Finite Group Databasebeta

← C22×#16812672 #28376253

Subgroups — C22×#16812672 #28376253

Order 13952.

2528 conjugacy classes of subgroups; only the 645 normal subgroups are shown.

normal subgroups only

This group has 2528 conjugacy classes of subgroups — too many to draw. Only its 645 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
1C23×D436C211yesmaximal, characteristic, complemented
2C23×C22⋊D109C211yesmaximal, characteristic, complemented
3C23×C22⋊D109C211yesmaximal, characteristic, complemented
4C23×C4×D109C211yesmaximal, characteristic, complemented
5C2×#16812672C2124yesmaximal, direct product factor
6C22×#13182008C211yesmaximal, characteristic, complemented
7C22×#13182006C211yesmaximal, characteristic, complemented
8C22×C22⋊C4×C109C211yesmaximal, characteristic, complemented
9C24×D109C2211yescharacteristic
10C24×D109C2211yescharacteristic
11C22×D436—28no—
12C22×D436C22112yescomplemented
13C22×C22⋊D109—28no—
14C22×C22⋊D109—28no—
15C22×C22⋊D109C22112yescomplemented
16C22×C22⋊D109C22112yescomplemented
17C23×Dic109C2211yescharacteristic
18C23×Dic109C2211yescharacteristic
19C22×C4×D109—28no—
20C22×C4×D109C22112yescomplemented
21#16812672C22164yesdirect product factor
22C2×#13182008C22112yescomplemented
23C2×#13182006C22112yescomplemented
24C25×C109C2211yescharacteristic
25C23×C4×C109C2211yescharacteristic
26C23×C4×C109C2211yescharacteristic
27C2×C22⋊C4×C109C22112yescomplemented
28C23×D109—24no—
29C23×D109—24no—
30C23×D109—24no—
31C23×D109—24no—
32C23×D109—24no—
33C23×D109D418yescomplemented
34C23×D109C2316yes—
35C23×D109C2316yes—
36C2×D436—248no—
37C2×D436C23116yescomplemented
38C2×C22⋊D109—248no—
39C2×C22⋊D109—248no—
40C2×C22⋊D109C23116yescomplemented
41C2×C22⋊D109C23116yescomplemented
42C22×Dic109—24no—
43C22×Dic109D418yescomplemented
44C22×Dic109C2316yes—
45C22×Dic109C2316yes—
46C2×C4×D109—248no—
47C2×C4×D109C23116yescomplemented
48#13182008C23116yescomplemented
49#13182006C23116yescomplemented
50C24×C109—24no—
51C24×C109—24no—
52C24×C109—24no—
53C24×C109C2316yes—
54C24×C109C2311yescharacteristic
55C22×C4×C109—24no—
56C22×C4×C109—24no—
57C22×C4×C109C2316yes—
58C22×C4×C109C2316yes—
59C22⋊C4×C109C23116yescomplemented
60C23×C109—212no—
61C23×C109—212no—
62C23×C109—212no—
63C23×C109—416no—
64C23×C109C2×D414yes—
65C23×C109C2×D414yes—
66C23×C109#4114yes—
67C23×C109C2414yes—
68C23×C109C2413yes—
69C22×D109—232no—
70C22×D109—212no—
71C22×D109—212no—
72C22×D109—212no—
73C22×D109—212no—
74C22×D109—212no—
75C22×D109—416no—
76C22×D109C2×D4124yescomplemented
77C22×D109C2414yes—
78C22×D109C2414yes—
79C2×C4×C109—212no—
80C2×C4×C109—212no—
81C2×C4×C109C2414yes—
82C2×C4×C109C2414yes—
83C22⋊D109—232no—
84C22⋊D109—232no—
85C2×Dic109—212no—
86C2×Dic109C2×D4124yescomplemented
87C2×Dic109C2414yes—
88C2×Dic109C2414yes—
89D436—232no—
90C4×D109—232no—
91C22×C109—24no—
92C22×C109—24no—
93C22×C109—24no—
94C22×C109—424no—
95C22×C109#121116yescomplemented
96C22×C109C22×D416yes—
97C22×C109C22×D416yes—
98C22×C109C2×#4116yes—
99C22×C109C2511yescharacteristic
100C2×D109—248no—
101C2×D109—24no—
102C2×D109—24no—
103C2×D109—24no—
104C2×D109—24no—
105C2×D109—24no—
106C2×D109—424no—
107C2×D109C22×D418yescomplemented
108C4×C109—24no—
109C4×C109—24no—
110Dic109—24no—
111Dic109C22×D418yescomplemented
112C2×C109—44no—
113C2×C109C2×#121112yescomplemented
114C2×C109C23×D411yescharacteristic
115C2×C109C23×D411yescharacteristic
116C2×C109C22×#4111yescharacteristic
117D109—28no—
118D109—44no—
119C22×#121—1091nomaximal, Sylow
120C109C22×#12111yescharacteristic, complemented, Sylow
121C23×D4—1091no—
122C23×D4—1091no—
123C23×D4—1091no—
124C24×C4—1091no—
125C2×#121—10924no—
126C22×C4⋊C4—1091no—
127C22×C22⋊C4—1091no—
128C22×C22⋊C4D10911yescharacteristic, complemented
129C25—1091no—
130C25—1091no—
131C25C2×D10911yescharacteristic
132C22×D4—10912no—
133C22×D4—10912no—
134C22×D4—10912no—
135C22×D4—2188no—
136C22×D4—2188no—
137C22×D4—2188no—
138C23×C4—1091no—
139C23×C4—1091no—
140C23×C4—10912no—
141C23×C4—2188no—
142C23×C4C2×D10911yescharacteristic
143C23×C4C2×D10911yescharacteristic
144#121—10964no—
145C2×C4⋊C4—10912no—
146C2×C22⋊C4—10912no—
147C2×C22⋊C4C2×D109112yescomplemented
148C24—24no—
149C24—24no—
150C24—24no—
151C24—1098no—
152C24—1096no—
153C24—1096no—
154C24—2184no—
155C24—2184no—
156C24—2184no—
157C24—2184no—
158C24—2184no—
159C24C22×D10916yes—
160C24C22×D10911yescharacteristic
161C2×D4—10916no—
162C2×D4—10916no—
163C2×D4—10916no—
164C2×D4—21848no—
165C2×D4—21848no—
166C2×D4—21848no—
167C22×C4—24no—
168C22×C4—24no—
169C22×C4—1098no—
170C22×C4—1096no—
171C22×C4—1096no—
172C22×C4—10916no—
173C22×C4—21848no—
174C22×C4—2184no—
175C22×C4C22×D10916yes—
176C22×C4C22×D10916yes—
177C4⋊C4—10916no—
178C22⋊C4—10916no—
179C22⋊C4C22×D109116yescomplemented
180C23—212no—
181C23—212no—
182C23—212no—
183C23—416no—
184C23—10924no—
185C23—1094no—
186C23—1094no—
187C23—43616no—
188C23—21812no—
189C23—21812no—
190C23—21812no—
191C23—21832no—
192C23—21812no—
193C23—21812no—
194C23D4×D10914yes—
195C23D4×D10914yes—
196C23#1318202914yes—
197C23C23×D10914yes—
198C23C23×D10913yes—
199D4—21832no—
200D4—21832no—
201D4—21832no—
202C2×C4—212no—
203C2×C4—212no—
204C2×C4—10924no—
205C2×C4—1094no—
206C2×C4—1094no—
207C2×C4—21832no—
208C2×C4—21812no—
209C2×C4C23×D10914yes—
210C2×C4C23×D10914yes—
211C22—24no—
212C22—24no—
213C22—24no—
214C22—424no—
215C22—1098no—
216C22—43624no—
217C22—2184no—
218C22—2184no—
219C22—2184no—
220C22—21848no—
221C22—2184no—
222C22—2184no—
223C22#16812672116yesdirect product factor
224C22C2×D4×D10916yes—
225C22C2×D4×D10916yes—
226C22C2×#1318202916yes—
227C22C24×D10911yescharacteristic
228C4—24no—
229C4—24no—
230C4—1098no—
231C4—2184no—
232C2—44no—
233C2—4364no—
234C2—2188no—
235C2C2×#16812672112yesdirect product factor
236C2C22×D4×D10911yescharacteristic
237C2C22×D4×D10911yescharacteristic
238C2C22×#1318202911yescharacteristic
subgroupG/H
centerC24C22×D109
commutator, cosocleC22×C109C25
FrattiniC22C24×D109
FittingC22×C22⋊C4×C109C2
socleC24×C109C23
hypercenterC22×C22⋊C4D109
hypocenterC109C22×#121
LCSG ▷ C22×C109 ▷ C109
UCSC1 ◁ C24 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C23×D436C2semidirect
2. C23×C22⋊D109C2semidirect
3. C23×C22⋊D109C2semidirect
4. C23×C4×D109C2semidirect
5. C2×#16812672C2direct (N × H)
6. C22×#13182008C2semidirect
7. C22×#13182006C2semidirect
8. C22×C22⋊C4×C109C2semidirect
12. C22×D436C22semidirect
15. C22×C22⋊D109C22semidirect
16. C22×C22⋊D109C22semidirect
20. C22×C4×D109C22semidirect
21. #16812672C22direct (N × H)
22. C2×#13182008C22semidirect
23. C2×#13182006C22semidirect
27. C2×C22⋊C4×C109C22semidirect
33. C23×D109D4semidirect
37. C2×D436C23semidirect
40. C2×C22⋊D109C23semidirect
41. C2×C22⋊D109C23semidirect
43. C22×Dic109D4semidirect
47. C2×C4×D109C23semidirect
48. #13182008C23semidirect
49. #13182006C23semidirect
59. C22⋊C4×C109C23semidirect
76. C22×D109C2×D4semidirect
86. C2×Dic109C2×D4semidirect
95. C22×C109#121semidirect
107. C2×D109C22×D4semidirect
111. Dic109C22×D4semidirect
113. C2×C109C2×#121semidirect
120. C109C22×#121semidirect
128. C22×C22⋊C4D109semidirect
147. C2×C22⋊C4C2×D109semidirect
179. C22⋊C4C22×D109semidirect
223. C22#16812672direct (N × H)
235. C2C2×#16812672direct (N × H)

Maximal subgroups

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

subgroupindex
C23×D4362
C23×C22⋊D1092
C23×C4×D1092
C2×#168126722
C22×#131820082
C22×#131820062
C22×C22⋊C4×C1092
C22×#121109