Finite Group Databasebeta

← C109×#12270 #42130264

Subgroups — C109×#12270 #42130264

Order 27904.

512 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#2555C211yesmaximal, characteristic
2C109×#2027C211yesmaximal, characteristic, complemented
3C109×#1605C211yesmaximal, characteristic, complemented
4C109×#1599C211yesmaximal, characteristic, complemented
5C109×#1354C211yesmaximal, characteristic, complemented
6C109×#1345C211yesmaximal, characteristic, complemented
7C109×#1318C211yesmaximal, characteristic, complemented
8#544×C109C2211yescharacteristic
9C2×#124×C109—21no—
10C2×#121×C109—21no—
11#451×C109—21no—
12#450×C109—21no—
13#418×C109—21no—
14#395×C109C2211yescharacteristic
15#394×C109C2211yescharacteristic
16#392×C109—21no—
17#390×C109C2211yescharacteristic
18#388×C109—21no—
19#386×C109—21no—
20C4×C22⋊C4×C109—21no—
21#343×C109C2211yescharacteristic
22#339×C109—21no—
23#328×C109—21no—
24#328×C109—21no—
25#324×C109C2211yescharacteristic
26#324×C109C2211yescharacteristic
27C22×Q8×C109D411yescharacteristic
28C22×D4×C109—21no—
29C22×D4×C109—21no—
30C22×D4×C109D411yescharacteristic
31C22×D4×C109D411yescharacteristic
32C23×C4×C109—21no—
33C2×Q16×C109—41no—
34C2×SD16×C109—41no—
35C2×M16×C109—21no—
36C4⋊Q8×C109—41no—
37#125×C109—41no—
38#124×C109—44no—
39C22⋊Q8×C109—21no—
40C22⋊Q8×C109—21no—
41#121×C109—44no—
42#117×C109—21no—
43C2×C4⋊C4×C109—21no—
44C2×C4⋊C4×C109C2311yescharacteristic
45C2×C22⋊C4×C109—21no—
46C2×C22⋊C4×C109—21no—
47C2×C22⋊C4×C109—21no—
48C2×C22⋊C4×C109—21no—
49C2×C22⋊C4×C109—21no—
50C2×C22⋊C4×C109—21no—
51C2×C22⋊C4×C109—21no—
52C2×C42×C109—21no—
53C2×C42×C109D411yescharacteristic
54#103×C109—41no—
55#102×C109—41no—
56#102×C109—41no—
57#102×C109—41no—
58C22⋊C8×C109—21no—
59C22⋊C8×C109—21no—
60#95×C109—21no—
61#95×C109—21no—
62#95×C109D411yescharacteristic
63#95×C109D411yescharacteristic
64C24×C109—21no—
65C24×C109—21no—
66C2×Q8×C109—21no—
67C2×Q8×C109—21no—
68C2×Q8×C109—42no—
69C2×Q8×C109—41no—
70C2×D4×C109—21no—
71C2×D4×C109—21no—
72C2×D4×C109—41no—
73C2×D4×C109—41no—
74C2×D4×C109—41no—
75C2×D4×C109—42no—
76C2×D4×C109—42no—
77C2×D4×C109—42no—
78C2×D4×C109—42no—
79C2×D4×C109—42no—
80C2×D4×C109—81no—
81C2×D4×C109—81no—
82C22×C4×C109—21no—
83C22×C4×C109—21no—
84C22×C4×C109—21no—
85C22×C4×C109—21no—
86C22×C4×C109—42no—
87C22×C4×C109—42no—
88C22×C4×C109—81no—
89C22×C4×C109C2×D411yescharacteristic
90C22×C4×C109C2×D411yescharacteristic
91C22×C4×C109C2×D411yescharacteristic
92Q16×C109—82no—
93SD16×C109—82no—
94M16×C109—41no—
95M16×C109—41no—
96C2×C8×C109—41no—
97C2×C8×C109—41no—
98C2×C8×C109—41no—
99C4⋊C4×C109—21no—
100C4⋊C4×C109—21no—
101C4⋊C4×C109—21no—
102C4⋊C4×C109—21no—
103C4⋊C4×C109—41no—
104C4⋊C4×C109—42no—
105C4⋊C4×C109—41no—
106C4⋊C4×C109—41no—
107C22⋊C4×C109—24no—
108C22⋊C4×C109—42no—
109C22⋊C4×C109—42no—
110C22⋊C4×C109—42no—
111C22⋊C4×C109—42no—
112C22⋊C4×C109—42no—
113C22⋊C4×C109—41no—
114C22⋊C4×C109—41no—
115C22⋊C4×C109—41no—
116C42×C109—41no—
117C42×C109—42no—
118C42×C109—41no—
119C23×C109—22no—
120C23×C109—41no—
121C23×C109—41no—
122C23×C109—41no—
123C23×C109—41no—
124C23×C109—41no—
125C23×C109—41no—
126C23×C109—41no—
127C23×C109—41no—
128C23×C109—41no—
129C23×C109—41no—
130C23×C109—41no—
131C23×C109—41no—
132C23×C109—41no—
133C23×C109#12011yescharacteristic
134Q8×C109—42no—
135Q8×C109—81no—
136Q8×C109—81no—
137Q8×C109—81no—
138D4×C109—42no—
139D4×C109—42no—
140D4×C109—81no—
141D4×C109—81no—
142D4×C109—84no—
143D4×C109—84no—
144C2×C4×C109—21no—
145C2×C4×C109—21no—
146C2×C4×C109—22no—
147C2×C4×C109—22no—
148C2×C4×C109—22no—
149C2×C4×C109—21no—
150C2×C4×C109—21no—
151C2×C4×C109—41no—
152C2×C4×C109—41no—
153C2×C4×C109—41no—
154C2×C4×C109—41no—
155C2×C4×C109—41no—
156C2×C4×C109—41no—
157C2×C4×C109—41no—
158C2×C4×C109—41no—
159C2×C4×C109—41no—
160C2×C4×C109—41no—
161C2×C4×C109—41no—
162C2×C4×C109—41no—
163C2×C4×C109—41no—
164C2×C4×C109—84no—
165C2×C4×C109#13511yescharacteristic
166C2×C4×C109#13711yescharacteristic
167C8×C109—81no—
168C8×C109—81no—
169C8×C109—81no—
170C22×C109—21no—
171C22×C109—21no—
172C22×C109—42no—
173C22×C109—42no—
174C22×C109—42no—
175C22×C109—81no—
176C22×C109—81no—
177C22×C109—81no—
178C22×C109—81no—
179C22×C109—81no—
180C22×C109—81no—
181C22×C109—81no—
182C22×C109—81no—
183C22×C109—81no—
184C22×C109—81no—
185C22×C109—81no—
186C22×C109#45211yescharacteristic
187C22×C109#45511yescharacteristic
188C22×C109B(4,𝔽2)11yescharacteristic
189C4×C109—21no—
190C4×C109—21no—
191C4×C109—42no—
192C4×C109—42no—
193C4×C109—41no—
194C4×C109—41no—
195C4×C109—42no—
196C4×C109—81no—
197C4×C109—81no—
198C4×C109—81no—
199#12270C10911yesmaximal, characteristic, direct product factor, Sylow
200C2×C109—21no—
201C2×C109—21no—
202C2×C109—82no—
203C2×C109—161no—
204C2×C109#160611yescharacteristic
205C2×C109#219611yescharacteristic
206C2×C109#220411yescharacteristic
207#2555C2×C10911yescharacteristic
208#2027C2×C10911yescharacteristic, complemented
209#1605C2×C10911yescharacteristic, complemented
210#1599C2×C10911yescharacteristic, complemented
211#1354C2×C10911yescharacteristic, complemented
212#1345C2×C10911yescharacteristic, complemented
213#1318C2×C10911yescharacteristic, complemented
214C109#1227011yescharacteristic, direct product factor, Sylow
215#544C22×C10911yescharacteristic
216C2×#124—21no—
217C2×#121—21no—
218#451—21no—
219#450—21no—
220#418—21no—
221#395C22×C10911yescharacteristic
222#394C22×C10911yescharacteristic
223#392—21no—
224#390C22×C10911yescharacteristic
225#388—21no—
226#386—21no—
227C4×C22⋊C4—21no—
228#343C22×C10911yescharacteristic
229#339—21no—
230#328—21no—
231#328—21no—
232#324C22×C10911yescharacteristic
233#324C22×C10911yescharacteristic
234C22×Q8D4×C10911yescharacteristic
235C22×D4—21no—
236C22×D4—21no—
237C22×D4D4×C10911yescharacteristic
238C22×D4D4×C10911yescharacteristic
239C23×C4—21no—
240C2×Q16—41no—
241C2×SD16—41no—
242C2×M16—21no—
243C4⋊Q8—41no—
244#125—41no—
245#124—44no—
246C22⋊Q8—21no—
247C22⋊Q8—21no—
248#121—44no—
249#117—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4C23×C10911yescharacteristic
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C42—21no—
260C2×C42D4×C10911yescharacteristic
261#103—41no—
262#102—41no—
263#102—41no—
264#102—41no—
265C22⋊C8—21no—
266C22⋊C8—21no—
267#95—21no—
268#95—21no—
269#95D4×C10911yescharacteristic
270#95D4×C10911yescharacteristic
271C24—21no—
272C24—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—42no—
276C2×Q8—41no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—42no—
286C2×D4—42no—
287C2×D4—81no—
288C2×D4—81no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—81no—
296C22×C4C2×D4×C10911yescharacteristic
297C22×C4C2×D4×C10911yescharacteristic
298C22×C4C2×D4×C10911yescharacteristic
299Q16—82no—
300SD16—82no—
301M16—41no—
302M16—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C22⋊C4—24no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C42—41no—
324C42—42no—
325C42—41no—
326C23—22no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23#120×C10911yescharacteristic
341Q8—42no—
342Q8—81no—
343Q8—81no—
344Q8—81no—
345D4—42no—
346D4—42no—
347D4—81no—
348D4—81no—
349D4—84no—
350D4—84no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—84no—
372C2×C4#135×C10911yescharacteristic
373C2×C4#137×C10911yescharacteristic
374C8—81no—
375C8—81no—
376C8—81no—
377C22—21no—
378C22—21no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22#452×C10911yescharacteristic
394C22#455×C10911yescharacteristic
395C22B(4,𝔽2)×C10911yescharacteristic
396C4—21no—
397C4—21no—
398C4—42no—
399C4—42no—
400C4—41no—
401C4—41no—
402C4—42no—
403C4—81no—
404C4—81no—
405C4—81no—
406C2—21no—
407C2—21no—
408C2—82no—
409C2—161no—
410C2C109×#160611yescharacteristic
411C2C109×#219611yescharacteristic
412C2C109×#220411yescharacteristic
subgroupG/H
center, socleC22×C109B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C109
prime radicalC109×#2555C2
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C109 ◁ C23×C109 ◁ C2×C4⋊C4×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C109×#2027C2semidirect
3. C109×#1605C2semidirect
4. C109×#1599C2semidirect
5. C109×#1354C2semidirect
6. C109×#1345C2semidirect
7. C109×#1318C2semidirect
199. #12270C109direct (N × H)
208. #2027C2×C109semidirect
209. #1605C2×C109semidirect
210. #1599C2×C109semidirect
211. #1354C2×C109semidirect
212. #1345C2×C109semidirect
213. #1318C2×C109semidirect
214. C109#12270direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#25552
C109×#20272
C109×#16052
C109×#15992
C109×#13542
C109×#13452
C109×#13182
#12270109