Finite Group Databasebeta

← #27297949

Subgroups — #27297949

Order 13568.

350 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 71 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#20276951C211yesmaximal, characteristic, complemented
2#20276867C211yesmaximal, characteristic, complemented
3#20274228C211yesmaximal, characteristic, complemented
4#20272758C211yesmaximal, characteristic, complemented
5#20272685C211yesmaximal, characteristic, complemented
6#20272673C211yesmaximal, characteristic, complemented
7C53×#3277C211yesmaximal, characteristic
8#16811028C2211yescharacteristic
9#16810828—21no—
10#16810448—21no—
11C4⋊#11719908C2211yescharacteristic
12#16810427—21no—
13#16810427—21no—
14#16810424—21no—
15#16810399C2211yescharacteristic
16#16810374C2211yescharacteristic
17#16810368—21no—
18C2×#137×C53—21no—
19C53×#548—21no—
20C53×#495C2211yescharacteristic
21C53×#468—21no—
22C53×#464—21no—
23C53×#451—21no—
24C53×#449—21no—
25C4×Q16×C53C2211yescharacteristic
26C53×#436C2211yescharacteristic
27C2×#11719923—41no—
28#13133217—21no—
29C2×#11719908—21no—
30C2×#11719908—21no—
31#13133151—21no—
32#13133151—21no—
33C4⋊#88322—21no—
34#13133140—41no—
35C2×#41×C53—21no—
36C22×Q8×C53—21no—
37#137×C53—42no—
38#137×C53—42no—
39C2×Q16×C53—21no—
40C2×Q16×C53D411yescharacteristic
41C2×Q16×C53D411yescharacteristic
42C2×SD16×C53—21no—
43C2×SD16×C53—21no—
44C2×D8×C53D411yescharacteristic
45C2×M16×C53—21no—
46#124×C53—21no—
47#124×C53—41no—
48C22⋊Q8×C53—21no—
49C22⋊Q8×C53—41no—
50#121×C53—21no—
51#121×C53—41no—
52C4×Q8×C53—21no—
53C4×D4×C53—41no—
54C4×D4×C53D411yescharacteristic
55#107×C53D411yescharacteristic
56C4⋊C8×C53D411yescharacteristic
57#103×C53—21no—
58#103×C53—21no—
59#102×C53—21no—
60C22⋊C8×C53—21no—
61C4×C8×C53C2311yescharacteristic
62#41×C53—42no—
63#41×C53—81no—
64C2×Q8×C53—21no—
65C2×Q8×C53—21no—
66C2×Q8×C53—41no—
67C2×Q8×C53—42no—
68C2×Q8×C53—81no—
69C2×D4×C53—21no—
70C2×D4×C53—41no—
71C2×D4×C53—41no—
72C2×D4×C53D811yescharacteristic
73C22×C4×C53—21no—
74C22×C4×C53—41no—
75C22×C4×C53—41no—
76C2×Dic106—41no—
77Q16×C53—21no—
78Q16×C53—41no—
79Q16×C53—42no—
80Q16×C53SD1612yescomplemented
81SD16×C53—42no—
82SD16×C53—41no—
83SD16×C53—41no—
84D8×C53—41no—
85M16×C53—41no—
86M16×C53—41no—
87C2×C8×C53—21no—
88C2×C8×C53C2×D411yescharacteristic
89C2×C8×C53C2×D411yescharacteristic
90C4⋊C4×C53—21no—
91C4⋊C4×C53—41no—
92C4⋊C4×C53—41no—
93C4⋊C4×C53D811yescharacteristic
94C22⋊C4×C53—21no—
95C22⋊C4×C53—41no—
96C22⋊C4×C53—41no—
97C22⋊C4×C53—41no—
98C22⋊C4×C53—41no—
99C42×C53—41no—
100C42×C53C2×D411yescharacteristic
101#11719923—82no—
102C4⋊Dic53—41no—
103C4⋊Dic53—41no—
104C2×#88322—41no—
105#11719908—41no—
106#11719908—41no—
107C23×C53—21no—
108C23×C53—41no—
109Q8×C53—22no—
110Q8×C53—41no—
111Q8×C53—41no—
112Q8×C53—41no—
113Q8×C53—82no—
114D4×C53—41no—
115D4×C53—41no—
116D4×C53—41no—
117D4×C53—82no—
118D4×C53—82no—
119C2×C4×C53—21no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53—82no—
126C2×C4×C53—82no—
127C2×C4×C53#12011yescharacteristic
128C2×C4×C53C2×D811yescharacteristic
129C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
130C2×Dic53—81no—
131Dic106—81no—
132Dic106—81no—
133C8×C53—21no—
134C8×C53—41no—
135C8×C53C2×SD1611yescharacteristic
136C8×C53AGL(1,ℤ/8)11yescharacteristic
137#88322—81no—
138#12379—531nomaximal, Sylow
139C22×C53—41no—
140C22×C53—41no—
141C22×C53—41no—
142C22×C53—81no—
143C22×C53—81no—
144C22×C53—81no—
145C22×C53#44711yescharacteristic
146C4×C53—21no—
147C4×C53—21no—
148C4×C53—41no—
149C4×C53—42no—
150C4×C53—81no—
151C4×C53—81no—
152C4×C53#45011yescharacteristic
153C4×C53#45511yescharacteristic
154Dic53—161no—
155#3277D5311yescharacteristic
156#2243—531no—
157#2227—531no—
158#1693—531no—
159#1360—531no—
160#1334—531no—
161#1330—531no—
162C2×C53—81no—
163C2×C53—161no—
164C2×C53#163911yescharacteristic
165C2×C53#219011yescharacteristic
166C2×C53#219511yescharacteristic
167C2×#137—21no—
168#548—21no—
169#500—531no—
170#495C2×D5311yescharacteristic
171#468—21no—
172#464—21no—
173#461—1061no—
174#451—21no—
175#449—21no—
176C4×Q16C2×D5311yescharacteristic
177#436C2×D5311yescharacteristic
178#366—1061no—
179C4⋊C16—531no—
180#358—1061no—
181#358—1061no—
182#357—1061no—
183#346—531no—
184#334—531no—
185#332—1061no—
186C53#1237911yescharacteristic, complemented, Sylow
187C2×#41—21no—
188C22×Q8—21no—
189#137—42no—
190#137—42no—
191C2×Q16—21no—
192C2×Q16C22⋊D5311yescharacteristic
193C2×Q16C22⋊D5311yescharacteristic
194C2×SD16—21no—
195C2×SD16—21no—
196C2×SD16—2121no—
197C2×D8C22⋊D5311yescharacteristic
198C2×M16—21no—
199C4⋊Q8—1061no—
200#124—21no—
201#124—41no—
202C22⋊Q8—21no—
203C22⋊Q8—41no—
204#121—21no—
205#121—41no—
206C4×Q8—21no—
207C4×D4—41no—
208C4×D4C22⋊D5311yescharacteristic
209C2×C16—1061no—
210C2×C16—1061no—
211#107—1061no—
212#107—1061no—
213#107C22⋊D5311yescharacteristic
214C4⋊C8—1061no—
215C4⋊C8C22⋊D5311yescharacteristic
216#103—21no—
217#103—21no—
218#103—2121no—
219#102—21no—
220C22⋊C8—21no—
221C4×C8C22×D5311yescharacteristic
222#41—42no—
223#41—81no—
224C2×Q8—21no—
225C2×Q8—21no—
226C2×Q8—41no—
227C2×Q8—42no—
228C2×Q8—81no—
229C2×Q8—2121no—
230C2×D4—21no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4#1171992211yescharacteristic
234C22×C4—21no—
235C22×C4—41no—
236C22×C4—41no—
237Q16—21no—
238Q16—41no—
239Q16—42no—
240Q16#1171992312yescomplemented
241SD16—42no—
242SD16—41no—
243SD16—41no—
244SD16—4242no—
245D8—41no—
246M16—41no—
247M16—41no—
248C2×C8—21no—
249C2×C8—2121no—
250C2×C8C2×C22⋊D5311yescharacteristic
251C2×C8C2×C22⋊D5311yescharacteristic
252C4⋊C4—21no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—2121no—
256C4⋊C4—2121no—
257C4⋊C4#1171992211yescharacteristic
258C22⋊C4—21no—
259C22⋊C4—41no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—41no—
263C42—41no—
264C42C2×C22⋊D5311yescharacteristic
265C16—2121no—
266C16—2121no—
267C23—21no—
268C23—41no—
269Q8—22no—
270Q8—41no—
271Q8—41no—
272Q8—41no—
273Q8—82no—
274Q8—4241no—
275Q8—4241no—
276D4—41no—
277D4—41no—
278D4—41no—
279D4—82no—
280D4—82no—
281C2×C4—21no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—4241no—
290C2×C4#1313318611yescharacteristic
291C2×C4C2×#1171992211yescharacteristic
292C2×C4#1313324311yescharacteristic
293C8—21no—
294C8—41no—
295C8—4241no—
296C8C2×#1171992311yescharacteristic
297C8#1313324311yescharacteristic
298C22—41no—
299C22—41no—
300C22—41no—
301C22—81no—
302C22—81no—
303C22—81no—
304C22#1681075711yescharacteristic
305C4—21no—
306C4—21no—
307C4—41no—
308C4—42no—
309C4—81no—
310C4—81no—
311C4—8481no—
312C4#1681077411yescharacteristic
313C4#1681080111yescharacteristic
314C2—81no—
315C2—161no—
316C2#2027389411yescharacteristic
317C2#2027666111yescharacteristic
318C2#2027669311yescharacteristic
subgroupG/H
centerC22#16810757
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
Fitting, prime radicalC53×#3277C2
socleC22×C53#447
hypercenter#3277D53
hypocenterC53#12379
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ #436 ◁ #3277

Complements

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

NH ≅ G/Ntype
1. #20276951C2semidirect
2. #20276867C2semidirect
3. #20274228C2semidirect
4. #20272758C2semidirect
5. #20272685C2semidirect
6. #20272673C2semidirect
80. Q16×C53SD16semidirect
186. C53#12379semidirect
240. Q16#11719923semidirect

Maximal subgroups

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

subgroupindex
#202769512
#202768672
#202742282
#202727582
#202726852
#202726732
C53×#32772
#1237953