Finite Group Databasebeta

← #28249074

Subgroups — #28249074

Order 13568.

1702 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283821C214yesmaximal, complemented
2#20283817C214yesmaximal, complemented
3#20283810C218yesmaximal, complemented
4Q8×#11719917C218yesmaximal, complemented
5#20283353C214yesmaximal
6#20283348C212yesmaximal
7C2×C53×#533C211yesmaximal, characteristic
8#16810754C22116yescomplemented
9#16810754C22132yescomplemented
10Q8×#88322C22116yescomplemented
11#16810684C22116yes—
12#16810684C22116yes—
13#16810680C2214yes—
14#16810680C2214yes—
15#16810676C2214yes—
16#16810676C2214yes—
17#16810647C2214yes—
18#16810647C2218yes—
19#16810587C2214yes—
20C4×#11719917C2214yes—
21C4×#11719917C2218yes—
22C53×#533C412yes—
23C53×#533C414yes—
24C53×#533C412yes—
25C53×#533C2214yes—
26C53×#533C2212yes—
27C53×#533C2212yes—
28C2×C4⋊Q8×C53C412yes—
29C2×#125×C53C412yes—
30C2×C22⋊Q8×C53C414yes—
31C2×C4×Q8×C53C2214yes—
32C2×#117×C53C2212yes—
33C22×C42×C53C2211yescharacteristic
34C2×#11719917C2318yes—
35C4⋊#88322C2318yes—
36C4⋊#88322C23132yes—
37C4⋊#88322C2318yes—
38C22⋊#88322C2318yes—
39C22⋊#88322C2318yes—
40#13133128C2318yes—
41#13133128C23116yes—
42C4×#88322C2318yes—
43C4×#88322C23116yes—
44C22×Q8×C53C2×C414yes—
45C23×C4×C53C2×C412yes—
46C23×C4×C53C2311yescharacteristic
47C4⋊Q8×C53C2×C414yes—
48C4⋊Q8×C53C2×C414yes—
49C4⋊Q8×C53C2×C414yes—
50C4⋊Q8×C53C2×C414yes—
51#125×C53C2×C418yes—
52#125×C53C2×C414yes—
53#125×C53C2×C414yes—
54C22⋊Q8×C53C2×C418yes—
55C22⋊Q8×C53C2×C418yes—
56C22⋊Q8×C53C2×C418yes—
57C22⋊Q8×C53C2×C418yes—
58C4×Q8×C53C2×C418yes—
59C4×Q8×C53C2×C418yes—
60C4×Q8×C53C2318yes—
61C4×Q8×C53C2318yes—
62#117×C53C2×C414yes—
63#117×C53C2×C414yes—
64#117×C53C2314yes—
65#117×C53C2314yes—
66C2×C4⋊C4×C53C2×C414yes—
67C2×C4⋊C4×C53C2×C418yes—
68C2×C4⋊C4×C53C2×C414yes—
69C2×C22⋊C4×C53C2×C414yes—
70C2×C42×C53C2×C411yescharacteristic
71C2×C42×C53C2×C411yescharacteristic
72C2×C42×C53C2×C411yescharacteristic
73C2×C42×C53C2×C412yes—
74C2×C42×C53C2×C411yescharacteristic
75C2×C42×C53C2312yes—
76C2×C42×C53C2311yescharacteristic
77C2×C42×C53C2312yes—
78C2×C42×C53C2311yescharacteristic
79C2×C42×C53C2314yes—
80C24×C53C22×C411yescharacteristic
81C2×Q8×C53—48no—
82C2×Q8×C53C22×C418yes—
83C2×Q8×C53C22×C418yes—
84C22×C4×C53—24no—
85C22×C4×C53—44no—
86C22×C4×C53C22×C411yescharacteristic
87C22×C4×C53C22×C411yescharacteristic
88C22×C4×C53C22×C411yescharacteristic
89C22×C4×C53C22×C412yes—
90C22×C4×C53C22×C412yes—
91C22×C4×C53C22×C412yes—
92C22×C4×C53C22×C412yes—
93C22×C4×C53C22×C412yes—
94C22×C4×C53C22×C412yes—
95C22×C4×C53C22×C418yes—
96C22×C4×C53C2411yescharacteristic
97C22×C4×C53C2411yescharacteristic
98C22×C4×C53C2411yescharacteristic
99C4⋊C4×C53C22×C418yes—
100C4⋊C4×C53C22×C418yes—
101C4⋊C4×C53C22×C4116yes—
102C4⋊C4×C53C22×C4116yes—
103C4⋊C4×C53C22×C418yes—
104C4⋊C4×C53C22×C418yes—
105C22⋊C4×C53C22×C418yes—
106C22⋊C4×C53C22×C418yes—
107C42×C53C22×C412yes—
108C42×C53C22×C412yes—
109C42×C53C22×C411yescharacteristic
110C42×C53C22×C412yes—
111C42×C53C22×C411yescharacteristic
112C42×C53C22×C411yescharacteristic
113C42×C53C22×C412yes—
114C42×C53C22×C411yescharacteristic
115C42×C53C22×C418yes—
116C42×C53C2412yes—
117C42×C53C2412yes—
118C42×C53C2418yes—
119#11719917—216no—
120C2×#88322C24116yes—
121C23×C53—22no—
122C23×C53—42no—
123C23×C53C23×C411yescharacteristic
124C23×C53C23×C411yescharacteristic
125C23×C53C23×C411yescharacteristic
126Q8×C53—216no—
127Q8×C53—48no—
128C2×C4×C53—22no—
129C2×C4×C53—22no—
130C2×C4×C53—22no—
131C2×C4×C53—24no—
132C2×C4×C53—24no—
133C2×C4×C53—24no—
134C2×C4×C53—22no—
135C2×C4×C53—22no—
136C2×C4×C53—42no—
137C2×C4×C53—42no—
138C2×C4×C53—42no—
139C2×C4×C53—48no—
140C2×C4×C53C23×C412yes—
141C2×C4×C53C23×C412yes—
142C2×C4×C53C23×C412yes—
143C2×C4×C53C23×C411yescharacteristic
144C2×C4×C53C23×C411yescharacteristic
145C2×C4×C53C23×C4116yes—
146C2×C4×C53C23×C411yescharacteristic
147C2×C4×C53C23×C412yes—
148C2×C4×C532−1+414yes—
149C2×C4×C53C2511yescharacteristic
150#88322—216no—
151#60249—531nomaximal, Sylow
152C22×C53—22no—
153C22×C53—21no—
154C22×C53—21no—
155C22×C53—22no—
156C22×C53—22no—
157C22×C53—44no—
158C22×C53#51912yes—
159C22×C53C24×C411yescharacteristic
160C4×C53—21no—
161C4×C53—21no—
162C4×C53—22no—
163C4×C53—22no—
164C4×C53—22no—
165C4×C53—216no—
166C4×C53#51912yes—
167C4×C53#56814yes—
168C4×C53C2×2−1+412yes—
169C2×#533D5311yescharacteristic
170#2987—534no—
171#2986—534no—
172#2985—538no—
173Q8×M16—538no—
174#2921—534no—
175#2920—532no—
176C2×C53—22no—
177C2×C53—41no—
178C2×C53—41no—
179C2×C53C2×#51911yescharacteristic
180C2×C53C2×#56812yes—
181#533Dic5312yes—
182#533Dic5314yes—
183#533Dic5312yes—
184#533C2×D5314yes—
185#533C2×D5312yes—
186#533C2×D5312yes—
187C2×C4⋊Q8Dic5312yes—
188C2×#125Dic5312yes—
189C2×C22⋊Q8Dic5314yes—
190C2×C4×Q8C2×D5314yes—
191C2×#117C2×D5312yes—
192C22×C42C2×D5311yescharacteristic
193#446—5316no—
194#446—5332no—
195C8×Q8—5316no—
196#433—5316no—
197#433—5316no—
198#432—534no—
199#432—534no—
200#431—534no—
201#431—534no—
202#423—534no—
203#423—538no—
204#407—534no—
205C4×M16—534no—
206C4×M16—538no—
207C53#6024911yescharacteristic, complemented, Sylow
208C22×Q8C2×Dic5314yes—
209C23×C4C2×Dic5312yes—
210C23×C4C22×D5311yescharacteristic
211C2×M16—538no—
212C4⋊Q8C2×Dic5314yes—
213C4⋊Q8C2×Dic5314yes—
214C4⋊Q8C2×Dic5314yes—
215C4⋊Q8C2×Dic5314yes—
216#125C2×Dic5318yes—
217#125C2×Dic5314yes—
218#125C2×Dic5314yes—
219C22⋊Q8C2×Dic5318yes—
220C22⋊Q8C2×Dic5318yes—
221C22⋊Q8C2×Dic5318yes—
222C22⋊Q8C2×Dic5318yes—
223C4×Q8C2×Dic5318yes—
224C4×Q8C2×Dic5318yes—
225C4×Q8C22×D5318yes—
226C4×Q8C22×D5318yes—
227#117C2×Dic5314yes—
228#117C2×Dic5314yes—
229#117C22×D5314yes—
230#117C22×D5314yes—
231C2×C4⋊C4C2×Dic5314yes—
232C2×C4⋊C4C2×Dic5318yes—
233C2×C4⋊C4C2×Dic5314yes—
234C2×C22⋊C4C2×Dic5314yes—
235C2×C42C2×Dic5311yescharacteristic
236C2×C42C2×Dic5311yescharacteristic
237C2×C42C2×Dic5311yescharacteristic
238C2×C42C2×Dic5312yes—
239C2×C42C2×Dic5311yescharacteristic
240C2×C42C22×D5314yes—
241C2×C42C22×D5312yes—
242C2×C42C22×D5311yescharacteristic
243C2×C42C22×D5312yes—
244C2×C42C22×D5311yescharacteristic
245C4⋊C8—538no—
246C4⋊C8—538no—
247C4⋊C8—5332no—
248C22⋊C8—538no—
249C22⋊C8—538no—
250#97—538no—
251#97—5316no—
252C4×C8—538no—
253C4×C8—5316no—
254C24C22×Dic5311yescharacteristic
255C2×Q8—48no—
256C2×Q8C22×Dic5318yes—
257C2×Q8C22×Dic5318yes—
258C22×C4—24no—
259C22×C4—44no—
260C22×C4C22×Dic5311yescharacteristic
261C22×C4C22×Dic5311yescharacteristic
262C22×C4C22×Dic5311yescharacteristic
263C22×C4C22×Dic5312yes—
264C22×C4C22×Dic5312yes—
265C22×C4C22×Dic5312yes—
266C22×C4C22×Dic5312yes—
267C22×C4C22×Dic5312yes—
268C22×C4C22×Dic5312yes—
269C22×C4C22×Dic5318yes—
270C22×C4C23×D5311yescharacteristic
271C22×C4C23×D5311yescharacteristic
272C22×C4C23×D5311yescharacteristic
273M16—10616no—
274C2×C8—5316no—
275C4⋊C4C22×Dic5318yes—
276C4⋊C4C22×Dic5318yes—
277C4⋊C4C22×Dic53116yes—
278C4⋊C4C22×Dic53116yes—
279C4⋊C4C22×Dic5318yes—
280C4⋊C4C22×Dic5318yes—
281C22⋊C4C22×Dic5318yes—
282C22⋊C4C22×Dic5318yes—
283C42C22×Dic5312yes—
284C42C22×Dic5312yes—
285C42C22×Dic5311yescharacteristic
286C42C22×Dic5312yes—
287C42C22×Dic5311yescharacteristic
288C42C22×Dic5311yescharacteristic
289C42C22×Dic5312yes—
290C42C22×Dic5311yescharacteristic
291C42C22×Dic5318yes—
292C42C23×D5312yes—
293C42C23×D5318yes—
294C42C23×D5312yes—
295C23—22no—
296C23—42no—
297C23C23×Dic5311yescharacteristic
298C23C23×Dic5311yescharacteristic
299C23C23×Dic5311yescharacteristic
300Q8—216no—
301Q8—48no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—48no—
314C2×C4C23×Dic5311yescharacteristic
315C2×C4C23×Dic5311yescharacteristic
316C2×C4C23×Dic5312yes—
317C2×C4C23×Dic5312yes—
318C2×C4C23×Dic5312yes—
319C2×C4C23×Dic53116yes—
320C2×C4C23×Dic5311yescharacteristic
321C2×C4C23×Dic5312yes—
322C2×C4#1313326514yes—
323C2×C4C24×D5311yescharacteristic
324C8—10616no—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—21no—
329C22—21no—
330C22—44no—
331C22#1681110112yes—
332C22C24×Dic5311yescharacteristic
333C4—22no—
334C4—22no—
335C4—22no—
336C4—216no—
337C4—21no—
338C4—21no—
339C4#1681110112yes—
340C4#1681139014yes—
341C4C2×#1313326512yes—
342C2—22no—
343C2—41no—
344C2—41no—
345C2C2×#1681110111yescharacteristic
346C2C2×#1681139012yes—
subgroupG/H
center, FrattiniC2×C4C24×D53
commutator, socleC22×C53C24×C4
FittingC2×C53×#533C2
prime radicalC24×C53C22×C4
cosocleC2×C4×C53C25
hypercenterC2×#533D53
hypocenterC53#60249
LCSG ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×#533

Complements

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

NH ≅ G/Ntype
1. #20283821C2semidirect
2. #20283817C2semidirect
3. #20283810C2semidirect
4. Q8×#11719917C2semidirect
8. #16810754C22semidirect
9. #16810754C22semidirect
10. Q8×#88322C22semidirect
207. C53#60249semidirect

Maximal subgroups

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

subgroupindex
#202838212
#202838172
#202838102
Q8×#117199172
#202833532
#202833482
C2×C53×#5332
#6024953