Finite Group Databasebeta

← C2×#20295239 #28372454

Subgroups — C2×#20295239 #28372454

Order 13952.

564 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 157 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
1C22×#13182017C211yesmaximal, characteristic, complemented
2C2×#16812686C211yesmaximal, characteristic, complemented
3C2×D4×Dic109C211yesmaximal, characteristic, complemented
4#20295239C218yesmaximal, direct product factor
5C2×#16812628C211yesmaximal, characteristic, complemented
6C2×#16812624C211yesmaximal, characteristic, complemented
7C2×#16812620C211yesmaximal, characteristic, complemented
8C22×D8×C109C211yesmaximal, characteristic
9C22×Dic218C2211yescharacteristic
10C23×Dic109—21no—
11C2×#13182017—24no—
12C2×#13182017C2214yescomplemented
13C22×#11724784C2211yescharacteristic
14#16812686—22no—
15#16812686C2214yescomplemented
16D4×Dic109—22no—
17D4×Dic109C2214yescomplemented
18C2×C4⋊Dic109C2211yescharacteristic
19C2×C22⋊Dic109—21no—
20C2×C22⋊Dic109—21no—
21C2×C22⋊Dic109—21no—
22C2×C4×Dic109C2211yescharacteristic
23#16812628C2214yescomplemented
24#16812624C2214yescomplemented
25#16812620C2214yescomplemented
26C22×D4×C109C2211yescharacteristic
27C22×D4×C109C2211yescharacteristic
28C2×D8×C109—24no—
29C2×D8×C109C2214yes—
30C22×C8×C109C2211yescharacteristic
31C2×Dic218—22no—
32C2×Dic218—22no—
33C2×Dic218—21no—
34C2×Dic218C2312yes—
35C22×Dic109—21no—
36C22×Dic109—21no—
37C22×Dic109—24no—
38C22×Dic109—44no—
39C22×Dic109D411yescharacteristic
40C22×Dic109D411yescharacteristic
41#13182017—28no—
42C2×#11724784—22no—
43C2×#11724784C2312yes—
44C4⋊Dic109—21no—
45C4⋊Dic109C2312yes—
46C22⋊Dic109—24no—
47C22⋊Dic109—24no—
48C22⋊Dic109—24no—
49C4×Dic109D412yes—
50C4×Dic109C2312yes—
51C24×C109—21no—
52C24×C109—21no—
53C2×D4×C109—21no—
54C2×D4×C109—22no—
55C2×D4×C109—21no—
56C2×D4×C109—22no—
57C2×D4×C109—22no—
58C2×D4×C109D414yes—
59C2×D4×C109C2312yes—
60C2×D4×C109C2312yes—
61C22×C4×C109C2311yescharacteristic
62D8×C109—28no—
63C2×C8×C109—22no—
64C2×C8×C109C2312yes—
65C23×C109—24no—
66C23×C109—22no—
67C23×C109—22no—
68C23×C109—42no—
69C23×C109—42no—
70C23×C109—42no—
71C23×C109—42no—
72C23×C109—42no—
73C23×C109C2×D411yescharacteristic
74D4×C109—24no—
75D4×C109—22no—
76D4×C109—22no—
77D4×C109—22no—
78D4×C109—42no—
79D4×C109C2×D414yes—
80C2×C4×C109C2×D411yescharacteristic
81C2×C4×C109C2×D412yes—
82C2×C4×C109#4112yes—
83C2×C4×C109C2411yescharacteristic
84C2×Dic109—22no—
85C2×Dic109—22no—
86C2×Dic109—22no—
87C2×Dic109—22no—
88C2×Dic109—48no—
89C2×Dic109—42no—
90C2×Dic109—42no—
91C2×Dic109C2×D412yes—
92C2×Dic109C2×D412yes—
93Dic218—22no—
94Dic218—22no—
95Dic218—42no—
96C8×C109—22no—
97#11724784—22no—
98C22×C109—24no—
99C22×C109—42no—
100C22×C109—42no—
101C22×C109—48no—
102C22×C109—42no—
103C22×C109—42no—
104C22×C109—42no—
105C22×C109—84no—
106C22×C109#12112yes—
107C22×C109#13512yes—
108C22×C109AGL(1,ℤ/8)12yes—
109C22×C109C22×D411yescharacteristic
110C4×C109#12112yes—
111C4×C109C22×D411yescharacteristic
112C4×C109C2×#4111yescharacteristic
113Dic109—22no—
114Dic109—22no—
115Dic109—42no—
116Dic109—42no—
117C2×C109—44no—
118C2×C109—82no—
119C2×C109#46314yescomplemented
120C2×C109C2×#12111yescharacteristic
121C2×C109C2×#13511yescharacteristic
122C2×C109C2×AGL(1,ℤ/8)11yescharacteristic
123C2×#463—1091nomaximal, Sylow
124C109C2×#46311yescharacteristic, complemented, Sylow
125C22×SD16—1091no—
126C22×D8D10911yescharacteristic
127C2×#124—1091no—
128C2×C4×D4—1091no—
129#463—1098no—
130C2×C4⋊C8—1091no—
131C2×#103—1091no—
132C2×#102—1091no—
133C22×Q8—1091no—
134C22×D4C2×D10911yescharacteristic
135C22×D4C2×D10911yescharacteristic
136C23×C4—2181no—
137C2×SD16—1094no—
138C2×SD16—2184no—
139C2×D8—24no—
140C2×D8C2×D10914yes—
141C22×C8—1091no—
142C22×C8C2×D10911yescharacteristic
143#124—1094no—
144#124—2182no—
145C4×D4—1094no—
146C4×D4—2182no—
147C2×C4⋊C4—1091no—
148C2×C22⋊C4—2181no—
149C2×C22⋊C4—2181no—
150C2×C22⋊C4—2181no—
151C2×C42—1091no—
152C4⋊C8—1094no—
153#103—1094no—
154#102—1094no—
155C24—21no—
156C24—21no—
157C2×Q8—1092no—
158C2×Q8—2182no—
159C2×Q8—2182no—
160C2×Q8—2181no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—22no—
164C2×D4—22no—
165C2×D4—22no—
166C2×D4C22⋊D10914yes—
167C2×D4C22×D10912yes—
168C2×D4C22×D10912yes—
169C22×C4—1091no—
170C22×C4—1091no—
171C22×C4—4364no—
172C22×C4—2184no—
173C22×C4—2181no—
174C22×C4—2181no—
175C22×C4C22×D10911yescharacteristic
176SD16—2188no—
177D8—28no—
178C2×C8—22no—
179C2×C8—1092no—
180C2×C8—2182no—
181C2×C8C22×D10912yes—
182C4⋊C4—1092no—
183C4⋊C4—2181no—
184C22⋊C4—2184no—
185C22⋊C4—2184no—
186C22⋊C4—2184no—
187C42—1092no—
188C42—1092no—
189C23—24no—
190C23—22no—
191C23—22no—
192C23—42no—
193C23—42no—
194C23—42no—
195C23—42no—
196C23—42no—
197C23D4×D10911yescharacteristic
198Q8—4362no—
199Q8—2182no—
200Q8—2182no—
201D4—24no—
202D4—22no—
203D4—22no—
204D4—22no—
205D4—42no—
206D4C2×C22⋊D10914yes—
207C2×C4—1092no—
208C2×C4—1092no—
209C2×C4—4362no—
210C2×C4—4362no—
211C2×C4—4368no—
212C2×C4—2182no—
213C2×C4—2182no—
214C2×C4—2182no—
215C2×C4—2182no—
216C2×C4D4×D10911yescharacteristic
217C2×C4C2×C22⋊D10912yes—
218C2×C4#1318202812yes—
219C2×C4C23×D10911yescharacteristic
220C8—22no—
221C8—2182no—
222C22—24no—
223C22—48no—
224C22—42no—
225C22—42no—
226C22—42no—
227C22—42no—
228C22—42no—
229C22—84no—
230C22#1681267112yes—
231C22#1681271812yes—
232C22#1681272312yes—
233C22C2×D4×D10911yescharacteristic
234C4—4362no—
235C4—4362no—
236C4—2182no—
237C4—2182no—
238C4#1681267112yes—
239C4C22×C22⋊D10911yescharacteristic
240C4C2×#1318202811yescharacteristic
241C2—44no—
242C2—82no—
243C2#2029523914yesdirect product factor
244C2C2×#1681267111yescharacteristic
245C2C2×#1681271811yescharacteristic
246C2C2×#1681272311yescharacteristic
subgroupG/H
centerC23D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting, prime radicalC22×D8×C109C2
socleC23×C109C2×D4
hypercenterC22×D8D109
hypocenterC109C2×#463
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×D8

Complements

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

NH ≅ G/Ntype
1. C22×#13182017C2semidirect
2. C2×#16812686C2semidirect
3. C2×D4×Dic109C2semidirect
4. #20295239C2direct (N × H)
5. C2×#16812628C2semidirect
6. C2×#16812624C2semidirect
7. C2×#16812620C2semidirect
12. C2×#13182017C22semidirect
15. #16812686C22semidirect
17. D4×Dic109C22semidirect
23. #16812628C22semidirect
24. #16812624C22semidirect
25. #16812620C22semidirect
119. C2×C109#463semidirect
124. C109C2×#463semidirect
243. C2#20295239direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131820172
C2×#168126862
C2×D4×Dic1092
#202952392
C2×#168126282
C2×#168126242
C2×#168126202
C22×D8×C1092
C2×#463109