Finite Group Databasebeta

← #27279172

Subgroups — #27279172

Order 13568.

670 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 111 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#20283744C211yesmaximal, characteristic, complemented
2C4×#13133134C211yesmaximal, characteristic, complemented
3#20273158C212yesmaximal, complemented
4#20272608C212yesmaximal, complemented
5C53×#2397C211yesmaximal, characteristic
6#16810695—24no—
7#16810680C2212yes—
8C2×#13133134—21no—
9#16810587—21no—
10C2×C22⋊#88322—21no—
11C4×#11719917C2212yes—
12#16810392—21no—
13D4⋊#88322—24no—
14C2×C4×D4×C53—21no—
15C2×C4×D4×C53—21no—
16C2×C4×D4×C53C2211yescharacteristic
17C2×C4×D4×C53C2211yescharacteristic
18C53×#399—21no—
19C53×#399C411yescharacteristic
20C53×#399C411yescharacteristic
21C53×#392—21no—
22C53×#392C411yescharacteristic
23C53×#392C411yescharacteristic
24C53×#376C2211yescharacteristic
25C2×#11719917—22no—
26C2×#11719917—22no—
27C22×#88322—22no—
28C4⋊#88322D414yescomplemented
29#13133134—24no—
30C22⋊#88322—22no—
31C22⋊#88322—22no—
32C22⋊#88322—22no—
33#13133128—22no—
34#13133128—22no—
35C4×#88322—22no—
36C22×D4×C53—21no—
37C22×D4×C53—21no—
38C22×D4×C53C2×C411yescharacteristic
39C22×D4×C53D411yescharacteristic
40C23×C4×C53—21no—
41C23×C4×C53D411yescharacteristic
42C23×C4×C53D411yescharacteristic
43C4×D4×C53—22no—
44C4×D4×C53—22no—
45C4×D4×C53—22no—
46C4×D4×C53—22no—
47C4×D4×C53—44no—
48C4×D4×C53—44no—
49C2×C4⋊C4×C53—21no—
50C2×C4⋊C4×C53—21no—
51C2×C4⋊C4×C53C2×C411yescharacteristic
52C2×C4⋊C4×C53D411yescharacteristic
53C2×C22⋊C4×C53—21no—
54C2×C22⋊C4×C53—21no—
55C2×C22⋊C4×C53—21no—
56C2×C22⋊C4×C53—21no—
57C2×C22⋊C4×C53—21no—
58C2×C22⋊C4×C53C2×C411yescharacteristic
59C2×C22⋊C4×C53C2×C411yescharacteristic
60C2×C42×C53—21no—
61C2×C42×C53C2311yescharacteristic
62#95×C53—21no—
63#95×C53C2×C411yescharacteristic
64#95×C53C2×C411yescharacteristic
65C24×C53—21no—
66C24×C53C22⋊C411yescharacteristic
67C24×C53C22⋊C411yescharacteristic
68C2×D4×C53—22no—
69C2×D4×C53—22no—
70C2×D4×C53—22no—
71C2×D4×C53—41no—
72C2×D4×C53—42no—
73C2×D4×C53—42no—
74C2×D4×C53—42no—
75C2×D4×C53—42no—
76C2×D4×C53—42no—
77C2×D4×C53—42no—
78C2×D4×C53—42no—
79C22×C4×C53—21no—
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—42no—
87C22×C4×C53—42no—
88C22×C4×C53—42no—
89C22×C4×C53—42no—
90C22×C4×C53—41no—
91C22×C4×C53—41no—
92C22×C4×C53C22⋊C411yescharacteristic
93C22×C4×C53C22⋊C411yescharacteristic
94C22×C4×C53C22×C411yescharacteristic
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53C2×D411yescharacteristic
97C4⋊C4×C53—22no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—42no—
100C4⋊C4×C53—42no—
101C22⋊C4×C53—22no—
102C22⋊C4×C53—22no—
103C22⋊C4×C53—42no—
104C22⋊C4×C53—42no—
105C22⋊C4×C53—42no—
106C22⋊C4×C53—42no—
107C22⋊C4×C53—42no—
108C42×C53—24no—
109C42×C53C2×D412yes—
110C42×C53#4112yes—
111#11719917—24no—
112#11719917—42no—
113#11719917—82no—
114C2×#88322—24no—
115C2×#88322—42no—
116C2×#88322—82no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53—22no—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53—21no—
123C23×C53—41no—
124C23×C53—41no—
125C23×C53—41no—
126C23×C53—41no—
127C23×C53—41no—
128C23×C53—41no—
129C23×C53—41no—
130C23×C53—41no—
131C23×C53—41no—
132C23×C53—41no—
133C23×C53#10412yes—
134C23×C53C2×C22⋊C411yescharacteristic
135D4×C53—24no—
136D4×C53—44no—
137D4×C53—42no—
138D4×C53—48no—
139D4×C53—48no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—22no—
143C2×C4×C53—22no—
144C2×C4×C53—22no—
145C2×C4×C53—21no—
146C2×C4×C53—21no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—42no—
153C2×C4×C53—42no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—81no—
160C2×C4×C53—81no—
161C2×C4×C53—82no—
162C2×C4×C53—82no—
163C2×C4×C53C4×D412yes—
164C2×C4×C53#12011yescharacteristic
165C2×C4×C53#12112yes—
166C2×C4×C53#12311yescharacteristic
167#88322—44no—
168#88322—44no—
169#9031—531nomaximal, Sylow
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—22no—
173C22×C53—42no—
174C22×C53—42no—
175C22×C53—42no—
176C22×C53—42no—
177C22×C53—41no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53—41no—
181C22×C53—42no—
182C22×C53—41no—
183C22×C53—81no—
184C22×C53—81no—
185C22×C53—81no—
186C22×C53—81no—
187C22×C53—81no—
188C22×C53—81no—
189C22×C53#38611yescharacteristic
190C22×C53C2×#10411yescharacteristic
191C22×C53#42111yescharacteristic
192C4×C53—22no—
193C4×C53—21no—
194C4×C53—22no—
195C4×C53—42no—
196C4×C53—42no—
197C4×C53—41no—
198C4×C53—41no—
199C4×C53—81no—
200C4×C53#45312yes—
201#2973—531no—
202#2397D5311yescharacteristic
203C4×#100—531no—
204#1490—532no—
205#1300—532no—
206C2×C53—21no—
207C2×C53—21no—
208C2×C53—42no—
209C2×C53—81no—
210C2×C53—82no—
211C2×C53#187111yescharacteristic
212C2×C53#188711yescharacteristic
213C2×C53#189111yescharacteristic
214C2×C4×D4—21no—
215C2×C4×D4—21no—
216C2×C4×D4C2×D5311yescharacteristic
217C2×C4×D4C2×D5311yescharacteristic
218#435—1064no—
219#432—532no—
220C2×#100—1061no—
221#407—1061no—
222C2×C22⋊C8—1061no—
223C4×M16—532no—
224#399—21no—
225#399Dic5311yescharacteristic
226#399Dic5311yescharacteristic
227#392—21no—
228#392Dic5311yescharacteristic
229#392Dic5311yescharacteristic
230#376C2×D5311yescharacteristic
231#343—1061no—
232D4⋊C8—1064no—
233C53#903111yescharacteristic, complemented, Sylow
234C22×D4—21no—
235C22×D4—21no—
236C22×D4C2×Dic5311yescharacteristic
237C22×D4C22⋊D5311yescharacteristic
238C23×C4—21no—
239C23×C4C22⋊D5311yescharacteristic
240C23×C4C22⋊D5311yescharacteristic
241C2×M16—1062no—
242C2×M16—1062no—
243C22×C8—1062no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—22no—
247C4×D4—22no—
248C4×D4—44no—
249C4×D4—44no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4C2×Dic5311yescharacteristic
253C2×C4⋊C4C22⋊D5311yescharacteristic
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4C2×Dic5311yescharacteristic
260C2×C22⋊C4C2×Dic5311yescharacteristic
261C2×C42—21no—
262C2×C42C22×D5311yescharacteristic
263C4⋊C8—534no—
264#100—1064no—
265C22⋊C8—1062no—
266C22⋊C8—1062no—
267C22⋊C8—1062no—
268#97—1062no—
269#97—1062no—
270C4×C8—1062no—
271#95—21no—
272#95C2×Dic5311yescharacteristic
273#95C2×Dic5311yescharacteristic
274C24—21no—
275C24C22⋊Dic5311yescharacteristic
276C24C22⋊Dic5311yescharacteristic
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—41no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—42no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4C22⋊Dic5311yescharacteristic
302C22×C4C22⋊Dic5311yescharacteristic
303C22×C4C22×Dic5311yescharacteristic
304C22×C4C2×C22⋊D5311yescharacteristic
305C22×C4C2×C22⋊D5311yescharacteristic
306M16—1064no—
307M16—4242no—
308M16—2122no—
309C2×C8—1064no—
310C2×C8—4242no—
311C2×C8—2122no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C22⋊C4—22no—
317C22⋊C4—22no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C42—24no—
324C42D4×D5312yes—
325C42#1171994712yes—
326C23—21no—
327C23—21no—
328C23—22no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23#1313314412yes—
343C23C2×C22⋊Dic5311yescharacteristic
344D4—24no—
345D4—44no—
346D4—42no—
347D4—48no—
348D4—48no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4D4×Dic5312yes—
373C2×C4#1313318511yescharacteristic
374C2×C4#1313318912yes—
375C2×C4#1313319811yescharacteristic
376C8—2124no—
377C8—2124no—
378C22—21no—
379C22—21no—
380C22—22no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—42no—
390C22—41no—
391C22—81no—
392C22—81no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—81no—
397C22#1681050311yescharacteristic
398C22C2×#1313314411yescharacteristic
399C22#1681064011yescharacteristic
400C4—22no—
401C4—21no—
402C4—22no—
403C4—42no—
404C4—42no—
405C4—41no—
406C4—41no—
407C4—81no—
408C4#1681079012yes—
409C2—21no—
410C2—21no—
411C2—42no—
412C2—81no—
413C2—82no—
414C2#2027501811yescharacteristic
415C2#2027511411yescharacteristic
416C2#2027513411yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#2397C2
socleC22×C53#386
cosocleC2×C42×C53C23
hypercenter#2397D53
hypocenterC53#9031
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2397

Complements

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

NH ≅ G/Ntype
1. #20283744C2semidirect
2. C4×#13133134C2semidirect
3. #20273158C2semidirect
4. #20272608C2semidirect
28. C4⋊#88322D4semidirect
233. C53#9031semidirect

Maximal subgroups

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

subgroupindex
#202837442
C4×#131331342
#202731582
#202726082
C53×#23972
#903153