Finite Group Databasebeta

← #27171515

Subgroups — #27171515

Order 12928.

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

normal subgroups only

This group has 332 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#20265723C211yesmaximal, characteristic
2#20265193C212yesmaximal, complemented
3#20265151C211yesmaximal, characteristic, complemented
4#20264938C212yesmaximal, complemented
5#568×C101C211yesmaximal, characteristic, complemented
6C2×#13108074C2211yescharacteristic
7#15692253C411yescharacteristic, complemented
8#15692251C411yescharacteristic, complemented
9#15692244C411yescharacteristic, complemented
10#15692242C411yescharacteristic, complemented
11#15692227—22no—
12C4×C22⋊D101—22no—
13#15692214C2212yes—
14C2×C4×Dic101C2211yescharacteristic
15#15692181—24no—
16#15692170—22no—
17C2×#41×C101C2211yescharacteristic
18#131×C101—24no—
19C2×M16×C101—22no—
20C2×M16×C101C2212yes—
21#13108074—22no—
22#13108074—22no—
23C2×Dic202C2×C411yescharacteristic
24C2×D404C2×C411yescharacteristic
25C2×C22⋊D101—22no—
26C2×C22⋊D101—21no—
27C22×Dic101—21no—
28C2×C4×D101—21no—
29C4⋊Dic101—22no—
30#13108053—22no—
31C22⋊Dic101—22no—
32C22⋊Dic101—22no—
33#13108051—22no—
34C4×Dic101—21no—
35C4×Dic101—22no—
36C4×Dic101C2×C411yescharacteristic
37C4×Dic101C2×C411yescharacteristic
38#41×C101—22no—
39#41×C101—22no—
40C2×Q8×C101C2×C411yescharacteristic
41C2×D4×C101C2×C411yescharacteristic
42C2×D4×C101D412yes—
43C22×C4×C101D412yes—
44C22×C4×C101C2311yescharacteristic
45M16×C101—24no—
46M16×C101—22no—
47M16×C101D414yescomplemented
48C2×C8×C101—24no—
49C2×C8×C101—22no—
50C23×C101C22⋊C412yes—
51C23×C101C2×D411yescharacteristic
52C22×D101—21no—
53Q8×C101—22no—
54D4×C101—24no—
55D4×C101—22no—
56C2×C4×C101—22no—
57C2×C4×C101—22no—
58C2×C4×C101C22⋊C412yes—
59C2×C4×C101C22×C411yescharacteristic
60C2×C4×C101C2×D412yes—
61C2×C4×C101C2×D411yescharacteristic
62C2×C4×C101#4112yes—
63C22⋊D101—44no—
64C22⋊D101—42no—
65C2×Dic101—21no—
66C2×Dic101—21no—
67C2×Dic101—21no—
68C2×Dic101—42no—
69C2×Dic101—42no—
70D404—22no—
71C4×D101—41no—
72C4×D101—41no—
73Dic202—22no—
74C8×C101—24no—
75C22×C101—22no—
76C22×C101—41no—
77C22×C101—42no—
78C22×C101C2×C22⋊C411yescharacteristic
79C22×C101#12112yes—
80C2×D101—42no—
81C2×D101—41no—
82C4×C101—22no—
83C4×C101C4×D412yes—
84C4×C101#12011yescharacteristic
85C4×C101#12311yescharacteristic
86Dic101—22no—
87Dic101—41no—
88Dic101—41no—
89Dic101—82no—
90C2×C101—21no—
91C2×C101—22no—
92C2×C101—42no—
93C2×C101#38611yescharacteristic
94D101—81no—
95#1884—1011nomaximal, Sylow
96C101#188411yescharacteristic, complemented, Sylow
97#568D10111yescharacteristic, complemented
98#532—1011no—
99#421—1012no—
100#410—1011no—
101#339—1012no—
102C2×#41—1011no—
103C2×#41C2×D10111yescharacteristic
104#131—24no—
105C2×M16—22no—
106C2×M16C2×D10112yes—
107C4⋊Q8—1011no—
108#127—1011no—
109#124—1011no—
110#124—1011no—
111#121—2022no—
112C4×D4—2022no—
113#117—1012no—
114C2×C42—1011no—
115#104—2024no—
116#99—2022no—
117#41—22no—
118#41—22no—
119#41—2022no—
120#41—2022no—
121C2×Q8—1011no—
122C2×Q8C4×D10111yescharacteristic
123C2×D4—1011no—
124C2×D4—2022no—
125C2×D4—2021no—
126C2×D4C4×D10111yescharacteristic
127C2×D4D40412yes—
128C22×C4—2021no—
129C22×C4—2021no—
130C22×C4C22⋊D10112yes—
131C22×C4C22×D10111yescharacteristic
132M16—24no—
133M16—22no—
134M16C22⋊D10114yescomplemented
135C2×C8—24no—
136C2×C8—22no—
137C4⋊C4—2022no—
138C4⋊C4—2022no—
139C22⋊C4—2022no—
140C22⋊C4—2022no—
141C22⋊C4—2022no—
142C42—1011no—
143C42—1011no—
144C42—2022no—
145C42—2021no—
146C23—2021no—
147C23#1310805112yes—
148C23C2×D40411yescharacteristic
149Q8—22no—
150Q8—2022no—
151D4—24no—
152D4—22no—
153D4—4042no—
154D4—4044no—
155D4—2022no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—4042no—
159C2×C4—4042no—
160C2×C4—4041no—
161C2×C4—4041no—
162C2×C4—2021no—
163C2×C4—2021no—
164C2×C4—2021no—
165C2×C4#1310805112yes—
166C2×C4C2×C4×D10111yescharacteristic
167C2×C4C2×C22⋊D10112yes—
168C2×C4C2×C22⋊D10111yescharacteristic
169C2×C4#1310807412yes—
170C8—24no—
171C22—22no—
172C22—41no—
173C22—42no—
174C22—4042no—
175C22—4041no—
176C22C2×#1310805111yescharacteristic
177C22#1569222912yes—
178C4—22no—
179C4—8082no—
180C4—4041no—
181C4—4041no—
182C4—2022no—
183C4C4×C22⋊D10112yes—
184C4#1569222411yescharacteristic
185C4#1569223811yescharacteristic
186C2—22no—
187C2—21no—
188C2—42no—
189C2—8081no—
190C2#2026505611yescharacteristic
subgroupG/H
centerC4#15692224
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#568×C101C2
socleC2×C101#386
prime radical#20265723C2
cosocleC22×C4×C101C23
hypercenter#568D101
hypocenterC101#1884
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C4 ◁ C22×C4 ◁ #568

Complements

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

NH ≅ G/Ntype
2. #20265193C2semidirect
3. #20265151C2semidirect
4. #20264938C2semidirect
5. #568×C101C2semidirect
7. #15692253C4semidirect
8. #15692251C4semidirect
9. #15692244C4semidirect
10. #15692242C4semidirect
47. M16×C101D4semidirect
96. C101#1884semidirect
97. #568D101semidirect
134. M16C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
#202657232
#202651932
#202651512
#202649382
#568×C1012
#1884101