Finite Group Databasebeta

← #16937481

Subgroups — #16937481

Order 4352.

466 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 95 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#14533795C211yesmaximal, characteristic
2#14529770C211yesmaximal, characteristic
3#14527356C211yesmaximal, characteristic
4#14526663C211yesmaximal, characteristic
5#14526652C211yesmaximal, characteristic
6#14526630C211yesmaximal, characteristic
7C2×C17×#328C211yesmaximal, characteristic
8C2×#10587610—21no—
9C2×C22⋊#3622C2211yescharacteristic
10#11762743—21no—
11#11762730—21no—
12#11762713—21no—
13#11762706—21no—
14#11762682—21no—
15#11762669C2211yescharacteristic
16#11762669C2211yescharacteristic
17C4×C22⋊Dic17C2211yescharacteristic
18#11762576—21no—
19#11762576—21no—
20#11762546—21no—
21#11762546—21no—
22C2×C17×C22⋊Q8C2211yescharacteristic
23C2×C17×C22⋊C8C2211yescharacteristic
24C2×C17×#95C2211yescharacteristic
25C17×#328—21no—
26C17×#328—21no—
27C22×Q8×C17Q811yescharacteristic, complemented
28C23×C4×C17—21no—
29C23×C4×C17C2311yescharacteristic
30C22×Dic34—21no—
31C22×C8×C17—21no—
32C17×C22⋊Q8—21no—
33C17×C22⋊Q8—21no—
34C17×C22⋊Q8—21no—
35C17×C22⋊Q8—21no—
36C2×C4⋊C4×C17—21no—
37C2×C4⋊C4×C17Q811yescharacteristic, complemented
38C2×C22⋊C4×C17—21no—
39C2×C22⋊Dic17—21no—
40C2×C22⋊Dic17—21no—
41C2×C22⋊Dic17D411yescharacteristic
42C2×C22⋊Dic17D411yescharacteristic
43C2×C4⋊Dic17—21no—
44C2×C4⋊Dic17—21no—
45C2×C4⋊Dic17—21no—
46#10587610—42no—
47#10587610—42no—
48C2×#63225—21no—
49C2×#63225—21no—
50C2×#63225—21no—
51C2×#63225—21no—
52C2×C4×Dic17—21no—
53C22×#3622—21no—
54C17×C22⋊C8—21no—
55C17×C22⋊C8—21no—
56C17×#95—21no—
57C17×#95D411yescharacteristic
58C17×#95D411yescharacteristic
59#10587502—21no—
60#10587502—21no—
61#10587502—21no—
62#10587502—21no—
63#10587502D411yescharacteristic
64#10587502D411yescharacteristic
65C22⋊#3622—21no—
66C22⋊#3622—21no—
67C24×C17C2×D411yescharacteristic
68C2×Q8×C17—21no—
69C2×Q8×C17—21no—
70C2×Q8×C17—81no—
71C22×C4×C17—21no—
72C22×C4×C17—21no—
73C22×C4×C17—21no—
74C22×C4×C17—21no—
75C22×C4×C17—42no—
76C22×C4×C17—41no—
77C22×C4×C17—41no—
78C22×C4×C17—81no—
79C22×C4×C17—81no—
80C22×C4×C17C2×D411yescharacteristic
81C22×C4×C17C2×D411yescharacteristic
82C22×C4×C17C2×Q811yescharacteristic
83C22×C4×C17#4111yescharacteristic
84C22×C4×C17#4111yescharacteristic
85C22×C4×C17#4111yescharacteristic
86C22×Dic17—21no—
87C22×Dic17—21no—
88C22×Dic17—21no—
89C22×Dic17—21no—
90C22×Dic17—41no—
91C2×Dic34—42no—
92C2×Dic34—42no—
93C2×Dic34—81no—
94C2×C8×C17—41no—
95C2×C8×C17—81no—
96C4⋊C4×C17—21no—
97C4⋊C4×C17—21no—
98C4⋊C4×C17—41no—
99C4⋊C4×C17—41no—
100C22⋊C4×C17—41no—
101C22⋊C4×C17—41no—
102C22⋊Dic17—21no—
103C22⋊Dic17—21no—
104C22⋊Dic17—41no—
105C22⋊Dic17—41no—
106C22⋊Dic17—41no—
107C22⋊Dic17—41no—
108C22⋊Dic17SD1612yes—
109C22⋊Dic17Q1612yes—
110C4⋊Dic17—42no—
111C4⋊Dic17—81no—
112C4⋊Dic17—81no—
113#63225—42no—
114#63225—42no—
115#63225—42no—
116#63225—81no—
117C4×Dic17—81no—
118C2×#3622—41no—
119C2×#3622—81no—
120#10375—171nomaximal, Sylow
121C23×C17—41no—
122C23×C17—41no—
123C23×C17#12011yescharacteristic
124C23×C17C22⋊Q811yescharacteristic
125C23×C17#12411yescharacteristic
126C23×C17C2×SD1611yescharacteristic
127C23×C17C2×Q1611yescharacteristic
128C23×C17AGL(1,ℤ/8)11yescharacteristic
129C23×C17#13711yescharacteristic
130Q8×C17—81no—
131Q8×C17—81no—
132C2×C4×C17—22no—
133C2×C4×C17—41no—
134C2×C4×C17—41no—
135C2×C4×C17—41no—
136C2×C4×C17—41no—
137C2×C4×C17—41no—
138C2×C4×C17—41no—
139C2×C4×C17—41no—
140C2×C4×C17—41no—
141C2×C4×C17—41no—
142C2×C4×C17—41no—
143C2×C4×C17—41no—
144C2×C4×C17—41no—
145C2×C4×C17—41no—
146C2×C4×C17—82no—
147C2×C4×C17—82no—
148C2×C4×C17—81no—
149C2×C4×C17—81no—
150C2×C4×C17C22⋊Q811yescharacteristic
151C2×C4×C17C22⋊Q811yescharacteristic
152C2×C4×C17#12411yescharacteristic
153C2×C4×C17#12411yescharacteristic
154C2×Dic17—22no—
155C2×Dic17—22no—
156C2×Dic17—41no—
157C2×Dic17—41no—
158C2×Dic17—41no—
159C2×Dic17—41no—
160C2×Dic17—41no—
161C2×Dic17—41no—
162C2×Dic17—41no—
163C2×Dic17—41no—
164C2×Dic17—41no—
165C2×Dic17—41no—
166C2×Dic17—81no—
167C2×Dic17—81no—
168C2×Dic17—81no—
169Dic34—84no—
170C8×C17—81no—
171#3622—81no—
172#2549—171no—
173#1996—171no—
174#1520—171no—
175C2×#328D1711yescharacteristic
176#1294—171no—
177#1289—171no—
178#1282—171no—
179C22×C17—21no—
180C22×C17—21no—
181C22×C17—21no—
182C22×C17—21no—
183C22×C17—21no—
184C22×C17—21no—
185C22×C17—41no—
186C22×C17—41no—
187C22×C17—41no—
188C22×C17—41no—
189C22×C17#39311yescharacteristic
190C22×C17#45011yescharacteristic
191C22×C17#45111yescharacteristic
192C22×C17#47511yescharacteristic
193C22×C17#47711yescharacteristic
194C22×C17#48711yescharacteristic
195C22×C17#48911yescharacteristic
196C4×C17—21no—
197C4×C17—21no—
198C4×C17—21no—
199C4×C17—21no—
200C4×C17—42no—
201C4×C17—81no—
202C4×C17—81no—
203C4×C17—81no—
204Dic17—42no—
205Dic17—42no—
206Dic17—81no—
207Dic17—81no—
208Dic17—161no—
209C2×C22⋊Q8—341no—
210C2×C22⋊Q8C2×D1711yescharacteristic
211C2×C22⋊C8—171no—
212C2×C22⋊C8C2×D1711yescharacteristic
213#398—341no—
214#395—341no—
215#391—341no—
216#389—341no—
217#385—341no—
218#380—171no—
219#380—171no—
220C4×C22⋊C4—171no—
221C2×#95C2×D1711yescharacteristic
222#342—341no—
223#342—341no—
224#328—21no—
225#328—21no—
226#328—341no—
227#328—341no—
228C2×C17—21no—
229C2×C17—21no—
230C2×C17—21no—
231C2×C17—21no—
232C2×C17—21no—
233C2×C17—21no—
234C2×C17#202411yescharacteristic
235C2×C17#203111yescharacteristic
236C2×C17#203211yescharacteristic
237C22×Q8—341no—
238C22×Q8Dic3411yescharacteristic, complemented
239C23×C4—21no—
240C23×C4C22×D1711yescharacteristic
241C22×C8—21no—
242C22×C8—341no—
243C22⋊Q8—21no—
244C22⋊Q8—21no—
245C22⋊Q8—21no—
246C22⋊Q8—21no—
247C22⋊Q8—682no—
248C22⋊Q8—682no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—341no—
251C2×C4⋊C4—341no—
252C2×C4⋊C4—341no—
253C2×C4⋊C4—341no—
254C2×C4⋊C4—341no—
255C2×C4⋊C4—341no—
256C2×C4⋊C4—341no—
257C2×C4⋊C4Dic3411yescharacteristic, complemented
258C2×C22⋊C4—21no—
259C2×C22⋊C4—171no—
260C2×C22⋊C4—171no—
261C2×C22⋊C4—341no—
262C2×C22⋊C4—341no—
263C2×C42—341no—
264C22⋊C8—21no—
265C22⋊C8—21no—
266C22⋊C8—341no—
267C22⋊C8—341no—
268#95—21no—
269#95—171no—
270#95—171no—
271#95—341no—
272#95—341no—
273#95—341no—
274#95—341no—
275#95C22⋊D1711yescharacteristic
276#95C22⋊D1711yescharacteristic
277C17#1037511yescharacteristic, complemented, Sylow
278C24D4×D1711yescharacteristic
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—81no—
282C2×Q8—682no—
283C2×Q8—682no—
284C2×Q8—1361no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—81no—
293C22×C4—81no—
294C22×C4—341no—
295C22×C4—341no—
296C22×C4—341no—
297C22×C4—341no—
298C22×C4—681no—
299C22×C4C2×Dic3411yescharacteristic
300C22×C4#6325211yescharacteristic
301C22×C4D4×D1711yescharacteristic
302C22×C4#6325411yescharacteristic
303C22×C4#6325411yescharacteristic
304C22×C4C2×C22⋊D1711yescharacteristic
305C2×C8—41no—
306C2×C8—81no—
307C2×C8—681no—
308C2×C8—1361no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—682no—
314C4⋊C4—682no—
315C4⋊C4—682no—
316C4⋊C4—682no—
317C4⋊C4—1361no—
318C4⋊C4—1361no—
319C4⋊C4—1361no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—172no—
323C22⋊C4—172no—
324C22⋊C4—341no—
325C22⋊C4—341no—
326C22⋊C4—681no—
327C22⋊C4—681no—
328C22⋊C4—681no—
329C22⋊C4—681no—
330C42—1361no—
331C23—41no—
332C23—41no—
333C23#1058756411yescharacteristic
334C23#1058757111yescharacteristic
335C23SD16×D1711yescharacteristic
336C23#1058760011yescharacteristic
337C23Q16×D1711yescharacteristic
338C23#1058760411yescharacteristic
339C23#1058762311yescharacteristic
340Q8—81no—
341Q8—81no—
342Q8—1364no—
343C2×C4—22no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—342no—
362C2×C4—342no—
363C2×C4—681no—
364C2×C4—681no—
365C2×C4—681no—
366C2×C4—681no—
367C2×C4—681no—
368C2×C4—681no—
369C2×C4—681no—
370C2×C4—681no—
371C2×C4—681no—
372C2×C4—681no—
373C2×C4—1361no—
374C2×C4—1361no—
375C2×C4—1361no—
376C2×C4#1058756411yescharacteristic
377C2×C4#1058757111yescharacteristic
378C2×C4#1058761011yescharacteristic
379C2×C4#1058762211yescharacteristic
380C8—81no—
381C8—1361no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22#1176272111yescharacteristic
393C22#1176295811yescharacteristic
394C22#1176296311yescharacteristic
395C22#1176309211yescharacteristic
396C22#1176310211yescharacteristic
397C22#1176315811yescharacteristic
398C22#1176317011yescharacteristic
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—42no—
404C4—81no—
405C4—81no—
406C4—81no—
407C4—2721no—
408C4—682no—
409C4—682no—
410C4—1361no—
411C4—1361no—
412C2—21no—
413C2—21no—
414C2—21no—
415C2—21no—
416C2—21no—
417C2—21no—
418C2#1452992311yescharacteristic
419C2#1452995611yescharacteristic
420C2#1452996111yescharacteristic
subgroupG/H
centerC22#11762721
commutator, cosocleC23×C4×C17C23
FrattiniC23×C4C22×D17
FittingC2×C17×#328C2
socleC22×C17#393
prime radicalC24×C17C2×D4
hypercenterC2×#328D17
hypocenterC17#10375
LCSG ▷ C23×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#328

Complements

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

NH ≅ G/Ntype
27. C22×Q8×C17Q8semidirect
37. C2×C4⋊C4×C17Q8semidirect
238. C22×Q8Dic34semidirect
257. C2×C4⋊C4Dic34semidirect
277. C17#10375semidirect

Maximal subgroups

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

subgroupindex
#145337952
#145297702
#145273562
#145266632
#145266522
#145266302
C2×C17×#3282
#1037517