Finite Group Databasebeta

← #16923378

Subgroups — #16923378

Order 4352.

462 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 87 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#14535155C211yesmaximal, characteristic
2#14530341C211yesmaximal, characteristic, complemented
3#14530188C211yesmaximal, characteristic
4#14526608C211yesmaximal, characteristic, complemented
5#14526607C211yesmaximal, characteristic, complemented
6#14526604C211yesmaximal, characteristic
7C17×#2084C211yesmaximal, characteristic, complemented
8C2×#10587583—21no—
9C2×#10587582—21no—
10C2×#10587480—21no—
11#11762731—21no—
12#11762731—21no—
13#11762730—21no—
14#11762711C2211yescharacteristic
15#11762707C2211yescharacteristic
16#11762687—21no—
17#11762571—21no—
18#11762571—21no—
19#11762571—21no—
20#11762564C2211yescharacteristic
21#11762564C2211yescharacteristic
22C2×C17×#106—21no—
23C17×#389C2211yescharacteristic
24C17×#384C2211yescharacteristic
25C17×#340—21no—
26C17×#336C2211yescharacteristic
27C22×D68D411yescharacteristic
28C22×C4×D17—21no—
29C22×Dic34—21no—
30C22×C8×C17—21no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17—21no—
33C2×C4⋊C4×C17—21no—
34C2×C4⋊C4×C17—21no—
35C2×C4⋊C4×C17D411yescharacteristic
36C2×C4⋊C4×C17D411yescharacteristic
37C2×C42×C17C2311yescharacteristic
38C2×#63227—21no—
39C2×#63227—21no—
40C2×#63227—21no—
41C2×#63227—21no—
42C2×#63227—21no—
43C2×C4⋊Dic17—21no—
44C2×C4⋊Dic17D411yescharacteristic
45C2×C4⋊Dic17D411yescharacteristic
46C2×C4⋊Dic17D411yescharacteristic
47C2×#63225—21no—
48C2×#63225—21no—
49C22×#3622—21no—
50C22×#3622—21no—
51#10587583—42no—
52#10587583—42no—
53#10587582—42no—
54#10587582—42no—
55C17×#106—42no—
56C17×#95—21no—
57#10587502—21no—
58#10587502—21no—
59#10587480—42no—
60C23×D17—21no—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—41no—
64C22×C4×C17C2×D411yescharacteristic
65C22×C4×C17C2×D411yescharacteristic
66C22×C4×C17C2×D411yescharacteristic
67C22×Dic17—21no—
68C22×Dic17—21no—
69C22×Dic17—21no—
70C2×D68—21no—
71C2×D68—41no—
72C2×D68—41no—
73C2×D68SD1612yes—
74C2×C4×D17—42no—
75C2×C4×D17—42no—
76C2×C4×D17—81no—
77C2×Dic34—42no—
78C2×Dic34—42no—
79C2×Dic34—81no—
80C2×C8×C17—22no—
81C2×C8×C17—41no—
82C2×C8×C17—41no—
83C4⋊C4×C17—21no—
84C4⋊C4×C17—41no—
85C4⋊C4×C17—42no—
86C4⋊C4×C17—41no—
87C4⋊C4×C17—41no—
88C4⋊C4×C17—41no—
89C4⋊C4×C17—41no—
90C4⋊C4×C17—41no—
91C4⋊C4×C17—41no—
92C4⋊C4×C17D812yes—
93C42×C17—41no—
94#63227—42no—
95#63227—42no—
96#63227—41no—
97#63227—41no—
98#63227—41no—
99#63227—41no—
100#63227—81no—
101C4⋊Dic17—21no—
102C4⋊Dic17—42no—
103C4⋊Dic17—41no—
104C4⋊Dic17—41no—
105C4⋊Dic17SD1612yes—
106#63225—42no—
107#63225—41no—
108#63225—41no—
109C2×#3622—41no—
110C2×#3622—41no—
111C2×#3622—41no—
112C2×#3622—41no—
113C2×#3622—41no—
114C2×#3622—41no—
115#7791—171nomaximal, Sylow
116C23×C17#12011yescharacteristic
117C22×D17—22no—
118C22×D17—41no—
119C22×D17—41no—
120C22×D17—41no—
121C22×D17—41no—
122C22×D17—41no—
123C22×D17—41no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—22no—
131C2×C4×C17—41no—
132C2×C4×C17—41no—
133C2×C4×C17—41no—
134C2×C4×C17—41no—
135C2×C4×C17—41no—
136C2×C4×C17—81no—
137C2×C4×C17—81no—
138C2×C4×C17—81no—
139C2×C4×C17C2×D811yescharacteristic
140C2×C4×C17C2×SD1611yescharacteristic
141C2×C4×C17C2×SD1611yescharacteristic
142C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
143C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C17#13711yescharacteristic
145C2×Dic17—22no—
146C2×Dic17—41no—
147C2×Dic17—41no—
148C2×Dic17—41no—
149C2×Dic17—41no—
150C2×Dic17—41no—
151C2×Dic17—41no—
152C2×Dic17—41no—
153C2×Dic17—41no—
154D68—42no—
155D68—81no—
156C4×D17—84no—
157Dic34—84no—
158C8×C17—42no—
159#3622—81no—
160#3622—81no—
161#2683—171no—
162#2099—171no—
163#2084D1711yescharacteristic, complemented
164#2071—171no—
165#1271—171no—
166#1271—171no—
167#1270—171no—
168C22×C17#44711yescharacteristic
169C22×C17#44811yescharacteristic
170C22×C17#45011yescharacteristic
171C22×C17#45411yescharacteristic
172C22×C17#45511yescharacteristic
173C22×C17#45611yescharacteristic
174C22×C17#45811yescharacteristic
175C2×D17—42no—
176C2×D17—42no—
177C2×D17—42no—
178C2×D17—81no—
179C2×D17—81no—
180C2×D17—81no—
181C2×D17—81no—
182C4×C17—21no—
183C4×C17—21no—
184C4×C17—21no—
185C4×C17—21no—
186C4×C17—21no—
187C4×C17—21no—
188C4×C17—42no—
189C4×C17—81no—
190C4×C17—161no—
191Dic17—42no—
192Dic17—81no—
193Dic17—81no—
194C2×C22⋊Q8—341no—
195C2×C22⋊Q8—341no—
196C2×#106—21no—
197C2×#102—341no—
198#396—341no—
199#396—341no—
200#395—341no—
201#390—171no—
202#389—171no—
203#389C2×D1711yescharacteristic
204#386—341no—
205#384C2×D1711yescharacteristic
206#340—21no—
207#340—341no—
208#340—341no—
209#340—341no—
210#336—171no—
211#336—171no—
212#336C2×D1711yescharacteristic
213C2×C17#160811yescharacteristic
214C2×C17#161011yescharacteristic
215C2×C17#161611yescharacteristic
216C2×C17#162711yescharacteristic
217C2×C17#163911yescharacteristic
218C2×C17#165111yescharacteristic
219C2×C17#166411yescharacteristic
220D17—82no—
221C22×Q8—341no—
222C22×D4—171no—
223C23×C4—341no—
224C22×C8—21no—
225C22×C8—341no—
226C22×C8—341no—
227C22⋊Q8—682no—
228C22⋊Q8—682no—
229C22⋊Q8—682no—
230C22⋊Q8—682no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—171no—
236C2×C4⋊C4—171no—
237C2×C4⋊C4—171no—
238C2×C4⋊C4—341no—
239C2×C4⋊C4—341no—
240C2×C4⋊C4—341no—
241C2×C4⋊C4C22⋊D1711yescharacteristic
242C2×C4⋊C4C22⋊D1711yescharacteristic
243C2×C22⋊C4—341no—
244C2×C22⋊C4—341no—
245C2×C22⋊C4—341no—
246C2×C22⋊C4—341no—
247C2×C22⋊C4—341no—
248C2×C42C22×D1711yescharacteristic
249#106—42no—
250#102—682no—
251#95—21no—
252#95—341no—
253#95—341no—
254C17#779111yescharacteristic, complemented, Sylow
255C24—341no—
256C2×Q8—682no—
257C2×Q8—682no—
258C2×Q8—1361no—
259C2×D4—172no—
260C2×D4—341no—
261C2×D4—681no—
262C2×D4—681no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—41no—
266C22×C4—341no—
267C22×C4—341no—
268C22×C4—341no—
269C22×C4—682no—
270C22×C4—682no—
271C22×C4—1361no—
272C22×C4D4×D1711yescharacteristic
273C22×C4D4×D1711yescharacteristic
274C22×C4C2×C22⋊D1711yescharacteristic
275C2×C8—22no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—681no—
279C2×C8—681no—
280C2×C8—681no—
281C2×C8—681no—
282C2×C8—681no—
283C2×C8—681no—
284C4⋊C4—21no—
285C4⋊C4—41no—
286C4⋊C4—42no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—41no—
293C4⋊C4—172no—
294C4⋊C4—341no—
295C4⋊C4—681no—
296C4⋊C4—681no—
297C4⋊C4—682no—
298C4⋊C4—682no—
299C4⋊C4—681no—
300C4⋊C4—681no—
301C4⋊C4#6322812yes—
302C22⋊C4—682no—
303C22⋊C4—682no—
304C22⋊C4—681no—
305C22⋊C4—681no—
306C22⋊C4—681no—
307C22⋊C4—681no—
308C22⋊C4—1361no—
309C42—41no—
310C23—342no—
311C23—681no—
312C23—681no—
313C23—681no—
314C23—681no—
315C23—681no—
316C23—681no—
317C23#1058762311yescharacteristic
318Q8—1364no—
319D4—682no—
320D4—1361no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—342no—
337C2×C4—681no—
338C2×C4—681no—
339C2×C4—681no—
340C2×C4—681no—
341C2×C4—681no—
342C2×C4—681no—
343C2×C4—681no—
344C2×C4—681no—
345C2×C4—1364no—
346C2×C4SD16×D1711yescharacteristic
347C2×C4SD16×D1711yescharacteristic
348C2×C4#1058760011yescharacteristic
349C2×C4#1058760111yescharacteristic
350C2×C4C2×#6322811yescharacteristic
351C2×C4#1058763511yescharacteristic
352C8—42no—
353C8—1361no—
354C8—1361no—
355C22—682no—
356C22—682no—
357C22—682no—
358C22—1361no—
359C22—1361no—
360C22—1361no—
361C22—1361no—
362C22#1176294211yescharacteristic
363C22#1176294611yescharacteristic
364C22#1176295811yescharacteristic
365C22#1176298011yescharacteristic
366C22#1176298511yescharacteristic
367C22#1176299111yescharacteristic
368C22#1176299811yescharacteristic
369C4—21no—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—81no—
377C4—161no—
378C4—682no—
379C4—1361no—
380C4—1361no—
381C2—1362no—
382C2#1452777311yescharacteristic
383C2#1452778611yescharacteristic
384C2#1452781911yescharacteristic
385C2#1452788511yescharacteristic
386C2#1452796111yescharacteristic
387C2#1452804411yescharacteristic
388C2#1452811611yescharacteristic
subgroupG/H
centerC23#10587623
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#2084C2
socleC23×C17#120
prime radicalC22×D68D4
hypercenter#2084D17
hypocenterC17#7791
LCSG ▷ C2×C42×C17 ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2084

Complements

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

NH ≅ G/Ntype
2. #14530341C2semidirect
4. #14526608C2semidirect
5. #14526607C2semidirect
7. C17×#2084C2semidirect
163. #2084D17semidirect
254. C17#7791semidirect

Maximal subgroups

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

subgroupindex
#145351552
#145303412
#145301882
#145266082
#145266072
#145266042
C17×#20842
#779117