Finite Group Databasebeta

← #27301050

Subgroups — #27301050

Order 13568.

476 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284435C211yesmaximal, characteristic
2#20276442C211yesmaximal, characteristic, complemented
3C2×#16810429C211yesmaximal, characteristic, complemented
4C2×#16810407C211yesmaximal, characteristic, complemented
5#20272826C211yesmaximal, characteristic, complemented
6#20272808C211yesmaximal, characteristic, complemented
7C53×#2916C211yesmaximal, characteristic, complemented
8#16811421—21no—
9C2×#13133239C2211yescharacteristic
10C2×#13133157C2211yescharacteristic
11C22×#11719908C2211yescharacteristic
12#16810898C412yes—
13#16810893C412yes—
14#16810886—22no—
15C2×#13133153C2211yescharacteristic
16#16810605—21no—
17#16810429—22no—
18#16810429—22no—
19#16810407—22no—
20C53×#568—21no—
21C2×#131×C53C2211yescharacteristic
22C53×#430—21no—
23C53×#430—22no—
24C2×#108×C53—22no—
25C2×#108×C53C2211yescharacteristic
26C2×#108×C53C2211yescharacteristic
27C2×#11719945—21no—
28#13133248—41no—
29#13133247—22no—
30#13133243—41no—
31#13133242—22no—
32#13133239—21no—
33#13133239—21no—
34#13133239—21no—
35#13133239—21no—
36#13133237—42no—
37C2×Dic212C2×C411yescharacteristic
38C2×#11719913—21no—
39C2×D424C2×C411yescharacteristic
40C2×#11719917—21no—
41#13133157—21no—
42#13133157—21no—
43C2×#11719908—21no—
44C2×#11719908—21no—
45C2×#11719908—21no—
46C2×#11719908—21no—
47#13133153—21no—
48#13133153C2×C412yes—
49#13133135—22no—
50#13133133—22no—
51C2×#41×C53D411yescharacteristic
52#131×C53—21no—
53#131×C53—21no—
54#131×C53—42no—
55#131×C53—42no—
56#131×C53C2×C412yes—
57#131×C53D412yes—
58C2×M16×C53—21no—
59C2×M16×C53—24no—
60C2×M16×C53—21no—
61C2×M16×C53—21no—
62C2×M16×C53D411yescharacteristic
63C22×C8×C53—21no—
64C22×C8×C53C2311yescharacteristic
65#108×C53—24no—
66#108×C53—21no—
67#108×C53—21no—
68#108×C53D412yes—
69#108×C53D412yes—
70#41×C53—21no—
71#41×C53—21no—
72#41×C53C22⋊C412yes—
73#41×C53D812yes—
74C2×Q8×C53SD1611yescharacteristic
75C2×D4×C53—21no—
76C2×D4×C53SD1611yescharacteristic
77C22×C4×C53—21no—
78C22×C4×C53C2×D411yescharacteristic
79C2×C22⋊D53—41no—
80#11719945—41no—
81#11719945—41no—
82#11719945—41no—
83#11719945—41no—
84C2×D212—21no—
85C2×C4×D53—41no—
86C2×Dic106—21no—
87M16×C53—21no—
88M16×C53—22no—
89M16×C53—22no—
90M16×C53—22no—
91M16×C53—22no—
92M16×C53—44no—
93M16×C53—41no—
94M16×C53—41no—
95M16×C53C22⋊C412yes—
96C2×C8×C53—21no—
97C2×C8×C53—22no—
98C2×C8×C53—22no—
99C2×C8×C53—44no—
100C2×C8×C53—41no—
101C2×C8×C53—41no—
102C2×C8×C53C22×C411yescharacteristic
103C2×C8×C53C2×D411yescharacteristic
104C2×C8×C53C2×D411yescharacteristic
105C2×C8×C53C2×D411yescharacteristic
106C2×C8×C53#4111yescharacteristic
107C2×C8×C53#4111yescharacteristic
108#11719925—42no—
109#11719924—42no—
110#11719923—42no—
111#11719922—42no—
112#11719917—22no—
113#11719917—41no—
114C2×#88322—41no—
115Dic212—21no—
116Dic212—21no—
117D424—21no—
118D424—21no—
119#11719913—41no—
120#11719913—41no—
121#11719908—41no—
122#11719908—41no—
123C23×C53—21no—
124C23×C53#13711yescharacteristic
125C22×D53—41no—
126Q8×C53—21no—
127Q8×C53#10212yes—
128D4×C53—22no—
129D4×C53—22no—
130D4×C53—21no—
131D4×C53#10212yes—
132C2×C4×C53—21no—
133C2×C4×C53—22no—
134C2×C4×C53—22no—
135C2×C4×C53C2×C22⋊C411yescharacteristic
136C2×C4×C53#12011yescharacteristic
137C2×C4×C53#12311yescharacteristic
138C2×C4×C53C2×D811yescharacteristic
139C2×C4×C53C2×SD1611yescharacteristic
140C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
141C22⋊D53—81no—
142C22⋊D53—81no—
143C2×Dic53—41no—
144D212—41no—
145D212—41no—
146C4×D53—81no—
147C4×D53—81no—
148Dic106—41no—
149Dic106—41no—
150C8×C53—22no—
151C8×C53—42no—
152C8×C53—42no—
153C8×C53C4×D411yescharacteristic
154C8×C53C4×D411yescharacteristic
155C8×C53#12111yescharacteristic
156C8×C53#12111yescharacteristic
157#88322—42no—
158#12995—531nomaximal, Sylow
159C22×C53—22no—
160C22×C53—41no—
161C22×C53—81no—
162C22×C53#41711yescharacteristic
163C22×C53#44811yescharacteristic
164C22×C53#48011yescharacteristic
165C2×D53—81no—
166C2×D53—161no—
167C4×C53—22no—
168C4×C53#38611yescharacteristic
169C4×C53C2×#10211yescharacteristic
170C4×C53#44711yescharacteristic
171C4×C53#48111yescharacteristic
172Dic53—81no—
173#3067—531no—
174#2916D5311yescharacteristic, complemented
175#2143—531no—
176C2×#359—531no—
177C2×#350—531no—
178#1388—531no—
179#1381—531no—
180C2×C53—21no—
181C2×C53—21no—
182C2×C53—21no—
183C2×C53—42no—
184C2×C53#187411yescharacteristic
185D53—161no—
186#575—1061no—
187C2×#135—531no—
188#568—21no—
189C2×#131C2×D5311yescharacteristic
190C2×M32—531no—
191C22×C16—531no—
192#473—532no—
193#472—532no—
194#471—1062no—
195#430—21no—
196#430—22no—
197C2×#108—22no—
198C2×#108—531no—
199C2×#108C2×D5311yescharacteristic
200C2×#108C2×D5311yescharacteristic
201#413—1061no—
202#359—1062no—
203#359—1062no—
204#350—1062no—
205C53#1299511yescharacteristic, complemented, Sylow
206C2×#41—1061no—
207C2×#41C22⋊D5311yescharacteristic
208#137—1062no—
209#137—2121no—
210AGL(1,ℤ/8)—1062no—
211AGL(1,ℤ/8)—2121no—
212#135—1061no—
213#135—1061no—
214#135—1061no—
215#135—1061no—
216#135—2122no—
217C2×Q16—531no—
218C2×SD16—1061no—
219C2×D8—531no—
220#131—21no—
221#131—21no—
222#131—42no—
223#131—42no—
224#131C4×D5312yes—
225#131D21212yes—
226C2×M16—21no—
227C2×M16—24no—
228C2×M16—21no—
229C2×M16—21no—
230C2×M16—1061no—
231C2×M16C22⋊D5311yescharacteristic
232C22×C8—21no—
233C22×C8C22×D5311yescharacteristic
234M32—1061no—
235M32—1061no—
236C2×C16—1061no—
237C2×C16—1061no—
238C2×C16—1061no—
239C2×C16—1061no—
240#108—24no—
241#108—21no—
242#108—21no—
243#108—532no—
244#108—1061no—
245#108C22⋊D5312yes—
246#108C22⋊D5312yes—
247#101—1062no—
248#100—1062no—
249#41—21no—
250#41—21no—
251#41—2121no—
252#41—2121no—
253#41—2121no—
254#41—2121no—
255#41#1171992112yes—
256#41#1171992212yes—
257C2×Q8—1061no—
258C2×Q8#1171992411yescharacteristic
259C2×D4—21no—
260C2×D4—1061no—
261C2×D4—2121no—
262C2×D4#1171992411yescharacteristic
263C22×C4—21no—
264C22×C4—2121no—
265C22×C4C2×C22⋊D5311yescharacteristic
266Q16—1061no—
267Q16—1061no—
268Q16—2122no—
269SD16—2122no—
270SD16—2122no—
271SD16—2121no—
272SD16—2121no—
273D8—1061no—
274D8—1061no—
275D8—2122no—
276M16—21no—
277M16—22no—
278M16—22no—
279M16—22no—
280M16—22no—
281M16—44no—
282M16—41no—
283M16—41no—
284M16—1062no—
285M16—2121no—
286M16#1171992112yes—
287C2×C8—21no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—44no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—2121no—
294C2×C8C2×C4×D5311yescharacteristic
295C2×C8C2×D21211yescharacteristic
296C2×C8#1171994511yescharacteristic
297C2×C8#1171994511yescharacteristic
298C2×C8C2×C22⋊D5311yescharacteristic
299C2×C8C2×C22⋊D5311yescharacteristic
300C16—2121no—
301C16—2121no—
302C23—21no—
303C23—2121no—
304C23#1313324811yescharacteristic
305Q8—21no—
306Q8—2121no—
307Q8—2121no—
308Q8#1313313712yes—
309D4—22no—
310D4—22no—
311D4—21no—
312D4—4241no—
313D4—4241no—
314D4—2121no—
315D4—2121no—
316D4#1313313712yes—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—4241no—
321C2×C4—4241no—
322C2×C4—2121no—
323C2×C4C2×#1171992111yescharacteristic
324C2×C4#1313318611yescharacteristic
325C2×C4#1313320011yescharacteristic
326C2×C4C2×#1171992211yescharacteristic
327C2×C4C2×#1171992411yescharacteristic
328C2×C4#1313324311yescharacteristic
329C8—22no—
330C8—42no—
331C8—42no—
332C8—2122no—
333C8C4×C22⋊D5311yescharacteristic
334C8C4×C22⋊D5311yescharacteristic
335C8#1313319111yescharacteristic
336C8#1313319111yescharacteristic
337C22—22no—
338C22—41no—
339C22—81no—
340C22—8481no—
341C22—4241no—
342C22#1681062211yescharacteristic
343C22#1681076211yescharacteristic
344C22#1681093611yescharacteristic
345C4—22no—
346C4—4241no—
347C4#1681050211yescharacteristic
348C4C2×#1313313711yescharacteristic
349C4#1681075711yescharacteristic
350C4#1681094111yescharacteristic
351C2—21no—
352C2—21no—
353C2—21no—
354C2—42no—
355C2—8481no—
356C2#2027503211yescharacteristic
subgroupG/H
centerC4#16810757
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC53×#2916C2
socleC2×C53#1874
prime radical#20284435C2
cosocleC22×C8×C53C23
hypercenter#2916D53
hypocenterC53#12995
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#131 ◁ #2916

Complements

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

NH ≅ G/Ntype
2. #20276442C2semidirect
3. C2×#16810429C2semidirect
4. C2×#16810407C2semidirect
5. #20272826C2semidirect
6. #20272808C2semidirect
7. C53×#2916C2semidirect
174. #2916D53semidirect
205. C53#12995semidirect

Maximal subgroups

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

subgroupindex
#202844352
#202764422
C2×#168104292
C2×#168104072
#202728262
#202728082
C53×#29162
#1299553