Finite Group Databasebeta

← #17037288

Subgroups — #17037288

Order 4352.

858 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#63231C211yesmaximal, characteristic
2#14531262C211yesmaximal, characteristic
3C4×#10587573C211yesmaximal, characteristic
4#14529421C212yesmaximal
5#14529358C212yesmaximal
6#14529122C212yesmaximal
7#14529053C212yesmaximal
8C4×#10587479C211yesmaximal, characteristic
9C4×#10587487C211yesmaximal, characteristic
10#14528529C211yesmaximal, characteristic
11C4×C17×#103C211yesmaximal, characteristic
12C22×#63231C2211yescharacteristic
13C2×C4×Dic34C2211yescharacteristic
14C2×#10587573C2212yes—
15C4×#63231C418yescomplemented
16C2×#10587478C2211yescharacteristic
17C2×#10587479C2212yes—
18C2×#10587487C2212yes—
19C2×#10587506C2211yescharacteristic
20C2×#10587481C2211yescharacteristic
21C2×C4×#3622C2211yescharacteristic
22#11762716C2212yes—
23#11762708C2212yes—
24#11762695—22no—
25C4⋊C4×Dic17C2212yes—
26C4×#63225—21no—
27C4×C4⋊Dic17—21no—
28C4×C4⋊Dic17C2211yescharacteristic
29#11762660—22no—
30#11762655—21no—
31C42×Dic17—21no—
32C42×Dic17C2211yescharacteristic
33#11762572C2212yes—
34#11762571C2212yes—
35#11762563C2212yes—
36#11762563C2212yes—
37C2×C4×Q8×C17C2211yescharacteristic
38C2×C17×#103C2212yes—
39C2×C4×C8×C17C2211yescharacteristic
40C4×C4⋊C4×C17C2211yescharacteristic
41C17×#340C2212yes—
42C22×Q8×C17C2311yescharacteristic
43C22×Dic34C2311yescharacteristic
44C22×C8×C17C2312yes—
45C4×Q8×C17—22no—
46C4×Q8×C17C2×C414yes—
47C2×C4⋊C4×C17—21no—
48C2×C4⋊C4×C17C2311yescharacteristic
49C2×C4⋊C4×C17C2312yes—
50C2×C42×C17—21no—
51C2×C42×C17—21no—
52C2×C42×C17C2311yescharacteristic
53C2×#63231—24no—
54C2×#63231C2×C414yes—
55C2×C4⋊Dic17—22no—
56C2×C4⋊Dic17C2312yes—
57C2×C4⋊Dic17C2311yescharacteristic
58C2×#63225—21no—
59C2×#63225—22no—
60C2×C4×Dic17—21no—
61C2×C4×Dic17—21no—
62C2×C4×Dic17—22no—
63C2×C4×Dic17—22no—
64C2×C4×Dic17D412yes—
65C2×C4×Dic17D411yescharacteristic
66C2×C4×Dic17D411yescharacteristic
67C2×C4×Dic17C2312yes—
68C22×#3622C2311yescharacteristic
69C22×#3622C2311yescharacteristic
70#10587573—24no—
71#10587573—24no—
72C4×Dic34—22no—
73C4×Dic34C2×C414yes—
74C17×#103—24no—
75C4×C8×C17—22no—
76C17×#95—21no—
77#10587506C2×C414yes—
78#10587502—21no—
79#10587502—22no—
80#10587502—22no—
81#10587502—22no—
82#10587502—22no—
83#10587487—24no—
84#10587481C2×C414yes—
85#10587479—24no—
86#10587478C2×C414yes—
87C4×#3622C2×C414yes—
88C2×Q8×C17—21no—
89C2×Q8×C17—22no—
90C2×Q8×C17—22no—
91C2×Q8×C17C22×C412yes—
92C22×C4×C17—21no—
93C22×C4×C17—21no—
94C22×C4×C17—22no—
95C22×C4×C17C2×D411yescharacteristic
96C22×C4×C17C2×D411yescharacteristic
97C22×C4×C17C2411yescharacteristic
98C22×Dic17—21no—
99C22×Dic17—21no—
100C22×Dic17—22no—
101C22×Dic17C2×D412yes—
102C22×Dic17C2×D412yes—
103C2×Dic34—21no—
104C2×Dic34—22no—
105C2×Dic34C22×C412yes—
106C2×Dic34#4114yes—
107C2×C8×C17—22no—
108C2×C8×C17—22no—
109C2×C8×C17—22no—
110C4⋊C4×C17—24no—
111C4⋊C4×C17—22no—
112C4⋊C4×C17—21no—
113C4⋊C4×C17C22×C412yes—
114C4⋊C4×C17#4114yes—
115C42×C17—24no—
116C42×C17—42no—
117C42×C17C22×C412yes—
118C42×C17#4112yes—
119#63231—28no—
120C4⋊Dic17—21no—
121C4⋊Dic17—22no—
122C4⋊Dic17—22no—
123C4⋊Dic17—44no—
124C4⋊Dic17C22×C412yes—
125#63225—28no—
126#63225—24no—
127C4×Dic17—28no—
128C4×Dic17—24no—
129C4×Dic17—22no—
130C4×Dic17—22no—
131C4×Dic17—22no—
132C4×Dic17—22no—
133C4×Dic17—24no—
134C4×Dic17—42no—
135C4×Dic17—44no—
136C2×#3622—22no—
137C2×#3622—22no—
138C2×#3622C22×C412yes—
139C2×#3622C22×C412yes—
140#21116—171nomaximal, Sylow
141C23×C17C22×D411yescharacteristic
142Q8×C17—22no—
143Q8×C17—22no—
144Q8×C17—42no—
145C2×C4×C17—22no—
146C2×C4×C17—22no—
147C2×C4×C17—22no—
148C2×C4×C17—24no—
149C2×C4×C17—42no—
150C2×C4×C17—42no—
151C2×C4×C17—42no—
152C2×C4×C17—42no—
153C2×C4×C17C4×D412yes—
154C2×C4×C17C4×D412yes—
155C2×C4×C17#13712yes—
156C2×C4×C17#13712yes—
157C2×C4×C17C23×C411yescharacteristic
158C2×C4×C17C2×#4112yes—
159C2×C4×C17C2×#4111yescharacteristic
160C2×C4×C17C2×#4112yes—
161C2×Dic17—22no—
162C2×Dic17—24no—
163C2×Dic17—22no—
164C2×Dic17—22no—
165C2×Dic17—22no—
166C2×Dic17—22no—
167C2×Dic17—22no—
168C2×Dic17—22no—
169C2×Dic17—42no—
170C2×Dic17—42no—
171C2×Dic17—42no—
172C2×Dic17—42no—
173C2×Dic17#13514yes—
174Dic34—22no—
175Dic34—24no—
176Dic34#11714yes—
177C8×C17—42no—
178#3622—22no—
179#3622—22no—
180C2×C4×Q16—171no—
181#2295—171no—
182C42×Q8—171no—
183#1927—172no—
184#1918—172no—
185#1878—172no—
186#1865—172no—
187C4×#106—171no—
188C4×#103—171no—
189C4×#103D1711yescharacteristic
190#1744—171no—
191C22×C17C2×C4×D411yescharacteristic
192C22×C17#52512yes—
193C22×C17C2×#13512yes—
194C22×C17C2×#13711yescharacteristic
195C22×C17C2×#13711yescharacteristic
196C4×C17—22no—
197C4×C17—44no—
198C4×C17—42no—
199C4×C17—42no—
200C4×C17#44114yes—
201C4×C17C2×#11711yescharacteristic
202C4×C17C4×#4111yescharacteristic
203C4×C17#52912yes—
204Dic17—24no—
205Dic17—24no—
206Dic17—42no—
207Dic17—42no—
208Dic17—82no—
209C22×Q16—171no—
210C2×C4×Q8—172no—
211C2×C4×Q8—171no—
212C2×C4×Q8C2×D1711yescharacteristic
213C4×Q16—178no—
214C2×#107—171no—
215C2×#106—172no—
216C2×#103—171no—
217C2×#103—171no—
218C2×#103—172no—
219C2×#103C2×D1712yes—
220C2×C4×C8—171no—
221C2×C4×C8C2×D1711yescharacteristic
222#391—172no—
223#389—172no—
224#387—342no—
225C4×C4⋊C4—172no—
226C4×C4⋊C4—171no—
227C4×C4⋊C4—342no—
228C4×C4⋊C4—341no—
229C4×C4⋊C4—341no—
230C4×C4⋊C4C2×D1711yescharacteristic
231#376—341no—
232C43—171no—
233C43—341no—
234#340—172no—
235#340—172no—
236#340C2×D1712yes—
237#336—172no—
238#336—172no—
239C2×C17#229011yescharacteristic
240C2×C17C4×#13511yescharacteristic
241C2×C17C2×#44111yescharacteristic
242C2×C17#310712yes—
243C2×C17#310812yes—
244C22×Q8—171no—
245C22×Q8C22×D1711yescharacteristic
246C2×Q16—174no—
247C2×Q16—344no—
248C22×C8—171no—
249C22×C8—171no—
250C22×C8C22×D1712yes—
251C4×Q8—22no—
252C4×Q8—174no—
253C4×Q8—344no—
254C4×Q8—344no—
255C4×Q8—342no—
256C4×Q8C4×D1714yes—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—171no—
259C2×C4⋊C4—172no—
260C2×C4⋊C4—342no—
261C2×C4⋊C4—341no—
262C2×C4⋊C4—342no—
263C2×C4⋊C4C22×D1711yescharacteristic
264C2×C4⋊C4C22×D1712yes—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—172no—
268C2×C42—171no—
269C2×C42—171no—
270C2×C42—172no—
271C2×C42—342no—
272C2×C42—341no—
273C2×C42—342no—
274C2×C42—341no—
275C2×C42C22×D1711yescharacteristic
276#107—174no—
277#106—344no—
278#103—24no—
279#103—174no—
280#103—174no—
281#103—344no—
282C4×C8—22no—
283C4×C8—174no—
284#95—21no—
285#95—342no—
286#95—342no—
287#95—342no—
288#95—342no—
289#95—341no—
290C17#2111611yescharacteristic, complemented, Sylow
291C2×Q8—21no—
292C2×Q8—22no—
293C2×Q8—22no—
294C2×Q8—172no—
295C2×Q8—174no—
296C2×Q8—342no—
297C2×Q8—341no—
298C2×Q8C2×C4×D1712yes—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—172no—
303C22×C4—172no—
304C22×C4—341no—
305C22×C4—341no—
306C22×C4—342no—
307C22×C4D4×D1711yescharacteristic
308C22×C4D4×D1711yescharacteristic
309C22×C4C23×D1711yescharacteristic
310Q16—348no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—172no—
315C2×C8—172no—
316C2×C8—342no—
317C2×C8—342no—
318C4⋊C4—21no—
319C4⋊C4—22no—
320C4⋊C4—24no—
321C4⋊C4—172no—
322C4⋊C4—348no—
323C4⋊C4—341no—
324C4⋊C4—344no—
325C4⋊C4—342no—
326C4⋊C4—342no—
327C4⋊C4—684no—
328C4⋊C4C2×C4×D1712yes—
329C4⋊C4#6325214yes—
330C42—24no—
331C42—42no—
332C42—344no—
333C42—348no—
334C42—342no—
335C42—342no—
336C42—344no—
337C42—342no—
338C42—342no—
339C42—684no—
340C42—682no—
341C42C2×C4×D1712yes—
342C42#6325412yes—
343C23C2×D4×D1711yescharacteristic
344Q8—22no—
345Q8—22no—
346Q8—42no—
347Q8—174no—
348Q8—342no—
349Q8—344no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—174no—
359C2×C4—342no—
360C2×C4—342no—
361C2×C4—344no—
362C2×C4—342no—
363C2×C4—342no—
364C2×C4—342no—
365C2×C4—342no—
366C2×C4—342no—
367C2×C4—682no—
368C2×C4—682no—
369C2×C4—682no—
370C2×C4—682no—
371C2×C4#1058756712yes—
372C2×C4#1058756712yes—
373C2×C4#1058760112yes—
374C2×C4#1058760412yes—
375C2×C4C22×C4×D1711yescharacteristic
376C2×C4C2×#6325212yes—
377C2×C4C2×#6325411yescharacteristic
378C2×C4#41×D1712yes—
379C8—42no—
380C8—342no—
381C8—342no—
382C22C2×#1058756711yescharacteristic
383C22#1176331812yes—
384C22#135×D1712yes—
385C22C2×#1058760111yescharacteristic
386C22C2×#1058760411yescharacteristic
387C4—22no—
388C4—42no—
389C4—44no—
390C4—42no—
391C4—344no—
392C4—344no—
393C4—682no—
394C4—682no—
395C4—1362no—
396C4#1176291314yes—
397C4#117×D1711yescharacteristic
398C4C4×#6325211yescharacteristic
399C4#1176334112yes—
400C2#1453123011yescharacteristic
401C2#1453755211yescharacteristic
402C2C2×#1176291311yescharacteristic
403C2#1453885112yes—
404C2#1453886812yes—
subgroupG/H
centerC22×C4D4×D17
commutatorC2×C4×C17C23×C4
FrattiniC22×C4C23×D17
FittingC4×C17×#103C2
socle, prime radicalC23×C17C22×D4
cosocleC22×C4×C17C24
hypercenterC4×#103D17
hypocenterC17#21116
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
15. C4×#63231C4semidirect
290. C17#21116semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#632312
#145312622
C4×#105875732
#145294212
#145293582
#145291222
#145290532
C4×#105874792
C4×#105874872
#145285292
C4×C17×#1032
#2111617