Finite Group Databasebeta

← #27207025

Subgroups — #27207025

Order 13184.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20268450C211yesmaximal, characteristic
2#20268148C211yesmaximal, characteristic, complemented
3#20268132C211yesmaximal, characteristic, complemented
4#20268104C211yesmaximal, characteristic, complemented
5#20268095C211yesmaximal, characteristic, complemented
6#20268030C211yesmaximal, characteristic, complemented
7#20268022C211yesmaximal, characteristic, complemented
8#20268013C211yesmaximal, characteristic, complemented
9#20267951C211yesmaximal, characteristic, complemented
10#20267946C211yesmaximal, characteristic, complemented
11C4×#13110095C211yesmaximal, characteristic, complemented
12D4×C103⋊C8C211yesmaximal, characteristic, complemented
13#20267785C211yesmaximal, characteristic, complemented
14C2×#15692944C211yesmaximal, characteristic, complemented
15#552×C103C211yesmaximal, characteristic, complemented
16C2×#13110105C2211yescharacteristic
17C22×Dic206C2211yescharacteristic
18C2×#13110095C2211yescharacteristic
19C2×#13110095C2211yescharacteristic
20C2×#13110092C2211yescharacteristic
21C22×C103⋊C8C2211yescharacteristic
22C22×C103⋊C8C2211yescharacteristic
23#15693009C2211yescharacteristic
24#15693007—21no—
25#15693000C2211yescharacteristic
26#15692999—21no—
27#15692995C2211yescharacteristic
28#15692993—21no—
29C4×Dic206C2211yescharacteristic
30(C22⋊D103)⋊C4—21no—
31#15692954C2211yescharacteristic
32#15692954C2211yescharacteristic
33#15692952C2211yescharacteristic
34C4⋊(C103⋊C8)C2211yescharacteristic
35#15692945C2211yescharacteristic
36#15692944C2211yescharacteristic
37#15692944C2211yescharacteristic
38#15692944C2211yescharacteristic
39#15692944C2211yescharacteristic
40#15692944C2211yescharacteristic
41#15692944C2211yescharacteristic
42#15692941C2211yescharacteristic
43#15692941C2211yescharacteristic
44#15692941C2211yescharacteristic
45C22⋊(C103⋊C8)C2211yescharacteristic
46C22⋊(C103⋊C8)C2211yescharacteristic
47C4×C103⋊C8C2211yescharacteristic
48#126×C103—21no—
49#125×C103—21no—
50#125×C103C2211yescharacteristic
51#123×C103—21no—
52C22⋊Q8×C103C2211yescharacteristic
53C22⋊Q8×C103C2211yescharacteristic
54C4×Q8×C103C2211yescharacteristic
55C4×D4×C103C2211yescharacteristic
56#117×C103—21no—
57#117×C103C2211yescharacteristic
58C2×C4⋊C4×C103C2211yescharacteristic
59#13110105—22no—
60#13110105—41no—
61C2×Dic206—22no—
62C2×Dic206—21no—
63C2×Dic206—41no—
64C2×Dic206C2311yescharacteristic
65C2×Dic206C2311yescharacteristic
66C2×D412C2311yescharacteristic
67C2×C22⋊D103—21no—
68C2×C22⋊D103—21no—
69C22×Dic103—21no—
70C2×C4×D103—21no—
71#13110095—22no—
72#13110095—41no—
73#13110092—41no—
74C2×C103⋊C8—41no—
75C2×C103⋊C8—41no—
76C2×C103⋊C8C2311yescharacteristic
77C2×C103⋊C8C2311yescharacteristic
78C2×C103⋊C8C2311yescharacteristic
79C2×C103⋊C8C2311yescharacteristic
80C4⋊Dic103C2311yescharacteristic
81Dic103⋊C4—21no—
82Dic103⋊C4—21no—
83C22⋊Dic103—21no—
84(C2×D103)⋊C4—21no—
85(C2×D103)⋊C4—21no—
86(C2×D103)⋊C4—21no—
87C4×Dic103—21no—
88C2×Q8×C103C2311yescharacteristic
89C2×D4×C103D411yescharacteristic
90C22×C4×C103—21no—
91C22×C4×C103C2311yescharacteristic
92C22×C4×C103C2311yescharacteristic
93C4⋊C4×C103—21no—
94C4⋊C4×C103—21no—
95C4⋊C4×C103—21no—
96C4⋊C4×C103—21no—
97C4⋊C4×C103—21no—
98C4⋊C4×C103—21no—
99C4⋊C4×C103D411yescharacteristic
100C4⋊C4×C103C2311yescharacteristic
101C4⋊C4×C103C2311yescharacteristic
102C4⋊C4×C103C2311yescharacteristic
103C22⋊C4×C103—21no—
104C22⋊C4×C103—21no—
105C22⋊C4×C103—21no—
106C22⋊C4×C103D411yescharacteristic
107C22⋊C4×C103D411yescharacteristic
108C42×C103—21no—
109C42×C103—21no—
110C42×C103C2311yescharacteristic
111C23×C103C2×D411yescharacteristic
112C23×C103C2×D411yescharacteristic
113C22×D103—21no—
114Q8×C103—21no—
115Q8×C103—21no—
116D4×C103—22no—
117C2×C4×C103—21no—
118C2×C4×C103—21no—
119C2×C4×C103—21no—
120C2×C4×C103—21no—
121C2×C4×C103—21no—
122C2×C4×C103—21no—
123C2×C4×C103—21no—
124C2×C4×C103—21no—
125C2×C4×C103—42no—
126C2×C4×C103C2×D411yescharacteristic
127C2×C4×C103C2×D411yescharacteristic
128C2×C4×C103C2×D411yescharacteristic
129C2×C4×C103C2×D411yescharacteristic
130C2×C4×C103C2411yescharacteristic
131C22⋊D103—42no—
132C22⋊D103—42no—
133C2×Dic103—21no—
134C2×Dic103—21no—
135C2×Dic103—21no—
136C2×Dic103—42no—
137D412—21no—
138D412—21no—
139C4×D103—42no—
140Dic206—21no—
141Dic206—21no—
142Dic206—21no—
143Dic206—42no—
144Dic206#4112yes—
145C103⋊C8—21no—
146C103⋊C8—21no—
147C103⋊C8—21no—
148C103⋊C8—21no—
149C22×C103—21no—
150C22×C103—21no—
151C22×C103—21no—
152C22×C103—21no—
153C22×C103—21no—
154C22×C103#13512yes—
155C22×C103C22×D411yescharacteristic
156C2×D103—41no—
157C2×D103—41no—
158C2×D103—41no—
159C4×C103—21no—
160C4×C103—21no—
161C4×C103—21no—
162C4×C103—21no—
163C4×C103—41no—
164C4×C103—41no—
165C4×C103—41no—
166C4×C103—41no—
167C4×C103C2×#4111yescharacteristic
168C4×C103H(5,𝔽2)11yescharacteristic
169Dic103—22no—
170Dic103—41no—
171Dic103—41no—
172C2×C103—22no—
173C2×C103—41no—
174C2×C103#54611yescharacteristic
175C2×C103C2×#13511yescharacteristic
176C2×C103#57811yescharacteristic
177D103—81no—
178#3300—1031nomaximal, Sylow
179C103#330011yescharacteristic, complemented, Sylow
180#552D10311yescharacteristic, complemented
181#548—1031no—
182#488—1031no—
183#485—1031no—
184#480—1031no—
185#479—1031no—
186#467—1031no—
187#465—1031no—
188#464—1031no—
189#452—1031no—
190#451—1031no—
191C4×Q16—1031no—
192C8×D4—1031no—
193#416—1031no—
194C2×#103—1031no—
195C2×#41—1031no—
196C22×Q8—1031no—
197C2×Q16—1031no—
198C2×Q16—1031no—
199C2×SD16—1031no—
200C22×C8—1031no—
201C22×C8—1031no—
202#126—21no—
203#125—21no—
204#125C2×D10311yescharacteristic
205#124—1031no—
206#124—2061no—
207#123—21no—
208C22⋊Q8—1031no—
209C22⋊Q8—2061no—
210C22⋊Q8C2×D10311yescharacteristic
211C22⋊Q8C2×D10311yescharacteristic
212#121—1031no—
213#121—2061no—
214C4×Q8—1031no—
215C4×Q8C2×D10311yescharacteristic
216C4×D4—2061no—
217C4×D4C2×D10311yescharacteristic
218#117—21no—
219#117C2×D10311yescharacteristic
220C2×C4⋊C4C2×D10311yescharacteristic
221#107—1031no—
222#107—1031no—
223#106—1031no—
224C4⋊C8—1031no—
225#103—1031no—
226#103—1031no—
227#103—1031no—
228#103—1031no—
229#103—1031no—
230#103—1031no—
231#103—1031no—
232#102—1031no—
233#102—1031no—
234#102—1031no—
235C22⋊C8—1031no—
236C22⋊C8—1031no—
237C4×C8—1031no—
238#41—4121no—
239#41—2062no—
240C2×Q8—1031no—
241C2×Q8—1031no—
242C2×Q8—4121no—
243C2×Q8—2062no—
244C2×Q8—2061no—
245C2×Q8C22×D10311yescharacteristic
246C2×D4—1031no—
247C2×D4—2061no—
248C2×D4—2061no—
249C2×D4C22⋊D10311yescharacteristic
250C22×C4—21no—
251C22×C4—2061no—
252C22×C4—2061no—
253C22×C4C22×D10311yescharacteristic
254C22×C4C22×D10311yescharacteristic
255Q16—4121no—
256Q16—2062no—
257SD16—4121no—
258C2×C8—1031no—
259C2×C8—1031no—
260C2×C8—1031no—
261C2×C8—1031no—
262C2×C8—4121no—
263C2×C8—4121no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—1031no—
271C4⋊C4—2061no—
272C4⋊C4—2061no—
273C4⋊C4C22⋊D10311yescharacteristic
274C4⋊C4C22×D10311yescharacteristic
275C4⋊C4C22×D10311yescharacteristic
276C4⋊C4C22×D10311yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—21no—
279C22⋊C4—21no—
280C22⋊C4—2061no—
281C22⋊C4—2061no—
282C22⋊C4—2061no—
283C22⋊C4—2061no—
284C22⋊C4C22⋊D10311yescharacteristic
285C22⋊C4C22⋊D10311yescharacteristic
286C42—21no—
287C42—21no—
288C42—2061no—
289C42C22×D10311yescharacteristic
290C23—2061no—
291C23C2×C22⋊D10311yescharacteristic
292C23C2×C22⋊D10311yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—1032no—
296Q8—4122no—
297Q8—2061no—
298Q8—2061no—
299Q8—2061no—
300D4—22no—
301D4—4122no—
302D4—4122no—
303D4—2061no—
304D4—2061no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—4122no—
315C2×C4—4122no—
316C2×C4—2061no—
317C2×C4—2061no—
318C2×C4—2061no—
319C2×C4C2×C22⋊D10311yescharacteristic
320C2×C4C2×C22⋊D10311yescharacteristic
321C2×C4C2×C22⋊D10311yescharacteristic
322C2×C4C2×C22⋊D10311yescharacteristic
323C2×C4C23×D10311yescharacteristic
324C8—2061no—
325C8—2061no—
326C8—2061no—
327C8—2061no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—4121no—
334C22—4121no—
335C22—4121no—
336C22#1569304112yes—
337C22C22×C22⋊D10311yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—4121no—
347C4—4121no—
348C4—2062no—
349C4#41×D10311yescharacteristic
350C4#1569306711yescharacteristic
351C2—22no—
352C2—41no—
353C2—8241no—
354C2#2026842911yescharacteristic
355C2C2×#1569304111yescharacteristic
356C2#2026860611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D103
commutator, cosocleC2×C4×C103C24
FrattiniC2×C4C23×D103
Fitting#552×C103C2
socleC22×C103C22×D4
prime radical#20268450C2
hypercenter#552D103
hypocenterC103#3300
LCSG ▷ C2×C4×C103 ▷ C2×C103 ▷ C103
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #20268148C2semidirect
3. #20268132C2semidirect
4. #20268104C2semidirect
5. #20268095C2semidirect
6. #20268030C2semidirect
7. #20268022C2semidirect
8. #20268013C2semidirect
9. #20267951C2semidirect
10. #20267946C2semidirect
11. C4×#13110095C2semidirect
12. D4×C103⋊C8C2semidirect
13. #20267785C2semidirect
14. C2×#15692944C2semidirect
15. #552×C103C2semidirect
179. C103#3300semidirect
180. #552D103semidirect

Maximal subgroups

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

subgroupindex
#202684502
#202681482
#202681322
#202681042
#202680952
#202680302
#202680222
#202680132
#202679512
#202679462
C4×#131100952
D4×C103⋊C82
#202677852
C2×#156929442
#552×C1032
#3300103