Finite Group Databasebeta

← #27171550

Subgroups — #27171550

Order 12928.

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

normal subgroups only

This group has 342 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#20265684C211yesmaximal, characteristic
2#20265161C211yesmaximal, characteristic, complemented
3#20265140C212yesmaximal, complemented
4#20264948C212yesmaximal, complemented
5#568×C101C211yesmaximal, characteristic, complemented
6C22×C4×D101C2211yescharacteristic
7C2×#13108058C2212yes—
8#15692236C412yes—
9#15692231—21no—
10#15692226—21no—
11#15692223C412yes—
12C4×C22⋊D101—22no—
13#15692214C2211yescharacteristic
14#15692174—21no—
15#15692172—21no—
16#15692167—24no—
17C2×#41×C101C2211yescharacteristic
18#131×C101—24no—
19C2×M16×C101—22no—
20C2×M16×C101C2212yes—
21C23×D101C2×C411yescharacteristic
22C2×C22⋊D101—22no—
23C22×Dic101C2×C411yescharacteristic
24C2×C4×D101—22no—
25C2×C4×D101—21no—
26C2×C4×D101—21no—
27C2×C4×D101—22no—
28#13108058—22no—
29#13108058—22no—
30C2×#11717579—22no—
31C4⋊Dic101—21no—
32#13108053—22no—
33C22⋊Dic101—22no—
34C22⋊Dic101C2×C412yes—
35#13108051—22no—
36C4×Dic101—21no—
37#41×C101—22no—
38#41×C101—22no—
39C2×Q8×C101D411yescharacteristic
40C2×D4×C101C2×C412yes—
41C2×D4×C101D411yescharacteristic
42C22×C4×C101D412yes—
43C22×C4×C101C2311yescharacteristic
44M16×C101—24no—
45M16×C101—22no—
46M16×C101—22no—
47C2×C8×C101—24no—
48C2×C8×C101D414yescomplemented
49C23×C101C22⋊C412yes—
50C23×C101C22×C411yescharacteristic
51C22×D101—21no—
52C22×D101—22no—
53C22×D101—42no—
54Q8×C101—22no—
55D4×C101—24no—
56D4×C101—22no—
57C2×C4×C101—22no—
58C2×C4×C101—22no—
59C2×C4×C101C22⋊C412yes—
60C2×C4×C101C2×D412yes—
61C2×C4×C101C2×D411yescharacteristic
62C2×C4×C101C2×D411yescharacteristic
63C2×C4×C101#4112yes—
64C22⋊D101—44no—
65C2×Dic101—22no—
66C2×Dic101—21no—
67C2×Dic101—22no—
68C4×D101—41no—
69C4×D101—42no—
70C4×D101—41no—
71C8×C101—24no—
72#11717579—42no—
73C22×C101—22no—
74C22×C101—41no—
75C22×C101—42no—
76C22×C101C2×C22⋊C411yescharacteristic
77C22×C101C4×D412yes—
78C2×D101—22no—
79C2×D101—24no—
80C2×D101—41no—
81C2×D101—42no—
82C2×D101—41no—
83C4×C101—22no—
84C4×C101#12011yescharacteristic
85C4×C101#12112yes—
86C4×C101#12311yescharacteristic
87Dic101—42no—
88Dic101—41no—
89C2×C101—21no—
90C2×C101—22no—
91C2×C101—42no—
92C2×C101#38611yescharacteristic
93D101—42no—
94#1891—1011nomaximal, Sylow
95C101#189111yescharacteristic, complemented, Sylow
96#568D10111yescharacteristic, complemented
97#525—1011no—
98#413—1011no—
99#407—1012no—
100#344—1012no—
101C2×#41C2×D10111yescharacteristic
102C23×C4—1011no—
103#131—24no—
104C2×M16—22no—
105C2×M16—1012no—
106C2×M16C2×D10112yes—
107#123—1012no—
108C22⋊Q8—2021no—
109#121—2021no—
110#120—1012no—
111C4×D4—2022no—
112#117—1011no—
113#101—2021no—
114#100—2021no—
115C22⋊C8—2024no—
116C24—1011no—
117#41—22no—
118#41—22no—
119C2×Q8D40411yescharacteristic
120C2×D4—2022no—
121C2×D4C4×D10112yes—
122C2×D4D40411yescharacteristic
123C22×C4—1011no—
124C22×C4—2022no—
125C22×C4—2021no—
126C22×C4—2021no—
127C22×C4—2022no—
128C22×C4C22⋊D10112yes—
129C22×C4C22×D10111yescharacteristic
130M16—24no—
131M16—22no—
132M16—22no—
133M16—2022no—
134M16—2022no—
135C2×C8—24no—
136C2×C8—2022no—
137C2×C8C22⋊D10114yescomplemented
138C4⋊C4—2022no—
139C4⋊C4—2021no—
140C22⋊C4—1012no—
141C22⋊C4—2022no—
142C22⋊C4—2022no—
143C42—2021no—
144C23—4042no—
145C23—2021no—
146C23—2022no—
147C23#1310805112yes—
148C23C2×C4×D10111yescharacteristic
149Q8—22no—
150D4—24no—
151D4—22no—
152D4—4044no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—4041no—
156C2×C4—4042no—
157C2×C4—4041no—
158C2×C4—2022no—
159C2×C4—2021no—
160C2×C4—2022no—
161C2×C4#1310805112yes—
162C2×C4C2×C22⋊D10112yes—
163C2×C4C2×C22⋊D10111yescharacteristic
164C2×C4C2×D40411yescharacteristic
165C2×C4#1310807412yes—
166C8—24no—
167C8—4042no—
168C22—22no—
169C22—41no—
170C22—42no—
171C22—4041no—
172C22—4042no—
173C22—4041no—
174C22—2022no—
175C22—2024no—
176C22C2×#1310805111yescharacteristic
177C22C4×C22⋊D10112yes—
178C4—22no—
179C4—4042no—
180C4—4041no—
181C4#1569222411yescharacteristic
182C4#1569222912yes—
183C4#1569223811yescharacteristic
184C2—21no—
185C2—22no—
186C2—42no—
187C2—4042no—
188C2#2026505611yescharacteristic
subgroupG/H
centerC4#15692224
commutatorC23×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#568×C101C2
socleC2×C101#386
prime radical#20265684C2
cosocleC22×C4×C101C23
hypercenter#568D101
hypocenterC101#1891
LCSG ▷ C23×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C4 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #20265161C2semidirect
3. #20265140C2semidirect
4. #20264948C2semidirect
5. #568×C101C2semidirect
48. C2×C8×C101D4semidirect
95. C101#1891semidirect
96. #568D101semidirect
137. C2×C8C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
#202656842
#202651612
#202651402
#202649482
#568×C1012
#1891101