Finite Group Databasebeta

← C109×#9713 #42127707

Subgroups — C109×#9713 #42127707

Order 27904.

742 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#2850C211yesmaximal, characteristic
2C109×#1797C211yesmaximal, characteristic, complemented
3C2×#328×C109C212yesmaximal, complemented
4C2×#327×C109C212yesmaximal
5C4×#95×C109C211yesmaximal, characteristic, complemented
6C2×#124×C109C411yescharacteristic, complemented
7C2×C22⋊Q8×C109C2211yescharacteristic
8C2×#121×C109C2211yescharacteristic
9C22×C42×C109—21no—
10C22×C42×C109C2211yescharacteristic
11C2×C4⋊C8×C109—21no—
12C2×C22⋊C8×C109C2212yes—
13#391×C109C411yescharacteristic, complemented
14#390×C109C411yescharacteristic, complemented
15#388×C109—21no—
16#388×C109—21no—
17#386×C109—21no—
18#386×C109—21no—
19#382×C109C411yescharacteristic, complemented
20C2×#95×C109—21no—
21C2×#95×C109C2212yes—
22#336×C109—21no—
23#328×C109—24no—
24#327×C109—24no—
25C22×Q8×C109C2×C411yescharacteristic
26C22×D4×C109—21no—
27C22×D4×C109C2×C411yescharacteristic
28C23×C4×C109—22no—
29C23×C4×C109D411yescharacteristic
30C23×C4×C109D411yescharacteristic
31C23×C4×C109C2311yescharacteristic
32C22×C8×C109—22no—
33#124×C109—42no—
34C22⋊Q8×C109—22no—
35C22⋊Q8×C109—22no—
36#121×C109—22no—
37#121×C109—22no—
38C2×C4⋊C4×C109—21no—
39C2×C4⋊C4×C109C2×C411yescharacteristic
40C2×C4⋊C4×C109C2×C411yescharacteristic
41C2×C22⋊C4×C109—21no—
42C2×C22⋊C4×C109—21no—
43C2×C22⋊C4×C109—21no—
44C2×C22⋊C4×C109—21no—
45C2×C22⋊C4×C109—21no—
46C2×C22⋊C4×C109—21no—
47C2×C42×C109—24no—
48C2×C42×C109—42no—
49C2×C42×C109—42no—
50C2×C42×C109—42no—
51C2×C42×C109C2×C411yescharacteristic
52C2×C42×C109C2×C411yescharacteristic
53C2×C42×C109D411yescharacteristic
54C2×C42×C109D411yescharacteristic
55C4⋊C8×C109—42no—
56C22⋊C8×C109—22no—
57C22⋊C8×C109—22no—
58#95×C109—24no—
59#95×C109—21no—
60#95×C109—21no—
61#95×C109—21no—
62#95×C109—21no—
63#95×C109—22no—
64#95×C109D414yes—
65C24×C109—21no—
66C24×C109C2×D411yescharacteristic
67C2×Q8×C109—22no—
68C2×Q8×C109—41no—
69C2×Q8×C109—41no—
70C2×D4×C109—22no—
71C2×D4×C109—41no—
72C2×D4×C109—41no—
73C2×D4×C109—42no—
74C2×D4×C109—42no—
75C2×D4×C109—41no—
76C2×D4×C109—41no—
77C22×C4×C109—21no—
78C22×C4×C109—21no—
79C22×C4×C109—22no—
80C22×C4×C109—22no—
81C22×C4×C109—24no—
82C22×C4×C109—24no—
83C22×C4×C109—24no—
84C22×C4×C109—22no—
85C22×C4×C109—22no—
86C22×C4×C109—21no—
87C22×C4×C109—21no—
88C22×C4×C109—21no—
89C22×C4×C109—41no—
90C22×C4×C109—41no—
91C22×C4×C109—41no—
92C22×C4×C109—42no—
93C22×C4×C109—42no—
94C22×C4×C109—42no—
95C22×C4×C109—42no—
96C22×C4×C109C22⋊C411yescharacteristic
97C22×C4×C109C22⋊C411yescharacteristic
98C22×C4×C109C22⋊C411yescharacteristic
99C22×C4×C109C22⋊C411yescharacteristic
100C22×C4×C109C22×C411yescharacteristic
101C22×C4×C109C2×D411yescharacteristic
102C22×C4×C109C2×D412yes—
103C22×C4×C109#4112yes—
104C2×C8×C109—42no—
105C2×C8×C109—42no—
106C2×C8×C109—42no—
107C4⋊C4×C109—22no—
108C4⋊C4×C109—22no—
109C4⋊C4×C109—41no—
110C4⋊C4×C109—41no—
111C22⋊C4×C109—42no—
112C22⋊C4×C109—42no—
113C22⋊C4×C109—42no—
114C22⋊C4×C109—42no—
115C22⋊C4×C109—42no—
116C22⋊C4×C109—41no—
117C22⋊C4×C109—41no—
118C42×C109—22no—
119C42×C109—22no—
120C42×C109—41no—
121C42×C109—41no—
122C42×C109—48no—
123C23×C109—21no—
124C23×C109—21no—
125C23×C109—21no—
126C23×C109—21no—
127C23×C109—41no—
128C23×C109—41no—
129C23×C109—41no—
130C23×C109—41no—
131C23×C109—41no—
132C23×C109—41no—
133C23×C109—41no—
134C23×C109C2×C22⋊C411yescharacteristic
135C23×C109#12112yes—
136C23×C109#13512yes—
137C23×C109AGL(1,ℤ/8)11yescharacteristic
138C23×C109#13711yescharacteristic
139Q8×C109—82no—
140D4×C109—82no—
141D4×C109—82no—
142D4×C109—82no—
143C2×C4×C109—24no—
144C2×C4×C109—24no—
145C2×C4×C109—21no—
146C2×C4×C109—21no—
147C2×C4×C109—21no—
148C2×C4×C109—21no—
149C2×C4×C109—21no—
150C2×C4×C109—21no—
151C2×C4×C109—21no—
152C2×C4×C109—21no—
153C2×C4×C109—21no—
154C2×C4×C109—21no—
155C2×C4×C109—28no—
156C2×C4×C109—21no—
157C2×C4×C109—21no—
158C2×C4×C109—21no—
159C2×C4×C109—42no—
160C2×C4×C109—42no—
161C2×C4×C109—44no—
162C2×C4×C109—44no—
163C2×C4×C109—44no—
164C2×C4×C109—44no—
165C2×C4×C109—44no—
166C2×C4×C109—44no—
167C2×C4×C109—44no—
168C2×C4×C109—41no—
169C2×C4×C109—41no—
170C2×C4×C109—41no—
171C2×C4×C109—41no—
172C2×C4×C109—41no—
173C2×C4×C109—41no—
174C2×C4×C109—41no—
175C2×C4×C109—41no—
176C2×C4×C109—41no—
177C2×C4×C109#9912yes—
178C2×C4×C109#10412yes—
179C2×C4×C109#10412yes—
180C2×C4×C109C4×D412yes—
181C2×C4×C109#12411yescharacteristic
182C2×C4×C109#12711yescharacteristic
183C8×C109—82no—
184C22×C109—22no—
185C22×C109—22no—
186C22×C109—22no—
187C22×C109—22no—
188C22×C109—21no—
189C22×C109—21no—
190C22×C109—21no—
191C22×C109—21no—
192C22×C109—81no—
193C22×C109—81no—
194C22×C109—81no—
195C22×C109—81no—
196C22×C109—81no—
197C22×C109—81no—
198C22×C109—81no—
199C22×C109#39011yescharacteristic
200C22×C109C2×#9911yescharacteristic
201C22×C109#41011yescharacteristic
202C22×C109#41611yescharacteristic
203C22×C109#41711yescharacteristic
204C22×C109C2×#10411yescharacteristic
205C22×C109C2×#10411yescharacteristic
206C22×C109#46312yes—
207C22×C109#46412yes—
208C4×C109—21no—
209C4×C109—22no—
210C4×C109—21no—
211C4×C109—22no—
212C4×C109—22no—
213C4×C109—22no—
214C4×C109—48no—
215C4×C109—41no—
216C4×C109—81no—
217C4×C109—81no—
218C4×C109—81no—
219#9713C10911yesmaximal, characteristic, direct product factor, Sylow
220C2×C109—22no—
221C2×C109—22no—
222C2×C109—161no—
223C2×C109#150011yescharacteristic
224C2×C109#150111yescharacteristic
225C2×C109#150411yescharacteristic
226C2×C109#150911yescharacteristic
227C2×C109#196211yescharacteristic
228C2×C109#196411yescharacteristic
229C2×C109#198311yescharacteristic
230#2850C2×C10911yescharacteristic
231#1797C2×C10911yescharacteristic, complemented
232C2×#328C2×C10912yescomplemented
233C2×#327C2×C10912yes—
234C4×#95C2×C10911yescharacteristic, complemented
235C109#971311yescharacteristic, direct product factor, Sylow
236C2×#124C4×C10911yescharacteristic, complemented
237C2×C22⋊Q8C22×C10911yescharacteristic
238C2×#121C22×C10911yescharacteristic
239C22×C42—21no—
240C22×C42C22×C10911yescharacteristic
241C2×C4⋊C8—21no—
242C2×C22⋊C8C22×C10912yes—
243#391C4×C10911yescharacteristic, complemented
244#390C4×C10911yescharacteristic, complemented
245#388—21no—
246#388—21no—
247#386—21no—
248#386—21no—
249#382C4×C10911yescharacteristic, complemented
250C2×#95—21no—
251C2×#95C22×C10912yes—
252#336—21no—
253#328—24no—
254#327—24no—
255C22×Q8C2×C4×C10911yescharacteristic
256C22×D4—21no—
257C22×D4C2×C4×C10911yescharacteristic
258C23×C4—22no—
259C23×C4D4×C10911yescharacteristic
260C23×C4D4×C10911yescharacteristic
261C23×C4C23×C10911yescharacteristic
262C22×C8—22no—
263#124—42no—
264C22⋊Q8—22no—
265C22⋊Q8—22no—
266#121—22no—
267#121—22no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4C2×C4×C10911yescharacteristic
270C2×C4⋊C4C2×C4×C10911yescharacteristic
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C42—24no—
278C2×C42—42no—
279C2×C42—42no—
280C2×C42—42no—
281C2×C42C2×C4×C10911yescharacteristic
282C2×C42C2×C4×C10911yescharacteristic
283C2×C42D4×C10911yescharacteristic
284C2×C42D4×C10911yescharacteristic
285C4⋊C8—42no—
286C22⋊C8—22no—
287C22⋊C8—22no—
288#95—24no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—22no—
294#95D4×C10914yes—
295C24—21no—
296C24C2×D4×C10911yescharacteristic
297C2×Q8—22no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×D4—22no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—41no—
306C2×D4—41no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4C22⋊C4×C10911yescharacteristic
327C22×C4C22⋊C4×C10911yescharacteristic
328C22×C4C22⋊C4×C10911yescharacteristic
329C22×C4C22⋊C4×C10911yescharacteristic
330C22×C4C22×C4×C10911yescharacteristic
331C22×C4C2×D4×C10911yescharacteristic
332C22×C4C2×D4×C10912yes—
333C22×C4#41×C10912yes—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C22⋊C4—42no—
342C22⋊C4—42no—
343C22⋊C4—42no—
344C22⋊C4—42no—
345C22⋊C4—42no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C42—22no—
349C42—22no—
350C42—41no—
351C42—41no—
352C42—48no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23C2×C22⋊C4×C10911yescharacteristic
365C23#121×C10912yes—
366C23#135×C10912yes—
367C23AGL(1,ℤ/8)×C10911yescharacteristic
368C23#137×C10911yescharacteristic
369Q8—82no—
370D4—82no—
371D4—82no—
372D4—82no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—28no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4#99×C10912yes—
408C2×C4#104×C10912yes—
409C2×C4#104×C10912yes—
410C2×C4C4×D4×C10912yes—
411C2×C4#124×C10911yescharacteristic
412C2×C4#127×C10911yescharacteristic
413C8—82no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22#390×C10911yescharacteristic
430C22C2×#99×C10911yescharacteristic
431C22#410×C10911yescharacteristic
432C22#416×C10911yescharacteristic
433C22#417×C10911yescharacteristic
434C22C2×#104×C10911yescharacteristic
435C22C2×#104×C10911yescharacteristic
436C22#463×C10912yes—
437C22#464×C10912yes—
438C4—21no—
439C4—22no—
440C4—21no—
441C4—22no—
442C4—22no—
443C4—22no—
444C4—48no—
445C4—41no—
446C4—81no—
447C4—81no—
448C4—81no—
449C2—22no—
450C2—22no—
451C2—161no—
452C2C109×#150011yescharacteristic
453C2C109×#150111yescharacteristic
454C2C109×#150411yescharacteristic
455C2C109×#150911yescharacteristic
456C2C109×#196211yescharacteristic
457C2C109×#196411yescharacteristic
458C2C109×#198311yescharacteristic
subgroupG/H
center, socleC23×C109C2×C22⋊C4
commutatorC22×C4C22×C4×C109
Frattini, cosocleC23×C4C23×C109
prime radicalC2×#121×C109C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C109 ◁ C22×C42×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C109×#1797C2semidirect
3. C2×#328×C109C2semidirect
5. C4×#95×C109C2semidirect
6. C2×#124×C109C4semidirect
13. #391×C109C4semidirect
14. #390×C109C4semidirect
19. #382×C109C4semidirect
219. #9713C109direct (N × H)
231. #1797C2×C109semidirect
232. C2×#328C2×C109semidirect
234. C4×#95C2×C109semidirect
235. C109#9713direct (N × H)
236. C2×#124C4×C109semidirect
243. #391C4×C109semidirect
244. #390C4×C109semidirect
249. #382C4×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#28502
C109×#17972
C2×#328×C1092
C2×#327×C1092
C4×#95×C1092
#9713109