Finite Group Databasebeta

← C109×#24985 #42142979

Subgroups — C109×#24985 #42142979

Order 27904.

646 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3067C211yesmaximal, characteristic
2C4×#135×C109C211yesmaximal, characteristic, complemented
3C109×#2080C212yesmaximal, complemented
4C109×#2053C212yesmaximal, complemented
5C109×#1976C211yesmaximal, characteristic, complemented
6C109×#1945C211yesmaximal, characteristic, complemented
7C109×#1906C212yesmaximal, complemented
8C109×#1833C211yesmaximal, characteristic, complemented
9C109×#1827C212yesmaximal, complemented
10C109×#1812C211yesmaximal, characteristic, complemented
11C109×#1764C211yesmaximal, characteristic, complemented
12#575×C109C2212yes—
13C2×#135×C109C2211yescharacteristic
14C2×#131×C109C2211yescharacteristic
15C4×#41×C109C2212yes—
16#473×C109—21no—
17#472×C109—21no—
18#471×C109—22no—
19C4×Q16×C109—21no—
20C4×SD16×C109—22no—
21C4×D8×C109—21no—
22C2×#108×C109C2211yescharacteristic
23#427×C109C2211yescharacteristic
24#427×C109C2211yescharacteristic
25#424×C109C2212yes—
26#421×C109C2212yes—
27C2×#104×C109C2211yescharacteristic
28#416×C109C2212yes—
29#416×C109C2211yescharacteristic
30#413×C109C2212yes—
31#410×C109C2212yes—
32#410×C109C2212yes—
33#405×C109C2211yescharacteristic
34C2×C4×C8×C109C2211yescharacteristic
35#344×C109C2212yes—
36#341×C109C2211yescharacteristic
37#339×C109C2212yes—
38#339×C109C2211yescharacteristic
39H(3,ℤ/4)×C109C2212yes—
40H(3,ℤ/4)×C109C2212yes—
41C2×#41×C109C2312yes—
42C2×#41×C109C2311yescharacteristic
43#137×C109—22no—
44#137×C109—22no—
45AGL(1,ℤ/8)×C109—22no—
46AGL(1,ℤ/8)×C109—22no—
47#135×C109—21no—
48#135×C109—21no—
49#135×C109—21no—
50#135×C109—21no—
51#135×C109—42no—
52C2×Q16×C109D411yescharacteristic
53C2×SD16×C109D412yes—
54C2×D8×C109D411yescharacteristic
55#131×C109—21no—
56#131×C109—21no—
57#131×C109—21no—
58#131×C109—21no—
59C2×M16×C109—21no—
60C2×M16×C109C2311yescharacteristic
61C2×M16×C109C2312yes—
62C2×M16×C109C2311yescharacteristic
63C22×C8×C109—21no—
64C22×C8×C109C2311yescharacteristic
65C22×C8×C109C2311yescharacteristic
66C4×Q8×C109—22no—
67C4×D4×C109—22no—
68C4×D4×C109—42no—
69#117×C109—22no—
70#117×C109C2312yes—
71#117×C109C2311yescharacteristic
72#117×C109C2312yes—
73C2×C42×C109—22no—
74C2×C42×C109C2311yescharacteristic
75#108×C109—21no—
76#108×C109—21no—
77#107×C109—21no—
78#107×C109—21no—
79#106×C109—21no—
80#106×C109—21no—
81C4⋊C8×C109—42no—
82#104×C109—21no—
83#104×C109—21no—
84#104×C109—42no—
85#103×C109—22no—
86#103×C109—21no—
87#102×C109—22no—
88#102×C109—21no—
89#101×C109—22no—
90#100×C109—22no—
91#99×C109—42no—
92#99×C109—42no—
93#97×C109—21no—
94C4×C8×C109—21no—
95C4×C8×C109—21no—
96C4×C8×C109—21no—
97#41×C109—21no—
98#41×C109—21no—
99#41×C109—21no—
100#41×C109—22no—
101#41×C109—22no—
102#41×C109—22no—
103#41×C109—22no—
104#41×C109—21no—
105C2×Q8×C109C2×D412yes—
106C2×Q8×C109#4111yescharacteristic
107C2×D4×C109—21no—
108C2×D4×C109—22no—
109C2×D4×C109C2×D412yes—
110C2×D4×C109#4111yescharacteristic
111C22×C4×C109—21no—
112C22×C4×C109—22no—
113C22×C4×C109—21no—
114C22×C4×C109—42no—
115C22×C4×C109C2411yescharacteristic
116Q16×C109—21no—
117Q16×C109—21no—
118Q16×C109—42no—
119SD16×C109—22no—
120SD16×C109—22no—
121SD16×C109—42no—
122SD16×C109—42no—
123D8×C109—21no—
124D8×C109—21no—
125D8×C109—42no—
126M16×C109—21no—
127M16×C109—22no—
128M16×C109—21no—
129M16×C109—21no—
130M16×C109—22no—
131M16×C109—21no—
132M16×C109—41no—
133M16×C109—41no—
134C2×C8×C109—21no—
135C2×C8×C109—21no—
136C2×C8×C109—21no—
137C2×C8×C109—21no—
138C2×C8×C109—21no—
139C2×C8×C109—22no—
140C2×C8×C109—41no—
141C2×C8×C109—41no—
142C2×C8×C109C2×D411yescharacteristic
143C2×C8×C109C2×D411yescharacteristic
144C2×C8×C109#4111yescharacteristic
145C2×C8×C109#4111yescharacteristic
146C2×C8×C109#4111yescharacteristic
147C2×C8×C109#4111yescharacteristic
148C4⋊C4×C109—22no—
149C4⋊C4×C109—21no—
150C4⋊C4×C109—22no—
151C4⋊C4×C109—42no—
152C22⋊C4×C109—22no—
153C22⋊C4×C109—21no—
154C22⋊C4×C109—22no—
155C22⋊C4×C109—42no—
156C42×C109—21no—
157C42×C109—21no—
158C42×C109—22no—
159C42×C109—22no—
160C42×C109—42no—
161C42×C109#4111yescharacteristic
162C42×C109#4111yescharacteristic
163C23×C109—22no—
164C23×C109—21no—
165C23×C109H(5,𝔽2)11yescharacteristic
166Q8×C109—22no—
167Q8×C109—21no—
168Q8×C109—21no—
169Q8×C109—22no—
170D4×C109—22no—
171D4×C109—21no—
172D4×C109—21no—
173D4×C109—22no—
174D4×C109—41no—
175D4×C109—42no—
176D4×C109—41no—
177D4×C109—42no—
178C2×C4×C109—21no—
179C2×C4×C109—21no—
180C2×C4×C109—22no—
181C2×C4×C109—21no—
182C2×C4×C109—42no—
183C2×C4×C109—41no—
184C2×C4×C109—42no—
185C2×C4×C109—42no—
186C2×C4×C109—41no—
187C2×C4×C109—42no—
188C2×C4×C109—42no—
189C2×C4×C109—41no—
190C2×C4×C109—84no—
191C2×C4×C109C22×D411yescharacteristic
192C2×C4×C109C2×#4111yescharacteristic
193C2×C4×C109C2×#4111yescharacteristic
194C2×C4×C109C2×#4111yescharacteristic
195C2×C4×C109C2×#4111yescharacteristic
196C2×C4×C109H(5,𝔽2)11yescharacteristic
197C8×C109—41no—
198C8×C109—41no—
199C8×C109—42no—
200C8×C109—41no—
201C8×C109#12311yescharacteristic
202C8×C109#12311yescharacteristic
203C8×C109#12311yescharacteristic
204C8×C109#12311yescharacteristic
205C22×C109—41no—
206C22×C109—41no—
207C22×C109—42no—
208C22×C109—82no—
209C22×C109—81no—
210C22×C109#54011yescharacteristic
211C22×C109#54611yescharacteristic
212C22×C109#55311yescharacteristic
213C4×C109—22no—
214C4×C109—41no—
215C4×C109—42no—
216C4×C109—41no—
217C4×C109—82no—
218C4×C109—81no—
219C4×C109—82no—
220C4×C109C2×#12311yescharacteristic
221C4×C109#52911yescharacteristic
222C4×C109#54611yescharacteristic
223C4×C109#55511yescharacteristic
224#24985C10911yesmaximal, characteristic, direct product factor, Sylow
225C2×C109—21no—
226C2×C109—21no—
227C2×C109—21no—
228C2×C109—82no—
229C2×C109—81no—
230C2×C109#242011yescharacteristic
231#3067C2×C10911yescharacteristic
232C4×#135C2×C10911yescharacteristic, complemented
233#2080C2×C10912yescomplemented
234#2053C2×C10912yescomplemented
235#1976C2×C10911yescharacteristic, complemented
236#1945C2×C10911yescharacteristic, complemented
237#1906C2×C10912yescomplemented
238#1833C2×C10911yescharacteristic, complemented
239#1827C2×C10912yescomplemented
240#1812C2×C10911yescharacteristic, complemented
241#1764C2×C10911yescharacteristic, complemented
242C109#2498511yescharacteristic, direct product factor, Sylow
243#575C22×C10912yes—
244C2×#135C22×C10911yescharacteristic
245C2×#131C22×C10911yescharacteristic
246C4×#41C22×C10912yes—
247#473—21no—
248#472—21no—
249#471—22no—
250C4×Q16—21no—
251C4×SD16—22no—
252C4×D8—21no—
253C2×#108C22×C10911yescharacteristic
254#427C22×C10911yescharacteristic
255#427C22×C10911yescharacteristic
256#424C22×C10912yes—
257#421C22×C10912yes—
258C2×#104C22×C10911yescharacteristic
259#416C22×C10911yescharacteristic
260#416C22×C10912yes—
261#413C22×C10912yes—
262#410C22×C10912yes—
263#410C22×C10912yes—
264#405C22×C10911yescharacteristic
265C2×C4×C8C22×C10911yescharacteristic
266#344C22×C10912yes—
267#341C22×C10911yescharacteristic
268#339C22×C10911yescharacteristic
269#339C22×C10912yes—
270H(3,ℤ/4)C22×C10912yes—
271H(3,ℤ/4)C22×C10912yes—
272C2×#41C23×C10912yes—
273C2×#41C23×C10911yescharacteristic
274#137—22no—
275#137—22no—
276AGL(1,ℤ/8)—22no—
277AGL(1,ℤ/8)—22no—
278#135—21no—
279#135—21no—
280#135—21no—
281#135—21no—
282#135—42no—
283C2×Q16D4×C10911yescharacteristic
284C2×SD16D4×C10912yes—
285C2×D8D4×C10911yescharacteristic
286#131—21no—
287#131—21no—
288#131—21no—
289#131—21no—
290C2×M16—21no—
291C2×M16C23×C10911yescharacteristic
292C2×M16C23×C10912yes—
293C2×M16C23×C10911yescharacteristic
294C22×C8—21no—
295C22×C8C23×C10911yescharacteristic
296C22×C8C23×C10911yescharacteristic
297C4×Q8—22no—
298C4×D4—22no—
299C4×D4—42no—
300#117—22no—
301#117C23×C10912yes—
302#117C23×C10911yescharacteristic
303#117C23×C10912yes—
304C2×C42—22no—
305C2×C42C23×C10911yescharacteristic
306#108—21no—
307#108—21no—
308#107—21no—
309#107—21no—
310#106—21no—
311#106—21no—
312C4⋊C8—42no—
313#104—21no—
314#104—21no—
315#104—42no—
316#103—22no—
317#103—21no—
318#102—22no—
319#102—21no—
320#101—22no—
321#100—22no—
322#99—42no—
323#99—42no—
324#97—21no—
325C4×C8—21no—
326C4×C8—21no—
327C4×C8—21no—
328#41—21no—
329#41—21no—
330#41—21no—
331#41—22no—
332#41—22no—
333#41—22no—
334#41—22no—
335#41—21no—
336C2×Q8C2×D4×C10912yes—
337C2×Q8#41×C10911yescharacteristic
338C2×D4—21no—
339C2×D4—22no—
340C2×D4C2×D4×C10912yes—
341C2×D4#41×C10911yescharacteristic
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—42no—
346C22×C4C24×C10911yescharacteristic
347Q16—21no—
348Q16—21no—
349Q16—42no—
350SD16—22no—
351SD16—22no—
352SD16—42no—
353SD16—42no—
354D8—21no—
355D8—21no—
356D8—42no—
357M16—21no—
358M16—22no—
359M16—21no—
360M16—21no—
361M16—22no—
362M16—21no—
363M16—41no—
364M16—41no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—22no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8C2×D4×C10911yescharacteristic
374C2×C8C2×D4×C10911yescharacteristic
375C2×C8#41×C10911yescharacteristic
376C2×C8#41×C10911yescharacteristic
377C2×C8#41×C10911yescharacteristic
378C2×C8#41×C10911yescharacteristic
379C4⋊C4—22no—
380C4⋊C4—21no—
381C4⋊C4—22no—
382C4⋊C4—42no—
383C22⋊C4—22no—
384C22⋊C4—21no—
385C22⋊C4—22no—
386C22⋊C4—42no—
387C42—21no—
388C42—21no—
389C42—22no—
390C42—22no—
391C42—42no—
392C42#41×C10911yescharacteristic
393C42#41×C10911yescharacteristic
394C23—22no—
395C23—21no—
396C23H(5,𝔽2)×C10911yescharacteristic
397Q8—22no—
398Q8—21no—
399Q8—21no—
400Q8—22no—
401D4—22no—
402D4—21no—
403D4—21no—
404D4—22no—
405D4—41no—
406D4—42no—
407D4—41no—
408D4—42no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—21no—
413C2×C4—42no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—41no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—84no—
422C2×C4C22×D4×C10911yescharacteristic
423C2×C4C2×#41×C10911yescharacteristic
424C2×C4C2×#41×C10911yescharacteristic
425C2×C4C2×#41×C10911yescharacteristic
426C2×C4C2×#41×C10911yescharacteristic
427C2×C4H(5,𝔽2)×C10911yescharacteristic
428C8—41no—
429C8—41no—
430C8—42no—
431C8—41no—
432C8#123×C10911yescharacteristic
433C8#123×C10911yescharacteristic
434C8#123×C10911yescharacteristic
435C8#123×C10911yescharacteristic
436C22—41no—
437C22—41no—
438C22—42no—
439C22—82no—
440C22—81no—
441C22#540×C10911yescharacteristic
442C22#546×C10911yescharacteristic
443C22#553×C10911yescharacteristic
444C4—22no—
445C4—41no—
446C4—42no—
447C4—41no—
448C4—82no—
449C4—82no—
450C4—81no—
451C4C2×#123×C10911yescharacteristic
452C4#529×C10911yescharacteristic
453C4#546×C10911yescharacteristic
454C4#555×C10911yescharacteristic
455C2—21no—
456C2—21no—
457C2—21no—
458C2—82no—
459C2—81no—
460C2C109×#242011yescharacteristic
subgroupG/H
centerC4×C109#555
commutator, Frattini, cosocleC22×C4C24×C109
socleC2×C109#2420
prime radicalC109×#3067C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C109 ◁ C22×C4×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C4×#135×C109C2semidirect
3. C109×#2080C2semidirect
4. C109×#2053C2semidirect
5. C109×#1976C2semidirect
6. C109×#1945C2semidirect
7. C109×#1906C2semidirect
8. C109×#1833C2semidirect
9. C109×#1827C2semidirect
10. C109×#1812C2semidirect
11. C109×#1764C2semidirect
224. #24985C109direct (N × H)
232. C4×#135C2×C109semidirect
233. #2080C2×C109semidirect
234. #2053C2×C109semidirect
235. #1976C2×C109semidirect
236. #1945C2×C109semidirect
237. #1906C2×C109semidirect
238. #1833C2×C109semidirect
239. #1827C2×C109semidirect
240. #1812C2×C109semidirect
241. #1764C2×C109semidirect
242. C109#24985direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#30672
C4×#135×C1092
C109×#20802
C109×#20532
C109×#19762
C109×#19452
C109×#19062
C109×#18332
C109×#18272
C109×#18122
C109×#17642
#24985109