Finite Group Databasebeta

← #28299456

Subgroups — #28299456

Order 13568.

604 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20290572C211yesmaximal, characteristic
2C4×#13133274C211yesmaximal, characteristic
3#20289534C212yesmaximal, complemented
4#20289533C212yesmaximal, complemented
5#389×D53C211yesmaximal, characteristic
6C2×#13133284—21no—
7C2×#13133274—21no—
8C2×#13133274—21no—
9C2×C4×#88324—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×C4⋊C4×D53—21no—
13C2×C4⋊C4×D53C411yescharacteristic
14C2×C4⋊C4×D53C411yescharacteristic
15C2×C42×D53C2211yescharacteristic
16#16810522C2211yescharacteristic
17#16810521C411yescharacteristic
18#16810520—21no—
19#16810519C2211yescharacteristic
20#16810518—21no—
21#95×D53—21no—
22C53×#389C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25#13133284—42no—
26#13133274—24no—
27#13133274—42no—
28C4×#88324—42no—
29C22×C4×D53—21no—
30C22×C4×D53—21no—
31C22×C4×D53C2×C411yescharacteristic
32C22×C4×D53D411yescharacteristic
33C22×C4×D53D411yescharacteristic
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53C2×C411yescharacteristic
36C2×C4⋊Dic53C2×C411yescharacteristic
37C2×#11719919—21no—
38C2×#11719919—21no—
39C2×#11719919—21no—
40C2×#11719919—21no—
41C4⋊C4×D53—22no—
42C4⋊C4×D53—42no—
43C4⋊C4×D53—42no—
44C4⋊C4×D53—42no—
45C4⋊C4×D53C814yescomplemented
46C2×C4×Dic53D411yescharacteristic
47C2×C4×Dic53D411yescharacteristic
48C2×C4×Dic53C2311yescharacteristic
49C42×D53—42no—
50#13133124—21no—
51#13133124—21no—
52#13133124—21no—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53C2×C411yescharacteristic
55C2×C4⋊C4×C53C2×C411yescharacteristic
56C2×C42×C53C2×C411yescharacteristic
57#95×C53—21no—
58C23×D53C22⋊C411yescharacteristic
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53C22⋊C411yescharacteristic
62C22×C4×C53C22⋊C411yescharacteristic
63C22×C4×C53C22×C411yescharacteristic
64C22×Dic53—21no—
65C22×Dic53—21no—
66C22×Dic53C22⋊C411yescharacteristic
67C22×Dic53C2×D411yescharacteristic
68C22×Dic53C2×D411yescharacteristic
69C2×C4×D53—22no—
70C2×C4×D53—22no—
71C2×C4×D53—24no—
72C2×C4×D53—22no—
73C2×C4×D53—22no—
74C2×C4×D53—41no—
75C2×C4×D53—41no—
76C2×C4×D53—41no—
77C2×C4×D53—42no—
78C2×C4×D53—42no—
79C2×C4×D53—42no—
80C2×C4×D53—42no—
81C2×C4×D53—42no—
82C2×C4×D53C2×C812yes—
83C2×C4×D53M1612yes—
84C2×C4×D53D812yes—
85C2×C4×D53SD1612yes—
86C2×#88324—24no—
87C2×#88324—42no—
88C2×#88324—42no—
89C2×#88324—42no—
90C2×#88324—42no—
91C2×#88324—42no—
92C4⋊C4×C53—21no—
93C4⋊C4×C53—41no—
94C4⋊C4×C53—42no—
95C4⋊C4×C53C2×C812yes—
96C42×C53—41no—
97C4⋊Dic53—21no—
98C4⋊Dic53—41no—
99C4⋊Dic53—41no—
100C4⋊Dic53—41no—
101C4⋊Dic53C2×C812yes—
102#11719919—24no—
103#11719919—42no—
104#11719919—42no—
105#11719919—41no—
106#11719919—41no—
107C4×Dic53—22no—
108C4×Dic53—22no—
109C4×Dic53—41no—
110C23×C53C2×C22⋊C411yescharacteristic
111C22×D53—21no—
112C22×D53—21no—
113C22×D53—21no—
114C22×D53—21no—
115C22×D53C22⋊C812yes—
116C22×D53#10212yes—
117C22×D53#10412yes—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—22no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53#10212yes—
132C2×C4×C53C22×C811yescharacteristic
133C2×C4×C53C2×M1611yescharacteristic
134C2×Dic53—21no—
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—22no—
139C2×Dic53—41no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C2×Dic53—41no—
143C2×Dic53—41no—
144C2×Dic53C22⋊C812yes—
145C2×Dic53#9912yes—
146C2×Dic53#10112yes—
147C2×Dic53C2×D811yescharacteristic
148C2×Dic53C2×SD1611yescharacteristic
149C2×Dic53#13712yes—
150C4×D53—24no—
151C4×D53—24no—
152C4×D53—42no—
153C4×D53—42no—
154C4×D53—44no—
155C4×D53—48no—
156C4×D53—84no—
157#88324—44no—
158#88324—82no—
159#7573—531nomaximal, Sylow
160C22×C53C2×C22⋊C811yescharacteristic
161C22×C53C2×#9911yescharacteristic
162C22×C53C2×#10111yescharacteristic
163C22×C53C2×#10211yescharacteristic
164C22×C53#41911yescharacteristic
165C22×C53C2×#10411yescharacteristic
166C22×C53#42111yescharacteristic
167C2×D53—22no—
168C2×D53—22no—
169C2×D53—22no—
170C2×D53—22no—
171C2×D53—22no—
172C2×D53—22no—
173C2×D53D4⋊C814yes—
174C4×C53—21no—
175C4×C53—21no—
176C4×C53—22no—
177C4×C53—42no—
178C4×C53—41no—
179C4×C53—81no—
180Dic53—22no—
181Dic53—21no—
182Dic53—21no—
183Dic53—42no—
184Dic53—81no—
185Dic53#32414yes—
186C2×#389—531no—
187#1797—531no—
188C4×C22⋊C8—531no—
189#1271—532no—
190#1271—532no—
191C2×C53C2×#32411yescharacteristic
192C2×C53C2×D4⋊C811yescharacteristic
193C2×C53#147911yescharacteristic
194C2×C53#150811yescharacteristic
195C2×C53#151911yescharacteristic
196C2×C53#152811yescharacteristic
197C2×C53#154611yescharacteristic
198D53—24no—
199C22×C4⋊C4—531no—
200C22×C4⋊C4—531no—
201C22×C4⋊C4—1061no—
202C22×C42—531no—
203C2×C4⋊C8—1061no—
204C2×C22⋊C8—1061no—
205C2×C22⋊C8—1061no—
206C2×C4×C8—1061no—
207#389—531no—
208#389—531no—
209#389—531no—
210#389—1061no—
211#389—1061no—
212#389#615711yescharacteristic
213C2×#95—1061no—
214#336—532no—
215#336—532no—
216C53#757311yescharacteristic, complemented, Sylow
217C23×C4—531no—
218C23×C4—531no—
219C23×C4—531no—
220C23×C4—1061no—
221C23×C4—1061no—
222C22×C8—1062no—
223C22×C8—1062no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—531no—
226C2×C4⋊C4—534no—
227C2×C4⋊C4—531no—
228C2×C4⋊C4—1062no—
229C2×C4⋊C4—1061no—
230C2×C4⋊C4—1061no—
231C2×C4⋊C4—1061no—
232C2×C4⋊C4—1061no—
233C2×C4⋊C4—1061no—
234C2×C4⋊C4—2122no—
235C2×C4⋊C4—2122no—
236C2×C4⋊C4—2122no—
237C2×C4⋊C4C2×#615711yescharacteristic
238C2×C4⋊C4C2×#615711yescharacteristic
239C2×C42—531no—
240C2×C42—531no—
241C2×C42—531no—
242C2×C42—2122no—
243C2×C42C2×#615711yescharacteristic
244C4⋊C8—2122no—
245C22⋊C8—1064no—
246C22⋊C8—2122no—
247C4×C8—2122no—
248#95—21no—
249#95—1061no—
250#95—1061no—
251#95—1061no—
252C24—531no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—531no—
256C22×C4—531no—
257C22×C4—531no—
258C22×C4—532no—
259C22×C4—532no—
260C22×C4—532no—
261C22×C4—532no—
262C22×C4—1062no—
263C22×C4—1062no—
264C22×C4—1062no—
265C22×C4—1062no—
266C22×C4—1064no—
267C22×C4—1061no—
268C22×C4—1061no—
269C22×C4—2121no—
270C22×C4—2121no—
271C22×C4—2121no—
272C22×C4—2122no—
273C22×C4—2122no—
274C22×C4—2122no—
275C22×C4—2122no—
276C22×C4—2122no—
277C22×C4C22⋊#615711yescharacteristic
278C22×C4C22⋊#615711yescharacteristic
279C22×C4C22×#615711yescharacteristic
280C2×C8—1064no—
281C2×C8—2122no—
282C2×C8—2122no—
283C2×C8—2122no—
284C2×C8—2122no—
285C2×C8—2122no—
286C4⋊C4—21no—
287C4⋊C4—41no—
288C4⋊C4—42no—
289C4⋊C4—532no—
290C4⋊C4—1064no—
291C4⋊C4—1061no—
292C4⋊C4—2121no—
293C4⋊C4—2121no—
294C4⋊C4—2121no—
295C4⋊C4—2121no—
296C4⋊C4—2121no—
297C4⋊C4—2122no—
298C4⋊C4—2122no—
299C4⋊C4D53⋊C812yes—
300C42—41no—
301C42—1062no—
302C42—1062no—
303C42—2121no—
304C23—532no—
305C23—532no—
306C23—532no—
307C23—1061no—
308C23—1061no—
309C23—1061no—
310C23—1061no—
311C23C2×C22⋊#615711yescharacteristic
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—532no—
326C2×C4—532no—
327C2×C4—532no—
328C2×C4—531no—
329C2×C4—531no—
330C2×C4—532no—
331C2×C4—1064no—
332C2×C4—1064no—
333C2×C4—1061no—
334C2×C4—1061no—
335C2×C4—1061no—
336C2×C4—1061no—
337C2×C4—1062no—
338C2×C4—4244no—
339C2×C4—2122no—
340C2×C4—2122no—
341C2×C4—2124no—
342C2×C4—2128no—
343C2×C4—2121no—
344C2×C4—2121no—
345C2×C4—2121no—
346C2×C4—2121no—
347C2×C4—2121no—
348C2×C4#1313327912yes—
349C2×C4C2×D53⋊C811yescharacteristic
350C2×C4D53⋊M1611yescharacteristic
351C8—4242no—
352C8—2124no—
353C22—534no—
354C22—1062no—
355C22—1062no—
356C22—1062no—
357C22—1062no—
358C22—1062no—
359C22—1062no—
360C22D53⋊(C22⋊C8)11yescharacteristic
361C22D53⋊#9911yescharacteristic
362C22D53⋊#10111yescharacteristic
363C22C2×#1313327911yescharacteristic
364C22#1681161611yescharacteristic
365C22D53⋊#10411yescharacteristic
366C22#1681162011yescharacteristic
367C4—22no—
368C4—21no—
369C4—21no—
370C4—41no—
371C4—42no—
372C4—81no—
373C4—534no—
374C4—1062no—
375C4—1061no—
376C4—1061no—
377C4—4241no—
378C4—2122no—
379C2—1064no—
380C2D53⋊#32411yescharacteristic
381C2D53⋊(D4⋊C8)11yescharacteristic
382C2#2029001811yescharacteristic
383C2#2029011311yescharacteristic
384C2#2029015011yescharacteristic
385C2#2029017811yescharacteristic
386C2#2029022511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC53×#389C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenter#389#6157
hypocenterC53#7573
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #389

Complements

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

NH ≅ G/Ntype
3. #20289534C2semidirect
4. #20289533C2semidirect
45. C4⋊C4×D53C8semidirect
216. C53#7573semidirect

Maximal subgroups

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

subgroupindex
#202905722
C4×#131332742
#202895342
#202895332
#389×D532
#757353