Finite Group Databasebeta

← #27412211

Subgroups — #27412211

Order 13568.

1074 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811254C211yesmaximal, characteristic
2#20283304C212yesmaximal
3#20276065C214yesmaximal, complemented
4#20276060C214yesmaximal, complemented
5C2×#16810392C211yesmaximal, characteristic
6#20274425C212yesmaximal
7C53×#2317C211yesmaximal, characteristic
8C22×#11719917C2212yes—
9#16811254—24no—
10#16811254—22no—
11#16811254C2214yes—
12C2×#13133216—22no—
13C2×#13133208—22no—
14C2×#13133196—22no—
15C2×#13133176C2212yes—
16#16810666—22no—
17#16810666—22no—
18#16810663C2214yes—
19#16810663C2214yes—
20#16810659—24no—
21#16810392C2214yes—
22#16810383C2218yes—
23C2×#117×C53C2211yescharacteristic
24C2×#117×C53C2211yescharacteristic
25C22×C4⋊C4×C53C2211yescharacteristic
26C53×#389—22no—
27C53×#385—22no—
28C53×#384C2212yes—
29C53×#382C2212yes—
30C22×Dic106—22no—
31C2×#11719917—22no—
32C2×#11719917—24no—
33C2×#11719917Q818yescomplemented
34C2×#11719917C2314yes—
35C22×#88322—22no—
36#13133216—28no—
37#13133216—28no—
38#13133208—28no—
39#13133208—28no—
40#13133196—28no—
41#13133196—28no—
42#13133176—22no—
43#13133176—22no—
44#13133176D414yes—
45#13133176C2314yes—
46C2×C4⋊Dic53—22no—
47C2×C4⋊Dic53D412yes—
48C2×C4⋊Dic53D412yes—
49C2×#11719919—24no—
50C2×C22⋊Dic53—22no—
51C2×C4×Dic53—22no—
52#13133150—24no—
53#13133150—24no—
54#13133148—24no—
55#13133148—24no—
56C23×C4×C53D412yes—
57C23×C4×C53C2311yescharacteristic
58#117×C53—22no—
59#117×C53—22no—
60#117×C53—22no—
61#117×C53Q814yescomplemented
62C2×C4⋊C4×C53—22no—
63C2×C4⋊C4×C53—22no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53—24no—
66C2×C4⋊C4×C53D412yes—
67C2×C4⋊C4×C53C2311yescharacteristic
68C2×C4⋊C4×C53C2311yescharacteristic
69C2×C4⋊C4×C53C2312yes—
70C2×C22⋊C4×C53—21no—
71C2×C22⋊C4×C53—21no—
72C2×C42×C53—21no—
73C2×C42×C53C2312yes—
74#95×C53—24no—
75C24×C53C2×D411yescharacteristic
76C22×C4×C53—22no—
77C22×C4×C53—22no—
78C22×C4×C53—22no—
79C22×C4×C53—21no—
80C22×C4×C53—21no—
81C22×C4×C53—82no—
82C22×C4×C53C2×D414yes—
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2×D412yes—
85C22×C4×C53C2×D412yes—
86C22×C4×C53C2×Q814yes—
87C22×C4×C53C2×Q812yes—
88C22×C4×C53#4112yes—
89C22×C4×C53C2411yescharacteristic
90C22×Dic53—22no—
91C22×Dic53—22no—
92C2×Dic106—28no—
93C2×Dic106—84no—
94C4⋊C4×C53—24no—
95C4⋊C4×C53—24no—
96C4⋊C4×C53—24no—
97C4⋊C4×C53—22no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—84no—
100C22⋊C4×C53—24no—
101C22⋊C4×C53—41no—
102C22⋊C4×C53—41no—
103C42×C53—22no—
104C42×C53—42no—
105C42×C53C2×Q814yescomplemented
106C22⋊Dic53—24no—
107C22⋊Dic53—24no—
108C4⋊Dic53—28no—
109C4⋊Dic53—22no—
110C4⋊Dic53—22no—
111C4⋊Dic53C2×D414yes—
112C4⋊Dic53C2×D414yes—
113#11719919—216no—
114C4×Dic53—24no—
115C4×Dic53—24no—
116#11719917—28no—
117#11719917—24no—
118#11719917C2×Q818yescomplemented
119C2×#88322—28no—
120C2×#88322—24no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53#12011yescharacteristic
124C23×C53#12011yescharacteristic
125C23×C53C22⋊Q812yes—
126C23×C53C4⋊Q812yes—
127C23×C53C22×D411yescharacteristic
128C2×C4×C53—24no—
129C2×C4×C53—24no—
130C2×C4×C53—44no—
131C2×C4×C53—42no—
132C2×C4×C53—42no—
133C2×C4×C53—44no—
134C2×C4×C53—42no—
135C2×C4×C53—42no—
136C2×C4×C53—41no—
137C2×C4×C53—41no—
138C2×C4×C53—82no—
139C2×C4×C53#12012yes—
140C2×C4×C53C22⋊Q814yes—
141C2×C4×C53C22⋊Q814yes—
142C2×C4×C53C22⋊Q812yes—
143C2×C4×C53C4⋊Q812yes—
144C2×C4×C53C4⋊Q814yes—
145C2×C4×C53C4⋊Q814yes—
146C2×C4×C53C22×D412yes—
147C2×C4×C53C22×Q812yes—
148C2×C4×C53C22×Q811yescharacteristic
149C2×C4×C53C2×#4111yescharacteristic
150C2×Dic53—24no—
151C2×Dic53—24no—
152C2×Dic53—44no—
153C2×Dic53—44no—
154Dic106—88no—
155#88322—28no—
156#23561—531nomaximal, Sylow
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—42no—
164C22×C53—42no—
165C22×C53#39512yes—
166C22×C53#39512yes—
167C22×C53C2×#12011yescharacteristic
168C22×C53C2×C22⋊Q811yescharacteristic
169C22×C53C2×C4⋊Q811yescharacteristic
170C4×C53—44no—
171C4×C53—44no—
172C4×C53—81no—
173C4×C53—81no—
174C4×C53#39514yes—
175C4×C53C2×C22⋊Q812yes—
176C4×C53C2×C4⋊Q812yes—
177Dic53—44no—
178Dic53—44no—
179C2×#544—531no—
180#2911—532no—
181#2317D5311yescharacteristic
182#2061—534no—
183#2060—534no—
184C2×#343—531no—
185#1733—532no—
186C2×C53—21no—
187C2×C53—21no—
188C2×C53—22no—
189C2×C53—22no—
190C2×C53C2×#39511yescharacteristic
191C2×C53#301911yescharacteristic
192C2×C53#302011yescharacteristic
193C22×M16—532no—
194#544—534no—
195#544—1064no—
196#544—1062no—
197C2×C4⋊Q8—1062no—
198C2×#125—1062no—
199C2×C22⋊Q8—1062no—
200C2×#117—532no—
201C2×#117C2×D5311yescharacteristic
202C2×#117C2×D5311yescharacteristic
203C22×C4⋊C4C2×D5311yescharacteristic
204#428—534no—
205#428—534no—
206#428—1062no—
207#428—1062no—
208#427—1064no—
209#389—22no—
210#385—22no—
211#384C2×D5312yes—
212#382C2×D5312yes—
213#343—534no—
214#339—538no—
215C53#2356111yescharacteristic, complemented, Sylow
216C22×Q8—1062no—
217C23×C4C22⋊D5312yes—
218C23×C4C22×D5311yescharacteristic
219C2×M16—538no—
220C2×M16—534no—
221C2×M16—1064no—
222C2×M16—1062no—
223C22×C8—1062no—
224C4⋊Q8—1068no—
225C4⋊Q8—1068no—
226#125—1068no—
227#125—1068no—
228C22⋊Q8—1068no—
229C22⋊Q8—1068no—
230#117—22no—
231#117—22no—
232#117—22no—
233#117—534no—
234#117—534no—
235#117—1062no—
236#117—1062no—
237#117Dic10614yescomplemented
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—532no—
243C2×C4⋊C4—532no—
244C2×C4⋊C4—1062no—
245C2×C4⋊C4—1064no—
246C2×C4⋊C4C22⋊D5312yes—
247C2×C4⋊C4C22×D5311yescharacteristic
248C2×C4⋊C4C22×D5311yescharacteristic
249C2×C4⋊C4C22×D5312yes—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—1062no—
253C2×C42—21no—
254C2×C42—1062no—
255C2×C42C22×D5312yes—
256#107—1064no—
257#107—1064no—
258#106—1064no—
259#106—1064no—
260#95—24no—
261C24C2×C22⋊D5311yescharacteristic
262C2×Q8—1068no—
263C2×Q8—4244no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—82no—
270C22×C4—1062no—
271C22×C4—1062no—
272C22×C4C2×Dic10612yes—
273C22×C4#1171994512yes—
274C22×C4D4×D5312yes—
275C22×C4D4×D5312yes—
276C22×C4Q8×D5314yes—
277C22×C4C2×C22⋊D5314yes—
278C22×C4C2×C22⋊D5311yescharacteristic
279C22×C4C23×D5311yescharacteristic
280M16—538no—
281M16—1068no—
282M16—1064no—
283C2×C8—1068no—
284C2×C8—1064no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—84no—
291C4⋊C4—534no—
292C4⋊C4—534no—
293C4⋊C4—1068no—
294C4⋊C4—10616no—
295C4⋊C4—1062no—
296C4⋊C4—1062no—
297C22⋊C4—24no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—1064no—
301C22⋊C4—1064no—
302C42—22no—
303C42—42no—
304C42—1064no—
305C42—1064no—
306C42C2×Dic10614yescomplemented
307C23—41no—
308C23—41no—
309C23#1313318511yescharacteristic
310C23#1313318511yescharacteristic
311C23#1313319612yes—
312C23#1313321512yes—
313C23C22×C22⋊D5311yescharacteristic
314Q8—4248no—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—82no—
326C2×C4—1064no—
327C2×C4—1064no—
328C2×C4—2124no—
329C2×C4—2124no—
330C2×C4#1313318512yes—
331C2×C4#1313319514yes—
332C2×C4#1313319514yes—
333C2×C4#1313319612yes—
334C2×C4#1313321512yes—
335C2×C4#1313321614yes—
336C2×C4#1313321614yes—
337C2×C4C2×D4×D5312yes—
338C2×C4C2×Q8×D5312yes—
339C2×C4C22×Dic10611yescharacteristic
340C2×C4C2×#1171994511yescharacteristic
341C8—1068no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—42no—
349C22—42no—
350C22#1681054312yes—
351C22#1681054312yes—
352C22C2×#1313318511yescharacteristic
353C22C2×#1313319611yescharacteristic
354C22C2×#1313321511yescharacteristic
355C4—44no—
356C4—44no—
357C4—81no—
358C4—81no—
359C4—2124no—
360C4—2124no—
361C4#1681054314yes—
362C4C2×#1313319512yes—
363C4C2×#1313321612yes—
364C2—21no—
365C2—22no—
366C2—22no—
367C2—21no—
368C2C2×#1681054311yescharacteristic
369C2#2028407111yescharacteristic
370C2#2028407811yescharacteristic
subgroupG/H
centerC22C2×#13133185
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#2317C2
socleC22×C53C2×#120
prime radicalC24×C53C2×D4
hypercenter#2317D53
hypocenterC53#23561
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2317

Complements

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

NH ≅ G/Ntype
3. #20276065C2semidirect
4. #20276060C2semidirect
33. C2×#11719917Q8semidirect
61. #117×C53Q8semidirect
105. C42×C53C2×Q8semidirect
118. #11719917C2×Q8semidirect
215. C53#23561semidirect
237. #117Dic106semidirect
306. C42C2×Dic106semidirect

Maximal subgroups

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

subgroupindex
C2×#168112542
#202833042
#202760652
#202760602
C2×#168103922
#202744252
C53×#23172
#2356153