Finite Group Databasebeta

← #26103705

Subgroups — #26103705

Order 12032.

228 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 43 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#20243597C211yesmaximal, characteristic
2#20241320C211yesmaximal, characteristic
3C47×#1270C211yesmaximal, characteristic
4#15689598C411yescharacteristic
5#15689163—21no—
6C2×#13093941C411yescharacteristic
7#15688883—21no—
8#15688849—21no—
9C47×#390C2211yescharacteristic
10C47×#336—21no—
11C2×#10625329—21no—
12D4×Dic47—41no—
13D4×Dic47—41no—
14#13093983—21no—
15C2×C22⋊Dic47—41no—
16#13093957—41no—
17#13093955—41no—
18#13093941—21no—
19#13093941C812yescomplemented
20C22⋊(C47⋊C8)—41no—
21C22×D4×C47C2×C411yescharacteristic
22C22×C8×C47—41no—
23C2×C4⋊C4×C47D411yescharacteristic
24C2×C22⋊C4×C47—41no—
25C2×C42×C47D411yescharacteristic
26C24×C47—21no—
27C2×D4×C47—21no—
28C2×D4×C47—41no—
29C2×D4×C47—41no—
30C2×D4×C47C2×C811yescharacteristic
31C2×D4×C47M1611yescharacteristic
32C22×C4×C47—21no—
33C22×C4×C47—21no—
34C22×C4×C47C22⋊C411yescharacteristic
35C22×Dic47—41no—
36C22×Dic47—41no—
37C2×C8×C47—42no—
38C2×C8×C47—81no—
39C2×C8×C47—81no—
40C4⋊C4×C47—41no—
41C22⋊C4×C47—81no—
42C22⋊C4×C47—81no—
43C42×C47—41no—
44C22⋊Dic47—42no—
45C22⋊Dic47—41no—
46C4⋊Dic47—21no—
47C4⋊Dic47—21no—
48C4×Dic47—41no—
49#10625329—22no—
50#10625329—41no—
51C2×C47⋊C8—41no—
52C2×C47⋊C8—81no—
53C23×C47—21no—
54C23×C47—21no—
55C23×C47—41no—
56C23×C47—41no—
57C23×C47—81no—
58C23×C47—81no—
59C23×C47#9911yescharacteristic
60D4×C47—41no—
61D4×C47—41no—
62D4×C47—42no—
63C2×C4×C47—21no—
64C2×C4×C47—21no—
65C2×C4×C47—21no—
66C2×C4×C47—21no—
67C2×C4×C47—41no—
68C2×C4×C47—41no—
69C2×C4×C47—41no—
70C2×C4×C47—81no—
71C2×C4×C47C22⋊C811yescharacteristic
72C2×C4×C47#10111yescharacteristic
73C2×Dic47—41no—
74C2×Dic47—41no—
75C2×Dic47—161no—
76C2×Dic47—161no—
77C8×C47—82no—
78C47⋊C8—42no—
79C47⋊C8—161no—
80#7375—471nomaximal, Sylow
81C22×C47—21no—
82C22×C47—21no—
83C22×C47—41no—
84C22×C47—41no—
85C22×C47—81no—
86C22×C47—81no—
87C22×C47—81no—
88C22×C47—81no—
89C22×C47—161no—
90C22×C47#32411yescharacteristic
91C22×C47AΣL(1,𝔽16)11yescharacteristic
92C22×C47#35211yescharacteristic
93C4×C47—21no—
94C4×C47—21no—
95C4×C47—41no—
96C4×C47—41no—
97C4×C47—81no—
98Dic47—81no—
99Dic47—81no—
100#1856—471no—
101#1318—471no—
102#1270D4711yescharacteristic
103C2×C47—21no—
104C2×C47—21no—
105C2×C47—81no—
106C2×C47—81no—
107C2×C47#132611yescharacteristic
108C2×C47#140211yescharacteristic
109C2×C47#140311yescharacteristic
110#518—471no—
111#428—941no—
112C2×#100—471no—
113#390C2×D4711yescharacteristic
114#339—941no—
115#336—21no—
116#324—941no—
117C47#737511yescharacteristic, complemented, Sylow
118C22×D4C4×D4711yescharacteristic
119C2×M16—941no—
120C22×C8—41no—
121C4×D4—1881no—
122C4×D4—1881no—
123#117—941no—
124C2×C4⋊C4D18811yescharacteristic
125C2×C22⋊C4—41no—
126C2×C22⋊C4—1881no—
127C2×C42C22⋊D4711yescharacteristic
128#107—1881no—
129#106—1881no—
130#100—472no—
131#100—941no—
132C22⋊C8—1881no—
133C24—21no—
134C2×D4—21no—
135C2×D4—41no—
136C2×D4—41no—
137C2×D4C8×D4711yescharacteristic
138C2×D4#1062532411yescharacteristic
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—1881no—
142C22×C4—1881no—
143C22×C4(C2×D47)⋊C411yescharacteristic
144M16—942no—
145M16—1881no—
146C2×C8—42no—
147C2×C8—81no—
148C2×C8—81no—
149C2×C8—3761no—
150C2×C8—1881no—
151C4⋊C4—41no—
152C4⋊C4—941no—
153C4⋊C4—941no—
154C22⋊C4—81no—
155C22⋊C4—81no—
156C22⋊C4—1882no—
157C22⋊C4—1881no—
158C42—41no—
159C42—1881no—
160C23—21no—
161C23—21no—
162C23—41no—
163C23—41no—
164C23—81no—
165C23—81no—
166C23#1309393811yescharacteristic
167D4—41no—
168D4—41no—
169D4—42no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—21no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—81no—
178C2×C4—1881no—
179C2×C4—1881no—
180C2×C4—7521no—
181C2×C4—7521no—
182C2×C4(C2×D47)⋊C811yescharacteristic
183C2×C4#1309394211yescharacteristic
184C8—82no—
185C8—1882no—
186C8—7521no—
187C22—21no—
188C22—21no—
189C22—41no—
190C22—41no—
191C22—81no—
192C22—81no—
193C22—81no—
194C22—81no—
195C22—161no—
196C22#1568884811yescharacteristic
197C22#1568891111yescharacteristic
198C22#1568891211yescharacteristic
199C4—21no—
200C4—21no—
201C4—41no—
202C4—41no—
203C4—81no—
204C4—3761no—
205C4—3761no—
206C2—21no—
207C2—21no—
208C2—81no—
209C2—81no—
210C2#2024133711yescharacteristic
211C2#2024153411yescharacteristic
212C2#2024153711yescharacteristic
subgroupG/H
centerC22#15688911
commutatorC2×D4×C47C2×C8
Frattini#390C2×D47
FittingC47×#1270C2
socleC22×C47AΣL(1,𝔽16)
prime radicalC22×D4×C47C2×C4
cosocleC47×#390C22
hypercenter#1270D47
hypocenterC47#7375
LCSG ▷ C2×D4×C47 ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #390 ◁ #1270

Complements

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

NH ≅ G/Ntype
19. #13093941C8semidirect
117. C47#7375semidirect

Maximal subgroups

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

subgroupindex
#202435972
#202413202
C47×#12702
#737547