Finite Group Databasebeta

← #16936747

Subgroups — #16936747

Order 4352.

556 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 105 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#14539928C211yesmaximal, characteristic, complemented
2#14538581C211yesmaximal, characteristic
3#14527456C211yesmaximal, characteristic, complemented
4#14527282C211yesmaximal, characteristic, complemented
5#14527175C211yesmaximal, characteristic
6#14526687C211yesmaximal, characteristic, complemented
7C17×#2547C211yesmaximal, characteristic
8#11763366C2211yescharacteristic
9#11763205—21no—
10#11763161—21no—
11#11763139—22no—
12#11763109—22no—
13#11763099—21no—
14#11763091—22no—
15#11763089—21no—
16#11762862C2211yescharacteristic
17#11762862C2211yescharacteristic
18#11762849—21no—
19#11762833C2211yescharacteristic
20#11762814—21no—
21#11762555—22no—
22#11762545—22no—
23D4⋊#3622—22no—
24C2×C17×C4⋊Q8C2211yescharacteristic
25C2×C17×#124—21no—
26C2×C4×D4×C17C2211yescharacteristic
27C17×#397—22no—
28C17×#391—21no—
29C17×#380—22no—
30C4×C4⋊C4×C17C2211yescharacteristic
31C22×Q8×C17—21no—
32C22×D4×C17D411yescharacteristic
33C23×C4×C17—21no—
34C17×C4⋊Q8—21no—
35C17×C4⋊Q8—21no—
36C17×C4⋊Q8—21no—
37C17×C4⋊Q8—21no—
38C17×#124—42no—
39C17×#124—42no—
40C4×D4×C17—21no—
41C4×D4×C17—21no—
42C4×D4×C17—21no—
43C4×D4×C17—21no—
44C2×C4⋊C4×C17—21no—
45C2×C4⋊C4×C17—21no—
46C2×C4⋊C4×C17D411yescharacteristic
47C2×C4⋊C4×C17D412yes—
48C2×C22⋊C4×C17—24no—
49C2×C22⋊C4×C17—21no—
50C2×C42×C17—21no—
51C2×C42×C17C2311yescharacteristic
52#10587612—21no—
53#10587610—21no—
54#10587610—21no—
55C2×#63223—21no—
56#10587556—21no—
57#10587555D411yescharacteristic
58#10587555D411yescharacteristic
59C4×Dic34—21no—
60C4×Dic34—21no—
61C17×#95—24no—
62C17×#95D412yes—
63#10587503—22no—
64#10587503—22no—
65C22⋊#3622—21no—
66C22⋊#3622—21no—
67#10587481—24no—
68#10587479—24no—
69#10587479—22no—
70#10587478—22no—
71C4⋊#3622—21no—
72C4⋊#3622—21no—
73C4⋊#3622D412yes—
74C4⋊#3622D412yes—
75C4×#3622—21no—
76C4×#3622—21no—
77C24×C17—21no—
78C2×Q8×C17—41no—
79C2×Q8×C17—41no—
80C2×Q8×C17—41no—
81C2×Q8×C17—41no—
82C2×Q8×C17—41no—
83C2×Q8×C17—41no—
84C2×D4×C17—21no—
85C2×D4×C17—21no—
86C2×D4×C17—21no—
87C2×D4×C17—21no—
88C2×D4×C17—21no—
89C2×D4×C17SD1612yes—
90C22×C4×C17—21no—
91C22×C4×C17—22no—
92C22×C4×C17—22no—
93C22×C4×C17—44no—
94C22×C4×C17—41no—
95C22×C4×C17—41no—
96C22×C4×C17C2×D411yescharacteristic
97C22×C4×C17C2×D411yescharacteristic
98C22×C4×C17C2×D411yescharacteristic
99C2×Dic34—21no—
100C2×Dic34—21no—
101C4⋊C4×C17—22no—
102C4⋊C4×C17—21no—
103C4⋊C4×C17—22no—
104C4⋊C4×C17—21no—
105C4⋊C4×C17—42no—
106C4⋊C4×C17—41no—
107C4⋊C4×C17—41no—
108C22⋊C4×C17—48no—
109C22⋊C4×C17—41no—
110C22⋊C4×C17—41no—
111C42×C17—41no—
112C42×C17—41no—
113C42×C17C2×D411yescharacteristic
114C42×C17C2×D411yescharacteristic
115C42×C17C2×D411yescharacteristic
116C42×C17#4111yescharacteristic
117C22⋊Dic17—41no—
118C4⋊Dic17—21no—
119C4⋊Dic17—21no—
120C4⋊Dic17—21no—
121C4⋊Dic17—21no—
122#63225—41no—
123#63225—41no—
124C4×Dic17—41no—
125#63223—41no—
126#63223—41no—
127C2×#3622—22no—
128C2×#3622—22no—
129C2×#3622—41no—
130#10247—171nomaximal, Sylow
131C23×C17—22no—
132C23×C17—42no—
133C23×C17—42no—
134C23×C17—41no—
135C23×C17—41no—
136C23×C17#12011yescharacteristic
137Q8×C17—81no—
138Q8×C17—81no—
139Q8×C17—81no—
140Q8×C17—81no—
141D4×C17—22no—
142D4×C17—22no—
143D4×C17—41no—
144D4×C17—41no—
145C2×C4×C17—22no—
146C2×C4×C17—41no—
147C2×C4×C17—42no—
148C2×C4×C17—42no—
149C2×C4×C17—42no—
150C2×C4×C17—41no—
151C2×C4×C17—42no—
152C2×C4×C17—42no—
153C2×C4×C17—42no—
154C2×C4×C17—42no—
155C2×C4×C17—82no—
156C2×C4×C17—82no—
157C2×C4×C17#12011yescharacteristic
158C2×C4×C17#12011yescharacteristic
159C2×C4×C17#12111yescharacteristic
160C2×C4×C17#12111yescharacteristic
161C2×C4×C17#12111yescharacteristic
162C2×C4×C17#12711yescharacteristic
163C2×C4×C17C2×SD1611yescharacteristic
164C2×C4×C17#13511yescharacteristic
165C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
166C2×C4×C17#13711yescharacteristic
167C2×Dic17—41no—
168C2×Dic17—41no—
169Dic34—42no—
170Dic34—81no—
171#3622—42no—
172#3622—42no—
173#3622—81no—
174#3229—171no—
175#3077—171no—
176#2547D1711yescharacteristic
177#1546—171no—
178#1503—171no—
179#1478—171no—
180#1303—171no—
181C22×C17—21no—
182C22×C17—21no—
183C22×C17—44no—
184C22×C17—42no—
185C22×C17—82no—
186C22×C17—81no—
187C22×C17—81no—
188C22×C17#39211yescharacteristic
189C22×C17#45011yescharacteristic
190C22×C17#45211yescharacteristic
191C4×C17—21no—
192C4×C17—21no—
193C4×C17—21no—
194C4×C17—82no—
195C4×C17—82no—
196C4×C17—81no—
197C4×C17#44811yescharacteristic
198C4×C17#44911yescharacteristic
199C4×C17#45412yes—
200C4×C17#46511yescharacteristic
201C4×C17#46811yescharacteristic
202Dic17—81no—
203Dic17—81no—
204#533—171no—
205C2×C4⋊Q8C2×D1711yescharacteristic
206C2×#124—21no—
207C2×C4×D4C2×D1711yescharacteristic
208#498—341no—
209#487—341no—
210#483—342no—
211#478—342no—
212#476—341no—
213#475—342no—
214#474—341no—
215#431—171no—
216#431—171no—
217#428—341no—
218#423—171no—
219#418—341no—
220#397—22no—
221#391—21no—
222#380—22no—
223C4×C4⋊C4C2×D1711yescharacteristic
224#332—342no—
225#328—342no—
226D4⋊C8—342no—
227C2×C17—21no—
228C2×C17—21no—
229C2×C17—82no—
230C2×C17#200011yescharacteristic
231C2×C17#200311yescharacteristic
232C2×C17#200911yescharacteristic
233C22×Q8—21no—
234C22×D4C22⋊D1711yescharacteristic
235C23×C4—21no—
236C2×M16—341no—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—171no—
242C4⋊Q8—171no—
243#125—341no—
244#124—42no—
245#124—42no—
246C22⋊Q8—341no—
247C22⋊Q8—341no—
248C4×Q8—341no—
249C4×Q8—341no—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4—21no—
253C4×D4—21no—
254#117—341no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4C22⋊D1711yescharacteristic
258C2×C4⋊C4C22⋊D1712yes—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—21no—
261C2×C42—21no—
262C2×C42C22×D1711yescharacteristic
263#107—342no—
264#106—342no—
265#106—344no—
266C4⋊C8—172no—
267C4⋊C8—172no—
268C4⋊C8—341no—
269C4⋊C8—341no—
270#103—344no—
271#102—342no—
272#102—342no—
273C22⋊C8—341no—
274C22⋊C8—341no—
275C4×C8—341no—
276C4×C8—341no—
277#95—24no—
278#95C22⋊D1712yes—
279C17#1024711yescharacteristic, complemented, Sylow
280C24—21no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×Q8—41no—
287C2×Q8—341no—
288C2×Q8—341no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4#6322912yes—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—44no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4C2×C22⋊D1711yescharacteristic
302C22×C4C2×C22⋊D1711yescharacteristic
303C22×C4C2×C22⋊D1711yescharacteristic
304M16—681no—
305M16—681no—
306C2×C8—342no—
307C2×C8—342no—
308C2×C8—681no—
309C4⋊C4—22no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—22no—
313C4⋊C4—42no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—341no—
317C4⋊C4—341no—
318C4⋊C4—341no—
319C4⋊C4—341no—
320C4⋊C4—681no—
321C4⋊C4—681no—
322C22⋊C4—48no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—681no—
326C42—41no—
327C42—41no—
328C42—681no—
329C42D4×D1711yescharacteristic
330C42D4×D1711yescharacteristic
331C42D4×D1711yescharacteristic
332C42#6325411yescharacteristic
333C23—22no—
334C23—42no—
335C23—42no—
336C23—41no—
337C23—41no—
338C23#1058763911yescharacteristic
339Q8—81no—
340Q8—81no—
341Q8—81no—
342Q8—81no—
343Q8—682no—
344Q8—1361no—
345D4—22no—
346D4—22no—
347D4—41no—
348D4—41no—
349C2×C4—22no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—681no—
362C2×C4—681no—
363C2×C4#1058761811yescharacteristic
364C2×C4C2×#6322911yescharacteristic
365C2×C4#1058762311yescharacteristic
366C2×C4#1058762311yescharacteristic
367C2×C4#1058762411yescharacteristic
368C2×C4#1058762511yescharacteristic
369C2×C4#1058762511yescharacteristic
370C2×C4#1058762611yescharacteristic
371C2×C4#1058763611yescharacteristic
372C2×C4#1058763711yescharacteristic
373C8—682no—
374C8—682no—
375C8—1361no—
376C22—21no—
377C22—21no—
378C22—44no—
379C22—42no—
380C22—82no—
381C22—81no—
382C22—81no—
383C22#1176271711yescharacteristic
384C22#1176296011yescharacteristic
385C22#1176297111yescharacteristic
386C4—21no—
387C4—21no—
388C4—21no—
389C4—82no—
390C4—81no—
391C4—82no—
392C4—1361no—
393C4—1361no—
394C4#1176294711yescharacteristic
395C4#1176295211yescharacteristic
396C4#1176298112yes—
397C4#1176303611yescharacteristic
398C4#1176305311yescharacteristic
399C2—21no—
400C2—21no—
401C2—82no—
402C2#1452978711yescharacteristic
403C2#1452980611yescharacteristic
404C2#1452984411yescharacteristic
subgroupG/H
centerC22#11762717
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#2547C2
socleC22×C17#392
prime radicalC22×D4×C17D4
hypercenter#2547D17
hypocenterC17#10247
LCSG ▷ C2×C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2547

Complements

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

NH ≅ G/Ntype
1. #14539928C2semidirect
3. #14527456C2semidirect
4. #14527282C2semidirect
6. #14526687C2semidirect
279. C17#10247semidirect

Maximal subgroups

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

subgroupindex
#145399282
#145385812
#145274562
#145272822
#145271752
#145266872
C17×#25472
#1024717