Finite Group Databasebeta

← #16932313

Subgroups — #16932313

Order 4352.

716 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 107 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#14532115C211yesmaximal, characteristic
2#14528712C211yesmaximal, characteristic
3#14527113C212yesmaximal, complemented
4#14526693C212yesmaximal, complemented
5C17×#2879C211yesmaximal, characteristic
6C2×#10587621C411yescharacteristic
7C2×#10587621C411yescharacteristic
8C22×#63225C2211yescharacteristic
9C2×#10587507—21no—
10C2×#10587504—21no—
11#11762773C2212yes—
12#11762728—21no—
13#11762723—21no—
14#11762689C411yescharacteristic
15#11762689C411yescharacteristic
16#11762682—21no—
17#11762682—21no—
18#11762672C2211yescharacteristic
19#11762534—24no—
20C22×#41×C17C2211yescharacteristic
21C22×M16×C17—21no—
22C17×#408—24no—
23C17×#407—21no—
24C17×#407C2212yes—
25C2×#41×C17—22no—
26C2×#41×C17—22no—
27C2×#41×C17—42no—
28C2×#41×C17—42no—
29C22×Q8×C17D411yescharacteristic
30C22×D4×C17C2×C411yescharacteristic
31C22×D4×C17C2×C411yescharacteristic
32C22×D4×C17D411yescharacteristic
33C23×C4×C17D411yescharacteristic
34C23×C4×C17D411yescharacteristic
35C23×C4×C17C2311yescharacteristic
36C23×Dic17C2×C411yescharacteristic
37C23×Dic17C2×C411yescharacteristic
38C2×M16×C17—22no—
39C2×M16×C17—22no—
40C2×M16×C17—42no—
41C2×M16×C17—42no—
42C22×C8×C17—22no—
43C2×C22⋊Dic17—21no—
44C2×C22⋊Dic17—21no—
45C2×C22⋊Dic17—21no—
46C2×C22⋊Dic17—21no—
47C2×C22⋊Dic17C2×C411yescharacteristic
48C2×C22⋊Dic17C2×C411yescharacteristic
49#10587621—42no—
50#10587621—42no—
51C2×#63225—21no—
52C2×#63225—21no—
53C2×#63225—21no—
54C2×#63225—22no—
55C2×#63225—21no—
56C2×#63225—22no—
57C2×#63223—22no—
58C17×C22⋊C8—24no—
59C17×C22⋊C8—22no—
60C17×C22⋊C8D414yes—
61#10587507—42no—
62#10587504—42no—
63#10587502—21no—
64#10587502—21no—
65#10587502—21no—
66#10587502—21no—
67C22⋊#3622—22no—
68C22⋊#3622—22no—
69C24×C17C22⋊C411yescharacteristic
70C24×C17C22⋊C411yescharacteristic
71C24×C17C22×C411yescharacteristic
72#41×C17—44no—
73#41×C17—44no—
74#41×C17—48no—
75C2×Q8×C17—22no—
76C2×Q8×C17—22no—
77C2×Q8×C17—41no—
78C2×D4×C17—22no—
79C2×D4×C17—22no—
80C2×D4×C17—22no—
81C2×D4×C17—22no—
82C2×D4×C17—22no—
83C2×D4×C17—22no—
84C2×D4×C17—41no—
85C2×D4×C17—41no—
86C2×D4×C17—41no—
87C22×C4×C17—22no—
88C22×C4×C17—22no—
89C22×C4×C17—21no—
90C22×C4×C17—21no—
91C22×C4×C17—21no—
92C22×C4×C17—21no—
93C22×C4×C17—41no—
94C22×C4×C17—41no—
95C22×C4×C17—41no—
96C22×C4×C17—41no—
97C22×C4×C17C22⋊C411yescharacteristic
98C22×C4×C17C22⋊C411yescharacteristic
99C22×C4×C17C2×D411yescharacteristic
100C22×C4×C17C2×D411yescharacteristic
101C22×C4×C17C2×D412yes—
102C22×C4×C17#4112yes—
103C22×Dic17—21no—
104C22×Dic17—22no—
105C22×Dic17—21no—
106C22×Dic17—21no—
107C22×Dic17—21no—
108C22×Dic17—22no—
109C22×Dic17—42no—
110C22×Dic17—41no—
111C22×Dic17—41no—
112M16×C17—44no—
113M16×C17—42no—
114M16×C17—42no—
115C2×C8×C17—24no—
116C2×C8×C17—44no—
117C22⋊Dic17—42no—
118C22⋊Dic17—42no—
119C22⋊Dic17—42no—
120C22⋊Dic17—42no—
121C22⋊Dic17—41no—
122C22⋊Dic17—41no—
123#63225—42no—
124#63225—42no—
125#63225—41no—
126#63225—42no—
127#63225—41no—
128#63223—42no—
129#63223—42no—
130C2×#3622—42no—
131#9513—171nomaximal, Sylow
132C23×C17—21no—
133C23×C17—21no—
134C23×C17—21no—
135C23×C17—21no—
136C23×C17—21no—
137C23×C17—21no—
138C23×C17—21no—
139C23×C17—41no—
140C23×C17—41no—
141C23×C17—41no—
142C23×C17—41no—
143C23×C17—41no—
144C23×C17—41no—
145C23×C17#9912yes—
146C23×C17C2×C22⋊C411yescharacteristic
147C23×C17C4×D412yes—
148Q8×C17—44no—
149D4×C17—44no—
150D4×C17—44no—
151D4×C17—44no—
152C2×C4×C17—22no—
153C2×C4×C17—22no—
154C2×C4×C17—24no—
155C2×C4×C17—21no—
156C2×C4×C17—21no—
157C2×C4×C17—21no—
158C2×C4×C17—21no—
159C2×C4×C17—21no—
160C2×C4×C17—21no—
161C2×C4×C17—21no—
162C2×C4×C17—21no—
163C2×C4×C17—21no—
164C2×C4×C17—42no—
165C2×C4×C17—42no—
166C2×C4×C17—42no—
167C2×C4×C17—42no—
168C2×C4×C17#9912yes—
169C2×C4×C17#12011yescharacteristic
170C2×C4×C17#12112yes—
171C2×C4×C17#12311yescharacteristic
172C2×Dic17—22no—
173C2×Dic17—24no—
174C2×Dic17—41no—
175C2×Dic17—42no—
176C2×Dic17—41no—
177C2×Dic17—41no—
178C2×Dic17—41no—
179C2×Dic17—41no—
180C2×Dic17—41no—
181C2×Dic17—41no—
182C2×Dic17—41no—
183C2×Dic17—41no—
184C2×Dic17—41no—
185C2×Dic17—41no—
186C2×Dic17—82no—
187C8×C17—44no—
188#3622—82no—
189#2879D1711yescharacteristic
190#2406—171no—
191#1785—171no—
192#1463—172no—
193#1306—172no—
194C22×C17—21no—
195C22×C17—21no—
196C22×C17—22no—
197C22×C17—22no—
198C22×C17—22no—
199C22×C17—22no—
200C22×C17—42no—
201C22×C17—42no—
202C22×C17—42no—
203C22×C17—42no—
204C22×C17—41no—
205C22×C17—41no—
206C22×C17—41no—
207C22×C17—41no—
208C22×C17—41no—
209C22×C17—81no—
210C22×C17—81no—
211C22×C17#38611yescharacteristic
212C22×C17C2×#9911yescharacteristic
213C22×C17C2×#9911yescharacteristic
214C4×C17—22no—
215C4×C17—21no—
216C4×C17—21no—
217C4×C17—22no—
218C4×C17—41no—
219Dic17—42no—
220Dic17—81no—
221Dic17—81no—
222Dic17—81no—
223C22×#41C2×D1711yescharacteristic
224C22×M16—21no—
225C2×#123—171no—
226C2×#123—171no—
227C22×C4⋊C4—171no—
228C2×#101—341no—
229C2×#100—341no—
230#408—24no—
231#407—21no—
232#407—172no—
233#407C2×D1712yes—
234#395—341no—
235#394—341no—
236#386—171no—
237#386—171no—
238#385—341no—
239#385—341no—
240#381—171no—
241#324—344no—
242C2×C17—21no—
243C2×C17—21no—
244C2×C17—42no—
245C2×C17—42no—
246C2×C17—81no—
247C2×C17#189111yescharacteristic
248C2×C17#189811yescharacteristic
249C2×C17#191511yescharacteristic
250C2×#41—22no—
251C2×#41—22no—
252C2×#41—42no—
253C2×#41—42no—
254C22×Q8D6811yescharacteristic
255C22×D4C4×D1711yescharacteristic
256C22×D4C4×D1711yescharacteristic
257C22×D4D6811yescharacteristic
258C23×C4—171no—
259C23×C4—171no—
260C23×C4C22⋊D1711yescharacteristic
261C23×C4C22⋊D1711yescharacteristic
262C23×C4C22×D1711yescharacteristic
263C2×M16—22no—
264C2×M16—22no—
265C2×M16—42no—
266C2×M16—42no—
267C2×M16—342no—
268C22×C8—22no—
269#123—682no—
270#123—682no—
271C2×C4⋊C4—341no—
272C2×C4⋊C4—342no—
273C2×C4⋊C4—342no—
274C2×C4⋊C4—341no—
275C2×C4⋊C4—341no—
276C2×C4⋊C4—341no—
277C2×C22⋊C4—171no—
278C2×C22⋊C4—171no—
279C2×C22⋊C4—341no—
280C2×C22⋊C4—341no—
281C2×C22⋊C4—341no—
282C2×C22⋊C4—341no—
283#101—682no—
284#100—682no—
285C22⋊C8—24no—
286C22⋊C8—22no—
287C22⋊C8—342no—
288C22⋊C8—342no—
289C22⋊C8C22⋊D1714yes—
290#95—341no—
291#95—341no—
292#95—341no—
293#95—341no—
294C17#951311yescharacteristic, complemented, Sylow
295C24#6322711yescharacteristic
296C24#6322711yescharacteristic
297C24C2×C4×D1711yescharacteristic
298#41—44no—
299#41—44no—
300#41—48no—
301C2×Q8—22no—
302C2×Q8—22no—
303C2×Q8—41no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—341no—
324C22×C4—342no—
325C22×C4—341no—
326C22×C4—341no—
327C22×C4—341no—
328C22×C4—342no—
329C22×C4—682no—
330C22×C4—681no—
331C22×C4—681no—
332C22×C4#6322711yescharacteristic
333C22×C4#6322711yescharacteristic
334C22×C4C2×D6811yescharacteristic
335C22×C4#6325212yes—
336C22×C4C2×C22⋊D1711yescharacteristic
337C22×C4C2×C22⋊D1712yes—
338M16—44no—
339M16—42no—
340M16—42no—
341M16—682no—
342M16—682no—
343C2×C8—24no—
344C2×C8—44no—
345C2×C8—682no—
346C4⋊C4—682no—
347C4⋊C4—682no—
348C4⋊C4—681no—
349C4⋊C4—682no—
350C4⋊C4—681no—
351C22⋊C4—682no—
352C22⋊C4—682no—
353C22⋊C4—682no—
354C22⋊C4—682no—
355C22⋊C4—681no—
356C22⋊C4—681no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23#1058747712yes—
371C23C2×#6322711yescharacteristic
372C23C4×C22⋊D1712yes—
373Q8—44no—
374D4—44no—
375D4—44no—
376D4—44no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—24no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—342no—
394C2×C4—344no—
395C2×C4—681no—
396C2×C4—682no—
397C2×C4—681no—
398C2×C4—681no—
399C2×C4—681no—
400C2×C4—681no—
401C2×C4—681no—
402C2×C4—681no—
403C2×C4—681no—
404C2×C4—681no—
405C2×C4—681no—
406C2×C4—681no—
407C2×C4—1362no—
408C2×C4#1058747712yes—
409C2×C4#1058761511yescharacteristic
410C2×C4#1058761612yes—
411C2×C4#1058763911yescharacteristic
412C8—44no—
413C8—1362no—
414C22—21no—
415C22—21no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—42no—
421C22—42no—
422C22—42no—
423C22—42no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—81no—
430C22—81no—
431C22#1176268811yescharacteristic
432C22C2×#1058747711yescharacteristic
433C22C2×#1058747711yescharacteristic
434C4—21no—
435C4—22no—
436C4—21no—
437C4—22no—
438C4—41no—
439C4—682no—
440C4—1361no—
441C4—1361no—
442C4—1361no—
443C2—21no—
444C2—21no—
445C2—42no—
446C2—42no—
447C2—81no—
448C2#1452920211yescharacteristic
449C2#1452923711yescharacteristic
450C2#1452933611yescharacteristic
subgroupG/H
centerC22#11762688
commutatorC24×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#2879C2
socleC22×C17#386
prime radicalC22×#41×C17C22
cosocleC23×C4×C17C23
hypercenter#2879D17
hypocenterC17#9513
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2879

Complements

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

NH ≅ G/Ntype
3. #14527113C2semidirect
4. #14526693C2semidirect
294. C17#9513semidirect

Maximal subgroups

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

subgroupindex
#145321152
#145287122
#145271132
#145266932
C17×#28792
#951317