Finite Group Databasebeta

← #28301593

Subgroups — #28301593

Order 13568.

542 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 138 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×#13133277C211yesmaximal, characteristic
2#20289954C211yesmaximal, characteristic
3C8×#11719940C212yesmaximal, complemented
4#20289560C212yesmaximal, complemented
5C2×#16810686C211yesmaximal, characteristic
6#16811640C2212yes—
7C2×#13133277C412yes—
8C4×#11719940C2212yes—
9#16811503C412yes—
10#16811467C412yes—
11#16811466C412yes—
12#88324⋊C8—22no—
13C8×#88324—22no—
14C23×#88322C411yescharacteristic
15C22×C8×D53C411yescharacteristic
16C2×C4×C22⋊D53C2211yescharacteristic
17#16810686—22no—
18#16810686—22no—
19C2×#13133146C2211yescharacteristic
20C2×#13133129C411yescharacteristic
21C2×C8×Dic53C2211yescharacteristic
22C2×C22⋊C8×C53C411yescharacteristic
23C2×#11719940C2×C414yes—
24#13133284—22no—
25#13133277C818yescomplemented
26C22⋊#88324C2×C414yes—
27#13133272—22no—
28C4×#88324—22no—
29C4×#88324—22no—
30C22×C22⋊D53C2×C411yescharacteristic
31C22×C4×D53C2×C411yescharacteristic
32C22×#88322—21no—
33C22×#88322—21no—
34C22×#88322—42no—
35C22×#88322C2×C411yescharacteristic
36C22×#88322C2×C411yescharacteristic
37C2×C8×D53—21no—
38C2×C8×D53—21no—
39C2×C8×D53—42no—
40C4×C22⋊D53—22no—
41C4×C22⋊D53—22no—
42C2×#11719919C2×C411yescharacteristic
43C2×C22⋊Dic53C2×C411yescharacteristic
44C2×#11719921C2×C411yescharacteristic
45C2×C4×Dic53C2311yescharacteristic
46#13133146—22no—
47#13133129—22no—
48C8×Dic53D412yes—
49C8×Dic53D412yes—
50C23×C4×C53C2×C411yescharacteristic
51C22×C8×C53C2×C411yescharacteristic
52C22×C8×C53C2×C411yescharacteristic
53C22⋊C8×C53—22no—
54C24×C53C4211yescharacteristic
55C23×D53C4211yescharacteristic
56C22×C4×C53—21no—
57C22×C4×C53—21no—
58C22×C4×C53—41no—
59C22×C4×C53—41no—
60C22×C4×C53C4211yescharacteristic
61C22×C4×C53C22×C411yescharacteristic
62C2×C22⋊D53—22no—
63C2×C22⋊D53—22no—
64C2×C22⋊D53C2×C814yes—
65C22×Dic53C4211yescharacteristic
66C22×Dic53C22×C411yescharacteristic
67C22×Dic53C22×C411yescharacteristic
68C2×C4×D53—21no—
69C2×C4×D53—21no—
70C2×C4×D53—41no—
71C2×C4×D53—41no—
72#11719940—24no—
73#11719940C2×C818yescomplemented
74C2×#88324—24no—
75C2×#88324—24no—
76C2×C8×C53—21no—
77C2×C8×C53—41no—
78C2×C8×C53—41no—
79C2×C8×C53C22⋊C412yes—
80C22⋊Dic53—22no—
81#11719921—22no—
82#11719919—22no—
83C4×Dic53C2×D412yes—
84C4×Dic53#4112yes—
85C2×#88322—22no—
86C2×#88322—21no—
87C2×#88322—42no—
88C2×#88322—42no—
89C2×#88322—41no—
90C2×#88322C22⋊C411yescharacteristic
91C2×#88322C22⋊C411yescharacteristic
92C8×D53—42no—
93C8×D53—42no—
94C23×C53—21no—
95C23×C53—21no—
96C23×C53—41no—
97C23×C53—41no—
98C23×C53C4×C812yes—
99C23×C53C2×C4211yescharacteristic
100C22×D53—21no—
101C22×D53—21no—
102C22×D53—41no—
103C22×D53—41no—
104C22×D53C4×C812yes—
105C2×C4×C53—21no—
106C2×C4×C53—21no—
107C2×C4×C53—21no—
108C2×C4×C53—42no—
109C2×C4×C53—41no—
110C2×C4×C53—41no—
111C2×C4×C53—41no—
112C2×C4×C53C2×C22⋊C411yescharacteristic
113C2×C4×C53#11711yescharacteristic
114C22⋊D53—28no—
115C2×Dic53—21no—
116C2×Dic53—41no—
117C2×Dic53—41no—
118C2×Dic53C4×D412yes—
119C2×Dic53C4×D412yes—
120C2×Dic53C22×C812yes—
121C2×Dic53#13112yes—
122C4×D53—42no—
123C4×D53—41no—
124C4×D53—41no—
125#88324—28no—
126#88324—44no—
127C8×C53—22no—
128C8×C53—41no—
129#88322—21no—
130#88322—41no—
131#88322#10012yescomplemented
132C8×#100—531nomaximal, Sylow
133C22×C53—21no—
134C22×C53—21no—
135C22×C53—22no—
136C22×C53—42no—
137C22×C53—42no—
138C22×C53—41no—
139C22×C53—41no—
140C22×C53C4×C22⋊C411yescharacteristic
141C22×C53C2×C4×C811yescharacteristic
142C22×C53#40511yescharacteristic
143C2×D53—22no—
144C2×D53—42no—
145C2×D53—41no—
146C2×D53—41no—
147C2×D53—42no—
148C4×C53—21no—
149C4×C53—41no—
150C4×C53C2×#10011yescharacteristic
151C4×C53#41311yescharacteristic
152Dic53—22no—
153Dic53—41no—
154Dic53C8×D414yes—
155C2×C8×D4—531no—
156C4×#100—531no—
157#1460—531no—
158C8×M16—532no—
159#1283—532no—
160C2×C53—21no—
161C2×C53—21no—
162C2×C53—42no—
163C2×C53C8×C22⋊C411yescharacteristic
164C2×C53C4×#10011yescharacteristic
165C2×C53#175711yescharacteristic
166D53—42no—
167C23×C8—531no—
168C23×C8—531no—
169C2×C4×D4—531no—
170C8×D4—1062no—
171C8×D4—1062no—
172#431—532no—
173C2×C4⋊C8—531no—
174C2×#100—532no—
175C2×C22⋊C8—531no—
176C2×C22⋊C8#615711yescharacteristic
177C4×M16—532no—
178C2×C4×C8—531no—
179#343—532no—
180#323—532no—
181#323—532no—
182#322—1062no—
183C82—1062no—
184C53C8×#10011yescharacteristic, complemented, Sylow
185C22×D4—531no—
186C23×C4—531no—
187C23×C4C2×#615711yescharacteristic
188C2×M16—534no—
189C22×C8—531no—
190C22×C8—531no—
191C22×C8—1061no—
192C22×C8—1061no—
193C22×C8—1061no—
194C22×C8—1061no—
195C22×C8—2122no—
196C22×C8—2122no—
197C22×C8C2×#615711yescharacteristic
198C22×C8C2×#615711yescharacteristic
199C4×D4—1062no—
200C4×D4—1062no—
201C2×C4⋊C4—531no—
202C2×C22⋊C4—531no—
203C2×C22⋊C4—531no—
204C2×C42—531no—
205C4⋊C8—1062no—
206C4⋊C8—1062no—
207#100—538no—
208C22⋊C8—22no—
209C22⋊C8—534no—
210C22⋊C8—1062no—
211#97—1062no—
212C4×C8—532no—
213C4×C8—532no—
214C4×C8—1062no—
215C4×C8—1062no—
216C24—531no—
217C24C4×#615711yescharacteristic
218C2×D4—534no—
219C2×D4—1062no—
220C2×D4—1062no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—531no—
226C22×C4—531no—
227C22×C4—531no—
228C22×C4—1061no—
229C22×C4—1061no—
230C22×C4—2121no—
231C22×C4—2121no—
232C22×C4C4×#615711yescharacteristic
233C22×C4C22×#615711yescharacteristic
234M16—538no—
235M16—1064no—
236C2×C8—21no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—531no—
240C2×C8—531no—
241C2×C8—1062no—
242C2×C8—1064no—
243C2×C8—1064no—
244C2×C8—1061no—
245C2×C8—2122no—
246C2×C8—2122no—
247C2×C8—2122no—
248C2×C8—2122no—
249C2×C8—2121no—
250C2×C8C22⋊#615712yes—
251C4⋊C4—1062no—
252C22⋊C4—1062no—
253C22⋊C4—1062no—
254C42—532no—
255C42—532no—
256C23—21no—
257C23—21no—
258C23—41no—
259C23—41no—
260C23—532no—
261C23—1061no—
262C23—1061no—
263C23—2121no—
264C23—2121no—
265C23C8×#615712yes—
266C23C2×C4×#615711yescharacteristic
267D4—1068no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—532no—
276C2×C4—532no—
277C2×C4—532no—
278C2×C4—532no—
279C2×C4—1061no—
280C2×C4—2122no—
281C2×C4—2121no—
282C2×C4—2121no—
283C2×C4—2121no—
284C2×C4—2121no—
285C2×C4C2×C22⋊#615711yescharacteristic
286C2×C4#1313329611yescharacteristic
287C8—22no—
288C8—41no—
289C8—532no—
290C8—1068no—
291C8—1061no—
292C8—2124no—
293C8—2121no—
294C22—21no—
295C22—21no—
296C22—22no—
297C22—42no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—1062no—
302C22—2122no—
303C22—2122no—
304C22—2121no—
305C22—2121no—
306C22C4×C22⋊#615711yescharacteristic
307C22C2×C8×#615711yescharacteristic
308C22#1681158611yescharacteristic
309C4—21no—
310C4—41no—
311C4—534no—
312C4—1062no—
313C4—2121no—
314C4D53⋊#10011yescharacteristic
315C4#1681161011yescharacteristic
316C2—21no—
317C2—21no—
318C2—42no—
319C2—2122no—
320C2C8×C22⋊#615711yescharacteristic
321C2#2029046511yescharacteristic
322C2#2029047111yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6157
commutatorC22×C53C2×C4×C8
FrattiniC22×C4C22×#6157
FittingC2×C22⋊C8×C53C4
socleC22×C53C4×C22⋊C4
prime radicalC22×C22⋊D53C2×C4
cosocleC2×C4×Dic53C23
hypercenterC2×C22⋊C8#6157
hypocenterC53C8×#100
LCSG ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
3. C8×#11719940C2semidirect
4. #20289560C2semidirect
25. #13133277C8semidirect
73. #11719940C2×C8semidirect
131. #88322#100semidirect
184. C53C8×#100semidirect

Maximal subgroups

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

subgroupindex
C4×#131332772
#202899542
C8×#117199402
#202895602
C2×#168106862
C8×#10053