Finite Group Databasebeta

← C109×#12239 #42130233

Subgroups — C109×#12239 #42130233

Order 27904.

378 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 70 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×#2785C211yesmaximal, characteristic
2C109×#2061C211yesmaximal, characteristic, complemented
3C109×#1605C211yesmaximal, characteristic, complemented
4C109×#1599C211yesmaximal, characteristic
5C109×#1354C211yesmaximal, characteristic, complemented
6C109×#1345C211yesmaximal, characteristic
7C109×#1324C211yesmaximal, characteristic, complemented
8#544×C109C2211yescharacteristic
9#451×C109—21no—
10#450×C109—21no—
11#428×C109—21no—
12#395×C109C2211yescharacteristic
13#394×C109—21no—
14#394×C109C2211yescharacteristic
15#393×C109—21no—
16#388×C109—21no—
17#388×C109—21no—
18#388×C109—21no—
19#382×C109—21no—
20#382×C109C2211yescharacteristic
21#343×C109C2211yescharacteristic
22#339×C109—21no—
23#328×C109—21no—
24#328×C109—21no—
25#324×C109C2211yescharacteristic
26#324×C109C2211yescharacteristic
27C22×Q8×C109D411yescharacteristic
28C22×D4×C109D411yescharacteristic
29C2×Q16×C109—41no—
30C2×SD16×C109—41no—
31C2×M16×C109—21no—
32C4⋊Q8×C109—41no—
33#125×C109—41no—
34C22⋊Q8×C109—21no—
35C22⋊Q8×C109—21no—
36#117×C109—21no—
37C2×C4⋊C4×C109—21no—
38C2×C4⋊C4×C109—21no—
39C2×C4⋊C4×C109D411yescharacteristic
40C2×C4⋊C4×C109C2311yescharacteristic
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×C42×C109—21no—
48C2×C42×C109D411yescharacteristic
49#107×C109—41no—
50#106×C109—41no—
51#103×C109—41no—
52#102×C109—41no—
53C22⋊C8×C109—21no—
54C22⋊C8×C109—21no—
55#95×C109—21no—
56#95×C109—21no—
57#95×C109—21no—
58#95×C109—21no—
59#95×C109—21no—
60#95×C109D411yescharacteristic
61#95×C109D411yescharacteristic
62C24×C109—21no—
63C2×Q8×C109—21no—
64C2×Q8×C109—21no—
65C2×Q8×C109—41no—
66C2×Q8×C109—81no—
67C2×D4×C109—41no—
68C2×D4×C109—81no—
69C22×C4×C109—21no—
70C22×C4×C109—21no—
71C22×C4×C109—21no—
72C22×C4×C109—21no—
73C22×C4×C109—21no—
74C22×C4×C109C2×D411yescharacteristic
75C22×C4×C109C2×D411yescharacteristic
76C22×C4×C109C2×D411yescharacteristic
77Q16×C109—82no—
78SD16×C109—82no—
79M16×C109—22no—
80M16×C109—41no—
81C2×C8×C109—41no—
82C2×C8×C109—41no—
83C2×C8×C109—41no—
84C4⋊C4×C109—21no—
85C4⋊C4×C109—21no—
86C4⋊C4×C109—21no—
87C4⋊C4×C109—21no—
88C4⋊C4×C109—41no—
89C4⋊C4×C109—41no—
90C4⋊C4×C109—41no—
91C4⋊C4×C109—41no—
92C4⋊C4×C109—81no—
93C4⋊C4×C109—81no—
94C22⋊C4×C109—42no—
95C22⋊C4×C109—42no—
96C22⋊C4×C109—41no—
97C22⋊C4×C109—41no—
98C22⋊C4×C109—41no—
99C22⋊C4×C109—81no—
100C22⋊C4×C109—81no—
101C22⋊C4×C109—81no—
102C42×C109—21no—
103C42×C109—21no—
104C42×C109—42no—
105C42×C109—41no—
106C23×C109—41no—
107C23×C109—41no—
108C23×C109—41no—
109C23×C109—41no—
110C23×C109—41no—
111C23×C109—41no—
112C23×C109—41no—
113C23×C109#12011yescharacteristic
114Q8×C109—42no—
115Q8×C109—81no—
116Q8×C109—81no—
117Q8×C109—81no—
118D4×C109—42no—
119D4×C109—81no—
120C2×C4×C109—21no—
121C2×C4×C109—21no—
122C2×C4×C109—22no—
123C2×C4×C109—21no—
124C2×C4×C109—21no—
125C2×C4×C109—41no—
126C2×C4×C109—41no—
127C2×C4×C109—41no—
128C2×C4×C109—41no—
129C2×C4×C109—41no—
130C2×C4×C109—41no—
131C2×C4×C109—41no—
132C2×C4×C109—41no—
133C2×C4×C109—41no—
134C2×C4×C109—41no—
135C2×C4×C109—41no—
136C2×C4×C109—41no—
137C2×C4×C109—41no—
138C2×C4×C109—41no—
139C2×C4×C109—41no—
140C2×C4×C109—41no—
141C2×C4×C109—41no—
142C2×C4×C109—41no—
143C2×C4×C109#13511yescharacteristic
144C2×C4×C109#13711yescharacteristic
145C8×C109—42no—
146C8×C109—81no—
147C8×C109—81no—
148C22×C109—21no—
149C22×C109—21no—
150C22×C109—81no—
151C22×C109—81no—
152C22×C109—81no—
153C22×C109—81no—
154C22×C109—81no—
155C22×C109—81no—
156C22×C109—81no—
157C22×C109#45211yescharacteristic
158C22×C109#45611yescharacteristic
159C22×C109B(4,𝔽2)11yescharacteristic
160C4×C109—21no—
161C4×C109—21no—
162C4×C109—42no—
163C4×C109—41no—
164C4×C109—41no—
165C4×C109—81no—
166C4×C109—81no—
167C4×C109—81no—
168C4×C109—81no—
169C4×C109—81no—
170C4×C109—81no—
171#12239C10911yesmaximal, characteristic, direct product factor, Sylow
172C2×C109—21no—
173C2×C109—21no—
174C2×C109—161no—
175C2×C109#160311yescharacteristic
176C2×C109#219911yescharacteristic
177C2×C109#220411yescharacteristic
178#2785C2×C10911yescharacteristic
179#2061C2×C10911yescharacteristic, complemented
180#1605C2×C10911yescharacteristic, complemented
181#1599C2×C10911yescharacteristic
182#1354C2×C10911yescharacteristic, complemented
183#1345C2×C10911yescharacteristic
184#1324C2×C10911yescharacteristic, complemented
185C109#1223911yescharacteristic, direct product factor, Sylow
186#544C22×C10911yescharacteristic
187#451—21no—
188#450—21no—
189#428—21no—
190#395C22×C10911yescharacteristic
191#394—21no—
192#394C22×C10911yescharacteristic
193#393—21no—
194#388—21no—
195#388—21no—
196#388—21no—
197#382—21no—
198#382C22×C10911yescharacteristic
199#343C22×C10911yescharacteristic
200#339—21no—
201#328—21no—
202#328—21no—
203#324C22×C10911yescharacteristic
204#324C22×C10911yescharacteristic
205C22×Q8D4×C10911yescharacteristic
206C22×D4D4×C10911yescharacteristic
207C2×Q16—41no—
208C2×SD16—41no—
209C2×M16—21no—
210C4⋊Q8—41no—
211#125—41no—
212C22⋊Q8—21no—
213C22⋊Q8—21no—
214#117—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4D4×C10911yescharacteristic
218C2×C4⋊C4C23×C10911yescharacteristic
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C42—21no—
226C2×C42D4×C10911yescharacteristic
227#107—41no—
228#106—41no—
229#103—41no—
230#102—41no—
231C22⋊C8—21no—
232C22⋊C8—21no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95D4×C10911yescharacteristic
239#95D4×C10911yescharacteristic
240C24—21no—
241C2×Q8—21no—
242C2×Q8—21no—
243C2×Q8—41no—
244C2×Q8—81no—
245C2×D4—41no—
246C2×D4—81no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4C2×D4×C10911yescharacteristic
253C22×C4C2×D4×C10911yescharacteristic
254C22×C4C2×D4×C10911yescharacteristic
255Q16—82no—
256SD16—82no—
257M16—22no—
258M16—41no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—41no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—81no—
271C4⋊C4—81no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—81no—
278C22⋊C4—81no—
279C22⋊C4—81no—
280C42—21no—
281C42—21no—
282C42—42no—
283C42—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—41no—
291C23#120×C10911yescharacteristic
292Q8—42no—
293Q8—81no—
294Q8—81no—
295Q8—81no—
296D4—42no—
297D4—81no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4#135×C10911yescharacteristic
322C2×C4#137×C10911yescharacteristic
323C8—42no—
324C8—81no—
325C8—81no—
326C22—21no—
327C22—21no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22—81no—
335C22#452×C10911yescharacteristic
336C22#456×C10911yescharacteristic
337C22B(4,𝔽2)×C10911yescharacteristic
338C4—21no—
339C4—21no—
340C4—42no—
341C4—41no—
342C4—41no—
343C4—81no—
344C4—81no—
345C4—81no—
346C4—81no—
347C4—81no—
348C4—81no—
349C2—21no—
350C2—21no—
351C2—161no—
352C2C109×#160311yescharacteristic
353C2C109×#219911yescharacteristic
354C2C109×#220411yescharacteristic
subgroupG/H
center, socleC22×C109B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C109
prime radicalC22×D4×C109D4
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C109 ◁ C23×C109 ◁ C2×C4⋊C4×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C109×#2061C2semidirect
3. C109×#1605C2semidirect
5. C109×#1354C2semidirect
7. C109×#1324C2semidirect
171. #12239C109direct (N × H)
179. #2061C2×C109semidirect
180. #1605C2×C109semidirect
182. #1354C2×C109semidirect
184. #1324C2×C109semidirect
185. C109#12239direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#27852
C109×#20612
C109×#16052
C109×#15992
C109×#13542
C109×#13452
C109×#13242
#12239109