Finite Group Databasebeta

← #16929708

Subgroups — #16929708

Order 4352.

628 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#11762862C211yesmaximal, characteristic
2C2×#11762693C211yesmaximal, characteristic
3C2×#11762534C212yesmaximal
4#14526626C212yesmaximal
5C17×#1817C211yesmaximal, characteristic
6C22×#63225C411yescharacteristic
7C22×#63225C2211yescharacteristic
8#11762862—24no—
9C2×C4⋊#3622—21no—
10C2×C22⋊#3622C2212yes—
11C2×C4×#3622—21no—
12#11762693—21no—
13#11762693—21no—
14#11762693—21no—
15#11762693—21no—
16C2×#10587502C411yescharacteristic
17C2×#10587502C411yescharacteristic
18C2×#10587502C411yescharacteristic
19C2×#10587502C2211yescharacteristic
20#11762534—24no—
21C22×C42×C17C2211yescharacteristic
22C2×C17×C22⋊C8C2212yes—
23C17×#336—21no—
24C17×#336—21no—
25C23×C4×C17C2×C411yescharacteristic
26C23×C4×C17C2×C411yescharacteristic
27C23×C4×C17C2311yescharacteristic
28C23×Dic17C2×C411yescharacteristic
29C23×Dic17C2×C411yescharacteristic
30C23×Dic17C2×C411yescharacteristic
31C23×Dic17C2×C411yescharacteristic
32C22×C8×C17—22no—
33C2×C42×C17—22no—
34C2×C42×C17—22no—
35C2×C42×C17D411yescharacteristic
36C2×C42×C17D411yescharacteristic
37C2×C42×C17D411yescharacteristic
38C2×C42×C17D411yescharacteristic
39C2×#63225—21no—
40C2×#63225—21no—
41C2×#63225—22no—
42C2×#63225—21no—
43C2×#63225—21no—
44C2×#63225—22no—
45C2×#63225—22no—
46C2×#63225C814yescomplemented
47C22×#3622—22no—
48C17×C22⋊C8—22no—
49C17×C22⋊C8—22no—
50#10587502—21no—
51#10587502—21no—
52#10587502—21no—
53#10587502—21no—
54#10587502—21no—
55#10587502—21no—
56#10587502—21no—
57#10587502—21no—
58C22⋊#3622—22no—
59C22⋊#3622D414yes—
60C4⋊#3622—24no—
61C4×#3622—24no—
62C24×C17C22×C411yescharacteristic
63C22×C4×C17—22no—
64C22×C4×C17—22no—
65C22×C4×C17—22no—
66C22×C4×C17—22no—
67C22×C4×C17—41no—
68C22×C4×C17—41no—
69C22×C4×C17—41no—
70C22×C4×C17C22⋊C411yescharacteristic
71C22×C4×C17C22⋊C411yescharacteristic
72C22×C4×C17C22⋊C411yescharacteristic
73C22×C4×C17C22⋊C411yescharacteristic
74C22×C4×C17C2×C812yes—
75C22×C4×C17M1612yes—
76C22×C4×C17C2×D411yescharacteristic
77C22×C4×C17C2×D412yes—
78C22×C4×C17C2×D411yescharacteristic
79C22×C4×C17#4112yes—
80C22×Dic17—21no—
81C22×Dic17—21no—
82C22×Dic17—22no—
83C22×Dic17—21no—
84C22×Dic17—21no—
85C22×Dic17—22no—
86C22×Dic17—21no—
87C22×Dic17—21no—
88C22×Dic17—22no—
89C22×Dic17—22no—
90C22×Dic17—41no—
91C22×Dic17—41no—
92C22×Dic17—41no—
93C22×Dic17—41no—
94C22×Dic17—41no—
95C22×Dic17—41no—
96C22×Dic17C2×C812yes—
97C22×Dic17C2×C812yes—
98C2×C8×C17—42no—
99C2×C8×C17—42no—
100C2×C8×C17—42no—
101C42×C17—21no—
102C42×C17—21no—
103C42×C17—22no—
104C42×C17—22no—
105C42×C17—21no—
106C42×C17—21no—
107#63225—24no—
108#63225—24no—
109#63225—41no—
110#63225—42no—
111#63225—41no—
112C2×#3622—24no—
113C2×#3622—42no—
114C2×#3622—42no—
115#9082—171nomaximal, Sylow
116C23×C17—21no—
117C23×C17—21no—
118C23×C17—21no—
119C23×C17—21no—
120C23×C17C2×C22⋊C411yescharacteristic
121C23×C17C4×D412yes—
122C23×C17C22×C811yescharacteristic
123C23×C17C2×M1611yescharacteristic
124C23×C17#13112yes—
125C2×C4×C17—21no—
126C2×C4×C17—21no—
127C2×C4×C17—21no—
128C2×C4×C17—21no—
129C2×C4×C17—21no—
130C2×C4×C17—21no—
131C2×C4×C17—21no—
132C2×C4×C17—21no—
133C2×C4×C17—21no—
134C2×C4×C17—21no—
135C2×C4×C17—24no—
136C2×C4×C17—24no—
137C2×C4×C17—21no—
138C2×C4×C17—21no—
139C2×C4×C17—21no—
140C2×C4×C17—42no—
141C2×C4×C17—42no—
142C2×C4×C17—44no—
143C2×C4×C17C22⋊C812yes—
144C2×C4×C17C22⋊C812yes—
145C2×C4×C17#10112yes—
146C2×C4×C17#12011yescharacteristic
147C2×C4×C17#12112yes—
148C2×C4×C17#12311yescharacteristic
149C2×Dic17—22no—
150C2×Dic17—22no—
151C2×Dic17—24no—
152C2×Dic17—42no—
153C2×Dic17—41no—
154C2×Dic17—41no—
155C2×Dic17—41no—
156C2×Dic17—41no—
157C2×Dic17—42no—
158C2×Dic17—42no—
159C2×Dic17—41no—
160C2×Dic17—41no—
161C2×Dic17—41no—
162C2×Dic17—41no—
163C2×Dic17—41no—
164C2×Dic17—41no—
165C2×Dic17—82no—
166C2×Dic17—82no—
167C8×C17—82no—
168#3622—44no—
169C2×#431—171no—
170C2×#387—171no—
171#1817D1711yescharacteristic
172C2×#324—172no—
173#1280—172no—
174C22×C17—22no—
175C22×C17—22no—
176C22×C17—21no—
177C22×C17—21no—
178C22×C17—21no—
179C22×C17—21no—
180C22×C17—22no—
181C22×C17—22no—
182C22×C17#38611yescharacteristic
183C22×C17C2×C22⋊C811yescharacteristic
184C22×C17#40811yescharacteristic
185C22×C17#41011yescharacteristic
186C22×C17#41011yescharacteristic
187C22×C17C2×#10111yescharacteristic
188C22×C17#41311yescharacteristic
189C22×C17C8×D412yes—
190C22×C17#43512yes—
191C4×C17—22no—
192C4×C17—22no—
193C4×C17—21no—
194C4×C17—22no—
195C4×C17—21no—
196C4×C17—22no—
197C4×C17—41no—
198Dic17—42no—
199Dic17—42no—
200Dic17—81no—
201Dic17—81no—
202C22×C4⋊C4—171no—
203C22×C4⋊C4—171no—
204C22×C42C2×D1711yescharacteristic
205#431—344no—
206C2×C4⋊C8—341no—
207C2×C22⋊C8—172no—
208C2×C22⋊C8C2×D1712yes—
209C2×C4×C8—341no—
210#387—341no—
211#387—341no—
212#387—341no—
213#387—341no—
214C2×#95—171no—
215C2×#95—171no—
216C2×#95—171no—
217C2×#95—171no—
218#336—21no—
219#336—21no—
220#324—344no—
221C2×C17—22no—
222C2×C17—22no—
223C2×C17#145811yescharacteristic
224C2×C17#146111yescharacteristic
225C2×C17#147011yescharacteristic
226C2×C17#147311yescharacteristic
227C2×C17#186911yescharacteristic
228C2×C17#189911yescharacteristic
229C2×C17#190111yescharacteristic
230C23×C4—171no—
231C23×C4—171no—
232C23×C4—171no—
233C23×C4—171no—
234C23×C4C4×D1711yescharacteristic
235C23×C4C4×D1711yescharacteristic
236C23×C4C22×D1711yescharacteristic
237C22×C8—22no—
238C22×C8—342no—
239C2×C4⋊C4—174no—
240C2×C4⋊C4—342no—
241C2×C4⋊C4—341no—
242C2×C4⋊C4—341no—
243C2×C4⋊C4—341no—
244C2×C4⋊C4—342no—
245C2×C4⋊C4—342no—
246C2×C4⋊C4—341no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42D6811yescharacteristic
250C2×C42D6811yescharacteristic
251C2×C42C22⋊D1711yescharacteristic
252C2×C42C22⋊D1711yescharacteristic
253C4⋊C8—344no—
254C22⋊C8—22no—
255C22⋊C8—22no—
256C22⋊C8—174no—
257C22⋊C8—342no—
258C4×C8—344no—
259#95—341no—
260#95—341no—
261#95—341no—
262#95—341no—
263#95—341no—
264#95—341no—
265#95—341no—
266#95—341no—
267C17#908211yescharacteristic, complemented, Sylow
268C24C2×C4×D1711yescharacteristic
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—172no—
277C22×C4—172no—
278C22×C4—342no—
279C22×C4—341no—
280C22×C4—341no—
281C22×C4—342no—
282C22×C4—341no—
283C22×C4—341no—
284C22×C4—341no—
285C22×C4—341no—
286C22×C4—342no—
287C22×C4—342no—
288C22×C4—681no—
289C22×C4—681no—
290C22×C4—681no—
291C22×C4—681no—
292C22×C4—681no—
293C22×C4—681no—
294C22×C4C8×D1712yes—
295C22×C4#6321812yes—
296C22×C4#6322711yescharacteristic
297C22×C4#6322711yescharacteristic
298C22×C4#6322711yescharacteristic
299C22×C4#6322711yescharacteristic
300C22×C4C2×D6811yescharacteristic
301C22×C4D4×D1712yes—
302C22×C4#6325412yes—
303C22×C4C2×C22⋊D1711yescharacteristic
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—344no—
308C2×C8—682no—
309C2×C8—682no—
310C4⋊C4—344no—
311C4⋊C4—344no—
312C4⋊C4—681no—
313C4⋊C4—681no—
314C4⋊C4—682no—
315C42—21no—
316C42—21no—
317C42—21no—
318C42—21no—
319C42—22no—
320C42—22no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23#1058756712yes—
326C23C2×C8×D1711yescharacteristic
327C23C2×#6321811yescharacteristic
328C23#1058759312yes—
329C23C2×#6322711yescharacteristic
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—342no—
349C2×C4—342no—
350C2×C4—344no—
351C2×C4—682no—
352C2×C4—681no—
353C2×C4—681no—
354C2×C4—681no—
355C2×C4—682no—
356C2×C4—681no—
357C2×C4—682no—
358C2×C4—681no—
359C2×C4—681no—
360C2×C4—681no—
361C2×C4—681no—
362C2×C4—681no—
363C2×C4—681no—
364C2×C4—1362no—
365C2×C4—1362no—
366C2×C4#1058749112yes—
367C2×C4#1058749112yes—
368C2×C4#1058749512yes—
369C2×C4#1058756811yescharacteristic
370C2×C4#1058757211yescharacteristic
371C2×C4#1058762512yes—
372C8—82no—
373C8—684no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22#1176268611yescharacteristic
383C22C2×#1058749111yescharacteristic
384C22#1176277711yescharacteristic
385C22#1176278211yescharacteristic
386C22#1176278211yescharacteristic
387C22C2×#1058749511yescharacteristic
388C22#1176279111yescharacteristic
389C22#1176287212yes—
390C22#1176287712yes—
391C4—22no—
392C4—22no—
393C4—22no—
394C4—21no—
395C4—21no—
396C4—22no—
397C4—41no—
398C4—682no—
399C4—682no—
400C4—1361no—
401C4—1361no—
402C2—22no—
403C2—22no—
404C2#1452709611yescharacteristic
405C2#1452710711yescharacteristic
406C2#1452714111yescharacteristic
407C2#1452715411yescharacteristic
408C2#1452907411yescharacteristic
409C2#1452924011yescharacteristic
410C2#1452925011yescharacteristic
subgroupG/H
centerC23C2×#63227
commutatorC23×C17C22×C8
FrattiniC23×C4C22×D17
FittingC17×#1817C2
socleC23×C17C2×C22⋊C4
prime radicalC24×C17C22×C4
cosocleC23×C4×C17C23
hypercenter#1817D17
hypocenterC17#9082
LCSG ▷ C23×C17 ▷ C2×C17 ▷ C17
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
46. C2×#63225C8semidirect
267. C17#9082semidirect

Maximal subgroups

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

subgroupindex
C2×#117628622
C2×#117626932
C2×#117625342
#145266262
C17×#18172
#908217