Finite Group Databasebeta

← #28299632

Subgroups — #28299632

Order 13568.

462 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 108 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#20290643C211yesmaximal, characteristic
2#20290500C211yesmaximal, characteristic
3#20289533C214yesmaximal
4C4⋊Q8×Dic53C211yesmaximal, characteristic
5C2×#13133284—21no—
6C2×#13133272—21no—
7#16811489—21no—
8#16811489—21no—
9#16811489C2212yes—
10#16811489C2212yes—
11C2×Q8×Dic53—21no—
12#16810530—21no—
13#16810522—21no—
14#16810522C2212yes—
15#16810493C411yescharacteristic
16C4⋊C4×Dic53—21no—
17C4⋊C4×Dic53C412yes—
18C42×Dic53C2211yescharacteristic
19C2×C4⋊Q8×C53C411yescharacteristic
20C22×#88324—22no—
21C22×#88324—22no—
22#13133284—24no—
23#13133272—42no—
24Q8×Dic53—44no—
25C2×C4⋊Dic53—21no—
26C2×C4⋊Dic53—21no—
27C2×C4⋊Dic53—22no—
28C2×C4⋊Dic53C2×C412yes—
29C2×C4×Dic53—22no—
30C2×C4×Dic53C2×C411yescharacteristic
31C2×C4×Dic53D411yescharacteristic
32C2×C4×Dic53D411yescharacteristic
33C2×C4×Dic53D411yescharacteristic
34C2×C4×Dic53D411yescharacteristic
35C2×C4×Dic53C2311yescharacteristic
36#13133124—22no—
37#13133124—22no—
38C22×Q8×C53—21no—
39C4⋊Q8×C53—24no—
40C2×C4⋊C4×C53—21no—
41C2×C4⋊C4×C53C2×C412yes—
42C2×C42×C53C2×C411yescharacteristic
43C2×Q8×C53—24no—
44C2×Q8×C53—42no—
45C2×Q8×C53—42no—
46C22×C4×C53—22no—
47C22×C4×C53C22⋊C411yescharacteristic
48C22×C4×C53C22⋊C411yescharacteristic
49C22×C4×C53C22×C411yescharacteristic
50C22×Dic53—22no—
51C22×Dic53C22⋊C411yescharacteristic
52C22×Dic53C22⋊C411yescharacteristic
53C22×Dic53C2×D411yescharacteristic
54C22×Dic53C2×D411yescharacteristic
55C2×#88324—24no—
56C2×#88324—42no—
57C2×#88324—42no—
58C2×#88324—42no—
59C2×#88324—42no—
60C2×#88324—42no—
61C4⋊C4×C53—24no—
62C4⋊C4×C53—22no—
63C4⋊C4×C53M1614yes—
64C42×C53—22no—
65C4⋊Dic53—42no—
66C4⋊Dic53—42no—
67C4⋊Dic53—44no—
68C4⋊Dic53—42no—
69C4×Dic53—22no—
70C4×Dic53—22no—
71C4×Dic53—22no—
72C4×Dic53—41no—
73C4×Dic53—41no—
74C4×Dic53—44no—
75C4×Dic53SD1612yes—
76C4×Dic53Q1612yes—
77C23×C53C2×C22⋊C411yescharacteristic
78Q8×C53—48no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—21no—
82C2×C4×C53—21no—
83C2×C4×C53—21no—
84C2×C4×C53—21no—
85C2×C4×C53—24no—
86C2×C4×C53—42no—
87C2×C4×C53—42no—
88C2×C4×C53#10312yes—
89C2×C4×C53#10412yes—
90C2×C4×C53C2×M1612yes—
91C2×Dic53—21no—
92C2×Dic53—21no—
93C2×Dic53—21no—
94C2×Dic53—21no—
95C2×Dic53—21no—
96C2×Dic53—21no—
97C2×Dic53—21no—
98C2×Dic53—21no—
99C2×Dic53—42no—
100C2×Dic53—42no—
101C2×Dic53—42no—
102C2×Dic53#10112yes—
103C2×Dic53#10312yes—
104C2×Dic53C2×SD1611yescharacteristic
105C2×Dic53C2×Q1611yescharacteristic
106C2×Dic53AGL(1,ℤ/8)12yes—
107#88324—44no—
108#88324—82no—
109#7622—531nomaximal, Sylow
110C22×C53#40711yescharacteristic
111C22×C53#41011yescharacteristic
112C22×C53C2×#10111yescharacteristic
113C22×C53C2×#10311yescharacteristic
114C22×C53#41811yescharacteristic
115C22×C53C2×#10411yescharacteristic
116C22×C53#42111yescharacteristic
117C4×C53—22no—
118C4×C53—22no—
119C4×C53—22no—
120C4×C53—44no—
121Dic53—22no—
122Dic53—21no—
123Dic53—21no—
124Dic53—22no—
125Dic53—22no—
126Dic53—82no—
127C4×C4⋊Q8—531no—
128#1839—531no—
129#1767—531no—
130#1271—534no—
131C2×C53#146711yescharacteristic
132C2×C53#148711yescharacteristic
133C2×C53#149011yescharacteristic
134C2×C53#150311yescharacteristic
135C2×C53#151411yescharacteristic
136C2×C53#153111yescharacteristic
137C2×C53#154911yescharacteristic
138C2×C4⋊Q8#615711yescharacteristic
139C2×C4×Q8—1061no—
140C2×C4⋊C8—1061no—
141C2×#97—1061no—
142#391—1061no—
143#389—532no—
144#389—1061no—
145#384—531no—
146C4×C4⋊C4—532no—
147C4×C4⋊C4—1061no—
148C43—531no—
149#336—532no—
150#336—532no—
151#336—1061no—
152#336—1061no—
153C53#762211yescharacteristic, complemented, Sylow
154C22×Q8—21no—
155C22×C8—1062no—
156C22×C8—1062no—
157C4⋊Q8—24no—
158C4×Q8—2124no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—532no—
161C2×C4⋊C4—1062no—
162C2×C4⋊C4—1061no—
163C2×C4⋊C4—1061no—
164C2×C4⋊C4C2×#615712yes—
165C2×C42—531no—
166C2×C42—531no—
167C2×C42—531no—
168C2×C42—531no—
169C2×C42—531no—
170C2×C42—531no—
171C2×C42—1062no—
172C2×C42C2×#615711yescharacteristic
173C4⋊C8—1064no—
174#97—2122no—
175#95—1062no—
176#95—1062no—
177C2×Q8—24no—
178C2×Q8—42no—
179C2×Q8—42no—
180C22×C4—22no—
181C22×C4—531no—
182C22×C4—531no—
183C22×C4—531no—
184C22×C4—531no—
185C22×C4—1062no—
186C22×C4C22⋊#615711yescharacteristic
187C22×C4C22⋊#615711yescharacteristic
188C22×C4C22×#615711yescharacteristic
189C2×C8—1064no—
190C2×C8—2122no—
191C2×C8—2122no—
192C2×C8—2122no—
193C2×C8—2122no—
194C2×C8—2122no—
195C4⋊C4—24no—
196C4⋊C4—22no—
197C4⋊C4—2122no—
198C4⋊C4—2122no—
199C4⋊C4—2122no—
200C4⋊C4—2124no—
201C4⋊C4#1171994014yes—
202C42—22no—
203C42—532no—
204C42—532no—
205C42—1062no—
206C42—1062no—
207C42—1062no—
208C42—2121no—
209C42—2121no—
210C42—2124no—
211C23C2×C22⋊#615711yescharacteristic
212Q8—48no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—24no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—532no—
223C2×C4—532no—
224C2×C4—531no—
225C2×C4—531no—
226C2×C4—532no—
227C2×C4—1061no—
228C2×C4—1061no—
229C2×C4—1061no—
230C2×C4—1061no—
231C2×C4—1061no—
232C2×C4—1061no—
233C2×C4—1061no—
234C2×C4—1061no—
235C2×C4—2122no—
236C2×C4—2122no—
237C2×C4—2122no—
238C2×C4#1313328012yes—
239C2×C4#1313328112yes—
240C2×C4C2×#1171994012yes—
241C8—4242no—
242C8—2124no—
243C22#1681159311yescharacteristic
244C22#1681160411yescharacteristic
245C22D53⋊#10111yescharacteristic
246C22C2×#1313328011yescharacteristic
247C22#1681161511yescharacteristic
248C22C2×#1313328111yescharacteristic
249C22#1681162211yescharacteristic
250C4—22no—
251C4—22no—
252C4—22no—
253C4—44no—
254C4—1061no—
255C4—1061no—
256C4—1062no—
257C4—1062no—
258C4—1062no—
259C4—4242no—
260C2#2028997811yescharacteristic
261C2#2029004311yescharacteristic
262C2#2029005411yescharacteristic
263C2#2029009811yescharacteristic
264C2#2029013611yescharacteristic
265C2#2029018811yescharacteristic
266C2#2029023111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC22×C4×C53C22×C4
FrattiniC22×C4C22×#6157
FittingC2×C4⋊Q8×C53C4
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×C4⋊Q8#6157
hypocenterC53#7622
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
153. C53#7622semidirect

Maximal subgroups

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

subgroupindex
#202906432
#202905002
#202895332
C4⋊Q8×Dic532
#762253