Finite Group Databasebeta

← #27173127

Subgroups — #27173127

Order 12928.

246 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 41 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#20265913C211yesmaximal, characteristic, complemented
2#20265367C212yesmaximal, complemented
3#20264970C211yesmaximal, characteristic, complemented
4#20264968C212yesmaximal, complemented
5#453×C101C211yesmaximal, characteristic
6#15692242—22no—
7#15692236—22no—
8#15692236—21no—
9#15692236C2212yes—
10C4×Dic202—21no—
11D4×Dic101—22no—
12#15692170—22no—
13#15692170C2212yes—
14H(5,𝔽2)×C101C2211yescharacteristic
15AGL(1,ℤ/8)×C101—21no—
16#127×C101C2211yescharacteristic
17#104×C101—21no—
18#100×C101C2211yescharacteristic
19C2×Dic202—21no—
20C22×Dic101—21no—
21C22×Dic101D412yes—
22C4⋊Dic101—21no—
23#13108053—22no—
24#13108053—42no—
25C22⋊Dic101—22no—
26C22⋊Dic101—21no—
27C22⋊Dic101—22no—
28C22⋊Dic101—22no—
29C22⋊Dic101—42no—
30C22⋊Dic101D412yes—
31C4×Dic101—22no—
32#41×C101—21no—
33#41×C101—41no—
34C2×D4×C101—22no—
35C2×D4×C101—22no—
36C2×D4×C101—41no—
37C2×D4×C101D411yescharacteristic
38C2×D4×C101C2311yescharacteristic
39SD16×C101—41no—
40D8×C101—41no—
41M16×C101—21no—
42C42×C101D411yescharacteristic
43C23×C101—22no—
44C23×C101—21no—
45C23×C101C2×D412yes—
46Q8×C101—21no—
47D4×C101—21no—
48D4×C101—41no—
49D4×C101—41no—
50D4×C101—44no—
51D4×C101—42no—
52D4×C101—41no—
53C2×C4×C101—21no—
54C2×C4×C101—22no—
55C2×C4×C101—41no—
56C2×C4×C101C2×D411yescharacteristic
57C2×Dic101—22no—
58C2×Dic101—22no—
59C2×Dic101—42no—
60C2×Dic101—42no—
61C2×Dic101—81no—
62Dic202—42no—
63C8×C101—41no—
64C22×C101—22no—
65C22×C101—42no—
66C22×C101—42no—
67C22×C101—41no—
68C22×C101—41no—
69C22×C101—82no—
70C22×C101#12011yescharacteristic
71C4×C101—22no—
72C4×C101—21no—
73C4×C101—41no—
74Dic101—42no—
75Dic101—42no—
76Dic101—82no—
77C2×C101—21no—
78C2×C101—42no—
79C2×C101—81no—
80C2×C101—81no—
81C2×C101B(4,𝔽2)11yescharacteristic
82#2198—1011nomaximal, Sylow
83C101#219811yescharacteristic, complemented, Sylow
84#559—1011no—
85#458—1012no—
86#453D10111yescharacteristic
87#353—1011no—
88#352—1012no—
89H(5,𝔽2)C2×D10111yescharacteristic
90AGL(1,ℤ/8)—21no—
91#127C2×D10111yescharacteristic
92#124—2022no—
93#123—1012no—
94#123—2022no—
95#123—2021no—
96C4×Q8—2021no—
97C4×D4—2022no—
98#104—21no—
99#100C2×D10111yescharacteristic
100#99—1012no—
101#99—2022no—
102#41—21no—
103#41—41no—
104C2×Q8—2021no—
105C2×D4—22no—
106C2×D4—22no—
107C2×D4—41no—
108C2×D4C22⋊D10111yescharacteristic
109C2×D4C22×D10111yescharacteristic
110C22×C4—1012no—
111C22×C4—2021no—
112SD16—41no—
113D8—41no—
114M16—21no—
115C4⋊C4—4042no—
116C4⋊C4—2021no—
117C4⋊C4—2022no—
118C22⋊C4—1012no—
119C22⋊C4—4042no—
120C22⋊C4—2022no—
121C22⋊C4—2022no—
122C22⋊C4—2021no—
123C22⋊C4—2022no—
124C42—2022no—
125C42C22⋊D10111yescharacteristic
126C23—22no—
127C23—21no—
128C23D4×D10112yes—
129Q8—21no—
130Q8—4042no—
131D4—21no—
132D4—41no—
133D4—41no—
134D4—44no—
135D4—42no—
136D4—41no—
137C2×C4—21no—
138C2×C4—22no—
139C2×C4—41no—
140C2×C4—8081no—
141C2×C4—4042no—
142C2×C4—4042no—
143C2×C4—2022no—
144C2×C4—2022no—
145C2×C4C2×C22⋊D10111yescharacteristic
146C8—41no—
147C22—22no—
148C22—42no—
149C22—42no—
150C22—41no—
151C22—41no—
152C22—82no—
153C22#1569222311yescharacteristic
154C4—22no—
155C4—21no—
156C4—41no—
157C4—8082no—
158C4—4042no—
159C4—4042no—
160C2—21no—
161C2—42no—
162C2—81no—
163C2—81no—
164C2#2026536311yescharacteristic
subgroupG/H
centerC2#20265363
commutator, cosocleC2×D4×C101C23
FrattiniC2×D4C22×D101
Fitting, prime radical#453×C101C2
socleC2×C101B(4,𝔽2)
hypercenter#453D101
hypocenterC101#2198
LCSG ▷ C2×D4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #453

Complements

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

NH ≅ G/Ntype
1. #20265913C2semidirect
2. #20265367C2semidirect
3. #20264970C2semidirect
4. #20264968C2semidirect
83. C101#2198semidirect

Maximal subgroups

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

subgroupindex
#202659132
#202653672
#202649702
#202649682
#453×C1012
#2198101