Finite Group Databasebeta

← #27280081

Subgroups — #27280081

Order 13568.

478 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 107 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#20283800C211yesmaximal, characteristic, complemented
2C4×#13133136C211yesmaximal, characteristic, complemented
3#20273313C212yesmaximal
4#20272579C212yesmaximal, complemented
5C53×#2632C211yesmaximal, characteristic
6#16810754—24no—
7#16810680—21no—
8#16810680C2212yes—
9#16810676—21no—
10C2×#13133136—21no—
11C4×#11719917C2212yes—
12#16810392—21no—
13#16810363—24no—
14C2×#124×C53C2211yescharacteristic
15C2×C4×Q8×C53C2211yescharacteristic
16C53×#399—21no—
17C53×#395C411yescharacteristic
18C53×#395C411yescharacteristic
19C53×#394C411yescharacteristic
20C53×#394C411yescharacteristic
21C53×#393—21no—
22C53×#388—21no—
23C53×#388—21no—
24C53×#383C2211yescharacteristic
25C2×#11719917—22no—
26C4⋊#88322—22no—
27C4⋊#88322—22no—
28C4⋊#88322—22no—
29#13133136—24no—
30C22⋊#88322—22no—
31#13133128—22no—
32#13133128—22no—
33#13133128D414yes—
34C4×#88322—22no—
35C4×#88322—22no—
36C22×Q8×C53C2×C411yescharacteristic
37C22×Q8×C53D411yescharacteristic
38C22×D4×C53C2×C411yescharacteristic
39#124×C53—22no—
40#124×C53—22no—
41C4×Q8×C53—22no—
42C4×Q8×C53—22no—
43C2×C4⋊C4×C53—21no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53C2×C411yescharacteristic
46C2×C4⋊C4×C53C2×C411yescharacteristic
47C2×C4⋊C4×C53D411yescharacteristic
48C2×C22⋊C4×C53—21no—
49C2×C22⋊C4×C53—21no—
50C2×C22⋊C4×C53—21no—
51C2×C22⋊C4×C53—21no—
52C2×C22⋊C4×C53—21no—
53C2×C22⋊C4×C53—21no—
54C2×C42×C53D411yescharacteristic
55C2×C42×C53D411yescharacteristic
56C2×C42×C53C2311yescharacteristic
57#95×C53—21no—
58#95×C53—21no—
59#95×C53—21no—
60#95×C53—21no—
61#95×C53C2×C411yescharacteristic
62#95×C53C2×C411yescharacteristic
63C24×C53—21no—
64C2×Q8×C53—22no—
65C2×Q8×C53—22no—
66C2×Q8×C53—22no—
67C2×Q8×C53—41no—
68C2×Q8×C53—81no—
69C2×D4×C53—22no—
70C2×D4×C53—81no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53C22⋊C411yescharacteristic
75C22×C4×C53C22⋊C411yescharacteristic
76C22×C4×C53C22⋊C411yescharacteristic
77C22×C4×C53C22⋊C411yescharacteristic
78C22×C4×C53C22×C411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C4⋊C4×C53—22no—
82C4⋊C4×C53—22no—
83C4⋊C4×C53—22no—
84C4⋊C4×C53—81no—
85C4⋊C4×C53—81no—
86C22⋊C4×C53—42no—
87C22⋊C4×C53—42no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53—42no—
90C22⋊C4×C53—41no—
91C22⋊C4×C53—81no—
92C22⋊C4×C53—81no—
93C42×C53—21no—
94C42×C53—21no—
95C42×C53—21no—
96C42×C53—21no—
97C42×C53C2×D412yes—
98C42×C53#4112yes—
99#11719917—24no—
100#11719917—42no—
101C2×#88322—24no—
102C2×#88322—42no—
103C23×C53—41no—
104C23×C53—41no—
105C23×C53—41no—
106C23×C53—41no—
107C23×C53—41no—
108C23×C53—41no—
109C23×C53—41no—
110C23×C53C2×C22⋊C411yescharacteristic
111Q8×C53—44no—
112Q8×C53—82no—
113D4×C53—82no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—21no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—21no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—41no—
126C2×C4×C53—41no—
127C2×C4×C53—41no—
128C2×C4×C53—41no—
129C2×C4×C53—41no—
130C2×C4×C53—41no—
131C2×C4×C53—41no—
132C2×C4×C53—41no—
133C2×C4×C53—41no—
134C2×C4×C53—41no—
135C2×C4×C53—41no—
136C2×C4×C53#10412yes—
137C2×C4×C53C4×D412yes—
138C2×C4×C53#12011yescharacteristic
139C2×C4×C53#12112yes—
140C2×C4×C53#12311yescharacteristic
141#88322—44no—
142#88322—44no—
143#9188—531nomaximal, Sylow
144C22×C53—21no—
145C22×C53—21no—
146C22×C53—81no—
147C22×C53—81no—
148C22×C53—81no—
149C22×C53—81no—
150C22×C53—81no—
151C22×C53—81no—
152C22×C53—81no—
153C22×C53#38611yescharacteristic
154C22×C53C2×#10411yescharacteristic
155C22×C53#42111yescharacteristic
156C4×C53—22no—
157C4×C53—22no—
158C4×C53—21no—
159C4×C53—21no—
160C4×C53—22no—
161C4×C53—41no—
162C4×C53—41no—
163C4×C53—41no—
164C4×C53—81no—
165C4×C53—81no—
166C4×C53—81no—
167#2983—531no—
168#2632D5311yescharacteristic
169C4×#101—531no—
170#1527—532no—
171#1288—532no—
172C2×C53—21no—
173C2×C53—21no—
174C2×C53—161no—
175C2×C53#187111yescharacteristic
176C2×C53#191211yescharacteristic
177C2×C53#191411yescharacteristic
178C2×#124C2×D5311yescharacteristic
179C2×C4×Q8C2×D5311yescharacteristic
180#446—1064no—
181#432—532no—
182#432—1061no—
183#431—1061no—
184C2×#101—1061no—
185C4×M16—532no—
186#399—21no—
187#395Dic5311yescharacteristic
188#395Dic5311yescharacteristic
189#394Dic5311yescharacteristic
190#394Dic5311yescharacteristic
191#393—21no—
192#388—21no—
193#388—21no—
194#383C2×D5311yescharacteristic
195#343—1061no—
196#329—1064no—
197C53#918811yescharacteristic, complemented, Sylow
198C22×Q8C2×Dic5311yescharacteristic
199C22×Q8C22⋊D5311yescharacteristic
200C22×D4C2×Dic5311yescharacteristic
201C2×M16—1062no—
202#124—22no—
203#124—22no—
204C4×Q8—22no—
205C4×Q8—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4C2×Dic5311yescharacteristic
209C2×C4⋊C4C2×Dic5311yescharacteristic
210C2×C4⋊C4C22⋊D5311yescharacteristic
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C42C22⋊D5311yescharacteristic
218C2×C42C22⋊D5311yescharacteristic
219C2×C42C22×D5311yescharacteristic
220C4⋊C8—1062no—
221C4⋊C8—1062no—
222C4⋊C8—1062no—
223#101—1064no—
224C22⋊C8—1062no—
225#97—534no—
226#97—1062no—
227#97—1062no—
228C4×C8—1062no—
229C4×C8—1062no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95C2×Dic5311yescharacteristic
235#95C2×Dic5311yescharacteristic
236C24—21no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—41no—
241C2×Q8—81no—
242C2×D4—22no—
243C2×D4—81no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4C22⋊Dic5311yescharacteristic
248C22×C4C22⋊Dic5311yescharacteristic
249C22×C4C22⋊Dic5311yescharacteristic
250C22×C4C22⋊Dic5311yescharacteristic
251C22×C4C22×Dic5311yescharacteristic
252C22×C4C2×C22⋊D5311yescharacteristic
253C22×C4C2×C22⋊D5311yescharacteristic
254M16—1064no—
255M16—2122no—
256C2×C8—1064no—
257C2×C8—2122no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—22no—
261C4⋊C4—81no—
262C4⋊C4—81no—
263C22⋊C4—42no—
264C22⋊C4—42no—
265C22⋊C4—41no—
266C22⋊C4—42no—
267C22⋊C4—41no—
268C22⋊C4—81no—
269C22⋊C4—81no—
270C42—21no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42D4×D5312yes—
275C42#1171994712yes—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23—41no—
282C23—41no—
283C23C2×C22⋊Dic5311yescharacteristic
284Q8—44no—
285Q8—82no—
286D4—82no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4#1313314412yes—
310C2×C4D4×Dic5312yes—
311C2×C4#1313318511yescharacteristic
312C2×C4#1313318912yes—
313C2×C4#1313319811yescharacteristic
314C8—2124no—
315C8—2124no—
316C22—21no—
317C22—21no—
318C22—81no—
319C22—81no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22—81no—
325C22#1681050311yescharacteristic
326C22C2×#1313314411yescharacteristic
327C22#1681064011yescharacteristic
328C4—21no—
329C4—21no—
330C4—22no—
331C4—22no—
332C4—22no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—81no—
337C4—81no—
338C4—81no—
339C2—21no—
340C2—21no—
341C2—161no—
342C2#2027501811yescharacteristic
343C2#2027525311yescharacteristic
344C2#2027526311yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2632C2
socleC22×C53#386
prime radicalC22×D4×C53C2×C4
cosocleC2×C42×C53C23
hypercenter#2632D53
hypocenterC53#9188
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2632

Complements

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

NH ≅ G/Ntype
1. #20283800C2semidirect
2. C4×#13133136C2semidirect
4. #20272579C2semidirect
197. C53#9188semidirect

Maximal subgroups

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

subgroupindex
#202838002
C4×#131331362
#202733132
#202725792
C53×#26322
#918853