Finite Group Databasebeta

← C109×#12224 #42130218

Subgroups — C109×#12224 #42130218

Order 27904.

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

normal subgroups only

This group has 380 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×#2811C211yesmaximal, characteristic
2C109×#2028C211yesmaximal, characteristic
3C109×#1605C212yesmaximal
4C109×#1354C212yesmaximal
5C109×#1319C211yesmaximal, characteristic
6#544×C109C2211yescharacteristic
7#451×C109—22no—
8#419×C109—21no—
9#395×C109—22no—
10#395×C109C2212yes—
11#391×C109—21no—
12#391×C109—21no—
13#391×C109—22no—
14#391×C109C2211yescharacteristic
15#343×C109C2211yescharacteristic
16#339×C109—21no—
17#328×C109—22no—
18#324×C109C2212yes—
19C22×Q8×C109—21no—
20C22×Q8×C109D411yescharacteristic
21C22×Q8×C109D412yes—
22C2×Q16×C109—42no—
23C2×M16×C109—21no—
24C4⋊Q8×C109—41no—
25#125×C109—41no—
26C22⋊Q8×C109—22no—
27#117×C109—21no—
28C2×C4⋊C4×C109—21no—
29C2×C4⋊C4×C109—21no—
30C2×C4⋊C4×C109—22no—
31C2×C4⋊C4×C109C2311yescharacteristic
32C2×C42×C109—21no—
33C2×C42×C109D411yescharacteristic
34#103×C109—41no—
35#103×C109—41no—
36#103×C109—42no—
37C22⋊C8×C109—22no—
38#95×C109—22no—
39#95×C109—22no—
40#95×C109—22no—
41#95×C109—22no—
42#95×C109D412yes—
43C2×Q8×C109—24no—
44C2×Q8×C109—21no—
45C2×Q8×C109—22no—
46C2×Q8×C109—41no—
47C2×Q8×C109—42no—
48C2×Q8×C109—41no—
49C2×Q8×C109—82no—
50C2×Q8×C109—81no—
51C2×Q8×C109—81no—
52C2×Q8×C109Q1612yescomplemented
53C22×C4×C109—21no—
54C22×C4×C109—21no—
55C22×C4×C109—22no—
56C22×C4×C109—22no—
57C22×C4×C109C2×D411yescharacteristic
58C22×C4×C109C2×D412yes—
59Q16×C109—84no—
60M16×C109—41no—
61M16×C109—41no—
62C2×C8×C109—42no—
63C2×C8×C109—41no—
64C4⋊C4×C109—21no—
65C4⋊C4×C109—21no—
66C4⋊C4×C109—22no—
67C4⋊C4×C109—41no—
68C4⋊C4×C109—42no—
69C4⋊C4×C109—82no—
70C4⋊C4×C109—81no—
71C4⋊C4×C109—81no—
72C22⋊C4×C109—41no—
73C42×C109—41no—
74C42×C109—41no—
75C42×C109—81no—
76C23×C109#12011yescharacteristic
77Q8×C109—42no—
78Q8×C109—44no—
79Q8×C109—81no—
80Q8×C109—84no—
81Q8×C109—82no—
82Q8×C109—82no—
83C2×C4×C109—21no—
84C2×C4×C109—21no—
85C2×C4×C109—22no—
86C2×C4×C109—24no—
87C2×C4×C109—22no—
88C2×C4×C109—22no—
89C2×C4×C109—41no—
90C2×C4×C109—42no—
91C2×C4×C109—41no—
92C2×C4×C109—41no—
93C2×C4×C109—42no—
94C2×C4×C109—41no—
95C2×C4×C109—42no—
96C2×C4×C109—42no—
97C2×C4×C109—42no—
98C2×C4×C109—41no—
99C2×C4×C109—41no—
100C2×C4×C109—42no—
101C2×C4×C109C2×Q1611yescharacteristic
102C2×C4×C109#13711yescharacteristic
103C8×C109—81no—
104C8×C109—82no—
105C22×C109—21no—
106C22×C109—21no—
107C22×C109#45111yescharacteristic
108C22×C109#45511yescharacteristic
109C22×C109B(4,𝔽2)11yescharacteristic
110C4×C109—21no—
111C4×C109—21no—
112C4×C109—42no—
113C4×C109—44no—
114C4×C109—42no—
115C4×C109—42no—
116C4×C109—82no—
117C4×C109—81no—
118C4×C109—81no—
119#12224C10911yesmaximal, characteristic, direct product factor, Sylow
120C2×C109—21no—
121C2×C109—21no—
122C2×C109#160211yescharacteristic
123C2×C109#219911yescharacteristic
124C2×C109#220511yescharacteristic
125#2811C2×C10911yescharacteristic
126#2028C2×C10911yescharacteristic
127#1605C2×C10912yes—
128#1354C2×C10912yes—
129#1319C2×C10911yescharacteristic
130C109#1222411yescharacteristic, direct product factor, Sylow
131#544C22×C10911yescharacteristic
132#451—22no—
133#419—21no—
134#395—22no—
135#395C22×C10912yes—
136#391—21no—
137#391—21no—
138#391—22no—
139#391C22×C10911yescharacteristic
140#343C22×C10911yescharacteristic
141#339—21no—
142#328—22no—
143#324C22×C10912yes—
144C22×Q8—21no—
145C22×Q8D4×C10911yescharacteristic
146C22×Q8D4×C10912yes—
147C2×Q16—42no—
148C2×M16—21no—
149C4⋊Q8—41no—
150#125—41no—
151C22⋊Q8—22no—
152#117—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4C23×C10911yescharacteristic
157C2×C42—21no—
158C2×C42D4×C10911yescharacteristic
159#103—41no—
160#103—41no—
161#103—42no—
162C22⋊C8—22no—
163#95—22no—
164#95—22no—
165#95—22no—
166#95—22no—
167#95D4×C10912yes—
168C2×Q8—24no—
169C2×Q8—21no—
170C2×Q8—22no—
171C2×Q8—41no—
172C2×Q8—42no—
173C2×Q8—41no—
174C2×Q8—82no—
175C2×Q8—81no—
176C2×Q8—81no—
177C2×Q8Q16×C10912yescomplemented
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—22no—
182C22×C4C2×D4×C10911yescharacteristic
183C22×C4C2×D4×C10912yes—
184Q16—84no—
185M16—41no—
186M16—41no—
187C2×C8—42no—
188C2×C8—41no—
189C4⋊C4—21no—
190C4⋊C4—21no—
191C4⋊C4—22no—
192C4⋊C4—41no—
193C4⋊C4—42no—
194C4⋊C4—82no—
195C4⋊C4—81no—
196C4⋊C4—81no—
197C22⋊C4—41no—
198C42—41no—
199C42—41no—
200C42—81no—
201C23#120×C10911yescharacteristic
202Q8—42no—
203Q8—44no—
204Q8—81no—
205Q8—84no—
206Q8—82no—
207Q8—82no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—24no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—42no—
226C2×C4C2×Q16×C10911yescharacteristic
227C2×C4#137×C10911yescharacteristic
228C8—81no—
229C8—82no—
230C22—21no—
231C22—21no—
232C22#451×C10911yescharacteristic
233C22#455×C10911yescharacteristic
234C22B(4,𝔽2)×C10911yescharacteristic
235C4—21no—
236C4—21no—
237C4—42no—
238C4—44no—
239C4—42no—
240C4—42no—
241C4—82no—
242C4—81no—
243C4—81no—
244C2—21no—
245C2—21no—
246C2C109×#160211yescharacteristic
247C2C109×#219911yescharacteristic
248C2C109×#220511yescharacteristic
subgroupG/H
center, socleC22×C109B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C109
prime radicalC23×C109#120
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
52. C2×Q8×C109Q16semidirect
119. #12224C109direct (N × H)
130. C109#12224direct (N × H)
177. C2×Q8Q16×C109semidirect

Maximal subgroups

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

subgroupindex
C109×#28112
C109×#20282
C109×#16052
C109×#13542
C109×#13192
#12224109