Finite Group Databasebeta

← #26103826

Subgroups — #26103826

Order 12032.

224 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 41 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#20244617C211yesmaximal, characteristic
2#20241332C211yesmaximal, characteristic, complemented
3C47×#1295C211yesmaximal, characteristic, complemented
4#15689704C411yescharacteristic
5C2×#13093952—21no—
6#15689122—21no—
7#15688892C411yescharacteristic
8#15688849—21no—
9C2×#97×C47—21no—
10C47×#382C2211yescharacteristic
11C2×#10625350—21no—
12C2×#10625329—21no—
13#13094004—41no—
14#13093999—41no—
15#13093998—41no—
16#13093952—41no—
17#13093952—41no—
18#13093947—41no—
19#13093944—41no—
20C22⋊(C47⋊C8)—41no—
21C22×C8×C47—41no—
22C2×C4⋊C4×C47C2×C411yescharacteristic
23C2×C4⋊C4×C47D411yescharacteristic
24C2×C42×C47D411yescharacteristic
25#97×C47—22no—
26#97×C47—41no—
27#95×C47—41no—
28C22×C4×C47—21no—
29C22×C4×C47—21no—
30C22×C4×C47—21no—
31C22×C4×C47C22⋊C411yescharacteristic
32C2×C22⋊D47—41no—
33#10625350—41no—
34#10625350—41no—
35#10625350—41no—
36#10625350—41no—
37C2×D188—21no—
38C2×D188—41no—
39C2×C4×D47—41no—
40C2×Dic94—21no—
41C2×C8×C47—42no—
42C2×C8×C47—42no—
43C2×C8×C47—42no—
44C4⋊C4×C47—41no—
45C4⋊C4×C47—41no—
46C42×C47—21no—
47C42×C47D811yescharacteristic
48C42×C47SD1611yescharacteristic
49(C2×D47)⋊C4—42no—
50C4⋊Dic47—41no—
51#10625329—41no—
52#10625329—41no—
53C2×C47⋊C8—41no—
54C2×C47⋊C8—81no—
55C23×C47#9911yescharacteristic
56C22×D47—41no—
57C2×C4×C47—21no—
58C2×C4×C47—21no—
59C2×C4×C47—21no—
60C2×C4×C47—21no—
61C2×C4×C47—41no—
62C2×C4×C47—41no—
63C2×C4×C47—41no—
64C2×C4×C47—41no—
65C2×C4×C47—81no—
66C2×C4×C47—81no—
67C2×C4×C47#10211yescharacteristic
68C2×C4×C47#10411yescharacteristic
69C22⋊D47—81no—
70C22⋊D47—81no—
71C2×Dic47—41no—
72D188—41no—
73D188—41no—
74D188—161no—
75C4×D47—81no—
76C4×D47—81no—
77Dic94—41no—
78Dic94—41no—
79C8×C47—44no—
80C47⋊C8—81no—
81C47⋊C8—161no—
82#7416—471nomaximal, Sylow
83C22×C47—21no—
84C22×C47—21no—
85C22×C47#32711yescharacteristic
86C22×C47AΣL(1,𝔽16)11yescharacteristic
87C22×C47#35511yescharacteristic
88C2×D47—81no—
89C2×D47—81no—
90C2×D47—81no—
91C4×C47—21no—
92C4×C47—21no—
93C4×C47—41no—
94C4×C47—41no—
95C4×C47—81no—
96C4×C47—81no—
97Dic47—81no—
98#2040—471no—
99#1324—471no—
100#1295D4711yescharacteristic, complemented
101C2×C47—21no—
102C2×C47—21no—
103C2×C47#134511yescharacteristic
104C2×C47#140311yescharacteristic
105C2×C47#140411yescharacteristic
106D47—161no—
107#537—471no—
108C2×#104—941no—
109#417—941no—
110C2×#97—21no—
111#382C2×D4711yescharacteristic
112#343—471no—
113#324—941no—
114C47#741611yescharacteristic, complemented, Sylow
115C2×#41—941no—
116C2×M16—941no—
117C22×C8—41no—
118C22⋊Q8—1881no—
119#121—1881no—
120#121—1881no—
121C2×C4⋊C4C4×D4711yescharacteristic
122C2×C4⋊C4D18811yescharacteristic
123C2×C42C22⋊D4711yescharacteristic
124#104—1881no—
125#104—1881no—
126#103—1881no—
127#102—1881no—
128C22⋊C8—1881no—
129#97—22no—
130#97—41no—
131#95—41no—
132#41—1881no—
133#41—1881no—
134#41—1881no—
135#41—1881no—
136C2×Q8—941no—
137C2×D4—941no—
138C2×D4—1881no—
139C2×D4—1881no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—1881no—
144C22×C4(C2×D47)⋊C411yescharacteristic
145M16—1881no—
146M16—1881no—
147C2×C8—42no—
148C2×C8—42no—
149C2×C8—42no—
150C2×C8—3761no—
151C2×C8—1881no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—1881no—
155C22⋊C4—1882no—
156C42—21no—
157C42#1062533411yescharacteristic
158C42#1062533611yescharacteristic
159C23—1881no—
160C23#1309393811yescharacteristic
161Q8—1881no—
162Q8—1881no—
163D4—3761no—
164D4—3761no—
165D4—1881no—
166D4—1881no—
167D4—7521no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—81no—
177C2×C4—81no—
178C2×C4—3761no—
179C2×C4—3761no—
180C2×C4—1881no—
181C2×C4#1309394411yescharacteristic
182C2×C4#1309395211yescharacteristic
183C8—44no—
184C8—3761no—
185C8—7521no—
186C22—21no—
187C22—21no—
188C22—3761no—
189C22—3761no—
190C22—3761no—
191C22#1568885611yescharacteristic
192C22#1568891111yescharacteristic
193C22#1568891911yescharacteristic
194C4—21no—
195C4—21no—
196C4—41no—
197C4—41no—
198C4—81no—
199C4—81no—
200C4—3761no—
201C2—21no—
202C2—21no—
203C2—7521no—
204C2#2024139011yescharacteristic
205C2#2024153711yescharacteristic
206C2#2024154011yescharacteristic
subgroupG/H
centerC22#15688911
commutatorC2×C4⋊C4×C47C2×C4
Frattini#382C2×D47
FittingC47×#1295C2
socleC22×C47AΣL(1,𝔽16)
prime radical#15689704C4
cosocleC47×#382C22
hypercenter#1295D47
hypocenterC47#7416
LCSG ▷ C2×C4⋊C4×C47 ▷ C22×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #382 ◁ #1295

Complements

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

NH ≅ G/Ntype
2. #20241332C2semidirect
3. C47×#1295C2semidirect
100. #1295D47semidirect
114. C47#7416semidirect

Maximal subgroups

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

subgroupindex
#202446172
#202413322
C47×#12952
#741647