Finite Group Databasebeta

← #46812579

Subgroups — #46812579

Order 34432.

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

normal subgroups only

This group has 230 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#31771176C211yesmaximal, characteristic
2#31770589C211yesmaximal, characteristic
3#31770403C214yesmaximal, complemented
4#413×C269C211yesmaximal, characteristic
5C22×Dic538C411yescharacteristic
6C2×#16864386—21no—
7#22575451C2212yes—
8#22575440C412yes—
9#22575434—21no—
10#22575434—21no—
11#22575432—21no—
12#22575432—21no—
13#22575412C411yescharacteristic
14#22575372C2212yes—
15#22575370—21no—
16C2×#41×C269C2211yescharacteristic
17C2×M16×C269—21no—
18#101×C269C2212yes—
19#100×C269—21no—
20C2×Dic538—21no—
21C2×Dic538—41no—
22C2×Dic538—41no—
23C2×Dic538C2×C412yes—
24C22×Dic269—21no—
25#16864386—22no—
26#16864386—41no—
27C2×#14524130—41no—
28C4⋊Dic269—21no—
29#16864381—22no—
30#16864381—22no—
31C22⋊Dic269—21no—
32C22⋊Dic269—22no—
33C22⋊Dic269—22no—
34C4×Dic269C2×C412yes—
35#41×C269—41no—
36#41×C269—41no—
37C2×Q8×C269C2311yescharacteristic
38C2×D4×C269C2×C411yescharacteristic
39C2×D4×C269D411yescharacteristic
40C2×D4×C269D411yescharacteristic
41C22×C4×C269C2×C411yescharacteristic
42C22×C4×C269D411yescharacteristic
43C22×C4×C269D411yescharacteristic
44M16×C269—22no—
45M16×C269—41no—
46C2×C8×C269—41no—
47Dic538—41no—
48Dic538—41no—
49Dic538—42no—
50C2×Dic269—22no—
51C2×Dic269—22no—
52C2×Dic269—42no—
53#14524130—42no—
54C23×C269C22⋊C411yescharacteristic
55C23×C269C22⋊C411yescharacteristic
56C23×C269C22⋊C411yescharacteristic
57Q8×C269—41no—
58D4×C269—41no—
59D4×C269—41no—
60D4×C269—41no—
61C2×C4×C269—21no—
62C2×C4×C269—21no—
63C2×C4×C269—21no—
64C2×C4×C269—21no—
65C2×C4×C269—21no—
66C2×C4×C269—21no—
67C2×C4×C269C22⋊C411yescharacteristic
68C2×C4×C269C22×C411yescharacteristic
69C2×C4×C269C2×D411yescharacteristic
70C2×C4×C269C2×D411yescharacteristic
71C8×C269—42no—
72C22×C269—21no—
73C22×C269—21no—
74C22×C269—21no—
75C22×C269—41no—
76C22×C269—41no—
77C22×C269—41no—
78C22×C269C2×C22⋊C411yescharacteristic
79C4×C269—21no—
80C4×C269—21no—
81C4×C269—21no—
82C4×C269#9911yescharacteristic
83C4×C269#9911yescharacteristic
84Dic269—42no—
85Dic269—42no—
86C2×C269—21no—
87C2×C269—41no—
88C2×C269—41no—
89C2×C269—41no—
90C2×C269C2×#9911yescharacteristic
91C269#213511yescharacteristic, complemented, Sylow
92#2135—2691nomaximal, Sylow
93#536—2691no—
94#413—2691no—
95#413D26911yescharacteristic
96#356—2694no—
97C2×#41C2×D26911yescharacteristic
98C22×Q8—2691no—
99C2×M16—21no—
100C2×M16—5381no—
101C4⋊Q8—2692no—
102#124—2692no—
103#123—5381no—
104#123—5381no—
105C22⋊Q8—5381no—
106C22⋊Q8—5381no—
107#117—2691no—
108#101—2692no—
109#101C2×D26912yes—
110#100—21no—
111#100—5381no—
112#41—41no—
113#41—41no—
114C2×Q8—2692no—
115C2×Q8—5381no—
116C2×Q8—10761no—
117C2×Q8—10761no—
118C2×Q8C22×D26911yescharacteristic
119C2×D4C4×D26911yescharacteristic
120C2×D4C22⋊D26911yescharacteristic
121C2×D4D107611yescharacteristic
122C22×C4—5381no—
123C22×C4C4×D26911yescharacteristic
124C22×C4C22⋊D26911yescharacteristic
125C22×C4D107611yescharacteristic
126M16—22no—
127M16—41no—
128M16—5382no—
129M16—10761no—
130C2×C8—41no—
131C2×C8—10761no—
132C4⋊C4—5381no—
133C4⋊C4—5382no—
134C4⋊C4—5382no—
135C22⋊C4—5381no—
136C22⋊C4—5382no—
137C22⋊C4—5382no—
138C42—2692no—
139C23#1686437911yescharacteristic
140C23#1686437911yescharacteristic
141C23#1686437911yescharacteristic
142Q8—41no—
143Q8—10762no—
144Q8—10761no—
145Q8—10761no—
146D4—41no—
147D4—41no—
148D4—41no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—5382no—
156C2×C4—5382no—
157C2×C4—10762no—
158C2×C4#1686437911yescharacteristic
159C2×C4C2×C4×D26911yescharacteristic
160C2×C4C2×C22⋊D26911yescharacteristic
161C2×C4C2×D107611yescharacteristic
162C8—42no—
163C8—10762no—
164C22—21no—
165C22—21no—
166C22—21no—
167C22—41no—
168C22—41no—
169C22—41no—
170C22C2×#1686437911yescharacteristic
171C4—21no—
172C4—21no—
173C4—21no—
174C4—10762no—
175C4—10762no—
176C4#2257536711yescharacteristic
177C4#2257536711yescharacteristic
178C2—21no—
179C2—41no—
180C2—41no—
181C2—41no—
182C2C2×#2257536711yescharacteristic
subgroupG/H
centerC2C2×#22575367
commutatorC2×C4×C269C22×C4
FrattiniC2×Q8C22×D269
Fitting#413×C269C2
socleC2×C269C2×#99
prime radicalC2×#41×C269C22
cosocleC2×Q8×C269C23
hypercenter#413D269
hypocenterC269#2135
LCSG ▷ C2×C4×C269 ▷ C22×C269 ▷ C2×C269 ▷ C269
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #413

Complements

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

NH ≅ G/Ntype
3. #31770403C2semidirect
91. C269#2135semidirect

Maximal subgroups

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

subgroupindex
#317711762
#317705892
#317704032
#413×C2692
#2135269