Finite Group Databasebeta

← #16932115

Subgroups — #16932115

Order 4352.

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

normal subgroups only

This group has 576 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#14533086C211yesmaximal, characteristic
3#14528870C211yesmaximal, characteristic, complemented
4#14528769C211yesmaximal, characteristic
5#14526652C211yesmaximal, characteristic, complemented
6#14526651C211yesmaximal, characteristic, complemented
7C17×#1823C211yesmaximal, characteristic, complemented
8C2×#10587615—21no—
9C2×C4×C22⋊D17C2211yescharacteristic
10#11762880—22no—
11#11762877—22no—
12#11762863—21no—
13C2×#10587477—21no—
14#11762771—21no—
15C2×C22⋊#3622C2211yescharacteristic
16#11762740C412yescomplemented
17#11762735C412yescomplemented
18#11762705—21no—
19#11762672—21no—
20#11762669C2211yescharacteristic
21#11762669C2211yescharacteristic
22#11762666—21no—
23#11762575—21no—
24#11762571—22no—
25C22×C4⋊C4×C17C2211yescharacteristic
26C2×C17×C22⋊C8C2211yescharacteristic
27C17×#380C2211yescharacteristic
28C17×#340—22no—
29C23×C4×C17—21no—
30C23×C4×C17C2311yescharacteristic
31C22×C22⋊D17D411yescharacteristic
32C22×C4×D17D411yescharacteristic
33C22×C8×C17D412yes—
34C2×C4⋊C4×C17—21no—
35C2×C4⋊C4×C17—21no—
36C2×C4⋊C4×C17—21no—
37C2×C4⋊C4×C17—21no—
38C2×C4⋊C4×C17—21no—
39C2×C4⋊C4×C17—21no—
40C2×C4⋊C4×C17C2×C412yes—
41C2×C22⋊C4×C17—21no—
42C2×C22⋊Dic17—21no—
43C2×C22⋊Dic17—21no—
44C2×C22⋊Dic17D411yescharacteristic
45#10587615—42no—
46#10587615—42no—
47C4×C22⋊D17—21no—
48C4×C22⋊D17—21no—
49C4×C22⋊D17—21no—
50C4×C22⋊D17—21no—
51C2×#63227—22no—
52C2×#63227C2×C411yescharacteristic
53C2×C4⋊Dic17—21no—
54C2×C4⋊Dic17—21no—
55C2×#63225—22no—
56C2×#63225C2×C411yescharacteristic
57C2×C4×Dic17D411yescharacteristic
58C22×#3622D412yes—
59C2×#63218—22no—
60C17×C22⋊C8—21no—
61C17×C22⋊C8—21no—
62C17×#95—21no—
63#10587502—22no—
64#10587502—22no—
65#10587502—21no—
66#10587502C2×C412yes—
67C22⋊#3622—21no—
68C22⋊#3622—21no—
69#10587491—22no—
70#10587486—22no—
71#10587485—22no—
72#10587477—42no—
73C24×C17C2×D411yescharacteristic
74C23×D17C22⋊C411yescharacteristic
75C22×C4×C17—22no—
76C22×C4×C17—42no—
77C22×C4×C17—41no—
78C22×C4×C17—41no—
79C22×C4×C17—41no—
80C22×C4×C17—41no—
81C22×C4×C17—41no—
82C22×C4×C17—41no—
83C22×C4×C17—41no—
84C22×C4×C17C22×C411yescharacteristic
85C22×C4×C17C2×D411yescharacteristic
86C22×C4×C17C2×D411yescharacteristic
87C22×C4×C17C2×D411yescharacteristic
88C22×C4×C17#4111yescharacteristic
89C22×C4×C17#4111yescharacteristic
90C2×C22⋊D17—21no—
91C2×C22⋊D17—21no—
92C2×C22⋊D17—21no—
93C2×C22⋊D17—21no—
94C2×C22⋊D17—21no—
95C2×C22⋊D17—21no—
96C22×Dic17—22no—
97C22×Dic17—41no—
98C22×Dic17C22⋊C411yescharacteristic
99C22×Dic17C22⋊C411yescharacteristic
100C22×Dic17C22⋊C411yescharacteristic
101C2×C4×D17—21no—
102C2×C4×D17—21no—
103C2×C4×D17—42no—
104C2×C8×C17—22no—
105C2×C8×C17—42no—
106C4⋊C4×C17—22no—
107C4⋊C4×C17—22no—
108C4⋊C4×C17—42no—
109C4⋊C4×C17—41no—
110C4⋊C4×C17—41no—
111C22⋊C4×C17—41no—
112C22⋊C4×C17—41no—
113C22⋊Dic17—21no—
114C22⋊Dic17—21no—
115C22⋊Dic17—41no—
116C22⋊Dic17—41no—
117C22⋊Dic17—41no—
118C22⋊Dic17—41no—
119#63227—21no—
120#63227—21no—
121#63227—44no—
122C4⋊Dic17—42no—
123C4⋊Dic17—42no—
124#63225—21no—
125#63225—21no—
126#63225—44no—
127C4×Dic17—21no—
128C4×Dic17—21no—
129C2×#3622—22no—
130C2×#3622—42no—
131#63218—82no—
132#9474—171nomaximal, Sylow
133C23×C17—41no—
134C23×C17—41no—
135C23×C17C2×C22⋊C411yescharacteristic
136C23×C17#12111yescharacteristic
137C23×C17#12111yescharacteristic
138C23×C17AGL(1,ℤ/8)11yescharacteristic
139C23×C17AGL(1,ℤ/8)11yescharacteristic
140C23×C17AGL(1,ℤ/8)11yescharacteristic
141C23×C17#13711yescharacteristic
142C22×D17—21no—
143C22×D17—21no—
144C22×D17—21no—
145C22×D17—41no—
146C22×D17—41no—
147C2×C4×C17—22no—
148C2×C4×C17—22no—
149C2×C4×C17—22no—
150C2×C4×C17—41no—
151C2×C4×C17—41no—
152C2×C4×C17—41no—
153C2×C4×C17—42no—
154C2×C4×C17—42no—
155C2×C4×C17—42no—
156C2×C4×C17—81no—
157C2×C4×C17—81no—
158C2×C4×C17—81no—
159C2×C4×C17—81no—
160C2×C4×C17—81no—
161C2×C4×C17—81no—
162C2×C4×C17—81no—
163C2×C4×C17C4×D411yescharacteristic
164C2×C4×C17C4×D411yescharacteristic
165C2×C4×C17#12011yescharacteristic
166C2×C4×C17#12311yescharacteristic
167C22⋊D17—41no—
168C22⋊D17—41no—
169C22⋊D17—41no—
170C22⋊D17—41no—
171C2×Dic17—21no—
172C2×Dic17—21no—
173C2×Dic17—21no—
174C2×Dic17—21no—
175C2×Dic17—21no—
176C2×Dic17—41no—
177C2×Dic17—41no—
178C2×Dic17—42no—
179C2×Dic17—42no—
180C2×Dic17—42no—
181C2×Dic17—81no—
182C2×Dic17—81no—
183C2×Dic17—81no—
184C4×D17—42no—
185C4×D17—81no—
186C8×C17—42no—
187#3622—42no—
188#2973—171no—
189#2486—171no—
190#1823—171no—
191#1823D1711yescharacteristic, complemented
192#1799—171no—
193#1289—171no—
194#1289—171no—
195C22×C17—21no—
196C22×C17—21no—
197C22×C17—21no—
198C22×C17—21no—
199C22×C17—21no—
200C22×C17—21no—
201C22×C17—41no—
202C22×C17—41no—
203C22×C17—41no—
204C22×C17—41no—
205C22×C17#38611yescharacteristic
206C22×C17#41711yescharacteristic
207C22×C17#41811yescharacteristic
208C22×C17#46811yescharacteristic
209C22×C17#46811yescharacteristic
210C22×C17#46911yescharacteristic
211C22×C17#46911yescharacteristic
212C2×D17—42no—
213C2×D17—41no—
214C2×D17—41no—
215C2×D17—41no—
216C2×D17—81no—
217C4×C17—21no—
218C4×C17—21no—
219C4×C17—21no—
220C4×C17—82no—
221C4×C17—161no—
222C4×C17#45812yes—
223Dic17—41no—
224Dic17—41no—
225Dic17—41no—
226Dic17—82no—
227Dic17—161no—
228C2×#123—341no—
229C2×C4×D4—171no—
230C22×C4⋊C4C2×D1711yescharacteristic
231#435—342no—
232#435—342no—
233#432—341no—
234C2×#99—341no—
235#407—341no—
236C2×C22⋊C8—171no—
237C2×C22⋊C8C2×D1711yescharacteristic
238#398—172no—
239#397—172no—
240#389—341no—
241#381—341no—
242#380—171no—
243#380—171no—
244#380—341no—
245#380C2×D1711yescharacteristic
246#342—341no—
247#340—22no—
248#340—342no—
249C2×C17—21no—
250C2×C17—21no—
251C2×C17—21no—
252C2×C17—21no—
253C2×C17—21no—
254C2×C17—21no—
255C2×C17#189011yescharacteristic
256C2×C17#189511yescharacteristic
257C2×C17#190911yescharacteristic
258D17—81no—
259C22×D4—171no—
260C23×C4—21no—
261C23×C4—171no—
262C23×C4C22×D1711yescharacteristic
263C2×M16—342no—
264C22×C8—172no—
265C22×C8C22⋊D1712yes—
266#123—682no—
267#123—682no—
268C4×D4—341no—
269C4×D4—341no—
270C4×D4—341no—
271C4×D4—341no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—171no—
279C2×C4⋊C4—342no—
280C2×C4⋊C4—341no—
281C2×C4⋊C4—341no—
282C2×C4⋊C4C4×D1712yes—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—171no—
285C2×C22⋊C4—171no—
286C2×C22⋊C4—342no—
287C2×C22⋊C4—341no—
288C2×C22⋊C4—341no—
289C2×C42—171no—
290C4⋊C8—342no—
291#99—682no—
292C22⋊C8—21no—
293C22⋊C8—21no—
294C22⋊C8—342no—
295C22⋊C8—341no—
296C22⋊C8—341no—
297#97—342no—
298#95—21no—
299#95—172no—
300#95—342no—
301#95—342no—
302#95—341no—
303C17#947411yescharacteristic, complemented, Sylow
304C24—171no—
305C24D4×D1711yescharacteristic
306C2×D4—341no—
307C2×D4—341no—
308C2×D4—341no—
309C2×D4—341no—
310C2×D4—341no—
311C2×D4—341no—
312C22×C4—22no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—171no—
322C22×C4—171no—
323C22×C4—171no—
324C22×C4—341no—
325C22×C4—341no—
326C22×C4—342no—
327C22×C4—682no—
328C22×C4—681no—
329C22×C4C2×C4×D1711yescharacteristic
330C22×C4#6325211yescharacteristic
331C22×C4D4×D1711yescharacteristic
332C22×C4D4×D1711yescharacteristic
333C22×C4#6325411yescharacteristic
334C22×C4C2×C22⋊D1711yescharacteristic
335M16—1362no—
336C2×C8—22no—
337C2×C8—42no—
338C2×C8—342no—
339C2×C8—682no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—42no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—341no—
346C4⋊C4—341no—
347C4⋊C4—682no—
348C4⋊C4—682no—
349C4⋊C4—684no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C22⋊C4—341no—
353C22⋊C4—341no—
354C22⋊C4—341no—
355C22⋊C4—341no—
356C22⋊C4—684no—
357C22⋊C4—681no—
358C22⋊C4—681no—
359C22⋊C4—681no—
360C22⋊C4—681no—
361C42—341no—
362C42—341no—
363C23—41no—
364C23—41no—
365C23—341no—
366C23—341no—
367C23—341no—
368C23—681no—
369C23—681no—
370C23C22⋊C4×D1711yescharacteristic
371C23#1058757011yescharacteristic
372C23#1058759711yescharacteristic
373C23#1058759711yescharacteristic
374C23#1058760011yescharacteristic
375C23#1058760111yescharacteristic
376C23#1058762411yescharacteristic
377D4—681no—
378D4—681no—
379D4—681no—
380D4—681no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4—341no—
398C2×C4—341no—
399C2×C4—341no—
400C2×C4—341no—
401C2×C4—341no—
402C2×C4—682no—
403C2×C4—681no—
404C2×C4—681no—
405C2×C4—682no—
406C2×C4—682no—
407C2×C4—682no—
408C2×C4—1361no—
409C2×C4—1361no—
410C2×C4—1361no—
411C2×C4—1361no—
412C2×C4#1058756711yescharacteristic
413C2×C4#1058756911yescharacteristic
414C2×C4C4×C22⋊D1711yescharacteristic
415C2×C4#1058762311yescharacteristic
416C8—42no—
417C8—682no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—682no—
429C22—681no—
430C22—681no—
431C22—681no—
432C22—1361no—
433C22#1176268511yescharacteristic
434C22#1176280711yescharacteristic
435C22#1176281211yescharacteristic
436C22#1176305211yescharacteristic
437C22#1176305411yescharacteristic
438C22#1176305911yescharacteristic
439C22#1176306111yescharacteristic
440C4—21no—
441C4—21no—
442C4—21no—
443C4—82no—
444C4—161no—
445C4—2721no—
446C4—681no—
447C4—681no—
448C4—681no—
449C4—1362no—
450C4#1176299812yes—
451C2—21no—
452C2—21no—
453C2—21no—
454C2—21no—
455C2—21no—
456C2—21no—
457C2—1361no—
458C2#1452919411yescharacteristic
459C2#1452921911yescharacteristic
460C2#1452930311yescharacteristic
subgroupG/H
centerC22#11762685
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#1823C2
socleC22×C17#386
prime radicalC22×C22⋊D17D4
cosocleC23×C4×C17C23
hypercenter#1823D17
hypocenterC17#9474
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
3. #14528870C2semidirect
5. #14526652C2semidirect
6. #14526651C2semidirect
7. C17×#1823C2semidirect
16. #11762740C4semidirect
17. #11762735C4semidirect
191. #1823D17semidirect
303. C17#9474semidirect

Maximal subgroups

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

subgroupindex
#145378102
#145330862
#145288702
#145287692
#145266522
#145266512
C17×#18232
#947417