Finite Group Databasebeta

← #27173512

Subgroups — #27173512

Order 12928.

1088 conjugacy classes of subgroups; only the 245 normal subgroups are shown.

normal subgroups only

This group has 1088 conjugacy classes of subgroups — too many to draw. Only its 245 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
1C23×C4×D101C211yesmaximal, characteristic
2C2×#15692211C212yesmaximal, complemented
3#20265040C211yesmaximal, characteristic, complemented
4#20265036C214yesmaximal, complemented
5#20265032C212yesmaximal
6#20265024C214yesmaximal, complemented
7#381×C101C211yesmaximal, characteristic, complemented
8C24×D101C2211yescharacteristic
9C23×Dic101C2211yescharacteristic
10C22×C4×D101—28no—
11C22×C4×D101C418yescomplemented
12C22×C4×D101C2212yes—
13C22×C4×D101C2212yes—
14#15692211—28no—
15C2×#13108053C2214yescomplemented
16C2×C22⋊Dic101C2212yes—
17C2×#13108051C2218yes—
18C2×C4×Dic101C2214yescomplemented
19#15692162C2214yescomplemented
20C23×C4×C101C2211yescharacteristic
21C2×C22⋊C4×C101C2212yes—
22#95×C101C2214yescomplemented
23C23×D101—24no—
24C23×D101—28no—
25C23×D101C2×C414yes—
26C23×D101C2312yes—
27C22×Dic101C2×C414yes—
28C22×Dic101D418yes—
29C22×Dic101C2314yes—
30C22×Dic101C2311yescharacteristic
31C22×Dic101C2311yescharacteristic
32C2×C4×D101—232no—
33C2×C4×D101—28no—
34C2×C4×D101—28no—
35C2×C4×D101C2×C418yes—
36C2×C4×D101C2×C418yes—
37#13108053—28no—
38C22⋊Dic101—24no—
39#13108051—216no—
40C4×Dic101—28no—
41C24×C101C2311yescharacteristic
42C22×C4×C101—24no—
43C22×C4×C101C2×C414yes—
44C22×C4×C101C2314yes—
45C22×C4×C101C2312yes—
46C22⋊C4×C101—24no—
47C23×C101—24no—
48C23×C101C22×C412yes—
49C23×C101C2×D414yes—
50C23×C101C2411yescharacteristic
51C22×D101—232no—
52C22×D101—28no—
53C22×D101—24no—
54C22×D101—24no—
55C22×D101—216no—
56C22×D101C22×C414yes—
57C22×D101#4118yes—
58C2×C4×C101—28no—
59C2×C4×C101—24no—
60C2×C4×C101—24no—
61C2×C4×C101—28no—
62C2×C4×C101C22×C414yes—
63C2×Dic101—24no—
64C2×Dic101—28no—
65C2×Dic101C22⋊C4116yescomplemented
66C2×Dic101C22×C412yes—
67C2×Dic101C22×C412yes—
68C2×Dic101C2×D414yes—
69C2×Dic101C2×D414yes—
70C4×D101—216no—
71C4×D101—216no—
72C22×C101—24no—
73C22×C101—28no—
74C22×C101C2×C22⋊C414yes—
75C22×C101C23×C411yescharacteristic
76C22×C101C22×D412yes—
77C22×C101C2×#4114yes—
78C2×D101—232no—
79C2×D101—24no—
80C2×D101—24no—
81C2×D101—216no—
82C2×D101#11718yescomplemented
83C4×C101—24no—
84C4×C101—44no—
85Dic101—44no—
86Dic101C2×C22⋊C414yes—
87Dic101C2×C22⋊C414yes—
88C2×C101—24no—
89C2×C101C22×C22⋊C411yescharacteristic
90C2×C101C2×#11712yes—
91C2×C101#52514yes—
92D101—28no—
93#2280—1011nomaximal, Sylow
94C101#228011yescharacteristic, complemented, Sylow
95C24×C4—1011no—
96C2×#117—1012no—
97#381—1011no—
98#381D10111yescharacteristic, complemented
99#380—1014no—
100#379—1012no—
101C4×C22⋊C4—1014no—
102C25—1011no—
103C23×C4—1011no—
104C23×C4—1018no—
105C23×C4—1012no—
106C23×C4—1012no—
107C23×C4—2028no—
108C23×C4C2×D10111yescharacteristic
109#117—2028no—
110C2×C4⋊C4—1014no—
111C2×C22⋊C4—1012no—
112C2×C22⋊C4—1018no—
113C2×C22⋊C4C2×D10112yes—
114C2×C42—1014no—
115#95—1014no—
116#95C2×D10114yescomplemented
117C24—1012no—
118C24—1014no—
119C24—2028no—
120C24—2024no—
121C24C22×D10111yescharacteristic
122C22×C4—24no—
123C22×C4—1014no—
124C22×C4—1018no—
125C22×C4—1011no—
126C22×C4—1011no—
127C22×C4—1018no—
128C22×C4—1018no—
129C22×C4—1014no—
130C22×C4—2028no—
131C22×C4—2028no—
132C22×C4—20232no—
133C22×C4C4×D10114yes—
134C22×C4C22×D10114yes—
135C22×C4C22×D10112yes—
136C4⋊C4—2028no—
137C22⋊C4—24no—
138C22⋊C4—20216no—
139C22⋊C4—2024no—
140C42—2028no—
141C23—24no—
142C23—1014no—
143C23—1018no—
144C23—20216no—
145C23—20232no—
146C23—2024no—
147C23—2024no—
148C23—2028no—
149C23C2×C4×D10112yes—
150C23D4×D10114yes—
151C23C23×D10111yescharacteristic
152C2×C4—24no—
153C2×C4—28no—
154C2×C4—24no—
155C2×C4—28no—
156C2×C4—1012no—
157C2×C4—1012no—
158C2×C4—1014no—
159C2×C4—1014no—
160C2×C4—10116no—
161C2×C4—20216no—
162C2×C4—20216no—
163C2×C4—2024no—
164C2×C4—2028no—
165C2×C4C2×C4×D10114yes—
166C22—24no—
167C22—28no—
168C22—1018no—
169C22—20232no—
170C22—2024no—
171C22—2024no—
172C22—20216no—
173C22C22⋊C4×D10114yes—
174C22C22×C4×D10111yescharacteristic
175C22C2×D4×D10112yes—
176C22#41×D10114yes—
177C4—24no—
178C4—44no—
179C4—1014no—
180C4—1014no—
181C4—4044no—
182C2—24no—
183C2—2028no—
184C2C2×C22⋊C4×D10111yescharacteristic
185C2#117×D10112yes—
186C2#2026568514yes—
subgroupG/H
center, FrattiniC23C23×D101
commutatorC22×C101C23×C4
Fitting#381×C101C2
socle, cosocleC23×C101C24
prime radicalC24×D101C22
hypercenter#381D101
hypocenterC101#2280
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C2×#15692211C2semidirect
3. #20265040C2semidirect
4. #20265036C2semidirect
6. #20265024C2semidirect
7. #381×C101C2semidirect
11. C22×C4×D101C4semidirect
15. C2×#13108053C22semidirect
18. C2×C4×Dic101C22semidirect
19. #15692162C22semidirect
22. #95×C101C22semidirect
65. C2×Dic101C22⋊C4semidirect
82. C2×D101#117semidirect
94. C101#2280semidirect
98. #381D101semidirect
116. #95C2×D101semidirect

Maximal subgroups

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

subgroupindex
C23×C4×D1012
C2×#156922112
#202650402
#202650362
#202650322
#202650242
#381×C1012
#2280101