Finite Group Databasebeta

← C109×#18143 #42136137

Subgroups — C109×#18143 #42136137

Order 27904.

1170 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#544×C109C211yesmaximal, characteristic, complemented
2C109×#3007C212yesmaximal
3C109×#2406C211yesmaximal, characteristic
4C109×#1605C214yesmaximal, complemented
5C109×#1599C214yesmaximal, complemented
6C109×#1511C212yesmaximal, complemented
7C109×#1463C211yesmaximal, characteristic, complemented
8C22×#41×C109C2211yescharacteristic
9C2×#137×C109—22no—
10#544×C109—24no—
11#544×C109—22no—
12#544×C109C2214yes—
13C2×C4⋊Q8×C109—22no—
14C2×#125×C109—22no—
15C2×#123×C109—21no—
16C2×C22⋊Q8×C109—21no—
17C2×C22⋊Q8×C109C2212yes—
18C2×#117×C109—21no—
19C22×C4⋊C4×C109C2211yescharacteristic
20#452×C109—24no—
21#451×C109C2214yes—
22#450×C109C2214yes—
23#417×C109—22no—
24#407×C109C2212yes—
25#395×C109C2212yes—
26#394×C109C2212yes—
27#386×C109—21no—
28#385×C109—22no—
29#381×C109C2211yescharacteristic
30#328×C109C2218yes—
31#324×C109C2214yes—
32C2×#41×C109—22no—
33C2×#41×C109—22no—
34C2×#41×C109—44no—
35C22×Q8×C109—21no—
36C22×Q8×C109D412yes—
37C22×Q8×C109C2311yescharacteristic
38C22×D4×C109—21no—
39C22×D4×C109C2311yescharacteristic
40C23×C4×C109—21no—
41C23×C4×C109D411yescharacteristic
42C23×C4×C109D411yescharacteristic
43C23×C4×C109C2311yescharacteristic
44#137×C109—48no—
45C2×Q16×C109—24no—
46C2×SD16×C109—24no—
47C2×M16×C109—22no—
48C4⋊Q8×C109—28no—
49C4⋊Q8×C109—28no—
50#125×C109—28no—
51#125×C109—28no—
52#123×C109—44no—
53C22⋊Q8×C109—24no—
54C22⋊Q8×C109—24no—
55C22⋊Q8×C109—22no—
56C22⋊Q8×C109—22no—
57C22⋊Q8×C109D414yes—
58C22⋊Q8×C109C2314yes—
59#117×C109—24no—
60#117×C109—24no—
61C2×C4⋊C4×C109—21no—
62C2×C4⋊C4×C109—21no—
63C2×C4⋊C4×C109—21no—
64C2×C4⋊C4×C109—22no—
65C2×C4⋊C4×C109—22no—
66C2×C4⋊C4×C109—22no—
67C2×C4⋊C4×C109—22no—
68C2×C4⋊C4×C109—22no—
69C2×C4⋊C4×C109D412yes—
70C2×C4⋊C4×C109D412yes—
71C2×C4⋊C4×C109C2312yes—
72C2×C22⋊C4×C109—22no—
73C2×C22⋊C4×C109—21no—
74C2×C22⋊C4×C109—22no—
75C2×C22⋊C4×C109—22no—
76C2×C42×C109—22no—
77#103×C109—24no—
78#102×C109—24no—
79C22⋊C8×C109—22no—
80C22⋊C8×C109C2314yes—
81#95×C109—21no—
82#95×C109—22no—
83#95×C109C2312yes—
84C24×C109—21no—
85C24×C109C2×D411yescharacteristic
86#41×C109—28no—
87#41×C109—44no—
88#41×C109—44no—
89#41×C109—44no—
90C2×Q8×C109—24no—
91C2×Q8×C109—22no—
92C2×Q8×C109—22no—
93C2×Q8×C109—42no—
94C2×Q8×C109—44no—
95C2×Q8×C109—82no—
96C2×Q8×C109C2×D414yes—
97C2×D4×C109—24no—
98C2×D4×C109—22no—
99C2×D4×C109—42no—
100C2×D4×C109—42no—
101C2×D4×C109—41no—
102C2×D4×C109—41no—
103C22×C4×C109—21no—
104C22×C4×C109—21no—
105C22×C4×C109—24no—
106C22×C4×C109—22no—
107C22×C4×C109—22no—
108C22×C4×C109—21no—
109C22×C4×C109—21no—
110C22×C4×C109—21no—
111C22×C4×C109—22no—
112C22×C4×C109—42no—
113C22×C4×C109—42no—
114C22×C4×C109—42no—
115C22×C4×C109—42no—
116C22×C4×C109—42no—
117C22×C4×C109C2×D412yes—
118C22×C4×C109C2×D412yes—
119C22×C4×C109C2×D412yes—
120C22×C4×C109C2×D411yescharacteristic
121C22×C4×C109C2×D412yes—
122C22×C4×C109C2411yescharacteristic
123Q16×C109—48no—
124SD16×C109—48no—
125M16×C109—44no—
126C2×C8×C109—24no—
127C4⋊C4×C109—24no—
128C4⋊C4×C109—24no—
129C4⋊C4×C109—24no—
130C4⋊C4×C109—28no—
131C4⋊C4×C109—24no—
132C4⋊C4×C109—24no—
133C4⋊C4×C109—22no—
134C4⋊C4×C109—24no—
135C4⋊C4×C109—24no—
136C4⋊C4×C109—44no—
137C4⋊C4×C109C2×D414yes—
138C22⋊C4×C109—24no—
139C22⋊C4×C109—24no—
140C22⋊C4×C109—44no—
141C22⋊C4×C109—42no—
142C22⋊C4×C109—44no—
143C42×C109—28no—
144C23×C109—22no—
145C23×C109—22no—
146C23×C109—22no—
147C23×C109—41no—
148C23×C109—41no—
149C23×C109—42no—
150C23×C109—41no—
151C23×C109—42no—
152C23×C109#12011yescharacteristic
153C23×C109#12011yescharacteristic
154C23×C109C22×D411yescharacteristic
155Q8×C109—24no—
156Q8×C109—44no—
157Q8×C109—42no—
158Q8×C109—82no—
159Q8×C109—84no—
160D4×C109—24no—
161D4×C109—42no—
162D4×C109—44no—
163D4×C109—44no—
164C2×C4×C109—24no—
165C2×C4×C109—24no—
166C2×C4×C109—22no—
167C2×C4×C109—22no—
168C2×C4×C109—24no—
169C2×C4×C109—22no—
170C2×C4×C109—24no—
171C2×C4×C109—24no—
172C2×C4×C109—22no—
173C2×C4×C109—41no—
174C2×C4×C109—42no—
175C2×C4×C109—41no—
176C2×C4×C109—48no—
177C2×C4×C109—44no—
178C2×C4×C109—41no—
179C2×C4×C109—41no—
180C2×C4×C109—42no—
181C2×C4×C109—44no—
182C2×C4×C109—42no—
183C2×C4×C109—42no—
184C2×C4×C109—42no—
185C2×C4×C109—42no—
186C2×C4×C109—81no—
187C2×C4×C109#12012yes—
188C2×C4×C109#13714yescomplemented
189C2×C4×C109C22×D412yes—
190C8×C109—44no—
191C22×C109—21no—
192C22×C109—21no—
193C22×C109—21no—
194C22×C109—21no—
195C22×C109—21no—
196C22×C109—21no—
197C22×C109—21no—
198C22×C109—44no—
199C22×C109—44no—
200C22×C109—42no—
201C22×C109—82no—
202C22×C109—81no—
203C22×C109—81no—
204C22×C109B(4,𝔽2)12yes—
205C22×C109C2×#12011yescharacteristic
206C22×C109C2×#13712yes—
207C4×C109—24no—
208C4×C109—22no—
209C4×C109—22no—
210C4×C109—44no—
211C4×C109—42no—
212C4×C109—42no—
213C4×C109—44no—
214C4×C109—82no—
215#18143C10911yesmaximal, characteristic, direct product factor, Sylow
216C2×C109—22no—
217C2×C109—22no—
218C2×C109—41no—
219C2×C109—82no—
220C2×C109#300411yescharacteristic
221C2×C109#302011yescharacteristic
222C2×C109C2×B(4,𝔽2)11yescharacteristic
223C2×#544C2×C10911yescharacteristic, complemented
224#3007C2×C10912yes—
225#2406C2×C10911yescharacteristic
226#1605C2×C10914yescomplemented
227#1599C2×C10914yescomplemented
228#1511C2×C10912yescomplemented
229#1463C2×C10911yescharacteristic, complemented
230C109#1814311yescharacteristic, direct product factor, Sylow
231C22×#41C22×C10911yescharacteristic
232C2×#137—22no—
233#544—24no—
234#544—22no—
235#544C22×C10914yes—
236C2×C4⋊Q8—22no—
237C2×#125—22no—
238C2×#123—21no—
239C2×C22⋊Q8—21no—
240C2×C22⋊Q8C22×C10912yes—
241C2×#117—21no—
242C22×C4⋊C4C22×C10911yescharacteristic
243#452—24no—
244#451C22×C10914yes—
245#450C22×C10914yes—
246#417—22no—
247#407C22×C10912yes—
248#395C22×C10912yes—
249#394C22×C10912yes—
250#386—21no—
251#385—22no—
252#381C22×C10911yescharacteristic
253#328C22×C10918yes—
254#324C22×C10914yes—
255C2×#41—22no—
256C2×#41—22no—
257C2×#41—44no—
258C22×Q8—21no—
259C22×Q8D4×C10912yes—
260C22×Q8C23×C10911yescharacteristic
261C22×D4—21no—
262C22×D4C23×C10911yescharacteristic
263C23×C4—21no—
264C23×C4D4×C10911yescharacteristic
265C23×C4D4×C10911yescharacteristic
266C23×C4C23×C10911yescharacteristic
267#137—48no—
268C2×Q16—24no—
269C2×SD16—24no—
270C2×M16—22no—
271C4⋊Q8—28no—
272C4⋊Q8—28no—
273#125—28no—
274#125—28no—
275#123—44no—
276C22⋊Q8—24no—
277C22⋊Q8—24no—
278C22⋊Q8—22no—
279C22⋊Q8—22no—
280C22⋊Q8D4×C10914yes—
281C22⋊Q8C23×C10914yes—
282#117—24no—
283#117—24no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—22no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4D4×C10912yes—
293C2×C4⋊C4D4×C10912yes—
294C2×C4⋊C4C23×C10912yes—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C42—22no—
300#103—24no—
301#102—24no—
302C22⋊C8—22no—
303C22⋊C8C23×C10914yes—
304#95—21no—
305#95—22no—
306#95C23×C10912yes—
307C24—21no—
308C24C2×D4×C10911yescharacteristic
309#41—28no—
310#41—44no—
311#41—44no—
312#41—44no—
313C2×Q8—24no—
314C2×Q8—22no—
315C2×Q8—22no—
316C2×Q8—42no—
317C2×Q8—44no—
318C2×Q8—82no—
319C2×Q8C2×D4×C10914yes—
320C2×D4—24no—
321C2×D4—22no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—41no—
325C2×D4—41no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4C2×D4×C10912yes—
341C22×C4C2×D4×C10912yes—
342C22×C4C2×D4×C10912yes—
343C22×C4C2×D4×C10911yescharacteristic
344C22×C4C2×D4×C10912yes—
345C22×C4C24×C10911yescharacteristic
346Q16—48no—
347SD16—48no—
348M16—44no—
349C2×C8—24no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—28no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—22no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—44no—
360C4⋊C4C2×D4×C10914yes—
361C22⋊C4—24no—
362C22⋊C4—24no—
363C22⋊C4—44no—
364C22⋊C4—42no—
365C22⋊C4—44no—
366C42—28no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—41no—
371C23—41no—
372C23—42no—
373C23—41no—
374C23—42no—
375C23#120×C10911yescharacteristic
376C23#120×C10911yescharacteristic
377C23C22×D4×C10911yescharacteristic
378Q8—24no—
379Q8—44no—
380Q8—42no—
381Q8—82no—
382Q8—84no—
383D4—24no—
384D4—42no—
385D4—44no—
386D4—44no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—24no—
392C2×C4—22no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—22no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—48no—
400C2×C4—41no—
401C2×C4—44no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—81no—
410C2×C4#120×C10912yes—
411C2×C4#137×C10914yescomplemented
412C2×C4C22×D4×C10912yes—
413C8—44no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—44no—
422C22—44no—
423C22—42no—
424C22—82no—
425C22—81no—
426C22—81no—
427C22B(4,𝔽2)×C10912yes—
428C22C2×#120×C10911yescharacteristic
429C22C2×#137×C10912yes—
430C4—24no—
431C4—22no—
432C4—22no—
433C4—44no—
434C4—42no—
435C4—42no—
436C4—44no—
437C4—82no—
438C2—22no—
439C2—22no—
440C2—41no—
441C2—82no—
442C2C109×#300411yescharacteristic
443C2C109×#302011yescharacteristic
444C2C2×B(4,𝔽2)×C10911yescharacteristic
subgroupG/H
center, socleC22×C109C2×#120
commutator, Frattini, cosocleC22×C4C24×C109
prime radicalC22×#41×C109C22
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
1. C2×#544×C109C2semidirect
4. C109×#1605C2semidirect
5. C109×#1599C2semidirect
6. C109×#1511C2semidirect
7. C109×#1463C2semidirect
188. C2×C4×C109#137semidirect
215. #18143C109direct (N × H)
223. C2×#544C2×C109semidirect
226. #1605C2×C109semidirect
227. #1599C2×C109semidirect
228. #1511C2×C109semidirect
229. #1463C2×C109semidirect
230. C109#18143direct (N × H)
411. C2×C4#137×C109semidirect

Maximal subgroups

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

subgroupindex
C2×#544×C1092
C109×#30072
C109×#24062
C109×#16052
C109×#15992
C109×#15112
C109×#14632
#18143109