Finite Group Databasebeta

← #16934895

Subgroups — #16934895

Order 4352.

658 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 113 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
1C2×#11763144C211yesmaximal, characteristic, complemented
2#14534717C211yesmaximal, characteristic
3C2×#11762547C211yesmaximal, characteristic, complemented
4C2×#11762543C211yesmaximal, characteristic
5#14527052C211yesmaximal, characteristic, complemented
6#14526626C211yesmaximal, characteristic, complemented
7C2×C17×#481C211yesmaximal, characteristic
8C2×#10587610C2211yescharacteristic
9#11763144—22no—
10#11763144—22no—
11C2×#10587478—21no—
12C2×#10587481—21no—
13C2×C22⋊#3622C2211yescharacteristic
14#11762747—21no—
15#11762734—21no—
16#11762723—21no—
17#11762721—21no—
18#11762689—22no—
19C2×#10587502—22no—
20C2×#10587502C2211yescharacteristic
21C2×#10587502C2211yescharacteristic
22#11762547—22no—
23#11762543—22no—
24C2×C17×#121C2211yescharacteristic
25C22×C4⋊C4×C17C2211yescharacteristic
26C17×#481—22no—
27C17×#481—22no—
28C2×C17×#106—21no—
29C2×C17×#102—21no—
30C2×C17×C22⋊C8C2211yescharacteristic
31C22×D4×C17—21no—
32C22×D4×C17D411yescharacteristic
33C23×C4×C17—21no—
34C23×C4×C17C2311yescharacteristic
35C23×Dic17—21no—
36C23×Dic17D412yes—
37C22×Dic34D411yescharacteristic
38C22×C8×C17—21no—
39C17×#121—21no—
40C17×#121—21no—
41C17×#121—21no—
42C17×#121—21no—
43C2×C4⋊C4×C17—21no—
44C2×C4⋊C4×C17—21no—
45C2×C4⋊C4×C17—21no—
46C2×C4⋊C4×C17—21no—
47C2×C4⋊C4×C17—21no—
48C2×C4⋊C4×C17D412yes—
49C2×C4⋊C4×C17D411yescharacteristic
50C2×C22⋊C4×C17—21no—
51C2×C22⋊Dic17—21no—
52C2×C22⋊Dic17—22no—
53C2×C22⋊Dic17—22no—
54C2×C4⋊Dic17D411yescharacteristic
55#10587610—21no—
56#10587610—21no—
57#10587610—21no—
58#10587610—21no—
59C2×#63225—21no—
60C22×#3622—21no—
61C17×#106—24no—
62C17×#102—24no—
63C17×C22⋊C8—21no—
64C17×C22⋊C8D412yes—
65#10587502—22no—
66#10587502—22no—
67#10587502—21no—
68#10587502—21no—
69#10587502—21no—
70#10587502—21no—
71#10587502—44no—
72C22⋊#3622—21no—
73C22⋊#3622D412yes—
74#10587481—24no—
75#10587478—24no—
76C24×C17—21no—
77C24×C17C2×D411yescharacteristic
78C2×D4×C17—21no—
79C2×D4×C17—21no—
80C2×D4×C17—42no—
81C2×D4×C17—41no—
82C2×D4×C17—41no—
83C2×D4×C17—42no—
84C2×D4×C17—42no—
85C22×C4×C17—21no—
86C22×C4×C17—22no—
87C22×C4×C17—42no—
88C22×C4×C17—42no—
89C22×C4×C17—42no—
90C22×C4×C17—41no—
91C22×C4×C17—41no—
92C22×C4×C17C2×D411yescharacteristic
93C22×C4×C17C2×D411yescharacteristic
94C22×C4×C17C2×D411yescharacteristic
95C22×C4×C17C2×D411yescharacteristic
96C22×C4×C17C2×D411yescharacteristic
97C22×C4×C17#4111yescharacteristic
98C22×Dic17—22no—
99C22×Dic17—21no—
100C22×Dic17—22no—
101C22×Dic17—21no—
102C22×Dic17—22no—
103C22×Dic17—22no—
104C22×Dic17—42no—
105C22×Dic17—42no—
106C22×Dic17—41no—
107C22×Dic17—42no—
108C22×Dic17—41no—
109C22×Dic17—42no—
110C2×Dic34—21no—
111C2×Dic34—21no—
112C2×Dic34—42no—
113C2×C8×C17—22no—
114C2×C8×C17—42no—
115C4⋊C4×C17—22no—
116C4⋊C4×C17—22no—
117C4⋊C4×C17—21no—
118C4⋊C4×C17—21no—
119C4⋊C4×C17—41no—
120C4⋊C4×C17—41no—
121C22⋊C4×C17—41no—
122C22⋊C4×C17—41no—
123C22⋊Dic17—41no—
124C22⋊Dic17—41no—
125C22⋊Dic17—82no—
126C22⋊Dic17—82no—
127C4⋊Dic17—21no—
128C4⋊Dic17—21no—
129#63225—41no—
130#63225—41no—
131C2×#3622—22no—
132C2×#3622—42no—
133#9969—171nomaximal, Sylow
134C23×C17—21no—
135C23×C17—21no—
136C23×C17—22no—
137C23×C17—41no—
138C23×C17—42no—
139C23×C17—41no—
140C23×C17—41no—
141C23×C17—42no—
142C23×C17#12011yescharacteristic
143C23×C17#12111yescharacteristic
144C23×C17#12711yescharacteristic
145C23×C17#13512yes—
146C23×C17#13712yes—
147D4×C17—42no—
148D4×C17—81no—
149D4×C17—82no—
150D4×C17—81no—
151D4×C17—81no—
152C2×C4×C17—22no—
153C2×C4×C17—22no—
154C2×C4×C17—41no—
155C2×C4×C17—41no—
156C2×C4×C17—41no—
157C2×C4×C17—41no—
158C2×C4×C17—44no—
159C2×C4×C17—41no—
160C2×C4×C17—42no—
161C2×C4×C17—42no—
162C2×C4×C17—82no—
163C2×C4×C17—81no—
164C2×C4×C17—81no—
165C2×C4×C17#12011yescharacteristic
166C2×C4×C17#12011yescharacteristic
167C2×C4×C17#12111yescharacteristic
168C2×C4×C17#12111yescharacteristic
169C2×Dic17—44no—
170C2×Dic17—42no—
171C2×Dic17—44no—
172C2×Dic17—41no—
173C2×Dic17—41no—
174C2×Dic17—42no—
175C2×Dic17—41no—
176C2×Dic17—42no—
177C2×Dic17—41no—
178C2×Dic17—42no—
179C2×Dic17—42no—
180C2×Dic17—82no—
181C2×Dic17—81no—
182C2×Dic17—81no—
183C2×Dic17—82no—
184Dic34—42no—
185Dic34—81no—
186C8×C17—42no—
187#3622—42no—
188C2×#484—171no—
189C2×#481D1711yescharacteristic
190#2639—171no—
191C2×#328—171no—
192C2×#327—171no—
193#1445—171no—
194#1280—171no—
195C22×C17—22no—
196C22×C17—22no—
197C22×C17—22no—
198C22×C17—22no—
199C22×C17—21no—
200C22×C17—21no—
201C22×C17—21no—
202C22×C17—21no—
203C22×C17—82no—
204C22×C17—82no—
205C22×C17—81no—
206C22×C17—81no—
207C22×C17—81no—
208C22×C17#39211yescharacteristic
209C22×C17#45212yes—
210C22×C17#45311yescharacteristic
211C22×C17#45411yescharacteristic
212C22×C17#45812yes—
213C22×C17#46412yes—
214C22×C17#49511yescharacteristic
215C22×C17#49711yescharacteristic
216C4×C17—21no—
217C4×C17—21no—
218C4×C17—21no—
219C4×C17—21no—
220C4×C17—82no—
221C4×C17—81no—
222Dic17—82no—
223Dic17—82no—
224Dic17—81no—
225Dic17—81no—
226C2×C22⋊Q8—171no—
227C2×#121C2×D1711yescharacteristic
228C22×C4⋊C4C2×D1711yescharacteristic
229#484—342no—
230#484—342no—
231#481—22no—
232#481—22no—
233C2×#107—341no—
234C2×#106—21no—
235C2×#103—341no—
236C2×#102—21no—
237C2×C22⋊C8—171no—
238C2×C22⋊C8C2×D1711yescharacteristic
239#400—341no—
240#397—341no—
241#394—341no—
242#393—341no—
243#386—342no—
244C2×#95—171no—
245C2×#95—171no—
246C2×#95—342no—
247#328—342no—
248#327—342no—
249C2×C17—21no—
250C2×C17—22no—
251C2×C17—21no—
252C2×C17—161no—
253C2×C17#161512yes—
254C2×C17#161812yes—
255C2×C17#200611yescharacteristic
256C2×C17#201511yescharacteristic
257C2×C17#202211yescharacteristic
258C22×Q8—171no—
259C22×D4—21no—
260C22×D4C22⋊D1711yescharacteristic
261C23×C4—21no—
262C23×C4—172no—
263C23×C4—341no—
264C23×C4C22×D1711yescharacteristic
265C22×C8—21no—
266C22×C8—341no—
267C22⋊Q8—341no—
268C22⋊Q8—341no—
269C22⋊Q8—341no—
270C22⋊Q8—341no—
271#121—21no—
272#121—21no—
273#121—21no—
274#121—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—171no—
281C2×C4⋊C4—341no—
282C2×C4⋊C4C22⋊D1712yes—
283C2×C4⋊C4C22⋊D1711yescharacteristic
284C2×C22⋊C4—21no—
285C2×C22⋊C4—341no—
286C2×C22⋊C4—342no—
287C2×C22⋊C4—342no—
288#107—344no—
289#106—24no—
290#103—344no—
291#102—24no—
292C22⋊C8—21no—
293C22⋊C8—172no—
294C22⋊C8—341no—
295C22⋊C8C22⋊D1712yes—
296#95—342no—
297#95—342no—
298#95—341no—
299#95—341no—
300#95—341no—
301#95—341no—
302#95—684no—
303C17#996911yescharacteristic, complemented, Sylow
304C24—21no—
305C24D4×D1711yescharacteristic
306C2×Q8—341no—
307C2×Q8—341no—
308C2×Q8—682no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—42no—
314C2×D4—41no—
315C2×D4—41no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—342no—
324C22×C4—342no—
325C22×C4—342no—
326C22×C4—341no—
327C22×C4—341no—
328C22×C4—342no—
329C22×C4—682no—
330C22×C4—682no—
331C22×C4—681no—
332C22×C4—682no—
333C22×C4—682no—
334C22×C4—681no—
335C22×C4D4×D1711yescharacteristic
336C22×C4D4×D1711yescharacteristic
337C22×C4#6325411yescharacteristic
338C22×C4C2×C22⋊D1711yescharacteristic
339C22×C4C2×C22⋊D1711yescharacteristic
340C22×C4C2×C22⋊D1711yescharacteristic
341C2×C8—22no—
342C2×C8—42no—
343C2×C8—342no—
344C2×C8—682no—
345C4⋊C4—22no—
346C4⋊C4—21no—
347C4⋊C4—22no—
348C4⋊C4—21no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—341no—
352C4⋊C4—341no—
353C4⋊C4—681no—
354C4⋊C4—681no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C22⋊C4—681no—
358C22⋊C4—681no—
359C22⋊C4—1362no—
360C22⋊C4—1362no—
361C23—21no—
362C23—21no—
363C23—22no—
364C23—42no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—42no—
369C23#1058760112yes—
370C23#1058760212yes—
371C23#1058762311yescharacteristic
372C23#1058762511yescharacteristic
373C23#1058762611yescharacteristic
374Q8—682no—
375Q8—1361no—
376D4—42no—
377D4—81no—
378D4—81no—
379D4—82no—
380D4—81no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—82no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—682no—
395C2×C4—684no—
396C2×C4—684no—
397C2×C4—682no—
398C2×C4—681no—
399C2×C4—681no—
400C2×C4—681no—
401C2×C4—682no—
402C2×C4—681no—
403C2×C4—682no—
404C2×C4—682no—
405C2×C4—1362no—
406C2×C4—1361no—
407C2×C4—1361no—
408C2×C4—1362no—
409C2×C4#1058762311yescharacteristic
410C2×C4#1058762411yescharacteristic
411C2×C4#1058762511yescharacteristic
412C2×C4#1058763911yescharacteristic
413C8—42no—
414C8—682no—
415C22—21no—
416C22—21no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—21no—
422C22—21no—
423C22—81no—
424C22—82no—
425C22—81no—
426C22—82no—
427C22—81no—
428C22#1176271711yescharacteristic
429C22#1176296812yes—
430C22#1176297511yescharacteristic
431C22#1176298011yescharacteristic
432C22#1176299812yes—
433C22#1176302912yes—
434C22#1176319011yescharacteristic
435C22#1176320011yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—82no—
441C4—81no—
442C4—1362no—
443C4—1361no—
444C4—1362no—
445C4—1361no—
446C2—21no—
447C2—21no—
448C2—22no—
449C2—161no—
450C2#1452781212yes—
451C2#1452782912yes—
452C2#1452982411yescharacteristic
453C2#1452987611yescharacteristic
454C2#1452991211yescharacteristic
subgroupG/H
centerC23#10587623
commutator, cosocleC23×C4×C17C23
FrattiniC23×C4C22×D17
FittingC2×C17×#481C2
socleC23×C17#120
prime radicalC2×C17×#121C22
hypercenterC2×#481D17
hypocenterC17#9969
LCSG ▷ C23×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#481

Complements

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

NH ≅ G/Ntype
1. C2×#11763144C2semidirect
3. C2×#11762547C2semidirect
5. #14527052C2semidirect
6. #14526626C2semidirect
303. C17#9969semidirect

Maximal subgroups

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

subgroupindex
C2×#117631442
#145347172
C2×#117625472
C2×#117625432
#145270522
#145266262
C2×C17×#4812
#996917