Finite Group Databasebeta

← #27296905

Subgroups — #27296905

Order 13568.

370 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 63 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#20275882C211yesmaximal, characteristic
2#20273739C211yesmaximal, characteristic
3#20273671C211yesmaximal, characteristic
4#20272739C211yesmaximal, characteristic
5#20272736C211yesmaximal, characteristic
6#20272645C211yesmaximal, characteristic
7C53×#2841C211yesmaximal, characteristic
8#16811249C2211yescharacteristic
9#16810958—21no—
10#16810779—21no—
11#16810486—21no—
12#16810390C2211yescharacteristic
13#16810389—21no—
14#16810359—21no—
15#16810359—21no—
16#16810347C2211yescharacteristic
17#16810347C2211yescharacteristic
18C53×#400—21no—
19C53×#399C2211yescharacteristic
20C53×#398—21no—
21C53×#398—21no—
22C53×#396—21no—
23C53×#395—21no—
24C53×#393C2211yescharacteristic
25C53×#381C2211yescharacteristic
26C53×#380—21no—
27C2×#11719925—41no—
28#13133196—21no—
29#13133196—21no—
30#13133194—41no—
31#13133194—41no—
32C2×C22⋊Dic53—21no—
33C2×C22⋊Dic53—21no—
34#13133150—41no—
35#13133141—41no—
36#13133140—41no—
37C22⋊#88322—21no—
38C22⋊#88322—21no—
39#13133124—41no—
40#13133124—41no—
41C22×Q8×C53D411yescharacteristic
42C23×C4×C53D411yescharacteristic
43C2×C4⋊C4×C53—21no—
44C2×C4⋊C4×C53—21no—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53—21no—
47C2×C4⋊C4×C53—21no—
48C2×C4⋊C4×C53—21no—
49C2×C4⋊C4×C53D411yescharacteristic
50C2×C22⋊C4×C53—21no—
51C2×C22⋊C4×C53—21no—
52C2×C22⋊C4×C53D411yescharacteristic
53C2×C22⋊C4×C53C2311yescharacteristic
54#95×C53—21no—
55#95×C53—21no—
56#95×C53—21no—
57#95×C53—21no—
58#95×C53—21no—
59#95×C53—21no—
60#95×C53D411yescharacteristic
61#95×C53D411yescharacteristic
62C24×C53C2×D411yescharacteristic
63C2×Q8×C53—41no—
64C2×Q8×C53—81no—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—21no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—41no—
73C22×C4×C53—41no—
74C22×C4×C53Q1612yescomplemented
75C22×C4×C53C2×D411yescharacteristic
76C22×C4×C53C2×D411yescharacteristic
77C22×Dic53—21no—
78C22×Dic53—21no—
79C22×Dic53—41no—
80C2×Dic106—21no—
81C2×Dic106—21no—
82C4⋊C4×C53—41no—
83C4⋊C4×C53—81no—
84C4⋊C4×C53—81no—
85C4⋊C4×C53—81no—
86C4⋊C4×C53—81no—
87C4⋊C4×C53—81no—
88C4⋊C4×C53—81no—
89C22⋊C4×C53—41no—
90C22⋊C4×C53—41no—
91C22⋊C4×C53—81no—
92C22⋊C4×C53—81no—
93C22⋊Dic53—41no—
94C22⋊Dic53—41no—
95C22⋊Dic53—41no—
96C22⋊Dic53—41no—
97#11719925—82no—
98C4⋊Dic53—21no—
99C4⋊Dic53—21no—
100#11719919—41no—
101#11719919—41no—
102C2×#88322—41no—
103C2×#88322—41no—
104C23×C53—21no—
105C23×C53—21no—
106C23×C53—41no—
107C23×C53—41no—
108C23×C53#12011yescharacteristic
109C23×C53C2×Q1611yescharacteristic
110C23×C53#13711yescharacteristic
111Q8×C53—42no—
112Q8×C53—81no—
113C2×C4×C53—22no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—42no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
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—81no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139C2×Dic53—81no—
140C2×Dic53—81no—
141C2×Dic53—81no—
142C2×Dic53—81no—
143C2×Dic53—81no—
144Dic106—81no—
145Dic106—81no—
146#88322—81no—
147#88322—81no—
148#12219—531nomaximal, Sylow
149C22×C53—21no—
150C22×C53—21no—
151C22×C53—21no—
152C22×C53—21no—
153C22×C53—41no—
154C22×C53—41no—
155C22×C53—41no—
156C22×C53—41no—
157C22×C53—81no—
158C22×C53#45111yescharacteristic
159C22×C53#45511yescharacteristic
160C22×C53B(4,𝔽2)11yescharacteristic
161C4×C53—42no—
162C4×C53—41no—
163C4×C53—41no—
164C4×C53—81no—
165C4×C53—81no—
166C4×C53—81no—
167C4×C53—81no—
168C4×C53—81no—
169Dic53—81no—
170Dic53—81no—
171Dic53—161no—
172#2841D5311yescharacteristic
173#2030—531no—
174#1614—531no—
175#1602—531no—
176#1352—531no—
177#1351—531no—
178#1317—531no—
179C2×C53—21no—
180C2×C53—21no—
181C2×C53—41no—
182C2×C53—41no—
183C2×C53#160211yescharacteristic
184C2×C53#220011yescharacteristic
185C2×C53#220111yescharacteristic
186#543—531no—
187#484—1061no—
188#451—1061no—
189#400—21no—
190#399C2×D5311yescharacteristic
191#398—21no—
192#398—21no—
193#396—21no—
194#395—21no—
195#393C2×D5311yescharacteristic
196#381—1061no—
197#381C2×D5311yescharacteristic
198#380—21no—
199#342—531no—
200#342—1061no—
201#328—1061no—
202#328—1061no—
203#323—531no—
204#323—531no—
205C53#1221911yescharacteristic, complemented, Sylow
206C22×Q8C22⋊D5311yescharacteristic
207C23×C4C22⋊D5311yescharacteristic
208C2×Q16—2121no—
209C22⋊Q8—1061no—
210C22⋊Q8—1061no—
211C22⋊Q8—2121no—
212C22⋊Q8—2121no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4C22⋊D5311yescharacteristic
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—1061no—
223C2×C22⋊C4—1061no—
224C2×C22⋊C4C22⋊D5311yescharacteristic
225C2×C22⋊C4C22×D5311yescharacteristic
226#107—2121no—
227#103—2121no—
228#103—2121no—
229C22⋊C8—1061no—
230C22⋊C8—1061no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—2121no—
238#95—2121no—
239#95C22⋊D5311yescharacteristic
240#95C22⋊D5311yescharacteristic
241C24C2×C22⋊D5311yescharacteristic
242C2×Q8—41no—
243C2×Q8—81no—
244C2×Q8—1061no—
245C2×Q8—1061no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—1061no—
256C22×C4—1061no—
257C22×C4—2121no—
258C22×C4#1171992512yescomplemented
259C22×C4C2×C22⋊D5311yescharacteristic
260C22×C4C2×C22⋊D5311yescharacteristic
261Q16—4242no—
262C2×C8—2121no—
263C2×C8—2121no—
264C4⋊C4—41no—
265C4⋊C4—81no—
266C4⋊C4—81no—
267C4⋊C4—81no—
268C4⋊C4—81no—
269C4⋊C4—81no—
270C4⋊C4—81no—
271C4⋊C4—1061no—
272C4⋊C4—1061no—
273C4⋊C4—2121no—
274C4⋊C4—2121no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—81no—
278C22⋊C4—81no—
279C22⋊C4—2121no—
280C22⋊C4—2121no—
281C22⋊C4—2121no—
282C22⋊C4—2121no—
283C23—21no—
284C23—21no—
285C23—41no—
286C23—41no—
287C23#1313318611yescharacteristic
288C23C2×#1171992511yescharacteristic
289C23#1313324811yescharacteristic
290Q8—42no—
291Q8—81no—
292Q8—4241no—
293Q8—4241no—
294C2×C4—22no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
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—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—81no—
318C2×C4—4241no—
319C2×C4—4241no—
320C2×C4—4241no—
321C2×C4—4241no—
322C2×C4—4241no—
323C2×C4—2121no—
324C2×C4—2121no—
325C8—4241no—
326C8—4241no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—81no—
336C22#1681077911yescharacteristic
337C22#1681080111yescharacteristic
338C22#1681081011yescharacteristic
339C4—42no—
340C4—41no—
341C4—41no—
342C4—81no—
343C4—81no—
344C4—81no—
345C4—81no—
346C4—81no—
347C4—8481no—
348C4—4241no—
349C4—4241no—
350C2—21no—
351C2—21no—
352C2—41no—
353C2—41no—
354C2#2027367211yescharacteristic
355C2#2027672211yescharacteristic
356C2#2027672811yescharacteristic
subgroupG/H
centerC22#16810810
commutator, cosocleC2×C22⋊C4×C53C23
FrattiniC2×C22⋊C4C22×D53
FittingC53×#2841C2
socleC22×C53B(4,𝔽2)
prime radicalC24×C53C2×D4
hypercenter#2841D53
hypocenterC53#12219
LCSG ▷ C2×C22⋊C4×C53 ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #2841

Complements

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

NH ≅ G/Ntype
74. C22×C4×C53Q16semidirect
205. C53#12219semidirect
258. C22×C4#11719925semidirect

Maximal subgroups

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

subgroupindex
#202758822
#202737392
#202736712
#202727392
#202727362
#202726452
C53×#28412
#1221953