Finite Group Databasebeta

← C109×#10189 #42128183

Subgroups — C109×#10189 #42128183

Order 27904.

518 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3227C211yesmaximal, characteristic
2C2×#475×C109C211yesmaximal, characteristic, complemented
3C109×#3037C211yesmaximal, characteristic
4C109×#1548C211yesmaximal, characteristic, complemented
5C109×#1480C211yesmaximal, characteristic
6C109×#1461C211yesmaximal, characteristic, complemented
7C109×#1301C211yesmaximal, characteristic, complemented
8C2×#137×C109—21no—
9#532×C109C2211yescharacteristic
10C2×C4⋊Q8×C109C2211yescharacteristic
11C2×C4×Q8×C109C2211yescharacteristic
12#498×C109—21no—
13#486×C109—21no—
14#481×C109—22no—
15#476×C109—22no—
16#475×C109—22no—
17#475×C109—22no—
18#464×C109—22no—
19#462×C109—21no—
20#460×C109—21no—
21#431×C109C2211yescharacteristic
22#431×C109C2211yescharacteristic
23C2×#106×C109—21no—
24#423×C109C2211yescharacteristic
25C2×C4⋊C8×C109C2211yescharacteristic
26#417×C109—21no—
27C2×#103×C109—21no—
28#333×C109—22no—
29D4⋊C8×C109—22no—
30#324×C109—22no—
31C2×#41×C109—21no—
32C22×Q8×C109—21no—
33C22×Q8×C109D411yescharacteristic
34#137×C109—42no—
35#137×C109—42no—
36C2×Q16×C109—22no—
37C2×SD16×C109—22no—
38C2×M16×C109—21no—
39C22×C8×C109—21no—
40C4⋊Q8×C109—21no—
41C4⋊Q8×C109—21no—
42C4⋊Q8×C109—21no—
43C4⋊Q8×C109D411yescharacteristic
44C4⋊Q8×C109D412yes—
45#127×C109D411yescharacteristic
46#124×C109—21no—
47#121×C109—21no—
48#121×C109—21no—
49C4×Q8×C109—21no—
50C4×Q8×C109—21no—
51C4×Q8×C109—21no—
52C4×Q8×C109—21no—
53C4×D4×C109—21no—
54C4×D4×C109—21no—
55C2×C4⋊C4×C109—21no—
56C2×C4⋊C4×C109—21no—
57C2×C4⋊C4×C109—21no—
58C2×C4⋊C4×C109D411yescharacteristic
59C2×C42×C109—21no—
60C2×C42×C109C2311yescharacteristic
61#106×C109—24no—
62#106×C109—24no—
63C4⋊C8×C109—21no—
64C4⋊C8×C109—21no—
65C4⋊C8×C109—21no—
66C4⋊C8×C109—21no—
67C4⋊C8×C109D412yes—
68C4⋊C8×C109D412yes—
69#103×C109—24no—
70#103×C109—22no—
71#102×C109—22no—
72#102×C109—24no—
73C22⋊C8×C109—21no—
74C22⋊C8×C109D412yes—
75C4×C8×C109—21no—
76C4×C8×C109—21no—
77#41×C109—41no—
78#41×C109—41no—
79#41×C109—41no—
80#41×C109—41no—
81C2×Q8×C109—21no—
82C2×Q8×C109—21no—
83C2×Q8×C109—21no—
84C2×Q8×C109—21no—
85C2×Q8×C109—22no—
86C2×Q8×C109—21no—
87C2×Q8×C109—21no—
88C2×Q8×C109—41no—
89C2×Q8×C109—82no—
90C2×Q8×C109SD1612yes—
91C2×D4×C109—21no—
92C2×D4×C109—21no—
93C2×D4×C109—21no—
94C2×D4×C109—41no—
95C22×C4×C109—21no—
96C22×C4×C109—21no—
97C22×C4×C109—41no—
98C22×C4×C109—41no—
99C22×C4×C109C2×D411yescharacteristic
100C22×C4×C109C2×D411yescharacteristic
101C22×C4×C109#4111yescharacteristic
102Q16×C109—44no—
103SD16×C109—44no—
104M16×C109—41no—
105M16×C109—41no—
106C2×C8×C109—22no—
107C2×C8×C109—22no—
108C2×C8×C109—22no—
109C2×C8×C109—41no—
110C2×C8×C109—41no—
111C2×C8×C109—41no—
112C4⋊C4×C109—21no—
113C4⋊C4×C109—22no—
114C4⋊C4×C109—22no—
115C4⋊C4×C109—21no—
116C4⋊C4×C109—21no—
117C4⋊C4×C109—21no—
118C4⋊C4×C109—41no—
119C4⋊C4×C109—41no—
120C4⋊C4×C109—41no—
121C4⋊C4×C109—41no—
122C22⋊C4×C109—41no—
123C22⋊C4×C109—41no—
124C42×C109—41no—
125C42×C109—41no—
126C42×C109C2×D411yescharacteristic
127C42×C109C2×D411yescharacteristic
128C42×C109C2×D411yescharacteristic
129C42×C109C2×D411yescharacteristic
130C23×C109—41no—
131C23×C109#12111yescharacteristic
132Q8×C109—22no—
133Q8×C109—22no—
134Q8×C109—41no—
135Q8×C109—41no—
136Q8×C109—41no—
137Q8×C109—41no—
138Q8×C109—82no—
139Q8×C109—82no—
140D4×C109—42no—
141D4×C109—41no—
142D4×C109—41no—
143D4×C109—42no—
144D4×C109—81no—
145D4×C109—81no—
146C2×C4×C109—22no—
147C2×C4×C109—22no—
148C2×C4×C109—41no—
149C2×C4×C109—41no—
150C2×C4×C109—41no—
151C2×C4×C109—41no—
152C2×C4×C109—41no—
153C2×C4×C109—42no—
154C2×C4×C109—82no—
155C2×C4×C109—81no—
156C2×C4×C109—81no—
157C2×C4×C109—81no—
158C2×C4×C109#12011yescharacteristic
159C2×C4×C109#12011yescharacteristic
160C2×C4×C109#12011yescharacteristic
161C2×C4×C109#12111yescharacteristic
162C2×C4×C109#12111yescharacteristic
163C2×C4×C109#12711yescharacteristic
164C2×C4×C109C2×SD1611yescharacteristic
165C2×C4×C109#13511yescharacteristic
166C2×C4×C109AGL(1,ℤ/8)11yescharacteristic
167C2×C4×C109#13711yescharacteristic
168C8×C109—42no—
169C8×C109—42no—
170C8×C109—42no—
171C8×C109—81no—
172C22×C109—21no—
173C22×C109—21no—
174C22×C109—81no—
175C22×C109—81no—
176C22×C109—81no—
177C22×C109#39211yescharacteristic
178C22×C109#46511yescharacteristic
179C22×C109#46811yescharacteristic
180C4×C109—21no—
181C4×C109—21no—
182C4×C109—21no—
183C4×C109—42no—
184C4×C109—81no—
185C4×C109—81no—
186C4×C109—82no—
187C4×C109#44811yescharacteristic
188C4×C109#44911yescharacteristic
189C4×C109#45011yescharacteristic
190C4×C109#45211yescharacteristic
191C4×C109#47112yes—
192#10189C10911yesmaximal, characteristic, direct product factor, Sylow
193C2×C109—21no—
194C2×C109—21no—
195C2×C109—161no—
196C2×C109#200011yescharacteristic
197C2×C109#200911yescharacteristic
198C2×C109#201911yescharacteristic
199#3227C2×C10911yescharacteristic
200C2×#475C2×C10911yescharacteristic, complemented
201#3037C2×C10911yescharacteristic
202#1548C2×C10911yescharacteristic, complemented
203#1480C2×C10911yescharacteristic
204#1461C2×C10911yescharacteristic, complemented
205#1301C2×C10911yescharacteristic, complemented
206C109#1018911yescharacteristic, direct product factor, Sylow
207C2×#137—21no—
208#532C22×C10911yescharacteristic
209C2×C4⋊Q8C22×C10911yescharacteristic
210C2×C4×Q8C22×C10911yescharacteristic
211#498—21no—
212#486—21no—
213#481—22no—
214#476—22no—
215#475—22no—
216#475—22no—
217#464—22no—
218#462—21no—
219#460—21no—
220#431C22×C10911yescharacteristic
221#431C22×C10911yescharacteristic
222C2×#106—21no—
223#423C22×C10911yescharacteristic
224C2×C4⋊C8C22×C10911yescharacteristic
225#417—21no—
226C2×#103—21no—
227#333—22no—
228D4⋊C8—22no—
229#324—22no—
230C2×#41—21no—
231C22×Q8—21no—
232C22×Q8D4×C10911yescharacteristic
233#137—42no—
234#137—42no—
235C2×Q16—22no—
236C2×SD16—22no—
237C2×M16—21no—
238C22×C8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8—21no—
242C4⋊Q8D4×C10912yes—
243C4⋊Q8D4×C10911yescharacteristic
244#127D4×C10911yescharacteristic
245#124—21no—
246#121—21no—
247#121—21no—
248C4×Q8—21no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×D4—21no—
253C4×D4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4D4×C10911yescharacteristic
258C2×C42—21no—
259C2×C42C23×C10911yescharacteristic
260#106—24no—
261#106—24no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8D4×C10912yes—
267C4⋊C8D4×C10912yes—
268#103—24no—
269#103—22no—
270#102—22no—
271#102—24no—
272C22⋊C8—21no—
273C22⋊C8D4×C10912yes—
274C4×C8—21no—
275C4×C8—21no—
276#41—41no—
277#41—41no—
278#41—41no—
279#41—41no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—22no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—41no—
288C2×Q8—82no—
289C2×Q8SD16×C10912yes—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—41no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4C2×D4×C10911yescharacteristic
299C22×C4C2×D4×C10911yescharacteristic
300C22×C4#41×C10911yescharacteristic
301Q16—44no—
302SD16—44no—
303M16—41no—
304M16—41no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C4⋊C4—21no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C42—41no—
324C42—41no—
325C42C2×D4×C10911yescharacteristic
326C42C2×D4×C10911yescharacteristic
327C42C2×D4×C10911yescharacteristic
328C42C2×D4×C10911yescharacteristic
329C23—41no—
330C23#121×C10911yescharacteristic
331Q8—22no—
332Q8—22no—
333Q8—41no—
334Q8—41no—
335Q8—41no—
336Q8—41no—
337Q8—82no—
338Q8—82no—
339D4—42no—
340D4—41no—
341D4—41no—
342D4—42no—
343D4—81no—
344D4—81no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—82no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4#120×C10911yescharacteristic
358C2×C4#120×C10911yescharacteristic
359C2×C4#120×C10911yescharacteristic
360C2×C4#121×C10911yescharacteristic
361C2×C4#121×C10911yescharacteristic
362C2×C4#127×C10911yescharacteristic
363C2×C4C2×SD16×C10911yescharacteristic
364C2×C4#135×C10911yescharacteristic
365C2×C4AGL(1,ℤ/8)×C10911yescharacteristic
366C2×C4#137×C10911yescharacteristic
367C8—42no—
368C8—42no—
369C8—42no—
370C8—81no—
371C22—21no—
372C22—21no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22#392×C10911yescharacteristic
377C22#465×C10911yescharacteristic
378C22#468×C10911yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—42no—
383C4—81no—
384C4—81no—
385C4—82no—
386C4#448×C10911yescharacteristic
387C4#449×C10911yescharacteristic
388C4#450×C10911yescharacteristic
389C4#452×C10911yescharacteristic
390C4#471×C10912yes—
391C2—21no—
392C2—21no—
393C2—161no—
394C2C109×#200011yescharacteristic
395C2C109×#200911yescharacteristic
396C2C109×#201911yescharacteristic
subgroupG/H
center, socleC22×C109#392
commutator, Frattini, cosocleC2×C42C23×C109
prime radical#532×C109C22
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
2. C2×#475×C109C2semidirect
4. C109×#1548C2semidirect
6. C109×#1461C2semidirect
7. C109×#1301C2semidirect
192. #10189C109direct (N × H)
200. C2×#475C2×C109semidirect
202. #1548C2×C109semidirect
204. #1461C2×C109semidirect
205. #1301C2×C109semidirect
206. C109#10189direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#32272
C2×#475×C1092
C109×#30372
C109×#15482
C109×#14802
C109×#14612
C109×#13012
#10189109