Finite Group Databasebeta

← #27186612

Subgroups — #27186612

Order 12928.

248 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 56 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#20266199C212yesmaximal, complemented
2#20266033C212yesmaximal, complemented
3#20266032C212yesmaximal, complemented
4#124×D101C211yesmaximal, characteristic
5C4⋊#11717581—22no—
6#15692312—22no—
7#15692310C2214yescomplemented
8C2×Q8×D101C411yescharacteristic
9C2×D4×D101C411yescharacteristic
10#15692244C2211yescharacteristic
11#15692243C411yescharacteristic
12#15692242C2211yescharacteristic
13#15692241—22no—
14C22⋊C4×D101—22no—
15C42×D101C2211yescharacteristic
16#124×C101C411yescharacteristic
17C2×#11717581—24no—
18C23×D101—21no—
19C2×Dic202—21no—
20C2×Dic202C2×C411yescharacteristic
21Q8×D101—42no—
22C2×D404C2×C411yescharacteristic
23C2×C22⋊D101—21no—
24D4×D101—42no—
25C2×C4×D101—21no—
26C2×C4×D101C2×C411yescharacteristic
27C2×C4×D101D412yes—
28C22⋊Dic101—22no—
29#13108051—22no—
30#13108051—22no—
31C4×Dic101D412yes—
32C4×Dic101C2311yescharacteristic
33C2×Q8×C101C2×C411yescharacteristic
34C2×D4×C101C2×C411yescharacteristic
35C22⋊C4×C101—22no—
36C42×C101C2×C411yescharacteristic
37C23×C101—21no—
38C22×D101—21no—
39C22×D101—44no—
40C22×D101—42no—
41C22×D101C22⋊C411yescharacteristic
42Q8×C101—41no—
43D4×C101—41no—
44C2×C4×C101—21no—
45C2×C4×C101C22⋊C412yes—
46C2×C4×C101C22×C411yescharacteristic
47C22⋊D101—42no—
48C2×Dic101—21no—
49C2×Dic101C22⋊C411yescharacteristic
50C2×Dic101C2×D412yes—
51D404—41no—
52C4×D101—22no—
53C4×D101—42no—
54C4×D101—42no—
55Dic202—42no—
56Dic202—41no—
57#11717581—44no—
58C22×C101—41no—
59C22×C101—42no—
60C22×C101C2×C22⋊C411yescharacteristic
61C2×D101—22no—
62C2×D101—41no—
63C2×D101—42no—
64C2×D101—84no—
65C2×D101#10012yes—
66C4×C101—21no—
67C4×C101—22no—
68C4×C101—41no—
69Dic101—21no—
70Dic101—22no—
71Dic101—41no—
72C2×C101—81no—
73C2×C101C2×#10011yescharacteristic
74C2×C101#42112yes—
75D101—22no—
76D101—81no—
77#1533—1011nomaximal, Sylow
78C101#153311yescharacteristic, complemented, Sylow
79C2×#124—1011no—
80#432—1012no—
81#329—1012no—
82#329—1012no—
83C22×Q8—1011no—
84C22×D4—1011no—
85#124—1011no—
86#124—1011no—
87#124—1011no—
88#124—2022no—
89#124#8793811yescharacteristic
90C2×C22⋊C4—2022no—
91C2×C42—1011no—
92C4⋊C8—2022no—
93C22⋊C8—2022no—
94#97—1014no—
95C24—2021no—
96C2×Q8—1011no—
97C2×Q8—4042no—
98C2×Q8—2021no—
99C2×Q8C2×#8793811yescharacteristic
100C2×D4—1011no—
101C2×D4—4042no—
102C2×D4—2021no—
103C2×D4C2×#8793811yescharacteristic
104C22×C4—1012no—
105C22×C4—1011no—
106C22×C4—2021no—
107C2×C8—2024no—
108C22⋊C4—22no—
109C22⋊C4—2022no—
110C22⋊C4—2022no—
111C22⋊C4—2022no—
112C42—1012no—
113C42—1011no—
114C42C2×#8793811yescharacteristic
115C23—21no—
116C23—1011no—
117C23—4042no—
118C23—4044no—
119C23—2021no—
120Q8—41no—
121Q8—4041no—
122Q8—4042no—
123D4—41no—
124D4—4041no—
125D4—4042no—
126C2×C4—21no—
127C2×C4—1012no—
128C2×C4—1011no—
129C2×C4—4042no—
130C2×C4—4042no—
131C2×C4—2022no—
132C2×C4—2021no—
133C2×C4C22⋊#8793812yes—
134C2×C4C22×#8793811yescharacteristic
135C8—4044no—
136C22—41no—
137C22—42no—
138C22—8084no—
139C22—1012no—
140C22—4041no—
141C22—4042no—
142C22—2022no—
143C22C2×C22⋊#8793811yescharacteristic
144C4—21no—
145C4—22no—
146C4—41no—
147C4—4041no—
148C4—2021no—
149C4—2022no—
150C2—81no—
151C2—8081no—
152C2—2022no—
153C2D101⋊#10011yescharacteristic
154C2#2026617412yes—
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
Fitting#124×C101C4
socleC22×C101C2×C22⋊C4
prime radicalC2×D4×D101C4
cosocleC4×Dic101C23
hypercenter#124#87938
hypocenterC101#1533
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #20266199C2semidirect
2. #20266033C2semidirect
3. #20266032C2semidirect
7. #15692310C22semidirect
78. C101#1533semidirect

Maximal subgroups

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

subgroupindex
#202661992
#202660332
#202660322
#124×D1012
#1533101