Finite Group Databasebeta

← #17953554

Subgroups — #17953554

Order 4352.

496 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 116 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#14544638C211yesmaximal, characteristic
2C4×C22⋊#3624C211yesmaximal, characteristic
3#14543620C212yesmaximal, complemented
4#14543620C212yesmaximal, complemented
5#14531179C211yesmaximal, characteristic
6C2×C4⋊#3624C2212yes—
7C2×C22⋊#3624—21no—
8C2×C22⋊#3624—21no—
9C2×C4×#3624C2212yes—
10#11763675—21no—
11#11763675—21no—
12C22×C4×Dic17C2211yescharacteristic
13#11762676—21no—
14#11762674C411yescharacteristic
15#11762674C411yescharacteristic
16#11762670—21no—
17#11762669C411yescharacteristic
18C4⋊C4×Dic17C2211yescharacteristic
19C4⋊C4×Dic17C2211yescharacteristic
20C22⋊C4×Dic17—21no—
21#11762654—21no—
22C17×#380C411yescharacteristic
23C23×C4×C17C2×C411yescharacteristic
24C23×Dic17D411yescharacteristic
25C23×Dic17D411yescharacteristic
26C22×#3624—22no—
27C22×#3624—22no—
28C2×C4⋊C4×C17C2×C411yescharacteristic
29C2×C4⋊C4×C17C2×C411yescharacteristic
30C2×C22⋊C4×C17—21no—
31C2×C22⋊Dic17—21no—
32C2×C4⋊Dic17C2×C411yescharacteristic
33C2×C4⋊Dic17C2×C411yescharacteristic
34C2×C4×Dic17—24no—
35C2×C4×Dic17—21no—
36C2×C4×Dic17—21no—
37C2×C4×Dic17C2×C411yescharacteristic
38C2×C4×Dic17D411yescharacteristic
39C2×C4×Dic17D411yescharacteristic
40C2×C4×Dic17C2311yescharacteristic
41C22⋊#3624—24no—
42C22⋊#3624—42no—
43C4⋊#3624—24no—
44C4×#3624—24no—
45C17×#95—21no—
46#10587502—21no—
47#10587502—21no—
48#10587502—21no—
49#10587502—21no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53C24×C17C22⋊C411yescharacteristic
54C22×C4×C17—22no—
55C22×C4×C17—22no—
56C22×C4×C17—22no—
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17C22⋊C411yescharacteristic
60C22×C4×C17C22×C411yescharacteristic
61C22×Dic17—22no—
62C22×Dic17—22no—
63C22×Dic17—22no—
64C22×Dic17—22no—
65C22×Dic17—21no—
66C22×Dic17—21no—
67C22×Dic17—21no—
68C22×Dic17—21no—
69C22×Dic17—41no—
70C22×Dic17C22⋊C411yescharacteristic
71C22×Dic17C22⋊C411yescharacteristic
72C22×Dic17C2×D411yescharacteristic
73C22×Dic17C2×D411yescharacteristic
74C2×#3624—24no—
75C2×#3624—42no—
76C2×#3624—42no—
77C2×#3624—42no—
78C2×#3624—42no—
79C2×#3624—42no—
80C4⋊C4×C17—22no—
81C4⋊C4×C17—22no—
82C22⋊C4×C17—42no—
83C22⋊Dic17—42no—
84C4⋊Dic17—22no—
85C4⋊Dic17—22no—
86C4×Dic17—22no—
87C4×Dic17—22no—
88C4×Dic17—22no—
89C4×Dic17—42no—
90C4×Dic17—41no—
91C4×Dic17—41no—
92C4×Dic17#4114yes—
93#8711—171nomaximal, Sylow
94C23×C17—21no—
95C23×C17—21no—
96C23×C17—21no—
97C23×C17—21no—
98C23×C17—21no—
99C23×C17—21no—
100C23×C17#10412yes—
101C23×C17C2×C22⋊C411yescharacteristic
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—44no—
106C2×C4×C17—41no—
107C2×C4×C17—41no—
108C2×C4×C17—41no—
109C2×C4×C17—41no—
110C2×C4×C17—41no—
111C2×C4×C17—41no—
112C2×C4×C17#11712yes—
113C2×C4×C17#13112yes—
114C2×C4×C17#13112yes—
115C2×Dic17—21no—
116C2×Dic17—21no—
117C2×Dic17—21no—
118C2×Dic17—21no—
119C2×Dic17—21no—
120C2×Dic17—21no—
121C2×Dic17—21no—
122C2×Dic17—21no—
123C2×Dic17—44no—
124C2×Dic17—42no—
125C2×Dic17—42no—
126C2×Dic17—41no—
127C2×Dic17—41no—
128C2×Dic17—41no—
129C2×Dic17—41no—
130C2×Dic17—41no—
131C2×Dic17—41no—
132C2×Dic17#12312yes—
133C2×Dic17#12312yes—
134C2×Dic17#13512yes—
135C2×Dic17AGL(1,ℤ/8)11yescharacteristic
136C2×Dic17#13711yescharacteristic
137#3624—44no—
138#3624—82no—
139#2283—171no—
140#1797—171no—
141C4×C22⋊C8—171no—
142#1279—172no—
143#1279—172no—
144C22×C17—22no—
145C22×C17—22no—
146C22×C17—22no—
147C22×C17—41no—
148C22×C17—41no—
149C22×C17—41no—
150C22×C17—41no—
151C22×C17#38111yescharacteristic
152C22×C17#40811yescharacteristic
153C22×C17#40811yescharacteristic
154C22×C17#41611yescharacteristic
155C22×C17#41711yescharacteristic
156C22×C17C2×#10411yescharacteristic
157C22×C17#42111yescharacteristic
158C4×C17—41no—
159C4×C17—81no—
160C4×C17—81no—
161C4×C17#43314yes—
162Dic17—22no—
163Dic17—41no—
164Dic17—41no—
165Dic17—81no—
166Dic17—81no—
167Dic17#48212yes—
168Dic17#48512yes—
169C22×C42—171no—
170C2×C4⋊C8—172no—
171C2×C22⋊C8—341no—
172C2×C22⋊C8—341no—
173C2×C4×C8—172no—
174#383—341no—
175#382—171no—
176#382—171no—
177#381—341no—
178#380—171no—
179#380#59511yescharacteristic
180C4×C4⋊C4—171no—
181C4×C4⋊C4—171no—
182C4×C22⋊C4—341no—
183#376—341no—
184#336—341no—
185#336—341no—
186C2×C17—42no—
187C2×C17#148211yescharacteristic
188C2×C17#148311yescharacteristic
189C2×C17#149411yescharacteristic
190C2×C17#149511yescharacteristic
191C2×C17#181811yescharacteristic
192C2×C17#182511yescharacteristic
193C2×C17#182611yescharacteristic
194C23×C4—171no—
195C23×C4—171no—
196C23×C4C2×#59511yescharacteristic
197C22×C8—342no—
198C22×C8—342no—
199C2×C4⋊C4—171no—
200C2×C4⋊C4—171no—
201C2×C4⋊C4C2×#59511yescharacteristic
202C2×C4⋊C4C2×#59511yescharacteristic
203C2×C22⋊C4—21no—
204C2×C22⋊C4—341no—
205C2×C42—171no—
206C2×C42—171no—
207C2×C42—171no—
208C2×C42—171no—
209C2×C42—344no—
210C2×C42—341no—
211C2×C42—341no—
212C4⋊C8—344no—
213C22⋊C8—344no—
214C22⋊C8—682no—
215C4×C8—344no—
216#95—21no—
217#95—341no—
218#95—341no—
219#95—341no—
220#95—341no—
221#95—341no—
222#95—341no—
223#95—341no—
224C17#871111yescharacteristic, complemented, Sylow
225C24C22⋊#59511yescharacteristic
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—171no—
232C22×C4—171no—
233C22×C4—171no—
234C22×C4—171no—
235C22×C4—341no—
236C22×C4—342no—
237C22×C4—342no—
238C22×C4—342no—
239C22×C4—341no—
240C22×C4—342no—
241C22×C4—341no—
242C22×C4—341no—
243C22×C4—681no—
244C22×C4C22⋊#59511yescharacteristic
245C22×C4C22×#59511yescharacteristic
246C2×C8—344no—
247C2×C8—682no—
248C2×C8—682no—
249C2×C8—682no—
250C2×C8—682no—
251C2×C8—682no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—342no—
255C4⋊C4—342no—
256C22⋊C4—42no—
257C22⋊C4—682no—
258C42—174no—
259C42—342no—
260C42—342no—
261C42—342no—
262C42—682no—
263C42—681no—
264C42—681no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23#1058755012yes—
272C23C2×C22⋊#59511yescharacteristic
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—44no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—172no—
284C2×C4—172no—
285C2×C4—171no—
286C2×C4—171no—
287C2×C4—172no—
288C2×C4—341no—
289C2×C4—341no—
290C2×C4—341no—
291C2×C4—341no—
292C2×C4—341no—
293C2×C4—341no—
294C2×C4—341no—
295C2×C4—341no—
296C2×C4—682no—
297C2×C4—682no—
298C2×C4—684no—
299C2×C4—681no—
300C2×C4—681no—
301C2×C4—681no—
302C2×C4—681no—
303C2×C4—681no—
304C2×C4—681no—
305C2×C4#1058767712yes—
306C2×C4#1058767812yes—
307C2×C4#1058767812yes—
308C8—684no—
309C8—1362no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—41no—
317C22#1176375211yescharacteristic
318C22#1176378211yescharacteristic
319C22#1176378211yescharacteristic
320C22#1176379911yescharacteristic
321C22#1176380011yescharacteristic
322C22C2×#1058755011yescharacteristic
323C22#1176380711yescharacteristic
324C4—41no—
325C4—81no—
326C4—81no—
327C4—172no—
328C4—172no—
329C4—342no—
330C4—681no—
331C4—681no—
332C4—1361no—
333C4—1361no—
334C4#1176383314yes—
335C2—42no—
336C2#1454409511yescharacteristic
337C2#1454410111yescharacteristic
338C2#1454413711yescharacteristic
339C2#1454413911yescharacteristic
340C2#1454467711yescharacteristic
341C2#1454468911yescharacteristic
342C2#1454469111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#595
commutatorC22×C4×C17C22×C4
FrattiniC22×C4C22×#595
FittingC17×#380C4
socleC23×C17C2×C22⋊C4
prime radicalC24×C17C22⋊C4
cosocleC2×C4×Dic17C23
hypercenter#380#595
hypocenterC17#8711
LCSG ▷ C22×C4×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #14543620C2semidirect
4. #14543620C2semidirect
224. C17#8711semidirect

Maximal subgroups

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

subgroupindex
#145446382
C4×C22⋊#36242
#145436202
#145311792
#871117