Finite Group Databasebeta

← C109×#12245 #42130239

Subgroups — C109×#12245 #42130239

Order 27904.

458 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C109×#2712C211yesmaximal, characteristic
2C109×#2062C211yesmaximal, characteristic, complemented
3C109×#1608C211yesmaximal, characteristic, complemented
4C109×#1596C211yesmaximal, characteristic
5C109×#1344C211yesmaximal, characteristic, complemented
6C109×#1343C211yesmaximal, characteristic
7C109×#1326C211yesmaximal, characteristic, complemented
8#543×C109C2211yescharacteristic
9C2×#124×C109—21no—
10#483×C109—21no—
11#450×C109—21no—
12#397×C109C2211yescharacteristic
13#394×C109—21no—
14#393×C109—21no—
15#392×C109C2211yescharacteristic
16#391×C109—21no—
17#388×C109—21no—
18#386×C109—21no—
19#385×C109—21no—
20#385×C109C2211yescharacteristic
21#342×C109—21no—
22#342×C109C2211yescharacteristic
23#328×C109—21no—
24#328×C109—21no—
25#323×C109C2211yescharacteristic
26#323×C109C2211yescharacteristic
27C22×Q8×C109—21no—
28C22×D4×C109—21no—
29C22×D4×C109D411yescharacteristic
30C23×C4×C109D411yescharacteristic
31C2×SD16×C109—41no—
32#124×C109—42no—
33#124×C109—42no—
34C22⋊Q8×C109—21no—
35C22⋊Q8×C109—21no—
36C22⋊Q8×C109—41no—
37C22⋊Q8×C109—41no—
38C2×C4⋊C4×C109—21no—
39C2×C4⋊C4×C109—41no—
40C2×C4⋊C4×C109D411yescharacteristic
41C2×C22⋊C4×C109—21no—
42C2×C22⋊C4×C109—21no—
43C2×C22⋊C4×C109—21no—
44C2×C22⋊C4×C109—21no—
45C2×C22⋊C4×C109—21no—
46C2×C22⋊C4×C109—21no—
47C2×C22⋊C4×C109—21no—
48C2×C22⋊C4×C109—21no—
49C2×C22⋊C4×C109D411yescharacteristic
50C2×C22⋊C4×C109C2311yescharacteristic
51C2×C42×C109—21no—
52#106×C109—41no—
53#103×C109—41no—
54#102×C109—41no—
55C22⋊C8×C109—21no—
56C22⋊C8×C109—21no—
57#95×C109—21no—
58#95×C109—21no—
59#95×C109—21no—
60#95×C109—21no—
61#95×C109—41no—
62#95×C109D411yescharacteristic
63#95×C109D411yescharacteristic
64C24×C109—21no—
65C24×C109C2×D411yescharacteristic
66C2×Q8×C109—21no—
67C2×Q8×C109—21no—
68C2×Q8×C109—42no—
69C2×Q8×C109—42no—
70C2×Q8×C109—81no—
71C2×D4×C109—41no—
72C2×D4×C109—42no—
73C2×D4×C109—41no—
74C2×D4×C109—41no—
75C2×D4×C109—81no—
76C2×D4×C109—81no—
77C22×C4×C109—21no—
78C22×C4×C109—21no—
79C22×C4×C109—21no—
80C22×C4×C109—21no—
81C22×C4×C109—21no—
82C22×C4×C109—21no—
83C22×C4×C109—21no—
84C22×C4×C109—41no—
85C22×C4×C109—41no—
86C22×C4×C109—81no—
87C22×C4×C109C2×D411yescharacteristic
88C22×C4×C109C2×D411yescharacteristic
89SD16×C109—82no—
90C2×C8×C109—41no—
91C2×C8×C109—41no—
92C4⋊C4×C109—21no—
93C4⋊C4×C109—21no—
94C4⋊C4×C109—41no—
95C4⋊C4×C109—41no—
96C4⋊C4×C109—41no—
97C4⋊C4×C109—81no—
98C4⋊C4×C109—82no—
99C22⋊C4×C109—21no—
100C22⋊C4×C109—22no—
101C22⋊C4×C109—41no—
102C22⋊C4×C109—42no—
103C22⋊C4×C109—42no—
104C22⋊C4×C109—42no—
105C22⋊C4×C109—41no—
106C22⋊C4×C109—41no—
107C22⋊C4×C109—41no—
108C22⋊C4×C109—41no—
109C22⋊C4×C109—41no—
110C22⋊C4×C109—81no—
111C22⋊C4×C109—81no—
112C22⋊C4×C109SD1612yescomplemented
113C42×C109—42no—
114C23×C109—21no—
115C23×C109—21no—
116C23×C109—41no—
117C23×C109—41no—
118C23×C109—41no—
119C23×C109—41no—
120C23×C109—41no—
121C23×C109—41no—
122C23×C109—41no—
123C23×C109—41no—
124C23×C109—41no—
125C23×C109#12011yescharacteristic
126C23×C109C2×SD1611yescharacteristic
127C23×C109AGL(1,ℤ/8)11yescharacteristic
128Q8×C109—84no—
129Q8×C109—81no—
130Q8×C109—81no—
131D4×C109—42no—
132D4×C109—82no—
133D4×C109—82no—
134D4×C109—81no—
135C2×C4×C109—22no—
136C2×C4×C109—22no—
137C2×C4×C109—21no—
138C2×C4×C109—21no—
139C2×C4×C109—41no—
140C2×C4×C109—41no—
141C2×C4×C109—41no—
142C2×C4×C109—41no—
143C2×C4×C109—41no—
144C2×C4×C109—41no—
145C2×C4×C109—42no—
146C2×C4×C109—41no—
147C2×C4×C109—41no—
148C2×C4×C109—41no—
149C2×C4×C109—41no—
150C2×C4×C109—42no—
151C2×C4×C109—41no—
152C2×C4×C109—41no—
153C2×C4×C109—41no—
154C2×C4×C109—41no—
155C2×C4×C109—41no—
156C2×C4×C109—41no—
157C2×C4×C109—41no—
158C2×C4×C109—81no—
159C2×C4×C109—81no—
160C2×C4×C109—81no—
161C2×C4×C109—81no—
162C2×C4×C109—81no—
163C8×C109—81no—
164C8×C109—81no—
165C22×C109—21no—
166C22×C109—21no—
167C22×C109—21no—
168C22×C109—21no—
169C22×C109—41no—
170C22×C109—41no—
171C22×C109—41no—
172C22×C109—41no—
173C22×C109—81no—
174C22×C109—81no—
175C22×C109—81no—
176C22×C109—81no—
177C22×C109—81no—
178C22×C109—81no—
179C22×C109—81no—
180C22×C109—81no—
181C22×C109#45011yescharacteristic
182C22×C109#45611yescharacteristic
183C22×C109B(4,𝔽2)11yescharacteristic
184C4×C109—42no—
185C4×C109—42no—
186C4×C109—41no—
187C4×C109—41no—
188C4×C109—81no—
189C4×C109—82no—
190C4×C109—81no—
191C4×C109—81no—
192C4×C109—81no—
193C4×C109—81no—
194#12245C10911yesmaximal, characteristic, direct product factor, Sylow
195C2×C109—21no—
196C2×C109—21no—
197C2×C109—41no—
198C2×C109—41no—
199C2×C109—161no—
200C2×C109#160411yescharacteristic
201C2×C109#219811yescharacteristic
202C2×C109#219911yescharacteristic
203#2712C2×C10911yescharacteristic
204#2062C2×C10911yescharacteristic, complemented
205#1608C2×C10911yescharacteristic, complemented
206#1596C2×C10911yescharacteristic
207#1344C2×C10911yescharacteristic, complemented
208#1343C2×C10911yescharacteristic
209#1326C2×C10911yescharacteristic, complemented
210C109#1224511yescharacteristic, direct product factor, Sylow
211#543C22×C10911yescharacteristic
212C2×#124—21no—
213#483—21no—
214#450—21no—
215#397C22×C10911yescharacteristic
216#394—21no—
217#393—21no—
218#392C22×C10911yescharacteristic
219#391—21no—
220#388—21no—
221#386—21no—
222#385—21no—
223#385C22×C10911yescharacteristic
224#342—21no—
225#342C22×C10911yescharacteristic
226#328—21no—
227#328—21no—
228#323C22×C10911yescharacteristic
229#323C22×C10911yescharacteristic
230C22×Q8—21no—
231C22×D4—21no—
232C22×D4D4×C10911yescharacteristic
233C23×C4D4×C10911yescharacteristic
234C2×SD16—41no—
235#124—42no—
236#124—42no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C22⋊Q8—41no—
240C22⋊Q8—41no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4D4×C10911yescharacteristic
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4D4×C10911yescharacteristic
253C2×C22⋊C4C23×C10911yescharacteristic
254C2×C42—21no—
255#106—41no—
256#103—41no—
257#102—41no—
258C22⋊C8—21no—
259C22⋊C8—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—41no—
265#95D4×C10911yescharacteristic
266#95D4×C10911yescharacteristic
267C24—21no—
268C24C2×D4×C10911yescharacteristic
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—81no—
274C2×D4—41no—
275C2×D4—42no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—81no—
279C2×D4—81no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—81no—
290C22×C4C2×D4×C10911yescharacteristic
291C22×C4C2×D4×C10911yescharacteristic
292SD16—82no—
293C2×C8—41no—
294C2×C8—41no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—81no—
301C4⋊C4—82no—
302C22⋊C4—21no—
303C22⋊C4—22no—
304C22⋊C4—41no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—81no—
314C22⋊C4—81no—
315C22⋊C4SD16×C10912yescomplemented
316C42—42no—
317C23—21no—
318C23—21no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23#120×C10911yescharacteristic
329C23C2×SD16×C10911yescharacteristic
330C23AGL(1,ℤ/8)×C10911yescharacteristic
331Q8—84no—
332Q8—81no—
333Q8—81no—
334D4—42no—
335D4—82no—
336D4—82no—
337D4—81no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C8—81no—
367C8—81no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22#450×C10911yescharacteristic
385C22#456×C10911yescharacteristic
386C22B(4,𝔽2)×C10911yescharacteristic
387C4—42no—
388C4—42no—
389C4—41no—
390C4—41no—
391C4—81no—
392C4—82no—
393C4—81no—
394C4—81no—
395C4—81no—
396C4—81no—
397C2—21no—
398C2—21no—
399C2—41no—
400C2—41no—
401C2—161no—
402C2C109×#160411yescharacteristic
403C2C109×#219811yescharacteristic
404C2C109×#219911yescharacteristic
subgroupG/H
center, socleC22×C109B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C109
prime radical#392×C109C22
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C109 ◁ C23×C109 ◁ C2×C22⋊C4×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C109×#2062C2semidirect
3. C109×#1608C2semidirect
5. C109×#1344C2semidirect
7. C109×#1326C2semidirect
112. C22⋊C4×C109SD16semidirect
194. #12245C109direct (N × H)
204. #2062C2×C109semidirect
205. #1608C2×C109semidirect
207. #1344C2×C109semidirect
209. #1326C2×C109semidirect
210. C109#12245direct (N × H)
315. C22⋊C4SD16×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#27122
C109×#20622
C109×#16082
C109×#15962
C109×#13442
C109×#13432
C109×#13262
#12245109