Finite Group Databasebeta

← #16923355

Subgroups — #16923355

Order 4352.

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

normal subgroups only

This group has 430 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#14535801C211yesmaximal, characteristic
2#14530343C211yesmaximal, characteristic, complemented
3#14530194C211yesmaximal, characteristic, complemented
4#14530187C211yesmaximal, characteristic
5#14526609C211yesmaximal, characteristic, complemented
6#14526608C211yesmaximal, characteristic, complemented
7C17×#1270C211yesmaximal, characteristic
8C2×#10587564—21no—
9C2×#10587487—21no—
10C2×#10587503—21no—
11#11762745—21no—
12#11762742—21no—
13#11762732—21no—
14#11762715C2211yescharacteristic
15#11762707C2211yescharacteristic
16#11762689—21no—
17#11762684—21no—
18#11762679C2211yescharacteristic
19#11762572—21no—
20#11762572—21no—
21#11762572—21no—
22#11762571—21no—
23#11762564C2211yescharacteristic
24C17×#390C2211yescharacteristic
25C17×#336C2211yescharacteristic
26C17×#336C2211yescharacteristic
27C22×D4×C17D411yescharacteristic
28C23×Dic17—21no—
29C22×Dic34D411yescharacteristic
30C22×C8×C17—21no—
31C22×C8×C17—21no—
32C2×C4⋊C4×C17D411yescharacteristic
33C2×C22⋊C4×C17—21no—
34C2×C22⋊C4×C17—21no—
35C2×C42×C17C2311yescharacteristic
36C2×C22⋊Dic17—21no—
37C2×C22⋊Dic17—21no—
38C2×C22⋊Dic17—21no—
39C2×C4⋊Dic17—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—
49C22×#3622—21no—
50#10587564—42no—
51#10587564—42no—
52#10587503—42no—
53#10587502—21no—
54#10587502—21no—
55#10587502—21no—
56#10587502—21no—
57#10587487—42no—
58C24×C17—21no—
59C2×D4×C17—21no—
60C2×D4×C17—41no—
61C2×D4×C17—41no—
62C2×D4×C17SD1612yes—
63C22×C4×C17—21no—
64C22×C4×C17C2×D411yescharacteristic
65C22×C4×C17C2×D411yescharacteristic
66C22×C4×C17C2×D411yescharacteristic
67C22×Dic17—21no—
68C22×Dic17—21no—
69C22×Dic17—21no—
70C22×Dic17—21no—
71C22×Dic17—42no—
72C22×Dic17—41no—
73C22×Dic17—81no—
74C22×Dic17—81no—
75C2×Dic34—21no—
76C2×Dic34—41no—
77C2×Dic34—41no—
78C2×Dic34D812yes—
79C2×C8×C17—41no—
80C2×C8×C17—41no—
81C2×C8×C17—41no—
82C2×C8×C17—41no—
83C2×C8×C17—41no—
84C2×C8×C17—41no—
85C4⋊C4×C17—41no—
86C22⋊C4×C17—41no—
87C22⋊C4×C17—41no—
88C22⋊C4×C17—41no—
89C22⋊C4×C17—41no—
90C42×C17—41no—
91C22⋊Dic17—42no—
92C22⋊Dic17—81no—
93C22⋊Dic17—81no—
94C4⋊Dic17—21no—
95C4⋊Dic17—21no—
96C4⋊Dic17—42no—
97C4⋊Dic17—41no—
98C4⋊Dic17—41no—
99C4⋊Dic17—42no—
100C4⋊Dic17—41no—
101C4⋊Dic17—41no—
102C4⋊Dic17—41no—
103C4⋊Dic17—41no—
104#63225—42no—
105#63225—41no—
106#63225—41no—
107#63225—81no—
108C2×#3622—41no—
109C2×#3622—41no—
110C2×#3622—41no—
111#7787—171nomaximal, Sylow
112C23×C17—22no—
113C23×C17—41no—
114C23×C17—41no—
115C23×C17—41no—
116C23×C17—41no—
117C23×C17—41no—
118C23×C17—41no—
119C23×C17#12011yescharacteristic
120D4×C17—42no—
121D4×C17—81no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—41no—
129C2×C4×C17—41no—
130C2×C4×C17—41no—
131C2×C4×C17C2×D811yescharacteristic
132C2×C4×C17C2×SD1611yescharacteristic
133C2×C4×C17#13511yescharacteristic
134C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
135C2×C4×C17#13711yescharacteristic
136C2×C4×C17#13711yescharacteristic
137C2×Dic17—22no—
138C2×Dic17—41no—
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—41no—
149C2×Dic17—84no—
150C2×Dic17—81no—
151C2×Dic17—81no—
152C2×Dic17—81no—
153Dic34—42no—
154Dic34—81no—
155C8×C17—81no—
156C8×C17—81no—
157#3622—81no—
158#2732—171no—
159#2099—171no—
160#2072—171no—
161#2071—171no—
162#1272—171no—
163#1271—171no—
164#1270D1711yescharacteristic
165C22×C17—42no—
166C22×C17—42no—
167C22×C17—42no—
168C22×C17—81no—
169C22×C17—81no—
170C22×C17—81no—
171C22×C17—81no—
172C22×C17#44711yescharacteristic
173C22×C17#44811yescharacteristic
174C22×C17#45211yescharacteristic
175C22×C17#45411yescharacteristic
176C22×C17#45511yescharacteristic
177C22×C17#45511yescharacteristic
178C22×C17#45811yescharacteristic
179C4×C17—21no—
180C4×C17—21no—
181C4×C17—21no—
182C4×C17—21no—
183C4×C17—21no—
184C4×C17—21no—
185C4×C17—81no—
186Dic17—42no—
187Dic17—81no—
188Dic17—81no—
189Dic17—81no—
190Dic17—161no—
191C2×C22⋊Q8—341no—
192C2×#103—341no—
193C2×#102—341no—
194#399—341no—
195#398—341no—
196#396—341no—
197#391—171no—
198#390C2×D1711yescharacteristic
199#389—171no—
200#386—341no—
201#385—341no—
202#384—171no—
203#340—341no—
204#340—341no—
205#340—341no—
206#340—341no—
207#336—171no—
208#336C2×D1711yescharacteristic
209#336C2×D1711yescharacteristic
210C2×C17—82no—
211C2×C17#160711yescharacteristic
212C2×C17#160811yescharacteristic
213C2×C17#161511yescharacteristic
214C2×C17#162711yescharacteristic
215C2×C17#164111yescharacteristic
216C2×C17#164311yescharacteristic
217C2×C17#166211yescharacteristic
218C22×Q8—171no—
219C22×D4D6811yescharacteristic
220C23×C4—341no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—341no—
224C22⋊Q8—682no—
225C22⋊Q8—682no—
226C2×C4⋊C4—171no—
227C2×C4⋊C4—171no—
228C2×C4⋊C4—171no—
229C2×C4⋊C4—341no—
230C2×C4⋊C4—341no—
231C2×C4⋊C4—341no—
232C2×C4⋊C4—341no—
233C2×C4⋊C4—341no—
234C2×C4⋊C4—341no—
235C2×C4⋊C4—341no—
236C2×C4⋊C4D6811yescharacteristic
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—341no—
240C2×C22⋊C4—341no—
241C2×C22⋊C4—341no—
242C2×C42C22×D1711yescharacteristic
243#103—682no—
244#102—682no—
245#95—341no—
246#95—341no—
247#95—341no—
248#95—341no—
249C17#778711yescharacteristic, complemented, Sylow
250C24—21no—
251C2×Q8—172no—
252C2×Q8—341no—
253C2×Q8—681no—
254C2×Q8—681no—
255C2×D4—21no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4#6321912yes—
259C22×C4—21no—
260C22×C4—341no—
261C22×C4—341no—
262C22×C4—341no—
263C22×C4—341no—
264C22×C4—682no—
265C22×C4—681no—
266C22×C4—1361no—
267C22×C4—1361no—
268C22×C4C2×D6811yescharacteristic
269C22×C4D4×D1711yescharacteristic
270C22×C4D4×D1711yescharacteristic
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—681no—
278C2×C8—681no—
279C2×C8—681no—
280C4⋊C4—41no—
281C4⋊C4—341no—
282C4⋊C4—341no—
283C4⋊C4—681no—
284C4⋊C4—681no—
285C4⋊C4—682no—
286C4⋊C4—682no—
287C4⋊C4—682no—
288C4⋊C4—681no—
289C4⋊C4—681no—
290C4⋊C4—681no—
291C4⋊C4—681no—
292C4⋊C4—681no—
293C4⋊C4—681no—
294C4⋊C4—1361no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—682no—
300C22⋊C4—1361no—
301C22⋊C4—1361no—
302C42—41no—
303C23—22no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23—41no—
310C23#1058756811yescharacteristic
311Q8—682no—
312Q8—1361no—
313D4—42no—
314D4—81no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—342no—
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—681no—
334C2×C4—681no—
335C2×C4—681no—
336C2×C4—1361no—
337C2×C4—1361no—
338C2×C4—1361no—
339C2×C4—1364no—
340C2×C4C2×#6321911yescharacteristic
341C2×C4#1058759511yescharacteristic
342C2×C4D8×D1711yescharacteristic
343C2×C4#1058759811yescharacteristic
344C2×C4#1058760011yescharacteristic
345C2×C4#1058760111yescharacteristic
346C8—81no—
347C8—81no—
348C8—1361no—
349C22—42no—
350C22—42no—
351C22—42no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22#1176294411yescharacteristic
357C22#1176295011yescharacteristic
358C22#1176297211yescharacteristic
359C22#1176298311yescharacteristic
360C22#1176298811yescharacteristic
361C22#1176298911yescharacteristic
362C22#1176300011yescharacteristic
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—81no—
370C4—2721no—
371C4—682no—
372C4—1361no—
373C4—1361no—
374C4—1361no—
375C2—82no—
376C2#1452777111yescharacteristic
377C2#1452777711yescharacteristic
378C2#1452781611yescharacteristic
379C2#1452789011yescharacteristic
380C2#1452797811yescharacteristic
381C2#1452799311yescharacteristic
382C2#1452810911yescharacteristic
subgroupG/H
centerC23#10587568
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#1270C2
socleC23×C17#120
prime radicalC22×D4×C17D4
hypercenter#1270D17
hypocenterC17#7787
LCSG ▷ C2×C42×C17 ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. #14530343C2semidirect
3. #14530194C2semidirect
5. #14526609C2semidirect
6. #14526608C2semidirect
249. C17#7787semidirect

Maximal subgroups

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

subgroupindex
#145358012
#145303432
#145301942
#145301872
#145266092
#145266082
C17×#12702
#778717