Finite Group Databasebeta

← C109×#10206 #42128200

Subgroups — C109×#10206 #42128200

Order 27904.

560 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3246C211yesmaximal, characteristic
2C109×#3229C211yesmaximal, characteristic, complemented
3C109×#3152C211yesmaximal, characteristic
4C109×#1988C211yesmaximal, characteristic, complemented
5C109×#1986C211yesmaximal, characteristic, complemented
6C109×#1502C211yesmaximal, characteristic
7C109×#1283C211yesmaximal, characteristic, complemented
8C2×#137×C109—22no—
9#533×C109C2211yescharacteristic
10#532×C109C2211yescharacteristic
11C2×C4⋊Q8×C109C2211yescharacteristic
12#498×C109—21no—
13#497×C109—22no—
14#494×C109—21no—
15#492×C109—21no—
16#487×C109—21no—
17#483×C109—22no—
18#476×C109—21no—
19#475×C109—22no—
20#475×C109—21no—
21#465×C109—22no—
22#461×C109—21no—
23#460×C109—21no—
24#431×C109C2211yescharacteristic
25C2×C4⋊C8×C109C2211yescharacteristic
26C4×M16×C109C2211yescharacteristic
27C4×C4⋊C4×C109C2211yescharacteristic
28#340×C109—22no—
29#339×C109—22no—
30#327×C109—22no—
31C2×#41×C109—21no—
32C22×Q8×C109—21no—
33#137×C109—44no—
34#137×C109—44no—
35C2×Q16×C109—24no—
36C2×SD16×C109—24no—
37C2×SD16×C109—22no—
38C2×M16×C109D412yes—
39C22×C8×C109D412yes—
40C4⋊Q8×C109—21no—
41C4⋊Q8×C109—21no—
42C4⋊Q8×C109—21no—
43C4⋊Q8×C109D411yescharacteristic
44C4⋊Q8×C109D412yes—
45C4⋊Q8×C109D411yescharacteristic
46C4⋊Q8×C109D411yescharacteristic
47#127×C109D411yescharacteristic
48#125×C109—21no—
49#124×C109—21no—
50C22⋊Q8×C109—21no—
51C22⋊Q8×C109—21no—
52#121×C109—21no—
53#121×C109—21no—
54C4×Q8×C109—21no—
55C4×Q8×C109—21no—
56C4×D4×C109—21no—
57C4×D4×C109—21no—
58#117×C109—21no—
59C2×C4⋊C4×C109—21no—
60C2×C4⋊C4×C109—21no—
61C2×C4⋊C4×C109—21no—
62C2×C42×C109—21no—
63C2×C42×C109C2311yescharacteristic
64#106×C109—24no—
65#106×C109—22no—
66C4⋊C8×C109—21no—
67C4⋊C8×C109—21no—
68C4⋊C8×C109—21no—
69#103×C109—22no—
70#103×C109—24no—
71#102×C109—22no—
72C22⋊C8×C109D412yes—
73#97×C109—21no—
74C4×C8×C109—21no—
75C4×C8×C109—21no—
76#95×C109—21no—
77#41×C109—42no—
78#41×C109—42no—
79C2×Q8×C109—21no—
80C2×Q8×C109—21no—
81C2×Q8×C109—21no—
82C2×Q8×C109—22no—
83C2×Q8×C109—41no—
84C2×Q8×C109—42no—
85C2×Q8×C109—42no—
86C2×D4×C109—21no—
87C2×D4×C109—21no—
88C2×D4×C109—21no—
89C2×D4×C109—41no—
90C22×C4×C109—41no—
91C22×C4×C109—41no—
92C22×C4×C109—41no—
93C22×C4×C109C2×D411yescharacteristic
94C22×C4×C109C2×D411yescharacteristic
95C22×C4×C109C2×D411yescharacteristic
96Q16×C109—48no—
97SD16×C109—48no—
98SD16×C109—82no—
99M16×C109—22no—
100M16×C109SD1614yescomplemented
101C2×C8×C109—22no—
102C2×C8×C109—22no—
103C2×C8×C109—22no—
104C2×C8×C109—42no—
105C4⋊C4×C109—22no—
106C4⋊C4×C109—21no—
107C4⋊C4×C109—22no—
108C4⋊C4×C109—21no—
109C4⋊C4×C109—21no—
110C4⋊C4×C109—22no—
111C4⋊C4×C109—21no—
112C4⋊C4×C109—21no—
113C4⋊C4×C109—21no—
114C4⋊C4×C109—41no—
115C4⋊C4×C109—41no—
116C4⋊C4×C109—41no—
117C4⋊C4×C109—41no—
118C4⋊C4×C109—41no—
119C22⋊C4×C109—41no—
120C22⋊C4×C109—41no—
121C22⋊C4×C109—41no—
122C42×C109—41no—
123C42×C109—42no—
124C42×C109C2×D411yescharacteristic
125C42×C109C2×D411yescharacteristic
126C42×C109C2×D411yescharacteristic
127C42×C109#4111yescharacteristic
128C23×C109—41no—
129C23×C109#12711yescharacteristic
130Q8×C109—42no—
131Q8×C109—44no—
132Q8×C109—42no—
133Q8×C109—41no—
134Q8×C109—41no—
135Q8×C109—82no—
136D4×C109—42no—
137D4×C109—41no—
138D4×C109—41no—
139D4×C109—81no—
140D4×C109—81no—
141D4×C109—81no—
142C2×C4×C109—21no—
143C2×C4×C109—21no—
144C2×C4×C109—41no—
145C2×C4×C109—42no—
146C2×C4×C109—41no—
147C2×C4×C109—41no—
148C2×C4×C109—41no—
149C2×C4×C109—42no—
150C2×C4×C109—81no—
151C2×C4×C109—81no—
152C2×C4×C109—81no—
153C2×C4×C109—81no—
154C2×C4×C109—82no—
155C2×C4×C109#12011yescharacteristic
156C2×C4×C109#12011yescharacteristic
157C2×C4×C109#12011yescharacteristic
158C2×C4×C109#12111yescharacteristic
159C2×C4×C109#12111yescharacteristic
160C2×C4×C109#12111yescharacteristic
161C2×C4×C109C2×SD1612yes—
162C2×C4×C109AGL(1,ℤ/8)12yes—
163C8×C109—24no—
164C8×C109—42no—
165C8×C109—42no—
166C22×C109—21no—
167C22×C109—21no—
168C22×C109—81no—
169C22×C109—81no—
170C22×C109—81no—
171C22×C109#39211yescharacteristic
172C22×C109#49211yescharacteristic
173C22×C109#49611yescharacteristic
174C4×C109—21no—
175C4×C109—21no—
176C4×C109—21no—
177C4×C109—82no—
178C4×C109—81no—
179C4×C109—81no—
180C4×C109—81no—
181C4×C109—82no—
182C4×C109#45012yes—
183C4×C109#45511yescharacteristic
184C4×C109#45611yescharacteristic
185C4×C109#46012yes—
186#10206C10911yesmaximal, characteristic, direct product factor, Sylow
187C2×C109—21no—
188C2×C109—21no—
189C2×C109—161no—
190C2×C109#200711yescharacteristic
191C2×C109#200911yescharacteristic
192C2×C109#201411yescharacteristic
193#3246C2×C10911yescharacteristic
194#3229C2×C10911yescharacteristic, complemented
195#3152C2×C10911yescharacteristic
196#1988C2×C10911yescharacteristic, complemented
197#1986C2×C10911yescharacteristic, complemented
198#1502C2×C10911yescharacteristic
199#1283C2×C10911yescharacteristic, complemented
200C109#1020611yescharacteristic, direct product factor, Sylow
201C2×#137—22no—
202#533C22×C10911yescharacteristic
203#532C22×C10911yescharacteristic
204C2×C4⋊Q8C22×C10911yescharacteristic
205#498—21no—
206#497—22no—
207#494—21no—
208#492—21no—
209#487—21no—
210#483—22no—
211#476—21no—
212#475—22no—
213#475—21no—
214#465—22no—
215#461—21no—
216#460—21no—
217#431C22×C10911yescharacteristic
218C2×C4⋊C8C22×C10911yescharacteristic
219C4×M16C22×C10911yescharacteristic
220C4×C4⋊C4C22×C10911yescharacteristic
221#340—22no—
222#339—22no—
223#327—22no—
224C2×#41—21no—
225C22×Q8—21no—
226#137—44no—
227#137—44no—
228C2×Q16—24no—
229C2×SD16—24no—
230C2×SD16—22no—
231C2×M16D4×C10912yes—
232C22×C8D4×C10912yes—
233C4⋊Q8—21no—
234C4⋊Q8—21no—
235C4⋊Q8—21no—
236C4⋊Q8D4×C10912yes—
237C4⋊Q8D4×C10911yescharacteristic
238C4⋊Q8D4×C10911yescharacteristic
239C4⋊Q8D4×C10911yescharacteristic
240#127D4×C10911yescharacteristic
241#125—21no—
242#124—21no—
243C22⋊Q8—21no—
244C22⋊Q8—21no—
245#121—21no—
246#121—21no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×D4—21no—
250C4×D4—21no—
251#117—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C42—21no—
256C2×C42C23×C10911yescharacteristic
257#106—22no—
258#106—24no—
259C4⋊C8—21no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262#103—22no—
263#103—24no—
264#102—22no—
265C22⋊C8D4×C10912yes—
266#97—21no—
267C4×C8—21no—
268C4×C8—21no—
269#95—21no—
270#41—42no—
271#41—42no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—22no—
276C2×Q8—41no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4C2×D4×C10911yescharacteristic
287C22×C4C2×D4×C10911yescharacteristic
288C22×C4C2×D4×C10911yescharacteristic
289Q16—48no—
290SD16—48no—
291SD16—82no—
292M16—22no—
293M16SD16×C10914yescomplemented
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—42no—
298C4⋊C4—22no—
299C4⋊C4—21no—
300C4⋊C4—22no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—22no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C42—41no—
316C42—42no—
317C42C2×D4×C10911yescharacteristic
318C42C2×D4×C10911yescharacteristic
319C42C2×D4×C10911yescharacteristic
320C42#41×C10911yescharacteristic
321C23—41no—
322C23#127×C10911yescharacteristic
323Q8—42no—
324Q8—44no—
325Q8—42no—
326Q8—41no—
327Q8—41no—
328Q8—82no—
329D4—42no—
330D4—41no—
331D4—41no—
332D4—81no—
333D4—81no—
334D4—81no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—82no—
348C2×C4#120×C10911yescharacteristic
349C2×C4#120×C10911yescharacteristic
350C2×C4#120×C10911yescharacteristic
351C2×C4#121×C10911yescharacteristic
352C2×C4#121×C10911yescharacteristic
353C2×C4#121×C10911yescharacteristic
354C2×C4C2×SD16×C10912yes—
355C2×C4AGL(1,ℤ/8)×C10912yes—
356C8—24no—
357C8—42no—
358C8—42no—
359C22—21no—
360C22—21no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22#392×C10911yescharacteristic
365C22#492×C10911yescharacteristic
366C22#496×C10911yescharacteristic
367C4—21no—
368C4—21no—
369C4—21no—
370C4—82no—
371C4—81no—
372C4—81no—
373C4—81no—
374C4—82no—
375C4#450×C10912yes—
376C4#455×C10911yescharacteristic
377C4#456×C10911yescharacteristic
378C4#460×C10912yes—
379C2—21no—
380C2—21no—
381C2—161no—
382C2C109×#200711yescharacteristic
383C2C109×#200911yescharacteristic
384C2C109×#201411yescharacteristic
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. C109×#3229C2semidirect
4. C109×#1988C2semidirect
5. C109×#1986C2semidirect
7. C109×#1283C2semidirect
100. M16×C109SD16semidirect
186. #10206C109direct (N × H)
194. #3229C2×C109semidirect
196. #1988C2×C109semidirect
197. #1986C2×C109semidirect
199. #1283C2×C109semidirect
200. C109#10206direct (N × H)
293. M16SD16×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#32462
C109×#32292
C109×#31522
C109×#19882
C109×#19862
C109×#15022
C109×#12832
#10206109