Finite Group Databasebeta

← #16930971

Subgroups — #16930971

Order 4352.

410 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 107 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#14539935C211yesmaximal, characteristic
2#14537721C211yesmaximal, characteristic
3#14529758C211yesmaximal, characteristic
4#14527349C211yesmaximal, characteristic
5#14526644C211yesmaximal, characteristic
6#14526614C211yesmaximal, characteristic
7C17×#1548C211yesmaximal, characteristic
8#11763366C2211yescharacteristic
9#11763212C412yescomplemented
10#11763161C412yescomplemented
11#11763144—22no—
12#11763101—22no—
13#11762938—22no—
14#11762938—22no—
15#11762862C2211yescharacteristic
16#11762837—21no—
17#11762767—21no—
18C2×#10587474C2211yescharacteristic
19#11762675C2211yescharacteristic
20#11762571—21no—
21#11762571—21no—
22#11762547—22no—
23C2×C17×C4⋊Q8C2211yescharacteristic
24C17×#431C2211yescharacteristic
25C17×#423C2211yescharacteristic
26C17×#333—22no—
27C22×Q8×C17—21no—
28C2×M16×C17—21no—
29C17×C4⋊Q8—21no—
30C17×C4⋊Q8—21no—
31C17×C4⋊Q8—21no—
32C17×C4⋊Q8C2×C412yes—
33C2×C4⋊C4×C17—21no—
34C2×C4⋊C4×C17—21no—
35C2×C42×C17C2311yescharacteristic
36#10587612—21no—
37C2×C4⋊Dic17—21no—
38#10587610—21no—
39#10587610D412yes—
40C22×#3622D411yescharacteristic
41C22×#3622D411yescharacteristic
42#10587556—21no—
43#10587555C2×C411yescharacteristic
44#10587555C2×C411yescharacteristic
45C4×Dic34—21no—
46C4×Dic34D412yes—
47C17×C4⋊C8—21no—
48C17×C4⋊C8—21no—
49C17×C4⋊C8D412yes—
50C17×C22⋊C8—21no—
51C4×C8×C17—21no—
52#10587502—21no—
53#10587502—21no—
54C22⋊#3622—21no—
55#10587486—22no—
56#10587485—24no—
57C8×Dic17—22no—
58#10587481—24no—
59#10587478—24no—
60C4⋊#3622—21no—
61#10587474—21no—
62#10587474—21no—
63C4×#3622C2×C412yes—
64C2×Q8×C17—22no—
65C2×Q8×C17—41no—
66C2×Q8×C17—82no—
67C22×C4×C17—41no—
68C22×C4×C17C2×D411yescharacteristic
69C22×C4×C17C2×D411yescharacteristic
70C22×C4×C17#4111yescharacteristic
71C22×Dic17—41no—
72C2×Dic34—21no—
73C2×Dic34C22⋊C412yes—
74M16×C17—41no—
75M16×C17—41no—
76C2×C8×C17—22no—
77C2×C8×C17—41no—
78C4⋊C4×C17—22no—
79C4⋊C4×C17—22no—
80C4⋊C4×C17—41no—
81C4⋊C4×C17—41no—
82C42×C17C22×C411yescharacteristic
83C42×C17C2×D411yescharacteristic
84C42×C17C2×D411yescharacteristic
85C42×C17#4111yescharacteristic
86C22⋊Dic17—22no—
87C4⋊Dic17—21no—
88C4⋊Dic17—21no—
89C4⋊Dic17—21no—
90C4⋊Dic17—41no—
91C4⋊Dic17—41no—
92C4⋊Dic17C22⋊C412yes—
93#63225—22no—
94#63225—22no—
95C4×Dic17—22no—
96C2×#3622—21no—
97C2×#3622—21no—
98C2×#3622—22no—
99C2×#3622—21no—
100C2×#3622—21no—
101C2×#3622—21no—
102C2×#3622Q1612yes—
103#9288—171nomaximal, Sylow
104C23×C17#12111yescharacteristic
105Q8×C17—82no—
106Q8×C17—82no—
107C2×C4×C17—41no—
108C2×C4×C17—41no—
109C2×C4×C17—42no—
110C2×C4×C17—82no—
111C2×C4×C17C2×C22⋊C411yescharacteristic
112C2×C4×C17C4×D411yescharacteristic
113C2×C4×C17C4×D411yescharacteristic
114C2×C4×C17#12011yescharacteristic
115C2×C4×C17#12111yescharacteristic
116C2×C4×C17#12311yescharacteristic
117C2×C4×C17C2×Q1611yescharacteristic
118C2×C4×C17#13511yescharacteristic
119C2×C4×C17#13711yescharacteristic
120C2×C4×C17#13711yescharacteristic
121C2×Dic17—22no—
122C2×Dic17—22no—
123C2×Dic17—81no—
124C2×Dic17—81no—
125C2×Dic17—81no—
126Dic34—42no—
127Dic34—81no—
128C8×C17—42no—
129C8×C17—81no—
130#3622—22no—
131#3622—42no—
132#3622—41no—
133#3230—171no—
134#2961—171no—
135#1993—171no—
136#1548D1711yescharacteristic
137#1518—171no—
138#1286—171no—
139#1275—171no—
140C22×C17—21no—
141C22×C17—21no—
142C22×C17#38611yescharacteristic
143C22×C17#46211yescharacteristic
144C22×C17#46411yescharacteristic
145C4×C17—21no—
146C4×C17—21no—
147C4×C17—21no—
148C4×C17—21no—
149C4×C17—82no—
150C4×C17C4×Q1611yescharacteristic
151C4×C17#44111yescharacteristic
152C4×C17#45111yescharacteristic
153C4×C17#45211yescharacteristic
154Dic17—42no—
155Dic17—42no—
156Dic17—161no—
157#533—171no—
158C2×C4⋊Q8C2×D1711yescharacteristic
159#500—172no—
160#487—172no—
161#484—342no—
162#477—342no—
163#446—342no—
164#446—342no—
165#431—171no—
166#431C2×D1711yescharacteristic
167#424—341no—
168#423C2×D1711yescharacteristic
169#405—341no—
170C2×#97—171no—
171#382—171no—
172#340—341no—
173#340—341no—
174#333—22no—
175#328—342no—
176C2×C17—21no—
177C2×C17—21no—
178C2×C17#187811yescharacteristic
179C2×C17#190311yescharacteristic
180C2×C17#190911yescharacteristic
181C22×Q8—21no—
182C2×M16—21no—
183C22×C8—171no—
184C22×C8—171no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—171no—
189C4⋊Q8—171no—
190C4⋊Q8C4×D1712yes—
191#125—341no—
192C22⋊Q8—172no—
193C22⋊Q8—341no—
194C4×Q8—172no—
195C4×Q8—341no—
196#117—341no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—341no—
200C2×C42C22×D1711yescharacteristic
201#107—344no—
202C4⋊C8—21no—
203C4⋊C8—21no—
204C4⋊C8—344no—
205C4⋊C8—341no—
206C4⋊C8C22⋊D1712yes—
207#103—344no—
208C22⋊C8—21no—
209C22⋊C8—341no—
210#97—342no—
211#97—341no—
212#97—341no—
213C4×C8—21no—
214C4×C8—172no—
215C4×C8—342no—
216#95—341no—
217#95—341no—
218C17#928811yescharacteristic, complemented, Sylow
219C2×Q8—22no—
220C2×Q8—41no—
221C2×Q8—82no—
222C2×Q8—172no—
223C2×Q8—341no—
224C22×C4—41no—
225C22×C4—681no—
226C22×C4#6325211yescharacteristic
227C22×C4D4×D1711yescharacteristic
228C22×C4D4×D1711yescharacteristic
229M16—41no—
230M16—41no—
231C2×C8—22no—
232C2×C8—41no—
233C2×C8—172no—
234C2×C8—341no—
235C2×C8—341no—
236C2×C8—342no—
237C2×C8—341no—
238C2×C8—341no—
239C2×C8—341no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—41no—
243C4⋊C4—41no—
244C4⋊C4—172no—
245C4⋊C4—341no—
246C4⋊C4—341no—
247C4⋊C4—342no—
248C4⋊C4—341no—
249C4⋊C4—342no—
250C4⋊C4—681no—
251C4⋊C4—681no—
252C22⋊C4—342no—
253C42—342no—
254C42C2×C4×D1711yescharacteristic
255C42D4×D1711yescharacteristic
256C42#6325411yescharacteristic
257C42C2×C22⋊D1711yescharacteristic
258C23#1058757011yescharacteristic
259Q8—82no—
260Q8—82no—
261Q8—682no—
262Q8—1361no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—82no—
267C2×C4—342no—
268C2×C4—342no—
269C2×C4—1361no—
270C2×C4—1361no—
271C2×C4—1361no—
272C2×C4C22⋊C4×D1711yescharacteristic
273C2×C4#1058756711yescharacteristic
274C2×C4#1058756911yescharacteristic
275C2×C4#1058760111yescharacteristic
276C2×C4#1058760211yescharacteristic
277C2×C4Q16×D1711yescharacteristic
278C2×C4#1058760411yescharacteristic
279C2×C4C4×C22⋊D1711yescharacteristic
280C2×C4#1058762311yescharacteristic
281C2×C4#1058762411yescharacteristic
282C8—42no—
283C8—81no—
284C8—342no—
285C8—682no—
286C8—681no—
287C22—21no—
288C22—21no—
289C22#1176268511yescharacteristic
290C22#1176301811yescharacteristic
291C22#1176303411yescharacteristic
292C4—21no—
293C4—21no—
294C4—21no—
295C4—21no—
296C4—82no—
297C4—2721no—
298C4—682no—
299C4—682no—
300C4#1176290111yescharacteristic
301C4#1176291311yescharacteristic
302C4#1176296311yescharacteristic
303C4#1176296811yescharacteristic
304C2—21no—
305C2—21no—
306C2#1452912211yescharacteristic
307C2#1452926711yescharacteristic
308C2#1452930311yescharacteristic
subgroupG/H
centerC22#11762685
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1548C2
socleC22×C17#386
prime radicalC23×C17#121
cosocleC2×C42×C17C23
hypercenter#1548D17
hypocenterC17#9288
LCSG ▷ C42×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
9. #11763212C4semidirect
10. #11763161C4semidirect
218. C17#9288semidirect

Maximal subgroups

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

subgroupindex
#145399352
#145377212
#145297582
#145273492
#145266442
#145266142
C17×#15482
#928817