Finite Group Databasebeta

← #46811303

Subgroups — #46811303

Order 34432.

342 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 89 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#31771112C211yesmaximal, characteristic
2#31770589C211yesmaximal, characteristic, complemented
3#31770568C212yesmaximal, complemented
4#31770376C212yesmaximal, complemented
5#568×C269C211yesmaximal, characteristic, complemented
6C22×C4×D269C2211yescharacteristic
7C2×#16864386C2212yes—
8#22575434C412yes—
9#22575429—21no—
10#22575424—21no—
11#22575421C412yes—
12C4×C22⋊D269—22no—
13#22575412C2211yescharacteristic
14#22575372—21no—
15#22575370—21no—
16#22575365—24no—
17C2×#41×C269C2211yescharacteristic
18#131×C269—24no—
19C2×M16×C269—22no—
20C2×M16×C269C2212yes—
21C23×D269C2×C411yescharacteristic
22C2×C22⋊D269—22no—
23C22×Dic269C2×C411yescharacteristic
24C2×C4×D269—22no—
25C2×C4×D269—21no—
26C2×C4×D269—21no—
27C2×C4×D269—22no—
28#16864386—22no—
29#16864386—22no—
30C2×#14524130—22no—
31C4⋊Dic269—21no—
32#16864381—22no—
33C22⋊Dic269—22no—
34C22⋊Dic269C2×C412yes—
35#16864379—22no—
36C4×Dic269—21no—
37#41×C269—22no—
38#41×C269—22no—
39C2×Q8×C269D411yescharacteristic
40C2×D4×C269C2×C412yes—
41C2×D4×C269D411yescharacteristic
42C22×C4×C269D412yes—
43C22×C4×C269C2311yescharacteristic
44M16×C269—24no—
45M16×C269—22no—
46M16×C269—22no—
47C2×C8×C269—24no—
48C2×C8×C269D414yescomplemented
49C22×D269—21no—
50C22×D269—22no—
51C22×D269—42no—
52C22⋊D269—44no—
53C2×Dic269—22no—
54C2×Dic269—21no—
55C2×Dic269—22no—
56C4×D269—41no—
57C4×D269—42no—
58C4×D269—41no—
59#14524130—42no—
60C23×C269C22⋊C412yes—
61C23×C269C22×C411yescharacteristic
62Q8×C269—22no—
63D4×C269—24no—
64D4×C269—22no—
65C2×C4×C269—22no—
66C2×C4×C269—22no—
67C2×C4×C269C22⋊C412yes—
68C2×C4×C269C2×D412yes—
69C2×C4×C269C2×D411yescharacteristic
70C2×C4×C269C2×D411yescharacteristic
71C2×C4×C269#4112yes—
72C8×C269—24no—
73C22×C269—22no—
74C22×C269—41no—
75C22×C269—42no—
76C22×C269C2×C22⋊C411yescharacteristic
77C22×C269C4×D412yes—
78C2×D269—22no—
79C2×D269—24no—
80C2×D269—41no—
81C2×D269—42no—
82C2×D269—41no—
83C4×C269—22no—
84C4×C269#12011yescharacteristic
85C4×C269#12112yes—
86C4×C269#12311yescharacteristic
87Dic269—42no—
88Dic269—41no—
89C2×C269—21no—
90C2×C269—22no—
91C2×C269—42no—
92C2×C269#38611yescharacteristic
93D269—42no—
94C269#189111yescharacteristic, complemented, Sylow
95#1891—2691nomaximal, Sylow
96#568D26911yescharacteristic, complemented
97#525—2691no—
98#413—2691no—
99#407—2692no—
100#344—2692no—
101C2×#41C2×D26911yescharacteristic
102C23×C4—2691no—
103#131—24no—
104C2×M16—22no—
105C2×M16—2692no—
106C2×M16C2×D26912yes—
107#123—2692no—
108C22⋊Q8—5381no—
109#121—5381no—
110#120—2692no—
111C4×D4—5382no—
112#117—2691no—
113#101—5381no—
114#100—5381no—
115C22⋊C8—5384no—
116C24—2691no—
117#41—22no—
118#41—22no—
119C2×Q8D107611yescharacteristic
120C2×D4—5382no—
121C2×D4C4×D26912yes—
122C2×D4D107611yescharacteristic
123C22×C4—2691no—
124C22×C4—5382no—
125C22×C4—5381no—
126C22×C4—5381no—
127C22×C4—5382no—
128C22×C4C22⋊D26912yes—
129C22×C4C22×D26911yescharacteristic
130M16—24no—
131M16—22no—
132M16—22no—
133M16—5382no—
134M16—5382no—
135C2×C8—24no—
136C2×C8—5382no—
137C2×C8C22⋊D26914yescomplemented
138C4⋊C4—5382no—
139C4⋊C4—5381no—
140C22⋊C4—2692no—
141C22⋊C4—5382no—
142C22⋊C4—5382no—
143C42—5381no—
144C23—5381no—
145C23—5382no—
146C23—10762no—
147C23#1686437912yes—
148C23C2×C4×D26911yescharacteristic
149Q8—22no—
150D4—24no—
151D4—22no—
152D4—10764no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—5382no—
156C2×C4—5381no—
157C2×C4—5382no—
158C2×C4—10761no—
159C2×C4—10762no—
160C2×C4—10761no—
161C2×C4#1686437912yes—
162C2×C4C2×C22⋊D26912yes—
163C2×C4C2×C22⋊D26911yescharacteristic
164C2×C4C2×D107611yescharacteristic
165C2×C4#1686440212yes—
166C8—24no—
167C8—10762no—
168C22—22no—
169C22—41no—
170C22—42no—
171C22—5382no—
172C22—5384no—
173C22—10761no—
174C22—10762no—
175C22—10761no—
176C22C2×#1686437911yescharacteristic
177C22C4×C22⋊D26912yes—
178C4—22no—
179C4—10762no—
180C4—10761no—
181C4#2257542211yescharacteristic
182C4#2257542712yes—
183C4#2257543611yescharacteristic
184C2—21no—
185C2—22no—
186C2—42no—
187C2—10762no—
188C2#3177048411yescharacteristic
subgroupG/H
centerC4#22575422
commutatorC23×C269C22×C4
FrattiniC22×C4C22×D269
Fitting#568×C269C2
socleC2×C269#386
prime radical#31771112C2
cosocleC22×C4×C269C23
hypercenter#568D269
hypocenterC269#1891
LCSG ▷ C23×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C4 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #31770589C2semidirect
3. #31770568C2semidirect
4. #31770376C2semidirect
5. #568×C269C2semidirect
48. C2×C8×C269D4semidirect
94. C269#1891semidirect
96. #568D269semidirect
137. C2×C8C22⋊D269semidirect

Maximal subgroups

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

subgroupindex
#317711122
#317705892
#317705682
#317703762
#568×C2692
#1891269