Finite Group Databasebeta

← #27302265

Subgroups — #27302265

Order 13568.

466 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810699C211yesmaximal, characteristic
2#20276470C211yesmaximal, characteristic, complemented
3C2×#16810405C211yesmaximal, characteristic, complemented
4C2×#16810404C211yesmaximal, characteristic, complemented
5C2×#16810404C211yesmaximal, characteristic, complemented
6M32×Dic53C211yesmaximal, characteristic, complemented
7C2×C53×#349C211yesmaximal, characteristic, complemented
8C22×#11719912C2211yescharacteristic
9C2×C4×C22⋊D53C411yescharacteristic
10C2×#13133157C2211yescharacteristic
11C2×#13133157C2211yescharacteristic
12#16810699—21no—
13#16810699—21no—
14#16810699—21no—
15#16810699—21no—
16C2×#13133146C411yescharacteristic
17C2×C22⋊#88322C411yescharacteristic
18C2×#13133129C411yescharacteristic
19C2×C8×Dic53C2211yescharacteristic
20#16810441—21no—
21#16810441—21no—
22#16810405—22no—
23#16810404—22no—
24#16810404—22no—
25#16810398—21no—
26C16×Dic53—21no—
27C22×M16×C53C2211yescharacteristic
28C2×M32×C53C2211yescharacteristic
29C2×M32×C53C2211yescharacteristic
30C53×#349—22no—
31C22×C22⋊D53C811yescharacteristic
32C22×C4×D53C2×C411yescharacteristic
33C2×#11719912—21no—
34C2×#11719912—21no—
35C2×#11719912—21no—
36C2×#11719912—21no—
37C2×#11719912—21no—
38C2×#11719912—21no—
39C22×#88322C2×C411yescharacteristic
40C22×#88322C2×C411yescharacteristic
41C4×C22⋊D53—21no—
42C4×C22⋊D53—21no—
43C4×C22⋊D53—21no—
44C4×C22⋊D53—21no—
45C2×#11719919C811yescharacteristic
46C2×C22⋊Dic53C811yescharacteristic
47C2×#11719921C811yescharacteristic
48C2×C4×Dic53C2×C411yescharacteristic
49#13133157—21no—
50#13133157—21no—
51#13133157—21no—
52#13133157D412yescomplemented
53C2×#11719908—21no—
54C2×#11719908—21no—
55#13133146—21no—
56#13133146—21no—
57C22⋊#88322—21no—
58C22⋊#88322—21no—
59#13133129—21no—
60#13133129—21no—
61C8×Dic53D411yescharacteristic
62C8×Dic53D411yescharacteristic
63C8×Dic53D411yescharacteristic
64C8×Dic53D411yescharacteristic
65C23×C4×C53C2×C411yescharacteristic
66C2×M16×C53—21no—
67C2×M16×C53—21no—
68C2×M16×C53—21no—
69C2×M16×C53—21no—
70C2×M16×C53—21no—
71C2×M16×C53—21no—
72C22×C8×C53C2×C411yescharacteristic
73C22×C8×C53C2311yescharacteristic
74M32×C53—21no—
75M32×C53—21no—
76M32×C53—21no—
77M32×C53D412yescomplemented
78C2×C16×C53—21no—
79C2×C16×C53—21no—
80C24×C53C2×C811yescharacteristic
81C23×D53C2×C811yescharacteristic
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53C2×C811yescharacteristic
89C22×C4×C53C22×C411yescharacteristic
90C2×C22⋊D53—21no—
91C2×C22⋊D53—21no—
92C2×C22⋊D53—42no—
93C22×Dic53C2×C811yescharacteristic
94C22×Dic53C2×C811yescharacteristic
95C22×Dic53C2×C811yescharacteristic
96C2×C4×D53—21no—
97C2×C4×D53—21no—
98C2×C4×D53—21no—
99C2×C4×D53—21no—
100C2×C4×D53—21no—
101C2×C4×D53—21no—
102M16×C53—41no—
103M16×C53—41no—
104M16×C53—41no—
105M16×C53—41no—
106C2×C8×C53—21no—
107C2×C8×C53—21no—
108C2×C8×C53—21no—
109C2×C8×C53C2×D411yescharacteristic
110C2×C8×C53C2×D411yescharacteristic
111C2×C8×C53C2×D411yescharacteristic
112C2×C8×C53C2×D411yescharacteristic
113C2×C8×C53#4111yescharacteristic
114C2×C8×C53#4111yescharacteristic
115C22⋊Dic53—21no—
116C22⋊Dic53—21no—
117#11719921—21no—
118#11719921—21no—
119#11719919—21no—
120#11719919—21no—
121C4×Dic53C22⋊C411yescharacteristic
122C4×Dic53C22⋊C411yescharacteristic
123C4×Dic53M1611yescharacteristic
124C4×Dic53M1611yescharacteristic
125C2×#88322—21no—
126C2×#88322—21no—
127C2×#88322—21no—
128C2×#88322—21no—
129C2×#88322—21no—
130C2×#88322C22⋊C411yescharacteristic
131C2×#88322C22⋊C411yescharacteristic
132#11719912—41no—
133#11719912—41no—
134#11719912—41no—
135#11719912—41no—
136C16×C53—22no—
137C16×C53—41no—
138#11719908—22no—
139#11719908—41no—
140C23×C53—21no—
141C23×C53—21no—
142C23×C53—21no—
143C23×C53—41no—
144C23×C53—41no—
145C23×C53C22×C811yescharacteristic
146C22×D53—21no—
147C22×D53—21no—
148C22×D53—21no—
149C22×D53—41no—
150C22×D53—41no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53C2×C22⋊C411yescharacteristic
159C2×C4×C53C4×D411yescharacteristic
160C2×C4×C53C4×D411yescharacteristic
161C2×C4×C53C2×M1611yescharacteristic
162C2×C4×C53#13111yescharacteristic
163C2×C4×C53#13111yescharacteristic
164C22⋊D53—42no—
165C22⋊D53—42no—
166C2×Dic53—22no—
167C2×Dic53—21no—
168C2×Dic53—21no—
169C2×Dic53—21no—
170C2×Dic53—41no—
171C2×Dic53C22⋊C811yescharacteristic
172C2×Dic53C22⋊C811yescharacteristic
173C2×Dic53C22⋊C811yescharacteristic
174C2×Dic53C22⋊C811yescharacteristic
175C4×D53—41no—
176C4×D53—41no—
177C4×D53—41no—
178C4×D53—41no—
179C8×C53—41no—
180C8×C53#12111yescharacteristic
181C8×C53#12111yescharacteristic
182C8×C53#12411yescharacteristic
183C8×C53#12711yescharacteristic
184#88322—21no—
185#88322—21no—
186#88322—41no—
187#13194—531nomaximal, Sylow
188C22×C53—21no—
189C22×C53—21no—
190C22×C53—41no—
191C22×C53—41no—
192C22×C53—41no—
193C22×C53—81no—
194C22×C53—81no—
195C22×C53C2×C22⋊C811yescharacteristic
196C22×C53C8×D411yescharacteristic
197C22×C53C8×D411yescharacteristic
198C2×D53—41no—
199C2×D53—41no—
200C2×D53—41no—
201C2×D53—81no—
202C2×D53—81no—
203C4×C53—41no—
204C4×C53#39011yescharacteristic
205C4×C53#40811yescharacteristic
206C4×C53#43611yescharacteristic
207C4×C53#43611yescharacteristic
208Dic53—21no—
209Dic53—21no—
210Dic53—21no—
211Dic53—41no—
212Dic53#34912yescomplemented
213C2×#436—531no—
214#2150—531no—
215C2×#349—531no—
216C2×#349—531no—
217C2×#349—531no—
218C2×#349D5311yescharacteristic, complemented
219C4×M32—531no—
220C2×C53—21no—
221C2×C53—21no—
222C2×C53—81no—
223C2×C53#195411yescharacteristic
224C2×C53C2×#34911yescharacteristic
225C2×C53#211511yescharacteristic
226D53—81no—
227C22×M16—531no—
228C22×M16C2×D5311yescharacteristic
229C2×C4×D4—531no—
230C2×M32—531no—
231C2×M32—531no—
232C2×M32C2×D5311yescharacteristic
233C2×M32C2×D5311yescharacteristic
234#436—1061no—
235#436—1061no—
236#436—1061no—
237#436—1061no—
238C2×C4⋊C8—531no—
239C2×C22⋊C8—531no—
240C2×C22⋊C8—531no—
241C2×C4×C8—531no—
242C4⋊C16—1061no—
243C4⋊C16—1061no—
244#349—22no—
245#349—1062no—
246#349—1062no—
247#349—1062no—
248#346—1061no—
249C4×C16—1061no—
250C53#1319411yescharacteristic, complemented, Sylow
251C22×D4—531no—
252C23×C4—531no—
253C23×C4C4×D5311yescharacteristic
254C2×M16—21no—
255C2×M16—21no—
256C2×M16—21no—
257C2×M16—21no—
258C2×M16—21no—
259C2×M16—21no—
260C2×M16—1061no—
261C2×M16—1061no—
262C2×M16—1061no—
263C2×M16—1061no—
264C2×M16—1061no—
265C2×M16—1061no—
266C22×C8—531no—
267C22×C8—531no—
268C22×C8C4×D5311yescharacteristic
269C22×C8C22×D5311yescharacteristic
270C4×D4—1061no—
271C4×D4—1061no—
272C4×D4—1061no—
273C4×D4—1061no—
274C2×C4⋊C4—531no—
275C2×C22⋊C4—531no—
276C2×C22⋊C4—531no—
277C2×C42—531no—
278M32—21no—
279M32—21no—
280M32—21no—
281M32—532no—
282M32—1061no—
283M32—1061no—
284M32—1061no—
285M32C22⋊D5312yescomplemented
286C2×C16—21no—
287C2×C16—21no—
288C2×C16—1061no—
289C2×C16—1061no—
290C4⋊C8—1061no—
291C4⋊C8—1061no—
292C22⋊C8—1061no—
293C22⋊C8—1061no—
294C22⋊C8—1061no—
295C22⋊C8—1061no—
296C4×C8—531no—
297C4×C8—531no—
298C4×C8—531no—
299C4×C8—531no—
300C24—531no—
301C24C8×D5311yescharacteristic
302C2×D4—1061no—
303C2×D4—1061no—
304C2×D4—2122no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—531no—
312C22×C4—531no—
313C22×C4—531no—
314C22×C4—1061no—
315C22×C4—1061no—
316C22×C4—1061no—
317C22×C4—1061no—
318C22×C4—1061no—
319C22×C4—1061no—
320C22×C4C8×D5311yescharacteristic
321C22×C4C2×C4×D5311yescharacteristic
322M16—41no—
323M16—41no—
324M16—41no—
325M16—41no—
326M16—2121no—
327M16—2121no—
328M16—2121no—
329M16—2121no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—531no—
334C2×C8—531no—
335C2×C8—1061no—
336C2×C8—1061no—
337C2×C8—1061no—
338C2×C8—1061no—
339C2×C8—1061no—
340C2×C8#1171994511yescharacteristic
341C2×C8D4×D5311yescharacteristic
342C2×C8D4×D5311yescharacteristic
343C2×C8D4×D5311yescharacteristic
344C2×C8#1171994711yescharacteristic
345C2×C8C2×C22⋊D5311yescharacteristic
346C4⋊C4—1061no—
347C4⋊C4—1061no—
348C22⋊C4—1061no—
349C22⋊C4—1061no—
350C22⋊C4—1061no—
351C22⋊C4—1061no—
352C42—531no—
353C42—531no—
354C42—531no—
355C42—531no—
356C16—22no—
357C16—41no—
358C16—1062no—
359C16—2121no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—41no—
364C23—41no—
365C23—1061no—
366C23—1061no—
367C23—1061no—
368C23—2121no—
369C23—2121no—
370C23C2×C8×D5311yescharacteristic
371D4—2122no—
372D4—2122no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—531no—
381C2×C4—531no—
382C2×C4—531no—
383C2×C4—531no—
384C2×C4—1062no—
385C2×C4—1061no—
386C2×C4—1061no—
387C2×C4—1061no—
388C2×C4—2121no—
389C2×C4—2121no—
390C2×C4—2121no—
391C2×C4—2121no—
392C2×C4—2121no—
393C2×C4C22⋊C4×D5311yescharacteristic
394C2×C4#1313317711yescharacteristic
395C2×C4C4×C22⋊D5311yescharacteristic
396C2×C4M16×D5311yescharacteristic
397C2×C4#1313322311yescharacteristic
398C2×C4#1313322511yescharacteristic
399C8—41no—
400C8—1061no—
401C8—1061no—
402C8—2121no—
403C8#1313318911yescharacteristic
404C8#1313319011yescharacteristic
405C8#1313320311yescharacteristic
406C8#1313321311yescharacteristic
407C22—21no—
408C22—21no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—81no—
413C22—81no—
414C22—4241no—
415C22—4241no—
416C22—2121no—
417C22—2121no—
418C22—2121no—
419C22C22⋊C8×D5311yescharacteristic
420C22#1681068611yescharacteristic
421C22C8×C22⋊D5311yescharacteristic
422C4—41no—
423C4—532no—
424C4—1061no—
425C4—1061no—
426C4—1061no—
427C4—2121no—
428C4#1681052311yescharacteristic
429C4#1681058811yescharacteristic
430C4#1681069811yescharacteristic
431C4#1681069911yescharacteristic
432C2—21no—
433C2—21no—
434C2—81no—
435C2—4241no—
436C2#2027550711yescharacteristic
437C2#349×D5311yescharacteristic
438C2#2027633411yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D53
commutatorC23×C53C22×C8
FrattiniC22×C8C22×D53
FittingC2×C53×#349C2
socleC22×C53C2×C22⋊C8
prime radicalC22×C22⋊D53C8
cosocleC22×C8×C53C23
hypercenterC2×#349D53
hypocenterC53#13194
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#349

Complements

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

NH ≅ G/Ntype
2. #20276470C2semidirect
3. C2×#16810405C2semidirect
4. C2×#16810404C2semidirect
5. C2×#16810404C2semidirect
6. M32×Dic53C2semidirect
7. C2×C53×#349C2semidirect
52. #13133157D4semidirect
77. M32×C53D4semidirect
212. Dic53#349semidirect
218. C2×#349D53semidirect
250. C53#13194semidirect
285. M32C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168106992
#202764702
C2×#168104052
C2×#168104042
M32×Dic532
C2×C53×#3492
#1319453