Finite Group Databasebeta

← #16923517

Subgroups — #16923517

Order 4352.

380 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 79 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#14536764C211yesmaximal, characteristic
2#14530343C211yesmaximal, characteristic, complemented
3#14530194C211yesmaximal, characteristic, complemented
4#14530187C211yesmaximal, characteristic
5#14526609C211yesmaximal, characteristic, complemented
6#14526606C211yesmaximal, characteristic, complemented
7C17×#1270C211yesmaximal, characteristic
8C2×#10587487—21no—
9C2×#10587503—21no—
10#11762734—21no—
11#11762723—21no—
12#11762721—21no—
13#11762715C2211yescharacteristic
14#11762707C2211yescharacteristic
15#11762702—21no—
16#11762702—21no—
17#11762690—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
28C22×Dic34D411yescharacteristic
29C22×C8×C17—21no—
30C22×C8×C17—21no—
31C2×C4⋊C4×C17D411yescharacteristic
32C2×C22⋊C4×C17—21no—
33C2×C22⋊C4×C17—21no—
34C2×C42×C17C2311yescharacteristic
35C2×C22⋊Dic17—21no—
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⋊Dic17D411yescharacteristic
43C2×C4⋊Dic17D411yescharacteristic
44C2×C4⋊Dic17D411yescharacteristic
45C2×#63225—21no—
46C2×C4×Dic17—21no—
47C22×#3622—21no—
48#10587503—42no—
49#10587502—21no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55#10587487—42no—
56C24×C17—21no—
57C2×D4×C17—21no—
58C2×D4×C17—21no—
59C2×D4×C17—41no—
60C2×D4×C17—41no—
61C22×C4×C17—21no—
62C22×C4×C17C2×D411yescharacteristic
63C22×C4×C17C2×D411yescharacteristic
64C22×C4×C17C2×D411yescharacteristic
65C22×Dic17—21no—
66C22×Dic17—21no—
67C22×Dic17—21no—
68C22×Dic17—21no—
69C22×Dic17—41no—
70C2×Dic34—21no—
71C2×Dic34—21no—
72C2×Dic34—41no—
73C2×Dic34—41no—
74C2×C8×C17—41no—
75C2×C8×C17—41no—
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—81no—
87C22⋊Dic17—81no—
88C22⋊Dic17—81no—
89C22⋊Dic17—81no—
90C4⋊Dic17—21no—
91C4⋊Dic17—21no—
92C4⋊Dic17—41no—
93C4⋊Dic17—41no—
94C4⋊Dic17—42no—
95C4⋊Dic17—41no—
96C4⋊Dic17—41no—
97C4⋊Dic17—41no—
98C4⋊Dic17—41no—
99#63225—41no—
100#63225—41no—
101C4×Dic17—81no—
102C2×#3622—41no—
103C2×#3622—41no—
104C2×#3622—41no—
105#7811—171nomaximal, Sylow
106C23×C17—41no—
107C23×C17—41no—
108C23×C17—41no—
109C23×C17—41no—
110C23×C17—41no—
111C23×C17—41no—
112C23×C17—41no—
113C23×C17#12011yescharacteristic
114D4×C17—42no—
115D4×C17—81no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17—41no—
125C2×C4×C17#13511yescharacteristic
126C2×C4×C17#13511yescharacteristic
127C2×C4×C17#13511yescharacteristic
128C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
129C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
130C2×C4×C17#13711yescharacteristic
131C2×Dic17—41no—
132C2×Dic17—41no—
133C2×Dic17—41no—
134C2×Dic17—41no—
135C2×Dic17—41no—
136C2×Dic17—41no—
137C2×Dic17—41no—
138C2×Dic17—41no—
139C2×Dic17—41no—
140C2×Dic17—41no—
141C2×Dic17—41no—
142C2×Dic17—41no—
143C2×Dic17—81no—
144C2×Dic17—81no—
145C2×Dic17—81no—
146Dic34—42no—
147Dic34—81no—
148C8×C17—81no—
149C8×C17—81no—
150#3622—81no—
151#2808—171no—
152#2099—171no—
153#2072—171no—
154#2071—171no—
155#1272—171no—
156#1271—171no—
157#1270D1711yescharacteristic
158C22×C17—81no—
159C22×C17—81no—
160C22×C17—81no—
161C22×C17—81no—
162C22×C17—81no—
163C22×C17—81no—
164C22×C17—81no—
165C22×C17#44911yescharacteristic
166C22×C17#44911yescharacteristic
167C22×C17#45211yescharacteristic
168C22×C17#45411yescharacteristic
169C22×C17#45511yescharacteristic
170C22×C17#45511yescharacteristic
171C22×C17#45811yescharacteristic
172C4×C17—21no—
173C4×C17—21no—
174C4×C17—21no—
175C4×C17—21no—
176C4×C17—21no—
177C4×C17—21no—
178C4×C17—81no—
179Dic17—81no—
180Dic17—81no—
181Dic17—81no—
182Dic17—81no—
183Dic17—161no—
184C2×#103—341no—
185C2×#102—341no—
186#397—341no—
187#394—341no—
188#393—341no—
189#391—171no—
190#390C2×D1711yescharacteristic
191#389—171no—
192#388—341no—
193#388—341no—
194#387—341no—
195#384—171no—
196#340—341no—
197#340—341no—
198#340—341no—
199#340—341no—
200#336—171no—
201#336C2×D1711yescharacteristic
202#336C2×D1711yescharacteristic
203C2×C17—161no—
204C2×C17#161211yescharacteristic
205C2×C17#161211yescharacteristic
206C2×C17#161511yescharacteristic
207C2×C17#162811yescharacteristic
208C2×C17#164011yescharacteristic
209C2×C17#164611yescharacteristic
210C2×C17#166211yescharacteristic
211C22×Q8—171no—
212C22×D4D6811yescharacteristic
213C22×C8—21no—
214C22×C8—21no—
215C22×C8—341no—
216C2×C4⋊C4—171no—
217C2×C4⋊C4—171no—
218C2×C4⋊C4—171no—
219C2×C4⋊C4—341no—
220C2×C4⋊C4—341no—
221C2×C4⋊C4—341no—
222C2×C4⋊C4—341no—
223C2×C4⋊C4D6811yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—341no—
227C2×C22⋊C4—341no—
228C2×C22⋊C4—341no—
229C2×C22⋊C4—341no—
230C2×C42—341no—
231C2×C42C22×D1711yescharacteristic
232#103—682no—
233#102—682no—
234#95—341no—
235#95—341no—
236#95—341no—
237#95—341no—
238#95—341no—
239#95—341no—
240C17#781111yescharacteristic, complemented, Sylow
241C24—21no—
242C2×Q8—341no—
243C2×Q8—341no—
244C2×Q8—681no—
245C2×Q8—681no—
246C2×D4—21no—
247C2×D4—21no—
248C2×D4—41no—
249C2×D4—41no—
250C22×C4—21no—
251C22×C4—341no—
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—681no—
266C2×C8—681no—
267C2×C8—681no—
268C4⋊C4—41no—
269C4⋊C4—341no—
270C4⋊C4—341no—
271C4⋊C4—681no—
272C4⋊C4—681no—
273C4⋊C4—682no—
274C4⋊C4—681no—
275C4⋊C4—681no—
276C4⋊C4—681no—
277C4⋊C4—681no—
278C4⋊C4—681no—
279C4⋊C4—681no—
280C22⋊C4—41no—
281C22⋊C4—41no—
282C22⋊C4—41no—
283C22⋊C4—41no—
284C22⋊C4—1361no—
285C22⋊C4—1361no—
286C22⋊C4—1361no—
287C22⋊C4—1361no—
288C42—41no—
289C42—1361no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23#1058756811yescharacteristic
298Q8—682no—
299Q8—1361no—
300D4—42no—
301D4—81no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—681no—
312C2×C4—681no—
313C2×C4—681no—
314C2×C4—681no—
315C2×C4—681no—
316C2×C4—681no—
317C2×C4—681no—
318C2×C4—681no—
319C2×C4—681no—
320C2×C4—681no—
321C2×C4—681no—
322C2×C4—681no—
323C2×C4—1361no—
324C2×C4—1361no—
325C2×C4—1361no—
326C2×C4#1058759011yescharacteristic
327C2×C4#1058759411yescharacteristic
328C2×C4#1058759711yescharacteristic
329C2×C4#1058759811yescharacteristic
330C2×C4#1058760111yescharacteristic
331C2×C4#1058760211yescharacteristic
332C8—81no—
333C8—81no—
334C8—1361no—
335C22—81no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22#1176295511yescharacteristic
343C22#1176295611yescharacteristic
344C22#1176297211yescharacteristic
345C22#1176298311yescharacteristic
346C22#1176298811yescharacteristic
347C22#1176298911yescharacteristic
348C22#1176300011yescharacteristic
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—81no—
356C4—2721no—
357C4—1361no—
358C4—1361no—
359C4—1361no—
360C4—1361no—
361C2—161no—
362C2#1452779911yescharacteristic
363C2#1452780011yescharacteristic
364C2#1452781611yescharacteristic
365C2#1452789611yescharacteristic
366C2#1452797111yescharacteristic
367C2#1452801411yescharacteristic
368C2#1452810911yescharacteristic
subgroupG/H
centerC23#10587568
commutator, cosocleC2×C42×C17C23
FrattiniC2×C42C22×D17
FittingC17×#1270C2
socleC23×C17#120
prime radicalC22×D4×C17D4
hypercenter#1270D17
hypocenterC17#7811
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. #14526606C2semidirect
240. C17#7811semidirect

Maximal subgroups

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

subgroupindex
#145367642
#145303432
#145301942
#145301872
#145266092
#145266062
C17×#12702
#781117