Finite Group Databasebeta

← #28299717

Subgroups — #28299717

Order 13568.

584 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 104 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#20290606C211yesmaximal, characteristic, complemented
2#20290571C211yesmaximal, characteristic, complemented
3#20289533C212yesmaximal, complemented
4#20289531C212yesmaximal, complemented
5#121×Dic53C211yesmaximal, characteristic
6C2×#13133284—21no—
7C2×C22⋊#88324—21no—
8C2×C22⋊#88324—21no—
9#16811489—21no—
10#16811489C2212yes—
11#16811489C2212yes—
12C2×D4×Dic53—21no—
13C2×D4×Dic53C411yescharacteristic
14C22×C4×Dic53C2211yescharacteristic
15#16810526C2211yescharacteristic
16#16810522C2211yescharacteristic
17#16810516—21no—
18#16810503—21no—
19#16810483C411yescharacteristic
20C4⋊C4×Dic53C411yescharacteristic
21C22⋊C4×Dic53—21no—
22C2×#121×C53C411yescharacteristic
23C22×#88324—22no—
24C22×#88324—22no—
25#13133284—42no—
26C22⋊#88324—42no—
27C22⋊#88324—42no—
28C23×Dic53—21no—
29C23×Dic53D411yescharacteristic
30C23×Dic53D411yescharacteristic
31D4×Dic53—42no—
32D4×Dic53—42no—
33D4×Dic53—42no—
34C2×C4⋊Dic53—21no—
35C2×C4⋊Dic53C2×C411yescharacteristic
36C2×C4⋊Dic53C2×C411yescharacteristic
37C2×C22⋊Dic53—21no—
38C2×C22⋊Dic53—21no—
39C2×C22⋊Dic53—21no—
40C2×C22⋊Dic53—21no—
41C2×C4×Dic53—21no—
42C2×C4×Dic53—42no—
43C2×C4×Dic53C2×C411yescharacteristic
44C2×C4×Dic53D411yescharacteristic
45C2×C4×Dic53D411yescharacteristic
46C2×C4×Dic53C2311yescharacteristic
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50C22×D4×C53—21no—
51C22×D4×C53C2×C411yescharacteristic
52C23×C4×C53C2×C411yescharacteristic
53#121×C53—22no—
54#121×C53—22no—
55C2×C4⋊C4×C53C2×C411yescharacteristic
56C2×C22⋊C4×C53—21no—
57C24×C53—21no—
58C24×C53C22⋊C411yescharacteristic
59C2×D4×C53—24no—
60C2×D4×C53—21no—
61C2×D4×C53—41no—
62C2×D4×C53—41no—
63C2×D4×C53—42no—
64C2×D4×C53—42no—
65C2×D4×C53M1612yes—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—41no—
70C22×C4×C53—41no—
71C22×C4×C53C22⋊C411yescharacteristic
72C22×C4×C53C22×C411yescharacteristic
73C22×Dic53—22no—
74C22×Dic53—22no—
75C22×Dic53—21no—
76C22×Dic53—21no—
77C22×Dic53—41no—
78C22×Dic53—41no—
79C22×Dic53—41no—
80C22×Dic53—42no—
81C22×Dic53—42no—
82C22×Dic53—42no—
83C22×Dic53C22⋊C411yescharacteristic
84C22×Dic53C22⋊C411yescharacteristic
85C22×Dic53D812yes—
86C22×Dic53SD1612yes—
87C22×Dic53C2×D411yescharacteristic
88C22×Dic53C2×D411yescharacteristic
89C2×#88324—42no—
90C2×#88324—42no—
91C2×#88324—42no—
92C2×#88324—42no—
93C2×#88324—42no—
94C2×#88324—42no—
95C4⋊C4×C53—21no—
96C4⋊C4×C53M1612yes—
97C22⋊C4×C53—24no—
98C22⋊Dic53—42no—
99C22⋊Dic53—42no—
100C22⋊Dic53—41no—
101C22⋊Dic53—41no—
102C22⋊Dic53—42no—
103C4⋊Dic53—41no—
104C4⋊Dic53—41no—
105C4⋊Dic53—41no—
106C4⋊Dic53—41no—
107C4×Dic53—22no—
108C4×Dic53—22no—
109C4×Dic53—41no—
110C4×Dic53—41no—
111C4×Dic53—42no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—21no—
118C23×C53—22no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53—41no—
124C23×C53—41no—
125C23×C53#10212yes—
126C23×C53C2×C22⋊C411yescharacteristic
127D4×C53—42no—
128D4×C53—42no—
129D4×C53—48no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—21no—
133C2×C4×C53—22no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—42no—
137C2×C4×C53—42no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53C2×M1611yescharacteristic
141C2×C4×C53C2×M1611yescharacteristic
142C2×Dic53—21no—
143C2×Dic53—21no—
144C2×Dic53—21no—
145C2×Dic53—21no—
146C2×Dic53—21no—
147C2×Dic53—21no—
148C2×Dic53—21no—
149C2×Dic53—21no—
150C2×Dic53—42no—
151C2×Dic53—42no—
152C2×Dic53—44no—
153C2×Dic53—41no—
154C2×Dic53—41no—
155C2×Dic53—41no—
156C2×Dic53—41no—
157C2×Dic53—41no—
158C2×Dic53—41no—
159C2×Dic53—84no—
160C2×Dic53#9912yes—
161C2×Dic53#10212yes—
162C2×Dic53C2×D811yescharacteristic
163C2×Dic53C2×SD1611yescharacteristic
164C2×Dic53#13512yes—
165#88324—82no—
166#88324—82no—
167#7647—531nomaximal, Sylow
168C22×C53—22no—
169C22×C53—22no—
170C22×C53—22no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53—41no—
174C22×C53—41no—
175C22×C53—42no—
176C22×C53—42no—
177C22×C53—42no—
178C22×C53—81no—
179C22×C53—81no—
180C22×C53—81no—
181C22×C53—81no—
182C22×C53#40711yescharacteristic
183C22×C53C2×#9911yescharacteristic
184C22×C53#41311yescharacteristic
185C22×C53C2×#10211yescharacteristic
186C22×C53#41611yescharacteristic
187C22×C53#42111yescharacteristic
188C22×C53#42111yescharacteristic
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—41no—
192C4×C53—42no—
193Dic53—22no—
194Dic53—22no—
195Dic53—21no—
196Dic53—22no—
197Dic53—21no—
198Dic53—81no—
199Dic53—81no—
200C4×#121—531no—
201#1817—531no—
202#1797—531no—
203#1271—532no—
204#1270—532no—
205C2×C53—42no—
206C2×C53—82no—
207C2×C53#146511yescharacteristic
208C2×C53#148611yescharacteristic
209C2×C53#148811yescharacteristic
210C2×C53#151611yescharacteristic
211C2×C53#151911yescharacteristic
212C2×C53#153511yescharacteristic
213C2×C53#154011yescharacteristic
214C2×#121#615711yescharacteristic
215C2×C4×D4—531no—
216C2×C4×D4—1061no—
217C22×C42—531no—
218C2×C4⋊C8—1061no—
219C2×C22⋊C8—1061no—
220C2×C22⋊C8—1061no—
221#390—531no—
222#389—531no—
223#388—1061no—
224#386—1061no—
225#380—531no—
226C4×C4⋊C4—531no—
227C4×C22⋊C4—1061no—
228#336—532no—
229#336—532no—
230#336—1061no—
231C53#764711yescharacteristic, complemented, Sylow
232C22×D4—21no—
233C22×D4C2×#615711yescharacteristic
234C23×C4—531no—
235C23×C4—531no—
236C23×C4—1061no—
237C23×C4C2×#615711yescharacteristic
238C22×C8—1062no—
239C22×C8—1062no—
240#121—22no—
241#121—22no—
242C4×D4—2122no—
243C4×D4—2122no—
244C4×D4—2122no—
245C2×C4⋊C4—531no—
246C2×C4⋊C4—531no—
247C2×C4⋊C4—1061no—
248C2×C4⋊C4C2×#615711yescharacteristic
249C2×C22⋊C4—21no—
250C2×C22⋊C4—1061no—
251C2×C22⋊C4—1061no—
252C2×C22⋊C4—1061no—
253C2×C22⋊C4—1061no—
254C2×C42—531no—
255C2×C42—531no—
256C2×C42—531no—
257C2×C42—531no—
258C2×C42—1061no—
259C2×C42—2122no—
260C4⋊C8—2122no—
261C22⋊C8—2122no—
262C22⋊C8—2122no—
263#95—1061no—
264#95—1061no—
265#95—1061no—
266C24—21no—
267C24C22⋊#615711yescharacteristic
268C2×D4—24no—
269C2×D4—21no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4#1171994012yes—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—531no—
281C22×C4—531no—
282C22×C4—531no—
283C22×C4—531no—
284C22×C4—532no—
285C22×C4—532no—
286C22×C4—1062no—
287C22×C4—1062no—
288C22×C4—1061no—
289C22×C4—1061no—
290C22×C4—2122no—
291C22×C4—2121no—
292C22×C4—2121no—
293C22×C4—2121no—
294C22×C4—2122no—
295C22×C4—2122no—
296C22×C4C22⋊#615711yescharacteristic
297C22×C4C22×#615711yescharacteristic
298C2×C8—2122no—
299C2×C8—2122no—
300C2×C8—2122no—
301C2×C8—2122no—
302C2×C8—2122no—
303C2×C8—2122no—
304C4⋊C4—21no—
305C4⋊C4—2121no—
306C4⋊C4—2121no—
307C4⋊C4—2121no—
308C4⋊C4—2121no—
309C4⋊C4#1171994012yes—
310C22⋊C4—24no—
311C22⋊C4—2121no—
312C22⋊C4—2121no—
313C22⋊C4—2122no—
314C22⋊C4—2122no—
315C22⋊C4—2122no—
316C42—1062no—
317C42—1062no—
318C42—2121no—
319C42—2121no—
320C42—2122no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—22no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23#1313327912yes—
335C23C2×C22⋊#615711yescharacteristic
336D4—42no—
337D4—42no—
338D4—48no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—532no—
350C2×C4—532no—
351C2×C4—531no—
352C2×C4—531no—
353C2×C4—532no—
354C2×C4—1061no—
355C2×C4—1061no—
356C2×C4—1061no—
357C2×C4—1061no—
358C2×C4—1061no—
359C2×C4—1061no—
360C2×C4—1061no—
361C2×C4—1061no—
362C2×C4—4244no—
363C2×C4—2122no—
364C2×C4—2122no—
365C2×C4—2124no—
366C2×C4—2121no—
367C2×C4—2121no—
368C2×C4—2121no—
369C2×C4—2121no—
370C2×C4—2121no—
371C2×C4—2121no—
372C2×C4C2×#1171994011yescharacteristic
373C2×C4C2×#1171994011yescharacteristic
374C8—4242no—
375C8—4242no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—42no—
384C22—42no—
385C22—42no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22#1681159311yescharacteristic
391C22D53⋊#9911yescharacteristic
392C22#1681161011yescharacteristic
393C22C2×#1313327911yescharacteristic
394C22#1681161311yescharacteristic
395C22#1681162011yescharacteristic
396C22#1681162111yescharacteristic
397C4—21no—
398C4—21no—
399C4—41no—
400C4—42no—
401C4—1062no—
402C4—1062no—
403C4—1061no—
404C4—1062no—
405C4—1061no—
406C4—4241no—
407C4—4241no—
408C2—42no—
409C2—82no—
410C2#2028997111yescharacteristic
411C2#2029004011yescharacteristic
412C2#2029004611yescharacteristic
413C2#2029014011yescharacteristic
414C2#2029015011yescharacteristic
415C2#2029020011yescharacteristic
416C2#2029021311yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
Fitting, prime radicalC2×#121×C53C4
socleC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×#121#6157
hypocenterC53#7647
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. #20290606C2semidirect
2. #20290571C2semidirect
3. #20289533C2semidirect
4. #20289531C2semidirect
231. C53#7647semidirect

Maximal subgroups

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

subgroupindex
#202906062
#202905712
#202895332
#202895312
#121×Dic532
#764753