Finite Group Databasebeta

← C109×#12209 #42130203

Subgroups — C109×#12209 #42130203

Order 27904.

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

normal subgroups only

This group has 404 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×#2724C211yesmaximal, characteristic
2C109×#2027C211yesmaximal, characteristic
3C109×#1611C212yesmaximal, complemented
4C109×#1346C212yesmaximal, complemented
5C109×#1318C211yesmaximal, characteristic
6#544×C109C2211yescharacteristic
7C2×#123×C109—21no—
8#483×C109—22no—
9#418×C109—21no—
10#398×C109C2212yes—
11#397×C109—22no—
12#390×C109C2211yescharacteristic
13#389×C109—21no—
14#385×C109—22no—
15#343×C109C2211yescharacteristic
16#339×C109—21no—
17#328×C109—22no—
18#324×C109C2212yes—
19C22×D4×C109D411yescharacteristic
20C23×C4×C109—21no—
21C2×M16×C109—21no—
22C4⋊Q8×C109—41no—
23#125×C109—41no—
24#123×C109—42no—
25#123×C109—42no—
26C22⋊Q8×C109—22no—
27#117×C109—21no—
28C2×C4⋊C4×C109—21no—
29C2×C4⋊C4×C109—22no—
30C2×C4⋊C4×C109—21no—
31C2×C4⋊C4×C109D412yes—
32C2×C4⋊C4×C109C2311yescharacteristic
33C2×C22⋊C4×C109—22no—
34C2×C22⋊C4×C109—21no—
35C2×C22⋊C4×C109—22no—
36C2×C42×C109D411yescharacteristic
37#106×C109—42no—
38#103×C109—42no—
39#102×C109—41no—
40#102×C109—41no—
41C22⋊C8×C109—22no—
42#95×C109—22no—
43#95×C109—22no—
44#95×C109D412yes—
45C24×C109—21no—
46C2×Q8×C109—22no—
47C2×D4×C109—21no—
48C2×D4×C109—41no—
49C2×D4×C109—41no—
50C2×D4×C109SD1612yes—
51C22×C4×C109—21no—
52C22×C4×C109—22no—
53C22×C4×C109—22no—
54C22×C4×C109—42no—
55C22×C4×C109—81no—
56C22×C4×C109—81no—
57C22×C4×C109C2×D411yescharacteristic
58C22×C4×C109C2×D412yes—
59M16×C109—41no—
60M16×C109—41no—
61C2×C8×C109—42no—
62C2×C8×C109—41no—
63C4⋊C4×C109—21no—
64C4⋊C4×C109—21no—
65C4⋊C4×C109—22no—
66C4⋊C4×C109—41no—
67C4⋊C4×C109—42no—
68C4⋊C4×C109—42no—
69C4⋊C4×C109—42no—
70C4⋊C4×C109—42no—
71C4⋊C4×C109—42no—
72C4⋊C4×C109—81no—
73C22⋊C4×C109—44no—
74C22⋊C4×C109—42no—
75C22⋊C4×C109—44no—
76C22⋊C4×C109—41no—
77C42×C109—41no—
78C42×C109—41no—
79C23×C109—22no—
80C23×C109—42no—
81C23×C109—42no—
82C23×C109—41no—
83C23×C109—41no—
84C23×C109#12011yescharacteristic
85Q8×C109—82no—
86D4×C109—42no—
87D4×C109—81no—
88C2×C4×C109—21no—
89C2×C4×C109—21no—
90C2×C4×C109—22no—
91C2×C4×C109—22no—
92C2×C4×C109—41no—
93C2×C4×C109—41no—
94C2×C4×C109—42no—
95C2×C4×C109—42no—
96C2×C4×C109—42no—
97C2×C4×C109—42no—
98C2×C4×C109—42no—
99C2×C4×C109—42no—
100C2×C4×C109—41no—
101C2×C4×C109—41no—
102C2×C4×C109—42no—
103C2×C4×C109—84no—
104C2×C4×C109C2×SD1611yescharacteristic
105C2×C4×C109AGL(1,ℤ/8)11yescharacteristic
106C8×C109—81no—
107C8×C109—82no—
108C22×C109—21no—
109C22×C109—21no—
110C22×C109—44no—
111C22×C109—42no—
112C22×C109—82no—
113C22×C109—81no—
114C22×C109—81no—
115C22×C109#45011yescharacteristic
116C22×C109#45511yescharacteristic
117C22×C109B(4,𝔽2)11yescharacteristic
118C4×C109—21no—
119C4×C109—21no—
120C4×C109—42no—
121C4×C109—42no—
122C4×C109—82no—
123C4×C109—82no—
124C4×C109—81no—
125C4×C109—81no—
126#12209C10911yesmaximal, characteristic, direct product factor, Sylow
127C2×C109—21no—
128C2×C109—21no—
129C2×C109—82no—
130C2×C109#160111yescharacteristic
131C2×C109#219811yescharacteristic
132C2×C109#220311yescharacteristic
133#2724C2×C10911yescharacteristic
134#2027C2×C10911yescharacteristic
135#1611C2×C10912yescomplemented
136#1346C2×C10912yescomplemented
137#1318C2×C10911yescharacteristic
138C109#1220911yescharacteristic, direct product factor, Sylow
139#544C22×C10911yescharacteristic
140C2×#123—21no—
141#483—22no—
142#418—21no—
143#398C22×C10912yes—
144#397—22no—
145#390C22×C10911yescharacteristic
146#389—21no—
147#385—22no—
148#343C22×C10911yescharacteristic
149#339—21no—
150#328—22no—
151#324C22×C10912yes—
152C22×D4D4×C10911yescharacteristic
153C23×C4—21no—
154C2×M16—21no—
155C4⋊Q8—41no—
156#125—41no—
157#123—42no—
158#123—42no—
159C22⋊Q8—22no—
160#117—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4D4×C10912yes—
165C2×C4⋊C4C23×C10911yescharacteristic
166C2×C22⋊C4—22no—
167C2×C22⋊C4—21no—
168C2×C22⋊C4—22no—
169C2×C42D4×C10911yescharacteristic
170#106—42no—
171#103—42no—
172#102—41no—
173#102—41no—
174C22⋊C8—22no—
175#95—22no—
176#95—22no—
177#95D4×C10912yes—
178C24—21no—
179C2×Q8—22no—
180C2×D4—21no—
181C2×D4—41no—
182C2×D4—41no—
183C2×D4SD16×C10912yes—
184C22×C4—21no—
185C22×C4—22no—
186C22×C4—22no—
187C22×C4—42no—
188C22×C4—81no—
189C22×C4—81no—
190C22×C4C2×D4×C10911yescharacteristic
191C22×C4C2×D4×C10912yes—
192M16—41no—
193M16—41no—
194C2×C8—42no—
195C2×C8—41no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—22no—
199C4⋊C4—41no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—42no—
203C4⋊C4—42no—
204C4⋊C4—42no—
205C4⋊C4—81no—
206C22⋊C4—44no—
207C22⋊C4—42no—
208C22⋊C4—44no—
209C22⋊C4—41no—
210C42—41no—
211C42—41no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23—41no—
216C23—41no—
217C23#120×C10911yescharacteristic
218Q8—82no—
219D4—42no—
220D4—81no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—84no—
237C2×C4C2×SD16×C10911yescharacteristic
238C2×C4AGL(1,ℤ/8)×C10911yescharacteristic
239C8—81no—
240C8—82no—
241C22—21no—
242C22—21no—
243C22—44no—
244C22—42no—
245C22—82no—
246C22—81no—
247C22—81no—
248C22#450×C10911yescharacteristic
249C22#455×C10911yescharacteristic
250C22B(4,𝔽2)×C10911yescharacteristic
251C4—21no—
252C4—21no—
253C4—42no—
254C4—42no—
255C4—82no—
256C4—81no—
257C4—81no—
258C4—82no—
259C2—21no—
260C2—21no—
261C2—82no—
262C2C109×#160111yescharacteristic
263C2C109×#219811yescharacteristic
264C2C109×#220311yescharacteristic
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
3. C109×#1611C2semidirect
4. C109×#1346C2semidirect
126. #12209C109direct (N × H)
135. #1611C2×C109semidirect
136. #1346C2×C109semidirect
138. C109#12209direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#27242
C109×#20272
C109×#16112
C109×#13462
C109×#13182
#12209109