Finite Group Databasebeta

← #17958288

Subgroups — #17958288

Order 4352.

488 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 100 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#14544024C211yesmaximal, characteristic
2#14544020C211yesmaximal, characteristic
3#14543728C212yesmaximal, complemented
4#14543710C212yesmaximal, complemented
5#123×Dic17C211yesmaximal, characteristic
6#11763825—21no—
7C2×C22⋊#3624—21no—
8#11763654—21no—
9#11763652—21no—
10#11763652C2212yes—
11#11763652C2212yes—
12C2×D4×Dic17C2211yescharacteristic
13C22×C4×Dic17C411yescharacteristic
14#11762702—21no—
15#11762695—21no—
16#11762689C411yescharacteristic
17#11762684C2211yescharacteristic
18#11762672C2211yescharacteristic
19C4⋊C4×Dic17—21no—
20C22⋊C4×Dic17—21no—
21C22⋊C4×Dic17C411yescharacteristic
22C2×C17×#123C411yescharacteristic
23C22×D4×C17C2×C411yescharacteristic
24C23×C4×C17C2×C411yescharacteristic
25C23×Dic17C2×C411yescharacteristic
26C23×Dic17C2×C411yescharacteristic
27C23×Dic17D411yescharacteristic
28C22×#3624—41no—
29C17×#123—22no—
30C17×#123C814yescomplemented
31C2×C4⋊C4×C17—21no—
32C2×C22⋊C4×C17—21no—
33C2×C22⋊C4×C17C2×C411yescharacteristic
34C2×C22⋊Dic17—21no—
35C2×C22⋊Dic17C2×C411yescharacteristic
36C2×C22⋊Dic17D411yescharacteristic
37C2×C22⋊Dic17C2311yescharacteristic
38D4×Dic17—41no—
39D4×Dic17—41no—
40C2×C4⋊Dic17—21no—
41C2×C4⋊Dic17D411yescharacteristic
42C2×C4×Dic17—21no—
43C2×C4×Dic17—21no—
44C2×C4×Dic17—21no—
45C2×C4×Dic17—41no—
46C2×C4×Dic17—41no—
47C2×C4×Dic17D411yescharacteristic
48C22⋊#3624—22no—
49C22⋊#3624—22no—
50C22⋊#3624—41no—
51C4⋊#3624—41no—
52C4×#3624—41no—
53#10587502—21no—
54#10587502—21no—
55#10587502—21no—
56#10587502—21no—
57#10587502—21no—
58C24×C17C22⋊C411yescharacteristic
59C24×C17C22×C411yescharacteristic
60C2×D4×C17—41no—
61C2×D4×C17—41no—
62C2×D4×C17C2×C812yes—
63C2×D4×C17M1612yes—
64C22×C4×C17—21no—
65C22×C4×C17—21no—
66C22×C4×C17—21no—
67C22×C4×C17—41no—
68C22×C4×C17—41no—
69C22×C4×C17C22⋊C411yescharacteristic
70C22×C4×C17C2×C812yescomplemented
71C22×Dic17—21no—
72C22×Dic17—21no—
73C22×Dic17—21no—
74C22×Dic17—21no—
75C22×Dic17—21no—
76C22×Dic17—21no—
77C22×Dic17—42no—
78C22×Dic17—42no—
79C22×Dic17—42no—
80C22×Dic17—41no—
81C22×Dic17C22⋊C411yescharacteristic
82C22×Dic17C22⋊C411yescharacteristic
83C22×Dic17C2×D411yescharacteristic
84C22×Dic17C2×D411yescharacteristic
85C2×#3624—42no—
86C2×#3624—42no—
87C2×#3624—82no—
88C4⋊C4×C17—24no—
89C22⋊C4×C17—24no—
90C22⋊C4×C17—21no—
91C22⋊C4×C17C2×C812yescomplemented
92C22⋊Dic17—41no—
93C22⋊Dic17—41no—
94C22⋊Dic17—42no—
95C22⋊Dic17—41no—
96C4⋊Dic17—41no—
97C4⋊Dic17—42no—
98C4×Dic17—21no—
99C4×Dic17—21no—
100C4×Dic17—42no—
101C4×Dic17—42no—
102C4×Dic17—41no—
103C4×Dic17—41no—
104#11472—171nomaximal, Sylow
105C23×C17—21no—
106C23×C17—21no—
107C23×C17—21no—
108C23×C17—21no—
109C23×C17—41no—
110C23×C17—41no—
111C23×C17—41no—
112C23×C17—41no—
113C23×C17C22⋊C812yes—
114C23×C17C2×C22⋊C411yescharacteristic
115C23×C17C22×C811yescharacteristic
116C23×C17C2×M1611yescharacteristic
117D4×C17—42no—
118D4×C17—42no—
119C2×C4×C17—21no—
120C2×C4×C17—21no—
121C2×C4×C17—22no—
122C2×C4×C17—22no—
123C2×C4×C17—42no—
124C2×C4×C17—41no—
125C2×C4×C17—41no—
126C2×C4×C17—41no—
127C2×C4×C17—41no—
128C2×C4×C17—42no—
129C2×C4×C17C22⋊C812yes—
130C2×Dic17—22no—
131C2×Dic17—22no—
132C2×Dic17—21no—
133C2×Dic17—21no—
134C2×Dic17—21no—
135C2×Dic17—42no—
136C2×Dic17—41no—
137C2×Dic17—42no—
138C2×Dic17—41no—
139C2×Dic17—42no—
140C2×Dic17—42no—
141C2×Dic17—41no—
142C2×Dic17—41no—
143C2×Dic17—41no—
144C2×Dic17—42no—
145C2×Dic17—41no—
146C2×Dic17—41no—
147C2×Dic17—41no—
148C2×Dic17—81no—
149C2×Dic17#9911yescharacteristic
150C2×Dic17#9911yescharacteristic
151#3624—82no—
152#3624—82no—
153C4×#123—171no—
154#1460—171no—
155#1459—171no—
156#1326—172no—
157#1317—172no—
158C22×C17—21no—
159C22×C17—21no—
160C22×C17—21no—
161C22×C17—21no—
162C22×C17—22no—
163C22×C17—42no—
164C22×C17—41no—
165C22×C17—41no—
166C22×C17—42no—
167C22×C17—41no—
168C22×C17—41no—
169C22×C17—41no—
170C22×C17—41no—
171C22×C17—81no—
172C22×C17C2×C22⋊C811yescharacteristic
173C22×C17C2×#9911yescharacteristic
174C22×C17#41311yescharacteristic
175C4×C17—22no—
176C4×C17—42no—
177C4×C17—42no—
178Dic17—21no—
179Dic17—41no—
180Dic17—41no—
181Dic17—41no—
182Dic17—42no—
183Dic17—81no—
184Dic17#35212yes—
185C2×#123#59511yescharacteristic
186C2×C4×D4—171no—
187C22×C42—171no—
188#431—341no—
189C2×C22⋊C8—341no—
190#388—341no—
191#387—341no—
192#386—171no—
193#385—171no—
194#381—171no—
195C4×C4⋊C4—341no—
196C4×C22⋊C4—171no—
197C4×C22⋊C4—341no—
198#324—341no—
199#323—172no—
200#323—172no—
201#323—341no—
202C2×C17—21no—
203C2×C17—21no—
204C2×C17—41no—
205C2×C17—41no—
206C2×C17—42no—
207C2×C17#145911yescharacteristic
208C2×C17C2×#35211yescharacteristic
209C2×C17#212011yescharacteristic
210C22×D4C2×#59511yescharacteristic
211C23×C4—171no—
212C23×C4—171no—
213C23×C4—171no—
214C23×C4C2×#59511yescharacteristic
215C22×C8—681no—
216#123—22no—
217#123#362414yescomplemented
218C4×D4—681no—
219C4×D4—681no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—171no—
222C2×C4⋊C4—341no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—171no—
225C2×C22⋊C4—171no—
226C2×C22⋊C4—171no—
227C2×C22⋊C4—341no—
228C2×C22⋊C4C2×#59511yescharacteristic
229C2×C42—171no—
230C2×C42—341no—
231C2×C42—341no—
232C2×C42—341no—
233C2×C42—681no—
234C2×C42—681no—
235C4⋊C8—681no—
236C22⋊C8—342no—
237C22⋊C8—342no—
238C22⋊C8—681no—
239C4×C8—681no—
240#95—341no—
241#95—341no—
242#95—341no—
243#95—341no—
244#95—341no—
245C17#1147211yescharacteristic, complemented, Sylow
246C24C22⋊#59511yescharacteristic
247C24C22×#59511yescharacteristic
248C2×D4—41no—
249C2×D4—41no—
250C2×D4C2×#362412yes—
251C2×D4#6324712yes—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—171no—
258C22×C4—171no—
259C22×C4—171no—
260C22×C4—171no—
261C22×C4—341no—
262C22×C4—341no—
263C22×C4—341no—
264C22×C4—341no—
265C22×C4—341no—
266C22×C4—341no—
267C22×C4—682no—
268C22×C4—682no—
269C22×C4—682no—
270C22×C4—681no—
271C22×C4C2×#362412yescomplemented
272C22×C4C22⋊#59511yescharacteristic
273C2×C8—682no—
274C2×C8—682no—
275C2×C8—1362no—
276C4⋊C4—24no—
277C4⋊C4—681no—
278C4⋊C4—682no—
279C22⋊C4—21no—
280C22⋊C4—24no—
281C22⋊C4—681no—
282C22⋊C4—681no—
283C22⋊C4—682no—
284C22⋊C4—681no—
285C22⋊C4C2×#362412yescomplemented
286C42—341no—
287C42—341no—
288C42—681no—
289C42—681no—
290C42—682no—
291C42—682no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23C22⋊#362412yes—
301C23C22×#362411yescharacteristic
302C23C2×#6324711yescharacteristic
303C23C2×C22⋊#59511yescharacteristic
304D4—42no—
305D4—42no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—171no—
317C2×C4—171no—
318C2×C4—342no—
319C2×C4—342no—
320C2×C4—341no—
321C2×C4—341no—
322C2×C4—341no—
323C2×C4—682no—
324C2×C4—682no—
325C2×C4—682no—
326C2×C4—681no—
327C2×C4—682no—
328C2×C4—681no—
329C2×C4—681no—
330C2×C4—681no—
331C2×C4—681no—
332C2×C4—682no—
333C2×C4—681no—
334C2×C4—681no—
335C2×C4—681no—
336C2×C4—1361no—
337C2×C4C22⋊#362412yes—
338C8—1362no—
339C8—1362no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—22no—
345C22—41no—
346C22—41no—
347C22—42no—
348C22—42no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—81no—
354C22C2×C22⋊#362411yescharacteristic
355C22D17⋊#9911yescharacteristic
356C22#1176379611yescharacteristic
357C4—22no—
358C4—42no—
359C4—42no—
360C4—172no—
361C4—341no—
362C4—681no—
363C4—681no—
364C4—681no—
365C4—682no—
366C4—1361no—
367C2—21no—
368C2—21no—
369C2—41no—
370C2—41no—
371C2—42no—
372C2#1454402011yescharacteristic
373C2D17⋊#35211yescharacteristic
374C2#1454503811yescharacteristic
subgroupG/H
centerC22D17⋊#99
commutatorC23×C17C22×C8
FrattiniC24C22×#595
FittingC2×C17×#123C4
socleC22×C17C2×#99
prime radicalC22×D4×C17C2×C4
cosocleC2×C22⋊Dic17C23
hypercenterC2×#123#595
hypocenterC17#11472
LCSG ▷ C23×C17 ▷ C22×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
3. #14543728C2semidirect
4. #14543710C2semidirect
30. C17×#123C8semidirect
70. C22×C4×C17C2×C8semidirect
91. C22⋊C4×C17C2×C8semidirect
217. #123#3624semidirect
245. C17#11472semidirect
271. C22×C4C2×#3624semidirect
285. C22⋊C4C2×#3624semidirect

Maximal subgroups

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

subgroupindex
#145440242
#145440202
#145437282
#145437102
#123×Dic172
#1147217