Finite Group Databasebeta

← #27171669

Subgroups — #27171669

Order 12928.

314 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 89 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#20265751C211yesmaximal, characteristic
2#20265195C212yesmaximal
3C2×#15692174C211yesmaximal, characteristic, complemented
4#20264933C212yesmaximal, complemented
5#519×C101C211yesmaximal, characteristic, complemented
6C2×#13108074C2211yescharacteristic
7C2×#13108058C2212yes—
8#15692253C412yescomplemented
9#15692244C412yescomplemented
10#15692238—22no—
11#15692231—22no—
12#15692214C2211yescharacteristic
13#15692183—24no—
14#15692174—22no—
15C22×Q8×C101C2211yescharacteristic
16C4×Q8×C101—24no—
17#117×C101—22no—
18#117×C101C2212yes—
19#13108074—22no—
20#13108074—22no—
21C2×Dic202C2×C411yescharacteristic
22C2×D404C2×C411yescharacteristic
23C2×C22⋊D101—21no—
24C2×C4×D101—21no—
25#13108058—22no—
26#13108058—22no—
27C2×#11717579—22no—
28C4⋊Dic101—21no—
29#13108053—24no—
30C22⋊Dic101—21no—
31#13108051—24no—
32C4×Dic101C2×C412yes—
33C2×Q8×C101—22no—
34C2×Q8×C101—22no—
35C2×Q8×C101C2×C412yes—
36C2×Q8×C101D412yes—
37C22×C4×C101D412yes—
38C22×C4×C101C2311yescharacteristic
39C4⋊C4×C101—24no—
40C4⋊C4×C101—22no—
41C22⋊C4×C101—22no—
42C42×C101—24no—
43C42×C101D414yes—
44C23×C101C2×D411yescharacteristic
45C22×D101—21no—
46Q8×C101—24no—
47Q8×C101—24no—
48C2×C4×C101—22no—
49C2×C4×C101—22no—
50C2×C4×C101—24no—
51C2×C4×C101C22⋊C414yes—
52C2×C4×C101C22×C411yescharacteristic
53C2×C4×C101C2×D412yes—
54C2×C4×C101C2×D411yescharacteristic
55C2×C4×C101#4112yes—
56C22⋊D101—41no—
57C22⋊D101—41no—
58C2×Dic101—22no—
59C2×Dic101—21no—
60D404—22no—
61C4×D101—42no—
62Dic202—22no—
63#11717579—42no—
64C22×C101—41no—
65C22×C101C2×C22⋊C411yescharacteristic
66C22×C101#12011yescharacteristic
67C22×C101#12311yescharacteristic
68C2×D101—42no—
69C2×D101—41no—
70C4×C101—24no—
71C4×C101—44no—
72C4×C101C4×D412yes—
73C4×C101#12112yes—
74Dic101—42no—
75Dic101—41no—
76C2×C101—21no—
77C2×C101—21no—
78C2×C101—21no—
79C2×C101#38611yescharacteristic
80D101—81no—
81#1912—1011nomaximal, Sylow
82C101#191211yescharacteristic, complemented, Sylow
83#536—1011no—
84#519D10111yescharacteristic, complemented
85#421—1012no—
86C2×#101—1011no—
87H(3,ℤ/4)—1012no—
88C2×#41—1011no—
89C22×Q8C2×D10111yescharacteristic
90C2×M16—1012no—
91C4⋊Q8—1012no—
92#124—1012no—
93#123—2022no—
94C22⋊Q8—2022no—
95C4×Q8—24no—
96#117—22no—
97#117—1011no—
98#117C2×D10112yes—
99#104—2024no—
100#101—2022no—
101#41—2022no—
102#41—2022no—
103C2×Q8—22no—
104C2×Q8—22no—
105C2×Q8—1011no—
106C2×Q8C4×D10112yes—
107C2×Q8D40412yes—
108C2×D4—1011no—
109C2×D4—2021no—
110C22×C4—2021no—
111C22×C4C22⋊D10112yes—
112C22×C4C22×D10111yescharacteristic
113M16—2022no—
114M16—2022no—
115C2×C8—2022no—
116C4⋊C4—24no—
117C4⋊C4—22no—
118C4⋊C4—2024no—
119C4⋊C4—2021no—
120C22⋊C4—22no—
121C22⋊C4—2021no—
122C22⋊C4—2024no—
123C42—24no—
124C42—1012no—
125C42C22⋊D10114yes—
126C23—2021no—
127C23C2×C22⋊D10111yescharacteristic
128Q8—24no—
129Q8—24no—
130Q8—2022no—
131D4—4041no—
132D4—4041no—
133D4—2022no—
134C2×C4—22no—
135C2×C4—22no—
136C2×C4—24no—
137C2×C4—4042no—
138C2×C4—2022no—
139C2×C4—2021no—
140C2×C4#1310805114yes—
141C2×C4C2×C4×D10111yescharacteristic
142C2×C4C2×C22⋊D10112yes—
143C2×C4C2×D40411yescharacteristic
144C2×C4#1310807412yes—
145C8—4042no—
146C22—41no—
147C22—4042no—
148C22—4041no—
149C22C2×#1310805111yescharacteristic
150C22#1569222411yescharacteristic
151C22#1569223811yescharacteristic
152C4—24no—
153C4—44no—
154C4—4042no—
155C4—4041no—
156C4C4×C22⋊D10112yes—
157C4#1569222912yes—
158C2—21no—
159C2—21no—
160C2—21no—
161C2—8081no—
162C2#2026505611yescharacteristic
subgroupG/H
centerC2#20265056
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#519×C101C2
socleC2×C101#386
prime radicalC2×#13108074C22
cosocleC22×C4×C101C23
hypercenter#519D101
hypocenterC101#1912
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22×C4 ◁ #519

Complements

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

NH ≅ G/Ntype
3. C2×#15692174C2semidirect
4. #20264933C2semidirect
5. #519×C101C2semidirect
8. #15692253C4semidirect
9. #15692244C4semidirect
82. C101#1912semidirect
84. #519D101semidirect

Maximal subgroups

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

subgroupindex
#202657512
#202651952
C2×#156921742
#202649332
#519×C1012
#1912101