Finite Group Databasebeta

← C109×#8897 #42126891

Subgroups — C109×#8897 #42126891

Order 27904.

530 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#2354C211yesmaximal, characteristic
2C109×#1947C211yesmaximal, characteristic, complemented
3C109×#1834C211yesmaximal, characteristic, complemented
4C109×#1802C211yesmaximal, characteristic
5C2×D4⋊C8×C109C211yesmaximal, characteristic
6C109×#1296C211yesmaximal, characteristic, complemented
7C109×#1279C211yesmaximal, characteristic, complemented
8C2×C4×D4×C109C2211yescharacteristic
9C2×#117×C109—21no—
10C2×#107×C109—21no—
11C2×C4⋊C8×C109C2211yescharacteristic
12C2×C4⋊C8×C109C2211yescharacteristic
13C2×#102×C109—21no—
14C2×C4×C8×C109C2211yescharacteristic
15#390×C109—21no—
16#388×C109—22no—
17#384×C109C2211yescharacteristic
18#384×C109C2211yescharacteristic
19#380×C109—21no—
20C4×C4⋊C4×C109—21no—
21C4×C4⋊C4×C109C2211yescharacteristic
22#340×C109—21no—
23#340×C109—21no—
24#340×C109—22no—
25D4⋊C8×C109C414yescomplemented
26C22×D4×C109D411yescharacteristic
27C23×C4×C109—21no—
28C22×C8×C109—21no—
29C22×C8×C109—21no—
30C22×C8×C109—21no—
31C4×D4×C109—21no—
32C4×D4×C109—21no—
33C4×D4×C109C2×C412yes—
34C4×D4×C109Q812yes—
35#117×C109—42no—
36#117×C109—42no—
37C2×C4⋊C4×C109—21no—
38C2×C4⋊C4×C109—21no—
39C2×C4⋊C4×C109—21no—
40C2×C4⋊C4×C109—21no—
41C2×C4⋊C4×C109—21no—
42C2×C4⋊C4×C109D411yescharacteristic
43C2×C4⋊C4×C109D411yescharacteristic
44C2×C4⋊C4×C109D411yescharacteristic
45C2×C4⋊C4×C109D412yes—
46C2×C22⋊C4×C109—21no—
47C2×C22⋊C4×C109—22no—
48C2×C22⋊C4×C109—22no—
49C2×C42×C109—22no—
50C2×C42×C109—21no—
51C2×C42×C109C2311yescharacteristic
52#107×C109—42no—
53C4⋊C8×C109—21no—
54C4⋊C8×C109—21no—
55C4⋊C8×C109—21no—
56C4⋊C8×C109C2×C412yes—
57#102×C109—42no—
58C4×C8×C109—21no—
59C4×C8×C109C2×C412yes—
60#95×C109—22no—
61#95×C109—21no—
62C24×C109—21no—
63C2×D4×C109—21no—
64C2×D4×C109—21no—
65C2×D4×C109—21no—
66C2×D4×C109—21no—
67C2×D4×C109—21no—
68C2×D4×C109C4⋊C412yes—
69C22×C4×C109—21no—
70C22×C4×C109—21no—
71C22×C4×C109—22no—
72C22×C4×C109—22no—
73C22×C4×C109—21no—
74C22×C4×C109—22no—
75C22×C4×C109—41no—
76C22×C4×C109—41no—
77C22×C4×C109—41no—
78C22×C4×C109—41no—
79C22×C4×C109—41no—
80C22×C4×C109C2×D411yescharacteristic
81C22×C4×C109C2×D411yescharacteristic
82C22×C4×C109C2×D411yescharacteristic
83C2×C8×C109—22no—
84C2×C8×C109—22no—
85C2×C8×C109—41no—
86C2×C8×C109—41no—
87C2×C8×C109—41no—
88C2×C8×C109—41no—
89C2×C8×C109—41no—
90C2×C8×C109—41no—
91C2×C8×C109—41no—
92C4⋊C4×C109—21no—
93C4⋊C4×C109—21no—
94C4⋊C4×C109—21no—
95C4⋊C4×C109—21no—
96C4⋊C4×C109—42no—
97C4⋊C4×C109—42no—
98C4⋊C4×C109—41no—
99C4⋊C4×C109—41no—
100C4⋊C4×C109—41no—
101C4⋊C4×C109—41no—
102C4⋊C4×C109—41no—
103C4⋊C4×C109—41no—
104C4⋊C4×C109—41no—
105C4⋊C4×C109—41no—
106C4⋊C4×C109C4⋊C412yes—
107C4⋊C4×C109D812yes—
108C22⋊C4×C109—22no—
109C22⋊C4×C109—22no—
110C22⋊C4×C109—44no—
111C22⋊C4×C109—82no—
112C42×C109—44no—
113C42×C109—41no—
114C42×C109—41no—
115C42×C109C22×C411yescharacteristic
116C42×C109C2×Q811yescharacteristic
117C42×C109#4111yescharacteristic
118C42×C109#4111yescharacteristic
119C23×C109—22no—
120C23×C109—41no—
121C23×C109—41no—
122C23×C109—41no—
123C23×C109—41no—
124C23×C109—41no—
125C23×C109—41no—
126C23×C109#12011yescharacteristic
127D4×C109—22no—
128D4×C109—22no—
129D4×C109—41no—
130D4×C109—41no—
131C2×C4×C109—21no—
132C2×C4×C109—21no—
133C2×C4×C109—22no—
134C2×C4×C109—22no—
135C2×C4×C109—41no—
136C2×C4×C109—41no—
137C2×C4×C109—41no—
138C2×C4×C109—41no—
139C2×C4×C109—42no—
140C2×C4×C109—42no—
141C2×C4×C109—42no—
142C2×C4×C109—42no—
143C2×C4×C109—42no—
144C2×C4×C109—41no—
145C2×C4×C109—41no—
146C2×C4×C109—42no—
147C2×C4×C109—41no—
148C2×C4×C109—42no—
149C2×C4×C109—81no—
150C2×C4×C109—81no—
151C2×C4×C109—81no—
152C2×C4×C109C2×C4⋊C411yescharacteristic
153C2×C4×C109C4×D411yescharacteristic
154C2×C4×C109C4×D411yescharacteristic
155C2×C4×C109C22⋊Q811yescharacteristic
156C2×C4×C109C22⋊Q811yescharacteristic
157C2×C4×C109#12311yescharacteristic
158C2×C4×C109C2×D811yescharacteristic
159C2×C4×C109#13511yescharacteristic
160C2×C4×C109#13511yescharacteristic
161C2×C4×C109#13511yescharacteristic
162C2×C4×C109AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C109AGL(1,ℤ/8)11yescharacteristic
164C2×C4×C109#13711yescharacteristic
165C2×C4×C109#13711yescharacteristic
166C8×C109—42no—
167C8×C109—42no—
168C8×C109—81no—
169C22×C109—42no—
170C22×C109—42no—
171C22×C109—42no—
172C22×C109—81no—
173C22×C109—81no—
174C22×C109—81no—
175C22×C109—81no—
176C22×C109#38511yescharacteristic
177C22×C109#44711yescharacteristic
178C22×C109#44911yescharacteristic
179C22×C109#45211yescharacteristic
180C22×C109#45211yescharacteristic
181C22×C109#45411yescharacteristic
182C22×C109#45511yescharacteristic
183C4×C109—21no—
184C4×C109—21no—
185C4×C109—42no—
186C4×C109—42no—
187C4×C109—82no—
188C4×C109—81no—
189C4×C109—161no—
190C4×C109#42711yescharacteristic
191C4×C109#42811yescharacteristic
192C4×C109C4×D811yescharacteristic
193C4×C109#44011yescharacteristic
194C4×C109#47411yescharacteristic
195C4×C109#47811yescharacteristic
196C4×C109#48211yescharacteristic
197C4×C109#48511yescharacteristic
198#8897C10911yesmaximal, characteristic, direct product factor, Sylow
199C2×C109—82no—
200C2×C109#162511yescharacteristic
201C2×C109#163811yescharacteristic
202C2×C109#164111yescharacteristic
203C2×C109#164611yescharacteristic
204C2×C109#185411yescharacteristic
205C2×C109#186011yescharacteristic
206C2×C109#186211yescharacteristic
207#2354C2×C10911yescharacteristic
208#1947C2×C10911yescharacteristic, complemented
209#1834C2×C10911yescharacteristic, complemented
210#1802C2×C10911yescharacteristic
211C2×D4⋊C8C2×C10911yescharacteristic
212#1296C2×C10911yescharacteristic, complemented
213#1279C2×C10911yescharacteristic, complemented
214C109#889711yescharacteristic, direct product factor, Sylow
215C2×C4×D4C22×C10911yescharacteristic
216C2×#117—21no—
217C2×#107—21no—
218C2×C4⋊C8C22×C10911yescharacteristic
219C2×C4⋊C8C22×C10911yescharacteristic
220C2×#102—21no—
221C2×C4×C8C22×C10911yescharacteristic
222#390—21no—
223#388—22no—
224#384C22×C10911yescharacteristic
225#384C22×C10911yescharacteristic
226#380—21no—
227C4×C4⋊C4—21no—
228C4×C4⋊C4C22×C10911yescharacteristic
229#340—21no—
230#340—22no—
231#340—21no—
232D4⋊C8C4×C10914yescomplemented
233C22×D4D4×C10911yescharacteristic
234C23×C4—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C4×D4—21no—
239C4×D4—21no—
240C4×D4C2×C4×C10912yes—
241C4×D4Q8×C10912yes—
242#117—42no—
243#117—42no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4D4×C10911yescharacteristic
250C2×C4⋊C4D4×C10911yescharacteristic
251C2×C4⋊C4D4×C10911yescharacteristic
252C2×C4⋊C4D4×C10912yes—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C42—22no—
257C2×C42—21no—
258C2×C42C23×C10911yescharacteristic
259#107—42no—
260C4⋊C8—21no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8C2×C4×C10912yes—
264#102—42no—
265C4×C8—21no—
266C4×C8C2×C4×C10912yes—
267#95—22no—
268#95—21no—
269C24—21no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4C4⋊C4×C10912yes—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4C2×D4×C10911yescharacteristic
288C22×C4C2×D4×C10911yescharacteristic
289C22×C4C2×D4×C10911yescharacteristic
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4C4⋊C4×C10912yes—
314C4⋊C4D8×C10912yes—
315C22⋊C4—22no—
316C22⋊C4—22no—
317C22⋊C4—44no—
318C22⋊C4—82no—
319C42—44no—
320C42—41no—
321C42—41no—
322C42C22×C4×C10911yescharacteristic
323C42C2×Q8×C10911yescharacteristic
324C42#41×C10911yescharacteristic
325C42#41×C10911yescharacteristic
326C23—22no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23#120×C10911yescharacteristic
334D4—22no—
335D4—22no—
336D4—41no—
337D4—41no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4C2×C4⋊C4×C10911yescharacteristic
360C2×C4C4×D4×C10911yescharacteristic
361C2×C4C4×D4×C10911yescharacteristic
362C2×C4C22⋊Q8×C10911yescharacteristic
363C2×C4C22⋊Q8×C10911yescharacteristic
364C2×C4#123×C10911yescharacteristic
365C2×C4C2×D8×C10911yescharacteristic
366C2×C4#135×C10911yescharacteristic
367C2×C4#135×C10911yescharacteristic
368C2×C4#135×C10911yescharacteristic
369C2×C4AGL(1,ℤ/8)×C10911yescharacteristic
370C2×C4AGL(1,ℤ/8)×C10911yescharacteristic
371C2×C4#137×C10911yescharacteristic
372C2×C4#137×C10911yescharacteristic
373C8—42no—
374C8—42no—
375C8—81no—
376C22—42no—
377C22—42no—
378C22—42no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22#385×C10911yescharacteristic
384C22#447×C10911yescharacteristic
385C22#449×C10911yescharacteristic
386C22#452×C10911yescharacteristic
387C22#452×C10911yescharacteristic
388C22#454×C10911yescharacteristic
389C22#455×C10911yescharacteristic
390C4—21no—
391C4—21no—
392C4—42no—
393C4—42no—
394C4—82no—
395C4—81no—
396C4—161no—
397C4#427×C10911yescharacteristic
398C4#428×C10911yescharacteristic
399C4C4×D8×C10911yescharacteristic
400C4#440×C10911yescharacteristic
401C4#474×C10911yescharacteristic
402C4#478×C10911yescharacteristic
403C4#482×C10911yescharacteristic
404C4#485×C10911yescharacteristic
405C2—82no—
406C2C109×#162511yescharacteristic
407C2C109×#163811yescharacteristic
408C2C109×#164111yescharacteristic
409C2C109×#164611yescharacteristic
410C2C109×#185411yescharacteristic
411C2C109×#186011yescharacteristic
412C2C109×#186211yescharacteristic
subgroupG/H
center, socleC23×C109#120
commutatorC42C22×C4×C109
Frattini, cosocleC2×C42C23×C109
prime radicalC22×D4×C109D4
LCSG ▷ 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. C109×#1947C2semidirect
3. C109×#1834C2semidirect
6. C109×#1296C2semidirect
7. C109×#1279C2semidirect
25. D4⋊C8×C109C4semidirect
198. #8897C109direct (N × H)
208. #1947C2×C109semidirect
209. #1834C2×C109semidirect
212. #1296C2×C109semidirect
213. #1279C2×C109semidirect
214. C109#8897direct (N × H)
232. D4⋊C8C4×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#23542
C109×#19472
C109×#18342
C109×#18022
C2×D4⋊C8×C1092
C109×#12962
C109×#12792
#8897109