Finite Group Databasebeta

← #16951388

Subgroups — #16951388

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×#10587499C211yesmaximal, characteristic
2C2×#11762613C211yesmaximal, characteristic, complemented
3C2×#11762609C211yesmaximal, characteristic
4#14529650C211yesmaximal, characteristic
5#14526875C211yesmaximal, characteristic, complemented
6#14526862C211yesmaximal, characteristic, complemented
7C2×C4×Q16×C17C211yesmaximal, characteristic, complemented
8C22×D136C2211yescharacteristic
9C2×#10587499—22no—
10C2×#10587499—22no—
11C2×#10587499C414yescomplemented
12C22×#63214C2211yescharacteristic
13C22×#63214C2211yescharacteristic
14C2×#10587489C2211yescharacteristic
15C2×#10587492—21no—
16#11762711—21no—
17#11762613—22no—
18#11762609—22no—
19C22×Q16×C17C2211yescharacteristic
20C2×C4×Q8×C17—21no—
21C4×Q16×C17—22no—
22C4×Q16×C17—22no—
23C2×C17×#107C2211yescharacteristic
24C2×C17×#103—21no—
25C2×C4×C8×C17C2211yescharacteristic
26C22×Q8×C17—21no—
27C22×D68—21no—
28C2×Q16×C17—21no—
29C2×Q16×C17—21no—
30C2×Q16×C17—21no—
31C2×Q16×C17C2×C412yes—
32C2×Q16×C17D412yes—
33C2×Q16×C17D412yes—
34C22×C8×C17D411yescharacteristic
35C22×C8×C17C2311yescharacteristic
36C4×Q8×C17—24no—
37C4×Q8×C17—42no—
38C2×C4⋊C4×C17—21no—
39C2×C4⋊C4×C17—41no—
40C2×C42×C17—41no—
41C2×C42×C17D411yescharacteristic
42C2×#63227—41no—
43C2×#63227—41no—
44C2×C4⋊Dic17—21no—
45C2×D136—21no—
46C2×D136—21no—
47C2×D136—21no—
48C2×D136—21no—
49C2×D136—21no—
50C2×D136C2×C412yes—
51C17×#107—21no—
52C17×#107D412yes—
53C17×#103—24no—
54C4×C8×C17—21no—
55C4×C8×C17—21no—
56#10587499—28no—
57#10587492—42no—
58#10587489—21no—
59#10587489—21no—
60C2×#63214—21no—
61C2×#63214—21no—
62C2×#63214—21no—
63C2×#63214—21no—
64C2×#63214—21no—
65C2×#63214C2×C412yes—
66C23×D17—41no—
67C2×Q8×C17—22no—
68C2×Q8×C17—22no—
69C2×Q8×C17—22no—
70C2×Q8×C17—41no—
71C2×Q8×C17—41no—
72C2×Q8×C17—41no—
73C22×C4×C17—41no—
74C22×C4×C17—41no—
75C22×C4×C17D811yescharacteristic
76C22×C4×C17D811yescharacteristic
77C22×C4×C17C2×D411yescharacteristic
78C22×Dic17—41no—
79C2×D68—22no—
80C2×D68—41no—
81C2×D68—41no—
82C2×D68—41no—
83C2×D68—41no—
84C2×D68—41no—
85Q16×C17—22no—
86Q16×C17—22no—
87Q16×C17—22no—
88Q16×C17C22⋊C414yescomplemented
89C2×C8×C17—21no—
90C2×C8×C17—21no—
91C2×C8×C17—21no—
92C2×C8×C17C22×C411yescharacteristic
93C2×C8×C17C2×D411yescharacteristic
94C2×C8×C17C2×D411yescharacteristic
95C2×C8×C17C2×D411yescharacteristic
96C2×C8×C17#4111yescharacteristic
97C2×C8×C17#4111yescharacteristic
98C4⋊C4×C17—22no—
99C4⋊C4×C17—44no—
100C4⋊C4×C17—41no—
101C42×C17—21no—
102C42×C17—21no—
103C42×C17—44no—
104#63227—82no—
105#63227—82no—
106C4⋊Dic17—41no—
107C4⋊Dic17—41no—
108D136—22no—
109D136—22no—
110D136—41no—
111D136—41no—
112#63214—22no—
113#63214—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—
123Q8×C17—24no—
124Q8×C17—42no—
125Q8×C17—42no—
126Q8×C17—42no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—42no—
133C2×C4×C17—42no—
134C2×C4×C17—42no—
135C2×C4×C17—42no—
136C2×C4×C17—81no—
137C2×C4×C17—81no—
138C2×C4×C17SD3212yescomplemented
139C2×C4×C17C4×D411yescharacteristic
140C2×C4×C17#12011yescharacteristic
141C2×C4×C17#12111yescharacteristic
142C2×C4×C17#13511yescharacteristic
143C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
145C2×Dic17—81no—
146C2×Dic17—81no—
147C2×Dic17—81no—
148D68—42no—
149D68—42no—
150D68—81no—
151D68—81no—
152C8×C17—41no—
153C8×C17C2×C22⋊C411yescharacteristic
154C8×C17C4×D411yescharacteristic
155C8×C17#12111yescharacteristic
156C8×C17#12311yescharacteristic
157C22×SD32—171no—
158C2×C4×Q16D1711yescharacteristic, complemented
159C2×#358—171no—
160C2×#357—171no—
161#1970—171no—
162#1380—171no—
163#1374—171no—
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—44no—
180C4×C17—41no—
181C4×C17—82no—
182C4×C17#38611yescharacteristic
183C4×C17#44211yescharacteristic
184C4×C17#44911yescharacteristic
185C4×C17#46311yescharacteristic
186Dic17—161no—
187C22×Q16C2×D1711yescharacteristic
188C22×D8—171no—
189C2×C4×Q8—21no—
190C2×SD32—174no—
191C2×SD32—342no—
192C2×SD32—342no—
193C22×C16—171no—
194C22×C16—171no—
195C4×Q16—22no—
196C4×Q16—22no—
197C2×#107—171no—
198C2×#107C2×D1711yescharacteristic
199C2×#103—21no—
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—21no—
214C22×D4—341no—
215C2×Q16—21no—
216C2×Q16—21no—
217C2×Q16—21no—
218C2×Q16C4×D1712yes—
219C2×Q16D6812yes—
220C2×Q16C22⋊D1712yes—
221C2×D8—172no—
222C2×D8—341no—
223C2×D8—341no—
224C2×D8—341no—
225C2×D8—341no—
226C2×D8—341no—
227C22×C8C22⋊D1711yescharacteristic
228C22×C8C22×D1711yescharacteristic
229C4×Q8—24no—
230C4×Q8—42no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—341no—
234C2×C22⋊C4—681no—
235C2×C22⋊C4—681no—
236C2×C42—41no—
237C2×C42C22⋊D1711yescharacteristic
238SD32—348no—
239C2×C16—172no—
240C2×C16—341no—
241C2×C16—341no—
242C2×C16—341no—
243C2×C16—341no—
244C2×C16—341no—
245#107—21no—
246#107—341no—
247#107—341no—
248#107C22⋊D1712yes—
249#103—24no—
250#102—682no—
251C4×C8—21no—
252C4×C8—21no—
253C17#1301311yescharacteristic, complemented, Sylow
254C24—681no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×D4—342no—
262C2×D4—681no—
263C2×D4—681no—
264C2×D4—681no—
265C2×D4—681no—
266C2×D4—681no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—681no—
270C22×C4#6322811yescharacteristic
271C22×C4#6322811yescharacteristic
272C22×C4C2×C22⋊D1711yescharacteristic
273Q16—22no—
274Q16—22no—
275Q16—22no—
276Q16#6322714yescomplemented
277D8—342no—
278D8—342no—
279D8—681no—
280D8—681no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8C2×C4×D1711yescharacteristic
285C2×C8C2×D6811yescharacteristic
286C2×C8#6325211yescharacteristic
287C2×C8#6325211yescharacteristic
288C2×C8C2×C22⋊D1711yescharacteristic
289C2×C8C2×C22⋊D1711yescharacteristic
290C4⋊C4—22no—
291C4⋊C4—44no—
292C4⋊C4—41no—
293C4⋊C4—681no—
294C4⋊C4—681no—
295C22⋊C4—1362no—
296C22⋊C4—1362no—
297C42—21no—
298C42—21no—
299C42—44no—
300C16—342no—
301C16—681no—
302C23—682no—
303C23—1361no—
304C23—1361no—
305C23—1361no—
306C23—1361no—
307C23—1361no—
308C23—1361no—
309C23C2×#6322811yescharacteristic
310Q8—24no—
311Q8—42no—
312Q8—42no—
313Q8—42no—
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—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—1361no—
330C2×C4—1361no—
331C2×C4—1361no—
332C2×C4#1058749912yescomplemented
333C2×C4C4×C22⋊D1711yescharacteristic
334C2×C4#1058761611yescharacteristic
335C2×C4#1058761811yescharacteristic
336C2×C4#1058763511yescharacteristic
337C2×C4#1058763611yescharacteristic
338C2×C4#1058763911yescharacteristic
339C8—41no—
340C8C2×#6322711yescharacteristic
341C8C4×C22⋊D1711yescharacteristic
342C8#1058761511yescharacteristic
343C8#1058761611yescharacteristic
344C22—2721no—
345C22—2721no—
346C22—2721no—
347C22—2721no—
348C22—1362no—
349C22—1362no—
350C22—1362no—
351C22C4×#6322811yescharacteristic
352C22#1176294311yescharacteristic
353C22#1176300411yescharacteristic
354C22C2×#1058749911yescharacteristic
355C22#1176323911yescharacteristic
356C22#1176324411yescharacteristic
357C22#1176325011yescharacteristic
358C4—22no—
359C4—44no—
360C4—41no—
361C4—82no—
362C4—2721no—
363C4#1176268811yescharacteristic
364C4#1176291911yescharacteristic
365C4#1176295411yescharacteristic
366C4#1176302411yescharacteristic
367C2—2722no—
368C2#1452913211yescharacteristic
369C2C4×#1058749911yescharacteristic
370C2#1453064611yescharacteristic
371C2#1453070111yescharacteristic
372C2#1453070711yescharacteristic
373C2#1453082411yescharacteristic
374C2#1453084611yescharacteristic
subgroupG/H
centerC23C2×#63228
commutatorC2×C8×C17C22×C4
FrattiniC22×C8C22×D17
FittingC2×C4×Q16×C17C2
socleC23×C17C2×D8
prime radicalC22×D136C22
cosocleC22×C8×C17C23
hypercenterC2×C4×Q16D17
hypocenterC17#13013
LCSG ▷ C2×C8×C17 ▷ C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×Q16

Complements

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

NH ≅ G/Ntype
2. C2×#11762613C2semidirect
5. #14526875C2semidirect
6. #14526862C2semidirect
7. C2×C4×Q16×C17C2semidirect
11. C2×#10587499C4semidirect
88. Q16×C17C22⋊C4semidirect
138. C2×C4×C17SD32semidirect
158. C2×C4×Q16D17semidirect
253. C17#13013semidirect
276. Q16#63227semidirect
332. C2×C4#10587499semidirect

Maximal subgroups

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

subgroupindex
C22×#105874992
C2×#117626132
C2×#117626092
#145296502
#145268752
#145268622
C2×C4×Q16×C172
#1301317