Finite Group Databasebeta

← #17949736

Subgroups — #17949736

Order 4352.

662 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 122 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, complemented
2#14544445C211yesmaximal, characteristic, complemented
3#14543603C212yesmaximal, complemented
4#14543600C212yesmaximal, complemented
5C4×#10587624C211yesmaximal, characteristic
6C22×#63242—21no—
7C2×#10587544—21no—
8C2×#10587543—21no—
9#11763675—21no—
10#11763675C2212yes—
11#11763675C2212yes—
12C2×#10587624C411yescharacteristic
13C2×C4×C22⋊D17—21no—
14C2×C42×D17C2211yescharacteristic
15#11762712C2211yescharacteristic
16#11762705C2211yescharacteristic
17#11762697—21no—
18#11762685—21no—
19#11762666C411yescharacteristic
20C4×C4⋊Dic17C411yescharacteristic
21C4×C22⋊Dic17—21no—
22C2×C4×D4×C17C411yescharacteristic
23C22×D4×C17C2×C411yescharacteristic
24C23×C4×C17—21no—
25C22×C22⋊D17—21no—
26C22×C4×D17C2×C411yescharacteristic
27C22×C4×D17D411yescharacteristic
28C22×C4×D17D411yescharacteristic
29C22×#3624—22no—
30C22×#3624—22no—
31C2×#63242—24no—
32C2×#63242—42no—
33C2×#63242—42no—
34C4×D4×C17—42no—
35C2×C4⋊C4×C17C2×C411yescharacteristic
36C2×C22⋊C4×C17—21no—
37C2×C42×C17C2×C411yescharacteristic
38C2×C22⋊Dic17—21no—
39C2×C22⋊Dic17—21no—
40C2×C22⋊Dic17—21no—
41#10587624—22no—
42#10587624C814yescomplemented
43C4×C22⋊D17—42no—
44C4×C22⋊D17—42no—
45C2×#63227—21no—
46C2×C4⋊Dic17C2×C411yescharacteristic
47C2×C4⋊Dic17C2×C411yescharacteristic
48C2×#63225—21no—
49C2×C4×Dic17—21no—
50C2×C4×Dic17D411yescharacteristic
51C2×C4×Dic17D411yescharacteristic
52C2×C4×Dic17C2311yescharacteristic
53C42×D17—42no—
54#10587544—42no—
55#10587543—42no—
56#10587502—21no—
57#10587502—21no—
58#10587502—21no—
59C24×C17—21no—
60C23×D17C22⋊C411yescharacteristic
61C2×D4×C17—21no—
62C2×D4×C17—41no—
63C2×D4×C17—41no—
64C2×D4×C17C2×C812yescomplemented
65C22×C4×C17—21no—
66C22×C4×C17—42no—
67C22×C4×C17—42no—
68C22×C4×C17—42no—
69C22×C4×C17C22⋊C411yescharacteristic
70C22×C4×C17C22⋊C411yescharacteristic
71C22×C4×C17C22×C411yescharacteristic
72C2×C22⋊D17—24no—
73C2×C22⋊D17—42no—
74C2×C22⋊D17—42no—
75C22×Dic17—21no—
76C22×Dic17—21no—
77C22×Dic17C22⋊C411yescharacteristic
78C22×Dic17C2×D411yescharacteristic
79C22×Dic17C2×D411yescharacteristic
80C2×C4×D17—22no—
81C2×C4×D17—22no—
82C2×C4×D17—21no—
83C2×C4×D17—21no—
84C2×C4×D17—41no—
85C2×C4×D17—41no—
86C2×C4×D17—41no—
87C2×C4×D17—41no—
88C2×C4×D17C2×C812yes—
89C2×C4×D17M1612yes—
90C2×C4×D17D812yes—
91C2×C4×D17SD1612yes—
92C2×#3624—24no—
93C2×#3624—42no—
94C2×#3624—42no—
95C2×#3624—42no—
96C2×#3624—42no—
97C2×#3624—42no—
98#63242—48no—
99C4⋊C4×C17—41no—
100C22⋊C4×C17—42no—
101C42×C17—22no—
102C22⋊Dic17—24no—
103C22⋊Dic17—41no—
104C22⋊Dic17—41no—
105C22⋊Dic17—41no—
106C22⋊Dic17—41no—
107#63227—42no—
108C4⋊Dic17—21no—
109C4⋊Dic17—41no—
110C4⋊Dic17C2×C812yescomplemented
111#63225—41no—
112#63225—41no—
113C4×Dic17—22no—
114C4×Dic17—41no—
115C4×Dic17—41no—
116C4×Dic17—42no—
117#7586—171nomaximal, Sylow
118C23×C17—22no—
119C23×C17—41no—
120C23×C17—41no—
121C23×C17—41no—
122C23×C17—41no—
123C23×C17—41no—
124C23×C17—41no—
125C23×C17C2×C22⋊C411yescharacteristic
126C22×D17—21no—
127C22×D17—21no—
128C22×D17—21no—
129C22×D17—21no—
130C22×D17C22⋊C812yes—
131C22×D17#10212yes—
132C22×D17#10412yes—
133D4×C17—42no—
134D4×C17—42no—
135C2×C4×C17—21no—
136C2×C4×C17—21no—
137C2×C4×C17—21no—
138C2×C4×C17—21no—
139C2×C4×C17—21no—
140C2×C4×C17—21no—
141C2×C4×C17—41no—
142C2×C4×C17—41no—
143C2×C4×C17—41no—
144C2×C4×C17—84no—
145C2×C4×C17#9912yes—
146C2×C4×C17#10212yes—
147C2×C4×C17C22×C811yescharacteristic
148C2×C4×C17C2×M1611yescharacteristic
149C22⋊D17—48no—
150C2×Dic17—21no—
151C2×Dic17—21no—
152C2×Dic17—21no—
153C2×Dic17—21no—
154C2×Dic17—21no—
155C2×Dic17—21no—
156C2×Dic17—22no—
157C2×Dic17—41no—
158C2×Dic17—41no—
159C2×Dic17—41no—
160C2×Dic17—41no—
161C2×Dic17—41no—
162C2×Dic17C22⋊C812yes—
163C2×Dic17C2×D811yescharacteristic
164C2×Dic17C2×SD1611yescharacteristic
165C2×Dic17AGL(1,ℤ/8)11yescharacteristic
166C2×Dic17#13711yescharacteristic
167C4×D17—24no—
168C4×D17—24no—
169C4×D17—42no—
170C4×D17—42no—
171C4×D17—42no—
172C4×D17—42no—
173#3624—44no—
174#3624—82no—
175C4×#121—171no—
176#1797—171no—
177C2×#336—171no—
178#1271—172no—
179#1270—172no—
180C22×C17—42no—
181C22×C17—42no—
182C22×C17—42no—
183C22×C17—81no—
184C22×C17—81no—
185C22×C17—81no—
186C22×C17—81no—
187C22×C17C2×C22⋊C811yescharacteristic
188C22×C17C2×#9911yescharacteristic
189C22×C17#41311yescharacteristic
190C22×C17C2×#10211yescharacteristic
191C22×C17#41711yescharacteristic
192C22×C17C2×#10411yescharacteristic
193C22×C17#42111yescharacteristic
194C2×D17—22no—
195C2×D17—22no—
196C2×D17—22no—
197C2×D17—22no—
198C2×D17—22no—
199C2×D17—22no—
200C2×D17D4⋊C814yes—
201C4×C17—21no—
202C4×C17—21no—
203C4×C17—22no—
204C4×C17—22no—
205C4×C17—81no—
206Dic17—22no—
207Dic17—21no—
208Dic17—21no—
209Dic17—21no—
210Dic17—21no—
211Dic17—42no—
212Dic17—81no—
213C23×C8—341no—
214C2×#121—171no—
215C2×C4×D4—341no—
216C2×C4×D4#59511yescharacteristic
217C22×C42—171no—
218C2×C4⋊C8—341no—
219C2×C22⋊C8—341no—
220#390—171no—
221#389—171no—
222#388—341no—
223#386—341no—
224#380—171no—
225C4×C4⋊C4—171no—
226C4×C22⋊C4—341no—
227#336—172no—
228#336—172no—
229#336—341no—
230C2×C17—82no—
231C2×C17#145911yescharacteristic
232C2×C17C2×D4⋊C811yescharacteristic
233C2×C17#147711yescharacteristic
234C2×C17#150411yescharacteristic
235C2×C17#151911yescharacteristic
236C2×C17#153211yescharacteristic
237C2×C17#154711yescharacteristic
238D17—24no—
239C22×D4—341no—
240C22×D4C2×#59511yescharacteristic
241C23×C4—21no—
242C23×C4—171no—
243C23×C4—171no—
244C23×C4—171no—
245C22×C8—342no—
246C22×C8—344no—
247C22×C8—342no—
248C22×C8—682no—
249C22×C8—682no—
250#121—174no—
251#121—342no—
252C4×D4—42no—
253C4×D4—682no—
254C4×D4—682no—
255C2×C4⋊C4—171no—
256C2×C4⋊C4—171no—
257C2×C4⋊C4—341no—
258C2×C4⋊C4C2×#59511yescharacteristic
259C2×C22⋊C4—21no—
260C2×C22⋊C4—341no—
261C2×C22⋊C4—341no—
262C2×C22⋊C4—341no—
263C2×C22⋊C4—341no—
264C2×C42—171no—
265C2×C42—171no—
266C2×C42—171no—
267C2×C42—341no—
268C2×C42—682no—
269C2×C42C2×#59511yescharacteristic
270C4⋊C8—682no—
271C22⋊C8—682no—
272#95—341no—
273#95—341no—
274#95—341no—
275C17#758611yescharacteristic, complemented, Sylow
276C24—21no—
277C24—171no—
278C2×D4—21no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—344no—
282C2×D4—682no—
283C2×D4—682no—
284C2×D4#6324212yescomplemented
285C22×C4—21no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—171no—
290C22×C4—171no—
291C22×C4—171no—
292C22×C4—172no—
293C22×C4—172no—
294C22×C4—172no—
295C22×C4—172no—
296C22×C4—341no—
297C22×C4—341no—
298C22×C4—342no—
299C22×C4—342no—
300C22×C4—341no—
301C22×C4—341no—
302C22×C4—681no—
303C22×C4—681no—
304C22×C4—681no—
305C22×C4—681no—
306C22×C4C22⋊#59511yescharacteristic
307C22×C4C22⋊#59511yescharacteristic
308C22×C4C22×#59511yescharacteristic
309C2×C8—344no—
310C2×C8—682no—
311C2×C8—688no—
312C2×C8—682no—
313C2×C8—682no—
314C2×C8—682no—
315C2×C8—682no—
316C4⋊C4—41no—
317C4⋊C4—172no—
318C4⋊C4—341no—
319C4⋊C4—681no—
320C4⋊C4—681no—
321C4⋊C4—681no—
322C22⋊C4—42no—
323C22⋊C4—344no—
324C22⋊C4—682no—
325C22⋊C4—681no—
326C22⋊C4—681no—
327C22⋊C4—681no—
328C22⋊C4—681no—
329C42—22no—
330C42—342no—
331C42—681no—
332C42—681no—
333C42—682no—
334C23—22no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—172no—
342C23—172no—
343C23—172no—
344C23—341no—
345C23—341no—
346C23—341no—
347C23—341no—
348C23C2×C22⋊#59511yescharacteristic
349D4—42no—
350D4—42no—
351D4—688no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—84no—
362C2×C4—172no—
363C2×C4—171no—
364C2×C4—171no—
365C2×C4—171no—
366C2×C4—171no—
367C2×C4—344no—
368C2×C4—344no—
369C2×C4—341no—
370C2×C4—341no—
371C2×C4—341no—
372C2×C4—341no—
373C2×C4—341no—
374C2×C4—341no—
375C2×C4—342no—
376C2×C4—682no—
377C2×C4—682no—
378C2×C4—682no—
379C2×C4—682no—
380C2×C4—681no—
381C2×C4—681no—
382C2×C4—681no—
383C2×C4—681no—
384C2×C4—681no—
385C2×C4#1058754712yes—
386C2×C4#1058755112yes—
387C2×C4C2×#6324211yescharacteristic
388C2×C4D17⋊M1611yescharacteristic
389C8—684no—
390C8—1362no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—174no—
399C22—342no—
400C22—342no—
401C22—342no—
402C22—342no—
403C22—342no—
404C22—342no—
405C22#1176377711yescharacteristic
406C22C2×#1058755111yescharacteristic
407C22#1176379511yescharacteristic
408C22C2×#1058754711yescharacteristic
409C22#1176380011yescharacteristic
410C22D17⋊#10411yescharacteristic
411C22#1176380711yescharacteristic
412C4—22no—
413C4—22no—
414C4—21no—
415C4—21no—
416C4—81no—
417C4—341no—
418C4—341no—
419C4—341no—
420C4—341no—
421C4—342no—
422C4—682no—
423C4—1361no—
424C2—82no—
425C2—344no—
426C2#1454401911yescharacteristic
427C2#1454407011yescharacteristic
428C2#1454408011yescharacteristic
429C2#1454417011yescharacteristic
430C2#1454422011yescharacteristic
431C2#1454425911yescharacteristic
432C2#1454429511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC2×C4×C17C22×C8
FrattiniC22×C4C22×#595
FittingC2×C4×D4×C17C4
socleC23×C17C2×C22⋊C4
prime radicalC2×#10587624C4
cosocleC2×C4×Dic17C23
hypercenterC2×C4×D4#595
hypocenterC17#7586
LCSG ▷ C2×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. #14544641C2semidirect
2. #14544445C2semidirect
3. #14543603C2semidirect
4. #14543600C2semidirect
42. #10587624C8semidirect
64. C2×D4×C17C2×C8semidirect
110. C4⋊Dic17C2×C8semidirect
275. C17#7586semidirect
284. C2×D4#63242semidirect

Maximal subgroups

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

subgroupindex
#145446412
#145444452
#145436032
#145436002
C4×#105876242
#758617