Finite Group Databasebeta

← #27297921

Subgroups — #27297921

Order 13568.

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

normal subgroups only

This group has 350 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#20276862C211yesmaximal, characteristic, complemented
3#20274114C211yesmaximal, characteristic, complemented
4#20272764C211yesmaximal, characteristic, complemented
5#20272676C211yesmaximal, characteristic, complemented
6D4⋊#11719908C211yesmaximal, characteristic, complemented
7C53×#3283C211yesmaximal, characteristic
8#16811028C2211yescharacteristic
9#16810913—21no—
10#16810450—21no—
11C4⋊#11719908C2211yescharacteristic
12#16810427—22no—
13#16810424—21no—
14C4×#11719908C2211yescharacteristic
15#16810376C2211yescharacteristic
16#16810371—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
25#13133217—21no—
26C2×#11719908—21no—
27C2×#11719908—21no—
28#13133151—21no—
29#13133151—21no—
30#13133148—41no—
31C4⋊#88322—21no—
32#13133138—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×C53SD1612yes—
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
84C4⋊Dic53—41no—
85C4⋊Dic53—41no—
86C2×#88322—41no—
87#11719908—41no—
88#11719908—41no—
89C23×C53—21no—
90C23×C53—41no—
91Q8×C53—42no—
92Q8×C53—42no—
93D4×C53—22no—
94D4×C53—21no—
95D4×C53—21no—
96D4×C53—41no—
97D4×C53—84no—
98C2×C4×C53—21no—
99C2×C4×C53—41no—
100C2×C4×C53—41no—
101C2×C4×C53—42no—
102C2×C4×C53—41no—
103C2×C4×C53—84no—
104C2×C4×C53—81no—
105C2×C4×C53—81no—
106C2×C4×C53#12011yescharacteristic
107C2×C4×C53C2×D811yescharacteristic
108C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
109C2×Dic53—81no—
110Dic106—161no—
111C8×C53—21no—
112C8×C53—41no—
113C8×C53C2×SD1611yescharacteristic
114C8×C53AGL(1,ℤ/8)11yescharacteristic
115#88322—81no—
116#12375—531nomaximal, Sylow
117C22×C53—41no—
118C22×C53—42no—
119C22×C53—41no—
120C22×C53—41no—
121C22×C53—81no—
122C22×C53—81no—
123C22×C53#44711yescharacteristic
124C4×C53—21no—
125C4×C53—41no—
126C4×C53—82no—
127C4×C53—81no—
128C4×C53#45011yescharacteristic
129C4×C53#45511yescharacteristic
130C4×C53#50811yescharacteristic
131C4×C53#51011yescharacteristic
132Dic53—161no—
133#3283D5311yescharacteristic
134#2241—531no—
135#2226—531no—
136#1675—531no—
137#1362—531no—
138#1331—531no—
139D4⋊C16—531no—
140C2×C53—81no—
141C2×C53—82no—
142C2×C53#163911yescharacteristic
143C2×C53#218911yescharacteristic
144C2×C53#219311yescharacteristic
145C2×#135—21no—
146#547—21no—
147#500—531no—
148#494C2×D5311yescharacteristic
149#476—1061no—
150#467—21no—
151#461—21no—
152#452—22no—
153C4×D8C2×D5311yescharacteristic
154C8×D4C2×D5311yescharacteristic
155#367—1061no—
156C4⋊C16—531no—
157#358—1062no—
158#357—1061no—
159C4×C16—531no—
160#335—531no—
161#333—1061no—
162C53#1237511yescharacteristic, complemented, Sylow
163C2×#41—22no—
164#135—42no—
165#135—42no—
166C2×Q16—21no—
167C2×Q16C22⋊D5312yes—
168C2×SD16—22no—
169C2×D8C22⋊D5311yescharacteristic
170C22×C8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—1061no—
173#123—42no—
174C22⋊Q8—22no—
175#121—41no—
176C4×D4—21no—
177C4×D4C22⋊D5311yescharacteristic
178C2×C4⋊C4—41no—
179C2×C16—1061no—
180C2×C16—1061no—
181#107—1061no—
182#107—1061no—
183#107C22⋊D5311yescharacteristic
184#106—2121no—
185C4⋊C8—1061no—
186C4⋊C8C22⋊D5311yescharacteristic
187#103—22no—
188#102—21no—
189#102—2121no—
190C22⋊C8—21no—
191C4×C8C22×D5311yescharacteristic
192#41—44no—
193#41—82no—
194C2×Q8—22no—
195C2×Q8—2121no—
196C2×D4—21no—
197C2×D4—42no—
198C2×D4#1171992211yescharacteristic
199C22×C4—21no—
200C22×C4—42no—
201C22×C4—41no—
202Q16—42no—
203Q16—41no—
204Q16—41no—
205SD16—44no—
206D8—21no—
207D8#1171992312yes—
208C2×C8—21no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—2121no—
212C2×C8C2×C22⋊D5311yescharacteristic
213C2×C8C2×C22⋊D5311yescharacteristic
214C4⋊C4—21no—
215C4⋊C4—42no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—2121no—
219C4⋊C4—2121no—
220C4⋊C4#1171992211yescharacteristic
221C22⋊C4—21no—
222C22⋊C4—42no—
223C22⋊C4—41no—
224C42C2×C22⋊D5311yescharacteristic
225C16—2121no—
226C16—2121no—
227C23—21no—
228C23—41no—
229Q8—42no—
230Q8—42no—
231Q8—8481no—
232D4—22no—
233D4—21no—
234D4—21no—
235D4—41no—
236D4—84no—
237C2×C4—21no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—42no—
242C2×C4—84no—
243C2×C4—81no—
244C2×C4—81no—
245C2×C4—4241no—
246C2×C4#1313318611yescharacteristic
247C2×C4C2×#1171992211yescharacteristic
248C2×C4#1313324311yescharacteristic
249C8—21no—
250C8—41no—
251C8—4241no—
252C8C2×#1171992311yescharacteristic
253C8#1313324311yescharacteristic
254C22—41no—
255C22—41no—
256C22—41no—
257C22—42no—
258C22—81no—
259C22—81no—
260C22#1681075711yescharacteristic
261C4—21no—
262C4—41no—
263C4—81no—
264C4—82no—
265C4—8481no—
266C4#1681077411yescharacteristic
267C4#1681080111yescharacteristic
268C4#1681105311yescharacteristic
269C4#1681106311yescharacteristic
270C2—81no—
271C2—82no—
272C2#2027389411yescharacteristic
273C2#2027665511yescharacteristic
274C2#2027667911yescharacteristic
subgroupG/H
centerC22#16810757
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
Fitting, prime radicalC53×#3283C2
socleC22×C53#447
hypercenter#3283D53
hypocenterC53#12375
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. #20276862C2semidirect
3. #20274114C2semidirect
4. #20272764C2semidirect
5. #20272676C2semidirect
6. D4⋊#11719908C2semidirect
162. C53#12375semidirect

Maximal subgroups

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

subgroupindex
#202769422
#202768622
#202741142
#202727642
#202726762
D4⋊#117199082
C53×#32832
#1237553