Finite Group Databasebeta

← #46809883

Subgroups — #46809883

Order 34432.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#31771200C211yesmaximal, characteristic
2#31770936C211yesmaximal, characteristic, complemented
3#31770916C211yesmaximal, characteristic
4#31770537C211yesmaximal, characteristic, complemented
5#31770341C211yesmaximal, characteristic, complemented
6#31770330C211yesmaximal, characteristic, complemented
7#327×C269C211yesmaximal, characteristic
8#22575446—21no—
9#22575443—21no—
10#22575434—21no—
11#22575434—21no—
12#22575432C2211yescharacteristic
13#22575430—21no—
14D4×Dic269—21no—
15C2×C4⋊Dic269—21no—
16C2×C4⋊Dic269C2211yescharacteristic
17C2×#16864381C2211yescharacteristic
18#22575386—21no—
19#22575385—21no—
20#22575378—21no—
21#22575374—21no—
22C22⋊#14524130C2211yescharacteristic
23#22575360—21no—
24#121×C269C2211yescharacteristic
25C22⋊C8×C269C2211yescharacteristic
26#95×C269C2211yescharacteristic
27C2×Dic538D411yescharacteristic
28C22×Dic269—21no—
29C22×Dic269—21no—
30C22×Dic269D411yescharacteristic
31C22×Dic269D411yescharacteristic
32C2×#14524130—21no—
33C4⋊Dic269—21no—
34C4⋊Dic269—21no—
35C4⋊Dic269—21no—
36C4⋊Dic269—42no—
37C4⋊Dic269D411yescharacteristic
38#16864381—21no—
39#16864381—21no—
40#16864381—21no—
41#16864381—21no—
42#16864381—21no—
43C22⋊Dic269—21no—
44C22⋊Dic269—21no—
45C22⋊Dic269—21no—
46C22⋊Dic269—21no—
47C4×Dic269—21no—
48C2×D4×C269—21no—
49C2×D4×C269D411yescharacteristic
50C22×C4×C269—21no—
51C22×C4×C269C2311yescharacteristic
52C2×C8×C269—21no—
53C4⋊C4×C269D411yescharacteristic
54C22⋊C4×C269—21no—
55Dic538—41no—
56C2×Dic269—21no—
57C2×Dic269—21no—
58C2×Dic269—21no—
59C2×Dic269—21no—
60C2×Dic269—21no—
61C2×Dic269—21no—
62C2×Dic269—41no—
63C2×Dic269—41no—
64C2×Dic269—41no—
65C2×Dic269—41no—
66C2×Dic269—81no—
67#14524130—41no—
68C23×C269—21no—
69C23×C269C2×D411yescharacteristic
70D4×C269—41no—
71D4×C269—41no—
72D4×C269—41no—
73C2×C4×C269—22no—
74C2×C4×C269—21no—
75C2×C4×C269—41no—
76C2×C4×C269—41no—
77C2×C4×C269—41no—
78C2×C4×C269C2×D411yescharacteristic
79C2×C4×C269C2×D411yescharacteristic
80C8×C269—41no—
81C22×C269—21no—
82C22×C269—21no—
83C22×C269—41no—
84C22×C269—41no—
85C22×C269—41no—
86C22×C269#12011yescharacteristic
87C22×C269#13511yescharacteristic
88C22×C269AGL(1,ℤ/8)11yescharacteristic
89C4×C269—21no—
90C4×C269—21no—
91C4×C269—42no—
92C4×C269—41no—
93Dic269—41no—
94Dic269—41no—
95Dic269—41no—
96Dic269—41no—
97Dic269—81no—
98C2×C269—21no—
99C2×C269—21no—
100C2×C269—81no—
101C2×C269#44911yescharacteristic
102C2×C269#45611yescharacteristic
103C2×C269#45811yescharacteristic
104C269#160911yescharacteristic, complemented, Sylow
105#1609—2691nomaximal, Sylow
106#539—2691no—
107#483—2691no—
108#480—2691no—
109#398—2691no—
110#328—2691no—
111#327D26911yescharacteristic
112#324—2691no—
113#126—5381no—
114#125—5381no—
115#123—5381no—
116#123—5381no—
117C22⋊Q8—2691no—
118C22⋊Q8—5381no—
119#121C2×D26911yescharacteristic
120C4×D4—5381no—
121C2×C4⋊C4—2691no—
122C2×C4⋊C4—2691no—
123C2×C4⋊C4—5381no—
124#107—5381no—
125#106—5381no—
126#103—5381no—
127#102—5381no—
128C22⋊C8—2691no—
129C22⋊C8C2×D26911yescharacteristic
130#95—5381no—
131#95C2×D26911yescharacteristic
132C2×Q8—2691no—
133C2×D4—21no—
134C2×D4D107611yescharacteristic
135C22×C4—21no—
136C22×C4—2691no—
137C22×C4—2691no—
138C22×C4—5381no—
139C22×C4—5381no—
140C22×C4C22×D26911yescharacteristic
141C2×C8—21no—
142C2×C8—5381no—
143C4⋊C4—2691no—
144C4⋊C4—5381no—
145C4⋊C4—5381no—
146C4⋊C4—5381no—
147C4⋊C4—5381no—
148C4⋊C4—5381no—
149C4⋊C4—5381no—
150C4⋊C4—5381no—
151C4⋊C4—5381no—
152C4⋊C4—10762no—
153C4⋊C4D107611yescharacteristic
154C22⋊C4—21no—
155C22⋊C4—5381no—
156C22⋊C4—5381no—
157C22⋊C4—5381no—
158C22⋊C4—5381no—
159C42—5381no—
160C23—21no—
161C23D4×D26911yescharacteristic
162Q8—10761no—
163D4—41no—
164D4—41no—
165D4—41no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—5381no—
172C2×C4—5381no—
173C2×C4—5381no—
174C2×C4—5381no—
175C2×C4—5381no—
176C2×C4—5381no—
177C2×C4—10761no—
178C2×C4—10761no—
179C2×C4—10761no—
180C2×C4—10761no—
181C2×C4—21521no—
182C2×C4D4×D26911yescharacteristic
183C2×C4C2×D107611yescharacteristic
184C8—41no—
185C8—10761no—
186C22—21no—
187C22—21no—
188C22—41no—
189C22—41no—
190C22—41no—
191C22#2257542311yescharacteristic
192C22#2257547411yescharacteristic
193C22#2257547711yescharacteristic
194C4—21no—
195C4—21no—
196C4—42no—
197C4—41no—
198C4—10761no—
199C4—10761no—
200C4—10761no—
201C4—10761no—
202C4—21521no—
203C2—21no—
204C2—21no—
205C2—81no—
206C2#3177075111yescharacteristic
207C2#3177078911yescharacteristic
208C2#3177079611yescharacteristic
subgroupG/H
centerC22#22575423
commutator, cosocleC22×C4×C269C23
FrattiniC22×C4C22×D269
Fitting#327×C269C2
socleC22×C269#120
prime radical#121×C269C22
hypercenter#327D269
hypocenterC269#1609
LCSG ▷ C22×C4×C269 ▷ C22×C269 ▷ C269
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #31770936C2semidirect
4. #31770537C2semidirect
5. #31770341C2semidirect
6. #31770330C2semidirect
104. C269#1609semidirect

Maximal subgroups

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

subgroupindex
#317712002
#317709362
#317709162
#317705372
#317703412
#317703302
#327×C2692
#1609269