Finite Group Databasebeta

← #27169754

Subgroups — #27169754

Order 12928.

248 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 59 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#124×D101C211yesmaximal, characteristic
2#20265232C212yesmaximal, complemented
3#20264917C211yesmaximal, characteristic, complemented
4#20264916C212yesmaximal, complemented
5#329×C101C211yesmaximal, characteristic, complemented
6C2×Q8×D101C411yescharacteristic
7C2×D4×D101C411yescharacteristic
8#15692244C411yescharacteristic
9#15692243C2211yescharacteristic
10#15692242C411yescharacteristic
11#15692241—22no—
12C22⋊C4×D101—22no—
13C42×D101C2211yescharacteristic
14#15692184—22no—
15#15692167—22no—
16#15692166C2212yescomplemented
17#124×C101C2211yescharacteristic
18#97×C101C2212yescomplemented
19C23×D101—21no—
20C2×Dic202—21no—
21C2×Dic202C2×C411yescharacteristic
22Q8×D101—42no—
23C2×D404C2×C411yescharacteristic
24C2×C22⋊D101—21no—
25D4×D101—42no—
26C2×C4×D101—21no—
27C2×C4×D101C2×C411yescharacteristic
28C2×C4×D101D412yes—
29C2×#11717579—22no—
30C22⋊Dic101—22no—
31#13108051—22no—
32#13108051—22no—
33C4×Dic101C2×C411yescharacteristic
34C4×Dic101D412yes—
35C2×Q8×C101C2×C411yescharacteristic
36C2×D4×C101C2×C411yescharacteristic
37C2×C8×C101—22no—
38C22⋊C4×C101—22no—
39C42×C101C2311yescharacteristic
40C23×C101—21no—
41C22×D101—21no—
42C22×D101—44no—
43C22×D101—42no—
44C22×D101C22⋊C411yescharacteristic
45Q8×C101—41no—
46D4×C101—41no—
47C2×C4×C101—21no—
48C2×C4×C101C22×C411yescharacteristic
49C2×C4×C101C2×D412yes—
50C22⋊D101—42no—
51C2×Dic101—21no—
52C2×Dic101C22⋊C412yes—
53C2×Dic101C22⋊C411yescharacteristic
54D404—41no—
55C4×D101—22no—
56C4×D101—42no—
57C4×D101—42no—
58Dic202—42no—
59Dic202—41no—
60C8×C101—42no—
61#11717579—42no—
62C22×C101—41no—
63C22×C101—42no—
64C22×C101C2×C22⋊C411yescharacteristic
65C2×D101—22no—
66C2×D101—41no—
67C2×D101—42no—
68C2×D101—84no—
69C2×D101#10012yes—
70C4×C101—21no—
71C4×C101—22no—
72C4×C101—41no—
73Dic101—21no—
74Dic101—22no—
75Dic101—41no—
76C2×C101—81no—
77C2×C101C2×#10011yescharacteristic
78C2×C101#42112yes—
79D101—22no—
80D101—81no—
81#1533—1011nomaximal, Sylow
82C101#153311yescharacteristic, complemented, Sylow
83C2×#124—1011no—
84#432—1012no—
85#329—1011no—
86#329—1012no—
87#329D10111yescharacteristic, complemented
88C22×Q8—1011no—
89C22×D4—1011no—
90#124—1011no—
91#124—1011no—
92#124—1011no—
93#124—2022no—
94#124C2×D10111yescharacteristic
95C2×C22⋊C4—2022no—
96C2×C42—1011no—
97C4⋊C8—2022no—
98C22⋊C8—2022no—
99#97—1012no—
100#97C2×D10112yescomplemented
101C24—2021no—
102C2×Q8—1011no—
103C2×Q8—4042no—
104C2×Q8—2021no—
105C2×Q8C4×D10111yescharacteristic
106C2×D4—1011no—
107C2×D4—4042no—
108C2×D4—2021no—
109C2×D4C4×D10111yescharacteristic
110C22×C4—1012no—
111C22×C4—1011no—
112C22×C4—2021no—
113C2×C8—22no—
114C2×C8—2022no—
115C22⋊C4—22no—
116C22⋊C4—2022no—
117C22⋊C4—2022no—
118C22⋊C4—2022no—
119C42—1011no—
120C42—1012no—
121C42C22×D10111yescharacteristic
122C23—21no—
123C23—1011no—
124C23—4042no—
125C23—4044no—
126C23—2021no—
127Q8—41no—
128Q8—4041no—
129Q8—4042no—
130D4—41no—
131D4—4041no—
132D4—4042no—
133C2×C4—21no—
134C2×C4—1012no—
135C2×C4—1011no—
136C2×C4—4042no—
137C2×C4—4042no—
138C2×C4—2022no—
139C2×C4—2021no—
140C2×C4C2×C4×D10111yescharacteristic
141C2×C4D4×D10112yes—
142C8—42no—
143C8—4042no—
144C22—41no—
145C22—42no—
146C22—8084no—
147C22—1012no—
148C22—4041no—
149C22—4042no—
150C22—2022no—
151C22C22⋊C4×D10111yescharacteristic
152C4—21no—
153C4—22no—
154C4—41no—
155C4—4041no—
156C4—2021no—
157C4—2022no—
158C2—81no—
159C2—8081no—
160C2—2022no—
161C2#100×D10111yescharacteristic
162C2#2026519212yes—
subgroupG/H
centerC22C22⋊C4×D101
commutatorC2×C4×C101C22×C4
FrattiniC42C22×D101
Fitting#329×C101C2
socleC22×C101C2×C22⋊C4
prime radicalC2×D4×D101C4
cosocleC42×C101C23
hypercenter#329D101
hypocenterC101#1533
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #20265232C2semidirect
3. #20264917C2semidirect
4. #20264916C2semidirect
5. #329×C101C2semidirect
16. #15692166C22semidirect
18. #97×C101C22semidirect
82. C101#1533semidirect
87. #329D101semidirect
100. #97C2×D101semidirect

Maximal subgroups

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

subgroupindex
#124×D1012
#202652322
#202649172
#202649162
#329×C1012
#1533101