Finite Group Databasebeta

← #27538918

Subgroups — #27538918

Order 13568.

632 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810859C211yesmaximal, characteristic
2#20276906C214yesmaximal, complemented
3#20276900C212yesmaximal, complemented
4#20276895C212yesmaximal, complemented
5#20276504C212yesmaximal, complemented
6#20276430C212yesmaximal, complemented
7#20276326C211yesmaximal, characteristic, complemented
8C2×C53×#427C211yesmaximal, characteristic, complemented
9C22×D424C2211yescharacteristic
10C2×#13133191—21no—
11C2×#13133157C2212yes—
12#16810859—21no—
13#16810859—21no—
14#16810859C2212yes—
15#16810859C2212yes—
16C2×#13133151C2211yescharacteristic
17C2×#13133139—21no—
18#16810450C2214yescomplemented
19#16810448C2214yescomplemented
20#16810423C2214yescomplemented
21#16810423C2214yescomplemented
22C22⋊#11719908C2212yescomplemented
23C22⋊#11719908C2212yescomplemented
24C23×C8×C53C2211yescharacteristic
25C2×#117×C53—21no—
26C53×#427—21no—
27C53×#427—21no—
28C53×#427C2212yes—
29C53×#427C2212yescomplemented
30C2×#107×C53C2211yescharacteristic, complemented
31C2×#107×C53C2211yescharacteristic, complemented
32C2×#106×C53—21no—
33C22×D212—21no—
34C22×C22⋊D53—41no—
35C2×D424—21no—
36C2×D424—24no—
37C2×D424C2312yes—
38#13133191—22no—
39#13133191—22no—
40#13133191—41no—
41#13133191—41no—
42C2×C4⋊Dic53—21no—
43C2×#11719921—41no—
44#13133157—24no—
45C2×#11719908C2314yes—
46#13133151—21no—
47#13133151C2312yes—
48#13133139—22no—
49#13133139—22no—
50C23×C4×C53D411yescharacteristic
51C22×C8×C53—24no—
52C22×C8×C53D411yescharacteristic
53C22×C8×C53D411yescharacteristic
54C22×C8×C53D411yescharacteristic
55C22×C8×C53C2311yescharacteristic
56C22×C8×C53C2311yescharacteristic
57C22×C8×C53C2311yescharacteristic
58#117×C53—22no—
59#117×C53—22no—
60#117×C53—41no—
61#117×C53—41no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C22⋊C4×C53—41no—
65C2×C42×C53—41no—
66#107×C53—21no—
67#107×C53—21no—
68#107×C53C2312yes—
69#107×C53C2312yes—
70#106×C53—22no—
71#106×C53—22no—
72C24×C53D811yescharacteristic
73C23×D53—41no—
74C22×C4×C53—22no—
75C22×C4×C53—22no—
76C22×C4×C53—41no—
77C22×C4×C53—41no—
78C22×C4×C53D811yescharacteristic
79C22×C4×C53D811yescharacteristic
80C22×C4×C53D811yescharacteristic
81C22×C4×C53C2×D411yescharacteristic
82C22×C4×C53C2×D411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C2×C22⋊D53—42no—
85C2×C22⋊D53—42no—
86C2×C22⋊D53—84no—
87C22×Dic53—41no—
88C2×D212—22no—
89C2×D212—42no—
90C2×D212—42no—
91C2×D212—41no—
92C2×C8×C53—22no—
93C2×C8×C53—22no—
94C2×C8×C53—22no—
95C2×C8×C53—22no—
96C2×C8×C53—22no—
97C2×C8×C53C2×D411yescharacteristic
98C2×C8×C53C2×D411yescharacteristic
99C2×C8×C53C2×D411yescharacteristic
100C2×C8×C53#4114yes—
101C2×C8×C53C2411yescharacteristic
102C4⋊C4×C53—22no—
103C4⋊C4×C53—22no—
104C4⋊C4×C53—41no—
105C4⋊C4×C53—41no—
106C22⋊C4×C53—42no—
107C22⋊C4×C53—42no—
108C42×C53—42no—
109C42×C53—42no—
110#11719921—42no—
111#11719921—42no—
112C4⋊Dic53—22no—
113C4⋊Dic53—41no—
114D424—24no—
115D424—42no—
116#11719908—24no—
117C23×C53—22no—
118C23×C53—22no—
119C23×C53—41no—
120C23×C53C2×D811yescharacteristic
121C23×C53C2×D811yescharacteristic
122C23×C53C2×D811yescharacteristic
123C22×D53—42no—
124C22×D53—82no—
125C22×D53—82no—
126C22×D53—82no—
127C2×C4×C53—22no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—22no—
133C2×C4×C53—22no—
134C2×C4×C53—42no—
135C2×C4×C53—42no—
136C2×C4×C53—82no—
137C2×C4×C53—82no—
138C2×C4×C53#12312yes—
139C2×C4×C53C2×D811yescharacteristic
140C2×C4×C53C2×D811yescharacteristic
141C2×C4×C53C2×D811yescharacteristic
142C2×C4×C53#13712yes—
143C2×C4×C53C22×D411yescharacteristic
144C22⋊D53—84no—
145C22⋊D53—84no—
146C2×Dic53—42no—
147C2×Dic53—82no—
148D212—42no—
149D212—42no—
150D212—82no—
151C8×C53—21no—
152C8×C53—21no—
153C8×C53#12312yes—
154C8×C53C2×#4112yes—
155#33648—531nomaximal, Sylow
156C22×C53—21no—
157C22×C53—21no—
158C22×C53—21no—
159C22×C53—21no—
160C22×C53—21no—
161C22×C53—21no—
162C22×C53—21no—
163C22×C53—44no—
164C22×C53#48012yes—
165C22×C53#50912yes—
166C22×C53C22×D811yescharacteristic
167C2×D53—82no—
168C2×D53—82no—
169C2×D53—82no—
170C2×D53—164no—
171C4×C53—21no—
172C4×C53—21no—
173C4×C53—82no—
174C4×C53—82no—
175C4×C53#48012yes—
176C4×C53C2×#12311yescharacteristic
177C4×C53C2×#13711yescharacteristic
178Dic53—82no—
179C2×#466—531no—
180C2×#427D5311yescharacteristic, complemented
181#2234—534no—
182#2233—532no—
183#2232—532no—
184#2159—532no—
185#2140—532no—
186#2112—531no—
187C2×C53—22no—
188C2×C53—22no—
189C2×C53—41no—
190C2×C53C2×#48011yescharacteristic
191C2×C53C2×#50911yescharacteristic
192C2×C53#341411yescharacteristic
193D53—162no—
194C22×D8—531no—
195C23×C8C2×D5311yescharacteristic
196C2×#121—1061no—
197C2×#117—21no—
198C2×M32—532no—
199#466—532no—
200#466—532no—
201#466—1061no—
202#466—1061no—
203#427—21no—
204#427—21no—
205#427C2×D5312yes—
206#427C2×D5312yescomplemented
207C2×#107—531no—
208C2×#107C2×D5311yescharacteristic, complemented
209C2×#107C2×D5311yescharacteristic, complemented
210C2×#106—21no—
211C2×#102—1061no—
212#367—534no—
213#366—534no—
214#357—534no—
215#357—534no—
216C22⋊C16—532no—
217C22⋊C16—532no—
218C53#3364811yescharacteristic, complemented, Sylow
219C22×D4—1061no—
220C22×D4—2121no—
221C23×C4C22⋊D5311yescharacteristic
222C2×D8—532no—
223C2×D8—1064no—
224C2×D8—1061no—
225C22×C8—24no—
226C22×C8C22⋊D5311yescharacteristic
227C22×C8C22⋊D5311yescharacteristic
228C22×C8C22⋊D5311yescharacteristic
229C22×C8C22×D5311yescharacteristic
230C22×C8C22×D5311yescharacteristic
231C22×C8C22×D5311yescharacteristic
232#121—1062no—
233#121—1062no—
234#121—2121no—
235#121—2121no—
236#117—22no—
237#117—22no—
238#117—41no—
239#117—41no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—1061no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—2121no—
245C2×C42—41no—
246M32—1064no—
247C2×C16—534no—
248#107—21no—
249#107—21no—
250#107—532no—
251#107—1061no—
252#107C22×D5312yes—
253#107C22×D5312yes—
254#106—22no—
255#106—22no—
256#102—1062no—
257#102—1062no—
258C24—2121no—
259C24#1171992211yescharacteristic
260C2×D4—1062no—
261C2×D4—4244no—
262C2×D4—2122no—
263C2×D4—2122no—
264C2×D4—2122no—
265C2×D4—2121no—
266C2×D4—2122no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—2121no—
272C22×C4#1171992211yescharacteristic
273C22×C4#1171992211yescharacteristic
274C22×C4#1171992211yescharacteristic
275C22×C4C2×C22⋊D5311yescharacteristic
276C22×C4C2×C22⋊D5311yescharacteristic
277C22×C4C2×C22⋊D5311yescharacteristic
278D8—1064no—
279D8—2122no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8#1171994514yes—
286C2×C8C2×C22⋊D5311yescharacteristic
287C2×C8C2×C22⋊D5311yescharacteristic
288C2×C8C2×C22⋊D5311yescharacteristic
289C2×C8C23×D5311yescharacteristic
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—1062no—
295C4⋊C4—2121no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—2122no—
299C22⋊C4—2122no—
300C42—42no—
301C42—42no—
302C16—1064no—
303C23—22no—
304C23—22no—
305C23—41no—
306C23—4242no—
307C23—4242no—
308C23—4242no—
309C23—2122no—
310C23C2×#1171992211yescharacteristic
311C23C2×#1171992211yescharacteristic
312C23C2×#1171992211yescharacteristic
313D4—4244no—
314D4—4244no—
315D4—4242no—
316D4—2122no—
317D4—2122no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—4242no—
330C2×C4—2122no—
331C2×C4#1313320012yes—
332C2×C4C2×#1171992211yescharacteristic
333C2×C4C2×#1171992211yescharacteristic
334C2×C4C2×#1171992211yescharacteristic
335C2×C4#1313324812yes—
336C2×C4C22×C22⋊D5311yescharacteristic
337C8—21no—
338C8—21no—
339C8#1313320012yes—
340C8C2×#1171994512yes—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—44no—
349C22—8484no—
350C22—4242no—
351C22—4242no—
352C22—4242no—
353C22#1681093612yes—
354C22#1681105812yes—
355C22C22×#1171992211yescharacteristic
356C4—21no—
357C4—21no—
358C4—82no—
359C4—82no—
360C4—4242no—
361C4#1681093612yes—
362C4C2×#1313320011yescharacteristic
363C4C2×#1313324811yescharacteristic
364C2—22no—
365C2—22no—
366C2—41no—
367C2—8482no—
368C2C2×#1681093611yescharacteristic
369C2C2×#1681105811yescharacteristic
370C2#2028784911yescharacteristic
subgroupG/H
centerC22C22×#11719922
commutator, cosocleC2×C8×C53C24
FrattiniC2×C8C23×D53
FittingC2×C53×#427C2
socleC22×C53C22×D8
prime radicalC2×#16810859C2
hypercenterC2×#427D53
hypocenterC53#33648
LCSG ▷ C2×C8×C53 ▷ C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C2×#427

Complements

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

NH ≅ G/Ntype
2. #20276906C2semidirect
3. #20276900C2semidirect
4. #20276895C2semidirect
5. #20276504C2semidirect
6. #20276430C2semidirect
7. #20276326C2semidirect
8. C2×C53×#427C2semidirect
18. #16810450C22semidirect
19. #16810448C22semidirect
20. #16810423C22semidirect
21. #16810423C22semidirect
22. C22⋊#11719908C22semidirect
23. C22⋊#11719908C22semidirect
29. C53×#427C22semidirect
30. C2×#107×C53C22semidirect
31. C2×#107×C53C22semidirect
180. C2×#427D53semidirect
206. #427C2×D53semidirect
208. C2×#107C2×D53semidirect
209. C2×#107C2×D53semidirect
218. C53#33648semidirect

Maximal subgroups

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

subgroupindex
C2×#168108592
#202769062
#202769002
#202768952
#202765042
#202764302
#202763262
C2×C53×#4272
#3364853