Finite Group Databasebeta

← C2×#20249107 #26383191

Subgroups — C2×#20249107 #26383191

Order 12032.

5194 conjugacy classes of subgroups; only the 907 normal subgroups are shown.

normal subgroups only

This group has 5194 conjugacy classes of subgroups — too many to draw. Only its 907 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×#13094007C213yesmaximal
2#20249107C2116yesmaximal, direct product factor
3C2×#15689042C214yesmaximal, complemented
4C2×#15689034C214yesmaximal, complemented
5C2×#15688986C213yesmaximal, complemented
6C24×Q8×C47C211yesmaximal, characteristic, complemented
7C23×C22⋊D47C2211yescharacteristic
8C2×#13094007—224no—
9C2×#13094007C22124yes—
10C22×Dic47⋊C4C2216yes—
11C22×C22⋊Dic47C2213yes—
12C22×(C2×D47)⋊C4C2216yes—
13#15689042C22132yescomplemented
14#15689034C22132yescomplemented
15#15688986C22124yescomplemented
16C2×C47⋊#95C22112yes—
17C23×Q8×C47—224no—
18C23×Q8×C47C2214yes—
19C23×Q8×C47C2218yescomplemented
20C24×C4×C47C2213yes—
21C24×D47C2311yescharacteristic
22C22×C22⋊D47—212no—
23C22×C22⋊D47C2314yes—
24C23×Dic47C2316yes—
25C23×Dic47C2311yescharacteristic
26#13094007—296no—
27C2×Dic47⋊C4—224no—
28C2×Dic47⋊C4C23124yes—
29C2×C22⋊Dic47—212no—
30C2×C22⋊Dic47C23112yes—
31C2×(C2×D47)⋊C4—224no—
32C2×(C2×D47)⋊C4C23124yes—
33C47⋊#95C23148yes—
34C25×C47C2311yescharacteristic
35C22×Q8×C47—2128no—
36C22×Q8×C47—248no—
37C22×Q8×C47—296no—
38C22×Q8×C47C23116yes—
39C23×C4×C47—224no—
40C23×C4×C47D4124yes—
41C23×C4×C47C2316yes—
42C23×C4×C47C23112yes—
43C24×C47—28no—
44C24×C47C2×D4112yes—
45C24×C47C2412yes—
46C24×C47C2411yescharacteristic
47C23×D47—26no—
48C23×D47—28no—
49C23×D47C2412yes—
50C2×Q8×C47—2128no—
51C2×Q8×C47—2256no—
52C2×Q8×C47—296no—
53C22×C4×C47—296no—
54C22×C4×C47—248no—
55C22×C4×C47—224no—
56C22×C4×C47C2×D4124yes—
57C22×C4×C47C2×D4148yes—
58C22×C4×C47C24112yes—
59C2×C22⋊D47—224no—
60C2×C22⋊D47—416no—
61C22×Dic47—212no—
62C22×Dic47—26no—
63C22×Dic47—224no—
64C22×Dic47C24112yes—
65C22×Dic47C2412yes—
66C22⋊Dic47—224no—
67(C2×D47)⋊C4—248no—
68Dic47⋊C4—248no—
69C23×C47—248no—
70C23×C47—28no—
71C23×C47#120116yes—
72C23×C47C22×D4112yes—
73C23×C47C22×D416yes—
74C23×C472−1+418yes—
75C23×C47C2511yescharacteristic
76C22×D47—26no—
77C22×D47—28no—
78C22×D47—44no—
79C22×D47—424no—
80Q8×C47—2128no—
81C2×C4×C47—296no—
82C2×C4×C47—248no—
83C2×C4×C47—224no—
84C2×C4×C47C22×D4124yes—
85C22⋊D47—416no—
86C2×Dic47—26no—
87C2×Dic47—212no—
88C2×Dic47—224no—
89C2×Dic47—44no—
90C2×Dic47—424no—
91C2×#2614—471nomaximal, Sylow
92C22×C47—232no—
93C22×C47—224no—
94C22×C47C2×#12018yes—
95C22×C47C2×#12014yes—
96C22×C47#536124yes—
97C22×C47C23×D413yes—
98C22×C47C2×2−1+414yes—
99C2×D47—42no—
100C2×D47—412no—
101C2×D47—88no—
102C4×C47—224no—
103Dic47—42no—
104Dic47—412no—
105C24×Q8D4711yescharacteristic, complemented
106C22×#123—473no—
107#2614—4716no—
108C2×#395—474no—
109C2×#393—474no—
110C2×#381—473no—
111C2×C47—28no—
112C2×C47#261418yescomplemented
113C2×C47C22×#12011yescharacteristic
114C2×C47C2×#53616yes—
115D47—82no—
116C23×Q8—224no—
117C23×Q8C2×D4714yes—
118C23×Q8C2×D4718yescomplemented
119C23×D4—471no—
120C24×C4C2×D4713yes—
121C2×#123—4724no—
122C2×#123—9424no—
123C22×C4⋊C4—476no—
124C22×C22⋊C4—473no—
125C22×C22⋊C4—476no—
126#395—4732no—
127#393—4732no—
128#381—4724no—
129C2×#95—4712no—
130C47C2×#261411yescharacteristic, complemented, Sylow
131C25—471no—
132C25C22×D4711yescharacteristic
133C22×Q8—2128no—
134C22×Q8—248no—
135C22×Q8—296no—
136C22×Q8C22×D47116yes—
137C22×D4—474no—
138C22×D4—9412no—
139C23×C4—224no—
140C23×C4—476no—
141C23×C4—471no—
142C23×C4C22⋊D47124yes—
143C23×C4C22×D47112yes—
144C23×C4C22×D4716yes—
145#123—9496no—
146C2×C4⋊C4—4724no—
147C2×C4⋊C4—9424no—
148C2×C22⋊C4—4712no—
149C2×C22⋊C4—4724no—
150C2×C22⋊C4—9412no—
151C2×C22⋊C4—9424no—
152#95—4748no—
153C24—28no—
154C24—472no—
155C24—946no—
156C24—948no—
157C24C2×C22⋊D47112yes—
158C24C23×D4712yes—
159C24C23×D4711yescharacteristic
160C2×Q8—2128no—
161C2×Q8—2256no—
162C2×Q8—296no—
163C2×D4—9424no—
164C2×D4—18816no—
165C22×C4—296no—
166C22×C4—248no—
167C22×C4—224no—
168C22×C4—4712no—
169C22×C4—472no—
170C22×C4—9412no—
171C22×C4—946no—
172C22×C4—9424no—
173C22×C4C2×C22⋊D47124yes—
174C22×C4C2×C22⋊D47148yes—
175C22×C4C23×D47112yes—
176C4⋊C4—9448no—
177C22⋊C4—9424no—
178C22⋊C4—9448no—
179C23—248no—
180C23—28no—
181C23—946no—
182C23—948no—
183C23—1884no—
184C23—18824no—
185C23#13093993116yes—
186C23C22×C22⋊D47112yes—
187C23C22×C22⋊D4716yes—
188C23#1309407218yes—
189C23C24×D4711yescharacteristic
190Q8—2128no—
191D4—18816no—
192C2×C4—296no—
193C2×C4—248no—
194C2×C4—224no—
195C2×C4—9412no—
196C2×C4—946no—
197C2×C4—9424no—
198C2×C4—1884no—
199C2×C4—18824no—
200C2×C4C22×C22⋊D47124yes—
201C22—232no—
202C22—224no—
203C22—3768no—
204C22—1882no—
205C22—18812no—
206C22C2×#1309399318yes—
207C22C2×#1309399314yes—
208C22#15689697124yes—
209C22C23×C22⋊D4713yes—
210C22C2×#1309407214yes—
211C4—224no—
212C4—1882no—
213C4—18812no—
214C2—28no—
215C2—3762no—
216C2#2024910718yesdirect product factor
217C2C22×#1309399311yescharacteristic
218C2C2×#1568969716yes—
subgroupG/H
centerC24C23×D47
commutator, cosocleC23×C47C25
FrattiniC23C24×D47
FittingC24×Q8×C47C2
socleC24×C47C24
prime radicalC23×C22⋊D47C22
hypercenterC24×Q8D47
hypocenterC47C2×#2614
LCSG ▷ C23×C47 ▷ C47
UCSC1 ◁ C24 ◁ C24×Q8

Complements

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

NH ≅ G/Ntype
2. #20249107C2direct (N × H)
3. C2×#15689042C2semidirect
4. C2×#15689034C2semidirect
5. C2×#15688986C2semidirect
6. C24×Q8×C47C2semidirect
13. #15689042C22semidirect
14. #15689034C22semidirect
15. #15688986C22semidirect
19. C23×Q8×C47C22semidirect
105. C24×Q8D47semidirect
112. C2×C47#2614semidirect
118. C23×Q8C2×D47semidirect
130. C47C2×#2614semidirect
216. C2#20249107direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130940072
#202491072
C2×#156890422
C2×#156890342
C2×#156889862
C24×Q8×C472
C2×#261447