Finite Group Databasebeta

← #27285056

Subgroups — #27285056

Order 13568.

636 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 636 conjugacy classes of subgroups — too many to draw. Only its 105 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#20284711C212yesmaximal, complemented
2#20275513C211yesmaximal, characteristic, complemented
3#20273114C212yesmaximal, complemented
4#20272546C211yesmaximal, characteristic, complemented
5C53×#3150C211yesmaximal, characteristic, complemented
6C2×C4×D212C2211yescharacteristic
7#16810934—22no—
8#16810900—22no—
9#16810815—22no—
10#16810756—22no—
11C4×#11719922—24no—
12#16810676C2212yes—
13C2×C4⋊#88322C2211yescharacteristic
14C2×#13133129—22no—
15D4⋊#88322—24no—
16C2×#135×C53—22no—
17C53×#532C2212yes—
18C53×#497—22no—
19C53×#496—22no—
20C2×#97×C53C2211yescharacteristic
21C22×D212D411yescharacteristic
22C22×C4×D53—21no—
23C2×#11719922—24no—
24C22×#88322—21no—
25C4×D212—22no—
26C4×D212—22no—
27C2×C4⋊Dic53D411yescharacteristic
28C2×#11719921—21no—
29#13133150—24no—
30C4⋊#88322—22no—
31C4⋊#88322—22no—
32#13133138—24no—
33#13133137—24no—
34C22⋊#88322—22no—
35#13133129—44no—
36C4×#88322D414yes—
37C2×#41×C53—22no—
38#135×C53—44no—
39#135×C53—44no—
40C2×Q16×C53—22no—
41C2×SD16×C53—24no—
42C2×D8×C53—22no—
43C22×C8×C53D412yes—
44C4⋊Q8×C53D412yes—
45#127×C53D412yes—
46#124×C53—22no—
47#121×C53—22no—
48#121×C53—22no—
49C4×D4×C53—22no—
50C4×D4×C53—22no—
51C2×C42×C53C2311yescharacteristic
52#97×C53—22no—
53C23×D53—21no—
54#41×C53—44no—
55#41×C53—44no—
56C2×Q8×C53—22no—
57C2×D4×C53—22no—
58C2×D4×C53—22no—
59C2×D4×C53—22no—
60C2×D4×C53—42no—
61C22×C4×C53—42no—
62C22×C4×C53C2×D411yescharacteristic
63C22×C4×C53C2×D411yescharacteristic
64C22×C4×C53#4111yescharacteristic
65C22×Dic53—21no—
66C2×D212—22no—
67C2×D212—42no—
68C2×C4×D53—41no—
69C2×C4×D53—41no—
70C2×C4×D53—42no—
71C2×C4×D53—42no—
72Q16×C53—44no—
73SD16×C53—48no—
74D8×C53—44no—
75C2×C8×C53—22no—
76C2×C8×C53—22no—
77C2×C8×C53D814yescomplemented
78C4⋊C4×C53—22no—
79C4⋊C4×C53—22no—
80C22⋊C4×C53—42no—
81C22⋊C4×C53—42no—
82C42×C53C2×D412yes—
83C42×C53C2×D412yes—
84#11719922—48no—
85#11719921—42no—
86C4⋊Dic53—22no—
87C2×#88322—24no—
88C2×#88322—41no—
89C2×#88322—42no—
90C23×C53—42no—
91C23×C53#12111yescharacteristic
92C22×D53—22no—
93C22×D53—41no—
94C22×D53—42no—
95C22×D53—41no—
96C22×D53—42no—
97Q8×C53—44no—
98D4×C53—44no—
99D4×C53—44no—
100D4×C53—44no—
101D4×C53—84no—
102C2×C4×C53—41no—
103C2×C4×C53—41no—
104C2×C4×C53—42no—
105C2×C4×C53—82no—
106C2×C4×C53—82no—
107C2×C4×C53#12012yes—
108C2×C4×C53#12011yescharacteristic
109C2×C4×C53#12112yes—
110C2×C4×C53#12711yescharacteristic
111C2×C4×C53C2×D812yes—
112C2×C4×C53AGL(1,ℤ/8)12yes—
113C2×Dic53—41no—
114C2×Dic53—42no—
115D212—24no—
116D212—42no—
117C4×D53—82no—
118C4×D53—81no—
119C4×D53—81no—
120C8×C53—24no—
121#88322—44no—
122#88322—81no—
123#10032—531nomaximal, Sylow
124C22×C53—21no—
125C22×C53—21no—
126C22×C53—82no—
127C22×C53—82no—
128C22×C53—82no—
129C22×C53#39211yescharacteristic
130C22×C53#45912yes—
131C2×D53—44no—
132C2×D53—42no—
133C2×D53—82no—
134C2×D53—81no—
135C2×D53—81no—
136C4×C53—22no—
137C4×C53—22no—
138C4×C53—82no—
139C4×C53#44712yes—
140C4×C53#49311yescharacteristic
141C4×C53#49611yescharacteristic
142Dic53—81no—
143#3150D5311yescharacteristic, complemented
144#3100—532no—
145#1955—531no—
146#1480—532no—
147#1275—531no—
148C2×C53—22no—
149C2×C53—162no—
150C2×C53#201011yescharacteristic
151C2×C53#201111yescharacteristic
152C2×C53#201711yescharacteristic
153D53—82no—
154C2×#135—22no—
155#532C2×D5312yes—
156C2×C4×D4—531no—
157#497—22no—
158#496—22no—
159#480—1062no—
160#474—1062no—
161#459—1062no—
162#447—1062no—
163C4×D8—1064no—
164#431—532no—
165C2×C4⋊C8—531no—
166C2×C22⋊C8—1062no—
167C2×#97C2×D5311yescharacteristic
168D4⋊C8—1064no—
169C53#1003211yescharacteristic, complemented, Sylow
170C2×#41—22no—
171C22×D4—531no—
172C23×C4—1061no—
173#135—44no—
174#135—44no—
175C2×Q16—22no—
176C2×SD16—24no—
177C2×D8—22no—
178C2×D8—1064no—
179C22×C8—1061no—
180C22×C8C22⋊D5312yes—
181C4⋊Q8C22⋊D5312yes—
182#127C22⋊D5312yes—
183#124—22no—
184#121—22no—
185#121—22no—
186C4×D4—22no—
187C4×D4—22no—
188C4×D4—1062no—
189C4×D4—1062no—
190C2×C4⋊C4—531no—
191C2×C22⋊C4—1061no—
192C2×C42C22×D5311yescharacteristic
193#107—1064no—
194C4⋊C8—1062no—
195C4⋊C8—1062no—
196#102—1064no—
197#102—1064no—
198C22⋊C8—1062no—
199C22⋊C8—2124no—
200#97—22no—
201C4×C8—534no—
202C24—1061no—
203#41—44no—
204#41—44no—
205C2×Q8—22no—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—42no—
210C2×D4—1062no—
211C2×D4—2122no—
212C22×C4—42no—
213C22×C4—1061no—
214C22×C4—2121no—
215C22×C4—2121no—
216C22×C4—2122no—
217C22×C4—2122no—
218C22×C4D4×D5311yescharacteristic
219C22×C4#1171994711yescharacteristic
220C22×C4C2×C22⋊D5311yescharacteristic
221Q16—44no—
222SD16—48no—
223D8—44no—
224D8—2128no—
225C2×C8—22no—
226C2×C8—22no—
227C2×C8—1064no—
228C2×C8—2121no—
229C2×C8—2122no—
230C2×C8#1171992214yescomplemented
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—1062no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—2122no—
237C42D4×D5312yes—
238C42C2×C22⋊D5312yes—
239C23—42no—
240C23—1062no—
241C23—2121no—
242C23—2122no—
243C23—2121no—
244C23—2122no—
245C23#1313318811yescharacteristic
246Q8—44no—
247D4—44no—
248D4—44no—
249D4—44no—
250D4—84no—
251D4—1064no—
252D4—2122no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—42no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4—4241no—
259C2×C4—4242no—
260C2×C4—4241no—
261C2×C4—2121no—
262C2×C4—2122no—
263C2×C4#1313318512yes—
264C2×C4#1313318611yescharacteristic
265C2×C4#1313318912yes—
266C2×C4#1313321311yescharacteristic
267C2×C4C2×#1171992212yes—
268C2×C4#1313324312yes—
269C8—24no—
270C8—4241no—
271C8—2124no—
272C22—21no—
273C22—21no—
274C22—82no—
275C22—82no—
276C22—82no—
277C22—4242no—
278C22—4241no—
279C22—4241no—
280C22—2124no—
281C22—2122no—
282C22#1681053111yescharacteristic
283C22#1681081512yes—
284C4—22no—
285C4—22no—
286C4—82no—
287C4—4241no—
288C4#1681075712yes—
289C4#1681099911yescharacteristic
290C4#1681101011yescharacteristic
291C2—22no—
292C2—162no—
293C2—4242no—
294C2#2027577911yescharacteristic
295C2#2027578611yescharacteristic
296C2#2027581811yescharacteristic
subgroupG/H
centerC2×C4#13133213
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#3150C2
socleC22×C53#392
hypercenter#3150D53
hypocenterC53#10032
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3150

Complements

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

NH ≅ G/Ntype
1. #20284711C2semidirect
2. #20275513C2semidirect
3. #20273114C2semidirect
4. #20272546C2semidirect
5. C53×#3150C2semidirect
77. C2×C8×C53D8semidirect
143. #3150D53semidirect
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
#202847112
#202755132
#202731142
#202725462
C53×#31502
#1003253