Finite Group Databasebeta

← #16929929

Subgroups — #16929929

Order 4352.

626 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14537094C211yesmaximal, characteristic
2#14528546C211yesmaximal, characteristic
3C2×#11762550C212yesmaximal, complemented
4#14526666C212yesmaximal, complemented
5C22×C17×#97C211yesmaximal, characteristic
6C2×#10587556C2211yescharacteristic
7C2×C22⋊#3622—21no—
8C2×#10587474C2212yes—
9#11762707C411yescharacteristic
10#11762707C411yescharacteristic
11#11762693—21no—
12#11762693—21no—
13#11762682—21no—
14#11762682—21no—
15#11762675C2211yescharacteristic
16#11762669C411yescharacteristic
17#11762669C411yescharacteristic
18#11762564—21no—
19#11762550—24no—
20C23×C8×C17—21no—
21C22×C42×C17C2211yescharacteristic
22C2×C17×#97—21no—
23C2×C17×#97—24no—
24C2×C17×#97C2212yes—
25C23×C4×C17C2×C411yescharacteristic
26C23×C4×C17D411yescharacteristic
27C23×C4×C17D411yescharacteristic
28C22×C8×C17—24no—
29C22×C8×C17—24no—
30C22×C8×C17—24no—
31C22×C8×C17—22no—
32C2×C42×C17—22no—
33C2×C42×C17—22no—
34C2×C42×C17C2×C411yescharacteristic
35C2×C42×C17D411yescharacteristic
36C2×C42×C17D411yescharacteristic
37C2×C42×C17C2311yescharacteristic
38C2×C22⋊Dic17—21no—
39C2×C22⋊Dic17—21no—
40C2×C4⋊Dic17C2×C411yescharacteristic
41C2×C4⋊Dic17C2×C411yescharacteristic
42C2×C4⋊Dic17C2×C411yescharacteristic
43C2×C4⋊Dic17C2×C411yescharacteristic
44C2×#63225—21no—
45C2×#63225—21no—
46C2×#63225—21no—
47C2×#63225—21no—
48C22×#3622—22no—
49#10587556—22no—
50#10587556—22no—
51C17×#97—24no—
52C17×#97—22no—
53C17×#97D414yes—
54#10587502—21no—
55#10587502—21no—
56#10587502—21no—
57#10587502—21no—
58#10587502—21no—
59#10587502—21no—
60C22⋊#3622—24no—
61#10587474—22no—
62#10587474—22no—
63C24×C17C22⋊C411yescharacteristic
64C22×C4×C17—22no—
65C22×C4×C17—22no—
66C22×C4×C17—21no—
67C22×C4×C17—21no—
68C22×C4×C17—22no—
69C22×C4×C17—22no—
70C22×C4×C17—22no—
71C22×C4×C17—22no—
72C22×C4×C17—41no—
73C22×C4×C17—41no—
74C22×C4×C17C22⋊C411yescharacteristic
75C22×C4×C17C22⋊C411yescharacteristic
76C22×C4×C17C22⋊C411yescharacteristic
77C22×C4×C17C22×C411yescharacteristic
78C22×C4×C17C2×D411yescharacteristic
79C22×C4×C17C2×D411yescharacteristic
80C22×Dic17—21no—
81C22×Dic17—21no—
82C22×Dic17—21no—
83C22×Dic17—21no—
84C2×C8×C17—216no—
85C2×C8×C17—24no—
86C2×C8×C17—24no—
87C2×C8×C17—24no—
88C42×C17—22no—
89C42×C17—21no—
90C42×C17—22no—
91C42×C17—21no—
92C42×C17C2×D412yes—
93C42×C17#4112yes—
94C22⋊Dic17—42no—
95C22⋊Dic17—42no—
96C4⋊Dic17—22no—
97C4⋊Dic17—22no—
98C4⋊Dic17—41no—
99C4⋊Dic17—41no—
100#63225—42no—
101#63225—42no—
102#63225—41no—
103#63225—42no—
104#63225—41no—
105C2×#3622—24no—
106C2×#3622—42no—
107C2×#3622—42no—
108#9121—171nomaximal, Sylow
109C23×C17—21no—
110C23×C17—21no—
111C23×C17—21no—
112C23×C17—21no—
113C23×C17#10112yes—
114C23×C17#10412yes—
115C23×C17#10412yes—
116C23×C17C2×C22⋊C411yescharacteristic
117C2×C4×C17—24no—
118C2×C4×C17—24no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17—44no—
134C2×C4×C17—44no—
135C2×C4×C17#10412yes—
136C2×C4×C17C4×D412yes—
137C2×C4×C17#12011yescharacteristic
138C2×C4×C17#12112yes—
139C2×C4×C17#12311yescharacteristic
140C2×Dic17—41no—
141C2×Dic17—41no—
142C2×Dic17—41no—
143C2×Dic17—41no—
144C2×Dic17—41no—
145C2×Dic17—41no—
146C2×Dic17—41no—
147C2×Dic17—41no—
148C2×Dic17—41no—
149C2×Dic17—41no—
150C2×Dic17—41no—
151C2×Dic17—41no—
152C8×C17—28no—
153#3622—44no—
154C22×#97D1711yescharacteristic
155#2851—171no—
156#1749—171no—
157C2×#330—172no—
158#1295—172no—
159C22×C17—22no—
160C22×C17—22no—
161C22×C17—22no—
162C22×C17—22no—
163C22×C17—22no—
164C22×C17—22no—
165C22×C17#33014yescomplemented
166C22×C17#38611yescharacteristic
167C22×C17C2×#10111yescharacteristic
168C22×C17#41311yescharacteristic
169C22×C17C2×#10411yescharacteristic
170C22×C17C2×#10411yescharacteristic
171C22×C17C2×#10411yescharacteristic
172C22×C17#42111yescharacteristic
173C4×C17—22no—
174C4×C17—21no—
175C4×C17—22no—
176C4×C17—21no—
177C4×C17—22no—
178C4×C17—41no—
179C4×C17—41no—
180Dic17—81no—
181Dic17—81no—
182Dic17—81no—
183Dic17—81no—
184C23×C8—21no—
185C2×#125—171no—
186C22×C42C2×D1711yescharacteristic
187C2×C22⋊C8—341no—
188C2×#97—21no—
189C2×#97—24no—
190C2×#97—172no—
191C2×#97C2×D1712yes—
192#389—171no—
193#389—171no—
194#387—341no—
195#387—341no—
196#385—341no—
197#385—341no—
198#382—171no—
199#380—171no—
200#380—171no—
201#336—341no—
202#330—344no—
203C2×C17—24no—
204C2×C17C2×#33011yescharacteristic
205C2×C17#152411yescharacteristic
206C2×C17#153111yescharacteristic
207C2×C17#153211yescharacteristic
208C2×C17#187111yescharacteristic
209C2×C17#189611yescharacteristic
210C2×C17#190111yescharacteristic
211C23×C4C4×D1711yescharacteristic
212C23×C4D6811yescharacteristic
213C23×C4C22⋊D1711yescharacteristic
214C22×C8—24no—
215C22×C8—24no—
216C22×C8—24no—
217C22×C8—22no—
218C22×C8—342no—
219#125—342no—
220#125—342no—
221C2×C4⋊C4—171no—
222C2×C4⋊C4—171no—
223C2×C4⋊C4—171no—
224C2×C4⋊C4—171no—
225C2×C4⋊C4—341no—
226C2×C4⋊C4—341no—
227C2×C4⋊C4—341no—
228C2×C4⋊C4—341no—
229C2×C22⋊C4—341no—
230C2×C22⋊C4—341no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42C4×D1711yescharacteristic
234C2×C42D6811yescharacteristic
235C2×C42C22⋊D1711yescharacteristic
236C2×C42C22×D1711yescharacteristic
237C22⋊C8—344no—
238#97—24no—
239#97—22no—
240#97—342no—
241#97—342no—
242#97C22⋊D1714yes—
243#95—341no—
244#95—341no—
245#95—341no—
246#95—341no—
247#95—341no—
248#95—341no—
249C17#912111yescharacteristic, complemented, Sylow
250C24#6322711yescharacteristic
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—341no—
262C22×C4—341no—
263C22×C4—341no—
264C22×C4—341no—
265C22×C4#6322711yescharacteristic
266C22×C4#6322711yescharacteristic
267C22×C4#6322711yescharacteristic
268C22×C4C2×C4×D1711yescharacteristic
269C22×C4C2×D6811yescharacteristic
270C22×C4C2×C22⋊D1711yescharacteristic
271C2×C8—216no—
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—344no—
276C2×C8—682no—
277C2×C8—682no—
278C4⋊C4—342no—
279C4⋊C4—342no—
280C4⋊C4—681no—
281C4⋊C4—681no—
282C4⋊C4—682no—
283C4⋊C4—682no—
284C4⋊C4—681no—
285C4⋊C4—682no—
286C4⋊C4—681no—
287C22⋊C4—682no—
288C22⋊C4—682no—
289C42—22no—
290C42—21no—
291C42—22no—
292C42—21no—
293C42#6325212yes—
294C42C2×C22⋊D1712yes—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23#1058747612yes—
300C23#1058749512yes—
301C23#1058749612yes—
302C23C2×#6322711yescharacteristic
303C2×C4—24no—
304C2×C4—24no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—681no—
322C2×C4—681no—
323C2×C4—681no—
324C2×C4—681no—
325C2×C4—681no—
326C2×C4—681no—
327C2×C4—681no—
328C2×C4—681no—
329C2×C4—681no—
330C2×C4—681no—
331C2×C4—681no—
332C2×C4—681no—
333C2×C4#1058747612yes—
334C2×C4C4×C22⋊D1712yes—
335C2×C4#1058761511yescharacteristic
336C2×C4#1058761612yes—
337C2×C4#1058763911yescharacteristic
338C8—28no—
339C8—684no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22#1176255014yescomplemented
347C22#1176268811yescharacteristic
348C22C2×#1058749511yescharacteristic
349C22#1176279111yescharacteristic
350C22C2×#1058749611yescharacteristic
351C22C2×#1058747611yescharacteristic
352C22C2×#1058747611yescharacteristic
353C22#1176282511yescharacteristic
354C4—22no—
355C4—21no—
356C4—22no—
357C4—21no—
358C4—22no—
359C4—41no—
360C4—41no—
361C4—1361no—
362C4—1361no—
363C4—1361no—
364C4—1361no—
365C2—24no—
366C2C2×#1176255011yescharacteristic
367C2#1452737011yescharacteristic
368C2#1452739811yescharacteristic
369C2#1452740211yescharacteristic
370C2#1452908611yescharacteristic
371C2#1452922711yescharacteristic
372C2#1452925211yescharacteristic
subgroupG/H
centerC23C2×#63227
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC22×C17×#97C2
socleC23×C17C2×C22⋊C4
prime radicalC24×C17C22⋊C4
cosocleC2×C42×C17C23
hypercenterC22×#97D17
hypocenterC17#9121
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×#97

Complements

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

NH ≅ G/Ntype
3. C2×#11762550C2semidirect
4. #14526666C2semidirect
165. C22×C17#330semidirect
249. C17#9121semidirect
346. C22#11762550semidirect

Maximal subgroups

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

subgroupindex
#145370942
#145285462
C2×#117625502
#145266662
C22×C17×#972
#912117