Finite Group Databasebeta

← #27279176

Subgroups — #27279176

Order 13568.

516 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 107 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
1#20277998C211yesmaximal, characteristic
2C4×#13133136C211yesmaximal, characteristic, complemented
3#20273157C212yesmaximal
4#20272607C212yesmaximal, complemented
5C53×#2983C211yesmaximal, characteristic, complemented
6C2×C4×D212C2211yescharacteristic
7C2×#13133136—21no—
8C4×#11719917C2212yes—
9#16810562C411yescharacteristic
10#16810562C411yescharacteristic
11#16810549—21no—
12#16810540C411yescharacteristic
13#16810540C411yescharacteristic
14#16810534—21no—
15C4×#11719921—21no—
16C4×#11719921—21no—
17#16810469C2211yescharacteristic
18#16810392—21no—
19#16810350—24no—
20C2×C4×Q8×C53C2211yescharacteristic
21C53×#446—24no—
22C53×#432—21no—
23C53×#432C2212yes—
24C53×#431—21no—
25C22×D212C2×C411yescharacteristic
26C22×C4×D53—21no—
27C2×#11719917—22no—
28C4×D212—22no—
29C4×D212—22no—
30C2×C4⋊Dic53C2×C411yescharacteristic
31C2×#11719921—21no—
32C2×#11719921—21no—
33C2×#11719921—21no—
34C2×#11719921—21no—
35C2×#11719921—21no—
36C2×C4×Dic53—21no—
37#13133136—24no—
38#13133128—22no—
39C4×#88322—22no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—21no—
45#13133124C2×C411yescharacteristic
46#13133124C2×C411yescharacteristic
47C22×Q8×C53D411yescharacteristic
48C4×Q8×C53—22no—
49C4×Q8×C53—22no—
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C4⋊C4×C53C2×C411yescharacteristic
52C2×C4⋊C4×C53D411yescharacteristic
53C2×C42×C53D411yescharacteristic
54C2×C42×C53D411yescharacteristic
55C2×C42×C53C2311yescharacteristic
56C4⋊C8×C53—22no—
57C4⋊C8×C53—22no—
58C4⋊C8×C53D414yes—
59C22⋊C8×C53—22no—
60#97×C53—22no—
61#97×C53—22no—
62C4×C8×C53—22no—
63C23×D53—21no—
64C2×Q8×C53—22no—
65C2×Q8×C53—22no—
66C2×Q8×C53—41no—
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C22⋊C411yescharacteristic
70C22×C4×C53C22⋊C411yescharacteristic
71C22×C4×C53C22×C411yescharacteristic
72C22×C4×C53C2×D411yescharacteristic
73C22×C4×C53C2×D411yescharacteristic
74C22×Dic53—21no—
75C22×Dic53—21no—
76C22×Dic53—21no—
77C2×D212—22no—
78C2×D212—42no—
79C2×C4×D53—42no—
80C2×C4×D53—42no—
81C2×C4×D53—41no—
82C2×C4×D53—41no—
83C2×C8×C53—24no—
84C4⋊C4×C53—22no—
85C4⋊C4×C53—22no—
86C4⋊C4×C53—22no—
87C42×C53—21no—
88C42×C53—21no—
89C42×C53—21no—
90C42×C53—21no—
91C42×C53C2×D412yes—
92C42×C53#4112yes—
93#11719921—42no—
94#11719921—42no—
95#11719921—42no—
96#11719921—42no—
97#11719921—42no—
98C4⋊Dic53—22no—
99C4×Dic53—42no—
100#11719917—24no—
101#11719917—42no—
102C2×#88322—42no—
103C23×C53C2×C22⋊C411yescharacteristic
104C22×D53—22no—
105C22×D53—41no—
106C22×D53—41no—
107C22×D53—41no—
108C22×D53—41no—
109C22×D53—41no—
110C22×D53—41no—
111Q8×C53—44no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—22no—
117C2×C4×C53—21no—
118C2×C4×C53—21no—
119C2×C4×C53—21no—
120C2×C4×C53—21no—
121C2×C4×C53—21no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53#10412yes—
126C2×C4×C53C4×D412yes—
127C2×C4×C53#12011yescharacteristic
128C2×C4×C53#12112yes—
129C2×C4×C53#12311yescharacteristic
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—41no—
138C2×Dic53—41no—
139D212—24no—
140D212—42no—
141C4×D53—82no—
142C4×D53—82no—
143C8×C53—44no—
144#88322—44no—
145#9032—531nomaximal, Sylow
146C22×C53—21no—
147C22×C53—21no—
148C22×C53#38611yescharacteristic
149C22×C53C2×#10411yescharacteristic
150C22×C53#42111yescharacteristic
151C2×D53—42no—
152C2×D53—42no—
153C2×D53—42no—
154C2×D53—81no—
155C2×D53—81no—
156C2×D53—81no—
157C2×D53—81no—
158C4×C53—22no—
159C4×C53—22no—
160C4×C53—21no—
161C4×C53—22no—
162C4×C53—41no—
163C4×C53—41no—
164C4×C53—41no—
165C4×C53#45412yes—
166Dic53—81no—
167Dic53—81no—
168Dic53—81no—
169#2983D5311yescharacteristic, complemented
170#2398—531no—
171C4×#101—531no—
172#1490—532no—
173#1300—532no—
174C2×C53—21no—
175C2×C53—21no—
176C2×C53#187111yescharacteristic
177C2×C53#188811yescharacteristic
178C2×C53#189111yescharacteristic
179D53—82no—
180C2×C4×Q8C2×D5311yescharacteristic
181C2×C4×D4—531no—
182#446—24no—
183#432—21no—
184#432C2×D5312yes—
185#431—21no—
186C2×#101—1061no—
187C4×M16—532no—
188#400—531no—
189#400—531no—
190#397—1061no—
191#394—531no—
192#394—531no—
193#393—1061no—
194C4×C22⋊C4—1061no—
195C4×C22⋊C4—1061no—
196#376—531no—
197#343—1061no—
198D4⋊C8—1064no—
199C53#903211yescharacteristic, complemented, Sylow
200C22×Q8D21211yescharacteristic
201C22×D4—531no—
202C23×C4—1061no—
203C2×M16—1062no—
204C4×Q8—22no—
205C4×Q8—22no—
206C4×D4—1062no—
207C4×D4—1062no—
208C2×C4⋊C4—531no—
209C2×C4⋊C4C4×D5311yescharacteristic
210C2×C4⋊C4C4×D5311yescharacteristic
211C2×C4⋊C4D21211yescharacteristic
212C2×C22⋊C4—1061no—
213C2×C22⋊C4—1061no—
214C2×C22⋊C4—1061no—
215C2×C22⋊C4—1061no—
216C2×C22⋊C4—1061no—
217C2×C42—1061no—
218C2×C42C22⋊D5311yescharacteristic
219C2×C42C22⋊D5311yescharacteristic
220C2×C42C22×D5311yescharacteristic
221C4⋊C8—22no—
222C4⋊C8—22no—
223C4⋊C8C22⋊D5314yes—
224#101—1064no—
225C22⋊C8—22no—
226#97—22no—
227#97—22no—
228#97—1062no—
229C4×C8—22no—
230C4×C8—1062no—
231#95—531no—
232#95—531no—
233#95—1061no—
234#95—1061no—
235#95—1061no—
236#95—1061no—
237#95—1061no—
238C24—1061no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—41no—
242C2×D4—1062no—
243C2×D4—2122no—
244C22×C4—1061no—
245C22×C4—1061no—
246C22×C4—1061no—
247C22×C4—2122no—
248C22×C4—2122no—
249C22×C4—2121no—
250C22×C4—2121no—
251C22×C4#1171992111yescharacteristic
252C22×C4#1171992111yescharacteristic
253C22×C4#1171992111yescharacteristic
254C22×C4#1171992111yescharacteristic
255C22×C4C2×C4×D5311yescharacteristic
256C22×C4C2×D21211yescharacteristic
257C22×C4C2×C22⋊D5311yescharacteristic
258M16—1064no—
259M16—2122no—
260C2×C8—24no—
261C2×C8—2122no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—1062no—
266C22⋊C4—2122no—
267C22⋊C4—2122no—
268C22⋊C4—2122no—
269C22⋊C4—2122no—
270C22⋊C4—2122no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42—21no—
275C42—2122no—
276C42#1171994512yes—
277C42C2×C22⋊D5312yes—
278C23—1062no—
279C23—2121no—
280C23—2121no—
281C23—2121no—
282C23—2121no—
283C23—2121no—
284C23—2121no—
285C23C2×#1171992111yescharacteristic
286Q8—44no—
287D4—1064no—
288D4—2122no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—4242no—
303C2×C4—4242no—
304C2×C4—2121no—
305C2×C4—2121no—
306C2×C4—2121no—
307C2×C4—2121no—
308C2×C4—2121no—
309C2×C4—2121no—
310C2×C4—2121no—
311C2×C4—2121no—
312C2×C4—2121no—
313C2×C4#1313314312yes—
314C2×C4C4×C22⋊D5312yes—
315C2×C4#1313318611yescharacteristic
316C2×C4#1313319112yes—
317C2×C4#1313320011yescharacteristic
318C8—44no—
319C8—2124no—
320C22—21no—
321C22—21no—
322C22—4241no—
323C22—4241no—
324C22—4241no—
325C22—4241no—
326C22—2122no—
327C22—2122no—
328C22—2122no—
329C22#1681050211yescharacteristic
330C22C2×#1313314311yescharacteristic
331C22#1681063911yescharacteristic
332C4—22no—
333C4—22no—
334C4—21no—
335C4—22no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—4241no—
340C4—4241no—
341C4—4241no—
342C4#1681079612yes—
343C2—21no—
344C2—21no—
345C2—4242no—
346C2#2027501711yescharacteristic
347C2#2027511811yescharacteristic
348C2#2027513311yescharacteristic
subgroupG/H
centerC2×C4#13133186
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2983C2
socleC22×C53#386
prime radicalC22×D212C2×C4
cosocleC2×C42×C53C23
hypercenter#2983D53
hypocenterC53#9032
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2983

Complements

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

NH ≅ G/Ntype
2. C4×#13133136C2semidirect
4. #20272607C2semidirect
5. C53×#2983C2semidirect
169. #2983D53semidirect
199. C53#9032semidirect

Maximal subgroups

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

subgroupindex
#202779982
C4×#131331362
#202731572
#202726072
C53×#29832
#903253