Finite Group Databasebeta

← #26128151

Subgroups — #26128151

Order 12032.

230 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 79 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#20242208C211yesmaximal, characteristic, complemented
3#20241453C211yesmaximal, characteristic, complemented
4#20241357C211yesmaximal, characteristic, complemented
5#20241355C211yesmaximal, characteristic
6#20241309C211yesmaximal, characteristic, complemented
7C47×#1580C211yesmaximal, characteristic
8C2×#13093968C411yescharacteristic
9C8×Dic94—21no—
10C4×Dic188C2211yescharacteristic
11#15688948C411yescharacteristic
12#15688926C411yescharacteristic
13#15688924C411yescharacteristic
14#15688899C2211yescharacteristic
15#15688899C2211yescharacteristic
16#15688874C2211yescharacteristic
17#15688855—21no—
18C4×D8×C47C2211yescharacteristic
19C8×D4×C47—21no—
20C47×#334C2211yescharacteristic
21C47×D4⋊C8—21no—
22C47×#322C2211yescharacteristic
23C2×Dic188C2×C411yescharacteristic
24C4×Dic94—21no—
25#13093968C814yescomplemented
26C2×C47⋊C16—21no—
27C2×C47⋊C16C2×C411yescharacteristic
28C2×C47⋊C16C2×C411yescharacteristic
29#13093958C2×C411yescharacteristic
30C4⋊(C47⋊C8)—21no—
31Dic47⋊C8—41no—
32#13093949—21no—
33C8×Dic47—41no—
34C2×D8×C47C2×C411yescharacteristic
35C22×C8×C47—41no—
36C4×D4×C47—21no—
37#107×C47C2×C411yescharacteristic
38C4⋊C8×C47—21no—
39#102×C47—21no—
40C22⋊C8×C47—41no—
41C4×C8×C47D411yescharacteristic
42C4×C8×C47D411yescharacteristic
43C4×C8×C47C2311yescharacteristic
44C2×D4×C47—21no—
45C22×C4×C47—41no—
46C2×Dic94—21no—
47D8×C47—21no—
48D8×C47C2×C812yes—
49C2×C8×C47—21no—
50C2×C8×C47—21no—
51C2×C8×C47—41no—
52C2×C8×C47—82no—
53C2×C8×C47C22×C411yescharacteristic
54C2×C8×C47#4111yescharacteristic
55C4⋊C4×C47—21no—
56C22⋊C4×C47—41no—
57C42×C47C2×D411yescharacteristic
58C4⋊Dic47—21no—
59Dic47⋊C4—41no—
60C4×Dic47—41no—
61C2×C47⋊C8—41no—
62Dic188—21no—
63Dic188C2×C812yescomplemented
64C47⋊C16—21no—
65C47⋊C16—41no—
66C47⋊C16C2×C812yescomplemented
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—22no—
79Dic94—41no—
80C8×C47—21no—
81C8×C47—41no—
82C8×C47—41no—
83C8×C47—81no—
84C8×C47C22×C811yescharacteristic
85C8×C47#13111yescharacteristic
86C47⋊C8—81no—
87#12065—471nomaximal, Sylow
88C22×C47—42no—
89C22×C47—81no—
90C22×C47—81no—
91C22×C47#44211yescharacteristic
92C4×C47—21no—
93C4×C47—21no—
94C4×C47—81no—
95C4×C47C8×D411yescharacteristic
96C4×C47#44411yescharacteristic
97Dic47—42no—
98Dic47—81no—
99#2175—471no—
100#1580D4711yescharacteristic
101#1579—471no—
102#1368—471no—
103#1334—471no—
104#1333—471no—
105#1313—471no—
106C2×C47—82no—
107C2×C47#158011yescharacteristic
108C2×C47#218011yescharacteristic
109C2×C47AGL(1,ℤ/16)11yescharacteristic
110C2×SD32—471no—
111C8×Q8—941no—
112C4×Q16—471no—
113C4×D8C2×D4711yescharacteristic
114C8×D4—21no—
115#366—471no—
116#358—471no—
117#357—471no—
118#346—471no—
119#346—471no—
120#334—471no—
121#334C2×D4711yescharacteristic
122Q8⋊C8—941no—
123D4⋊C8—21no—
124#322C2×D4711yescharacteristic
125C47#1206511yescharacteristic, complemented, Sylow
126C2×Q16—471no—
127C2×D8C4×D4711yescharacteristic
128C22×C8—41no—
129C4×Q8—941no—
130C4×D4—21no—
131SD32—474no—
132C2×C16—471no—
133C2×C16—471no—
134C2×C16—941no—
135#107—471no—
136#107C4×D4711yescharacteristic
137C4⋊C8—21no—
138C4⋊C8—941no—
139C4⋊C8—1881no—
140#103—941no—
141#102—21no—
142C22⋊C8—41no—
143C4×C8—1881no—
144C4×C8C22⋊D4711yescharacteristic
145C4×C8C22⋊D4711yescharacteristic
146C4×C8C22×D4711yescharacteristic
147C2×Q8—941no—
148C2×D4—21no—
149C22×C4—41no—
150Q16—472no—
151Q16—941no—
152D8—21no—
153D8C8×D4712yes—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—41no—
157C2×C8—82no—
158C2×C8—1881no—
159C2×C8C2×C4×D4711yescharacteristic
160C2×C8#1062535011yescharacteristic
161C4⋊C4—21no—
162C4⋊C4—941no—
163C4⋊C4—1881no—
164C22⋊C4—41no—
165C42—1881no—
166C42C2×C22⋊D4711yescharacteristic
167C16—472no—
168C16—941no—
169C16—1881no—
170C23—41no—
171Q8—942no—
172Q8—1881no—
173D4—22no—
174D4—41no—
175C2×C4—41no—
176C2×C4—81no—
177C2×C4—81no—
178C2×C4—1881no—
179C2×C4—1881no—
180C2×C4C4×C22⋊D4711yescharacteristic
181C2×C4#1309404911yescharacteristic
182C2×C4#1309405011yescharacteristic
183C8—21no—
184C8—41no—
185C8—41no—
186C8—81no—
187C8—3761no—
188C8C2×C8×D4711yescharacteristic
189C8#1309403011yescharacteristic
190C22—42no—
191C22—81no—
192C22—81no—
193C22#1568923311yescharacteristic
194C4—21no—
195C4—21no—
196C4—81no—
197C4—3761no—
198C4—1882no—
199C4C8×C22⋊D4711yescharacteristic
200C4#1568924311yescharacteristic
201C2—82no—
202C2#2024221011yescharacteristic
203C2#2024527911yescharacteristic
204C2#2024528411yescharacteristic
subgroupG/H
centerC22#15689233
commutatorC8×C47C22×C8
FrattiniC4×C8C22×D47
FittingC47×#1580C2
socleC22×C47#442
prime radicalC2×D8×C47C2×C4
cosocleC4×C8×C47C23
hypercenter#1580D47
hypocenterC47#12065
LCSG ▷ C8×C47 ▷ C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C42 ◁ #322 ◁ #1580

Complements

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

NH ≅ G/Ntype
2. #20242208C2semidirect
3. #20241453C2semidirect
4. #20241357C2semidirect
6. #20241309C2semidirect
25. #13093968C8semidirect
63. Dic188C2×C8semidirect
66. C47⋊C16C2×C8semidirect
125. C47#12065semidirect

Maximal subgroups

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

subgroupindex
#202452522
#202422082
#202414532
#202413572
#202413552
#202413092
C47×#15802
#1206547