Finite Group Databasebeta

← #27299972

Subgroups — #27299972

Order 13568.

740 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 131 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×#16810422C212yesmaximal, complemented
2C2×#16810420C211yesmaximal, characteristic
3#20275119C211yesmaximal, characteristic
4#20275109C212yesmaximal, complemented
5C2×C53×#563C211yesmaximal, characteristic
6#16811101C2211yescharacteristic
7#16810632—22no—
8#16810628—21no—
9C2×#13133136C2212yes—
10C2×#13133136C2211yescharacteristic
11#16810422—24no—
12#16810420—22no—
13#16810383—22no—
14#16810383—21no—
15C53×#563—24no—
16C53×#563—22no—
17C53×#563C414yescomplemented
18C2×C4⋊Q8×C53C2212yes—
19C2×#126×C53—22no—
20C2×#125×C53—21no—
21C2×#124×C53C2211yescharacteristic
22C2×#123×C53—21no—
23C2×C22⋊Q8×C53—22no—
24C2×#11719917—21no—
25C2×#11719917—22no—
26Q8×Dic53—42no—
27#13133176—21no—
28#13133176—22no—
29#13133141—44no—
30#13133138—42no—
31#13133136—22no—
32#13133136—22no—
33#13133136—22no—
34C22×Q8×C53D412yes—
35C22×Q8×C53C2311yescharacteristic
36C22×D4×C53D411yescharacteristic
37C23×C4×C53—21no—
38C4⋊Q8×C53—22no—
39C4⋊Q8×C53—22no—
40C4⋊Q8×C53C2×C414yes—
41C4⋊Q8×C53D414yes—
42#126×C53—28no—
43#126×C53—28no—
44#125×C53—24no—
45#125×C53—24no—
46#124×C53—22no—
47#124×C53C2×C412yes—
48#124×C53D412yes—
49#123×C53—24no—
50#123×C53—24no—
51C22⋊Q8×C53—28no—
52C22⋊Q8×C53—28no—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53—22no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53—22no—
57C2×C4⋊C4×C53—22no—
58C2×C22⋊C4×C53—22no—
59C2×C22⋊C4×C53—21no—
60C2×C22⋊C4×C53—22no—
61C2×C42×C53D411yescharacteristic
62C2×C42×C53D412yes—
63C24×C53—21no—
64C2×Q8×C53—22no—
65C2×Q8×C53—41no—
66C2×Q8×C53—41no—
67C2×Q8×C53—42no—
68C2×Q8×C53—42no—
69C2×Q8×C53C22⋊C414yes—
70C2×Q8×C53C22×C411yescharacteristic
71C2×Q8×C53C2×D412yes—
72C2×Q8×C53C2×D411yescharacteristic
73C2×D4×C53—21no—
74C2×D4×C53—41no—
75C2×D4×C53—41no—
76C2×D4×C53C22⋊C412yes—
77C22×C4×C53—21no—
78C22×C4×C53—22no—
79C22×C4×C53—22no—
80C22×C4×C53—24no—
81C22×C4×C53—81no—
82C22×C4×C53—81no—
83C22×C4×C53C2×D412yes—
84C22×C4×C53C2×D411yescharacteristic
85C4⋊C4×C53—24no—
86C4⋊C4×C53—28no—
87C4⋊C4×C53—24no—
88C4⋊C4×C53—24no—
89C4⋊C4×C53—24no—
90C4⋊C4×C53—24no—
91C4⋊C4×C53—24no—
92C22⋊C4×C53—28no—
93C22⋊C4×C53—24no—
94C22⋊C4×C53—24no—
95C22⋊C4×C53—24no—
96C42×C53—21no—
97C42×C53—22no—
98C42×C53C22⋊C412yes—
99C42×C53C22⋊C414yes—
100C22⋊Dic53—41no—
101C4⋊Dic53—22no—
102C4⋊Dic53—24no—
103C4×Dic53—41no—
104C4×Dic53—42no—
105#11719917—41no—
106#11719917—42no—
107#11719917—41no—
108#11719917—42no—
109C2×#88322—41no—
110C2×#88322—42no—
111C23×C53—22no—
112C23×C53—44no—
113C23×C53—41no—
114C23×C53—41no—
115C23×C53#12011yescharacteristic
116Q8×C53—41no—
117Q8×C53—42no—
118Q8×C53—41no—
119Q8×C53—44no—
120Q8×C53—82no—
121D4×C53—42no—
122D4×C53—81no—
123C2×C4×C53—21no—
124C2×C4×C53—22no—
125C2×C4×C53—21no—
126C2×C4×C53—22no—
127C2×C4×C53—24no—
128C2×C4×C53—22no—
129C2×C4×C53—24no—
130C2×C4×C53—42no—
131C2×C4×C53—44no—
132C2×C4×C53—42no—
133C2×C4×C53—42no—
134C2×C4×C53—84no—
135C2×C4×C53C2×C22⋊C411yescharacteristic
136C2×C4×C53C2×C22⋊C412yes—
137C2×C4×C53#12011yescharacteristic
138C2×C4×C53#12012yes—
139C2×Dic53—42no—
140#88322—81no—
141#88322—82no—
142#12775—531nomaximal, Sylow
143C22×C53—22no—
144C22×C53—44no—
145C22×C53—42no—
146C22×C53—82no—
147C22×C53—82no—
148C22×C53#37911yescharacteristic
149C22×C53B(4,𝔽2)11yescharacteristic
150C22×C53#45811yescharacteristic
151C4×C53—21no—
152C4×C53—22no—
153C4×C53—21no—
154C4×C53—22no—
155C4×C53—42no—
156C4×C53—44no—
157C4×C53—44no—
158Dic53—82no—
159C2×#563D5311yescharacteristic
160C2×#356—532no—
161C2×#355—531no—
162#1888—531no—
163#1886—532no—
164C2×C53—21no—
165C2×C53—21no—
166C2×C53—82no—
167C2×C53#179011yescharacteristic
168C2×C53#220312yes—
169#563—24no—
170#563—22no—
171#563Dic5314yescomplemented
172C2×C4⋊Q8C2×D5312yes—
173C2×#126—22no—
174C2×#125—21no—
175C2×#124C2×D5311yescharacteristic
176C2×#123—21no—
177C2×C22⋊Q8—22no—
178#519—531no—
179#419—1062no—
180#418—1061no—
181C2×#101—532no—
182C2×#101—531no—
183#356—1064no—
184#355—1062no—
185#339—1062no—
186#339—1061no—
187C53#1277511yescharacteristic, complemented, Sylow
188C22×Q8C22⋊D5312yes—
189C22×Q8C22×D5311yescharacteristic
190C22×D4C22⋊D5311yescharacteristic
191C23×C4—21no—
192C2×M16—1061no—
193C2×M16—1062no—
194C4⋊Q8—22no—
195C4⋊Q8—22no—
196C4⋊Q8C2×Dic5314yes—
197C4⋊Q8C22⋊D5314yes—
198#126—28no—
199#126—28no—
200#125—24no—
201#125—24no—
202#124—22no—
203#124C2×Dic5312yes—
204#124C22⋊D5312yes—
205#123—24no—
206#123—24no—
207C22⋊Q8—28no—
208C22⋊Q8—28no—
209C4×Q8—2122no—
210#117—1061no—
211#117—1062no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—22no—
220C2×C42C22⋊D5311yescharacteristic
221C2×C42C22⋊D5312yes—
222#103—2124no—
223#102—2122no—
224#101—1062no—
225#101—1062no—
226#101—1062no—
227C24—21no—
228C2×Q8—22no—
229C2×Q8—41no—
230C2×Q8—41no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8C22⋊Dic5314yes—
234C2×Q8C22×Dic5311yescharacteristic
235C2×Q8C2×C22⋊D5312yes—
236C2×Q8C2×C22⋊D5311yescharacteristic
237C2×D4—21no—
238C2×D4—41no—
239C2×D4—41no—
240C2×D4C22⋊Dic5312yes—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—81no—
246C22×C4—81no—
247C22×C4C2×C22⋊D5312yes—
248C22×C4C2×C22⋊D5311yescharacteristic
249M16—2121no—
250M16—2122no—
251M16—2122no—
252M16—2121no—
253C2×C8—2121no—
254C2×C8—2122no—
255C4⋊C4—24no—
256C4⋊C4—28no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—1062no—
263C4⋊C4—1064no—
264C22⋊C4—28no—
265C22⋊C4—24no—
266C22⋊C4—24no—
267C22⋊C4—24no—
268C22⋊C4—2121no—
269C42—21no—
270C42—22no—
271C42—2121no—
272C42—2122no—
273C42C22⋊Dic5312yes—
274C42C22⋊Dic5314yes—
275C23—22no—
276C23—44no—
277C23—41no—
278C23—41no—
279C23#1313318611yescharacteristic
280Q8—41no—
281Q8—42no—
282Q8—41no—
283Q8—44no—
284Q8—82no—
285D4—42no—
286D4—81no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—84no—
299C2×C4—2122no—
300C2×C4C2×C22⋊Dic5312yes—
301C2×C4C2×C22⋊Dic5311yescharacteristic
302C2×C4#1313318612yes—
303C2×C4#1313318611yescharacteristic
304C8—4241no—
305C8—4242no—
306C22—22no—
307C22—44no—
308C22—42no—
309C22—82no—
310C22—82no—
311C22#1681047911yescharacteristic
312C22#1681081011yescharacteristic
313C22#1681081311yescharacteristic
314C4—21no—
315C4—22no—
316C4—21no—
317C4—22no—
318C4—42no—
319C4—44no—
320C4—44no—
321C4—4242no—
322C2—21no—
323C2—21no—
324C2—82no—
325C2#2027466311yescharacteristic
326C2#2027673612yes—
subgroupG/H
centerC22#16810810
commutatorC2×Q8×C53C22×C4
FrattiniC22×Q8C22×D53
FittingC2×C53×#563C2
socleC22×C53B(4,𝔽2)
prime radicalC22×D4×C53D4
cosocleC22×Q8×C53C23
hypercenterC2×#563D53
hypocenterC53#12775
LCSG ▷ C2×Q8×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#563

Complements

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

NH ≅ G/Ntype
1. C2×#16810422C2semidirect
4. #20275109C2semidirect
17. C53×#563C4semidirect
171. #563Dic53semidirect
187. C53#12775semidirect

Maximal subgroups

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

subgroupindex
C2×#168104222
C2×#168104202
#202751192
#202751092
C2×C53×#5632
#1277553