Finite Group Databasebeta

← #27172362

Subgroups — #27172362

Order 12928.

290 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 67 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
2C2×#15692181C212yesmaximal, complemented
3#20265179C212yesmaximal, complemented
4#20265042C211yesmaximal, characteristic, complemented
5#343×C101C211yesmaximal, characteristic, complemented
6C2×#13108074C2212yes—
7#15692235—22no—
8#15692230—22no—
9#15692229—22no—
10C2×C4⋊Dic101C2211yescharacteristic
11C2×C4×Dic101C2211yescharacteristic
12#15692181—24no—
13#15692180—22no—
14#15692177—22no—
15#15692162—22no—
16C2×M16×C101C2212yes—
17C2×C4⋊C4×C101C2211yescharacteristic
18#13108074—22no—
19#13108074—22no—
20#13108074—22no—
21#13108074—22no—
22C2×Dic202D412yes—
23C2×D404—22no—
24C2×D404D412yes—
25C2×C22⋊D101—22no—
26C22×Dic101—21no—
27C22×Dic101—21no—
28C2×C4×D101—22no—
29C4⋊Dic101—22no—
30C4⋊Dic101—22no—
31#13108051—24no—
32C4×Dic101—21no—
33C4×Dic101D412yescomplemented
34C22×C4×C101D412yes—
35C22×C4×C101C2311yescharacteristic
36M16×C101—22no—
37M16×C101—22no—
38C2×C8×C101—22no—
39C4⋊C4×C101—22no—
40C23×C101C2×D411yescharacteristic
41C22×D101—22no—
42C2×C4×C101—22no—
43C2×C4×C101—42no—
44C2×C4×C101C2×D411yescharacteristic
45C2×C4×C101C2×D412yes—
46C2×C4×C101#4112yes—
47C2×C4×C101#4111yescharacteristic
48C22⋊D101—42no—
49C22⋊D101—42no—
50C2×Dic101—22no—
51C2×Dic101—22no—
52C2×Dic101—41no—
53C2×Dic101—41no—
54C2×Dic101—42no—
55C2×Dic101—41no—
56D404—22no—
57D404—22no—
58D404—82no—
59C4×D101—42no—
60C4×D101—42no—
61Dic202—22no—
62Dic202—22no—
63C8×C101—42no—
64C22×C101—21no—
65C22×C101—21no—
66C22×C101#12011yescharacteristic
67C22×C101#12111yescharacteristic
68C22×C101#12411yescharacteristic
69C2×D101—42no—
70C2×D101—42no—
71C2×D101—42no—
72C4×C101—42no—
73C4×C101#12112yes—
74C4×C101#12312yes—
75Dic101—42no—
76Dic101—42no—
77Dic101—81no—
78C2×C101—21no—
79C2×C101—21no—
80C2×C101#39411yescharacteristic
81C2×C101#45411yescharacteristic
82C2×C101#45511yescharacteristic
83D101—82no—
84#2040—1011nomaximal, Sylow
85C101#204011yescharacteristic, complemented, Sylow
86#537—1011no—
87C2×#104—1012no—
88#417—1012no—
89#382—1011no—
90#343D10111yescharacteristic, complemented
91C2×#41—1012no—
92C2×M16C2×D10112yes—
93C22⋊Q8—2022no—
94#121—2022no—
95#121—2022no—
96C2×C4⋊C4—1011no—
97C2×C4⋊C4C2×D10111yescharacteristic
98C2×C42—1011no—
99#104—2024no—
100#103—2022no—
101#102—2022no—
102#95—2022no—
103#41—2022no—
104#41—2022no—
105#41—2022no—
106#41—2022no—
107C2×Q8—1012no—
108C2×D4—1012no—
109C2×D4—2022no—
110C2×D4—2022no—
111C22×C4—2021no—
112C22×C4—2021no—
113C22×C4—2022no—
114C22×C4D40412yes—
115C22×C4C22×D10111yescharacteristic
116M16—22no—
117M16—22no—
118C2×C8—22no—
119C4⋊C4—22no—
120C4⋊C4—2022no—
121C4⋊C4—2022no—
122C22⋊C4—2024no—
123C42—1012no—
124C42—2021no—
125C23—2022no—
126C23C2×D40411yescharacteristic
127Q8—2022no—
128Q8—2022no—
129D4—8082no—
130D4—4042no—
131D4—4042no—
132D4—2022no—
133D4—2022no—
134C2×C4—22no—
135C2×C4—42no—
136C2×C4—4042no—
137C2×C4—4041no—
138C2×C4—4041no—
139C2×C4—4041no—
140C2×C4—4042no—
141C2×C4—4042no—
142C2×C4—2022no—
143C2×C4—2022no—
144C2×C4D4×D10111yescharacteristic
145C2×C4D4×D10112yes—
146C2×C4#1310807412yes—
147C2×C4#1310807511yescharacteristic
148C8—42no—
149C22—21no—
150C22—21no—
151C22—4042no—
152C22—4042no—
153C22—4042no—
154C22#1569222511yescharacteristic
155C22#1569223011yescharacteristic
156C22#1569224311yescharacteristic
157C4—42no—
158C4—8081no—
159C4—4042no—
160C4—4042no—
161C4#1569222812yes—
162C4#1569224012yes—
163C2—21no—
164C2—21no—
165C2—8082no—
166C2#2026509511yescharacteristic
167C2#2026535211yescharacteristic
168C2#2026535711yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#343×C101C2
socleC22×C101#120
prime radical#20265758C2
hypercenter#343D101
hypocenterC101#2040
LCSG ▷ C22×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #343

Complements

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

NH ≅ G/Ntype
2. C2×#15692181C2semidirect
3. #20265179C2semidirect
4. #20265042C2semidirect
5. #343×C101C2semidirect
33. C4×Dic101D4semidirect
85. C101#2040semidirect
90. #343D101semidirect

Maximal subgroups

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

subgroupindex
#202657582
C2×#156921812
#202651792
#202650422
#343×C1012
#2040101