Finite Group Databasebeta

← C109×#12803 #42130797

Subgroups — C109×#12803 #42130797

Order 27904.

480 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 130 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×#2326C211yesmaximal, characteristic
2C2×#355×C109C212yesmaximal
3C109×#1801C212yesmaximal, complemented
4C109×#1394C212yesmaximal, complemented
5C2×#124×C109C2211yescharacteristic
6C2×C4×Q8×C109C2211yescharacteristic
7C2×C4×D4×C109C411yescharacteristic
8#423×C109—22no—
9C2×#101×C109C2214yes—
10#391×C109C2211yescharacteristic
11#390×C109C411yescharacteristic
12#388×C109—22no—
13#385×C109—22no—
14#384×C109C411yescharacteristic
15#376×C109C411yescharacteristic
16#355×C109—24no—
17#343×C109—22no—
18C22×Q8×C109C2311yescharacteristic
19C22×D4×C109C2×C411yescharacteristic
20C23×C4×C109—21no—
21C2×M16×C109—24no—
22#124×C109—22no—
23#124×C109D412yes—
24#124×C109Q812yes—
25C4×Q8×C109—42no—
26C4×D4×C109—42no—
27C2×C4⋊C4×C109—22no—
28C2×C4⋊C4×C109C2×C411yescharacteristic
29C2×C4⋊C4×C109C2×C411yescharacteristic
30C2×C4⋊C4×C109D412yes—
31C2×C22⋊C4×C109—22no—
32C2×C22⋊C4×C109—21no—
33C2×C22⋊C4×C109—22no—
34C2×C42×C109C2×C411yescharacteristic
35C2×C42×C109C2×C411yescharacteristic
36C2×C42×C109C2×C411yescharacteristic
37C2×C42×C109D412yes—
38C4⋊C8×C109—44no—
39#101×C109—24no—
40#101×C109—24no—
41#95×C109—21no—
42#95×C109—21no—
43#95×C109—22no—
44C24×C109—21no—
45C2×Q8×C109—41no—
46C2×Q8×C109—41no—
47C2×Q8×C109C2×D411yescharacteristic
48C2×Q8×C109C2×Q811yescharacteristic
49C2×Q8×C109#4112yes—
50C2×D4×C109—21no—
51C2×D4×C109—41no—
52C2×D4×C109—41no—
53C2×D4×C109C4⋊C412yes—
54C22×C4×C109—21no—
55C22×C4×C109—21no—
56C22×C4×C109—21no—
57C22×C4×C109—42no—
58C22×C4×C109—82no—
59C22×C4×C109C22⋊C411yescharacteristic
60C22×C4×C109C22⋊C412yes—
61C22×C4×C109C22⋊C411yescharacteristic
62C22×C4×C109C22×C411yescharacteristic
63C22×C4×C109C2×D412yes—
64M16×C109—44no—
65M16×C109—44no—
66C2×C8×C109—44no—
67C4⋊C4×C109—41no—
68C4⋊C4×C109—42no—
69C4⋊C4×C109—41no—
70C4⋊C4×C109—82no—
71C22⋊C4×C109—24no—
72C22⋊C4×C109—24no—
73C22⋊C4×C109—42no—
74C22⋊C4×C109—82no—
75C42×C109—24no—
76C42×C109—22no—
77C42×C109—21no—
78C42×C109—41no—
79C42×C109C4⋊C412yes—
80C23×C109—22no—
81C23×C109—42no—
82C23×C109—42no—
83C23×C109—41no—
84C23×C109—41no—
85C23×C109C2×C22⋊C411yescharacteristic
86Q8×C109—41no—
87Q8×C109—41no—
88Q8×C109—42no—
89D4×C109—42no—
90D4×C109—42no—
91C2×C4×C109—21no—
92C2×C4×C109—22no—
93C2×C4×C109—21no—
94C2×C4×C109—22no—
95C2×C4×C109—22no—
96C2×C4×C109—21no—
97C2×C4×C109—22no—
98C2×C4×C109—41no—
99C2×C4×C109—41no—
100C2×C4×C109—41no—
101C2×C4×C109—42no—
102C2×C4×C109—41no—
103C2×C4×C109—41no—
104C2×C4×C109—42no—
105C2×C4×C109—41no—
106C2×C4×C109—41no—
107C2×C4×C109—41no—
108C2×C4×C109—84no—
109C2×C4×C109#9912yes—
110C2×C4×C109C2×C4⋊C411yescharacteristic
111C2×C4×C109#11711yescharacteristic
112C2×C4×C109#12112yes—
113C2×C4×C109C22⋊Q812yes—
114C8×C109—84no—
115C22×C109—21no—
116C22×C109—21no—
117C22×C109—44no—
118C22×C109—42no—
119C22×C109—82no—
120C22×C109—81no—
121C22×C109—81no—
122C22×C109#38011yescharacteristic
123C22×C109C2×#9911yescharacteristic
124C22×C109#41011yescharacteristic
125C4×C109—22no—
126C4×C109—21no—
127C4×C109—21no—
128C4×C109—22no—
129C4×C109—22no—
130C4×C109—42no—
131C4×C109—42no—
132C4×C109—41no—
133C4×C109—81no—
134C4×C109—81no—
135#12803C10911yesmaximal, characteristic, direct product factor, Sylow
136C2×C109—21no—
137C2×C109—21no—
138C2×C109—82no—
139C2×C109#179911yescharacteristic
140C2×C109#213311yescharacteristic
141C2×C109#213411yescharacteristic
142#2326C2×C10911yescharacteristic
143C2×#355C2×C10912yes—
144#1801C2×C10912yescomplemented
145#1394C2×C10912yescomplemented
146C109#1280311yescharacteristic, direct product factor, Sylow
147C2×#124C22×C10911yescharacteristic
148C2×C4×Q8C22×C10911yescharacteristic
149C2×C4×D4C4×C10911yescharacteristic
150#423—22no—
151C2×#101C22×C10914yes—
152#391C22×C10911yescharacteristic
153#390C4×C10911yescharacteristic
154#388—22no—
155#385—22no—
156#384C4×C10911yescharacteristic
157#376C4×C10911yescharacteristic
158#355—24no—
159#343—22no—
160C22×Q8C23×C10911yescharacteristic
161C22×D4C2×C4×C10911yescharacteristic
162C23×C4—21no—
163C2×M16—24no—
164#124—22no—
165#124D4×C10912yes—
166#124Q8×C10912yes—
167C4×Q8—42no—
168C4×D4—42no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4C2×C4×C10911yescharacteristic
171C2×C4⋊C4C2×C4×C10911yescharacteristic
172C2×C4⋊C4D4×C10912yes—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—22no—
176C2×C42C2×C4×C10911yescharacteristic
177C2×C42C2×C4×C10911yescharacteristic
178C2×C42C2×C4×C10911yescharacteristic
179C2×C42D4×C10912yes—
180C4⋊C8—44no—
181#101—24no—
182#101—24no—
183#95—21no—
184#95—21no—
185#95—22no—
186C24—21no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8C2×D4×C10911yescharacteristic
190C2×Q8C2×Q8×C10911yescharacteristic
191C2×Q8#41×C10912yes—
192C2×D4—21no—
193C2×D4—41no—
194C2×D4—41no—
195C2×D4C4⋊C4×C10912yes—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—42no—
200C22×C4—82no—
201C22×C4C22⋊C4×C10911yescharacteristic
202C22×C4C22⋊C4×C10912yes—
203C22×C4C22⋊C4×C10911yescharacteristic
204C22×C4C22×C4×C10911yescharacteristic
205C22×C4C2×D4×C10912yes—
206M16—44no—
207M16—44no—
208C2×C8—44no—
209C4⋊C4—41no—
210C4⋊C4—42no—
211C4⋊C4—41no—
212C4⋊C4—82no—
213C22⋊C4—24no—
214C22⋊C4—24no—
215C22⋊C4—42no—
216C22⋊C4—82no—
217C42—24no—
218C42—22no—
219C42—21no—
220C42—41no—
221C42C4⋊C4×C10912yes—
222C23—22no—
223C23—42no—
224C23—42no—
225C23—41no—
226C23—41no—
227C23C2×C22⋊C4×C10911yescharacteristic
228Q8—41no—
229Q8—41no—
230Q8—42no—
231D4—42no—
232D4—42no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—21no—
239C2×C4—22no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—84no—
251C2×C4#99×C10912yes—
252C2×C4C2×C4⋊C4×C10911yescharacteristic
253C2×C4#117×C10911yescharacteristic
254C2×C4#121×C10912yes—
255C2×C4C22⋊Q8×C10912yes—
256C8—84no—
257C22—21no—
258C22—21no—
259C22—44no—
260C22—42no—
261C22—82no—
262C22—81no—
263C22—81no—
264C22#380×C10911yescharacteristic
265C22C2×#99×C10911yescharacteristic
266C22#410×C10911yescharacteristic
267C4—22no—
268C4—21no—
269C4—21no—
270C4—22no—
271C4—22no—
272C4—42no—
273C4—41no—
274C4—42no—
275C4—81no—
276C4—81no—
277C2—21no—
278C2—21no—
279C2—82no—
280C2C109×#179911yescharacteristic
281C2C109×#213311yescharacteristic
282C2C109×#213411yescharacteristic
subgroupG/H
center, socleC22×C109C2×#99
commutatorC22×C4C22×C4×C109
Frattini, cosocleC22×Q8C23×C109
prime radicalC22×D4×C109C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C109 ◁ C22×C4×C109 ◁ C2×C4×Q8×C109 ◁ G

Complements

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

NH ≅ G/Ntype
3. C109×#1801C2semidirect
4. C109×#1394C2semidirect
135. #12803C109direct (N × H)
144. #1801C2×C109semidirect
145. #1394C2×C109semidirect
146. C109#12803direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#23262
C2×#355×C1092
C109×#18012
C109×#13942
#12803109