Finite Group Databasebeta

← #16931035

Subgroups — #16931035

Order 4352.

454 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 113 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#14539732C211yesmaximal, characteristic
2#14531533C211yesmaximal, characteristic
3#14529764C211yesmaximal, characteristic
4#14527349C211yesmaximal, characteristic
5#14526644C211yesmaximal, characteristic
6#14526612C211yesmaximal, characteristic
7C17×#1548C211yesmaximal, characteristic
8#11763366C2211yescharacteristic
9#11763188C412yescomplemented
10#11763159C412yescomplemented
11#11763016—22no—
12#11762965—22no—
13#11762862C2211yescharacteristic
14#11762716—22no—
15#11762708—21no—
16#11762695—22no—
17#11762675C2211yescharacteristic
18C4⋊C4×Dic17—21no—
19#11762655C2211yescharacteristic
20#11762570—21no—
21#11762570—21no—
22#11762547—22no—
23C2×C17×C4⋊Q8C2211yescharacteristic
24C17×#431C2211yescharacteristic
25C17×#423C2211yescharacteristic
26C17×#333—22no—
27C22×Q8×C17D412yes—
28C2×M16×C17—21no—
29C17×C4⋊Q8—21no—
30C17×C4⋊Q8—21no—
31C17×C4⋊Q8C2×C412yes—
32C17×C4⋊Q8D412yes—
33C2×C4⋊C4×C17—21no—
34C2×C4⋊C4×C17—21no—
35C2×C42×C17C2311yescharacteristic
36C2×#63231—24no—
37#10587612—21no—
38C2×C4⋊Dic17—22no—
39C2×C4⋊Dic17—21no—
40#10587610—21no—
41#10587610—21no—
42C2×C4×Dic17—22no—
43C2×C4×Dic17D411yescharacteristic
44C2×C4×Dic17D411yescharacteristic
45#10587556—21no—
46#10587555C2×C411yescharacteristic
47#10587555C2×C411yescharacteristic
48C4×Dic34—21no—
49C4×Dic34—21no—
50C17×C4⋊C8—21no—
51C17×C4⋊C8—21no—
52C17×C4⋊C8—21no—
53C17×C22⋊C8—21no—
54C4×C8×C17—21no—
55#10587506—24no—
56#10587502—22no—
57#10587502—22no—
58#10587502—21no—
59#10587502—21no—
60#10587502—21no—
61#10587502D412yes—
62C22⋊#3622—21no—
63C4⋊#3622—21no—
64C4×#3622C2×C412yes—
65C2×Q8×C17—22no—
66C2×Q8×C17—82no—
67C2×Q8×C17C22⋊C414yes—
68C22×C4×C17—22no—
69C22×C4×C17C2×D411yescharacteristic
70C22×C4×C17C2×D411yescharacteristic
71C22×C4×C17C2×D411yescharacteristic
72C22×Dic17—21no—
73C22×Dic17—21no—
74C22×Dic17—22no—
75C2×Dic34—21no—
76C2×Dic34—21no—
77M16×C17—41no—
78M16×C17—41no—
79C2×C8×C17—41no—
80C2×C8×C17—41no—
81C4⋊C4×C17—22no—
82C4⋊C4×C17—22no—
83C4⋊C4×C17—22no—
84C4⋊C4×C17—22no—
85C42×C17C22×C411yescharacteristic
86C42×C17C2×D411yescharacteristic
87C42×C17#4111yescharacteristic
88C42×C17#4111yescharacteristic
89C22⋊Dic17—41no—
90#63231—84no—
91C4⋊Dic17—21no—
92C4⋊Dic17—21no—
93C4⋊Dic17—21no—
94C4⋊Dic17—21no—
95C4⋊Dic17—41no—
96C4⋊Dic17—41no—
97C4⋊Dic17—82no—
98#63225—41no—
99#63225—41no—
100C4×Dic17—21no—
101C4×Dic17—21no—
102C4×Dic17—21no—
103C4×Dic17—41no—
104C4×Dic17—82no—
105C4×Dic17Q1612yescomplemented
106C2×#3622—22no—
107#9297—171nomaximal, Sylow
108C23×C17#12011yescharacteristic
109Q8×C17—44no—
110Q8×C17—82no—
111C2×C4×C17—24no—
112C2×C4×C17—41no—
113C2×C4×C17—41no—
114C2×C4×C17—42no—
115C2×C4×C17—42no—
116C2×C4×C17C2×C22⋊C411yescharacteristic
117C2×C4×C17C4×D411yescharacteristic
118C2×C4×C17C4×D411yescharacteristic
119C2×C4×C17#12111yescharacteristic
120C2×C4×C17#12111yescharacteristic
121C2×C4×C17#12311yescharacteristic
122C2×C4×C17C2×Q1611yescharacteristic
123C2×C4×C17#13511yescharacteristic
124C2×C4×C17#13711yescharacteristic
125C2×C4×C17#13711yescharacteristic
126C2×Dic17—22no—
127C2×Dic17—41no—
128C2×Dic17—41no—
129C2×Dic17—41no—
130C2×Dic17—42no—
131C2×Dic17—42no—
132C2×Dic17—41no—
133C2×Dic17—41no—
134C2×Dic17—41no—
135C2×Dic17—41no—
136C2×Dic17—42no—
137Dic34—42no—
138Dic34—81no—
139C8×C17—81no—
140C8×C17—81no—
141#3622—42no—
142#3209—171no—
143#2331—171no—
144#1995—171no—
145#1548D1711yescharacteristic
146#1518—171no—
147#1286—171no—
148#1274—171no—
149C22×C17—21no—
150C22×C17—21no—
151C22×C17#38611yescharacteristic
152C22×C17#45111yescharacteristic
153C22×C17#45211yescharacteristic
154C4×C17—21no—
155C4×C17—21no—
156C4×C17—21no—
157C4×C17—21no—
158C4×C17—44no—
159C4×C17C4×Q1611yescharacteristic
160C4×C17#44111yescharacteristic
161C4×C17#46211yescharacteristic
162C4×C17#46411yescharacteristic
163Dic17—42no—
164Dic17—82no—
165Dic17—81no—
166Dic17—81no—
167Dic17—81no—
168#533—171no—
169C2×C4⋊Q8C2×D1711yescharacteristic
170#494—172no—
171#487—172no—
172#462—342no—
173#451—342no—
174#431—171no—
175#431C2×D1711yescharacteristic
176#423C2×D1711yescharacteristic
177#391—342no—
178#389—341no—
179#387—342no—
180#382—171no—
181C4×C4⋊C4—341no—
182#376—171no—
183#339—341no—
184#339—341no—
185#333—22no—
186#328—342no—
187C2×C17—21no—
188C2×C17—21no—
189C2×C17#187811yescharacteristic
190C2×C17#190511yescharacteristic
191C2×C17#191111yescharacteristic
192C22×Q8D6812yes—
193C2×Q16—344no—
194C2×M16—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—171no—
198C4⋊Q8—171no—
199C4⋊Q8C4×D1712yes—
200C4⋊Q8C22⋊D1712yes—
201#125—341no—
202C22⋊Q8—341no—
203C22⋊Q8—341no—
204C4×Q8—341no—
205C4×Q8—341no—
206#117—341no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—341no—
210C2×C4⋊C4—342no—
211C2×C42—171no—
212C2×C42—171no—
213C2×C42—342no—
214C2×C42C22×D1711yescharacteristic
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—341no—
219#103—344no—
220C22⋊C8—21no—
221C22⋊C8—341no—
222C4×C8—21no—
223C4×C8—172no—
224#95—172no—
225#95—342no—
226#95—342no—
227#95—341no—
228#95—341no—
229#95—341no—
230C17#929711yescharacteristic, complemented, Sylow
231C2×Q8—22no—
232C2×Q8—82no—
233C2×Q8—341no—
234C2×Q8—341no—
235C2×Q8#6322714yes—
236C22×C4—22no—
237C22×C4—341no—
238C22×C4—342no—
239C22×C4—341no—
240C22×C4C2×D6811yescharacteristic
241C22×C4D4×D1711yescharacteristic
242C22×C4D4×D1711yescharacteristic
243Q16—1364no—
244M16—41no—
245M16—41no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—342no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—341no—
254C4⋊C4—341no—
255C4⋊C4—341no—
256C4⋊C4—341no—
257C4⋊C4—681no—
258C4⋊C4—681no—
259C4⋊C4—681no—
260C4⋊C4—681no—
261C4⋊C4—1362no—
262C22⋊C4—681no—
263C42—172no—
264C42—341no—
265C42—341no—
266C42—341no—
267C42—681no—
268C42—1362no—
269C42C2×C4×D1711yescharacteristic
270C42#6325411yescharacteristic
271C42#6325411yescharacteristic
272C42C2×C22⋊D1711yescharacteristic
273C23#1058756811yescharacteristic
274Q8—44no—
275Q8—82no—
276Q8—682no—
277Q8—1361no—
278C2×C4—24no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—342no—
284C2×C4—681no—
285C2×C4—681no—
286C2×C4—682no—
287C2×C4—681no—
288C2×C4—682no—
289C2×C4—681no—
290C2×C4—681no—
291C2×C4—682no—
292C2×C4—681no—
293C2×C4—681no—
294C2×C4#1058756711yescharacteristic
295C2×C4#1058756711yescharacteristic
296C2×C4#1058757211yescharacteristic
297C2×C4#1058760111yescharacteristic
298C2×C4#1058760211yescharacteristic
299C2×C4Q16×D1711yescharacteristic
300C2×C4#1058760411yescharacteristic
301C2×C4C2×#6322711yescharacteristic
302C2×C4#1058762511yescharacteristic
303C2×C4#1058762511yescharacteristic
304C8—81no—
305C8—81no—
306C8—682no—
307C22—21no—
308C22—21no—
309C22#1176268611yescharacteristic
310C22#1176296611yescharacteristic
311C22#1176297411yescharacteristic
312C4—21no—
313C4—21no—
314C4—21no—
315C4—21no—
316C4—44no—
317C4—682no—
318C4—1362no—
319C4—1361no—
320C4—1361no—
321C4—1361no—
322C4#1176290111yescharacteristic
323C4#1176291311yescharacteristic
324C4#1176301711yescharacteristic
325C4#1176302911yescharacteristic
326C2—21no—
327C2—21no—
328C2#1452912311yescharacteristic
329C2#1452927611yescharacteristic
330C2#1452931411yescharacteristic
subgroupG/H
centerC22#11762686
commutatorC42×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1548C2
socleC22×C17#386
prime radicalC23×C17#120
cosocleC2×C42×C17C23
hypercenter#1548D17
hypocenterC17#9297
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. #11763188C4semidirect
10. #11763159C4semidirect
105. C4×Dic17Q16semidirect
230. C17#9297semidirect

Maximal subgroups

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

subgroupindex
#145397322
#145315332
#145297642
#145273492
#145266442
#145266122
C17×#15482
#929717