Finite Group Databasebeta

← #28299449

Subgroups — #28299449

Order 13568.

516 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 146 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
1C4×#13133284C212yesmaximal
2#20289536C212yesmaximal
3#20289535C212yesmaximal
4C4×#13133215C211yesmaximal, characteristic
5C2×#13133284—22no—
6C2×C4×#88324—22no—
7#16811489C2214yes—
8C2×#13133215C411yescharacteristic
9C2×C4×Dic106C411yescharacteristic
10#16810530C2211yescharacteristic
11#16810528C2211yescharacteristic
12#16810518—22no—
13#16810492C411yescharacteristic
14C4×#11719919—22no—
15C42×Dic53C2211yescharacteristic
16C2×C4×Q8×C53C411yescharacteristic
17C22×#88324—24no—
18#13133284—28no—
19C4×#88324—44no—
20C22×Dic106C2×C411yescharacteristic
21#13133215—22no—
22#13133215C814yescomplemented
23C4×Dic106—42no—
24C2×C4⋊Dic53C2×C411yescharacteristic
25C2×#11719919—21no—
26C2×#11719919—22no—
27C2×#11719919—22no—
28C2×C4×Dic53—21no—
29C2×C4×Dic53C2×C411yescharacteristic
30C2×C4×Dic53D412yes—
31C2×C4×Dic53D412yes—
32C2×C4×Dic53C2311yescharacteristic
33#13133124—22no—
34#13133124—22no—
35C22×Q8×C53C2×C411yescharacteristic
36C4×Q8×C53—42no—
37C2×C4⋊C4×C53—21no—
38C2×C4⋊C4×C53C2×C411yescharacteristic
39C2×C42×C53—21no—
40C2×C42×C53C2×C411yescharacteristic
41C2×Q8×C53—21no—
42C2×Q8×C53—41no—
43C2×Q8×C53—41no—
44C2×Q8×C53C2×C812yes—
45C22×C4×C53—21no—
46C22×C4×C53—21no—
47C22×C4×C53C22⋊C411yescharacteristic
48C22×C4×C53C22⋊C411yescharacteristic
49C22×C4×C53C22×C411yescharacteristic
50C22×Dic53—21no—
51C22×Dic53—21no—
52C22×Dic53C22⋊C412yes—
53C22×Dic53C2×D412yes—
54C2×Dic106—21no—
55C2×Dic106—41no—
56C2×Dic106—41no—
57C2×Dic106C2×C812yes—
58C2×#88324—28no—
59C2×#88324—44no—
60C2×#88324—44no—
61C4⋊C4×C53—42no—
62C4⋊C4×C53—41no—
63C42×C53—22no—
64C42×C53—42no—
65C4⋊Dic53—41no—
66#11719919—28no—
67#11719919—42no—
68#11719919—44no—
69C4×Dic53—24no—
70C4×Dic53—41no—
71C4×Dic53—42no—
72C4×Dic53C2×C812yes—
73C4×Dic53M1612yes—
74C4×Dic53SD1614yes—
75C4×Dic53Q1614yes—
76C23×C53C2×C22⋊C411yescharacteristic
77Q8×C53—42no—
78Q8×C53—42no—
79C2×C4×C53—21no—
80C2×C4×C53—21no—
81C2×C4×C53—22no—
82C2×C4×C53—21no—
83C2×C4×C53—22no—
84C2×C4×C53—41no—
85C2×C4×C53—41no—
86C2×C4×C53—41no—
87C2×C4×C53—42no—
88C2×C4×C53#10012yes—
89C2×C4×C53#10314yes—
90C2×C4×C53C22×C811yescharacteristic
91C2×C4×C53C2×M1611yescharacteristic
92C2×Dic53—22no—
93C2×Dic53—22no—
94C2×Dic53—22no—
95C2×Dic53—41no—
96C2×Dic53—41no—
97C2×Dic53—41no—
98C2×Dic53—42no—
99C2×Dic53C22⋊C814yes—
100C2×Dic53#10314yes—
101C2×Dic53#10414yes—
102C2×Dic53C2×SD1612yes—
103C2×Dic53C2×Q1612yes—
104Dic106—42no—
105Dic106—42no—
106#88324—48no—
107#7571—531nomaximal, Sylow
108C22×C53C2×C22⋊C811yescharacteristic
109C22×C53#41011yescharacteristic
110C22×C53C2×#10011yescharacteristic
111C22×C53C2×#10312yes—
112C22×C53C2×#10412yes—
113C4×C53—21no—
114C4×C53—21no—
115C4×C53—42no—
116C4×C53—41no—
117C4×C53—81no—
118C4×C53#33314yes—
119Dic53—22no—
120Dic53—22no—
121Dic53—42no—
122Dic53—81no—
123Dic53Q8⋊C818yes—
124C4×C4⋊Q8—531no—
125C4×C4⋊C8—532no—
126#1272—532no—
127#1272—532no—
128C2×C53#146011yescharacteristic
129C2×C53C2×Q8⋊C812yes—
130C2×C53#150312yes—
131C2×C53#152511yescharacteristic
132C2×C53C2×#33311yescharacteristic
133C2×C4⋊Q8—531no—
134C2×C4×Q8—531no—
135C2×C4×Q8#615711yescharacteristic
136C2×C4⋊C8—1062no—
137C2×C4×C8—1062no—
138#391—531no—
139#391—531no—
140#389—1062no—
141#384—531no—
142C4×C4⋊C4—1062no—
143C43—531no—
144#336—534no—
145C53#757111yescharacteristic, complemented, Sylow
146C22×Q8—531no—
147C22×Q8C2×#615711yescharacteristic
148C22×C8—1064no—
149C4⋊Q8—534no—
150C4⋊Q8—1062no—
151C4×Q8—42no—
152C4×Q8—2122no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—531no—
155C2×C4⋊C4—1062no—
156C2×C4⋊C4—1062no—
157C2×C4⋊C4—1061no—
158C2×C4⋊C4C2×#615711yescharacteristic
159C2×C42—21no—
160C2×C42—532no—
161C2×C42—531no—
162C2×C42—532no—
163C2×C42—531no—
164C2×C42—1061no—
165C2×C42C2×#615711yescharacteristic
166C4⋊C8—1068no—
167C4×C8—2124no—
168#95—1062no—
169#95—1062no—
170C2×Q8—21no—
171C2×Q8—41no—
172C2×Q8—41no—
173C2×Q8—532no—
174C2×Q8—1061no—
175C2×Q8—2121no—
176C2×Q8—2121no—
177C2×Q8D53⋊C812yes—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—532no—
181C22×C4—532no—
182C22×C4—1061no—
183C22×C4—1061no—
184C22×C4C22⋊#615711yescharacteristic
185C22×C4C22⋊#615711yescharacteristic
186C22×C4C22×#615711yescharacteristic
187C2×C8—1068no—
188C2×C8—2124no—
189C2×C8—2124no—
190C4⋊C4—41no—
191C4⋊C4—42no—
192C4⋊C4—1068no—
193C4⋊C4—2121no—
194C4⋊C4—2122no—
195C4⋊C4—2124no—
196C42—22no—
197C42—42no—
198C42—534no—
199C42—534no—
200C42—532no—
201C42—532no—
202C42—1064no—
203C42—2121no—
204C42—2122no—
205C23C2×C22⋊#615711yescharacteristic
206Q8—42no—
207Q8—42no—
208Q8—2122no—
209Q8—2122no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—21no—
214C2×C4—22no—
215C2×C4—41no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—534no—
220C2×C4—534no—
221C2×C4—534no—
222C2×C4—532no—
223C2×C4—532no—
224C2×C4—1062no—
225C2×C4—1062no—
226C2×C4—1062no—
227C2×C4—2121no—
228C2×C4—2122no—
229C2×C4—2121no—
230C2×C4—2121no—
231C2×C4#1313327712yes—
232C2×C4#1313328014yes—
233C2×C4C2×D53⋊C811yescharacteristic
234C2×C4D53⋊M1611yescharacteristic
235C8—2128no—
236C22D53⋊(C22⋊C8)11yescharacteristic
237C22#1681160211yescharacteristic
238C22C2×#1313327711yescharacteristic
239C22C2×#1313328012yes—
240C22D53⋊#10412yes—
241C4—21no—
242C4—21no—
243C4—41no—
244C4—42no—
245C4—81no—
246C4—538no—
247C4—1062no—
248C4—1062no—
249C4—4241no—
250C4—2122no—
251C4#1681148314yes—
252C2#2028995411yescharacteristic
253C2D53⋊(Q8⋊C8)12yes—
254C2#2029009712yes—
255C2#2029016911yescharacteristic
256C2C2×#1681148311yescharacteristic
subgroupG/H
centerC22×C4C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC2×C4×Q8×C53C4
socle, prime radicalC23×C53C2×C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC2×C4×Q8#6157
hypocenterC53#7571
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
22. #13133215C8semidirect
145. C53#7571semidirect

Maximal subgroups

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

subgroupindex
C4×#131332842
#202895362
#202895352
C4×#131332152
#757153