Finite Group Databasebeta

← #28311573

Subgroups — #28311573

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×#16811682C211yesmaximal, characteristic, complemented
2C2×#16811522C211yesmaximal, characteristic, complemented
3C2×#16811521C211yesmaximal, characteristic, complemented
4#20290945C211yesmaximal, characteristic, complemented
5#20289759C211yesmaximal, characteristic, complemented
6#20289753C211yesmaximal, characteristic, complemented
7C2×#16810737C211yesmaximal, characteristic
8#16811682—22no—
9#16811682—22no—
10C2×#13133292—21no—
11C2×#13133291C2211yescharacteristic
12C2×#13133291C2211yescharacteristic
13C2×D53⋊C16—21no—
14C22×#11719910C2211yescharacteristic
15C22×#11719910C2211yescharacteristic
16#16811522—22no—
17#16811521—22no—
18C22⋊#11719910—21no—
19C22⋊#11719910—21no—
20C2×#13133237C411yescharacteristic
21C2×#13133225C2211yescharacteristic
22C2×#13133224C2211yescharacteristic
23#16810737—22no—
24#16810737—22no—
25C2×#13133152C411yescharacteristic
26C2×#13133145C411yescharacteristic
27C2×C4×#88322C2211yescharacteristic
28C2×#104×C53C411yescharacteristic
29#13133292—24no—
30#13133291—21no—
31#13133291—21no—
32#13133291—21no—
33#13133291D412yes—
34D53⋊C16—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
44C2×#11719945C2×C411yescharacteristic
45#13133237—22no—
46#13133237—22no—
47C2×#11719925C811yescharacteristic
48C2×#11719923C811yescharacteristic
49C2×#11719924C811yescharacteristic
50C2×#11719922C811yescharacteristic
51#13133225—21no—
52#13133225—21no—
53#13133225D412yes—
54#13133225D412yes—
55#13133224—21no—
56#13133224—21no—
57#13133224—21no—
58#13133224—21no—
59C2×#11719917—21no—
60C2×#11719917C2×C411yescharacteristic
61C2×#11719912—21no—
62C22×#88322—21no—
63C22×#88322C2×C411yescharacteristic
64C22×#88322C2311yescharacteristic
65C2×C8×D53—21no—
66#13133152—22no—
67#13133145—22no—
68C4×#88322—21no—
69C4×#88322—21no—
70C2×#41×C53C2×C411yescharacteristic
71C2×M16×C53C2×C411yescharacteristic
72C2×C42×C53C2×C411yescharacteristic
73#104×C53—22no—
74#41×C53—21no—
75#41×C53—21no—
76#41×C53—21no—
77#41×C53—21no—
78C2×Q8×C53C2×C811yescharacteristic
79C2×D4×C53—21no—
80C2×D4×C53C2×C811yescharacteristic
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53C22×C411yescharacteristic
84C2×C22⋊D53—21no—
85#11719945—21no—
86#11719945—21no—
87#11719945C22⋊C412yes—
88#11719945M1612yes—
89C2×D212C2×C811yescharacteristic
90C2×C4×D53—21no—
91C2×Dic106C2×C811yescharacteristic
92M16×C53—21no—
93M16×C53C22⋊C412yes—
94C2×C8×C53—21no—
95C42×C53—21no—
96C42×C53—21no—
97#11719925—22no—
98#11719924—22no—
99#11719923—22no—
100#11719922—22no—
101#11719917—21no—
102#11719917—21no—
103#11719917—41no—
104#11719917—41no—
105C2×#88322—21no—
106C2×#88322—21no—
107C2×#88322—21no—
108C2×#88322—41no—
109C2×#88322—41no—
110C2×#88322C2×C811yescharacteristic
111C2×#88322C2×C811yescharacteristic
112C2×#88322C2×D411yescharacteristic
113C2×#88322C2×D411yescharacteristic
114C2×#88322C2×D411yescharacteristic
115C2×#88322C2×D411yescharacteristic
116C2×#88322#4111yescharacteristic
117C2×#88322#4111yescharacteristic
118#11719912—22no—
119#11719912—22no—
120C8×D53—22no—
121C8×D53—22no—
122#11719910—21no—
123#11719910—21no—
124#11719910—22no—
125#11719910—41no—
126#11719910—41no—
127C23×C53—21no—
128C23×C53#13111yescharacteristic
129C22×D53—21no—
130Q8×C53—21no—
131Q8×C53—21no—
132D4×C53—21no—
133D4×C53—21no—
134D4×C53—41no—
135D4×C53—41no—
136C2×C4×C53—22no—
137C2×C4×C53—21no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53C2×C22⋊C411yescharacteristic
143C2×C4×C53C4×D411yescharacteristic
144C2×C4×C53C4×D411yescharacteristic
145C2×C4×C53C22×C811yescharacteristic
146C2×C4×C53C2×M1611yescharacteristic
147C2×C4×C53#13111yescharacteristic
148C22⋊D53—22no—
149C22⋊D53—22no—
150C2×Dic53—21no—
151D212—21no—
152D212C22⋊C812yes—
153C4×D53—22no—
154C4×D53—22no—
155Dic106—21no—
156Dic106C22⋊C812yes—
157C8×C53—22no—
158#88322—21no—
159#88322—21no—
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—41no—
169C22×C53—41no—
170C22×C53—41no—
171C22×C53#40811yescharacteristic
172C22×C53C8×D411yescharacteristic
173C22×C53#43511yescharacteristic
174C2×D53—22no—
175C2×D53—41no—
176C2×D53—41no—
177C4×C53—42no—
178C4×C53—41no—
179C4×C53#38611yescharacteristic
180C4×C53C2×C22⋊C811yescharacteristic
181C4×C53C8×D411yescharacteristic
182C4×C53#43511yescharacteristic
183Dic53—22no—
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—81no—
194C2×C53#186911yescharacteristic
195C2×C53#217011yescharacteristic
196C2×C53#217111yescharacteristic
197D53—42no—
198C2×#135—531no—
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#615711yescharacteristic
214C2×C4×C8—531no—
215#350—1062no—
216#350—1062no—
217C22⋊C16—1061no—
218C22⋊C16—1061no—
219C53#1298011yescharacteristic, complemented, Sylow
220C2×#41—531no—
221C2×#41C2×#615711yescharacteristic
222#135—1062no—
223#135—1062no—
224C2×Q16—531no—
225C2×SD16—531no—
226C2×SD16—531no—
227C2×D8—531no—
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—1061no—
238C2×M16—1061no—
239C2×M16C2×#615711yescharacteristic
240C22×C8—531no—
241C22×C8—531no—
242C22×C8—1061no—
243C22×C8—1061no—
244C2×C42C2×#615711yescharacteristic
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—22no—
262#104—1062no—
263C4×C8—1061no—
264C4×C8—1061no—
265#41—21no—
266#41—21no—
267#41—21no—
268#41—21no—
269#41—532no—
270#41—532no—
271#41—1061no—
272#41—1061no—
273C2×Q8—531no—
274C2×Q8D53⋊C811yescharacteristic
275C2×D4—21no—
276C2×D4—531no—
277C2×D4—1061no—
278C2×D4D53⋊C811yescharacteristic
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—1061no—
282C22×C4C22×#615711yescharacteristic
283Q16—1062no—
284SD16—1062no—
285SD16—1062no—
286D8—1062no—
287M16—21no—
288M16—1062no—
289M16—1062no—
290M16—1061no—
291M16—1061no—
292M16—2121no—
293M16—2121no—
294M16C22⋊#615712yes—
295C2×C8—21no—
296C2×C8—531no—
297C2×C8—531no—
298C2×C8—531no—
299C2×C8—531no—
300C2×C8—531no—
301C2×C8—531no—
302C2×C8—531no—
303C2×C8—531no—
304C2×C8—1062no—
305C2×C8—1062no—
306C2×C8—1061no—
307C2×C8—1061no—
308C2×C8—1061no—
309C2×C8—2121no—
310C2×C8—2121no—
311C42—21no—
312C42—21no—
313C16—1061no—
314C16—1061no—
315C16—1062no—
316C16—2121no—
317C16—2121no—
318C23—21no—
319C23—1061no—
320C23#1313330411yescharacteristic
321Q8—21no—
322Q8—21no—
323Q8—532no—
324Q8—1061no—
325D4—21no—
326D4—21no—
327D4—41no—
328D4—41no—
329D4—532no—
330D4—1062no—
331D4—1062no—
332D4—1061no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—1061no—
340C2×C4—1062no—
341C2×C4—1062no—
342C2×C4C2×C22⋊#615711yescharacteristic
343C2×C4D4×#615711yescharacteristic
344C2×C4D4×#615711yescharacteristic
345C2×C4C2×D53⋊C811yescharacteristic
346C2×C4D53⋊M1611yescharacteristic
347C2×C4#1313330411yescharacteristic
348C8—22no—
349C8—531no—
350C8—531no—
351C8—531no—
352C8—531no—
353C8—1061no—
354C8—1061no—
355C8—2121no—
356C22—21no—
357C22—21no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—1062no—
362C22—2121no—
363C22—2121no—
364C22#1681159711yescharacteristic
365C22#1681165011yescharacteristic
366C22#1681165511yescharacteristic
367C4—42no—
368C4—41no—
369C4—1062no—
370C4#1681157111yescharacteristic
371C4D53⋊(C22⋊C8)11yescharacteristic
372C4#1681165011yescharacteristic
373C4#1681165511yescharacteristic
374C2—21no—
375C2—21no—
376C2—81no—
377C2—2122no—
378C2#2029068911yescharacteristic
379C2#2029110311yescharacteristic
380C2#2029110611yescharacteristic
subgroupG/H
centerC2×C4D4×#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC2×#104×C53C4
socleC22×C53C8×D4
prime radicalC2×#13133237C4
cosocleC22×#88322C23
hypercenterC2×#104#6157
hypocenterC53#12980
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
1. C2×#16811682C2semidirect
2. C2×#16811522C2semidirect
3. C2×#16811521C2semidirect
4. #20290945C2semidirect
5. #20289759C2semidirect
6. #20289753C2semidirect
219. C53#12980semidirect

Maximal subgroups

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

subgroupindex
C2×#168116822
C2×#168115222
C2×#168115212
#202909452
#202897592
#202897532
C2×#168107372
#1298053