Finite Group Databasebeta

← C109×#12705 #42130699

Subgroups — C109×#12705 #42130699

Order 27904.

1186 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1186 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3484C211yesmaximal, characteristic
2C109×#2121C214yesmaximal, complemented
3C109×#1783C211yesmaximal, characteristic, complemented
4C109×#1781C211yesmaximal, characteristic, complemented
5C2×H(5,𝔽2)×C109C411yescharacteristic, complemented
6C23×D4×C109C2211yescharacteristic
7#546×C109—24no—
8D42×C109—24no—
9#535×C109—21no—
10#534×C109—21no—
11#534×C109C2212yes—
12#525×C109—21no—
13C2×#120×C109—21no—
14C2×#120×C109C412yescomplemented
15#518×C109C411yescharacteristic, complemented
16C2×#100×C109—21no—
17C2×#100×C109C2212yescomplemented
18C2×#99×C109—21no—
19C2×#99×C109C2212yescomplemented
20#407×C109—21no—
21#379×C109—21no—
22#352×C109—24no—
23AΣL(1,𝔽16)×C109—24no—
24C25×C109C2×C411yescharacteristic
25C25×C109D411yescharacteristic
26H(5,𝔽2)×C109—42no—
27H(5,𝔽2)×C109—42no—
28C2×#41×C109—21no—
29C2×#41×C109—22no—
30C22×D4×C109—21no—
31C22×D4×C109—22no—
32C22×D4×C109—22no—
33C22×D4×C109—22no—
34C22×D4×C109—24no—
35C22×D4×C109—42no—
36C22×D4×C109—42no—
37C22×D4×C109C2×C411yescharacteristic
38C22×D4×C109C2×C412yes—
39C22×D4×C109D411yescharacteristic
40C22×D4×C109D411yescharacteristic
41C22×D4×C109C2311yescharacteristic
42C23×C4×C109D411yescharacteristic
43C2×M16×C109—22no—
44#127×C109—22no—
45#124×C109—22no—
46#123×C109—22no—
47#123×C109—22no—
48#123×C109D414yes—
49C22⋊Q8×C109—22no—
50#121×C109—22no—
51#121×C109—24no—
52#121×C109—24no—
53#121×C109—24no—
54#120×C109—24no—
55#120×C109—22no—
56#120×C109—22no—
57#120×C109—22no—
58#120×C109—22no—
59#120×C109—24no—
60#120×C109D414yes—
61C4×D4×C109—24no—
62C4×D4×C109—24no—
63#117×C109—21no—
64C2×C22⋊C4×C109—22no—
65C2×C22⋊C4×C109—21no—
66C2×C22⋊C4×C109—22no—
67C2×C22⋊C4×C109—22no—
68C2×C22⋊C4×C109—42no—
69C2×C22⋊C4×C109C2×C412yes—
70#100×C109—24no—
71#100×C109—42no—
72#99×C109—24no—
73#99×C109—42no—
74C22⋊C8×C109—42no—
75C24×C109—21no—
76C24×C109—22no—
77C24×C109—22no—
78C24×C109—22no—
79C24×C109—41no—
80C24×C109—41no—
81C24×C109—41no—
82C24×C109—82no—
83C24×C109—82no—
84C24×C109C22⋊C411yescharacteristic
85C24×C109C22⋊C411yescharacteristic
86C24×C109C22⋊C411yescharacteristic
87C24×C109C22⋊C412yescomplemented
88C24×C109C22×C411yescharacteristic
89C24×C109C2×D411yescharacteristic
90#41×C109—42no—
91#41×C109—48no—
92#41×C109—42no—
93C2×Q8×C109—21no—
94C2×D4×C109—22no—
95C2×D4×C109—22no—
96C2×D4×C109—24no—
97C2×D4×C109—22no—
98C2×D4×C109—21no—
99C2×D4×C109—22no—
100C2×D4×C109—22no—
101C2×D4×C109—22no—
102C2×D4×C109—22no—
103C2×D4×C109—24no—
104C2×D4×C109—24no—
105C2×D4×C109—22no—
106C2×D4×C109—22no—
107C2×D4×C109—22no—
108C2×D4×C109—41no—
109C2×D4×C109—41no—
110C2×D4×C109—41no—
111C2×D4×C109—41no—
112C2×D4×C109—416no—
113C2×D4×C109—48no—
114C2×D4×C109—42no—
115C2×D4×C109—42no—
116C2×D4×C109—42no—
117C2×D4×C109—42no—
118C2×D4×C109—84no—
119C2×D4×C109—84no—
120C2×D4×C109C2×D414yes—
121C22×C4×C109—21no—
122C22×C4×C109—22no—
123C22×C4×C109—22no—
124C22×C4×C109—22no—
125C22×C4×C109—22no—
126C22×C4×C109—22no—
127C22×C4×C109—41no—
128C22×C4×C109—42no—
129C22×C4×C109C22⋊C411yescharacteristic
130C22×C4×C109C22⋊C412yescomplemented
131C22×C4×C109C2×D411yescharacteristic
132M16×C109—42no—
133M16×C109—42no—
134C2×C8×C109—42no—
135C4⋊C4×C109—22no—
136C4⋊C4×C109—24no—
137C22⋊C4×C109—24no—
138C22⋊C4×C109—22no—
139C22⋊C4×C109—24no—
140C22⋊C4×C109—22no—
141C22⋊C4×C109—24no—
142C22⋊C4×C109—24no—
143C22⋊C4×C109—42no—
144C22⋊C4×C109—42no—
145C22⋊C4×C109—84no—
146C22⋊C4×C109C22⋊C414yescomplemented
147C42×C109—22no—
148C23×C109—21no—
149C23×C109—21no—
150C23×C109—21no—
151C23×C109—21no—
152C23×C109—21no—
153C23×C109—21no—
154C23×C109—21no—
155C23×C109—21no—
156C23×C109—21no—
157C23×C109—21no—
158C23×C109—22no—
159C23×C109—21no—
160C23×C109—21no—
161C23×C109—42no—
162C23×C109—44no—
163C23×C109—44no—
164C23×C109—44no—
165C23×C109—44no—
166C23×C109—44no—
167C23×C109—42no—
168C23×C109—42no—
169C23×C109—82no—
170C23×C109—82no—
171C23×C109—82no—
172C23×C109—81no—
173C23×C109—81no—
174C23×C109—81no—
175C23×C109—81no—
176C23×C109—82no—
177C23×C109—82no—
178C23×C109—82no—
179C23×C109—82no—
180C23×C109—82no—
181C23×C109—82no—
182C23×C109—82no—
183C23×C109—82no—
184C23×C109#9912yes—
185C23×C109C2×C22⋊C412yes—
186C23×C109C2×C22⋊C411yescharacteristic
187C23×C109#12012yes—
188Q8×C109—42no—
189D4×C109—28no—
190D4×C109—44no—
191D4×C109—44no—
192D4×C109—44no—
193D4×C109—42no—
194D4×C109—44no—
195D4×C109—44no—
196D4×C109—48no—
197D4×C109—42no—
198D4×C109—42no—
199D4×C109—84no—
200D4×C109—84no—
201C2×C4×C109—21no—
202C2×C4×C109—21no—
203C2×C4×C109—21no—
204C2×C4×C109—21no—
205C2×C4×C109—24no—
206C2×C4×C109—22no—
207C2×C4×C109—44no—
208C2×C4×C109—44no—
209C2×C4×C109—42no—
210C2×C4×C109—42no—
211C2×C4×C109—82no—
212C2×C4×C109—82no—
213C2×C4×C109—82no—
214C2×C4×C109—84no—
215C2×C4×C109#9912yes—
216C2×C4×C109#12012yes—
217C8×C109—82no—
218C22×C109—22no—
219C22×C109—22no—
220C22×C109—22no—
221C22×C109—22no—
222C22×C109—22no—
223C22×C109—22no—
224C22×C109—41no—
225C22×C109—42no—
226C22×C109—42no—
227C22×C109—42no—
228C22×C109—42no—
229C22×C109—42no—
230C22×C109—41no—
231C22×C109—41no—
232C22×C109—41no—
233C22×C109—44no—
234C22×C109—42no—
235C22×C109—41no—
236C22×C109—42no—
237C22×C109—41no—
238C22×C109—81no—
239C22×C109—81no—
240C22×C109—81no—
241C22×C109—81no—
242C22×C109—81no—
243C22×C109—81no—
244C22×C109—88no—
245C22×C109—88no—
246C22×C109—162no—
247C22×C109—162no—
248C22×C109#37911yescharacteristic
249C22×C109C2×#9911yescharacteristic
250C22×C109C2×#9911yescharacteristic
251C4×C109—22no—
252C4×C109—22no—
253C4×C109—44no—
254C4×C109—42no—
255C4×C109—162no—
256#12705C10911yesmaximal, characteristic, direct product factor, Sylow
257C2×C109—21no—
258C2×C109—21no—
259C2×C109—21no—
260C2×C109—42no—
261C2×C109—42no—
262C2×C109—42no—
263C2×C109—42no—
264C2×C109—81no—
265C2×C109—82no—
266C2×C109#178111yescharacteristic
267#3484C2×C10911yescharacteristic
268#2121C2×C10914yescomplemented
269#1783C2×C10911yescharacteristic, complemented
270#1781C2×C10911yescharacteristic, complemented
271C109#1270511yescharacteristic, direct product factor, Sylow
272C2×H(5,𝔽2)C4×C10911yescharacteristic, complemented
273C23×D4C22×C10911yescharacteristic
274#546—24no—
275D42—24no—
276#535—21no—
277#534—21no—
278#534C22×C10912yes—
279#525—21no—
280C2×#120—21no—
281C2×#120C4×C10912yescomplemented
282#518C4×C10911yescharacteristic, complemented
283C2×#100—21no—
284C2×#100C22×C10912yescomplemented
285C2×#99—21no—
286C2×#99C22×C10912yescomplemented
287#407—21no—
288#379—21no—
289#352—24no—
290AΣL(1,𝔽16)—24no—
291C25C2×C4×C10911yescharacteristic
292C25D4×C10911yescharacteristic
293H(5,𝔽2)—42no—
294H(5,𝔽2)—42no—
295C2×#41—21no—
296C2×#41—22no—
297C22×D4—21no—
298C22×D4—22no—
299C22×D4—22no—
300C22×D4—22no—
301C22×D4—24no—
302C22×D4—42no—
303C22×D4—42no—
304C22×D4C2×C4×C10911yescharacteristic
305C22×D4C2×C4×C10912yes—
306C22×D4D4×C10911yescharacteristic
307C22×D4D4×C10911yescharacteristic
308C22×D4C23×C10911yescharacteristic
309C23×C4D4×C10911yescharacteristic
310C2×M16—22no—
311#127—22no—
312#124—22no—
313#123—22no—
314#123—22no—
315#123D4×C10914yes—
316C22⋊Q8—22no—
317#121—22no—
318#121—24no—
319#121—24no—
320#121—24no—
321#120—24no—
322#120—22no—
323#120—22no—
324#120—22no—
325#120—22no—
326#120—24no—
327#120D4×C10914yes—
328C4×D4—24no—
329C4×D4—24no—
330#117—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—42no—
336C2×C22⋊C4C2×C4×C10912yes—
337#100—24no—
338#100—42no—
339#99—24no—
340#99—42no—
341C22⋊C8—42no—
342C24—21no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—41no—
347C24—41no—
348C24—41no—
349C24—82no—
350C24—82no—
351C24C22⋊C4×C10911yescharacteristic
352C24C22⋊C4×C10911yescharacteristic
353C24C22⋊C4×C10911yescharacteristic
354C24C22⋊C4×C10912yescomplemented
355C24C22×C4×C10911yescharacteristic
356C24C2×D4×C10911yescharacteristic
357#41—42no—
358#41—48no—
359#41—42no—
360C2×Q8—21no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—24no—
364C2×D4—22no—
365C2×D4—21no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—24no—
371C2×D4—24no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4—416no—
380C2×D4—48no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—84no—
386C2×D4—84no—
387C2×D4C2×D4×C10914yes—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—41no—
395C22×C4—42no—
396C22×C4C22⋊C4×C10911yescharacteristic
397C22×C4C22⋊C4×C10912yescomplemented
398C22×C4C2×D4×C10911yescharacteristic
399M16—42no—
400M16—42no—
401C2×C8—42no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C22⋊C4—24no—
405C22⋊C4—22no—
406C22⋊C4—24no—
407C22⋊C4—22no—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—84no—
413C22⋊C4C22⋊C4×C10914yescomplemented
414C42—22no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—22no—
426C23—21no—
427C23—21no—
428C23—42no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—42no—
435C23—42no—
436C23—82no—
437C23—82no—
438C23—82no—
439C23—81no—
440C23—81no—
441C23—81no—
442C23—81no—
443C23—82no—
444C23—82no—
445C23—82no—
446C23—82no—
447C23—82no—
448C23—82no—
449C23—82no—
450C23—82no—
451C23#99×C10912yes—
452C23C2×C22⋊C4×C10912yes—
453C23C2×C22⋊C4×C10911yescharacteristic
454C23#120×C10912yes—
455Q8—42no—
456D4—28no—
457D4—44no—
458D4—44no—
459D4—44no—
460D4—42no—
461D4—44no—
462D4—44no—
463D4—48no—
464D4—42no—
465D4—42no—
466D4—84no—
467D4—84no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—84no—
482C2×C4#99×C10912yes—
483C2×C4#120×C10912yes—
484C8—82no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—41no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—44no—
501C22—42no—
502C22—41no—
503C22—42no—
504C22—41no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—88no—
512C22—88no—
513C22—162no—
514C22—162no—
515C22#379×C10911yescharacteristic
516C22C2×#99×C10911yescharacteristic
517C22C2×#99×C10911yescharacteristic
518C4—22no—
519C4—22no—
520C4—44no—
521C4—42no—
522C4—162no—
523C2—21no—
524C2—21no—
525C2—21no—
526C2—42no—
527C2—42no—
528C2—42no—
529C2—42no—
530C2—81no—
531C2—82no—
532C2C109×#178111yescharacteristic
subgroupG/H
center, socleC2×C109#1781
commutatorC24C22×C4×C109
Frattini, cosocleC22×D4C23×C109
prime radicalC109×#3484C2
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C109 ◁ C23×C109 ◁ C23×D4×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C109×#2121C2semidirect
3. C109×#1783C2semidirect
4. C109×#1781C2semidirect
5. C2×H(5,𝔽2)×C109C4semidirect
14. C2×#120×C109C4semidirect
15. #518×C109C4semidirect
17. C2×#100×C109C22semidirect
19. C2×#99×C109C22semidirect
87. C24×C109C22⋊C4semidirect
130. C22×C4×C109C22⋊C4semidirect
146. C22⋊C4×C109C22⋊C4semidirect
256. #12705C109direct (N × H)
268. #2121C2×C109semidirect
269. #1783C2×C109semidirect
270. #1781C2×C109semidirect
271. C109#12705direct (N × H)
272. C2×H(5,𝔽2)C4×C109semidirect
281. C2×#120C4×C109semidirect
282. #518C4×C109semidirect
284. C2×#100C22×C109semidirect
286. C2×#99C22×C109semidirect
354. C24C22⋊C4×C109semidirect
397. C22×C4C22⋊C4×C109semidirect
413. C22⋊C4C22⋊C4×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#34842
C109×#21212
C109×#17832
C109×#17812
#12705109