Finite Group Databasebeta

← C109×#10268 #42128262

Subgroups — C109×#10268 #42128262

Order 27904.

658 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#492×C109C211yesmaximal, characteristic
2C2×#475×C109C211yesmaximal, characteristic, complemented
3C2×#460×C109C211yesmaximal, characteristic
4C109×#1984C211yesmaximal, characteristic, complemented
5C2×#332×C109C211yesmaximal, characteristic, complemented
6C2×#331×C109C211yesmaximal, characteristic
7C109×#1277C211yesmaximal, characteristic, complemented
8C22×SD16×C109—21no—
9C22×SD16×C109—22no—
10C2×C4⋊Q8×C109C2211yescharacteristic
11C2×#127×C109C2211yescharacteristic
12C2×C4×Q8×C109C2211yescharacteristic
13#492×C109—22no—
14#492×C109—22no—
15#475×C109—22no—
16#475×C109—22no—
17#460×C109—22no—
18#460×C109—22no—
19C2×#106×C109—21no—
20C2×C4⋊C8×C109C2211yescharacteristic
21C2×C4⋊C8×C109C2211yescharacteristic
22C2×C4⋊C8×C109C2211yescharacteristic
23C2×#103×C109—21no—
24C2×#102×C109—21no—
25C2×C4×C8×C109C2211yescharacteristic
26#336×C109—22no—
27#332×C109—22no—
28#331×C109—22no—
29C22×Q8×C109—21no—
30C22×Q8×C109D411yescharacteristic
31C22×D4×C109—21no—
32C22×D4×C109—21no—
33C22×D4×C109—21no—
34C2×SD16×C109—24no—
35C2×SD16×C109—28no—
36C2×SD16×C109—44no—
37C2×SD16×C109—42no—
38C2×SD16×C109—42no—
39C2×SD16×C109—44no—
40C22×C8×C109—21no—
41C22×C8×C109—21no—
42C22×C8×C109—21no—
43C22×C8×C109D412yes—
44C4⋊Q8×C109—21no—
45C4⋊Q8×C109—21no—
46C4⋊Q8×C109—21no—
47C4⋊Q8×C109D412yes—
48#127×C109—21no—
49#127×C109—21no—
50#127×C109—21no—
51#127×C109D412yes—
52C4×Q8×C109—21no—
53C4×Q8×C109—21no—
54C4×Q8×C109—21no—
55C4×Q8×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×C109C2311yescharacteristic
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#102×C109—24no—
71C4×C8×C109—21no—
72C4×C8×C109—21no—
73C24×C109—41no—
74C2×Q8×C109—21no—
75C2×Q8×C109—21no—
76C2×Q8×C109—21no—
77C2×Q8×C109—22no—
78C2×Q8×C109—21no—
79C2×Q8×C109—21no—
80C2×Q8×C109—41no—
81C2×Q8×C109—42no—
82C2×Q8×C109—42no—
83C2×Q8×C109SD1612yescomplemented
84C2×D4×C109—22no—
85C2×D4×C109—22no—
86C2×D4×C109—22no—
87C2×D4×C109—41no—
88C2×D4×C109—41no—
89C2×D4×C109—41no—
90C2×D4×C109—41no—
91C2×D4×C109—42no—
92C2×D4×C109—41no—
93C2×D4×C109—41no—
94C2×D4×C109—42no—
95C2×D4×C109—41no—
96C2×D4×C109—82no—
97C22×C4×C109—21no—
98C22×C4×C109—21no—
99C22×C4×C109—41no—
100C22×C4×C109C2×D411yescharacteristic
101C22×C4×C109C2×D411yescharacteristic
102C22×C4×C109#4111yescharacteristic
103SD16×C109—416no—
104SD16×C109—48no—
105C2×C8×C109—22no—
106C2×C8×C109—22no—
107C2×C8×C109—22no—
108C2×C8×C109—22no—
109C2×C8×C109—41no—
110C2×C8×C109—41no—
111C2×C8×C109—41no—
112C2×C8×C109—41no—
113C2×C8×C109—41no—
114C2×C8×C109—42no—
115C2×C8×C109SD1614yescomplemented
116C4⋊C4×C109—21no—
117C4⋊C4×C109—22no—
118C4⋊C4×C109—21no—
119C4⋊C4×C109—22no—
120C4⋊C4×C109—41no—
121C4⋊C4×C109—41no—
122C4⋊C4×C109—41no—
123C4⋊C4×C109—41no—
124C42×C109—41no—
125C42×C109—41no—
126C42×C109C2×D411yescharacteristic
127C42×C109C2×D411yescharacteristic
128C42×C109C2×D411yescharacteristic
129C42×C109C2×D411yescharacteristic
130C23×C109—42no—
131C23×C109—82no—
132C23×C109—81no—
133C23×C109—81no—
134C23×C109—82no—
135C23×C109#12111yescharacteristic
136Q8×C109—22no—
137Q8×C109—22no—
138Q8×C109—41no—
139Q8×C109—41no—
140Q8×C109—44no—
141Q8×C109—81no—
142Q8×C109—81no—
143D4×C109—42no—
144D4×C109—44no—
145D4×C109—42no—
146D4×C109—82no—
147D4×C109—82no—
148D4×C109—81no—
149D4×C109—81no—
150D4×C109—81no—
151D4×C109—81no—
152C2×C4×C109—22no—
153C2×C4×C109—22no—
154C2×C4×C109—41no—
155C2×C4×C109—41no—
156C2×C4×C109—41no—
157C2×C4×C109—42no—
158C2×C4×C109—42no—
159C2×C4×C109—82no—
160C2×C4×C109#12011yescharacteristic
161C2×C4×C109#12011yescharacteristic
162C2×C4×C109#12011yescharacteristic
163C2×C4×C109#12111yescharacteristic
164C2×C4×C109#12111yescharacteristic
165C2×C4×C109#12711yescharacteristic
166C2×C4×C109C2×SD1611yescharacteristic
167C2×C4×C109C2×SD1612yes—
168C2×C4×C109#13511yescharacteristic
169C2×C4×C109AGL(1,ℤ/8)11yescharacteristic
170C2×C4×C109AGL(1,ℤ/8)12yes—
171C2×C4×C109#13711yescharacteristic
172C8×C109—24no—
173C8×C109—42no—
174C8×C109—42no—
175C8×C109—81no—
176C22×C109—82no—
177C22×C109—82no—
178C22×C109—82no—
179C22×C109—162no—
180C22×C109—162no—
181C22×C109#39211yescharacteristic
182C22×C109#46012yes—
183C22×C109#46511yescharacteristic
184C22×C109#46811yescharacteristic
185C22×C109#47112yes—
186C4×C109—21no—
187C4×C109—21no—
188C4×C109—42no—
189C4×C109—82no—
190C4×C109—81no—
191C4×C109#44811yescharacteristic
192C4×C109#44911yescharacteristic
193C4×C109#45011yescharacteristic
194C4×C109#45012yes—
195C4×C109#45211yescharacteristic
196C4×C109#49211yescharacteristic
197C4×C109#49611yescharacteristic
198#10268C10911yesmaximal, characteristic, direct product factor, Sylow
199C2×C109—162no—
200C2×C109#166612yes—
201C2×C109#168612yes—
202C2×C109#200011yescharacteristic
203C2×C109#201411yescharacteristic
204C2×C109#201911yescharacteristic
205C2×#492C2×C10911yescharacteristic
206C2×#475C2×C10911yescharacteristic, complemented
207C2×#460C2×C10911yescharacteristic
208#1984C2×C10911yescharacteristic, complemented
209C2×#332C2×C10911yescharacteristic, complemented
210C2×#331C2×C10911yescharacteristic
211#1277C2×C10911yescharacteristic, complemented
212C109#1026811yescharacteristic, direct product factor, Sylow
213C22×SD16—21no—
214C22×SD16—22no—
215C2×C4⋊Q8C22×C10911yescharacteristic
216C2×#127C22×C10911yescharacteristic
217C2×C4×Q8C22×C10911yescharacteristic
218#492—22no—
219#492—22no—
220#475—22no—
221#475—22no—
222#460—22no—
223#460—22no—
224C2×#106—21no—
225C2×C4⋊C8C22×C10911yescharacteristic
226C2×C4⋊C8C22×C10911yescharacteristic
227C2×C4⋊C8C22×C10911yescharacteristic
228C2×#103—21no—
229C2×#102—21no—
230C2×C4×C8C22×C10911yescharacteristic
231#336—22no—
232#332—22no—
233#331—22no—
234C22×Q8—21no—
235C22×Q8D4×C10911yescharacteristic
236C22×D4—21no—
237C22×D4—21no—
238C22×D4—21no—
239C2×SD16—24no—
240C2×SD16—28no—
241C2×SD16—44no—
242C2×SD16—42no—
243C2×SD16—42no—
244C2×SD16—44no—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8—21no—
248C22×C8D4×C10912yes—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8D4×C10912yes—
253#127—21no—
254#127—21no—
255#127—21no—
256#127D4×C10912yes—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×Q8—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4D4×C10911yescharacteristic
265C2×C42—21no—
266C2×C42C23×C10911yescharacteristic
267#106—24no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8D4×C10912yes—
273C4⋊C8D4×C10912yes—
274#103—24no—
275#102—24no—
276C4×C8—21no—
277C4×C8—21no—
278C24—41no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—22no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—41no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8SD16×C10912yescomplemented
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—22no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—42no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—42no—
300C2×D4—41no—
301C2×D4—82no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—41no—
305C22×C4C2×D4×C10911yescharacteristic
306C22×C4C2×D4×C10911yescharacteristic
307C22×C4#41×C10911yescharacteristic
308SD16—416no—
309SD16—48no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—42no—
320C2×C8SD16×C10914yescomplemented
321C4⋊C4—21no—
322C4⋊C4—22no—
323C4⋊C4—21no—
324C4⋊C4—22no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C42—41no—
330C42—41no—
331C42C2×D4×C10911yescharacteristic
332C42C2×D4×C10911yescharacteristic
333C42C2×D4×C10911yescharacteristic
334C42C2×D4×C10911yescharacteristic
335C23—42no—
336C23—82no—
337C23—81no—
338C23—81no—
339C23—82no—
340C23#121×C10911yescharacteristic
341Q8—22no—
342Q8—22no—
343Q8—41no—
344Q8—41no—
345Q8—44no—
346Q8—81no—
347Q8—81no—
348D4—42no—
349D4—44no—
350D4—42no—
351D4—82no—
352D4—82no—
353D4—81no—
354D4—81no—
355D4—81no—
356D4—81no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—82no—
365C2×C4#120×C10911yescharacteristic
366C2×C4#120×C10911yescharacteristic
367C2×C4#120×C10911yescharacteristic
368C2×C4#121×C10911yescharacteristic
369C2×C4#121×C10911yescharacteristic
370C2×C4#127×C10911yescharacteristic
371C2×C4C2×SD16×C10911yescharacteristic
372C2×C4C2×SD16×C10912yes—
373C2×C4#135×C10911yescharacteristic
374C2×C4AGL(1,ℤ/8)×C10911yescharacteristic
375C2×C4AGL(1,ℤ/8)×C10912yes—
376C2×C4#137×C10911yescharacteristic
377C8—24no—
378C8—42no—
379C8—42no—
380C8—81no—
381C22—82no—
382C22—82no—
383C22—82no—
384C22—162no—
385C22—162no—
386C22#392×C10911yescharacteristic
387C22#460×C10912yes—
388C22#465×C10911yescharacteristic
389C22#468×C10911yescharacteristic
390C22#471×C10912yes—
391C4—21no—
392C4—21no—
393C4—42no—
394C4—82no—
395C4—81no—
396C4#448×C10911yescharacteristic
397C4#449×C10911yescharacteristic
398C4#450×C10911yescharacteristic
399C4#450×C10912yes—
400C4#452×C10911yescharacteristic
401C4#492×C10911yescharacteristic
402C4#496×C10911yescharacteristic
403C2—162no—
404C2C109×#166612yes—
405C2C109×#168612yes—
406C2C109×#200011yescharacteristic
407C2C109×#201411yescharacteristic
408C2C109×#201911yescharacteristic
subgroupG/H
center, socleC23×C109#121
commutator, Frattini, cosocleC2×C42C23×C109
prime radicalC2×#127×C109C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×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×#1984C2semidirect
5. C2×#332×C109C2semidirect
7. C109×#1277C2semidirect
83. C2×Q8×C109SD16semidirect
115. C2×C8×C109SD16semidirect
198. #10268C109direct (N × H)
206. C2×#475C2×C109semidirect
208. #1984C2×C109semidirect
209. C2×#332C2×C109semidirect
211. #1277C2×C109semidirect
212. C109#10268direct (N × H)
288. C2×Q8SD16×C109semidirect
320. C2×C8SD16×C109semidirect

Maximal subgroups

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

subgroupindex
C2×#492×C1092
C2×#475×C1092
C2×#460×C1092
C109×#19842
C2×#332×C1092
C2×#331×C1092
C109×#12772
#10268109