Finite Group Databasebeta

← #28307928

Subgroups — #28307928

Order 13568.

526 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 104 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#20289946C212yesmaximal
2#20289646C214yesmaximal, complemented
3#20277299C211yesmaximal, characteristic
4#16811639—22no—
5#16811468C2214yes—
6#16811466—22no—
7C23×Dic106C411yescharacteristic
8C2×#13133176C2211yescharacteristic
9#16810529C412yes—
10#16810529C2212yes—
11#16810528—22no—
12#16810484—22no—
13C2×#117×C53C411yescharacteristic
14C4⋊#88324—42no—
15C22⋊#88324—24no—
16C4×#88324—42no—
17C22×Dic106—21no—
18C22×Dic106—22no—
19C22×Dic106—42no—
20C22×Dic106—42no—
21C22×Dic106C814yescomplemented
22C22×Dic106C2×C412yes—
23C23×Dic53—21no—
24#13133176—42no—
25C2×C4⋊Dic53C2×C411yescharacteristic
26C2×C4⋊Dic53C2311yescharacteristic
27C2×C22⋊Dic53D412yes—
28C2×C4×Dic53D412yes—
29#13133124—24no—
30#13133124—24no—
31C23×C4×C53C2×C411yescharacteristic
32#117×C53—42no—
33C2×C4⋊C4×C53—21no—
34C2×C22⋊C4×C53—21no—
35C2×C42×C53C2×C412yes—
36C24×C53C22⋊C411yescharacteristic
37C22×C4×C53—22no—
38C22×C4×C53—22no—
39C22×C4×C53—22no—
40C22×C4×C53C22⋊C411yescharacteristic
41C22×C4×C53C2×C812yes—
42C22×C4×C53M1612yes—
43C22×C4×C53C22×C411yescharacteristic
44C22×Dic53—24no—
45C22×Dic53—22no—
46C22×Dic53—42no—
47C22×Dic53—42no—
48C22×Dic53C22⋊C412yes—
49C22×Dic53C2×D412yes—
50C2×Dic106—24no—
51C2×Dic106—22no—
52C2×Dic106—48no—
53C2×Dic106—42no—
54C2×Dic106—42no—
55C2×Dic106—42no—
56C2×Dic106—42no—
57C2×Dic106—48no—
58C2×Dic106C2×C814yes—
59C2×#88324—44no—
60C4⋊C4×C53—42no—
61C22⋊C4×C53—42no—
62C42×C53—42no—
63C22⋊Dic53—42no—
64C4⋊Dic53—41no—
65C4⋊Dic53—41no—
66C4×Dic53—22no—
67C4×Dic53—22no—
68C23×C53—21no—
69C23×C53—21no—
70C23×C53—41no—
71C23×C53—41no—
72C23×C53C22⋊C812yes—
73C23×C53C2×C22⋊C411yescharacteristic
74C2×C4×C53—21no—
75C2×C4×C53—21no—
76C2×C4×C53—24no—
77C2×C4×C53—24no—
78C2×C4×C53—42no—
79C2×C4×C53—42no—
80C2×C4×C53—42no—
81C2×C4×C53C22⋊C812yes—
82C2×C4×C53#9912yes—
83C2×C4×C53#10012yes—
84C2×C4×C53C22×C811yescharacteristic
85C2×C4×C53C2×M1611yescharacteristic
86C2×Dic53—24no—
87C2×Dic53—22no—
88C2×Dic53—22no—
89C2×Dic53—48no—
90C2×Dic53—42no—
91C2×Dic53—42no—
92C2×Dic53—42no—
93C2×Dic53—42no—
94Dic106—44no—
95Dic106—48no—
96Dic106—44no—
97#88324—84no—
98#11434—531nomaximal, Sylow
99C22×C53—21no—
100C22×C53—21no—
101C22×C53—22no—
102C22×C53—42no—
103C22×C53—42no—
104C22×C53—41no—
105C22×C53—41no—
106C22×C53C2×C22⋊C811yescharacteristic
107C22×C53C2×#9911yescharacteristic
108C22×C53C2×#10011yescharacteristic
109C4×C53—21no—
110C4×C53—21no—
111C4×C53—82no—
112C4×C53#32314yes—
113Dic53—42no—
114Dic53—42no—
115Dic53—44no—
116#2310—531no—
117#1458—532no—
118#1319—534no—
119C2×C53—21no—
120C2×C53—21no—
121C2×C53—42no—
122C2×C53C2×#32311yescharacteristic
123C2×C53#212911yescharacteristic
124C2×C53#213511yescharacteristic
125C23×Q8—531no—
126C2×#117—531no—
127C2×#117#615711yescharacteristic
128#431—1062no—
129#391—532no—
130#391—532no—
131#391—1062no—
132#381—1062no—
133#324—534no—
134#323—1062no—
135C53#1143411yescharacteristic, complemented, Sylow
136C22×Q8—532no—
137C22×Q8—534no—
138C22×Q8—1062no—
139C22×Q8—1061no—
140C22×Q8—2122no—
141C22×Q8—2122no—
142C23×C4—1061no—
143C23×C4C2×#615711yescharacteristic
144#117—42no—
145#117—2122no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—531no—
148C2×C4⋊C4—531no—
149C2×C22⋊C4—21no—
150C2×C22⋊C4—532no—
151C2×C42—532no—
152C2×C42C2×#615712yes—
153C4⋊C8—2122no—
154C22⋊C8—1064no—
155C4×C8—2122no—
156#95—1064no—
157#95—1064no—
158C24C22⋊#615711yescharacteristic
159C2×Q8—534no—
160C2×Q8—1064no—
161C2×Q8—1062no—
162C2×Q8—2128no—
163C2×Q8—2128no—
164C2×Q8—2122no—
165C2×Q8—2122no—
166C2×Q8—2122no—
167C2×Q8—2122no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—532no—
172C22×C4—532no—
173C22×C4—1064no—
174C22×C4—1062no—
175C22×C4—2122no—
176C22×C4—2122no—
177C22×C4D53⋊C812yes—
178C22×C4#1171993612yes—
179C22×C4C22⋊#615711yescharacteristic
180C22×C4C22×#615711yescharacteristic
181C2×C8—2124no—
182C4⋊C4—42no—
183C4⋊C4—2121no—
184C4⋊C4—2121no—
185C22⋊C4—42no—
186C22⋊C4—2122no—
187C42—42no—
188C42—1062no—
189C42—1062no—
190C23—21no—
191C23—21no—
192C23—41no—
193C23—41no—
194C23#1313327412yes—
195C23C2×C22⋊#615711yescharacteristic
196Q8—2124no—
197Q8—2124no—
198Q8—2128no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—24no—
202C2×C4—24no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—1064no—
207C2×C4—1062no—
208C2×C4—1062no—
209C2×C4—2128no—
210C2×C4—2122no—
211C2×C4—2122no—
212C2×C4—2122no—
213C2×C4—2122no—
214C2×C4#1313327412yes—
215C2×C4#1313327612yes—
216C2×C4#1313327712yes—
217C2×C4C2×D53⋊C811yescharacteristic
218C2×C4C2×#1171993611yescharacteristic
219C8—4244no—
220C22—21no—
221C22—21no—
222C22—22no—
223C22—42no—
224C22—42no—
225C22—41no—
226C22—41no—
227C22C2×#1313327411yescharacteristic
228C22C2×#1313327611yescharacteristic
229C22C2×#1313327711yescharacteristic
230C4—21no—
231C4—21no—
232C4—82no—
233C4—2124no—
234C4—2122no—
235C4—2122no—
236C4#1681146714yes—
237C2—21no—
238C2—21no—
239C2—42no—
240C2C2×#1681146711yescharacteristic
241C2#2029099911yescharacteristic
242C2#2029101911yescharacteristic
subgroupG/H
centerC22C2×#13133276
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC2×#117×C53C4
socleC22×C53C2×#99
prime radicalC24×C53C22⋊C4
cosocleC2×C4⋊Dic53C23
hypercenterC2×#117#6157
hypocenterC53#11434
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #20289646C2semidirect
21. C22×Dic106C8semidirect
135. C53#11434semidirect

Maximal subgroups

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

subgroupindex
#202899462
#202896462
#202772992
#1143453