Finite Group Databasebeta

← #26107206

Subgroups — #26107206

Order 12032.

444 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 141 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#125×Dic47C211yesmaximal, characteristic
2#20243136C212yesmaximal
3C2×#15688865C211yesmaximal, characteristic
4#20241270C212yesmaximal
5C2×C47×#330C211yesmaximal, characteristic
6C2×#13093935C2212yes—
7#15689022C412yes—
8#15689009—22no—
9#15688989C2211yescharacteristic
10C4⋊C4×Dic47—22no—
11C4⋊C4×Dic47C412yes—
12C42×Dic47C2211yescharacteristic
13#15688877—22no—
14#15688877—22no—
15#15688865C414yescomplemented
16C2×#125×C47C2211yescharacteristic
17C2×#97×C47C2212yes—
18C47×#330C414yescomplemented
19C22×C47⋊C8—22no—
20C2×C4⋊Dic47—22no—
21C2×C4⋊Dic47C2×C412yes—
22C2×C4×Dic47—22no—
23C2×C4×Dic47C2×C411yescharacteristic
24C2×C4×Dic47C2×C411yescharacteristic
25C2×C4×Dic47D412yes—
26C2×C4×Dic47D412yes—
27#13093935—22no—
28#13093935C2×C414yes—
29C47⋊#95—22no—
30C47⋊#95—22no—
31C47⋊#95—22no—
32C22×C8×C47—22no—
33#125×C47—22no—
34#125×C47C2×C412yes—
35#125×C47C2×C412yes—
36C2×C4⋊C4×C47—22no—
37C2×C4⋊C4×C47C2×C412yes—
38C2×C42×C47C2311yescharacteristic
39#97×C47—22no—
40#97×C47C2×C414yes—
41C22×C4×C47—22no—
42C22×C4×C47C22×C411yescharacteristic
43C22×C4×C47C2×D412yes—
44C22×Dic47—22no—
45C22×Dic47C22⋊C411yescharacteristic
46C22×Dic47C22⋊C412yes—
47C22×Dic47C22⋊C411yescharacteristic
48C2×C8×C47—24no—
49C2×C8×C47—42no—
50C2×C8×C47—42no—
51C4⋊C4×C47—22no—
52C4⋊C4×C47—24no—
53C4⋊C4×C47—24no—
54C4⋊C4×C47C4214yes—
55C42×C47C22×C411yescharacteristic
56C42×C47C22×C411yescharacteristic
57C42×C47#4112yes—
58C4⋊Dic47—44no—
59C4⋊Dic47—42no—
60C4×Dic47—22no—
61C4×Dic47—41no—
62C4×Dic47—42no—
63C4×Dic47—42no—
64C4×Dic47—44no—
65C2×C47⋊C8—24no—
66C2×C47⋊C8—42no—
67C2×C47⋊C8—42no—
68C23×C47C2×C22⋊C411yescharacteristic
69C2×C4×C47—21no—
70C2×C4×C47—21no—
71C2×C4×C47—22no—
72C2×C4×C47—22no—
73C2×C4×C47—24no—
74C2×C4×C47—42no—
75C2×C4×C47—42no—
76C2×C4×C47C2×C4211yescharacteristic
77C2×C4×C47#11711yescharacteristic
78C2×C4×C47C4×D412yes—
79C2×C4×C47C4×D412yes—
80C2×Dic47—22no—
81C2×Dic47—21no—
82C2×Dic47—22no—
83C2×Dic47—22no—
84C2×Dic47—22no—
85C2×Dic47—22no—
86C2×Dic47—42no—
87C2×Dic47—42no—
88C2×Dic47—42no—
89C2×Dic47#10112yes—
90C8×C47—44no—
91C47⋊C8—44no—
92#8225—471nomaximal, Sylow
93C22×C47C4×C22⋊C411yescharacteristic
94C22×C47C2×#10111yescharacteristic
95C22×C47#41311yescharacteristic
96C22×C47#42112yes—
97C22×C47#42112yes—
98C4×C47—21no—
99C4×C47—21no—
100C4×C47—22no—
101C4×C47—22no—
102C4×C47—44no—
103Dic47—22no—
104Dic47—41no—
105Dic47—42no—
106Dic47—82no—
107C4×#125—471no—
108#1745—472no—
109C2×#330—471no—
110C2×#330D4711yescharacteristic
111#1295—472no—
112C2×C47#153411yescharacteristic
113C2×C47#153512yes—
114C2×C47#153711yescharacteristic
115C2×C47C4×#10111yescharacteristic
116C2×C47#175912yes—
117C2×#125C2×D4711yescharacteristic
118C2×#97—472no—
119C2×#97C2×D4712yes—
120#389—472no—
121#387—942no—
122#382—471no—
123C4×C4⋊C4—472no—
124C4×C4⋊C4—942no—
125C43—471no—
126#336—942no—
127#336—942no—
128#330—474no—
129#330Dic4714yescomplemented
130C47#822511yescharacteristic, complemented, Sylow
131C22×C8—22no—
132C22×C8—942no—
133#125—22no—
134#125C4×D4712yes—
135#125C2×Dic4712yes—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—472no—
138C2×C4⋊C4—942no—
139C2×C4⋊C4C4×D4712yes—
140C2×C42—471no—
141C2×C42—471no—
142C2×C42—472no—
143C2×C42—472no—
144C2×C42—942no—
145C2×C42C22×D4711yescharacteristic
146#97—22no—
147#97—474no—
148#97—942no—
149#97C2×Dic4714yes—
150#95—942no—
151#95—942no—
152#95—942no—
153C22×C4—22no—
154C22×C4—471no—
155C22×C4—471no—
156C22×C4—472no—
157C22×C4—942no—
158C22×C4C2×C4×D4711yescharacteristic
159C22×C4D4×D4712yes—
160C2×C8—24no—
161C2×C8—42no—
162C2×C8—42no—
163C2×C8—944no—
164C2×C8—1882no—
165C2×C8—1882no—
166C4⋊C4—22no—
167C4⋊C4—24no—
168C4⋊C4—24no—
169C4⋊C4—1882no—
170C4⋊C4—1884no—
171C4⋊C4C4×Dic4714yes—
172C42—942no—
173C42—1881no—
174C42—1882no—
175C42—1882no—
176C42—1884no—
177C42C2×C4×D4711yescharacteristic
178C42#1062535212yes—
179C42C22×Dic4711yescharacteristic
180C23C22⋊C4×D4711yescharacteristic
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—24no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—472no—
189C2×C4—942no—
190C2×C4—941no—
191C2×C4—942no—
192C2×C4—942no—
193C2×C4—942no—
194C2×C4—942no—
195C2×C4—1882no—
196C2×C4—1882no—
197C2×C4—1882no—
198C2×C4C2×C4×Dic4711yescharacteristic
199C2×C4#1309398011yescharacteristic
200C2×C4(C22⋊D47)⋊C412yes—
201C2×C4D4×Dic4712yes—
202C8—44no—
203C8—1884no—
204C22C22⋊C4×Dic4711yescharacteristic
205C22#101×D4711yescharacteristic
206C22#1568910611yescharacteristic
207C22#1568913812yes—
208C22#1568913812yes—
209C4—22no—
210C4—22no—
211C4—21no—
212C4—21no—
213C4—44no—
214C4—942no—
215C4—3762no—
216C4—1882no—
217C4—1881no—
218C2#2024201311yescharacteristic
219C2#2024201812yes—
220C2#2024202411yescharacteristic
221C2#101×Dic4711yescharacteristic
222C2#2024320012yes—
subgroupG/H
centerC23C22⋊C4×D47
commutatorC2×C4×C47C2×C42
FrattiniC2×C42C22×D47
FittingC2×C47×#330C2
socle, prime radicalC23×C47C2×C22⋊C4
cosocleC2×C42×C47C23
hypercenterC2×#330D47
hypocenterC47#8225
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#330

Complements

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

NH ≅ G/Ntype
15. #15688865C4semidirect
18. C47×#330C4semidirect
129. #330Dic47semidirect
130. C47#8225semidirect

Maximal subgroups

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

subgroupindex
#125×Dic472
#202431362
C2×#156888652
#202412702
C2×C47×#3302
#822547