Finite Group Databasebeta

← C109×#18171 #42136165

Subgroups — C109×#18171 #42136165

Order 27904.

1042 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3492C211yesmaximal, characteristic
2C109×#3092C212yesmaximal, complemented
3C2×#398×C109C211yesmaximal, characteristic, complemented
4C109×#1611C212yesmaximal, complemented
5C109×#1610C212yesmaximal, complemented
6C109×#1609C214yesmaximal, complemented
7C109×#1504C212yesmaximal, complemented
8C109×#1463C211yesmaximal, characteristic, complemented
9#554×C109—22no—
10#553×C109—22no—
11#552×C109—22no—
12#547×C109—22no—
13#544×C109C2211yescharacteristic
14#539×C109—21no—
15#539×C109C2212yes—
16#537×C109C2211yescharacteristic
17#525×C109—21no—
18#525×C109—21no—
19#525×C109C2212yes—
20#520×C109—21no—
21C22×C4⋊C4×C109—21no—
22C22×C4⋊C4×C109C2211yescharacteristic
23C22×C4⋊C4×C109C2211yescharacteristic
24#484×C109—22no—
25#483×C109C2214yes—
26#481×C109—22no—
27#480×C109C2214yes—
28#428×C109—22no—
29#417×C109—22no—
30#407×C109C2212yes—
31#398×C109—21no—
32#398×C109—21no—
33#398×C109C2212yes—
34#398×C109C2212yes—
35C2×#95×C109—21no—
36C2×#95×C109C2211yescharacteristic
37#328×C109C2214yes—
38#327×C109C2214yes—
39#324×C109C2212yes—
40#324×C109C2212yes—
41C2×#41×C109—21no—
42C2×#41×C109—21no—
43C2×#41×C109D412yes—
44C23×C4×C109—21no—
45C23×C4×C109—21no—
46C23×C4×C109D411yescharacteristic
47C23×C4×C109D411yescharacteristic
48C23×C4×C109C2311yescharacteristic
49C2×M16×C109—22no—
50C4⋊Q8×C109—22no—
51#126×C109—22no—
52#126×C109—22no—
53#125×C109—22no—
54#125×C109—22no—
55#125×C109—22no—
56#123×C109—22no—
57#123×C109—22no—
58#123×C109—22no—
59#123×C109—22no—
60#123×C109—22no—
61C22⋊Q8×C109—21no—
62C22⋊Q8×C109—21no—
63C22⋊Q8×C109—21no—
64C22⋊Q8×C109—22no—
65C22⋊Q8×C109—22no—
66C22⋊Q8×C109—21no—
67C22⋊Q8×C109—22no—
68C22⋊Q8×C109D412yes—
69C22⋊Q8×C109C2312yes—
70#121×C109—21no—
71#121×C109—21no—
72#121×C109—21no—
73#121×C109—22no—
74#121×C109—21no—
75#121×C109D412yes—
76#121×C109C2312yes—
77#120×C109—22no—
78C4×Q8×C109—22no—
79C4×D4×C109—22no—
80C4×D4×C109—22no—
81C4×D4×C109—22no—
82C4×D4×C109—22no—
83C4×D4×C109—22no—
84C4×D4×C109—22no—
85C4×D4×C109—22no—
86#117×C109—22no—
87#117×C109—21no—
88#117×C109—21no—
89#117×C109D412yes—
90C2×C4⋊C4×C109—22no—
91C2×C4⋊C4×C109—22no—
92C2×C4⋊C4×C109—22no—
93C2×C4⋊C4×C109—22no—
94C2×C4⋊C4×C109—22no—
95C2×C4⋊C4×C109—22no—
96C2×C4⋊C4×C109—22no—
97C2×C4⋊C4×C109—21no—
98C2×C4⋊C4×C109—44no—
99C2×C4⋊C4×C109D411yescharacteristic
100C2×C4⋊C4×C109D411yescharacteristic
101C2×C4⋊C4×C109C2312yes—
102C2×C4⋊C4×C109C2311yescharacteristic
103C2×C4⋊C4×C109C2311yescharacteristic
104#107×C109—24no—
105#106×C109—24no—
106#103×C109—24no—
107#102×C109—24no—
108C22⋊C8×C109—22no—
109C22⋊C8×C109C2314yes—
110#95×C109—22no—
111#95×C109—22no—
112#95×C109—21no—
113#95×C109C2312yes—
114C24×C109C2×D411yescharacteristic
115#41×C109—42no—
116#41×C109—44no—
117#41×C109—42no—
118#41×C109—82no—
119C2×Q8×C109—21no—
120C2×Q8×C109—21no—
121C2×Q8×C109C2×D412yes—
122C2×D4×C109—21no—
123C2×D4×C109—22no—
124C2×D4×C109—22no—
125C2×D4×C109—22no—
126C2×D4×C109—21no—
127C2×D4×C109C2×D412yes—
128C22×C4×C109—22no—
129C22×C4×C109—24no—
130C22×C4×C109—21no—
131C22×C4×C109—21no—
132C22×C4×C109—21no—
133C22×C4×C109—22no—
134C22×C4×C109—22no—
135C22×C4×C109—21no—
136C22×C4×C109—22no—
137C22×C4×C109—22no—
138C22×C4×C109—22no—
139C22×C4×C109—41no—
140C22×C4×C109—42no—
141C22×C4×C109—42no—
142C22×C4×C109—42no—
143C22×C4×C109—42no—
144C22×C4×C109—42no—
145C22×C4×C109—42no—
146C22×C4×C109—41no—
147C22×C4×C109—41no—
148C22×C4×C109C2×D411yescharacteristic
149C22×C4×C109C2×D412yes—
150C22×C4×C109C2×D411yescharacteristic
151C22×C4×C109C2×D411yescharacteristic
152C22×C4×C109C2×D411yescharacteristic
153C22×C4×C109C2×D411yescharacteristic
154C22×C4×C109C2×D412yes—
155C22×C4×C109C2411yescharacteristic
156M16×C109—44no—
157C2×C8×C109—24no—
158C4⋊C4×C109—21no—
159C4⋊C4×C109—22no—
160C4⋊C4×C109—21no—
161C4⋊C4×C109—21no—
162C4⋊C4×C109—21no—
163C4⋊C4×C109—22no—
164C4⋊C4×C109—22no—
165C4⋊C4×C109—22no—
166C4⋊C4×C109—22no—
167C4⋊C4×C109—22no—
168C4⋊C4×C109—22no—
169C4⋊C4×C109—22no—
170C4⋊C4×C109—22no—
171C4⋊C4×C109—22no—
172C4⋊C4×C109—24no—
173C4⋊C4×C109—44no—
174C4⋊C4×C109—42no—
175C4⋊C4×C109—44no—
176C4⋊C4×C109C2×D412yes—
177C4⋊C4×C109C2×D412yes—
178C22⋊C4×C109—22no—
179C22⋊C4×C109—22no—
180C22⋊C4×C109—22no—
181C22⋊C4×C109—22no—
182C22⋊C4×C109—22no—
183C22⋊C4×C109—22no—
184C22⋊C4×C109—22no—
185C22⋊C4×C109—22no—
186C22⋊C4×C109—22no—
187C42×C109—22no—
188C42×C109—22no—
189C42×C109—22no—
190C23×C109—21no—
191C23×C109—21no—
192C23×C109—22no—
193C23×C109—22no—
194C23×C109—22no—
195C23×C109#12011yescharacteristic
196C23×C109#12011yescharacteristic
197C23×C109C22×D411yescharacteristic
198Q8×C109—42no—
199Q8×C109—42no—
200Q8×C109—81no—
201D4×C109—42no—
202D4×C109—42no—
203D4×C109—42no—
204D4×C109—42no—
205D4×C109—44no—
206D4×C109—44no—
207D4×C109—81no—
208C2×C4×C109—21no—
209C2×C4×C109—21no—
210C2×C4×C109—24no—
211C2×C4×C109—24no—
212C2×C4×C109—22no—
213C2×C4×C109—22no—
214C2×C4×C109—21no—
215C2×C4×C109—22no—
216C2×C4×C109—21no—
217C2×C4×C109—21no—
218C2×C4×C109—21no—
219C2×C4×C109—21no—
220C2×C4×C109—42no—
221C2×C4×C109—41no—
222C2×C4×C109—41no—
223C2×C4×C109—42no—
224C2×C4×C109—42no—
225C2×C4×C109—44no—
226C2×C4×C109—48no—
227C2×C4×C109—42no—
228C2×C4×C109—42no—
229C2×C4×C109—42no—
230C2×C4×C109—41no—
231C2×C4×C109—41no—
232C2×C4×C109—42no—
233C2×C4×C109—42no—
234C2×C4×C109—42no—
235C2×C4×C109—82no—
236C2×C4×C109—81no—
237C2×C4×C109—82no—
238C2×C4×C109—82no—
239C2×C4×C109—82no—
240C2×C4×C109#12012yes—
241C2×C4×C109AGL(1,ℤ/8)12yes—
242C2×C4×C109C22×D412yes—
243C8×C109—44no—
244C22×C109—24no—
245C22×C109—24no—
246C22×C109—21no—
247C22×C109—21no—
248C22×C109—21no—
249C22×C109—21no—
250C22×C109—21no—
251C22×C109—21no—
252C22×C109—42no—
253C22×C109—42no—
254C22×C109—42no—
255C22×C109#45612yes—
256C22×C109#45812yes—
257C22×C109C2×#12011yescharacteristic
258C22×C109C2×AGL(1,ℤ/8)11yescharacteristic
259C22×C109#57511yescharacteristic
260C4×C109—22no—
261C4×C109—22no—
262C4×C109—22no—
263C4×C109—44no—
264C4×C109—41no—
265C4×C109—42no—
266C4×C109—42no—
267C4×C109—41no—
268C4×C109—41no—
269C4×C109—42no—
270C4×C109—82no—
271#18171C10911yesmaximal, characteristic, direct product factor, Sylow
272C2×C109—21no—
273C2×C109—21no—
274C2×C109—22no—
275C2×C109—22no—
276C2×C109—82no—
277C2×C109#300611yescharacteristic
278C2×C109C2×#45611yescharacteristic
279C2×C109C2×#45811yescharacteristic
280#3492C2×C10911yescharacteristic
281#3092C2×C10912yescomplemented
282C2×#398C2×C10911yescharacteristic, complemented
283#1611C2×C10912yescomplemented
284#1610C2×C10912yescomplemented
285#1609C2×C10914yescomplemented
286#1504C2×C10912yescomplemented
287#1463C2×C10911yescharacteristic, complemented
288C109#1817111yescharacteristic, direct product factor, Sylow
289#554—22no—
290#553—22no—
291#552—22no—
292#547—22no—
293#544C22×C10911yescharacteristic
294#539—21no—
295#539C22×C10912yes—
296#537C22×C10911yescharacteristic
297#525—21no—
298#525—21no—
299#525C22×C10912yes—
300#520—21no—
301C22×C4⋊C4—21no—
302C22×C4⋊C4C22×C10911yescharacteristic
303C22×C4⋊C4C22×C10911yescharacteristic
304#484—22no—
305#483C22×C10914yes—
306#481—22no—
307#480C22×C10914yes—
308#428—22no—
309#417—22no—
310#407C22×C10912yes—
311#398—21no—
312#398—21no—
313#398C22×C10912yes—
314#398C22×C10912yes—
315C2×#95—21no—
316C2×#95C22×C10911yescharacteristic
317#328C22×C10914yes—
318#327C22×C10914yes—
319#324C22×C10912yes—
320#324C22×C10912yes—
321C2×#41—21no—
322C2×#41—21no—
323C2×#41D4×C10912yes—
324C23×C4—21no—
325C23×C4—21no—
326C23×C4D4×C10911yescharacteristic
327C23×C4D4×C10911yescharacteristic
328C23×C4C23×C10911yescharacteristic
329C2×M16—22no—
330C4⋊Q8—22no—
331#126—22no—
332#126—22no—
333#125—22no—
334#125—22no—
335#125—22no—
336#123—22no—
337#123—22no—
338#123—22no—
339#123—22no—
340#123—22no—
341C22⋊Q8—21no—
342C22⋊Q8—21no—
343C22⋊Q8—21no—
344C22⋊Q8—22no—
345C22⋊Q8—22no—
346C22⋊Q8—21no—
347C22⋊Q8—22no—
348C22⋊Q8D4×C10912yes—
349C22⋊Q8C23×C10912yes—
350#121—21no—
351#121—21no—
352#121—21no—
353#121—22no—
354#121—21no—
355#121D4×C10912yes—
356#121C23×C10912yes—
357#120—22no—
358C4×Q8—22no—
359C4×D4—22no—
360C4×D4—22no—
361C4×D4—22no—
362C4×D4—22no—
363C4×D4—22no—
364C4×D4—22no—
365C4×D4—22no—
366#117—22no—
367#117—21no—
368#117—21no—
369#117D4×C10912yes—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—22no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—22no—
374C2×C4⋊C4—22no—
375C2×C4⋊C4—22no—
376C2×C4⋊C4—22no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—44no—
379C2×C4⋊C4D4×C10911yescharacteristic
380C2×C4⋊C4D4×C10911yescharacteristic
381C2×C4⋊C4C23×C10912yes—
382C2×C4⋊C4C23×C10911yescharacteristic
383C2×C4⋊C4C23×C10911yescharacteristic
384#107—24no—
385#106—24no—
386#103—24no—
387#102—24no—
388C22⋊C8—22no—
389C22⋊C8C23×C10914yes—
390#95—22no—
391#95—22no—
392#95—21no—
393#95C23×C10912yes—
394C24C2×D4×C10911yescharacteristic
395#41—42no—
396#41—44no—
397#41—42no—
398#41—82no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8C2×D4×C10912yes—
402C2×D4—21no—
403C2×D4—22no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—21no—
407C2×D4C2×D4×C10912yes—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—41no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—42no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4C2×D4×C10911yescharacteristic
429C22×C4C2×D4×C10912yes—
430C22×C4C2×D4×C10911yescharacteristic
431C22×C4C2×D4×C10911yescharacteristic
432C22×C4C2×D4×C10911yescharacteristic
433C22×C4C2×D4×C10911yescharacteristic
434C22×C4C2×D4×C10912yes—
435C22×C4C24×C10911yescharacteristic
436M16—44no—
437C2×C8—24no—
438C4⋊C4—21no—
439C4⋊C4—22no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—22no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—24no—
453C4⋊C4—44no—
454C4⋊C4—42no—
455C4⋊C4—44no—
456C4⋊C4C2×D4×C10912yes—
457C4⋊C4C2×D4×C10912yes—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C42—22no—
468C42—22no—
469C42—22no—
470C23—21no—
471C23—21no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23#120×C10911yescharacteristic
476C23#120×C10911yescharacteristic
477C23C22×D4×C10911yescharacteristic
478Q8—42no—
479Q8—42no—
480Q8—81no—
481D4—42no—
482D4—42no—
483D4—42no—
484D4—42no—
485D4—44no—
486D4—44no—
487D4—81no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—24no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—42no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—48no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—82no—
516C2×C4—81no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4#120×C10912yes—
521C2×C4AGL(1,ℤ/8)×C10912yes—
522C2×C4C22×D4×C10912yes—
523C8—44no—
524C22—24no—
525C22—24no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22#456×C10912yes—
536C22#458×C10912yes—
537C22C2×#120×C10911yescharacteristic
538C22C2×AGL(1,ℤ/8)×C10911yescharacteristic
539C22#575×C10911yescharacteristic
540C4—22no—
541C4—22no—
542C4—22no—
543C4—44no—
544C4—41no—
545C4—42no—
546C4—42no—
547C4—41no—
548C4—41no—
549C4—42no—
550C4—82no—
551C2—21no—
552C2—21no—
553C2—22no—
554C2—22no—
555C2—82no—
556C2C109×#300611yescharacteristic
557C2C2×#456×C10911yescharacteristic
558C2C2×#458×C10911yescharacteristic
subgroupG/H
center, socleC22×C109C2×#120
commutator, Frattini, cosocleC22×C4C24×C109
prime radicalC109×#3492C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C109 ◁ C23×C4×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C109×#3092C2semidirect
3. C2×#398×C109C2semidirect
4. C109×#1611C2semidirect
5. C109×#1610C2semidirect
6. C109×#1609C2semidirect
7. C109×#1504C2semidirect
8. C109×#1463C2semidirect
271. #18171C109direct (N × H)
281. #3092C2×C109semidirect
282. C2×#398C2×C109semidirect
283. #1611C2×C109semidirect
284. #1610C2×C109semidirect
285. #1609C2×C109semidirect
286. #1504C2×C109semidirect
287. #1463C2×C109semidirect
288. C109#18171direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#34922
C109×#30922
C2×#398×C1092
C109×#16112
C109×#16102
C109×#16092
C109×#15042
C109×#14632
#18171109