Finite Group Databasebeta

← #27170134

Subgroups — #27170134

Order 12928.

254 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 53 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#20265758C211yesmaximal, characteristic
2#20265490C212yesmaximal, complemented
3#20264910C212yesmaximal, complemented
4#20264903C211yesmaximal, characteristic, complemented
5#398×C101C211yesmaximal, characteristic, complemented
6C2×#13108074—21no—
7#15692235—22no—
8#15692230—22no—
9#15692229—21no—
10#15692229C2212yes—
11#15692188—22no—
12#15692175—22no—
13C22⋊#11717579C2212yes—
14C2×C4⋊C4×C101—21no—
15C2×C4⋊C4×C101C2211yescharacteristic
16C2×C4⋊C4×C101C2211yescharacteristic
17#95×C101—21no—
18#95×C101C2211yescharacteristic
19#13108074—42no—
20#13108074—42no—
21C2×Dic202—21no—
22C2×D404—21no—
23C2×D404—21no—
24C2×D404D412yes—
25C2×C22⋊D101—22no—
26C2×C4×D101—22no—
27C2×#11717579—22no—
28C4⋊Dic101—21no—
29C4⋊Dic101D412yes—
30#13108051—22no—
31#13108051—22no—
32C22×C4×C101—21no—
33C22×C4×C101—21no—
34C22×C4×C101D411yescharacteristic
35C22×C4×C101D411yescharacteristic
36C22×C4×C101C2311yescharacteristic
37C4⋊C4×C101—22no—
38C4⋊C4×C101—22no—
39C4⋊C4×C101—42no—
40C23×C101C2×D411yescharacteristic
41C22×D101—22no—
42C2×C4×C101—21no—
43C2×C4×C101—21no—
44C2×C4×C101—22no—
45C2×C4×C101—21no—
46C2×C4×C101—41no—
47C2×C4×C101—41no—
48C2×C4×C101—41no—
49C2×C4×C101—41no—
50C2×C4×C101—41no—
51C2×C4×C101—41no—
52C2×C4×C101—41no—
53C2×C4×C101D812yes—
54C2×C4×C101C2×D412yes—
55C22⋊D101—42no—
56C22⋊D101—42no—
57C2×Dic101—22no—
58D404—42no—
59D404—42no—
60D404—81no—
61C4×D101—44no—
62Dic202—42no—
63#11717579—42no—
64C22×C101—21no—
65C22×C101—21no—
66C22×C101#12011yescharacteristic
67C22×C101C2×D811yescharacteristic
68C22×C101AGL(1,ℤ/8)11yescharacteristic
69C2×D101—42no—
70C2×D101—42no—
71C2×D101—42no—
72C4×C101—22no—
73C4×C101—22no—
74C4×C101—42no—
75C4×C101—41no—
76C4×C101—81no—
77Dic101—42no—
78C2×C101—21no—
79C2×C101—21no—
80C2×C101#44711yescharacteristic
81C2×C101#45611yescharacteristic
82C2×C101#45811yescharacteristic
83D101—82no—
84#1610—1011nomaximal, Sylow
85C101#161011yescharacteristic, complemented, Sylow
86#537—1011no—
87#480—1012no—
88#398D10111yescharacteristic, complemented
89#327—1012no—
90#324—1011no—
91C2×#41—2021no—
92C22⋊Q8—2022no—
93#121—1012no—
94#121—2022no—
95#121—2021no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4C2×D10111yescharacteristic
98C2×C4⋊C4C2×D10111yescharacteristic
99#107—2022no—
100#102—2022no—
101C22⋊C8—1012no—
102#95—21no—
103#95C2×D10111yescharacteristic
104#41—4042no—
105#41—4042no—
106C2×Q8—2021no—
107C2×D4—1012no—
108C2×D4—2021no—
109C2×D4—2021no—
110C2×D4—2022no—
111C22×C4—21no—
112C22×C4—21no—
113C22×C4—2022no—
114C22×C4C22⋊D10111yescharacteristic
115C22×C4C22⋊D10111yescharacteristic
116C22×C4C22×D10111yescharacteristic
117C2×C8—2022no—
118C4⋊C4—22no—
119C4⋊C4—22no—
120C4⋊C4—42no—
121C4⋊C4—1012no—
122C4⋊C4—2021no—
123C22⋊C4—2022no—
124C22⋊C4—2022no—
125C23—2022no—
126C23C2×C22⋊D10111yescharacteristic
127Q8—4042no—
128D4—8081no—
129D4—4042no—
130D4—4042no—
131D4—4042no—
132D4—4042no—
133C2×C4—21no—
134C2×C4—21no—
135C2×C4—22no—
136C2×C4—21no—
137C2×C4—41no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—41no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—4044no—
145C2×C4—2022no—
146C2×C4#1310805912yes—
147C2×C4D4×D10112yes—
148C8—4042no—
149C22—21no—
150C22—21no—
151C22—4042no—
152C22—4042no—
153C22—4042no—
154C22#1569222311yescharacteristic
155C22C2×#1310805911yescharacteristic
156C22#1569228011yescharacteristic
157C4—22no—
158C4—22no—
159C4—42no—
160C4—41no—
161C4—81no—
162C4—4042no—
163C2—21no—
164C2—21no—
165C2—8082no—
166C2#2026531011yescharacteristic
167C2#2026535911yescharacteristic
168C2#2026536611yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#398×C101C2
socleC22×C101#120
prime radical#20265758C2
hypercenter#398D101
hypocenterC101#1610
LCSG ▷ C22×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #398

Complements

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

NH ≅ G/Ntype
2. #20265490C2semidirect
3. #20264910C2semidirect
4. #20264903C2semidirect
5. #398×C101C2semidirect
85. C101#1610semidirect
88. #398D101semidirect

Maximal subgroups

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

subgroupindex
#202657582
#202654902
#202649102
#202649032
#398×C1012
#1610101