Finite Group Databasebeta

← #17952559

Subgroups — #17952559

Order 4352.

1030 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 1030 conjugacy classes of subgroups — too many to draw. Only its 152 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#14544641C211yesmaximal, characteristic
2#14544641C211yesmaximal, characteristic
3C2×#11763668C214yesmaximal, complemented
4C2×C4⋊Q8×D17C211yesmaximal, characteristic
5C2×#10587544C2212yes—
6C2×#10587544C2212yes—
7C2×#10587543—21no—
8C2×#10587543—21no—
9#11763675—21no—
10#11763675—21no—
11#11763668—28no—
12C22×Q8×D17—21no—
13C2×#10587555C411yescharacteristic
14C2×#10587574—21no—
15C2×#10587574C2212yes—
16C2×#10587630—21no—
17C4⋊Q8×D17—28no—
18C2×C4⋊C4×D17—21no—
19C2×C4⋊C4×D17C412yes—
20C2×C42×D17C2211yescharacteristic
21C2×C17×C4⋊Q8C411yescharacteristic
22C22×Q8×C17—21no—
23C2×Q8×D17—28no—
24C2×Q8×D17—44no—
25C2×Q8×D17—42no—
26C2×Q8×D17—42no—
27C22×C4×D17—22no—
28C22×C4×D17C2×C411yescharacteristic
29C22×C4×D17D411yescharacteristic
30C22×C4×D17D411yescharacteristic
31C22×Dic34—21no—
32C22×Dic34—22no—
33C22×#3624—22no—
34C22×#3624—22no—
35C17×C4⋊Q8—24no—
36C2×C4⋊C4×C17—21no—
37C2×C4⋊C4×C17C2×C412yes—
38C2×C42×C17C2×C411yescharacteristic
39#10587630—28no—
40C2×C4⋊Dic17—21no—
41C2×C4⋊Dic17C2×C412yes—
42C2×#63225—22no—
43C2×#63225—22no—
44C2×C4×Dic17D411yescharacteristic
45C2×C4×Dic17D411yescharacteristic
46C2×C4×Dic17C2311yescharacteristic
47C4⋊C4×D17—28no—
48C4⋊C4×D17—24no—
49C4⋊C4×D17—24no—
50#10587574—28no—
51#10587574—24no—
52#10587574D418yes—
53C42×D17—24no—
54#10587555—24no—
55#10587544—24no—
56#10587544—24no—
57#10587543—42no—
58#10587543—42no—
59C23×D17C22⋊C411yescharacteristic
60C2×Q8×C17—24no—
61C2×Q8×C17—42no—
62C2×Q8×C17—42no—
63C22×C4×C17—22no—
64C22×C4×C17C22⋊C411yescharacteristic
65C22×C4×C17C22⋊C411yescharacteristic
66C22×C4×C17C22×C411yescharacteristic
67C22×Dic17—22no—
68C22×Dic17C22⋊C411yescharacteristic
69C22×Dic17C2×D411yescharacteristic
70C22×Dic17C2×D411yescharacteristic
71Q8×D17—416no—
72Q8×D17—416no—
73C2×C4×D17—22no—
74C2×C4×D17—22no—
75C2×C4×D17—22no—
76C2×C4×D17—22no—
77C2×C4×D17—22no—
78C2×C4×D17—22no—
79C2×C4×D17—22no—
80C2×C4×D17—28no—
81C2×C4×D17—44no—
82C2×C4×D17—44no—
83C2×C4×D17D812yes—
84C2×C4×D17SD1612yes—
85C2×C4×D17SD1612yes—
86C2×C4×D17Q1612yes—
87C2×Dic34—24no—
88C2×Dic34—28no—
89C2×Dic34—42no—
90C2×Dic34—42no—
91C2×Dic34—42no—
92C2×Dic34—42no—
93C2×Dic34—44no—
94C2×#3624—42no—
95C2×#3624—42no—
96C2×#3624—42no—
97C2×#3624—42no—
98C2×#3624—42no—
99C2×#3624—42no—
100C4⋊C4×C17—24no—
101C4⋊C4×C17—22no—
102C4⋊C4×C17C22⋊C414yes—
103C42×C17—22no—
104C4⋊Dic17—24no—
105C4⋊Dic17—22no—
106C4⋊Dic17C22⋊C414yes—
107#63225—28no—
108#63225—28no—
109C4×Dic17—22no—
110C4×Dic17—22no—
111C4×Dic17C2×D414yes—
112#8340—171nomaximal, Sylow
113C23×C17C2×C22⋊C411yescharacteristic
114C22×D17—21no—
115C22×D17—21no—
116C22×D17—21no—
117C22×D17—21no—
118C22×D17#10112yes—
119C22×D17#10212yes—
120C22×D17#10312yes—
121Q8×C17—48no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—24no—
129C2×C4×C17—42no—
130C2×C4×C17—42no—
131C2×C4×C17#10212yes—
132C2×C4×C17#10312yes—
133C2×C4×C17C2×C22⋊C412yes—
134C2×Dic17—21no—
135C2×Dic17—21no—
136C2×Dic17—24no—
137C2×Dic17—42no—
138C2×Dic17—42no—
139C2×Dic17#10112yes—
140C2×Dic17#12012yes—
141C2×Dic17#12012yes—
142C2×Dic17C2×D811yescharacteristic
143C2×Dic17C2×SD1611yescharacteristic
144C2×Dic17C2×SD1611yescharacteristic
145C2×Dic17C2×Q1611yescharacteristic
146C2×Dic17AGL(1,ℤ/8)12yes—
147C2×Dic17#13712yes—
148C4×D17—28no—
149C4×D17—24no—
150C4×D17—24no—
151C4×D17—24no—
152C4×D17—24no—
153C4×D17—416no—
154Dic34—48no—
155Dic34—48no—
156Dic34—48no—
157#3624—82no—
158#3624—82no—
159C22×C4⋊Q8—171no—
160#1797—171no—
161#1797—171no—
162C2×#332—174no—
163C22×C17#37911yescharacteristic
164C22×C17C2×#10111yescharacteristic
165C22×C17C2×#10111yescharacteristic
166C22×C17C2×#10211yescharacteristic
167C22×C17C2×#10311yescharacteristic
168C22×C17#41811yescharacteristic
169C22×C17#41911yescharacteristic
170C2×D17—22no—
171C2×D17—22no—
172C2×D17—22no—
173C2×D17—22no—
174C2×D17—22no—
175C2×D17—22no—
176C2×D17#33214yescomplemented
177C4×C17—22no—
178C4×C17—22no—
179C4×C17—22no—
180C4×C17—44no—
181Dic17—22no—
182Dic17—44no—
183Dic17#44712yes—
184Dic17#44812yes—
185Dic17#45012yes—
186Dic17#45112yes—
187C23×Q8—341no—
188C2×C4⋊Q8—172no—
189C2×C4⋊Q8—171no—
190C2×C4⋊Q8—341no—
191C2×C4⋊Q8—341no—
192C2×C4⋊Q8—348no—
193C2×C4⋊Q8#59511yescharacteristic
194C22×C4⋊C4—172no—
195C22×C4⋊C4—341no—
196C22×C42—171no—
197C2×C4⋊C8—172no—
198C2×C4⋊C8—172no—
199C2×C22⋊C8—341no—
200C2×C22⋊C8—341no—
201#336—341no—
202#336—341no—
203#332—348no—
204C2×C17C2×#33211yescharacteristic
205C2×C17#154611yescharacteristic
206C2×C17#154911yescharacteristic
207C2×C17#155611yescharacteristic
208C2×C17#178411yescharacteristic
209C2×C17#178611yescharacteristic
210C2×C17#178711yescharacteristic
211D17—24no—
212C22×Q8—21no—
213C22×Q8—342no—
214C22×Q8—348no—
215C22×Q8—341no—
216C22×Q8—682no—
217C22×Q8—682no—
218C22×Q8—684no—
219C23×C4—171no—
220C23×C4—171no—
221C23×C4—171no—
222C23×C4—342no—
223C22×C8—342no—
224C22×C8—342no—
225C4⋊Q8—24no—
226C4⋊Q8—178no—
227C4⋊Q8—344no—
228C4⋊Q8—344no—
229C4⋊Q8—348no—
230C4⋊Q8—348no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—172no—
233C2×C4⋊C4—344no—
234C2×C4⋊C4—348no—
235C2×C4⋊C4—342no—
236C2×C4⋊C4—342no—
237C2×C4⋊C4—341no—
238C2×C4⋊C4—344no—
239C2×C4⋊C4C2×#59512yes—
240C2×C42—171no—
241C2×C42—171no—
242C2×C42—171no—
243C2×C42—344no—
244C2×C42C2×#59511yescharacteristic
245C4⋊C8—344no—
246C4⋊C8—344no—
247C22⋊C8—682no—
248C22⋊C8—682no—
249C17#834011yescharacteristic, complemented, Sylow
250C24—171no—
251C2×Q8—24no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—348no—
255C2×Q8—344no—
256C2×Q8—682no—
257C2×Q8—682no—
258C2×Q8—682no—
259C2×Q8—684no—
260C2×Q8—6816no—
261C2×Q8—6816no—
262C2×Q8—682no—
263C22×C4—22no—
264C22×C4—171no—
265C22×C4—171no—
266C22×C4—171no—
267C22×C4—172no—
268C22×C4—172no—
269C22×C4—172no—
270C22×C4—172no—
271C22×C4—342no—
272C22×C4—342no—
273C22×C4—342no—
274C22×C4—342no—
275C22×C4—342no—
276C22×C4—342no—
277C22×C4—348no—
278C22×C4—342no—
279C22×C4—342no—
280C22×C4—684no—
281C22×C4—684no—
282C22×C4C22⋊#59511yescharacteristic
283C22×C4C22⋊#59511yescharacteristic
284C22×C4C22×#59511yescharacteristic
285C2×C8—682no—
286C2×C8—682no—
287C2×C8—682no—
288C2×C8—682no—
289C2×C8—682no—
290C2×C8—682no—
291C4⋊C4—24no—
292C4⋊C4—22no—
293C4⋊C4—174no—
294C4⋊C4—348no—
295C4⋊C4—348no—
296C4⋊C4—344no—
297C4⋊C4—342no—
298C4⋊C4C22⋊#59514yes—
299C42—22no—
300C42—174no—
301C42—342no—
302C42—342no—
303C23—172no—
304C23—172no—
305C23—172no—
306C23—341no—
307C23—341no—
308C23—341no—
309C23—341no—
310C23C2×C22⋊#59511yescharacteristic
311Q8—48no—
312Q8—688no—
313Q8—688no—
314Q8—688no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—24no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—172no—
325C2×C4—172no—
326C2×C4—171no—
327C2×C4—171no—
328C2×C4—171no—
329C2×C4—171no—
330C2×C4—172no—
331C2×C4—172no—
332C2×C4—172no—
333C2×C4—344no—
334C2×C4—344no—
335C2×C4—344no—
336C2×C4—344no—
337C2×C4—348no—
338C2×C4—341no—
339C2×C4—341no—
340C2×C4—344no—
341C2×C4—6816no—
342C2×C4—682no—
343C2×C4—682no—
344C2×C4#1058754712yes—
345C2×C4#1058754912yes—
346C2×C4C2×C22⋊#59512yes—
347C8—1362no—
348C8—1362no—
349C22—174no—
350C22—342no—
351C22—342no—
352C22—342no—
353C22—342no—
354C22—342no—
355C22—342no—
356C22#1176375011yescharacteristic
357C22D17⋊#10111yescharacteristic
358C22D17⋊#10111yescharacteristic
359C22C2×#1058754711yescharacteristic
360C22C2×#1058754911yescharacteristic
361C22#1176380111yescharacteristic
362C22#1176380211yescharacteristic
363C4—22no—
364C4—22no—
365C4—22no—
366C4—44no—
367C4—172no—
368C4—172no—
369C4—172no—
370C4—172no—
371C4—342no—
372C4—684no—
373C2—344no—
374C2D17⋊#33211yescharacteristic
375C2#1454429411yescharacteristic
376C2#1454430011yescharacteristic
377C2#1454431411yescharacteristic
378C2#1454461911yescharacteristic
379C2#1454462211yescharacteristic
380C2#1454462311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC2×C17×C4⋊Q8C4
socleC23×C17C2×C22⋊C4
prime radicalC23×D17C22⋊C4
cosocleC2×C4×Dic17C23
hypercenterC2×C4⋊Q8#595
hypocenterC17#8340
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×#11763668C2semidirect
176. C2×D17#332semidirect
249. C17#8340semidirect

Maximal subgroups

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

subgroupindex
#145446412
C2×#117636682
C2×C4⋊Q8×D172
#834017