Finite Group Databasebeta

← #28299457

Subgroups — #28299457

Order 13568.

604 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 146 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#20290569C211yesmaximal, characteristic
2C4×C22⋊#88324C211yesmaximal, characteristic
3#20289534C214yesmaximal, complemented
4C2×#16810519C211yesmaximal, characteristic
5C2×C4⋊#88324—21no—
6C2×C22⋊#88324—21no—
7C2×C22⋊#88324—21no—
8C2×C4×#88324—21no—
9#16811489C2212yes—
10#16811489C2212yes—
11C22×C4⋊Dic53C411yescharacteristic
12C22×#11719919—21no—
13C22×C4×Dic53C2211yescharacteristic
14#16810519—21no—
15#16810519—21no—
16#16810519C412yes—
17#16810519C2212yes—
18C2×#13133124—21no—
19C22×C4⋊C4×C53C411yescharacteristic
20C22×#88324—22no—
21C22×#88324—22no—
22C4⋊#88324—42no—
23C22⋊#88324—24no—
24C22⋊#88324—42no—
25C4×#88324—42no—
26C23×Dic53—21no—
27C23×Dic53D411yescharacteristic
28C23×Dic53D411yescharacteristic
29C2×C4⋊Dic53—21no—
30C2×C4⋊Dic53—22no—
31C2×C4⋊Dic53C814yescomplemented
32C2×C4⋊Dic53C2×C412yes—
33C2×#11719919—22no—
34C2×#11719919—22no—
35C2×#11719919—42no—
36C2×#11719919—42no—
37C2×C4×Dic53—42no—
38C2×C4×Dic53C2×C411yescharacteristic
39C2×C4×Dic53D411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41C2×C4×Dic53C2311yescharacteristic
42#13133124—22no—
43#13133124—22no—
44C23×C4×C53—21no—
45C23×C4×C53C2×C411yescharacteristic
46C2×C4⋊C4×C53—21no—
47C2×C4⋊C4×C53—42no—
48C2×C4⋊C4×C53C2×C412yes—
49C24×C53C22⋊C411yescharacteristic
50C22×C4×C53—22no—
51C22×C4×C53—22no—
52C22×C4×C53—22no—
53C22×C4×C53—42no—
54C22×C4×C53—42no—
55C22×C4×C53—42no—
56C22×C4×C53C22⋊C411yescharacteristic
57C22×C4×C53C2×C812yes—
58C22×C4×C53M1612yes—
59C22×C4×C53C22×C411yescharacteristic
60C22×Dic53—24no—
61C22×Dic53—22no—
62C22×Dic53—22no—
63C22×Dic53—22no—
64C22×Dic53—41no—
65C22×Dic53—41no—
66C22×Dic53—41no—
67C22×Dic53—42no—
68C22×Dic53—42no—
69C22×Dic53C22⋊C411yescharacteristic
70C22×Dic53C22⋊C411yescharacteristic
71C22×Dic53D812yes—
72C22×Dic53SD1612yes—
73C22×Dic53C2×D411yescharacteristic
74C22×Dic53C2×D411yescharacteristic
75C2×#88324—24no—
76C2×#88324—42no—
77C2×#88324—42no—
78C2×#88324—42no—
79C2×#88324—42no—
80C2×#88324—42no—
81C4⋊C4×C53—42no—
82C4⋊C4×C53—42no—
83C4⋊Dic53—24no—
84C4⋊Dic53—22no—
85C4⋊Dic53C2×C814yes—
86#11719919—44no—
87#11719919—42no—
88#11719919—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×C53C22⋊C812yes—
98C23×C53#10212yes—
99C23×C53#10412yes—
100C23×C53C2×C22⋊C411yescharacteristic
101C2×C4×C53—21no—
102C2×C4×C53—21no—
103C2×C4×C53—24no—
104C2×C4×C53—24no—
105C2×C4×C53—42no—
106C2×C4×C53—42no—
107C2×C4×C53—42no—
108C2×C4×C53—84no—
109C2×C4×C53C22⋊C812yes—
110C2×C4×C53#9912yes—
111C2×C4×C53#10112yes—
112C2×C4×C53C22×C811yescharacteristic
113C2×C4×C53C2×M1611yescharacteristic
114C2×Dic53—21no—
115C2×Dic53—21no—
116C2×Dic53—21no—
117C2×Dic53—21no—
118C2×Dic53—21no—
119C2×Dic53—21no—
120C2×Dic53—24no—
121C2×Dic53—21no—
122C2×Dic53—21no—
123C2×Dic53—21no—
124C2×Dic53—42no—
125C2×Dic53—42no—
126C2×Dic53—44no—
127C2×Dic53—48no—
128C2×Dic53—42no—
129C2×Dic53—42no—
130C2×Dic53#10212yes—
131C2×Dic53C2×D811yescharacteristic
132C2×Dic53C2×SD1611yescharacteristic
133C2×Dic53#13712yes—
134#88324—44no—
135#88324—82no—
136#7573—531nomaximal, Sylow
137C22×C53—22no—
138C22×C53—22no—
139C22×C53—22no—
140C22×C53—22no—
141C22×C53—22no—
142C22×C53—22no—
143C22×C53D4⋊C814yes—
144C22×C53C2×C22⋊C811yescharacteristic
145C22×C53C2×#9911yescharacteristic
146C22×C53C2×#10111yescharacteristic
147C22×C53C2×#10211yescharacteristic
148C22×C53#41911yescharacteristic
149C22×C53C2×#10411yescharacteristic
150C22×C53#42111yescharacteristic
151C4×C53—21no—
152C4×C53—21no—
153C4×C53—82no—
154C4×C53#32414yes—
155Dic53—22no—
156Dic53—21no—
157Dic53—22no—
158Dic53—21no—
159Dic53—44no—
160Dic53—41no—
161C2×#389—531no—
162#1797—531no—
163C4×C22⋊C8—531no—
164#1271—534no—
165C2×C53—24no—
166C2×C53C2×#32411yescharacteristic
167C2×C53C2×D4⋊C811yescharacteristic
168C2×C53#147911yescharacteristic
169C2×C53#150811yescharacteristic
170C2×C53#151911yescharacteristic
171C2×C53#152811yescharacteristic
172C2×C53#154611yescharacteristic
173C22×C4⋊C4—531no—
174C22×C4⋊C4—1061no—
175C22×C4⋊C4#615711yescharacteristic
176C22×C42—531no—
177C2×C4⋊C8—1061no—
178C2×C22⋊C8—1061no—
179C2×C22⋊C8—1061no—
180C2×C4×C8—1061no—
181#389—532no—
182#389—532no—
183#389—1061no—
184#389—1061no—
185C2×#95—1061no—
186#336—532no—
187#336—532no—
188C53#757311yescharacteristic, complemented, Sylow
189C23×C4—21no—
190C23×C4—531no—
191C23×C4—531no—
192C23×C4—1061no—
193C23×C4C2×#615711yescharacteristic
194C22×C8—1062no—
195C22×C8—1062no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—532no—
199C2×C4⋊C4—534no—
200C2×C4⋊C4—1062no—
201C2×C4⋊C4—1061no—
202C2×C4⋊C4—1062no—
203C2×C4⋊C4—1062no—
204C2×C4⋊C4—2122no—
205C2×C4⋊C4—2122no—
206C2×C4⋊C4C2×#615712yes—
207C2×C42—531no—
208C2×C42—531no—
209C2×C42—531no—
210C2×C42—531no—
211C2×C42—2122no—
212C4⋊C8—2122no—
213C22⋊C8—1064no—
214C22⋊C8—2122no—
215C4×C8—2122no—
216#95—1062no—
217#95—1062no—
218C24C22⋊#615711yescharacteristic
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—531no—
226C22×C4—531no—
227C22×C4—531no—
228C22×C4—532no—
229C22×C4—532no—
230C22×C4—531no—
231C22×C4—1062no—
232C22×C4—1064no—
233C22×C4—1062no—
234C22×C4—1062no—
235C22×C4—2121no—
236C22×C4—2121no—
237C22×C4—2121no—
238C22×C4—2122no—
239C22×C4—2122no—
240C22×C4D53⋊C812yes—
241C22×C4#1171993612yes—
242C22×C4C22⋊#615711yescharacteristic
243C22×C4C22×#615711yescharacteristic
244C2×C8—1064no—
245C2×C8—2122no—
246C2×C8—2122no—
247C2×C8—2122no—
248C2×C8—2122no—
249C2×C8—2122no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—534no—
253C4⋊C4—1064no—
254C4⋊C4—1062no—
255C4⋊C4—2124no—
256C4⋊C4—2122no—
257C4⋊C4—2122no—
258C42—1062no—
259C42—1062no—
260C42—2121no—
261C42—2121no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23#1313327412yes—
267C23#1313327912yes—
268C23#1313328212yes—
269C23C2×C22⋊#615711yescharacteristic
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—24no—
273C2×C4—24no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—84no—
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—1064no—
289C2×C4—1061no—
290C2×C4—1061no—
291C2×C4—1061no—
292C2×C4—2122no—
293C2×C4—2122no—
294C2×C4—2124no—
295C2×C4—2122no—
296C2×C4—2122no—
297C2×C4—2128no—
298C2×C4#1313327412yes—
299C2×C4#1313327512yes—
300C2×C4#1313327812yes—
301C2×C4C2×D53⋊C811yescharacteristic
302C2×C4C2×#1171993611yescharacteristic
303C8—4242no—
304C8—2124no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22#1681147114yes—
312C22C2×#1313327411yescharacteristic
313C22C2×#1313327511yescharacteristic
314C22C2×#1313327811yescharacteristic
315C22C2×#1313327911yescharacteristic
316C22#1681161611yescharacteristic
317C22C2×#1313328211yescharacteristic
318C22#1681162111yescharacteristic
319C4—21no—
320C4—21no—
321C4—82no—
322C4—1062no—
323C4—1061no—
324C4—1062no—
325C4—1061no—
326C4—2121no—
327C4—2124no—
328C4#1681146914yes—
329C2—24no—
330C2C2×#1681146911yescharacteristic
331C2C2×#1681147111yescharacteristic
332C2#2029001911yescharacteristic
333C2#2029011511yescharacteristic
334C2#2029015111yescharacteristic
335C2#2029017911yescharacteristic
336C2#2029022411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC2×C4×C53C22×C8
FrattiniC22×C4C22×#6157
FittingC22×C4⋊C4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC24×C53C22⋊C4
cosocleC2×C4×Dic53C23
hypercenterC22×C4⋊C4#6157
hypocenterC53#7573
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #20289534C2semidirect
31. C2×C4⋊Dic53C8semidirect
188. C53#7573semidirect

Maximal subgroups

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

subgroupindex
#202905692
C4×C22⋊#883242
#202895342
C2×#168105192
#757353