋊Dic17—21no—
118#63223—41no—
119#63223—41no—
120C2×#3622—22no—
121C2×#3622—41no—
122C2×#3622—41no—
123#9238—171nomaximal, Sylow
124C23×C17#12011yescharacteristic
125C22×D17—41no—
126Q8×C17—22no—
127Q8×C17—22no—
128Q8×C17—42no—
129C2×C4×C17—22no—
130C2×C4×C17—22no—
131C2×C4×C17—22no—
132C2×C4×C17—22no—
133C2×C4×C17—22no—
134C2×C4×C17—41no—
135C2×C4×C17—41no—
136C2×C4×C17—41no—
137C2×C4×C17—41no—
138C2×C4×C17—42no—
139C2×C4×C17—42no—
140C2×C4×C17—41no—
141C2×C4×C17—42no—
142C2×C4×C17—42no—
143C2×C4×C17—42no—
144C2×C4×C17C2×C22⋊C411yescharacteristic
145C2×C4×C17C4×D411yescharacteristic
146C2×C4×C17C4×D411yescharacteristic
147C2×C4×C17#12111yescharacteristic
148C2×C4×C17#12111yescharacteristic
149C2×C4×C17#12311yescharacteristic
150C2×C4×C17C2×SD1611yescharacteristic
151C2×C4×C17#13511yescharacteristic
152C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C17#13711yescharacteristic
154C22⋊D17—81no—
155C22⋊D17—81no—
156C2×Dic17—41no—
157D68—42no—
158D68—41no—
159D68—41no—
160D68—81no—
161C4×D17—81no—
162C4×D17—81no—
163Dic34—41no—
164Dic34—41no—
165#3622—42no—
166#3622—81no—
167#3622—81no—
168#3103—171no—
169#2295D1711yescharacteristic, complemented
170#1962—171no—
171#1502—171no—
172#1487—171no—
173#1303—171no—
174#1276—171no—
175C22×C17—21no—
176C22×C17—21no—
177C22×C17#38611yescharacteristic
178C22×C17#44811yescharacteristic
179C22×C17#44911yescharacteristic
180C2×D17—81no—
181C2×D17—81no—
182C2×D17—81no—
183C4×C17—21no—
184C4×C17—21no—
185C4×C17—21no—
186C4×C17—42no—
187C4×C17—42no—
188C4×C17—42no—
189C4×C17—42no—
190C4×C17—82no—
191C4×C17#42112yes—
192C4×C17C4×SD1611yescharacteristic
193C4×C17#44011yescharacteristic
194C4×C17#46011yescharacteristic
195C4×C17#46411yescharacteristic
196Dic17—81no—
197#532—171no—
198C2×C4×Q8C2×D1711yescharacteristic
199#481—341no—
200#475—172no—
201#460—172no—
202#448—341no—
203C4×SD16—342no—
204#431—171no—
205#431—171no—
206#423—171no—
207C2×#104—341no—
208C2×#104—341no—
209#391—21no—
210#389—21no—
211#387—22no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4C2×D1711yescharacteristic
214#376—21no—
215C43C2×D1711yescharacteristic
216#327—342no—
217Q8⋊C8—342no—
218C2×C17—21no—
219C2×C17—21no—
220C2×C17#188011yescharacteristic
221C2×C17#189711yescharacteristic
222C2×C17#190611yescharacteristic
223D17—161no—
224C2×#41—341no—
225C22×Q8C22⋊D1711yescharacteristic
226C2×SD16—342no—
227C2×M16—341no—
228C4⋊Q8—171no—
229#127—171no—
230#124—341no—
231#121—341no—
232#121—341no—
233C4×Q8—21no—
234C4×Q8—21no—
235C4×Q8C4×D1712yes—
236C4×Q8D6812yes—
237C4×D4—341no—
238C4×D4—341no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4C22⋊D1711yescharacteristic
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42C22⋊D1711yescharacteristic
248C2×C42C22⋊D1711yescharacteristic
249C2×C42C22×D1711yescharacteristic
250#106—342no—
251C4⋊C8—172no—
252C4⋊C8—341no—
253C4⋊C8—341no—
254C4⋊C8—341no—
255#104—682no—
256#104—682no—
257#103—342no—
258#102—342no—
259C22⋊C8—341no—
260C22⋊C8—341no—
261C4×C8—341no—
262C4×C8—341no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95C22⋊D1712yes—
267C17#923811yescharacteristic, complemented, Sylow
268#41—681no—
269#41—681no—
270#41—681no—
271#41—681no—
272C2×Q8—21no—
273C2×Q8—42no—
274C2×Q8—341no—
275C2×Q8#6322712yes—
276C2×D4—341no—
277C2×D4—341no—
278C2×D4—341no—
279C2×D4—681no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—681no—
286C22×C4C2×C22⋊D1711yescharacteristic
287C22×C4C2×C22⋊D1711yescharacteristic
288C22×C4C2×C22⋊D1711yescharacteristic
289SD16—684no—
290M16—681no—
291M16—681no—
292C2×C8—342no—
293C2×C8—681no—
294C2×C8—681no—
295C4⋊C4—22no—
296C4⋊C4—21no—
297C4⋊C4—22no—
298C4⋊C4—42no—
299C4⋊C4—44no—
300C4⋊C4—341no—
301C4⋊C4—341no—
302C4⋊C4#6322712yes—
303C22⋊C4—681no—
304C22⋊C4—681no—
305C42—22no—
306C42—22no—
307C42—21no—
308C42—21no—
309C42—21no—
310C42—42no—
311C42—42no—
312C42—41no—
313C42—41no—
314C42#6323012yescomplemented
315C42C2×C4×D1711yescharacteristic
316C42C2×D6811yescharacteristic
317C42#6325211yescharacteristic
318C42#6325211yescharacteristic
319C23—681no—
320C23#1058763911yescharacteristic
321Q8—22no—
322Q8—22no—
323Q8—42no—
324Q8—681no—
325Q8—681no—
326D4—682no—
327D4—681no—
328D4—681no—
329D4—1361no—
330D4—1361no—
331D4—1361no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—681no—
348C2×C4—1361no—
349C2×C4—1361no—
350C2×C4C2×#6322711yescharacteristic
351C2×C4C4×C22⋊D1711yescharacteristic
352C2×C4C4×C22⋊D1711yescharacteristic
353C2×C4#1058761511yescharacteristic
354C2×C4#1058761611yescharacteristic
355C2×C4#1058761611yescharacteristic
356C2×C4C2×#6323011yescharacteristic
357C2×C4#1058762811yescharacteristic
358C2×C4#1058763511yescharacteristic
359C2×C4#1058763611yescharacteristic
360C8—682no—
361C8—1361no—
362C8—1361no—
363C22—21no—
364C22—21no—
365C22—1361no—
366C22—1361no—
367C22—1361no—
368C22#1176268811yescharacteristic
369C22#1176294811yescharacteristic
370C22#1176295411yescharacteristic
371C4—21no—
372C4—21no—
373C4—21no—
374C4—42no—
375C4—42no—
376C4—42no—
377C4—42no—
378C4—82no—
379C4—1361no—
380C4#1176282712yes—
381C4C4×#6323011yescharacteristic
382C4#1176290711yescharacteristic
383C4#1176300911yescharacteristic
384C4#1176303111yescharacteristic
385C2—21no—
386C2—21no—
387C2—2721no—
388C2#1452913911yescharacteristic
389C2#1452923211yescharacteristic
390C2#1452928511yescharacteristic
subgroupG/H
centerC2×C4#10587616
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#2295C2
socleC22×C17#386
prime radical#11763359C22
cosocleC2×C42×C17C23
hypercenter#2295D17
hypocenterC17#9238
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2295

Complements

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

NH ≅ G/Ntype
4. #14527213C2semidirect
5. #14526687C2semidirect
6. #14526617C2semidirect
7. C17×#2295C2semidirect
10. #11763094C4semidirect
11. #11763009C4semidirect
108. C42×C17SD16semidirect
169. #2295D17semidirect
267. C17#9238semidirect
314. C42#63230semidirect

Maximal subgroups

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

subgroupindex
#145388152
#145296122
#145272782
#145272132
#145266872
#145266172
C17×#22952
#923817