Finite Group Databasebeta

← C109×#12235 #42130229

Subgroups — C109×#12235 #42130229

Order 27904.

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

normal subgroups only

This group has 382 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×#2782C211yesmaximal, characteristic
2C109×#2061C211yesmaximal, characteristic, complemented
3C109×#1617C211yesmaximal, characteristic, complemented
4C109×#1599C211yesmaximal, characteristic
5C109×#1353C211yesmaximal, characteristic, complemented
6C109×#1345C211yesmaximal, characteristic
7C109×#1324C211yesmaximal, characteristic, complemented
8#544×C109C2211yescharacteristic
9#484×C109—21no—
10#450×C109—21no—
11#428×C109—21no—
12#400×C109—21no—
13#400×C109C2211yescharacteristic
14#397×C109—21no—
15#394×C109C2211yescharacteristic
16#388×C109—21no—
17#386×C109—21no—
18#385×C109—21no—
19#385×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×D4×C109D411yescharacteristic
28C23×C4×C109—21no—
29C2×SD16×C109—41no—
30C2×M16×C109—21no—
31C4⋊Q8×C109—41no—
32#125×C109—41no—
33C22⋊Q8×C109—21no—
34C22⋊Q8×C109—21no—
35#117×C109—21no—
36C2×C4⋊C4×C109—21no—
37C2×C4⋊C4×C109—21no—
38C2×C4⋊C4×C109D411yescharacteristic
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×C42×C109D411yescharacteristic
47#107×C109—41no—
48#107×C109—41no—
49#106×C109—41no—
50#103×C109—41no—
51#102×C109—41no—
52C22⋊C8×C109—21no—
53C22⋊C8×C109—21no—
54#95×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×D4×C109—41no—
66C2×D4×C109—42no—
67C22×C4×C109—21no—
68C22×C4×C109—21no—
69C22×C4×C109—21no—
70C22×C4×C109—21no—
71C22×C4×C109—21no—
72C22×C4×C109—81no—
73C22×C4×C109—81no—
74C22×C4×C109—81no—
75C22×C4×C109C2×D411yescharacteristic
76C22×C4×C109C2×D411yescharacteristic
77C22×C4×C109C2×D411yescharacteristic
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—41no—
93C4⋊C4×C109—81no—
94C4⋊C4×C109—81no—
95C22⋊C4×C109—42no—
96C22⋊C4×C109—42no—
97C22⋊C4×C109—41no—
98C22⋊C4×C109—41no—
99C22⋊C4×C109—41no—
100C22⋊C4×C109—81no—
101C22⋊C4×C109—81no—
102C42×C109—21no—
103C42×C109—21no—
104C42×C109—41no—
105C23×C109—22no—
106C23×C109—41no—
107C23×C109—41no—
108C23×C109—41no—
109C23×C109—41no—
110C23×C109—41no—
111C23×C109—41no—
112C23×C109#12011yescharacteristic
113Q8×C109—81no—
114Q8×C109—81no—
115D4×C109—42no—
116D4×C109—81no—
117C2×C4×C109—21no—
118C2×C4×C109—21no—
119C2×C4×C109—21no—
120C2×C4×C109—21no—
121C2×C4×C109—41no—
122C2×C4×C109—41no—
123C2×C4×C109—41no—
124C2×C4×C109—41no—
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—84no—
141C2×C4×C109#13511yescharacteristic
142C2×C4×C109#13711yescharacteristic
143C8×C109—42no—
144C8×C109—81no—
145C8×C109—81no—
146C22×C109—21no—
147C22×C109—21no—
148C22×C109—42no—
149C22×C109—42no—
150C22×C109—42no—
151C22×C109—81no—
152C22×C109—81no—
153C22×C109—81no—
154C22×C109—81no—
155C22×C109#45211yescharacteristic
156C22×C109#45611yescharacteristic
157C22×C109B(4,𝔽2)11yescharacteristic
158C4×C109—21no—
159C4×C109—21no—
160C4×C109—41no—
161C4×C109—41no—
162C4×C109—81no—
163C4×C109—81no—
164C4×C109—81no—
165C4×C109—81no—
166C4×C109—81no—
167C4×C109—81no—
168C4×C109—81no—
169#12235C10911yesmaximal, characteristic, direct product factor, Sylow
170C2×C109—21no—
171C2×C109—21no—
172C2×C109—82no—
173C2×C109#160311yescharacteristic
174C2×C109#219711yescharacteristic
175C2×C109#220211yescharacteristic
176#2782C2×C10911yescharacteristic
177#2061C2×C10911yescharacteristic, complemented
178#1617C2×C10911yescharacteristic, complemented
179#1599C2×C10911yescharacteristic
180#1353C2×C10911yescharacteristic, complemented
181#1345C2×C10911yescharacteristic
182#1324C2×C10911yescharacteristic, complemented
183C109#1223511yescharacteristic, direct product factor, Sylow
184#544C22×C10911yescharacteristic
185#484—21no—
186#450—21no—
187#428—21no—
188#400—21no—
189#400C22×C10911yescharacteristic
190#397—21no—
191#394C22×C10911yescharacteristic
192#388—21no—
193#386—21no—
194#385—21no—
195#385—21no—
196#382C22×C10911yescharacteristic
197#343C22×C10911yescharacteristic
198#339—21no—
199#328—21no—
200#328—21no—
201#324C22×C10911yescharacteristic
202#324C22×C10911yescharacteristic
203C22×D4D4×C10911yescharacteristic
204C23×C4—21no—
205C2×SD16—41no—
206C2×M16—21no—
207C4⋊Q8—41no—
208#125—41no—
209C22⋊Q8—21no—
210C22⋊Q8—21no—
211#117—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4D4×C10911yescharacteristic
215C2×C4⋊C4D4×C10911yescharacteristic
216C2×C4⋊C4C23×C10911yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C42D4×C10911yescharacteristic
223#107—41no—
224#107—41no—
225#106—41no—
226#103—41no—
227#102—41no—
228C22⋊C8—21no—
229C22⋊C8—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95D4×C10911yescharacteristic
237#95D4×C10911yescharacteristic
238C24—21no—
239C2×Q8—21no—
240C2×Q8—21no—
241C2×D4—41no—
242C2×D4—42no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—81no—
249C22×C4—81no—
250C22×C4—81no—
251C22×C4C2×D4×C10911yescharacteristic
252C22×C4C2×D4×C10911yescharacteristic
253C22×C4C2×D4×C10911yescharacteristic
254SD16—82no—
255M16—22no—
256M16—41no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—41no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—81no—
270C4⋊C4—81no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—81no—
277C22⋊C4—81no—
278C42—21no—
279C42—21no—
280C42—41no—
281C23—22no—
282C23—41no—
283C23—41no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23#120×C10911yescharacteristic
289Q8—81no—
290Q8—81no—
291D4—42no—
292D4—81no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
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—84no—
317C2×C4#135×C10911yescharacteristic
318C2×C4#137×C10911yescharacteristic
319C8—42no—
320C8—81no—
321C8—81no—
322C22—21no—
323C22—21no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22—81no—
328C22—81no—
329C22—81no—
330C22—81no—
331C22#452×C10911yescharacteristic
332C22#456×C10911yescharacteristic
333C22B(4,𝔽2)×C10911yescharacteristic
334C4—21no—
335C4—21no—
336C4—41no—
337C4—41no—
338C4—81no—
339C4—81no—
340C4—81no—
341C4—81no—
342C4—81no—
343C4—81no—
344C4—81no—
345C2—21no—
346C2—21no—
347C2—82no—
348C2C109×#160311yescharacteristic
349C2C109×#219711yescharacteristic
350C2C109×#220211yescharacteristic
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×#1617C2semidirect
5. C109×#1353C2semidirect
7. C109×#1324C2semidirect
169. #12235C109direct (N × H)
177. #2061C2×C109semidirect
178. #1617C2×C109semidirect
180. #1353C2×C109semidirect
182. #1324C2×C109semidirect
183. C109#12235direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#27822
C109×#20612
C109×#16172
C109×#15992
C109×#13532
C109×#13452
C109×#13242
#12235109