Finite Group Databasebeta

← #27300886

Subgroups — #27300886

Order 13568.

672 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 99 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×#16810416C211yesmaximal, characteristic
2#20275114C211yesmaximal, characteristic
3#20274995C212yesmaximal, complemented
4#20272833C212yesmaximal, complemented
5C53×#2366C211yesmaximal, characteristic
6#16811098C2211yescharacteristic
7#16810628—21no—
8C2×#13133134C2211yescharacteristic
9C2×#13133132C2212yes—
10#16810496—22no—
11#16810416C414yescomplemented
12#16810389—22no—
13#16810383—21no—
14C2×#127×C53C2211yescharacteristic
15C2×C4×D4×C53—22no—
16C53×#390—22no—
17C53×#386—21no—
18C53×#386C2212yes—
19C4×C4⋊C4×C53—21no—
20C2×#11719917—21no—
21D4×Dic53—41no—
22D4×Dic53—41no—
23#13133176—21no—
24C2×#11719919—42no—
25C2×C22⋊Dic53—22no—
26C2×C22⋊Dic53—22no—
27#13133138—41no—
28#13133138—41no—
29#13133134—22no—
30#13133132—22no—
31#13133132C2×C414yescomplemented
32#13133124—42no—
33C22×D4×C53—22no—
34C22×D4×C53D411yescharacteristic
35C22×D4×C53C2311yescharacteristic
36C23×C4×C53—21no—
37C23×C4×C53D412yes—
38#127×C53—22no—
39#127×C53C2×C412yes—
40#127×C53Q812yes—
41C4×D4×C53—48no—
42C2×C4⋊C4×C53—21no—
43C2×C4⋊C4×C53—21no—
44C2×C22⋊C4×C53—22no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—22no—
47C2×C22⋊C4×C53D412yes—
48C2×C42×C53—22no—
49C2×C42×C53D411yescharacteristic
50#95×C53—22no—
51C24×C53—21no—
52C24×C53C2×D412yes—
53C2×D4×C53—24no—
54C2×D4×C53—24no—
55C2×D4×C53—21no—
56C2×D4×C53—41no—
57C2×D4×C53—41no—
58C2×D4×C53—44no—
59C2×D4×C53—44no—
60C2×D4×C53—41no—
61C2×D4×C53—41no—
62C2×D4×C53C4⋊C412yescomplemented
63C2×D4×C53C22×C411yescharacteristic
64C2×D4×C53C2×Q811yescharacteristic
65C2×D4×C53#4112yes—
66C22×C4×C53—21no—
67C22×C4×C53—22no—
68C22×C4×C53—22no—
69C22×C4×C53—44no—
70C22×C4×C53—44no—
71C22×C4×C53—42no—
72C22×C4×C53—81no—
73C22×C4×C53C2×D411yescharacteristic
74C22×Dic53—22no—
75C22×Dic53—22no—
76C22×Dic53—42no—
77C4⋊C4×C53—24no—
78C4⋊C4×C53—42no—
79C22⋊C4×C53—44no—
80C22⋊C4×C53—44no—
81C22⋊C4×C53—42no—
82C22⋊C4×C53—81no—
83C42×C53—21no—
84C42×C53—44no—
85C42×C53C4⋊C412yescomplemented
86C22⋊Dic53—24no—
87C22⋊Dic53—42no—
88C22⋊Dic53—42no—
89C22⋊Dic53—41no—
90C4⋊Dic53—21no—
91C4⋊Dic53—21no—
92#11719919—84no—
93C4×Dic53—41no—
94#11719917—41no—
95#11719917—41no—
96C2×#88322—41no—
97C23×C53—22no—
98C23×C53—22no—
99C23×C53—22no—
100C23×C53—42no—
101C23×C53—42no—
102C23×C53—42no—
103C23×C53—42no—
104C23×C53—41no—
105C23×C53—41no—
106C23×C53C4×D412yes—
107C23×C53#12011yescharacteristic
108C23×C53C22⋊Q812yes—
109D4×C53—41no—
110D4×C53—41no—
111D4×C53—42no—
112D4×C53—42no—
113D4×C53—48no—
114D4×C53—48no—
115D4×C53—81no—
116C2×C4×C53—22no—
117C2×C4×C53—22no—
118C2×C4×C53—22no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—24no—
122C2×C4×C53—42no—
123C2×C4×C53—42no—
124C2×C4×C53—44no—
125C2×C4×C53—42no—
126C2×C4×C53—42no—
127C2×C4×C53—42no—
128C2×C4×C53—82no—
129C2×C4×C53—84no—
130C2×C4×C53C2×C4⋊C411yescharacteristic
131C2×C4×C53#12311yescharacteristic
132C2×Dic53—44no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—82no—
136C2×Dic53—82no—
137C2×Dic53—82no—
138C2×Dic53—82no—
139#88322—81no—
140#12964—531nomaximal, Sylow
141C22×C53—21no—
142C22×C53—21no—
143C22×C53—22no—
144C22×C53—22no—
145C22×C53—42no—
146C22×C53—42no—
147C22×C53—44no—
148C22×C53—42no—
149C22×C53—42no—
150C22×C53—42no—
151C22×C53—82no—
152C22×C53—82no—
153C22×C53—81no—
154C22×C53—81no—
155C22×C53#38511yescharacteristic
156C22×C53B(4,𝔽2)11yescharacteristic
157C22×C53#45811yescharacteristic
158C4×C53—24no—
159C4×C53—21no—
160C4×C53—21no—
161C4×C53—44no—
162C4×C53—82no—
163Dic53—84no—
164Dic53—81no—
165Dic53—81no—
166#2366D5311yescharacteristic
167C2×#353—531no—
168#1887—531no—
169#1867—532no—
170#1390—532no—
171C2×C53—21no—
172C2×C53—21no—
173C2×C53—42no—
174C2×C53—42no—
175C2×C53—82no—
176C2×C53#186711yescharacteristic
177C2×C53#219611yescharacteristic
178C2×C53#219811yescharacteristic
179C2×#127C2×D5311yescharacteristic
180#518—531no—
181C2×C4×D4—22no—
182#418—1061no—
183C2×#100—531no—
184C2×#99—532no—
185#390—22no—
186#386—21no—
187#386C2×D5312yes—
188#385—1062no—
189C4×C4⋊C4—21no—
190#353—534no—
191#342—1062no—
192#339—1061no—
193C53#1296411yescharacteristic, complemented, Sylow
194C22×D4—22no—
195C22×D4C22⋊D5311yescharacteristic
196C22×D4C22×D5311yescharacteristic
197C23×C4—21no—
198C23×C4C22⋊D5312yes—
199C2×M16—1061no—
200#127—22no—
201#127Dic10612yes—
202#127C4×D5312yes—
203C4×D4—48no—
204C4×D4—2121no—
205C4×D4—2121no—
206#117—1061no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—2122no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—1062no—
214C2×C22⋊C4—1062no—
215C2×C22⋊C4C22⋊D5312yes—
216C2×C42—22no—
217C2×C42C22⋊D5311yescharacteristic
218#102—2121no—
219#102—2121no—
220#100—1062no—
221#99—534no—
222#99—1062no—
223#95—22no—
224#95—2122no—
225C24—21no—
226C24C2×C22⋊D5312yes—
227C2×D4—24no—
228C2×D4—24no—
229C2×D4—21no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—44no—
233C2×D4—44no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4#1171991912yescomplemented
237C2×D4C2×Dic10611yescharacteristic
238C2×D4C2×C4×D5311yescharacteristic
239C2×D4#1171994512yes—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—44no—
244C22×C4—44no—
245C22×C4—42no—
246C22×C4—81no—
247C22×C4—1062no—
248C22×C4—1062no—
249C22×C4—2122no—
250C22×C4C2×C22⋊D5311yescharacteristic
251M16—2121no—
252M16—2121no—
253C2×C8—2121no—
254C4⋊C4—24no—
255C4⋊C4—42no—
256C4⋊C4—1061no—
257C4⋊C4—1061no—
258C4⋊C4—4244no—
259C22⋊C4—44no—
260C22⋊C4—44no—
261C22⋊C4—42no—
262C22⋊C4—81no—
263C22⋊C4—1064no—
264C22⋊C4—2122no—
265C22⋊C4—2122no—
266C22⋊C4—2121no—
267C42—21no—
268C42—44no—
269C42—2121no—
270C42#1171991912yescomplemented
271C23—22no—
272C23—22no—
273C23—22no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23—41no—
279C23—41no—
280C23C4×C22⋊D5312yes—
281C23#1313318611yescharacteristic
282C23#1313319612yes—
283D4—41no—
284D4—41no—
285D4—42no—
286D4—42no—
287D4—48no—
288D4—48no—
289D4—81no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—24no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—44no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—82no—
303C2×C4—84no—
304C2×C4—4242no—
305C2×C4—4242no—
306C2×C4—4242no—
307C2×C4—4242no—
308C2×C4—2124no—
309C2×C4—2121no—
310C2×C4—2121no—
311C2×C4C2×#1171991911yescharacteristic
312C2×C4#1313320011yescharacteristic
313C8—4241no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—22no—
318C22—42no—
319C22—42no—
320C22—44no—
321C22—42no—
322C22—42no—
323C22—42no—
324C22—82no—
325C22—82no—
326C22—81no—
327C22—81no—
328C22#1681049611yescharacteristic
329C22#1681081011yescharacteristic
330C22#1681081311yescharacteristic
331C4—24no—
332C4—21no—
333C4—21no—
334C4—44no—
335C4—82no—
336C4—4244no—
337C4—4241no—
338C4—4241no—
339C2—21no—
340C2—21no—
341C2—42no—
342C2—42no—
343C2—82no—
344C2#2027499611yescharacteristic
345C2#2027669911yescharacteristic
346C2#2027671111yescharacteristic
subgroupG/H
centerC22#16810810
commutatorC2×D4×C53C22×C4
FrattiniC22×D4C22×D53
FittingC53×#2366C2
socleC22×C53B(4,𝔽2)
prime radicalC2×#127×C53C22
cosocleC22×D4×C53C23
hypercenter#2366D53
hypocenterC53#12964
LCSG ▷ C2×D4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #2366

Complements

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

NH ≅ G/Ntype
3. #20274995C2semidirect
4. #20272833C2semidirect
11. #16810416C4semidirect
31. #13133132C2×C4semidirect
62. C2×D4×C53C4⋊C4semidirect
85. C42×C53C4⋊C4semidirect
193. C53#12964semidirect
236. C2×D4#11719919semidirect
270. C42#11719919semidirect

Maximal subgroups

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

subgroupindex
C2×#168104162
#202751142
#202749952
#202728332
C53×#23662
#1296453