Finite Group Databasebeta

← #16930596

Subgroups — #16930596

Order 4352.

604 conjugacy classes of subgroups.

The subgroup lattice has 398 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
7C42×Q8×C17C211yesmaximal, 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×C17—21no—
22C2×C4×Q8×C17C2211yescharacteristic
23C4×C4⋊C4×C17—21no—
24C4×C4⋊C4×C17—21no—
25C4×C4⋊C4×C17—22no—
26C4×C4⋊C4×C17C2211yescharacteristic
27C43×C17—21no—
28C43×C17C2211yescharacteristic
29C22×Q8×C17D411yescharacteristic
30C2×#63252—21no—
31C4×Q8×C17—28no—
32C4×Q8×C17—21no—
33C4×Q8×C17—21no—
34C4×Q8×C17C2×C412yes—
35C4×Q8×C17D412yes—
36C2×C4⋊C4×C17—22no—
37C2×C4⋊C4×C17—21no—
38C2×C4⋊C4×C17—21no—
39C2×C4⋊C4×C17D411yescharacteristic
40C2×C42×C17—22no—
41C2×C42×C17—21no—
42C2×C42×C17—21no—
43C2×C42×C17—22no—
44C2×C42×C17—21no—
45C2×C42×C17—21no—
46C2×C42×C17D411yescharacteristic
47C2×C42×C17D411yescharacteristic
48C2×C42×C17C2311yescharacteristic
49C2×#63230—22no—
50#10587616—21no—
51#10587616—21no—
52C2×#63223—21no—
53#10587561—21no—
54#10587560C2×C411yescharacteristic
55C4×D68—21no—
56C4×D68—21no—
57#10587555C2×C411yescharacteristic
58C17×#95—22no—
59C17×#95D412yes—
60#10587506—22no—
61C22⋊#3622—21no—
62C22⋊#3622—21no—
63#10587480—22no—
64#10587479—22no—
65#10587476—42no—
66#10587476—42no—
67C4⋊#3622—21no—
68C4⋊#3622—21no—
69C4⋊#3622—21no—
70C4⋊#3622C2×C412yes—
71C4×#3622—21no—
72C4×#3622—21no—
73C2×Q8×C17—24no—
74C2×Q8×C17—21no—
75C2×Q8×C17C22⋊C412yes—
76C22×C4×C17—21no—
77C22×C4×C17—21no—
78C22×C4×C17—21no—
79C22×C4×C17—21no—
80C22×C4×C17—22no—
81C22×C4×C17C2×D411yescharacteristic
82C22×C4×C17C2×D411yescharacteristic
83C22×C4×C17C2×D411yescharacteristic
84C2×C22⋊D17—41no—
85#63252—41no—
86#63252—41no—
87#63252—41no—
88#63252—41no—
89C2×D68—21no—
90C2×D68—21no—
91C2×D68—21no—
92C2×C4×D17—41no—
93C2×Dic34—21no—
94C4⋊C4×C17—28no—
95C4⋊C4×C17—24no—
96C4⋊C4×C17—22no—
97C4⋊C4×C17—22no—
98C4⋊C4×C17—21no—
99C4⋊C4×C17C22⋊C412yes—
100C42×C17—28no—
101C42×C17—24no—
102C42×C17—22no—
103C42×C17—22no—
104C42×C17—21no—
105C42×C17—21no—
106C42×C17—21no—
107C42×C17—42no—
108C42×C17—44no—
109C42×C17—41no—
110C42×C17—41no—
111C42×C17SD1612yescomplemented
112C42×C17C22×C411yescharacteristic
113C42×C17C2×D411yescharacteristic
114C42×C17#4111yescharacteristic
115C42×C17#4111yescharacteristic
116#63230—44no—
117#63227—41no—
118#63227—41no—
119C4⋊Dic17—21no—
120C4⋊Dic17—21no—
121#63223—41no—
122#63223—41no—
123C2×#3622—22no—
124C2×#3622—41no—
125C2×#3622—41no—
126#9234—171nomaximal, Sylow
127C23×C17#12011yescharacteristic
128C22×D17—41no—
129Q8×C17—24no—
130Q8×C17—22no—
131Q8×C17#10414yescomplemented
132C2×C4×C17—22no—
133C2×C4×C17—24no—
134C2×C4×C17—22no—
135C2×C4×C17—22no—
136C2×C4×C17—22no—
137C2×C4×C17—22no—
138C2×C4×C17—22no—
139C2×C4×C17—24no—
140C2×C4×C17—22no—
141C2×C4×C17—41no—
142C2×C4×C17—41no—
143C2×C4×C17—42no—
144C2×C4×C17—42no—
145C2×C4×C17—41no—
146C2×C4×C17—42no—
147C2×C4×C17C2×C22⋊C411yescharacteristic
148C2×C4×C17C4×D411yescharacteristic
149C2×C4×C17C4×D411yescharacteristic
150C2×C4×C17#12111yescharacteristic
151C2×C4×C17#12111yescharacteristic
152C2×C4×C17#12311yescharacteristic
153C2×C4×C17C2×SD1611yescharacteristic
154C2×C4×C17#13511yescharacteristic
155C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
156C2×C4×C17#13711yescharacteristic
157C22⋊D17—81no—
158C22⋊D17—81no—
159C2×Dic17—41no—
160D68—42no—
161D68—41no—
162D68—41no—
163D68—81no—
164C4×D17—81no—
165C4×D17—81no—
166Dic34—41no—
167Dic34—41no—
168#3622—42no—
169#3622—81no—
170#3622—81no—
171#3103—171no—
172C42×Q8D1711yescharacteristic, complemented
173#1962—171no—
174#1502—171no—
175#1487—171no—
176#1303—171no—
177#1276—171no—
178C22×C17—21no—
179C22×C17—21no—
180C22×C17#38611yescharacteristic
181C22×C17#44811yescharacteristic
182C22×C17#44911yescharacteristic
183C2×D17—81no—
184C2×D17—81no—
185C2×D17—81no—
186C4×C17—24no—
187C4×C17—24no—
188C4×C17—21no—
189C4×C17—21no—
190C4×C17—21no—
191C4×C17—42no—
192C4×C17—44no—
193C4×C17—42no—
194C4×C17C2×#10412yes—
195C4×C17C4×SD1611yescharacteristic
196C4×C17#44011yescharacteristic
197C4×C17#46011yescharacteristic
198C4×C17#46411yescharacteristic
199Dic17—81no—
200#532—171no—
201C2×C4×Q8—21no—
202C2×C4×Q8C2×D1711yescharacteristic
203#481—341no—
204#475—172no—
205#460—172no—
206#448—341no—
207C4×SD16—342no—
208#431—171no—
209#431—171no—
210#423—171no—
211C2×#104—341no—
212C2×#104—341no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4C2×D1711yescharacteristic
217C43—21no—
218C43C2×D1711yescharacteristic
219#327—342no—
220Q8⋊C8—342no—
221C2×C17—21no—
222C2×C17—21no—
223C2×C17#188011yescharacteristic
224C2×C17#189611yescharacteristic
225C2×C17#190611yescharacteristic
226D17—161no—
227C2×#41—341no—
228C22×Q8C22⋊D1711yescharacteristic
229C2×SD16—342no—
230C2×M16—341no—
231C4⋊Q8—171no—
232#127—171no—
233#124—341no—
234#121—341no—
235#121—341no—
236C4×Q8—28no—
237C4×Q8—21no—
238C4×Q8—21no—
239C4×Q8C4×D1712yes—
240C4×Q8D6812yes—
241C4×D4—341no—
242C4×D4—341no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4C22⋊D1711yescharacteristic
247C2×C42—22no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—22no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42C22⋊D1711yescharacteristic
254C2×C42C22⋊D1711yescharacteristic
255C2×C42C22×D1711yescharacteristic
256#106—342no—
257C4⋊C8—172no—
258C4⋊C8—341no—
259C4⋊C8—341no—
260C4⋊C8—341no—
261#104—682no—
262#104—682no—
263#103—342no—
264#102—342no—
265C22⋊C8—341no—
266C22⋊C8—341no—
267C4×C8—341no—
268C4×C8—341no—
269#95—22no—
270#95C22⋊D1712yes—
271C17#923411yescharacteristic, complemented, Sylow
272#41—681no—
273#41—681no—
274#41—681no—
275#41—681no—
276C2×Q8—24no—
277C2×Q8—21no—
278C2×Q8—341no—
279C2×Q8#6322712yes—
280C2×D4—341no—
281C2×D4—341no—
282C2×D4—341no—
283C2×D4—681no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—681no—
290C22×C4C2×C22⋊D1711yescharacteristic
291C22×C4C2×C22⋊D1711yescharacteristic
292C22×C4C2×C22⋊D1711yescharacteristic
293SD16—684no—
294M16—681no—
295M16—681no—
296C2×C8—342no—
297C2×C8—681no—
298C2×C8—681no—
299C4⋊C4—28no—
300C4⋊C4—24no—
301C4⋊C4—22no—
302C4⋊C4—21no—
303C4⋊C4—22no—
304C4⋊C4—341no—
305C4⋊C4—341no—
306C4⋊C4#6322712yes—
307C22⋊C4—681no—
308C22⋊C4—681no—
309C42—28no—
310C42—24no—
311C42—22no—
312C42—22no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—42no—
317C42—44no—
318C42—41no—
319C42—41no—
320C42#6323012yescomplemented
321C42C2×C4×D1711yescharacteristic
322C42C2×D6811yescharacteristic
323C42#6325211yescharacteristic
324C42#6325211yescharacteristic
325C23—681no—
326C23#1058763911yescharacteristic
327Q8—24no—
328Q8—22no—
329Q8—681no—
330Q8—681no—
331Q8#1058747614yescomplemented
332D4—682no—
333D4—681no—
334D4—681no—
335D4—1361no—
336D4—1361no—
337D4—1361no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—681no—
354C2×C4—1361no—
355C2×C4—1361no—
356C2×C4C2×#6322711yescharacteristic
357C2×C4C4×C22⋊D1711yescharacteristic
358C2×C4C4×C22⋊D1711yescharacteristic
359C2×C4#1058761511yescharacteristic
360C2×C4#1058761611yescharacteristic
361C2×C4#1058761611yescharacteristic
362C2×C4C2×#6323011yescharacteristic
363C2×C4#1058762811yescharacteristic
364C2×C4#1058763511yescharacteristic
365C2×C4#1058763611yescharacteristic
366C8—682no—
367C8—1361no—
368C8—1361no—
369C22—21no—
370C22—21no—
371C22—1361no—
372C22—1361no—
373C22—1361no—
374C22#1176268811yescharacteristic
375C22#1176294811yescharacteristic
376C22#1176295411yescharacteristic
377C4—24no—
378C4—21no—
379C4—21no—
380C4—24no—
381C4—21no—
382C4—42no—
383C4—44no—
384C4—42no—
385C4—1361no—
386C4C2×#1058747612yes—
387C4C4×#6323011yescharacteristic
388C4#1176290711yescharacteristic
389C4#1176300911yescharacteristic
390C4#1176303111yescharacteristic
391C2—21no—
392C2—21no—
393C2—2721no—
394C2#1452913911yescharacteristic
395C2#1452922711yescharacteristic
396C2#1452928511yescharacteristic
subgroupG/H
centerC2×C4#10587616
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC42×Q8×C17C2
socleC22×C17#386
prime radical#11763359C22
cosocleC2×C42×C17C23
hypercenterC42×Q8D17
hypocenterC17#9234
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C42×Q8

Complements

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

NH ≅ G/Ntype
4. #14527213C2semidirect
5. #14526687C2semidirect
6. #14526617C2semidirect
7. C42×Q8×C17C2semidirect
10. #11763094C4semidirect
11. #11763009C4semidirect
111. C42×C17SD16semidirect
131. Q8×C17#104semidirect
172. C42×Q8D17semidirect
271. C17#9234semidirect
320. C42#63230semidirect
331. Q8#10587476semidirect

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
C42×Q8×C172
#923417