Finite Group Databasebeta

← #28305741

Subgroups — #28305741

Order 13568.

640 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 126 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#20290571C211yesmaximal, characteristic
2#20290441C211yesmaximal, characteristic
3C2×#16811468C214yesmaximal
4C2×#16810522C211yesmaximal, characteristic
5C2×#13133284—21no—
6C2×C22⋊#88324C2212yes—
7C2×C22⋊#88324C2212yes—
8C2×#13133272—21no—
9#16811489—21no—
10#16811489—21no—
11#16811468—28no—
12C22×C4⋊Dic53C411yescharacteristic
13C22×C4⋊Dic53C2212yes—
14C22×C4×Dic53C2211yescharacteristic
15#16810522—21no—
16#16810522—21no—
17#16810522—21no—
18#16810522—21no—
19C2×#13133124C412yes—
20C22×C4⋊C4×C53C411yescharacteristic
21C22×#88324—22no—
22C22×#88324—22no—
23#13133284—42no—
24C22⋊#88324—22no—
25C22⋊#88324—24no—
26C22⋊#88324D414yes—
27#13133272—42no—
28C23×Dic53C2×C411yescharacteristic
29C23×Dic53C2×C412yes—
30C23×Dic53C2311yescharacteristic
31C2×C4⋊Dic53—21no—
32C2×C4⋊Dic53—21no—
33C2×C4⋊Dic53—22no—
34C2×C4⋊Dic53—22no—
35C2×C4⋊Dic53—24no—
36C2×C4⋊Dic53—24no—
37C2×C4⋊Dic53—42no—
38C2×C4×Dic53—42no—
39C2×C4×Dic53D411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41C2×C4×Dic53D411yescharacteristic
42C2×C4×Dic53D411yescharacteristic
43#13133124—22no—
44#13133124—22no—
45C23×C4×C53C2×C411yescharacteristic
46C23×C4×C53C2×C412yes—
47C2×C4⋊C4×C53—21no—
48C2×C4⋊C4×C53—21no—
49C2×C4⋊C4×C53—24no—
50C24×C53C22×C411yescharacteristic
51C22×C4×C53—22no—
52C22×C4×C53—22no—
53C22×C4×C53—22no—
54C22×C4×C53—22no—
55C22×C4×C53—24no—
56C22×C4×C53—22no—
57C22×C4×C53—24no—
58C22×C4×C53C22⋊C411yescharacteristic
59C22×C4×C53C22⋊C411yescharacteristic
60C22×C4×C53M1614yes—
61C22×Dic53—22no—
62C22×Dic53—22no—
63C22×Dic53—22no—
64C22×Dic53—24no—
65C22×Dic53—41no—
66C22×Dic53—41no—
67C22×Dic53—41no—
68C22×Dic53—42no—
69C22×Dic53—42no—
70C22×Dic53C22⋊C411yescharacteristic
71C22×Dic53C22⋊C411yescharacteristic
72C22×Dic53C2×D411yescharacteristic
73C22×Dic53C2×D411yescharacteristic
74C22×Dic53C2×D412yes—
75C22×Dic53#4112yes—
76C2×#88324—24no—
77C2×#88324—42no—
78C2×#88324—42no—
79C2×#88324—42no—
80C2×#88324—42no—
81C2×#88324—42no—
82C4⋊C4×C53—24no—
83C4⋊C4×C53—24no—
84C4⋊Dic53—42no—
85C4⋊Dic53—42no—
86C4⋊Dic53—44no—
87C4⋊Dic53—42no—
88C4⋊Dic53—42no—
89C4×Dic53—22no—
90C4×Dic53—22no—
91C4×Dic53—41no—
92C4×Dic53—41no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53—21no—
96C23×C53—21no—
97C23×C53C2×C22⋊C411yescharacteristic
98C23×C53C4×D412yes—
99C23×C53C2×M1612yes—
100C23×C53#13112yes—
101C2×C4×C53—28no—
102C2×C4×C53—21no—
103C2×C4×C53—21no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—24no—
107C2×C4×C53—28no—
108C2×C4×C53—42no—
109C2×C4×C53—42no—
110C2×C4×C53—44no—
111C2×C4×C53#9912yes—
112C2×C4×C53#10112yes—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—21no—
116C2×Dic53—21no—
117C2×Dic53—21no—
118C2×Dic53—21no—
119C2×Dic53—21no—
120C2×Dic53—21no—
121C2×Dic53—21no—
122C2×Dic53—42no—
123C2×Dic53—42no—
124C2×Dic53—44no—
125C2×Dic53—42no—
126C2×Dic53—42no—
127C2×Dic53—42no—
128C2×Dic53—84no—
129C2×Dic53#10112yes—
130C2×Dic53#12112yes—
131C2×Dic53#12411yescharacteristic
132C2×Dic53#12711yescharacteristic
133#88324—44no—
134#88324—82no—
135#9654—531nomaximal, Sylow
136C22×C53—22no—
137C22×C53—22no—
138C22×C53—22no—
139C22×C53—22no—
140C22×C53—22no—
141C22×C53—22no—
142C22×C53#39011yescharacteristic
143C22×C53#40711yescharacteristic
144C22×C53#40811yescharacteristic
145C22×C53C2×#9911yescharacteristic
146C22×C53#41011yescharacteristic
147C22×C53C2×#10111yescharacteristic
148C22×C53C2×#10111yescharacteristic
149C22×C53#43514yes—
150C4×C53—22no—
151C4×C53—22no—
152C4×C53—44no—
153Dic53—22no—
154Dic53—21no—
155Dic53—21no—
156Dic53—22no—
157Dic53—41no—
158Dic53—82no—
159C2×#389—531no—
160#1797—531no—
161#1749—531no—
162C2×#324—534no—
163C2×C53—24no—
164C2×C53#146311yescharacteristic
165C2×C53#146711yescharacteristic
166C2×C53#147011yescharacteristic
167C2×C53#147211yescharacteristic
168C2×C53#195511yescharacteristic
169C2×C53#196411yescharacteristic
170C2×C53#196611yescharacteristic
171C22×C4⋊C4—532no—
172C22×C4⋊C4—531no—
173C22×C4⋊C4#615711yescharacteristic
174C22×C42—531no—
175C2×C4⋊C8—1061no—
176C2×C22⋊C8—532no—
177C2×C22⋊C8—532no—
178C2×#97—1061no—
179#389—1061no—
180#389—1061no—
181#389—1061no—
182#389—1061no—
183C2×#95—532no—
184#336—1061no—
185#336—1061no—
186#324—1068no—
187C53#965411yescharacteristic, complemented, Sylow
188C23×C4—531no—
189C23×C4—531no—
190C23×C4—532no—
191C23×C4C2×#615712yes—
192C23×C4C2×#615711yescharacteristic
193C22×C8—1062no—
194C22×C8—1062no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—1064no—
199C2×C4⋊C4—1064no—
200C2×C4⋊C4—1062no—
201C2×C4⋊C4—1062no—
202C2×C4⋊C4—1061no—
203C2×C4⋊C4—1061no—
204C2×C4⋊C4—2122no—
205C2×C42—531no—
206C2×C42—531no—
207C2×C42—531no—
208C2×C42—531no—
209C2×C42—2122no—
210C4⋊C8—2122no—
211C22⋊C8—534no—
212C22⋊C8—1062no—
213C22⋊C8—1064no—
214#97—2122no—
215#95—1062no—
216#95—1062no—
217C24C22×#615711yescharacteristic
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—531no—
226C22×C4—531no—
227C22×C4—531no—
228C22×C4—532no—
229C22×C4—532no—
230C22×C4—531no—
231C22×C4—1062no—
232C22×C4—1062no—
233C22×C4—1064no—
234C22×C4—1062no—
235C22×C4—2121no—
236C22×C4—2121no—
237C22×C4—2121no—
238C22×C4—2122no—
239C22×C4—2122no—
240C22×C4#1171994014yes—
241C22×C4C22⋊#615711yescharacteristic
242C22×C4C22⋊#615711yescharacteristic
243C2×C8—1064no—
244C2×C8—2122no—
245C2×C8—2122no—
246C2×C8—2122no—
247C2×C8—2122no—
248C2×C8—2122no—
249C4⋊C4—24no—
250C4⋊C4—24no—
251C4⋊C4—2122no—
252C4⋊C4—2124no—
253C4⋊C4—2122no—
254C4⋊C4—2122no—
255C4⋊C4—2122no—
256C42—1062no—
257C42—1062no—
258C42—2121no—
259C42—2121no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23C2×C22⋊#615711yescharacteristic
265C23D4×#615712yes—
266C23C2×#1171994012yes—
267C23#1313330412yes—
268C2×C4—28no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—24no—
274C2×C4—28no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—44no—
278C2×C4—532no—
279C2×C4—531no—
280C2×C4—531no—
281C2×C4—532no—
282C2×C4—1061no—
283C2×C4—1061no—
284C2×C4—1061no—
285C2×C4—1061no—
286C2×C4—1061no—
287C2×C4—1061no—
288C2×C4—1061no—
289C2×C4—1061no—
290C2×C4—1061no—
291C2×C4—4244no—
292C2×C4—2122no—
293C2×C4—2122no—
294C2×C4—2124no—
295C2×C4—2122no—
296C2×C4—2122no—
297C2×C4—2122no—
298C2×C4#1313327612yes—
299C2×C4#1313327812yes—
300C8—4242no—
301C8—2124no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22#1681157511yescharacteristic
309C22#1681159311yescharacteristic
310C22#1681159711yescharacteristic
311C22C2×#1313327611yescharacteristic
312C22#1681160411yescharacteristic
313C22C2×#1313327811yescharacteristic
314C22D53⋊#10111yescharacteristic
315C22#1681165414yes—
316C4—22no—
317C4—22no—
318C4—44no—
319C4—1061no—
320C4—1062no—
321C4—1061no—
322C4—1062no—
323C4—4242no—
324C4—2121no—
325C2—24no—
326C2#2028996311yescharacteristic
327C2#2028997811yescharacteristic
328C2#2028998811yescharacteristic
329C2#2028999311yescharacteristic
330C2#2029083911yescharacteristic
331C2#2029085511yescharacteristic
332C2#2029085811yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutator, prime radicalC24×C53C22×C4
FrattiniC24C22×#6157
FittingC22×C4⋊C4×C53C4
socleC23×C53C2×C22⋊C4
cosocleC23×Dic53C23
hypercenterC22×C4⋊C4#6157
hypocenterC53#9654
LCSG ▷ C24×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
187. C53#9654semidirect

Maximal subgroups

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

subgroupindex
#202905712
#202904412
C2×#168114682
C2×#168105222
#965453