Finite Group Databasebeta

← #16923398

Subgroups — #16923398

Order 4352.

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

normal subgroups only

This group has 420 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#14535743C211yesmaximal, characteristic
2#14530312C211yesmaximal, characteristic
3#14530262C211yesmaximal, characteristic, complemented
4#14530257C211yesmaximal, characteristic, complemented
5#14526608C211yesmaximal, characteristic, complemented
6#14526606C211yesmaximal, characteristic, complemented
7C17×#1270C211yesmaximal, characteristic
8C2×#10587565—21no—
9C2×#10587489—21no—
10C2×#10587478—21no—
11C2×#10587488—21no—
12C2×#10587503—21no—
13#11762745—21no—
14#11762732—21no—
15#11762723—21no—
16#11762707C2211yescharacteristic
17#11762707C2211yescharacteristic
18#11762704—21no—
19#11762702—21no—
20#11762679C2211yescharacteristic
21#11762572—21no—
22#11762572—21no—
23#11762564C2211yescharacteristic
24C17×#390C2211yescharacteristic
25C17×#336C2211yescharacteristic
26C17×#336C2211yescharacteristic
27C22×D4×C17D411yescharacteristic
28C22×C8×C17—21no—
29C22×C8×C17—21no—
30C2×C4⋊C4×C17D411yescharacteristic
31C2×C22⋊C4×C17—21no—
32C2×C22⋊C4×C17—21no—
33C2×C42×C17C2311yescharacteristic
34C2×C22⋊Dic17—21no—
35C2×C22⋊Dic17—21no—
36C2×C22⋊Dic17—21no—
37C2×C22⋊Dic17—21no—
38C2×C4⋊Dic17—21no—
39C2×C4⋊Dic17—21no—
40C2×C4⋊Dic17—21no—
41C2×C4⋊Dic17D411yescharacteristic
42C2×C4⋊Dic17D411yescharacteristic
43C2×C4⋊Dic17D411yescharacteristic
44C2×C4⋊Dic17D411yescharacteristic
45C2×#63225—21no—
46C2×#63225—21no—
47C2×#63225—21no—
48C2×#63225—21no—
49C2×C4×Dic17—21no—
50C22×#3622—21no—
51#10587565—42no—
52#10587565—42no—
53#10587503—42no—
54#10587502—21no—
55#10587502—21no—
56#10587502—21no—
57#10587489—42no—
58#10587488—42no—
59#10587478—42no—
60C24×C17—21no—
61C2×D4×C17—21no—
62C2×D4×C17—21no—
63C2×D4×C17—41no—
64C2×D4×C17—41no—
65C22×C4×C17—21no—
66C22×C4×C17C2×D411yescharacteristic
67C22×C4×C17C2×D411yescharacteristic
68C22×C4×C17C2×D411yescharacteristic
69C22×Dic17—21no—
70C22×Dic17—21no—
71C22×Dic17—21no—
72C22×Dic17—21no—
73C22×Dic17—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—41no—
81C22⋊C4×C17—41no—
82C22⋊C4×C17—41no—
83C22⋊C4×C17—41no—
84C22⋊C4×C17—41no—
85C42×C17—41no—
86C22⋊Dic17—42no—
87C22⋊Dic17—42no—
88C22⋊Dic17—81no—
89C22⋊Dic17—81no—
90C4⋊Dic17—21no—
91C4⋊Dic17—21no—
92C4⋊Dic17—21no—
93C4⋊Dic17—41no—
94C4⋊Dic17—41no—
95C4⋊Dic17—41no—
96C4⋊Dic17—41no—
97C4⋊Dic17—41no—
98C4⋊Dic17—41no—
99C4⋊Dic17—41no—
100C4⋊Dic17—41no—
101C4⋊Dic17D812yes—
102#63225—42no—
103#63225—41no—
104#63225—41no—
105#63225—41no—
106#63225—41no—
107#63225—81no—
108C4×Dic17—42no—
109C2×#3622—22no—
110C2×#3622—41no—
111C2×#3622—41no—
112#7793—171nomaximal, Sylow
113C23×C17—41no—
114C23×C17—41no—
115C23×C17—41no—
116C23×C17—41no—
117C23×C17—41no—
118C23×C17—41no—
119C23×C17—41no—
120C23×C17#12011yescharacteristic
121D4×C17—42no—
122D4×C17—81no—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—41no—
130C2×C4×C17—41no—
131C2×C4×C17—41no—
132C2×C4×C17C2×D811yescharacteristic
133C2×C4×C17#13511yescharacteristic
134C2×C4×C17#13511yescharacteristic
135C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
136C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
137C2×C4×C17#13711yescharacteristic
138C2×Dic17—22no—
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—41no—
150C2×Dic17—81no—
151C2×Dic17—81no—
152C2×Dic17—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#1271—171no—
162#1270D1711yescharacteristic
163C22×C17—81no—
164C22×C17—81no—
165C22×C17—81no—
166C22×C17—81no—
167C22×C17—81no—
168C22×C17—81no—
169C22×C17—81no—
170C22×C17#44711yescharacteristic
171C22×C17#44911yescharacteristic
172C22×C17#45211yescharacteristic
173C22×C17#45311yescharacteristic
174C22×C17#45611yescharacteristic
175C22×C17#45611yescharacteristic
176C22×C17#45811yescharacteristic
177C4×C17—21no—
178C4×C17—21no—
179C4×C17—21no—
180C4×C17—21no—
181C4×C17—21no—
182C4×C17—21no—
183C4×C17—81no—
184Dic17—42no—
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#390C2×D1711yescharacteristic
198#389—171no—
199#389—171no—
200#389—341no—
201#388—341no—
202#384—171no—
203#340—341no—
204#340—341no—
205#336—171no—
206#336C2×D1711yescharacteristic
207#336C2×D1711yescharacteristic
208C2×C17—161no—
209C2×C17#160911yescharacteristic
210C2×C17#161011yescharacteristic
211C2×C17#161811yescharacteristic
212C2×C17#162111yescharacteristic
213C2×C17#164711yescharacteristic
214C2×C17#165211yescharacteristic
215C2×C17#166011yescharacteristic
216C22×D4D6811yescharacteristic
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—341no—
220#126—682no—
221#126—682no—
222C2×C4⋊C4—171no—
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⋊C4—341no—
233C2×C4⋊C4D6811yescharacteristic
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
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—341no—
247#95—341no—
248#95—341no—
249C17#779311yescharacteristic, complemented, Sylow
250C24—21no—
251C2×D4—21no—
252C2×D4—21no—
253C2×D4—41no—
254C2×D4—41no—
255C22×C4—21no—
256C22×C4—341no—
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—41no—
274C4⋊C4—172no—
275C4⋊C4—341no—
276C4⋊C4—341no—
277C4⋊C4—341no—
278C4⋊C4—681no—
279C4⋊C4—681no—
280C4⋊C4—681no—
281C4⋊C4—681no—
282C4⋊C4—682no—
283C4⋊C4—681no—
284C4⋊C4—681no—
285C4⋊C4—681no—
286C4⋊C4—681no—
287C4⋊C4—681no—
288C4⋊C4—681no—
289C4⋊C4—681no—
290C4⋊C4—681no—
291C4⋊C4—1361no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—682no—
297C22⋊C4—682no—
298C22⋊C4—1361no—
299C22⋊C4—1361no—
300C42—41no—
301C42—682no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23#1058756811yescharacteristic
310D4—42no—
311D4—81no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—342no—
322C2×C4—681no—
323C2×C4—681no—
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×C4#1058759011yescharacteristic
337C2×C4#1058759511yescharacteristic
338C2×C4D8×D1711yescharacteristic
339C2×C4#1058759711yescharacteristic
340C2×C4#1058760011yescharacteristic
341C2×C4#1058760211yescharacteristic
342C8—42no—
343C8—42no—
344C8—682no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22#1176294411yescharacteristic
353C22#1176295511yescharacteristic
354C22#1176297311yescharacteristic
355C22#1176297911yescharacteristic
356C22#1176299311yescharacteristic
357C22#1176299311yescharacteristic
358C22#1176300011yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—81no—
366C4—2721no—
367C4—682no—
368C4—1361no—
369C4—1361no—
370C4—1361no—
371C2—161no—
372C2#1452778211yescharacteristic
373C2#1452778811yescharacteristic
374C2#1452783411yescharacteristic
375C2#1452785011yescharacteristic
376C2#1452802211yescharacteristic
377C2#1452805711yescharacteristic
378C2#1452809611yescharacteristic
subgroupG/H
centerC23#10587568
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#1270C2
socleC23×C17#120
prime radicalC22×D4×C17D4
hypercenter#1270D17
hypocenterC17#7793
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
3. #14530262C2semidirect
4. #14530257C2semidirect
5. #14526608C2semidirect
6. #14526606C2semidirect
249. C17#7793semidirect

Maximal subgroups

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

subgroupindex
#145357432
#145303122
#145302622
#145302572
#145266082
#145266062
C17×#12702
#779317