Finite Group Databasebeta

← #27172802

Subgroups — #27172802

Order 12928.

270 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 55 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#20265740C211yesmaximal, characteristic
2#20265149C212yesmaximal, complemented
3#20264970C211yesmaximal, characteristic, complemented
4#20264969C212yesmaximal, complemented
5#353×C101C211yesmaximal, characteristic, complemented
6C2×#13108073C2211yescharacteristic
7C2×D4×D101C411yescharacteristic, complemented
8#15692251C2211yescharacteristic
9#15692243C411yescharacteristic, complemented
10#15692242C411yescharacteristic, complemented
11#15692227—22no—
12#15692223—22no—
13#15692212C411yescharacteristic, complemented
14#15692211—22no—
15#15692170C2212yescomplemented
16#15692169—22no—
17#127×C101C2211yescharacteristic
18#99×C101C2212yescomplemented
19C23×D101—21no—
20#13108073—42no—
21C2×Dic202C2×C411yescharacteristic
22C2×D404C2×C411yescharacteristic
23C2×C22⋊D101—22no—
24C2×C22⋊D101—21no—
25C2×C22⋊D101D412yes—
26D4×D101—42no—
27C22×Dic101D412yes—
28C2×C4×D101C2×C411yescharacteristic
29#13108053—21no—
30#13108053—42no—
31C22⋊Dic101—22no—
32C22⋊Dic101—22no—
33#13108051—21no—
34#13108051—22no—
35C4×Dic101—42no—
36C4×Dic101C2×C411yescharacteristic
37C2×D4×C101—22no—
38C2×D4×C101C2×C411yescharacteristic
39C2×D4×C101C2311yescharacteristic
40C22⋊C4×C101—22no—
41C42×C101C2×C411yescharacteristic
42C23×C101—21no—
43C23×C101C2×D412yes—
44C22×D101—21no—
45C22×D101—42no—
46C22×D101—82no—
47C22×D101C22⋊C411yescharacteristic
48D4×C101—41no—
49D4×C101—41no—
50D4×C101—82no—
51C2×C4×C101—21no—
52C2×C4×C101—42no—
53C2×C4×C101C22×C411yescharacteristic
54C22⋊D101—42no—
55C22⋊D101—42no—
56C22⋊D101—82no—
57C2×Dic101—22no—
58C2×Dic101—22no—
59C2×Dic101—21no—
60C2×Dic101—42no—
61C2×Dic101C22⋊C412yes—
62C2×Dic101C22⋊C411yescharacteristic
63D404—41no—
64C4×D101—22no—
65Dic202—41no—
66C22×C101—22no—
67C22×C101—42no—
68C22×C101—42no—
69C22×C101—41no—
70C22×C101C2×C22⋊C411yescharacteristic
71C2×D101—21no—
72C2×D101—41no—
73C2×D101—42no—
74C2×D101—41no—
75C2×D101—84no—
76C4×C101—21no—
77C4×C101—41no—
78C4×C101—82no—
79Dic101—22no—
80Dic101—41no—
81Dic101—82no—
82Dic101#9912yes—
83C2×C101—21no—
84C2×C101—42no—
85C2×C101—81no—
86C2×C101C2×#9911yescharacteristic
87D101—41no—
88D101—81no—
89#2128—1011nomaximal, Sylow
90C101#212811yescharacteristic, complemented, Sylow
91#535—1011no—
92#410—1012no—
93#353—1011no—
94#353—1012no—
95#353D10111yescharacteristic, complemented
96C2×#41—1011no—
97C22×D4—1011no—
98#127—1011no—
99#127C2×D10111yescharacteristic
100#124—1011no—
101#124—1011no—
102#121—2022no—
103#120—2022no—
104#117—1011no—
105#117—2022no—
106#99—1012no—
107#99—2022no—
108#99C2×D10112yescomplemented
109C24—2021no—
110#41—4042no—
111C2×Q8—1011no—
112C2×D4—22no—
113C2×D4—1012no—
114C2×D4—1011no—
115C2×D4—4042no—
116C2×D4—2021no—
117C2×D4—2022no—
118C2×D4C4×D10111yescharacteristic
119C2×D4C22×D10111yescharacteristic
120C22×C4—1012no—
121C22×C4—1011no—
122C4⋊C4—4042no—
123C4⋊C4—2021no—
124C22⋊C4—22no—
125C22⋊C4—2022no—
126C22⋊C4—2021no—
127C22⋊C4—2022no—
128C22⋊C4—2022no—
129C42—1011no—
130C42—4042no—
131C42C4×D10111yescharacteristic
132C23—21no—
133C23—8082no—
134C23—1011no—
135C23—4042no—
136C23—2021no—
137C23D4×D10112yes—
138Q8—4041no—
139D4—41no—
140D4—41no—
141D4—82no—
142D4—8082no—
143D4—4041no—
144D4—4042no—
145D4—4042no—
146C2×C4—21no—
147C2×C4—42no—
148C2×C4—1011no—
149C2×C4—1012no—
150C2×C4—4042no—
151C2×C4—2022no—
152C2×C4—2022no—
153C2×C4—2022no—
154C2×C4—2021no—
155C2×C4C2×C4×D10111yescharacteristic
156C22—22no—
157C22—42no—
158C22—42no—
159C22—41no—
160C22—8084no—
161C22—4042no—
162C22—4041no—
163C22—4041no—
164C22—2021no—
165C22C22⋊C4×D10111yescharacteristic
166C4—21no—
167C4—41no—
168C4—82no—
169C4—8082no—
170C4—1012no—
171C4—4041no—
172C4—2022no—
173C2—21no—
174C2—42no—
175C2—81no—
176C2—8081no—
177C2—4041no—
178C2#99×D10111yescharacteristic
subgroupG/H
centerC2#99×D101
commutatorC2×C4×C101C22×C4
FrattiniC2×D4C22×D101
Fitting#353×C101C2
socleC2×C101C2×#99
prime radical#20265740C2
cosocleC2×D4×C101C23
hypercenter#353D101
hypocenterC101#2128
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
2. #20265149C2semidirect
3. #20264970C2semidirect
4. #20264969C2semidirect
5. #353×C101C2semidirect
7. C2×D4×D101C4semidirect
9. #15692243C4semidirect
10. #15692242C4semidirect
13. #15692212C4semidirect
15. #15692170C22semidirect
18. #99×C101C22semidirect
90. C101#2128semidirect
95. #353D101semidirect
108. #99C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202657402
#202651492
#202649702
#202649692
#353×C1012
#2128101