Finite Group Databasebeta

← #27297970

Subgroups — #27297970

Order 13568.

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

normal subgroups only

This group has 288 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
2#20276857C211yesmaximal, characteristic
3#20274109C211yesmaximal, characteristic
4#20272764C211yesmaximal, characteristic
5Q8⋊#11719908C211yesmaximal, characteristic
6#20272679C211yesmaximal, characteristic
7C53×#3395C211yesmaximal, characteristic
8#16811028C2211yescharacteristic
9#16810833—21no—
10#16810450—21no—
11C4⋊#11719908C2211yescharacteristic
12#16810427—21no—
13#16810427—22no—
14C4×#11719908C2211yescharacteristic
15#16810376C2211yescharacteristic
16#16810368—21no—
17C53×#557—21no—
18C53×#494—21no—
19C53×#494C2211yescharacteristic
20C53×#462—22no—
21C53×#462—21no—
22C8×Q8×C53C2211yescharacteristic
23C4×Q16×C53—21no—
24C4×Q16×C53C2211yescharacteristic
25C2×#11719925—41no—
26#13133217—21no—
27C2×#11719908—21no—
28C2×#11719908—21no—
29#13133151—21no—
30#13133151—21no—
31C4⋊#88322—21no—
32#13133140—41no—
33C2×Q16×C53—22no—
34C2×Q16×C53—21no—
35C2×Q16×C53D411yescharacteristic
36C2×Q16×C53D412yes—
37C4⋊Q8×C53—22no—
38C4⋊Q8×C53—21no—
39#125×C53—42no—
40#125×C53—41no—
41C4×Q8×C53—22no—
42C4×Q8×C53—21no—
43C4×Q8×C53—41no—
44C4×Q8×C53D411yescharacteristic
45#107×C53D411yescharacteristic
46C4⋊C8×C53—21no—
47C4⋊C8×C53D411yescharacteristic
48#103×C53—22no—
49#103×C53—21no—
50C4×C8×C53—21no—
51C4×C8×C53C2311yescharacteristic
52C2×Q8×C53—21no—
53C2×Q8×C53—22no—
54C2×Q8×C53D811yescharacteristic
55C2×Dic106—41no—
56Q16×C53—21no—
57Q16×C53—21no—
58Q16×C53—42no—
59Q16×C53—41no—
60Q16×C53—41no—
61Q16×C53—82no—
62C2×C8×C53—21no—
63C2×C8×C53C2×D411yescharacteristic
64C2×C8×C53C2×D411yescharacteristic
65C4⋊C4×C53—21no—
66C4⋊C4×C53—21no—
67C4⋊C4×C53—42no—
68C4⋊C4×C53—42no—
69C4⋊C4×C53—42no—
70C4⋊C4×C53—41no—
71C4⋊C4×C53—41no—
72C4⋊C4×C53—41no—
73C4⋊C4×C53D811yescharacteristic
74C42×C53—21no—
75C42×C53—42no—
76C42×C53—41no—
77C42×C53C2×D411yescharacteristic
78#11719925—82no—
79C4⋊Dic53—41no—
80C4⋊Dic53—41no—
81C2×#88322—41no—
82#11719908—41no—
83#11719908—41no—
84Q8×C53—21no—
85Q8×C53—41no—
86Q8×C53—42no—
87Q8×C53—42no—
88Q8×C53—41no—
89Q8×C53SD3212yes—
90C2×C4×C53—21no—
91C2×C4×C53—21no—
92C2×C4×C53—42no—
93C2×C4×C53—41no—
94C2×C4×C53—41no—
95C2×C4×C53#12011yescharacteristic
96C2×C4×C53C2×D811yescharacteristic
97C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
98C2×Dic53—81no—
99Dic106—81no—
100Dic106—81no—
101C8×C53—21no—
102C8×C53—41no—
103C8×C53#13511yescharacteristic
104C8×C53#13711yescharacteristic
105#88322—81no—
106#12382—531nomaximal, Sylow
107C22×C53#44711yescharacteristic
108C4×C53—22no—
109C4×C53—21no—
110C4×C53—41no—
111C4×C53—81no—
112C4×C53—82no—
113C4×C53—81no—
114C4×C53#45211yescharacteristic
115C4×C53#45511yescharacteristic
116C4×C53C2×SD3211yescharacteristic
117C4×C53#51011yescharacteristic
118Dic53—161no—
119#3395D5311yescharacteristic
120#2241—531no—
121#2225—531no—
122#1674—531no—
123#1362—531no—
124Q8⋊C16—531no—
125#1332—531no—
126C2×C53#164111yescharacteristic
127C2×C53#218511yescharacteristic
128C2×C53#219511yescharacteristic
129#557—21no—
130#500—531no—
131#494—21no—
132#494C2×D5311yescharacteristic
133#462—22no—
134#462—21no—
135#462—1061no—
136C8×Q8C2×D5311yescharacteristic
137C4×Q16—21no—
138C4×Q16C2×D5311yescharacteristic
139#367—1061no—
140C4⋊C16—531no—
141#358—1061no—
142#358—1062no—
143C4×C16—531no—
144#335—531no—
145#332—1061no—
146C53#1238211yescharacteristic, complemented, Sylow
147C2×Q16—22no—
148C2×Q16—21no—
149C2×Q16—2121no—
150C2×Q16C22⋊D5311yescharacteristic
151C2×Q16C22⋊D5312yes—
152C4⋊Q8—22no—
153C4⋊Q8—21no—
154C4⋊Q8—1061no—
155#125—42no—
156#125—41no—
157C4×Q8—22no—
158C4×Q8—21no—
159C4×Q8—41no—
160C4×Q8C22⋊D5311yescharacteristic
161C2×C16—1061no—
162C2×C16—1061no—
163#107—1061no—
164#107—1061no—
165#107C22⋊D5311yescharacteristic
166C4⋊C8—21no—
167C4⋊C8—1061no—
168C4⋊C8C22⋊D5311yescharacteristic
169#103—22no—
170#103—21no—
171#103—2121no—
172C4×C8—21no—
173C4×C8C22×D5311yescharacteristic
174C2×Q8—21no—
175C2×Q8—22no—
176C2×Q8—2121no—
177C2×Q8#1171992211yescharacteristic
178Q16—21no—
179Q16—21no—
180Q16—42no—
181Q16—41no—
182Q16—41no—
183Q16—82no—
184Q16—4242no—
185C2×C8—21no—
186C2×C8—2121no—
187C2×C8C2×C22⋊D5311yescharacteristic
188C2×C8C2×C22⋊D5311yescharacteristic
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—42no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—2121no—
198C4⋊C4—2121no—
199C4⋊C4#1171992211yescharacteristic
200C42—21no—
201C42—42no—
202C42—41no—
203C42C2×C22⋊D5311yescharacteristic
204C16—2121no—
205C16—2121no—
206Q8—21no—
207Q8—41no—
208Q8—42no—
209Q8—42no—
210Q8—41no—
211Q8—4241no—
212Q8—4241no—
213Q8#1313316012yes—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—41no—
217C2×C4—42no—
218C2×C4—41no—
219C2×C4—4241no—
220C2×C4#1313318611yescharacteristic
221C2×C4C2×#1171992211yescharacteristic
222C2×C4#1313324311yescharacteristic
223C8—21no—
224C8—41no—
225C8—4241no—
226C8#1313323711yescharacteristic
227C8#1313324711yescharacteristic
228C22#1681075711yescharacteristic
229C4—22no—
230C4—21no—
231C4—41no—
232C4—81no—
233C4—81no—
234C4—82no—
235C4—8481no—
236C4#1681078511yescharacteristic
237C4#1681080111yescharacteristic
238C4C2×#1313316011yescharacteristic
239C4#1681106411yescharacteristic
240C2#2027390811yescharacteristic
241C2#2027663211yescharacteristic
242C2#2027669311yescharacteristic
subgroupG/H
centerC22#16810757
commutator, cosocleC4×C8×C53C23
FrattiniC4×C8C22×D53
FittingC53×#3395C2
socle, prime radicalC22×C53#447
hypercenter#3395D53
hypocenterC53#12382
LCSG ▷ C4×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #3395

Complements

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

NH ≅ G/Ntype
146. C53#12382semidirect

Maximal subgroups

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

subgroupindex
#202769422
#202768572
#202741092
#202727642
Q8⋊#117199082
#202726792
C53×#33952
#1238253