Finite Group Databasebeta

← #27301513

Subgroups — #27301513

Order 13568.

456 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
2C2×#16810434C211yesmaximal, characteristic, complemented
3#20276444C211yesmaximal, characteristic, complemented
4#20274774C211yesmaximal, characteristic, complemented
5#20272827C211yesmaximal, characteristic, complemented
6#20272814C211yesmaximal, characteristic, complemented
7C53×#3407C211yesmaximal, characteristic, complemented
8#16811421—21no—
9C2×#13133239C2211yescharacteristic
10#16810898C412yescomplemented
11#16810893C412yescomplemented
12#16810886—22no—
13C2×#13133153C2211yescharacteristic
14C2×#13133153C2211yescharacteristic
15C2×#13133149C2211yescharacteristic
16#16810624—21no—
17#16810605—21no—
18#16810434—22no—
19#16810431—22no—
20#16810391—21no—
21C2×#131×C53C2211yescharacteristic
22C53×#504—22no—
23C53×#504—22no—
24C2×M32×C53—22no—
25C2×M32×C53C2211yescharacteristic
26C2×M32×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—
41C2×#11719917—21no—
42C2×C4⋊Dic53—21no—
43#13133153—21no—
44#13133153—21no—
45#13133153—21no—
46#13133153C2×C412yes—
47#13133149—21no—
48#13133149—21no—
49#13133141—41no—
50#13133138—41no—
51#13133135—22no—
52#13133133—22no—
53C2×#41×C53D411yescharacteristic
54#131×C53—21no—
55#131×C53—21no—
56#131×C53C2×C412yes—
57#131×C53D412yes—
58C2×M16×C53—21no—
59C2×M16×C53D411yescharacteristic
60C22×C8×C53—21no—
61C22×C8×C53C2311yescharacteristic
62M32×C53—24no—
63M32×C53—21no—
64M32×C53—21no—
65M32×C53—21no—
66M32×C53D412yes—
67C2×C16×C53—24no—
68C2×C16×C53—21no—
69C2×C16×C53D412yes—
70#41×C53—21no—
71#41×C53—21no—
72#41×C53C22⋊C412yes—
73#41×C53SD1612yes—
74C2×Q8×C53D811yescharacteristic
75C2×D4×C53—21no—
76C2×D4×C53D811yescharacteristic
77C22×C4×C53—21no—
78C22×C4×C53C2×D411yescharacteristic
79C2×C22⋊D53—41no—
80C22×Dic53—41no—
81#11719945—41no—
82#11719945—41no—
83#11719945—41no—
84#11719945—41no—
85C2×D212—21no—
86C2×C4×D53—41no—
87C2×Dic106—21no—
88M16×C53—21no—
89M16×C53—22no—
90M16×C53—22no—
91M16×C53C22⋊C412yes—
92C2×C8×C53—21no—
93C2×C8×C53—22no—
94C2×C8×C53—22no—
95C2×C8×C53C22×C411yescharacteristic
96C2×C8×C53C2×D411yescharacteristic
97C2×C8×C53C2×D411yescharacteristic
98C2×C8×C53C2×D411yescharacteristic
99C2×C8×C53#4111yescharacteristic
100C2×C8×C53#4111yescharacteristic
101#11719925—42no—
102#11719924—42no—
103#11719923—42no—
104#11719922—42no—
105C4⋊Dic53—41no—
106C4⋊Dic53—41no—
107#11719917—22no—
108#11719917—41no—
109#11719917—41no—
110#11719917—41no—
111C2×#88322—41no—
112C2×#88322—41no—
113Dic212—21no—
114Dic212—21no—
115D424—21no—
116D424—21no—
117#11719913—41no—
118#11719913—41no—
119C16×C53—22no—
120C16×C53—22no—
121C23×C53—21no—
122C23×C53#13511yescharacteristic
123C22×D53—41no—
124Q8×C53—21no—
125Q8×C53#10212yes—
126D4×C53—22no—
127D4×C53—22no—
128D4×C53—21no—
129D4×C53#10212yes—
130C2×C4×C53—21no—
131C2×C4×C53—22no—
132C2×C4×C53—22no—
133C2×C4×C53C2×C22⋊C411yescharacteristic
134C2×C4×C53#12111yescharacteristic
135C2×C4×C53#12111yescharacteristic
136C2×C4×C53C2×D811yescharacteristic
137C2×C4×C53C2×SD1611yescharacteristic
138C2×C4×C53#13511yescharacteristic
139C22⋊D53—81no—
140C22⋊D53—81no—
141C2×Dic53—41no—
142C2×Dic53—81no—
143C2×Dic53—161no—
144D212—41no—
145D212—41no—
146C4×D53—81no—
147C4×D53—81no—
148Dic106—41no—
149Dic106—41no—
150C8×C53—22no—
151C8×C53C4×D411yescharacteristic
152C8×C53C4×D411yescharacteristic
153C8×C53#12011yescharacteristic
154C8×C53#12311yescharacteristic
155#88322—42no—
156#88322—81no—
157#13070—531nomaximal, Sylow
158C22×C53—22no—
159C22×C53—41no—
160C22×C53—81no—
161C22×C53#41611yescharacteristic
162C22×C53#46511yescharacteristic
163C22×C53#46611yescharacteristic
164C2×D53—81no—
165C2×D53—161no—
166C4×C53—22no—
167C4×C53#38611yescharacteristic
168C4×C53C2×#10211yescharacteristic
169C4×C53#46511yescharacteristic
170C4×C53#46611yescharacteristic
171Dic53—81no—
172Dic53—161no—
173#3407D5311yescharacteristic, complemented
174#3067—531no—
175C2×#360—531no—
176#2143—531no—
177#1815—531no—
178#1388—531no—
179#1383—531no—
180C2×C53—21no—
181C2×C53—21no—
182C2×C53—21no—
183C2×C53—42no—
184C2×C53#189311yescharacteristic
185D53—161no—
186#575—1061no—
187C2×#135—531no—
188C2×#131C2×D5311yescharacteristic
189#504—22no—
190#504—22no—
191C2×M32—22no—
192C2×M32C2×D5311yescharacteristic
193C2×M32C2×D5311yescharacteristic
194#473—532no—
195#472—532no—
196#471—1062no—
197C2×#108—531no—
198C2×#108—531no—
199C2×#106—531no—
200#417—1061no—
201#413—1061no—
202#360—1062no—
203#359—1062no—
204#343—1061no—
205C53#1307011yescharacteristic, complemented, Sylow
206C2×#41—1061no—
207C2×#41D21211yescharacteristic
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#131C4×D5312yes—
223#131C22⋊D5312yes—
224C2×M16—21no—
225C2×M16—1061no—
226C2×M16—1061no—
227C2×M16D21211yescharacteristic
228C22×C8—21no—
229C22×C8C22×D5311yescharacteristic
230C2×C4⋊C4—1061no—
231M32—24no—
232M32—21no—
233M32—21no—
234M32—21no—
235M32C22⋊D5312yes—
236C2×C16—24no—
237C2×C16—21no—
238C2×C16C22⋊D5312yes—
239#108—532no—
240#108—1061no—
241#108—1061no—
242#108—1061no—
243#106—1061no—
244#106—1061no—
245#103—2121no—
246#102—2121no—
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#1171991312yes—
256#41#1171992112yes—
257C2×Q8—1061no—
258C2×Q8D42411yescharacteristic
259C2×D4—21no—
260C2×D4—1061no—
261C2×D4—2121no—
262C2×D4D42411yescharacteristic
263C22×C4—21no—
264C22×C4—2121no—
265C22×C4—2121no—
266C22×C4C2×D21211yescharacteristic
267Q16—1061no—
268Q16—1061no—
269Q16—2122no—
270SD16—2122no—
271SD16—2122no—
272SD16—2121no—
273SD16—2121no—
274D8—1061no—
275D8—1061no—
276D8—2122no—
277M16—21no—
278M16—22no—
279M16—22no—
280M16—1062no—
281M16—2121no—
282M16—2121no—
283M16—2121no—
284M16#1171992112yes—
285C2×C8—21no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—2121no—
289C2×C8—2121no—
290C2×C8C2×C4×D5311yescharacteristic
291C2×C8#1171994511yescharacteristic
292C2×C8#1171994511yescharacteristic
293C2×C8C2×C22⋊D5311yescharacteristic
294C2×C8C2×C22⋊D5311yescharacteristic
295C2×C8C2×C22⋊D5311yescharacteristic
296C4⋊C4—2121no—
297C4⋊C4—2121no—
298C16—22no—
299C16—22no—
300C23—21no—
301C23—2121no—
302C23#1313323911yescharacteristic
303Q8—21no—
304Q8—2121no—
305Q8—2121no—
306Q8#1313313912yes—
307D4—22no—
308D4—22no—
309D4—21no—
310D4—4241no—
311D4—4241no—
312D4—2121no—
313D4—2121no—
314D4#1313313912yes—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—8481no—
319C2×C4—4241no—
320C2×C4—4241no—
321C2×C4—4241no—
322C2×C4—2121no—
323C2×C4C2×#1171992111yescharacteristic
324C2×C4#1313319111yescharacteristic
325C2×C4#1313319111yescharacteristic
326C2×C4C2×D42411yescharacteristic
327C2×C4C2×#1171991311yescharacteristic
328C2×C4#1313323911yescharacteristic
329C8—22no—
330C8—4241no—
331C8—2122no—
332C8C4×C22⋊D5311yescharacteristic
333C8C4×C22⋊D5311yescharacteristic
334C8#1313318611yescharacteristic
335C8#1313320011yescharacteristic
336C22—22no—
337C22—41no—
338C22—81no—
339C22—8481no—
340C22—4241no—
341C22#1681062011yescharacteristic
342C22#1681085211yescharacteristic
343C22#1681085911yescharacteristic
344C4—22no—
345C4—8481no—
346C4—4241no—
347C4#1681050211yescharacteristic
348C4C2×#1313313911yescharacteristic
349C4#1681085211yescharacteristic
350C4#1681085911yescharacteristic
351C2—21no—
352C2—21no—
353C2—21no—
354C2—42no—
355C2—8481no—
356C2#2027514311yescharacteristic
subgroupG/H
centerC2#20275143
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC53×#3407C2
socleC2×C53#1893
prime radical#20284435C2
cosocleC22×C8×C53C23
hypercenter#3407D53
hypocenterC53#13070
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ #3407

Complements

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

NH ≅ G/Ntype
2. C2×#16810434C2semidirect
3. #20276444C2semidirect
4. #20274774C2semidirect
5. #20272827C2semidirect
6. #20272814C2semidirect
7. C53×#3407C2semidirect
10. #16810898C4semidirect
11. #16810893C4semidirect
173. #3407D53semidirect
205. C53#13070semidirect

Maximal subgroups

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

subgroupindex
#202844352
C2×#168104342
#202764442
#202747742
#202728272
#202728142
C53×#34072
#1307053