Finite Group Databasebeta

← #27172800

Subgroups — #27172800

Order 12928.

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

normal subgroups only

This group has 286 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#20265745C211yesmaximal, characteristic
2#99×D101C212yesmaximal, complemented
3#20264972C211yesmaximal, characteristic, complemented
4#20264969C212yesmaximal, complemented
5#354×C101C211yesmaximal, characteristic, complemented
6C2×#13108075C411yescharacteristic, complemented
7C2×D4×D101C2211yescharacteristic
8#15692252C411yescharacteristic, complemented
9#15692251C2211yescharacteristic
10#15692244C411yescharacteristic, complemented
11#15692228—22no—
12#15692225—22no—
13#15692212C411yescharacteristic, complemented
14C22⋊C4×D101—22no—
15#15692170C2212yes—
16#15692169—22no—
17#124×C101C2211yescharacteristic
18#99×C101C2212yes—
19C23×D101D412yes—
20#13108075—42no—
21C2×D404—21no—
22C2×D404—22no—
23C2×D404C2×C411yescharacteristic
24C2×D404C2×C411yescharacteristic, complemented
25C2×C22⋊D101—22no—
26C2×C22⋊D101D412yes—
27D4×D101—42no—
28C2×C4×D101—21no—
29C2×C4×D101—42no—
30C2×C4×D101C2×C411yescharacteristic
31#13108053—21no—
32C22⋊Dic101—22no—
33#13108051—21no—
34#13108051—22no—
35#13108051—42no—
36C4×Dic101C2×C411yescharacteristic, complemented
37C2×Q8×C101C2×C411yescharacteristic, complemented
38C2×D4×C101C2311yescharacteristic
39C22⋊C4×C101—22no—
40C22⋊C4×C101—22no—
41C42×C101C2×C411yescharacteristic, complemented
42C23×C101C2×D412yes—
43C22×D101—22no—
44C22×D101—22no—
45C22×D101—21no—
46C22×D101—82no—
47C22×D101C22⋊C412yes—
48C22×D101C22⋊C411yescharacteristic
49Q8×C101—41no—
50D4×C101—41no—
51C2×C4×C101—21no—
52C2×C4×C101—21no—
53C2×C4×C101—42no—
54C2×C4×C101C22×C411yescharacteristic
55C22⋊D101—42no—
56C22⋊D101—82no—
57C2×Dic101—21no—
58C2×Dic101—42no—
59C2×Dic101C22⋊C411yescharacteristic
60D404—41no—
61D404—41no—
62D404—42no—
63D404—82no—
64C4×D101—22no—
65C4×D101—42no—
66C4×D101—84no—
67C22×C101—22no—
68C22×C101—42no—
69C22×C101C2×C22⋊C411yescharacteristic
70C2×D101—22no—
71C2×D101—41no—
72C2×D101—44no—
73C2×D101—42no—
74C2×D101—42no—
75C2×D101—41no—
76C2×D101—82no—
77C2×D101#9912yes—
78C4×C101—21no—
79C4×C101—41no—
80C4×C101—41no—
81C4×C101—82no—
82Dic101—21no—
83Dic101—41no—
84Dic101—82no—
85C2×C101—21no—
86C2×C101—42no—
87C2×C101C2×#9911yescharacteristic
88D101—22no—
89D101—42no—
90D101—81no—
91#2127—1011nomaximal, Sylow
92C101#212711yescharacteristic, complemented, Sylow
93#535—1011no—
94C2×#99—1012no—
95#354—1011no—
96#354D10111yescharacteristic, complemented
97#353—1012no—
98C2×#41—1011no—
99C22×D4—1011no—
100#127—1011no—
101#127—1011no—
102#124—1011no—
103#124C2×D10111yescharacteristic
104#121—2022no—
105#120—2022no—
106#117—1011no—
107C2×C22⋊C4—2022no—
108#99—1012no—
109#99—2022no—
110#99C2×D10112yes—
111C24—1012no—
112#41—4042no—
113C2×Q8C4×D10111yescharacteristic, complemented
114C2×D4—1012no—
115C2×D4—1011no—
116C2×D4—1011no—
117C2×D4—4042no—
118C2×D4—2022no—
119C2×D4—2021no—
120C2×D4—2022no—
121C2×D4C22×D10111yescharacteristic
122C22×C4—1011no—
123C22×C4—4042no—
124C22×C4—2021no—
125C4⋊C4—2021no—
126C22⋊C4—22no—
127C22⋊C4—22no—
128C22⋊C4—4042no—
129C22⋊C4—2021no—
130C22⋊C4—2022no—
131C22⋊C4—2022no—
132C42—1011no—
133C42C4×D10111yescharacteristic, complemented
134C23—8082no—
135C23—1011no—
136C23—1012no—
137C23—2022no—
138C23—2022no—
139C23—2021no—
140C23D4×D10112yes—
141Q8—41no—
142D4—41no—
143D4—8082no—
144D4—8082no—
145D4—4041no—
146D4—4041no—
147D4—4042no—
148D4—4042no—
149C2×C4—21no—
150C2×C4—21no—
151C2×C4—42no—
152C2×C4—8084no—
153C2×C4—1011no—
154C2×C4—4042no—
155C2×C4—4042no—
156C2×C4—2022no—
157C2×C4—2021no—
158C2×C4C2×C4×D10111yescharacteristic
159C22—22no—
160C22—42no—
161C22—8082no—
162C22—1012no—
163C22—4042no—
164C22—4042no—
165C22—4041no—
166C22—4044no—
167C22—4041no—
168C22—2022no—
169C22C22⋊C4×D10111yescharacteristic
170C4—21no—
171C4—41no—
172C4—41no—
173C4—82no—
174C4—8082no—
175C4—4041no—
176C4—2021no—
177C2—21no—
178C2—42no—
179C2—8081no—
180C2—4042no—
181C2—2022no—
182C2#99×D10111yescharacteristic
subgroupG/H
centerC2#99×D101
commutatorC2×C4×C101C22×C4
FrattiniC2×D4C22×D101
Fitting#354×C101C2
socleC2×C101C2×#99
prime radical#20265745C2
cosocleC2×D4×C101C23
hypercenter#354D101
hypocenterC101#2127
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
2. #99×D101C2semidirect
3. #20264972C2semidirect
4. #20264969C2semidirect
5. #354×C101C2semidirect
6. C2×#13108075C4semidirect
8. #15692252C4semidirect
10. #15692244C4semidirect
13. #15692212C4semidirect
24. C2×D404C2×C4semidirect
36. C4×Dic101C2×C4semidirect
37. C2×Q8×C101C2×C4semidirect
41. C42×C101C2×C4semidirect
92. C101#2127semidirect
96. #354D101semidirect
113. C2×Q8C4×D101semidirect
133. C42C4×D101semidirect

Maximal subgroups

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

subgroupindex
#202657452
#99×D1012
#202649722
#202649692
#354×C1012
#2127101