Finite Group Databasebeta

← #27301014

Subgroups — #27301014

Order 13568.

466 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287817C211yesmaximal, characteristic, complemented
2#20283434C211yesmaximal, characteristic
3#20276344C211yesmaximal, characteristic, complemented
4#20276342C211yesmaximal, characteristic, complemented
5#20276335C211yesmaximal, characteristic, complemented
6#20272795C211yesmaximal, characteristic, complemented
7C53×#1375C211yesmaximal, characteristic, complemented
8C2×#13133223C2211yescharacteristic
9C2×#13133223C2211yescharacteristic
10#16811173C411yescharacteristic
11#16811041—22no—
12#16811039—22no—
13C2×#13133157C2211yescharacteristic
14C2×#13133157C2211yescharacteristic
15M32×D53—22no—
16#16810696—22no—
17#16810696—22no—
18#16810647C411yescharacteristic
19#16810589C411yescharacteristic
20#16810589C411yescharacteristic
21#16810408—22no—
22#16810406—22no—
23#16810404—22no—
24C2×M32×C53C2211yescharacteristic
25C2×M32×C53C2211yescharacteristic
26C2×#97×C53C2211yescharacteristic
27C2×#11719945C2×C411yescharacteristic
28C2×#11719945C2×C411yescharacteristic
29#13133223—21no—
30#13133223—21no—
31#13133223—21no—
32#13133223—21no—
33#13133223—21no—
34#13133223—21no—
35#13133223—21no—
36#13133223—21no—
37C2×#11719917C2×C411yescharacteristic
38C2×#11719917C2×C411yescharacteristic
39C2×#11719912—21no—
40C2×#11719912D412yes—
41C2×C8×D53—21no—
42C2×C8×D53D412yes—
43#13133217C811yescharacteristic
44#13133214C811yescharacteristic
45#13133205C811yescharacteristic
46#13133205C811yescharacteristic
47#13133191—22no—
48C4×D212—22no—
49#13133157—21no—
50#13133157—21no—
51#13133157—21no—
52#13133157—21no—
53#13133156—24no—
54C2×#11719908—21no—
55C2×#11719908D412yescomplemented
56C16×D53—24no—
57C4⋊#88322—22no—
58#13133129—22no—
59#13133129—22no—
60C22×C8×C53C2×C411yescharacteristic
61C22×C8×C53C2311yescharacteristic
62C2×C42×C53C2×C411yescharacteristic
63M32×C53—21no—
64M32×C53—21no—
65M32×C53—21no—
66M32×C53—21no—
67C2×C16×C53—21no—
68C2×C16×C53D412yescomplemented
69#97×C53—21no—
70#97×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53C22×C411yescharacteristic
73C2×C22⋊D53—21no—
74C2×C22⋊D53M1612yes—
75#11719945—21no—
76#11719945—21no—
77#11719945—21no—
78#11719945—21no—
79#11719945—21no—
80#11719945—21no—
81#11719945—21no—
82#11719945—21no—
83C2×D212—22no—
84C2×D212C2×C811yescharacteristic
85C2×D212C2×C811yescharacteristic
86C2×C4×D53—21no—
87C2×C4×D53C22⋊C412yes—
88C2×Dic106C2×C811yescharacteristic
89C2×Dic106C2×C811yescharacteristic
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
99C42×C53—21no—
100C42×C53C2×C811yescharacteristic
101C42×C53C2×C811yescharacteristic
102#11719921—22no—
103#11719921—22no—
104C4⋊Dic53—22no—
105#11719917—21no—
106#11719917—21no—
107#11719917—21no—
108#11719917—21no—
109C2×#88322—21no—
110C2×#88322C22⋊C412yes—
111#11719912—22no—
112#11719912—22no—
113#11719912—41no—
114#11719912—41no—
115C8×D53—22no—
116C8×D53—22no—
117C8×D53—41no—
118C8×D53—41no—
119C16×C53—22no—
120C16×C53—41no—
121#11719908—22no—
122#11719908—41no—
123C23×C53C2×M1611yescharacteristic
124C22×D53—21no—
125C22×D53C22⋊C812yes—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—81no—
129C2×C4×C53C2×C22⋊C411yescharacteristic
130C2×C4×C53C4×D411yescharacteristic
131C2×C4×C53C4×D411yescharacteristic
132C2×C4×C53C22×C811yescharacteristic
133C2×C4×C53#13111yescharacteristic
134C2×C4×C53#13111yescharacteristic
135C22⋊D53—22no—
136C22⋊D53—22no—
137C22⋊D53—41no—
138C22⋊D53—41no—
139C2×Dic53—21no—
140C2×Dic53C22⋊C812yes—
141D212—21no—
142D212—21no—
143D212—21no—
144D212—21no—
145D212—82no—
146C4×D53—22no—
147C4×D53—22no—
148C4×D53—41no—
149C4×D53—41no—
150Dic106—21no—
151Dic106—21no—
152Dic106—21no—
153Dic106—21no—
154C8×C53—41no—
155C8×C53#12011yescharacteristic
156C8×C53#12111yescharacteristic
157C8×C53#12111yescharacteristic
158C8×C53#12311yescharacteristic
159#88322—22no—
160#88322—41no—
161#12988—531nomaximal, Sylow
162C22×C53—41no—
163C22×C53C2×C22⋊C811yescharacteristic
164C22×C53#43511yescharacteristic
165C22×C53#43511yescharacteristic
166C2×D53—22no—
167C2×D53—42no—
168C2×D53—42no—
169C2×D53—41no—
170C4×C53—41no—
171C4×C53—41no—
172C4×C53#38611yescharacteristic
173C4×C53#40811yescharacteristic
174C4×C53C8×D411yescharacteristic
175C4×C53C8×D411yescharacteristic
176Dic53—22no—
177Dic53—41no—
178#3407—531no—
179#2932—531no—
180#2117—531no—
181#2117—531no—
182#2115—531no—
183#1375—531no—
184#1375D5311yescharacteristic, complemented
185C2×C53—21no—
186C2×C53—21no—
187C2×C53—21no—
188C2×C53#186911yescharacteristic
189D53—42no—
190D53—81no—
191C2×#131—531no—
192C2×#131—531no—
193#532—531no—
194#504—1062no—
195#504—1062no—
196C2×M32—531no—
197C2×M32—531no—
198C2×M32—1062no—
199C2×M32C2×D5311yescharacteristic
200C2×M32C2×D5311yescharacteristic
201#435—1062no—
202#435—1062no—
203#423—531no—
204#408—531no—
205#408—531no—
206C2×#97C2×D5311yescharacteristic
207#350—1062no—
208#350—1062no—
209#349—1062no—
210C53#1298811yescharacteristic, complemented, Sylow
211C2×#41—531no—
212C2×#41—531no—
213#131—1061no—
214#131—1061no—
215#131—1061no—
216#131—1061no—
217#131—1061no—
218#131—1061no—
219#131—1061no—
220#131—1061no—
221C2×M16—531no—
222C2×M16—532no—
223C2×M16—531no—
224C2×M16—1061no—
225C22×C8—532no—
226C22×C8—1061no—
227C22×C8C4×D5311yescharacteristic
228C22×C8C22×D5311yescharacteristic
229C4⋊Q8—531no—
230#127—531no—
231#124—531no—
232#124—531no—
233#121—1062no—
234C4×D4—1062no—
235C2×C42C4×D5311yescharacteristic
236M32—21no—
237M32—21no—
238M32—21no—
239M32—21no—
240M32—1064no—
241M32—1061no—
242M32—1061no—
243M32—1061no—
244M32—1061no—
245C2×C16—21no—
246C2×C16—532no—
247C2×C16—1064no—
248C2×C16—1061no—
249C2×C16D21212yescomplemented
250C4⋊C8—1062no—
251C22⋊C8—1062no—
252C22⋊C8—1062no—
253#97—21no—
254#97—21no—
255#41—1061no—
256#41—1061no—
257#41—1061no—
258#41—1061no—
259#41—1061no—
260#41—1061no—
261#41—1061no—
262#41—1061no—
263C2×Q8—531no—
264C2×Q8—531no—
265C2×D4—532no—
266C2×D4—531no—
267C2×D4—531no—
268C2×D4—1062no—
269C2×D4—1061no—
270C22×C4—21no—
271C22×C4—532no—
272C22×C4—1061no—
273C22×C4C2×C4×D5311yescharacteristic
274M16—1061no—
275M16—1061no—
276M16—1062no—
277M16—1062no—
278M16—1061no—
279M16—1061no—
280M16—2121no—
281M16—2121no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—532no—
286C2×C8—1062no—
287C2×C8—1062no—
288C2×C8—1061no—
289C2×C8—2121no—
290C2×C8—2121no—
291C2×C8C2×D21211yescharacteristic
292C2×C8#1171994511yescharacteristic
293C2×C8D4×D5311yescharacteristic
294C2×C8D4×D5311yescharacteristic
295C2×C8D4×D5311yescharacteristic
296C2×C8#1171994911yescharacteristic
297C4⋊C4—1062no—
298C22⋊C4—1062no—
299C22⋊C4—1062no—
300C42—21no—
301C42C8×D5311yescharacteristic
302C42C8×D5311yescharacteristic
303C16—22no—
304C16—41no—
305C16—1062no—
306C16—2121no—
307C23—532no—
308C23—1061no—
309C23M16×D5311yescharacteristic
310Q8—1061no—
311Q8—1061no—
312Q8—1061no—
313Q8—1061no—
314D4—1062no—
315D4—1062no—
316D4—1061no—
317D4—1061no—
318D4—1061no—
319D4—1061no—
320D4—4242no—
321D4—2121no—
322D4—2121no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—81no—
326C2×C4—532no—
327C2×C4—1062no—
328C2×C4—1062no—
329C2×C4—1061no—
330C2×C4—2121no—
331C2×C4—2121no—
332C2×C4C22⋊C4×D5311yescharacteristic
333C2×C4#1313317911yescharacteristic
334C2×C4C4×D21211yescharacteristic
335C2×C4C2×C8×D5311yescharacteristic
336C2×C4#1313322311yescharacteristic
337C2×C4#1313322511yescharacteristic
338C8—41no—
339C8—1062no—
340C8—2121no—
341C8#1313318711yescharacteristic
342C8#1313319011yescharacteristic
343C8#1313319211yescharacteristic
344C8#1313320211yescharacteristic
345C22—41no—
346C22—1062no—
347C22—2122no—
348C22—2122no—
349C22—2121no—
350C22C22⋊C8×D5311yescharacteristic
351C22#1681069311yescharacteristic
352C22#1681069611yescharacteristic
353C4—41no—
354C4—41no—
355C4—1062no—
356C4—2121no—
357C4#1681050111yescharacteristic
358C4#1681058811yescharacteristic
359C4#1681068811yescharacteristic
360C4C8×D21211yescharacteristic
361C2—21no—
362C2—21no—
363C2—21no—
364C2—4241no—
365C2—2122no—
366C2#2027500611yescharacteristic
subgroupG/H
centerC4#16810501
commutatorC2×C4×C53C22×C8
FrattiniC22×C8C22×D53
FittingC53×#1375C2
socleC2×C53#1869
prime radical#16811173C4
cosocleC22×C8×C53C23
hypercenter#1375D53
hypocenterC53#12988
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C8 ◁ #1375

Complements

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

NH ≅ G/Ntype
1. #20287817C2semidirect
3. #20276344C2semidirect
4. #20276342C2semidirect
5. #20276335C2semidirect
6. #20272795C2semidirect
7. C53×#1375C2semidirect
55. C2×#11719908D4semidirect
68. C2×C16×C53D4semidirect
184. #1375D53semidirect
210. C53#12988semidirect
249. C2×C16D212semidirect

Maximal subgroups

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

subgroupindex
#202878172
#202834342
#202763442
#202763422
#202763352
#202727952
C53×#13752
#1298853