Finite Group Databasebeta

← #27301268

Subgroups — #27301268

Order 13568.

512 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811058C211yesmaximal, characteristic
2C2×#16810435C211yesmaximal, characteristic, complemented
3#20276427C211yesmaximal, characteristic, complemented
4#20275581C211yesmaximal, characteristic, complemented
5#20272825C211yesmaximal, characteristic, complemented
6#20272797C211yesmaximal, characteristic, complemented
7C2×C53×#443C211yesmaximal, characteristic, complemented
8C22×D424C2211yescharacteristic
9#16811058—22no—
10#16811058C414yescomplemented
11C2×#13133160—21no—
12C2×#13133158—21no—
13C2×#13133157C2211yescharacteristic
14C2×#13133157C2211yescharacteristic
15C2×#13133151C2211yescharacteristic
16C2×#13133139—21no—
17#16810525—21no—
18#16810435—22no—
19#16810427—21no—
20#16810424—21no—
21C2×#135×C53C2211yescharacteristic
22C2×#131×C53—21no—
23C53×#443—22no—
24C53×#443—22no—
25C2×#108×C53C2211yescharacteristic
26C2×#104×C53—21no—
27C2×C4×C8×C53C2211yescharacteristic
28C22×D212—21no—
29C2×D424—21no—
30C2×D424—21no—
31C2×D424—21no—
32C2×D424—21no—
33C2×D424—21no—
34C2×D424C2×C412yes—
35C2×C4⋊Dic53—21no—
36C2×#11719921—42no—
37#13133160—24no—
38#13133158—24no—
39#13133157—21no—
40#13133157—21no—
41#13133157—21no—
42#13133157C2×C412yes—
43C2×#11719908—21no—
44C2×#11719908—21no—
45#13133151—21no—
46#13133151—21no—
47#13133139—41no—
48#13133139—41no—
49C2×#41×C53—21no—
50#135×C53—21no—
51#135×C53—21no—
52#135×C53C2×C412yes—
53#135×C53D412yes—
54C2×Q16×C53D411yescharacteristic
55C2×SD16×C53—21no—
56C2×D8×C53D411yescharacteristic
57#131×C53—24no—
58#131×C53—42no—
59C2×M16×C53—21no—
60C2×M16×C53—41no—
61C22×C8×C53—41no—
62C22×C8×C53D411yescharacteristic
63C22×C8×C53C2311yescharacteristic
64C2×C42×C53D411yescharacteristic
65#108×C53—21no—
66#108×C53D412yes—
67#104×C53—24no—
68C4×C8×C53—21no—
69C4×C8×C53—21no—
70C23×D53—41no—
71#41×C53—22no—
72#41×C53—22no—
73#41×C53—41no—
74#41×C53—41no—
75C2×Q8×C53—21no—
76C2×D4×C53—21no—
77C2×D4×C53—41no—
78C22×C4×C53—21no—
79C22×C4×C53—41no—
80C22×C4×C53C2×D411yescharacteristic
81C22×Dic53—41no—
82C2×D212—22no—
83C2×D212—41no—
84C2×D212—41no—
85C2×D212—41no—
86C2×D212—41no—
87C2×D212—41no—
88Q16×C53—21no—
89Q16×C53C22⋊C412yescomplemented
90SD16×C53—22no—
91SD16×C53—22no—
92D8×C53—21no—
93D8×C53C22⋊C412yescomplemented
94M16×C53—22no—
95M16×C53—44no—
96M16×C53—41no—
97C2×C8×C53—21no—
98C2×C8×C53—21no—
99C2×C8×C53—44no—
100C2×C8×C53—41no—
101C2×C8×C53D811yescharacteristic
102C2×C8×C53D811yescharacteristic
103C2×C8×C53C22×C411yescharacteristic
104C2×C8×C53C2×D411yescharacteristic
105C2×C8×C53C2×D411yescharacteristic
106C2×C8×C53C2×D411yescharacteristic
107C2×C8×C53#4111yescharacteristic
108C2×C8×C53#4111yescharacteristic
109C42×C53—21no—
110C42×C53—21no—
111#11719921—84no—
112C4⋊Dic53—41no—
113C4⋊Dic53—41no—
114D424—22no—
115D424—22no—
116D424—41no—
117D424—41no—
118#11719908—22no—
119#11719908—41no—
120C23×C53—41no—
121C23×C53AGL(1,ℤ/8)11yescharacteristic
122C22×D53—42no—
123C22×D53—82no—
124C22×D53—82no—
125C22×D53—81no—
126C22×D53—81no—
127Q8×C53—22no—
128Q8×C53—41no—
129D4×C53—22no—
130D4×C53—42no—
131D4×C53—42no—
132D4×C53—41no—
133C2×C4×C53—22no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—42no—
138C2×C4×C53—42no—
139C2×C4×C53C4×D411yescharacteristic
140C2×C4×C53#12011yescharacteristic
141C2×C4×C53#12111yescharacteristic
142C2×C4×C53C2×D811yescharacteristic
143C2×C4×C53#13511yescharacteristic
144C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
145C2×Dic53—82no—
146C2×Dic53—81no—
147D212—42no—
148D212—42no—
149D212—81no—
150D212—81no—
151C8×C53—21no—
152C8×C53—21no—
153C8×C53—42no—
154C8×C53C2×C22⋊C411yescharacteristic
155C8×C53C4×D411yescharacteristic
156C8×C53#12111yescharacteristic
157C8×C53#12311yescharacteristic
158#13029—531nomaximal, Sylow
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—42no—
162C22×C53—81no—
163C22×C53—81no—
164C22×C53#44211yescharacteristic
165C22×C53#44911yescharacteristic
166C22×C53#45911yescharacteristic
167C2×D53—84no—
168C2×D53—82no—
169C2×D53—162no—
170C2×D53—161no—
171C2×D53—161no—
172C4×C53—42no—
173C4×C53—42no—
174C4×C53#38611yescharacteristic
175C4×C53C4×D811yescharacteristic
176C4×C53#44711yescharacteristic
177C4×C53#46311yescharacteristic
178Dic53—161no—
179C2×#509—531no—
180C2×#443D5311yescharacteristic, complemented
181C2×#361—531no—
182#2139—531no—
183#1970—531no—
184#1387—531no—
185#1376—531no—
186C2×C53—21no—
187C2×C53—21no—
188C2×C53—82no—
189C2×C53#187911yescharacteristic
190C2×C53#221211yescharacteristic
191C2×C53#221311yescharacteristic
192D53—162no—
193C2×#135C2×D5311yescharacteristic
194C22×D8—531no—
195C2×#131—21no—
196#509—534no—
197#509—1062no—
198C2×SD32—1061no—
199C2×D16—1061no—
200C2×M32—531no—
201C2×M32—531no—
202#443—22no—
203#443—22no—
204C2×#108C2×D5311yescharacteristic
205C2×#107—531no—
206C2×#104—21no—
207C2×#102—1061no—
208C2×C4×C8C2×D5311yescharacteristic
209#390—1061no—
210#361—1062no—
211#358—1061no—
212#357—1061no—
213C53#1302911yescharacteristic, complemented, Sylow
214C2×#41—21no—
215C22×D4—1061no—
216#135—21no—
217#135—21no—
218#135C4×D5312yes—
219#135C22⋊D5312yes—
220C2×Q16D21211yescharacteristic
221C2×SD16—21no—
222C2×D8—532no—
223C2×D8—1061no—
224C2×D8—1061no—
225C2×D8—1061no—
226C2×D8—1061no—
227C2×D8—1061no—
228C2×D8D21211yescharacteristic
229#131—24no—
230#131—42no—
231C2×M16—21no—
232C2×M16—41no—
233C22×C8—41no—
234C22×C8C22⋊D5311yescharacteristic
235C22×C8C22×D5311yescharacteristic
236C2×C4⋊C4—1061no—
237C2×C22⋊C4—2122no—
238C2×C42C22⋊D5311yescharacteristic
239SD32—1064no—
240D16—1064no—
241M32—532no—
242M32—1061no—
243M32—1061no—
244M32—1061no—
245C2×C16—1061no—
246C2×C16—1061no—
247#108—21no—
248#108C22⋊D5312yes—
249#107—1061no—
250#107—1061no—
251#104—24no—
252#102—2121no—
253#102—2121no—
254C4×C8—21no—
255C4×C8—21no—
256C24—2121no—
257#41—22no—
258#41—22no—
259#41—41no—
260#41—41no—
261C2×Q8—21no—
262C2×D4—21no—
263C2×D4—41no—
264C2×D4—1062no—
265C2×D4—2121no—
266C2×D4—2121no—
267C2×D4—2121no—
268C2×D4—2121no—
269C2×D4—2121no—
270C22×C4—21no—
271C22×C4—41no—
272C22×C4—2121no—
273C22×C4C2×C22⋊D5311yescharacteristic
274Q16—21no—
275Q16#1171992112yescomplemented
276SD16—22no—
277SD16—22no—
278D8—21no—
279D8—1062no—
280D8—1062no—
281D8—2121no—
282D8—2121no—
283D8#1171992112yescomplemented
284M16—22no—
285M16—44no—
286M16—41no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—44no—
290C2×C8—41no—
291C2×C8#1171992211yescharacteristic
292C2×C8#1171992211yescharacteristic
293C2×C8C2×C4×D5311yescharacteristic
294C2×C8C2×D21211yescharacteristic
295C2×C8#1171994511yescharacteristic
296C2×C8#1171994511yescharacteristic
297C2×C8C2×C22⋊D5311yescharacteristic
298C2×C8C2×C22⋊D5311yescharacteristic
299C4⋊C4—2121no—
300C4⋊C4—2121no—
301C22⋊C4—4244no—
302C42—21no—
303C42—21no—
304C16—1062no—
305C16—2121no—
306C23—41no—
307C23—4242no—
308C23—4242no—
309C23—4241no—
310C23—4241no—
311C23—2122no—
312C23#1313324211yescharacteristic
313Q8—22no—
314Q8—41no—
315D4—22no—
316D4—42no—
317D4—42no—
318D4—41no—
319D4—4241no—
320D4—4241no—
321D4—2122no—
322D4—2122no—
323C2×C4—22no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—4242no—
330C2×C4—4241no—
331C2×C4C4×C22⋊D5311yescharacteristic
332C2×C4#1313318611yescharacteristic
333C2×C4#1313319111yescharacteristic
334C2×C4C2×#1171992211yescharacteristic
335C2×C4#1313323711yescharacteristic
336C2×C4#1313324311yescharacteristic
337C8—21no—
338C8—21no—
339C8—42no—
340C8C2×#1171992111yescharacteristic
341C8C4×C22⋊D5311yescharacteristic
342C8#1313319111yescharacteristic
343C8#1313320011yescharacteristic
344C22—21no—
345C22—21no—
346C22—42no—
347C22—81no—
348C22—81no—
349C22—8482no—
350C22—8481no—
351C22—8481no—
352C22—4244no—
353C22—4242no—
354C22#1681073311yescharacteristic
355C22#1681076811yescharacteristic
356C22#1681081811yescharacteristic
357C4—42no—
358C4—42no—
359C4—8481no—
360C4#1681050211yescharacteristic
361C4C4×#1171992211yescharacteristic
362C4#1681075711yescharacteristic
363C4#1681083811yescharacteristic
364C2—21no—
365C2—21no—
366C2—82no—
367C2—8482no—
368C2#2027506311yescharacteristic
369C2#2027678411yescharacteristic
370C2#2027679111yescharacteristic
subgroupG/H
centerC22#16810818
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC2×C53×#443C2
socleC22×C53#459
prime radicalC2×#16811058C2
cosocleC22×C8×C53C23
hypercenterC2×#443D53
hypocenterC53#13029
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×#443

Complements

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

NH ≅ G/Ntype
2. C2×#16810435C2semidirect
3. #20276427C2semidirect
4. #20275581C2semidirect
5. #20272825C2semidirect
6. #20272797C2semidirect
7. C2×C53×#443C2semidirect
10. #16811058C4semidirect
89. Q16×C53C22⋊C4semidirect
93. D8×C53C22⋊C4semidirect
180. C2×#443D53semidirect
213. C53#13029semidirect
275. Q16#11719921semidirect
283. D8#11719921semidirect

Maximal subgroups

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

subgroupindex
C2×#168110582
C2×#168104352
#202764272
#202755812
#202728252
#202727972
C2×C53×#4432
#1302953