Finite Group Databasebeta

← C109×#12299 #42130293

Subgroups — C109×#12299 #42130293

Order 27904.

324 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 70 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×#3195C211yesmaximal, characteristic
2C109×#2239C212yesmaximal, complemented
3C109×#2091C211yesmaximal, characteristic
4C109×#1342C212yesmaximal, complemented
5C109×#1320C211yesmaximal, characteristic
6C2×C22⋊Q8×C109—21no—
7#483×C109—22no—
8#470×C109—21no—
9#470×C109C2212yes—
10#451×C109—22no—
11#430×C109—21no—
12#430×C109C2211yescharacteristic
13#407×C109—21no—
14#407×C109C2211yescharacteristic
15#365×C109—22no—
16#362×C109—22no—
17#349×C109C2212yes—
18#338×C109C2211yescharacteristic
19C22×Q8×C109—21no—
20C23×C4×C109D411yescharacteristic
21C2×Q16×C109—21no—
22C2×Q16×C109D412yes—
23C2×M16×C109—21no—
24C2×M16×C109—21no—
25C2×M16×C109D411yescharacteristic
26C2×M16×C109C2311yescharacteristic
27C22⋊Q8×C109—22no—
28C22⋊Q8×C109—22no—
29C22⋊Q8×C109—41no—
30C22⋊Q8×C109—41no—
31C2×C4⋊C4×C109—21no—
32C2×C4⋊C4×C109—41no—
33C2×C22⋊C4×C109—41no—
34M32×C109—22no—
35#108×C109—22no—
36#108×C109—42no—
37#106×C109—21no—
38#106×C109D412yes—
39#103×C109—22no—
40#103×C109—22no—
41C22⋊C8×C109—22no—
42C22⋊C8×C109—42no—
43C24×C109SD1611yescharacteristic
44C2×Q8×C109—22no—
45C2×Q8×C109—41no—
46C2×Q8×C109—81no—
47C2×Q8×C109—81no—
48C22×C4×C109—21no—
49C22×C4×C109—21no—
50C22×C4×C109—41no—
51C22×C4×C109—41no—
52C22×C4×C109—41no—
53C22×C4×C109—41no—
54C22×C4×C109SD1611yescharacteristic
55C22×C4×C109C2×D411yescharacteristic
56Q16×C109—42no—
57Q16×C109—81no—
58M16×C109—41no—
59M16×C109—41no—
60M16×C109—41no—
61M16×C109—41no—
62M16×C109—41no—
63M16×C109—41no—
64C2×C8×C109—22no—
65C2×C8×C109—21no—
66C2×C8×C109—41no—
67C2×C8×C109C2×D412yes—
68C4⋊C4×C109—22no—
69C4⋊C4×C109—41no—
70C4⋊C4×C109—42no—
71C4⋊C4×C109—42no—
72C22⋊C4×C109—42no—
73C22⋊C4×C109—42no—
74C16×C109—42no—
75C23×C109—21no—
76C23×C109—41no—
77C23×C109—42no—
78C23×C109C2×SD1611yescharacteristic
79Q8×C109—42no—
80Q8×C109—42no—
81Q8×C109—82no—
82C2×C4×C109—21no—
83C2×C4×C109—21no—
84C2×C4×C109—21no—
85C2×C4×C109—41no—
86C2×C4×C109—41no—
87C2×C4×C109—41no—
88C2×C4×C109—42no—
89C2×C4×C109—42no—
90C2×C4×C109—81no—
91C2×C4×C109—82no—
92C2×C4×C109—82no—
93C2×C4×C109#12011yescharacteristic
94C2×C4×C109AGL(1,ℤ/8)11yescharacteristic
95C8×C109—22no—
96C8×C109—42no—
97C8×C109—41no—
98C8×C109—81no—
99C22×C109—22no—
100C22×C109—21no—
101C22×C109—41no—
102C22×C109—41no—
103C22×C109—42no—
104C22×C109—41no—
105C22×C109—81no—
106C22×C109#45011yescharacteristic
107C4×C109—21no—
108C4×C109—41no—
109C4×C109—82no—
110C4×C109—82no—
111C4×C109#45311yescharacteristic
112C4×C109#45811yescharacteristic
113#12299C10911yesmaximal, characteristic, direct product factor, Sylow
114C2×C109—21no—
115C2×C109—42no—
116C2×C109—41no—
117C2×C109#161311yescharacteristic
118#3195C2×C10911yescharacteristic
119#2239C2×C10912yescomplemented
120#2091C2×C10911yescharacteristic
121#1342C2×C10912yescomplemented
122#1320C2×C10911yescharacteristic
123C109#1229911yescharacteristic, direct product factor, Sylow
124C2×C22⋊Q8—21no—
125#483—22no—
126#470—21no—
127#470C22×C10912yes—
128#451—22no—
129#430—21no—
130#430C22×C10911yescharacteristic
131#407—21no—
132#407C22×C10911yescharacteristic
133#365—22no—
134#362—22no—
135#349C22×C10912yes—
136#338C22×C10911yescharacteristic
137C22×Q8—21no—
138C23×C4D4×C10911yescharacteristic
139C2×Q16—21no—
140C2×Q16D4×C10912yes—
141C2×M16—21no—
142C2×M16—21no—
143C2×M16D4×C10911yescharacteristic
144C2×M16C23×C10911yescharacteristic
145C22⋊Q8—22no—
146C22⋊Q8—22no—
147C22⋊Q8—41no—
148C22⋊Q8—41no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—41no—
151C2×C22⋊C4—41no—
152M32—22no—
153#108—22no—
154#108—42no—
155#106—21no—
156#106D4×C10912yes—
157#103—22no—
158#103—22no—
159C22⋊C8—22no—
160C22⋊C8—42no—
161C24SD16×C10911yescharacteristic
162C2×Q8—22no—
163C2×Q8—41no—
164C2×Q8—81no—
165C2×Q8—81no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—41no—
172C22×C4SD16×C10911yescharacteristic
173C22×C4C2×D4×C10911yescharacteristic
174Q16—42no—
175Q16—81no—
176M16—41no—
177M16—41no—
178M16—41no—
179M16—41no—
180M16—41no—
181M16—41no—
182C2×C8—22no—
183C2×C8—21no—
184C2×C8—41no—
185C2×C8C2×D4×C10912yes—
186C4⋊C4—22no—
187C4⋊C4—41no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C22⋊C4—42no—
191C22⋊C4—42no—
192C16—42no—
193C23—21no—
194C23—41no—
195C23—42no—
196C23C2×SD16×C10911yescharacteristic
197Q8—42no—
198Q8—42no—
199Q8—82no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—81no—
209C2×C4—82no—
210C2×C4—82no—
211C2×C4#120×C10911yescharacteristic
212C2×C4AGL(1,ℤ/8)×C10911yescharacteristic
213C8—22no—
214C8—42no—
215C8—41no—
216C8—81no—
217C22—22no—
218C22—21no—
219C22—41no—
220C22—41no—
221C22—42no—
222C22—41no—
223C22—81no—
224C22#450×C10911yescharacteristic
225C4—21no—
226C4—41no—
227C4—82no—
228C4—82no—
229C4#453×C10911yescharacteristic
230C4#458×C10911yescharacteristic
231C2—21no—
232C2—42no—
233C2—41no—
234C2C109×#161311yescharacteristic
subgroupG/H
center, socleC2×C109#1613
commutator, Frattini, cosocleC2×M16C23×C109
prime radicalC24×C109SD16
LCSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C109 ◁ C2×C4×C109 ◁ C2×M16×C109 ◁ G

Complements

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

NH ≅ G/Ntype
2. C109×#2239C2semidirect
4. C109×#1342C2semidirect
113. #12299C109direct (N × H)
119. #2239C2×C109semidirect
121. #1342C2×C109semidirect
123. C109#12299direct (N × H)

Maximal subgroups

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

subgroupindex
C109×#31952
C109×#22392
C109×#20912
C109×#13422
C109×#13202
#12299109