Finite Group Databasebeta

← #27274627

Subgroups — #27274627

Order 13568.

400 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810685C211yesmaximal, characteristic
2#20273027C211yesmaximal, characteristic
3#20273027C211yesmaximal, characteristic
4#20273008C211yesmaximal, characteristic, complemented
5#20272999C211yesmaximal, characteristic, complemented
6#20272560C211yesmaximal, characteristic, complemented
7C53×#1281C211yesmaximal, characteristic, complemented
8C2×#13133174C2211yescharacteristic
9#16810685—21no—
10#16810685—21no—
11#16810685—21no—
12#16810685—21no—
13#16810676C2211yescharacteristic
14#16810676C2211yescharacteristic
15C2×#13133146C411yescharacteristic
16C2×#13133146C411yescharacteristic
17C2×#13133129C411yescharacteristic
18C2×#13133129C411yescharacteristic
19C2×C4×#88322C2211yescharacteristic
20#16810348C412yes—
21#16810348C412yes—
22#16810346C412yes—
23#16810346C412yes—
24#16810345—21no—
25#16810345—21no—
26#16810344—21no—
27C8×#88322—21no—
28C53×#431C2211yescharacteristic
29C53×#431C2211yescharacteristic
30C2×#97×C53C2211yescharacteristic
31C22×C4×D53C2×C411yescharacteristic
32C22×#88322C2×C411yescharacteristic
33C22×#88322C2×C411yescharacteristic
34#13133174—21no—
35#13133174—21no—
36#13133174—21no—
37#13133174—21no—
38C2×#11719919C2×C411yescharacteristic
39C2×#11719919C2×C411yescharacteristic
40C2×#11719921C2×C411yescharacteristic
41C2×#11719921C2×C411yescharacteristic
42C2×C4×Dic53C2×C411yescharacteristic
43C4⋊#88322—21no—
44C4⋊#88322—21no—
45#13133146—21no—
46#13133146—21no—
47#13133146—21no—
48#13133146—21no—
49C22⋊#88322C2×C412yes—
50C22⋊#88322C2×C412yes—
51#13133129—21no—
52#13133129—21no—
53#13133129—21no—
54#13133129—21no—
55C4×#88322—21no—
56C4×#88322—21no—
57C4×#88322D411yescharacteristic
58C4×#88322D411yescharacteristic
59C4×#88322D411yescharacteristic
60C4×#88322D411yescharacteristic
61C22×C8×C53C2×C411yescharacteristic
62C22×C8×C53C2×C411yescharacteristic
63C2×C42×C53C2311yescharacteristic
64C4⋊C8×C53—21no—
65C4⋊C8×C53—21no—
66C22⋊C8×C53C2×C412yes—
67C22⋊C8×C53C2×C412yes—
68#97×C53—21no—
69#97×C53—21no—
70C4×C8×C53—21no—
71C4×C8×C53—21no—
72C23×D53C4211yescharacteristic
73C22×C4×C53C22×C411yescharacteristic
74C22×C4×C53C22×C411yescharacteristic
75C22×C4×C53C22×C411yescharacteristic
76C22×Dic53C4211yescharacteristic
77C22×Dic53C4211yescharacteristic
78C22×Dic53C4211yescharacteristic
79C2×C4×D53—21no—
80C2×C4×D53—21no—
81C2×C4×D53—21no—
82C2×C4×D53—21no—
83C2×C4×D53—21no—
84C2×C4×D53—21no—
85C2×C8×C53—21no—
86C2×C8×C53—21no—
87C2×C8×C53—22no—
88C2×C8×C53—22no—
89C2×C8×C53—21no—
90C2×C8×C53—21no—
91C2×C8×C53—21no—
92C2×C8×C53—21no—
93C42×C53C2×D411yescharacteristic
94C42×C53C2×D411yescharacteristic
95C42×C53#4111yescharacteristic
96C42×C53#4111yescharacteristic
97#11719921—21no—
98#11719921—21no—
99#11719921—21no—
100#11719921—21no—
101#11719919—21no—
102#11719919—21no—
103#11719919—21no—
104#11719919—21no—
105C4×Dic53—21no—
106C4×Dic53—21no—
107C2×#88322—21no—
108C2×#88322—21no—
109C2×#88322—21no—
110C2×#88322—21no—
111C2×#88322—22no—
112C2×#88322—22no—
113C2×#88322C22⋊C411yescharacteristic
114C2×#88322C22⋊C411yescharacteristic
115C2×#88322C22⋊C411yescharacteristic
116C2×#88322C22⋊C411yescharacteristic
117C23×C53C2×C4211yescharacteristic
118C22×D53—21no—
119C22×D53—21no—
120C22×D53—21no—
121C22×D53—41no—
122C22×D53—41no—
123C2×C4×C53—41no—
124C2×C4×C53—41no—
125C2×C4×C53C2×C22⋊C411yescharacteristic
126C2×C4×C53#11711yescharacteristic
127C2×C4×C53C4×D411yescharacteristic
128C2×C4×C53C4×D411yescharacteristic
129C2×C4×C53C4×D411yescharacteristic
130C2×C4×C53C4×D411yescharacteristic
131C2×C4×C53#13111yescharacteristic
132C2×C4×C53#13111yescharacteristic
133C2×C4×C53#13111yescharacteristic
134C2×C4×C53#13111yescharacteristic
135C2×Dic53—21no—
136C2×Dic53—21no—
137C2×Dic53—21no—
138C2×Dic53—21no—
139C2×Dic53—21no—
140C2×Dic53—41no—
141C2×Dic53—41no—
142C4×D53—41no—
143C4×D53—41no—
144C4×D53—41no—
145C4×D53—41no—
146C8×C53—42no—
147C8×C53—42no—
148C8×C53—41no—
149C8×C53—41no—
150#88322—21no—
151#88322—21no—
152#88322—21no—
153#88322—21no—
154#88322—42no—
155#88322—42no—
156#8158—531nomaximal, Sylow
157C22×C53—21no—
158C22×C53—21no—
159C22×C53C4×C22⋊C411yescharacteristic
160C22×C53#40511yescharacteristic
161C22×C53#40511yescharacteristic
162C2×D53—41no—
163C2×D53—41no—
164C2×D53—41no—
165C2×D53—81no—
166C2×D53—81no—
167C4×C53—21no—
168C4×C53—21no—
169C4×C53—21no—
170C4×C53—21no—
171C4×C53#40811yescharacteristic
172C4×C53#40811yescharacteristic
173C4×C53#43311yescharacteristic
174C4×C53#43311yescharacteristic
175Dic53—41no—
176Dic53—41no—
177Dic53—41no—
178C2×#433—531no—
179#1458—531no—
180#1458—531no—
181#1452—531no—
182#1450—531no—
183#1281—531no—
184#1281D5311yescharacteristic, complemented
185C2×C53—21no—
186C2×C53—21no—
187C2×C53#175011yescharacteristic
188C2×C53#175711yescharacteristic
189C2×C53#175711yescharacteristic
190D53—81no—
191C2×#117—531no—
192#433—1061no—
193#433—1061no—
194#433—1061no—
195#433—1061no—
196#431—531no—
197#431—531no—
198#431C2×D5311yescharacteristic
199#431C2×D5311yescharacteristic
200C2×C4⋊C8—531no—
201C2×C4⋊C8—531no—
202C2×C22⋊C8—531no—
203C2×C22⋊C8—531no—
204C2×#97C2×D5311yescharacteristic
205C2×C4×C8—531no—
206#324—532no—
207#324—532no—
208#323—532no—
209#323—532no—
210#322—1061no—
211#322—1061no—
212#322—1061no—
213C82—1061no—
214C53#815811yescharacteristic, complemented, Sylow
215C23×C4—531no—
216C22×C8—531no—
217C22×C8—531no—
218C22×C8C4×D5311yescharacteristic
219C22×C8C4×D5311yescharacteristic
220#117—1061no—
221#117—1061no—
222#117—1061no—
223#117—1061no—
224C2×C4⋊C4—531no—
225C2×C4⋊C4—531no—
226C2×C22⋊C4—531no—
227C2×C22⋊C4—531no—
228C2×C42—531no—
229C2×C42C22×D5311yescharacteristic
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—1061no—
233C4⋊C8—1061no—
234C4⋊C8—1061no—
235C4⋊C8—1061no—
236C4⋊C8—1061no—
237C4⋊C8—1061no—
238C22⋊C8—532no—
239C22⋊C8—532no—
240C22⋊C8—1061no—
241C22⋊C8—1061no—
242C22⋊C8—1061no—
243C22⋊C8—1061no—
244C22⋊C8C4×D5312yes—
245C22⋊C8C4×D5312yes—
246#97—21no—
247#97—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—531no—
251C4×C8—531no—
252C4×C8—531no—
253C4×C8—531no—
254C4×C8—1061no—
255C4×C8—1061no—
256C24—531no—
257C22×C4—531no—
258C22×C4—531no—
259C22×C4—531no—
260C22×C4—1061no—
261C22×C4—1061no—
262C22×C4—1061no—
263C22×C4—1061no—
264C22×C4—1061no—
265C22×C4—1061no—
266C22×C4C2×C4×D5311yescharacteristic
267C22×C4C2×C4×D5311yescharacteristic
268C22×C4C2×C4×D5311yescharacteristic
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—531no—
278C2×C8—531no—
279C2×C8—531no—
280C2×C8—531no—
281C2×C8—1061no—
282C2×C8—1061no—
283C2×C8—1061no—
284C2×C8—1061no—
285C2×C8—1062no—
286C2×C8—1062no—
287C4⋊C4—1061no—
288C4⋊C4—1061no—
289C4⋊C4—1061no—
290C4⋊C4—1061no—
291C22⋊C4—1061no—
292C22⋊C4—1061no—
293C22⋊C4—1061no—
294C22⋊C4—1061no—
295C42—1061no—
296C42—1061no—
297C42D4×D5311yescharacteristic
298C42D4×D5311yescharacteristic
299C42#1171994711yescharacteristic
300C42#1171994911yescharacteristic
301C23—1061no—
302C23—1061no—
303C23—1061no—
304C23—2121no—
305C23—2121no—
306C23C42×D5311yescharacteristic
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—1061no—
310C2×C4—1061no—
311C2×C4—1061no—
312C2×C4—1061no—
313C2×C4—1061no—
314C2×C4—2121no—
315C2×C4—2121no—
316C2×C4—2121no—
317C2×C4—2121no—
318C2×C4—2121no—
319C2×C4—2121no—
320C2×C4C22⋊C4×D5311yescharacteristic
321C2×C4#1313317511yescharacteristic
322C2×C4#1313317711yescharacteristic
323C2×C4#1313317711yescharacteristic
324C2×C4#1313317911yescharacteristic
325C2×C4#1313317911yescharacteristic
326C2×C4#1313322311yescharacteristic
327C2×C4#1313322311yescharacteristic
328C2×C4#1313322511yescharacteristic
329C2×C4#1313322511yescharacteristic
330C8—42no—
331C8—42no—
332C8—41no—
333C8—41no—
334C8—1061no—
335C8—1061no—
336C8—1061no—
337C8—1061no—
338C8—2122no—
339C8—2122no—
340C22—21no—
341C22—21no—
342C22—4241no—
343C22—4241no—
344C22—2121no—
345C22—2121no—
346C22—2121no—
347C22#1681047011yescharacteristic
348C22#1681057711yescharacteristic
349C22#1681058011yescharacteristic
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—2121no—
355C4—2121no—
356C4—2121no—
357C4#1681058811yescharacteristic
358C4#1681058811yescharacteristic
359C4#1681068311yescharacteristic
360C4#1681068311yescharacteristic
361C2—21no—
362C2—21no—
363C2—4241no—
364C2#2027448011yescharacteristic
365C2#2027451411yescharacteristic
366C2#2027451411yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D53
commutatorC23×C53C2×C42
FrattiniC2×C42C22×D53
FittingC53×#1281C2
socleC22×C53C4×C22⋊C4
prime radicalC23×D53C42
cosocleC2×C42×C53C23
hypercenter#1281D53
hypocenterC53#8158
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1281

Complements

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

NH ≅ G/Ntype
4. #20273008C2semidirect
5. #20272999C2semidirect
6. #20272560C2semidirect
7. C53×#1281C2semidirect
184. #1281D53semidirect
214. C53#8158semidirect

Maximal subgroups

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

subgroupindex
C2×#168106852
#202730272
#202730082
#202729992
#202725602
C53×#12812
#815853