Finite Group Databasebeta

← #17949808

Subgroups — #17949808

Order 4352.

612 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 110 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#14544676C211yesmaximal, characteristic
2#14544448C211yesmaximal, characteristic
3#14543603C212yesmaximal, complemented
4#14543602C212yesmaximal, complemented
5D17×#389C211yesmaximal, characteristic
6C22×#63243—21no—
7C2×#10587543—21no—
8#11763675—21no—
9#11763675—21no—
10#11763675C2212yes—
11#11763675C2212yes—
12C2×C4⋊C4×D17—21no—
13C2×C4⋊C4×D17C411yescharacteristic
14C2×C4⋊C4×D17C411yescharacteristic
15C2×C42×D17C2211yescharacteristic
16#11762708C2211yescharacteristic
17#11762707C411yescharacteristic
18#11762706—21no—
19#11762705C2211yescharacteristic
20#11762704—21no—
21#95×D17—21no—
22C17×#389C411yescharacteristic
23C22×C4×D17—21no—
24C22×C4×D17—21no—
25C22×C4×D17C2×C411yescharacteristic
26C22×C4×D17D411yescharacteristic
27C22×C4×D17D411yescharacteristic
28C22×#3624—22no—
29C22×#3624—22no—
30C2×#63243—24no—
31C2×#63243—42no—
32C2×#63243—42no—
33C2×C4⋊C4×C17—21no—
34C2×C4⋊C4×C17C2×C411yescharacteristic
35C2×C4⋊C4×C17C2×C411yescharacteristic
36C2×C42×C17C2×C411yescharacteristic
37C2×C4⋊Dic17—21no—
38C2×C4⋊Dic17C2×C411yescharacteristic
39C2×C4⋊Dic17C2×C411yescharacteristic
40C2×#63225—21no—
41C2×#63225—21no—
42C2×#63225—21no—
43C2×#63225—21no—
44C2×C4×Dic17D411yescharacteristic
45C2×C4×Dic17D411yescharacteristic
46C2×C4×Dic17C2311yescharacteristic
47C4⋊C4×D17—22no—
48C4⋊C4×D17—22no—
49C4⋊C4×D17—44no—
50C4⋊C4×D17—42no—
51C42×D17—42no—
52#10587543—42no—
53C17×#95—21no—
54#10587502—21no—
55#10587502—21no—
56#10587502—21no—
57C23×D17C22⋊C411yescharacteristic
58C22×C4×C17—21no—
59C22×C4×C17—21no—
60C22×C4×C17C22⋊C411yescharacteristic
61C22×C4×C17C22⋊C411yescharacteristic
62C22×C4×C17C22×C411yescharacteristic
63C22×Dic17—21no—
64C22×Dic17—21no—
65C22×Dic17C22⋊C411yescharacteristic
66C22×Dic17C2×D411yescharacteristic
67C22×Dic17C2×D411yescharacteristic
68C2×C4×D17—22no—
69C2×C4×D17—22no—
70C2×C4×D17—22no—
71C2×C4×D17—21no—
72C2×C4×D17—21no—
73C2×C4×D17—22no—
74C2×C4×D17—24no—
75C2×C4×D17—22no—
76C2×C4×D17—22no—
77C2×C4×D17—41no—
78C2×C4×D17—42no—
79C2×C4×D17—42no—
80C2×C4×D17—42no—
81C2×C4×D17—42no—
82C2×C4×D17—42no—
83C2×C4×D17SD1612yes—
84C2×C4×D17Q1612yes—
85C2×#3624—24no—
86C2×#3624—42no—
87C2×#3624—42no—
88C2×#3624—42no—
89C2×#3624—42no—
90C2×#3624—42no—
91#63243—48no—
92C4⋊C4×C17—21no—
93C4⋊C4×C17—41no—
94C4⋊C4×C17—42no—
95C4⋊C4×C17M1612yescomplemented
96C42×C17—41no—
97C4⋊Dic17—21no—
98C4⋊Dic17—42no—
99C4⋊Dic17—41no—
100C4⋊Dic17M1612yescomplemented
101#63225—24no—
102#63225—42no—
103#63225—42no—
104#63225—42no—
105C4×Dic17—41no—
106C4×Dic17—41no—
107C4×Dic17—41no—
108#7607—171nomaximal, Sylow
109C23×C17C2×C22⋊C411yescharacteristic
110C22×D17—21no—
111C22×D17—21no—
112C22×D17—21no—
113C22×D17—21no—
114C22×D17#9912yes—
115C22×D17#10312yes—
116C22×D17#10412yes—
117C2×C4×C17—21no—
118C2×C4×C17—21no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—22no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—21no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17—41no—
128C2×C4×C17—41no—
129C2×C4×C17—41no—
130C2×C4×C17#10312yes—
131C2×C4×C17C2×M1611yescharacteristic
132C2×C4×C17C2×M1611yescharacteristic
133C2×Dic17—21no—
134C2×Dic17—21no—
135C2×Dic17—21no—
136C2×Dic17—21no—
137C2×Dic17—21no—
138C2×Dic17—21no—
139C2×Dic17—22no—
140C2×Dic17—41no—
141C2×Dic17—41no—
142C2×Dic17—41no—
143C2×Dic17—41no—
144C2×Dic17—41no—
145C2×Dic17#10112yes—
146C2×Dic17C2×SD1611yescharacteristic
147C2×Dic17C2×Q1611yescharacteristic
148C2×Dic17AGL(1,ℤ/8)12yes—
149C4×D17—24no—
150C4×D17—24no—
151C4×D17—24no—
152C4×D17—24no—
153C4×D17—44no—
154C4×D17—48no—
155C4×D17—84no—
156#3624—44no—
157#3624—82no—
158C2×#389—171no—
159#1817—171no—
160#1728—171no—
161#1271—172no—
162#1271—172no—
163C22×C17#40711yescharacteristic
164C22×C17C2×#9911yescharacteristic
165C22×C17C2×#10111yescharacteristic
166C22×C17C2×#10311yescharacteristic
167C22×C17#41811yescharacteristic
168C22×C17C2×#10411yescharacteristic
169C22×C17#42111yescharacteristic
170C2×D17—22no—
171C2×D17—22no—
172C2×D17—22no—
173C2×D17—22no—
174C2×D17—22no—
175C2×D17—22no—
176C2×D17#32814yes—
177C4×C17—21no—
178C4×C17—21no—
179C4×C17—22no—
180C4×C17—42no—
181C4×C17—41no—
182C4×C17—81no—
183Dic17—22no—
184Dic17—21no—
185Dic17—21no—
186Dic17—22no—
187Dic17—42no—
188Dic17—81no—
189C22×M16—341no—
190C22×C4⋊C4—171no—
191C22×C4⋊C4—171no—
192C22×C4⋊C4—341no—
193C22×C42—171no—
194C2×C22⋊C8—341no—
195#389—171no—
196#389—171no—
197#389—171no—
198#389—341no—
199#389—341no—
200#389#59511yescharacteristic
201C2×#95—341no—
202#336—172no—
203#336—172no—
204#336—341no—
205#336—341no—
206C2×C17#146311yescharacteristic
207C2×C17#148711yescharacteristic
208C2×C17#149011yescharacteristic
209C2×C17C2×#32811yescharacteristic
210C2×C17#151011yescharacteristic
211C2×C17#152811yescharacteristic
212C2×C17#154911yescharacteristic
213D17—24no—
214C23×C4—171no—
215C23×C4—171no—
216C23×C4—171no—
217C23×C4—341no—
218C23×C4—341no—
219C2×M16—344no—
220C2×M16—682no—
221C2×M16—682no—
222C22×C8—342no—
223C22×C8—342no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—171no—
226C2×C4⋊C4—171no—
227C2×C4⋊C4—342no—
228C2×C4⋊C4—341no—
229C2×C4⋊C4—342no—
230C2×C4⋊C4—341no—
231C2×C4⋊C4—341no—
232C2×C4⋊C4—341no—
233C2×C4⋊C4—341no—
234C2×C4⋊C4—682no—
235C2×C4⋊C4—684no—
236C2×C4⋊C4C2×#59511yescharacteristic
237C2×C4⋊C4C2×#59511yescharacteristic
238C2×C42—171no—
239C2×C42—171no—
240C2×C42—171no—
241C2×C42—682no—
242C2×C42C2×#59511yescharacteristic
243C22⋊C8—682no—
244#95—21no—
245#95—341no—
246#95—341no—
247#95—341no—
248C17#760711yescharacteristic, complemented, Sylow
249C24—171no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—171no—
253C22×C4—171no—
254C22×C4—171no—
255C22×C4—172no—
256C22×C4—172no—
257C22×C4—342no—
258C22×C4—342no—
259C22×C4—342no—
260C22×C4—342no—
261C22×C4—342no—
262C22×C4—342no—
263C22×C4—344no—
264C22×C4—341no—
265C22×C4—341no—
266C22×C4—341no—
267C22×C4—341no—
268C22×C4—681no—
269C22×C4—682no—
270C22×C4—682no—
271C22×C4—682no—
272C22×C4—682no—
273C22×C4—682no—
274C22×C4C22⋊#59511yescharacteristic
275C22×C4C22⋊#59511yescharacteristic
276C22×C4C22×#59511yescharacteristic
277M16—688no—
278C2×C8—344no—
279C2×C8—682no—
280C2×C8—682no—
281C2×C8—682no—
282C2×C8—682no—
283C2×C8—682no—
284C4⋊C4—21no—
285C4⋊C4—41no—
286C4⋊C4—42no—
287C4⋊C4—172no—
288C4⋊C4—344no—
289C4⋊C4—341no—
290C4⋊C4—681no—
291C4⋊C4—682no—
292C4⋊C4—682no—
293C4⋊C4—682no—
294C4⋊C4—682no—
295C4⋊C4#6324312yescomplemented
296C42—41no—
297C42—681no—
298C42—681no—
299C42—681no—
300C23—172no—
301C23—172no—
302C23—172no—
303C23—341no—
304C23—341no—
305C23—341no—
306C23—341no—
307C23C2×C22⋊#59511yescharacteristic
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—172no—
322C2×C4—171no—
323C2×C4—171no—
324C2×C4—172no—
325C2×C4—344no—
326C2×C4—344no—
327C2×C4—344no—
328C2×C4—344no—
329C2×C4—341no—
330C2×C4—341no—
331C2×C4—341no—
332C2×C4—341no—
333C2×C4—341no—
334C2×C4—341no—
335C2×C4—342no—
336C2×C4—684no—
337C2×C4—688no—
338C2×C4—681no—
339C2×C4—681no—
340C2×C4—681no—
341C2×C4—681no—
342C2×C4—681no—
343C2×C4—1364no—
344C2×C4#1058754912yes—
345C2×C4C2×#6324311yescharacteristic
346C2×C4D17⋊M1611yescharacteristic
347C8—684no—
348C8—1362no—
349C22—174no—
350C22—342no—
351C22—342no—
352C22—342no—
353C22—342no—
354C22—342no—
355C22—342no—
356C22#1176378011yescharacteristic
357C22D17⋊#9911yescharacteristic
358C22D17⋊#10111yescharacteristic
359C22C2×#1058754911yescharacteristic
360C22#1176380111yescharacteristic
361C22D17⋊#10411yescharacteristic
362C22#1176380611yescharacteristic
363C4—22no—
364C4—21no—
365C4—21no—
366C4—41no—
367C4—42no—
368C4—81no—
369C4—341no—
370C4—341no—
371C4—342no—
372C4—342no—
373C4—682no—
374C4—1361no—
375C2—344no—
376C2#1454403311yescharacteristic
377C2#1454411311yescharacteristic
378C2#1454412411yescharacteristic
379C2D17⋊#32811yescharacteristic
380C2#1454419011yescharacteristic
381C2#1454424711yescharacteristic
382C2#1454430011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC17×#389C4
socleC23×C17C2×C22⋊C4
prime radicalC23×D17C22⋊C4
cosocleC2×C4×Dic17C23
hypercenter#389#595
hypocenterC17#7607
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
3. #14543603C2semidirect
4. #14543602C2semidirect
95. C4⋊C4×C17M16semidirect
100. C4⋊Dic17M16semidirect
248. C17#7607semidirect
295. C4⋊C4#63243semidirect

Maximal subgroups

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

subgroupindex
#145446762
#145444482
#145436032
#145436022
D17×#3892
#760717