Finite Group Databasebeta

← #27297893

Subgroups — #27297893

Order 13568.

356 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 75 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#20276942C211yesmaximal, characteristic, complemented
2#20276852C211yesmaximal, characteristic, complemented
3#20274104C211yesmaximal, characteristic, complemented
4#20272764C211yesmaximal, characteristic, complemented
5#20272676C211yesmaximal, characteristic, complemented
6D4⋊#11719908C211yesmaximal, characteristic, complemented
7C53×#3283C211yesmaximal, characteristic
8#16811028C2211yescharacteristic
9#16810828—21no—
10#16810448—21no—
11C4⋊#11719908C2211yescharacteristic
12#16810427—22no—
13#16810424—21no—
14C4×#11719908C2211yescharacteristic
15#16810376C2211yescharacteristic
16#16810368—21no—
17C2×#135×C53—21no—
18C53×#547—21no—
19C53×#494C2211yescharacteristic
20C53×#467—21no—
21C53×#461—21no—
22C53×#452—22no—
23C4×D8×C53C2211yescharacteristic
24C8×D4×C53C2211yescharacteristic
25C2×#11719923—41no—
26#13133217—21no—
27C2×#11719908—21no—
28C2×#11719908—21no—
29#13133151—21no—
30#13133151—21no—
31C4⋊#88322—21no—
32#13133140—41no—
33C2×#41×C53—22no—
34#135×C53—42no—
35#135×C53—42no—
36C2×Q16×C53—21no—
37C2×Q16×C53D412yes—
38C2×SD16×C53—22no—
39C2×D8×C53D411yescharacteristic
40C22×C8×C53—21no—
41C4⋊Q8×C53—21no—
42#123×C53—42no—
43C22⋊Q8×C53—22no—
44#121×C53—41no—
45C4×D4×C53—21no—
46C4×D4×C53D411yescharacteristic
47C2×C4⋊C4×C53—41no—
48#107×C53D411yescharacteristic
49C4⋊C8×C53D411yescharacteristic
50#103×C53—22no—
51#102×C53—21no—
52C22⋊C8×C53—21no—
53C4×C8×C53C2311yescharacteristic
54#41×C53—44no—
55#41×C53—82no—
56C2×Q8×C53—22no—
57C2×D4×C53—21no—
58C2×D4×C53—42no—
59C2×D4×C53D811yescharacteristic
60C22×C4×C53—21no—
61C22×C4×C53—42no—
62C22×C4×C53—41no—
63C2×Dic106—41no—
64Q16×C53—42no—
65Q16×C53—41no—
66Q16×C53—41no—
67SD16×C53—44no—
68D8×C53—21no—
69D8×C53SD1612yescomplemented
70C2×C8×C53—21no—
71C2×C8×C53—41no—
72C2×C8×C53—41no—
73C2×C8×C53C2×D411yescharacteristic
74C2×C8×C53C2×D411yescharacteristic
75C4⋊C4×C53—21no—
76C4⋊C4×C53—42no—
77C4⋊C4×C53—41no—
78C4⋊C4×C53—41no—
79C4⋊C4×C53D811yescharacteristic
80C22⋊C4×C53—21no—
81C22⋊C4×C53—42no—
82C22⋊C4×C53—41no—
83C42×C53C2×D411yescharacteristic
84#11719923—82no—
85C4⋊Dic53—41no—
86C4⋊Dic53—41no—
87C2×#88322—41no—
88#11719908—41no—
89#11719908—41no—
90C23×C53—21no—
91C23×C53—41no—
92Q8×C53—42no—
93Q8×C53—42no—
94D4×C53—22no—
95D4×C53—21no—
96D4×C53—21no—
97D4×C53—41no—
98D4×C53—84no—
99C2×C4×C53—21no—
100C2×C4×C53—41no—
101C2×C4×C53—41no—
102C2×C4×C53—42no—
103C2×C4×C53—41no—
104C2×C4×C53—84no—
105C2×C4×C53—81no—
106C2×C4×C53—81no—
107C2×C4×C53#12011yescharacteristic
108C2×C4×C53C2×D811yescharacteristic
109C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
110C2×Dic53—81no—
111Dic106—81no—
112Dic106—81no—
113C8×C53—21no—
114C8×C53—41no—
115C8×C53C2×SD1611yescharacteristic
116C8×C53AGL(1,ℤ/8)11yescharacteristic
117#88322—81no—
118#12371—531nomaximal, Sylow
119C22×C53—41no—
120C22×C53—42no—
121C22×C53—41no—
122C22×C53—41no—
123C22×C53—81no—
124C22×C53—81no—
125C22×C53#44711yescharacteristic
126C4×C53—21no—
127C4×C53—41no—
128C4×C53—82no—
129C4×C53—81no—
130C4×C53#45011yescharacteristic
131C4×C53#45511yescharacteristic
132C4×C53#50811yescharacteristic
133C4×C53#50911yescharacteristic
134Dic53—161no—
135#3283D5311yescharacteristic
136#2241—531no—
137#2224—531no—
138#1673—531no—
139#1362—531no—
140#1331—531no—
141D4⋊C16—531no—
142C2×C53—81no—
143C2×C53—82no—
144C2×C53#163911yescharacteristic
145C2×C53#218611yescharacteristic
146C2×C53#219511yescharacteristic
147C2×#135—21no—
148#547—21no—
149#500—531no—
150#494C2×D5311yescharacteristic
151#467—21no—
152#461—21no—
153#461—1061no—
154#452—22no—
155C4×D8C2×D5311yescharacteristic
156C8×D4C2×D5311yescharacteristic
157#366—1061no—
158C4⋊C16—531no—
159#358—1062no—
160#357—1061no—
161C4×C16—531no—
162#335—531no—
163#332—1061no—
164C53#1237111yescharacteristic, complemented, Sylow
165C2×#41—22no—
166#135—42no—
167#135—42no—
168C2×Q16—21no—
169C2×Q16C22⋊D5312yes—
170C2×SD16—22no—
171C2×SD16—2121no—
172C2×D8C22⋊D5311yescharacteristic
173C22×C8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—1061no—
176#123—42no—
177C22⋊Q8—22no—
178#121—41no—
179C4×D4—21no—
180C4×D4C22⋊D5311yescharacteristic
181C2×C4⋊C4—41no—
182C2×C16—1061no—
183C2×C16—1061no—
184#107—1061no—
185#107—1061no—
186#107C22⋊D5311yescharacteristic
187C4⋊C8—1061no—
188C4⋊C8C22⋊D5311yescharacteristic
189#103—22no—
190#103—2121no—
191#102—21no—
192C22⋊C8—21no—
193C4×C8C22×D5311yescharacteristic
194#41—44no—
195#41—82no—
196C2×Q8—22no—
197C2×Q8—2121no—
198C2×D4—21no—
199C2×D4—42no—
200C2×D4#1171992211yescharacteristic
201C22×C4—21no—
202C22×C4—42no—
203C22×C4—41no—
204Q16—42no—
205Q16—41no—
206Q16—41no—
207SD16—44no—
208SD16—4242no—
209D8—21no—
210D8#1171992312yescomplemented
211C2×C8—21no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—2121no—
215C2×C8C2×C22⋊D5311yescharacteristic
216C2×C8C2×C22⋊D5311yescharacteristic
217C4⋊C4—21no—
218C4⋊C4—42no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—2121no—
222C4⋊C4—2121no—
223C4⋊C4#1171992211yescharacteristic
224C22⋊C4—21no—
225C22⋊C4—42no—
226C22⋊C4—41no—
227C42C2×C22⋊D5311yescharacteristic
228C16—2121no—
229C16—2121no—
230C23—21no—
231C23—41no—
232Q8—42no—
233Q8—42no—
234Q8—4241no—
235Q8—4241no—
236D4—22no—
237D4—21no—
238D4—21no—
239D4—41no—
240D4—84no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—84no—
247C2×C4—81no—
248C2×C4—81no—
249C2×C4—4241no—
250C2×C4#1313318611yescharacteristic
251C2×C4C2×#1171992211yescharacteristic
252C2×C4#1313324311yescharacteristic
253C8—21no—
254C8—41no—
255C8—4241no—
256C8C2×#1171992311yescharacteristic
257C8#1313324311yescharacteristic
258C22—41no—
259C22—41no—
260C22—41no—
261C22—42no—
262C22—81no—
263C22—81no—
264C22#1681075711yescharacteristic
265C4—21no—
266C4—41no—
267C4—81no—
268C4—82no—
269C4—8481no—
270C4#1681077411yescharacteristic
271C4#1681080111yescharacteristic
272C4#1681105311yescharacteristic
273C4#1681105811yescharacteristic
274C2—81no—
275C2—82no—
276C2#2027389411yescharacteristic
277C2#2027663811yescharacteristic
278C2#2027669311yescharacteristic
subgroupG/H
centerC22#16810757
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
Fitting, prime radicalC53×#3283C2
socleC22×C53#447
hypercenter#3283D53
hypocenterC53#12371
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C8×D4 ◁ #3283

Complements

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

NH ≅ G/Ntype
1. #20276942C2semidirect
2. #20276852C2semidirect
3. #20274104C2semidirect
4. #20272764C2semidirect
5. #20272676C2semidirect
6. D4⋊#11719908C2semidirect
69. D8×C53SD16semidirect
164. C53#12371semidirect
210. D8#11719923semidirect

Maximal subgroups

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

subgroupindex
#202769422
#202768522
#202741042
#202727642
#202726762
D4⋊#117199082
C53×#32832
#1237153