Finite Group Databasebeta

← C109×#12688 #42130682

Subgroups — C109×#12688 #42130682

Order 27904.

1220 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1220 conjugacy classes of subgroups — too many to draw. Only its 158 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#123×C109C211yesmaximal, characteristic
2C2×#352×C109C214yesmaximal, complemented
3C109×#1783C211yesmaximal, characteristic
4C109×#1781C211yesmaximal, characteristic
5C23×D4×C109C2211yescharacteristic
6C24×C4×C109C411yescharacteristic, complemented
7C2×#123×C109—21no—
8C2×#123×C109—21no—
9C2×#123×C109—28no—
10C2×#123×C109C412yescomplemented
11C2×#123×C109C2212yes—
12C22×C4⋊C4×C109—21no—
13C22×C22⋊C4×C109—21no—
14C22×C22⋊C4×C109C411yescharacteristic, complemented
15C2×#100×C109—21no—
16C2×#100×C109C2212yes—
17C2×#99×C109—21no—
18C2×#99×C109C2212yes—
19#407×C109—21no—
20#379×C109—21no—
21#352×C109—28no—
22C25×C109C2×C411yescharacteristic
23C25×C109D411yescharacteristic
24C22×D4×C109—24no—
25C22×D4×C109—42no—
26C22×D4×C109—42no—
27C22×D4×C109C2×C411yescharacteristic
28C22×D4×C109D411yescharacteristic
29C22×D4×C109D411yescharacteristic
30C22×D4×C109C2311yescharacteristic
31C23×C4×C109—21no—
32C23×C4×C109—21no—
33C23×C4×C109—21no—
34C23×C4×C109—24no—
35C23×C4×C109—42no—
36C23×C4×C109—42no—
37C23×C4×C109C2×C412yes—
38C23×C4×C109D411yescharacteristic
39C2×M16×C109—22no—
40#123×C109—28no—
41#123×C109—28no—
42#123×C109—24no—
43#123×C109—24no—
44#123×C109—24no—
45#123×C109D418yes—
46C2×C4⋊C4×C109—28no—
47C2×C4⋊C4×C109—22no—
48C2×C4⋊C4×C109—22no—
49C2×C22⋊C4×C109—28no—
50C2×C22⋊C4×C109—22no—
51C2×C22⋊C4×C109—21no—
52C2×C22⋊C4×C109—22no—
53C2×C22⋊C4×C109—24no—
54C2×C22⋊C4×C109—22no—
55C2×C22⋊C4×C109—42no—
56C2×C22⋊C4×C109C2×C412yes—
57#100×C109—24no—
58#100×C109—42no—
59#99×C109—24no—
60#99×C109—42no—
61C22⋊C8×C109—42no—
62C24×C109—22no—
63C24×C109—22no—
64C24×C109—22no—
65C24×C109—22no—
66C24×C109—22no—
67C24×C109—41no—
68C24×C109—44no—
69C24×C109—44no—
70C24×C109C22⋊C411yescharacteristic
71C24×C109C22⋊C411yescharacteristic
72C24×C109C22⋊C411yescharacteristic
73C24×C109C22×C411yescharacteristic
74C24×C109C2×D411yescharacteristic
75C2×D4×C109—22no—
76C2×D4×C109—22no—
77C2×D4×C109—22no—
78C2×D4×C109—22no—
79C2×D4×C109—22no—
80C2×D4×C109—41no—
81C2×D4×C109—41no—
82C2×D4×C109—41no—
83C2×D4×C109—41no—
84C2×D4×C109—41no—
85C2×D4×C109—41no—
86C2×D4×C109—416no—
87C2×D4×C109C2×D414yes—
88C22×C4×C109—24no—
89C22×C4×C109—24no—
90C22×C4×C109—24no—
91C22×C4×C109—22no—
92C22×C4×C109—22no—
93C22×C4×C109—22no—
94C22×C4×C109—22no—
95C22×C4×C109—22no—
96C22×C4×C109—41no—
97C22×C4×C109—44no—
98C22×C4×C109—416no—
99C22×C4×C109—42no—
100C22×C4×C109—42no—
101C22×C4×C109—42no—
102C22×C4×C109—42no—
103C22×C4×C109—42no—
104C22×C4×C109—42no—
105C22×C4×C109—42no—
106C22×C4×C109C22⋊C411yescharacteristic
107C22×C4×C109C22⋊C414yescomplemented
108C22×C4×C109C2×D411yescharacteristic
109M16×C109—42no—
110M16×C109—42no—
111C2×C8×C109—42no—
112C4⋊C4×C109—28no—
113C4⋊C4×C109—28no—
114C22⋊C4×C109—24no—
115C22⋊C4×C109—28no—
116C22⋊C4×C109—22no—
117C22⋊C4×C109—28no—
118C22⋊C4×C109—42no—
119C22⋊C4×C109—42no—
120C22⋊C4×C109—84no—
121C22⋊C4×C109C22⋊C414yescomplemented
122C23×C109—24no—
123C23×C109—21no—
124C23×C109—21no—
125C23×C109—28no—
126C23×C109—21no—
127C23×C109—21no—
128C23×C109—21no—
129C23×C109—21no—
130C23×C109—21no—
131C23×C109—21no—
132C23×C109—21no—
133C23×C109—21no—
134C23×C109—21no—
135C23×C109—44no—
136C23×C109—42no—
137C23×C109—42no—
138C23×C109—48no—
139C23×C109—48no—
140C23×C109—44no—
141C23×C109—42no—
142C23×C109—42no—
143C23×C109—48no—
144C23×C109—44no—
145C23×C109—44no—
146C23×C109—82no—
147C23×C109—84no—
148C23×C109#9912yes—
149C23×C109#9912yes—
150C23×C109C2×C22⋊C411yescharacteristic
151C23×C109C2×C22⋊C412yes—
152C23×C109#12012yes—
153D4×C109—44no—
154D4×C109—44no—
155D4×C109—44no—
156D4×C109—44no—
157C2×C4×C109—21no—
158C2×C4×C109—21no—
159C2×C4×C109—24no—
160C2×C4×C109—24no—
161C2×C4×C109—21no—
162C2×C4×C109—21no—
163C2×C4×C109—21no—
164C2×C4×C109—44no—
165C2×C4×C109—48no—
166C2×C4×C109—44no—
167C2×C4×C109—48no—
168C2×C4×C109—48no—
169C2×C4×C109—44no—
170C2×C4×C109—42no—
171C2×C4×C109—42no—
172C2×C4×C109—44no—
173C2×C4×C109—82no—
174C2×C4×C109—84no—
175C2×C4×C109#12012yes—
176C8×C109—82no—
177C22×C109—24no—
178C22×C109—24no—
179C22×C109—22no—
180C22×C109—22no—
181C22×C109—22no—
182C22×C109—22no—
183C22×C109—22no—
184C22×C109—22no—
185C22×C109—42no—
186C22×C109—42no—
187C22×C109—42no—
188C22×C109—42no—
189C22×C109—416no—
190C22×C109—42no—
191C22×C109—42no—
192C22×C109—42no—
193C22×C109—42no—
194C22×C109—42no—
195C22×C109—42no—
196C22×C109—41no—
197C22×C109—81no—
198C22×C109—88no—
199C22×C109—81no—
200C22×C109#35214yescomplemented
201C22×C109#37911yescharacteristic
202C22×C109C2×#9911yescharacteristic
203C22×C109C2×#9911yescharacteristic
204C4×C109—22no—
205C4×C109—44no—
206C4×C109—41no—
207C4×C109—44no—
208C4×C109—162no—
209#12688C10911yesmaximal, characteristic, direct product factor, Sylow
210C2×C109—21no—
211C2×C109—24no—
212C2×C109—21no—
213C2×C109—42no—
214C2×C109—42no—
215C2×C109—42no—
216C2×C109—81no—
217C2×C109#178111yescharacteristic
218C2×C109C2×#35212yes—
219C22×#123C2×C10911yescharacteristic
220C2×#352C2×C10914yescomplemented
221#1783C2×C10911yescharacteristic
222#1781C2×C10911yescharacteristic
223C109#1268811yescharacteristic, direct product factor, Sylow
224C23×D4C22×C10911yescharacteristic
225C24×C4C4×C10911yescharacteristic, complemented
226C2×#123—21no—
227C2×#123—21no—
228C2×#123—28no—
229C2×#123C4×C10912yescomplemented
230C2×#123C22×C10912yes—
231C22×C4⋊C4—21no—
232C22×C22⋊C4—21no—
233C22×C22⋊C4C4×C10911yescharacteristic, complemented
234C2×#100—21no—
235C2×#100C22×C10912yes—
236C2×#99—21no—
237C2×#99C22×C10912yes—
238#407—21no—
239#379—21no—
240#352—28no—
241C25C2×C4×C10911yescharacteristic
242C25D4×C10911yescharacteristic
243C22×D4—24no—
244C22×D4—42no—
245C22×D4—42no—
246C22×D4C2×C4×C10911yescharacteristic
247C22×D4D4×C10911yescharacteristic
248C22×D4D4×C10911yescharacteristic
249C22×D4C23×C10911yescharacteristic
250C23×C4—21no—
251C23×C4—21no—
252C23×C4—21no—
253C23×C4—24no—
254C23×C4—42no—
255C23×C4—42no—
256C23×C4C2×C4×C10912yes—
257C23×C4D4×C10911yescharacteristic
258C2×M16—22no—
259#123—28no—
260#123—28no—
261#123—24no—
262#123—24no—
263#123—24no—
264#123D4×C10918yes—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4C2×C4×C10912yes—
276#100—24no—
277#100—42no—
278#99—24no—
279#99—42no—
280C22⋊C8—42no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—41no—
287C24—44no—
288C24—44no—
289C24C22⋊C4×C10911yescharacteristic
290C24C22⋊C4×C10911yescharacteristic
291C24C22⋊C4×C10911yescharacteristic
292C24C22×C4×C10911yescharacteristic
293C24C2×D4×C10911yescharacteristic
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—416no—
306C2×D4C2×D4×C10914yes—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—41no—
316C22×C4—44no—
317C22×C4—416no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4C22⋊C4×C10911yescharacteristic
326C22×C4C22⋊C4×C10914yescomplemented
327C22×C4C2×D4×C10911yescharacteristic
328M16—42no—
329M16—42no—
330C2×C8—42no—
331C4⋊C4—28no—
332C4⋊C4—28no—
333C22⋊C4—24no—
334C22⋊C4—28no—
335C22⋊C4—22no—
336C22⋊C4—28no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—84no—
340C22⋊C4C22⋊C4×C10914yescomplemented
341C23—24no—
342C23—21no—
343C23—21no—
344C23—28no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—44no—
355C23—42no—
356C23—42no—
357C23—48no—
358C23—48no—
359C23—44no—
360C23—42no—
361C23—42no—
362C23—48no—
363C23—44no—
364C23—44no—
365C23—82no—
366C23—84no—
367C23#99×C10912yes—
368C23#99×C10912yes—
369C23C2×C22⋊C4×C10911yescharacteristic
370C23C2×C22⋊C4×C10912yes—
371C23#120×C10912yes—
372D4—44no—
373D4—44no—
374D4—44no—
375D4—44no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—44no—
384C2×C4—48no—
385C2×C4—48no—
386C2×C4—44no—
387C2×C4—48no—
388C2×C4—44no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—82no—
393C2×C4—84no—
394C2×C4#120×C10912yes—
395C8—82no—
396C22—24no—
397C22—24no—
398C22—22no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—42no—
405C22—42no—
406C22—42no—
407C22—42no—
408C22—416no—
409C22—42no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—42no—
414C22—41no—
415C22—42no—
416C22—81no—
417C22—88no—
418C22—81no—
419C22#352×C10914yescomplemented
420C22#379×C10911yescharacteristic
421C22C2×#99×C10911yescharacteristic
422C22C2×#99×C10911yescharacteristic
423C4—22no—
424C4—44no—
425C4—41no—
426C4—44no—
427C4—162no—
428C2—21no—
429C2—24no—
430C2—21no—
431C2—42no—
432C2—42no—
433C2—42no—
434C2—81no—
435C2C109×#178111yescharacteristic
436C2C2×#352×C10912yes—
subgroupG/H
center, socleC22×C109C2×#99
commutatorC24C22×C4×C109
Frattini, cosocleC22×D4C23×C109
prime radicalC23×D4×C109C22
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C109 ◁ C24×C109 ◁ C23×D4×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#352×C109C2semidirect
6. C24×C4×C109C4semidirect
10. C2×#123×C109C4semidirect
14. C22×C22⋊C4×C109C4semidirect
107. C22×C4×C109C22⋊C4semidirect
121. C22⋊C4×C109C22⋊C4semidirect
200. C22×C109#352semidirect
209. #12688C109direct (N × H)
220. C2×#352C2×C109semidirect
223. C109#12688direct (N × H)
225. C24×C4C4×C109semidirect
229. C2×#123C4×C109semidirect
233. C22×C22⋊C4C4×C109semidirect
326. C22×C4C22⋊C4×C109semidirect
340. C22⋊C4C22⋊C4×C109semidirect
419. C22#352×C109semidirect

Maximal subgroups

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

subgroupindex
C22×#123×C1092
C2×#352×C1092
C109×#17832
C109×#17812
#12688109