Finite Group Databasebeta

← #17953155

Subgroups — #17953155

Order 4352.

588 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#14544716C211yesmaximal, characteristic
2C4×C4⋊#3624C211yesmaximal, characteristic
3C2×#11763659C212yesmaximal
4#14543620C212yesmaximal
5C4×#10587573C211yesmaximal, characteristic
6C2×C4⋊#3624—21no—
7C2×C4⋊#3624—21no—
8C2×C4⋊#3624—21no—
9C2×C4⋊#3624C2212yes—
10C2×C4×#3624C2212yes—
11#11763675—21no—
12#11763659—24no—
13C2×C4×Dic34C411yescharacteristic
14C2×#10587573C411yescharacteristic
15C2×#10587573C2211yescharacteristic
16C4⋊C4×Dic17C411yescharacteristic
17C4⋊C4×Dic17C2211yescharacteristic
18C4×#63225—21no—
19#11762660—21no—
20#11762660—21no—
21C42×Dic17—21no—
22C42×Dic17C2211yescharacteristic
23C4×C4⋊C4×C17C411yescharacteristic
24C22×Dic34C2×C411yescharacteristic
25C22×#3624—22no—
26C22×#3624—22no—
27C2×C4⋊C4×C17C2×C411yescharacteristic
28C2×C4⋊C4×C17C2×C411yescharacteristic
29C2×C42×C17—21no—
30C2×C42×C17C2×C411yescharacteristic
31C2×C4⋊Dic17C2×C411yescharacteristic
32C2×#63225—21no—
33C2×#63225—21no—
34C2×#63225—21no—
35C2×C4×Dic17—21no—
36C2×C4×Dic17—21no—
37C2×C4×Dic17—21no—
38C2×C4×Dic17—21no—
39C2×C4×Dic17—21no—
40C2×C4×Dic17C2×C411yescharacteristic
41C2×C4×Dic17D411yescharacteristic
42C2×C4×Dic17D411yescharacteristic
43C2×C4×Dic17D411yescharacteristic
44C2×C4×Dic17D411yescharacteristic
45C2×C4×Dic17C2311yescharacteristic
46#10587573—21no—
47#10587573—21no—
48#10587573—22no—
49#10587573—22no—
50#10587573D412yes—
51#10587573Q812yes—
52C4×Dic34—22no—
53C4×Dic34C814yescomplemented
54C4⋊#3624—24no—
55C4⋊#3624—22no—
56C4⋊#3624—22no—
57C4⋊#3624—42no—
58C4⋊#3624—42no—
59C4×#3624—22no—
60C4×#3624—22no—
61C17×#95—21no—
62#10587502—21no—
63#10587502—21no—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22×C411yescharacteristic
69C22×Dic17—21no—
70C22×Dic17—21no—
71C22×Dic17C22⋊C411yescharacteristic
72C22×Dic17C22⋊C411yescharacteristic
73C22×Dic17C2×D411yescharacteristic
74C22×Dic17C2×D411yescharacteristic
75C2×Dic34—21no—
76C2×Dic34—21no—
77C2×Dic34—21no—
78C2×Dic34C4⋊C412yes—
79C2×Dic34C2×C812yes—
80C2×Dic34M1612yes—
81C2×#3624—24no—
82C2×#3624—42no—
83C2×#3624—42no—
84C2×#3624—42no—
85C2×#3624—42no—
86C2×#3624—42no—
87C4⋊C4×C17—21no—
88C4⋊C4×C17—21no—
89C4⋊C4×C17C4⋊C412yes—
90C4⋊C4×C17M1612yes—
91C42×C17—42no—
92C42×C17C2×C812yes—
93C42×C17M1612yes—
94C4⋊Dic17—21no—
95C4⋊Dic17C2×C812yes—
96#63225—24no—
97#63225—24no—
98#63225—22no—
99#63225—22no—
100C4×Dic17—24no—
101C4×Dic17—24no—
102C4×Dic17—22no—
103C4×Dic17—21no—
104C4×Dic17—21no—
105C4×Dic17—22no—
106C4×Dic17—22no—
107C4×Dic17—21no—
108C4×Dic17—21no—
109C4×Dic17—41no—
110C4×Dic17—42no—
111C4×Dic17—41no—
112C4×Dic17—41no—
113C4×Dic17SD1612yes—
114C4×Dic17Q1612yes—
115C4×Dic17C2×D411yescharacteristic
116C4×Dic17C2×Q811yescharacteristic
117C4×Dic17#4111yescharacteristic
118C4×Dic17#4111yescharacteristic
119#8564—171nomaximal, Sylow
120C23×C17C2×C22⋊C411yescharacteristic
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—22no—
124C2×C4×C17—22no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17—41no—
128C2×C4×C17—41no—
129C2×C4×C17C22⋊C812yes—
130C2×C4×C17C22⋊C812yes—
131C2×C4×C17#10312yes—
132C2×C4×C17#10412yes—
133C2×C4×C17C2×C4⋊C411yescharacteristic
134C2×C4×C17#11711yescharacteristic
135C2×C4×C17C22×C811yescharacteristic
136C2×C4×C17C2×M1611yescharacteristic
137C2×C4×C17C2×M1611yescharacteristic
138C2×C4×C17C2×M1611yescharacteristic
139C2×Dic17—22no—
140C2×Dic17—22no—
141C2×Dic17—22no—
142C2×Dic17—21no—
143C2×Dic17—21no—
144C2×Dic17—21no—
145C2×Dic17—21no—
146C2×Dic17—22no—
147C2×Dic17—21no—
148C2×Dic17—21no—
149C2×Dic17—21no—
150C2×Dic17—41no—
151C2×Dic17—41no—
152C2×Dic17#10312yes—
153C2×Dic17#12111yescharacteristic
154C2×Dic17#12111yescharacteristic
155C2×Dic17C22⋊Q811yescharacteristic
156C2×Dic17C22⋊Q811yescharacteristic
157C2×Dic17C2×SD1611yescharacteristic
158C2×Dic17C2×Q1611yescharacteristic
159C2×Dic17AGL(1,ℤ/8)11yescharacteristic
160C2×Dic17#13711yescharacteristic
161Dic34—22no—
162Dic34—22no—
163Dic34—22no—
164Dic34C4⋊C814yescomplemented
165#3624—44no—
166#3624—82no—
167C42×Q8—171no—
168#1842—171no—
169C4×C4⋊C8—171no—
170C2×Q8⋊C8—172no—
171#1279—172no—
172C22×C17#38011yescharacteristic
173C22×C17C2×C22⋊C811yescharacteristic
174C22×C17#40711yescharacteristic
175C22×C17C2×#10311yescharacteristic
176C22×C17#41711yescharacteristic
177C22×C17C2×#10411yescharacteristic
178C22×C17#42111yescharacteristic
179C4×C17—22no—
180C4×C17—42no—
181C4×C17—42no—
182C4×C17Q8⋊C814yescomplemented
183C4×C17C2×C4⋊C811yescharacteristic
184C4×C17#42311yescharacteristic
185C4×C17#43111yescharacteristic
186C4×C17#43211yescharacteristic
187Dic17—24no—
188Dic17—21no—
189Dic17—21no—
190Dic17—22no—
191Dic17—42no—
192Dic17—41no—
193Dic17#46011yescharacteristic
194Dic17#46211yescharacteristic
195Dic17#47511yescharacteristic
196Dic17#47711yescharacteristic
197C2×C4×Q8—171no—
198C2×C4×Q8—171no—
199C2×C4×Q8—171no—
200C2×C4⋊C8—172no—
201C2×C4⋊C8—341no—
202C2×C4⋊C8—341no—
203C2×C4⋊C8—341no—
204C2×C4×C8—172no—
205C4×C4⋊C4—171no—
206C4×C4⋊C4—171no—
207C4×C4⋊C4—341no—
208C4×C4⋊C4—341no—
209C4×C4⋊C4—341no—
210C4×C4⋊C4#59511yescharacteristic
211C43—171no—
212C43—341no—
213#336—341no—
214Q8⋊C8—344no—
215C2×C17C2×Q8⋊C811yescharacteristic
216C2×C17#147711yescharacteristic
217C2×C17#148711yescharacteristic
218C2×C17#148911yescharacteristic
219C2×C17#179711yescharacteristic
220C2×C17#180311yescharacteristic
221C2×C17#180611yescharacteristic
222C22×Q8—171no—
223C22×C8—342no—
224C22×C8—342no—
225C4×Q8—172no—
226C4×Q8—172no—
227C4×Q8—174no—
228C4×Q8—341no—
229C4×Q8—341no—
230C4×Q8—342no—
231C4×Q8—342no—
232C4×Q8—342no—
233C2×C4⋊C4—171no—
234C2×C4⋊C4—341no—
235C2×C4⋊C4—341no—
236C2×C4⋊C4—341no—
237C2×C4⋊C4C2×#59511yescharacteristic
238C2×C4⋊C4C2×#59511yescharacteristic
239C2×C42—21no—
240C2×C42—171no—
241C2×C42—171no—
242C2×C42—171no—
243C2×C42—171no—
244C2×C42—171no—
245C2×C42—171no—
246C2×C42—341no—
247C2×C42—341no—
248C2×C42—341no—
249C2×C42—341no—
250C2×C42—341no—
251C2×C42C2×#59511yescharacteristic
252C4⋊C8—342no—
253C4⋊C8—342no—
254C4⋊C8—344no—
255C4⋊C8—682no—
256C4⋊C8—682no—
257C4×C8—342no—
258C4×C8—342no—
259#95—21no—
260#95—341no—
261#95—341no—
262C17#856411yescharacteristic, complemented, Sylow
263C2×Q8—172no—
264C2×Q8—172no—
265C2×Q8—172no—
266C2×Q8—341no—
267C2×Q8—341no—
268C2×Q8—341no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—171no—
272C22×C4—171no—
273C22×C4—171no—
274C22×C4—171no—
275C22×C4—341no—
276C22×C4—341no—
277C22×C4C22⋊#59511yescharacteristic
278C22×C4C22⋊#59511yescharacteristic
279C22×C4C22×#59511yescharacteristic
280C2×C8—344no—
281C2×C8—682no—
282C2×C8—682no—
283C2×C8—682no—
284C2×C8—682no—
285C2×C8—682no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—172no—
289C4⋊C4—342no—
290C4⋊C4—342no—
291C4⋊C4—344no—
292C4⋊C4—341no—
293C4⋊C4—344no—
294C4⋊C4#6324312yes—
295C4⋊C4C4⋊#59512yes—
296C42—42no—
297C42—171no—
298C42—171no—
299C42—171no—
300C42—171no—
301C42—172no—
302C42—172no—
303C42—341no—
304C42—341no—
305C42—342no—
306C42—342no—
307C42—344no—
308C42—341no—
309C42—341no—
310C42—344no—
311C42—342no—
312C42—681no—
313C42—681no—
314C42—681no—
315C42—682no—
316C42#6324212yes—
317C42#6324312yes—
318C23C2×C22⋊#59511yescharacteristic
319Q8—174no—
320Q8—342no—
321Q8—342no—
322Q8—342no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—171no—
332C2×C4—171no—
333C2×C4—171no—
334C2×C4—171no—
335C2×C4—171no—
336C2×C4—171no—
337C2×C4—171no—
338C2×C4—171no—
339C2×C4—172no—
340C2×C4—342no—
341C2×C4—342no—
342C2×C4—341no—
343C2×C4—342no—
344C2×C4—341no—
345C2×C4—341no—
346C2×C4—341no—
347C2×C4—342no—
348C2×C4—341no—
349C2×C4—341no—
350C2×C4—341no—
351C2×C4—681no—
352C2×C4—681no—
353C2×C4#1058754312yes—
354C2×C4#1058754312yes—
355C2×C4#1058754812yes—
356C2×C4#1058754912yes—
357C2×C4C2×#6324211yescharacteristic
358C2×C4C2×#6324311yescharacteristic
359C2×C4C2×#6324311yescharacteristic
360C2×C4D17⋊M1611yescharacteristic
361C2×C4C2×C4⋊#59511yescharacteristic
362C2×C4#1058767711yescharacteristic
363C8—684no—
364C8—1362no—
365C22#1176375111yescharacteristic
366C22C2×#1058754311yescharacteristic
367C22#1176378011yescharacteristic
368C22C2×#1058754911yescharacteristic
369C22#1176380011yescharacteristic
370C22C2×#1058754811yescharacteristic
371C22#1176380611yescharacteristic
372C4—22no—
373C4—42no—
374C4—42no—
375C4—171no—
376C4—171no—
377C4—171no—
378C4—171no—
379C4—341no—
380C4—341no—
381C4—342no—
382C4—344no—
383C4—682no—
384C4—681no—
385C4#1176365914yescomplemented
386C4#1176381011yescharacteristic
387C4#1176381311yescharacteristic
388C4#1176382811yescharacteristic
389C4#1176383111yescharacteristic
390C2C2×#1176365911yescharacteristic
391C2#1454408111yescharacteristic
392C2#1454411311yescharacteristic
393C2#1454412011yescharacteristic
394C2#1454463911yescharacteristic
395C2#1454465311yescharacteristic
396C2#1454465611yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC4×C4⋊C4×C17C4
socle, prime radicalC23×C17C2×C22⋊C4
cosocleC2×C4×Dic17C23
hypercenterC4×C4⋊C4#595
hypocenterC17#8564
LCSG ▷ C2×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
53. C4×Dic34C8semidirect
164. Dic34C4⋊C8semidirect
182. C4×C17Q8⋊C8semidirect
262. C17#8564semidirect
385. C4#11763659semidirect

Maximal subgroups

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

subgroupindex
#145447162
C4×C4⋊#36242
C2×#117636592
#145436202
C4×#105875732
#856417