Finite Group Databasebeta

← #16952621

Subgroups — #16952621

Order 4352.

540 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C2×#11762599C214yesmaximal
2#14528776C211yesmaximal, characteristic
3C4×#10587505C211yesmaximal, characteristic
4C17×#2486C211yesmaximal, characteristic
5#11762833—21no—
6C2×#10587504C2212yes—
7C2×#10587505C2212yes—
8C4×C22⋊Dic17—21no—
9#11762599—28no—
10#11762578—21no—
11#11762576—21no—
12C2×C17×#123C2212yes—
13C2×C4×D4×C17C2211yescharacteristic
14C22×C4⋊C4×C17C411yescharacteristic, complemented
15C17×#398—21no—
16C17×#397—21no—
17C17×#389—21no—
18C17×#381C412yescomplemented
19C17×#380—21no—
20C17×#380C411yescharacteristic, complemented
21C22×D4×C17C2311yescharacteristic
22C23×C4×C17C2×C412yes—
23C23×C4×C17C2×C411yescharacteristic
24C23×C4×C17D411yescharacteristic
25C17×#123—24no—
26C17×#123—24no—
27C4×D4×C17—41no—
28C4×D4×C17—41no—
29C2×C4⋊C4×C17—21no—
30C2×C4⋊C4×C17—21no—
31C2×C4⋊C4×C17—21no—
32C2×C4⋊C4×C17—22no—
33C2×C4⋊C4×C17—41no—
34C2×C4⋊C4×C17—41no—
35C2×C4⋊C4×C17D411yescharacteristic
36C2×C22⋊C4×C17—22no—
37C2×C22⋊C4×C17C2×C412yes—
38C2×C22⋊C4×C17C2×C411yescharacteristic
39C2×C22⋊C4×C17D411yescharacteristic
40C2×C42×C17D411yescharacteristic
41C2×C22⋊Dic17—22no—
42C2×C4×Dic17—41no—
43C2×#63223—22no—
44C17×#95—22no—
45C17×#95—21no—
46C17×#95—22no—
47#10587505—22no—
48#10587505D414yes—
49#10587504—24no—
50#10587502—41no—
51C4⋊#3622—41no—
52C4⋊#3622—41no—
53C24×C17C22×C411yescharacteristic
54C24×C17C2×D411yescharacteristic
55C2×D4×C17—41no—
56C2×D4×C17—41no—
57C2×D4×C17C2×D412yes—
58C2×D4×C17#4112yes—
59C22×C4×C17—21no—
60C22×C4×C17—21no—
61C22×C4×C17—21no—
62C22×C4×C17—21no—
63C22×C4×C17—22no—
64C22×C4×C17—22no—
65C22×C4×C17—24no—
66C22×C4×C17—42no—
67C22×C4×C17—42no—
68C22×C4×C17—42no—
69C22×C4×C17—42no—
70C22×C4×C17C22⋊C411yescharacteristic
71C22×C4×C17C22⋊C411yescharacteristic
72C22×C4×C17C22⋊C411yescharacteristic
73C22×C4×C17C22⋊C411yescharacteristic
74C22×C4×C17C2×D411yescharacteristic
75C22×Dic17—42no—
76C4⋊C4×C17—41no—
77C4⋊C4×C17—42no—
78C4⋊C4×C17—42no—
79C4⋊C4×C17—41no—
80C4⋊C4×C17—41no—
81C4⋊C4×C17—42no—
82C4⋊C4×C17—42no—
83C22⋊C4×C17—24no—
84C22⋊C4×C17—41no—
85C22⋊C4×C17—41no—
86C22⋊C4×C17—44no—
87C42×C17—21no—
88C42×C17—21no—
89C22⋊Dic17—24no—
90C22⋊Dic17—42no—
91C4×Dic17—82no—
92#63223—42no—
93#63223—42no—
94C2×#3622—42no—
95#13220—171nomaximal, Sylow
96C23×C17—21no—
97C23×C17—21no—
98C23×C17—21no—
99C23×C17—21no—
100C23×C17—41no—
101C23×C17—41no—
102C23×C17—41no—
103C23×C17—41no—
104C23×C17C2×C22⋊C411yescharacteristic
105C23×C17C4×D412yes—
106C23×C17#12112yes—
107D4×C17—42no—
108D4×C17—42no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—22no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—42no—
116C2×C4×C17—42no—
117C2×C4×C17—42no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17—42no—
122C2×C4×C17—42no—
123C2×C4×C17—42no—
124C2×C4×C17—42no—
125C2×C4×C17—42no—
126C2×C4×C17—42no—
127C2×C4×C17—81no—
128C2×C4×C17—84no—
129C2×C4×C17#9911yescharacteristic
130C2×C4×C17#9911yescharacteristic
131C2×C4×C17#12411yescharacteristic
132C2×C4×C17#12711yescharacteristic
133C2×Dic17—44no—
134C2×Dic17—82no—
135C2×Dic17—82no—
136#3622—82no—
137#2486D1711yescharacteristic
138C2×#352—174no—
139#1800—171no—
140C4×#99—171no—
141C22×C17—21no—
142C22×C17—21no—
143C22×C17—22no—
144C22×C17—22no—
145C22×C17—42no—
146C22×C17—42no—
147C22×C17—42no—
148C22×C17—41no—
149C22×C17—41no—
150C22×C17—41no—
151C22×C17—41no—
152C22×C17—81no—
153C22×C17#39011yescharacteristic
154C22×C17C2×#9911yescharacteristic
155C22×C17#41011yescharacteristic
156C4×C17—21no—
157C4×C17—21no—
158C4×C17—21no—
159C4×C17—41no—
160C4×C17—41no—
161C4×C17—41no—
162C4×C17—82no—
163C4×C17—82no—
164C4×C17#35212yescomplemented
165Dic17—84no—
166C2×#123C2×D1712yes—
167C2×C4×D4C2×D1711yescharacteristic
168C22×C4⋊C4Dic1711yescharacteristic, complemented
169#423—341no—
170C2×#100—172no—
171C2×#99—172no—
172#398—21no—
173#397—21no—
174#389—21no—
175#381Dic1712yescomplemented
176#380—21no—
177#380Dic1711yescharacteristic, complemented
178C4×C22⋊C4—341no—
179#352—348no—
180#343—341no—
181#342—341no—
182C2×C17—21no—
183C2×C17—21no—
184C2×C17—42no—
185C2×C17—42no—
186C2×C17#196411yescharacteristic
187C2×C17C2×#35211yescharacteristic
188C2×C17#212311yescharacteristic
189C22×D4C22×D1711yescharacteristic
190C23×C4C2×Dic1712yes—
191C23×C4C2×Dic1711yescharacteristic
192C23×C4C22⋊D1711yescharacteristic
193C2×M16—342no—
194#123—24no—
195#123—24no—
196C4×D4—41no—
197C4×D4—41no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4C22⋊D1711yescharacteristic
205C2×C22⋊C4—22no—
206C2×C22⋊C4—342no—
207C2×C22⋊C4C2×Dic1712yes—
208C2×C22⋊C4C2×Dic1711yescharacteristic
209C2×C22⋊C4C22⋊D1711yescharacteristic
210C2×C42—681no—
211C2×C42C22⋊D1711yescharacteristic
212C4⋊C8—681no—
213C4⋊C8—681no—
214#100—344no—
215#99—174no—
216#99—342no—
217#95—22no—
218#95—21no—
219#95—22no—
220#95—681no—
221C17#1322011yescharacteristic, complemented, Sylow
222C24C22×Dic1711yescharacteristic
223C24C2×C22⋊D1711yescharacteristic
224C2×D4—41no—
225C2×D4—41no—
226C2×D4D4×D1712yes—
227C2×D4#6325412yes—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—682no—
240C22×C4C22⋊Dic1711yescharacteristic
241C22×C4C22⋊Dic1711yescharacteristic
242C22×C4C22⋊Dic1711yescharacteristic
243C22×C4C22⋊Dic1711yescharacteristic
244C22×C4C2×C22⋊D1711yescharacteristic
245M16—682no—
246M16—682no—
247C2×C8—682no—
248C4⋊C4—41no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C22⋊C4—24no—
256C22⋊C4—41no—
257C22⋊C4—41no—
258C22⋊C4—44no—
259C22⋊C4—344no—
260C22⋊C4—682no—
261C42—21no—
262C42—21no—
263C42—1362no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23—41no—
269C23—41no—
270C23—41no—
271C23—41no—
272C23D4×Dic1712yes—
273C23#1058762412yes—
274C23C2×C22⋊Dic1711yescharacteristic
275D4—42no—
276D4—42no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—81no—
296C2×C4—84no—
297C2×C4—684no—
298C2×C4—1362no—
299C2×C4—1362no—
300C2×C4#1058750511yescharacteristic
301C2×C4#1058750511yescharacteristic
302C2×C4#1058762211yescharacteristic
303C2×C4#1058762611yescharacteristic
304C8—1362no—
305C22—21no—
306C22—21no—
307C22—22no—
308C22—22no—
309C22—42no—
310C22—42no—
311C22—42no—
312C22—41no—
313C22—41no—
314C22—41no—
315C22—41no—
316C22—81no—
317C22#1176271211yescharacteristic
318C22C2×#1058750511yescharacteristic
319C22#1176278311yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—41no—
324C4—41no—
325C4—41no—
326C4—82no—
327C4—82no—
328C4—1364no—
329C4#1176259912yescomplemented
330C2—21no—
331C2—21no—
332C2—42no—
333C2—42no—
334C2#1452962111yescharacteristic
335C2C2×#1176259911yescharacteristic
336C2#1453043811yescharacteristic
subgroupG/H
centerC22C2×#10587505
commutatorC24×C17C22×C4
FrattiniC22×D4C22×D17
FittingC17×#2486C2
socleC22×C17C2×#99
prime radical, cosocleC22×D4×C17C23
hypercenter#2486D17
hypocenterC17#13220
LCSG ▷ C24×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2486

Complements

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

NH ≅ G/Ntype
14. C22×C4⋊C4×C17C4semidirect
18. C17×#381C4semidirect
20. C17×#380C4semidirect
164. C4×C17#352semidirect
168. C22×C4⋊C4Dic17semidirect
175. #381Dic17semidirect
177. #380Dic17semidirect
221. C17#13220semidirect
329. C4#11762599semidirect

Maximal subgroups

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

subgroupindex
C2×#117625992
#145287762
C4×#105875052
C17×#24862
#1322017