Finite Group Databasebeta

← #27170301

Subgroups — #27170301

Order 12928.

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

normal subgroups only

This group has 194 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#20265901C211yesmaximal, characteristic
2#20265469C211yesmaximal, characteristic, complemented
3#20265464C211yesmaximal, characteristic
4#20264923C211yesmaximal, characteristic, complemented
5#20264909C211yesmaximal, characteristic, complemented
6#20264906C211yesmaximal, characteristic
7#487×C101C211yesmaximal, characteristic, complemented
8#15692250—22no—
9#15692244—21no—
10#15692235—21no—
11Q8×Dic101—21no—
12C4×Dic202C2211yescharacteristic
13C4×D404C2211yescharacteristic
14#15692213—21no—
15#15692188—21no—
16#15692186—21no—
17C4⋊#11717579C2211yescharacteristic
18C4⋊#11717579C2211yescharacteristic
19#15692178—21no—
20#15692175—21no—
21C4⋊Q8×C101C2211yescharacteristic
22C4⋊Q8×C101C2211yescharacteristic
23#103×C101—22no—
24C4×C8×C101C2211yescharacteristic
25C2×Dic202D411yescharacteristic
26C2×D404D411yescharacteristic
27C2×C4×D101—21no—
28C2×#11717579—21no—
29C2×#11717579—21no—
30C4⋊Dic101—22no—
31C4⋊Dic101D411yescharacteristic
32C4⋊Dic101D411yescharacteristic
33#13108053—21no—
34#13108051—22no—
35#13108051—22no—
36#13108051—21no—
37C4×Dic101—22no—
38C4×Dic101—21no—
39C2×Q8×C101—21no—
40C2×Q8×C101—21no—
41C2×C8×C101—21no—
42C4⋊C4×C101—21no—
43C4⋊C4×C101—21no—
44C4⋊C4×C101—21no—
45C4⋊C4×C101D412yes—
46C42×C101C2311yescharacteristic
47C22×D101—21no—
48Q8×C101—42no—
49Q8×C101—41no—
50Q8×C101—41no—
51C2×C4×C101—22no—
52C2×C4×C101—41no—
53C2×C4×C101C2×D411yescharacteristic
54C2×C4×C101C2×D411yescharacteristic
55C2×C4×C101C2×D411yescharacteristic
56C2×Dic101—21no—
57C2×Dic101—21no—
58C2×Dic101—21no—
59D404—21no—
60D404—21no—
61C4×D101—41no—
62C4×D101—41no—
63Dic202—21no—
64Dic202—21no—
65C8×C101—41no—
66#11717579—41no—
67#11717579—41no—
68C22×C101#12011yescharacteristic
69C2×D101—41no—
70C2×D101—41no—
71C2×D101—41no—
72C4×C101—21no—
73C4×C101—42no—
74C4×C101—81no—
75C4×C101#13511yescharacteristic
76C4×C101#13511yescharacteristic
77C4×C101#13711yescharacteristic
78C4×C101#13711yescharacteristic
79Dic101—41no—
80Dic101—41no—
81Dic101—41no—
82C2×C101#45211yescharacteristic
83C2×C101#45211yescharacteristic
84C2×C101#45411yescharacteristic
85D101—81no—
86#1638—1011nomaximal, Sylow
87C101#163811yescharacteristic, complemented, Sylow
88#556—1011no—
89#487D10111yescharacteristic, complemented
90#477—1011no—
91#476—1011no—
92#332—1011no—
93Q8⋊C8—1011no—
94D4⋊C8—1011no—
95C4⋊Q8C2×D10111yescharacteristic
96C4⋊Q8C2×D10111yescharacteristic
97#126—2022no—
98#124—2021no—
99C22⋊Q8—2021no—
100C4×Q8—1011no—
101C4×Q8—2021no—
102C4×D4—1011no—
103#117—2021no—
104#107—2021no—
105#106—2021no—
106C4⋊C8—1011no—
107C4⋊C8—1011no—
108#103—22no—
109#103—2021no—
110#102—2021no—
111C4×C8C2×D10111yescharacteristic
112C2×Q8—21no—
113C2×Q8—21no—
114C2×Q8—1011no—
115C2×D4—1011no—
116C22×C4—2021no—
117C2×C8—21no—
118C2×C8—2021no—
119C2×C8—2021no—
120C4⋊C4—21no—
121C4⋊C4—21no—
122C4⋊C4—21no—
123C4⋊C4—1011no—
124C4⋊C4—1011no—
125C4⋊C4—2022no—
126C4⋊C4—2021no—
127C4⋊C4C22⋊D10112yes—
128C22⋊C4—2022no—
129C22⋊C4—2022no—
130C22⋊C4—2021no—
131C42—2022no—
132C42—2021no—
133C42C22×D10111yescharacteristic
134C23—2021no—
135Q8—42no—
136Q8—41no—
137Q8—41no—
138Q8—2021no—
139Q8—2021no—
140D4—2021no—
141D4—2021no—
142C2×C4—22no—
143C2×C4—41no—
144C2×C4—4041no—
145C2×C4—4041no—
146C2×C4—2021no—
147C2×C4—2021no—
148C2×C4—2021no—
149C2×C4D4×D10111yescharacteristic
150C2×C4D4×D10111yescharacteristic
151C2×C4C2×C22⋊D10111yescharacteristic
152C8—41no—
153C8—4041no—
154C8—4041no—
155C22—4041no—
156C22—4041no—
157C22—4041no—
158C22#1569222311yescharacteristic
159C4—21no—
160C4—42no—
161C4—81no—
162C4—4041no—
163C4—4041no—
164C4—4041no—
165C4#1569227611yescharacteristic
166C4#1569227811yescharacteristic
167C4#1569228411yescharacteristic
168C4#1569228711yescharacteristic
169C2—8081no—
170C2#2026533611yescharacteristic
171C2#2026533811yescharacteristic
172C2#2026534911yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#487×C101C2
socleC22×C101#120
prime radicalC2×D404D4
hypercenter#487D101
hypocenterC101#1638
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #487

Complements

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

NH ≅ G/Ntype
2. #20265469C2semidirect
4. #20264923C2semidirect
5. #20264909C2semidirect
7. #487×C101C2semidirect
87. C101#1638semidirect
89. #487D101semidirect

Maximal subgroups

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

subgroupindex
#202659012
#202654692
#202654642
#202649232
#202649092
#202649062
#487×C1012
#1638101