Finite Group Databasebeta

← #16927876

Subgroups — #16927876

Order 4352.

530 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 123 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#14531694C211yesmaximal, characteristic
2#14528772C211yesmaximal, characteristic
3#14526664C212yesmaximal, complemented
4#14526652C212yesmaximal, complemented
5C17×#1459C211yesmaximal, characteristic
6C2×D4×Dic17—21no—
7C2×#10587505—21no—
8#11762708—21no—
9#11762684—21no—
10#11762684—21no—
11#11762675C411yescharacteristic, complemented
12#11762669C411yescharacteristic, complemented
13#11762669C2212yes—
14#11762669C2211yescharacteristic
15C4×C4⋊Dic17C411yescharacteristic, complemented
16C2×#10587502C411yescharacteristic, complemented
17C2×#10587502C2211yescharacteristic
18#11762576—21no—
19C2×C4×D4×C17C2211yescharacteristic
20C2×C17×C22⋊C8C2212yes—
21C17×#324—21no—
22C17×#323—21no—
23C22×D4×C17D411yescharacteristic
24C23×C4×C17C2×C411yescharacteristic
25C23×C4×C17C2311yescharacteristic
26C23×Dic17—21no—
27C23×Dic17—21no—
28C22×C8×C17—22no—
29C4×D4×C17—22no—
30C4×D4×C17—22no—
31C2×C4⋊C4×C17D411yescharacteristic
32C2×C22⋊C4×C17D411yescharacteristic
33C2×C22⋊C4×C17D411yescharacteristic
34C2×C42×C17C2×C411yescharacteristic
35C2×C22⋊Dic17—21no—
36C2×C22⋊Dic17—21no—
37C2×C22⋊Dic17—22no—
38D4×Dic17—44no—
39C2×C4⋊Dic17—21no—
40C2×C4⋊Dic17—21no—
41C2×C4⋊Dic17—21no—
42C2×C4⋊Dic17—21no—
43C2×C4⋊Dic17—22no—
44C2×C4×Dic17—21no—
45C17×C22⋊C8—22no—
46C17×C22⋊C8—22no—
47#10587505—42no—
48#10587502—21no—
49#10587502—21no—
50#10587502—21no—
51#10587502—22no—
52#10587502C2×C411yescharacteristic
53#10587502C2×C411yescharacteristic
54#10587502C2×C411yescharacteristic
55#10587502C2×C411yescharacteristic
56C24×C17C22⋊C411yescharacteristic
57C24×C17C2×D411yescharacteristic
58C2×D4×C17—22no—
59C2×D4×C17—22no—
60C2×D4×C17SD1612yes—
61C2×D4×C17Q1612yes—
62C22×C4×C17—21no—
63C22×C4×C17—21no—
64C22×C4×C17—81no—
65C22×C4×C17—81no—
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22×C411yescharacteristic
70C22×C4×C17C2×D411yescharacteristic
71C22×C4×C17#4112yes—
72C22×C4×C17#4112yes—
73C22×Dic17—21no—
74C22×Dic17—21no—
75C22×Dic17—21no—
76C22×Dic17—21no—
77C22×Dic17—42no—
78C22×Dic17—42no—
79C22×Dic17—42no—
80C22×Dic17—42no—
81C22×Dic17—42no—
82C22×Dic17—81no—
83C22×Dic17—81no—
84C2×C8×C17—42no—
85C2×C8×C17—82no—
86C4⋊C4×C17—22no—
87C22⋊C4×C17—22no—
88C22⋊C4×C17—22no—
89C42×C17—21no—
90C42×C17—21no—
91C22⋊Dic17—42no—
92C22⋊Dic17—42no—
93C22⋊Dic17—42no—
94C22⋊Dic17—42no—
95C4⋊Dic17—42no—
96C4⋊Dic17—42no—
97C4⋊Dic17—81no—
98C4⋊Dic17—81no—
99C4⋊Dic17—82no—
100C4×Dic17—42no—
101#8761—171nomaximal, Sylow
102C23×C17—21no—
103C23×C17—21no—
104C23×C17—41no—
105C23×C17—41no—
106C23×C17—41no—
107C23×C17—41no—
108C23×C17#10312yes—
109C23×C17C2×C22⋊C411yescharacteristic
110C23×C17#12312yes—
111C23×C17C2×SD1611yescharacteristic
112C23×C17C2×Q1611yescharacteristic
113C23×C17AGL(1,ℤ/8)12yes—
114D4×C17—24no—
115D4×C17—24no—
116C2×C4×C17—21no—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—42no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17—42no—
124C2×C4×C17—41no—
125C2×C4×C17—41no—
126C2×C4×C17—81no—
127C2×C4×C17—81no—
128C2×C4×C17#10312yes—
129C2×C4×C17#11711yescharacteristic
130C2×C4×C17#11711yescharacteristic
131C2×C4×C17#12312yes—
132C2×Dic17—22no—
133C2×Dic17—41no—
134C2×Dic17—41no—
135C2×Dic17—41no—
136C2×Dic17—41no—
137C2×Dic17—41no—
138C2×Dic17—41no—
139C2×Dic17—41no—
140C2×Dic17—41no—
141C2×Dic17—41no—
142C2×Dic17—41no—
143C2×Dic17—41no—
144C2×Dic17—84no—
145C2×Dic17—84no—
146C2×Dic17—82no—
147C2×Dic17—82no—
148C2×Dic17—82no—
149C8×C17—82no—
150#2349—171no—
151#1799—171no—
152#1459D1711yescharacteristic
153#1294—172no—
154#1289—172no—
155C22×C17—21no—
156C22×C17—22no—
157C22×C17—21no—
158C22×C17—22no—
159C22×C17—22no—
160C22×C17—42no—
161C22×C17—41no—
162C22×C17—41no—
163C22×C17—42no—
164C22×C17—42no—
165C22×C17—41no—
166C22×C17—41no—
167C22×C17#38111yescharacteristic
168C22×C17C2×#10311yescharacteristic
169C22×C17#41811yescharacteristic
170C22×C17#48112yes—
171C22×C17#48412yes—
172C4×C17—22no—
173C4×C17—22no—
174C4×C17—21no—
175C4×C17—21no—
176C4×C17—41no—
177C4×C17—41no—
178Dic17—42no—
179Dic17—81no—
180Dic17—81no—
181Dic17—81no—
182Dic17—162no—
183C2×C4×D4—341no—
184C2×C4×D4C2×D1711yescharacteristic
185C2×#99—341no—
186C2×C22⋊C8C2×D1712yes—
187#389—341no—
188#385—341no—
189#385—341no—
190#382—171no—
191#380—171no—
192#380—171no—
193#380—172no—
194C4×C4⋊C4—171no—
195C2×#95—171no—
196C2×#95—171no—
197#342—341no—
198#324—21no—
199#323—21no—
200C2×C17—21no—
201C2×C17—22no—
202C2×C17—21no—
203C2×C17—22no—
204C2×C17—42no—
205C2×C17#182411yescharacteristic
206C2×C17#182711yescharacteristic
207C2×C17#182811yescharacteristic
208C22×D4D6811yescharacteristic
209C23×C4—341no—
210C23×C4—341no—
211C23×C4C4×D1711yescharacteristic
212C23×C4C22×D1711yescharacteristic
213C22×C8—22no—
214C4×D4—22no—
215C4×D4—22no—
216C4×D4—684no—
217C2×C4⋊C4—341no—
218C2×C4⋊C4—341no—
219C2×C4⋊C4—341no—
220C2×C4⋊C4—341no—
221C2×C4⋊C4—342no—
222C2×C4⋊C4C22⋊D1711yescharacteristic
223C2×C22⋊C4—341no—
224C2×C22⋊C4—341no—
225C2×C22⋊C4—342no—
226C2×C22⋊C4D6811yescharacteristic
227C2×C22⋊C4C22⋊D1711yescharacteristic
228C2×C42—341no—
229C2×C42C4×D1711yescharacteristic
230#99—682no—
231C22⋊C8—22no—
232C22⋊C8—22no—
233#95—171no—
234#95—171no—
235#95—171no—
236#95—171no—
237#95—341no—
238#95—341no—
239#95—341no—
240#95—342no—
241C17#876111yescharacteristic, complemented, Sylow
242C24#6322711yescharacteristic
243C24C2×D6811yescharacteristic
244C2×D4—22no—
245C2×D4—22no—
246C2×D4#6321912yes—
247C2×D4Dic6812yes—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—81no—
251C22×C4—81no—
252C22×C4—341no—
253C22×C4—341no—
254C22×C4—341no—
255C22×C4—341no—
256C22×C4—682no—
257C22×C4—682no—
258C22×C4—682no—
259C22×C4—682no—
260C22×C4—682no—
261C22×C4—1361no—
262C22×C4—1361no—
263C22×C4#6322711yescharacteristic
264C22×C4#6322711yescharacteristic
265C22×C4#6322711yescharacteristic
266C22×C4C2×C4×D1711yescharacteristic
267C22×C4#6325412yes—
268C22×C4#6325412yes—
269C22×C4C2×C22⋊D1711yescharacteristic
270C2×C8—42no—
271C2×C8—82no—
272C4⋊C4—22no—
273C4⋊C4—682no—
274C4⋊C4—682no—
275C4⋊C4—1361no—
276C4⋊C4—1361no—
277C4⋊C4—1362no—
278C22⋊C4—22no—
279C22⋊C4—22no—
280C22⋊C4—682no—
281C22⋊C4—682no—
282C22⋊C4—682no—
283C22⋊C4—682no—
284C42—21no—
285C42—21no—
286C42—682no—
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23#1058748712yes—
294C23#1058757212yes—
295C23C2×#6321911yescharacteristic
296C23C2×Dic6811yescharacteristic
297C23#1058759412yes—
298C23C2×#6322711yescharacteristic
299D4—24no—
300D4—24no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—342no—
314C2×C4—681no—
315C2×C4—681no—
316C2×C4—681no—
317C2×C4—681no—
318C2×C4—681no—
319C2×C4—681no—
320C2×C4—681no—
321C2×C4—681no—
322C2×C4—681no—
323C2×C4—681no—
324C2×C4—681no—
325C2×C4—1362no—
326C2×C4—1362no—
327C2×C4—1362no—
328C2×C4—1364no—
329C2×C4—1364no—
330C2×C4#1058748712yes—
331C2×C4#1058756311yescharacteristic
332C2×C4#1058756311yescharacteristic
333C2×C4#1058762112yes—
334C8—82no—
335C22—21no—
336C22—22no—
337C22—21no—
338C22—22no—
339C22—22no—
340C22—42no—
341C22—41no—
342C22—41no—
343C22—42no—
344C22—42no—
345C22—41no—
346C22—41no—
347C22#1176267011yescharacteristic
348C22C2×#1058748711yescharacteristic
349C22#1176281511yescharacteristic
350C22#1176312812yes—
351C22#1176314512yes—
352C4—22no—
353C4—22no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—2722no—
359C4—682no—
360C4—1361no—
361C4—1361no—
362C4—1361no—
363C2—22no—
364C2—21no—
365C2—21no—
366C2—22no—
367C2—42no—
368C2#1452887511yescharacteristic
369C2#1452888911yescharacteristic
370C2#1452889011yescharacteristic
subgroupG/H
centerC22#11762670
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#1459C2
socleC22×C17#381
prime radicalC22×D4×C17D4
cosocleC23×C4×C17C23
hypercenter#1459D17
hypocenterC17#8761
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. #14526664C2semidirect
4. #14526652C2semidirect
11. #11762675C4semidirect
12. #11762669C4semidirect
15. C4×C4⋊Dic17C4semidirect
16. C2×#10587502C4semidirect
241. C17#8761semidirect

Maximal subgroups

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

subgroupindex
#145316942
#145287722
#145266642
#145266522
C17×#14592
#876117