Finite Group Databasebeta

← C109×#10269 #42128263

Subgroups — C109×#10269 #42128263

Order 27904.

594 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#486×C109C211yesmaximal, characteristic
2C2×#477×C109C211yesmaximal, characteristic, complemented
3C2×#460×C109C211yesmaximal, characteristic
4C109×#1984C211yesmaximal, characteristic, complemented
5C2×#333×C109C211yesmaximal, characteristic, complemented
6C2×#331×C109C211yesmaximal, characteristic
7C109×#1277C211yesmaximal, characteristic, complemented
8C22×SD16×C109—21no—
9C2×C4⋊Q8×C109C2211yescharacteristic
10C2×#127×C109C2211yescharacteristic
11C2×C4×Q8×C109C2211yescharacteristic
12#486×C109—22no—
13#486×C109—22no—
14#477×C109—22no—
15#477×C109—22no—
16#460×C109—22no—
17#460×C109—22no—
18C2×#107×C109—21no—
19C2×C4⋊C8×C109C2211yescharacteristic
20C2×C4⋊C8×C109C2211yescharacteristic
21C2×C4⋊C8×C109C2211yescharacteristic
22C2×#103×C109—21no—
23C2×#102×C109—22no—
24C2×#102×C109—21no—
25C2×C4×C8×C109C2211yescharacteristic
26#336×C109—22no—
27#333×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—42no—
36C2×SD16×C109—42no—
37C22×C8×C109—21no—
38C22×C8×C109—21no—
39C22×C8×C109—21no—
40C22×C8×C109D412yes—
41C4⋊Q8×C109—21no—
42C4⋊Q8×C109—21no—
43C4⋊Q8×C109—21no—
44C4⋊Q8×C109D412yes—
45#127×C109—21no—
46#127×C109—21no—
47#127×C109—21no—
48#127×C109D412yes—
49C4×Q8×C109—21no—
50C4×Q8×C109—21no—
51C4×Q8×C109—21no—
52C4×Q8×C109—21no—
53C2×C4⋊C4×C109—21no—
54C2×C4⋊C4×C109—21no—
55C2×C4⋊C4×C109—21no—
56C2×C4⋊C4×C109D411yescharacteristic
57C2×C42×C109—21no—
58C2×C42×C109C2311yescharacteristic
59#107×C109—24no—
60C4⋊C8×C109—21no—
61C4⋊C8×C109—21no—
62C4⋊C8×C109—21no—
63C4⋊C8×C109—21no—
64C4⋊C8×C109D412yes—
65C4⋊C8×C109D412yes—
66#103×C109—24no—
67#102×C109—24no—
68#102×C109—28no—
69C4×C8×C109—21no—
70C4×C8×C109—21no—
71C24×C109—41no—
72C2×Q8×C109—21no—
73C2×Q8×C109—21no—
74C2×Q8×C109—21no—
75C2×Q8×C109—22no—
76C2×Q8×C109—21no—
77C2×Q8×C109—21no—
78C2×Q8×C109—41no—
79C2×Q8×C109—82no—
80C2×Q8×C109SD1612yes—
81C2×D4×C109—22no—
82C2×D4×C109—22no—
83C2×D4×C109—22no—
84C2×D4×C109—41no—
85C2×D4×C109—41no—
86C2×D4×C109—41no—
87C2×D4×C109—41no—
88C2×D4×C109—41no—
89C2×D4×C109—42no—
90C2×D4×C109—41no—
91C2×D4×C109—41no—
92C2×D4×C109—42no—
93C2×D4×C109—82no—
94C22×C4×C109—21no—
95C22×C4×C109—21no—
96C22×C4×C109—41no—
97C22×C4×C109C2×D411yescharacteristic
98C22×C4×C109C2×D411yescharacteristic
99C22×C4×C109#4111yescharacteristic
100SD16×C109—48no—
101C2×C8×C109—22no—
102C2×C8×C109—22no—
103C2×C8×C109—22no—
104C2×C8×C109—22no—
105C2×C8×C109—22no—
106C2×C8×C109—41no—
107C2×C8×C109—41no—
108C2×C8×C109—41no—
109C2×C8×C109—41no—
110C2×C8×C109—41no—
111C2×C8×C109—42no—
112C4⋊C4×C109—22no—
113C4⋊C4×C109—21no—
114C4⋊C4×C109—22no—
115C4⋊C4×C109—21no—
116C4⋊C4×C109—41no—
117C4⋊C4×C109—41no—
118C4⋊C4×C109—41no—
119C4⋊C4×C109—41no—
120C42×C109—41no—
121C42×C109—41no—
122C42×C109C2×D411yescharacteristic
123C42×C109C2×D411yescharacteristic
124C42×C109C2×D411yescharacteristic
125C42×C109C2×D411yescharacteristic
126C23×C109—42no—
127C23×C109—82no—
128C23×C109—81no—
129C23×C109—81no—
130C23×C109—82no—
131C23×C109#12111yescharacteristic
132Q8×C109—22no—
133Q8×C109—22no—
134Q8×C109—41no—
135Q8×C109—41no—
136Q8×C109—82no—
137Q8×C109—82no—
138D4×C109—42no—
139D4×C109—42no—
140D4×C109—44no—
141D4×C109—82no—
142D4×C109—82no—
143D4×C109—81no—
144D4×C109—81no—
145D4×C109—81no—
146D4×C109—81no—
147C2×C4×C109—22no—
148C2×C4×C109—22no—
149C2×C4×C109—41no—
150C2×C4×C109—41no—
151C2×C4×C109—41no—
152C2×C4×C109—42no—
153C2×C4×C109—42no—
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#12711yescharacteristic
161C2×C4×C109C2×SD1611yescharacteristic
162C2×C4×C109#13511yescharacteristic
163C2×C4×C109#13512yes—
164C2×C4×C109AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C109#13711yescharacteristic
166C2×C4×C109#13712yes—
167C8×C109—42no—
168C8×C109—42no—
169C8×C109—42no—
170C8×C109—81no—
171C22×C109—82no—
172C22×C109—82no—
173C22×C109—82no—
174C22×C109—162no—
175C22×C109—162no—
176C22×C109#39211yescharacteristic
177C22×C109#46412yes—
178C22×C109#46511yescharacteristic
179C22×C109#46811yescharacteristic
180C22×C109#47212yes—
181C4×C109—21no—
182C4×C109—21no—
183C4×C109—42no—
184C4×C109—81no—
185C4×C109—82no—
186C4×C109#44811yescharacteristic
187C4×C109#44911yescharacteristic
188C4×C109#45011yescharacteristic
189C4×C109#45211yescharacteristic
190C4×C109#45212yes—
191C4×C109#49511yescharacteristic
192C4×C109#49711yescharacteristic
193#10269C10911yesmaximal, characteristic, direct product factor, Sylow
194C2×C109—162no—
195C2×C109#167812yes—
196C2×C109#169812yes—
197C2×C109#200011yescharacteristic
198C2×C109#201511yescharacteristic
199C2×C109#201811yescharacteristic
200C2×#486C2×C10911yescharacteristic
201C2×#477C2×C10911yescharacteristic, complemented
202C2×#460C2×C10911yescharacteristic
203#1984C2×C10911yescharacteristic, complemented
204C2×#333C2×C10911yescharacteristic, complemented
205C2×#331C2×C10911yescharacteristic
206#1277C2×C10911yescharacteristic, complemented
207C109#1026911yescharacteristic, direct product factor, Sylow
208C22×SD16—21no—
209C2×C4⋊Q8C22×C10911yescharacteristic
210C2×#127C22×C10911yescharacteristic
211C2×C4×Q8C22×C10911yescharacteristic
212#486—22no—
213#486—22no—
214#477—22no—
215#477—22no—
216#460—22no—
217#460—22no—
218C2×#107—21no—
219C2×C4⋊C8C22×C10911yescharacteristic
220C2×C4⋊C8C22×C10911yescharacteristic
221C2×C4⋊C8C22×C10911yescharacteristic
222C2×#103—21no—
223C2×#102—22no—
224C2×#102—21no—
225C2×C4×C8C22×C10911yescharacteristic
226#336—22no—
227#333—22no—
228#331—22no—
229C22×Q8—21no—
230C22×Q8D4×C10911yescharacteristic
231C22×D4—21no—
232C22×D4—21no—
233C22×D4—21no—
234C2×SD16—24no—
235C2×SD16—42no—
236C2×SD16—42no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8D4×C10912yes—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8D4×C10912yes—
245#127—21no—
246#127—21no—
247#127—21no—
248#127D4×C10912yes—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4D4×C10911yescharacteristic
257C2×C42—21no—
258C2×C42C23×C10911yescharacteristic
259#107—24no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8—21no—
264C4⋊C8D4×C10912yes—
265C4⋊C8D4×C10912yes—
266#103—24no—
267#102—24no—
268#102—28no—
269C4×C8—21no—
270C4×C8—21no—
271C24—41no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—22no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—41no—
279C2×Q8—82no—
280C2×Q8SD16×C10912yes—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—42no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—42no—
293C2×D4—82no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—41no—
297C22×C4C2×D4×C10911yescharacteristic
298C22×C4C2×D4×C10911yescharacteristic
299C22×C4#41×C10911yescharacteristic
300SD16—48no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—42no—
312C4⋊C4—22no—
313C4⋊C4—21no—
314C4⋊C4—22no—
315C4⋊C4—21no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C42—41no—
321C42—41no—
322C42C2×D4×C10911yescharacteristic
323C42C2×D4×C10911yescharacteristic
324C42C2×D4×C10911yescharacteristic
325C42C2×D4×C10911yescharacteristic
326C23—42no—
327C23—82no—
328C23—81no—
329C23—81no—
330C23—82no—
331C23#121×C10911yescharacteristic
332Q8—22no—
333Q8—22no—
334Q8—41no—
335Q8—41no—
336Q8—82no—
337Q8—82no—
338D4—42no—
339D4—42no—
340D4—44no—
341D4—82no—
342D4—82no—
343D4—81no—
344D4—81no—
345D4—81no—
346D4—81no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—82no—
355C2×C4#120×C10911yescharacteristic
356C2×C4#120×C10911yescharacteristic
357C2×C4#120×C10911yescharacteristic
358C2×C4#121×C10911yescharacteristic
359C2×C4#121×C10911yescharacteristic
360C2×C4#127×C10911yescharacteristic
361C2×C4C2×SD16×C10911yescharacteristic
362C2×C4#135×C10911yescharacteristic
363C2×C4#135×C10912yes—
364C2×C4AGL(1,ℤ/8)×C10911yescharacteristic
365C2×C4#137×C10911yescharacteristic
366C2×C4#137×C10912yes—
367C8—42no—
368C8—42no—
369C8—42no—
370C8—81no—
371C22—82no—
372C22—82no—
373C22—82no—
374C22—162no—
375C22—162no—
376C22#392×C10911yescharacteristic
377C22#464×C10912yes—
378C22#465×C10911yescharacteristic
379C22#468×C10911yescharacteristic
380C22#472×C10912yes—
381C4—21no—
382C4—21no—
383C4—42no—
384C4—81no—
385C4—82no—
386C4#448×C10911yescharacteristic
387C4#449×C10911yescharacteristic
388C4#450×C10911yescharacteristic
389C4#452×C10911yescharacteristic
390C4#452×C10912yes—
391C4#495×C10911yescharacteristic
392C4#497×C10911yescharacteristic
393C2—162no—
394C2C109×#167812yes—
395C2C109×#169812yes—
396C2C109×#200011yescharacteristic
397C2C109×#201511yescharacteristic
398C2C109×#201811yescharacteristic
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×#477×C109C2semidirect
4. C109×#1984C2semidirect
5. C2×#333×C109C2semidirect
7. C109×#1277C2semidirect
193. #10269C109direct (N × H)
201. C2×#477C2×C109semidirect
203. #1984C2×C109semidirect
204. C2×#333C2×C109semidirect
206. #1277C2×C109semidirect
207. C109#10269direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#486×C1092
C2×#477×C1092
C2×#460×C1092
C109×#19842
C2×#333×C1092
C2×#331×C1092
C109×#12772
#10269109