Finite Group Databasebeta

← #27286665

Subgroups — #27286665

Order 13568.

594 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810910C211yesmaximal, characteristic
2C2×#16810825C211yesmaximal, characteristic
3#20275509C211yesmaximal, characteristic
4C2×#16810372C211yesmaximal, characteristic, complemented
5C2×#16810369C211yesmaximal, characteristic, complemented
6#20272550C211yesmaximal, characteristic, complemented
7C2×C53×#498C211yesmaximal, characteristic, complemented
8C22×#11719924—21no—
9C2×C4×D212C2211yescharacteristic
10#16810910—22no—
11#16810910—22no—
12#16810825—22no—
13#16810825—22no—
14C2×#13133148—21no—
15C2×C4⋊#88322C2211yescharacteristic
16C2×C4⋊#88322C2211yescharacteristic
17C2×C4⋊#88322C2211yescharacteristic
18C2×#13133141—21no—
19C2×#13133137—21no—
20C2×#13133129—22no—
21#16810372—22no—
22#16810369—22no—
23C2×C4⋊Q8×C53C2211yescharacteristic
24C2×C4⋊Q8×C53C2211yescharacteristic
25C53×#498—22no—
26C53×#498—22no—
27C2×#106×C53—22no—
28C2×C4×C8×C53C2211yescharacteristic
29C22×D212D411yescharacteristic
30C22×C4×D53—21no—
31C2×#11719924—24no—
32C2×#11719924—42no—
33C2×#11719924—42no—
34C22×#88322—21no—
35C22×#88322—21no—
36C22×#88322—21no—
37C4×D212—21no—
38C4×D212—21no—
39C4×D212—21no—
40C4×D212—21no—
41C2×C4⋊Dic53D411yescharacteristic
42C2×#11719921—21no—
43#13133148—24no—
44C4⋊#88322—21no—
45C4⋊#88322—21no—
46C4⋊#88322—21no—
47C4⋊#88322—21no—
48C4⋊#88322D412yes—
49C4⋊#88322D412yes—
50#13133141—24no—
51#13133137—24no—
52#13133129—44no—
53C22×Q8×C53—21no—
54C22×Q8×C53—21no—
55C22×C8×C53D412yes—
56C4⋊Q8×C53—21no—
57C4⋊Q8×C53—21no—
58C4⋊Q8×C53—21no—
59C4⋊Q8×C53—21no—
60C4⋊Q8×C53—21no—
61C4⋊Q8×C53—21no—
62C4⋊Q8×C53D412yes—
63C4⋊Q8×C53D412yes—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53—21no—
67C2×C4⋊C4×C53—21no—
68C2×C42×C53C2311yescharacteristic
69#106×C53—28no—
70C4×C8×C53—21no—
71C4×C8×C53—21no—
72C23×D53—21no—
73C2×Q8×C53—22no—
74C2×Q8×C53—22no—
75C2×Q8×C53—41no—
76C2×Q8×C53—41no—
77C2×Q8×C53—41no—
78C2×Q8×C53—41no—
79C2×Q8×C53—41no—
80C2×Q8×C53—41no—
81C2×Q8×C53—82no—
82C22×C4×C53—41no—
83C22×C4×C53—41no—
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53#4111yescharacteristic
87C22×Dic53—21no—
88C2×D212—21no—
89C2×D212—21no—
90C2×D212—21no—
91C2×D212—21no—
92C2×D212—21no—
93C2×D212SD1612yes—
94C2×C4×D53—41no—
95C2×C4×D53—41no—
96C2×C4×D53—42no—
97C2×C4×D53—42no—
98C2×C8×C53—22no—
99C2×C8×C53—22no—
100C2×C8×C53SD1614yescomplemented
101C4⋊C4×C53—22no—
102C4⋊C4×C53—22no—
103C4⋊C4×C53—22no—
104C4⋊C4×C53—22no—
105C4⋊C4×C53—41no—
106C4⋊C4×C53—41no—
107C4⋊C4×C53—41no—
108C4⋊C4×C53—41no—
109C42×C53C2×D411yescharacteristic
110C42×C53C2×D411yescharacteristic
111C42×C53C2×D411yescharacteristic
112C42×C53C2×D411yescharacteristic
113#11719924—48no—
114#11719921—41no—
115#11719921—41no—
116C4⋊Dic53—21no—
117C4⋊Dic53—21no—
118C2×#88322—22no—
119C2×#88322—22no—
120C2×#88322—41no—
121C2×#88322—41no—
122C2×#88322—41no—
123C2×#88322—41no—
124C2×#88322—41no—
125C2×#88322—42no—
126C23×C53#12111yescharacteristic
127C22×D53—22no—
128C22×D53—41no—
129C22×D53—42no—
130C22×D53—41no—
131C22×D53—42no—
132Q8×C53—42no—
133Q8×C53—42no—
134Q8×C53—82no—
135Q8×C53—82no—
136Q8×C53—81no—
137Q8×C53—81no—
138C2×C4×C53—22no—
139C2×C4×C53—42no—
140C2×C4×C53—42no—
141C2×C4×C53—81no—
142C2×C4×C53—81no—
143C2×C4×C53—82no—
144C2×C4×C53#12011yescharacteristic
145C2×C4×C53#12011yescharacteristic
146C2×C4×C53#12011yescharacteristic
147C2×C4×C53#12111yescharacteristic
148C2×C4×C53#12111yescharacteristic
149C2×C4×C53#12711yescharacteristic
150C2×C4×C53C2×SD1611yescharacteristic
151C2×C4×C53C2×SD1612yes—
152C2×C4×C53#13511yescharacteristic
153C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C53#13711yescharacteristic
155C2×C4×C53#13712yes—
156C2×Dic53—41no—
157C2×Dic53—42no—
158D212—22no—
159D212—22no—
160D212—41no—
161D212—41no—
162C4×D53—82no—
163C4×D53—81no—
164C4×D53—81no—
165C8×C53—24no—
166#88322—42no—
167#88322—42no—
168#88322—81no—
169#10271—531nomaximal, Sylow
170C22×C53#39211yescharacteristic
171C22×C53#46112yes—
172C22×C53#46511yescharacteristic
173C22×C53#46811yescharacteristic
174C22×C53#47112yes—
175C2×D53—44no—
176C2×D53—42no—
177C2×D53—82no—
178C2×D53—81no—
179C2×D53—81no—
180C4×C53—21no—
181C4×C53—21no—
182C4×C53—82no—
183C4×C53—82no—
184C4×C53#44811yescharacteristic
185C4×C53#44812yes—
186C4×C53#44911yescharacteristic
187C4×C53#45011yescharacteristic
188C4×C53#45211yescharacteristic
189C4×C53#49211yescharacteristic
190C4×C53#49711yescharacteristic
191Dic53—81no—
192C2×#498D5311yescharacteristic, complemented
193C2×#476—531no—
194C2×#461—531no—
195#1954—531no—
196C2×#333—531no—
197C2×#332—531no—
198#1277—531no—
199C2×C53#167112yes—
200C2×C53#169112yes—
201C2×C53#200011yescharacteristic
202C2×C53#201311yescharacteristic
203C2×C53#201911yescharacteristic
204D53—82no—
205C22×SD16—1061no—
206C2×C4⋊Q8C2×D5311yescharacteristic
207C2×C4⋊Q8C2×D5311yescharacteristic
208C2×C4×D4—531no—
209#498—22no—
210#498—22no—
211#476—1062no—
212#476—1062no—
213#461—1062no—
214#461—1062no—
215C2×#106—22no—
216C2×#106—1061no—
217C2×C4⋊C8—531no—
218C2×C4⋊C8—531no—
219C2×C4⋊C8—531no—
220C2×#103—1061no—
221C2×#102—1061no—
222C2×C22⋊C8—1062no—
223C2×C4×C8C2×D5311yescharacteristic
224#333—1062no—
225#332—1062no—
226C53#1027111yescharacteristic, complemented, Sylow
227C22×Q8—21no—
228C22×Q8—21no—
229C22×D4—531no—
230C23×C4—1061no—
231C2×SD16—1064no—
232C2×SD16—2122no—
233C2×SD16—2122no—
234C22×C8—1061no—
235C22×C8—1061no—
236C22×C8—1061no—
237C22×C8C22⋊D5312yes—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8C22⋊D5312yes—
245C4⋊Q8C22⋊D5312yes—
246C4×D4—1061no—
247C4×D4—1061no—
248C4×D4—1061no—
249C4×D4—1061no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—531no—
255C2×C22⋊C4—1061no—
256C2×C42C22×D5311yescharacteristic
257#106—28no—
258#106—1064no—
259C4⋊C8—532no—
260C4⋊C8—532no—
261C4⋊C8—1061no—
262C4⋊C8—1061no—
263C4⋊C8—1061no—
264C4⋊C8—1061no—
265#103—1064no—
266#102—1064no—
267C22⋊C8—2124no—
268C4×C8—21no—
269C4×C8—21no—
270C24—1061no—
271C2×Q8—22no—
272C2×Q8—22no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—82no—
280C2×D4—532no—
281C2×D4—1061no—
282C2×D4—1061no—
283C2×D4—1061no—
284C2×D4—1061no—
285C2×D4—1061no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—1061no—
289C22×C4—2121no—
290C22×C4—2121no—
291C22×C4—2122no—
292C22×C4—2122no—
293C22×C4D4×D5311yescharacteristic
294C22×C4#1171994711yescharacteristic
295C22×C4C2×C22⋊D5311yescharacteristic
296SD16—2128no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—1062no—
300C2×C8—1062no—
301C2×C8—2121no—
302C2×C8—2122no—
303C2×C8—2121no—
304C2×C8—2121no—
305C2×C8—2121no—
306C2×C8—2121no—
307C2×C8#1171992414yescomplemented
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—1061no—
317C4⋊C4—1061no—
318C22⋊C4—2121no—
319C22⋊C4—2121no—
320C42D4×D5311yescharacteristic
321C42D4×D5311yescharacteristic
322C42C2×C22⋊D5311yescharacteristic
323C42C2×C22⋊D5311yescharacteristic
324C23—1062no—
325C23—2121no—
326C23—2122no—
327C23—2121no—
328C23—2122no—
329C23#1313318811yescharacteristic
330Q8—42no—
331Q8—42no—
332Q8—82no—
333Q8—82no—
334Q8—81no—
335Q8—81no—
336D4—1062no—
337D4—1062no—
338D4—2121no—
339D4—2121no—
340C2×C4—22no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—82no—
346C2×C4—4241no—
347C2×C4—4242no—
348C2×C4—4241no—
349C2×C4—2121no—
350C2×C4—2122no—
351C2×C4#1313318511yescharacteristic
352C2×C4#1313318511yescharacteristic
353C2×C4#1313318611yescharacteristic
354C2×C4#1313318911yescharacteristic
355C2×C4#1313318911yescharacteristic
356C2×C4#1313321311yescharacteristic
357C2×C4SD16×D5311yescharacteristic
358C2×C4C2×#1171992412yes—
359C2×C4#1313323811yescharacteristic
360C2×C4#1313324411yescharacteristic
361C2×C4#1313324611yescharacteristic
362C2×C4#1313324812yes—
363C8—24no—
364C8—4241no—
365C8—2122no—
366C8—2122no—
367C22—4242no—
368C22—4241no—
369C22—4241no—
370C22—2122no—
371C22—2124no—
372C22#1681053111yescharacteristic
373C22#1681082512yes—
374C22#1681084911yescharacteristic
375C22#1681086611yescharacteristic
376C22#1681088312yes—
377C4—21no—
378C4—21no—
379C4—82no—
380C4—82no—
381C4—4241no—
382C4#1681076011yescharacteristic
383C4#1681076212yes—
384C4#1681076711yescharacteristic
385C4#1681077211yescharacteristic
386C4#1681078211yescharacteristic
387C4#1681099711yescharacteristic
388C4#1681101511yescharacteristic
389C2—4242no—
390C2#2027408712yes—
391C2#2027421112yes—
392C2#2027571511yescharacteristic
393C2#2027579811yescharacteristic
394C2#2027582711yescharacteristic
subgroupG/H
centerC23#13133188
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC2×C53×#498C2
socleC23×C53#121
prime radicalC22×D212D4
hypercenterC2×#498D53
hypocenterC53#10271
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#498

Complements

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

NH ≅ G/Ntype
4. C2×#16810372C2semidirect
5. C2×#16810369C2semidirect
6. #20272550C2semidirect
7. C2×C53×#498C2semidirect
100. C2×C8×C53SD16semidirect
192. C2×#498D53semidirect
226. C53#10271semidirect
307. C2×C8#11719924semidirect

Maximal subgroups

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

subgroupindex
C2×#168109102
C2×#168108252
#202755092
C2×#168103722
C2×#168103692
#202725502
C2×C53×#4982
#1027153