Finite Group Databasebeta

← C109×#12181 #42130175

Subgroups — C109×#12181 #42130175

Order 27904.

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

normal subgroups only

This group has 372 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×#2571C211yesmaximal, characteristic, complemented
2C109×#2039C211yesmaximal, characteristic
3C109×#1616C212yesmaximal
4C109×#1353C212yesmaximal, complemented
5C109×#1325C211yesmaximal, characteristic, complemented
6#537×C109C2211yescharacteristic
7#481×C109—22no—
8C2×#104×C109—21no—
9#417×C109—21no—
10#400×C109—21no—
11#400×C109C2212yes—
12#389×C109—22no—
13#384×C109C2211yescharacteristic
14#382×C109—22no—
15C4×C4⋊C4×C109—21no—
16#343×C109C2211yescharacteristic
17#327×C109—22no—
18#324×C109C2212yes—
19C2×#41×C109—21no—
20C2×M16×C109—21no—
21C22⋊Q8×C109—41no—
22#121×C109—22no—
23#121×C109—41no—
24C2×C4⋊C4×C109—21no—
25C2×C4⋊C4×C109—21no—
26C2×C4⋊C4×C109—22no—
27C2×C4⋊C4×C109—21no—
28C2×C4⋊C4×C109D411yescharacteristic
29C2×C4⋊C4×C109D412yes—
30C2×C4⋊C4×C109C2311yescharacteristic
31C2×C42×C109—22no—
32C2×C42×C109D411yescharacteristic
33#106×C109—42no—
34#104×C109—42no—
35#103×C109—41no—
36#102×C109—41no—
37#102×C109—42no—
38C22⋊C8×C109—22no—
39#95×C109—21no—
40#95×C109—22no—
41#95×C109—22no—
42#95×C109—22no—
43#95×C109D412yes—
44#41×C109—41no—
45#41×C109—41no—
46#41×C109—41no—
47#41×C109—41no—
48C2×Q8×C109—21no—
49C2×D4×C109—21no—
50C2×D4×C109—22no—
51C2×D4×C109—41no—
52C22×C4×C109—21no—
53C22×C4×C109—22no—
54C22×C4×C109—22no—
55C22×C4×C109—21no—
56C22×C4×C109—41no—
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—22no—
64C4⋊C4×C109—41no—
65C4⋊C4×C109—41no—
66C4⋊C4×C109—42no—
67C4⋊C4×C109—42no—
68C4⋊C4×C109—44no—
69C4⋊C4×C109—42no—
70C4⋊C4×C109—81no—
71C22⋊C4×C109—42no—
72C42×C109—21no—
73C42×C109—44no—
74C42×C109SD1612yes—
75C23×C109—41no—
76C23×C109#12011yescharacteristic
77Q8×C109—41no—
78Q8×C109—41no—
79D4×C109—41no—
80D4×C109—41no—
81D4×C109—82no—
82D4×C109—81no—
83D4×C109—81no—
84C2×C4×C109—22no—
85C2×C4×C109—22no—
86C2×C4×C109—22no—
87C2×C4×C109—21no—
88C2×C4×C109—21no—
89C2×C4×C109—22no—
90C2×C4×C109—42no—
91C2×C4×C109—42no—
92C2×C4×C109—42no—
93C2×C4×C109—42no—
94C2×C4×C109—42no—
95C2×C4×C109—42no—
96C2×C4×C109—42no—
97C2×C4×C109—41no—
98C2×C4×C109—41no—
99C2×C4×C109—42no—
100C2×C4×C109—81no—
101C2×C4×C109—81no—
102C2×C4×C109C2×SD1611yescharacteristic
103C2×C4×C109#13711yescharacteristic
104C8×C109—81no—
105C8×C109—82no—
106C22×C109—21no—
107C22×C109—21no—
108C22×C109—81no—
109C22×C109—81no—
110C22×C109—81no—
111C22×C109#44811yescharacteristic
112C22×C109#45411yescharacteristic
113C22×C109B(4,𝔽2)11yescharacteristic
114C4×C109—24no—
115C4×C109—21no—
116C4×C109—21no—
117C4×C109—42no—
118C4×C109—82no—
119C4×C109—81no—
120C4×C109—81no—
121C4×C109—82no—
122#12181C10911yesmaximal, characteristic, direct product factor, Sylow
123C2×C109—21no—
124C2×C109—21no—
125C2×C109—161no—
126C2×C109#159911yescharacteristic
127C2×C109#219811yescharacteristic
128C2×C109#220411yescharacteristic
129#2571C2×C10911yescharacteristic, complemented
130#2039C2×C10911yescharacteristic
131#1616C2×C10912yes—
132#1353C2×C10912yescomplemented
133#1325C2×C10911yescharacteristic, complemented
134C109#1218111yescharacteristic, direct product factor, Sylow
135#537C22×C10911yescharacteristic
136#481—22no—
137C2×#104—21no—
138#417—21no—
139#400—21no—
140#400C22×C10912yes—
141#389—22no—
142#384C22×C10911yescharacteristic
143#382—22no—
144C4×C4⋊C4—21no—
145#343C22×C10911yescharacteristic
146#327—22no—
147#324C22×C10912yes—
148C2×#41—21no—
149C2×M16—21no—
150C22⋊Q8—41no—
151#121—22no—
152#121—41no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4D4×C10911yescharacteristic
158C2×C4⋊C4D4×C10912yes—
159C2×C4⋊C4C23×C10911yescharacteristic
160C2×C42—22no—
161C2×C42D4×C10911yescharacteristic
162#106—42no—
163#104—42no—
164#103—41no—
165#102—41no—
166#102—42no—
167C22⋊C8—22no—
168#95—21no—
169#95—22no—
170#95—22no—
171#95—22no—
172#95D4×C10912yes—
173#41—41no—
174#41—41no—
175#41—41no—
176#41—41no—
177C2×Q8—21no—
178C2×D4—21no—
179C2×D4—22no—
180C2×D4—41no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—21no—
185C22×C4—41no—
186C22×C4C2×D4×C10911yescharacteristic
187C22×C4C2×D4×C10912yes—
188M16—41no—
189M16—41no—
190C2×C8—42no—
191C2×C8—41no—
192C4⋊C4—22no—
193C4⋊C4—41no—
194C4⋊C4—41no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—44no—
198C4⋊C4—42no—
199C4⋊C4—81no—
200C22⋊C4—42no—
201C42—21no—
202C42—44no—
203C42SD16×C10912yes—
204C23—41no—
205C23#120×C10911yescharacteristic
206Q8—41no—
207Q8—41no—
208D4—41no—
209D4—41no—
210D4—82no—
211D4—81no—
212D4—81no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—41no—
227C2×C4—41no—
228C2×C4—42no—
229C2×C4—81no—
230C2×C4—81no—
231C2×C4C2×SD16×C10911yescharacteristic
232C2×C4#137×C10911yescharacteristic
233C8—81no—
234C8—82no—
235C22—21no—
236C22—21no—
237C22—81no—
238C22—81no—
239C22—81no—
240C22#448×C10911yescharacteristic
241C22#454×C10911yescharacteristic
242C22B(4,𝔽2)×C10911yescharacteristic
243C4—24no—
244C4—21no—
245C4—21no—
246C4—42no—
247C4—82no—
248C4—81no—
249C4—82no—
250C4—81no—
251C2—21no—
252C2—21no—
253C2—161no—
254C2C109×#159911yescharacteristic
255C2C109×#219811yescharacteristic
256C2C109×#220411yescharacteristic
subgroupG/H
center, socleC22×C109B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C109
prime radical#537×C109C22
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
1. C109×#2571C2semidirect
4. C109×#1353C2semidirect
5. C109×#1325C2semidirect
122. #12181C109direct (N × H)
129. #2571C2×C109semidirect
132. #1353C2×C109semidirect
133. #1325C2×C109semidirect
134. C109#12181direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#25712
C109×#20392
C109×#16162
C109×#13532
C109×#13252
#12181109