Finite Group Databasebeta

← #27172357

Subgroups — #27172357

Order 12928.

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

normal subgroups only

This group has 306 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#20265046C211yesmaximal, characteristic, complemented
5#343×C101C211yesmaximal, characteristic, complemented
6C2×#13108074C2212yes—
7#15692235—22no—
8#15692230—22no—
9#15692229—22no—
10C2×C4⋊Dic101C2211yescharacteristic
11C2×#13108053—22no—
12C2×C4×Dic101C2211yescharacteristic
13#15692181—24no—
14#15692180—22no—
15#15692177—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#13108053—44no—
32#13108051—24no—
33C4×Dic101—21no—
34C4×Dic101D412yescomplemented
35C22×C4×C101D412yes—
36C22×C4×C101C2311yescharacteristic
37M16×C101—22no—
38M16×C101—22no—
39C2×C8×C101—22no—
40C4⋊C4×C101—22no—
41C23×C101C2×D411yescharacteristic
42C22×D101—22no—
43C2×C4×C101—22no—
44C2×C4×C101—42no—
45C2×C4×C101C2×D411yescharacteristic
46C2×C4×C101C2×D412yes—
47C2×C4×C101#4112yes—
48C2×C4×C101#4111yescharacteristic
49C22⋊D101—42no—
50C22⋊D101—42no—
51C2×Dic101—22no—
52C2×Dic101—22no—
53C2×Dic101—22no—
54C2×Dic101—22no—
55C2×Dic101—42no—
56C2×Dic101—41no—
57D404—22no—
58D404—22no—
59D404—82no—
60C4×D101—42no—
61C4×D101—42no—
62Dic202—22no—
63Dic202—22no—
64C8×C101—42no—
65C22×C101—21no—
66C22×C101—21no—
67C22×C101#12011yescharacteristic
68C22×C101#12111yescharacteristic
69C22×C101#12411yescharacteristic
70C2×D101—42no—
71C2×D101—42no—
72C2×D101—42no—
73C4×C101—42no—
74C4×C101#12112yes—
75C4×C101#12312yes—
76Dic101—24no—
77Dic101—42no—
78Dic101—81no—
79C2×C101—21no—
80C2×C101—21no—
81C2×C101#39411yescharacteristic
82C2×C101#45311yescharacteristic
83C2×C101#45611yescharacteristic
84D101—82no—
85#2039—1011nomaximal, Sylow
86C101#203911yescharacteristic, complemented, Sylow
87#537—1011no—
88C2×#104—1012no—
89#417—1012no—
90#384—1011no—
91#343D10111yescharacteristic, complemented
92C2×#41—1012no—
93C2×M16C2×D10112yes—
94C22⋊Q8—2022no—
95#121—2022no—
96#121—2022no—
97C2×C4⋊C4—1011no—
98C2×C4⋊C4—2022no—
99C2×C4⋊C4C2×D10111yescharacteristic
100C2×C42—1011no—
101#104—2024no—
102#103—2022no—
103#102—2022no—
104#41—2022no—
105#41—2022no—
106#41—2022no—
107#41—2022no—
108C2×Q8—1012no—
109C2×D4—1012no—
110C2×D4—2022no—
111C2×D4—2022no—
112C22×C4—2021no—
113C22×C4—2022no—
114C22×C4—2021no—
115C22×C4D40412yes—
116C22×C4C22×D10111yescharacteristic
117M16—22no—
118M16—22no—
119C2×C8—22no—
120C4⋊C4—22no—
121C4⋊C4—4044no—
122C4⋊C4—2022no—
123C4⋊C4—2022no—
124C22⋊C4—2024no—
125C42—1012no—
126C42—2021no—
127C23—2022no—
128C23C2×D40411yescharacteristic
129Q8—2022no—
130Q8—2022no—
131D4—8082no—
132D4—4042no—
133D4—4042no—
134D4—2022no—
135D4—2022no—
136C2×C4—22no—
137C2×C4—42no—
138C2×C4—4042no—
139C2×C4—4041no—
140C2×C4—4042no—
141C2×C4—4042no—
142C2×C4—2022no—
143C2×C4—2022no—
144C2×C4—2022no—
145C2×C4—2022no—
146C2×C4D4×D10111yescharacteristic
147C2×C4D4×D10112yes—
148C2×C4#1310807412yes—
149C2×C4#1310807511yescharacteristic
150C8—42no—
151C22—21no—
152C22—21no—
153C22—4042no—
154C22—4042no—
155C22—4042no—
156C22#1569222511yescharacteristic
157C22#1569223011yescharacteristic
158C22#1569224311yescharacteristic
159C4—42no—
160C4—8081no—
161C4—4042no—
162C4—2024no—
163C4#1569222812yes—
164C4#1569224012yes—
165C2—21no—
166C2—21no—
167C2—8082no—
168C2#2026509511yescharacteristic
169C2#2026534711yescharacteristic
170C2#2026536211yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#343×C101C2
socleC22×C101#120
prime radical#20265758C2
hypercenter#343D101
hypocenterC101#2039
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. #20265046C2semidirect
5. #343×C101C2semidirect
34. C4×Dic101D4semidirect
86. C101#2039semidirect
91. #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
#202650462
#343×C1012
#2039101