Finite Group Databasebeta

← #16935553

Subgroups — #16935553

Order 4352.

654 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11763157C211yesmaximal, characteristic
2#14533783C211yesmaximal, characteristic
3#14529708C211yesmaximal, characteristic, complemented
4C2×#11762552C211yesmaximal, characteristic
5C2×Q8⋊#3622C211yesmaximal, characteristic, complemented
6#14526623C211yesmaximal, characteristic, complemented
7C2×C17×#475C211yesmaximal, characteristic, complemented
8C2×#10587560C2211yescharacteristic
9C2×#10587633—21no—
10#11763157—22no—
11#11763157—22no—
12C2×C4⋊#3622C2211yescharacteristic
13C2×#10587506—22no—
14C2×#10587480—22no—
15C2×C4×#3622C2211yescharacteristic
16#11762726—22no—
17#11762711—22no—
18#11762710—21no—
19C4×C4⋊Dic17C2211yescharacteristic
20#11762552—22no—
21Q8⋊#3622—22no—
22C2×C17×C4⋊Q8C2211yescharacteristic
23C2×C4×Q8×C17C2211yescharacteristic
24C17×#475—22no—
25C17×#475—22no—
26C2×C17×#106—21no—
27C2×C17×C4⋊C8C2211yescharacteristic
28C2×C17×#103—21no—
29C22×Q8×C17—21no—
30C22×Q8×C17D411yescharacteristic
31C22×D68—21no—
32C22×D68—21no—
33C22×D68D412yes—
34C22×C8×C17—21no—
35C17×C4⋊Q8—21no—
36C17×C4⋊Q8—21no—
37C17×C4⋊Q8—21no—
38C17×C4⋊Q8D412yes—
39C4×Q8×C17—21no—
40C4×Q8×C17—21no—
41C4×Q8×C17—21no—
42C4×Q8×C17—21no—
43C2×C4⋊C4×C17—21no—
44C2×C4⋊C4×C17—21no—
45C2×C4⋊C4×C17—21no—
46C2×C4⋊C4×C17D411yescharacteristic
47C2×C42×C17—21no—
48C2×C42×C17C2311yescharacteristic
49#10587633—42no—
50#10587633—42no—
51C2×#63227—24no—
52C2×#63227—24no—
53C2×C4⋊Dic17D412yes—
54C2×C4×Dic17—21no—
55C22×#3622—21no—
56C22×#3622—21no—
57#10587560—21no—
58#10587560—21no—
59#10587560—21no—
60#10587560—21no—
61C17×#106—24no—
62C17×C4⋊C8—21no—
63C17×C4⋊C8D412yes—
64C17×#103—24no—
65#10587506—44no—
66#10587502—21no—
67#10587480—28no—
68C4⋊#3622—21no—
69C4⋊#3622—21no—
70C4×#3622—21no—
71C4×#3622D412yes—
72C23×D17—22no—
73C2×Q8×C17—21no—
74C2×Q8×C17—21no—
75C2×Q8×C17—21no—
76C2×Q8×C17—22no—
77C2×Q8×C17—21no—
78C2×Q8×C17—21no—
79C2×Q8×C17—41no—
80C2×Q8×C17—42no—
81C2×Q8×C17—42no—
82C2×Q8×C17SD1612yes—
83C22×C4×C17—21no—
84C22×C4×C17—21no—
85C22×C4×C17—41no—
86C22×C4×C17C2×D411yescharacteristic
87C22×C4×C17C2×D411yescharacteristic
88C22×C4×C17C2×D411yescharacteristic
89C22×Dic17—22no—
90C2×D68—22no—
91C2×D68—22no—
92C2×D68—44no—
93C2×D68—41no—
94C2×D68—41no—
95C2×D68—41no—
96C2×D68—41no—
97C2×D68—41no—
98C2×D68—42no—
99C2×D68—41no—
100C2×D68—42no—
101C2×D68—41no—
102C2×D68—41no—
103C2×C8×C17—22no—
104C2×C8×C17—41no—
105C2×C8×C17—41no—
106C4⋊C4×C17—22no—
107C4⋊C4×C17—22no—
108C4⋊C4×C17—21no—
109C4⋊C4×C17—21no—
110C4⋊C4×C17—41no—
111C4⋊C4×C17—41no—
112C4⋊C4×C17—41no—
113C4⋊C4×C17—41no—
114C42×C17—41no—
115C42×C17—41no—
116C42×C17C2×D411yescharacteristic
117C42×C17C2×D411yescharacteristic
118C42×C17C2×D411yescharacteristic
119C42×C17#4111yescharacteristic
120#63227—48no—
121#63227—84no—
122C4⋊Dic17—22no—
123C4⋊Dic17—22no—
124C4×Dic17—41no—
125C4×Dic17—41no—
126C2×#3622—22no—
127C2×#3622—42no—
128C2×#3622—41no—
129C2×#3622—42no—
130#10071—171nomaximal, Sylow
131C23×C17#12011yescharacteristic
132C22×D17—42no—
133C22×D17—44no—
134C22×D17—42no—
135C22×D17—42no—
136C22×D17—44no—
137Q8×C17—22no—
138Q8×C17—22no—
139Q8×C17—41no—
140Q8×C17—41no—
141Q8×C17—44no—
142Q8×C17—81no—
143Q8×C17—81no—
144C2×C4×C17—22no—
145C2×C4×C17—22no—
146C2×C4×C17—41no—
147C2×C4×C17—41no—
148C2×C4×C17—41no—
149C2×C4×C17—42no—
150C2×C4×C17—42no—
151C2×C4×C17—82no—
152C2×C4×C17#12011yescharacteristic
153C2×C4×C17#12011yescharacteristic
154C2×C4×C17#12111yescharacteristic
155C2×C4×C17#12111yescharacteristic
156C2×C4×C17#12111yescharacteristic
157C2×C4×C17#12711yescharacteristic
158C2×C4×C17C2×SD1611yescharacteristic
159C2×C4×C17#13512yes—
160C2×C4×C17#13511yescharacteristic
161C2×C4×C17AGL(1,ℤ/8)12yes—
162C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C17#13711yescharacteristic
164C2×Dic17—42no—
165C2×Dic17—42no—
166C2×Dic17—42no—
167D68—44no—
168D68—82no—
169D68—81no—
170D68—81no—
171D68—82no—
172D68—81no—
173D68—81no—
174D68—81no—
175D68—81no—
176C8×C17—42no—
177#3622—42no—
178#3622—81no—
179C2×#486—171no—
180C2×#475D1711yescharacteristic, complemented
181#2548—171no—
182#1981—171no—
183C2×#331—171no—
184C2×Q8⋊C8—171no—
185#1279—171no—
186C22×C17#39211yescharacteristic
187C22×C17#44912yes—
188C22×C17#45011yescharacteristic
189C22×C17#45211yescharacteristic
190C22×C17#45512yes—
191C2×D17—84no—
192C2×D17—84no—
193C2×D17—82no—
194C2×D17—82no—
195C2×D17—82no—
196C4×C17—21no—
197C4×C17—21no—
198C4×C17—42no—
199C4×C17—81no—
200C4×C17—82no—
201C4×C17#44811yescharacteristic
202C4×C17#44911yescharacteristic
203C4×C17#46312yes—
204C4×C17#46511yescharacteristic
205C4×C17#46811yescharacteristic
206C4×C17#49511yescharacteristic
207C4×C17#49611yescharacteristic
208Dic17—82no—
209C2×C4⋊Q8C2×D1711yescharacteristic
210C2×#127—171no—
211C2×#124—341no—
212C2×C4×Q8C2×D1711yescharacteristic
213#486—342no—
214#486—342no—
215#475—22no—
216#475—22no—
217C2×#106—21no—
218C2×C4⋊C8—171no—
219C2×C4⋊C8C2×D1711yescharacteristic
220C2×#103—21no—
221C2×#103—342no—
222C2×#102—342no—
223C2×C4×C8—171no—
224#394—342no—
225#390—342no—
226#390—341no—
227C4×C4⋊C4—171no—
228#331—342no—
229Q8⋊C8—342no—
230C2×C17#164012yes—
231C2×C17#164212yes—
232C2×C17#200011yescharacteristic
233C2×C17#200411yescharacteristic
234C2×C17#201211yescharacteristic
235D17—162no—
236C22×Q8—21no—
237C22×Q8C22⋊D1711yescharacteristic
238C22×D4—172no—
239C22×D4—341no—
240C22×D4—341no—
241C22×C8—21no—
242C22×C8—341no—
243C22×C8—341no—
244C4⋊Q8—21no—
245C4⋊Q8—21no—
246C4⋊Q8—21no—
247C4⋊Q8C22⋊D1712yes—
248#127—341no—
249#127—341no—
250#127—341no—
251#127—341no—
252#124—682no—
253#124—682no—
254C4×Q8—21no—
255C4×Q8—21no—
256C4×Q8—21no—
257C4×Q8—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—172no—
262C2×C4⋊C4C22⋊D1711yescharacteristic
263C2×C22⋊C4—344no—
264C2×C22⋊C4—344no—
265C2×C42—21no—
266C2×C42—341no—
267C2×C42C22×D1711yescharacteristic
268#106—24no—
269C4⋊C8—21no—
270C4⋊C8—341no—
271C4⋊C8—341no—
272C4⋊C8C22⋊D1712yes—
273#103—24no—
274#103—684no—
275#102—348no—
276C4×C8—172no—
277C4×C8—341no—
278#95—341no—
279C17#1007111yescharacteristic, complemented, Sylow
280C24—342no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—22no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—41no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8#6323012yes—
291C2×D4—342no—
292C2×D4—342no—
293C2×D4—684no—
294C2×D4—682no—
295C2×D4—682no—
296C2×D4—681no—
297C2×D4—681no—
298C2×D4—681no—
299C2×D4—681no—
300C2×D4—681no—
301C2×D4—681no—
302C2×D4—681no—
303C2×D4—681no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—41no—
307C22×C4—342no—
308C22×C4D4×D1711yescharacteristic
309C22×C4D4×D1711yescharacteristic
310C22×C4C2×C22⋊D1711yescharacteristic
311C2×C8—22no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—342no—
315C2×C8—682no—
316C2×C8—682no—
317C2×C8—681no—
318C4⋊C4—22no—
319C4⋊C4—21no—
320C4⋊C4—22no—
321C4⋊C4—21no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—342no—
327C4⋊C4—342no—
328C22⋊C4—688no—
329C22⋊C4—1364no—
330C42—41no—
331C42—41no—
332C42—681no—
333C42—681no—
334C42D4×D1711yescharacteristic
335C42#6325411yescharacteristic
336C42C2×C22⋊D1711yescharacteristic
337C42C2×C22⋊D1711yescharacteristic
338C23—684no—
339C23—682no—
340C23—682no—
341C23—682no—
342C23—684no—
343C23#1058762311yescharacteristic
344Q8—22no—
345Q8—22no—
346Q8—41no—
347Q8—41no—
348Q8—44no—
349Q8—81no—
350Q8—81no—
351D4—684no—
352D4—1362no—
353D4—1361no—
354D4—1361no—
355D4—1362no—
356D4—1361no—
357D4—1361no—
358D4—1361no—
359D4—1361no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—82no—
368C2×C4—682no—
369C2×C4—682no—
370C2×C4—682no—
371C2×C4#1058760012yes—
372C2×C4#1058760212yes—
373C2×C4#1058762311yescharacteristic
374C2×C4#1058762411yescharacteristic
375C2×C4#1058762511yescharacteristic
376C2×C4#1058762511yescharacteristic
377C2×C4#1058762611yescharacteristic
378C2×C4C2×#6323011yescharacteristic
379C2×C4#1058762811yescharacteristic
380C2×C4#1058763511yescharacteristic
381C2×C4#1058763611yescharacteristic
382C2×C4#1058763911yescharacteristic
383C8—42no—
384C8—682no—
385C8—1361no—
386C22—1362no—
387C22—1362no—
388C22—1364no—
389C22—1364no—
390C22—1362no—
391C22#1176271711yescharacteristic
392C22#1176295312yes—
393C22#1176295911yescharacteristic
394C22#1176296911yescharacteristic
395C22#1176298512yes—
396C4—21no—
397C4—21no—
398C4—42no—
399C4—81no—
400C4—82no—
401C4—1362no—
402C4#1176294811yescharacteristic
403C4#1176295411yescharacteristic
404C4#1176302612yes—
405C4#1176304011yescharacteristic
406C4#1176305711yescharacteristic
407C4#1176319011yescharacteristic
408C4#1176319511yescharacteristic
409C2—2722no—
410C2#1452796912yes—
411C2#1452798312yes—
412C2#1452978511yescharacteristic
413C2#1452981011yescharacteristic
414C2#1452985811yescharacteristic
subgroupG/H
centerC23#10587623
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC2×C17×#475C2
socleC23×C17#120
prime radicalC2×#10587560C22
hypercenterC2×#475D17
hypocenterC17#10071
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
3. #14529708C2semidirect
5. C2×Q8⋊#3622C2semidirect
6. #14526623C2semidirect
7. C2×C17×#475C2semidirect
180. C2×#475D17semidirect
279. C17#10071semidirect

Maximal subgroups

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

subgroupindex
C2×#117631572
#145337832
#145297082
C2×#117625522
C2×Q8⋊#36222
#145266232
C2×C17×#4752
#1007117