Finite Group Databasebeta

← #16951387

Subgroups — #16951387

Order 4352.

526 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#10587471C211yesmaximal, characteristic
2C2×C4×Dic68C211yesmaximal, characteristic, complemented
3C2×#11762614C211yesmaximal, characteristic, complemented
4C2×#11762611C211yesmaximal, characteristic
5#14529650C211yesmaximal, characteristic
6#14526876C211yesmaximal, characteristic, complemented
7C17×#1374C211yesmaximal, characteristic, complemented
8C22×Dic68C2211yescharacteristic
9C22×D136C2211yescharacteristic
10C2×C4×Dic34—21no—
11C2×#10587471—22no—
12C2×#10587471—22no—
13C2×#10587471C414yescomplemented
14C4×Dic68—22no—
15C4×Dic68—22no—
16C2×#10587489C2211yescharacteristic
17C2×#10587489C2211yescharacteristic
18C2×#10587487—21no—
19C2×#10587492—21no—
20#11762711—21no—
21#11762614—22no—
22#11762611—22no—
23C22×C16×C17C2211yescharacteristic
24C22×C16×C17C2211yescharacteristic
25C2×C4×C8×C17C2211yescharacteristic
26C22×D68—21no—
27C22×Dic34—21no—
28C22×C8×C17D411yescharacteristic
29C22×C8×C17C2311yescharacteristic
30C2×C42×C17D411yescharacteristic
31C2×#63227—41no—
32C2×#63227—41no—
33C2×C4⋊Dic17—21no—
34C2×C4⋊Dic17—21no—
35C2×#63225—41no—
36C2×C4×Dic17—41no—
37C2×Dic68—21no—
38C2×Dic68—21no—
39C2×Dic68—21no—
40C2×Dic68C2×C412yes—
41C2×Dic68D412yes—
42C2×Dic68D412yes—
43C2×D136—21no—
44C2×D136—21no—
45C2×D136—21no—
46C2×D136—21no—
47C2×D136—21no—
48C2×D136C2×C412yes—
49C4×Dic34—24no—
50C4×Dic34—42no—
51C2×C16×C17—21no—
52C2×C16×C17—21no—
53C2×C16×C17—21no—
54C2×C16×C17—21no—
55C2×C16×C17—21no—
56C2×C16×C17C2×C412yes—
57C4×C8×C17—21no—
58C4×C8×C17—21no—
59#10587492—42no—
60#10587489—21no—
61#10587489—21no—
62#10587489—21no—
63#10587489D412yes—
64#10587487—24no—
65#10587471—28no—
66C23×D17—41no—
67C22×C4×C17D811yescharacteristic
68C22×C4×C17D811yescharacteristic
69C22×C4×C17C2×D411yescharacteristic
70C22×Dic17—41no—
71C22×Dic17—41no—
72C22×Dic17—41no—
73C2×D68—22no—
74C2×D68—41no—
75C2×D68—41no—
76C2×D68—41no—
77C2×D68—41no—
78C2×D68—41no—
79C2×Dic34—22no—
80C2×Dic34—22no—
81C2×Dic34—22no—
82C2×Dic34—41no—
83C2×Dic34—41no—
84C2×Dic34—41no—
85C2×C8×C17—21no—
86C2×C8×C17—21no—
87C2×C8×C17—21no—
88C2×C8×C17C22×C411yescharacteristic
89C2×C8×C17C2×D411yescharacteristic
90C2×C8×C17C2×D411yescharacteristic
91C2×C8×C17C2×D411yescharacteristic
92C2×C8×C17#4111yescharacteristic
93C2×C8×C17#4111yescharacteristic
94C42×C17—21no—
95C42×C17—21no—
96#63227—82no—
97#63227—82no—
98C4⋊Dic17—22no—
99C4⋊Dic17—41no—
100C4⋊Dic17—41no—
101C4⋊Dic17—41no—
102#63225—44no—
103C4×Dic17—44no—
104Dic68—22no—
105Dic68—22no—
106Dic68—22no—
107Dic68C22⋊C414yescomplemented
108D136—22no—
109D136—22no—
110D136—41no—
111D136—41no—
112C16×C17—22no—
113C16×C17—41no—
114#13013—171nomaximal, Sylow
115C23×C17C2×D811yescharacteristic
116C22×D17—42no—
117C22×D17—81no—
118C22×D17—81no—
119C22×D17—81no—
120C22×D17—81no—
121C22×D17—81no—
122C22×D17—81no—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17SD3212yescomplemented
129C2×C4×C17C4×D411yescharacteristic
130C2×C4×C17#12011yescharacteristic
131C2×C4×C17#12111yescharacteristic
132C2×C4×C17#13511yescharacteristic
133C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
135C2×Dic17—42no—
136C2×Dic17—42no—
137C2×Dic17—42no—
138C2×Dic17—42no—
139C2×Dic17—81no—
140C2×Dic17—81no—
141C2×Dic17—81no—
142C2×Dic17—81no—
143C2×Dic17—81no—
144D68—42no—
145D68—42no—
146D68—81no—
147D68—81no—
148Dic34—24no—
149Dic34—42no—
150Dic34—42no—
151Dic34—42no—
152C8×C17—41no—
153C8×C17C2×C22⋊C411yescharacteristic
154C8×C17C4×D411yescharacteristic
155C8×C17#12111yescharacteristic
156C8×C17#12311yescharacteristic
157C22×SD32—171no—
158C2×C4×Q16—171no—
159C2×#358—171no—
160C2×#357—171no—
161#1970—171no—
162#1380—171no—
163#1374D1711yescharacteristic, complemented
164C22×C17C4×D811yescharacteristic
165C22×C17#44711yescharacteristic
166C22×C17#45911yescharacteristic
167C22×C17C2×SD3211yescharacteristic
168C22×C17#50811yescharacteristic
169C22×C17#50911yescharacteristic
170C22×C17#51011yescharacteristic
171C2×D17—82no—
172C2×D17—82no—
173C2×D17—82no—
174C2×D17—161no—
175C2×D17—161no—
176C2×D17—161no—
177C2×D17—161no—
178C4×C17—22no—
179C4×C17—41no—
180C4×C17#38611yescharacteristic
181C4×C17#44211yescharacteristic
182C4×C17#44911yescharacteristic
183C4×C17#46311yescharacteristic
184Dic17—44no—
185Dic17—82no—
186Dic17—161no—
187C22×Q16—171no—
188C22×D8—171no—
189C2×C4×Q8—341no—
190C2×SD32—174no—
191C2×SD32—342no—
192C2×SD32—342no—
193C22×C16C2×D1711yescharacteristic
194C22×C16C2×D1711yescharacteristic
195C4×Q16—342no—
196C4×Q16—342no—
197C2×#107—171no—
198C2×#107—171no—
199C2×#103—341no—
200C2×#102—341no—
201C2×C4×C8C2×D1711yescharacteristic
202#390—341no—
203#358—342no—
204#357—342no—
205C2×C17#187911yescharacteristic
206C2×C17C4×SD3211yescharacteristic
207C2×C17#217511yescharacteristic
208C2×C17#218511yescharacteristic
209C2×C17#218611yescharacteristic
210C2×C17#220711yescharacteristic
211C2×C17#221111yescharacteristic
212D17—162no—
213C22×Q8—341no—
214C22×D4—341no—
215C2×Q16—172no—
216C2×Q16—172no—
217C2×Q16—172no—
218C2×Q16—341no—
219C2×Q16—341no—
220C2×Q16—341no—
221C2×D8—172no—
222C2×D8—341no—
223C2×D8—341no—
224C2×D8—341no—
225C2×D8—341no—
226C2×D8—341no—
227C22×C8D6811yescharacteristic
228C22×C8C22×D1711yescharacteristic
229C4×Q8—344no—
230C4×Q8—682no—
231C2×C4⋊C4—341no—
232C2×C4⋊C4—341no—
233C2×C4⋊C4—681no—
234C2×C22⋊C4—681no—
235C2×C22⋊C4—681no—
236C2×C42—681no—
237C2×C42D6811yescharacteristic
238SD32—348no—
239C2×C16—21no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—21no—
243C2×C16—21no—
244C2×C16C4×D1712yes—
245#107—172no—
246#107—341no—
247#107—341no—
248#107—341no—
249#103—344no—
250#102—682no—
251C4×C8—21no—
252C4×C8—21no—
253C17#1301311yescharacteristic, complemented, Sylow
254C24—681no—
255C2×Q8—342no—
256C2×Q8—342no—
257C2×Q8—342no—
258C2×Q8—681no—
259C2×Q8—681no—
260C2×Q8—681no—
261C2×D4—342no—
262C2×D4—681no—
263C2×D4—681no—
264C2×D4—681no—
265C2×D4—681no—
266C2×D4—681no—
267C22×C4—681no—
268C22×C4—681no—
269C22×C4—681no—
270C22×C4D13611yescharacteristic
271C22×C4D13611yescharacteristic
272C22×C4C2×D6811yescharacteristic
273Q16—174no—
274Q16—342no—
275Q16—342no—
276Q16—342no—
277D8—342no—
278D8—342no—
279D8—681no—
280D8—681no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8C2×C4×D1711yescharacteristic
285C2×C8#6325211yescharacteristic
286C2×C8D4×D1711yescharacteristic
287C2×C8D4×D1711yescharacteristic
288C2×C8D4×D1711yescharacteristic
289C2×C8#6325611yescharacteristic
290C4⋊C4—342no—
291C4⋊C4—684no—
292C4⋊C4—681no—
293C4⋊C4—681no—
294C4⋊C4—681no—
295C22⋊C4—1362no—
296C22⋊C4—1362no—
297C42—21no—
298C42—21no—
299C42—684no—
300C16—22no—
301C16—41no—
302C23—682no—
303C23—1361no—
304C23—1361no—
305C23—1361no—
306C23—1361no—
307C23—1361no—
308C23—1361no—
309C23C2×D13611yescharacteristic
310Q8—344no—
311Q8—682no—
312Q8—682no—
313Q8—682no—
314D4—682no—
315D4—682no—
316D4—1361no—
317D4—1361no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—682no—
324C2×C4—682no—
325C2×C4—682no—
326C2×C4—682no—
327C2×C4—1361no—
328C2×C4—1361no—
329C2×C4—1361no—
330C2×C4—1361no—
331C2×C4—1361no—
332C2×C4#1058747112yescomplemented
333C2×C4C4×D6811yescharacteristic
334C2×C4#1058756811yescharacteristic
335C2×C4#1058758111yescharacteristic
336C2×C4#1058759011yescharacteristic
337C2×C4#1058759411yescharacteristic
338C2×C4#1058759411yescharacteristic
339C8—41no—
340C8C22⋊C4×D1711yescharacteristic
341C8#1058757011yescharacteristic
342C8#1058757911yescharacteristic
343C8#1058758011yescharacteristic
344C22—2721no—
345C22—2721no—
346C22—2721no—
347C22—2721no—
348C22—1362no—
349C22—1362no—
350C22—1362no—
351C22C4×D13611yescharacteristic
352C22#1176294511yescharacteristic
353C22#1176300511yescharacteristic
354C22C2×#1058747111yescharacteristic
355C22#1176324111yescharacteristic
356C22#1176324711yescharacteristic
357C22#1176325211yescharacteristic
358C4—22no—
359C4—41no—
360C4—2721no—
361C4—684no—
362C4—1362no—
363C4#1176268711yescharacteristic
364C4#1176292211yescharacteristic
365C4#1176295611yescharacteristic
366C4#1176302511yescharacteristic
367C2—2722no—
368C2#1452913111yescharacteristic
369C2C4×#1058747111yescharacteristic
370C2#1453065011yescharacteristic
371C2#1453070311yescharacteristic
372C2#1453070911yescharacteristic
373C2#1453082511yescharacteristic
374C2#1453084711yescharacteristic
subgroupG/H
centerC23C2×D136
commutatorC2×C8×C17C22×C4
FrattiniC22×C8C22×D17
FittingC17×#1374C2
socleC23×C17C2×D8
prime radicalC22×D136C22
cosocleC22×C8×C17C23
hypercenter#1374D17
hypocenterC17#13013
LCSG ▷ C2×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ #1374

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic68C2semidirect
3. C2×#11762614C2semidirect
6. #14526876C2semidirect
7. C17×#1374C2semidirect
13. C2×#10587471C4semidirect
107. Dic68C22⋊C4semidirect
128. C2×C4×C17SD32semidirect
163. #1374D17semidirect
253. C17#13013semidirect
332. C2×C4#10587471semidirect

Maximal subgroups

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

subgroupindex
C22×#105874712
C2×C4×Dic682
C2×#117626142
C2×#117626112
#145296502
#145268762
C17×#13742
#1301317