Finite Group Databasebeta

← #28313758

Subgroups — #28313758

Order 13568.

914 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811640C211yesmaximal, characteristic
2C2×C4×#11719940C211yesmaximal, characteristic
3#20289920C214yesmaximal, complemented
4#20289916C214yesmaximal, complemented
5C8×#11719940C214yesmaximal, complemented
6C2×#16810673C211yesmaximal, characteristic
7C22×#11719940C412yes—
8#16811640C412yes—
9#16811640C412yes—
10#16811640C2212yes—
11#16811640C2212yes—
12C2×#13133284C412yes—
13C2×C22⋊#88324C412yes—
14C4×#11719940C412yes—
15C4×#11719940C412yes—
16C4×#11719940C2212yes—
17C4×#11719940C2212yes—
18C2×#13133272C412yes—
19C2×C4×#88324C2212yes—
20C2×C4×#88324C2212yes—
21#88324⋊C8C2218yescomplemented
22C8×#88324C2218yescomplemented
23C22×C4×Dic53C2211yescharacteristic
24#16810673C412yes—
25#16810673C412yes—
26#16810673C2212yes—
27#16810673C2212yes—
28C2×#13133146C412yes—
29C2×C22⋊#88322C411yescharacteristic
30C2×C8×Dic53C2212yes—
31C2×C22⋊C8×C53C411yescharacteristic
32C2×#11719940C8116yescomplemented
33C2×#11719940C2×C414yes—
34C2×#11719940C2×C414yes—
35C22×#88324C2×C414yes—
36C22×#88324C2×C414yes—
37#13133284C2×C414yes—
38#13133284C2×C414yes—
39C22⋊#88324C2×C414yes—
40C22⋊#88324C2×C414yes—
41#13133272C2×C414yes—
42#13133272C2×C414yes—
43C4×#88324C2×C414yes—
44C4×#88324C2×C414yes—
45C4×#88324C2314yes—
46C4×#88324C2314yes—
47C23×Dic53C2×C411yescharacteristic
48C23×Dic53C2×C411yescharacteristic
49C22×#88322C2×C412yes—
50C2×C4×Dic53C2×C411yescharacteristic
51C2×C4×Dic53C2×C411yescharacteristic
52C2×C4×Dic53C2×C411yescharacteristic
53C2×C4×Dic53C2×C411yescharacteristic
54C2×C4×Dic53C2×C411yescharacteristic
55C2×C4×Dic53C2×C411yescharacteristic
56C2×C4×Dic53C2×C411yescharacteristic
57C2×C4×Dic53C2×C411yescharacteristic
58C2×C4×Dic53C2×C411yescharacteristic
59C2×C4×Dic53C2311yescharacteristic
60C2×C4×Dic53C2311yescharacteristic
61C2×C4×Dic53C2311yescharacteristic
62#13133146C2×C414yes—
63#13133146C2×C414yes—
64C22⋊#88322C2×C412yes—
65C22⋊#88322C2×C412yes—
66C8×Dic53C2×C414yes—
67C8×Dic53C2314yes—
68C23×C4×C53C2×C411yescharacteristic
69C22×C8×C53C2×C412yes—
70C22⋊C8×C53C2×C412yes—
71C22⋊C8×C53C2×C412yes—
72C24×C53C4211yescharacteristic
73C22×C4×C53—42no—
74C22×C4×C53C4211yescharacteristic
75C22×C4×C53C4211yescharacteristic
76C22×C4×C53C4211yescharacteristic
77C22×C4×C53C22×C411yescharacteristic
78C22×C4×C53C22×C411yescharacteristic
79C22×C4×C53C22×C411yescharacteristic
80C22×Dic53—42no—
81C22×Dic53C4211yescharacteristic
82C22×Dic53C4211yescharacteristic
83C22×Dic53C4211yescharacteristic
84C22×Dic53C4211yescharacteristic
85C22×Dic53C4211yescharacteristic
86C22×Dic53C4211yescharacteristic
87C22×Dic53C2×C818yes—
88C22×Dic53C22×C411yescharacteristic
89C22×Dic53C22×C411yescharacteristic
90C22×Dic53C22×C411yescharacteristic
91C22×Dic53C22×C411yescharacteristic
92C22×Dic53C22×C411yescharacteristic
93C22×Dic53C22×C411yescharacteristic
94#11719940C2×C8116yescomplemented
95#11719940C2×C8116yescomplemented
96C2×#88324—44no—
97C2×#88324C2×C8116yescomplemented
98C2×#88324C22×C418yes—
99C2×#88324C22×C418yes—
100C2×C8×C53—42no—
101C2×C8×C53C22×C414yes—
102C4×Dic53C4211yescharacteristic
103C4×Dic53C4211yescharacteristic
104C4×Dic53C4211yescharacteristic
105C4×Dic53C4211yescharacteristic
106C4×Dic53C4211yescharacteristic
107C4×Dic53C4211yescharacteristic
108C4×Dic53C22×C411yescharacteristic
109C4×Dic53C22×C411yescharacteristic
110C4×Dic53C22×C411yescharacteristic
111C4×Dic53C22×C411yescharacteristic
112C4×Dic53C22×C411yescharacteristic
113C4×Dic53C22×C411yescharacteristic
114C4×Dic53C22×C411yescharacteristic
115C4×Dic53C22×C411yescharacteristic
116C4×Dic53C22×C411yescharacteristic
117C4×Dic53C2411yescharacteristic
118C2×#88322—42no—
119C2×#88322C22×C414yes—
120C23×C53—41no—
121C23×C53—41no—
122C23×C53C4×C814yes—
123C23×C53C2×C4211yescharacteristic
124C23×C53C2×C4211yescharacteristic
125C23×C53C2×C4211yescharacteristic
126C2×C4×C53—22no—
127C2×C4×C53—22no—
128C2×C4×C53—22no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53C2×C4211yescharacteristic
133C2×C4×C53C2×C4211yescharacteristic
134C2×C4×C53C2×C4211yescharacteristic
135C2×C4×C53C23×C411yescharacteristic
136C2×Dic53—22no—
137C2×Dic53—22no—
138C2×Dic53—22no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53C4×C814yes—
143C2×Dic53C4×C814yes—
144C2×Dic53C4×C814yes—
145C2×Dic53C2×C4211yescharacteristic
146C2×Dic53C2×C4211yescharacteristic
147C2×Dic53C2×C4211yescharacteristic
148C2×Dic53C2×C4211yescharacteristic
149C2×Dic53C2×C4211yescharacteristic
150C2×Dic53C2×C4211yescharacteristic
151C2×Dic53C22×C814yes—
152C2×Dic53C22×C814yes—
153C2×Dic53C22×C814yes—
154C2×Dic53C23×C411yescharacteristic
155C2×Dic53C23×C411yescharacteristic
156#88324—28no—
157#88324C22×C8116yes—
158C8×C53—24no—
159#88322—24no—
160#17314—531nomaximal, Sylow
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—44no—
168C22×C53C2×C4×C812yes—
169C22×C53C2×C4×C812yes—
170C22×C53C2×C4×C812yes—
171C22×C53C22×C4211yescharacteristic
172C4×C53—21no—
173C4×C53—21no—
174C4×C53—21no—
175C4×C53#56812yes—
176Dic53—21no—
177Dic53—21no—
178Dic53—21no—
179Dic53C2×C4×C812yes—
180Dic53C2×C4×C812yes—
181Dic53C2×C4×C812yes—
182Dic53C23×C812yes—
183Dic53#56812yes—
184C2×#431—531no—
185C2×#431—531no—
186C2×C4×M16—531no—
187#1450—534no—
188C8×M16—534no—
189C8×M16—534no—
190C2×C53—22no—
191C2×C53—22no—
192C2×C53—22no—
193C2×C53C22×C4×C811yescharacteristic
194C2×C53#287311yescharacteristic
195C2×C53#287311yescharacteristic
196C22×M16—532no—
197C22×C42—531no—
198#431—532no—
199#431—532no—
200#431—532no—
201#431—532no—
202#431—532no—
203#431—532no—
204#431—532no—
205#431—532no—
206C2×C4⋊C8—532no—
207C2×C4⋊C8—532no—
208C2×C22⋊C8—532no—
209C2×C22⋊C8—531no—
210C2×C22⋊C8#615711yescharacteristic
211C4×M16—532no—
212C4×M16—532no—
213C4×M16—532no—
214C4×M16—532no—
215C2×#97—532no—
216C2×C4×C8—532no—
217C2×C4×C8—532no—
218C2×C4×C8—532no—
219#322—538no—
220C82—538no—
221C53#1731411yescharacteristic, complemented, Sylow
222C23×C4—531no—
223C23×C4—531no—
224C23×C4C2×#615711yescharacteristic
225C2×M16—534no—
226C2×M16—534no—
227C2×M16—5316no—
228C22×C8—534no—
229C22×C8—534no—
230C22×C8—532no—
231C22×C8C2×#615712yes—
232C2×C42—531no—
233C2×C42—531no—
234C2×C42—531no—
235C2×C42—531no—
236C2×C42—531no—
237C2×C42—531no—
238C2×C42—531no—
239C2×C42—531no—
240C2×C42—531no—
241C2×C42—531no—
242C2×C42—531no—
243C2×C42—531no—
244C4⋊C8—534no—
245C4⋊C8—534no—
246C4⋊C8—534no—
247C4⋊C8—534no—
248C22⋊C8—534no—
249C22⋊C8—534no—
250C22⋊C8—532no—
251C22⋊C8—532no—
252C22⋊C8C2×#615712yes—
253C22⋊C8C2×#615712yes—
254#97—534no—
255#97—534no—
256C4×C8—534no—
257C4×C8—534no—
258C4×C8—534no—
259C4×C8—534no—
260C4×C8—534no—
261C4×C8—534no—
262C24C4×#615711yescharacteristic
263C22×C4—42no—
264C22×C4—538no—
265C22×C4—531no—
266C22×C4—531no—
267C22×C4—531no—
268C22×C4—531no—
269C22×C4—531no—
270C22×C4—531no—
271C22×C4—531no—
272C22×C4—531no—
273C22×C4—531no—
274C22×C4—531no—
275C22×C4—531no—
276C22×C4—531no—
277C22×C4—2122no—
278C22×C4C4×#615711yescharacteristic
279C22×C4C4×#615711yescharacteristic
280C22×C4C4×#615711yescharacteristic
281C22×C4C22×#615711yescharacteristic
282C22×C4C22×#615711yescharacteristic
283C22×C4C22×#615711yescharacteristic
284M16—5316no—
285M16—5316no—
286C2×C8—42no—
287C2×C8—5316no—
288C2×C8—538no—
289C2×C8—538no—
290C2×C8—534no—
291C2×C8—2122no—
292C2×C8—2124no—
293C2×C8C22×#615714yes—
294C42—531no—
295C42—531no—
296C42—531no—
297C42—531no—
298C42—531no—
299C42—531no—
300C42—531no—
301C42—531no—
302C42—531no—
303C42—531no—
304C42—531no—
305C42—531no—
306C42—531no—
307C42—531no—
308C42—531no—
309C42—531no—
310C23—41no—
311C23—41no—
312C23C8×#615714yes—
313C23C2×C4×#615711yescharacteristic
314C23C2×C4×#615711yescharacteristic
315C23C2×C4×#615711yescharacteristic
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—534no—
323C2×C4—534no—
324C2×C4—534no—
325C2×C4—534no—
326C2×C4—534no—
327C2×C4—534no—
328C2×C4—531no—
329C2×C4—531no—
330C2×C4—531no—
331C2×C4—531no—
332C2×C4—531no—
333C2×C4—531no—
334C2×C4—531no—
335C2×C4—531no—
336C2×C4—1062no—
337C2×C4—1062no—
338C2×C4—1062no—
339C2×C4—2121no—
340C2×C4—2121no—
341C2×C4—2121no—
342C2×C4C2×C4×#615711yescharacteristic
343C2×C4C2×C4×#615711yescharacteristic
344C2×C4C2×C4×#615711yescharacteristic
345C2×C4C23×#615711yescharacteristic
346C8—24no—
347C8—5316no—
348C8—1068no—
349C8—1064no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—44no—
357C22C2×C8×#615712yes—
358C22C2×C8×#615712yes—
359C22C2×C8×#615712yes—
360C22C22×C4×#615711yescharacteristic
361C4—21no—
362C4—21no—
363C4—21no—
364C4—532no—
365C4—532no—
366C4—532no—
367C4—532no—
368C4—532no—
369C4—1061no—
370C4—1061no—
371C4—1061no—
372C4#1681170112yes—
373C2—22no—
374C2—22no—
375C2—22no—
376C2C22×C8×#615711yescharacteristic
377C2#2029127811yescharacteristic
378C2#2029127911yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×#6157
commutatorC2×C53C22×C4×C8
FittingC2×C22⋊C8×C53C4
socleC22×C53C22×C42
prime radicalC24×C53C42
cosocleC4×Dic53C24
hypercenterC2×C22⋊C8#6157
hypocenterC53#17314
LCSG ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
3. #20289920C2semidirect
4. #20289916C2semidirect
5. C8×#11719940C2semidirect
21. #88324⋊C8C22semidirect
22. C8×#88324C22semidirect
32. C2×#11719940C8semidirect
94. #11719940C2×C8semidirect
95. #11719940C2×C8semidirect
97. C2×#88324C2×C8semidirect
221. C53#17314semidirect

Maximal subgroups

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

subgroupindex
C2×#168116402
C2×C4×#117199402
#202899202
#202899162
C8×#117199402
C2×#168106732
#1731453