Finite Group Databasebeta

← #28308010

Subgroups — #28308010

Order 13568.

374 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 94 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#20291077C212yesmaximal, complemented
2#20289651C212yesmaximal, complemented
3#20289636C212yesmaximal
4C2×#16810675C211yesmaximal, characteristic
5#16811544—22no—
6C22⋊#11719910C2214yes—
7#16811512—22no—
8C2×C42×D53C411yescharacteristic
9#16810675—22no—
10#16810675—22no—
11C2×C4⋊#88322C2211yescharacteristic
12C2×#13133129C411yescharacteristic
13C2×#13133129C2211yescharacteristic
14C2×C8×Dic53C411yescharacteristic
15C2×C8×Dic53C2211yescharacteristic
16C2×C4⋊C8×C53C411yescharacteristic
17C2×#11719910—24no—
18C22×C4×D53C811yescharacteristic
19C22×C4×D53C811yescharacteristic
20C22×C4×D53C2×C411yescharacteristic
21C22×#88322C2×C411yescharacteristic
22C22×#88322C2311yescharacteristic
23C2×C4×Dic53C811yescharacteristic
24C2×C4×Dic53C811yescharacteristic
25C2×C4×Dic53C2×C411yescharacteristic
26C42×D53—21no—
27C42×D53—21no—
28C42×D53—21no—
29C42×D53—21no—
30C4⋊#88322—22no—
31#13133129—22no—
32#13133129—22no—
33C8×Dic53—21no—
34C8×Dic53—21no—
35C8×Dic53D412yes—
36C8×Dic53D412yes—
37C22×C8×C53C2×C411yescharacteristic
38C22×C8×C53C2×C411yescharacteristic
39C2×C42×C53C2×C411yescharacteristic
40C4⋊C8×C53—22no—
41C23×D53C2×C811yescharacteristic
42C22×C4×C53C2×C811yescharacteristic
43C22×C4×C53C2×C811yescharacteristic
44C22×C4×C53C22×C411yescharacteristic
45C22×Dic53C2×C811yescharacteristic
46C22×Dic53C2×C811yescharacteristic
47C22×Dic53C2×C811yescharacteristic
48C2×C4×D53—22no—
49C2×C4×D53—22no—
50C2×C4×D53—21no—
51C2×C4×D53—21no—
52C2×C4×D53—21no—
53C2×C4×D53—21no—
54C2×C4×D53—21no—
55C2×C4×D53—21no—
56C2×C4×D53—42no—
57C2×C4×D53—41no—
58C2×C4×D53—41no—
59C2×C8×C53—21no—
60C2×C8×C53—41no—
61C2×C8×C53—41no—
62C2×C8×C53C22⋊C412yes—
63C42×C53—21no—
64C42×C53—21no—
65C4×Dic53—21no—
66C4×Dic53—21no—
67C4×Dic53—21no—
68C4×Dic53—21no—
69C4×Dic53C22⋊C411yescharacteristic
70C4×Dic53C22⋊C411yescharacteristic
71C4×Dic53M1611yescharacteristic
72C4×Dic53M1611yescharacteristic
73C2×#88322—21no—
74C2×#88322—41no—
75C2×#88322—41no—
76C2×#88322C2×D412yes—
77#11719910—44no—
78C23×C53C22×C811yescharacteristic
79C22×D53—21no—
80C22×D53—21no—
81C22×D53—41no—
82C22×D53—41no—
83C22×D53M3212yes—
84C2×C4×C53—21no—
85C2×C4×C53—21no—
86C2×C4×C53—21no—
87C2×C4×C53—21no—
88C2×C4×C53—21no—
89C2×C4×C53—41no—
90C2×C4×C53M3212yes—
91C2×C4×C53C2×C22⋊C411yescharacteristic
92C2×C4×C53C2×M1611yescharacteristic
93C2×Dic53—21no—
94C2×Dic53—21no—
95C2×Dic53—22no—
96C2×Dic53—21no—
97C2×Dic53—41no—
98C2×Dic53C22⋊C811yescharacteristic
99C2×Dic53C22⋊C811yescharacteristic
100C2×Dic53C22⋊C811yescharacteristic
101C2×Dic53C22⋊C811yescharacteristic
102C4×D53—24no—
103C4×D53—24no—
104C4×D53—42no—
105C4×D53—42no—
106C4×D53—41no—
107C4×D53—41no—
108C4×D53—42no—
109C8×C53—22no—
110C8×C53—41no—
111#88322—22no—
112#88322—41no—
113#11460—531nomaximal, Sylow
114C22×C53—21no—
115C22×C53—21no—
116C22×C53C2×C22⋊C811yescharacteristic
117C22×C53C2×M3211yescharacteristic
118C22×C53C2×M3211yescharacteristic
119C2×D53—22no—
120C2×D53—42no—
121C2×D53—41no—
122C2×D53—41no—
123C2×D53—42no—
124C4×C53—22no—
125C4×C53—21no—
126C4×C53—41no—
127C4×C53#41011yescharacteristic
128C4×C53#41311yescharacteristic
129Dic53—21no—
130Dic53—21no—
131Dic53—21no—
132Dic53—41no—
133Dic53#34912yes—
134C2×#431—531no—
135#2163—532no—
136#1322—532no—
137#1314—532no—
138C2×C53—21no—
139C2×C53—21no—
140C2×C53#145811yescharacteristic
141C2×C53#211211yescharacteristic
142C2×C53C2×#34911yescharacteristic
143D53—42no—
144C22×C42—531no—
145#431—1062no—
146#431—1062no—
147C2×C4⋊C8—531no—
148C2×C4⋊C8#615711yescharacteristic
149C2×C22⋊C8—531no—
150C2×C22⋊C8—531no—
151C2×C4×C8—531no—
152C2×C4×C8—531no—
153C4⋊C16—1062no—
154C22⋊C16—534no—
155#346—1062no—
156C53#1146011yescharacteristic, complemented, Sylow
157C23×C4—531no—
158C23×C4—531no—
159C23×C4—531no—
160C22×C8—531no—
161C22×C8—531no—
162C22×C8C2×#615711yescharacteristic
163C22×C8C2×#615711yescharacteristic
164C2×C42—531no—
165C2×C42—531no—
166C2×C42—531no—
167C2×C42—1061no—
168C2×C42—1061no—
169C2×C42—1061no—
170C2×C42—1061no—
171C2×C42C2×#615711yescharacteristic
172C2×C16—1064no—
173C4⋊C8—22no—
174C4⋊C8—1062no—
175C22⋊C8—1062no—
176C22⋊C8—1062no—
177C4×C8—532no—
178C4×C8—532no—
179C4×C8—1061no—
180C4×C8—1061no—
181C24—531no—
182C22×C4—531no—
183C22×C4—531no—
184C22×C4—531no—
185C22×C4—1061no—
186C22×C4—1061no—
187C22×C4—1061no—
188C22×C4—1061no—
189C22×C4—1062no—
190C22×C4—1062no—
191C22×C4—1061no—
192C22×C4—1061no—
193C22×C4—2121no—
194C22×C4—2121no—
195C22×C4—2122no—
196C22×C4D53⋊C811yescharacteristic
197C22×C4D53⋊C811yescharacteristic
198C22×C4C22×#615711yescharacteristic
199C2×C8—21no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—532no—
203C2×C8—1061no—
204C2×C8—2121no—
205C2×C8—2121no—
206C2×C8C22⋊#615712yes—
207C42—21no—
208C42—21no—
209C42—531no—
210C42—531no—
211C42—531no—
212C42—531no—
213C42—1061no—
214C42—1061no—
215C42—1061no—
216C42—1061no—
217C16—2124no—
218C23—532no—
219C23—1061no—
220C23—1061no—
221C23—2121no—
222C23—2121no—
223C23C2×D53⋊C811yescharacteristic
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—41no—
230C2×C4—531no—
231C2×C4—531no—
232C2×C4—531no—
233C2×C4—531no—
234C2×C4—1064no—
235C2×C4—1064no—
236C2×C4—1062no—
237C2×C4—1061no—
238C2×C4—1061no—
239C2×C4—1061no—
240C2×C4—2122no—
241C2×C4—2122no—
242C2×C4—2122no—
243C2×C4—2121no—
244C2×C4—2121no—
245C2×C4—2121no—
246C2×C4#1313329212yes—
247C2×C4C2×C22⋊#615711yescharacteristic
248C2×C4D53⋊M1611yescharacteristic
249C8—22no—
250C8—41no—
251C8—1062no—
252C8—2121no—
253C22—21no—
254C22—21no—
255C22—1062no—
256C22—2122no—
257C22—2122no—
258C22—2121no—
259C22—2121no—
260C22D53⋊(C22⋊C8)11yescharacteristic
261C22D53⋊M3211yescharacteristic
262C22C2×#1313329211yescharacteristic
263C4—22no—
264C4—21no—
265C4—41no—
266C4—532no—
267C4—1061no—
268C4—1061no—
269C4—1061no—
270C4—2121no—
271C4#1681160411yescharacteristic
272C4#1681161011yescharacteristic
273C2—21no—
274C2—21no—
275C2—2122no—
276C2#2028994911yescharacteristic
277C2#2029094311yescharacteristic
278C2D53⋊#34911yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6157
commutatorC23×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC2×C4⋊C8×C53C4
socleC22×C53C2×C22⋊C8
prime radicalC23×D53C2×C8
cosocleC22×#88322C23
hypercenterC2×C4⋊C8#6157
hypocenterC53#11460
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4⋊C8

Complements

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

NH ≅ G/Ntype
1. #20291077C2semidirect
2. #20289651C2semidirect
156. C53#11460semidirect

Maximal subgroups

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

subgroupindex
#202910772
#202896512
#202896362
C2×#168106752
#1146053