Finite Group Databasebeta

← #16932112

Subgroups — #16932112

Order 4352.

556 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14537810C211yesmaximal, characteristic
2#14533102C211yesmaximal, characteristic
3#14528875C211yesmaximal, characteristic, complemented
4#14528872C211yesmaximal, characteristic, complemented
5#14528769C211yesmaximal, characteristic
6#14526651C211yesmaximal, characteristic, complemented
7C17×#1289C211yesmaximal, characteristic, complemented
8C2×C4×C22⋊D17C2211yescharacteristic
9C22×C4⋊Dic17C2211yescharacteristic
10#11762880—22no—
11#11762877—22no—
12#11762863—21no—
13C2×#10587477—21no—
14#11762771—21no—
15C2×C22⋊#3622C2211yescharacteristic
16#11762749C412yescomplemented
17#11762746C412yescomplemented
18#11762708—21no—
19#11762686—21no—
20#11762684—21no—
21#11762669C2211yescharacteristic
22#11762666—21no—
23#11762575—21no—
24#11762572—22no—
25#11762571—22no—
26C2×C17×C22⋊C8C2211yescharacteristic
27C17×#380C2211yescharacteristic
28C17×#380C2211yescharacteristic
29C23×C4×C17C2311yescharacteristic
30C22×C22⋊D17D411yescharacteristic
31C23×Dic17—21no—
32C22×C4×D17D411yescharacteristic
33C22×C8×C17D412yes—
34C2×C4⋊C4×C17—21no—
35C2×C4⋊C4×C17—21no—
36C2×C22⋊C4×C17—21no—
37C2×C22⋊C4×C17—21no—
38C2×C22⋊Dic17—21no—
39C2×C22⋊Dic17D411yescharacteristic
40C4×C22⋊D17—21no—
41C4×C22⋊D17—21no—
42C4×C22⋊D17—21no—
43C4×C22⋊D17—21no—
44C2×#63227—22no—
45C2×#63227C2×C411yescharacteristic
46C2×C4⋊Dic17—21no—
47C2×C4⋊Dic17—22no—
48C2×C4⋊Dic17—21no—
49C2×C4⋊Dic17—21no—
50C2×C4⋊Dic17—21no—
51C2×C4⋊Dic17—21no—
52C2×C4⋊Dic17—21no—
53C2×C4⋊Dic17C2×C412yes—
54C2×#63225C2×C411yescharacteristic
55C2×C4×Dic17D411yescharacteristic
56C22×#3622D412yes—
57C2×#63218—22no—
58C17×C22⋊C8—21no—
59C17×C22⋊C8—21no—
60C17×#95—21no—
61C17×#95C2×C412yes—
62#10587502—22no—
63#10587502—22no—
64#10587502—21no—
65C22⋊#3622—21no—
66C22⋊#3622—21no—
67#10587491—22no—
68#10587486—22no—
69#10587485—22no—
70#10587477—42no—
71C24×C17C2×D411yescharacteristic
72C23×D17C22⋊C411yescharacteristic
73C22×C4×C17—22no—
74C22×C4×C17—42no—
75C22×C4×C17—41no—
76C22×C4×C17C22×C411yescharacteristic
77C22×C4×C17C2×D411yescharacteristic
78C22×C4×C17C2×D411yescharacteristic
79C22×C4×C17C2×D411yescharacteristic
80C22×C4×C17#4111yescharacteristic
81C22×C4×C17#4111yescharacteristic
82C2×C22⋊D17—21no—
83C2×C22⋊D17—21no—
84C2×C22⋊D17—21no—
85C2×C22⋊D17—21no—
86C2×C22⋊D17—21no—
87C2×C22⋊D17—21no—
88C22×Dic17—22no—
89C22×Dic17—41no—
90C22×Dic17—41no—
91C22×Dic17—41no—
92C22×Dic17—82no—
93C22×Dic17C22⋊C411yescharacteristic
94C22×Dic17C22⋊C411yescharacteristic
95C22×Dic17C22⋊C411yescharacteristic
96C2×C4×D17—21no—
97C2×C4×D17—21no—
98C2×C4×D17—42no—
99C2×C8×C17—22no—
100C2×C8×C17—42no—
101C4⋊C4×C17—42no—
102C4⋊C4×C17—42no—
103C22⋊C4×C17—41no—
104C22⋊C4×C17—41no—
105C22⋊C4×C17—41no—
106C22⋊C4×C17—41no—
107C22⋊Dic17—21no—
108C22⋊Dic17—21no—
109C22⋊Dic17—41no—
110C22⋊Dic17—41no—
111#63227—21no—
112#63227—21no—
113#63227—82no—
114C4⋊Dic17—22no—
115C4⋊Dic17—22no—
116C4⋊Dic17—41no—
117C4⋊Dic17—41no—
118C4⋊Dic17—42no—
119C4⋊Dic17—82no—
120#63225—21no—
121#63225—21no—
122C4×Dic17—21no—
123C4×Dic17—21no—
124C2×#3622—22no—
125C2×#3622—42no—
126#63218—82no—
127#9473—171nomaximal, Sylow
128C23×C17—41no—
129C23×C17—41no—
130C23×C17C2×C22⋊C411yescharacteristic
131C23×C17#12111yescharacteristic
132C23×C17#12111yescharacteristic
133C23×C17AGL(1,ℤ/8)11yescharacteristic
134C23×C17#13711yescharacteristic
135C23×C17#13711yescharacteristic
136C23×C17#13711yescharacteristic
137C22×D17—21no—
138C22×D17—21no—
139C22×D17—21no—
140C22×D17—41no—
141C22×D17—41no—
142C2×C4×C17—22no—
143C2×C4×C17—22no—
144C2×C4×C17—22no—
145C2×C4×C17—41no—
146C2×C4×C17—41no—
147C2×C4×C17—41no—
148C2×C4×C17—42no—
149C2×C4×C17—42no—
150C2×C4×C17—42no—
151C2×C4×C17—81no—
152C2×C4×C17—81no—
153C2×C4×C17—81no—
154C2×C4×C17C4×D411yescharacteristic
155C2×C4×C17C4×D411yescharacteristic
156C2×C4×C17#12011yescharacteristic
157C2×C4×C17#12311yescharacteristic
158C22⋊D17—41no—
159C22⋊D17—41no—
160C22⋊D17—41no—
161C22⋊D17—41no—
162C2×Dic17—21no—
163C2×Dic17—21no—
164C2×Dic17—21no—
165C2×Dic17—21no—
166C2×Dic17—21no—
167C2×Dic17—41no—
168C2×Dic17—41no—
169C2×Dic17—42no—
170C2×Dic17—42no—
171C2×Dic17—42no—
172C2×Dic17—82no—
173C2×Dic17—82no—
174C2×Dic17—81no—
175C2×Dic17—81no—
176C2×Dic17—81no—
177C4×D17—42no—
178C4×D17—81no—
179C8×C17—42no—
180#3622—42no—
181#2973—171no—
182#2487—171no—
183#1824—171no—
184#1824—171no—
185#1799—171no—
186#1289—171no—
187#1289D1711yescharacteristic, complemented
188C22×C17—21no—
189C22×C17—21no—
190C22×C17—21no—
191C22×C17—21no—
192C22×C17—21no—
193C22×C17—21no—
194C22×C17—41no—
195C22×C17—41no—
196C22×C17—41no—
197C22×C17—41no—
198C22×C17#38611yescharacteristic
199C22×C17#41711yescharacteristic
200C22×C17#41911yescharacteristic
201C22×C17#46811yescharacteristic
202C22×C17#46811yescharacteristic
203C22×C17#47011yescharacteristic
204C22×C17#47011yescharacteristic
205C2×D17—42no—
206C2×D17—41no—
207C2×D17—41no—
208C2×D17—41no—
209C2×D17—81no—
210C4×C17—21no—
211C4×C17—21no—
212C4×C17—21no—
213C4×C17—82no—
214C4×C17—161no—
215C4×C17#45812yes—
216Dic17—41no—
217Dic17—41no—
218Dic17—41no—
219Dic17—82no—
220Dic17—161no—
221C2×C4×D4—171no—
222C22×C4⋊C4—171no—
223#435—342no—
224#435—342no—
225#432—341no—
226C2×#99—341no—
227#407—341no—
228C2×C22⋊C8—171no—
229C2×C22⋊C8C2×D1711yescharacteristic
230#400—172no—
231#399—172no—
232#389—341no—
233#386—341no—
234#385—341no—
235#380—171no—
236#380—341no—
237#380C2×D1711yescharacteristic
238#380C2×D1711yescharacteristic
239#342—341no—
240#340—342no—
241#340—342no—
242C2×C17—21no—
243C2×C17—21no—
244C2×C17—21no—
245C2×C17—21no—
246C2×C17—21no—
247C2×C17—21no—
248C2×C17#189011yescharacteristic
249C2×C17#189411yescharacteristic
250C2×C17#191011yescharacteristic
251D17—81no—
252C22×D4—171no—
253C23×C4—171no—
254C23×C4—341no—
255C23×C4C22×D1711yescharacteristic
256C2×M16—342no—
257C22×C8—172no—
258C22×C8D6812yes—
259C4×D4—341no—
260C4×D4—341no—
261C4×D4—341no—
262C4×D4—341no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—171no—
266C2×C4⋊C4—172no—
267C2×C4⋊C4—341no—
268C2×C4⋊C4—341no—
269C2×C4⋊C4—341no—
270C2×C4⋊C4—341no—
271C2×C4⋊C4—341no—
272C2×C4⋊C4—341no—
273C2×C4⋊C4—342no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—171no—
277C2×C22⋊C4—171no—
278C2×C22⋊C4—341no—
279C2×C22⋊C4—342no—
280C2×C42—171no—
281C4⋊C8—342no—
282#99—682no—
283C22⋊C8—21no—
284C22⋊C8—21no—
285C22⋊C8—342no—
286C22⋊C8—341no—
287C22⋊C8—341no—
288#97—342no—
289#95—21no—
290#95—342no—
291#95—342no—
292#95—341no—
293#95C4×D1712yes—
294C17#947311yescharacteristic, complemented, Sylow
295C24—171no—
296C24D4×D1711yescharacteristic
297C2×D4—341no—
298C2×D4—341no—
299C2×D4—341no—
300C2×D4—341no—
301C2×D4—341no—
302C2×D4—341no—
303C22×C4—22no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—171no—
307C22×C4—171no—
308C22×C4—171no—
309C22×C4—341no—
310C22×C4—341no—
311C22×C4—342no—
312C22×C4—682no—
313C22×C4—681no—
314C22×C4—681no—
315C22×C4—681no—
316C22×C4—1362no—
317C22×C4C2×C4×D1711yescharacteristic
318C22×C4C2×D6811yescharacteristic
319C22×C4#6325211yescharacteristic
320C22×C4D4×D1711yescharacteristic
321C22×C4D4×D1711yescharacteristic
322C22×C4#6325611yescharacteristic
323M16—1362no—
324C2×C8—22no—
325C2×C8—42no—
326C2×C8—342no—
327C2×C8—682no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—341no—
331C4⋊C4—341no—
332C4⋊C4—342no—
333C4⋊C4—342no—
334C4⋊C4—682no—
335C4⋊C4—681no—
336C4⋊C4—681no—
337C4⋊C4—1362no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—341no—
343C22⋊C4—341no—
344C22⋊C4—341no—
345C22⋊C4—341no—
346C22⋊C4—681no—
347C22⋊C4—681no—
348C22⋊C4—1362no—
349C42—341no—
350C42—341no—
351C23—41no—
352C23—41no—
353C23—341no—
354C23—341no—
355C23—341no—
356C23—681no—
357C23—681no—
358C23C22⋊C4×D1711yescharacteristic
359C23#1058757011yescharacteristic
360C23#1058758111yescharacteristic
361C23#1058759711yescharacteristic
362C23#1058760111yescharacteristic
363C23#1058760111yescharacteristic
364C23#1058760411yescharacteristic
365D4—681no—
366D4—681no—
367D4—681no—
368D4—681no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—341no—
382C2×C4—341no—
383C2×C4—341no—
384C2×C4—341no—
385C2×C4—341no—
386C2×C4—682no—
387C2×C4—681no—
388C2×C4—681no—
389C2×C4—682no—
390C2×C4—682no—
391C2×C4—682no—
392C2×C4—1361no—
393C2×C4—1361no—
394C2×C4—1361no—
395C2×C4—1361no—
396C2×C4—1362no—
397C2×C4—1362no—
398C2×C4C4×D6811yescharacteristic
399C2×C4#1058756811yescharacteristic
400C2×C4#1058757911yescharacteristic
401C2×C4#1058758011yescharacteristic
402C8—42no—
403C8—682no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—682no—
415C22—681no—
416C22—681no—
417C22—681no—
418C22—1361no—
419C22#1176268711yescharacteristic
420C22#1176280711yescharacteristic
421C22#1176281611yescharacteristic
422C22#1176305411yescharacteristic
423C22#1176305611yescharacteristic
424C22#1176306611yescharacteristic
425C22#1176306811yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—82no—
430C4—161no—
431C4—2721no—
432C4—681no—
433C4—681no—
434C4—681no—
435C4—1362no—
436C4#1176300012yes—
437C2—21no—
438C2—21no—
439C2—21no—
440C2—21no—
441C2—21no—
442C2—21no—
443C2—1361no—
444C2#1452919611yescharacteristic
445C2#1452921611yescharacteristic
446C2#1452931011yescharacteristic
subgroupG/H
centerC22#11762687
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#1289C2
socleC22×C17#386
prime radicalC22×C22⋊D17D4
cosocleC23×C4×C17C23
hypercenter#1289D17
hypocenterC17#9473
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1289

Complements

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

NH ≅ G/Ntype
3. #14528875C2semidirect
4. #14528872C2semidirect
6. #14526651C2semidirect
7. C17×#1289C2semidirect
16. #11762749C4semidirect
17. #11762746C4semidirect
187. #1289D17semidirect
294. C17#9473semidirect

Maximal subgroups

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

subgroupindex
#145378102
#145331022
#145288752
#145288722
#145287692
#145266512
C17×#12892
#947317