Finite Group Databasebeta

← C109×#12798 #42130792

Subgroups — C109×#12798 #42130792

Order 27904.

544 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 134 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
1C4×#124×C109C211yesmaximal, characteristic
2C2×#354×C109C212yesmaximal
3C109×#1799C212yesmaximal
4C109×#1393C212yesmaximal
5C2×#124×C109C2211yescharacteristic
6C2×C4×Q8×C109C411yescharacteristic, complemented
7C2×C4×D4×C109C2211yescharacteristic
8C2×#99×C109C2214yes—
9#391×C109C411yescharacteristic, complemented
10#390×C109C2211yescharacteristic
11#388×C109—22no—
12#382×C109C411yescharacteristic, complemented
13#380×C109—22no—
14C4×C22⋊C4×C109—22no—
15C43×C109C411yescharacteristic, complemented
16#354×C109—24no—
17#342×C109—22no—
18C22×Q8×C109C2×C411yescharacteristic
19C22×D4×C109C2311yescharacteristic
20C23×C4×C109D412yes—
21#124×C109—22no—
22#124×C109D412yes—
23#124×C109Q812yes—
24C4×Q8×C109—41no—
25C4×Q8×C109—41no—
26C4×D4×C109—41no—
27C4×D4×C109—41no—
28C2×C4⋊C4×C109—21no—
29C2×C4⋊C4×C109—42no—
30C2×C4⋊C4×C109C2×C411yescharacteristic
31C2×C4⋊C4×C109C2×C411yescharacteristic
32C2×C22⋊C4×C109—22no—
33C2×C22⋊C4×C109—22no—
34C2×C22⋊C4×C109—24no—
35C2×C22⋊C4×C109D412yes—
36C2×C42×C109—21no—
37C2×C42×C109—21no—
38C2×C42×C109—21no—
39C2×C42×C109C2×C411yescharacteristic
40C2×C42×C109C2×C411yescharacteristic
41C2×C42×C109C2×C411yescharacteristic
42#99×C109—24no—
43#99×C109—24no—
44#95×C109—22no—
45#95×C109—22no—
46#95×C109—42no—
47C24×C109C2×D412yes—
48C2×Q8×C109—21no—
49C2×Q8×C109—41no—
50C2×Q8×C109—41no—
51C2×Q8×C109C4⋊C412yescomplemented
52C2×D4×C109—41no—
53C2×D4×C109—41no—
54C2×D4×C109C2×D411yescharacteristic
55C2×D4×C109C2×Q811yescharacteristic
56C2×D4×C109#4112yes—
57C22×C4×C109—21no—
58C22×C4×C109—21no—
59C22×C4×C109—21no—
60C22×C4×C109—22no—
61C22×C4×C109—22no—
62C22×C4×C109—21no—
63C22×C4×C109—44no—
64C22×C4×C109—44no—
65C22×C4×C109C22⋊C412yes—
66C22×C4×C109C22⋊C411yescharacteristic
67C22×C4×C109C22⋊C411yescharacteristic
68C22×C4×C109C22×C411yescharacteristic
69C4⋊C4×C109—41no—
70C4⋊C4×C109—41no—
71C4⋊C4×C109—42no—
72C4⋊C4×C109—84no—
73C22⋊C4×C109—24no—
74C22⋊C4×C109—24no—
75C22⋊C4×C109—42no—
76C22⋊C4×C109—44no—
77C22⋊C4×C109—44no—
78C22⋊C4×C109—44no—
79C42×C109—21no—
80C42×C109—21no—
81C42×C109—21no—
82C42×C109—41no—
83C42×C109—42no—
84C42×C109—42no—
85C42×C109—41no—
86C42×C109—41no—
87C42×C109C4⋊C412yescomplemented
88C23×C109—22no—
89C23×C109—22no—
90C23×C109—42no—
91C23×C109—42no—
92C23×C109C2×C22⋊C411yescharacteristic
93C23×C109#12112yes—
94C23×C109C22⋊Q812yes—
95Q8×C109—42no—
96Q8×C109—42no—
97D4×C109—41no—
98D4×C109—41no—
99D4×C109—42no—
100C2×C4×C109—21no—
101C2×C4×C109—21no—
102C2×C4×C109—22no—
103C2×C4×C109—22no—
104C2×C4×C109—22no—
105C2×C4×C109—22no—
106C2×C4×C109—21no—
107C2×C4×C109—22no—
108C2×C4×C109—41no—
109C2×C4×C109—41no—
110C2×C4×C109—41no—
111C2×C4×C109—41no—
112C2×C4×C109—41no—
113C2×C4×C109—41no—
114C2×C4×C109—44no—
115C2×C4×C109—41no—
116C2×C4×C109—42no—
117C2×C4×C109—42no—
118C2×C4×C109—41no—
119C2×C4×C109—82no—
120C2×C4×C109—84no—
121C2×C4×C109—84no—
122C2×C4×C109—84no—
123C2×C4×C109#9911yescharacteristic
124C2×C4×C109#9911yescharacteristic
125C2×C4×C109C2×C4⋊C411yescharacteristic
126C2×C4×C109#11711yescharacteristic
127C22×C109—21no—
128C22×C109—21no—
129C22×C109—22no—
130C22×C109—22no—
131C22×C109—42no—
132C22×C109—42no—
133C22×C109—42no—
134C22×C109—42no—
135C22×C109—82no—
136C22×C109#38011yescharacteristic
137C22×C109C2×#9911yescharacteristic
138C22×C109#41011yescharacteristic
139C4×C109—21no—
140C4×C109—21no—
141C4×C109—21no—
142C4×C109—42no—
143C4×C109—42no—
144C4×C109—42no—
145C4×C109—42no—
146C4×C109—41no—
147C4×C109—81no—
148C4×C109—164no—
149C4×C109#35412yescomplemented
150#12798C10911yesmaximal, characteristic, direct product factor, Sylow
151C2×C109—21no—
152C2×C109—21no—
153C2×C109—42no—
154C2×C109—42no—
155C2×C109#179911yescharacteristic
156C2×C109C2×#35411yescharacteristic
157C2×C109#212611yescharacteristic
158C4×#124C2×C10911yescharacteristic
159C2×#354C2×C10912yes—
160#1799C2×C10912yes—
161#1393C2×C10912yes—
162C109#1279811yescharacteristic, direct product factor, Sylow
163C2×#124C22×C10911yescharacteristic
164C2×C4×Q8C4×C10911yescharacteristic, complemented
165C2×C4×D4C22×C10911yescharacteristic
166C2×#99C22×C10914yes—
167#391C4×C10911yescharacteristic, complemented
168#390C22×C10911yescharacteristic
169#388—22no—
170#382C4×C10911yescharacteristic, complemented
171#380—22no—
172C4×C22⋊C4—22no—
173C43C4×C10911yescharacteristic, complemented
174#354—24no—
175#342—22no—
176C22×Q8C2×C4×C10911yescharacteristic
177C22×D4C23×C10911yescharacteristic
178C23×C4D4×C10912yes—
179#124—22no—
180#124D4×C10912yes—
181#124Q8×C10912yes—
182C4×Q8—41no—
183C4×Q8—41no—
184C4×D4—41no—
185C4×D4—41no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4C2×C4×C10911yescharacteristic
189C2×C4⋊C4C2×C4×C10911yescharacteristic
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4D4×C10912yes—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42C2×C4×C10911yescharacteristic
198C2×C42C2×C4×C10911yescharacteristic
199C2×C42C2×C4×C10911yescharacteristic
200#99—24no—
201#99—24no—
202#95—22no—
203#95—22no—
204#95—42no—
205C24C2×D4×C10912yes—
206C2×Q8—21no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8C4⋊C4×C10912yescomplemented
210C2×D4—41no—
211C2×D4—41no—
212C2×D4C2×D4×C10911yescharacteristic
213C2×D4C2×Q8×C10911yescharacteristic
214C2×D4#41×C10912yes—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—21no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4C22⋊C4×C10911yescharacteristic
224C22×C4C22⋊C4×C10912yes—
225C22×C4C22⋊C4×C10911yescharacteristic
226C22×C4C22×C4×C10911yescharacteristic
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—84no—
231C22⋊C4—24no—
232C22⋊C4—24no—
233C22⋊C4—42no—
234C22⋊C4—44no—
235C22⋊C4—44no—
236C22⋊C4—44no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—41no—
241C42—42no—
242C42—42no—
243C42—41no—
244C42—41no—
245C42C4⋊C4×C10912yescomplemented
246C23—22no—
247C23—22no—
248C23—42no—
249C23—42no—
250C23C2×C22⋊C4×C10911yescharacteristic
251C23#121×C10912yes—
252C23C22⋊Q8×C10912yes—
253Q8—42no—
254Q8—42no—
255D4—41no—
256D4—41no—
257D4—42no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—44no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—82no—
278C2×C4—84no—
279C2×C4—84no—
280C2×C4—84no—
281C2×C4#99×C10911yescharacteristic
282C2×C4#99×C10911yescharacteristic
283C2×C4C2×C4⋊C4×C10911yescharacteristic
284C2×C4#117×C10911yescharacteristic
285C22—21no—
286C22—21no—
287C22—22no—
288C22—22no—
289C22—42no—
290C22—42no—
291C22—42no—
292C22—42no—
293C22—82no—
294C22#380×C10911yescharacteristic
295C22C2×#99×C10911yescharacteristic
296C22#410×C10911yescharacteristic
297C4—21no—
298C4—21no—
299C4—21no—
300C4—42no—
301C4—42no—
302C4—42no—
303C4—42no—
304C4—41no—
305C4—81no—
306C4—164no—
307C4#354×C10912yescomplemented
308C2—21no—
309C2—21no—
310C2—42no—
311C2—42no—
312C2C109×#179911yescharacteristic
313C2C2×#354×C10911yescharacteristic
314C2C109×#212611yescharacteristic
subgroupG/H
center, socleC22×C109C2×#99
commutatorC22×C4C22×C4×C109
Frattini, cosocleC22×D4C23×C109
prime radicalC22×D4×C109C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C109 ◁ C22×C4×C109 ◁ C2×C4×D4×C109 ◁ G

Complements

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

NH ≅ G/Ntype
6. C2×C4×Q8×C109C4semidirect
9. #391×C109C4semidirect
12. #382×C109C4semidirect
15. C43×C109C4semidirect
51. C2×Q8×C109C4⋊C4semidirect
87. C42×C109C4⋊C4semidirect
149. C4×C109#354semidirect
150. #12798C109direct (N × H)
162. C109#12798direct (N × H)
164. C2×C4×Q8C4×C109semidirect
167. #391C4×C109semidirect
170. #382C4×C109semidirect
173. C43C4×C109semidirect
209. C2×Q8C4⋊C4×C109semidirect
245. C42C4⋊C4×C109semidirect
307. C4#354×C109semidirect

Maximal subgroups

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

subgroupindex
C4×#124×C1092
C2×#354×C1092
C109×#17992
C109×#13932
#12798109