Finite Group Databasebeta

← #27301019

Subgroups — #27301019

Order 13568.

470 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20287817C211yesmaximal, characteristic, complemented
2#20283420C211yesmaximal, characteristic
3#20276335C211yesmaximal, characteristic, complemented
4C2×#16810404C211yesmaximal, characteristic, complemented
5C2×#16810404C211yesmaximal, characteristic, complemented
6#20272801C211yesmaximal, characteristic, complemented
7C53×#1378C211yesmaximal, characteristic, complemented
8C2×#13133223C2211yescharacteristic
9C22×#11719912C2211yescharacteristic
10#16811135C411yescharacteristic
11#16811041—22no—
12#16811039—22no—
13C2×#13133157C2211yescharacteristic
14C2×#13133157C2211yescharacteristic
15M32×D53—22no—
16#16810702—22no—
17#16810696—22no—
18#16810650C411yescharacteristic
19#16810591C411yescharacteristic
20#16810586C411yescharacteristic
21#16810404—22no—
22#16810404—22no—
23#16810404—22no—
24C2×M32×C53C2211yescharacteristic
25C2×M32×C53C2211yescharacteristic
26C53×#405C2211yescharacteristic
27C2×#11719945C2×C411yescharacteristic
28C22×C4×D53C2×C411yescharacteristic
29#13133223—21no—
30#13133223—21no—
31#13133223—21no—
32#13133223—21no—
33C2×#11719917C2×C411yescharacteristic
34C2×#11719912—21no—
35C2×#11719912—21no—
36C2×#11719912—21no—
37C2×#11719912—21no—
38C2×#11719912—21no—
39C2×#11719912—21no—
40C2×#11719912D412yes—
41C22×#88322C2×C411yescharacteristic
42C2×C8×D53D412yes—
43#13133201C812yes—
44#13133197—21no—
45#13133192—21no—
46#13133187C812yes—
47C4×D212—22no—
48#13133157—21no—
49#13133157—21no—
50#13133157—21no—
51#13133157D412yescomplemented
52#13133156—24no—
53C2×#11719908—21no—
54C2×#11719908—21no—
55C16×D53—24no—
56C4⋊#88322—22no—
57#13133129—22no—
58#13133129—22no—
59C2×M16×C53C2×C411yescharacteristic
60C22×C8×C53C2311yescharacteristic
61#117×C53C2×C411yescharacteristic
62M32×C53—21no—
63M32×C53—21no—
64M32×C53—21no—
65M32×C53D412yescomplemented
66C2×C16×C53—21no—
67C2×C16×C53—21no—
68#97×C53—21no—
69C4×C8×C53—21no—
70C23×D53C2×C811yescharacteristic
71C22×C4×C53C22×C411yescharacteristic
72C2×C22⋊D53C2×C812yes—
73C22×Dic53C2×C811yescharacteristic
74#11719945—21no—
75#11719945—21no—
76#11719945—21no—
77#11719945—21no—
78C2×D212—22no—
79C2×D212M1611yescharacteristic
80C2×C4×D53—21no—
81C2×C4×D53—21no—
82C2×C4×D53—21no—
83C2×C4×D53—21no—
84C2×C4×D53—21no—
85C2×C4×D53—21no—
86C2×C4×D53C22⋊C412yes—
87C2×Dic106M1611yescharacteristic
88M16×C53—21no—
89M16×C53—21no—
90C2×C8×C53—21no—
91C2×C8×C53C2×D411yescharacteristic
92C2×C8×C53C2×D411yescharacteristic
93C2×C8×C53C2×D411yescharacteristic
94C2×C8×C53C2×D411yescharacteristic
95C2×C8×C53#4111yescharacteristic
96C2×C8×C53#4111yescharacteristic
97C4⋊C4×C53—21no—
98C22⋊C4×C53C2×C812yes—
99C42×C53—21no—
100#11719921—22no—
101#11719921—22no—
102C4⋊Dic53—22no—
103#11719917—21no—
104#11719917—21no—
105C2×#88322—21no—
106C2×#88322—21no—
107C2×#88322—21no—
108C2×#88322C22⋊C412yes—
109#11719912—22no—
110#11719912—22no—
111#11719912—41no—
112#11719912—41no—
113#11719912—41no—
114#11719912—41no—
115C8×D53—22no—
116C8×D53—22no—
117C16×C53—22no—
118C16×C53—41no—
119#11719908—22no—
120#11719908—41no—
121C23×C53C22×C811yescharacteristic
122C22×D53—21no—
123C22×D53—21no—
124C22×D53—21no—
125C22×D53—81no—
126C22×D53C22⋊C812yes—
127C2×C4×C53—22no—
128C2×C4×C53C2×C22⋊C411yescharacteristic
129C2×C4×C53C4×D411yescharacteristic
130C2×C4×C53C4×D411yescharacteristic
131C2×C4×C53C2×M1611yescharacteristic
132C2×C4×C53#13111yescharacteristic
133C2×C4×C53#13111yescharacteristic
134C22⋊D53—22no—
135C22⋊D53—22no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53C22⋊C812yes—
140D212—21no—
141D212—21no—
142D212—82no—
143C4×D53—22no—
144C4×D53—22no—
145C4×D53—41no—
146C4×D53—41no—
147C4×D53—41no—
148C4×D53—41no—
149Dic106—21no—
150Dic106—21no—
151C8×C53—41no—
152C8×C53#12011yescharacteristic
153C8×C53#12111yescharacteristic
154C8×C53#12111yescharacteristic
155C8×C53#12311yescharacteristic
156#88322—22no—
157#88322—41no—
158#12989—531nomaximal, Sylow
159C22×C53—41no—
160C22×C53C2×C22⋊C811yescharacteristic
161C22×C53C8×D411yescharacteristic
162C22×C53C8×D411yescharacteristic
163C2×D53—22no—
164C2×D53—42no—
165C2×D53—41no—
166C2×D53—41no—
167C2×D53—41no—
168C2×D53—81no—
169C2×D53—81no—
170C4×C53—42no—
171C4×C53#38611yescharacteristic
172C4×C53#40811yescharacteristic
173C4×C53#43511yescharacteristic
174C4×C53#43511yescharacteristic
175Dic53—22no—
176Dic53—41no—
177#3407—531no—
178#2930—531no—
179#2115—531no—
180C2×#349—531no—
181C2×#349—531no—
182#1378—531no—
183#1378D5311yescharacteristic, complemented
184C2×C53—21no—
185C2×C53—21no—
186C2×C53—21no—
187C2×C53#186911yescharacteristic
188D53—42no—
189D53—81no—
190C2×#131—531no—
191C22×M16—531no—
192#525—531no—
193#504—1062no—
194#504—1062no—
195C2×M32—531no—
196C2×M32—531no—
197C2×M32—1062no—
198C2×M32C2×D5311yescharacteristic
199C2×M32C2×D5311yescharacteristic
200#436—1062no—
201#435—1062no—
202#424—531no—
203#408—531no—
204#407—531no—
205#405C2×D5311yescharacteristic
206#349—1062no—
207#349—1062no—
208#349—1062no—
209C53#1298911yescharacteristic, complemented, Sylow
210C2×#41—531no—
211C23×C4—531no—
212#131—1061no—
213#131—1061no—
214#131—1061no—
215#131—1061no—
216C2×M16—531no—
217C2×M16—532no—
218C2×M16—1061no—
219C2×M16—1061no—
220C2×M16—1061no—
221C2×M16—1061no—
222C2×M16—1061no—
223C2×M16—1061no—
224C2×M16C4×D5311yescharacteristic
225C22×C8—532no—
226C22×C8—531no—
227C22×C8C22×D5311yescharacteristic
228#123—532no—
229C22⋊Q8—1061no—
230#121—1061no—
231#120—532no—
232C4×D4—1062no—
233#117C4×D5311yescharacteristic
234M32—21no—
235M32—21no—
236M32—21no—
237M32—532no—
238M32—1064no—
239M32—1061no—
240M32—1061no—
241M32—1061no—
242M32D21212yescomplemented
243C2×C16—21no—
244C2×C16—21no—
245C2×C16—1064no—
246C2×C16—1061no—
247C2×C16—1061no—
248C4⋊C8—1062no—
249C22⋊C8—1062no—
250C22⋊C8—1062no—
251#97—21no—
252C4×C8—21no—
253C24—531no—
254#41—1061no—
255#41—1061no—
256#41—1061no—
257#41—1061no—
258C2×Q8—531no—
259C2×D4—532no—
260C2×D4—531no—
261C2×D4—1062no—
262C22×C4—532no—
263C22×C4—531no—
264C22×C4—1061no—
265C22×C4—1061no—
266C22×C4—1061no—
267C22×C4—1061no—
268C22×C4—1061no—
269C22×C4—1061no—
270C22×C4C2×C4×D5311yescharacteristic
271M16—21no—
272M16—21no—
273M16—1061no—
274M16—1061no—
275M16—1062no—
276M16—1062no—
277M16—2121no—
278M16—2121no—
279M16—2121no—
280M16—2121no—
281C2×C8—21no—
282C2×C8—532no—
283C2×C8—1062no—
284C2×C8—1062no—
285C2×C8—1061no—
286C2×C8—1061no—
287C2×C8—1061no—
288C2×C8C2×D21211yescharacteristic
289C2×C8#1171994511yescharacteristic
290C2×C8D4×D5311yescharacteristic
291C2×C8D4×D5311yescharacteristic
292C2×C8D4×D5311yescharacteristic
293C2×C8#1171994911yescharacteristic
294C4⋊C4—21no—
295C4⋊C4—1062no—
296C22⋊C4—1062no—
297C22⋊C4—1062no—
298C22⋊C4C8×D5312yes—
299C42—21no—
300C16—22no—
301C16—41no—
302C16—1062no—
303C16—2121no—
304C23—532no—
305C23—1061no—
306C23—1061no—
307C23—1061no—
308C23—4241no—
309C23C2×C8×D5311yescharacteristic
310Q8—1061no—
311Q8—1061no—
312D4—1062no—
313D4—1062no—
314D4—1061no—
315D4—1061no—
316D4—4242no—
317C2×C4—22no—
318C2×C4—532no—
319C2×C4—1062no—
320C2×C4—1062no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—2121no—
325C2×C4—2121no—
326C2×C4—2121no—
327C2×C4—2121no—
328C2×C4C22⋊C4×D5311yescharacteristic
329C2×C4#1313317911yescharacteristic
330C2×C4C4×D21211yescharacteristic
331C2×C4M16×D5311yescharacteristic
332C2×C4#1313322311yescharacteristic
333C2×C4#1313322511yescharacteristic
334C8—41no—
335C8—1062no—
336C8—2121no—
337C8#1313318711yescharacteristic
338C8#1313319011yescharacteristic
339C8#1313319211yescharacteristic
340C8#1313320211yescharacteristic
341C22—41no—
342C22—1062no—
343C22—4241no—
344C22—4241no—
345C22—2121no—
346C22—2121no—
347C22—2122no—
348C22—2121no—
349C22C22⋊C8×D5311yescharacteristic
350C22#1681068811yescharacteristic
351C22C8×D21211yescharacteristic
352C4—42no—
353C4—1062no—
354C4—2121no—
355C4#1681050111yescharacteristic
356C4#1681058811yescharacteristic
357C4#1681069311yescharacteristic
358C4#1681069611yescharacteristic
359C2—21no—
360C2—21no—
361C2—21no—
362C2—4241no—
363C2—2122no—
364C2#2027500611yescharacteristic
subgroupG/H
centerC4#16810501
commutatorC23×C53C22×C8
FrattiniC22×C8C22×D53
FittingC53×#1378C2
socleC2×C53#1869
prime radical#16811135C4
cosocleC22×C8×C53C23
hypercenter#1378D53
hypocenterC53#12989
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C4 ◁ C22×C8 ◁ #1378

Complements

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

NH ≅ G/Ntype
1. #20287817C2semidirect
3. #20276335C2semidirect
4. C2×#16810404C2semidirect
5. C2×#16810404C2semidirect
6. #20272801C2semidirect
7. C53×#1378C2semidirect
51. #13133157D4semidirect
65. M32×C53D4semidirect
183. #1378D53semidirect
209. C53#12989semidirect
242. M32D212semidirect

Maximal subgroups

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

subgroupindex
#202878172
#202834202
#202763352
C2×#168104042
#202728012
C53×#13782
#1298953