Finite Group Databasebeta

← #17949818

Subgroups — #17949818

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—42no—
50C4⋊C4×D17—42no—
51C4⋊C4×D17—42no—
52C42×D17—42no—
53#10587543—42no—
54C17×#95—21no—
55#10587502—21no—
56#10587502—21no—
57#10587502—21no—
58C23×D17C22⋊C411yescharacteristic
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17C22⋊C411yescharacteristic
62C22×C4×C17C22⋊C411yescharacteristic
63C22×C4×C17C22×C411yescharacteristic
64C22×Dic17—21no—
65C22×Dic17—21no—
66C22×Dic17C22⋊C411yescharacteristic
67C22×Dic17C2×D411yescharacteristic
68C22×Dic17C2×D411yescharacteristic
69C2×C4×D17—22no—
70C2×C4×D17—22no—
71C2×C4×D17—22no—
72C2×C4×D17—21no—
73C2×C4×D17—21no—
74C2×C4×D17—22no—
75C2×C4×D17—24no—
76C2×C4×D17—22no—
77C2×C4×D17—22no—
78C2×C4×D17—41no—
79C2×C4×D17—42no—
80C2×C4×D17—42no—
81C2×C4×D17—42no—
82C2×C4×D17—42no—
83C2×C4×D17—42no—
84C2×C4×D17D812yes—
85C2×C4×D17SD1612yes—
86C2×#3624—24no—
87C2×#3624—42no—
88C2×#3624—42no—
89C2×#3624—42no—
90C2×#3624—42no—
91C2×#3624—42no—
92#63243—48no—
93C4⋊C4×C17—21no—
94C4⋊C4×C17—41no—
95C4⋊C4×C17—42no—
96C4⋊C4×C17M1612yescomplemented
97C42×C17—41no—
98C4⋊Dic17—21no—
99C4⋊Dic17—41no—
100C4⋊Dic17—41no—
101C4⋊Dic17—41no—
102C4⋊Dic17M1612yescomplemented
103#63225—24no—
104#63225—42no—
105#63225—42no—
106#63225—41no—
107#63225—41no—
108C4×Dic17—41no—
109C4×Dic17—41no—
110C4×Dic17—41no—
111#7610—171nomaximal, Sylow
112C23×C17C2×C22⋊C411yescharacteristic
113C22×D17—21no—
114C22×D17—21no—
115C22×D17—21no—
116C22×D17—21no—
117C22×D17#9912yes—
118C22×D17#10212yes—
119C22×D17#10412yes—
120C2×C4×C17—21no—
121C2×C4×C17—21no—
122C2×C4×C17—21no—
123C2×C4×C17—21no—
124C2×C4×C17—22no—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—41no—
129C2×C4×C17—41no—
130C2×C4×C17—41no—
131C2×C4×C17—41no—
132C2×C4×C17—41no—
133C2×C4×C17#10212yes—
134C2×C4×C17C2×M1611yescharacteristic
135C2×C4×C17C2×M1611yescharacteristic
136C2×Dic17—21no—
137C2×Dic17—21no—
138C2×Dic17—21no—
139C2×Dic17—21no—
140C2×Dic17—21no—
141C2×Dic17—21no—
142C2×Dic17—22no—
143C2×Dic17—41no—
144C2×Dic17—41no—
145C2×Dic17—41no—
146C2×Dic17—41no—
147C2×Dic17—41no—
148C2×Dic17#10112yes—
149C2×Dic17C2×D811yescharacteristic
150C2×Dic17C2×SD1611yescharacteristic
151C2×Dic17#13712yes—
152C4×D17—24no—
153C4×D17—24no—
154C4×D17—24no—
155C4×D17—24no—
156C4×D17—44no—
157C4×D17—48no—
158C4×D17—84no—
159#3624—44no—
160#3624—82no—
161C2×#389—171no—
162#1817—171no—
163#1728—171no—
164#1271—172no—
165#1271—172no—
166C22×C17#40711yescharacteristic
167C22×C17C2×#9911yescharacteristic
168C22×C17C2×#10111yescharacteristic
169C22×C17C2×#10211yescharacteristic
170C22×C17#41911yescharacteristic
171C22×C17C2×#10411yescharacteristic
172C22×C17#42111yescharacteristic
173C2×D17—22no—
174C2×D17—22no—
175C2×D17—22no—
176C2×D17—22no—
177C2×D17—22no—
178C2×D17—22no—
179C2×D17#32714yes—
180C4×C17—21no—
181C4×C17—21no—
182C4×C17—22no—
183C4×C17—42no—
184C4×C17—41no—
185C4×C17—81no—
186Dic17—22no—
187Dic17—21no—
188Dic17—21no—
189Dic17—22no—
190Dic17—42no—
191Dic17—81no—
192C22×M16—341no—
193C22×C4⋊C4—171no—
194C22×C4⋊C4—171no—
195C22×C4⋊C4—341no—
196C22×C42—171no—
197C2×C22⋊C8—341no—
198#389—171no—
199#389—171no—
200#389—171no—
201#389—341no—
202#389—341no—
203#389#59511yescharacteristic
204C2×#95—341no—
205#336—172no—
206#336—172no—
207#336—341no—
208#336—341no—
209C2×C17#146311yescharacteristic
210C2×C17#148611yescharacteristic
211C2×C17#149111yescharacteristic
212C2×C17C2×#32711yescharacteristic
213C2×C17#151111yescharacteristic
214C2×C17#152811yescharacteristic
215C2×C17#154611yescharacteristic
216D17—24no—
217C23×C4—171no—
218C23×C4—171no—
219C23×C4—171no—
220C23×C4—341no—
221C23×C4—341no—
222C2×M16—344no—
223C2×M16—682no—
224C2×M16—682no—
225C22×C8—342no—
226C22×C8—342no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—171no—
229C2×C4⋊C4—171no—
230C2×C4⋊C4—342no—
231C2×C4⋊C4—341no—
232C2×C4⋊C4—342no—
233C2×C4⋊C4—341no—
234C2×C4⋊C4—341no—
235C2×C4⋊C4—341no—
236C2×C4⋊C4—341no—
237C2×C4⋊C4—682no—
238C2×C4⋊C4—682no—
239C2×C4⋊C4—682no—
240C2×C4⋊C4C2×#59511yescharacteristic
241C2×C4⋊C4C2×#59511yescharacteristic
242C2×C42—171no—
243C2×C42—171no—
244C2×C42—171no—
245C2×C42—682no—
246C2×C42C2×#59511yescharacteristic
247C22⋊C8—682no—
248#95—21no—
249#95—341no—
250#95—341no—
251#95—341no—
252C17#761011yescharacteristic, complemented, Sylow
253C24—171no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—171no—
257C22×C4—171no—
258C22×C4—171no—
259C22×C4—172no—
260C22×C4—172no—
261C22×C4—342no—
262C22×C4—342no—
263C22×C4—342no—
264C22×C4—342no—
265C22×C4—342no—
266C22×C4—342no—
267C22×C4—344no—
268C22×C4—341no—
269C22×C4—341no—
270C22×C4—341no—
271C22×C4—341no—
272C22×C4—681no—
273C22×C4—682no—
274C22×C4—682no—
275C22×C4—682no—
276C22×C4—682no—
277C22×C4—682no—
278C22×C4C22⋊#59511yescharacteristic
279C22×C4C22⋊#59511yescharacteristic
280C22×C4C22×#59511yescharacteristic
281M16—688no—
282C2×C8—344no—
283C2×C8—682no—
284C2×C8—682no—
285C2×C8—682no—
286C2×C8—682no—
287C2×C8—682no—
288C4⋊C4—21no—
289C4⋊C4—41no—
290C4⋊C4—42no—
291C4⋊C4—172no—
292C4⋊C4—344no—
293C4⋊C4—341no—
294C4⋊C4—681no—
295C4⋊C4—681no—
296C4⋊C4—681no—
297C4⋊C4—681no—
298C4⋊C4—681no—
299C4⋊C4—682no—
300C4⋊C4—682no—
301C4⋊C4#6324312yescomplemented
302C42—41no—
303C42—681no—
304C42—681no—
305C42—681no—
306C23—172no—
307C23—172no—
308C23—172no—
309C23—341no—
310C23—341no—
311C23—341no—
312C23—341no—
313C23C2×C22⋊#59511yescharacteristic
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—172no—
328C2×C4—171no—
329C2×C4—171no—
330C2×C4—172no—
331C2×C4—344no—
332C2×C4—344no—
333C2×C4—344no—
334C2×C4—344no—
335C2×C4—341no—
336C2×C4—341no—
337C2×C4—341no—
338C2×C4—341no—
339C2×C4—341no—
340C2×C4—341no—
341C2×C4—342no—
342C2×C4—684no—
343C2×C4—688no—
344C2×C4—681no—
345C2×C4—681no—
346C2×C4—681no—
347C2×C4—681no—
348C2×C4—681no—
349C2×C4—1364no—
350C2×C4#1058754712yes—
351C2×C4C2×#6324311yescharacteristic
352C2×C4D17⋊M1611yescharacteristic
353C8—684no—
354C8—1362no—
355C22—174no—
356C22—342no—
357C22—342no—
358C22—342no—
359C22—342no—
360C22—342no—
361C22—342no—
362C22#1176378011yescharacteristic
363C22D17⋊#9911yescharacteristic
364C22D17⋊#10111yescharacteristic
365C22C2×#1058754711yescharacteristic
366C22#1176380211yescharacteristic
367C22D17⋊#10411yescharacteristic
368C22#1176380611yescharacteristic
369C4—22no—
370C4—21no—
371C4—21no—
372C4—41no—
373C4—42no—
374C4—81no—
375C4—341no—
376C4—341no—
377C4—342no—
378C4—342no—
379C4—682no—
380C4—1361no—
381C2—344no—
382C2#1454403311yescharacteristic
383C2#1454411011yescharacteristic
384C2#1454412711yescharacteristic
385C2D17⋊#32711yescharacteristic
386C2#1454419311yescharacteristic
387C2#1454424711yescharacteristic
388C2#1454429411yescharacteristic
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#7610
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
96. C4⋊C4×C17M16semidirect
102. C4⋊Dic17M16semidirect
252. C17#7610semidirect
301. 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
#761017