Finite Group Databasebeta

← #16923396

Subgroups — #16923396

Order 4352.

420 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 83 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#14535745C211yesmaximal, characteristic
2#14530311C211yesmaximal, characteristic
3#14530260C211yesmaximal, characteristic, complemented
4#14530257C211yesmaximal, characteristic, complemented
5#14526606C211yesmaximal, characteristic, complemented
6#14526604C211yesmaximal, characteristic
7C17×#1271C211yesmaximal, characteristic, complemented
8C2×#10587584—21no—
9C2×#10587489—21no—
10C2×#10587489—21no—
11C2×#10587479—21no—
12C2×#10587492—21no—
13#11762746—21no—
14#11762731—21no—
15#11762726—21no—
16#11762711C2211yescharacteristic
17#11762708—21no—
18#11762707C2211yescharacteristic
19#11762701—21no—
20#11762679C2211yescharacteristic
21#11762572—21no—
22#11762571—21no—
23#11762564C2211yescharacteristic
24C17×#389C2211yescharacteristic
25C17×#336C2211yescharacteristic
26C17×#336C2211yescharacteristic
27C22×D68D411yescharacteristic
28C22×C8×C17—21no—
29C22×C8×C17—21no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17D411yescharacteristic
32C2×C4⋊C4×C17D411yescharacteristic
33C2×C42×C17C2311yescharacteristic
34C2×#63227—21no—
35C2×#63227—21no—
36C2×#63227—21no—
37C2×#63227—21no—
38C2×#63227—21no—
39C2×#63227—21no—
40C2×C4⋊Dic17—21no—
41C2×C4⋊Dic17—21no—
42C2×C4⋊Dic17—21no—
43C2×C4⋊Dic17—21no—
44C2×C4⋊Dic17—21no—
45C2×C4⋊Dic17D411yescharacteristic
46C2×C4⋊Dic17D411yescharacteristic
47C2×C4⋊Dic17D411yescharacteristic
48C2×#63225—21no—
49C2×C4×Dic17—21no—
50C22×#3622—21no—
51#10587584—42no—
52#10587584—42no—
53C17×#95—21no—
54#10587502—21no—
55#10587502—21no—
56#10587492—42no—
57#10587489—42no—
58#10587489—42no—
59#10587479—42no—
60C23×D17—21no—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17C2×D411yescharacteristic
64C22×C4×C17C2×D411yescharacteristic
65C22×C4×C17C2×D411yescharacteristic
66C22×Dic17—21no—
67C22×Dic17—21no—
68C22×Dic17—21no—
69C22×Dic17—41no—
70C2×D68—21no—
71C2×D68—21no—
72C2×D68—41no—
73C2×D68—41no—
74C2×C8×C17—22no—
75C2×C8×C17—22no—
76C2×C8×C17—41no—
77C2×C8×C17—41no—
78C2×C8×C17—41no—
79C2×C8×C17—41no—
80C4⋊C4×C17—21no—
81C4⋊C4×C17—41no—
82C4⋊C4×C17—41no—
83C4⋊C4×C17—41no—
84C4⋊C4×C17D812yes—
85C42×C17—41no—
86#63227—42no—
87#63227—42no—
88#63227—41no—
89#63227—41no—
90#63227—41no—
91#63227—41no—
92#63227—81no—
93#63227—81no—
94C4⋊Dic17—21no—
95C4⋊Dic17—21no—
96C4⋊Dic17—42no—
97C4⋊Dic17—41no—
98C4⋊Dic17—41no—
99C4⋊Dic17—41no—
100C4⋊Dic17—41no—
101C4⋊Dic17—41no—
102C4⋊Dic17—41no—
103C4⋊Dic17—41no—
104C4⋊Dic17—41no—
105C4⋊Dic17—81no—
106#63225—41no—
107#63225—41no—
108C4×Dic17—42no—
109C2×#3622—22no—
110C2×#3622—41no—
111C2×#3622—41no—
112#7793—171nomaximal, Sylow
113C23×C17#12011yescharacteristic
114C22×D17—41no—
115C22×D17—41no—
116C22×D17—41no—
117C22×D17—41no—
118C22×D17—41no—
119C22×D17—41no—
120C22×D17—41no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—22no—
128C2×C4×C17—41no—
129C2×C4×C17—41no—
130C2×C4×C17—41no—
131C2×C4×C17—41no—
132C2×C4×C17—41no—
133C2×C4×C17C2×D811yescharacteristic
134C2×C4×C17#13511yescharacteristic
135C2×C4×C17#13511yescharacteristic
136C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
138C2×C4×C17#13711yescharacteristic
139C2×Dic17—41no—
140C2×Dic17—41no—
141C2×Dic17—41no—
142C2×Dic17—41no—
143C2×Dic17—41no—
144C2×Dic17—41no—
145C2×Dic17—41no—
146C2×Dic17—41no—
147C2×Dic17—41no—
148C2×Dic17—81no—
149C2×Dic17—81no—
150C2×Dic17—81no—
151D68—42no—
152D68—81no—
153C8×C17—42no—
154C8×C17—42no—
155#3622—42no—
156#2727—171no—
157#2093—171no—
158#2084—171no—
159#2083—171no—
160#1271—171no—
161#1271D1711yescharacteristic, complemented
162#1270—171no—
163C22×C17#44711yescharacteristic
164C22×C17#44911yescharacteristic
165C22×C17#45211yescharacteristic
166C22×C17#45311yescharacteristic
167C22×C17#45611yescharacteristic
168C22×C17#45611yescharacteristic
169C22×C17#45811yescharacteristic
170C2×D17—81no—
171C2×D17—81no—
172C2×D17—81no—
173C2×D17—81no—
174C2×D17—81no—
175C2×D17—81no—
176C2×D17—81no—
177C4×C17—21no—
178C4×C17—21no—
179C4×C17—21no—
180C4×C17—21no—
181C4×C17—21no—
182C4×C17—21no—
183C4×C17—42no—
184C4×C17—81no—
185Dic17—81no—
186Dic17—81no—
187Dic17—81no—
188Dic17—161no—
189C2×#126—341no—
190C2×#107—341no—
191C2×#107—341no—
192C2×#106—341no—
193C2×#102—341no—
194#399—341no—
195#396—341no—
196#394—341no—
197#390—171no—
198#389—171no—
199#389—341no—
200#389C2×D1711yescharacteristic
201#388—341no—
202#384—171no—
203#340—341no—
204#340—341no—
205#336—171no—
206#336C2×D1711yescharacteristic
207#336C2×D1711yescharacteristic
208C2×C17#160911yescharacteristic
209C2×C17#161011yescharacteristic
210C2×C17#161811yescharacteristic
211C2×C17#162111yescharacteristic
212C2×C17#164711yescharacteristic
213C2×C17#165211yescharacteristic
214C2×C17#166011yescharacteristic
215D17—161no—
216C22×D4—171no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—341no—
220#126—682no—
221#126—682no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—171no—
224C2×C4⋊C4—171no—
225C2×C4⋊C4—171no—
226C2×C4⋊C4—341no—
227C2×C4⋊C4—341no—
228C2×C4⋊C4—341no—
229C2×C4⋊C4—341no—
230C2×C4⋊C4—341no—
231C2×C4⋊C4—341no—
232C2×C4⋊C4D6811yescharacteristic
233C2×C4⋊C4D6811yescharacteristic
234C2×C22⋊C4—341no—
235C2×C22⋊C4—341no—
236C2×C22⋊C4—341no—
237C2×C22⋊C4—341no—
238C2×C22⋊C4—341no—
239C2×C22⋊C4—341no—
240C2×C42—341no—
241C2×C42C22×D1711yescharacteristic
242#107—682no—
243#107—682no—
244#106—682no—
245#102—682no—
246#95—21no—
247#95—341no—
248#95—341no—
249C17#779311yescharacteristic, complemented, Sylow
250C24—341no—
251C2×D4—341no—
252C2×D4—341no—
253C2×D4—681no—
254C2×D4—681no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—341no—
258C22×C4—341no—
259C22×C4—341no—
260C22×C4—681no—
261C22×C4C2×D6811yescharacteristic
262C22×C4D4×D1711yescharacteristic
263C22×C4D4×D1711yescharacteristic
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—342no—
271C2×C8—681no—
272C2×C8—681no—
273C4⋊C4—21no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—341no—
278C4⋊C4—341no—
279C4⋊C4—681no—
280C4⋊C4—681no—
281C4⋊C4—682no—
282C4⋊C4—681no—
283C4⋊C4—681no—
284C4⋊C4—681no—
285C4⋊C4—681no—
286C4⋊C4—681no—
287C4⋊C4—681no—
288C4⋊C4—681no—
289C4⋊C4—681no—
290C4⋊C4—1361no—
291C4⋊C4D13612yes—
292C22⋊C4—682no—
293C22⋊C4—682no—
294C22⋊C4—681no—
295C22⋊C4—681no—
296C22⋊C4—681no—
297C22⋊C4—681no—
298C22⋊C4—1361no—
299C22⋊C4—1361no—
300C42—41no—
301C42—682no—
302C23—681no—
303C23—681no—
304C23—681no—
305C23—681no—
306C23—681no—
307C23—681no—
308C23—681no—
309C23#1058756811yescharacteristic
310D4—682no—
311D4—1361no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—681no—
325C2×C4—681no—
326C2×C4—681no—
327C2×C4—681no—
328C2×C4—681no—
329C2×C4—681no—
330C2×C4—681no—
331C2×C4—681no—
332C2×C4—681no—
333C2×C4—1361no—
334C2×C4—1361no—
335C2×C4—1361no—
336C2×C4C2×D13611yescharacteristic
337C2×C4#1058759411yescharacteristic
338C2×C4#1058759811yescharacteristic
339C2×C4#1058760011yescharacteristic
340C2×C4#1058760111yescharacteristic
341C2×C4#1058760511yescharacteristic
342C8—42no—
343C8—42no—
344C8—682no—
345C22—1361no—
346C22—1361no—
347C22—1361no—
348C22—1361no—
349C22—1361no—
350C22—1361no—
351C22—1361no—
352C22#1176294511yescharacteristic
353C22#1176295711yescharacteristic
354C22#1176297211yescharacteristic
355C22#1176297811yescharacteristic
356C22#1176299311yescharacteristic
357C22#1176299411yescharacteristic
358C22#1176300011yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—42no—
366C4—81no—
367C4—2721no—
368C4—1361no—
369C4—1361no—
370C4—1361no—
371C2—2721no—
372C2#1452778411yescharacteristic
373C2#1452778911yescharacteristic
374C2#1452783311yescharacteristic
375C2#1452785111yescharacteristic
376C2#1452802111yescharacteristic
377C2#1452805511yescharacteristic
378C2#1452809711yescharacteristic
subgroupG/H
centerC23#10587568
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#1271C2
socleC23×C17#120
prime radicalC22×D68D4
hypercenter#1271D17
hypocenterC17#7793
LCSG ▷ C2×C42×C17 ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
3. #14530260C2semidirect
4. #14530257C2semidirect
5. #14526606C2semidirect
7. C17×#1271C2semidirect
161. #1271D17semidirect
249. C17#7793semidirect

Maximal subgroups

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

subgroupindex
#145357452
#145303112
#145302602
#145302572
#145266062
#145266042
C17×#12712
#779317