Finite Group Databasebeta

← #27302301

Subgroups — #27302301

Order 13568.

420 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283418C211yesmaximal, characteristic
2#20276480C211yesmaximal, characteristic, complemented
3#20276344C211yesmaximal, characteristic, complemented
4#20276342C211yesmaximal, characteristic, complemented
5C2×#16810405C211yesmaximal, characteristic, complemented
6#20276319C211yesmaximal, characteristic, complemented
7C2×C53×#349C211yesmaximal, characteristic, complemented
8C2×#13133223C2211yescharacteristic
9#16811134C411yescharacteristic
10C2×#13133157C2211yescharacteristic
11C2×#13133157C2211yescharacteristic
12#16810699—22no—
13#16810692—22no—
14#16810649C411yescharacteristic
15#16810589C411yescharacteristic
16#16810587C411yescharacteristic
17#16810578C2211yescharacteristic
18#16810443—22no—
19#16810408—22no—
20#16810406—22no—
21#16810405—22no—
22#16810400—22no—
23C22×M16×C53C2211yescharacteristic
24C2×M32×C53C2211yescharacteristic
25C2×M32×C53C2211yescharacteristic
26C53×#349—22no—
27C2×#11719945C2×C411yescharacteristic
28#13133223—21no—
29#13133223—21no—
30#13133223—21no—
31#13133223—21no—
32C2×#11719917C2×C411yescharacteristic
33C2×#11719917C2×C411yescharacteristic
34C2×#11719912—21no—
35C2×C8×D53—21no—
36#13133200—21no—
37#13133196—21no—
38#13133191—21no—
39#13133186—21no—
40C4×C22⋊D53—22no—
41#13133176C2×C411yescharacteristic
42#13133157—21no—
43#13133157—21no—
44#13133157—21no—
45#13133157D412yescomplemented
46C2×#11719908—21no—
47C2×#11719908—21no—
48#13133146—22no—
49C22⋊#88322—22no—
50#13133129—22no—
51#13133127D412yes—
52C8×Dic53D412yes—
53C23×C4×C53C2×C411yescharacteristic
54C2×M16×C53—21no—
55C2×M16×C53—21no—
56C2×M16×C53—21no—
57C2×M16×C53—21no—
58C2×M16×C53—21no—
59C2×M16×C53—21no—
60C22×C8×C53C2×C411yescharacteristic
61C22×C8×C53C2311yescharacteristic
62M32×C53—21no—
63M32×C53—21no—
64M32×C53—21no—
65M32×C53D412yescomplemented
66C2×C16×C53—21no—
67C2×C16×C53—21no—
68C24×C53M1611yescharacteristic
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53M1611yescharacteristic
76C22×C4×C53C22×C411yescharacteristic
77C2×C22⋊D53—21no—
78C2×C22⋊D53—22no—
79#11719945—21no—
80#11719945—21no—
81#11719945—21no—
82#11719945—21no—
83C2×D212M1611yescharacteristic
84C2×C4×D53—21no—
85C2×Dic106M1611yescharacteristic
86M16×C53—41no—
87M16×C53—41no—
88M16×C53—41no—
89M16×C53—41no—
90C2×C8×C53—21no—
91C2×C8×C53—21no—
92C2×C8×C53—21no—
93C2×C8×C53C2×D411yescharacteristic
94C2×C8×C53C2×D411yescharacteristic
95C2×C8×C53C2×D411yescharacteristic
96C2×C8×C53C2×D411yescharacteristic
97C2×C8×C53#4111yescharacteristic
98C2×C8×C53#4111yescharacteristic
99C22⋊Dic53—22no—
100C22⋊Dic53—21no—
101#11719921—22no—
102C4⋊Dic53—21no—
103#11719919—22no—
104C4×Dic53C22⋊C412yes—
105#11719917—21no—
106#11719917—21no—
107#11719917—21no—
108#11719917—21no—
109C2×#88322—21no—
110C2×#88322C22⋊C412yes—
111#11719912—41no—
112#11719912—41no—
113C8×D53—41no—
114C8×D53—41no—
115C16×C53—22no—
116C16×C53—41no—
117#11719908—22no—
118#11719908—41no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53—81no—
123C23×C53C2×M1611yescharacteristic
124C22×D53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53C2×C22⋊C411yescharacteristic
133C2×C4×C53C4×D411yescharacteristic
134C2×C4×C53C4×D411yescharacteristic
135C2×C4×C53C2×M1611yescharacteristic
136C2×C4×C53#13111yescharacteristic
137C2×C4×C53#13111yescharacteristic
138C22⋊D53—41no—
139C22⋊D53—41no—
140C22⋊D53—82no—
141C2×Dic53—21no—
142C2×Dic53—22no—
143D212—21no—
144D212—21no—
145C4×D53—41no—
146C4×D53—41no—
147Dic106—21no—
148Dic106—21no—
149C8×C53—41no—
150C8×C53#12111yescharacteristic
151C8×C53#12111yescharacteristic
152C8×C53#12411yescharacteristic
153C8×C53#12711yescharacteristic
154#88322—22no—
155#88322—41no—
156#13201—531nomaximal, Sylow
157C22×C53—41no—
158C22×C53—41no—
159C22×C53—41no—
160C22×C53—41no—
161C22×C53—81no—
162C22×C53—81no—
163C22×C53#40711yescharacteristic
164C22×C53#43511yescharacteristic
165C22×C53#43511yescharacteristic
166C2×D53—42no—
167C2×D53—41no—
168C4×C53—41no—
169C4×C53#39011yescharacteristic
170C4×C53#40811yescharacteristic
171C4×C53#43511yescharacteristic
172C4×C53#43511yescharacteristic
173Dic53—42no—
174Dic53—41no—
175#2930—531no—
176#2153—531no—
177#2117—531no—
178#2117—531no—
179C2×#349—531no—
180C2×#349D5311yescharacteristic, complemented
181#2110—531no—
182C2×C53—21no—
183C2×C53—21no—
184C2×C53—21no—
185C2×C53—81no—
186C2×C53#195511yescharacteristic
187D53—81no—
188C2×#131—531no—
189C22×M16C2×D5311yescharacteristic
190#525—531no—
191C2×M32—531no—
192C2×M32—531no—
193C2×M32C2×D5311yescharacteristic
194C2×M32C2×D5311yescharacteristic
195#436—1062no—
196#435—1062no—
197#424—531no—
198#408—531no—
199#407—531no—
200#405—531no—
201#364—1062no—
202#350—1062no—
203#350—1062no—
204#349—22no—
205#349—1062no—
206#347—1062no—
207C53#1320111yescharacteristic, complemented, Sylow
208C2×#41—531no—
209C23×C4C4×D5311yescharacteristic
210#131—1061no—
211#131—1061no—
212#131—1061no—
213#131—1061no—
214C2×M16—21no—
215C2×M16—21no—
216C2×M16—21no—
217C2×M16—21no—
218C2×M16—21no—
219C2×M16—21no—
220C2×M16—531no—
221C2×M16—531no—
222C2×M16—1061no—
223C22×C8—1061no—
224C22×C8C4×D5311yescharacteristic
225C22×C8C22×D5311yescharacteristic
226#123—1061no—
227C22⋊Q8—1061no—
228#121—1061no—
229#120—1061no—
230C4×D4—1062no—
231#117—531no—
232M32—21no—
233M32—21no—
234M32—21no—
235M32—532no—
236M32—1061no—
237M32—1061no—
238M32—1061no—
239M32C22⋊D5312yescomplemented
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—1061no—
243C2×C16—1061no—
244C4⋊C8—1062no—
245C22⋊C8—1062no—
246C22⋊C8—1062no—
247#97—532no—
248C4×C8—532no—
249C24#1171991211yescharacteristic
250#41—1061no—
251#41—1061no—
252#41—1061no—
253#41—1061no—
254C2×Q8—531no—
255C2×D4—531no—
256C2×D4—1062no—
257C2×D4—1061no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—1061no—
265C22×C4#1171991211yescharacteristic
266C22×C4C2×C4×D5311yescharacteristic
267M16—41no—
268M16—41no—
269M16—41no—
270M16—41no—
271M16—1061no—
272M16—1061no—
273M16—1061no—
274M16—1061no—
275M16—2121no—
276M16—2121no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—532no—
281C2×C8—1061no—
282C2×C8—2121no—
283C2×C8—2121no—
284C2×C8#1171994511yescharacteristic
285C2×C8D4×D5311yescharacteristic
286C2×C8D4×D5311yescharacteristic
287C2×C8D4×D5311yescharacteristic
288C2×C8#1171994711yescharacteristic
289C2×C8C2×C22⋊D5311yescharacteristic
290C4⋊C4—1062no—
291C4⋊C4—1061no—
292C22⋊C4—1062no—
293C22⋊C4—1062no—
294C22⋊C4—1061no—
295C42—532no—
296C16—22no—
297C16—41no—
298C16—1062no—
299C16—2121no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—81no—
304C23—1061no—
305C23C2×#1171991211yescharacteristic
306Q8—1061no—
307Q8—1061no—
308D4—1061no—
309D4—1061no—
310D4—4242no—
311D4—2121no—
312D4—2121no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—1062no—
321C2×C4—1061no—
322C2×C4—2121no—
323C2×C4—2121no—
324C2×C4C22⋊C4×D5311yescharacteristic
325C2×C4#1313317711yescharacteristic
326C2×C4C4×C22⋊D5311yescharacteristic
327C2×C4M16×D5311yescharacteristic
328C2×C4#1313322311yescharacteristic
329C2×C4#1313322511yescharacteristic
330C8—41no—
331C8—1062no—
332C8—2121no—
333C8#1313318911yescharacteristic
334C8#1313319011yescharacteristic
335C8#1313320311yescharacteristic
336C8#1313321311yescharacteristic
337C22—41no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—81no—
342C22—81no—
343C22—2122no—
344C22—2121no—
345C22#1681058511yescharacteristic
346C22#1681069111yescharacteristic
347C22#1681069411yescharacteristic
348C4—41no—
349C4—2122no—
350C4—2121no—
351C4#1681052311yescharacteristic
352C4#1681058811yescharacteristic
353C4#1681069211yescharacteristic
354C4#1681069711yescharacteristic
355C2—21no—
356C2—21no—
357C2—21no—
358C2—81no—
359C2—4241no—
360C2#2027551111yescharacteristic
subgroupG/H
centerC4#16810523
commutatorC22×C4×C53C22×C4
FrattiniC22×C8C22×D53
FittingC2×C53×#349C2
socleC2×C53#1955
prime radical#16811134C4
cosocleC22×C8×C53C23
hypercenterC2×#349D53
hypocenterC53#13201
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C8 ◁ C2×#349

Complements

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

NH ≅ G/Ntype
2. #20276480C2semidirect
3. #20276344C2semidirect
4. #20276342C2semidirect
5. C2×#16810405C2semidirect
6. #20276319C2semidirect
7. C2×C53×#349C2semidirect
45. #13133157D4semidirect
65. M32×C53D4semidirect
180. C2×#349D53semidirect
207. C53#13201semidirect
239. M32C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
#202834182
#202764802
#202763442
#202763422
C2×#168104052
#202763192
C2×C53×#3492
#1320153