Finite Group Databasebeta

← #17953599

Subgroups — #17953599

Order 4352.

514 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 114 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#14544640C211yesmaximal, characteristic
2#14544640C211yesmaximal, characteristic
3#14543662C212yesmaximal, complemented
4#14543662C212yesmaximal, complemented
5#14531186C211yesmaximal, characteristic
6C2×#10587544C2212yes—
7C2×#10587544C2212yes—
8C2×C22⋊#3624—21no—
9C2×C22⋊#3624—21no—
10#11763675—21no—
11#11763675—21no—
12C2×#10587563—21no—
13C22×C4×Dic17C2211yescharacteristic
14#11762678C2211yescharacteristic
15#11762678C2211yescharacteristic
16#11762673—21no—
17#11762669C411yescharacteristic
18#11762667—21no—
19C4⋊C4×Dic17C411yescharacteristic
20C4⋊C4×Dic17C411yescharacteristic
21#11762660—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×#63225—21no—
35C2×#63225—21no—
36C2×#63225—21no—
37C2×#63225—21no—
38C2×C4×Dic17—24no—
39C2×C4×Dic17—21no—
40C2×C4×Dic17—21no—
41C2×C4×Dic17C2×C411yescharacteristic
42C2×C4×Dic17D411yescharacteristic
43C2×C4×Dic17D411yescharacteristic
44C2×C4×Dic17C2311yescharacteristic
45#10587563—42no—
46#10587563—42no—
47C22⋊#3624—42no—
48C22⋊#3624—42no—
49#10587544—24no—
50#10587544—24no—
51C17×#95—21no—
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55C24×C17C22⋊C411yescharacteristic
56C22×C4×C17—22no—
57C22×C4×C17—21no—
58C22×C4×C17—21no—
59C22×C4×C17—41no—
60C22×C4×C17—41no—
61C22×C4×C17C22⋊C411yescharacteristic
62C22×C4×C17C22×C411yescharacteristic
63C22×Dic17—22no—
64C22×Dic17—22no—
65C22×Dic17—22no—
66C22×Dic17—22no—
67C22×Dic17—21no—
68C22×Dic17—21no—
69C22×Dic17—21no—
70C22×Dic17—21no—
71C22×Dic17C22⋊C411yescharacteristic
72C22×Dic17C22⋊C411yescharacteristic
73C22×Dic17D812yes—
74C22×Dic17SD1612yes—
75C22×Dic17C2×D411yescharacteristic
76C22×Dic17C2×D411yescharacteristic
77C2×#3624—42no—
78C2×#3624—42no—
79C2×#3624—42no—
80C2×#3624—42no—
81C2×#3624—42no—
82C2×#3624—42no—
83C4⋊C4×C17—41no—
84C4⋊C4×C17—41no—
85C22⋊C4×C17—42no—
86C22⋊Dic17—42no—
87C4⋊Dic17—41no—
88C4⋊Dic17—41no—
89#63225—41no—
90#63225—41no—
91#63225—41no—
92#63225—41no—
93#63225—42no—
94#63225—41no—
95#63225—41no—
96C4×Dic17—22no—
97C4×Dic17—22no—
98C4×Dic17—22no—
99C4×Dic17—42no—
100C4×Dic17—42no—
101C4×Dic17#4114yes—
102#8726—171nomaximal, Sylow
103C23×C17—21no—
104C23×C17—21no—
105C23×C17—21no—
106C23×C17—21no—
107C23×C17—21no—
108C23×C17—21no—
109C23×C17#10212yes—
110C23×C17C2×C22⋊C411yescharacteristic
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—44no—
117C2×C4×C17—41no—
118C2×C4×C17—41no—
119C2×C4×C17—41no—
120C2×C4×C17—41no—
121C2×C4×C17—41no—
122C2×C4×C17—41no—
123C2×C4×C17#11712yes—
124C2×Dic17—21no—
125C2×Dic17—21no—
126C2×Dic17—24no—
127C2×Dic17—24no—
128C2×Dic17—21no—
129C2×Dic17—21no—
130C2×Dic17—21no—
131C2×Dic17—42no—
132C2×Dic17—42no—
133C2×Dic17—41no—
134C2×Dic17—41no—
135C2×Dic17—41no—
136C2×Dic17—41no—
137C2×Dic17—41no—
138C2×Dic17—41no—
139C2×Dic17#10212yes—
140C2×Dic17#12312yes—
141C2×Dic17#12312yes—
142C2×Dic17C2×D811yescharacteristic
143C2×Dic17C2×SD1611yescharacteristic
144C2×Dic17#13512yes—
145C2×Dic17AGL(1,ℤ/8)11yescharacteristic
146C2×Dic17#13712yes—
147C2×Dic17#13711yescharacteristic
148#3624—82no—
149#3624—82no—
150#2285—171no—
151#1797—171no—
152#1797—171no—
153#1296—172no—
154#1296—172no—
155C22×C17—22no—
156C22×C17—22no—
157C22×C17—22no—
158C22×C17—41no—
159C22×C17—41no—
160C22×C17—41no—
161C22×C17—41no—
162C22×C17#38111yescharacteristic
163C22×C17#41311yescharacteristic
164C22×C17#41311yescharacteristic
165C22×C17C2×#10211yescharacteristic
166C22×C17#41611yescharacteristic
167C22×C17#41711yescharacteristic
168C22×C17#41911yescharacteristic
169C4×C17—22no—
170C4×C17—41no—
171C4×C17—81no—
172C4×C17—81no—
173Dic17—22no—
174Dic17—41no—
175Dic17—81no—
176Dic17—81no—
177Dic17#48012yes—
178Dic17#48212yes—
179Dic17#48312yes—
180Dic17#48512yes—
181C2×#117—341no—
182C22×C42—171no—
183C2×C4⋊C8—172no—
184C2×C4⋊C8—172no—
185C2×C22⋊C8—341no—
186C2×C22⋊C8—341no—
187#384—171no—
188#384—171no—
189#382—341no—
190#380—171no—
191#380—341no—
192#380#59511yescharacteristic
193C4×C4⋊C4—171no—
194C4×C4⋊C4—171no—
195C4×C4⋊C4—341no—
196#336—341no—
197#336—341no—
198C2×C17—42no—
199C2×C17#154011yescharacteristic
200C2×C17#154711yescharacteristic
201C2×C17#155411yescharacteristic
202C2×C17#155911yescharacteristic
203C2×C17#182211yescharacteristic
204C2×C17#182311yescharacteristic
205C2×C17#182511yescharacteristic
206C23×C4—171no—
207C23×C4—171no—
208C23×C4C2×#59511yescharacteristic
209C22×C8—342no—
210C22×C8—342no—
211#117—682no—
212#117—682no—
213C2×C4⋊C4—171no—
214C2×C4⋊C4—171no—
215C2×C4⋊C4—341no—
216C2×C4⋊C4—341no—
217C2×C4⋊C4—341no—
218C2×C4⋊C4—341no—
219C2×C4⋊C4C2×#59511yescharacteristic
220C2×C4⋊C4C2×#59511yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—341no—
223C2×C42—171no—
224C2×C42—171no—
225C2×C42—171no—
226C2×C42—171no—
227C2×C42—344no—
228C2×C42—341no—
229C2×C42—341no—
230C4⋊C8—344no—
231C4⋊C8—344no—
232C22⋊C8—682no—
233C22⋊C8—682no—
234#95—21no—
235#95—341no—
236#95—341no—
237#95—341no—
238C17#872611yescharacteristic, complemented, Sylow
239C24C22⋊#59511yescharacteristic
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—171no—
246C22×C4—171no—
247C22×C4—171no—
248C22×C4—172no—
249C22×C4—172no—
250C22×C4—171no—
251C22×C4—342no—
252C22×C4—342no—
253C22×C4—341no—
254C22×C4—342no—
255C22×C4—342no—
256C22×C4—341no—
257C22×C4—341no—
258C22×C4—341no—
259C22×C4C22⋊#59511yescharacteristic
260C22×C4C22×#59511yescharacteristic
261C2×C8—682no—
262C2×C8—682no—
263C2×C8—682no—
264C2×C8—682no—
265C2×C8—682no—
266C2×C8—682no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—681no—
270C4⋊C4—681no—
271C4⋊C4—681no—
272C4⋊C4—681no—
273C4⋊C4—681no—
274C4⋊C4—681no—
275C4⋊C4—682no—
276C4⋊C4—681no—
277C4⋊C4—681no—
278C22⋊C4—42no—
279C22⋊C4—682no—
280C42—174no—
281C42—342no—
282C42—342no—
283C42—342no—
284C42—682no—
285C42—682no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23#1058754712yes—
293C23C2×C22⋊#59511yescharacteristic
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—44no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—172no—
307C2×C4—172no—
308C2×C4—171no—
309C2×C4—171no—
310C2×C4—172no—
311C2×C4—171no—
312C2×C4—171no—
313C2×C4—172no—
314C2×C4—172no—
315C2×C4—344no—
316C2×C4—344no—
317C2×C4—341no—
318C2×C4—341no—
319C2×C4—341no—
320C2×C4—341no—
321C2×C4—341no—
322C2×C4—682no—
323C2×C4—682no—
324C2×C4—681no—
325C2×C4—681no—
326C2×C4—681no—
327C2×C4—681no—
328C2×C4—681no—
329C2×C4—681no—
330C2×C4#1058767712yes—
331C8—1362no—
332C8—1362no—
333C22—22no—
334C22—22no—
335C22—22no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22#1176375211yescharacteristic
341C22#1176379611yescharacteristic
342C22#1176379611yescharacteristic
343C22C2×#1058754711yescharacteristic
344C22#1176379911yescharacteristic
345C22#1176380011yescharacteristic
346C22#1176380211yescharacteristic
347C4—22no—
348C4—41no—
349C4—81no—
350C4—81no—
351C4—172no—
352C4—172no—
353C4—172no—
354C4—172no—
355C4—342no—
356C4—681no—
357C4—1361no—
358C4—1361no—
359C2—42no—
360C2#1454428211yescharacteristic
361C2#1454429611yescharacteristic
362C2#1454431011yescharacteristic
363C2#1454432011yescharacteristic
364C2#1454468611yescharacteristic
365C2#1454468711yescharacteristic
366C2#1454468911yescharacteristic
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#8726
LCSG ▷ C22×C4×C17 ▷ C22×C17 ▷ C17
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #14543662C2semidirect
4. #14543662C2semidirect
238. C17#8726semidirect

Maximal subgroups

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

subgroupindex
#145446402
#145436622
#145311862
#872617