Finite Group Databasebeta

← C109×#18202 #42136196

Subgroups — C109×#18202 #42136196

Order 27904.

1282 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C109×#3496C211yesmaximal, characteristic
2C2×#480×C109C212yesmaximal, complemented
3C109×#2486C211yesmaximal, characteristic, complemented
4C109×#1610C214yesmaximal, complemented
5C109×#1607C214yesmaximal, complemented
6C109×#1519C212yesmaximal, complemented
7C109×#1459C211yesmaximal, characteristic, complemented
8C2×H(5,𝔽2)×C109—21no—
9#547×C109—22no—
10#546×C109—24no—
11D42×C109—22no—
12#537×C109C2212yes—
13#534×C109—21no—
14#534×C109C2212yes—
15#532×C109—21no—
16C2×#123×C109—21no—
17C2×#121×C109—21no—
18C2×#121×C109C2212yes—
19C2×C4×D4×C109C2211yescharacteristic
20C22×C4⋊C4×C109C2211yescharacteristic, complemented
21#480×C109—24no—
22#480×C109C2218yescomplemented
23C2×#107×C109—22no—
24C2×#102×C109—22no—
25C2×C22⋊C8×C109C2212yescomplemented
26#398×C109C2212yescomplemented
27#397×C109C2212yescomplemented
28#389×C109—21no—
29#381×C109—21no—
30#380×C109—21no—
31#380×C109C2211yescharacteristic, complemented
32#327×C109C2218yescomplemented
33#324×C109C2212yescomplemented
34#323×C109C2212yescomplemented
35H(5,𝔽2)×C109—44no—
36H(5,𝔽2)×C109—44no—
37C2×#41×C109—24no—
38C2×#41×C109—22no—
39C22×D4×C109—24no—
40C22×D4×C109—22no—
41C22×D4×C109—22no—
42C22×D4×C109—21no—
43C22×D4×C109D411yescharacteristic
44C22×D4×C109D412yes—
45C23×C4×C109—21no—
46C23×C4×C109D411yescharacteristic
47C23×C4×C109C2311yescharacteristic
48C22×C8×C109—22no—
49C4⋊Q8×C109—21no—
50#127×C109—21no—
51#124×C109—22no—
52#123×C109—24no—
53#123×C109—24no—
54#123×C109—44no—
55C22⋊Q8×C109—24no—
56#121×C109—22no—
57#121×C109—22no—
58#121×C109—24no—
59#121×C109—22no—
60#121×C109—22no—
61#121×C109—22no—
62#121×C109—22no—
63#121×C109D414yes—
64#121×C109C2314yes—
65#120×C109—24no—
66#120×C109—24no—
67C4×D4×C109—22no—
68C4×D4×C109—22no—
69C4×D4×C109—22no—
70C4×D4×C109—22no—
71C2×C4⋊C4×C109—21no—
72C2×C4⋊C4×C109—22no—
73C2×C4⋊C4×C109—22no—
74C2×C4⋊C4×C109—21no—
75C2×C4⋊C4×C109—22no—
76C2×C4⋊C4×C109—21no—
77C2×C4⋊C4×C109D412yes—
78C2×C4⋊C4×C109C2312yes—
79C2×C4⋊C4×C109C2311yescharacteristic
80C2×C22⋊C4×C109—22no—
81C2×C22⋊C4×C109—22no—
82C2×C22⋊C4×C109—22no—
83C2×C22⋊C4×C109—21no—
84C2×C22⋊C4×C109D411yescharacteristic
85C2×C22⋊C4×C109C2311yescharacteristic
86C2×C42×C109D411yescharacteristic
87#107×C109—28no—
88#102×C109—28no—
89C22⋊C8×C109—22no—
90C22⋊C8×C109C2314yes—
91#95×C109—22no—
92#95×C109—22no—
93#95×C109C2312yes—
94C24×C109—22no—
95C24×C109—22no—
96C24×C109C2×D411yescharacteristic
97C24×C109C2×D411yescharacteristic
98#41×C109—44no—
99#41×C109—416no—
100#41×C109—44no—
101C2×Q8×C109—22no—
102C2×D4×C109—22no—
103C2×D4×C109—22no—
104C2×D4×C109—24no—
105C2×D4×C109—22no—
106C2×D4×C109—24no—
107C2×D4×C109—24no—
108C2×D4×C109—22no—
109C2×D4×C109—22no—
110C2×D4×C109—22no—
111C2×D4×C109—44no—
112C2×D4×C109—416no—
113C2×D4×C109—44no—
114C2×D4×C109—44no—
115C2×D4×C109—44no—
116C2×D4×C109—44no—
117C2×D4×C109—82no—
118C2×D4×C109D814yes—
119C2×D4×C109C2×D414yes—
120C22×C4×C109—22no—
121C22×C4×C109—24no—
122C22×C4×C109—22no—
123C22×C4×C109—21no—
124C22×C4×C109—21no—
125C22×C4×C109—22no—
126C22×C4×C109—22no—
127C22×C4×C109—42no—
128C22×C4×C109—42no—
129C22×C4×C109—42no—
130C22×C4×C109—42no—
131C22×C4×C109—42no—
132C22×C4×C109C2×D411yescharacteristic
133C22×C4×C109C2×D412yes—
134C22×C4×C109C2×D411yescharacteristic
135C22×C4×C109C2×D411yescharacteristic
136C22×C4×C109C2×D411yescharacteristic
137C22×C4×C109C2×D412yes—
138C22×C4×C109C2411yescharacteristic
139C2×C8×C109—24no—
140C2×C8×C109—44no—
141C4⋊C4×C109—22no—
142C4⋊C4×C109—22no—
143C4⋊C4×C109—22no—
144C4⋊C4×C109—22no—
145C4⋊C4×C109—24no—
146C4⋊C4×C109—42no—
147C4⋊C4×C109—41no—
148C4⋊C4×C109—41no—
149C4⋊C4×C109—44no—
150C4⋊C4×C109C2×D414yes—
151C22⋊C4×C109—24no—
152C22⋊C4×C109—24no—
153C22⋊C4×C109—22no—
154C22⋊C4×C109—24no—
155C22⋊C4×C109—24no—
156C22⋊C4×C109—22no—
157C22⋊C4×C109—42no—
158C22⋊C4×C109—44no—
159C42×C109—21no—
160C42×C109—21no—
161C23×C109—21no—
162C23×C109—21no—
163C23×C109—24no—
164C23×C109—41no—
165C23×C109—41no—
166C23×C109—41no—
167C23×C109—41no—
168C23×C109—44no—
169C23×C109—44no—
170C23×C109—82no—
171C23×C109—82no—
172C23×C109—82no—
173C23×C109—82no—
174C23×C109#12012yes—
175C23×C109C2×D812yes—
176C23×C109C2×D812yes—
177C23×C109AGL(1,ℤ/8)12yes—
178C23×C109C22×D411yescharacteristic
179Q8×C109—44no—
180D4×C109—28no—
181D4×C109—44no—
182D4×C109—44no—
183D4×C109—48no—
184D4×C109—44no—
185D4×C109—44no—
186D4×C109—48no—
187D4×C109—44no—
188D4×C109—82no—
189D4×C109—82no—
190C2×C4×C109—21no—
191C2×C4×C109—21no—
192C2×C4×C109—22no—
193C2×C4×C109—24no—
194C2×C4×C109—21no—
195C2×C4×C109—21no—
196C2×C4×C109—24no—
197C2×C4×C109—42no—
198C2×C4×C109—41no—
199C2×C4×C109—48no—
200C2×C4×C109—44no—
201C2×C4×C109—44no—
202C2×C4×C109—42no—
203C2×C4×C109—41no—
204C2×C4×C109—41no—
205C2×C4×C109—42no—
206C2×C4×C109—42no—
207C2×C4×C109—42no—
208C2×C4×C109—42no—
209C2×C4×C109—42no—
210C2×C4×C109—42no—
211C2×C4×C109—81no—
212C2×C4×C109—84no—
213C2×C4×C109#12011yescharacteristic
214C2×C4×C109#12011yescharacteristic
215C2×C4×C109C2×D812yes—
216C2×C4×C109C22×D412yes—
217C8×C109—44no—
218C22×C109—21no—
219C22×C109—22no—
220C22×C109—21no—
221C22×C109—22no—
222C22×C109—22no—
223C22×C109—42no—
224C22×C109—42no—
225C22×C109—42no—
226C22×C109—41no—
227C22×C109—41no—
228C22×C109—44no—
229C22×C109—44no—
230C22×C109—44no—
231C22×C109—81no—
232C22×C109—164no—
233C22×C109—164no—
234C22×C109#44714yes—
235C22×C109C2×#12011yescharacteristic
236C22×C109C22×D811yescharacteristic
237C22×C109C2×AGL(1,ℤ/8)11yescharacteristic
238C4×C109—22no—
239C4×C109—22no—
240C4×C109—21no—
241C4×C109—44no—
242C4×C109—41no—
243C4×C109—41no—
244C4×C109—82no—
245C4×C109—82no—
246C4×C109#45812yes—
247#18202C10911yesmaximal, characteristic, direct product factor, Sylow
248C2×C109—21no—
249C2×C109—21no—
250C2×C109—24no—
251C2×C109—42no—
252C2×C109—84no—
253C2×C109C2×#44711yescharacteristic
254C2×C109#302211yescharacteristic
255C2×C109C2×#45811yescharacteristic
256#3496C2×C10911yescharacteristic
257C2×#480C2×C10912yescomplemented
258#2486C2×C10911yescharacteristic, complemented
259#1610C2×C10914yescomplemented
260#1607C2×C10914yescomplemented
261#1519C2×C10912yescomplemented
262#1459C2×C10911yescharacteristic, complemented
263C109#1820211yescharacteristic, direct product factor, Sylow
264C2×H(5,𝔽2)—21no—
265#547—22no—
266#546—24no—
267D42—22no—
268#537C22×C10912yes—
269#534—21no—
270#534C22×C10912yes—
271#532—21no—
272C2×#123—21no—
273C2×#121—21no—
274C2×#121C22×C10912yes—
275C2×C4×D4C22×C10911yescharacteristic
276C22×C4⋊C4C22×C10911yescharacteristic, complemented
277#480—24no—
278#480C22×C10918yescomplemented
279C2×#107—22no—
280C2×#102—22no—
281C2×C22⋊C8C22×C10912yescomplemented
282#398C22×C10912yescomplemented
283#397C22×C10912yescomplemented
284#389—21no—
285#381—21no—
286#380—21no—
287#380C22×C10911yescharacteristic, complemented
288#327C22×C10918yescomplemented
289#324C22×C10912yescomplemented
290#323C22×C10912yescomplemented
291H(5,𝔽2)—44no—
292H(5,𝔽2)—44no—
293C2×#41—24no—
294C2×#41—22no—
295C22×D4—24no—
296C22×D4—22no—
297C22×D4—22no—
298C22×D4—21no—
299C22×D4D4×C10911yescharacteristic
300C22×D4D4×C10912yes—
301C23×C4—21no—
302C23×C4D4×C10911yescharacteristic
303C23×C4C23×C10911yescharacteristic
304C22×C8—22no—
305C4⋊Q8—21no—
306#127—21no—
307#124—22no—
308#123—24no—
309#123—24no—
310#123—44no—
311C22⋊Q8—24no—
312#121—22no—
313#121—22no—
314#121—24no—
315#121—22no—
316#121—22no—
317#121—22no—
318#121—22no—
319#121D4×C10914yes—
320#121C23×C10914yes—
321#120—24no—
322#120—24no—
323C4×D4—22no—
324C4×D4—22no—
325C4×D4—22no—
326C4×D4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4D4×C10912yes—
334C2×C4⋊C4C23×C10912yes—
335C2×C4⋊C4C23×C10911yescharacteristic
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—22no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4D4×C10911yescharacteristic
341C2×C22⋊C4C23×C10911yescharacteristic
342C2×C42D4×C10911yescharacteristic
343#107—28no—
344#102—28no—
345C22⋊C8—22no—
346C22⋊C8C23×C10914yes—
347#95—22no—
348#95—22no—
349#95C23×C10912yes—
350C24—22no—
351C24—22no—
352C24C2×D4×C10911yescharacteristic
353C24C2×D4×C10911yescharacteristic
354#41—44no—
355#41—416no—
356#41—44no—
357C2×Q8—22no—
358C2×D4—22no—
359C2×D4—22no—
360C2×D4—24no—
361C2×D4—22no—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—44no—
368C2×D4—416no—
369C2×D4—44no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—82no—
374C2×D4D8×C10914yes—
375C2×D4C2×D4×C10914yes—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4C2×D4×C10911yescharacteristic
389C22×C4C2×D4×C10912yes—
390C22×C4C2×D4×C10911yescharacteristic
391C22×C4C2×D4×C10911yescharacteristic
392C22×C4C2×D4×C10911yescharacteristic
393C22×C4C2×D4×C10912yes—
394C22×C4C24×C10911yescharacteristic
395C2×C8—24no—
396C2×C8—44no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—24no—
402C4⋊C4—42no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—44no—
406C4⋊C4C2×D4×C10914yes—
407C22⋊C4—24no—
408C22⋊C4—24no—
409C22⋊C4—22no—
410C22⋊C4—24no—
411C22⋊C4—24no—
412C22⋊C4—22no—
413C22⋊C4—42no—
414C22⋊C4—44no—
415C42—21no—
416C42—21no—
417C23—21no—
418C23—21no—
419C23—24no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—44no—
425C23—44no—
426C23—82no—
427C23—82no—
428C23—82no—
429C23—82no—
430C23#120×C10912yes—
431C23C2×D8×C10912yes—
432C23C2×D8×C10912yes—
433C23AGL(1,ℤ/8)×C10912yes—
434C23C22×D4×C10911yescharacteristic
435Q8—44no—
436D4—28no—
437D4—44no—
438D4—44no—
439D4—48no—
440D4—44no—
441D4—44no—
442D4—48no—
443D4—44no—
444D4—82no—
445D4—82no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—24no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—48no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—81no—
468C2×C4—84no—
469C2×C4#120×C10911yescharacteristic
470C2×C4#120×C10911yescharacteristic
471C2×C4C2×D8×C10912yes—
472C2×C4C22×D4×C10912yes—
473C8—44no—
474C22—21no—
475C22—22no—
476C22—21no—
477C22—22no—
478C22—22no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—41no—
483C22—41no—
484C22—44no—
485C22—44no—
486C22—44no—
487C22—81no—
488C22—164no—
489C22—164no—
490C22#447×C10914yes—
491C22C2×#120×C10911yescharacteristic
492C22C22×D8×C10911yescharacteristic
493C22C2×AGL(1,ℤ/8)×C10911yescharacteristic
494C4—22no—
495C4—22no—
496C4—21no—
497C4—44no—
498C4—41no—
499C4—41no—
500C4—82no—
501C4—82no—
502C4#458×C10912yes—
503C2—21no—
504C2—21no—
505C2—24no—
506C2—42no—
507C2—84no—
508C2C2×#447×C10911yescharacteristic
509C2C109×#302211yescharacteristic
510C2C2×#458×C10911yescharacteristic
subgroupG/H
center, socleC22×C109C2×#120
commutator, Frattini, cosocleC22×C4C24×C109
prime radicalC109×#3496C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C109 ◁ C23×C4×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#480×C109C2semidirect
3. C109×#2486C2semidirect
4. C109×#1610C2semidirect
5. C109×#1607C2semidirect
6. C109×#1519C2semidirect
7. C109×#1459C2semidirect
20. C22×C4⋊C4×C109C22semidirect
22. #480×C109C22semidirect
25. C2×C22⋊C8×C109C22semidirect
26. #398×C109C22semidirect
27. #397×C109C22semidirect
31. #380×C109C22semidirect
32. #327×C109C22semidirect
33. #324×C109C22semidirect
34. #323×C109C22semidirect
247. #18202C109direct (N × H)
257. C2×#480C2×C109semidirect
258. #2486C2×C109semidirect
259. #1610C2×C109semidirect
260. #1607C2×C109semidirect
261. #1519C2×C109semidirect
262. #1459C2×C109semidirect
263. C109#18202direct (N × H)
276. C22×C4⋊C4C22×C109semidirect
278. #480C22×C109semidirect
281. C2×C22⋊C8C22×C109semidirect
282. #398C22×C109semidirect
283. #397C22×C109semidirect
287. #380C22×C109semidirect
288. #327C22×C109semidirect
289. #324C22×C109semidirect
290. #323C22×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#34962
C2×#480×C1092
C109×#24862
C109×#16102
C109×#16072
C109×#15192
C109×#14592
#18202109