Finite Group Databasebeta

← C109×#18760 #42136754

Subgroups — C109×#18760 #42136754

Order 27904.

932 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#557×C109C211yesmaximal, characteristic, complemented
2C109×#3226C211yesmaximal, characteristic
3C109×#3103C211yesmaximal, characteristic
4C109×#3037C211yesmaximal, characteristic
5C109×#1648C214yesmaximal, complemented
6C109×#1626C214yesmaximal, complemented
7C109×#1546C211yesmaximal, characteristic, complemented
8C109×#1487C211yesmaximal, characteristic, complemented
9C109×#1479C211yesmaximal, characteristic, complemented
10C2×#137×C109—21no—
11#557×C109—22no—
12#557×C109—22no—
13#557×C109—22no—
14#557×C109C2214yes—
15#532×C109C2211yescharacteristic
16C2×C4⋊Q8×C109—21no—
17C2×C4⋊Q8×C109C2211yescharacteristic
18C2×#125×C109—22no—
19C2×#125×C109—21no—
20C2×C4×Q8×C109—21no—
21C2×C4×Q8×C109—21no—
22C2×C4×Q8×C109C2211yescharacteristic
23C2×C4×Q8×C109C2211yescharacteristic
24#500×C109C2212yes—
25#486×C109C2212yes—
26#481×C109—21no—
27#480×C109—22no—
28#475×C109C2212yes—
29#474×C109—22no—
30#464×C109—22no—
31#462×C109C2212yes—
32#460×C109C2212yes—
33#460×C109C2212yes—
34#448×C109—21no—
35C4×SD16×C109—22no—
36#431×C109C2211yescharacteristic
37#431×C109C2211yescharacteristic
38#423×C109C2211yescharacteristic
39#417×C109—21no—
40#332×C109C2214yes—
41Q8⋊C8×C109C2214yes—
42Q8⋊C8×C109C2214yes—
43C2×#41×C109—21no—
44C22×Q8×C109—21no—
45C22×Q8×C109D411yescharacteristic
46C22×Q8×C109D411yescharacteristic
47#137×C109—44no—
48C2×Q16×C109—22no—
49C2×SD16×C109—22no—
50C2×SD16×C109—22no—
51C2×M16×C109—21no—
52C4⋊Q8×C109—24no—
53C4⋊Q8×C109—24no—
54C4⋊Q8×C109—21no—
55C4⋊Q8×C109—21no—
56C4⋊Q8×C109D412yes—
57C4⋊Q8×C109C2311yescharacteristic
58C4⋊Q8×C109C2312yes—
59#127×C109C2311yescharacteristic
60#125×C109—28no—
61#125×C109—24no—
62#125×C109—28no—
63#125×C109—24no—
64#124×C109—21no—
65#121×C109—21no—
66#121×C109—21no—
67C4×Q8×C109—24no—
68C4×Q8×C109—24no—
69C4×Q8×C109—24no—
70C4×Q8×C109—24no—
71C4×Q8×C109—21no—
72C4×Q8×C109—21no—
73C4×Q8×C109—21no—
74C4×Q8×C109—21no—
75C4×Q8×C109D412yes—
76C4×Q8×C109D412yes—
77C4×Q8×C109C2312yes—
78C4×Q8×C109C2312yes—
79C4×D4×C109—21no—
80C4×D4×C109—21no—
81C2×C4⋊C4×C109—22no—
82C2×C4⋊C4×C109—22no—
83C2×C4⋊C4×C109—22no—
84C2×C4⋊C4×C109—21no—
85C2×C4⋊C4×C109—21no—
86C2×C4⋊C4×C109—21no—
87C2×C4⋊C4×C109D411yescharacteristic
88C2×C4⋊C4×C109D411yescharacteristic
89C2×C4⋊C4×C109D412yes—
90C2×C42×C109—22no—
91C2×C42×C109—21no—
92C2×C42×C109—21no—
93C2×C42×C109C2311yescharacteristic
94#107×C109—24no—
95#106×C109—22no—
96C4⋊C8×C109—21no—
97C4⋊C8×C109—21no—
98C4⋊C8×C109C2312yes—
99C4⋊C8×C109C2312yes—
100#103×C109—22no—
101#103×C109—22no—
102#102×C109—22no—
103#102×C109—22no—
104#102×C109—24no—
105C22⋊C8×C109—21no—
106C22⋊C8×C109—21no—
107C4×C8×C109—21no—
108C4×C8×C109C2312yes—
109#41×C109—42no—
110#41×C109—42no—
111C2×Q8×C109—24no—
112C2×Q8×C109—21no—
113C2×Q8×C109—21no—
114C2×Q8×C109—22no—
115C2×Q8×C109—22no—
116C2×Q8×C109—22no—
117C2×Q8×C109—22no—
118C2×Q8×C109—21no—
119C2×Q8×C109—42no—
120C2×Q8×C109—42no—
121C2×Q8×C109C2×D412yes—
122C2×Q8×C109C2×D412yes—
123C2×D4×C109—21no—
124C2×D4×C109—21no—
125C2×D4×C109—21no—
126C2×D4×C109—41no—
127C22×C4×C109—21no—
128C22×C4×C109—21no—
129C22×C4×C109—21no—
130C22×C4×C109—21no—
131C22×C4×C109—22no—
132C22×C4×C109—41no—
133C22×C4×C109C2×D411yescharacteristic
134C22×C4×C109C2×D411yescharacteristic
135C22×C4×C109C2×D411yescharacteristic
136Q16×C109—44no—
137SD16×C109—44no—
138SD16×C109—44no—
139M16×C109—42no—
140C2×C8×C109—22no—
141C2×C8×C109—22no—
142C2×C8×C109—22no—
143C4⋊C4×C109—28no—
144C4⋊C4×C109—28no—
145C4⋊C4×C109—28no—
146C4⋊C4×C109—22no—
147C4⋊C4×C109—22no—
148C4⋊C4×C109—22no—
149C4⋊C4×C109—21no—
150C4⋊C4×C109—21no—
151C4⋊C4×C109—22no—
152C4⋊C4×C109—22no—
153C4⋊C4×C109—22no—
154C4⋊C4×C109—22no—
155C4⋊C4×C109—21no—
156C4⋊C4×C109—21no—
157C4⋊C4×C109C2×D412yes—
158C4⋊C4×C109C2×D412yes—
159C4⋊C4×C109C2×D414yes—
160C22⋊C4×C109—41no—
161C22⋊C4×C109—41no—
162C42×C109—28no—
163C42×C109—22no—
164C42×C109—22no—
165C42×C109—22no—
166C42×C109—22no—
167C42×C109C2×D411yescharacteristic
168C42×C109C2×D411yescharacteristic
169C42×C109C2×D411yescharacteristic
170C42×C109C2411yescharacteristic
171C23×C109—41no—
172C23×C109#12011yescharacteristic
173Q8×C109—24no—
174Q8×C109—22no—
175Q8×C109—22no—
176Q8×C109—22no—
177Q8×C109—42no—
178Q8×C109—42no—
179Q8×C109—42no—
180Q8×C109—44no—
181Q8×C109—41no—
182Q8×C109—41no—
183Q8×C109#13714yescomplemented
184D4×C109—42no—
185D4×C109—41no—
186D4×C109—41no—
187D4×C109—42no—
188D4×C109—82no—
189C2×C4×C109—24no—
190C2×C4×C109—22no—
191C2×C4×C109—22no—
192C2×C4×C109—22no—
193C2×C4×C109—22no—
194C2×C4×C109—22no—
195C2×C4×C109—24no—
196C2×C4×C109—41no—
197C2×C4×C109—42no—
198C2×C4×C109—42no—
199C2×C4×C109—42no—
200C2×C4×C109—44no—
201C2×C4×C109—41no—
202C2×C4×C109—82no—
203C2×C4×C109#12011yescharacteristic
204C2×C4×C109#12011yescharacteristic
205C2×C4×C109#12011yescharacteristic
206C2×C4×C109#13512yes—
207C2×C4×C109AGL(1,ℤ/8)12yes—
208C2×C4×C109C22×D411yescharacteristic
209C2×C4×C109C22×D411yescharacteristic
210C2×C4×C109C22×D411yescharacteristic
211C8×C109—42no—
212C8×C109—42no—
213C8×C109—42no—
214C22×C109—21no—
215C22×C109—21no—
216C22×C109—81no—
217C22×C109—81no—
218C22×C109—81no—
219C22×C109#44912yes—
220C22×C109C2×#12011yescharacteristic
221C4×C109—24no—
222C4×C109—21no—
223C4×C109—21no—
224C4×C109—21no—
225C4×C109—42no—
226C4×C109—44no—
227C4×C109—42no—
228C4×C109—42no—
229C4×C109—81no—
230C4×C109#44912yes—
231C4×C109C2×#13511yescharacteristic
232C4×C109C2×AGL(1,ℤ/8)11yescharacteristic
233C4×C109C2×#13712yes—
234#18760C10911yesmaximal, characteristic, direct product factor, Sylow
235C2×C109—22no—
236C2×C109—161no—
237C2×C109C2×#44911yescharacteristic
238C2×C109#300411yescharacteristic
239C2×C109#302011yescharacteristic
240C2×#557C2×C10911yescharacteristic, complemented
241#3226C2×C10911yescharacteristic
242#3103C2×C10911yescharacteristic
243#3037C2×C10911yescharacteristic
244#1648C2×C10914yescomplemented
245#1626C2×C10914yescomplemented
246#1546C2×C10911yescharacteristic, complemented
247#1487C2×C10911yescharacteristic, complemented
248#1479C2×C10911yescharacteristic, complemented
249C109#1876011yescharacteristic, direct product factor, Sylow
250C2×#137—21no—
251#557—22no—
252#557—22no—
253#557—22no—
254#557C22×C10914yes—
255#532C22×C10911yescharacteristic
256C2×C4⋊Q8—21no—
257C2×C4⋊Q8C22×C10911yescharacteristic
258C2×#125—22no—
259C2×#125—21no—
260C2×C4×Q8—21no—
261C2×C4×Q8—21no—
262C2×C4×Q8C22×C10911yescharacteristic
263C2×C4×Q8C22×C10911yescharacteristic
264#500C22×C10912yes—
265#486C22×C10912yes—
266#481—21no—
267#480—22no—
268#475C22×C10912yes—
269#474—22no—
270#464—22no—
271#462C22×C10912yes—
272#460C22×C10912yes—
273#460C22×C10912yes—
274#448—21no—
275C4×SD16—22no—
276#431C22×C10911yescharacteristic
277#431C22×C10911yescharacteristic
278#423C22×C10911yescharacteristic
279#417—21no—
280#332C22×C10914yes—
281Q8⋊C8C22×C10914yes—
282Q8⋊C8C22×C10914yes—
283C2×#41—21no—
284C22×Q8—21no—
285C22×Q8D4×C10911yescharacteristic
286C22×Q8D4×C10911yescharacteristic
287#137—44no—
288C2×Q16—22no—
289C2×SD16—22no—
290C2×SD16—22no—
291C2×M16—21no—
292C4⋊Q8—24no—
293C4⋊Q8—24no—
294C4⋊Q8—21no—
295C4⋊Q8—21no—
296C4⋊Q8D4×C10912yes—
297C4⋊Q8C23×C10911yescharacteristic
298C4⋊Q8C23×C10912yes—
299#127C23×C10911yescharacteristic
300#125—28no—
301#125—24no—
302#125—28no—
303#125—24no—
304#124—21no—
305#121—21no—
306#121—21no—
307C4×Q8—24no—
308C4×Q8—24no—
309C4×Q8—24no—
310C4×Q8—24no—
311C4×Q8—21no—
312C4×Q8—21no—
313C4×Q8—21no—
314C4×Q8—21no—
315C4×Q8D4×C10912yes—
316C4×Q8D4×C10912yes—
317C4×Q8C23×C10912yes—
318C4×Q8C23×C10912yes—
319C4×D4—21no—
320C4×D4—21no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4D4×C10911yescharacteristic
328C2×C4⋊C4D4×C10911yescharacteristic
329C2×C4⋊C4D4×C10912yes—
330C2×C42—22no—
331C2×C42—21no—
332C2×C42—21no—
333C2×C42C23×C10911yescharacteristic
334#107—24no—
335#106—22no—
336C4⋊C8—21no—
337C4⋊C8—21no—
338C4⋊C8C23×C10912yes—
339C4⋊C8C23×C10912yes—
340#103—22no—
341#103—22no—
342#102—22no—
343#102—22no—
344#102—24no—
345C22⋊C8—21no—
346C22⋊C8—21no—
347C4×C8—21no—
348C4×C8C23×C10912yes—
349#41—42no—
350#41—42no—
351C2×Q8—24no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—21no—
359C2×Q8—42no—
360C2×Q8—42no—
361C2×Q8C2×D4×C10912yes—
362C2×Q8C2×D4×C10912yes—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—41no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—41no—
373C22×C4C2×D4×C10911yescharacteristic
374C22×C4C2×D4×C10911yescharacteristic
375C22×C4C2×D4×C10911yescharacteristic
376Q16—44no—
377SD16—44no—
378SD16—44no—
379M16—42no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C4⋊C4—28no—
384C4⋊C4—28no—
385C4⋊C4—28no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4C2×D4×C10912yes—
398C4⋊C4C2×D4×C10912yes—
399C4⋊C4C2×D4×C10914yes—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C42—28no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—22no—
407C42C2×D4×C10911yescharacteristic
408C42C2×D4×C10911yescharacteristic
409C42C2×D4×C10911yescharacteristic
410C42C24×C10911yescharacteristic
411C23—41no—
412C23#120×C10911yescharacteristic
413Q8—24no—
414Q8—22no—
415Q8—22no—
416Q8—22no—
417Q8—42no—
418Q8—42no—
419Q8—42no—
420Q8—44no—
421Q8—41no—
422Q8—41no—
423Q8#137×C10914yescomplemented
424D4—42no—
425D4—41no—
426D4—41no—
427D4—42no—
428D4—82no—
429C2×C4—24no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—44no—
441C2×C4—41no—
442C2×C4—82no—
443C2×C4#120×C10911yescharacteristic
444C2×C4#120×C10911yescharacteristic
445C2×C4#120×C10911yescharacteristic
446C2×C4#135×C10912yes—
447C2×C4AGL(1,ℤ/8)×C10912yes—
448C2×C4C22×D4×C10911yescharacteristic
449C2×C4C22×D4×C10911yescharacteristic
450C2×C4C22×D4×C10911yescharacteristic
451C8—42no—
452C8—42no—
453C8—42no—
454C22—21no—
455C22—21no—
456C22—81no—
457C22—81no—
458C22—81no—
459C22#449×C10912yes—
460C22C2×#120×C10911yescharacteristic
461C4—24no—
462C4—21no—
463C4—21no—
464C4—21no—
465C4—42no—
466C4—44no—
467C4—42no—
468C4—42no—
469C4—81no—
470C4#449×C10912yes—
471C4C2×#135×C10911yescharacteristic
472C4C2×AGL(1,ℤ/8)×C10911yescharacteristic
473C4C2×#137×C10912yes—
474C2—22no—
475C2—161no—
476C2C2×#449×C10911yescharacteristic
477C2C109×#300411yescharacteristic
478C2C109×#302011yescharacteristic
subgroupG/H
center, socleC22×C109C2×#120
commutator, Frattini, cosocleC42C24×C109
prime radical#532×C109C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C109 ◁ C2×C42×C109 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#557×C109C2semidirect
5. C109×#1648C2semidirect
6. C109×#1626C2semidirect
7. C109×#1546C2semidirect
8. C109×#1487C2semidirect
9. C109×#1479C2semidirect
183. Q8×C109#137semidirect
234. #18760C109direct (N × H)
240. C2×#557C2×C109semidirect
244. #1648C2×C109semidirect
245. #1626C2×C109semidirect
246. #1546C2×C109semidirect
247. #1487C2×C109semidirect
248. #1479C2×C109semidirect
249. C109#18760direct (N × H)
423. Q8#137×C109semidirect

Maximal subgroups

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

subgroupindex
C2×#557×C1092
C109×#32262
C109×#31032
C109×#30372
C109×#16482
C109×#16262
C109×#15462
C109×#14872
C109×#14792
#18760109