Finite Group Databasebeta

← #27278128

Subgroups — #27278128

Order 13568.

418 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 103 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#20283744C211yesmaximal, characteristic
2#20273498C212yesmaximal, complemented
3#20273036C211yesmaximal, characteristic
4#20272614C212yesmaximal, complemented
5C53×#1955C211yesmaximal, characteristic
6#16810695—22no—
7#16810695—22no—
8#16810680C2212yes—
9#16810676C2212yes—
10#16810587—21no—
11C2×C22⋊#88322—21no—
12#16810375—22no—
13#16810373—22no—
14#16810347C412yes—
15#16810347C412yes—
16C2×C4×D4×C53C2211yescharacteristic
17C2×C4⋊C8×C53C2211yescharacteristic
18C2×C22⋊C8×C53—21no—
19C2×C22⋊C8×C53—21no—
20C2×#97×C53C2211yescharacteristic
21C2×#11719917—21no—
22C2×#11719917—21no—
23C22×#88322—21no—
24C22×#88322—21no—
25C4⋊#88322—22no—
26C4⋊#88322D412yes—
27C4⋊#88322D412yes—
28C22⋊#88322—22no—
29C22⋊#88322—22no—
30C22⋊#88322—22no—
31C22⋊#88322C2×C414yes—
32#13133128—22no—
33C4×#88322—22no—
34C22×D4×C53Q811yescharacteristic
35C23×C4×C53D411yescharacteristic
36C23×C4×C53D411yescharacteristic
37C22×C8×C53—21no—
38C22×C8×C53—21no—
39C4×D4×C53—22no—
40C4×D4×C53—22no—
41C2×C4⋊C4×C53Q811yescharacteristic
42C2×C22⋊C4×C53C2×C411yescharacteristic
43C2×C22⋊C4×C53C2×C411yescharacteristic
44C2×C42×C53C2311yescharacteristic
45C4⋊C8×C53—22no—
46C22⋊C8×C53—42no—
47C22⋊C8×C53—42no—
48#97×C53—22no—
49C24×C53C4⋊C411yescharacteristic
50C24×C53C4⋊C411yescharacteristic
51C2×D4×C53—22no—
52C2×D4×C53—22no—
53C2×D4×C53—41no—
54C22×C4×C53—21no—
55C22×C4×C53—21no—
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—41no—
59C22×C4×C53—41no—
60C22×C4×C53—41no—
61C22×C4×C53—41no—
62C22×C4×C53C4⋊C411yescharacteristic
63C22×C4×C53C4⋊C411yescharacteristic
64C22×C4×C53C22×C411yescharacteristic
65C22×C4×C53C2×D411yescharacteristic
66C22×C4×C53C2×Q811yescharacteristic
67C2×C8×C53—22no—
68C2×C8×C53—41no—
69C2×C8×C53—41no—
70C2×C8×C53—41no—
71C2×C8×C53—41no—
72C2×C8×C53—41no—
73C4⋊C4×C53—22no—
74C22⋊C4×C53—22no—
75C22⋊C4×C53—22no—
76C42×C53C2×D412yes—
77C42×C53#4112yes—
78#11719917—81no—
79#11719917—81no—
80C2×#88322—24no—
81C2×#88322—22no—
82C2×#88322—22no—
83C2×#88322—81no—
84C2×#88322—81no—
85C23×C53—21no—
86C23×C53—21no—
87C23×C53—21no—
88C23×C53—21no—
89C23×C53—21no—
90C23×C53—41no—
91C23×C53—41no—
92C23×C53—41no—
93C23×C53—41no—
94C23×C53#10812yes—
95C23×C53C2×C4⋊C411yescharacteristic
96D4×C53—44no—
97C2×C4×C53—21no—
98C2×C4×C53—21no—
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—41no—
104C2×C4×C53—41no—
105C2×C4×C53—41no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—41no—
109C2×C4×C53—41no—
110C2×C4×C53—81no—
111C2×C4×C53—81no—
112C2×C4×C53C4×D412yes—
113C2×C4×C53#12011yescharacteristic
114C2×C4×C53C22⋊Q812yes—
115C2×C4×C53#12311yescharacteristic
116C8×C53—42no—
117C8×C53—81no—
118#88322—44no—
119#88322—42no—
120#88322—42no—
121#8864—531nomaximal, Sylow
122C22×C53—21no—
123C22×C53—21no—
124C22×C53—22no—
125C22×C53—42no—
126C22×C53—42no—
127C22×C53—41no—
128C22×C53—41no—
129C22×C53—41no—
130C22×C53—41no—
131C22×C53—41no—
132C22×C53—81no—
133C22×C53—81no—
134C22×C53#38511yescharacteristic
135C22×C53C2×#10811yescharacteristic
136C22×C53#43011yescharacteristic
137C4×C53—22no—
138C4×C53—21no—
139C4×C53—22no—
140C4×C53—41no—
141C4×C53—41no—
142C4×C53#45311yescharacteristic
143C4×C53#45511yescharacteristic
144#2973—531no—
145#1955D5311yescharacteristic
146#1573—532no—
147#1460—531no—
148#1302—532no—
149C2×C53—21no—
150C2×C53—21no—
151C2×C53—42no—
152C2×C53—81no—
153C2×C53#185511yescharacteristic
154C2×C53#185611yescharacteristic
155C2×C53#186811yescharacteristic
156C2×C4×D4C2×D5311yescharacteristic
157#435—1062no—
158#435—1062no—
159#432—532no—
160#431—532no—
161C2×C4⋊C8C2×D5311yescharacteristic
162#407—1061no—
163C2×C22⋊C8—21no—
164C2×C22⋊C8—21no—
165C2×C22⋊C8—1061no—
166C2×#97C2×D5311yescharacteristic
167#335—1062no—
168#334—1062no—
169#323—532no—
170#323—532no—
171C53#886411yescharacteristic, complemented, Sylow
172C22×D4Dic10611yescharacteristic
173C23×C4C22⋊D5311yescharacteristic
174C23×C4C22⋊D5311yescharacteristic
175C2×M16—1061no—
176C2×M16—1061no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—1061no—
180C22×C8—1061no—
181C4×D4—22no—
182C4×D4—22no—
183C2×C4⋊C4Dic10611yescharacteristic
184C2×C22⋊C4C4×D5311yescharacteristic
185C2×C22⋊C4C4×D5311yescharacteristic
186C2×C42C22×D5311yescharacteristic
187C4⋊C8—22no—
188C4⋊C8—532no—
189C4⋊C8—532no—
190C4⋊C8—1062no—
191C22⋊C8—42no—
192C22⋊C8—42no—
193C22⋊C8—534no—
194C22⋊C8—1062no—
195C22⋊C8—1062no—
196C22⋊C8—1062no—
197#97—22no—
198#97—1062no—
199C4×C8—1062no—
200C24#1171991911yescharacteristic
201C24#1171991911yescharacteristic
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—41no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4#1171991911yescharacteristic
214C22×C4#1171991911yescharacteristic
215C22×C4C2×Dic10611yescharacteristic
216C22×C4C2×C4×D5311yescharacteristic
217C22×C4C2×C22⋊D5311yescharacteristic
218M16—4241no—
219M16—4241no—
220C2×C8—22no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—1064no—
227C2×C8—1062no—
228C2×C8—1062no—
229C2×C8—4241no—
230C2×C8—4241no—
231C4⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C42D4×D5312yes—
235C42#1171994712yes—
236C23—21no—
237C23—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23#1313315212yes—
246C23C2×#1171991911yescharacteristic
247D4—44no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—81no—
262C2×C4—81no—
263C2×C4#1313317712yes—
264C2×C4#1313318511yescharacteristic
265C2×C4#1313319412yes—
266C2×C4#1313319811yescharacteristic
267C8—42no—
268C8—81no—
269C8—2124no—
270C8—2122no—
271C8—2122no—
272C22—21no—
273C22—21no—
274C22—22no—
275C22—42no—
276C22—42no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—41no—
282C22—81no—
283C22—81no—
284C22#1681049511yescharacteristic
285C22C2×#1313315211yescharacteristic
286C22#1681067111yescharacteristic
287C4—22no—
288C4—22no—
289C4—21no—
290C4—41no—
291C4—41no—
292C4#1681079011yescharacteristic
293C4#1681080011yescharacteristic
294C2—21no—
295C2—21no—
296C2—42no—
297C2—81no—
298C2#2027491911yescharacteristic
299C2#2027492411yescharacteristic
300C2#2027500011yescharacteristic
subgroupG/H
centerC2×C4#13133185
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1955C2
socleC22×C53#385
prime radicalC22×D4×C53Q8
cosocleC2×C42×C53C23
hypercenter#1955D53
hypocenterC53#8864
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1955

Complements

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

NH ≅ G/Ntype
2. #20273498C2semidirect
4. #20272614C2semidirect
171. C53#8864semidirect

Maximal subgroups

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

subgroupindex
#202837442
#202734982
#202730362
#202726142
C53×#19552
#886453