Finite Group Databasebeta

← #26110661

Subgroups — #26110661

Order 12032.

396 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 111 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
1#20246574C211yesmaximal, characteristic
2#20243523C212yesmaximal
3C2×#15688865C211yesmaximal, characteristic
4#20241270C212yesmaximal
5C47×#1992C211yesmaximal, characteristic
6C2×#13093935C2212yes—
7#15689022—22no—
8#15689009—22no—
9#15689009—22no—
10#15688989C2212yes—
11#15688885—22no—
12#15688885—22no—
13#15688865C414yescomplemented
14C2×#125×C47C2211yescharacteristic
15C2×#97×C47C2211yescharacteristic
16C47×#384C2211yescharacteristic
17C47×#340—22no—
18C22×C47⋊C8—22no—
19C2×C4⋊Dic47—22no—
20C2×C4⋊Dic47D412yes—
21C2×C4⋊Dic47D412yes—
22C2×C4×Dic47—22no—
23#13093935—22no—
24#13093935C2×C414yes—
25C47⋊#95—22no—
26C47⋊#95—22no—
27C47⋊#95—22no—
28C47⋊#95—22no—
29C22×C8×C47—21no—
30#125×C47—22no—
31#125×C47C2×C412yes—
32#125×C47Q812yes—
33C2×C4⋊C4×C47—22no—
34C2×C4⋊C4×C47—21no—
35C2×C4⋊C4×C47—22no—
36C2×C4⋊C4×C47D412yes—
37C2×C42×C47C2311yescharacteristic
38#97×C47—22no—
39C22×C4×C47—22no—
40C22×C4×C47—41no—
41C22×C4×C47C2×D411yescharacteristic
42C22×C4×C47C2×D412yes—
43C22×Dic47—22no—
44C22×Dic47—22no—
45C2×C8×C47—22no—
46C2×C8×C47—41no—
47C2×C8×C47—41no—
48C4⋊C4×C47—24no—
49C4⋊C4×C47—24no—
50C4⋊C4×C47—22no—
51C4⋊C4×C47—44no—
52C4⋊C4×C47—42no—
53C4⋊C4×C47C4⋊C414yes—
54C42×C47C22×C411yescharacteristic
55C42×C47C2×Q811yescharacteristic
56C42×C47#4112yes—
57C4⋊Dic47—42no—
58C4⋊Dic47—42no—
59C4⋊Dic47—82no—
60C4×Dic47—82no—
61C2×C47⋊C8—24no—
62C2×C47⋊C8—42no—
63C2×C47⋊C8—42no—
64C23×C47#12011yescharacteristic
65C2×C4×C47—22no—
66C2×C4×C47—21no—
67C2×C4×C47—22no—
68C2×C4×C47—21no—
69C2×C4×C47—24no—
70C2×C4×C47—42no—
71C2×C4×C47—42no—
72C2×C4×C47—82no—
73C2×C4×C47—81no—
74C2×C4×C47C2×C4⋊C411yescharacteristic
75C2×C4×C47C4×D412yes—
76C2×C4×C47C22⋊Q812yes—
77C2×C4×C47#12311yescharacteristic
78C2×Dic47—42no—
79C2×Dic47—42no—
80C2×Dic47—42no—
81C2×Dic47—42no—
82C2×Dic47—42no—
83C2×Dic47—42no—
84C8×C47—42no—
85C47⋊C8—44no—
86#8915—471nomaximal, Sylow
87C22×C47#38511yescharacteristic
88C22×C47#45412yes—
89C22×C47#45411yescharacteristic
90C22×C47#45512yes—
91C22×C47#45611yescharacteristic
92C4×C47—21no—
93C4×C47—21no—
94C4×C47—22no—
95C4×C47—22no—
96C4×C47—44no—
97C4×C47—161no—
98Dic47—82no—
99Dic47—82no—
100#2382—471no—
101#1992D4711yescharacteristic
102#1836—472no—
103C2×#330—471no—
104#1295—472no—
105C2×C47#165811yescharacteristic
106C2×C47#166211yescharacteristic
107C2×C47#166412yes—
108C2×C47#185711yescharacteristic
109C2×C47#186012yes—
110C2×#125C2×D4711yescharacteristic
111C2×#97—472no—
112C2×#97C2×D4711yescharacteristic
113#389—942no—
114#387—942no—
115#387—942no—
116#384C2×D4711yescharacteristic
117#382—472no—
118#340—22no—
119#340—942no—
120#340—942no—
121#330—474no—
122C47#891511yescharacteristic, complemented, Sylow
123C22×C8—21no—
124C22×C8—942no—
125#125—22no—
126#125Dic9412yes—
127#125C4×D4712yes—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—472no—
132C2×C4⋊C4—472no—
133C2×C4⋊C4—942no—
134C2×C4⋊C4C22⋊D4712yes—
135C2×C42—942no—
136C2×C42C22×D4711yescharacteristic
137#97—22no—
138#97—474no—
139#97—942no—
140#95—942no—
141#95—942no—
142#95—942no—
143#95—942no—
144C22×C4—22no—
145C22×C4—41no—
146C22×C4—942no—
147C22×C4—942no—
148C22×C4D4×D4712yes—
149C22×C4C2×C22⋊D4711yescharacteristic
150C2×C8—22no—
151C2×C8—41no—
152C2×C8—41no—
153C2×C8—944no—
154C2×C8—1882no—
155C2×C8—1882no—
156C4⋊C4—24no—
157C4⋊C4—24no—
158C4⋊C4—22no—
159C4⋊C4—44no—
160C4⋊C4—42no—
161C4⋊C4—3762no—
162C4⋊C4—1882no—
163C4⋊C4—1882no—
164C4⋊C4Dic47⋊C414yes—
165C42—3762no—
166C42C2×Dic9411yescharacteristic
167C42C2×C4×D4711yescharacteristic
168C42#1062535212yes—
169C23#1309399211yescharacteristic
170C2×C4—22no—
171C2×C4—21no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—24no—
175C2×C4—42no—
176C2×C4—42no—
177C2×C4—82no—
178C2×C4—81no—
179C2×C4—1882no—
180C2×C4—1882no—
181C2×C4—1882no—
182C2×C4—1882no—
183C2×C4—1882no—
184C2×C4—1882no—
185C2×C4C2×Dic47⋊C411yescharacteristic
186C2×C4(C22⋊D47)⋊C412yes—
187C2×C4#1309400112yes—
188C2×C4#1309400511yescharacteristic
189C8—42no—
190C8—1884no—
191C22#1568899511yescharacteristic
192C22#1568929412yes—
193C22#1568929511yescharacteristic
194C22#1568929912yes—
195C22#1568930511yescharacteristic
196C4—21no—
197C4—21no—
198C4—22no—
199C4—22no—
200C4—44no—
201C4—161no—
202C4—3762no—
203C4—3762no—
204C2#2024268911yescharacteristic
205C2#2024270911yescharacteristic
206C2#2024272012yes—
207C2#2024360211yescharacteristic
208C2#2024361912yes—
subgroupG/H
centerC23#13093992
commutatorC42×C47C22×C4
FrattiniC2×C42C22×D47
FittingC47×#1992C2
socle, prime radicalC23×C47#120
cosocleC2×C42×C47C23
hypercenter#1992D47
hypocenterC47#8915
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1992

Complements

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

NH ≅ G/Ntype
13. #15688865C4semidirect
122. C47#8915semidirect

Maximal subgroups

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

subgroupindex
#202465742
#202435232
C2×#156888652
#202412702
C47×#19922
#891547