Finite Group Databasebeta

← #16932173

Subgroups — #16932173

Order 4352.

626 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 131 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#14533681C211yesmaximal, characteristic
2#14528776C211yesmaximal, characteristic, complemented
3#14527210C212yesmaximal, complemented
4#14526643C212yesmaximal, complemented
5C17×#2972C211yesmaximal, characteristic, complemented
6C2×#10587624—21no—
7C2×C4×D68C2211yescharacteristic
8#11762833C2212yes—
9C2×#10587504—21no—
10#11762734C412yes—
11#11762718C412yes—
12#11762688—22no—
13#11762675C2211yescharacteristic
14#11762666—21no—
15#11762578—21no—
16#11762538—24no—
17C2×C4×D4×C17C2211yescharacteristic
18C8×D4×C17—24no—
19C17×#431C2212yes—
20C2×C17×C22⋊C8—22no—
21C22×D4×C17D411yescharacteristic
22C23×C4×C17D412yes—
23C22×C22⋊D17—22no—
24C22×D68C2×C411yescharacteristic
25C22×C4×D17—21no—
26C22×C8×C17—24no—
27C4×D4×C17—22no—
28C4×D4×C17—22no—
29C2×C4⋊C4×C17D411yescharacteristic
30C2×C22⋊C4×C17C2×C412yes—
31C2×C42×C17C2311yescharacteristic
32C2×C22⋊Dic17—22no—
33#10587624—42no—
34#10587624—42no—
35C2×#63227—22no—
36C2×#63227—21no—
37C2×C4⋊Dic17—21no—
38C2×C4⋊Dic17C2×C411yescharacteristic
39C2×#63223—22no—
40C4×D68—22no—
41C4×D68—22no—
42C17×C4⋊C8—22no—
43C17×C22⋊C8—24no—
44C17×C22⋊C8—22no—
45C4×C8×C17D414yescomplemented
46#10587504—42no—
47#10587502—22no—
48#10587502—21no—
49#10587502C2×C412yes—
50C4⋊#3622—22no—
51C4⋊#3622—22no—
52C24×C17C22⋊C412yes—
53C23×D17—21no—
54C2×D4×C17—22no—
55C2×D4×C17—22no—
56C2×D4×C17D812yes—
57C2×D4×C17SD1612yes—
58C22×C4×C17—22no—
59C22×C4×C17—22no—
60C22×C4×C17—82no—
61C22×C4×C17C22⋊C412yes—
62C22×C4×C17C22×C411yescharacteristic
63C22×C4×C17C2×D411yescharacteristic
64C22×C4×C17C2×D411yescharacteristic
65C2×C22⋊D17—44no—
66C2×C22⋊D17—44no—
67C2×C22⋊D17—82no—
68C22×Dic17—21no—
69C22×Dic17—22no—
70C2×D68—22no—
71C2×D68—42no—
72C2×C4×D17—42no—
73C2×C4×D17—41no—
74C2×C4×D17—41no—
75C2×C4×D17—42no—
76C2×C8×C17—24no—
77C2×C8×C17—84no—
78C4⋊C4×C17—22no—
79C22⋊C4×C17—24no—
80C42×C17C2×D412yes—
81C42×C17#4112yes—
82C22⋊Dic17—44no—
83#63227—41no—
84#63227—41no—
85#63227—82no—
86C4⋊Dic17—22no—
87C4⋊Dic17—41no—
88C4⋊Dic17—41no—
89#63223—42no—
90#63223—42no—
91C2×#3622—42no—
92#9485—171nomaximal, Sylow
93C23×C17—22no—
94C23×C17—22no—
95C23×C17—42no—
96C23×C17—42no—
97C23×C17#10214yes—
98C23×C17C2×C22⋊C411yescharacteristic
99C22×D17—22no—
100C22×D17—41no—
101C22×D17—41no—
102C22×D17—41no—
103C22×D17—41no—
104C22×D17—41no—
105C22×D17—41no—
106D4×C17—24no—
107D4×C17—24no—
108C2×C4×C17—22no—
109C2×C4×C17—41no—
110C2×C4×C17—41no—
111C2×C4×C17—44no—
112C2×C4×C17—42no—
113C2×C4×C17—82no—
114C2×C4×C17C4×D412yes—
115C2×C4×C17#12011yescharacteristic
116C2×C4×C17#12112yes—
117C2×C4×C17#12311yescharacteristic
118C2×C4×C17C2×D811yescharacteristic
119C2×C4×C17C2×SD1611yescharacteristic
120C2×C4×C17#13512yes—
121C22⋊D17—88no—
122C2×Dic17—42no—
123C2×Dic17—42no—
124C2×Dic17—41no—
125C2×Dic17—41no—
126C2×Dic17—42no—
127C2×Dic17—41no—
128D68—24no—
129D68—42no—
130C4×D17—82no—
131C4×D17—82no—
132C8×C17—44no—
133#3622—82no—
134#2972D1711yescharacteristic, complemented
135#2535—171no—
136#1800—171no—
137#1486—172no—
138#1286—172no—
139C22×C17—21no—
140C22×C17—21no—
141C22×C17—24no—
142C22×C17—44no—
143C22×C17—42no—
144C22×C17—42no—
145C22×C17—82no—
146C22×C17#38611yescharacteristic
147C22×C17C2×#10211yescharacteristic
148C22×C17#41611yescharacteristic
149C2×D17—42no—
150C2×D17—42no—
151C2×D17—42no—
152C2×D17—81no—
153C2×D17—81no—
154C2×D17—81no—
155C2×D17—81no—
156C4×C17—22no—
157C4×C17—21no—
158C4×C17—21no—
159C4×C17—42no—
160C4×C17#46512yes—
161C4×C17#46612yes—
162Dic17—82no—
163Dic17—81no—
164C2×#121—341no—
165C2×C4×D4—171no—
166C2×C4×D4C2×D1711yescharacteristic
167C8×D4—24no—
168#431C2×D1712yes—
169#423—172no—
170C2×#100—341no—
171C2×C22⋊C8—22no—
172#397—172no—
173#392—172no—
174#386—342no—
175#382—171no—
176#380—341no—
177#343—341no—
178D4⋊C8—344no—
179C2×C17—21no—
180C2×C17—21no—
181C2×C17—44no—
182C2×C17#189111yescharacteristic
183C2×C17#189311yescharacteristic
184C2×C17#191111yescharacteristic
185D17—82no—
186C22×D4—171no—
187C22×D4—342no—
188C22×D4D6811yescharacteristic
189C23×C4—341no—
190C23×C4C22⋊D1712yes—
191C2×M16—342no—
192C22×C8—24no—
193#121—682no—
194#121—682no—
195C4×D4—22no—
196C4×D4—22no—
197C4×D4—342no—
198C4×D4—342no—
199C2×C4⋊C4—171no—
200C2×C4⋊C4—341no—
201C2×C4⋊C4D6811yescharacteristic
202C2×C22⋊C4—342no—
203C2×C22⋊C4—341no—
204C2×C22⋊C4—342no—
205C2×C22⋊C4C4×D1712yes—
206C2×C42C22×D1711yescharacteristic
207C4⋊C8—22no—
208C4⋊C8—342no—
209C4⋊C8—342no—
210#100—682no—
211C22⋊C8—24no—
212C22⋊C8—22no—
213C4×C8C22⋊D1714yescomplemented
214#95—172no—
215#95—341no—
216#95—342no—
217C17#948511yescharacteristic, complemented, Sylow
218C24—341no—
219C24#6322712yes—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—342no—
223C2×D4—682no—
224C2×D4—684no—
225C2×D4—684no—
226C2×D4—1362no—
227C2×D4#6321912yes—
228C2×D4D13612yes—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—82no—
232C22×C4—342no—
233C22×C4—341no—
234C22×C4—681no—
235C22×C4—681no—
236C22×C4—682no—
237C22×C4—682no—
238C22×C4#6322712yes—
239C22×C4C2×C4×D1711yescharacteristic
240C22×C4C2×D6811yescharacteristic
241C22×C4C2×C22⋊D1711yescharacteristic
242M16—682no—
243M16—682no—
244C2×C8—24no—
245C2×C8—84no—
246C2×C8—682no—
247C4⋊C4—22no—
248C4⋊C4—342no—
249C4⋊C4—681no—
250C4⋊C4—681no—
251C22⋊C4—24no—
252C22⋊C4—684no—
253C22⋊C4—681no—
254C22⋊C4—681no—
255C22⋊C4—1362no—
256C42#6325212yes—
257C42C2×C22⋊D1712yes—
258C23—22no—
259C23—22no—
260C23—42no—
261C23—42no—
262C23—342no—
263C23—681no—
264C23—681no—
265C23—681no—
266C23—681no—
267C23—681no—
268C23—681no—
269C23#1058749214yes—
270C23C2×#6322711yescharacteristic
271D4—24no—
272D4—24no—
273D4—344no—
274D4—682no—
275D4—1368no—
276C2×C4—22no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—44no—
280C2×C4—42no—
281C2×C4—82no—
282C2×C4—682no—
283C2×C4—682no—
284C2×C4—681no—
285C2×C4—681no—
286C2×C4—682no—
287C2×C4—681no—
288C2×C4—1362no—
289C2×C4—1362no—
290C2×C4C2×#6321911yescharacteristic
291C2×C4C2×D13611yescharacteristic
292C2×C4#1058759012yes—
293C2×C4C4×C22⋊D1712yes—
294C2×C4#1058761511yescharacteristic
295C2×C4#1058761612yes—
296C2×C4#1058763911yescharacteristic
297C8—44no—
298C8—1362no—
299C22—21no—
300C22—21no—
301C22—24no—
302C22—44no—
303C22—42no—
304C22—42no—
305C22—82no—
306C22—682no—
307C22—682no—
308C22—682no—
309C22—1361no—
310C22—1361no—
311C22—1361no—
312C22—1361no—
313C22#1176268811yescharacteristic
314C22C2×#1058749211yescharacteristic
315C22#1176280611yescharacteristic
316C4—21no—
317C4—21no—
318C4—22no—
319C4—42no—
320C4—1362no—
321C4—1361no—
322C4#1176303812yes—
323C4#1176304512yes—
324C2—21no—
325C2—21no—
326C2—44no—
327C2—1362no—
328C2#1452920211yescharacteristic
329C2#1452921211yescharacteristic
330C2#1452931611yescharacteristic
subgroupG/H
centerC22#11762688
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#2972C2
socleC22×C17#386
prime radical#14533681C2
cosocleC2×C42×C17C23
hypercenter#2972D17
hypocenterC17#9485
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2972

Complements

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

NH ≅ G/Ntype
2. #14528776C2semidirect
3. #14527210C2semidirect
4. #14526643C2semidirect
5. C17×#2972C2semidirect
45. C4×C8×C17D4semidirect
134. #2972D17semidirect
213. C4×C8C22⋊D17semidirect
217. C17#9485semidirect

Maximal subgroups

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

subgroupindex
#145336812
#145287762
#145272102
#145266432
C17×#29722
#948517