Finite Group Databasebeta

← #16932091

Subgroups — #16932091

Order 4352.

578 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 131 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#14537808C211yesmaximal, characteristic
2#14533101C211yesmaximal, characteristic
3#14528872C212yesmaximal, complemented
4#14526651C212yesmaximal, complemented
5C17×#1799C211yesmaximal, characteristic
6C22×C4⋊Dic17C2211yescharacteristic
7D4×#3622—24no—
8#11762862—21no—
9C2×C22⋊#3622—21no—
10C2×C22⋊#3622C2212yes—
11#11762748C412yescomplemented
12#11762745C412yescomplemented
13#11762707—21no—
14#11762689—21no—
15#11762681—21no—
16#11762669—21no—
17#11762669C2211yescharacteristic
18#11762571—24no—
19C2×C4×D4×C17C2211yescharacteristic
20C2×C17×#99—21no—
21C17×#380C2212yes—
22C17×#342—21no—
23C22×D4×C17D411yescharacteristic
24C23×C4×C17D411yescharacteristic
25C23×C4×C17C2311yescharacteristic
26C23×Dic17—21no—
27C4×D4×C17—22no—
28C4×D4×C17—22no—
29C2×C4⋊C4×C17—22no—
30C2×C4⋊C4×C17C2×C411yescharacteristic
31C2×C22⋊C4×C17—22no—
32C2×C22⋊C4×C17C2×C411yescharacteristic
33C2×C22⋊C4×C17D411yescharacteristic
34C2×C42×C17D411yescharacteristic
35C2×C22⋊Dic17—21no—
36C2×C22⋊Dic17—22no—
37C2×C4⋊Dic17—21no—
38C2×C4⋊Dic17—21no—
39C2×C4⋊Dic17—22no—
40C2×C4⋊Dic17—22no—
41C2×C4⋊Dic17C2×C412yes—
42C2×#63225—22no—
43C22×#3622—22no—
44C22×#3622D414yes—
45C17×#99—42no—
46C17×#95—22no—
47#10587502—22no—
48#10587502—22no—
49#10587502C2×C412yes—
50C22⋊#3622—22no—
51C22⋊#3622—22no—
52C22⋊#3622—22no—
53C4⋊#3622—22no—
54C4×#3622—22no—
55C24×C17C22⋊C411yescharacteristic
56C24×C17C2×D411yescharacteristic
57C2×D4×C17—22no—
58C2×D4×C17—22no—
59C2×D4×C17SD1612yes—
60C2×D4×C17Q1612yes—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—42no—
64C22×C4×C17—42no—
65C22×C4×C17—42no—
66C22×C4×C17C22⋊C411yescharacteristic
67C22×C4×C17C22⋊C411yescharacteristic
68C22×C4×C17C22⋊C411yescharacteristic
69C22×C4×C17C22×C411yescharacteristic
70C22×C4×C17C2×D411yescharacteristic
71C22×C4×C17C2×D412yes—
72C22×C4×C17#4112yes—
73C22×Dic17—22no—
74C22×Dic17—22no—
75C22×Dic17—42no—
76C22×Dic17—82no—
77C4⋊C4×C17—22no—
78C4⋊C4×C17—44no—
79C22⋊C4×C17—22no—
80C22⋊C4×C17—22no—
81C22⋊C4×C17—42no—
82C22⋊C4×C17—42no—
83C42×C17—21no—
84C42×C17—21no—
85C22⋊Dic17—42no—
86C22⋊Dic17—82no—
87C4⋊Dic17—24no—
88C4⋊Dic17—42no—
89C4⋊Dic17—41no—
90C4⋊Dic17—41no—
91#63225—82no—
92C2×#3622—24no—
93C2×#3622—44no—
94C2×#3622—82no—
95#9469—171nomaximal, Sylow
96C23×C17—21no—
97C23×C17—21no—
98C23×C17—41no—
99C23×C17—41no—
100C23×C17—41no—
101C23×C17—41no—
102C23×C17#10312yes—
103C23×C17C2×C22⋊C411yescharacteristic
104C23×C17#12112yes—
105C23×C17C2×SD1611yescharacteristic
106C23×C17C2×Q1611yescharacteristic
107C23×C17#13512yes—
108D4×C17—24no—
109D4×C17—24no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—24no—
113C2×C4×C17—22no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—41no—
117C2×C4×C17—42no—
118C2×C4×C17—42no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17—81no—
122C2×C4×C17—82no—
123C2×C4×C17—82no—
124C2×C4×C17—82no—
125C2×C4×C17#10312yes—
126C2×C4×C17C4×D412yes—
127C2×C4×C17#12011yescharacteristic
128C2×C4×C17#12311yescharacteristic
129C2×Dic17—42no—
130C2×Dic17—42no—
131C2×Dic17—42no—
132C2×Dic17—42no—
133C2×Dic17—42no—
134C2×Dic17—42no—
135C2×Dic17—84no—
136#3622—44no—
137#2972—171no—
138#2487—171no—
139#1824—172no—
140#1799D1711yescharacteristic
141#1289—172no—
142C22×C17—21no—
143C22×C17—22no—
144C22×C17—21no—
145C22×C17—22no—
146C22×C17—22no—
147C22×C17—41no—
148C22×C17—41no—
149C22×C17—42no—
150C22×C17—42no—
151C22×C17—41no—
152C22×C17—41no—
153C22×C17—81no—
154C22×C17#38611yescharacteristic
155C22×C17C2×#10311yescharacteristic
156C22×C17#41611yescharacteristic
157C22×C17#46512yes—
158C22×C17#46712yes—
159C4×C17—21no—
160C4×C17—22no—
161C4×C17—22no—
162C4×C17—41no—
163C4×C17—41no—
164C4×C17—162no—
165C4×C17#45812yes—
166Dic17—82no—
167Dic17—82no—
168C2×C4×D4C2×D1711yescharacteristic
169C22×C4⋊C4—171no—
170C8×D4—344no—
171#431—341no—
172C2×#99—21no—
173C2×C22⋊C8—172no—
174C2×C22⋊C8—341no—
175#400—172no—
176#399—172no—
177#389—341no—
178#386—341no—
179#385—341no—
180#380—171no—
181#380—341no—
182#380C2×D1712yes—
183#342—21no—
184#340—344no—
185C2×C17—21no—
186C2×C17—22no—
187C2×C17—21no—
188C2×C17—22no—
189C2×C17—42no—
190C2×C17#189011yescharacteristic
191C2×C17#189211yescharacteristic
192C2×C17#191011yescharacteristic
193C22×D4C22⋊D1711yescharacteristic
194C23×C4—341no—
195C23×C4D6811yescharacteristic
196C23×C4C22×D1711yescharacteristic
197C22×C8—174no—
198C22×C8—342no—
199C4×D4—22no—
200C4×D4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—172no—
203C2×C4⋊C4—342no—
204C2×C4⋊C4—342no—
205C2×C4⋊C4—341no—
206C2×C4⋊C4—341no—
207C2×C4⋊C4—342no—
208C2×C4⋊C4C4×D1711yescharacteristic
209C2×C22⋊C4—22no—
210C2×C22⋊C4—341no—
211C2×C22⋊C4—342no—
212C2×C22⋊C4C4×D1711yescharacteristic
213C2×C22⋊C4C22⋊D1711yescharacteristic
214C2×C42D6811yescharacteristic
215C4⋊C8—342no—
216#99—42no—
217C22⋊C8—342no—
218C22⋊C8—342no—
219C22⋊C8—342no—
220C4×C8—342no—
221#95—22no—
222#95—172no—
223#95—342no—
224#95—342no—
225C17#946911yescharacteristic, complemented, Sylow
226C24#6322711yescharacteristic
227C24C2×C22⋊D1711yescharacteristic
228C2×D4—22no—
229C2×D4—22no—
230C2×D4#6322912yes—
231C2×D4#6323112yes—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—342no—
238C22×C4—342no—
239C22×C4—682no—
240C22×C4—1362no—
241C22×C4#6322711yescharacteristic
242C22×C4#6322711yescharacteristic
243C22×C4#6322711yescharacteristic
244C22×C4C2×C4×D1711yescharacteristic
245C22×C4C2×D6811yescharacteristic
246C22×C4D4×D1712yes—
247C22×C4#6325412yes—
248C2×C8—344no—
249C2×C8—684no—
250C2×C8—1362no—
251C4⋊C4—22no—
252C4⋊C4—44no—
253C4⋊C4—344no—
254C4⋊C4—682no—
255C4⋊C4—681no—
256C4⋊C4—681no—
257C4⋊C4—1362no—
258C22⋊C4—22no—
259C22⋊C4—22no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—682no—
263C22⋊C4—1362no—
264C42—21no—
265C42—21no—
266C23—21no—
267C23—21no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23#1058748112yes—
273C23C2×#6322711yescharacteristic
274C23C2×#6322911yescharacteristic
275C23#1058762512yes—
276C23C2×#6323111yescharacteristic
277C23#1058763612yes—
278D4—24no—
279D4—24no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—81no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—682no—
296C2×C4—682no—
297C2×C4—682no—
298C2×C4—682no—
299C2×C4—682no—
300C2×C4—682no—
301C2×C4—1364no—
302C2×C4#1058748112yes—
303C2×C4#1058756712yes—
304C2×C4#1058756811yescharacteristic
305C2×C4#1058757211yescharacteristic
306C8—684no—
307C22—21no—
308C22—22no—
309C22—21no—
310C22—22no—
311C22—22no—
312C22—41no—
313C22—41no—
314C22—42no—
315C22—42no—
316C22—41no—
317C22—41no—
318C22—81no—
319C22#1176268611yescharacteristic
320C22C2×#1058748111yescharacteristic
321C22#1176280311yescharacteristic
322C22#1176303612yes—
323C22#1176304812yes—
324C4—21no—
325C4—22no—
326C4—22no—
327C4—41no—
328C4—41no—
329C4—162no—
330C4—1362no—
331C4—1362no—
332C4#1176300012yes—
333C2—22no—
334C2—21no—
335C2—21no—
336C2—22no—
337C2—42no—
338C2#1452919511yescharacteristic
339C2#1452920511yescharacteristic
340C2#1452930911yescharacteristic
subgroupG/H
centerC22#11762686
commutatorC22×C4×C17C22×C4
FrattiniC23×C4C22×D17
FittingC17×#1799C2
socleC22×C17#386
prime radicalC22×D4×C17D4
cosocleC23×C4×C17C23
hypercenter#1799D17
hypocenterC17#9469
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
3. #14528872C2semidirect
4. #14526651C2semidirect
11. #11762748C4semidirect
12. #11762745C4semidirect
225. C17#9469semidirect

Maximal subgroups

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

subgroupindex
#145378082
#145331012
#145288722
#145266512
C17×#17992
#946917