Finite Group Databasebeta

← C109×#10238 #42128232

Subgroups — C109×#10238 #42128232

Order 27904.

794 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3205C211yesmaximal, characteristic, complemented
2C109×#3036C211yesmaximal, characteristic, complemented
3C109×#2541C211yesmaximal, characteristic, complemented
4C109×#1545C211yesmaximal, characteristic, complemented
5C109×#1502C211yesmaximal, characteristic, complemented
6C109×#1478C211yesmaximal, characteristic, complemented
7C109×#1303C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C109—21no—
9#532×C109C2211yescharacteristic
10C2×#127×C109—21no—
11C2×#127×C109C2211yescharacteristic
12C2×#121×C109—22no—
13C2×C4×D4×C109C2211yescharacteristic
14#493×C109—21no—
15#486×C109—21no—
16#463×C109—22no—
17#461×C109—21no—
18#459×C109—21no—
19#459×C109—22no—
20#447×C109—22no—
21#431×C109C2211yescharacteristic
22#431×C109C2211yescharacteristic
23#423×C109C2211yescharacteristic
24#417×C109—21no—
25#392×C109—22no—
26#390×C109—21no—
27C4×C4⋊C4×C109C2211yescharacteristic
28#331×C109—22no—
29#327×C109—22no—
30D4⋊C8×C109—22no—
31C2×#41×C109—21no—
32C22×D4×C109—24no—
33C22×D4×C109—21no—
34C22×D4×C109—21no—
35C22×D4×C109D411yescharacteristic
36C22×D4×C109D412yes—
37C23×C4×C109—21no—
38AGL(1,ℤ/8)×C109—42no—
39AGL(1,ℤ/8)×C109—42no—
40C2×SD16×C109—22no—
41C2×D8×C109—22no—
42C2×D8×C109—24no—
43C2×M16×C109—21no—
44C4⋊Q8×C109D411yescharacteristic, complemented
45#127×C109—21no—
46#127×C109—21no—
47#127×C109—21no—
48#127×C109—21no—
49#127×C109—42no—
50#127×C109—42no—
51#127×C109D411yescharacteristic, complemented
52#124×C109—21no—
53#121×C109—21no—
54#121×C109—21no—
55#121×C109—48no—
56C4×D4×C109—21no—
57C4×D4×C109—21no—
58C4×D4×C109—21no—
59C4×D4×C109—21no—
60C4×D4×C109—21no—
61C4×D4×C109—21no—
62C2×C4⋊C4×C109—21no—
63C2×C4⋊C4×C109D411yescharacteristic
64C2×C22⋊C4×C109—24no—
65C2×C22⋊C4×C109—21no—
66C2×C42×C109—21no—
67C2×C42×C109C2311yescharacteristic
68C4⋊C8×C109—21no—
69C4⋊C8×C109—21no—
70C4⋊C8×C109D412yescomplemented
71C4⋊C8×C109D412yescomplemented
72#103×C109—22no—
73#102×C109—22no—
74#102×C109—24no—
75C22⋊C8×C109—21no—
76C22⋊C8×C109—21no—
77C4×C8×C109—21no—
78C4×C8×C109—21no—
79#95×C109D412yes—
80C24×C109—21no—
81C24×C109—22no—
82#41×C109—41no—
83#41×C109—41no—
84#41×C109—41no—
85#41×C109—41no—
86C2×Q8×C109—21no—
87C2×D4×C109—21no—
88C2×D4×C109—21no—
89C2×D4×C109—22no—
90C2×D4×C109—21no—
91C2×D4×C109—21no—
92C2×D4×C109—21no—
93C2×D4×C109—22no—
94C2×D4×C109—21no—
95C2×D4×C109—21no—
96C2×D4×C109—21no—
97C2×D4×C109—44no—
98C2×D4×C109—48no—
99C2×D4×C109—48no—
100C2×D4×C109—41no—
101C2×D4×C109—41no—
102C2×D4×C109—41no—
103C2×D4×C109—41no—
104C2×D4×C109—41no—
105C2×D4×C109—41no—
106C2×D4×C109—41no—
107C2×D4×C109—41no—
108C2×D4×C109—41no—
109C2×D4×C109—82no—
110C2×D4×C109—84no—
111C2×D4×C109D812yescomplemented
112C22×C4×C109—22no—
113C22×C4×C109—21no—
114C22×C4×C109—44no—
115C22×C4×C109—41no—
116C22×C4×C109—41no—
117C22×C4×C109—41no—
118C22×C4×C109C2×D411yescharacteristic
119C22×C4×C109C2×D411yescharacteristic
120C22×C4×C109C2×D411yescharacteristic
121SD16×C109—44no—
122D8×C109—44no—
123D8×C109—84no—
124M16×C109—41no—
125M16×C109—41no—
126C2×C8×C109—22no—
127C2×C8×C109—22no—
128C2×C8×C109—41no—
129C4⋊C4×C109—24no—
130C4⋊C4×C109—21no—
131C4⋊C4×C109—21no—
132C4⋊C4×C109—21no—
133C4⋊C4×C109—21no—
134C22⋊C4×C109—48no—
135C22⋊C4×C109—41no—
136C22⋊C4×C109—41no—
137C22⋊C4×C109—41no—
138C22⋊C4×C109—41no—
139C42×C109—41no—
140C42×C109—41no—
141C42×C109C2×D411yescharacteristic
142C42×C109C2×D411yescharacteristic
143C42×C109C2×D411yescharacteristic
144C42×C109#4111yescharacteristic
145C23×C109—22no—
146C23×C109—42no—
147C23×C109—42no—
148C23×C109—44no—
149C23×C109—44no—
150C23×C109—41no—
151C23×C109—41no—
152C23×C109—41no—
153C23×C109—42no—
154C23×C109—42no—
155C23×C109—42no—
156C23×C109#12011yescharacteristic
157Q8×C109—41no—
158Q8×C109—41no—
159D4×C109—22no—
160D4×C109—22no—
161D4×C109—41no—
162D4×C109—41no—
163D4×C109—41no—
164D4×C109—42no—
165D4×C109—44no—
166D4×C109—41no—
167D4×C109—81no—
168D4×C109—82no—
169D4×C109—82no—
170D4×C109—82no—
171D4×C109—816no—
172D4×C109—81no—
173D4×C109—81no—
174D4×C109—81no—
175D4×C109—81no—
176D4×C109—81no—
177D4×C109—81no—
178C2×C4×C109—22no—
179C2×C4×C109—24no—
180C2×C4×C109—41no—
181C2×C4×C109—42no—
182C2×C4×C109—42no—
183C2×C4×C109—42no—
184C2×C4×C109—41no—
185C2×C4×C109—41no—
186C2×C4×C109—82no—
187C2×C4×C109—82no—
188C2×C4×C109—81no—
189C2×C4×C109—81no—
190C2×C4×C109#12011yescharacteristic
191C2×C4×C109#12011yescharacteristic
192C2×C4×C109#12111yescharacteristic
193C2×C4×C109#12111yescharacteristic
194C2×C4×C109#12111yescharacteristic
195C2×C4×C109#12711yescharacteristic
196C2×C4×C109C2×D811yescharacteristic
197C2×C4×C109#13511yescharacteristic
198C2×C4×C109AGL(1,ℤ/8)11yescharacteristic
199C2×C4×C109AGL(1,ℤ/8)11yescharacteristic
200C8×C109—42no—
201C8×C109—42no—
202C8×C109—81no—
203C22×C109—21no—
204C22×C109—21no—
205C22×C109—44no—
206C22×C109—42no—
207C22×C109—82no—
208C22×C109—84no—
209C22×C109—84no—
210C22×C109—81no—
211C22×C109—81no—
212C22×C109—81no—
213C22×C109—82no—
214C22×C109—82no—
215C22×C109—81no—
216C22×C109—82no—
217C22×C109—81no—
218C22×C109#39211yescharacteristic
219C22×C109#44711yescharacteristic
220C22×C109#44911yescharacteristic
221C4×C109—21no—
222C4×C109—21no—
223C4×C109—21no—
224C4×C109—44no—
225C4×C109—81no—
226C4×C109—81no—
227C4×C109#44711yescharacteristic
228C4×C109#44911yescharacteristic
229C4×C109#45312yes—
230C4×C109#46611yescharacteristic
231C4×C109#46911yescharacteristic
232#10238C10911yesmaximal, characteristic, direct product factor, Sylow
233C2×C109—21no—
234C2×C109—21no—
235C2×C109—82no—
236C2×C109—161no—
237C2×C109—162no—
238C2×C109#199911yescharacteristic
239C2×C109#200211yescharacteristic
240C2×C109#200811yescharacteristic
241#3205C2×C10911yescharacteristic, complemented
242#3036C2×C10911yescharacteristic, complemented
243#2541C2×C10911yescharacteristic, complemented
244#1545C2×C10911yescharacteristic, complemented
245#1502C2×C10911yescharacteristic, complemented
246#1478C2×C10911yescharacteristic, complemented
247#1303C2×C10911yescharacteristic, complemented
248C109#1023811yescharacteristic, direct product factor, Sylow
249C2×AGL(1,ℤ/8)—21no—
250#532C22×C10911yescharacteristic
251C2×#127—21no—
252C2×#127C22×C10911yescharacteristic
253C2×#121—22no—
254C2×C4×D4C22×C10911yescharacteristic
255#493—21no—
256#486—21no—
257#463—22no—
258#461—21no—
259#459—21no—
260#459—22no—
261#447—22no—
262#431C22×C10911yescharacteristic
263#431C22×C10911yescharacteristic
264#423C22×C10911yescharacteristic
265#417—21no—
266#392—22no—
267#390—21no—
268C4×C4⋊C4C22×C10911yescharacteristic
269#331—22no—
270#327—22no—
271D4⋊C8—22no—
272C2×#41—21no—
273C22×D4—24no—
274C22×D4—21no—
275C22×D4—21no—
276C22×D4D4×C10911yescharacteristic
277C22×D4D4×C10912yes—
278C23×C4—21no—
279AGL(1,ℤ/8)—42no—
280AGL(1,ℤ/8)—42no—
281C2×SD16—22no—
282C2×D8—22no—
283C2×D8—24no—
284C2×M16—21no—
285C4⋊Q8D4×C10911yescharacteristic, complemented
286#127—21no—
287#127—21no—
288#127—21no—
289#127—21no—
290#127—42no—
291#127—42no—
292#127D4×C10911yescharacteristic, complemented
293#124—21no—
294#121—21no—
295#121—21no—
296#121—48no—
297C4×D4—21no—
298C4×D4—21no—
299C4×D4—21no—
300C4×D4—21no—
301C4×D4—21no—
302C4×D4—21no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4D4×C10911yescharacteristic
305C2×C22⋊C4—24no—
306C2×C22⋊C4—21no—
307C2×C42—21no—
308C2×C42C23×C10911yescharacteristic
309C4⋊C8—21no—
310C4⋊C8—21no—
311C4⋊C8D4×C10912yescomplemented
312C4⋊C8D4×C10912yescomplemented
313#103—22no—
314#102—22no—
315#102—24no—
316C22⋊C8—21no—
317C22⋊C8—21no—
318C4×C8—21no—
319C4×C8—21no—
320#95D4×C10912yes—
321C24—21no—
322C24—22no—
323#41—41no—
324#41—41no—
325#41—41no—
326#41—41no—
327C2×Q8—21no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—22no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—22no—
335C2×D4—21no—
336C2×D4—21no—
337C2×D4—21no—
338C2×D4—44no—
339C2×D4—48no—
340C2×D4—48no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—82no—
351C2×D4—84no—
352C2×D4D8×C10912yescomplemented
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—44no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4C2×D4×C10911yescharacteristic
360C22×C4C2×D4×C10911yescharacteristic
361C22×C4C2×D4×C10911yescharacteristic
362SD16—44no—
363D8—44no—
364D8—84no—
365M16—41no—
366M16—41no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—41no—
370C4⋊C4—24no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C22⋊C4—48no—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C22⋊C4—41no—
380C42—41no—
381C42—41no—
382C42C2×D4×C10911yescharacteristic
383C42C2×D4×C10911yescharacteristic
384C42C2×D4×C10911yescharacteristic
385C42#41×C10911yescharacteristic
386C23—22no—
387C23—42no—
388C23—42no—
389C23—44no—
390C23—44no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23#120×C10911yescharacteristic
398Q8—41no—
399Q8—41no—
400D4—22no—
401D4—22no—
402D4—41no—
403D4—41no—
404D4—41no—
405D4—42no—
406D4—44no—
407D4—41no—
408D4—82no—
409D4—81no—
410D4—82no—
411D4—82no—
412D4—816no—
413D4—81no—
414D4—81no—
415D4—81no—
416D4—81no—
417D4—81no—
418D4—81no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4#120×C10911yescharacteristic
432C2×C4#120×C10911yescharacteristic
433C2×C4#121×C10911yescharacteristic
434C2×C4#121×C10911yescharacteristic
435C2×C4#121×C10911yescharacteristic
436C2×C4#127×C10911yescharacteristic
437C2×C4C2×D8×C10911yescharacteristic
438C2×C4#135×C10911yescharacteristic
439C2×C4AGL(1,ℤ/8)×C10911yescharacteristic
440C2×C4AGL(1,ℤ/8)×C10911yescharacteristic
441C8—42no—
442C8—42no—
443C8—81no—
444C22—21no—
445C22—21no—
446C22—44no—
447C22—42no—
448C22—82no—
449C22—84no—
450C22—84no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22—82no—
455C22—82no—
456C22—81no—
457C22—82no—
458C22—81no—
459C22#392×C10911yescharacteristic
460C22#447×C10911yescharacteristic
461C22#449×C10911yescharacteristic
462C4—21no—
463C4—21no—
464C4—21no—
465C4—44no—
466C4—81no—
467C4—81no—
468C4#447×C10911yescharacteristic
469C4#449×C10911yescharacteristic
470C4#453×C10912yes—
471C4#466×C10911yescharacteristic
472C4#469×C10911yescharacteristic
473C2—21no—
474C2—21no—
475C2—82no—
476C2—162no—
477C2—161no—
478C2C109×#199911yescharacteristic
479C2C109×#200211yescharacteristic
480C2C109×#200811yescharacteristic
subgroupG/H
center, socleC22×C109#392
commutator, Frattini, cosocleC2×C42C23×C109
LCSG ▷ C2×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. C109×#3205C2semidirect
2. C109×#3036C2semidirect
3. C109×#2541C2semidirect
4. C109×#1545C2semidirect
5. C109×#1502C2semidirect
6. C109×#1478C2semidirect
7. C109×#1303C2semidirect
44. C4⋊Q8×C109D4semidirect
51. #127×C109D4semidirect
70. C4⋊C8×C109D4semidirect
71. C4⋊C8×C109D4semidirect
111. C2×D4×C109D8semidirect
232. #10238C109direct (N × H)
241. #3205C2×C109semidirect
242. #3036C2×C109semidirect
243. #2541C2×C109semidirect
244. #1545C2×C109semidirect
245. #1502C2×C109semidirect
246. #1478C2×C109semidirect
247. #1303C2×C109semidirect
248. C109#10238direct (N × H)
285. C4⋊Q8D4×C109semidirect
292. #127D4×C109semidirect
311. C4⋊C8D4×C109semidirect
312. C4⋊C8D4×C109semidirect
352. C2×D4D8×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#32052
C109×#30362
C109×#25412
C109×#15452
C109×#15022
C109×#14782
C109×#13032
#10238109