Finite Group Databasebeta

← C109×#10302 #42128296

Subgroups — C109×#10302 #42128296

Order 27904.

516 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 106 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×#3255C211yesmaximal, characteristic, complemented
2C109×#3254C212yesmaximal
3C109×#1994C212yesmaximal, complemented
4C109×#1962C211yesmaximal, characteristic
5C109×#1309C211yesmaximal, characteristic, complemented
6#533×C109C2212yes—
7#532×C109C2211yescharacteristic
8#477×C109—22no—
9#475×C109—22no—
10#474×C109—22no—
11#470×C109—22no—
12#468×C109—24no—
13#462×C109—22no—
14#461×C109—22no—
15#460×C109—22no—
16#423×C109C2211yescharacteristic
17#423×C109C2212yes—
18C2×#104×C109—22no—
19C43×C109C2211yescharacteristic
20#339×C109—24no—
21C2×#41×C109—21no—
22C2×Q16×C109—22no—
23C2×SD16×C109—24no—
24C2×M16×C109D414yes—
25C2×M16×C109D412yes—
26C4⋊Q8×C109D411yescharacteristic
27C4⋊Q8×C109D412yes—
28C4⋊Q8×C109D412yes—
29#127×C109D411yescharacteristic
30#125×C109—22no—
31#124×C109—21no—
32C22⋊Q8×C109—22no—
33C22⋊Q8×C109—22no—
34#121×C109—22no—
35C4×Q8×C109—22no—
36C4×Q8×C109—22no—
37C4×D4×C109—22no—
38#117×C109—22no—
39C2×C42×C109—22no—
40C2×C42×C109—21no—
41C2×C42×C109C2311yescharacteristic
42#107×C109—24no—
43#106×C109—22no—
44C4⋊C8×C109—22no—
45C4⋊C8×C109—22no—
46C4⋊C8×C109—22no—
47#104×C109—44no—
48#103×C109—24no—
49#103×C109—24no—
50#102×C109—24no—
51#41×C109—42no—
52#41×C109—42no—
53C2×Q8×C109—22no—
54C2×Q8×C109—22no—
55C2×Q8×C109—21no—
56C2×D4×C109—22no—
57C2×D4×C109—21no—
58C2×D4×C109—41no—
59C22×C4×C109—41no—
60C22×C4×C109—41no—
61C22×C4×C109C2×D412yes—
62C22×C4×C109C2×D411yescharacteristic
63Q16×C109—82no—
64SD16×C109—84no—
65M16×C109—42no—
66M16×C109—44no—
67C2×C8×C109—24no—
68C2×C8×C109—22no—
69C4⋊C4×C109—22no—
70C4⋊C4×C109—22no—
71C4⋊C4×C109—22no—
72C4⋊C4×C109—22no—
73C4⋊C4×C109—22no—
74C4⋊C4×C109—42no—
75C4⋊C4×C109—42no—
76C22⋊C4×C109—42no—
77C22⋊C4×C109—42no—
78C42×C109—22no—
79C42×C109—24no—
80C42×C109—41no—
81C42×C109—42no—
82C42×C109—42no—
83C42×C109C2×D412yes—
84C42×C109C2×D411yescharacteristic
85C42×C109#4111yescharacteristic
86C23×C109—41no—
87C23×C109#12711yescharacteristic
88Q8×C109—42no—
89Q8×C109—42no—
90Q8×C109—42no—
91Q8×C109—42no—
92Q8×C109—42no—
93D4×C109—42no—
94D4×C109—42no—
95D4×C109—42no—
96D4×C109—82no—
97C2×C4×C109—21no—
98C2×C4×C109—21no—
99C2×C4×C109—22no—
100C2×C4×C109—22no—
101C2×C4×C109—42no—
102C2×C4×C109—44no—
103C2×C4×C109—42no—
104C2×C4×C109—42no—
105C2×C4×C109—41no—
106C2×C4×C109—81no—
107C2×C4×C109—81no—
108C2×C4×C109#12011yescharacteristic
109C2×C4×C109#12012yes—
110C2×C4×C109#12112yes—
111C2×C4×C109#12111yescharacteristic
112C8×C109—42no—
113C8×C109—44no—
114C22×C109—21no—
115C22×C109—22no—
116C22×C109—81no—
117C22×C109—82no—
118C22×C109#39211yescharacteristic
119C4×C109—21no—
120C4×C109—22no—
121C4×C109—44no—
122C4×C109—81no—
123C4×C109—82no—
124C4×C109—82no—
125C4×C109#45311yescharacteristic
126C4×C109#45411yescharacteristic
127C4×C109#45512yes—
128C4×C109#45612yes—
129#10302C10911yesmaximal, characteristic, direct product factor, Sylow
130C2×C109—41no—
131C2×C109—161no—
132C2×C109#200611yescharacteristic
133C2×C109#200712yes—
134#3255C2×C10911yescharacteristic, complemented
135#3254C2×C10912yes—
136#1994C2×C10912yescomplemented
137#1962C2×C10911yescharacteristic
138#1309C2×C10911yescharacteristic, complemented
139C109#1030211yescharacteristic, direct product factor, Sylow
140#533C22×C10912yes—
141#532C22×C10911yescharacteristic
142#477—22no—
143#475—22no—
144#474—22no—
145#470—22no—
146#468—24no—
147#462—22no—
148#461—22no—
149#460—22no—
150#423C22×C10911yescharacteristic
151#423C22×C10912yes—
152C2×#104—22no—
153C43C22×C10911yescharacteristic
154#339—24no—
155C2×#41—21no—
156C2×Q16—22no—
157C2×SD16—24no—
158C2×M16D4×C10914yes—
159C2×M16D4×C10912yes—
160C4⋊Q8D4×C10911yescharacteristic
161C4⋊Q8D4×C10912yes—
162C4⋊Q8D4×C10912yes—
163#127D4×C10911yescharacteristic
164#125—22no—
165#124—21no—
166C22⋊Q8—22no—
167C22⋊Q8—22no—
168#121—22no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×D4—22no—
172#117—22no—
173C2×C42—22no—
174C2×C42—21no—
175C2×C42C23×C10911yescharacteristic
176#107—24no—
177#106—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181#104—44no—
182#103—24no—
183#103—24no—
184#102—24no—
185#41—42no—
186#41—42no—
187C2×Q8—22no—
188C2×Q8—22no—
189C2×Q8—21no—
190C2×D4—22no—
191C2×D4—21no—
192C2×D4—41no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4C2×D4×C10912yes—
196C22×C4C2×D4×C10911yescharacteristic
197Q16—82no—
198SD16—84no—
199M16—42no—
200M16—44no—
201C2×C8—24no—
202C2×C8—22no—
203C4⋊C4—22no—
204C4⋊C4—22no—
205C4⋊C4—22no—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—42no—
209C4⋊C4—42no—
210C22⋊C4—42no—
211C22⋊C4—42no—
212C42—22no—
213C42—24no—
214C42—41no—
215C42—42no—
216C42—42no—
217C42C2×D4×C10912yes—
218C42C2×D4×C10911yescharacteristic
219C42#41×C10911yescharacteristic
220C23—41no—
221C23#127×C10911yescharacteristic
222Q8—42no—
223Q8—42no—
224Q8—42no—
225Q8—42no—
226Q8—42no—
227D4—42no—
228D4—42no—
229D4—42no—
230D4—82no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—42no—
236C2×C4—44no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—81no—
241C2×C4—81no—
242C2×C4#120×C10911yescharacteristic
243C2×C4#120×C10912yes—
244C2×C4#121×C10912yes—
245C2×C4#121×C10911yescharacteristic
246C8—42no—
247C8—44no—
248C22—21no—
249C22—22no—
250C22—81no—
251C22—82no—
252C22#392×C10911yescharacteristic
253C4—21no—
254C4—22no—
255C4—44no—
256C4—81no—
257C4—82no—
258C4—82no—
259C4#453×C10911yescharacteristic
260C4#454×C10911yescharacteristic
261C4#455×C10912yes—
262C4#456×C10912yes—
263C2—41no—
264C2—161no—
265C2C109×#200611yescharacteristic
266C2C109×#200712yes—
subgroupG/H
center, socleC22×C109#392
commutator, Frattini, cosocleC2×C42C23×C109
prime radical#532×C109C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C109 ◁ C2×C42×C109 ◁ G

Complements

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

NH ≅ G/Ntype
1. C109×#3255C2semidirect
3. C109×#1994C2semidirect
5. C109×#1309C2semidirect
129. #10302C109direct (N × H)
134. #3255C2×C109semidirect
136. #1994C2×C109semidirect
138. #1309C2×C109semidirect
139. C109#10302direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#32552
C109×#32542
C109×#19942
C109×#19622
C109×#13092
#10302109