Finite Group Databasebeta

← #26128226

Subgroups — #26128226

Order 12032.

220 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 69 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#20245252C211yesmaximal, characteristic
2#20242296C211yesmaximal, characteristic, complemented
3#20242212C211yesmaximal, characteristic
4#20241453C211yesmaximal, characteristic, complemented
5#20241359C211yesmaximal, characteristic, complemented
6#20241308C211yesmaximal, characteristic, complemented
7C47×#1333C211yesmaximal, characteristic
8C2×#13093968C411yescharacteristic
9#15689250—21no—
10C4×Dic188C2211yescharacteristic
11D4×C47⋊C8—21no—
12#15688948C411yescharacteristic
13#15688926C411yescharacteristic
14#15688924C411yescharacteristic
15#15688899C2211yescharacteristic
16#15688874C2211yescharacteristic
17#15688853—21no—
18D4⋊(C47⋊C8)—21no—
19#15688844C2211yescharacteristic
20C4×D8×C47C2211yescharacteristic
21C47×#346C2211yescharacteristic
22C47×#334C2211yescharacteristic
23C2×Dic188C2×C411yescharacteristic
24C22×C47⋊C8—41no—
25C4×Dic94—21no—
26#13093968—22no—
27C2×C47⋊C16C2×C411yescharacteristic
28C2×C47⋊C16C2×C411yescharacteristic
29#13093958C2×C411yescharacteristic
30C4⋊(C47⋊C8)—21no—
31Dic47⋊C8—41no—
32#13093949—21no—
33C22⋊(C47⋊C8)—41no—
34#13093934—41no—
35C4×C47⋊C8D411yescharacteristic
36C4×C47⋊C8D411yescharacteristic
37C2×D8×C47C2×C411yescharacteristic
38C4×D4×C47—21no—
39C2×C16×C47—21no—
40#107×C47C2×C411yescharacteristic
41C4⋊C8×C47—21no—
42#102×C47—21no—
43C4×C8×C47C2311yescharacteristic
44C2×D4×C47—21no—
45C22×C4×C47—41no—
46C2×Dic94—21no—
47D8×C47—21no—
48D8×C47M1612yes—
49C2×C8×C47—41no—
50C2×C8×C47C22×C411yescharacteristic
51C2×C8×C47#4111yescharacteristic
52C4⋊C4×C47—21no—
53C22⋊C4×C47—41no—
54C42×C47C2×D411yescharacteristic
55C4⋊Dic47—21no—
56Dic47⋊C4—41no—
57C4×Dic47—41no—
58C2×C47⋊C8—21no—
59C2×C47⋊C8—21no—
60C2×C47⋊C8—41no—
61C2×C47⋊C8—82no—
62Dic188—21no—
63Dic188—21no—
64C16×C47—22no—
65C47⋊C16—21no—
66C47⋊C16—21no—
67C23×C47—41no—
68D4×C47—22no—
69D4×C47—41no—
70C2×C4×C47—41no—
71C2×C4×C47—81no—
72C2×C4×C47—81no—
73C2×C4×C47C4×D411yescharacteristic
74C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
75C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
76C2×Dic47—41no—
77C2×Dic47—41no—
78Dic94—41no—
79Dic94—41no—
80C8×C47—21no—
81C8×C47—81no—
82C8×C47C2×M1611yescharacteristic
83C8×C47#13111yescharacteristic
84C47⋊C8—41no—
85C47⋊C8—41no—
86C47⋊C8—81no—
87#12077—471nomaximal, Sylow
88C22×C47—42no—
89C22×C47—81no—
90C22×C47—81no—
91C22×C47#44211yescharacteristic
92C4×C47—21no—
93C4×C47—21no—
94C4×C47—81no—
95C4×C47#43511yescharacteristic
96C4×C47#44311yescharacteristic
97Dic47—81no—
98Dic47—81no—
99#2175—471no—
100#1593—471no—
101#1580—471no—
102#1368—471no—
103#1334—471no—
104#1333D4711yescharacteristic
105#1313—471no—
106C2×C47—82no—
107C2×C47#158611yescharacteristic
108C2×C47#218011yescharacteristic
109C2×C47AGL(1,ℤ/16)11yescharacteristic
110C2×SD32—471no—
111#446—941no—
112C4×Q16—471no—
113C4×D8C2×D4711yescharacteristic
114C8×D4—941no—
115#366—471no—
116#358—471no—
117#357—471no—
118#346—471no—
119#346C2×D4711yescharacteristic
120#334—471no—
121#334C2×D4711yescharacteristic
122Q8⋊C8—941no—
123D4⋊C8—941no—
124#322—471no—
125C47#1207711yescharacteristic, complemented, Sylow
126C2×Q16—471no—
127C2×D8C4×D4711yescharacteristic
128C22×C8—1881no—
129C4×Q8—941no—
130C4×D4—21no—
131SD32—942no—
132C2×C16—21no—
133C2×C16—471no—
134C2×C16—471no—
135#107—471no—
136#107C4×D4711yescharacteristic
137C4⋊C8—21no—
138C4⋊C8—941no—
139C4⋊C8—1881no—
140#103—941no—
141#102—21no—
142C22⋊C8—1881no—
143#97—1881no—
144C4×C8—471no—
145C4×C8—471no—
146C4×C8C22×D4711yescharacteristic
147C2×Q8—941no—
148C2×D4—21no—
149C22×C4—41no—
150Q16—941no—
151Q16—941no—
152D8—21no—
153D8#1062532412yes—
154C2×C8—41no—
155C2×C8—941no—
156C2×C8—941no—
157C2×C8—3762no—
158C2×C8—1881no—
159C2×C8C2×C4×D4711yescharacteristic
160C2×C8#1062535211yescharacteristic
161C4⋊C4—21no—
162C4⋊C4—941no—
163C4⋊C4—1881no—
164C22⋊C4—41no—
165C42—1881no—
166C42D4×D4711yescharacteristic
167C16—22no—
168C16—941no—
169C16—941no—
170C23—41no—
171Q8—1881no—
172Q8—1881no—
173D4—22no—
174D4—41no—
175C2×C4—41no—
176C2×C4—81no—
177C2×C4—81no—
178C2×C4—1881no—
179C2×C4—1881no—
180C2×C4(C22⋊D47)⋊C411yescharacteristic
181C2×C4#1309404811yescharacteristic
182C2×C4#1309405111yescharacteristic
183C8—21no—
184C8—81no—
185C8—3761no—
186C8—1881no—
187C8—1881no—
188C8C2×#1062532411yescharacteristic
189C8#1309403211yescharacteristic
190C22—42no—
191C22—81no—
192C22—81no—
193C22#1568923211yescharacteristic
194C4—21no—
195C4—21no—
196C4—81no—
197C4—3761no—
198C4—3761no—
199C4#1568919111yescharacteristic
200C4#1568923711yescharacteristic
201C2—82no—
202C2#2024224911yescharacteristic
203C2#2024527611yescharacteristic
204C2#2024528911yescharacteristic
subgroupG/H
centerC22#15689232
commutatorC2×C8×C47C22×C4
FrattiniC4×C8C22×D47
FittingC47×#1333C2
socleC22×C47#442
prime radicalC2×D8×C47C2×C4
cosocleC4×C8×C47C23
hypercenter#1333D47
hypocenterC47#12077
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1333

Complements

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

NH ≅ G/Ntype
2. #20242296C2semidirect
4. #20241453C2semidirect
5. #20241359C2semidirect
6. #20241308C2semidirect
125. C47#12077semidirect

Maximal subgroups

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

subgroupindex
#202452522
#202422962
#202422122
#202414532
#202413592
#202413082
C47×#13332
#1207747