Finite Group Databasebeta

← #27284813

Subgroups — #27284813

Order 13568.

510 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 89 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
1C2×#16810956C211yesmaximal, characteristic
2C2×#16810951C211yesmaximal, characteristic
3#20274793C211yesmaximal, characteristic
4#20273079C211yesmaximal, characteristic
5#20273079C211yesmaximal, characteristic
6#20272602C211yesmaximal, characteristic
7C53×#3195C211yesmaximal, characteristic
8C22×C4⋊Dic53C2211yescharacteristic
9#16810956—22no—
10#16810956—22no—
11#16810951—22no—
12#16810951—22no—
13C2×#13133150—21no—
14C2×#13133148—21no—
15C2×#13133141—21no—
16C2×#13133141—21no—
17#16810587C2211yescharacteristic
18C2×C22⋊#88322C2211yescharacteristic
19C2×C22⋊#88322C2211yescharacteristic
20#16810391—22no—
21#16810349—22no—
22#16810349—22no—
23C2×C22⋊Q8×C53C2211yescharacteristic
24C2×C22⋊Q8×C53C2211yescharacteristic
25C53×#483—21no—
26C53×#483—21no—
27C53×#470—22no—
28C53×#451—21no—
29C53×#451—21no—
30C53×#407C2211yescharacteristic
31C23×Dic53—21no—
32C2×#11719917—21no—
33C22×#88322—21no—
34C22×#88322—21no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53—21no—
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53—21no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53D412yes—
41#13133150—24no—
42#13133148—24no—
43#13133141—24no—
44#13133141—24no—
45C22⋊#88322—21no—
46C22⋊#88322—21no—
47C22⋊#88322—21no—
48C22⋊#88322—21no—
49C22⋊#88322D412yes—
50C22⋊#88322D412yes—
51C22×Q8×C53D411yescharacteristic
52C22×Q8×C53D411yescharacteristic
53C23×C4×C53C2311yescharacteristic
54C2×Q16×C53—22no—
55C2×M16×C53D412yes—
56C22⋊Q8×C53—21no—
57C22⋊Q8×C53—21no—
58C22⋊Q8×C53—21no—
59C22⋊Q8×C53—21no—
60C22⋊Q8×C53—21no—
61C22⋊Q8×C53—21no—
62C22⋊Q8×C53—21no—
63C22⋊Q8×C53—21no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53D411yescharacteristic
67C2×C4⋊C4×C53D411yescharacteristic
68C2×C22⋊C4×C53—21no—
69C2×C22⋊C4×C53—21no—
70#106×C53—22no—
71#103×C53—22no—
72#103×C53—22no—
73C22⋊C8×C53—21no—
74C22⋊C8×C53—21no—
75C24×C53#4111yescharacteristic
76C2×Q8×C53—21no—
77C2×Q8×C53—21no—
78C2×Q8×C53—21no—
79C2×Q8×C53—21no—
80C2×Q8×C53—42no—
81C2×Q8×C53—42no—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—42no—
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53C2×D411yescharacteristic
90C22×C4×C53C2×D411yescharacteristic
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53C2×D411yescharacteristic
93C22×Dic53—22no—
94C22×Dic53—41no—
95C22×Dic53—41no—
96C22×Dic53—82no—
97Q16×C53—82no—
98M16×C53—42no—
99C2×C8×C53—22no—
100C4⋊C4×C53—21no—
101C4⋊C4×C53—21no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—41no—
105C4⋊C4×C53—41no—
106C4⋊C4×C53—41no—
107C4⋊C4×C53—41no—
108C22⋊C4×C53—41no—
109C22⋊C4×C53—41no—
110C22⋊C4×C53—41no—
111C22⋊C4×C53—41no—
112C4⋊Dic53—22no—
113C4⋊Dic53—22no—
114C4⋊Dic53—41no—
115C4⋊Dic53—41no—
116#11719917—42no—
117C2×#88322—22no—
118C2×#88322—22no—
119C2×#88322—42no—
120C2×#88322—42no—
121C2×#88322—41no—
122C23×C53—22no—
123C23×C53—22no—
124C23×C53—41no—
125C23×C53#12111yescharacteristic
126C23×C53#12111yescharacteristic
127C23×C53#12111yescharacteristic
128Q8×C53—42no—
129Q8×C53—42no—
130Q8×C53—42no—
131Q8×C53—42no—
132C2×C4×C53—22no—
133C2×C4×C53—22no—
134C2×C4×C53—22no—
135C2×C4×C53—41no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—41no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—42no—
143C2×C4×C53—42no—
144C2×C4×C53—81no—
145C2×C4×C53—81no—
146C2×C4×C53#12011yescharacteristic
147C2×C4×C53#12011yescharacteristic
148C2×C4×C53#12011yescharacteristic
149C2×C4×C53#12711yescharacteristic
150C2×Dic53—42no—
151C2×Dic53—82no—
152C2×Dic53—82no—
153C2×Dic53—82no—
154C8×C53—42no—
155#88322—42no—
156#88322—42no—
157#88322—81no—
158#9995—531nomaximal, Sylow
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—44no—
167C22×C53#39211yescharacteristic
168C22×C53#47112yes—
169C22×C53#47312yes—
170C4×C53—21no—
171C4×C53—21no—
172C4×C53—21no—
173C4×C53—81no—
174C4×C53—81no—
175C4×C53—81no—
176C4×C53—81no—
177C4×C53#45812yes—
178Dic53—82no—
179#3195D5311yescharacteristic
180C2×#484—531no—
181C2×#483—531no—
182#1821—531no—
183#1472—531no—
184#1472—531no—
185#1298—531no—
186C2×C53—22no—
187C2×C53—22no—
188C2×C53—41no—
189C2×C53#201911yescharacteristic
190C2×C53#202011yescharacteristic
191C2×C53#202211yescharacteristic
192C2×C22⋊Q8C2×D5311yescharacteristic
193C2×C22⋊Q8C2×D5311yescharacteristic
194C22×C4⋊C4—531no—
195#484—1062no—
196#484—1062no—
197#483—21no—
198#483—21no—
199#483—1062no—
200#483—1062no—
201#470—22no—
202#451—21no—
203#451—21no—
204C2×#107—1061no—
205C2×#106—1061no—
206C2×#103—1061no—
207C2×#103—1061no—
208#407—531no—
209#407C2×D5311yescharacteristic
210C2×C22⋊C8—531no—
211C2×C22⋊C8—531no—
212#343—1062no—
213#324—1062no—
214#324—1062no—
215C53#999511yescharacteristic, complemented, Sylow
216C22×Q8C22⋊D5311yescharacteristic
217C22×Q8C22⋊D5311yescharacteristic
218C23×C4—1061no—
219C23×C4C22×D5311yescharacteristic
220C2×Q16—22no—
221C2×M16—1061no—
222C2×M16C22⋊D5312yes—
223C22×C8—1061no—
224C22×C8—1061no—
225C22⋊Q8—21no—
226C22⋊Q8—21no—
227C22⋊Q8—21no—
228C22⋊Q8—21no—
229C22⋊Q8—21no—
230C22⋊Q8—21no—
231C22⋊Q8—21no—
232C22⋊Q8—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—532no—
236C2×C4⋊C4—1061no—
237C2×C4⋊C4—1061no—
238C2×C4⋊C4—1061no—
239C2×C4⋊C4—1061no—
240C2×C4⋊C4—1061no—
241C2×C4⋊C4C22⋊D5311yescharacteristic
242C2×C4⋊C4C22⋊D5311yescharacteristic
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245#107—1064no—
246#106—22no—
247#106—1064no—
248#103—22no—
249#103—22no—
250#103—1064no—
251#103—1064no—
252C22⋊C8—21no—
253C22⋊C8—21no—
254C22⋊C8—532no—
255C22⋊C8—532no—
256C22⋊C8—1061no—
257C22⋊C8—1061no—
258C22⋊C8—1061no—
259C22⋊C8—1061no—
260C24#1171994711yescharacteristic
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—42no—
266C2×Q8—42no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—42no—
272C22×C4—1062no—
273C22×C4—4242no—
274C22×C4—2121no—
275C22×C4—2121no—
276C22×C4D4×D5311yescharacteristic
277C22×C4D4×D5311yescharacteristic
278C22×C4D4×D5311yescharacteristic
279C22×C4C2×C22⋊D5311yescharacteristic
280C22×C4C2×C22⋊D5311yescharacteristic
281C22×C4C2×C22⋊D5311yescharacteristic
282Q16—82no—
283M16—42no—
284M16—2122no—
285C2×C8—22no—
286C2×C8—1062no—
287C2×C8—1062no—
288C2×C8—2121no—
289C2×C8—2122no—
290C2×C8—2122no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—1062no—
300C4⋊C4—1062no—
301C4⋊C4—2121no—
302C4⋊C4—2121no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C23—22no—
308C23—22no—
309C23—41no—
310C23#1313318811yescharacteristic
311C23#1313318911yescharacteristic
312C23#1313318911yescharacteristic
313Q8—42no—
314Q8—42no—
315Q8—42no—
316Q8—42no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—4242no—
332C2×C4—4242no—
333C2×C4—4242no—
334C2×C4—2122no—
335C2×C4#1313318511yescharacteristic
336C2×C4#1313318511yescharacteristic
337C2×C4#1313318611yescharacteristic
338C2×C4#1313321311yescharacteristic
339C8—42no—
340C8—4241no—
341C8—2122no—
342C8—2122no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—44no—
351C22#1681053111yescharacteristic
352C22#1681088812yes—
353C22#1681089612yes—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—81no—
358C4—81no—
359C4—81no—
360C4—81no—
361C4—4242no—
362C4#1681081312yes—
363C2—22no—
364C2—22no—
365C2—41no—
366C2#2027582611yescharacteristic
367C2#2027583311yescharacteristic
368C2#2027584411yescharacteristic
subgroupG/H
centerC22#16810531
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
FittingC53×#3195C2
socleC22×C53#392
prime radicalC24×C53#41
hypercenter#3195D53
hypocenterC53#9995
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3195

Complements

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

NH ≅ G/Ntype
215. C53#9995semidirect

Maximal subgroups

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

subgroupindex
C2×#168109562
C2×#168109512
#202747932
#202730792
#202726022
C53×#31952
#999553