Finite Group Databasebeta

← #27285058

Subgroups — #27285058

Order 13568.

636 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 103 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#20285161C211yesmaximal, characteristic, complemented
2#20284715C212yesmaximal, complemented
3#20273113C212yesmaximal, complemented
4#20272546C211yesmaximal, characteristic, complemented
5C53×#1955C211yesmaximal, characteristic, complemented
6C2×#13133239—22no—
7#16811173C2212yes—
8#16811017—22no—
9#16811012—22no—
10#16810936—22no—
11#16810903—22no—
12#16810818—22no—
13#16810757—22no—
14C4×#11719922—24no—
15#16810676C2212yes—
16#16810350—24no—
17C2×C4×D4×C53C2211yescharacteristic
18C2×C4⋊C8×C53C2211yescharacteristic
19C2×C22⋊C8×C53—22no—
20C2×#97×C53C2211yescharacteristic
21C2×#11719945—22no—
22#13133239—44no—
23#13133239—44no—
24C2×Dic212—22no—
25C2×#11719913—24no—
26C2×D424—22no—
27C2×#11719922—24no—
28#13133217D412yes—
29#13133214D412yes—
30#13133205—22no—
31#13133191—22no—
32#13133191—22no—
33C4×D212—22no—
34C4×D212—22no—
35#13133150—24no—
36C4⋊#88322—22no—
37#13133138—24no—
38#13133137—24no—
39C22⋊#88322—22no—
40C4×#88322D414yes—
41C22×D4×C53D411yescharacteristic
42C23×C4×C53—21no—
43C22×C8×C53—21no—
44C22×C8×C53D412yes—
45C4×D4×C53—22no—
46C4×D4×C53—22no—
47C2×C4⋊C4×C53D411yescharacteristic
48C2×C22⋊C4×C53—21no—
49C2×C42×C53C2311yescharacteristic
50C4⋊C8×C53—22no—
51C22⋊C8×C53—44no—
52#97×C53—22no—
53C24×C53—21no—
54C2×D4×C53—22no—
55C2×D4×C53—42no—
56C22×C4×C53—21no—
57C22×C4×C53—41no—
58C22×C4×C53—41no—
59C22×C4×C53—42no—
60C22×C4×C53—42no—
61C22×C4×C53C2×D411yescharacteristic
62C22×C4×C53C2×D411yescharacteristic
63C22×C4×C53#4111yescharacteristic
64C2×C22⋊D53—42no—
65#11719945—44no—
66#11719945—44no—
67C2×D212—22no—
68C2×D212—22no—
69C2×D212—22no—
70C2×C4×D53—42no—
71C2×Dic106—22no—
72C2×C8×C53—22no—
73C2×C8×C53—22no—
74C2×C8×C53—41no—
75C2×C8×C53—42no—
76C2×C8×C53D814yescomplemented
77C4⋊C4×C53—22no—
78C22⋊C4×C53—42no—
79C42×C53C2×D412yes—
80C42×C53C2×D412yes—
81#11719922—48no—
82#11719921—42no—
83#11719921—42no—
84C4⋊Dic53—22no—
85C4⋊Dic53—22no—
86C2×#88322—24no—
87Dic212—44no—
88D424—44no—
89#11719913—48no—
90C23×C53—22no—
91C23×C53—41no—
92C23×C53—42no—
93C23×C53—41no—
94C23×C53—42no—
95C23×C53#12111yescharacteristic
96C22×D53—42no—
97D4×C53—24no—
98D4×C53—42no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53—42no—
103C2×C4×C53—82no—
104C2×C4×C53—81no—
105C2×C4×C53—81no—
106C2×C4×C53#12011yescharacteristic
107C2×C4×C53#12012yes—
108C2×C4×C53#12112yes—
109C2×C4×C53#12711yescharacteristic
110C2×C4×C53C2×D812yes—
111C2×C4×C53AGL(1,ℤ/8)12yes—
112C22⋊D53—84no—
113C2×Dic53—42no—
114D212—44no—
115D212—44no—
116D212—44no—
117C4×D53—82no—
118C4×D53—82no—
119Dic106—44no—
120C8×C53—24no—
121C8×C53—81no—
122#88322—44no—
123#10032—531nomaximal, Sylow
124C22×C53—21no—
125C22×C53—21no—
126C22×C53—44no—
127C22×C53—42no—
128C22×C53—82no—
129C22×C53—81no—
130C22×C53—81no—
131C22×C53#39211yescharacteristic
132C22×C53#45912yes—
133C2×D53—82no—
134C2×D53—82no—
135C2×D53—82no—
136C4×C53—22no—
137C4×C53—22no—
138C4×C53—81no—
139C4×C53#44712yes—
140C4×C53#49311yescharacteristic
141C4×C53#49611yescharacteristic
142Dic53—82no—
143#3150—531no—
144#3100—532no—
145#1955D5311yescharacteristic, complemented
146#1480—532no—
147#1275—531no—
148C2×C53—22no—
149C2×C53—82no—
150C2×C53#201011yescharacteristic
151C2×C53#201111yescharacteristic
152C2×C53#201711yescharacteristic
153D53—162no—
154C2×#135—1062no—
155#532—532no—
156C2×C4×D4C2×D5311yescharacteristic
157#497—1062no—
158#496—1062no—
159#480—1062no—
160#474—1062no—
161#459—1062no—
162#447—1062no—
163C4×D8—1064no—
164#431—532no—
165C2×C4⋊C8C2×D5311yescharacteristic
166C2×C22⋊C8—22no—
167C2×#97C2×D5311yescharacteristic
168D4⋊C8—1064no—
169C53#1003211yescharacteristic, complemented, Sylow
170C2×#41—1062no—
171C22×D4D21211yescharacteristic
172C23×C4—21no—
173#135—2124no—
174#135—2124no—
175C2×Q16—1062no—
176C2×SD16—1064no—
177C2×D8—1062no—
178C2×D8—1064no—
179C22×C8—21no—
180C22×C8C22⋊D5312yes—
181C4⋊Q8—532no—
182#127—532no—
183#124—1062no—
184#121—1062no—
185#121—1062no—
186C4×D4—22no—
187C4×D4—22no—
188C4×D4—1062no—
189C4×D4—1062no—
190C2×C4⋊C4D21211yescharacteristic
191C2×C22⋊C4—21no—
192C2×C42C22×D5311yescharacteristic
193#107—1064no—
194C4⋊C8—22no—
195C4⋊C8—1062no—
196#102—1064no—
197#102—1064no—
198C22⋊C8—44no—
199C22⋊C8—1062no—
200#97—22no—
201C4×C8—534no—
202C24—21no—
203#41—2124no—
204#41—2124no—
205C2×Q8—1062no—
206C2×D4—22no—
207C2×D4—42no—
208C2×D4—1062no—
209C2×D4—1062no—
210C2×D4—1062no—
211C2×D4—2122no—
212C22×C4—21no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—2122no—
218C22×C4C2×D21211yescharacteristic
219C22×C4#1171994511yescharacteristic
220C22×C4C2×C22⋊D5311yescharacteristic
221Q16—2124no—
222SD16—2128no—
223D8—2124no—
224D8—2128no—
225C2×C8—22no—
226C2×C8—22no—
227C2×C8—41no—
228C2×C8—42no—
229C2×C8—1064no—
230C2×C8#1171992214yescomplemented
231C4⋊C4—22no—
232C4⋊C4—1062no—
233C4⋊C4—1062no—
234C22⋊C4—42no—
235C22⋊C4—2122no—
236C22⋊C4—2122no—
237C42D4×D5312yes—
238C42D4×D5312yes—
239C23—22no—
240C23—41no—
241C23—42no—
242C23—41no—
243C23—42no—
244C23—2122no—
245C23#1313319111yescharacteristic
246Q8—2124no—
247D4—24no—
248D4—42no—
249D4—4244no—
250D4—2124no—
251D4—2124no—
252D4—2124no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—82no—
258C2×C4—81no—
259C2×C4—81no—
260C2×C4—4242no—
261C2×C4—4242no—
262C2×C4—2122no—
263C2×C4#1313318511yescharacteristic
264C2×C4#1313318712yes—
265C2×C4#1313319012yes—
266C2×C4#1313321311yescharacteristic
267C2×C4C2×#1171992212yes—
268C2×C4#1313324212yes—
269C8—24no—
270C8—81no—
271C8—2124no—
272C22—21no—
273C22—21no—
274C22—44no—
275C22—42no—
276C22—82no—
277C22—81no—
278C22—81no—
279C22—4242no—
280C22—4242no—
281C22—4242no—
282C22#1681053211yescharacteristic
283C22#1681081812yes—
284C4—22no—
285C4—22no—
286C4—81no—
287C4—4242no—
288C4#1681075612yes—
289C4#1681099911yescharacteristic
290C4#1681101311yescharacteristic
291C2—22no—
292C2—82no—
293C2—8482no—
294C2#2027578211yescharacteristic
295C2#2027578511yescharacteristic
296C2#2027581911yescharacteristic
subgroupG/H
centerC2×C4#13133213
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1955C2
socleC22×C53#392
hypercenter#1955D53
hypocenterC53#10032
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1955

Complements

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

NH ≅ G/Ntype
1. #20285161C2semidirect
2. #20284715C2semidirect
3. #20273113C2semidirect
4. #20272546C2semidirect
5. C53×#1955C2semidirect
76. C2×C8×C53D8semidirect
145. #1955D53semidirect
169. C53#10032semidirect
230. C2×C8#11719922semidirect

Maximal subgroups

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

subgroupindex
#202851612
#202847152
#202731132
#202725462
C53×#19552
#1003253