Finite Group Databasebeta

← #16930352

Subgroups — #16930352

Order 4352.

478 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 115 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#14534550C211yesmaximal, characteristic
2#14527457C212yesmaximal, complemented
3#14527098C211yesmaximal, characteristic
4#14526644C212yesmaximal, complemented
5C17×#2921C211yesmaximal, characteristic
6C2×#10587555C2211yescharacteristic
7C2×#10587610—21no—
8#11762862C2212yes—
9#11762739—21no—
10#11762739C412yes—
11#11762734—21no—
12#11762734C412yes—
13#11762715—21no—
14#11762675C2211yescharacteristic
15#11762556—24no—
16#11762535—21no—
17#11762533—21no—
18C2×C17×#117C2211yescharacteristic
19C17×#433—24no—
20C17×#423C2212yes—
21C17×#407—21no—
22C4×M16×C17—21no—
23C23×C4×C17D411yescharacteristic
24C22×Dic34—21no—
25C2×M16×C17—22no—
26C17×#117—22no—
27C17×#117—22no—
28C2×C4⋊C4×C17C2×C411yescharacteristic
29C2×C4⋊C4×C17D411yescharacteristic
30C2×C22⋊C4×C17C2×C411yescharacteristic
31C2×C22⋊C4×C17D411yescharacteristic
32C2×C42×C17D411yescharacteristic
33C2×C42×C17C2311yescharacteristic
34C2×C22⋊Dic17—22no—
35C2×C4⋊Dic17—21no—
36C2×C4⋊Dic17—21no—
37C2×C4⋊Dic17C2×C412yes—
38#10587610—44no—
39C2×#63225—22no—
40#10587555—22no—
41#10587555—22no—
42C17×C4⋊C8—22no—
43C17×C4⋊C8D414yes—
44C17×C22⋊C8—24no—
45C17×#97—22no—
46C4×C8×C17—22no—
47#10587502—22no—
48#10587502—22no—
49#10587502—21no—
50#10587502C2×C412yes—
51C22⋊#3622—22no—
52C4⋊#3622—22no—
53C4×#3622—22no—
54C24×C17C22⋊C411yescharacteristic
55C22×C4×C17—21no—
56C22×C4×C17—21no—
57C22×C4×C17—22no—
58C22×C4×C17—22no—
59C22×C4×C17C22⋊C411yescharacteristic
60C22×C4×C17C22⋊C411yescharacteristic
61C22×C4×C17C22⋊C411yescharacteristic
62C22×C4×C17C22×C411yescharacteristic
63C22×C4×C17C2×D411yescharacteristic
64C22×C4×C17C2×D411yescharacteristic
65C22×Dic17—22no—
66C22×Dic17—22no—
67C2×Dic34—41no—
68C2×Dic34—41no—
69C2×Dic34—42no—
70C2×Dic34—42no—
71M16×C17—42no—
72M16×C17—42no—
73C2×C8×C17—24no—
74C4⋊C4×C17—22no—
75C4⋊C4×C17—22no—
76C22⋊C4×C17—22no—
77C22⋊C4×C17—22no—
78C42×C17—21no—
79C42×C17—21no—
80C42×C17C2×D412yes—
81C42×C17#4112yes—
82C22⋊Dic17—44no—
83C4⋊Dic17—24no—
84C4⋊Dic17—42no—
85C4⋊Dic17—42no—
86#63225—44no—
87C2×#3622—42no—
88#9196—171nomaximal, Sylow
89C23×C17—21no—
90C23×C17—21no—
91C23×C17—21no—
92C23×C17—41no—
93C23×C17—41no—
94C23×C17C2×C22⋊C411yescharacteristic
95C2×C4×C17—21no—
96C2×C4×C17—21no—
97C2×C4×C17—21no—
98C2×C4×C17—21no—
99C2×C4×C17—21no—
100C2×C4×C17—41no—
101C2×C4×C17—41no—
102C2×C4×C17—42no—
103C2×C4×C17—42no—
104C2×C4×C17—41no—
105C2×C4×C17—41no—
106C2×C4×C17C4×D412yes—
107C2×C4×C17#12011yescharacteristic
108C2×C4×C17#12112yes—
109C2×C4×C17#12311yescharacteristic
110C2×C4×C17#13512yes—
111C2×C4×C17AGL(1,ℤ/8)12yes—
112C2×Dic17—42no—
113C2×Dic17—42no—
114C2×Dic17—42no—
115C2×Dic17—42no—
116C2×Dic17—42no—
117C2×Dic17—42no—
118Dic34—82no—
119Dic34—82no—
120C8×C17—44no—
121#3622—82no—
122#2921D1711yescharacteristic
123#2621—171no—
124#1546—172no—
125#1458—171no—
126#1286—172no—
127C22×C17—21no—
128C22×C17—21no—
129C22×C17—42no—
130C22×C17—41no—
131C22×C17—41no—
132C22×C17—41no—
133C22×C17—81no—
134C22×C17#38611yescharacteristic
135C22×C17#41611yescharacteristic
136C22×C17#41811yescharacteristic
137C4×C17—22no—
138C4×C17—22no—
139C4×C17—41no—
140C4×C17—41no—
141C4×C17—41no—
142C4×C17#44912yes—
143C4×C17#48212yes—
144Dic17—82no—
145Dic17—82no—
146C2×C4⋊Q8—171no—
147C2×C22⋊Q8—341no—
148C2×#117C2×D1711yescharacteristic
149#433—24no—
150#431—172no—
151#423C2×D1712yes—
152#407—21no—
153C4×M16—21no—
154#398—172no—
155#398—341no—
156#397—172no—
157#397—341no—
158#391—341no—
159#382—171no—
160#332—344no—
161#324—341no—
162#323—341no—
163C2×C17—21no—
164C2×C17—21no—
165C2×C17—81no—
166C2×C17#187511yescharacteristic
167C2×C17#190911yescharacteristic
168C2×C17#191511yescharacteristic
169C22×Q8—341no—
170C23×C4C22⋊D1711yescharacteristic
171C2×M16—22no—
172C4⋊Q8—342no—
173C4⋊Q8—342no—
174C22⋊Q8—684no—
175#117—22no—
176#117—22no—
177C2×C4⋊C4—172no—
178C2×C4⋊C4—341no—
179C2×C4⋊C4—341no—
180C2×C4⋊C4—342no—
181C2×C4⋊C4C4×D1711yescharacteristic
182C2×C4⋊C4D6811yescharacteristic
183C2×C22⋊C4—342no—
184C2×C22⋊C4C4×D1711yescharacteristic
185C2×C22⋊C4D6811yescharacteristic
186C2×C42C22⋊D1711yescharacteristic
187C2×C42C22×D1711yescharacteristic
188C4⋊C8—22no—
189C4⋊C8—342no—
190C4⋊C8C22⋊D1714yes—
191C22⋊C8—24no—
192C22⋊C8—342no—
193#97—22no—
194C4×C8—22no—
195C4×C8—342no—
196#95—172no—
197#95—341no—
198#95—342no—
199#95—342no—
200C17#919611yescharacteristic, complemented, Sylow
201C24#6322711yescharacteristic
202C2×Q8—681no—
203C2×Q8—682no—
204C2×Q8—682no—
205C2×Q8—681no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—342no—
211C22×C4—342no—
212C22×C4#6322711yescharacteristic
213C22×C4#6322711yescharacteristic
214C22×C4#6322711yescharacteristic
215C22×C4C2×C4×D1711yescharacteristic
216C22×C4C2×D6811yescharacteristic
217C22×C4C2×C22⋊D1711yescharacteristic
218M16—42no—
219M16—42no—
220C2×C8—24no—
221C2×C8—682no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—344no—
225C4⋊C4—684no—
226C4⋊C4—682no—
227C4⋊C4—682no—
228C22⋊C4—22no—
229C22⋊C4—22no—
230C22⋊C4—684no—
231C42—21no—
232C42—21no—
233C42#6325212yes—
234C42C2×C22⋊D1712yes—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—41no—
239C23—41no—
240C23C2×#6322711yescharacteristic
241Q8—1362no—
242Q8—1362no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—682no—
255C2×C4—682no—
256C2×C4—682no—
257C2×C4—682no—
258C2×C4—682no—
259C2×C4—682no—
260C2×C4C4×C22⋊D1712yes—
261C2×C4#1058761511yescharacteristic
262C2×C4#1058761612yes—
263C2×C4#1058763512yes—
264C2×C4#1058763612yes—
265C2×C4#1058763911yescharacteristic
266C8—44no—
267C8—1362no—
268C22—21no—
269C22—21no—
270C22—42no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—81no—
275C22#1176268811yescharacteristic
276C22#1176280311yescharacteristic
277C22#1176281311yescharacteristic
278C4—22no—
279C4—22no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—1362no—
284C4—1362no—
285C4#1176295412yes—
286C4#1176313312yes—
287C2—21no—
288C2—21no—
289C2—81no—
290C2#1452910611yescharacteristic
291C2#1452930611yescharacteristic
292C2#1452933611yescharacteristic
subgroupG/H
centerC22#11762688
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#2921C2
socleC22×C17#386
prime radicalC24×C17C22⋊C4
cosocleC2×C42×C17C23
hypercenter#2921D17
hypocenterC17#9196
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2921

Complements

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

NH ≅ G/Ntype
2. #14527457C2semidirect
4. #14526644C2semidirect
200. C17#9196semidirect

Maximal subgroups

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

subgroupindex
#145345502
#145274572
#145270982
#145266442
C17×#29212
#919617