Finite Group Databasebeta

← #16923497

Subgroups — #16923497

Order 4352.

404 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 81 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#14535744C211yesmaximal, characteristic
2#14530343C211yesmaximal, characteristic, complemented
3#14530260C211yesmaximal, characteristic, complemented
4#14530189C211yesmaximal, characteristic
5#14526608C211yesmaximal, characteristic, complemented
6#14526604C211yesmaximal, characteristic
7C17×#1271C211yesmaximal, characteristic, complemented
8C2×#10587584—21no—
9C2×#10587479—21no—
10C2×#10587492—21no—
11#11762744—21no—
12#11762731—21no—
13#11762727—21no—
14#11762711C2211yescharacteristic
15#11762707C2211yescharacteristic
16#11762706—21no—
17#11762699—21no—
18#11762679C2211yescharacteristic
19#11762572—21no—
20#11762572—21no—
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⋊Dic17D411yescharacteristic
44C2×C4⋊Dic17D411yescharacteristic
45C2×C4⋊Dic17D411yescharacteristic
46C2×#63225—21no—
47C2×#63225—21no—
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#10587479—42no—
58C23×D17—21no—
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17C2×D411yescharacteristic
62C22×C4×C17C2×D411yescharacteristic
63C22×C4×C17C2×D411yescharacteristic
64C22×Dic17—21no—
65C22×Dic17—21no—
66C22×Dic17—21no—
67C22×Dic17—41no—
68C2×D68—21no—
69C2×D68—21no—
70C2×D68—41no—
71C2×D68—41no—
72C2×C8×C17—41no—
73C2×C8×C17—41no—
74C2×C8×C17—41no—
75C2×C8×C17—41no—
76C2×C8×C17—41no—
77C2×C8×C17—41no—
78C4⋊C4×C17—21no—
79C4⋊C4×C17—21no—
80C4⋊C4×C17—41no—
81C4⋊C4×C17—41no—
82C4⋊C4×C17—41no—
83C42×C17—41no—
84#63227—42no—
85#63227—42no—
86#63227—41no—
87#63227—41no—
88#63227—41no—
89#63227—41no—
90#63227—81no—
91#63227—81no—
92C4⋊Dic17—21no—
93C4⋊Dic17—41no—
94C4⋊Dic17—41no—
95C4⋊Dic17—42no—
96C4⋊Dic17—41no—
97C4⋊Dic17—41no—
98C4⋊Dic17—41no—
99C4⋊Dic17—41no—
100C4⋊Dic17SD1612yes—
101#63225—42no—
102#63225—41no—
103#63225—41no—
104#63225—81no—
105C4×Dic17—42no—
106C2×#3622—22no—
107C2×#3622—41no—
108C2×#3622—41no—
109#7808—171nomaximal, Sylow
110C23×C17#12011yescharacteristic
111C22×D17—41no—
112C22×D17—41no—
113C22×D17—41no—
114C22×D17—41no—
115C22×D17—41no—
116C22×D17—41no—
117C22×D17—41no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—41no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17—41no—
128C2×C4×C17—41no—
129C2×C4×C17—41no—
130C2×C4×C17C2×SD1611yescharacteristic
131C2×C4×C17#13511yescharacteristic
132C2×C4×C17#13511yescharacteristic
133C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
134C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
135C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
136C2×Dic17—22no—
137C2×Dic17—41no—
138C2×Dic17—41no—
139C2×Dic17—41no—
140C2×Dic17—41no—
141C2×Dic17—41no—
142C2×Dic17—41no—
143C2×Dic17—41no—
144C2×Dic17—41no—
145C2×Dic17—81no—
146C2×Dic17—81no—
147C2×Dic17—81no—
148D68—42no—
149D68—81no—
150C8×C17—81no—
151C8×C17—81no—
152#3622—42no—
153#2727—171no—
154#2099—171no—
155#2084—171no—
156#2071—171no—
157#1271—171no—
158#1271D1711yescharacteristic, complemented
159#1270—171no—
160C22×C17#44911yescharacteristic
161C22×C17#44911yescharacteristic
162C22×C17#45011yescharacteristic
163C22×C17#45411yescharacteristic
164C22×C17#45511yescharacteristic
165C22×C17#45611yescharacteristic
166C22×C17#45811yescharacteristic
167C2×D17—81no—
168C2×D17—81no—
169C2×D17—81no—
170C2×D17—81no—
171C2×D17—81no—
172C2×D17—81no—
173C2×D17—81no—
174C4×C17—21no—
175C4×C17—21no—
176C4×C17—21no—
177C4×C17—21no—
178C4×C17—21no—
179C4×C17—21no—
180C4×C17—81no—
181C4×C17—81no—
182Dic17—42no—
183Dic17—81no—
184Dic17—81no—
185Dic17—161no—
186C2×#126—341no—
187C2×#106—341no—
188C2×#102—341no—
189#399—341no—
190#396—341no—
191#394—341no—
192#390—171no—
193#389—171no—
194#389—341no—
195#389C2×D1711yescharacteristic
196#388—341no—
197#384—171no—
198#340—341no—
199#340—341no—
200#340—341no—
201#340—341no—
202#336—171no—
203#336C2×D1711yescharacteristic
204#336C2×D1711yescharacteristic
205C2×C17#160911yescharacteristic
206C2×C17#161211yescharacteristic
207C2×C17#161611yescharacteristic
208C2×C17#162811yescharacteristic
209C2×C17#164211yescharacteristic
210C2×C17#165411yescharacteristic
211C2×C17#166411yescharacteristic
212D17—161no—
213C22×D4—171no—
214C22×C8—21no—
215C22×C8—21no—
216C22×C8—341no—
217#126—682no—
218#126—682no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—171no—
221C2×C4⋊C4—171no—
222C2×C4⋊C4—171no—
223C2×C4⋊C4—341no—
224C2×C4⋊C4—341no—
225C2×C4⋊C4—341no—
226C2×C4⋊C4—341no—
227C2×C4⋊C4—341no—
228C2×C4⋊C4—341no—
229C2×C4⋊C4D6811yescharacteristic
230C2×C4⋊C4D6811yescharacteristic
231C2×C22⋊C4—341no—
232C2×C22⋊C4—341no—
233C2×C22⋊C4—341no—
234C2×C22⋊C4—341no—
235C2×C22⋊C4—341no—
236C2×C22⋊C4—341no—
237C2×C42—341no—
238C2×C42C22×D1711yescharacteristic
239#106—682no—
240#102—682no—
241#95—21no—
242#95—341no—
243#95—341no—
244C17#780811yescharacteristic, complemented, Sylow
245C24—341no—
246C2×D4—341no—
247C2×D4—341no—
248C2×D4—681no—
249C2×D4—681no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—341no—
253C22×C4—341no—
254C22×C4—341no—
255C22×C4—681no—
256C22×C4C2×D6811yescharacteristic
257C22×C4D4×D1711yescharacteristic
258C22×C4D4×D1711yescharacteristic
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—342no—
266C2×C8—681no—
267C2×C8—681no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—172no—
274C4⋊C4—341no—
275C4⋊C4—681no—
276C4⋊C4—681no—
277C4⋊C4—682no—
278C4⋊C4—682no—
279C4⋊C4—681no—
280C4⋊C4—681no—
281C4⋊C4—681no—
282C4⋊C4—681no—
283C4⋊C4—681no—
284C4⋊C4—681no—
285C4⋊C4—1361no—
286C22⋊C4—682no—
287C22⋊C4—682no—
288C22⋊C4—681no—
289C22⋊C4—681no—
290C22⋊C4—681no—
291C22⋊C4—681no—
292C22⋊C4—1361no—
293C22⋊C4—1361no—
294C42—41no—
295C42—682no—
296C23—681no—
297C23—681no—
298C23—681no—
299C23—681no—
300C23—681no—
301C23—681no—
302C23—681no—
303C23#1058756811yescharacteristic
304D4—682no—
305D4—1361no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—342no—
319C2×C4—681no—
320C2×C4—681no—
321C2×C4—681no—
322C2×C4—681no—
323C2×C4—681no—
324C2×C4—681no—
325C2×C4—681no—
326C2×C4—681no—
327C2×C4—1361no—
328C2×C4—1361no—
329C2×C4—1361no—
330C2×C4#1058759011yescharacteristic
331C2×C4#1058759411yescharacteristic
332C2×C4#1058759711yescharacteristic
333C2×C4SD16×D1711yescharacteristic
334C2×C4#1058760011yescharacteristic
335C2×C4#1058760511yescharacteristic
336C8—81no—
337C8—81no—
338C8—682no—
339C22—1361no—
340C22—1361no—
341C22—1361no—
342C22—1361no—
343C22—1361no—
344C22—1361no—
345C22—1361no—
346C22#1176295611yescharacteristic
347C22#1176295711yescharacteristic
348C22#1176296111yescharacteristic
349C22#1176298311yescharacteristic
350C22#1176298811yescharacteristic
351C22#1176299411yescharacteristic
352C22#1176300011yescharacteristic
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—81no—
360C4—81no—
361C4—2721no—
362C4—682no—
363C4—1361no—
364C4—1361no—
365C2—2721no—
366C2#1452778311yescharacteristic
367C2#1452780111yescharacteristic
368C2#1452782211yescharacteristic
369C2#1452789711yescharacteristic
370C2#1452798611yescharacteristic
371C2#1452806911yescharacteristic
372C2#1452812011yescharacteristic
subgroupG/H
centerC23#10587568
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#1271C2
socleC23×C17#120
prime radicalC22×D68D4
hypercenter#1271D17
hypocenterC17#7808
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
2. #14530343C2semidirect
3. #14530260C2semidirect
5. #14526608C2semidirect
7. C17×#1271C2semidirect
158. #1271D17semidirect
244. C17#7808semidirect

Maximal subgroups

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

subgroupindex
#145357442
#145303432
#145302602
#145301892
#145266082
#145266042
C17×#12712
#780817