Finite Group Databasebeta

← #17949785

Subgroups — #17949785

Order 4352.

504 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 118 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#14544674C211yesmaximal, characteristic
2#14544638C211yesmaximal, characteristic
3#14543604C212yesmaximal, complemented
4#14543602C212yesmaximal, complemented
5C22⋊Q8×Dic17C211yesmaximal, characteristic
6C2×C4⋊#3624—21no—
7C2×C22⋊#3624—21no—
8C2×C22⋊#3624—21no—
9#11763675—21no—
10#11763675C2212yes—
11#11763675C2212yes—
12C2×Q8×Dic17C411yescharacteristic
13C22×C4×Dic17C2211yescharacteristic
14#11762716C2211yescharacteristic
15#11762708C2211yescharacteristic
16#11762695—21no—
17#11762684—21no—
18#11762669C411yescharacteristic
19C4⋊C4×Dic17—21no—
20C4⋊C4×Dic17C411yescharacteristic
21C22⋊C4×Dic17—21no—
22C2×C17×C22⋊Q8C411yescharacteristic
23C22×Q8×C17C2×C411yescharacteristic
24C23×C4×C17C2×C411yescharacteristic
25C23×Dic17D411yescharacteristic
26C23×Dic17D411yescharacteristic
27C22×#3624—22no—
28C22×#3624—22no—
29C17×C22⋊Q8—22no—
30C17×C22⋊Q8C814yescomplemented
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17C2×C411yescharacteristic
33C2×C22⋊C4×C17—21no—
34C2×C22⋊Dic17—21no—
35Q8×Dic17—42no—
36C2×C4⋊Dic17—21no—
37C2×C4⋊Dic17—21no—
38C2×C4⋊Dic17C2×C411yescharacteristic
39C2×C4⋊Dic17C2×C411yescharacteristic
40C2×C4×Dic17—21no—
41C2×C4×Dic17—21no—
42C2×C4×Dic17—42no—
43C2×C4×Dic17C2×C411yescharacteristic
44C2×C4×Dic17D411yescharacteristic
45C2×C4×Dic17D411yescharacteristic
46C2×C4×Dic17C2311yescharacteristic
47C22⋊#3624—42no—
48C22⋊#3624—42no—
49C4⋊#3624—42no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55C24×C17C22⋊C411yescharacteristic
56C2×Q8×C17—21no—
57C2×Q8×C17—41no—
58C2×Q8×C17—41no—
59C2×Q8×C17C2×C812yes—
60C22×C4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17—41no—
63C22×C4×C17—41no—
64C22×C4×C17C22⋊C411yescharacteristic
65C22×C4×C17C2×C812yes—
66C22×C4×C17M1612yes—
67C22×C4×C17C22×C411yescharacteristic
68C22×Dic17—22no—
69C22×Dic17—21no—
70C22×Dic17—22no—
71C22×Dic17—21no—
72C22×Dic17—21no—
73C22×Dic17—21no—
74C22×Dic17—41no—
75C22×Dic17—41no—
76C22×Dic17—41no—
77C22×Dic17C22⋊C411yescharacteristic
78C22×Dic17C22⋊C411yescharacteristic
79C22×Dic17C2×D411yescharacteristic
80C22×Dic17C2×D411yescharacteristic
81C2×#3624—42no—
82C2×#3624—42no—
83C2×#3624—42no—
84C2×#3624—42no—
85C2×#3624—42no—
86C2×#3624—42no—
87C4⋊C4×C17—24no—
88C4⋊C4×C17—21no—
89C4⋊C4×C17C2×C812yes—
90C22⋊C4×C17—24no—
91C22⋊Dic17—42no—
92C4⋊Dic17—42no—
93C4⋊Dic17—41no—
94C4⋊Dic17—41no—
95C4⋊Dic17—41no—
96C4⋊Dic17—41no—
97C4×Dic17—22no—
98C4×Dic17—22no—
99C4×Dic17—41no—
100C4×Dic17—41no—
101C4×Dic17—42no—
102C4×Dic17—42no—
103#7600—171nomaximal, Sylow
104C23×C17—21no—
105C23×C17—21no—
106C23×C17—21no—
107C23×C17—21no—
108C23×C17—21no—
109C23×C17—21no—
110C23×C17C22⋊C812yes—
111C23×C17C2×C22⋊C411yescharacteristic
112Q8×C17—42no—
113Q8×C17—42no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—22no—
119C2×C4×C17—22no—
120C2×C4×C17—42no—
121C2×C4×C17—42no—
122C2×C4×C17—41no—
123C2×C4×C17—41no—
124C2×C4×C17—41no—
125C2×C4×C17—41no—
126C2×C4×C17C22⋊C812yes—
127C2×C4×C17C22×C811yescharacteristic
128C2×C4×C17C2×M1611yescharacteristic
129C2×Dic17—21no—
130C2×Dic17—21no—
131C2×Dic17—21no—
132C2×Dic17—21no—
133C2×Dic17—21no—
134C2×Dic17—21no—
135C2×Dic17—21no—
136C2×Dic17—21no—
137C2×Dic17—21no—
138C2×Dic17—44no—
139C2×Dic17—42no—
140C2×Dic17—42no—
141C2×Dic17—41no—
142C2×Dic17—41no—
143C2×Dic17—41no—
144C2×Dic17—41no—
145C2×Dic17—41no—
146C2×Dic17—41no—
147C2×Dic17#9912yes—
148C2×Dic17AGL(1,ℤ/8)11yescharacteristic
149C2×Dic17#13711yescharacteristic
150C2×Dic17#13712yes—
151#3624—82no—
152#3624—82no—
153C4×C22⋊Q8—171no—
154#1817—171no—
155#1797—171no—
156#1272—172no—
157#1271—172no—
158C22×C17—22no—
159C22×C17—22no—
160C22×C17—22no—
161C22×C17—41no—
162C22×C17—41no—
163C22×C17—41no—
164C22×C17—41no—
165C22×C17C2×C22⋊C811yescharacteristic
166C22×C17C2×#9911yescharacteristic
167C22×C17#41311yescharacteristic
168C22×C17#41711yescharacteristic
169C22×C17#41911yescharacteristic
170C22×C17#42111yescharacteristic
171C22×C17#42111yescharacteristic
172C4×C17—22no—
173C4×C17—21no—
174C4×C17—21no—
175C4×C17—42no—
176C4×C17—42no—
177Dic17—22no—
178Dic17—22no—
179Dic17—21no—
180Dic17—21no—
181Dic17—41no—
182Dic17—81no—
183Dic17—81no—
184C2×C22⋊Q8#59511yescharacteristic
185C2×C4×Q8—171no—
186C22×C42—171no—
187C2×C4⋊C8—341no—
188C2×C22⋊C8—341no—
189C2×C22⋊C8—341no—
190#391—171no—
191#389—171no—
192#387—341no—
193#385—341no—
194#380—171no—
195C4×C4⋊C4—171no—
196C4×C4⋊C4—341no—
197C4×C22⋊C4—341no—
198#336—172no—
199#336—172no—
200#336—341no—
201C2×C17—42no—
202C2×C17#145911yescharacteristic
203C2×C17#147711yescharacteristic
204C2×C17#147911yescharacteristic
205C2×C17#151111yescharacteristic
206C2×C17#151311yescharacteristic
207C2×C17#153511yescharacteristic
208C2×C17#155911yescharacteristic
209C22×Q8C2×#59511yescharacteristic
210C23×C4—171no—
211C23×C4—171no—
212C23×C4C2×#59511yescharacteristic
213C22×C8—342no—
214C22×C8—342no—
215C22⋊Q8—22no—
216C22⋊Q8#362414yescomplemented
217C4×Q8—682no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—171no—
220C2×C4⋊C4—171no—
221C2×C4⋊C4—341no—
222C2×C4⋊C4—341no—
223C2×C4⋊C4C2×#59511yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—341no—
226C2×C42—171no—
227C2×C42—171no—
228C2×C42—171no—
229C2×C42—171no—
230C2×C42—341no—
231C2×C42—341no—
232C2×C42—682no—
233C4⋊C8—682no—
234C22⋊C8—682no—
235C22⋊C8—682no—
236#95—341no—
237#95—341no—
238#95—341no—
239#95—341no—
240#95—341no—
241C17#760011yescharacteristic, complemented, Sylow
242C24C22⋊#59511yescharacteristic
243C2×Q8—21no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8C2×#362412yes—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—171no—
252C22×C4—171no—
253C22×C4—171no—
254C22×C4—171no—
255C22×C4—341no—
256C22×C4—342no—
257C22×C4—341no—
258C22×C4—342no—
259C22×C4—341no—
260C22×C4—341no—
261C22×C4—681no—
262C22×C4—681no—
263C22×C4—681no—
264C22×C4C2×#362412yes—
265C22×C4#6324712yes—
266C22×C4C22⋊#59511yescharacteristic
267C22×C4C22×#59511yescharacteristic
268C2×C8—682no—
269C2×C8—682no—
270C2×C8—682no—
271C2×C8—682no—
272C2×C8—682no—
273C2×C8—682no—
274C4⋊C4—21no—
275C4⋊C4—24no—
276C4⋊C4—681no—
277C4⋊C4—681no—
278C4⋊C4—681no—
279C4⋊C4—681no—
280C4⋊C4—682no—
281C4⋊C4C2×#362412yes—
282C22⋊C4—24no—
283C22⋊C4—682no—
284C42—342no—
285C42—342no—
286C42—681no—
287C42—681no—
288C42—682no—
289C42—682no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23C22⋊#362412yes—
297C23C2×C22⋊#59511yescharacteristic
298Q8—42no—
299Q8—42no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—172no—
313C2×C4—171no—
314C2×C4—171no—
315C2×C4—172no—
316C2×C4—341no—
317C2×C4—341no—
318C2×C4—341no—
319C2×C4—341no—
320C2×C4—341no—
321C2×C4—341no—
322C2×C4—341no—
323C2×C4—341no—
324C2×C4—341no—
325C2×C4—684no—
326C2×C4—682no—
327C2×C4—682no—
328C2×C4—681no—
329C2×C4—681no—
330C2×C4—681no—
331C2×C4—681no—
332C2×C4—681no—
333C2×C4—681no—
334C2×C4C22⋊#362412yes—
335C2×C4C22×#362411yescharacteristic
336C2×C4C2×#6324711yescharacteristic
337C8—1362no—
338C8—1362no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22C2×C22⋊#362411yescharacteristic
347C22D17⋊#9911yescharacteristic
348C22#1176379611yescharacteristic
349C22#1176380011yescharacteristic
350C22#1176380211yescharacteristic
351C22#1176380611yescharacteristic
352C22#1176380811yescharacteristic
353C4—21no—
354C4—21no—
355C4—22no—
356C4—42no—
357C4—42no—
358C4—342no—
359C4—342no—
360C4—341no—
361C4—341no—
362C4—681no—
363C4—1361no—
364C4—1361no—
365C2—42no—
366C2#1454402011yescharacteristic
367C2#1454407911yescharacteristic
368C2#1454408611yescharacteristic
369C2#1454419311yescharacteristic
370C2#1454420211yescharacteristic
371C2#1454427111yescharacteristic
372C2#1454432011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC2×C17×C22⋊Q8C4
socleC23×C17C2×C22⋊C4
prime radicalC24×C17C22⋊C4
cosocleC2×C4×Dic17C23
hypercenterC2×C22⋊Q8#595
hypocenterC17#7600
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #14543604C2semidirect
4. #14543602C2semidirect
30. C17×C22⋊Q8C8semidirect
216. C22⋊Q8#3624semidirect
241. C17#7600semidirect

Maximal subgroups

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

subgroupindex
#145446742
#145446382
#145436042
#145436022
C22⋊Q8×Dic172
#760017