Finite Group Databasebeta

← #16922277

Subgroups — #16922277

Order 4352.

550 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14531265C211yesmaximal, characteristic
2#14528541C211yesmaximal, characteristic
3#14528541C211yesmaximal, characteristic
4#14526608C211yesmaximal, characteristic, complemented
5#14526607C211yesmaximal, characteristic, complemented
6#14526606C211yesmaximal, characteristic, complemented
7C17×#1271C211yesmaximal, characteristic, complemented
8C2×#10587578C411yescharacteristic
9C2×#10587578C411yescharacteristic
10C2×C42×D17C2211yescharacteristic
11C2×#10587491—21no—
12C2×#10587491—21no—
13C2×C8×Dic17—21no—
14C2×C8×Dic17—21no—
15#11762708C411yescharacteristic
16#11762707C2211yescharacteristic
17#11762705C411yescharacteristic
18#11762683—21no—
19#11762680—21no—
20C4⋊C4×Dic17—21no—
21#11762660—21no—
22#11762564C2211yescharacteristic
23#11762564C2211yescharacteristic
24C17×#389C2211yescharacteristic
25C17×#336C2211yescharacteristic
26C17×#336C2211yescharacteristic
27C22×C4×D17C2×C411yescharacteristic
28C22×C4×D17D411yescharacteristic
29C22×C4×D17D411yescharacteristic
30C22×C8×C17—21no—
31C22×C8×C17—21no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17C2×C411yescharacteristic
34C2×C4⋊C4×C17C2×C411yescharacteristic
35C2×C42×C17C2311yescharacteristic
36C2×#63227—21no—
37C2×#63227—21no—
38C2×C4⋊Dic17—21no—
39C2×C4⋊Dic17C2×C411yescharacteristic
40C2×C4⋊Dic17C2×C411yescharacteristic
41C2×#63225—21no—
42C2×#63225—21no—
43C2×C4×Dic17—21no—
44C2×C4×Dic17—21no—
45C2×C4×Dic17C2×C411yescharacteristic
46C2×C4×Dic17D411yescharacteristic
47C2×C4×Dic17D411yescharacteristic
48C22×#3622—21no—
49C22×#3622—21no—
50#10587578—22no—
51#10587578—42no—
52#10587578C814yescomplemented
53C42×D17—42no—
54C17×#95—21no—
55#10587502—21no—
56#10587502—21no—
57#10587502—21no—
58#10587491—24no—
59#10587491—24no—
60C8×Dic17—42no—
61C8×Dic17—42no—
62C23×D17C22⋊C411yescharacteristic
63C22×C4×C17—21no—
64C22×C4×C17—21no—
65C22×C4×C17C22×C411yescharacteristic
66C22×C4×C17C2×D411yescharacteristic
67C22×C4×C17C2×D411yescharacteristic
68C22×Dic17—21no—
69C22×Dic17—21no—
70C22×Dic17C22⋊C411yescharacteristic
71C22×Dic17C22⋊C411yescharacteristic
72C22×Dic17C22⋊C411yescharacteristic
73C2×C4×D17—22no—
74C2×C4×D17—22no—
75C2×C4×D17—21no—
76C2×C4×D17—21no—
77C2×C4×D17—21no—
78C2×C4×D17—21no—
79C2×C4×D17—22no—
80C2×C4×D17—22no—
81C2×C4×D17—41no—
82C2×C4×D17—41no—
83C2×C4×D17C2×C812yes—
84C2×C4×D17M1612yes—
85C2×C8×C17—22no—
86C2×C8×C17—22no—
87C2×C8×C17—41no—
88C2×C8×C17—41no—
89C2×C8×C17—41no—
90C2×C8×C17—41no—
91C4⋊C4×C17—21no—
92C4⋊C4×C17—41no—
93C4⋊C4×C17—41no—
94C4⋊C4×C17—41no—
95C4⋊C4×C17C2×C812yes—
96C42×C17—41no—
97#63227—24no—
98#63227—42no—
99C4⋊Dic17—21no—
100C4⋊Dic17—42no—
101C4⋊Dic17—41no—
102C4⋊Dic17C2×C812yes—
103#63225—41no—
104#63225—41no—
105#63225—42no—
106C4×Dic17—24no—
107C4×Dic17—22no—
108C4×Dic17—22no—
109C4×Dic17—41no—
110C4×Dic17—42no—
111C2×#3622—22no—
112C2×#3622—22no—
113C2×#3622—41no—
114C2×#3622—41no—
115C2×#3622—41no—
116C2×#3622—41no—
117#7564—171nomaximal, Sylow
118C23×C17C2×C22⋊C411yescharacteristic
119C22×D17—21no—
120C22×D17—21no—
121C22×D17—21no—
122C22×D17—21no—
123C22×D17—21no—
124C22×D17—21no—
125C22×D17C22⋊C812yes—
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—22no—
133C2×C4×C17—41no—
134C2×C4×C17—41no—
135C2×C4×C17—41no—
136C2×C4×C17—41no—
137C2×C4×C17—41no—
138C2×C4×C17C22×C811yescharacteristic
139C2×C4×C17C2×M1611yescharacteristic
140C2×C4×C17#13511yescharacteristic
141C2×C4×C17#13511yescharacteristic
142C2×C4×C17#13511yescharacteristic
143C2×C4×C17#13511yescharacteristic
144C2×Dic17—21no—
145C2×Dic17—21no—
146C2×Dic17—21no—
147C2×Dic17—21no—
148C2×Dic17—22no—
149C2×Dic17—41no—
150C2×Dic17—41no—
151C2×Dic17—41no—
152C2×Dic17—41no—
153C2×Dic17—41no—
154C2×Dic17C22⋊C812yes—
155C2×Dic17#9912yes—
156C2×Dic17#10112yes—
157C2×Dic17#10412yes—
158C2×Dic17#10412yes—
159C4×D17—24no—
160C4×D17—24no—
161C4×D17—42no—
162C4×D17—42no—
163C4×D17—42no—
164C4×D17—42no—
165C8×C17—42no—
166C8×C17—42no—
167#3622—42no—
168#3622—42no—
169#2296—171no—
170C4×C22⋊C8—171no—
171C4×C22⋊C8—171no—
172#1271—171no—
173#1271—171no—
174#1271—171no—
175#1271D1711yescharacteristic, complemented
176C22×C17C2×C22⋊C811yescharacteristic
177C22×C17C2×#9911yescharacteristic
178C22×C17C2×#10111yescharacteristic
179C22×C17#41611yescharacteristic
180C22×C17#41611yescharacteristic
181C22×C17C2×#10411yescharacteristic
182C22×C17C2×#10411yescharacteristic
183C2×D17—22no—
184C2×D17—22no—
185C2×D17—22no—
186C2×D17—41no—
187C2×D17—41no—
188C2×D17—41no—
189C2×D17—41no—
190C4×C17—21no—
191C4×C17—21no—
192C4×C17—21no—
193C4×C17—21no—
194C4×C17—21no—
195C4×C17—21no—
196C4×C17—42no—
197C4×C17—81no—
198Dic17—22no—
199Dic17—22no—
200Dic17—42no—
201Dic17—81no—
202Dic17#32414yes—
203C2×#117—171no—
204C2×#117—171no—
205C22×C42—171no—
206C2×C22⋊C8—341no—
207C2×C22⋊C8—341no—
208C2×C4×C8—341no—
209C2×C4×C8—341no—
210#389—171no—
211#389—171no—
212#389—171no—
213#389C2×D1711yescharacteristic
214#385—341no—
215#385—341no—
216C4×C4⋊C4—341no—
217C4×C4⋊C4—341no—
218#336—171no—
219#336—171no—
220#336C2×D1711yescharacteristic
221#336C2×D1711yescharacteristic
222C2×C17C2×#32411yescharacteristic
223C2×C17#147611yescharacteristic
224C2×C17#147611yescharacteristic
225C2×C17#150111yescharacteristic
226C2×C17#150111yescharacteristic
227C2×C17#152611yescharacteristic
228C2×C17#154411yescharacteristic
229D17—42no—
230C23×C4—171no—
231C23×C4—171no—
232C23×C4—171no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—341no—
236C22×C8—341no—
237#117—174no—
238#117—342no—
239#117—682no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—171no—
242C2×C4⋊C4—171no—
243C2×C4⋊C4—341no—
244C2×C4⋊C4—341no—
245C2×C4⋊C4—341no—
246C2×C4⋊C4C4×D1711yescharacteristic
247C2×C4⋊C4C4×D1711yescharacteristic
248C2×C22⋊C4—341no—
249C2×C22⋊C4—341no—
250C2×C42—171no—
251C2×C42—171no—
252C2×C42—171no—
253C2×C42—341no—
254C2×C42—341no—
255C2×C42—682no—
256C2×C42C22×D1711yescharacteristic
257C22⋊C8—344no—
258C22⋊C8—344no—
259C4×C8—682no—
260C4×C8—682no—
261#95—21no—
262#95—341no—
263#95—341no—
264#95—341no—
265C17#756411yescharacteristic, complemented, Sylow
266C24—171no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—171no—
270C22×C4—171no—
271C22×C4—171no—
272C22×C4—172no—
273C22×C4—172no—
274C22×C4—342no—
275C22×C4—342no—
276C22×C4—342no—
277C22×C4—342no—
278C22×C4—341no—
279C22×C4—341no—
280C22×C4—341no—
281C22×C4—341no—
282C22×C4—341no—
283C22×C4—341no—
284C22×C4—681no—
285C22×C4—681no—
286C22×C4C2×C4×D1711yescharacteristic
287C22×C4D4×D1711yescharacteristic
288C22×C4D4×D1711yescharacteristic
289C2×C8—22no—
290C2×C8—22no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—342no—
296C2×C8—342no—
297C2×C8—681no—
298C2×C8—681no—
299C2×C8—681no—
300C2×C8—681no—
301C4⋊C4—21no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—172no—
306C4⋊C4—341no—
307C4⋊C4—681no—
308C4⋊C4—682no—
309C4⋊C4—682no—
310C4⋊C4—681no—
311C4⋊C4—681no—
312C4⋊C4C8×D1712yes—
313C22⋊C4—344no—
314C22⋊C4—682no—
315C42—41no—
316C42—342no—
317C42—342no—
318C42—344no—
319C42—681no—
320C42—682no—
321C23—172no—
322C23—341no—
323C23—341no—
324C23—341no—
325C23—341no—
326C23—341no—
327C23—341no—
328C23C22⋊C4×D1711yescharacteristic
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—172no—
342C2×C4—172no—
343C2×C4—172no—
344C2×C4—172no—
345C2×C4—172no—
346C2×C4—344no—
347C2×C4—344no—
348C2×C4—341no—
349C2×C4—341no—
350C2×C4—341no—
351C2×C4—341no—
352C2×C4—342no—
353C2×C4—682no—
354C2×C4—682no—
355C2×C4—682no—
356C2×C4—682no—
357C2×C4—681no—
358C2×C4—681no—
359C2×C4—681no—
360C2×C4—681no—
361C2×C4—681no—
362C2×C4C2×C8×D1711yescharacteristic
363C2×C4M16×D1711yescharacteristic
364C2×C4#1058759811yescharacteristic
365C2×C4#1058760211yescharacteristic
366C2×C4#1058760211yescharacteristic
367C2×C4#1058760511yescharacteristic
368C8—42no—
369C8—42no—
370C8—682no—
371C8—682no—
372C22—342no—
373C22—342no—
374C22—342no—
375C22—681no—
376C22—681no—
377C22—681no—
378C22—681no—
379C22C22⋊C8×D1711yescharacteristic
380C22#99×D1711yescharacteristic
381C22#101×D1711yescharacteristic
382C22#1176280211yescharacteristic
383C22#1176280411yescharacteristic
384C22#104×D1711yescharacteristic
385C22#104×D1711yescharacteristic
386C4—21no—
387C4—21no—
388C4—21no—
389C4—21no—
390C4—21no—
391C4—21no—
392C4—42no—
393C4—81no—
394C4—174no—
395C4—342no—
396C4—342no—
397C4—682no—
398C4—1361no—
399C2—682no—
400C2D17×#32411yescharacteristic
401C2#1452716311yescharacteristic
402C2#1452716511yescharacteristic
403C2#1452727211yescharacteristic
404C2#1452727411yescharacteristic
405C2#1452737711yescharacteristic
406C2#1452744711yescharacteristic
subgroupG/H
centerC23C22⋊C4×D17
commutatorC2×C4×C17C22×C8
FrattiniC2×C42C22×D17
FittingC17×#1271C2
socleC23×C17C2×C22⋊C4
prime radicalC23×D17C22⋊C4
cosocleC2×C42×C17C23
hypercenter#1271D17
hypocenterC17#7564
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
4. #14526608C2semidirect
5. #14526607C2semidirect
6. #14526606C2semidirect
7. C17×#1271C2semidirect
52. #10587578C8semidirect
175. #1271D17semidirect
265. C17#7564semidirect

Maximal subgroups

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

subgroupindex
#145312652
#145285412
#145266082
#145266072
#145266062
C17×#12712
#756417