Finite Group Databasebeta

← #28311572

Subgroups — #28311572

Order 13568.

470 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811681C211yesmaximal, characteristic, complemented
2C2×#16811521C211yesmaximal, characteristic, complemented
3C2×#16811521C211yesmaximal, characteristic, complemented
4#20290945C211yesmaximal, characteristic, complemented
5#20289760C211yesmaximal, characteristic, complemented
6#20289754C211yesmaximal, characteristic, complemented
7C2×#16810740C211yesmaximal, characteristic
8#16811681—22no—
9#16811681—22no—
10C2×#13133291—21no—
11C2×#13133291C2211yescharacteristic
12C2×#13133291C2211yescharacteristic
13C22×#11719910—21no—
14C22×#11719910C2211yescharacteristic
15C22×#11719910C2211yescharacteristic
16#16811521—22no—
17#16811521—22no—
18C22⋊#11719910—21no—
19C22⋊#11719910—21no—
20C2×#13133224C2211yescharacteristic
21C2×#13133224C2211yescharacteristic
22#16810740—22no—
23#16810740—22no—
24C2×#13133153C411yescharacteristic
25C2×#13133144C411yescharacteristic
26C2×#13133144C411yescharacteristic
27C2×C8×Dic53C2211yescharacteristic
28C2×#135×C53C411yescharacteristic
29#13133291—24no—
30#13133291—21no—
31#13133291—21no—
32#13133291—21no—
33#13133291D412yes—
34C2×#11719910—24no—
35C2×#11719910—21no—
36C2×#11719910—21no—
37C2×#11719910—21no—
38C2×#11719910—21no—
39C2×#11719910—21no—
40C2×#11719910—21no—
41C2×#11719910—21no—
42C2×#11719910D411yescharacteristic
43C2×#11719910D411yescharacteristic
44#13133224—21no—
45#13133224—21no—
46#13133224—21no—
47#13133224—21no—
48#13133224—21no—
49#13133224—21no—
50#13133224D412yes—
51#13133224D412yes—
52C2×#11719917—21no—
53C2×#11719917—21no—
54C2×#11719917C2×C411yescharacteristic
55C2×#11719917C2×C411yescharacteristic
56C22×#88322—21no—
57C22×#88322—21no—
58C22×#88322C2311yescharacteristic
59C2×C4×Dic53C2×C411yescharacteristic
60#13133153—22no—
61#13133144—22no—
62#13133144—22no—
63C8×Dic53—21no—
64C8×Dic53—21no—
65C2×#41×C53C2×C411yescharacteristic
66C2×#41×C53C2×C411yescharacteristic
67#135×C53—22no—
68#135×C53—22no—
69C2×Q16×C53C811yescharacteristic
70C2×SD16×C53C811yescharacteristic
71C2×SD16×C53C811yescharacteristic
72C2×D8×C53C811yescharacteristic
73C22×C8×C53C2×C411yescharacteristic
74#41×C53—21no—
75#41×C53—21no—
76#41×C53—21no—
77#41×C53—21no—
78#41×C53—21no—
79#41×C53—21no—
80#41×C53C22⋊C412yes—
81#41×C53M1612yes—
82C2×Q8×C53C2×C811yescharacteristic
83C2×Q8×C53C2×C811yescharacteristic
84C2×D4×C53—21no—
85C2×D4×C53—21no—
86C2×D4×C53C2×C811yescharacteristic
87C2×D4×C53C2×C811yescharacteristic
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53C22×C411yescharacteristic
91C22×Dic53—21no—
92Q16×C53—22no—
93SD16×C53—22no—
94SD16×C53—22no—
95D8×C53—22no—
96C2×C8×C53—21no—
97C2×C8×C53—21no—
98C2×C8×C53C2×C811yescharacteristic
99C2×C8×C53C2×C811yescharacteristic
100C4×Dic53—21no—
101C4×Dic53—21no—
102#11719917—21no—
103#11719917—22no—
104#11719917—22no—
105#11719917—21no—
106#11719917—21no—
107#11719917—41no—
108#11719917—41no—
109#11719917C22⋊C412yes—
110C2×#88322—21no—
111C2×#88322—22no—
112C2×#88322—22no—
113C2×#88322—21no—
114C2×#88322—41no—
115C2×#88322—41no—
116C2×#88322C2×D411yescharacteristic
117C2×#88322C2×D411yescharacteristic
118C2×#88322C2×D411yescharacteristic
119C2×#88322C2×D411yescharacteristic
120C2×#88322#4111yescharacteristic
121C2×#88322#4111yescharacteristic
122#11719910—21no—
123#11719910—22no—
124#11719910—21no—
125#11719910—41no—
126#11719910—41no—
127C23×C53—21no—
128C23×C53—21no—
129C23×C53#13111yescharacteristic
130Q8×C53—21no—
131Q8×C53—21no—
132Q8×C53—21no—
133Q8×C53C22⋊C812yes—
134D4×C53—22no—
135D4×C53—22no—
136D4×C53—21no—
137D4×C53—21no—
138D4×C53—21no—
139D4×C53—41no—
140D4×C53—41no—
141D4×C53C22⋊C812yes—
142C2×C4×C53—21no—
143C2×C4×C53—22no—
144C2×C4×C53—22no—
145C2×C4×C53—21no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53C2×C22⋊C411yescharacteristic
149C2×C4×C53C4×D411yescharacteristic
150C2×C4×C53C4×D411yescharacteristic
151C2×C4×C53C22×C811yescharacteristic
152C2×C4×C53C2×M1611yescharacteristic
153C2×C4×C53#13111yescharacteristic
154C2×Dic53—22no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C8×C53—21no—
158C8×C53—21no—
159#88322—22no—
160#88322—41no—
161#88322#12011yescharacteristic
162#88322#12111yescharacteristic
163#88322#12111yescharacteristic
164#88322#12311yescharacteristic
165#12980—531nomaximal, Sylow
166C22×C53—21no—
167C22×C53—21no—
168C22×C53—22no—
169C22×C53—41no—
170C22×C53—41no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53—41no—
174C22×C53#40811yescharacteristic
175C22×C53C8×D411yescharacteristic
176C22×C53#43511yescharacteristic
177C4×C53—22no—
178C4×C53—41no—
179C4×C53#38611yescharacteristic
180C4×C53C2×C22⋊C811yescharacteristic
181C4×C53C8×D411yescharacteristic
182C4×C53#43511yescharacteristic
183Dic53—42no—
184C2×#504—531no—
185C2×#443—531no—
186C2×#350—531no—
187C2×#350—531no—
188#2113—531no—
189#1376—531no—
190#1374—531no—
191C2×C53—21no—
192C2×C53—21no—
193C2×C53—42no—
194C2×C53—81no—
195C2×C53#186911yescharacteristic
196C2×C53#217011yescharacteristic
197C2×C53#217111yescharacteristic
198C2×#135#615711yescharacteristic
199C2×#131—531no—
200C2×#131—531no—
201#504—1062no—
202#504—1062no—
203C2×M32—531no—
204C2×M32—531no—
205C2×M32—1061no—
206C22×C16—531no—
207C22×C16—531no—
208C22×C16—1061no—
209#443—1062no—
210#443—1062no—
211C2×#108—531no—
212C2×#104—531no—
213C2×#104—531no—
214C2×C4×C8—531no—
215#350—1062no—
216#350—1062no—
217C22⋊C16—1061no—
218C22⋊C16—1061no—
219C53#1298011yescharacteristic, complemented, Sylow
220C2×#41C2×#615711yescharacteristic
221C2×#41C2×#615711yescharacteristic
222#135—22no—
223#135—22no—
224C2×Q16#8832411yescharacteristic
225C2×SD16#8832411yescharacteristic
226C2×SD16#8832411yescharacteristic
227C2×D8#8832411yescharacteristic
228#131—532no—
229#131—532no—
230#131—1061no—
231#131—1061no—
232#131—1061no—
233#131—1061no—
234#131—1061no—
235#131—1061no—
236C2×M16—531no—
237C2×M16—531no—
238C2×M16—1061no—
239C2×M16—1061no—
240C22×C8—531no—
241C22×C8—1061no—
242C22×C8—1061no—
243C22×C8C2×#615711yescharacteristic
244C2×C42—531no—
245M32—532no—
246M32—1061no—
247M32—1064no—
248M32—1061no—
249M32—1061no—
250C2×C16—531no—
251C2×C16—531no—
252C2×C16—1061no—
253C2×C16—1061no—
254C2×C16—1061no—
255C2×C16—1064no—
256C2×C16—1061no—
257C2×C16—1061no—
258C2×C16—1061no—
259C2×C16—1061no—
260#108—1062no—
261#104—1062no—
262#104—1062no—
263C4×C8—1061no—
264C4×C8—1061no—
265#41—21no—
266#41—21no—
267#41—21no—
268#41—21no—
269#41—21no—
270#41—21no—
271#41#1171994012yes—
272#41C22⋊#615712yes—
273C2×Q8C2×#8832411yescharacteristic
274C2×Q8C2×#8832411yescharacteristic
275C2×D4—21no—
276C2×D4—21no—
277C2×D4C2×#8832411yescharacteristic
278C2×D4C2×#8832411yescharacteristic
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—1061no—
282C22×C4C22×#615711yescharacteristic
283Q16—22no—
284SD16—22no—
285SD16—22no—
286D8—22no—
287M16—532no—
288M16—1062no—
289M16—1062no—
290M16—1061no—
291M16—1061no—
292M16—1061no—
293M16—2121no—
294M16—2121no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—531no—
298C2×C8—531no—
299C2×C8—531no—
300C2×C8—531no—
301C2×C8—531no—
302C2×C8—531no—
303C2×C8—1062no—
304C2×C8—1062no—
305C2×C8—1061no—
306C2×C8—1061no—
307C2×C8—2121no—
308C2×C8—2121no—
309C2×C8C2×#8832411yescharacteristic
310C2×C8C2×#8832411yescharacteristic
311C42—1061no—
312C42—1061no—
313C16—1061no—
314C16—1061no—
315C16—1062no—
316C16—2121no—
317C16—2121no—
318C23—21no—
319C23—21no—
320C23#1313330411yescharacteristic
321Q8—21no—
322Q8—21no—
323Q8—21no—
324Q8C22⋊#8832412yes—
325D4—22no—
326D4—22no—
327D4—21no—
328D4—21no—
329D4—21no—
330D4—41no—
331D4—41no—
332D4C22⋊#8832412yes—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—1062no—
340C2×C4—2121no—
341C2×C4—2121no—
342C2×C4C2×C22⋊#615711yescharacteristic
343C2×C4D4×#615711yescharacteristic
344C2×C4D4×#615711yescharacteristic
345C2×C4C22×#8832411yescharacteristic
346C2×C4C2×#1171994011yescharacteristic
347C2×C4#1313330411yescharacteristic
348C8—21no—
349C8—21no—
350C8—531no—
351C8—531no—
352C8—531no—
353C8—531no—
354C8—1062no—
355C8—2121no—
356C22—21no—
357C22—21no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22#1681159611yescharacteristic
365C22D4×#8832411yescharacteristic
366C22#1681165411yescharacteristic
367C4—22no—
368C4—41no—
369C4—2122no—
370C4#1681157111yescharacteristic
371C4C2×C22⋊#8832411yescharacteristic
372C4D4×#8832411yescharacteristic
373C4#1681165411yescharacteristic
374C2—21no—
375C2—21no—
376C2—42no—
377C2—81no—
378C2#2029068811yescharacteristic
379C2#2029110211yescharacteristic
380C2#2029110511yescharacteristic
subgroupG/H
centerC2×C4D4×#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
Fitting, prime radicalC2×#135×C53C4
socleC22×C53C8×D4
cosocleC22×#88322C23
hypercenterC2×#135#6157
hypocenterC53#12980
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#16811681C2semidirect
2. C2×#16811521C2semidirect
3. C2×#16811521C2semidirect
4. #20290945C2semidirect
5. #20289760C2semidirect
6. #20289754C2semidirect
219. C53#12980semidirect

Maximal subgroups

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

subgroupindex
C2×#168116812
C2×#168115212
#202909452
#202897602
#202897542
C2×#168107402
#1298053