Finite Group Databasebeta

← #27301004

Subgroups — #27301004

Order 13568.

456 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287817C211yesmaximal, characteristic, complemented
2C2×#16810743C211yesmaximal, characteristic
3#20276344C211yesmaximal, characteristic, complemented
4#20276343C211yesmaximal, characteristic, complemented
5#20272798C211yesmaximal, characteristic, complemented
6#20272794C211yesmaximal, characteristic, complemented
7C53×#2115C211yesmaximal, characteristic, complemented
8C2×#13133237C411yescharacteristic
9C2×#13133223C2211yescharacteristic
10#16811041—22no—
11#16811039—22no—
12C2×#13133157C2211yescharacteristic
13C2×#13133157C2211yescharacteristic
14M32×D53—22no—
15#16810743—22no—
16#16810743—22no—
17C2×#13133152C411yescharacteristic
18C2×#13133144C411yescharacteristic
19C2×#13133143C411yescharacteristic
20C2×#13133127C2211yescharacteristic
21#16810408—22no—
22#16810407—22no—
23C2×#131×C53C2211yescharacteristic
24C2×M32×C53C2211yescharacteristic
25C2×M32×C53C2211yescharacteristic
26C53×#349—22no—
27C2×#11719945C2×C411yescharacteristic
28#13133237—22no—
29#13133237—22no—
30C2×#11719925C811yescharacteristic
31C2×#11719923C811yescharacteristic
32C2×#11719924C811yescharacteristic
33C2×#11719922C811yescharacteristic
34#13133223—21no—
35#13133223—21no—
36#13133223D412yes—
37#13133223D412yes—
38C2×#11719917C2×C411yescharacteristic
39C2×#11719912—21no—
40C22×#88322C2×C411yescharacteristic
41C2×C8×D53—21no—
42C2×C4×Dic53C2×C411yescharacteristic
43#13133157—21no—
44#13133157—21no—
45#13133157—21no—
46#13133157—21no—
47#13133156—24no—
48C2×#11719908—21no—
49C2×#11719908D412yes—
50C16×D53—24no—
51#13133152—22no—
52#13133144—22no—
53#13133143—22no—
54#13133127—21no—
55#13133127—21no—
56C2×#41×C53C2×C411yescharacteristic
57#131×C53—21no—
58#131×C53—21no—
59#131×C53—21no—
60#131×C53—21no—
61C2×M16×C53—21no—
62C2×M16×C53C2×C411yescharacteristic
63C22×C8×C53—21no—
64C22×C8×C53C2311yescharacteristic
65M32×C53—21no—
66M32×C53—21no—
67M32×C53—21no—
68M32×C53D412yes—
69C2×C16×C53—21no—
70C2×C16×C53—21no—
71#41×C53—21no—
72#41×C53—21no—
73#41×C53—21no—
74#41×C53—21no—
75C2×Q8×C53C2×C811yescharacteristic
76C2×D4×C53—21no—
77C2×D4×C53C2×C811yescharacteristic
78C22×C4×C53—21no—
79C22×C4×C53C22×C411yescharacteristic
80C2×C22⋊D53—21no—
81C22×Dic53—21no—
82#11719945—21no—
83#11719945—21no—
84#11719945C22⋊C412yes—
85#11719945M1612yes—
86C2×D212C2×C811yescharacteristic
87C2×C4×D53—21no—
88C2×Dic106C2×C811yescharacteristic
89M16×C53—21no—
90M16×C53—21no—
91M16×C53—41no—
92M16×C53—41no—
93C2×C8×C53—21no—
94C2×C8×C53—41no—
95C2×C8×C53—41no—
96C2×C8×C53C2×D411yescharacteristic
97C2×C8×C53C2×D411yescharacteristic
98C2×C8×C53C2×D411yescharacteristic
99C2×C8×C53C2×D411yescharacteristic
100C2×C8×C53#4111yescharacteristic
101C2×C8×C53#4111yescharacteristic
102#11719925—22no—
103#11719924—22no—
104#11719923—22no—
105#11719922—22no—
106C4×Dic53—21no—
107C4×Dic53—21no—
108#11719917—21no—
109#11719917C22⋊C412yes—
110C2×#88322—21no—
111C2×#88322—21no—
112C2×#88322—21no—
113C2×#88322C2×C811yescharacteristic
114C2×#88322C2×C811yescharacteristic
115#11719912—22no—
116#11719912—22no—
117C8×D53—22no—
118C8×D53—22no—
119C16×C53—22no—
120C16×C53—41no—
121#11719908—22no—
122#11719908—41no—
123C23×C53—21no—
124C23×C53#13111yescharacteristic
125C22×D53—21no—
126Q8×C53—21no—
127Q8×C53—21no—
128D4×C53—21no—
129D4×C53—21no—
130D4×C53—41no—
131D4×C53—41no—
132C2×C4×C53—21no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53C2×C22⋊C411yescharacteristic
136C2×C4×C53C4×D411yescharacteristic
137C2×C4×C53C4×D411yescharacteristic
138C2×C4×C53C22×C811yescharacteristic
139C2×C4×C53C2×M1611yescharacteristic
140C2×C4×C53#13111yescharacteristic
141C22⋊D53—22no—
142C22⋊D53—22no—
143C2×Dic53—21no—
144C2×Dic53—41no—
145C2×Dic53—81no—
146D212—21no—
147D212C22⋊C812yes—
148C4×D53—22no—
149C4×D53—22no—
150Dic106—21no—
151Dic106C22⋊C812yes—
152C8×C53—41no—
153C8×C53#12011yescharacteristic
154C8×C53#12111yescharacteristic
155C8×C53#12111yescharacteristic
156C8×C53#12311yescharacteristic
157#88322—22no—
158#88322—21no—
159#88322—21no—
160#12986—531nomaximal, Sylow
161C22×C53—41no—
162C22×C53—41no—
163C22×C53—81no—
164C22×C53#40811yescharacteristic
165C22×C53C8×D411yescharacteristic
166C22×C53#43511yescharacteristic
167C2×D53—22no—
168C2×D53—81no—
169C4×C53—41no—
170C4×C53#38611yescharacteristic
171C4×C53C2×C22⋊C811yescharacteristic
172C4×C53C8×D411yescharacteristic
173C4×C53#43511yescharacteristic
174Dic53—22no—
175Dic53—81no—
176#3407—531no—
177C2×#444—531no—
178#2117—531no—
179#2117—531no—
180#2115D5311yescharacteristic, complemented
181#1377—531no—
182#1375—531no—
183C2×C53—21no—
184C2×C53—21no—
185C2×C53—21no—
186C2×C53—81no—
187C2×C53#186911yescharacteristic
188D53—42no—
189C2×#135—531no—
190C2×#131—531no—
191C2×#131C2×D5311yescharacteristic
192#504—1062no—
193#504—1062no—
194C2×M32—531no—
195C2×M32—531no—
196C2×M32—1062no—
197C2×M32C2×D5311yescharacteristic
198C2×M32C2×D5311yescharacteristic
199#444—1062no—
200#444—1062no—
201C2×#108—531no—
202C2×#104—531no—
203C2×#104—531no—
204C2×#97—531no—
205#350—1062no—
206#350—1062no—
207#349—22no—
208C53#1298611yescharacteristic, complemented, Sylow
209C2×#41—531no—
210C2×#41C4×D5311yescharacteristic
211#135—1062no—
212#135—1062no—
213C2×Q16—531no—
214C2×SD16—531no—
215C2×SD16—531no—
216C2×D8—531no—
217#131—21no—
218#131—21no—
219#131—21no—
220#131—21no—
221#131—532no—
222#131—532no—
223#131—1061no—
224#131—1061no—
225C2×M16—21no—
226C2×M16—531no—
227C2×M16—1061no—
228C2×M16C4×D5311yescharacteristic
229C22×C8—21no—
230C22×C8—531no—
231C22×C8—1061no—
232C22×C8C22×D5311yescharacteristic
233C2×C42—531no—
234M32—21no—
235M32—21no—
236M32—21no—
237M32—1064no—
238M32—1061no—
239M32—1061no—
240M32—1061no—
241M32—1061no—
242M32C22⋊D5312yes—
243C2×C16—21no—
244C2×C16—21no—
245C2×C16—532no—
246C2×C16—1064no—
247C2×C16—1061no—
248#108—1062no—
249#104—1062no—
250#104—1062no—
251#97—1061no—
252#97—1061no—
253#41—21no—
254#41—21no—
255#41—21no—
256#41—21no—
257#41—532no—
258#41—532no—
259#41—1061no—
260#41—1061no—
261C2×Q8—531no—
262C2×Q8C8×D5311yescharacteristic
263C2×D4—21no—
264C2×D4—531no—
265C2×D4—1061no—
266C2×D4C8×D5311yescharacteristic
267C22×C4—21no—
268C22×C4—1061no—
269C22×C4—1061no—
270C22×C4C2×C4×D5311yescharacteristic
271Q16—1062no—
272SD16—1062no—
273SD16—1062no—
274D8—1062no—
275M16—21no—
276M16—21no—
277M16—41no—
278M16—41no—
279M16—532no—
280M16—1061no—
281M16—1062no—
282M16—1062no—
283C2×C8—21no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—531no—
287C2×C8—531no—
288C2×C8—1061no—
289C2×C8—1061no—
290C2×C8—1061no—
291C2×C8—1062no—
292C2×C8—1062no—
293C2×C8#1171994511yescharacteristic
294C2×C8D4×D5311yescharacteristic
295C2×C8D4×D5311yescharacteristic
296C2×C8D4×D5311yescharacteristic
297C2×C8#1171994711yescharacteristic
298C2×C8C2×C22⋊D5311yescharacteristic
299C42—1061no—
300C42—1061no—
301C16—22no—
302C16—41no—
303C16—1062no—
304C16—2121no—
305C23—21no—
306C23—1061no—
307C23#1313322311yescharacteristic
308Q8—21no—
309Q8—21no—
310Q8—532no—
311Q8—1061no—
312D4—21no—
313D4—21no—
314D4—41no—
315D4—41no—
316D4—532no—
317D4—1062no—
318D4—1062no—
319D4—1061no—
320C2×C4—21no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—1061no—
324C2×C4—1062no—
325C2×C4—1062no—
326C2×C4—4241no—
327C2×C4—2121no—
328C2×C4C22⋊C4×D5311yescharacteristic
329C2×C4#1313317711yescharacteristic
330C2×C4C4×C22⋊D5311yescharacteristic
331C2×C4C2×C8×D5311yescharacteristic
332C2×C4M16×D5311yescharacteristic
333C2×C4#1313322511yescharacteristic
334C8—41no—
335C8—1062no—
336C8—1061no—
337C8—1061no—
338C8#1313318511yescharacteristic
339C8#1313318811yescharacteristic
340C8#1313319011yescharacteristic
341C8#1313319911yescharacteristic
342C22—41no—
343C22—41no—
344C22—81no—
345C22—1062no—
346C22—4241no—
347C22#1681058811yescharacteristic
348C22C8×C22⋊D5311yescharacteristic
349C22#1681069711yescharacteristic
350C4—41no—
351C4—1062no—
352C4—4241no—
353C4#1681049911yescharacteristic
354C4C22⋊C8×D5311yescharacteristic
355C4#1681068611yescharacteristic
356C4#1681069211yescharacteristic
357C2—21no—
358C2—21no—
359C2—21no—
360C2—81no—
361C2—2122no—
362C2#2027500411yescharacteristic
subgroupG/H
centerC4#16810499
commutatorC2×C4×C53C22×C8
FrattiniC22×C8C22×D53
FittingC53×#2115C2
socleC2×C53#1869
prime radicalC2×#13133237C4
cosocleC22×C8×C53C23
hypercenter#2115D53
hypocenterC53#12986
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C8 ◁ #2115

Complements

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

NH ≅ G/Ntype
1. #20287817C2semidirect
3. #20276344C2semidirect
4. #20276343C2semidirect
5. #20272798C2semidirect
6. #20272794C2semidirect
7. C53×#2115C2semidirect
180. #2115D53semidirect
208. C53#12986semidirect

Maximal subgroups

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

subgroupindex
#202878172
C2×#168107432
#202763442
#202763432
#202727982
#202727942
C53×#21152
#1298653