Finite Group Databasebeta

← #28299467

Subgroups — #28299467

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—44no—
36C2×C4×Dic53—42no—
37C2×C4×Dic53C2×C411yescharacteristic
38C2×C4×Dic53D411yescharacteristic
39C2×C4×Dic53D411yescharacteristic
40C2×C4×Dic53C2311yescharacteristic
41#13133124—22no—
42#13133124—22no—
43C23×C4×C53—21no—
44C23×C4×C53C2×C411yescharacteristic
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53—42no—
47C2×C4⋊C4×C53C2×C412yes—
48C24×C53C22⋊C411yescharacteristic
49C22×C4×C53—22no—
50C22×C4×C53—22no—
51C22×C4×C53—22no—
52C22×C4×C53—42no—
53C22×C4×C53—42no—
54C22×C4×C53—42no—
55C22×C4×C53C22⋊C411yescharacteristic
56C22×C4×C53C2×C812yes—
57C22×C4×C53M1612yes—
58C22×C4×C53C22×C411yescharacteristic
59C22×Dic53—24no—
60C22×Dic53—22no—
61C22×Dic53—22no—
62C22×Dic53—22no—
63C22×Dic53—41no—
64C22×Dic53—41no—
65C22×Dic53—41no—
66C22×Dic53—42no—
67C22×Dic53—42no—
68C22×Dic53C22⋊C411yescharacteristic
69C22×Dic53C22⋊C411yescharacteristic
70C22×Dic53SD1612yes—
71C22×Dic53Q1612yes—
72C22×Dic53C2×D411yescharacteristic
73C22×Dic53C2×D411yescharacteristic
74C2×#88324—24no—
75C2×#88324—42no—
76C2×#88324—42no—
77C2×#88324—42no—
78C2×#88324—42no—
79C2×#88324—42no—
80C4⋊C4×C53—42no—
81C4⋊C4×C53—42no—
82C4⋊Dic53—24no—
83C4⋊Dic53—22no—
84C4⋊Dic53C2×C814yes—
85#11719919—44no—
86#11719919—44no—
87C4×Dic53—22no—
88C4×Dic53—22no—
89C4×Dic53—41no—
90C4×Dic53—41no—
91C23×C53—21no—
92C23×C53—21no—
93C23×C53—21no—
94C23×C53—21no—
95C23×C53C22⋊C812yes—
96C23×C53#10312yes—
97C23×C53#10412yes—
98C23×C53C2×C22⋊C411yescharacteristic
99C2×C4×C53—21no—
100C2×C4×C53—21no—
101C2×C4×C53—24no—
102C2×C4×C53—24no—
103C2×C4×C53—42no—
104C2×C4×C53—42no—
105C2×C4×C53—42no—
106C2×C4×C53—84no—
107C2×C4×C53C22⋊C812yes—
108C2×C4×C53#9912yes—
109C2×C4×C53#10112yes—
110C2×C4×C53C22×C811yescharacteristic
111C2×C4×C53C2×M1611yescharacteristic
112C2×Dic53—21no—
113C2×Dic53—21no—
114C2×Dic53—21no—
115C2×Dic53—21no—
116C2×Dic53—21no—
117C2×Dic53—21no—
118C2×Dic53—24no—
119C2×Dic53—21no—
120C2×Dic53—21no—
121C2×Dic53—21no—
122C2×Dic53—42no—
123C2×Dic53—42no—
124C2×Dic53—44no—
125C2×Dic53—48no—
126C2×Dic53—42no—
127C2×Dic53—42no—
128C2×Dic53#10312yes—
129C2×Dic53C2×SD1611yescharacteristic
130C2×Dic53C2×Q1611yescharacteristic
131C2×Dic53AGL(1,ℤ/8)12yes—
132#88324—44no—
133#88324—82no—
134#7576—531nomaximal, Sylow
135C22×C53—22no—
136C22×C53—22no—
137C22×C53—22no—
138C22×C53—22no—
139C22×C53—22no—
140C22×C53—22no—
141C22×C53Q8⋊C814yes—
142C22×C53C2×C22⋊C811yescharacteristic
143C22×C53C2×#9911yescharacteristic
144C22×C53C2×#10111yescharacteristic
145C22×C53C2×#10311yescharacteristic
146C22×C53#41811yescharacteristic
147C22×C53C2×#10411yescharacteristic
148C22×C53#42111yescharacteristic
149C4×C53—21no—
150C4×C53—21no—
151C4×C53—82no—
152C4×C53#32414yes—
153Dic53—22no—
154Dic53—21no—
155Dic53—22no—
156Dic53—21no—
157Dic53—44no—
158Dic53—41no—
159C2×#389—531no—
160#1797—531no—
161C4×C22⋊C8—531no—
162#1271—534no—
163C2×C53—24no—
164C2×C53C2×#32411yescharacteristic
165C2×C53C2×Q8⋊C811yescharacteristic
166C2×C53#147811yescharacteristic
167C2×C53#150711yescharacteristic
168C2×C53#152011yescharacteristic
169C2×C53#152811yescharacteristic
170C2×C53#154911yescharacteristic
171C22×C4⋊C4—531no—
172C22×C4⋊C4—1061no—
173C22×C4⋊C4#615711yescharacteristic
174C22×C42—531no—
175C2×C4⋊C8—1061no—
176C2×C22⋊C8—1061no—
177C2×C22⋊C8—1061no—
178C2×C4×C8—1061no—
179#389—532no—
180#389—532no—
181#389—1061no—
182#389—1061no—
183C2×#95—1061no—
184#336—532no—
185#336—532no—
186C53#757611yescharacteristic, complemented, Sylow
187C23×C4—21no—
188C23×C4—531no—
189C23×C4—531no—
190C23×C4—1061no—
191C23×C4C2×#615711yescharacteristic
192C22×C8—1062no—
193C22×C8—1062no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—532no—
197C2×C4⋊C4—534no—
198C2×C4⋊C4—1062no—
199C2×C4⋊C4—1061no—
200C2×C4⋊C4—1062no—
201C2×C4⋊C4—1062no—
202C2×C4⋊C4—2124no—
203C2×C4⋊C4C2×#615712yes—
204C2×C42—531no—
205C2×C42—531no—
206C2×C42—531no—
207C2×C42—531no—
208C2×C42—2122no—
209C4⋊C8—2122no—
210C22⋊C8—1064no—
211C22⋊C8—2122no—
212C4×C8—2122no—
213#95—1062no—
214#95—1062no—
215C24C22⋊#615711yescharacteristic
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—531no—
223C22×C4—531no—
224C22×C4—531no—
225C22×C4—532no—
226C22×C4—532no—
227C22×C4—531no—
228C22×C4—1062no—
229C22×C4—1064no—
230C22×C4—1062no—
231C22×C4—1062no—
232C22×C4—2121no—
233C22×C4—2121no—
234C22×C4—2121no—
235C22×C4—2122no—
236C22×C4—2122no—
237C22×C4D53⋊C812yes—
238C22×C4#1171993612yes—
239C22×C4C22⋊#615711yescharacteristic
240C22×C4C22×#615711yescharacteristic
241C2×C8—1064no—
242C2×C8—2122no—
243C2×C8—2122no—
244C2×C8—2122no—
245C2×C8—2122no—
246C2×C8—2122no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—534no—
250C4⋊C4—1064no—
251C4⋊C4—1062no—
252C4⋊C4—2124no—
253C4⋊C4—2124no—
254C42—1062no—
255C42—1062no—
256C42—2121no—
257C42—2121no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23#1313327412yes—
263C23#1313328012yes—
264C23#1313328112yes—
265C23C2×C22⋊#615711yescharacteristic
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—84no—
274C2×C4—532no—
275C2×C4—531no—
276C2×C4—531no—
277C2×C4—532no—
278C2×C4—1061no—
279C2×C4—1061no—
280C2×C4—1061no—
281C2×C4—1061no—
282C2×C4—1061no—
283C2×C4—1061no—
284C2×C4—1064no—
285C2×C4—1061no—
286C2×C4—1061no—
287C2×C4—1061no—
288C2×C4—2122no—
289C2×C4—2122no—
290C2×C4—2124no—
291C2×C4—2128no—
292C2×C4—2122no—
293C2×C4—2122no—
294C2×C4#1313327412yes—
295C2×C4#1313327512yes—
296C2×C4#1313327812yes—
297C2×C4C2×D53⋊C811yescharacteristic
298C2×C4C2×#1171993611yescharacteristic
299C8—4242no—
300C8—2124no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—22no—
307C22#1681147314yes—
308C22C2×#1313327411yescharacteristic
309C22C2×#1313327511yescharacteristic
310C22C2×#1313327811yescharacteristic
311C22C2×#1313328011yescharacteristic
312C22#1681161511yescharacteristic
313C22C2×#1313328111yescharacteristic
314C22#1681162211yescharacteristic
315C4—21no—
316C4—21no—
317C4—82no—
318C4—1062no—
319C4—1061no—
320C4—1062no—
321C4—1061no—
322C4—2121no—
323C4—2124no—
324C4#1681146914yes—
325C2—24no—
326C2C2×#1681146911yescharacteristic
327C2C2×#1681147311yescharacteristic
328C2#2029001611yescharacteristic
329C2#2029011111yescharacteristic
330C2#2029015511yescharacteristic
331C2#2029017711yescharacteristic
332C2#2029023011yescharacteristic
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#7576
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
186. C53#7576semidirect

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
#757653