Finite Group Databasebeta

← C109×#12751 #42130745

Subgroups — C109×#12751 #42130745

Order 27904.

876 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#578×C109C211yesmaximal, characteristic
2C109×#2148C211yesmaximal, characteristic, complemented
3C2×#362×C109C211yesmaximal, characteristic, complemented
4C109×#2145C211yesmaximal, characteristic, complemented
5C109×#2141C211yesmaximal, characteristic, complemented
6C109×#2117C211yesmaximal, characteristic, complemented
7C109×#1814C211yesmaximal, characteristic, complemented
8C2×2−1+4×C109—21no—
9#578×C109—22no—
10#578×C109—22no—
11#578×C109—22no—
12#578×C109C414yescomplemented
13C2×#137×C109—22no—
14C2×#137×C109—21no—
15C2×#135×C109—21no—
16C2×#135×C109C2211yescharacteristic
17C22×Q16×C109—21no—
18C22×Q16×C109C2211yescharacteristic
19C2×#131×C109C2211yescharacteristic
20C2×M32×C109C2211yescharacteristic
21C2×M32×C109C2211yescharacteristic
22C2×#108×C109C2211yescharacteristic
23#427×C109C2211yescharacteristic
24#421×C109—21no—
25#362×C109—22no—
26#362×C109—21no—
27#361×C109—21no—
28#360×C109—22no—
29#358×C109—22no—
30#350×C109—22no—
31#344×C109—21no—
322−1+4×C109—24no—
332−1+4×C109—24no—
342−1+4×C109—42no—
352−1+4×C109—42no—
36C2×#41×C109—22no—
37C2×#41×C109—21no—
38C2×#41×C109—42no—
39C2×#41×C109—41no—
40C2×#41×C109D411yescharacteristic
41C22×Q8×C109—22no—
42C22×Q8×C109—21no—
43C22×Q8×C109—41no—
44#137×C109—24no—
45#137×C109—28no—
46#137×C109—28no—
47#137×C109—24no—
48#135×C109—24no—
49#135×C109—24no—
50#135×C109—21no—
51#135×C109—21no—
52#135×C109C2×C412yes—
53#135×C109D412yes—
54C2×Q16×C109—22no—
55C2×Q16×C109—24no—
56C2×Q16×C109—24no—
57C2×Q16×C109—21no—
58C2×Q16×C109—21no—
59C2×Q16×C109—21no—
60C2×Q16×C109C2×C412yes—
61C2×Q16×C109D411yescharacteristic
62C2×Q16×C109D412yes—
63C2×Q16×C109D412yes—
64C2×SD16×C109—22no—
65C2×SD16×C109—21no—
66C2×D8×C109D411yescharacteristic
67#131×C109—21no—
68#131×C109—21no—
69#131×C109C2×C412yes—
70#131×C109D412yes—
71C2×M16×C109—21no—
72C2×M16×C109—21no—
73C2×M16×C109D411yescharacteristic
74C22×C8×C109—21no—
75C22×C8×C109C2311yescharacteristic
76#117×C109—21no—
77M32×C109—21no—
78M32×C109—21no—
79M32×C109—21no—
80M32×C109—21no—
81C2×C16×C109—21no—
82C2×C16×C109—21no—
83#108×C109—21no—
84#108×C109—21no—
85#107×C109—21no—
86#106×C109—21no—
87#104×C109—42no—
88#41×C109—21no—
89#41×C109—21no—
90#41×C109—28no—
91#41×C109—22no—
92#41×C109—22no—
93#41×C109—44no—
94#41×C109—48no—
95#41×C109—48no—
96#41×C109—44no—
97#41×C109—44no—
98#41×C109—41no—
99#41×C109—41no—
100#41×C109C22⋊C412yes—
101#41×C109D812yes—
102C2×Q8×C109—28no—
103C2×Q8×C109—22no—
104C2×Q8×C109—22no—
105C2×Q8×C109—21no—
106C2×Q8×C109—22no—
107C2×Q8×C109—42no—
108C2×Q8×C109—44no—
109C2×Q8×C109—44no—
110C2×Q8×C109—44no—
111C2×Q8×C109—41no—
112C2×Q8×C109—41no—
113C2×Q8×C109—41no—
114C2×Q8×C109SD1611yescharacteristic
115C2×D4×C109—21no—
116C2×D4×C109—21no—
117C2×D4×C109—42no—
118C2×D4×C109—41no—
119C2×D4×C109SD1611yescharacteristic
120C22×C4×C109—21no—
121C22×C4×C109—42no—
122C22×C4×C109—42no—
123C22×C4×C109—41no—
124C22×C4×C109—41no—
125C22×C4×C109C2×D411yescharacteristic
126Q16×C109—28no—
127Q16×C109—22no—
128Q16×C109—21no—
129Q16×C109—22no—
130Q16×C109—22no—
131Q16×C109C22⋊C412yes—
132Q16×C109C22⋊C414yes—
133SD16×C109—28no—
134SD16×C109—22no—
135SD16×C109—22no—
136D8×C109—21no—
137D8×C109C22⋊C412yes—
138M16×C109—21no—
139M16×C109—22no—
140M16×C109—22no—
141M16×C109—41no—
142M16×C109—41no—
143M16×C109C22⋊C412yes—
144C2×C8×C109—21no—
145C2×C8×C109—22no—
146C2×C8×C109—22no—
147C2×C8×C109—41no—
148C2×C8×C109C22×C411yescharacteristic
149C2×C8×C109C2×D411yescharacteristic
150C2×C8×C109C2×D411yescharacteristic
151C2×C8×C109C2×D411yescharacteristic
152C2×C8×C109C2×D411yescharacteristic
153C2×C8×C109C2×D411yescharacteristic
154C4⋊C4×C109—41no—
155C22⋊C4×C109—41no—
156C42×C109—41no—
157C16×C109—41no—
158C16×C109—41no—
159C23×C109—21no—
160C23×C109—41no—
161C23×C109AGL(1,ℤ/8)11yescharacteristic
162Q8×C109—21no—
163Q8×C109—24no—
164Q8×C109—22no—
165Q8×C109—48no—
166Q8×C109—42no—
167Q8×C109—41no—
168Q8×C109—42no—
169Q8×C109—42no—
170Q8×C109#10212yes—
171D4×C109—22no—
172D4×C109—22no—
173D4×C109—22no—
174D4×C109—21no—
175D4×C109—48no—
176D4×C109—42no—
177D4×C109—42no—
178D4×C109—41no—
179D4×C109#10212yes—
180C2×C4×C109—21no—
181C2×C4×C109—22no—
182C2×C4×C109—22no—
183C2×C4×C109—48no—
184C2×C4×C109—48no—
185C2×C4×C109—42no—
186C2×C4×C109—42no—
187C2×C4×C109—42no—
188C2×C4×C109—41no—
189C2×C4×C109—42no—
190C2×C4×C109—42no—
191C2×C4×C109—81no—
192C2×C4×C109C2×C22⋊C411yescharacteristic
193C2×C4×C109#12011yescharacteristic
194C2×C4×C109#12011yescharacteristic
195C2×C4×C109C2×D811yescharacteristic
196C2×C4×C109C2×SD1611yescharacteristic
197C2×C4×C109AGL(1,ℤ/8)11yescharacteristic
198C8×C109—22no—
199C8×C109—81no—
200C8×C109C2×C22⋊C411yescharacteristic
201C8×C109C2×C22⋊C411yescharacteristic
202C8×C109#12011yescharacteristic
203C8×C109#12011yescharacteristic
204C22×C109—22no—
205C22×C109—41no—
206C22×C109—42no—
207C22×C109—42no—
208C22×C109—82no—
209C22×C109#41811yescharacteristic
210C22×C109#44711yescharacteristic
211C22×C109#45011yescharacteristic
212C4×C109—22no—
213C4×C109—44no—
214C4×C109—42no—
215C4×C109—161no—
216C4×C109#37911yescharacteristic
217C4×C109C2×#10211yescharacteristic
218C4×C109#44711yescharacteristic
219C4×C109#45011yescharacteristic
220#12751C10911yesmaximal, characteristic, direct product factor, Sylow
221C2×C109—21no—
222C2×C109—21no—
223C2×C109—21no—
224C2×C109—42no—
225C2×C109—82no—
226C2×C109#178611yescharacteristic
227C2×#578C2×C10911yescharacteristic
228#2148C2×C10911yescharacteristic, complemented
229C2×#362C2×C10911yescharacteristic, complemented
230#2145C2×C10911yescharacteristic, complemented
231#2141C2×C10911yescharacteristic, complemented
232#2117C2×C10911yescharacteristic, complemented
233#1814C2×C10911yescharacteristic, complemented
234C109#1275111yescharacteristic, direct product factor, Sylow
235C2×2−1+4—21no—
236#578—22no—
237#578—22no—
238#578—22no—
239#578C4×C10914yescomplemented
240C2×#137—22no—
241C2×#137—21no—
242C2×#135—21no—
243C2×#135C22×C10911yescharacteristic
244C22×Q16—21no—
245C22×Q16C22×C10911yescharacteristic
246C2×#131C22×C10911yescharacteristic
247C2×M32C22×C10911yescharacteristic
248C2×M32C22×C10911yescharacteristic
249C2×#108C22×C10911yescharacteristic
250#427C22×C10911yescharacteristic
251#421—21no—
252#362—22no—
253#362—21no—
254#361—21no—
255#360—22no—
256#358—22no—
257#350—22no—
258#344—21no—
2592−1+4—24no—
2602−1+4—24no—
2612−1+4—42no—
2622−1+4—42no—
263C2×#41—22no—
264C2×#41—21no—
265C2×#41—42no—
266C2×#41—41no—
267C2×#41D4×C10911yescharacteristic
268C22×Q8—22no—
269C22×Q8—21no—
270C22×Q8—41no—
271#137—24no—
272#137—28no—
273#137—28no—
274#137—24no—
275#135—24no—
276#135—24no—
277#135—21no—
278#135—21no—
279#135C2×C4×C10912yes—
280#135D4×C10912yes—
281C2×Q16—22no—
282C2×Q16—24no—
283C2×Q16—24no—
284C2×Q16—21no—
285C2×Q16—21no—
286C2×Q16—21no—
287C2×Q16C2×C4×C10912yes—
288C2×Q16D4×C10911yescharacteristic
289C2×Q16D4×C10912yes—
290C2×Q16D4×C10912yes—
291C2×SD16—22no—
292C2×SD16—21no—
293C2×D8D4×C10911yescharacteristic
294#131—21no—
295#131—21no—
296#131C2×C4×C10912yes—
297#131D4×C10912yes—
298C2×M16—21no—
299C2×M16—21no—
300C2×M16D4×C10911yescharacteristic
301C22×C8—21no—
302C22×C8C23×C10911yescharacteristic
303#117—21no—
304M32—21no—
305M32—21no—
306M32—21no—
307M32—21no—
308C2×C16—21no—
309C2×C16—21no—
310#108—21no—
311#108—21no—
312#107—21no—
313#106—21no—
314#104—42no—
315#41—21no—
316#41—21no—
317#41—28no—
318#41—22no—
319#41—22no—
320#41—44no—
321#41—48no—
322#41—48no—
323#41—44no—
324#41—44no—
325#41—41no—
326#41—41no—
327#41C22⋊C4×C10912yes—
328#41D8×C10912yes—
329C2×Q8—28no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—21no—
333C2×Q8—22no—
334C2×Q8—42no—
335C2×Q8—44no—
336C2×Q8—44no—
337C2×Q8—44no—
338C2×Q8—41no—
339C2×Q8—41no—
340C2×Q8—41no—
341C2×Q8SD16×C10911yescharacteristic
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—42no—
345C2×D4—41no—
346C2×D4SD16×C10911yescharacteristic
347C22×C4—21no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4C2×D4×C10911yescharacteristic
353Q16—28no—
354Q16—22no—
355Q16—21no—
356Q16—22no—
357Q16—22no—
358Q16C22⋊C4×C10912yes—
359Q16C22⋊C4×C10914yes—
360SD16—28no—
361SD16—22no—
362SD16—22no—
363D8—21no—
364D8C22⋊C4×C10912yes—
365M16—21no—
366M16—22no—
367M16—22no—
368M16—41no—
369M16—41no—
370M16C22⋊C4×C10912yes—
371C2×C8—21no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—41no—
375C2×C8C22×C4×C10911yescharacteristic
376C2×C8C2×D4×C10911yescharacteristic
377C2×C8C2×D4×C10911yescharacteristic
378C2×C8C2×D4×C10911yescharacteristic
379C2×C8C2×D4×C10911yescharacteristic
380C2×C8C2×D4×C10911yescharacteristic
381C4⋊C4—41no—
382C22⋊C4—41no—
383C42—41no—
384C16—41no—
385C16—41no—
386C23—21no—
387C23—41no—
388C23AGL(1,ℤ/8)×C10911yescharacteristic
389Q8—21no—
390Q8—24no—
391Q8—22no—
392Q8—48no—
393Q8—42no—
394Q8—41no—
395Q8—42no—
396Q8—42no—
397Q8#102×C10912yes—
398D4—22no—
399D4—22no—
400D4—22no—
401D4—21no—
402D4—48no—
403D4—42no—
404D4—42no—
405D4—41no—
406D4#102×C10912yes—
407C2×C4—21no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—48no—
411C2×C4—48no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—81no—
419C2×C4C2×C22⋊C4×C10911yescharacteristic
420C2×C4#120×C10911yescharacteristic
421C2×C4#120×C10911yescharacteristic
422C2×C4C2×D8×C10911yescharacteristic
423C2×C4C2×SD16×C10911yescharacteristic
424C2×C4AGL(1,ℤ/8)×C10911yescharacteristic
425C8—22no—
426C8—81no—
427C8C2×C22⋊C4×C10911yescharacteristic
428C8C2×C22⋊C4×C10911yescharacteristic
429C8#120×C10911yescharacteristic
430C8#120×C10911yescharacteristic
431C22—22no—
432C22—41no—
433C22—42no—
434C22—42no—
435C22—82no—
436C22#418×C10911yescharacteristic
437C22#447×C10911yescharacteristic
438C22#450×C10911yescharacteristic
439C4—22no—
440C4—44no—
441C4—42no—
442C4—161no—
443C4#379×C10911yescharacteristic
444C4C2×#102×C10911yescharacteristic
445C4#447×C10911yescharacteristic
446C4#450×C10911yescharacteristic
447C2—21no—
448C2—21no—
449C2—21no—
450C2—42no—
451C2—82no—
452C2C109×#178611yescharacteristic
subgroupG/H
center, socleC2×C109#1786
commutatorC2×C8C22×C4×C109
Frattini, cosocleC22×C8C23×C109
prime radicalC2×#578×C109C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C109 ◁ C22×C4×C109 ◁ C2×#131×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C109×#2148C2semidirect
3. C2×#362×C109C2semidirect
4. C109×#2145C2semidirect
5. C109×#2141C2semidirect
6. C109×#2117C2semidirect
7. C109×#1814C2semidirect
12. #578×C109C4semidirect
220. #12751C109direct (N × H)
228. #2148C2×C109semidirect
229. C2×#362C2×C109semidirect
230. #2145C2×C109semidirect
231. #2141C2×C109semidirect
232. #2117C2×C109semidirect
233. #1814C2×C109semidirect
234. C109#12751direct (N × H)
239. #578C4×C109semidirect

Maximal subgroups

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

subgroupindex
C2×#578×C1092
C109×#21482
C2×#362×C1092
C109×#21452
C109×#21412
C109×#21172
C109×#18142
#12751109