Finite Group Databasebeta

← #28309730

Subgroups — #28309730

Order 13568.

394 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 92 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#20291073C211yesmaximal, characteristic
2#20290272C211yesmaximal, characteristic
3#20289730C214yesmaximal, complemented
4C2×#16811001C211yesmaximal, characteristic
5#16811626—21no—
6#16811544C2212yes—
7C22⋊#11719910—21no—
8#16811509—21no—
9#16811487C2212yes—
10#16811486—21no—
11C22×Dic212C411yescharacteristic
12C22×#11719925—21no—
13C2×#13133215—21no—
14#16811001—21no—
15#16811001—21no—
16#16811001C412yes—
17#16811001C2212yes—
18C2×C8×Dic53C2211yescharacteristic
19C22×Q16×C53C411yescharacteristic
20C2×#11719940—41no—
21C2×#11719910—22no—
22#13133284—41no—
23C4⋊#88324—22no—
24C22×Dic106—21no—
25C2×Dic212—21no—
26C2×Dic212—42no—
27C2×Dic212C2×C412yes—
28C2×#11719925—22no—
29C2×#11719925—22no—
30C2×#11719925—44no—
31C22×#88322D411yescharacteristic
32#13133215—22no—
33#13133215—22no—
34#13133215—41no—
35#13133215—41no—
36C2×#11719919—41no—
37C2×#11719919—41no—
38C2×C4×Dic53D411yescharacteristic
39C8×Dic53C2×C411yescharacteristic
40C8×Dic53D411yescharacteristic
41C8×Dic53D411yescharacteristic
42C8×Dic53C2311yescharacteristic
43C22×Q8×C53—21no—
44C2×Q16×C53—21no—
45C2×Q16×C53—42no—
46C2×Q16×C53C2×C412yes—
47C22×C8×C53C2×C411yescharacteristic
48C2×Q8×C53—22no—
49C2×Q8×C53—42no—
50C2×Q8×C53—42no—
51C2×Q8×C53—41no—
52C22×C4×C53—41no—
53C22×C4×C53C22⋊C411yescharacteristic
54C22×Dic53—41no—
55C22×Dic53D811yescharacteristic
56C22×Dic53SD1611yescharacteristic
57C2×Dic106—22no—
58C2×Dic106—42no—
59C2×Dic106—42no—
60C2×Dic106—41no—
61#11719940—82no—
62C2×#88324—42no—
63Q16×C53—42no—
64Q16×C53—42no—
65C2×C8×C53—22no—
66C2×C8×C53C22⋊C411yescharacteristic
67C2×C8×C53C22×C411yescharacteristic
68#11719925—44no—
69#11719925—44no—
70#11719919—42no—
71#11719919—42no—
72#11719919—42no—
73#11719919—42no—
74C4×Dic53C22⋊C411yescharacteristic
75C4×Dic53D811yescharacteristic
76C4×Dic53SD1611yescharacteristic
77C4×Dic53C2×D411yescharacteristic
78C2×#88322—22no—
79C2×#88322C22⋊C411yescharacteristic
80C2×#88322C2×D411yescharacteristic
81Dic212—42no—
82Dic212—42no—
83#11719910—42no—
84C23×C53#10211yescharacteristic
85Q8×C53—42no—
86Q8×C53—42no—
87Q8×C53—82no—
88C2×C4×C53—21no—
89C2×C4×C53—21no—
90C2×C4×C53—42no—
91C2×C4×C53—82no—
92C2×C4×C53#10211yescharacteristic
93C2×C4×C53C2×C22⋊C411yescharacteristic
94C2×Dic53—22no—
95C2×Dic53—42no—
96C2×Dic53—82no—
97C2×Dic53#10211yescharacteristic
98C2×Dic53#10211yescharacteristic
99C2×Dic53SD3212yes—
100C2×Dic53Q3212yes—
101C2×Dic53C2×D811yescharacteristic
102C2×Dic53C2×SD1611yescharacteristic
103Dic106—42no—
104Dic106—42no—
105Dic106—82no—
106#88324—82no—
107C8×C53—21no—
108C8×C53#10012yes—
109#88322—21no—
110#88322AGL(1,ℤ/8)12yes—
111#11962—531nomaximal, Sylow
112C22×C53—21no—
113C22×C53—21no—
114C22×C53#35812yes—
115C22×C53C2×#10211yescharacteristic
116C4×C53—21no—
117C4×C53—82no—
118C4×C53C2×#10011yescharacteristic
119C4×C53#41811yescharacteristic
120Dic53—21no—
121Dic53—82no—
122Dic53#35812yes—
123Dic53#36212yes—
124Dic53C2×SD3211yescharacteristic
125Dic53C2×Q3211yescharacteristic
126C2×#494—531no—
127#2162—531no—
128#1567—531no—
129#1362—534no—
130C2×C53—22no—
131C2×C53#154511yescharacteristic
132C2×C53C2×#35811yescharacteristic
133C2×C53C2×#36211yescharacteristic
134C22×Q16—531no—
135C22×Q16—1061no—
136C22×Q16#615711yescharacteristic
137C2×C4⋊Q8—1061no—
138#494—532no—
139#494—532no—
140#494—1061no—
141#494—1061no—
142#423—1061no—
143C2×C4×C8—531no—
144C4⋊C16—532no—
145C22⋊C16—1061no—
146C4×C16—1061no—
147#335—532no—
148#335—1061no—
149C53#1196211yescharacteristic, complemented, Sylow
150C22×Q8—21no—
151C22×Q8—1061no—
152C2×Q16—21no—
153C2×Q16—42no—
154C2×Q16—532no—
155C2×Q16—1062no—
156C2×Q16—1062no—
157C2×Q16—1061no—
158C2×Q16—2122no—
159C2×Q16—2124no—
160C2×Q16C2×#615712yes—
161C2×M16—2121no—
162C22×C8—531no—
163C22×C8C2×#615711yescharacteristic
164C4⋊Q8—1062no—
165C4⋊Q8—1062no—
166C4⋊Q8—2121no—
167C4⋊Q8—2121no—
168C2×C4⋊C4—2121no—
169C2×C4⋊C4—2121no—
170C2×C42—531no—
171C2×C16—1062no—
172C4⋊C8—1062no—
173C4⋊C8—2121no—
174C4×C8—531no—
175C4×C8—531no—
176C4×C8—531no—
177C4×C8—531no—
178C2×Q8—22no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—41no—
182C2×Q8—1062no—
183C2×Q8—2122no—
184C2×Q8—2122no—
185C2×Q8—2121no—
186C22×C4—41no—
187C22×C4—531no—
188C22×C4—531no—
189C22×C4—2121no—
190C22×C4C22⋊#615711yescharacteristic
191Q16—42no—
192Q16—42no—
193Q16—2122no—
194Q16—2124no—
195Q16—2124no—
196Q16—2122no—
197M16—4242no—
198C2×C8—22no—
199C2×C8—531no—
200C2×C8—531no—
201C2×C8—1062no—
202C2×C8—2122no—
203C2×C8C22⋊#615711yescharacteristic
204C2×C8C22×#615711yescharacteristic
205C4⋊C4—2122no—
206C4⋊C4—2122no—
207C4⋊C4—2122no—
208C4⋊C4—2122no—
209C42—531no—
210C42—531no—
211C42—531no—
212C42—531no—
213C16—2122no—
214C23#1313327911yescharacteristic
215Q8—42no—
216Q8—42no—
217Q8—82no—
218Q8—4242no—
219Q8—2122no—
220Q8—2122no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—42no—
224C2×C4—82no—
225C2×C4—532no—
226C2×C4—532no—
227C2×C4—531no—
228C2×C4—531no—
229C2×C4—531no—
230C2×C4—531no—
231C2×C4—1062no—
232C2×C4—4242no—
233C2×C4—2122no—
234C2×C4#1313327911yescharacteristic
235C2×C4C2×C22⋊#615711yescharacteristic
236C8—21no—
237C8—532no—
238C8—1061no—
239C8—4242no—
240C8#1313327712yes—
241C22—21no—
242C22—21no—
243C22#1681153612yes—
244C22C2×#1313327911yescharacteristic
245C4—21no—
246C4—82no—
247C4—531no—
248C4—532no—
249C4—531no—
250C4—532no—
251C4—1061no—
252C4—4242no—
253C4C2×#1313327711yescharacteristic
254C4#1681161511yescharacteristic
255C2—22no—
256C2#2029022211yescharacteristic
257C2C2×#1681153611yescharacteristic
258C2D53⋊#36211yescharacteristic
subgroupG/H
centerC22C2×#13133279
commutatorC2×C8×C53C22×C4
FrattiniC2×C8C22×#6157
FittingC22×Q16×C53C4
socleC22×C53C2×#102
prime radicalC23×C53#102
cosocleC8×Dic53C23
hypercenterC22×Q16#6157
hypocenterC53#11962
LCSG ▷ C2×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
3. #20289730C2semidirect
149. C53#11962semidirect

Maximal subgroups

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

subgroupindex
#202910732
#202902722
#202897302
C2×#168110012
#1196253