Finite Group Databasebeta

← #16923147

Subgroups — #16923147

Order 4352.

504 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 85 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#14535802C211yesmaximal, characteristic
2#14530113C211yesmaximal, characteristic, complemented
3#14529921C211yesmaximal, characteristic
4#14529883C211yesmaximal, characteristic, complemented
5#14526606C211yesmaximal, characteristic, complemented
6#14526604C211yesmaximal, characteristic
7C17×#1272C211yesmaximal, characteristic, complemented
8C22×#63231—21no—
9C2×#10587555C2211yescharacteristic
10C2×#10587632—21no—
11C2×#10587489—21no—
12C2×#10587481—21no—
13C2×#10587492—21no—
14#11762746—21no—
15#11762741—21no—
16#11762731—21no—
17#11762711C2211yescharacteristic
18#11762707C2211yescharacteristic
19#11762687—21no—
20#11762683—21no—
21#11762572—21no—
22#11762572—21no—
23#11762564C2211yescharacteristic
24C17×#391C2211yescharacteristic
25C17×#336C2211yescharacteristic
26C17×#336C2211yescharacteristic
27C22×Q8×C17D411yescharacteristic
28C22×D68D411yescharacteristic
29C22×C4×D17—21no—
30C22×Dic34—21no—
31C22×C8×C17—21no—
32C22×C8×C17—21no—
33C2×C4⋊C4×C17D411yescharacteristic
34C2×C42×C17C2311yescharacteristic
35#10587632—42no—
36#10587632—42no—
37C2×#63231—42no—
38C2×#63231—42no—
39C2×#63231—42no—
40C2×#63227—21no—
41C2×#63227—21no—
42C2×#63227—21no—
43C2×#63227—21no—
44C2×#63227—21no—
45C2×C4⋊Dic17—21no—
46C2×C4⋊Dic17—21no—
47C2×C4⋊Dic17—21no—
48C2×C4⋊Dic17—21no—
49C2×C4⋊Dic17D411yescharacteristic
50C2×C4⋊Dic17D411yescharacteristic
51C2×C4⋊Dic17D411yescharacteristic
52C2×#63225—21no—
53C2×#63225—21no—
54C22×#3622—21no—
55#10587555—42no—
56C17×#95—21no—
57C17×#95—21no—
58#10587502—21no—
59#10587502—21no—
60#10587492—42no—
61#10587489—42no—
62#10587481—42no—
63C23×D17—21no—
64C2×Q8×C17—21no—
65C2×Q8×C17—41no—
66C2×Q8×C17—41no—
67C2×Q8×C17D812yes—
68C22×C4×C17—21no—
69C22×C4×C17—21no—
70C22×C4×C17C2×D411yescharacteristic
71C22×C4×C17C2×D411yescharacteristic
72C22×C4×C17C2×D411yescharacteristic
73C22×Dic17—21no—
74C22×Dic17—21no—
75C22×Dic17—21no—
76C22×Dic17—41no—
77C2×D68—21no—
78C2×D68—41no—
79C2×D68—41no—
80C2×D68Q1612yescomplemented
81C2×C4×D17—42no—
82C2×C4×D17—81no—
83C2×C4×D17—81no—
84C2×Dic34—22no—
85C2×Dic34—41no—
86C2×Dic34—41no—
87C2×Dic34—41no—
88C2×Dic34—41no—
89C2×Dic34—41no—
90C2×C8×C17—22no—
91C2×C8×C17—41no—
92C2×C8×C17—41no—
93C2×C8×C17—41no—
94C2×C8×C17—41no—
95C2×C8×C17—41no—
96C4⋊C4×C17—41no—
97C42×C17—41no—
98#63231—48no—
99#63227—42no—
100#63227—41no—
101#63227—41no—
102#63227—41no—
103#63227—41no—
104#63227—81no—
105#63227—81no—
106C4⋊Dic17—22no—
107C4⋊Dic17—22no—
108C4⋊Dic17—21no—
109C4⋊Dic17—21no—
110C4⋊Dic17—41no—
111C4⋊Dic17—41no—
112C4⋊Dic17—42no—
113C4⋊Dic17—41no—
114C4⋊Dic17—41no—
115C4⋊Dic17—81no—
116#63225—42no—
117#63225—41no—
118#63225—41no—
119C2×#3622—22no—
120C2×#3622—41no—
121C2×#3622—41no—
122#7756—171nomaximal, Sylow
123C23×C17#12011yescharacteristic
124C22×D17—22no—
125C22×D17—41no—
126C22×D17—41no—
127C22×D17—41no—
128C22×D17—41no—
129C22×D17—41no—
130C22×D17—41no—
131Q8×C17—42no—
132Q8×C17—81no—
133C2×C4×C17—21no—
134C2×C4×C17—21no—
135C2×C4×C17—21no—
136C2×C4×C17—21no—
137C2×C4×C17—21no—
138C2×C4×C17—21no—
139C2×C4×C17—22no—
140C2×C4×C17—41no—
141C2×C4×C17—41no—
142C2×C4×C17—41no—
143C2×C4×C17—41no—
144C2×C4×C17—41no—
145C2×C4×C17C2×D811yescharacteristic
146C2×C4×C17C2×Q1611yescharacteristic
147C2×C4×C17#13511yescharacteristic
148C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C17#13711yescharacteristic
150C2×C4×C17#13711yescharacteristic
151C2×Dic17—41no—
152C2×Dic17—41no—
153C2×Dic17—41no—
154C2×Dic17—41no—
155C2×Dic17—41no—
156C2×Dic17—41no—
157C2×Dic17—41no—
158C2×Dic17—41no—
159C2×Dic17—42no—
160C2×Dic17—41no—
161C2×Dic17—81no—
162C2×Dic17—81no—
163D68—42no—
164D68—81no—
165C4×D17—84no—
166Dic34—42no—
167Dic34—42no—
168Dic34—81no—
169Dic34—81no—
170C8×C17—42no—
171C8×C17—81no—
172#3622—42no—
173#2732—171no—
174#2057—171no—
175#2023—171no—
176#2016—171no—
177#1272D1711yescharacteristic, complemented
178#1271—171no—
179#1270—171no—
180C22×C17#44711yescharacteristic
181C22×C17#45111yescharacteristic
182C22×C17#45211yescharacteristic
183C22×C17#45311yescharacteristic
184C22×C17#45511yescharacteristic
185C22×C17#45611yescharacteristic
186C22×C17B(4,𝔽2)11yescharacteristic
187C2×D17—42no—
188C2×D17—42no—
189C2×D17—42no—
190C2×D17—81no—
191C2×D17—81no—
192C2×D17—81no—
193C2×D17—81no—
194C4×C17—21no—
195C4×C17—21no—
196C4×C17—21no—
197C4×C17—21no—
198C4×C17—21no—
199C4×C17—21no—
200C4×C17—42no—
201C4×C17—81no—
202Dic17—82no—
203Dic17—81no—
204Dic17—81no—
205Dic17—81no—
206C22×Q16—341no—
207C2×C4⋊Q8—171no—
208C2×C22⋊Q8—341no—
209C2×#107—341no—
210C2×#103—341no—
211C2×#102—341no—
212#399—341no—
213#398—341no—
214#396—341no—
215#391C2×D1711yescharacteristic
216#390—171no—
217#389—171no—
218#386—341no—
219#385—341no—
220#340—341no—
221#340—341no—
222#336—171no—
223#336C2×D1711yescharacteristic
224#336C2×D1711yescharacteristic
225C2×C17#159511yescharacteristic
226C2×C17#160211yescharacteristic
227C2×C17#160311yescharacteristic
228C2×C17#163211yescharacteristic
229C2×C17#164111yescharacteristic
230C2×C17#164911yescharacteristic
231C2×C17#166111yescharacteristic
232D17—82no—
233C22×Q8—341no—
234C22×Q8D6811yescharacteristic
235C22×D4—171no—
236C23×C4—341no—
237C2×Q16—682no—
238C2×Q16—682no—
239C2×Q16—682no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—341no—
243C4⋊Q8—682no—
244C22⋊Q8—682no—
245C22⋊Q8—682no—
246C2×C4⋊C4—171no—
247C2×C4⋊C4—171no—
248C2×C4⋊C4—171no—
249C2×C4⋊C4—341no—
250C2×C4⋊C4—341no—
251C2×C4⋊C4—341no—
252C2×C4⋊C4—341no—
253C2×C4⋊C4—341no—
254C2×C4⋊C4—341no—
255C2×C4⋊C4D6811yescharacteristic
256C2×C22⋊C4—341no—
257C2×C22⋊C4—341no—
258C2×C22⋊C4—341no—
259C2×C22⋊C4—341no—
260C2×C22⋊C4—341no—
261C2×C42C22×D1711yescharacteristic
262#107—682no—
263#103—682no—
264#102—682no—
265#95—21no—
266#95—21no—
267#95—341no—
268#95—341no—
269C17#775611yescharacteristic, complemented, Sylow
270C24—341no—
271C2×Q8—21no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—342no—
275C2×Q8—681no—
276C2×Q8—681no—
277C2×Q8—681no—
278C2×Q8—681no—
279C2×Q8—681no—
280C2×Q8D13612yes—
281C2×D4—172no—
282C2×D4—341no—
283C2×D4—681no—
284C2×D4—681no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—341no—
288C22×C4—341no—
289C22×C4—341no—
290C22×C4—682no—
291C22×C4—681no—
292C22×C4—1361no—
293C22×C4—1361no—
294C22×C4C2×D6811yescharacteristic
295C22×C4D4×D1711yescharacteristic
296C22×C4D4×D1711yescharacteristic
297Q16—688no—
298C2×C8—22no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—342no—
305C2×C8—681no—
306C2×C8—681no—
307C4⋊C4—41no—
308C4⋊C4—342no—
309C4⋊C4—342no—
310C4⋊C4—341no—
311C4⋊C4—341no—
312C4⋊C4—681no—
313C4⋊C4—681no—
314C4⋊C4—682no—
315C4⋊C4—682no—
316C4⋊C4—681no—
317C4⋊C4—681no—
318C4⋊C4—681no—
319C4⋊C4—681no—
320C4⋊C4—1361no—
321C22⋊C4—682no—
322C22⋊C4—681no—
323C22⋊C4—681no—
324C22⋊C4—681no—
325C22⋊C4—681no—
326C22⋊C4—1361no—
327C22⋊C4—1361no—
328C42—41no—
329C23—342no—
330C23—681no—
331C23—681no—
332C23—681no—
333C23—681no—
334C23—681no—
335C23—681no—
336C23#1058756811yescharacteristic
337Q8—42no—
338Q8—81no—
339Q8—682no—
340Q8—682no—
341Q8—1361no—
342Q8—1361no—
343D4—682no—
344D4—1361no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—681no—
358C2×C4—681no—
359C2×C4—681no—
360C2×C4—681no—
361C2×C4—681no—
362C2×C4—681no—
363C2×C4—681no—
364C2×C4—681no—
365C2×C4—682no—
366C2×C4—681no—
367C2×C4—1364no—
368C2×C4—1361no—
369C2×C4—1361no—
370C2×C4C2×D13611yescharacteristic
371C2×C4#1058759411yescharacteristic
372C2×C4#1058760111yescharacteristic
373C2×C4#1058760211yescharacteristic
374C2×C4Q16×D1711yescharacteristic
375C2×C4#1058760411yescharacteristic
376C8—42no—
377C8—81no—
378C8—682no—
379C22—682no—
380C22—682no—
381C22—682no—
382C22—1361no—
383C22—1361no—
384C22—1361no—
385C22—1361no—
386C22#1176294511yescharacteristic
387C22#1176296611yescharacteristic
388C22#1176297411yescharacteristic
389C22#1176297911yescharacteristic
390C22#1176298811yescharacteristic
391C22#1176299311yescharacteristic
392C22#1176299711yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—42no—
400C4—81no—
401C4—1361no—
402C4—1362no—
403C4—1361no—
404C4—1361no—
405C2—1362no—
406C2#1452770411yescharacteristic
407C2#1452774211yescharacteristic
408C2#1452775011yescharacteristic
409C2#1452791911yescharacteristic
410C2#1452797911yescharacteristic
411C2#1452803511yescharacteristic
412C2#1452810211yescharacteristic
subgroupG/H
centerC23#10587568
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#1272C2
socleC23×C17#120
prime radicalC22×D68D4
hypercenter#1272D17
hypocenterC17#7756
LCSG ▷ C2×C42×C17 ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
2. #14530113C2semidirect
4. #14529883C2semidirect
5. #14526606C2semidirect
7. C17×#1272C2semidirect
80. C2×D68Q16semidirect
177. #1272D17semidirect
269. C17#7756semidirect

Maximal subgroups

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

subgroupindex
#145358022
#145301132
#145299212
#145298832
#145266062
#145266042
C17×#12722
#775617