Finite Group Databasebeta

← #28299432

Subgroups — #28299432

Order 13568.

596 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13133274C211yesmaximal, characteristic
2#20290376C211yesmaximal, characteristic
3#20289536C212yesmaximal, complemented
4#20289533C212yesmaximal, complemented
5C4×#13133197C211yesmaximal, characteristic
6C22×D53⋊C8—21no—
7C2×#13133274—21no—
8C2×C4×#88324—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×#13133197C411yescharacteristic
13C2×C4×Dic106C411yescharacteristic
14C2×C42×D53C2211yescharacteristic
15#16810528C2211yescharacteristic
16#16810522C2211yescharacteristic
17#16810508—21no—
18#16810497—21no—
19#16810480C411yescharacteristic
20C4×C4⋊Dic53—21no—
21C4×#11719921—21no—
22C4×C4⋊C4×C53C411yescharacteristic
23C2×D53⋊C8—24no—
24C2×D53⋊C8—42no—
25C2×D53⋊C8—42no—
26C22×#88324—22no—
27C22×#88324—22no—
28#13133274—24no—
29C4×#88324—42no—
30C22×Dic106C2×C411yescharacteristic
31C22×C4×D53C2×C411yescharacteristic
32C22×C4×D53D411yescharacteristic
33C22×C4×D53D411yescharacteristic
34#13133197—22no—
35#13133197C814yescomplemented
36C4×Dic106—42no—
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53—21no—
39C2×C4⋊Dic53C2×C411yescharacteristic
40C2×#11719919—21no—
41C2×#11719921—21no—
42C2×#11719921—21no—
43C2×C4×Dic53—21no—
44C2×C4×Dic53D411yescharacteristic
45C2×C4×Dic53D411yescharacteristic
46C2×C4×Dic53C2311yescharacteristic
47C42×D53—42no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51#13133124—21no—
52C2×C4⋊C4×C53C2×C411yescharacteristic
53C2×C4⋊C4×C53C2×C411yescharacteristic
54C2×C42×C53—21no—
55C2×C42×C53C2×C411yescharacteristic
56#95×C53—21no—
57C23×D53C22⋊C411yescharacteristic
58C22×C4×C53—21no—
59C22×C4×C53—21no—
60C22×C4×C53C22⋊C411yescharacteristic
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22×C411yescharacteristic
63C22×Dic53—21no—
64C22×Dic53—21no—
65C22×Dic53C22⋊C411yescharacteristic
66C22×Dic53C2×D411yescharacteristic
67C22×Dic53C2×D411yescharacteristic
68C2×C4×D53—22no—
69C2×C4×D53—22no—
70C2×C4×D53—22no—
71C2×C4×D53—22no—
72C2×C4×D53—41no—
73C2×C4×D53—41no—
74C2×C4×D53—41no—
75C2×C4×D53C2×C812yes—
76C2×C4×D53M1612yes—
77C2×C4×D53SD1612yes—
78C2×C4×D53Q1612yes—
79C2×Dic106—21no—
80C2×Dic106—41no—
81C2×Dic106—41no—
82C2×Dic106C2×C812yescomplemented
83C2×#88324—24no—
84C2×#88324—24no—
85C2×#88324—42no—
86C2×#88324—42no—
87C2×#88324—42no—
88C2×#88324—42no—
89D53⋊C8—48no—
90C4⋊C4×C53—21no—
91C4⋊C4×C53—41no—
92C4⋊C4×C53C2×C812yescomplemented
93C42×C53—22no—
94C42×C53—42no—
95#11719921—24no—
96#11719921—42no—
97C4⋊Dic53—24no—
98C4⋊Dic53—42no—
99C4⋊Dic53—41no—
100#11719919—42no—
101C4×Dic53—22no—
102C4×Dic53—41no—
103C4×Dic53—41no—
104C4×Dic53—42no—
105C23×C53C2×C22⋊C411yescharacteristic
106C22×D53—21no—
107C22×D53—21no—
108C22×D53—21no—
109C22×D53—21no—
110C22×D53C22⋊C812yes—
111C22×D53#10312yes—
112C22×D53#10412yes—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—22no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53#9912yes—
125C2×C4×C53#10312yes—
126C2×C4×C53#10412yes—
127C2×C4×C53C22×C811yescharacteristic
128C2×C4×C53C2×M1611yescharacteristic
129C2×Dic53—21no—
130C2×Dic53—21no—
131C2×Dic53—21no—
132C2×Dic53—21no—
133C2×Dic53—21no—
134C2×Dic53—21no—
135C2×Dic53—22no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53C22⋊C812yes—
142C2×Dic53C2×SD1611yescharacteristic
143C2×Dic53C2×Q1611yescharacteristic
144C2×Dic53#13512yes—
145C4×D53—24no—
146C4×D53—24no—
147C4×D53—42no—
148C4×D53—42no—
149C4×D53—44no—
150Dic106—42no—
151Dic106—42no—
152#88324—44no—
153#88324—44no—
154#7566—531nomaximal, Sylow
155C22×C53C2×C22⋊C811yescharacteristic
156C22×C53C2×#9911yescharacteristic
157C22×C53#41311yescharacteristic
158C22×C53C2×#10311yescharacteristic
159C22×C53#41611yescharacteristic
160C22×C53C2×#10411yescharacteristic
161C22×C53C2×#10411yescharacteristic
162C2×D53—22no—
163C2×D53—22no—
164C2×D53—22no—
165C2×D53—22no—
166C2×D53—22no—
167C2×D53—22no—
168C2×D53Q8⋊C814yes—
169C4×C53—21no—
170C4×C53—21no—
171C4×C53—42no—
172C4×C53—41no—
173C4×C53—81no—
174C4×C53#32814yes—
175Dic53—22no—
176Dic53—21no—
177Dic53—21no—
178Dic53—22no—
179Dic53—42no—
180Dic53—81no—
181C4×C22⋊Q8—531no—
182C4×C22⋊C8—531no—
183C2×#336—531no—
184#1272—532no—
185#1271—532no—
186C2×C53#145911yescharacteristic
187C2×C53C2×Q8⋊C811yescharacteristic
188C2×C53#147611yescharacteristic
189C2×C53C2×#32811yescharacteristic
190C2×C53#150111yescharacteristic
191C2×C53#152411yescharacteristic
192C2×C53#154211yescharacteristic
193D53—24no—
194C23×C8—1061no—
195C2×C22⋊Q8—531no—
196C2×C4×Q8—531no—
197C22×C42—531no—
198C2×C22⋊C8—1061no—
199C2×C4×C8—1061no—
200#391—531no—
201#389—531no—
202#387—1061no—
203#385—1061no—
204#380—531no—
205C4×C4⋊C4—1061no—
206C4×C4⋊C4#615711yescharacteristic
207C4×C22⋊C4—1061no—
208#336—532no—
209#336—532no—
210#336—1061no—
211C53#756611yescharacteristic, complemented, Sylow
212C22×Q8—531no—
213C23×C4—531no—
214C23×C4—531no—
215C23×C4—531no—
216C22×C8—1062no—
217C22×C8—1062no—
218C22×C8—1064no—
219C22×C8—2122no—
220C22×C8—2122no—
221C22⋊Q8—534no—
222C22⋊Q8—1062no—
223C4×Q8—2122no—
224C2×C4⋊C4—531no—
225C2×C4⋊C4—1061no—
226C2×C4⋊C4—1061no—
227C2×C4⋊C4—1061no—
228C2×C4⋊C4C2×#615711yescharacteristic
229C2×C4⋊C4C2×#615711yescharacteristic
230C2×C22⋊C4—1061no—
231C2×C22⋊C4—1061no—
232C2×C42—21no—
233C2×C42—531no—
234C2×C42—531no—
235C2×C42—531no—
236C2×C42—1061no—
237C2×C42—2122no—
238C2×C42C2×#615711yescharacteristic
239C22⋊C8—1064no—
240C4×C8—2122no—
241#95—21no—
242#95—1061no—
243#95—1061no—
244#95—1061no—
245#95—1061no—
246C24—531no—
247C2×Q8—532no—
248C2×Q8—1061no—
249C2×Q8—2121no—
250C2×Q8—2121no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—531no—
254C22×C4—531no—
255C22×C4—531no—
256C22×C4—532no—
257C22×C4—532no—
258C22×C4—532no—
259C22×C4—532no—
260C22×C4—1062no—
261C22×C4—1062no—
262C22×C4—1062no—
263C22×C4—1062no—
264C22×C4—1061no—
265C22×C4—1061no—
266C22×C4—2121no—
267C22×C4—2121no—
268C22×C4—2121no—
269C22×C4C22⋊#615711yescharacteristic
270C22×C4C22⋊#615711yescharacteristic
271C22×C4C22×#615711yescharacteristic
272C2×C8—1064no—
273C2×C8—1064no—
274C2×C8—2122no—
275C2×C8—2122no—
276C2×C8—2122no—
277C2×C8—2122no—
278C2×C8—2128no—
279C4⋊C4—21no—
280C4⋊C4—41no—
281C4⋊C4—1064no—
282C4⋊C4—2121no—
283C4⋊C4—2122no—
284C4⋊C4—2122no—
285C4⋊C4D53⋊C812yescomplemented
286C22⋊C4—1064no—
287C22⋊C4—2122no—
288C42—22no—
289C42—42no—
290C42—1062no—
291C42—2121no—
292C42—2121no—
293C42—2122no—
294C23—532no—
295C23—532no—
296C23—532no—
297C23—1061no—
298C23—1061no—
299C23—1061no—
300C23—1061no—
301C23C2×C22⋊#615711yescharacteristic
302Q8—2122no—
303Q8—2122no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—532no—
316C2×C4—531no—
317C2×C4—531no—
318C2×C4—532no—
319C2×C4—1061no—
320C2×C4—1061no—
321C2×C4—1061no—
322C2×C4—1061no—
323C2×C4—1061no—
324C2×C4—1061no—
325C2×C4—1064no—
326C2×C4—1064no—
327C2×C4—1062no—
328C2×C4—2122no—
329C2×C4—2122no—
330C2×C4—2124no—
331C2×C4—2121no—
332C2×C4—2121no—
333C2×C4—2121no—
334C2×C4—2121no—
335C2×C4—2121no—
336C2×C4#1313327612yes—
337C2×C4#1313328012yes—
338C2×C4#1313328112yes—
339C2×C4C2×D53⋊C811yescharacteristic
340C2×C4D53⋊M1611yescharacteristic
341C8—2124no—
342C8—2124no—
343C22—534no—
344C22—1062no—
345C22—1062no—
346C22—1062no—
347C22—1062no—
348C22—1062no—
349C22—1062no—
350C22D53⋊(C22⋊C8)11yescharacteristic
351C22C2×#1313327611yescharacteristic
352C22#1681160911yescharacteristic
353C22C2×#1313328011yescharacteristic
354C22#1681161311yescharacteristic
355C22C2×#1313328111yescharacteristic
356C22D53⋊#10411yescharacteristic
357C4—21no—
358C4—21no—
359C4—41no—
360C4—42no—
361C4—81no—
362C4—1061no—
363C4—1061no—
364C4—1062no—
365C4—1062no—
366C4—4241no—
367C4—2122no—
368C4#1681147614yes—
369C2—1064no—
370C2#2028995211yescharacteristic
371C2D53⋊(Q8⋊C8)11yescharacteristic
372C2#2029000711yescharacteristic
373C2C2×#1681147611yescharacteristic
374C2#2029009211yescharacteristic
375C2#2029016511yescharacteristic
376C2#2029021611yescharacteristic
subgroupG/H
centerC22×C4C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC4×C4⋊C4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC4×C4⋊C4#6157
hypocenterC53#7566
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #20289536C2semidirect
4. #20289533C2semidirect
35. #13133197C8semidirect
82. C2×Dic106C2×C8semidirect
92. C4⋊C4×C53C2×C8semidirect
211. C53#7566semidirect
285. C4⋊C4D53⋊C8semidirect

Maximal subgroups

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

subgroupindex
C4×#131332742
#202903762
#202895362
#202895332
C4×#131331972
#756653