Finite Group Databasebeta

← #26128292

Subgroups — #26128292

Order 12032.

212 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 77 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#20242302C211yesmaximal, characteristic
2#20242297C211yesmaximal, characteristic, complemented
3#20241454C211yesmaximal, characteristic, complemented
4#20241358C211yesmaximal, characteristic, complemented
5#20241355C211yesmaximal, characteristic
6#20241308C211yesmaximal, characteristic, complemented
7C47×#2175C211yesmaximal, characteristic
8#15689254—21no—
9#15689201—21no—
10#15688899C2211yescharacteristic
11#15688874C2211yescharacteristic
12#15688874C2211yescharacteristic
13Q8⋊(C47⋊C8)—21no—
14D4⋊(C47⋊C8)—21no—
15#15688844C2211yescharacteristic
16C2×SD32×C47C411yescharacteristic
17C4×Q16×C47C2211yescharacteristic
18C4×D8×C47C2211yescharacteristic
19C47×#366C411yescharacteristic
20C47×#358C411yescharacteristic
21C47×#357C411yescharacteristic
22C47×#346C2211yescharacteristic
23C2×#10625329—41no—
24C2×C47⋊C16—21no—
25C4⋊(C47⋊C8)—21no—
26C4⋊(C47⋊C8)—21no—
27C4⋊(C47⋊C8)—41no—
28C22⋊(C47⋊C8)—41no—
29#13093935—41no—
30C4×C47⋊C8D411yescharacteristic
31C4×C47⋊C8D411yescharacteristic
32C2×Q16×C47C2×C411yescharacteristic
33C2×D8×C47C2×C411yescharacteristic
34C4×Q8×C47—21no—
35C4×D4×C47—21no—
36SD32×C47—22no—
37C2×C16×C47C2×C411yescharacteristic
38C2×C16×C47C2×C411yescharacteristic
39#107×C47C2×C411yescharacteristic
40#107×C47C2×C411yescharacteristic
41#103×C47—21no—
42#102×C47—21no—
43C4×C8×C47C2311yescharacteristic
44C2×Q8×C47—21no—
45C2×D4×C47—21no—
46C22×C4×C47—41no—
47Q16×C47—21no—
48Q16×C47—21no—
49D8×C47—21no—
50D8×C47—21no—
51C2×C8×C47C22×C411yescharacteristic
52C2×C8×C47#4111yescharacteristic
53C4⋊C4×C47—21no—
54C4⋊C4×C47—21no—
55C4⋊C4×C47—41no—
56C22⋊C4×C47—41no—
57C42×C47—41no—
58C42×C47C2×D411yescharacteristic
59#10625329—82no—
60C2×C47⋊C8—21no—
61C2×C47⋊C8—21no—
62C2×C47⋊C8—41no—
63C2×C47⋊C8—41no—
64C16×C47—21no—
65C16×C47M1612yescomplemented
66C47⋊C16—41no—
67C23×C47—41no—
68Q8×C47—41no—
69Q8×C47—41no—
70D4×C47—41no—
71D4×C47—41no—
72C2×C4×C47—41no—
73C2×C4×C47—41no—
74C2×C4×C47—41no—
75C2×C4×C47—81no—
76C2×C4×C47—81no—
77C2×C4×C47C4×D411yescharacteristic
78C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
79C2×C4×C47AGL(1,ℤ/8)11yescharacteristic
80C8×C47—21no—
81C8×C47C2×M1611yescharacteristic
82C8×C47#13111yescharacteristic
83C47⋊C8—41no—
84C47⋊C8—41no—
85C47⋊C8—81no—
86C47⋊C8—81no—
87#12087—471nomaximal, Sylow
88C22×C47—81no—
89C22×C47—81no—
90C22×C47—81no—
91C22×C47#44211yescharacteristic
92C4×C47—21no—
93C4×C47—21no—
94C4×C47—81no—
95C4×C47—81no—
96C4×C47—81no—
97C4×C47#43611yescharacteristic
98C4×C47#44411yescharacteristic
99#2175D4711yescharacteristic
100#1594—471no—
101#1593—471no—
102#1368—471no—
103#1334—471no—
104#1333—471no—
105#1313—471no—
106C2×C47—161no—
107C2×C47#159411yescharacteristic
108C2×C47#218011yescharacteristic
109C2×C47AGL(1,ℤ/16)11yescharacteristic
110C2×SD32Dic4711yescharacteristic
111#446—941no—
112C4×Q16C2×D4711yescharacteristic
113C4×D8C2×D4711yescharacteristic
114#436—941no—
115#366Dic4711yescharacteristic
116#358Dic4711yescharacteristic
117#357Dic4711yescharacteristic
118#346—471no—
119#346C2×D4711yescharacteristic
120#334—471no—
121#334—471no—
122Q8⋊C8—941no—
123D4⋊C8—941no—
124#322—471no—
125C47#1208711yescharacteristic, complemented, Sylow
126C2×Q16C2×Dic4711yescharacteristic
127C2×D8C2×Dic4711yescharacteristic
128C2×M16—1881no—
129C4×Q8—21no—
130C4×D4—21no—
131SD32—22no—
132C2×C16—941no—
133C2×C16C2×Dic4711yescharacteristic
134C2×C16C2×Dic4711yescharacteristic
135#107C2×Dic4711yescharacteristic
136#107C2×Dic4711yescharacteristic
137C4⋊C8—941no—
138C4⋊C8—941no—
139C4⋊C8—1881no—
140#103—21no—
141#102—21no—
142C22⋊C8—1881no—
143#97—1881no—
144C4×C8—471no—
145C4×C8—471no—
146C4×C8C22×D4711yescharacteristic
147C2×Q8—21no—
148C2×D4—21no—
149C22×C4—41no—
150Q16—21no—
151Q16—21no—
152D8—21no—
153D8—21no—
154M16—3762no—
155C2×C8—941no—
156C2×C8—941no—
157C2×C8—1881no—
158C2×C8—1881no—
159C2×C8#1062535211yescharacteristic
160C2×C8C22×Dic4711yescharacteristic
161C4⋊C4—21no—
162C4⋊C4—21no—
163C4⋊C4—41no—
164C22⋊C4—41no—
165C42—41no—
166C42D4×D4711yescharacteristic
167C16—21no—
168C16—1881no—
169C16#1062532912yescomplemented
170C23—41no—
171Q8—41no—
172Q8—41no—
173D4—41no—
174D4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—41no—
178C2×C4—81no—
179C2×C4—81no—
180C2×C4D4×Dic4711yescharacteristic
181C2×C4#1309404811yescharacteristic
182C2×C4#1309404811yescharacteristic
183C8—21no—
184C8—3761no—
185C8—3761no—
186C8—1881no—
187C8—1881no—
188C8C2×#1062532911yescharacteristic
189C8#1309403111yescharacteristic
190C22—81no—
191C22—81no—
192C22—81no—
193C22#1568923511yescharacteristic
194C4—21no—
195C4—21no—
196C4—81no—
197C4—81no—
198C4—81no—
199C4#1568920111yescharacteristic
200C4#1568924511yescharacteristic
201C2—161no—
202C2#2024230211yescharacteristic
203C2#2024528011yescharacteristic
204C2#2024528711yescharacteristic
subgroupG/H
centerC22#15689235
commutatorC2×C8×C47C22×C4
FrattiniC4×C8C22×D47
FittingC47×#2175C2
socleC22×C47#442
prime radicalC2×D8×C47C2×C4
cosocleC4×C8×C47C23
hypercenter#2175D47
hypocenterC47#12087
LCSG ▷ C2×C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2175

Complements

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

NH ≅ G/Ntype
2. #20242297C2semidirect
3. #20241454C2semidirect
4. #20241358C2semidirect
6. #20241308C2semidirect
65. C16×C47M16semidirect
125. C47#12087semidirect
169. C16#10625329semidirect

Maximal subgroups

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

subgroupindex
#202423022
#202422972
#202414542
#202413582
#202413552
#202413082
C47×#21752
#1208747