Finite Group Databasebeta

← C109×#10244 #42128238

Subgroups — C109×#10244 #42128238

Order 27904.

554 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3206C211yesmaximal, characteristic
2C109×#3037C211yesmaximal, characteristic
3C109×#2568C211yesmaximal, characteristic, complemented
4C109×#1548C211yesmaximal, characteristic, complemented
5C109×#1502C211yesmaximal, characteristic
6C109×#1479C211yesmaximal, characteristic, complemented
7C109×#1303C211yesmaximal, characteristic, complemented
8C2×#137×C109—21no—
9#532×C109C2211yescharacteristic
10C2×C4⋊Q8×C109C2211yescharacteristic
11C2×C4×Q8×C109C2211yescharacteristic
12#492×C109—21no—
13#487×C109—21no—
14#464×C109—22no—
15#462×C109—21no—
16#461×C109—22no—
17#460×C109—21no—
18#448×C109—22no—
19#431×C109C2211yescharacteristic
20#431×C109C2211yescharacteristic
21#423×C109C2211yescharacteristic
22#417×C109—21no—
23#400×C109—22no—
24#391×C109—22no—
25#391×C109—21no—
26#382×C109—21no—
27C4×C4⋊C4×C109C2211yescharacteristic
28#333×C109—22no—
29#327×C109—22no—
30Q8⋊C8×C109—22no—
31C2×#41×C109—21no—
32C22×Q8×C109D411yescharacteristic
33C22×Q8×C109D412yes—
34#137×C109—42no—
35#137×C109—42no—
36C2×Q16×C109—22no—
37C2×SD16×C109—22no—
38C2×SD16×C109—24no—
39C2×M16×C109—21no—
40C4⋊Q8×C109—21no—
41C4⋊Q8×C109—21no—
42C4⋊Q8×C109—21no—
43C4⋊Q8×C109—21no—
44C4⋊Q8×C109D411yescharacteristic
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×C109—21no—
59C2×C4⋊C4×C109D411yescharacteristic
60C2×C42×C109—21no—
61C2×C42×C109—21no—
62C2×C42×C109C2311yescharacteristic
63C4⋊C8×C109—21no—
64C4⋊C8×C109—21no—
65C4⋊C8×C109D412yes—
66C4⋊C8×C109D412yes—
67#103×C109—22no—
68#103×C109—24no—
69#102×C109—22no—
70C22⋊C8×C109—21no—
71C22⋊C8×C109—21no—
72C4×C8×C109—21no—
73C4×C8×C109—21no—
74#95×C109—24no—
75#95×C109—24no—
76#95×C109D412yes—
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—22no—
85C2×Q8×C109—21no—
86C2×Q8×C109—22no—
87C2×Q8×C109—21no—
88C2×Q8×C109—21no—
89C2×Q8×C109—44no—
90C2×Q8×C109SD1612yescomplemented
91C2×D4×C109—21no—
92C2×D4×C109—21no—
93C2×D4×C109—21no—
94C2×D4×C109—41no—
95C22×C4×C109—21no—
96C22×C4×C109—22no—
97C22×C4×C109—21no—
98C22×C4×C109—22no—
99C22×C4×C109—41no—
100C22×C4×C109C2×D411yescharacteristic
101C22×C4×C109C2×D411yescharacteristic
102C22×C4×C109C2×D411yescharacteristic
103Q16×C109—44no—
104SD16×C109—44no—
105SD16×C109—84no—
106M16×C109—41no—
107M16×C109—41no—
108C2×C8×C109—22no—
109C2×C8×C109—22no—
110C2×C8×C109—41no—
111C4⋊C4×C109—21no—
112C4⋊C4×C109—21no—
113C4⋊C4×C109—21no—
114C4⋊C4×C109—21no—
115C4⋊C4×C109—42no—
116C4⋊C4×C109—41no—
117C4⋊C4×C109—41no—
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×C109—41no—
127C42×C109—41no—
128C42×C109C2×D411yescharacteristic
129C42×C109C2×D411yescharacteristic
130C42×C109C2×D411yescharacteristic
131C42×C109#4111yescharacteristic
132C23×C109—41no—
133C23×C109#12011yescharacteristic
134Q8×C109—22no—
135Q8×C109—22no—
136Q8×C109—41no—
137Q8×C109—41no—
138Q8×C109—41no—
139Q8×C109—41no—
140Q8×C109—44no—
141Q8×C109—82no—
142D4×C109—41no—
143D4×C109—41no—
144D4×C109—42no—
145D4×C109—81no—
146D4×C109—81no—
147D4×C109—81no—
148C2×C4×C109—22no—
149C2×C4×C109—22no—
150C2×C4×C109—41no—
151C2×C4×C109—41no—
152C2×C4×C109—41no—
153C2×C4×C109—42no—
154C2×C4×C109—42no—
155C2×C4×C109—42no—
156C2×C4×C109—44no—
157C2×C4×C109—41no—
158C2×C4×C109—41no—
159C2×C4×C109—42no—
160C2×C4×C109—42no—
161C2×C4×C109—81no—
162C2×C4×C109—81no—
163C2×C4×C109#12011yescharacteristic
164C2×C4×C109#12011yescharacteristic
165C2×C4×C109#12111yescharacteristic
166C2×C4×C109#12111yescharacteristic
167C2×C4×C109#12111yescharacteristic
168C2×C4×C109#12711yescharacteristic
169C2×C4×C109C2×SD1611yescharacteristic
170C2×C4×C109#13511yescharacteristic
171C2×C4×C109AGL(1,ℤ/8)11yescharacteristic
172C2×C4×C109#13711yescharacteristic
173C8×C109—42no—
174C8×C109—42no—
175C8×C109—81no—
176C22×C109—21no—
177C22×C109—21no—
178C22×C109—81no—
179C22×C109—81no—
180C22×C109—81no—
181C22×C109#39211yescharacteristic
182C22×C109#44811yescharacteristic
183C22×C109#44911yescharacteristic
184C4×C109—21no—
185C4×C109—21no—
186C4×C109—21no—
187C4×C109—42no—
188C4×C109—82no—
189C4×C109—81no—
190C4×C109—82no—
191C4×C109—81no—
192C4×C109#45011yescharacteristic
193C4×C109#45211yescharacteristic
194C4×C109#45512yes—
195C4×C109#46511yescharacteristic
196C4×C109#46811yescharacteristic
197#10244C10911yesmaximal, characteristic, direct product factor, Sylow
198C2×C109—21no—
199C2×C109—21no—
200C2×C109—161no—
201C2×C109#200011yescharacteristic
202C2×C109#200411yescharacteristic
203C2×C109#200911yescharacteristic
204#3206C2×C10911yescharacteristic
205#3037C2×C10911yescharacteristic
206#2568C2×C10911yescharacteristic, complemented
207#1548C2×C10911yescharacteristic, complemented
208#1502C2×C10911yescharacteristic
209#1479C2×C10911yescharacteristic, complemented
210#1303C2×C10911yescharacteristic, complemented
211C109#1024411yescharacteristic, direct product factor, Sylow
212C2×#137—21no—
213#532C22×C10911yescharacteristic
214C2×C4⋊Q8C22×C10911yescharacteristic
215C2×C4×Q8C22×C10911yescharacteristic
216#492—21no—
217#487—21no—
218#464—22no—
219#462—21no—
220#461—22no—
221#460—21no—
222#448—22no—
223#431C22×C10911yescharacteristic
224#431C22×C10911yescharacteristic
225#423C22×C10911yescharacteristic
226#417—21no—
227#400—22no—
228#391—22no—
229#391—21no—
230#382—21no—
231C4×C4⋊C4C22×C10911yescharacteristic
232#333—22no—
233#327—22no—
234Q8⋊C8—22no—
235C2×#41—21no—
236C22×Q8D4×C10911yescharacteristic
237C22×Q8D4×C10912yes—
238#137—42no—
239#137—42no—
240C2×Q16—22no—
241C2×SD16—22no—
242C2×SD16—24no—
243C2×M16—21no—
244C4⋊Q8—21no—
245C4⋊Q8—21no—
246C4⋊Q8—21no—
247C4⋊Q8—21no—
248C4⋊Q8D4×C10911yescharacteristic
249#127D4×C10911yescharacteristic
250#124—21no—
251#121—21no—
252#121—21no—
253C4×Q8—21no—
254C4×Q8—21no—
255C4×Q8—21no—
256C4×Q8—21no—
257C4×D4—21no—
258C4×D4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4D4×C10911yescharacteristic
264C2×C42—21no—
265C2×C42—21no—
266C2×C42C23×C10911yescharacteristic
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8D4×C10912yes—
270C4⋊C8D4×C10912yes—
271#103—22no—
272#103—24no—
273#102—22no—
274C22⋊C8—21no—
275C22⋊C8—21no—
276C4×C8—21no—
277C4×C8—21no—
278#95—24no—
279#95—24no—
280#95D4×C10912yes—
281#41—41no—
282#41—41no—
283#41—41no—
284#41—41no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—22no—
289C2×Q8—21no—
290C2×Q8—22no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—44no—
294C2×Q8SD16×C10912yescomplemented
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—41no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—41no—
304C22×C4C2×D4×C10911yescharacteristic
305C22×C4C2×D4×C10911yescharacteristic
306C22×C4C2×D4×C10911yescharacteristic
307Q16—44no—
308SD16—44no—
309SD16—84no—
310M16—41no—
311M16—41no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—41no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—42no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C42—41no—
329C42—41no—
330C42—41no—
331C42—41no—
332C42C2×D4×C10911yescharacteristic
333C42C2×D4×C10911yescharacteristic
334C42C2×D4×C10911yescharacteristic
335C42#41×C10911yescharacteristic
336C23—41no—
337C23#120×C10911yescharacteristic
338Q8—22no—
339Q8—22no—
340Q8—41no—
341Q8—41no—
342Q8—41no—
343Q8—41no—
344Q8—44no—
345Q8—82no—
346D4—41no—
347D4—41no—
348D4—42no—
349D4—81no—
350D4—81no—
351D4—81no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4#120×C10911yescharacteristic
368C2×C4#120×C10911yescharacteristic
369C2×C4#121×C10911yescharacteristic
370C2×C4#121×C10911yescharacteristic
371C2×C4#121×C10911yescharacteristic
372C2×C4#127×C10911yescharacteristic
373C2×C4C2×SD16×C10911yescharacteristic
374C2×C4#135×C10911yescharacteristic
375C2×C4AGL(1,ℤ/8)×C10911yescharacteristic
376C2×C4#137×C10911yescharacteristic
377C8—42no—
378C8—42no—
379C8—81no—
380C22—21no—
381C22—21no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22#392×C10911yescharacteristic
386C22#448×C10911yescharacteristic
387C22#449×C10911yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—42no—
392C4—82no—
393C4—81no—
394C4—82no—
395C4—81no—
396C4#450×C10911yescharacteristic
397C4#452×C10911yescharacteristic
398C4#455×C10912yes—
399C4#465×C10911yescharacteristic
400C4#468×C10911yescharacteristic
401C2—21no—
402C2—21no—
403C2—161no—
404C2C109×#200011yescharacteristic
405C2C109×#200411yescharacteristic
406C2C109×#200911yescharacteristic
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
3. C109×#2568C2semidirect
4. C109×#1548C2semidirect
6. C109×#1479C2semidirect
7. C109×#1303C2semidirect
90. C2×Q8×C109SD16semidirect
197. #10244C109direct (N × H)
206. #2568C2×C109semidirect
207. #1548C2×C109semidirect
209. #1479C2×C109semidirect
210. #1303C2×C109semidirect
211. C109#10244direct (N × H)
294. C2×Q8SD16×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#32062
C109×#30372
C109×#25682
C109×#15482
C109×#15022
C109×#14792
C109×#13032
#10244109