Finite Group Databasebeta

← C109×#10599 #42128593

Subgroups — C109×#10599 #42128593

Order 27904.

454 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#1982C211yesmaximal, characteristic
2C109×#1957C211yesmaximal, characteristic
3C109×#1948C211yesmaximal, characteristic, complemented
4C109×#1939C211yesmaximal, characteristic, complemented
5C2×#335×C109C211yesmaximal, characteristic, complemented
6C2×#331×C109C211yesmaximal, characteristic
7C109×#1296C211yesmaximal, characteristic, complemented
8C2×#127×C109C2211yescharacteristic
9C2×#106×C109—21no—
10C2×C4⋊C8×C109—21no—
11C2×C4⋊C8×C109C2211yescharacteristic
12C2×C4⋊C8×C109C2211yescharacteristic
13C2×C4⋊C8×C109C2211yescharacteristic
14C2×#102×C109—21no—
15C2×#102×C109—21no—
16C2×#102×C109—21no—
17C2×#102×C109—21no—
18C2×C4×C8×C109C2211yescharacteristic
19#384×C109C2211yescharacteristic
20C4×C4⋊C4×C109C2211yescharacteristic
21#340×C109—21no—
22#336×C109—21no—
23#335×C109—22no—
24#331×C109—22no—
25C22×D4×C109—21no—
26C22×D4×C109—21no—
27C22×D4×C109—21no—
28C22×C8×C109—21no—
29C22×C8×C109—21no—
30C22×C8×C109—21no—
31C22×C8×C109D411yescharacteristic
32C22×C8×C109D411yescharacteristic
33#127×C109—21no—
34#127×C109—21no—
35#127×C109—21no—
36#127×C109Q812yes—
37C2×C4⋊C4×C109—21no—
38C2×C4⋊C4×C109—21no—
39C2×C4⋊C4×C109—21no—
40C2×C4⋊C4×C109D411yescharacteristic
41C2×C4⋊C4×C109D411yescharacteristic
42C2×C42×C109—21no—
43C2×C42×C109C2311yescharacteristic
44#106×C109—42no—
45C4⋊C8×C109—21no—
46C4⋊C8×C109—21no—
47C4⋊C8×C109—21no—
48C4⋊C8×C109—21no—
49C4⋊C8×C109—21no—
50C4⋊C8×C109—42no—
51C4⋊C8×C109D412yes—
52#102×C109—42no—
53#102×C109—42no—
54#102×C109—42no—
55#102×C109—42no—
56C4×C8×C109—21no—
57C4×C8×C109—21no—
58#95×C109—21no—
59C24×C109—41no—
60C2×D4×C109—22no—
61C2×D4×C109—22no—
62C2×D4×C109—22no—
63C2×D4×C109—41no—
64C2×D4×C109—41no—
65C2×D4×C109—41no—
66C2×D4×C109—41no—
67C2×D4×C109—41no—
68C2×D4×C109—41no—
69C2×D4×C109—41no—
70C2×D4×C109—41no—
71C2×D4×C109—41no—
72C2×D4×C109—41no—
73C2×D4×C109—41no—
74C2×D4×C109—81no—
75C2×D4×C109—81no—
76C22×C4×C109—21no—
77C22×C4×C109—21no—
78C22×C4×C109—41no—
79C22×C4×C109C2×D411yescharacteristic
80C22×C4×C109C2×D411yescharacteristic
81C22×C4×C109#4111yescharacteristic
82C2×C8×C109—21no—
83C2×C8×C109—21no—
84C2×C8×C109—21no—
85C2×C8×C109—22no—
86C2×C8×C109—21no—
87C2×C8×C109—21no—
88C2×C8×C109—41no—
89C2×C8×C109—41no—
90C2×C8×C109—41no—
91C2×C8×C109—41no—
92C2×C8×C109—41no—
93C2×C8×C109—41no—
94C2×C8×C109—41no—
95C2×C8×C109—41no—
96C2×C8×C109SD1612yes—
97C4⋊C4×C109—21no—
98C4⋊C4×C109—21no—
99C4⋊C4×C109—21no—
100C4⋊C4×C109—41no—
101C4⋊C4×C109—41no—
102C4⋊C4×C109—41no—
103C4⋊C4×C109—41no—
104C4⋊C4×C109—81no—
105C4⋊C4×C109D812yes—
106C42×C109—41no—
107C42×C109—41no—
108C42×C109C2×D411yescharacteristic
109C42×C109C2×Q811yescharacteristic
110C42×C109#4111yescharacteristic
111C42×C109#4111yescharacteristic
112C23×C109—42no—
113C23×C109—81no—
114C23×C109—81no—
115C23×C109—81no—
116C23×C109—81no—
117C23×C109—81no—
118C23×C109—81no—
119C23×C109#12111yescharacteristic
120D4×C109—42no—
121D4×C109—42no—
122D4×C109—42no—
123D4×C109—42no—
124D4×C109—82no—
125D4×C109—82no—
126D4×C109—81no—
127D4×C109—81no—
128D4×C109—81no—
129D4×C109—81no—
130C2×C4×C109—21no—
131C2×C4×C109—21no—
132C2×C4×C109—22no—
133C2×C4×C109—41no—
134C2×C4×C109—41no—
135C2×C4×C109—41no—
136C2×C4×C109—41no—
137C2×C4×C109—41no—
138C2×C4×C109—81no—
139C2×C4×C109—81no—
140C2×C4×C109—81no—
141C2×C4×C109#12111yescharacteristic
142C2×C4×C109#12111yescharacteristic
143C2×C4×C109C22⋊Q811yescharacteristic
144C2×C4×C109C22⋊Q811yescharacteristic
145C2×C4×C109C22⋊Q811yescharacteristic
146C2×C4×C109#12611yescharacteristic
147C2×C4×C109C2×D811yescharacteristic
148C2×C4×C109C2×SD1611yescharacteristic
149C2×C4×C109#13511yescharacteristic
150C2×C4×C109#13511yescharacteristic
151C2×C4×C109AGL(1,ℤ/8)11yescharacteristic
152C2×C4×C109AGL(1,ℤ/8)11yescharacteristic
153C2×C4×C109AGL(1,ℤ/8)11yescharacteristic
154C2×C4×C109#13711yescharacteristic
155C8×C109—22no—
156C8×C109—42no—
157C8×C109—41no—
158C8×C109—81no—
159C8×C109—81no—
160C22×C109—82no—
161C22×C109—82no—
162C22×C109—82no—
163C22×C109—161no—
164C22×C109—161no—
165C22×C109—161no—
166C22×C109—161no—
167C22×C109#39611yescharacteristic
168C22×C109#46011yescharacteristic
169C22×C109#46311yescharacteristic
170C22×C109#46611yescharacteristic
171C22×C109#46811yescharacteristic
172C22×C109#47111yescharacteristic
173C22×C109#47211yescharacteristic
174C4×C109—21no—
175C4×C109—21no—
176C4×C109—42no—
177C4×C109—81no—
178C4×C109—161no—
179C4×C109#45911yescharacteristic
180C4×C109#46411yescharacteristic
181C4×C109#46511yescharacteristic
182C4×C109#46911yescharacteristic
183C4×C109#47411yescharacteristic
184C4×C109#47611yescharacteristic
185C4×C109#47811yescharacteristic
186C4×C109#47811yescharacteristic
187#10599C10911yesmaximal, characteristic, direct product factor, Sylow
188C2×C109—162no—
189C2×C109#166811yescharacteristic
190C2×C109#168111yescharacteristic
191C2×C109#168611yescharacteristic
192C2×C109#169711yescharacteristic
193C2×C109#206311yescharacteristic
194C2×C109#206511yescharacteristic
195C2×C109#206711yescharacteristic
196#1982C2×C10911yescharacteristic
197#1957C2×C10911yescharacteristic
198#1948C2×C10911yescharacteristic, complemented
199#1939C2×C10911yescharacteristic, complemented
200C2×#335C2×C10911yescharacteristic, complemented
201C2×#331C2×C10911yescharacteristic
202#1296C2×C10911yescharacteristic, complemented
203C109#1059911yescharacteristic, direct product factor, Sylow
204C2×#127C22×C10911yescharacteristic
205C2×#106—21no—
206C2×C4⋊C8—21no—
207C2×C4⋊C8C22×C10911yescharacteristic
208C2×C4⋊C8C22×C10911yescharacteristic
209C2×C4⋊C8C22×C10911yescharacteristic
210C2×#102—21no—
211C2×#102—21no—
212C2×#102—21no—
213C2×#102—21no—
214C2×C4×C8C22×C10911yescharacteristic
215#384C22×C10911yescharacteristic
216C4×C4⋊C4C22×C10911yescharacteristic
217#340—21no—
218#336—21no—
219#335—22no—
220#331—22no—
221C22×D4—21no—
222C22×D4—21no—
223C22×D4—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8D4×C10911yescharacteristic
228C22×C8D4×C10911yescharacteristic
229#127—21no—
230#127—21no—
231#127—21no—
232#127Q8×C10912yes—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4D4×C10911yescharacteristic
237C2×C4⋊C4D4×C10911yescharacteristic
238C2×C42—21no—
239C2×C42C23×C10911yescharacteristic
240#106—42no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—42no—
247C4⋊C8D4×C10912yes—
248#102—42no—
249#102—42no—
250#102—42no—
251#102—42no—
252C4×C8—21no—
253C4×C8—21no—
254#95—21no—
255C24—41no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—81no—
271C2×D4—81no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—41no—
275C22×C4C2×D4×C10911yescharacteristic
276C22×C4C2×D4×C10911yescharacteristic
277C22×C4#41×C10911yescharacteristic
278C2×C8—21no—
279C2×C8—21no—
280C2×C8—21no—
281C2×C8—22no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8SD16×C10912yes—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—81no—
301C4⋊C4D8×C10912yes—
302C42—41no—
303C42—41no—
304C42C2×D4×C10911yescharacteristic
305C42C2×Q8×C10911yescharacteristic
306C42#41×C10911yescharacteristic
307C42#41×C10911yescharacteristic
308C23—42no—
309C23—81no—
310C23—81no—
311C23—81no—
312C23—81no—
313C23—81no—
314C23—81no—
315C23#121×C10911yescharacteristic
316D4—42no—
317D4—42no—
318D4—42no—
319D4—42no—
320D4—82no—
321D4—82no—
322D4—81no—
323D4—81no—
324D4—81no—
325D4—81no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4#121×C10911yescharacteristic
338C2×C4#121×C10911yescharacteristic
339C2×C4C22⋊Q8×C10911yescharacteristic
340C2×C4C22⋊Q8×C10911yescharacteristic
341C2×C4C22⋊Q8×C10911yescharacteristic
342C2×C4#126×C10911yescharacteristic
343C2×C4C2×D8×C10911yescharacteristic
344C2×C4C2×SD16×C10911yescharacteristic
345C2×C4#135×C10911yescharacteristic
346C2×C4#135×C10911yescharacteristic
347C2×C4AGL(1,ℤ/8)×C10911yescharacteristic
348C2×C4AGL(1,ℤ/8)×C10911yescharacteristic
349C2×C4AGL(1,ℤ/8)×C10911yescharacteristic
350C2×C4#137×C10911yescharacteristic
351C8—22no—
352C8—42no—
353C8—41no—
354C8—81no—
355C8—81no—
356C22—82no—
357C22—82no—
358C22—82no—
359C22—161no—
360C22—161no—
361C22—161no—
362C22—161no—
363C22#396×C10911yescharacteristic
364C22#460×C10911yescharacteristic
365C22#463×C10911yescharacteristic
366C22#466×C10911yescharacteristic
367C22#468×C10911yescharacteristic
368C22#471×C10911yescharacteristic
369C22#472×C10911yescharacteristic
370C4—21no—
371C4—21no—
372C4—42no—
373C4—81no—
374C4—161no—
375C4#459×C10911yescharacteristic
376C4#464×C10911yescharacteristic
377C4#465×C10911yescharacteristic
378C4#469×C10911yescharacteristic
379C4#474×C10911yescharacteristic
380C4#476×C10911yescharacteristic
381C4#478×C10911yescharacteristic
382C4#478×C10911yescharacteristic
383C2—162no—
384C2C109×#166811yescharacteristic
385C2C109×#168111yescharacteristic
386C2C109×#168611yescharacteristic
387C2C109×#169711yescharacteristic
388C2C109×#206311yescharacteristic
389C2C109×#206511yescharacteristic
390C2C109×#206711yescharacteristic
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
3. C109×#1948C2semidirect
4. C109×#1939C2semidirect
5. C2×#335×C109C2semidirect
7. C109×#1296C2semidirect
187. #10599C109direct (N × H)
198. #1948C2×C109semidirect
199. #1939C2×C109semidirect
200. C2×#335C2×C109semidirect
202. #1296C2×C109semidirect
203. C109#10599direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#19822
C109×#19572
C109×#19482
C109×#19392
C2×#335×C1092
C2×#331×C1092
C109×#12962
#10599109