Finite Group Databasebeta

← #27174412

Subgroups — #27174412

Order 12928.

840 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 163 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
1C2×#15692217C213yesmaximal, complemented
2C2×#15692215C213yesmaximal, complemented
3#20265112C213yesmaximal, complemented
4#20265087C211yesmaximal, characteristic, complemented
5#20265086C213yesmaximal, complemented
6#20265022C211yesmaximal, characteristic, complemented
7#399×C101C211yesmaximal, characteristic, complemented
8C22×D404C2213yes—
9C22×C22⋊D101C2213yes—
10C23×Dic101C2211yescharacteristic
11C22×C4×D101C2213yes—
12#15692217—212no—
13#15692215—212no—
14C2×#13108053C2213yes—
15C2×#13108051C2216yes—
16C2×#13108051C2213yes—
17C2×C4×Dic101C2213yes—
18#15692162C2213yes—
19C2×C4⋊C4×C101C2213yes—
20C2×C22⋊C4×C101C2213yes—
21#95×C101C2211yescharacteristic
22C23×D101C2313yes—
23C2×D404—212no—
24C2×D404—26no—
25C2×C22⋊D101—26no—
26C2×C22⋊D101—26no—
27C2×C22⋊D101—26no—
28C22×Dic101—26no—
29C22×Dic101C2311yescharacteristic
30C22×Dic101C2313yes—
31C22×Dic101C2311yescharacteristic
32C2×C4×D101—26no—
33C2×C4×D101—212no—
34#13108053—26no—
35#13108051—26no—
36#13108051—212no—
37C4×Dic101D4112yescomplemented
38C24×C101C2311yescharacteristic
39C22×C4×C101C2313yes—
40C22×C4×C101C2313yes—
41C4⋊C4×C101—26no—
42C22⋊C4×C101—26no—
43C23×C101—23no—
44C23×C101—23no—
45C23×C101#4112yes—
46C23×C101C2411yescharacteristic
47C22×D101—23no—
48C22×D101—26no—
49C22×D101—23no—
50C22×D101—26no—
51C22×D101#4116yes—
52C2×C4×C101—26no—
53C2×C4×C101—23no—
54C2×C4×C101—26no—
55C2×C4×C101C2×D416yes—
56C22⋊D101—224no—
57C2×Dic101—23no—
58C2×Dic101—23no—
59C2×Dic101—23no—
60C2×Dic101—44no—
61C2×Dic101C2×D416yes—
62C2×Dic101C2×D416yes—
63D404—224no—
64C4×D101—412no—
65C22×C101—26no—
66C22×C101—43no—
67C22×C101—41no—
68C22×C101C22×D413yes—
69C22×C101C2×#4113yes—
70C22×C101C2×#4111yescharacteristic
71C2×D101—212no—
72C2×D101—26no—
73C2×D101—43no—
74C2×D101—46no—
75C2×D101—43no—
76C4×C101—26no—
77C4×C101—43no—
78Dic101—26no—
79Dic101—41no—
80Dic101#12714yes—
81C2×C101—42no—
82C2×C101#52513yes—
83C2×C101#52513yes—
84C2×C101C2×#12711yescharacteristic
85D101—46no—
86#2397—1011nomaximal, Sylow
87C101#239711yescharacteristic, complemented, Sylow
88C2×C4×D4—1013no—
89C2×C4×D4—1013no—
90#399—1013no—
91#399D10111yescharacteristic, complemented
92#392—1013no—
93#392—1011no—
94#376—1011no—
95C22×D4—1013no—
96C22×D4—1013no—
97C23×C4—1011no—
98C23×C4—1013no—
99C4×D4—20212no—
100C4×D4—20212no—
101C2×C4⋊C4—1013no—
102C2×C4⋊C4C2×D10113yes—
103C2×C22⋊C4—1016no—
104C2×C22⋊C4—1013no—
105C2×C22⋊C4C2×D10113yes—
106C2×C42—1013no—
107#95—1013no—
108#95C2×D10111yescharacteristic
109C24—1013no—
110C24C22×D10111yescharacteristic
111C2×D4—20212no—
112C2×D4—2026no—
113C2×D4—2026no—
114C2×D4—2026no—
115C2×D4—2026no—
116C22×C4—1011no—
117C22×C4—1013no—
118C22×C4—1011no—
119C22×C4—2026no—
120C22×C4—20212no—
121C22×C4—2026no—
122C22×C4C22×D10113yes—
123C22×C4C22×D10113yes—
124C4⋊C4—26no—
125C4⋊C4—2026no—
126C22⋊C4—26no—
127C22⋊C4—20212no—
128C22⋊C4—2026no—
129C42—10112no—
130C23—23no—
131C23—23no—
132C23—1016no—
133C23—2023no—
134C23—2026no—
135C23—2026no—
136C23—2023no—
137C23#1310807512yes—
138C23C23×D10111yescharacteristic
139D4—20224no—
140D4—20224no—
141C2×C4—26no—
142C2×C4—23no—
143C2×C4—26no—
144C2×C4—1016no—
145C2×C4—1016no—
146C2×C4—40412no—
147C2×C4—4044no—
148C2×C4—2023no—
149C2×C4—2023no—
150C2×C4—2023no—
151C2×C4D4×D10116yes—
152C22—26no—
153C22—43no—
154C22—41no—
155C22—4043no—
156C22—4046no—
157C22—4043no—
158C22—20212no—
159C22—2026no—
160C22C2×D4×D10113yes—
161C22#41×D10113yes—
162C22C2×#1310807511yescharacteristic
163C4—26no—
164C4—43no—
165C4—1014no—
166C4—4041no—
167C4—2026no—
168C2—42no—
169C2—4046no—
170C2#2026568513yes—
171C2#2026569013yes—
172C2#127×D10111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting#399×C101C2
hypercenter#399D101
hypocenterC101#2397
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#15692217C2semidirect
2. C2×#15692215C2semidirect
3. #20265112C2semidirect
4. #20265087C2semidirect
5. #20265086C2semidirect
6. #20265022C2semidirect
7. #399×C101C2semidirect
37. C4×Dic101D4semidirect
87. C101#2397semidirect
91. #399D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922172
C2×#156922152
#202651122
#202650872
#202650862
#202650222
#399×C1012
#2397101