Finite Group Databasebeta

← #27491274

Subgroups — #27491274

Order 13568.

738 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284651C212yesmaximal, complemented
2#20283278C211yesmaximal, characteristic, complemented
3#20276226C212yesmaximal, complemented
4#20276073C214yesmaximal, complemented
5#20275968C212yesmaximal, complemented
6#20275179C212yesmaximal, complemented
7#20274757C211yesmaximal, characteristic, complemented
8C2×C53×#469C211yesmaximal, characteristic
9C22×#11719917C2211yescharacteristic
10C2×#13133176C2211yescharacteristic
11#16810944—24no—
12#16810939—22no—
13#16810934—22no—
14C2×#13133153C2212yes—
15#16810663C2214yes—
16#16810651—22no—
17#16810649C2212yes—
18#16810649C2212yes—
19#16810646—22no—
20#16810628C2214yes—
21C2×#13133134C2214yes—
22#16810587C2212yes—
23#16810392C2214yes—
24#16810388C2212yes—
25C22×D8×C53C2211yescharacteristic
26C22×M16×C53C2211yescharacteristic
27C2×#121×C53C2212yes—
28C53×#469—22no—
29C53×#469—22no—
30C2×#106×C53C2211yescharacteristic
31C2×#102×C53C2212yes—
32C2×#11719917—21no—
33C2×#11719917—42no—
34C2×#11719917C2314yes—
35C2×#11719917C2312yes—
36C22×#88322—21no—
37#13133176—21no—
38#13133176—21no—
39#13133176D412yes—
40#13133176C2312yes—
41C2×C4⋊Dic53D412yes—
42C2×C22⋊Dic53—21no—
43C2×C4×Dic53—21no—
44#13133153—42no—
45#13133150—24no—
46#13133148—24no—
47#13133146—24no—
48#13133146—24no—
49#13133138—24no—
50#13133138—24no—
51#13133134—44no—
52C22⋊#88322—22no—
53C22⋊#88322—22no—
54C22×D4×C53—22no—
55C22×D4×C53C2312yes—
56C23×C4×C53C2311yescharacteristic
57C2×D8×C53—22no—
58C2×D8×C53—42no—
59C2×M16×C53—42no—
60C2×M16×C53D412yes—
61C2×M16×C53D412yes—
62C22×C8×C53C2311yescharacteristic
63C22×C8×C53C2311yescharacteristic
64#121×C53—22no—
65#121×C53—22no—
66#121×C53—22no—
67#121×C53—22no—
68C2×C4⋊C4×C53C2312yes—
69C2×C22⋊C4×C53—22no—
70#106×C53—22no—
71#102×C53—24no—
72C24×C53—22no—
73C24×C53#4111yescharacteristic
74C2×D4×C53—22no—
75C2×D4×C53—22no—
76C2×D4×C53—24no—
77C2×D4×C53—24no—
78C2×D4×C53—42no—
79C2×D4×C53—42no—
80C2×D4×C53—84no—
81C22×C4×C53—22no—
82C22×C4×C53—22no—
83C22×C4×C53—22no—
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53#4111yescharacteristic
89C22×C4×C53C2411yescharacteristic
90C22×Dic53—22no—
91D8×C53—44no—
92M16×C53—42no—
93M16×C53—42no—
94C2×C8×C53—22no—
95C2×C8×C53—22no—
96C2×C8×C53C2×D412yes—
97C2×C8×C53C2×D412yes—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—22no—
100C22⋊C4×C53—42no—
101C22⋊C4×C53—42no—
102C22⋊Dic53—22no—
103C22⋊Dic53—22no—
104C4⋊Dic53—22no—
105C4⋊Dic53C2×D414yes—
106C4×Dic53—22no—
107C4×Dic53—22no—
108#11719917—24no—
109#11719917—24no—
110#11719917—24no—
111#11719917—42no—
112C2×#88322—24no—
113C2×#88322—22no—
114C2×#88322—42no—
115C23×C53—22no—
116C23×C53—42no—
117C23×C53—42no—
118C23×C53—44no—
119C23×C53—82no—
120C23×C53—82no—
121C23×C53#12111yescharacteristic
122C23×C53#12111yescharacteristic
123C23×C53C2×#4111yescharacteristic
124D4×C53—42no—
125D4×C53—42no—
126D4×C53—44no—
127D4×C53—84no—
128D4×C53—84no—
129C2×C4×C53—22no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—22no—
135C2×C4×C53—22no—
136C2×C4×C53—42no—
137C2×C4×C53—82no—
138C2×C4×C53#12111yescharacteristic
139C2×C4×C53#12111yescharacteristic
140C2×C4×C53C22×D411yescharacteristic
141C2×C4×C53C22×D411yescharacteristic
142C2×C4×C53H(5,𝔽2)12yes—
143C2×C4×C532−1+412yes—
144C2×Dic53—24no—
145C2×Dic53—82no—
146C8×C53—22no—
147C8×C53—22no—
148#88322—42no—
149#88322—44no—
150#29791—531nomaximal, Sylow
151C22×C53—22no—
152C22×C53—21no—
153C22×C53—21no—
154C22×C53—22no—
155C22×C53—22no—
156C22×C53—82no—
157C22×C53—82no—
158C22×C53—82no—
159C22×C53—82no—
160C22×C53—164no—
161C22×C53C2×#12111yescharacteristic
162C22×C53#53912yes—
163C4×C53—21no—
164C4×C53—21no—
165C4×C53—82no—
166C4×C53#53411yescharacteristic
167C4×C53#53611yescharacteristic
168C4×C53#53712yes—
169Dic53—44no—
170#3094—532no—
171C2×#469D5311yescharacteristic
172#2905—531no—
173#2090—532no—
174#2063—534no—
175#2045—532no—
176#1900—532no—
177#1811—531no—
178C2×C53—22no—
179C2×C53—41no—
180C2×C53—41no—
181C2×C53—162no—
182C2×C53#261811yescharacteristic
183C2×C53#306812yes—
184C22×D8C2×D5311yescharacteristic
185C22×M16—531no—
186C22×M16C2×D5311yescharacteristic
187C2×#121C2×D5312yes—
188C2×#117—531no—
189#482—1064no—
190#481—1062no—
191#480—1062no—
192#469—22no—
193#469—22no—
194C2×#108—532no—
195#428—534no—
196C2×#106C2×D5311yescharacteristic
197#424—532no—
198#424—532no—
199#424—1062no—
200#423—1062no—
201#418—534no—
202C2×#102C2×D5312yes—
203C2×#100—534no—
204#407—532no—
205#343—534no—
206#341—532no—
207C53#2979111yescharacteristic, complemented, Sylow
208C22×D4—22no—
209C22×D4C22×D5312yes—
210C23×C4C22×D5311yescharacteristic
211C2×D8—22no—
212C2×D8—42no—
213C2×M16—42no—
214C2×M16—532no—
215C2×M16—534no—
216C2×M16—1061no—
217C2×M16—2122no—
218C2×M16C22⋊D5312yes—
219C2×M16C22⋊D5312yes—
220C22×C8—1061no—
221C22×C8C22×D5311yescharacteristic
222C22×C8C22×D5311yescharacteristic
223#121—22no—
224#121—22no—
225#121—22no—
226#121—22no—
227#117—532no—
228#117—532no—
229#117—1061no—
230#117—1061no—
231C2×C4⋊C4—532no—
232C2×C4⋊C4C22×D5312yes—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—1061no—
235C2×C42—1061no—
236#108—2122no—
237#107—1064no—
238#106—22no—
239#106—1064no—
240C4⋊C8—1064no—
241C4⋊C8—1064no—
242#102—24no—
243#102—1064no—
244#102—1064no—
245#100—2124no—
246C22⋊C8—1062no—
247C22⋊C8—1062no—
248C24—22no—
249C24#1171994711yescharacteristic
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—24no—
253C2×D4—24no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—84no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—1062no—
261C22×C4D4×D5311yescharacteristic
262C22×C4D4×D5311yescharacteristic
263C22×C4#1171994711yescharacteristic
264C22×C4C2×C22⋊D5311yescharacteristic
265C22×C4C2×C22⋊D5311yescharacteristic
266C22×C4C23×D5311yescharacteristic
267D8—44no—
268M16—42no—
269M16—42no—
270M16—1064no—
271M16—1064no—
272M16—1064no—
273M16—2122no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—1064no—
277C2×C8—1062no—
278C2×C8—2122no—
279C2×C8C2×C22⋊D5312yes—
280C2×C8C2×C22⋊D5312yes—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—534no—
284C4⋊C4—1062no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—1062no—
288C22⋊C4—1062no—
289C42—1062no—
290C42—1062no—
291C23—22no—
292C23—42no—
293C23—42no—
294C23—44no—
295C23—82no—
296C23—82no—
297C23#1313318811yescharacteristic
298C23#1313318811yescharacteristic
299C23C2×#1171994711yescharacteristic
300D4—42no—
301D4—42no—
302D4—44no—
303D4—84no—
304D4—84no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—42no—
313C2×C4—82no—
314C2×C4—1064no—
315C2×C4—4242no—
316C2×C4#1313318811yescharacteristic
317C2×C4#1313318811yescharacteristic
318C2×C4C2×D4×D5311yescharacteristic
319C2×C4C22×C22⋊D5311yescharacteristic
320C2×C4#1313326212yes—
321C2×C4#1313326412yes—
322C8—22no—
323C8—22no—
324C8—2122no—
325C8—2124no—
326C22—22no—
327C22—22no—
328C22—22no—
329C22—21no—
330C22—21no—
331C22—82no—
332C22—82no—
333C22—82no—
334C22—82no—
335C22—164no—
336C22C2×#1313318811yescharacteristic
337C22#1681121812yes—
338C4—21no—
339C4—21no—
340C4—82no—
341C4—2124no—
342C4#1681118211yescharacteristic
343C4#1681119411yescharacteristic
344C4#1681120012yes—
345C2—22no—
346C2—41no—
347C2—41no—
348C2—162no—
349C2#2028045911yescharacteristic
350C2#2028443912yes—
subgroupG/H
centerC22C2×#13133188
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#469C2
socleC22×C53C2×#121
hypercenterC2×#469D53
hypocenterC53#29791
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#469

Complements

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

NH ≅ G/Ntype
1. #20284651C2semidirect
2. #20283278C2semidirect
3. #20276226C2semidirect
4. #20276073C2semidirect
5. #20275968C2semidirect
6. #20275179C2semidirect
7. #20274757C2semidirect
207. C53#29791semidirect

Maximal subgroups

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

subgroupindex
#202846512
#202832782
#202762262
#202760732
#202759682
#202751792
#202747572
C2×C53×#4692
#2979153