Finite Group Databasebeta

← #27206648

Subgroups — #27206648

Order 13184.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×SD16×D103C211yesmaximal, characteristic, complemented
2Q8×C22⋊D103C211yesmaximal, characteristic, complemented
3#20268429C211yesmaximal, characteristic
4#20268184C211yesmaximal, characteristic, complemented
5#20268037C211yesmaximal, characteristic, complemented
6#20268029C211yesmaximal, characteristic, complemented
7#20268015C211yesmaximal, characteristic, complemented
8#20267994C211yesmaximal, characteristic, complemented
9#20267953C211yesmaximal, characteristic, complemented
10#20267945C211yesmaximal, characteristic, complemented
11#20267940C211yesmaximal, characteristic, complemented
12#20267930C211yesmaximal, characteristic, complemented
13#20267892C211yesmaximal, characteristic, complemented
14#20267860C211yesmaximal, characteristic, complemented
15C2×#137×C103C211yesmaximal, characteristic, complemented
16C2×Q8×D103C2211yescharacteristic
17C2×D4×D103C2211yescharacteristic
18C2×#13110095C2211yescharacteristic, complemented
19C2×#13110094C2211yescharacteristic
20C2×#13110093C2211yescharacteristic
21C2×#13110092C2211yescharacteristic
22C2×#13110092C2211yescharacteristic
23SD16×D103—24no—
24C2×C8×D103C2211yescharacteristic
25#15693019—21no—
26#15693019C2211yescharacteristic, complemented
27#15693010C2211yescharacteristic
28#15693004—21no—
29#15693000—21no—
30#15693000C2211yescharacteristic, complemented
31#15692997—21no—
32#15692997C2211yescharacteristic
33#15692997C2211yescharacteristic
34#15692995C2211yescharacteristic
35#15692993—21no—
36#15692989—21no—
37Q8×Dic103C2211yescharacteristic, complemented
38D4×Dic103C2211yescharacteristic
39C4×C22⋊D103—21no—
40C4×C22⋊D103C2211yescharacteristic
41C2×(C2×D103)⋊C4—21no—
42#15692955C2211yescharacteristic, complemented
43Dic103⋊C8C2211yescharacteristic, complemented
44#15692946C2211yescharacteristic, complemented
45#15692945C2211yescharacteristic, complemented
46#15692945C2211yescharacteristic, complemented
47#15692945C2211yescharacteristic, complemented
48#15692943C2211yescharacteristic
49#15692942C2211yescharacteristic
50C22⋊(C103⋊C8)C2211yescharacteristic, complemented
51(C2×D103)⋊C8C2211yescharacteristic
52#15692931C2211yescharacteristic, complemented
53C2×#41×C103C2211yescharacteristic
54C22×Q8×C103C2211yescharacteristic, complemented
55#137×C103—24no—
56C2×Q16×C103C2211yescharacteristic, complemented
57C2×Q16×C103C2211yescharacteristic, complemented
58C2×SD16×C103C2211yescharacteristic
59C2×SD16×C103C2211yescharacteristic
60C2×M16×C103C2211yescharacteristic, complemented
61C23×D103—21no—
62C2×Dic206—21no—
63C2×Dic206C2311yescharacteristic
64Q8×D103—22no—
65Q8×D103—41no—
66C2×D412C2311yescharacteristic
67C2×C22⋊D103—21no—
68C2×C22⋊D103—21no—
69C2×C22⋊D103D411yescharacteristic
70D4×D103—22no—
71D4×D103—41no—
72C22×Dic103—21no—
73C2×C4×D103—21no—
74C2×C4×D103C2311yescharacteristic
75#13110095—41no—
76#13110094—22no—
77#13110093—22no—
78#13110092—22no—
79#13110092—41no—
80C2×C103⋊C8C2311yescharacteristic
81C2×C103⋊C8C2311yescharacteristic
82C8×D103—22no—
83C4⋊Dic103—21no—
84C4⋊Dic103C2311yescharacteristic
85C4⋊Dic103C2311yescharacteristic
86Dic103⋊C4—21no—
87Dic103⋊C4—21no—
88Dic103⋊C4—21no—
89Dic103⋊C4—21no—
90Dic103⋊C4—21no—
91Dic103⋊C4D411yescharacteristic
92C22⋊Dic103—21no—
93C22⋊Dic103—21no—
94C22⋊Dic103D411yescharacteristic
95(C2×D103)⋊C4—21no—
96(C2×D103)⋊C4—21no—
97(C2×D103)⋊C4—21no—
98(C2×D103)⋊C4—21no—
99(C2×D103)⋊C4—21no—
100(C2×D103)⋊C4D411yescharacteristic
101C4×Dic103—21no—
102C4×Dic103C2311yescharacteristic
103#41×C103—22no—
104#41×C103—41no—
105C2×Q8×C103—21no—
106C2×Q8×C103—22no—
107C2×Q8×C103—41no—
108C2×Q8×C103C2311yescharacteristic
109C2×Q8×C103C2311yescharacteristic
110C2×Q8×C103C2311yescharacteristic
111C2×D4×C103—21no—
112C2×D4×C103C2311yescharacteristic
113C22×C4×C103—21no—
114C22×C4×C103—21no—
115C22×C4×C103C2311yescharacteristic
116Q16×C103—22no—
117Q16×C103—22no—
118SD16×C103—22no—
119SD16×C103D414yescomplemented
120M16×C103—22no—
121C2×C8×C103C2311yescharacteristic
122C2×C8×C103C2311yescharacteristic
123C23×C103—21no—
124C23×C103C2×D411yescharacteristic
125C22×D103—21no—
126C22×D103—42no—
127C22×D103—42no—
128C22×D103—81no—
129C22×D103C2×D411yescharacteristic
130Q8×C103—21no—
131Q8×C103—21no—
132Q8×C103—21no—
133Q8×C103—21no—
134Q8×C103—21no—
135Q8×C103—42no—
136Q8×C103C2×D412yes—
137D4×C103—21no—
138D4×C103—42no—
139D4×C103C2×D412yes—
140C2×C4×C103—21no—
141C2×C4×C103—21no—
142C2×C4×C103—21no—
143C2×C4×C103—21no—
144C2×C4×C103—21no—
145C2×C4×C103—42no—
146C2×C4×C103—42no—
147C2×C4×C103C2×D411yescharacteristic
148C2×C4×C103C2411yescharacteristic
149C22⋊D103—22no—
150C22⋊D103—42no—
151C22⋊D103—81no—
152C2×Dic103—21no—
153C2×Dic103—21no—
154C2×Dic103—21no—
155C2×Dic103—81no—
156C2×Dic103C2×D411yescharacteristic
157C2×Dic103C2×D411yescharacteristic
158C2×Dic103C2×D411yescharacteristic
159D412—21no—
160D412—21no—
161C4×D103—21no—
162C4×D103—21no—
163C4×D103—42no—
164Dic206—21no—
165Dic206—21no—
166Dic206—42no—
167C8×C103—21no—
168C8×C103C2×D412yes—
169C103⋊C8—21no—
170C103⋊C8—21no—
171C22×C103—22no—
172C22×C103—21no—
173C22×C103—21no—
174C22×C103—21no—
175C22×C103—41no—
176C22×C103—41no—
177C22×C103C22×D411yescharacteristic
178C2×D103—21no—
179C2×D103—21no—
180C2×D103—41no—
181C2×D103—41no—
182C2×D103—41no—
183C2×D103—84no—
184C2×D103#13712yescomplemented
185C4×C103—22no—
186C4×C103—21no—
187C4×C103—41no—
188C4×C103—41no—
189C4×C103C22×D411yescharacteristic
190C4×C103H(5,𝔽2)11yescharacteristic
191Dic103—21no—
192Dic103—21no—
193Dic103—21no—
194Dic103—41no—
195Dic103—41no—
196Dic103—41no—
197C2×C103—42no—
198C2×C103—41no—
199C2×C103D4211yescharacteristic
200C2×C103C2×#13711yescharacteristic
201C2×C103#57711yescharacteristic
202D103—22no—
203D103—81no—
204#3266—1031nomaximal, Sylow
205C103#326611yescharacteristic, complemented, Sylow
206C2×#137D10311yescharacteristic, complemented
207C22×SD16—1031no—
208D4×Q8—1031no—
209#546—1031no—
210#497—1031no—
211#468—1031no—
212#467—1031no—
213#464—1031no—
214#461—1031no—
215#452—1031no—
216#451—1031no—
217#450—1031no—
218#448—1031no—
219#440—1031no—
220#435—1031no—
221C2×#41C2×D10311yescharacteristic
222C22×Q8—1031no—
223C22×Q8C2×D10311yescharacteristic, complemented
224C22×D4—1031no—
225#137—24no—
226C2×Q16—1031no—
227C2×Q16C2×D10311yescharacteristic, complemented
228C2×Q16C2×D10311yescharacteristic, complemented
229C2×SD16—1031no—
230C2×SD16—1031no—
231C2×SD16—1031no—
232C2×SD16—1031no—
233C2×SD16—2064no—
234C2×SD16C2×D10311yescharacteristic
235C2×SD16C2×D10311yescharacteristic
236C2×M16C2×D10311yescharacteristic, complemented
237C22×C8—1031no—
238C4⋊Q8—1031no—
239C4⋊Q8—2061no—
240#124—1031no—
241#123—2061no—
242C22⋊Q8—1031no—
243C22⋊Q8—1031no—
244C22⋊Q8—1031no—
245C22⋊Q8—2061no—
246C22⋊Q8—2061no—
247#121—1031no—
248#121—2061no—
249#120—2061no—
250C4×Q8—1031no—
251C4×D4—1031no—
252C4×D4—1031no—
253C4×D4—2061no—
254C2×C22⋊C4—2061no—
255#107—1031no—
256C4⋊C8—1031no—
257#103—1031no—
258#103—1031no—
259#103—1031no—
260#103—1031no—
261#102—1031no—
262#102—1031no—
263C22⋊C8—1031no—
264C22⋊C8—1031no—
265#97—1031no—
266C24—2061no—
267#41—22no—
268#41—41no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—1031no—
273C2×Q8—4121no—
274C2×Q8—2061no—
275C2×Q8—2062no—
276C2×Q8C22×D10311yescharacteristic
277C2×Q8C22×D10311yescharacteristic
278C2×Q8C22×D10311yescharacteristic
279C2×D4—21no—
280C2×D4—1031no—
281C2×D4—1031no—
282C2×D4—4121no—
283C2×D4—2061no—
284C2×D4—2061no—
285C2×D4—2062no—
286C2×D4C22×D10311yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—1031no—
290C22×C4—2061no—
291C22×C4—2061no—
292C22×C4C22×D10311yescharacteristic
293Q16—22no—
294Q16—22no—
295Q16—4121no—
296SD16—22no—
297SD16—4121no—
298SD16—2062no—
299SD16—2062no—
300SD16—2062no—
301SD16C22⋊D10314yescomplemented
302M16—22no—
303C2×C8—1031no—
304C2×C8—1031no—
305C2×C8—2062no—
306C2×C8C22×D10311yescharacteristic
307C2×C8C22×D10311yescharacteristic
308C4⋊C4—1031no—
309C4⋊C4—1031no—
310C4⋊C4—1031no—
311C4⋊C4—2061no—
312C4⋊C4—2061no—
313C4⋊C4—2061no—
314C4⋊C4—2061no—
315C4⋊C4—2061no—
316C4⋊C4—2061no—
317C22⋊C4—1031no—
318C22⋊C4—1031no—
319C22⋊C4—2061no—
320C22⋊C4—2061no—
321C22⋊C4—2061no—
322C22⋊C4—2061no—
323C22⋊C4—2061no—
324C22⋊C4—2061no—
325C22⋊C4—2061no—
326C42—1031no—
327C42—2061no—
328C23—21no—
329C23—8241no—
330C23—1031no—
331C23—4122no—
332C23—4122no—
333C23—2061no—
334C23D4×D10311yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8—4122no—
342Q8—2061no—
343Q8—2061no—
344Q8C2×C22⋊D10312yes—
345D4—21no—
346D4—42no—
347D4—8241no—
348D4—4122no—
349D4—2062no—
350D4—2061no—
351D4—2061no—
352D4C2×C22⋊D10312yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—8241no—
361C2×C4—1031no—
362C2×C4—1031no—
363C2×C4—1031no—
364C2×C4—4122no—
365C2×C4—2061no—
366C2×C4—2061no—
367C2×C4—2061no—
368C2×C4—2061no—
369C2×C4—2061no—
370C2×C4D4×D10311yescharacteristic
371C2×C4C23×D10311yescharacteristic
372C8—21no—
373C8—2061no—
374C8—2061no—
375C8C2×C22⋊D10312yes—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—22no—
380C22—41no—
381C22—41no—
382C22—8244no—
383C22—1032no—
384C22—4121no—
385C22—4121no—
386C22—4121no—
387C22—2061no—
388C22—2061no—
389C22C2×D4×D10311yescharacteristic
390C4—21no—
391C4—22no—
392C4—41no—
393C4—41no—
394C4—4121no—
395C4—4121no—
396C4—4121no—
397C4—2061no—
398C4—2061no—
399C4—2061no—
400C4C22×C22⋊D10311yescharacteristic
401C4#1569306611yescharacteristic
402C2—41no—
403C2—42no—
404C2—8241no—
405C2—2062no—
406C2D4×C22⋊D10311yescharacteristic
407C2#137×D10311yescharacteristic
408C2#2026860011yescharacteristic
subgroupG/H
centerC22C2×D4×D103
commutator, cosocleC2×C4×C103C24
FrattiniC2×C4C23×D103
FittingC2×#137×C103C2
socleC22×C103C22×D4
prime radical#20268429C2
hypercenterC2×#137D103
hypocenterC103#3266
LCSG ▷ C2×C4×C103 ▷ C2×C103 ▷ C103
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×SD16×D103C2semidirect
2. Q8×C22⋊D103C2semidirect
4. #20268184C2semidirect
5. #20268037C2semidirect
6. #20268029C2semidirect
7. #20268015C2semidirect
8. #20267994C2semidirect
9. #20267953C2semidirect
10. #20267945C2semidirect
11. #20267940C2semidirect
12. #20267930C2semidirect
13. #20267892C2semidirect
14. #20267860C2semidirect
15. C2×#137×C103C2semidirect
18. C2×#13110095C22semidirect
26. #15693019C22semidirect
30. #15693000C22semidirect
37. Q8×Dic103C22semidirect
42. #15692955C22semidirect
43. Dic103⋊C8C22semidirect
44. #15692946C22semidirect
45. #15692945C22semidirect
46. #15692945C22semidirect
47. #15692945C22semidirect
50. C22⋊(C103⋊C8)C22semidirect
52. #15692931C22semidirect
54. C22×Q8×C103C22semidirect
56. C2×Q16×C103C22semidirect
57. C2×Q16×C103C22semidirect
60. C2×M16×C103C22semidirect
119. SD16×C103D4semidirect
184. C2×D103#137semidirect
205. C103#3266semidirect
206. C2×#137D103semidirect
223. C22×Q8C2×D103semidirect
227. C2×Q16C2×D103semidirect
228. C2×Q16C2×D103semidirect
236. C2×M16C2×D103semidirect
301. SD16C22⋊D103semidirect

Maximal subgroups

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

subgroupindex
C2×SD16×D1032
Q8×C22⋊D1032
#202684292
#202681842
#202680372
#202680292
#202680152
#202679942
#202679532
#202679452
#202679402
#202679302
#202678922
#202678602
C2×#137×C1032
#3266103