Finite Group Databasebeta

← #46809614

Subgroups — #46809614

Order 34432.

202 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 65 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#31771160C211yesmaximal, characteristic
2#31770661C211yesmaximal, characteristic
3#31770627C211yesmaximal, characteristic
4#31770353C211yesmaximal, characteristic, complemented
5#31770353C211yesmaximal, characteristic, complemented
6#31770351C211yesmaximal, characteristic, complemented
7#332×C269C211yesmaximal, characteristic, complemented
8C2×#16864385—21no—
9#22575453C2211yescharacteristic
10#22575444—21no—
11#22575433C412yes—
12#22575429—21no—
13Q8×Dic269—21no—
14#22575418C412yes—
15#22575411—21no—
16C42×D269C2211yescharacteristic
17C4⋊#14524130C2211yescharacteristic
18C4⋊#14524130C2211yescharacteristic
19#22575382—21no—
20#22575365—21no—
21#22575363—21no—
22C4⋊Q8×C269C2211yescharacteristic
23C4⋊C8×C269C2211yescharacteristic
24C4⋊C8×C269C2211yescharacteristic
25C2×Dic538C2×C412yes—
26C2×C4×D269C2×C411yescharacteristic
27C2×C4×D269D411yescharacteristic
28C2×C4×D269D411yescharacteristic
29#16864385—42no—
30C2×#14524130—21no—
31C2×#14524130—21no—
32C4⋊Dic269—22no—
33C4⋊Dic269—21no—
34C4⋊Dic269—21no—
35#16864381—22no—
36#16864379—22no—
37C4×Dic269—22no—
38C4×Dic269C2×C411yescharacteristic
39C4×Dic269D411yescharacteristic
40C4×Dic269D411yescharacteristic
41C2×Q8×C269—21no—
42C2×C8×C269—21no—
43C2×C8×C269—21no—
44C4⋊C4×C269—21no—
45C4⋊C4×C269C2×C412yes—
46C42×C269C2311yescharacteristic
47C22×D269C22⋊C411yescharacteristic
48Dic538—42no—
49C2×Dic269—22no—
50C2×Dic269C22⋊C411yescharacteristic
51C2×Dic269C22⋊C411yescharacteristic
52C2×Dic269C22⋊C411yescharacteristic
53C4×D269—21no—
54C4×D269—21no—
55C4×D269—21no—
56C4×D269—21no—
57C4×D269—41no—
58#14524130—41no—
59#14524130—41no—
60Q8×C269—41no—
61Q8×C269—41no—
62C2×C4×C269—22no—
63C2×C4×C269C22×C411yescharacteristic
64C2×C4×C269C2×D411yescharacteristic
65C2×C4×C269C2×D411yescharacteristic
66C8×C269—41no—
67C8×C269—41no—
68C22×C269C2×C22⋊C411yescharacteristic
69C2×D269—21no—
70C2×D269—21no—
71C2×D269—21no—
72C4×C269—21no—
73C4×C269—42no—
74C4×C269AGL(1,ℤ/8)11yescharacteristic
75C4×C269#13711yescharacteristic
76C4×C269#13711yescharacteristic
77C4×C269#13711yescharacteristic
78Dic269—21no—
79Dic269—21no—
80Dic269—21no—
81Dic269—42no—
82C2×C269#41311yescharacteristic
83C2×C269#41711yescharacteristic
84C2×C269#41911yescharacteristic
85D269—41no—
86C269#155911yescharacteristic, complemented, Sylow
87#1559—2691nomaximal, Sylow
88#533—2691no—
89#432—2691no—
90#423—2691no—
91#333—2691no—
92#333—2691no—
93#332—2691no—
94#332D26911yescharacteristic, complemented
95C2×M16—5381no—
96C4⋊Q8—2691no—
97C4⋊Q8C2×D26911yescharacteristic
98#125—5381no—
99C22⋊Q8—2692no—
100C22⋊Q8—5381no—
101C4×Q8—2692no—
102C4×Q8—5381no—
103#117—5381no—
104C2×C42—2691no—
105C4⋊C8—2691no—
106C4⋊C8—2691no—
107C4⋊C8—5381no—
108C4⋊C8C2×D26911yescharacteristic
109C4⋊C8C2×D26911yescharacteristic
110C22⋊C8—5381no—
111#97—5381no—
112C2×Q8—21no—
113C2×Q8—2692no—
114C22×C4—2691no—
115C22×C4—2691no—
116C22×C4—2691no—
117M16—10762no—
118C2×C8—21no—
119C2×C8—21no—
120C2×C8—5381no—
121C2×C8—5381no—
122C4⋊C4—21no—
123C4⋊C4—5382no—
124C4⋊C4—5382no—
125C4⋊C4—5381no—
126C4⋊C4—5381no—
127C4⋊C4C4×D26912yes—
128C22⋊C4—5382no—
129C42—2691no—
130C42—2691no—
131C42—2691no—
132C42—5382no—
133C42C22×D26911yescharacteristic
134C23—2691no—
135Q8—41no—
136Q8—41no—
137Q8—10762no—
138C2×C4—22no—
139C2×C4—2691no—
140C2×C4—2691no—
141C2×C4—2691no—
142C2×C4—5381no—
143C2×C4—5381no—
144C2×C4—5381no—
145C2×C4—5381no—
146C2×C4—5382no—
147C2×C4—10761no—
148C2×C4C2×C4×D26911yescharacteristic
149C2×C4D4×D26911yescharacteristic
150C2×C4D4×D26911yescharacteristic
151C8—41no—
152C8—41no—
153C8—10761no—
154C8—10761no—
155C22—5381no—
156C22—5381no—
157C22—5381no—
158C22C22⋊C4×D26911yescharacteristic
159C4—21no—
160C4—42no—
161C4—5381no—
162C4—5381no—
163C4—5381no—
164C4—10762no—
165C4#2257547711yescharacteristic
166C4#2257548211yescharacteristic
167C4#2257548211yescharacteristic
168C4#2257548511yescharacteristic
169C2—10761no—
170C2#3177058811yescharacteristic
171C2#3177060311yescharacteristic
172C2#3177061211yescharacteristic
subgroupG/H
centerC22C22⋊C4×D269
commutatorC2×C4×C269C22×C4
FrattiniC42C22×D269
Fitting#332×C269C2
socleC22×C269C2×C22⋊C4
prime radicalC22×D269C22⋊C4
cosocleC42×C269C23
hypercenter#332D269
hypocenterC269#1559
LCSG ▷ C2×C4×C269 ▷ C22×C269 ▷ C269
UCSC1 ◁ C22 ◁ C42 ◁ #332

Complements

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

NH ≅ G/Ntype
4. #31770353C2semidirect
5. #31770353C2semidirect
6. #31770351C2semidirect
7. #332×C269C2semidirect
86. C269#1559semidirect
94. #332D269semidirect

Maximal subgroups

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

subgroupindex
#317711602
#317706612
#317706272
#317703532
#317703512
#332×C2692
#1559269