Finite Group Databasebeta

← #16932162

Subgroups — #16932162

Order 4352.

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

normal subgroups only

This group has 542 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#14537807C211yesmaximal, characteristic
2#14534143C211yesmaximal, characteristic
3#14528773C211yesmaximal, characteristic
4#14527214C211yesmaximal, characteristic, complemented
5#14527212C211yesmaximal, characteristic, complemented
6#14526642C211yesmaximal, characteristic, complemented
7C17×#1286C211yesmaximal, characteristic, complemented
8C2×C4×Dic34C2211yescharacteristic
9C2×C4×D68C2211yescharacteristic
10C8×D68—22no—
11#11762874—22no—
12#11762862C2211yescharacteristic
13#11762833C2211yescharacteristic
14C2×#10587494—21no—
15C22⋊C8×D17—22no—
16#11762746C412yes—
17#11762722C412yes—
18#11762710—21no—
19#11762708—21no—
20#11762683—22no—
21#11762577—21no—
22#11762541—22no—
23#11762539—22no—
24C17×#431C2211yescharacteristic
25C17×#423C2211yescharacteristic
26C17×#382C2211yescharacteristic
27C22×D68D411yescharacteristic
28C22×C4×D17D412yes—
29C22×Dic34C2×C411yescharacteristic
30C2×M16×C17—21no—
31C2×C4⋊C4×C17—21no—
32C2×C42×C17C2311yescharacteristic
33C2×#63227—22no—
34C2×#63227—22no—
35C2×#63227C2×C412yes—
36C2×C4⋊Dic17—22no—
37C2×C4⋊Dic17C2×C411yescharacteristic
38C2×C4⋊Dic17D411yescharacteristic
39C2×#63225—21no—
40C2×C4×Dic17—21no—
41C2×#63223—21no—
42C2×C8×D17—24no—
43C4×D68—21no—
44C4×D68—21no—
45C4×D68—21no—
46C4×D68—21no—
47C4×Dic34—21no—
48C4×Dic34—21no—
49C4×Dic34—21no—
50C4×Dic34—21no—
51C17×C4⋊C8—21no—
52C17×C4⋊C8—21no—
53C17×C4⋊C8—21no—
54C17×C22⋊C8—21no—
55C4×C8×C17D412yes—
56C17×#95—21no—
57C17×#95C2×C412yes—
58#10587502—22no—
59C22⋊#3622—21no—
60#10587494—42no—
61#10587491—24no—
62C4⋊#3622—21no—
63C4⋊#3622—21no—
64C4⋊#3622—21no—
65C4×#3622D412yes—
66C23×D17C22⋊C412yes—
67C22×C4×C17—22no—
68C22×C4×C17C22×C411yescharacteristic
69C22×C4×C17C2×D411yescharacteristic
70C22×C4×C17C2×D411yescharacteristic
71C22×Dic17—21no—
72C22×Dic17—21no—
73C22×Dic17C22⋊C412yes—
74C2×D68—21no—
75C2×D68—21no—
76C2×D68—21no—
77C2×D68—21no—
78C2×D68SD1612yes—
79C2×D68Q1612yes—
80C2×C4×D17—22no—
81C2×C4×D17—22no—
82C2×C4×D17—82no—
83C2×Dic34—21no—
84C2×Dic34—21no—
85C2×Dic34—42no—
86M16×C17—41no—
87M16×C17—41no—
88C2×C8×C17—22no—
89C2×C8×C17—41no—
90C4⋊C4×C17—41no—
91C4⋊C4×C17—41no—
92C42×C17C2×D411yescharacteristic
93C42×C17C2×D411yescharacteristic
94C42×C17#4111yescharacteristic
95C42×C17#4111yescharacteristic
96#63227—22no—
97#63227—22no—
98#63227—44no—
99#63227—82no—
100C4⋊Dic17—21no—
101C4⋊Dic17—21no—
102C4⋊Dic17—21no—
103C4⋊Dic17—21no—
104C4⋊Dic17—82no—
105#63225—41no—
106#63225—41no—
107C4×Dic17—41no—
108C4×Dic17—41no—
109#63223—41no—
110#63223—41no—
111C2×#3622—22no—
112C2×#3622—41no—
113C8×D17—84no—
114#9483—171nomaximal, Sylow
115C23×C17C2×C22⋊C411yescharacteristic
116C22×D17—22no—
117C22×D17—22no—
118C22×D17—42no—
119C22×D17—42no—
120C22×D17#10314yes—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17—42no—
124C2×C4×C17—42no—
125C2×C4×C17—42no—
126C2×C4×C17C4×D411yescharacteristic
127C2×C4×C17C4×D411yescharacteristic
128C2×C4×C17#12011yescharacteristic
129C2×C4×C17#12111yescharacteristic
130C2×C4×C17#12111yescharacteristic
131C2×C4×C17#12311yescharacteristic
132C2×C4×C17C2×SD1611yescharacteristic
133C2×C4×C17C2×Q1611yescharacteristic
134C2×C4×C17#13511yescharacteristic
135C2×C4×C17#13511yescharacteristic
136C2×Dic17—22no—
137C2×Dic17—22no—
138C2×Dic17—41no—
139C2×Dic17—41no—
140C2×Dic17—42no—
141C2×Dic17—41no—
142C2×Dic17—41no—
143C2×Dic17—41no—
144D68—22no—
145D68—22no—
146D68—22no—
147D68—22no—
148C4×D17—44no—
149C4×D17—82no—
150Dic34—22no—
151Dic34—22no—
152Dic34—41no—
153Dic34—41no—
154C8×C17—42no—
155C8×C17—81no—
156#3622—42no—
157#3622—81no—
158#2972—171no—
159#2580—171no—
160#1800—171no—
161#1487—171no—
162#1487—171no—
163#1286—171no—
164#1286D1711yescharacteristic, complemented
165C22×C17—21no—
166C22×C17—21no—
167C22×C17#38611yescharacteristic
168C22×C17C2×#10311yescharacteristic
169C22×C17#41611yescharacteristic
170C2×D17—24no—
171C2×D17—42no—
172C2×D17—42no—
173C2×D17—44no—
174C2×D17—82no—
175C4×C17—21no—
176C4×C17—21no—
177C4×C17—21no—
178C4×C17—21no—
179C4×C17—82no—
180C4×C17#46511yescharacteristic
181C4×C17#46511yescharacteristic
182C4×C17#46711yescharacteristic
183C4×C17#46711yescharacteristic
184Dic17—42no—
185Dic17—42no—
186Dic17—81no—
187C2×C4×Q8—171no—
188C2×C4×D4—171no—
189C8×D4—342no—
190C8×D4—342no—
191#431—171no—
192#431C2×D1711yescharacteristic
193#423—171no—
194#423C2×D1711yescharacteristic
195C2×#100—341no—
196C2×C22⋊C8—342no—
197#399—172no—
198#393—172no—
199#390—341no—
200#389—341no—
201#385—342no—
202#382C2×D1711yescharacteristic
203#343—341no—
204Q8⋊C8—342no—
205Q8⋊C8—342no—
206C2×C17—21no—
207C2×C17—21no—
208C2×C17#189111yescharacteristic
209C2×C17#189211yescharacteristic
210C2×C17#191111yescharacteristic
211D17—44no—
212C22×Q8—171no—
213C22×D4—171no—
214C23×C4—172no—
215C2×M16—21no—
216C2×M16—341no—
217C22×C8—344no—
218C4×Q8—341no—
219C4×Q8—341no—
220C4×Q8—341no—
221C4×Q8—341no—
222C4×D4—341no—
223C4×D4—341no—
224C4×D4—341no—
225C4×D4—341no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—171no—
228C2×C4⋊C4—171no—
229C2×C4⋊C4—341no—
230C2×C4⋊C4—342no—
231C2×C22⋊C4—172no—
232C2×C22⋊C4—342no—
233C2×C22⋊C4—342no—
234C2×C42—341no—
235C2×C42C22×D1711yescharacteristic
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—341no—
240C4⋊C8—341no—
241C4⋊C8—341no—
242#100—682no—
243C22⋊C8—21no—
244C22⋊C8—344no—
245C22⋊C8—341no—
246C4×C8—172no—
247C4×C8D6812yes—
248#95—21no—
249#95—342no—
250#95C4×D1712yes—
251C17#948311yescharacteristic, complemented, Sylow
252C24—172no—
253C2×Q8—341no—
254C2×Q8—341no—
255C2×Q8—682no—
256C2×D4—172no—
257C2×D4—172no—
258C2×D4—341no—
259C2×D4—341no—
260C2×D4—341no—
261C2×D4—341no—
262C22×C4—22no—
263C22×C4—172no—
264C22×C4—341no—
265C22×C4—342no—
266C22×C4—342no—
267C22×C4—341no—
268C22×C4—1362no—
269C22×C4C2×C4×D1711yescharacteristic
270C22×C4D4×D1711yescharacteristic
271C22×C4D4×D1711yescharacteristic
272M16—41no—
273M16—41no—
274M16—681no—
275M16—681no—
276C2×C8—22no—
277C2×C8—41no—
278C2×C8—342no—
279C2×C8—681no—
280C2×C8—1364no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—341no—
284C4⋊C4—341no—
285C4⋊C4—341no—
286C4⋊C4—341no—
287C4⋊C4—681no—
288C4⋊C4—681no—
289C4⋊C4—1362no—
290C22⋊C4—342no—
291C22⋊C4—342no—
292C22⋊C4—684no—
293C22⋊C4—1362no—
294C42—681no—
295C42—681no—
296C42C2×D6811yescharacteristic
297C42#6325211yescharacteristic
298C42D4×D1711yescharacteristic
299C42#6325611yescharacteristic
300C23—174no—
301C23—342no—
302C23—342no—
303C23—682no—
304C23—682no—
305C23C22⋊C4×D1711yescharacteristic
306Q8—342no—
307Q8—342no—
308Q8—681no—
309Q8—681no—
310D4—342no—
311D4—342no—
312D4—342no—
313D4—342no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—342no—
320C2×C4—342no—
321C2×C4—684no—
322C2×C4—681no—
323C2×C4—681no—
324C2×C4—682no—
325C2×C4—681no—
326C2×C4—681no—
327C2×C4—681no—
328C2×C4—1362no—
329C2×C4C4×D6811yescharacteristic
330C2×C4#1058756811yescharacteristic
331C2×C4#1058757011yescharacteristic
332C2×C4#1058757911yescharacteristic
333C2×C4#1058758011yescharacteristic
334C2×C4#1058758111yescharacteristic
335C2×C4#1058759811yescharacteristic
336C2×C4SD16×D1711yescharacteristic
337C2×C4#1058760211yescharacteristic
338C2×C4Q16×D1711yescharacteristic
339C8—42no—
340C8—81no—
341C8—682no—
342C8—1361no—
343C22—21no—
344C22—21no—
345C22—344no—
346C22—684no—
347C22—682no—
348C22—682no—
349C22—1362no—
350C22#1176268711yescharacteristic
351C22#103×D1711yescharacteristic
352C22#1176280211yescharacteristic
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—82no—
358C4—682no—
359C4—682no—
360C4—1361no—
361C4#1176303711yescharacteristic
362C4#1176303911yescharacteristic
363C4#1176304911yescharacteristic
364C4#1176305111yescharacteristic
365C2—21no—
366C2—21no—
367C2—684no—
368C2#1452920111yescharacteristic
369C2#1452920611yescharacteristic
370C2#1452931511yescharacteristic
subgroupG/H
centerC22#11762687
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1286C2
socleC22×C17#386
prime radicalC22×D68D4
cosocleC2×C42×C17C23
hypercenter#1286D17
hypocenterC17#9483
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1286

Complements

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

NH ≅ G/Ntype
4. #14527214C2semidirect
5. #14527212C2semidirect
6. #14526642C2semidirect
7. C17×#1286C2semidirect
164. #1286D17semidirect
251. C17#9483semidirect

Maximal subgroups

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

subgroupindex
#145378072
#145341432
#145287732
#145272142
#145272122
#145266422
C17×#12862
#948317