Finite Group Databasebeta

← #27169732

Subgroups — #27169732

Order 12928.

212 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 61 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#125×D101C211yesmaximal, characteristic
2#20265232C211yesmaximal, characteristic
3#20265129C211yesmaximal, characteristic
4#20264919C211yesmaximal, characteristic, complemented
5#20264918C211yesmaximal, characteristic, complemented
6#20264918C211yesmaximal, characteristic, complemented
7#330×C101C211yesmaximal, characteristic, complemented
8C2×#13108057—21no—
9#15692247C2211yescharacteristic
10#15692246C411yescharacteristic
11#15692246C411yescharacteristic
12#15692245—21no—
13#15692245—21no—
14C4⋊C4×D101—21no—
15C4⋊C4×D101—21no—
16C4⋊C4×D101C411yescharacteristic
17C4⋊C4×D101C411yescharacteristic
18C42×D101C2211yescharacteristic
19#15692184—21no—
20#15692167—21no—
21#15692166C2211yescharacteristic
22#15692166C2211yescharacteristic
23C8×Dic101—21no—
24#125×C101C2211yescharacteristic
25#97×C101C2211yescharacteristic
26#97×C101C2211yescharacteristic
27C2×C4×D101—21no—
28C2×C4×D101—21no—
29C2×C4×D101C2×C411yescharacteristic
30C2×C4×D101D411yescharacteristic
31C2×C4×D101D411yescharacteristic
32#13108057—24no—
33C2×#11717579—21no—
34C2×#11717579—21no—
35C4⋊Dic101—21no—
36C4⋊Dic101—21no—
37C4⋊Dic101C2×C411yescharacteristic
38C4⋊Dic101C2×C411yescharacteristic
39#13108053—21no—
40#13108053—21no—
41#13108053—21no—
42#13108053—21no—
43#13108053—21no—
44#13108053—21no—
45C4×Dic101C2×C411yescharacteristic
46C4×Dic101D411yescharacteristic
47C4×Dic101D411yescharacteristic
48C2×C8×C101—21no—
49C2×C8×C101—21no—
50C4⋊C4×C101—21no—
51C4⋊C4×C101—21no—
52C4⋊C4×C101C2×C411yescharacteristic
53C4⋊C4×C101C2×C411yescharacteristic
54C42×C101C2311yescharacteristic
55C22×D101C22⋊C411yescharacteristic
56C2×C4×C101—21no—
57C2×C4×C101—21no—
58C2×C4×C101C22×C411yescharacteristic
59C2×C4×C101C2×D411yescharacteristic
60C2×C4×C101C2×D411yescharacteristic
61C2×Dic101—21no—
62C2×Dic101—21no—
63C2×Dic101C22⋊C411yescharacteristic
64C2×Dic101C22⋊C411yescharacteristic
65C2×Dic101C22⋊C411yescharacteristic
66C4×D101—22no—
67C4×D101—22no—
68C4×D101—41no—
69C4×D101—42no—
70C4×D101—42no—
71C8×C101—22no—
72C8×C101—41no—
73#11717579—22no—
74#11717579—41no—
75C22×C101C2×C22⋊C411yescharacteristic
76C2×D101—21no—
77C2×D101—21no—
78C2×D101#10412yes—
79C4×C101—21no—
80C4×C101—21no—
81C4×C101—21no—
82C4×C101—41no—
83C4×C101—41no—
84Dic101—21no—
85Dic101—21no—
86Dic101—41no—
87Dic101—41no—
88Dic101#10112yes—
89C2×C101C2×#10111yescharacteristic
90C2×C101C2×#10411yescharacteristic
91C2×C101#42111yescharacteristic
92D101—22no—
93#1528—1011nomaximal, Sylow
94C101#152811yescharacteristic, complemented, Sylow
95C2×#125—1011no—
96#432—1011no—
97C4×M16—1011no—
98#330—1011no—
99#330—1011no—
100#330—1011no—
101#330D10111yescharacteristic, complemented
102C2×M16—2021no—
103#125—1011no—
104#125—1011no—
105#125—1011no—
106#125—2021no—
107#125—2021no—
108#125C2×D10111yescharacteristic
109C2×C4⋊C4—1011no—
110C2×C4⋊C4—1011no—
111C2×C4⋊C4—2021no—
112C2×C4⋊C4—2021no—
113C2×C42—1011no—
114C4⋊C8—2021no—
115C22⋊C8—2021no—
116#97—1011no—
117#97—1011no—
118#97C2×D10111yescharacteristic
119#97C2×D10111yescharacteristic
120C4×C8—2021no—
121C22×C4—1011no—
122C22×C4—1011no—
123C22×C4—1011no—
124C22×C4—2021no—
125C22×C4—2021no—
126M16—2024no—
127C2×C8—21no—
128C2×C8—21no—
129C2×C8—2021no—
130C2×C8—2021no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—1011no—
134C4⋊C4—1011no—
135C4⋊C4—2021no—
136C4⋊C4—2021no—
137C4⋊C4—2021no—
138C4⋊C4—2021no—
139C4⋊C4—2021no—
140C4⋊C4—2021no—
141C4⋊C4—2021no—
142C4⋊C4—2021no—
143C4⋊C4C4×D10111yescharacteristic
144C4⋊C4C4×D10111yescharacteristic
145C42—1011no—
146C42—1011no—
147C42—1011no—
148C42C22×D10111yescharacteristic
149C23—1011no—
150C2×C4—21no—
151C2×C4—21no—
152C2×C4—1011no—
153C2×C4—1011no—
154C2×C4—1011no—
155C2×C4—4041no—
156C2×C4—4042no—
157C2×C4—4042no—
158C2×C4—2022no—
159C2×C4—2022no—
160C2×C4—2021no—
161C2×C4—2021no—
162C2×C4C2×C4×D10111yescharacteristic
163C2×C4D4×D10111yescharacteristic
164C2×C4D4×D10111yescharacteristic
165C8—22no—
166C8—41no—
167C8—4041no—
168C8—2022no—
169C22—1012no—
170C22—2021no—
171C22—2021no—
172C22C22⋊C4×D10111yescharacteristic
173C4—21no—
174C4—21no—
175C4—21no—
176C4—41no—
177C4—41no—
178C4—1012no—
179C4—4041no—
180C4—4041no—
181C4—2021no—
182C4—2021no—
183C2—2022no—
184C2#101×D10111yescharacteristic
185C2#104×D10111yescharacteristic
186C2#2026519211yescharacteristic
subgroupG/H
centerC22C22⋊C4×D101
commutatorC2×C4×C101C22×C4
FrattiniC42C22×D101
Fitting#330×C101C2
socleC22×C101C2×C22⋊C4
prime radicalC22×D101C22⋊C4
cosocleC42×C101C23
hypercenter#330D101
hypocenterC101#1528
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
4. #20264919C2semidirect
5. #20264918C2semidirect
6. #20264918C2semidirect
7. #330×C101C2semidirect
94. C101#1528semidirect
101. #330D101semidirect

Maximal subgroups

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

subgroupindex
#125×D1012
#202652322
#202651292
#202649192
#202649182
#330×C1012
#1528101