Finite Group Databasebeta

← #16929849

Subgroups — #16929849

Order 4352.

602 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 119 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#14531506C211yesmaximal, characteristic
2#14527176C212yesmaximal, complemented
3#14527105C211yesmaximal, characteristic, complemented
4#14526612C212yesmaximal, complemented
5C17×#2975C211yesmaximal, characteristic, complemented
6C2×#10587626C411yescharacteristic
7C2×C4×D68C2211yescharacteristic
8#11762862C2212yes—
9#11762712C411yescharacteristic
10#11762710C411yescharacteristic
11#11762688—22no—
12#11762685—22no—
13#11762674C411yescharacteristic
14#11762655C2211yescharacteristic
15#11762538—24no—
16#11762533—22no—
17C2×C4×D4×C17C2211yescharacteristic
18C17×#436—24no—
19C17×#431C2212yes—
20C17×#407—22no—
21C22×D4×C17C2×C411yescharacteristic
22C23×C4×C17D412yes—
23C22×C22⋊D17—22no—
24C22×D68C2×C411yescharacteristic
25C22×C4×D17—21no—
26C2×M16×C17—24no—
27C4×D4×C17—22no—
28C4×D4×C17—22no—
29C2×C4⋊C4×C17C2×C411yescharacteristic
30C2×C22⋊C4×C17D412yes—
31C2×C42×C17C2311yescharacteristic
32C2×C22⋊Dic17—22no—
33#10587626—22no—
34#10587626—22no—
35C2×#63227—22no—
36C2×#63227—21no—
37C2×C4⋊Dic17C2×C411yescharacteristic
38C2×C4×Dic17C2×C411yescharacteristic
39C2×C4×Dic17C2×C411yescharacteristic
40C4×D68—22no—
41C4×D68—22no—
42C17×C4⋊C8—22no—
43C17×C22⋊C8—24no—
44C17×C22⋊C8—22no—
45C4×C8×C17D414yescomplemented
46#10587502—22no—
47#10587502—21no—
48#10587502—21no—
49C22⋊#3622—22no—
50C4⋊#3622—22no—
51C4×#3622—22no—
52C24×C17C22⋊C412yes—
53C23×D17—21no—
54C2×D4×C17—22no—
55C2×D4×C17—21no—
56C2×D4×C17—21no—
57C2×D4×C17—22no—
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—28no—
66C2×C22⋊D17—82no—
67C2×C22⋊D17—82no—
68C22×Dic17—21no—
69C22×Dic17—21no—
70C22×Dic17—21no—
71C2×D68—22no—
72C2×D68—22no—
73C2×D68—21no—
74C2×D68M1612yescomplemented
75C2×C4×D17—42no—
76C2×C4×D17—82no—
77M16×C17—84no—
78C2×C8×C17—24no—
79C4⋊C4×C17—22no—
80C22⋊C4×C17—24no—
81C42×C17C2×D412yes—
82C42×C17#4112yes—
83C22⋊Dic17—82no—
84#63227—41no—
85#63227—41no—
86#63227—82no—
87C4⋊Dic17—22no—
88C4×Dic17—21no—
89C4×Dic17—22no—
90C4×Dic17M1612yescomplemented
91C2×#3622—42no—
92#9105—171nomaximal, Sylow
93C23×C17—22no—
94C23×C17—22no—
95C23×C17—22no—
96C23×C17—42no—
97C23×C17—42no—
98C23×C17C2×C22⋊C411yescharacteristic
99C22×D17—22no—
100C22×D17—42no—
101C22×D17—42no—
102C22×D17—41no—
103C22×D17—41no—
104D4×C17—42no—
105D4×C17—42no—
106C2×C4×C17—22no—
107C2×C4×C17—41no—
108C2×C4×C17—41no—
109C2×C4×C17—44no—
110C2×C4×C17—42no—
111C2×C4×C17—82no—
112C2×C4×C17C4×D412yes—
113C2×C4×C17#12011yescharacteristic
114C2×C4×C17#12112yes—
115C2×C4×C17#12311yescharacteristic
116C2×C4×C17C2×M1611yescharacteristic
117C2×C4×C17C2×M1611yescharacteristic
118C2×C4×C17#13112yes—
119C22⋊D17—88no—
120C2×Dic17—22no—
121C2×Dic17—41no—
122C2×Dic17—41no—
123C2×Dic17—41no—
124C2×Dic17—42no—
125C2×Dic17—41no—
126C2×Dic17—41no—
127C2×Dic17—41no—
128D68—24no—
129D68—42no—
130C4×D17—84no—
131C8×C17—44no—
132#3622—82no—
133#2975D1711yescharacteristic, complemented
134#2328—171no—
135#1478—172no—
136#1460—171no—
137#1274—172no—
138C22×C17—21no—
139C22×C17—21no—
140C22×C17—44no—
141C22×C17—42no—
142C22×C17—42no—
143C22×C17—42no—
144C22×C17—82no—
145C22×C17#38611yescharacteristic
146C22×C17#40711yescharacteristic
147C22×C17#40811yescharacteristic
148C2×D17—44no—
149C2×D17—42no—
150C2×D17—82no—
151C2×D17—81no—
152C2×D17—81no—
153C4×C17—22no—
154C4×C17—21no—
155C4×C17—21no—
156C4×C17—42no—
157C4×C17#43512yes—
158C4×C17#43612yes—
159Dic17—42no—
160Dic17—81no—
161Dic17—81no—
162C2×#127—171no—
163C2×C4×D4—171no—
164C2×C4×D4C2×D1711yescharacteristic
165#436—24no—
166#431—172no—
167#431C2×D1712yes—
168#407—22no—
169#390—171no—
170#390—171no—
171#386—342no—
172#386—342no—
173#382—171no—
174#376—171no—
175D4⋊C8—344no—
176#323—342no—
177C2×C17—21no—
178C2×C17—21no—
179C2×C17—82no—
180C2×C17#187011yescharacteristic
181C2×C17#191011yescharacteristic
182C2×C17#191111yescharacteristic
183D17—82no—
184C22×D4—171no—
185C22×D4—342no—
186C22×D4C4×D1711yescharacteristic
187C23×C4—341no—
188C23×C4C22⋊D1712yes—
189C2×M16—24no—
190#127—342no—
191#127—342no—
192C4×D4—22no—
193C4×D4—22no—
194C4×D4—342no—
195C4×D4—342no—
196C2×C4⋊C4—171no—
197C2×C4⋊C4C4×D1711yescharacteristic
198C2×C22⋊C4—341no—
199C2×C22⋊C4—342no—
200C2×C22⋊C4—342no—
201C2×C22⋊C4D6812yes—
202C2×C42—171no—
203C2×C42—171no—
204C2×C42C22×D1711yescharacteristic
205C4⋊C8—22no—
206C4⋊C8—342no—
207C22⋊C8—24no—
208C22⋊C8—22no—
209C22⋊C8—342no—
210C4×C8—342no—
211C4×C8C22⋊D1714yescomplemented
212#95—341no—
213#95—341no—
214#95—342no—
215C17#910511yescharacteristic, complemented, Sylow
216C24—341no—
217C24#6322712yes—
218C2×D4—21no—
219C2×D4—21no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—172no—
223C2×D4—348no—
224C2×D4—341no—
225C2×D4—342no—
226C2×D4—342no—
227C2×D4—1362no—
228C2×D4—1362no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—82no—
232C22×C4—341no—
233C22×C4—341no—
234C22×C4—341no—
235C22×C4—682no—
236C22×C4—1362no—
237C22×C4#6322712yes—
238C22×C4C2×C4×D1711yescharacteristic
239C22×C4C2×D6811yescharacteristic
240C22×C4C2×C22⋊D1711yescharacteristic
241M16—84no—
242C2×C8—24no—
243C2×C8—682no—
244C4⋊C4—22no—
245C4⋊C4—342no—
246C22⋊C4—24no—
247C22⋊C4—681no—
248C22⋊C4—681no—
249C22⋊C4—1362no—
250C22⋊C4—1362no—
251C42—172no—
252C42—341no—
253C42—342no—
254C42#6325212yes—
255C42C2×C22⋊D1712yes—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—42no—
260C23—42no—
261C23—342no—
262C23—682no—
263C23—682no—
264C23—681no—
265C23—681no—
266C23C2×#6322711yescharacteristic
267D4—42no—
268D4—42no—
269D4—344no—
270D4—682no—
271D4—1368no—
272C2×C4—22no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—44no—
276C2×C4—42no—
277C2×C4—82no—
278C2×C4—342no—
279C2×C4—681no—
280C2×C4—681no—
281C2×C4—681no—
282C2×C4—681no—
283C2×C4—682no—
284C2×C4—681no—
285C2×C4—681no—
286C2×C4—1364no—
287C2×C4M16×D1711yescharacteristic
288C2×C4M16×D1711yescharacteristic
289C2×C4#1058759312yes—
290C2×C4C4×C22⋊D1712yes—
291C2×C4#1058761511yescharacteristic
292C2×C4#1058761612yes—
293C2×C4#1058763911yescharacteristic
294C8—44no—
295C8—1362no—
296C22—21no—
297C22—21no—
298C22—44no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—82no—
303C22—684no—
304C22—682no—
305C22—1362no—
306C22—1361no—
307C22—1361no—
308C22#1176268811yescharacteristic
309C22#1176277211yescharacteristic
310C22#1176277711yescharacteristic
311C4—22no—
312C4—21no—
313C4—21no—
314C4—42no—
315C4—682no—
316C4—1361no—
317C4—1361no—
318C4#1176287812yes—
319C4#1176288512yes—
320C2—21no—
321C2—21no—
322C2—82no—
323C2—1362no—
324C2#1452908111yescharacteristic
325C2#1452931111yescharacteristic
326C2#1452931611yescharacteristic
subgroupG/H
centerC22#11762688
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#2975C2
socleC22×C17#386
prime radicalC2×#10587626C4
cosocleC2×C42×C17C23
hypercenter#2975D17
hypocenterC17#9105
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2975

Complements

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

NH ≅ G/Ntype
2. #14527176C2semidirect
3. #14527105C2semidirect
4. #14526612C2semidirect
5. C17×#2975C2semidirect
45. C4×C8×C17D4semidirect
74. C2×D68M16semidirect
90. C4×Dic17M16semidirect
133. #2975D17semidirect
211. C4×C8C22⋊D17semidirect
215. C17#9105semidirect

Maximal subgroups

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

subgroupindex
#145315062
#145271762
#145271052
#145266122
C17×#29752
#910517