Finite Group Databasebeta

← #16922979

Subgroups — #16922979

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#14537135C211yesmaximal, characteristic
2#14528851C211yesmaximal, characteristic
3#14528763C211yesmaximal, characteristic
4#14526608C211yesmaximal, characteristic, complemented
5#14526607C211yesmaximal, characteristic, complemented
6#14526606C211yesmaximal, characteristic, complemented
7C17×#1271C211yesmaximal, characteristic, complemented
8C2×#10587574C411yescharacteristic
9C2×#10587582—21no—
10C2×C42×D17C2211yescharacteristic
11C2×#10587485—21no—
12C2×#10587491—21no—
13C2×#10587491—21no—
14#11762713—21no—
15#11762707C2211yescharacteristic
16#11762704—21no—
17#11762678C411yescharacteristic
18#11762668—21no—
19#11762666C411yescharacteristic
20#11762666C411yescharacteristic
21#11762564C2211yescharacteristic
22#11762564C2211yescharacteristic
23#11762563—21no—
24C17×#389C2211yescharacteristic
25C17×#336C2211yescharacteristic
26C17×#336C2211yescharacteristic
27C22×C4×D17C2×C411yescharacteristic
28C22×C4×D17D411yescharacteristic
29C22×C4×D17D411yescharacteristic
30C22×Dic34—21no—
31C22×C8×C17—21no—
32C22×C8×C17—21no—
33C2×C4⋊C4×C17—21no—
34C2×C4⋊C4×C17C2×C411yescharacteristic
35C2×C4⋊C4×C17C2×C411yescharacteristic
36C2×C42×C17C2311yescharacteristic
37C2×#63227—21no—
38C2×#63227—21no—
39C2×C4⋊Dic17C2×C411yescharacteristic
40C2×C4⋊Dic17C2×C411yescharacteristic
41C2×#63225—21no—
42C2×#63225—21no—
43C2×#63225—21no—
44C2×#63225—21no—
45C2×C4×Dic17C2×C411yescharacteristic
46C2×C4×Dic17D411yescharacteristic
47C2×C4×Dic17D411yescharacteristic
48C22×#3622—21no—
49C22×#3622—21no—
50#10587582—42no—
51#10587582—42no—
52#10587574—42no—
53C42×D17—42no—
54C17×#95—21no—
55#10587502—21no—
56#10587502—21no—
57#10587502—21no—
58#10587491—42no—
59#10587491—42no—
60#10587485—42no—
61C23×D17C22⋊C411yescharacteristic
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17C22×C411yescharacteristic
65C22×C4×C17C2×D411yescharacteristic
66C22×C4×C17C2×D411yescharacteristic
67C22×Dic17—21no—
68C22×Dic17—21no—
69C22×Dic17C22⋊C411yescharacteristic
70C22×Dic17C22⋊C411yescharacteristic
71C22×Dic17C22⋊C411yescharacteristic
72C2×C4×D17—21no—
73C2×C4×D17—21no—
74C2×C4×D17—21no—
75C2×C4×D17—21no—
76C2×C4×D17—22no—
77C2×C4×D17—22no—
78C2×C4×D17—41no—
79C2×C4×D17—41no—
80C2×C4×D17—41no—
81C2×C4×D17—41no—
82C2×C4×D17SD1612yes—
83C2×C4×D17Q1612yes—
84C2×Dic34—42no—
85C2×Dic34—42no—
86C2×Dic34—42no—
87C2×C8×C17—41no—
88C2×C8×C17—41no—
89C2×C8×C17—41no—
90C2×C8×C17—41no—
91C2×C8×C17—41no—
92C2×C8×C17—41no—
93C4⋊C4×C17—21no—
94C4⋊C4×C17—21no—
95C4⋊C4×C17—41no—
96C4⋊C4×C17—41no—
97C4⋊C4×C17—41no—
98C42×C17—41no—
99#63227—42no—
100#63227—42no—
101C4⋊Dic17—21no—
102C4⋊Dic17—21no—
103C4⋊Dic17—41no—
104#63225—42no—
105#63225—42no—
106#63225—42no—
107#63225—41no—
108#63225—41no—
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#7725—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—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17—41no—
134C2×C4×C17—41no—
135C2×C4×C17—41no—
136C2×C4×C17—41no—
137C2×C4×C17—41no—
138C2×C4×C17—41no—
139C2×C4×C17#13111yescharacteristic
140C2×C4×C17#13111yescharacteristic
141C2×C4×C17C2×SD1611yescharacteristic
142C2×C4×C17C2×Q1611yescharacteristic
143C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
144C2×C4×C17AGL(1,ℤ/8)11yescharacteristic
145C2×Dic17—21no—
146C2×Dic17—21no—
147C2×Dic17—21no—
148C2×Dic17—21no—
149C2×Dic17—21no—
150C2×Dic17—21no—
151C2×Dic17—41no—
152C2×Dic17—41no—
153C2×Dic17—41no—
154C2×Dic17—41no—
155C2×Dic17—41no—
156C2×Dic17—41no—
157C2×Dic17#9912yes—
158C2×Dic17#10112yes—
159C2×Dic17#10312yes—
160C4×D17—42no—
161C4×D17—42no—
162C4×D17—42no—
163C4×D17—42no—
164C4×D17—42no—
165C4×D17—42no—
166Dic34—84no—
167C8×C17—81no—
168C8×C17—81no—
169#3622—81no—
170#3622—81no—
171#2857—171no—
172#1817—171no—
173#1797—171no—
174#1271—171no—
175#1271—171no—
176#1271—171no—
177#1271D1711yescharacteristic, complemented
178C22×C17#40811yescharacteristic
179C22×C17C2×#9911yescharacteristic
180C22×C17C2×#10111yescharacteristic
181C22×C17C2×#10311yescharacteristic
182C22×C17#41811yescharacteristic
183C22×C17#42111yescharacteristic
184C22×C17#42111yescharacteristic
185C2×D17—22no—
186C2×D17—22no—
187C2×D17—22no—
188C2×D17—41no—
189C2×D17—41no—
190C2×D17—41no—
191C2×D17—41no—
192C4×C17—21no—
193C4×C17—21no—
194C4×C17—21no—
195C4×C17—21no—
196C4×C17—21no—
197C4×C17—21no—
198C4×C17—81no—
199C4×C17—81no—
200Dic17—22no—
201Dic17—22no—
202Dic17—22no—
203Dic17—81no—
204Dic17—81no—
205C2×C4⋊Q8—171no—
206C2×C22⋊Q8—341no—
207C22×C42—171no—
208C2×C4⋊C8—341no—
209C2×C22⋊C8—341no—
210C2×C22⋊C8—341no—
211#391—341no—
212#389—171no—
213#389—341no—
214#389C2×D1711yescharacteristic
215#384—171no—
216#380—171no—
217#380—171no—
218#380—341no—
219#336—171no—
220#336—171no—
221#336—341no—
222#336C2×D1711yescharacteristic
223#336C2×D1711yescharacteristic
224C2×C17#147211yescharacteristic
225C2×C17#148111yescharacteristic
226C2×C17#149611yescharacteristic
227C2×C17#151011yescharacteristic
228C2×C17#152011yescharacteristic
229C2×C17#153711yescharacteristic
230C2×C17#154911yescharacteristic
231D17—42no—
232C22×Q8—341no—
233C23×C4—171no—
234C23×C4—171no—
235C23×C4—171no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—341no—
239C22×C8—341no—
240C4⋊Q8—682no—
241C22⋊Q8—682no—
242C22⋊Q8—682no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—171no—
245C2×C4⋊C4—171no—
246C2×C4⋊C4—341no—
247C2×C4⋊C4—341no—
248C2×C4⋊C4—341no—
249C2×C4⋊C4—341no—
250C2×C4⋊C4C4×D1711yescharacteristic
251C2×C4⋊C4C4×D1711yescharacteristic
252C2×C22⋊C4—341no—
253C2×C22⋊C4—341no—
254C2×C42—171no—
255C2×C42—171no—
256C2×C42—171no—
257C2×C42—682no—
258C2×C42C22×D1711yescharacteristic
259C4⋊C8—682no—
260C22⋊C8—682no—
261C22⋊C8—682no—
262#95—21no—
263#95—341no—
264#95—341no—
265#95—341no—
266C17#772511yescharacteristic, complemented, Sylow
267C24—171no—
268C2×Q8—682no—
269C2×Q8—682no—
270C2×Q8—682no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—171no—
274C22×C4—171no—
275C22×C4—171no—
276C22×C4—172no—
277C22×C4—172no—
278C22×C4—342no—
279C22×C4—342no—
280C22×C4—341no—
281C22×C4—341no—
282C22×C4—341no—
283C22×C4—341no—
284C22×C4—341no—
285C22×C4—341no—
286C22×C4—681no—
287C22×C4—681no—
288C22×C4—681no—
289C22×C4—681no—
290C22×C4C2×C4×D1711yescharacteristic
291C22×C4D4×D1711yescharacteristic
292C22×C4D4×D1711yescharacteristic
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—681no—
300C2×C8—681no—
301C2×C8—681no—
302C2×C8—681no—
303C2×C8—681no—
304C2×C8—681no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—341no—
311C4⋊C4—341no—
312C4⋊C4—681no—
313C4⋊C4—682no—
314C4⋊C4—682no—
315C4⋊C4—682no—
316C4⋊C4—681no—
317C4⋊C4—681no—
318C22⋊C4—682no—
319C22⋊C4—682no—
320C42—41no—
321C42—342no—
322C42—342no—
323C42—681no—
324C23—172no—
325C23—341no—
326C23—341no—
327C23—341no—
328C23—341no—
329C23—341no—
330C23—341no—
331C23C22⋊C4×D1711yescharacteristic
332Q8—1364no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—172no—
346C2×C4—172no—
347C2×C4—172no—
348C2×C4—341no—
349C2×C4—341no—
350C2×C4—341no—
351C2×C4—341no—
352C2×C4—341no—
353C2×C4—341no—
354C2×C4—682no—
355C2×C4—682no—
356C2×C4—682no—
357C2×C4—682no—
358C2×C4—682no—
359C2×C4—682no—
360C2×C4—681no—
361C2×C4—681no—
362C2×C4—681no—
363C2×C4—681no—
364C2×C4—681no—
365C2×C4—681no—
366C2×C4#1058758711yescharacteristic
367C2×C4#1058759311yescharacteristic
368C2×C4#1058759711yescharacteristic
369C2×C4SD16×D1711yescharacteristic
370C2×C4#1058760011yescharacteristic
371C2×C4Q16×D1711yescharacteristic
372C8—81no—
373C8—81no—
374C8—1361no—
375C8—1361no—
376C22—342no—
377C22—342no—
378C22—342no—
379C22—681no—
380C22—681no—
381C22—681no—
382C22—681no—
383C22#1176277411yescharacteristic
384C22#99×D1711yescharacteristic
385C22#101×D1711yescharacteristic
386C22#103×D1711yescharacteristic
387C22#1176281211yescharacteristic
388C22#1176282411yescharacteristic
389C22#1176282411yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—81no—
397C4—81no—
398C4—342no—
399C4—342no—
400C4—342no—
401C4—1361no—
402C4—1361no—
403C2—682no—
404C2#1452714611yescharacteristic
405C2#1452718511yescharacteristic
406C2#1452725111yescharacteristic
407C2#1452731111yescharacteristic
408C2#1452735411yescharacteristic
409C2#1452742011yescharacteristic
410C2#1452746711yescharacteristic
subgroupG/H
centerC23C22⋊C4×D17
commutatorC22×C4×C17C22×C4
FrattiniC2×C42C22×D17
FittingC17×#1271C2
socleC23×C17C2×C22⋊C4
prime radicalC23×D17C22⋊C4
cosocleC2×C42×C17C23
hypercenter#1271D17
hypocenterC17#7725
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1271

Complements

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

NH ≅ G/Ntype
4. #14526608C2semidirect
5. #14526607C2semidirect
6. #14526606C2semidirect
7. C17×#1271C2semidirect
177. #1271D17semidirect
266. C17#7725semidirect

Maximal subgroups

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

subgroupindex
#145371352
#145288512
#145287632
#145266082
#145266072
#145266062
C17×#12712
#772517