Finite Group Databasebeta

← #27172798

Subgroups — #27172798

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#20265740C211yesmaximal, characteristic
2#99×D101C212yesmaximal, complemented
3#20264971C212yesmaximal, complemented
4#20264970C211yesmaximal, characteristic, complemented
5#353×C101C211yesmaximal, characteristic, complemented
6C2×#13108073C411yescharacteristic, complemented
7C2×D4×D101C2211yescharacteristic
8#15692251C411yescharacteristic, complemented
9#15692243C411yescharacteristic, complemented
10#15692242C2211yescharacteristic
11#15692227—22no—
12#15692223—22no—
13#15692212C411yescharacteristic, complemented
14C22⋊C4×D101—22no—
15#15692170C2212yes—
16#15692169—22no—
17#127×C101C2211yescharacteristic
18#99×C101C2212yes—
19C23×D101D412yes—
20#13108073—42no—
21C2×Dic202C2×C411yescharacteristic, complemented
22C2×D404C2×C411yescharacteristic
23C2×C22⋊D101—21no—
24C2×C22⋊D101—22no—
25C2×C22⋊D101D412yes—
26D4×D101—42no—
27C22×Dic101—21no—
28C2×C4×D101—42no—
29C2×C4×D101C2×C411yescharacteristic
30#13108053—21no—
31C22⋊Dic101—22no—
32C22⋊Dic101—22no—
33#13108051—21no—
34#13108051—22no—
35#13108051—42no—
36C4×Dic101C2×C411yescharacteristic, complemented
37C2×D4×C101—22no—
38C2×D4×C101C2×C411yescharacteristic, complemented
39C2×D4×C101C2311yescharacteristic
40C22⋊C4×C101—22no—
41C42×C101C2×C411yescharacteristic, complemented
42C23×C101—21no—
43C23×C101C2×D412yes—
44C22×D101—22no—
45C22×D101—22no—
46C22×D101—82no—
47C22×D101C22⋊C412yes—
48C22×D101C22⋊C411yescharacteristic
49D4×C101—41no—
50D4×C101—41no—
51D4×C101—82no—
52C2×C4×C101—21no—
53C2×C4×C101—42no—
54C2×C4×C101C22×C411yescharacteristic
55C22⋊D101—42no—
56C22⋊D101—42no—
57C22⋊D101—82no—
58C2×Dic101—21no—
59C2×Dic101—21no—
60C2×Dic101—42no—
61C2×Dic101—42no—
62C2×Dic101C22⋊C411yescharacteristic
63D404—41no—
64C4×D101—22no—
65C4×D101—84no—
66Dic202—41no—
67C22×C101—22no—
68C22×C101—42no—
69C22×C101—42no—
70C22×C101—41no—
71C22×C101C2×C22⋊C411yescharacteristic
72C2×D101—22no—
73C2×D101—41no—
74C2×D101—44no—
75C2×D101—42no—
76C2×D101—82no—
77C2×D101#9912yes—
78C4×C101—21no—
79C4×C101—41no—
80C4×C101—82no—
81Dic101—21no—
82Dic101—41no—
83Dic101—41no—
84Dic101—82no—
85C2×C101—21no—
86C2×C101—42no—
87C2×C101—81no—
88C2×C101C2×#9911yescharacteristic
89D101—22no—
90D101—42no—
91#2127—1011nomaximal, Sylow
92C101#212711yescharacteristic, complemented, Sylow
93#535—1011no—
94C2×#99—1012no—
95#354—1012no—
96#353—1011no—
97#353D10111yescharacteristic, complemented
98C2×#41—1011no—
99C22×D4—1011no—
100#127—1011no—
101#127C2×D10111yescharacteristic
102#124—1011no—
103#124—1011no—
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×Q8—1011no—
114C2×D4—22no—
115C2×D4—1012no—
116C2×D4—1011no—
117C2×D4—4042no—
118C2×D4—2022no—
119C2×D4—2021no—
120C2×D4C4×D10111yescharacteristic, complemented
121C2×D4C22×D10111yescharacteristic
122C22×C4—1011no—
123C22×C4—4042no—
124C22×C4—2021no—
125C4⋊C4—2021no—
126C22⋊C4—22no—
127C22⋊C4—4042no—
128C22⋊C4—2021no—
129C22⋊C4—2022no—
130C22⋊C4—2022no—
131C22⋊C4—2022no—
132C42—1011no—
133C42C4×D10111yescharacteristic, complemented
134C23—21no—
135C23—8082no—
136C23—1011no—
137C23—1012no—
138C23—2022no—
139C23—2022no—
140C23D4×D10112yes—
141Q8—4041no—
142D4—41no—
143D4—41no—
144D4—82no—
145D4—8082no—
146D4—4041no—
147D4—4042no—
148D4—4042no—
149C2×C4—21no—
150C2×C4—42no—
151C2×C4—8084no—
152C2×C4—1011no—
153C2×C4—4042no—
154C2×C4—4042no—
155C2×C4—2022no—
156C2×C4—2021no—
157C2×C4—2021no—
158C2×C4C2×C4×D10111yescharacteristic
159C22—22no—
160C22—42no—
161C22—42no—
162C22—41no—
163C22—8082no—
164C22—1012no—
165C22—4042no—
166C22—4041no—
167C22—4044no—
168C22—2022no—
169C22C22⋊C4×D10111yescharacteristic
170C4—21no—
171C4—41no—
172C4—82no—
173C4—8082no—
174C4—4041no—
175C4—4041no—
176C4—2021no—
177C2—21no—
178C2—42no—
179C2—81no—
180C2—4042no—
181C2—2022no—
182C2#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#2127
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. #99×D101C2semidirect
3. #20264971C2semidirect
4. #20264970C2semidirect
5. #353×C101C2semidirect
6. C2×#13108073C4semidirect
8. #15692251C4semidirect
9. #15692243C4semidirect
13. #15692212C4semidirect
21. C2×Dic202C2×C4semidirect
36. C4×Dic101C2×C4semidirect
38. C2×D4×C101C2×C4semidirect
41. C42×C101C2×C4semidirect
92. C101#2127semidirect
97. #353D101semidirect
120. C2×D4C4×D101semidirect
133. C42C4×D101semidirect

Maximal subgroups

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

subgroupindex
#202657402
#99×D1012
#202649712
#202649702
#353×C1012
#2127101