Finite Group Databasebeta

← #27302276

Subgroups — #27302276

Order 13568.

430 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810738C211yesmaximal, characteristic
2C2×#16810443C211yesmaximal, characteristic, complemented
3C2×#16810406C211yesmaximal, characteristic, complemented
4#20276330C211yesmaximal, characteristic, complemented
5#20276329C211yesmaximal, characteristic, complemented
6C2×#16810398C211yesmaximal, characteristic, complemented
7C2×C53×#350C211yesmaximal, characteristic, complemented
8C2×#13133237C411yescharacteristic
9C2×#13133223C2211yescharacteristic
10C2×#13133157C2211yescharacteristic
11C22×#11719908C2211yescharacteristic
12#16810738—22no—
13#16810738—22no—
14C2×#13133152C411yescharacteristic
15C2×#13133144C411yescharacteristic
16C2×#13133143C411yescharacteristic
17C2×C8×Dic53C2211yescharacteristic
18#16810443—22no—
19#16810406—22no—
20C22⋊#11719908—22no—
21#16810402—22no—
22#16810398—22no—
23C2×#131×C53C2211yescharacteristic
24C2×M32×C53C2211yescharacteristic
25C22×C16×C53C2211yescharacteristic
26C53×#350—22no—
27C2×#11719945C2×C411yescharacteristic
28#13133237—22no—
29#13133237—22no—
30C2×#11719925C811yescharacteristic
31C2×#11719923C811yescharacteristic
32C2×#11719924C811yescharacteristic
33C2×#11719922C811yescharacteristic
34#13133223—21no—
35#13133223—21no—
36#13133223—21no—
37#13133223—21no—
38C2×#11719917C2×C411yescharacteristic
39C2×#11719912—21no—
40C22×#88322C2×C411yescharacteristic
41C2×C8×D53—21no—
42C2×C4×Dic53C2×C411yescharacteristic
43#13133157—21no—
44#13133157—21no—
45C2×#11719908—21no—
46C2×#11719908—21no—
47C2×#11719908—21no—
48C2×#11719908D412yes—
49#13133152—22no—
50#13133144—22no—
51#13133143—22no—
52C8×Dic53D412yes—
53C8×Dic53D412yes—
54C2×#41×C53C2×C411yescharacteristic
55#131×C53—21no—
56#131×C53—21no—
57#131×C53—21no—
58#131×C53—21no—
59C2×M16×C53—21no—
60C2×M16×C53C2×C411yescharacteristic
61C22×C8×C53—21no—
62C22×C8×C53C2311yescharacteristic
63M32×C53—21no—
64M32×C53—21no—
65C2×C16×C53—21no—
66C2×C16×C53—21no—
67C2×C16×C53—21no—
68C2×C16×C53D412yes—
69#41×C53—21no—
70#41×C53—21no—
71#41×C53—21no—
72#41×C53—21no—
73C2×Q8×C53C2×C811yescharacteristic
74C2×D4×C53—21no—
75C2×D4×C53C2×C811yescharacteristic
76C22×C4×C53—21no—
77C22×C4×C53C22×C411yescharacteristic
78C2×C22⋊D53—21no—
79C22×Dic53—21no—
80#11719945—21no—
81#11719945—21no—
82#11719945—21no—
83#11719945—21no—
84C2×D212C2×C811yescharacteristic
85C2×C4×D53—21no—
86C2×Dic106C2×C811yescharacteristic
87M16×C53—21no—
88M16×C53—21no—
89M16×C53—41no—
90M16×C53—41no—
91C2×C8×C53—21no—
92C2×C8×C53—41no—
93C2×C8×C53—41no—
94C2×C8×C53C2×D411yescharacteristic
95C2×C8×C53C2×D411yescharacteristic
96C2×C8×C53C2×D411yescharacteristic
97C2×C8×C53C2×D411yescharacteristic
98C2×C8×C53#4111yescharacteristic
99C2×C8×C53#4111yescharacteristic
100#11719925—22no—
101#11719924—22no—
102#11719923—22no—
103#11719922—22no—
104C4×Dic53—21no—
105C4×Dic53C22⋊C412yes—
106#11719917—21no—
107#11719917—21no—
108C2×#88322—21no—
109C2×#88322C22⋊C412yes—
110C2×#88322C2×C811yescharacteristic
111C2×#88322C2×C811yescharacteristic
112C2×#88322M1612yes—
113#11719912—41no—
114#11719912—41no—
115C8×D53—41no—
116C8×D53—41no—
117C16×C53—22no—
118C16×C53—41no—
119#11719908—22no—
120#11719908—41no—
121C23×C53—21no—
122C23×C53#13111yescharacteristic
123C22×D53—21no—
124Q8×C53—21no—
125Q8×C53—21no—
126D4×C53—21no—
127D4×C53—21no—
128D4×C53—41no—
129D4×C53—41no—
130C2×C4×C53—21no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53C2×C22⋊C411yescharacteristic
134C2×C4×C53C4×D411yescharacteristic
135C2×C4×C53C4×D411yescharacteristic
136C2×C4×C53C22×C811yescharacteristic
137C2×C4×C53C2×M1611yescharacteristic
138C2×C4×C53#13111yescharacteristic
139C22⋊D53—41no—
140C22⋊D53—41no—
141C2×Dic53—22no—
142C2×Dic53—21no—
143C2×Dic53—42no—
144D212—21no—
145D212—21no—
146C4×D53—41no—
147C4×D53—41no—
148Dic106—21no—
149Dic106—21no—
150C8×C53—41no—
151C8×C53#12111yescharacteristic
152C8×C53#12111yescharacteristic
153C8×C53#12411yescharacteristic
154C8×C53#12711yescharacteristic
155#88322—41no—
156#88322C22⋊C812yes—
157#88322C22⋊C812yes—
158#13196—531nomaximal, Sylow
159C22×C53—22no—
160C22×C53—42no—
161C22×C53—41no—
162C22×C53#40811yescharacteristic
163C22×C53C8×D411yescharacteristic
164C22×C53#43611yescharacteristic
165C2×D53—42no—
166C2×D53—41no—
167C4×C53—41no—
168C4×C53#39011yescharacteristic
169C4×C53C2×C22⋊C811yescharacteristic
170C4×C53C8×D411yescharacteristic
171C4×C53#43611yescharacteristic
172Dic53—42no—
173Dic53—41no—
174C2×#443—531no—
175C2×#364—531no—
176C2×#350—531no—
177C2×#350D5311yescharacteristic, complemented
178#2113—531no—
179#2113—531no—
180C2×#346—531no—
181C2×C53—21no—
182C2×C53—21no—
183C2×C53—81no—
184C2×C53#195411yescharacteristic
185C2×C53#217112yes—
186D53—81no—
187C2×#135—531no—
188C2×#131—531no—
189C2×#131C2×D5311yescharacteristic
190C2×M32—531no—
191C2×M32C2×D5311yescharacteristic
192C22×C16—531no—
193C22×C16C2×D5311yescharacteristic
194#443—1062no—
195#443—1062no—
196C2×#108—531no—
197C2×#104—531no—
198C2×#104—531no—
199C2×C4×C8—531no—
200#364—1062no—
201#350—22no—
202#350—1062no—
203C22⋊C16—1062no—
204C22⋊C16—1062no—
205#346—1062no—
206C53#1319611yescharacteristic, complemented, Sylow
207C2×#41—531no—
208C2×#41C4×D5311yescharacteristic
209#135—1062no—
210#135—1062no—
211C2×Q16—531no—
212C2×SD16—531no—
213C2×SD16—531no—
214C2×D8—531no—
215#131—21no—
216#131—21no—
217#131—21no—
218#131—21no—
219#131—1061no—
220#131—1061no—
221#131—1061no—
222#131—1061no—
223C2×M16—21no—
224C2×M16—531no—
225C2×M16—1061no—
226C2×M16C4×D5311yescharacteristic
227C22×C8—21no—
228C22×C8—531no—
229C22×C8—1061no—
230C22×C8C22×D5311yescharacteristic
231C2×C42—531no—
232M32—21no—
233M32—21no—
234M32—1061no—
235M32—1061no—
236C2×C16—21no—
237C2×C16—21no—
238C2×C16—21no—
239C2×C16—532no—
240C2×C16—1061no—
241C2×C16—1061no—
242C2×C16—1061no—
243C2×C16C22⋊D5312yes—
244#108—1062no—
245#104—1062no—
246#104—1062no—
247C4×C8—532no—
248C4×C8—532no—
249#41—21no—
250#41—21no—
251#41—21no—
252#41—21no—
253#41—1061no—
254#41—1061no—
255#41—1061no—
256#41—1061no—
257C2×Q8—531no—
258C2×Q8C8×D5311yescharacteristic
259C2×D4—21no—
260C2×D4—531no—
261C2×D4—1061no—
262C2×D4C8×D5311yescharacteristic
263C22×C4—21no—
264C22×C4—1061no—
265C22×C4—1061no—
266C22×C4C2×C4×D5311yescharacteristic
267Q16—1062no—
268SD16—1062no—
269SD16—1062no—
270D8—1062no—
271M16—21no—
272M16—21no—
273M16—41no—
274M16—41no—
275M16—1061no—
276M16—1061no—
277M16—2121no—
278M16—2121no—
279C2×C8—21no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—532no—
283C2×C8—531no—
284C2×C8—531no—
285C2×C8—532no—
286C2×C8—1061no—
287C2×C8—2121no—
288C2×C8—2121no—
289C2×C8#1171994511yescharacteristic
290C2×C8D4×D5311yescharacteristic
291C2×C8D4×D5311yescharacteristic
292C2×C8D4×D5311yescharacteristic
293C2×C8#1171994711yescharacteristic
294C2×C8C2×C22⋊D5311yescharacteristic
295C42—532no—
296C42—1061no—
297C16—22no—
298C16—41no—
299C16—1062no—
300C16—2121no—
301C23—21no—
302C23—1061no—
303C23#1313322511yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—1061no—
307Q8—1061no—
308D4—21no—
309D4—21no—
310D4—41no—
311D4—41no—
312D4—1061no—
313D4—1061no—
314D4—2121no—
315D4—2121no—
316C2×C4—21no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—1062no—
320C2×C4—1061no—
321C2×C4—2122no—
322C2×C4—2121no—
323C2×C4—2121no—
324C2×C4C22⋊C4×D5311yescharacteristic
325C2×C4#1313317711yescharacteristic
326C2×C4C4×C22⋊D5311yescharacteristic
327C2×C4C2×C8×D5311yescharacteristic
328C2×C4M16×D5311yescharacteristic
329C2×C4#1313322311yescharacteristic
330C8—41no—
331C8—532no—
332C8—532no—
333C8—2121no—
334C8#1313318911yescharacteristic
335C8#1313319011yescharacteristic
336C8#1313320311yescharacteristic
337C8#1313321311yescharacteristic
338C22—22no—
339C22—41no—
340C22—42no—
341C22—2121no—
342C22—2122no—
343C22#1681058811yescharacteristic
344C22#1681068611yescharacteristic
345C22#1681069911yescharacteristic
346C4—41no—
347C4—2122no—
348C4—2121no—
349C4#1681052311yescharacteristic
350C4C22⋊C8×D5311yescharacteristic
351C4C8×C22⋊D5311yescharacteristic
352C4#1681069811yescharacteristic
353C2—21no—
354C2—21no—
355C2—81no—
356C2—4241no—
357C2#2027550711yescharacteristic
358C2#2027655412yes—
subgroupG/H
centerC2×C8D4×D53
commutatorC2×C4×C53C22×C8
FrattiniC22×C8C22×D53
FittingC2×C53×#350C2
socleC22×C53C8×D4
prime radicalC2×#13133237C4
cosocleC22×C8×C53C23
hypercenterC2×#350D53
hypocenterC53#13196
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C8 ◁ C22×C8 ◁ C2×#350

Complements

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

NH ≅ G/Ntype
2. C2×#16810443C2semidirect
3. C2×#16810406C2semidirect
4. #20276330C2semidirect
5. #20276329C2semidirect
6. C2×#16810398C2semidirect
7. C2×C53×#350C2semidirect
177. C2×#350D53semidirect
206. C53#13196semidirect

Maximal subgroups

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

subgroupindex
C2×#168107382
C2×#168104432
C2×#168104062
#202763302
#202763292
C2×#168103982
C2×C53×#3502
#1319653