Finite Group Databasebeta

← #27279295

Subgroups — #27279295

Order 13568.

476 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284723C211yesmaximal, characteristic
2#20283655C211yesmaximal, characteristic, complemented
3#20273373C211yesmaximal, characteristic, complemented
4#20273140C211yesmaximal, characteristic
5#20272612C211yesmaximal, characteristic, complemented
6#20272560C211yesmaximal, characteristic, complemented
7C53×#1955C211yesmaximal, characteristic
8#16811180C2211yescharacteristic
9#16810953C412yes—
10#16810913—21no—
11#16810828—21no—
12#16810774C412yes—
13#16810751—22no—
14#16810751—22no—
15C4×#11719923—22no—
16#16810676C2211yescharacteristic
17#16810676C2211yescharacteristic
18#16810649—21no—
19#16810647C2211yescharacteristic
20#16810578—21no—
21#16810371—21no—
22#16810368—21no—
23D4⋊#88322—22no—
24C2×C4×D4×C53C2211yescharacteristic
25C2×C4⋊C8×C53C2211yescharacteristic
26C2×C22⋊C8×C53—21no—
27C2×C22⋊C8×C53—21no—
28C2×#97×C53C2211yescharacteristic
29C2×#11719923—22no—
30C2×#11719917—21no—
31#13133217D411yescharacteristic
32#13133217D411yescharacteristic
33#13133209—21no—
34#13133196—21no—
35#13133196C2×C412yes—
36C4×Dic106—21no—
37C4×Dic106D412yes—
38#13133176—21no—
39#13133148—22no—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42C4⋊#88322—21no—
43C4⋊#88322D412yes—
44#13133146—24no—
45#13133140—22no—
46#13133138—22no—
47C22⋊#88322—21no—
48C22⋊#88322C2×C412yes—
49#13133127—22no—
50C4×#88322—21no—
51C4×#88322—21no—
52C8×Dic53—22no—
53C22×D4×C53C2×C411yescharacteristic
54C23×C4×C53—21no—
55C22×C8×C53—21no—
56C22×C8×C53D411yescharacteristic
57C22×C8×C53D411yescharacteristic
58C4×D4×C53—21no—
59C4×D4×C53—21no—
60C4×D4×C53—21no—
61C4×D4×C53—21no—
62C2×C4⋊C4×C53C2×C411yescharacteristic
63C2×C22⋊C4×C53—21no—
64C2×C42×C53C2311yescharacteristic
65C4⋊C8×C53—21no—
66C4⋊C8×C53—21no—
67C22⋊C8×C53—42no—
68C22⋊C8×C53—42no—
69#97×C53—21no—
70#97×C53—21no—
71C24×C53—21no—
72C2×D4×C53—21no—
73C2×D4×C53—21no—
74C2×D4×C53—42no—
75C22×C4×C53—21no—
76C22×C4×C53—41no—
77C22×C4×C53—41no—
78C22×C4×C53—41no—
79C22×C4×C53—41no—
80C22×C4×C53—41no—
81C22×C4×C53—41no—
82C22×C4×C53C22×C411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53#4111yescharacteristic
85C2×Dic106—21no—
86C2×Dic106C22⋊C412yes—
87C2×C8×C53—21no—
88C2×C8×C53—21no—
89C2×C8×C53—21no—
90C2×C8×C53—21no—
91C2×C8×C53—21no—
92C2×C8×C53—41no—
93C2×C8×C53—41no—
94C2×C8×C53—41no—
95C2×C8×C53SD1612yescomplemented
96C4⋊C4×C53—21no—
97C4⋊C4×C53—21no—
98C22⋊C4×C53—41no—
99C22⋊C4×C53—41no—
100C42×C53C2×D411yescharacteristic
101C42×C53C2×D411yescharacteristic
102C42×C53C2×D411yescharacteristic
103C42×C53#4111yescharacteristic
104C22⋊Dic53—22no—
105#11719923—44no—
106C4⋊Dic53—21no—
107C4⋊Dic53—21no—
108C4⋊Dic53—21no—
109C4⋊Dic53C22⋊C412yes—
110#11719919—22no—
111#11719919—22no—
112C4×Dic53—22no—
113#11719917—41no—
114#11719917—41no—
115C2×#88322—22no—
116C2×#88322—22no—
117C2×#88322—41no—
118C23×C53—22no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53—41no—
124C23×C53—41no—
125C23×C53C4×D411yescharacteristic
126D4×C53—22no—
127D4×C53—22no—
128D4×C53—41no—
129D4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—81no—
136C2×C4×C53—81no—
137C2×C4×C53—81no—
138C2×C4×C53—81no—
139C2×C4×C53C2×C22⋊C411yescharacteristic
140C2×C4×C53C4×D411yescharacteristic
141C2×C4×C53#12011yescharacteristic
142C2×C4×C53#12111yescharacteristic
143C2×C4×C53#12111yescharacteristic
144C2×C4×C53#12311yescharacteristic
145C2×C4×C53C2×SD1611yescharacteristic
146C2×C4×C53#13511yescharacteristic
147C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C53#13711yescharacteristic
149C2×Dic53—22no—
150C2×Dic53—22no—
151Dic106—42no—
152Dic106—81no—
153C8×C53—22no—
154C8×C53—41no—
155C8×C53—81no—
156#88322—42no—
157#88322—42no—
158#88322—81no—
159#9050—531nomaximal, Sylow
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—42no—
163C22×C53—42no—
164C22×C53—42no—
165C22×C53—81no—
166C22×C53—81no—
167C22×C53—81no—
168C22×C53—81no—
169C22×C53#38611yescharacteristic
170C22×C53C4×SD1611yescharacteristic
171C22×C53#44011yescharacteristic
172C4×C53—21no—
173C4×C53—21no—
174C4×C53—21no—
175C4×C53—21no—
176C4×C53—81no—
177C4×C53#44811yescharacteristic
178C4×C53#44911yescharacteristic
179C4×C53#46011yescharacteristic
180C4×C53#46411yescharacteristic
181Dic53—42no—
182Dic53—42no—
183#3101—531no—
184#2961—531no—
185#1955D5311yescharacteristic
186#1542—531no—
187#1486—531no—
188#1301—531no—
189#1281—531no—
190C2×C53—21no—
191C2×C53—21no—
192C2×C53—82no—
193C2×C53#188011yescharacteristic
194C2×C53#188111yescharacteristic
195C2×C53#189111yescharacteristic
196#533—531no—
197C2×C4×D4C2×D5311yescharacteristic
198#483—532no—
199#476—1061no—
200#461—1061no—
201#450—532no—
202#446—1062no—
203#446—1062no—
204C4×SD16—1062no—
205#431—531no—
206#431—531no—
207#424—1061no—
208#423—531no—
209C2×C4⋊C8C2×D5311yescharacteristic
210C2×C22⋊C8—21no—
211C2×C22⋊C8—21no—
212#405—1061no—
213C2×#97C2×D5311yescharacteristic
214#333—1061no—
215#332—1061no—
216D4⋊C8—1062no—
217C53#905011yescharacteristic, complemented, Sylow
218C22×D4C4×D5311yescharacteristic
219C23×C4—21no—
220C2×SD16—1062no—
221C2×M16—1061no—
222C22×C8—21no—
223C22×C8C22⋊D5311yescharacteristic
224C22×C8C22⋊D5311yescharacteristic
225C4⋊Q8—531no—
226C4⋊Q8—531no—
227#125—1061no—
228C22⋊Q8—532no—
229C22⋊Q8—1061no—
230C4×Q8—532no—
231C4×Q8—1061no—
232C4×D4—21no—
233C4×D4—21no—
234C4×D4—21no—
235C4×D4—21no—
236#117—1061no—
237C2×C4⋊C4C4×D5311yescharacteristic
238C2×C22⋊C4—21no—
239C2×C42C22×D5311yescharacteristic
240#106—1062no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—532no—
244C4⋊C8—1064no—
245C4⋊C8—1061no—
246C4⋊C8—1061no—
247C4⋊C8—1061no—
248#103—1062no—
249#102—1062no—
250C22⋊C8—42no—
251C22⋊C8—42no—
252C22⋊C8—532no—
253C22⋊C8—1061no—
254#97—21no—
255#97—21no—
256#97—1062no—
257C4×C8—1062no—
258C4×C8—1061no—
259C4×C8—1061no—
260C24—21no—
261C2×Q8—532no—
262C2×Q8—1061no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—42no—
266C22×C4—21no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4C2×C4×D5311yescharacteristic
274C22×C4#1171994511yescharacteristic
275C22×C4C2×C22⋊D5311yescharacteristic
276SD16—2124no—
277M16—2121no—
278M16—2121no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—1062no—
288C2×C8—1062no—
289C2×C8—2121no—
290C2×C8#1171992312yescomplemented
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—532no—
294C4⋊C4—1062no—
295C4⋊C4—1061no—
296C4⋊C4—1061no—
297C4⋊C4—1061no—
298C4⋊C4—1062no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—1062no—
302C42—1062no—
303C42D4×D5311yescharacteristic
304C42D4×D5311yescharacteristic
305C42D4×D5311yescharacteristic
306C42#1171994711yescharacteristic
307C23—22no—
308C23—41no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23C4×C22⋊D5311yescharacteristic
315Q8—4241no—
316Q8—2122no—
317D4—22no—
318D4—22no—
319D4—41no—
320D4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4—1062no—
331C2×C4—1062no—
332C2×C4C22⋊C4×D5311yescharacteristic
333C2×C4#1313317711yescharacteristic
334C2×C4#1313318511yescharacteristic
335C2×C4#1313318811yescharacteristic
336C2×C4#1313319011yescharacteristic
337C2×C4#1313319911yescharacteristic
338C2×C4C2×#1171992311yescharacteristic
339C2×C4#1313323711yescharacteristic
340C2×C4#1313324311yescharacteristic
341C2×C4#1313324711yescharacteristic
342C8—22no—
343C8—41no—
344C8—81no—
345C8—4241no—
346C8—2122no—
347C8—2122no—
348C22—21no—
349C22—21no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—81no—
354C22—81no—
355C22—81no—
356C22—81no—
357C22#1681049911yescharacteristic
358C22C4×#1171992311yescharacteristic
359C22#1681072311yescharacteristic
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—81no—
365C4—2122no—
366C4—2122no—
367C4#1681076111yescharacteristic
368C4#1681076611yescharacteristic
369C4#1681082011yescharacteristic
370C4#1681084211yescharacteristic
371C2—21no—
372C2—21no—
373C2—82no—
374C2#2027507311yescharacteristic
375C2#2027508011yescharacteristic
376C2#2027513011yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1955C2
socleC22×C53#386
prime radicalC22×D4×C53C2×C4
cosocleC2×C42×C53C23
hypercenter#1955D53
hypocenterC53#9050
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1955

Complements

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

NH ≅ G/Ntype
2. #20283655C2semidirect
3. #20273373C2semidirect
5. #20272612C2semidirect
6. #20272560C2semidirect
95. C2×C8×C53SD16semidirect
217. C53#9050semidirect
290. C2×C8#11719923semidirect

Maximal subgroups

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

subgroupindex
#202847232
#202836552
#202733732
#202731402
#202726122
#202725602
C53×#19552
#905053