Finite Group Databasebeta

← #16922988

Subgroups — #16922988

Order 4352.

494 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 103 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#14537133C211yesmaximal, characteristic
2#14528851C211yesmaximal, characteristic
3#14528763C211yesmaximal, characteristic
4#14526610C211yesmaximal, characteristic, complemented
5#14526609C211yesmaximal, characteristic, complemented
6#14526609C211yesmaximal, characteristic, complemented
7C17×#1272C211yesmaximal, characteristic, complemented
8C2×#10587630C411yescharacteristic
9C2×#10587583C411yescharacteristic
10C2×#10587632C411yescharacteristic
11C2×C42×D17C2211yescharacteristic
12C2×#10587485—21no—
13C2×#10587491—21no—
14C2×#10587491—21no—
15#11762715C2211yescharacteristic
16#11762706—21no—
17#11762706—21no—
18#11762678C411yescharacteristic
19#11762668—21no—
20#11762668—21no—
21#11762564C2211yescharacteristic
22#11762564C2211yescharacteristic
23#11762563—21no—
24C17×#391C2211yescharacteristic
25C17×#336C2211yescharacteristic
26C17×#336C2211yescharacteristic
27C22×Q8×C17C2×C411yescharacteristic
28C22×C4×D17C2×C411yescharacteristic
29C22×C4×D17D411yescharacteristic
30C22×C4×D17D411yescharacteristic
31C22×Dic34C2×C411yescharacteristic
32C22×C8×C17—21no—
33C22×C8×C17—21no—
34C2×C4⋊C4×C17C2×C411yescharacteristic
35C2×C42×C17C2311yescharacteristic
36#10587632—42no—
37#10587630—42no—
38C2×#63227—21no—
39C2×#63227—21no—
40C2×C4⋊Dic17—21no—
41C2×C4⋊Dic17C2×C411yescharacteristic
42C2×#63225—21no—
43C2×#63225—21no—
44C2×#63225—21no—
45C2×#63225—21no—
46C2×#63225—21no—
47C2×C4×Dic17C2×C411yescharacteristic
48C2×C4×Dic17D411yescharacteristic
49C2×C4×Dic17D411yescharacteristic
50C22×#3622—21no—
51C22×#3622—21no—
52#10587583—42no—
53C42×D17—42no—
54C17×#95—21no—
55C17×#95—21no—
56#10587502—21no—
57#10587502—21no—
58#10587491—42no—
59#10587491—42no—
60#10587485—42no—
61C23×D17C22⋊C411yescharacteristic
62C2×Q8×C17—21no—
63C2×Q8×C17—21no—
64C2×Q8×C17—41no—
65C2×Q8×C17—41no—
66C22×C4×C17—21no—
67C22×C4×C17—21no—
68C22×C4×C17C22×C411yescharacteristic
69C22×C4×C17C2×D411yescharacteristic
70C22×C4×C17C2×D411yescharacteristic
71C22×Dic17—21no—
72C22×Dic17—21no—
73C22×Dic17C22⋊C411yescharacteristic
74C22×Dic17C22⋊C411yescharacteristic
75C22×Dic17C22⋊C411yescharacteristic
76C2×C4×D17—21no—
77C2×C4×D17—21no—
78C2×C4×D17—21no—
79C2×C4×D17—21no—
80C2×C4×D17—22no—
81C2×C4×D17—22no—
82C2×C4×D17—41no—
83C2×C4×D17—41no—
84C2×C4×D17—41no—
85C2×C4×D17—41no—
86C2×C4×D17SD1612yes—
87C2×C4×D17Q1612yes—
88C2×Dic34—21no—
89C2×Dic34—21no—
90C2×Dic34—41no—
91C2×Dic34—41no—
92C2×C8×C17—41no—
93C2×C8×C17—41no—
94C2×C8×C17—41no—
95C2×C8×C17—41no—
96C2×C8×C17—41no—
97C2×C8×C17—41no—
98C4⋊C4×C17—41no—
99C42×C17—41no—
100#63227—42no—
101#63227—42no—
102C4⋊Dic17—42no—
103C4⋊Dic17—41no—
104#63225—42no—
105#63225—42no—
106#63225—42no—
107#63225—42no—
108#63225—42no—
109C4×Dic17—22no—
110C4×Dic17—22no—
111C4×Dic17—41no—
112C2×#3622—41no—
113C2×#3622—41no—
114C2×#3622—41no—
115C2×#3622—41no—
116C2×#3622—41no—
117C2×#3622—41no—
118#7727—171nomaximal, Sylow
119C23×C17C2×C22⋊C411yescharacteristic
120C22×D17—21no—
121C22×D17—21no—
122C22×D17—21no—
123C22×D17—21no—
124C22×D17—21no—
125C22×D17—21no—
126C22×D17#10312yes—
127Q8×C17—81no—
128Q8×C17—81no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17—21no—
134C2×C4×C17—21no—
135C2×C4×C17—41no—
136C2×C4×C17—41no—
137C2×C4×C17—41no—
138C2×C4×C17—41no—
139C2×C4×C17—41no—
140C2×C4×C17—41no—
141C2×C4×C17#13111yescharacteristic
142C2×C4×C17#13111yescharacteristic
143C2×C4×C17C2×SD1611yescharacteristic
144C2×C4×C17C2×Q1611yescharacteristic
145C2×C4×C17#13711yescharacteristic
146C2×C4×C17#13711yescharacteristic
147C2×Dic17—21no—
148C2×Dic17—21no—
149C2×Dic17—21no—
150C2×Dic17—21no—
151C2×Dic17—21no—
152C2×Dic17—21no—
153C2×Dic17—41no—
154C2×Dic17—41no—
155C2×Dic17—41no—
156C2×Dic17—41no—
157C2×Dic17—41no—
158C2×Dic17—41no—
159C2×Dic17#9912yes—
160C2×Dic17#10012yes—
161C2×Dic17#10312yes—
162C4×D17—42no—
163C4×D17—42no—
164C4×D17—42no—
165C4×D17—42no—
166C4×D17—42no—
167C4×D17—42no—
168Dic34—81no—
169Dic34—81no—
170C8×C17—81no—
171C8×C17—81no—
172#3622—81no—
173#3622—81no—
174#2857—171no—
175#1817—171no—
176#1797—171no—
177#1272—171no—
178#1272—171no—
179#1272—171no—
180#1272D1711yescharacteristic, complemented
181C22×C17#40811yescharacteristic
182C22×C17C2×#9911yescharacteristic
183C22×C17C2×#10011yescharacteristic
184C22×C17C2×#10311yescharacteristic
185C22×C17#41911yescharacteristic
186C22×C17#42111yescharacteristic
187C22×C17#42111yescharacteristic
188C2×D17—22no—
189C2×D17—22no—
190C2×D17—22no—
191C2×D17—41no—
192C2×D17—41no—
193C2×D17—41no—
194C2×D17—41no—
195C4×C17—21no—
196C4×C17—21no—
197C4×C17—21no—
198C4×C17—21no—
199C4×C17—21no—
200C4×C17—21no—
201C4×C17—81no—
202C4×C17—81no—
203Dic17—22no—
204Dic17—22no—
205Dic17—22no—
206Dic17—81no—
207Dic17—81no—
208C2×C4⋊Q8—171no—
209C2×C22⋊Q8—171no—
210C2×C22⋊Q8—171no—
211C22×C42—171no—
212C2×C4⋊C8—341no—
213C2×C22⋊C8—341no—
214C2×C22⋊C8—341no—
215#391—171no—
216#391C2×D1711yescharacteristic
217#389—341no—
218#389—341no—
219#384—171no—
220#380—341no—
221#380—341no—
222#336—171no—
223#336—171no—
224#336—341no—
225#336C2×D1711yescharacteristic
226#336C2×D1711yescharacteristic
227C2×C17#147111yescharacteristic
228C2×C17#148111yescharacteristic
229C2×C17#149711yescharacteristic
230C2×C17#151111yescharacteristic
231C2×C17#152011yescharacteristic
232C2×C17#153611yescharacteristic
233C2×C17#154811yescharacteristic
234D17—42no—
235C22×Q8—171no—
236C22×Q8C4×D1711yescharacteristic
237C23×C4—171no—
238C23×C4—171no—
239C23×C4—171no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—341no—
243C22×C8—341no—
244C4⋊Q8—682no—
245C22⋊Q8—682no—
246C22⋊Q8—682no—
247C2×C4⋊C4—171no—
248C2×C4⋊C4—341no—
249C2×C4⋊C4—341no—
250C2×C4⋊C4—341no—
251C2×C4⋊C4—341no—
252C2×C4⋊C4—341no—
253C2×C4⋊C4—341no—
254C2×C4⋊C4C4×D1711yescharacteristic
255C2×C22⋊C4—341no—
256C2×C22⋊C4—341no—
257C2×C42—171no—
258C2×C42—171no—
259C2×C42—171no—
260C2×C42—682no—
261C2×C42C22×D1711yescharacteristic
262C4⋊C8—682no—
263C22⋊C8—682no—
264C22⋊C8—682no—
265#95—21no—
266#95—21no—
267#95—341no—
268#95—341no—
269C17#772711yescharacteristic, complemented, Sylow
270C24—171no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—341no—
276C2×Q8—341no—
277C2×Q8—681no—
278C2×Q8—681no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—171no—
282C22×C4—171no—
283C22×C4—171no—
284C22×C4—172no—
285C22×C4—172no—
286C22×C4—342no—
287C22×C4—342no—
288C22×C4—341no—
289C22×C4—341no—
290C22×C4—341no—
291C22×C4—341no—
292C22×C4—341no—
293C22×C4—341no—
294C22×C4—681no—
295C22×C4—681no—
296C22×C4—681no—
297C22×C4—681no—
298C22×C4C2×C4×D1711yescharacteristic
299C22×C4D4×D1711yescharacteristic
300C22×C4D4×D1711yescharacteristic
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—681no—
308C2×C8—681no—
309C2×C8—681no—
310C2×C8—681no—
311C2×C8—681no—
312C2×C8—681no—
313C4⋊C4—41no—
314C4⋊C4—681no—
315C4⋊C4—682no—
316C4⋊C4—682no—
317C4⋊C4—682no—
318C4⋊C4—682no—
319C4⋊C4—682no—
320C4⋊C4—682no—
321C22⋊C4—682no—
322C22⋊C4—682no—
323C42—41no—
324C42—342no—
325C42—342no—
326C42—681no—
327C23—172no—
328C23—341no—
329C23—341no—
330C23—341no—
331C23—341no—
332C23—341no—
333C23—341no—
334C23C22⋊C4×D1711yescharacteristic
335Q8—81no—
336Q8—81no—
337Q8—1361no—
338Q8—1361no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—172no—
352C2×C4—172no—
353C2×C4—172no—
354C2×C4—341no—
355C2×C4—341no—
356C2×C4—341no—
357C2×C4—341no—
358C2×C4—341no—
359C2×C4—341no—
360C2×C4—682no—
361C2×C4—682no—
362C2×C4—682no—
363C2×C4—682no—
364C2×C4—682no—
365C2×C4—682no—
366C2×C4—681no—
367C2×C4—681no—
368C2×C4—681no—
369C2×C4—681no—
370C2×C4—681no—
371C2×C4—681no—
372C2×C4#1058758711yescharacteristic
373C2×C4#1058759311yescharacteristic
374C2×C4SD16×D1711yescharacteristic
375C2×C4#1058760111yescharacteristic
376C2×C4Q16×D1711yescharacteristic
377C2×C4#1058760411yescharacteristic
378C8—81no—
379C8—81no—
380C8—1361no—
381C8—1361no—
382C22—342no—
383C22—342no—
384C22—342no—
385C22—681no—
386C22—681no—
387C22—681no—
388C22—681no—
389C22#1176277411yescharacteristic
390C22#99×D1711yescharacteristic
391C22#100×D1711yescharacteristic
392C22#103×D1711yescharacteristic
393C22#1176281611yescharacteristic
394C22#1176282411yescharacteristic
395C22#1176282411yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—81no—
403C4—81no—
404C4—342no—
405C4—342no—
406C4—342no—
407C4—1361no—
408C4—1361no—
409C2—682no—
410C2#1452714211yescharacteristic
411C2#1452718511yescharacteristic
412C2#1452725511yescharacteristic
413C2#1452731511yescharacteristic
414C2#1452735411yescharacteristic
415C2#1452741711yescharacteristic
416C2#1452746311yescharacteristic
subgroupG/H
centerC23C22⋊C4×D17
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1272C2
socleC23×C17C2×C22⋊C4
prime radicalC23×D17C22⋊C4
cosocleC2×C42×C17C23
hypercenter#1272D17
hypocenterC17#7727
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
4. #14526610C2semidirect
5. #14526609C2semidirect
6. #14526609C2semidirect
7. C17×#1272C2semidirect
180. #1272D17semidirect
269. C17#7727semidirect

Maximal subgroups

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

subgroupindex
#145371332
#145288512
#145287632
#145266102
#145266092
C17×#12722
#772717